\\frac{1}{4}$ .","Our main goal in this paper is to introduce an iterative procedure to improve this regularity threshold for almost sure local well-posedness.","Furthermore, we study a critical regularity associated with this iterative procedure.","By introducing a modified iterative approach, we then prove almost sure local well-posedness of (REF ) in an almost optimal range with respect to the original iterative procedure (Theorem REF ).","Beyond the concrete results in this paper, we believe that the iterative procedures based on (modified) partial power series expansions are themselves of interest for further development in the well-posedness theory of dispersive equations with random initial data and/or random forcing.","Such a probabilistic construction of solutions to dispersive PDEs first appeared in the works by McKean [34] and Bourgain [5] in the study of invariant Gibbs measures for the cubic NLS on $d$ , $d = 1, 2$ .","In particular, they established almost sure local well-posedness with respect to particular random initial data, basically corresponding to the Brownian motion.","(These local-in-time solutions were then extended globally in time by invariance of the Gibbs measures.","In the following, however, we restrict our attention to local-in-time solutions.)","In [8], Burq-Tzvetkov further elaborated this idea and considered a randomization for any rough initial condition via the Fourier series expansion.","More precisely, they studied the cubic nonlinear wave equation (NLW) on a three-dimensional compact Riemannian manifold below the scaling critical regularity.","By introducing a randomization via the multiplication of the Fourier coefficients by independent random variables, they established almost sure local well-posedness below the critical regularity.","Such randomization via the Fourier series expansion is natural on compact domains [15], [35] and more generally in situations where the associated elliptic operators have discrete spectra [47], [45].","See [2], [3] for more references therein.","In the following, we go over a randomization suitable for our problem on $\\mathbb {R}^3$ .","Recall the Wiener decomposition of the frequency space [48]: $\\mathbb {R}^3_\\xi = \\bigcup _{n \\in \\mathbb {Z}^3} (n+ (-\\frac{1}{2}, \\frac{1}{2}]^3)$ .","We employ a randomization adapted to this Wiener decomposition.","Let $\\psi \\in \\mathcal {S}(\\mathbb {R}^3)$ satisfy $\\operatornamewithlimits{supp}\\psi \\subset [-1,1]^3\\qquad \\text{and}\\qquad \\sum _{n \\in \\mathbb {Z}^3} \\psi (\\xi -n) =1 \\quad \\text{for any $\\xi \\in \\mathbb {R}^3$}.$ Then, given a function $\\phi $ on $\\mathbb {R}^3$ , we have $ \\phi = \\sum _{n \\in \\mathbb {Z}^3} \\psi (D-n) \\phi .$ We define the Wiener randomization $\\phi ^\\omega $ of $\\phi $ by $ \\phi ^{\\omega } := \\sum _{n \\in \\mathbb {Z}^3} g_n (\\omega ) \\psi (D-n) \\phi ,$ where $\\lbrace g_n \\rbrace _{n \\in \\mathbb {Z}^3} $ is a sequence of independent mean-zero complex-valued random variables on a probability space $(\\Omega , \\mathcal {F} ,P)$ .","The randomization (REF ) based on the Wiener decomposition of the frequency space is natural in view of time-frequency analysis; see [1] for a further discussion.","Almost simultaneously with [1], Lührmann-Mendelson [32] also considered a randomization of the form (REF ) (with cubes being substituted by appropriately localized balls) in the study of NLW on $\\mathbb {R}^3$ .","For a similar randomization used in the study of the Navier-Stokes equations, see also the work of Zhang and Fang [50], preceding [1], [32].","In [50], the decomposition of the frequency space is not explicitly provided but their randomization certainly includes randomization based on the decomposition of the frequency space by unit cubes or balls.","In the following, we assume that the probability distribution $\\mu _n$ of $g_n$ satisfies the following exponential moment bound: $\\int _{\\mathbb {R}^2} e^{\\kappa \\cdot x} d \\mu _{n} (x) \\le e^{c |\\kappa | ^2}$ for all $\\kappa \\in \\mathbb {R}^2$ and $n \\in \\mathbb {Z}^3$ .","This condition is satisfied by the standard complex-valued Gaussian random variables and the uniform distribution on the circle in the complex plane.","We now recall the almost sure local well-posedness result from [2] which is of interest to us.For simplicity, we only consider positive times.","By the time reversibility of the equation, the same analysis applies to negative times.","Theorem A Let $\\frac{1}{4} < s < \\frac{1}{2}$ .","Given $\\phi \\in H^s (\\mathbb {R}^3)$ , let $\\phi ^{\\omega }$ be its Wiener randomization defined in (REF ).","Then, the Cauchy problem (REF ) is almost surely locally well-posed with respect to the random initial data $\\phi ^\\omega $ .","More precisely, there exists a set $\\Sigma = \\Sigma (\\phi ) \\subset \\Omega $ with $P(\\Sigma ) = 1$ such that, for any $\\omega \\in \\Sigma $ , there exists a unique function $u = u^\\omega $ in the class: $S(t)\\phi ^\\omega + X^\\frac{1}{2}([0, T]) \\subset S(t) \\phi ^{\\omega } + C([0,T] ; H^\\frac{1}{2}(\\mathbb {R}^3)) \\subset C([0,T] ; H^s(\\mathbb {R}^3))$ with $T = T(\\phi , \\omega ) >0$ such that $u$ is a solution to (REF ) on $[0, T]$ .","Here, $S(t) := e^{it\\Delta }$ denotes the linear Schrödinger operator and the function space $X^\\frac{1}{2}([0, T]) \\subset C([0, T]; H^\\frac{1}{2}(\\mathbb {R}^3))$ is defined in Section below.","Note that the uniqueness statement of Theorem REF in the class: $S(t)\\phi ^\\omega + X^\\frac{1}{2}([0, T])$ is to be interpreted as follows; by setting $v = u - S(t) \\phi ^\\omega $ (see (REF ) below), uniqueness for the residual term $v$ holds in $X^\\frac{1}{2}([0, T])$ .","See also Remark REF below.","Recall that while the Wiener randomization (REF ) does not improve differentiability, it improves integrability (see Lemma 4 in [1]).","See [42], [27], [8] for the corresponding statements in the context of the random Fourier series.","The main idea for proving Theorem REF is to exploit this gain of integrability.","More precisely, let $z_1(t) := S(t) \\phi ^{\\omega }$ denote the random linear solution with $\\phi ^\\omega $ as initial data and write $u = z_1 + v.$ Then, we see that $v: = u - z_1$ satisfies the following perturbed NLS: ${\\left\\lbrace \\begin{array}{ll}i \\partial _tv + \\Delta v = |v + z_1|^2(v+z_1)\\\\v|_{t = 0} = 0,\\end{array}\\right.","}$ where $z_1$ is viewed as a given (random) source term.","The main point is that the gain of space-time integrability of the random linear solution $z_1$ (Lemma REF ) makes this problem subcriticalThe scaling critical Sobolev regularity $s_\\text{crit} = \\frac{1}{2}$ for (REF ) is defined by the fact that the $\\dot{H}^\\frac{1}{2}$ -norm remains invariant under the scaling symmetry (REF ).","Given $1 \\le r \\le \\infty $ , we can also define the scaling critical Sobolev regularity $s_\\text{crit}(r)$ in terms of the $L^r$ -based Sobolev space $\\dot{W}^{s, r}$ .","A direct computation gives $s_\\text{crit}(r) = \\frac{3}{r} - 1\\, (\\rightarrow -1$ as $r\\rightarrow \\infty $ ).","In particular, the gain of integrability of the random linear solution $z_1$ stated in Lemma REF implies that $z_1$ in (REF ) gives rise only to a subcritical perturbation.","See [3] for a further discussion on this issue.","Note, however, that if we consider a non-zero initial condition for $v$ , then the critical nature of the equation comes into play through the initial condition.","See Remark REF .","This plays an important role in studying the global-in-time behavior of solutions.","See, for example, [37].","and hence we can solve it by a standard fixed point argument.","Over the last several years, there have been many results on probabilistic well-posedness of nonlinear dispersive PDEs, using this change of viewpoint.In the field of stochastic parabolic PDEs, this change of viewpoint and solving the fixed point problem for the residual term $v$ is called the Da Prato-Debussche trick [16], [17].","See, for example, [34], [5], [8], [15], [9], [35], [32], [1], [2], [43], [38], [25], [33], [7], [30], [37].","In [2], we studied the Duhamel formulation for (REF ): $v(t) = - i \\int _0^t S(t-t^{\\prime }) |v + z_1|^2(v+z_1)(t^{\\prime }) dt^{\\prime },$ by carrying out case-by-case analysis on terms of the form $v \\overline{v} v$ , $v \\overline{v} z_1$ , $v \\overline{z}_1 z_1$ , etc.","in $X^\\frac{1}{2}([0, T])$ .","The main tools were (i) the gain of space-time integrability of $z_1$ and (ii) the bilinear refinement of the Strichartz estimate (Lemma REF ).","This yields Theorem REF .","By examining the case-by-case analysis in [2], we see that the regularity restriction $s > \\frac{1}{4}$ in Theorem REF comes from the cubic interaction of the random linear solution: $z_3(t) := -i \\int _0^t S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime }) dt^{\\prime }.$ See Proposition REF below.","In the following, we discuss an iterative approach to lower this regularity threshold by studying further expansions in terms of the random linear solution $z_1$ .","We will also discuss the limitation of this iterative procedure.","Remark 1.1 (i) The proof of Theorem REF , presented in [2], is based on a standard contraction argument for the residual term $v = u - S(t) \\phi ^\\omega $ in a ball $B$ of radius $O(1)$ in $X^\\frac{1}{2}([0, T])$ .","As such, the argument in [2] only yields uniqueness of $v$ in the ball $B$ .","Noting that $v \\in C([0, T]; H^{\\frac{1}{2}}(\\mathbb {R}^3)) \\cap X^\\frac{1}{2}([0, T])$ ,By convention, our definition of the $X^s([0, T])$ -space already assumes that functions in $X^s([0, T])$ belong to $ C([0, T]; H^s(\\mathbb {R}^3))$ .","See Definition REF below.","it follows from Lemma A.8 in [2] that the $X^\\frac{1}{2}([0, t])$ -norm of $v$ is continuous in $t \\in [0, T]$ .","Then, by possibly shrinking the local existence time $T>0$ , we can easily upgrade the uniqueness of $v$ in the ball $B$ to uniqueness of $v$ in the entire $X^\\frac{1}{2}([0, T])$ .","Namely, uniqueness of $u$ holds in the class (REF ).","(ii) With the exponential moment assumption (REF ), the proof of Theorem REF , presented in [2], allows us to conclude that the set $\\Sigma $ of full probability in Theorem REF has the following decomposition: $ \\Sigma = \\bigcup _{0 < T \\ll 1} \\Sigma _T$ such that (a) there exist $c, C >0$ and $C_0 = C_0 (\\Vert \\phi \\Vert _{H^s})>0$ such that $P(\\Sigma _T^c) < C \\exp \\Big (-\\frac{C_0(\\Vert \\phi \\Vert _{H^s})}{T^c }\\Big )$ for each $0 < T \\ll 1$ , (b) for each $\\omega \\in \\Sigma _T\\subset \\Sigma $ , $0 < T \\ll 1$ , the function $u = u^\\omega $ constructed in Theorem REF is a solution to (REF ) on $[0, T]$ .","See the statement of Theorem 1.1 in [2].","A similar decomposition of the set $\\Sigma $ of full probability applies to Theorems REF and REF below.","We, however, do not state it in an explicit manner in the following."],["Improved almost sure local well-posedness","Let us first state the following proposition on the regularity property of the cubic term $z_3$ defined above.","Proposition 1.2 Given $0 \\le s < 1$ , let $\\phi ^\\omega $ be the Wiener randomization of $\\phi \\in H^s(\\mathbb {R}^3)$ defined in (REF ) and set $z_1 = S(t) \\phi ^\\omega $ .","(i) For any $\\sigma < 2s$ , we have $z_3 \\in X^\\sigma _\\textup {loc}\\subset C(\\mathbb {R}; H^\\sigma (\\mathbb {R}^3))$ almost surely.","More precisely, there exists an almost surely finite constant $ C(\\omega , \\Vert \\phi \\Vert _{H^s} ) > 0$ and $\\theta > 0$ such that $\\Vert z_3 \\Vert _{X^{\\sigma }([0, T])} = \\bigg \\Vert \\int _0^t S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime }) dt^{\\prime }\\bigg \\Vert _{X^\\sigma ([0, T])}\\le T^\\theta C(\\omega , \\Vert \\phi \\Vert _{H^s})$ for any $T > 0$ .","In particular, by taking $\\sigma = 2s-\\varepsilon $ for small $\\varepsilon > 0$ , (REF ) shows that the second orderHere, we referred to $z_3$ as the second order term since it corresponds to the second order term appearing in the (formal) power series expansion of solutions to (REF ) in terms of the random initial data.","For the same reason, we refer to $z_5$ in (REF ) and $z_7$ in (REF ) as the third and fourth order terms in the following.","See Subsection REF below.","term $z_3$ is smoother (by $s - \\varepsilon $ ) than the first order term $z_1$ , provided $s > 0$ .","(ii) When $s = 0$ , there is no smoothing in the second order term $z_3$ in general; there exists $\\phi \\in L^2(\\mathbb {R}^3)$ such that the estimate (REF ) with $\\sigma = \\varepsilon > 0$ fails for any $\\varepsilon > 0$ .","In [2], we already proved (REF ) when $\\sigma = \\frac{1}{2}$ and $s > \\frac{1}{4}$ , giving the regularity restriction in Theorem REF .","See Section for the proof of Proposition REF for a general value of $\\sigma $ .","In the proof of Theorem REF , in order to carry out the case-by-case analysis for (REF ), we need to have $z_3 \\in X^\\frac{1}{2}_\\text{loc}$ , (where we have a deterministic local well-posedness for (REF )).","In view of Proposition REF , this imposes the regularity restriction $s > \\frac{1}{4}$ .","Note, however, that even when $s \\le \\frac{1}{4}$ , $z_3$ is still a well defined space-time function of spatial regularity $2s- < \\frac{1}{2} $ .","This motivates us to consider the following second order expansion: $u = z_1 + z_3 + v$ and remove the second order interaction $z_1 \\overline{z}_1 z_1$ .","Indeed, the residual term $v := u - z_1 - z_3$ satisfies the following equation: ${\\left\\lbrace \\begin{array}{ll}i \\partial _tv + \\Delta v = \\mathcal {N}(v + z_1+z_3) - \\mathcal {N}(z_1)\\\\v|_{t = 0} = 0,\\end{array}\\right.","}$ where $\\mathcal {N}(u) = |u|^2 u$ .","In terms of the Duhamel formulation, we have $v(t) = - i \\int _0^t S(t-t^{\\prime })\\big \\lbrace \\mathcal {N}(v + z_1 + z_3) - \\mathcal {N}(z_1)\\big \\rbrace (t^{\\prime }) dt^{\\prime }.$ Then, by studying the fixed point problem (REF ) for $v$ , we have the following improved almost sure local well-posedness of (REF ).","Theorem 1.3 Given $ \\frac{1}{2} \\le \\sigma \\le 1$ , let $ \\frac{2}{5} \\sigma < s < \\frac{1}{2}$ .","Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).","Then, the cubic NLS (REF ) on $\\mathbb {R}^3$ is almost surely locally well-posed with respect to the random initial data $\\phi ^\\omega $ .","More precisely, there exists a set $\\Sigma = \\Sigma (\\phi , \\sigma ) \\subset \\Omega $ with $P(\\Sigma ) = 1$ such that, for any $\\omega \\in \\Sigma $ , there exists a unique function $u = u^\\omega $ in the class: $z_1 + z_3+ X^\\sigma ([0, T])& \\subset z_1 + z_3+ C([0, T]; H^{\\sigma } (\\mathbb {R}^3))\\\\\\ &\\subset C([0, T];H^s(\\mathbb {R}^3))$ with $T = T(\\phi , \\omega ) >0$ such that $u$ is a solution to (REF ) on $[0, T]$ .","As in Theorem REF , the uniqueness of $u$ in the class $z_1 + z_3+ X^\\sigma ([0, T])$ is to be interpreted as uniqueness of the residual term $v = u - z_1 - z_3$ in $X^\\sigma ([0, T])$ .","See also Remark REF .","By taking $\\sigma = \\frac{1}{2}$ , Theorem REF states that (REF ) is almost surely locally well-posed in $H^s(\\mathbb {R}^3)$ , provided $s > \\frac{1}{5}$ .","This in particular improves the almost sure local well-posedness in Theorem REF .","On the other hand, by taking $\\sigma = 1$ , Theorem REF allows us to construct a solution $v = u - z_1 - z_3\\in X^1([0, T]) \\subset C([0,T] ; H^1(\\mathbb {R}^3))$ , provided that $s > \\frac{2}{5}$ .","In particular, we can take random initial data below the scaling critical regularity $s_\\text{crit} = \\frac{1}{2}$ , while we construct the residual part $v$ in $X^1([0, T])$ .","This opens up a possibility of studying the global-in-time behavior of $v$ , using the (non-conserved) energy of $v$ : $E(v)(t) = \\frac{1}{2} \\int _{\\mathbb {R}^3} |\\nabla v(t, x)|^2 dx+\\frac{1}{4} \\int _{\\mathbb {R}^3} |v(t, x)|^4 dx$ with random initial data below the scaling critical regularity.","We remark that by inspecting the argument in [2], a modification of (the proof of) Theorem REF yields almost sure local well-posedness of (REF ) in the class: $ S(t)\\phi ^\\omega + X^1([0, T]) \\subset S(t) \\phi ^{\\omega } + C([0,T] ; H^1(\\mathbb {R}^3))$ only for $s > \\frac{1}{2}$ .","(This restriction on $s$ can be easily seen by setting $\\sigma = 1$ in Proposition REF .)","In particular, Theorem REF does not allow us to take random initial data below the scaling critical regularity $s_\\text{crit} = \\frac{1}{2}$ in studying the global-in-time behavior of $v \\in X^1([0, T])$ , namely at the level of the energy $E(v)$ .","As our focus in this paper is the local-in-time analysis, we do not pursue further this issue on almost sure global well-posedness of (REF ) below the scaling critical regularity in this paper.","We, however, point out two recent results [30], [37] on almost sure global well-posedness below the energy space for the defocusing energy-critical NLS in higher dimensions.","The main strategy for proving Theorem REF is to study the fixed point problem (REF ) by carrying out case-by-case analysis on $w_1\\overline{w_2} w_3,\\quad \\text{for $w_i = v,$ $z_1$, or $z_3$, $i = 1, 2, 3$, but not all $w_i$ equal to $z_1$}$ in $N^\\sigma ([0, T])$ , where the dual norm is defined by $\\Vert F\\Vert _{N^\\sigma ([0, T])} = \\bigg \\Vert \\int _{0}^t S(t - t^{\\prime }) F(t^{\\prime }) dt^{\\prime }\\bigg \\Vert _{X^\\sigma ([0, T])}.$ Note that the number of cases has increased from the case-by-case analysis in the proof of Theorem REF , where we had $w_i = v$ or $z_1$ .","One of the main ingredients is the smoothing on $z_3$ stated in Proposition REF above.","Note, however, that in order to exploit this smoothing, we need to measure $z_3$ in the $X^{2s-}([0, T])$ -norm, which imposes a certain rigidity on the space-time integrability.Namely, we need to measure $z_3$ in $L^q_t([0, T]; W^{2s-, r}(\\mathbb {R}^3))$ for admissible pairs $(q, r)$ .","See Lemma REF .","In order to prove Theorem REF , we also need to exploit a gain of integrability on $z_3$ .","In Lemma REF , we use the dispersive estimate (see (REF ) below) and the gain of integrability on each $z_1$ of the three factors in (REF ) and show that $z_3$ also enjoys a gain of integrability by giving up some differentiability.","Remark 1.4 Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).","Given $v_0 \\in H^\\frac{1}{2}(\\mathbb {R}^3)$ , we can also consider (REF ) with the random initial data of the form $u_0^\\omega = v_0 + \\phi ^\\omega $ : ${\\left\\lbrace \\begin{array}{ll}i \\partial _tu + \\Delta u = |u|^2 u\\\\u|_{t = 0} = v_0+ \\phi ^\\omega .\\end{array}\\right.","}$ Then, by slightly modifying the proofs, we see that the analogues of Theorems A and REF (with $\\sigma = \\frac{1}{2}$ ) also hold for (REF ).","Namely, (REF ) is almost surely locally well-posed, provided $s > \\frac{1}{5}$ .","This amounts to considering the following Cauchy problems: ${\\left\\lbrace \\begin{array}{ll}i \\partial _tv + \\Delta v = |v + z_1|^2(v+z_1)\\\\v|_{t = 0} = v_0,\\end{array}\\right.","}$ when $\\frac{1}{4} < s < \\frac{1}{2}$ and ${\\left\\lbrace \\begin{array}{ll}i \\partial _tv + \\Delta v = \\mathcal {N}(v + z_1+z_3) - \\mathcal {N}(z_1)\\\\v|_{t = 0} = v_0,\\end{array}\\right.","}$ when $\\frac{1}{5} < s \\le \\frac{1}{4}$ .","In this case, the critical nature of the problem appears through the $v \\overline{v} v$ interaction in the case-by-case analysis (REF ) due to the deterministic (non-zero) initial data $v_0$ at the critical regularity.","The required modification is straightforward and thus we omit details.","See Proposition 6.3 in [2] and Lemma 6.2 in [37].","We point out that the discussion in the next subsection also applies to (REF ).","Remark 1.5 (i) Let $I(u_1, u_2, u_3)$ denote the trilinear operator defined by $I(u_1, u_2, u_3) (t):= -i \\int _0^t S(t - t^{\\prime }) u_1 \\overline{u_2} u_3 (t^{\\prime }) dt^{\\prime }.$ Then, for $\\sigma > 3s-1$ , there is no finite constant $C > 0$ such that $\\big \\Vert I( u_1, u_2, u_3)\\big \\Vert _{X^\\sigma ([0, 1])}\\le C \\prod _{j = 1}^3 \\Vert \\phi _j \\Vert _{H^s}$ for all $\\phi _j \\in H^s(\\mathbb {R}^3)$ , where $u_j = S(t) \\phi _j$ .","See Appendix .","In particular, this shows that when $s \\le \\frac{1}{2}$ , there is no deterministic smoothing for $I$ , i.e.","(REF ) does not hold for $\\sigma > s$ .","(ii) The proof of Proposition REF only exploits “the randomization at the linear level”.","Given $s \\in \\mathbb {R}$ , let $\\mathcal {R}^s$ denote the class of functions defined by $\\mathcal {R}^s = \\big \\lbrace u \\text{ on } \\mathbb {R}\\times \\mathbb {R}^3:\\ & i \\partial _tu + \\Delta u = 0 \\text{ and } \\\\&u \\in L^q_{t, \\text{loc}} W^{s, r}_x(\\mathbb {R}^3)\\text{ for any } 2 \\le q, r < \\infty \\big \\rbrace .$ Under the regularity assumption: $\\sigma < 2s$ and $0 \\le s < 1$ , it follows from the proof of Proposition REF that the left-hand side of (REF ) is finite for any $u_1, u_2, u_3 \\in \\mathcal {R}^s$ .Here, we only need finiteness of the $A_3^s$ -norm defined in (REF ) for each $u_j$ , $j = 1, 2, 3$ .","See Remark REF .","In other words, the proof of Proposition REF only uses the fact that the random linear solution $z_1 = S(t) \\phi ^\\omega $ belongs to $\\mathcal {R}^s$ almost surely; see the probabilistic Strichartz estimate (Lemma REF ).","The multilinear random structure of $z_3$ in terms of the random linear solution $z_1$ yields further cancellation.","See, for example, Lemma 3.6 in [15].","Such extra cancellation seems to improve only space-time integrability and we do not know how to use it to improve the regularity threshold (i.e.","differentiability) at this point.","A similar comment applies to the unbalanced higher order terms $\\zeta _{2k-1}$ (including $z_5$ below) studied in Proposition REF below."],["Partial power series expansion\nand the associated critical regularity","In this subsection, we discuss possible improvements over Theorem REF by considering further expansions.","For this purpose, we fix $ \\sigma = \\frac{1}{2}$ in the following.","By examining the proof of Theorem REF , we see that the regularity restriction $s > \\frac{1}{5}$ comes from the following third order term: $z_5 (t) := -i \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 5\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime })z_{j_1} \\overline{z_{j_2}}z_{j_3} (t^{\\prime }) dt^{\\prime }.$ Namely, we have $(j_1, j_2, j_3) = (1, 1, 3)$ up to permutations.","In Lemma REF , we show that given $0< s< \\frac{1}{2}$ , we have $ z_5 \\in X^{\\frac{5}{2} s -}_\\textup {loc}$ .","In particular, we have $ z_5 \\in X^\\frac{1}{2}_\\textup {loc}$ , provided $s > \\frac{1}{5} $ , yielding the regularity threshold in Theorem REF .","A natural next step is to remove this non-desirable third order interaction: $ \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 5\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3\\rbrace \\end{array}}z_{j_1} \\overline{z_{j_2}}z_{j_3}$ in the case-by-case analysis in (REF ) by considering the following third order expansion: $u = z_1 + z_3 + z_5+ v.$ In this case, the residual term $v := u - z_1 - z_3- z_5$ satisfies the following equation: ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}(v + z_1+z_3+z_5)- \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 \\in \\lbrace 3, 5\\rbrace \\\\j_1, j_2, j_3 \\in \\lbrace 1, 3\\rbrace \\end{array}}z_{j_1} \\overline{z_{j_2}}z_{j_3}\\\\v|_{t = 0} = 0.\\end{array}\\right.","}$ We expect that (REF ) is almost surely locally well-posed for $s > \\frac{2}{11}$ , which would be an improvement over Theorem REF .","The proof will be once again based on case-by-case analysis: $w_1\\overline{w_2} w_3\\quad & \\text{for $w_i = v, z_1, z_3$, or $z_5$, $i = 1, 2, 3$, such that } \\\\& \\text{it is not of the form $z_{j_1}\\overline{z_{j_2}}z_{j_3} $ with $j_1 + j_2 + j_3 \\in \\lbrace 3, 5\\rbrace $}$ in $N^\\frac{1}{2}([0, T])$ .","Note the increasing number of combinations.","In the following, however, we do not discuss details of this particular improvement over Theorem REF .","Instead, we consider further iterative steps and discuss a possible limitation of this procedure.","Remark 1.6 We point out that the expansions (REF ), (REF ), and (REF ) correspond to partial power series expansions of the first, second, and third orders,A (partial) power series expansion of a solution to (REF ) in terms of the random initial data can be expressed as a summation of certain multilinear operators over ternary trees.","See, for example, [11], [36].","Here, the term “order” in our context corresponds to “generation (of the associated trees) $+1$ ” with the convention that the trivial tree consisting only of the root node is of the zeroth generation.","For example, the third order term $z_5$ in (REF ) appears as the summation over all the multilinear operators associated to the ternary trees of the third generation.","In terms of the graphical representation in [36], we have $z_5 \\,\\text{``}\\!=\\!\\text{''} \\,[baseline=-3,scale=0.15]{(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (0.9, -1) node[dot] {};(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (-0.9, -1) node[dot] {};(0,0)node[ddot] {} -- (1.3,1) node[ddot] {}-- (1.3, 0) node[dot] {};(1.3,1) node[ddot] {}-- (2.6, 0) node[dot] {};}\\,+ \\,[baseline=-3,scale=0.15]{(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (0.9, -1) node[dot] {};(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (-0.9, -1) node[dot] {};(0,0)node[ddot] {} -- (0,1) node[ddot] {}-- (0.9, 0) node[dot] {};(0,0)node[ddot] {} -- (0,1) node[ddot] {}-- (-0.9, 0) node[dot] {};}\\,+\\, [baseline=-3,scale=0.15]{(2.6,-1)node[dot] {} -- (2.6,0) node[ddot] {}-- (3.5, -1) node[dot] {};(2.6,-1)node[dot] {} -- (2.6,0) node[ddot] {}-- (1.7, -1) node[dot] {};(0,0)node[dot] {} -- (1.3,1) node[ddot] {}-- (1.3, 0) node[dot] {};(1.3,1) node[ddot] {}-- (2.6, 0) node[ddot] {};} \\ ,$ where “ $[baseline=-2.7,scale=0.15]{(0,0) node[dot]{};}$ ” denotes the random linear solution $z_1 = S(t) \\phi ^\\omega $ and “ $[baseline=-2.7,scale=0.15]{(0,0) node[ddot]{};}$ ” denotes the trilinear Duhamel integral operator $ I(u_1, u_2, u_3)$ defined in (REF ) with its three children as its arguments $u_1, u_2$ , and $u_3$ .","respectively, of a solution to (REF ) in terms of the random initial data.","Then, by considering the associated equations (REF ), (REF ), and (REF ) for the residual term $v$ , we are recasting the original problem (REF ) as a fixed point problem centered at the partial power series expansions of the first, second, and third orders, respectively.","By drawing an analogy to the previous steps, we expect that the worst contribution comes from the following fourth order terms: $z_7 (t): = -i \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 7\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3, 5\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) z_{j_1} \\overline{z_{j_2}}z_{j_3} (t^{\\prime }) dt^{\\prime }.$ There are basically two contributions to (REF ): $(j_1, j_2, j_3) = (1, 3, 3)$ or $(1, 1, 5)$ up to permutations.","In Lemma REF , we show that the contribution from $(j_1, j_2, j_3) = (1, 1, 5)$ is worse, being responsible for the expected regularity restriction $s > \\frac{2}{11}$ .","In order to remove this term, we can consider the following fourth order expansion: $u = z_1 + z_3 + z_5+z_7+ v$ as in the previous steps and try to solve the following equation for the residual term $v = u - z_1 - z_3- z_5-z_7$ : ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}(v + z_1+z_3+z_5+z_7)- \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 \\in \\lbrace 3, 5, 7\\rbrace \\\\j_1, j_2, j_3 \\in \\lbrace 1, 3, 5\\rbrace \\end{array}}z_{j_1} \\overline{z_{j_2}}z_{j_3}\\\\v|_{t = 0} = 0.\\end{array}\\right.","}$ We can obviously iterate this argument and consider the following $k$ th order expansion: $u = \\sum _{\\ell = 1}^{k} z_{2\\ell -1}+ v.$ In this case, we need to consider the following equation for the residual term $v := u - \\sum _{\\ell = 1}^{k} z_{2\\ell -1}$ : ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}\\bigg (v +\\sum _{\\ell = 1}^{k} z_{2\\ell -1}\\bigg )-\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 \\in \\lbrace 3, 5, \\dots , 2k-1\\rbrace \\\\j_1, j_2, j_3 \\in \\lbrace 1, 3, \\dots , 2k-3\\rbrace \\end{array}}z_{j_1} \\overline{z_{j_2}}z_{j_3}\\\\v|_{t = 0} = 0\\end{array}\\right.","}$ and hope to construct a solution $v \\in X^\\frac{1}{2}([0, T])$ by carrying out the following case-by-case analysis: $\\begin{split}w_1\\overline{w_2} w_3,\\quad & \\text{for $w_i = v, z_j,$ $j \\in \\lbrace 1, 3, \\dots , 2k-1\\rbrace $,$i = 1, 2, 3$, such that } \\\\& \\text{it is not of the form $z_{j_1}\\overline{z_{j_2}}z_{j_3} $ with$j_1 + j_2 + j_3\\in \\lbrace 3, 5, \\dots , 2k-1\\rbrace $}\\end{split}$ in $N^\\frac{1}{2}([0, T])$ .","Then, a natural question to ask is “Does this iterative procedure work indefinitely, allowing us to arbitrarily lower the regularity threshold for almost sure local well-posedness for (REF )?","Or is there any limitation to it?","We now consider a “critical” regularity $s_* < \\frac{1}{2}$ with respect to this iterative procedure for proving almost sure local well-posedness of (REF ).","We simply define the critical regularity $s_* <\\frac{1}{2} $ for this problem to be the infimum of the values of $s< \\frac{1}{2}$ such that given any $\\phi \\in H^s(\\mathbb {R}^3)$ , the above iterative procedureNamely, we solve (REF ) for $v \\in X^\\frac{1}{2}([0, T])$ with some finite (or infinite) number of steps.","shows that (REF ) is almost surely locally well-posed with respect to the Wiener randomization $\\phi ^\\omega $ of $\\phi $ .","This is an empirical notion of criticality; unlike the scaling criticality, we can not a priori compute this critical regularity $s_*$ .","Moreover, our discussion will be based on the estimates on the stochastic multilinear terms (Proposition REF and Proposition REF below).","In the following, we discuss a (possible) lower bound on $s_*$ , presenting a limitation to our iterative procedure based on partial power series expansions.","Within the framework of the iterative procedure discussed above, a necessary condition for carrying out the case-by-case analysis (REF ) to study (REF ) in $X^\\frac{1}{2}([0, T])$ is that the $(k+1)$ st order term $ z_{2k+1} = z_{2(k+1)-1}$ defined in a recursive manner: $z_{2k+1}(t) : = -i \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = {2k+1}\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3, \\dots , 2k-1\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) z_{j_1} \\overline{z_{j_2}}z_{j_3} (t^{\\prime }) dt^{\\prime }$ belongs to $X^\\frac{1}{2}([0, T])$ .","By the nature of this iterative procedure, we may assume that the lower order terms $z_{2\\ell -1}$ , $\\ell \\in \\lbrace 1, 3, \\dots , k\\rbrace $ , belong to $X^{ s_\\ell }([0, T])$ for some $ s_\\ell < \\frac{1}{2}$ but not in $X^\\frac{1}{2}([0, T])$ , since if any of the lower order terms, say $z_{2\\ell -1}$ for some $\\ell \\in \\lbrace 1, 3, \\dots , k\\rbrace $ , were in $X^\\frac{1}{2}([0, T])$ , then we would have stopped the iterative procedure at the $(\\ell -1)$ th step.","As in the previous steps, we expect that $z_{2k+1}$ is responsible for a regularity restriction at the $k$ th step of this iterative approach.","It could be a cumbersome task to study the regularity property of $z_{2k+1}$ due to the increasing number of combinations for $z_{j_1}\\overline{z_{j_2}} z_{j_3}$ , satisfying $j_1 + j_2 + j_3 = {2k+1}$ , $j_1, j_2, j_3 \\in \\lbrace 1, 3, \\dots , 2k-1\\rbrace $ .","In the following, we instead study the regularity property of the $(k+1)$ st order termIn fact, we study the $k$ th order term $\\zeta _{2k-1}$ in Proposition REF .","of a particular form, corresponding to $(j_1, j_2, j_3) = (1, 1, 2k - 1)$ up to permutations in (REF ); given an integer $k \\ge 0$ , define $\\zeta _{2k+1}$ by setting $\\zeta _1 := z_1 = S(t) \\phi ^\\omega $ and $\\zeta _{2k+1}(t) := -i\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = {2k+1}\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2k-1\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) \\zeta _{j_1} \\overline{\\zeta _{j_2}} \\zeta _{j_3} (t^{\\prime }) dt^{\\prime }.$ As mentioned above, there are only three terms in this sum: $(j_1, j_2, j_3) = (1, 1, 2k - 1)$ up to permutations.","Hence, $\\zeta _{2k+1}$ consists of the “unbalanced”By associating the $(2k+1)$ -linear terms appearing in the summation in (REF ) with ternary trees of the $k$ th generation as in [36], the summands in (REF ) correspond to the “unbalanced” trees of the $k$ th generation, where two of the three children of the root node are terminal.","$(2k+1)$ -linear terms appearing in the definition (REF ) of $z_{2k+1}$ .","We claim that the $(k+1)$ st order term $\\zeta _{2k + 1}$ is responsible for the regularity restriction at the $k$ th step of the iterative procedure.","See Theorem REF below.","In anticipating the alternative expansion (REF ) below, let us study the regularity property of the unbalanced $k$ th order term $\\zeta _{2k-1}$ : $\\zeta _{2k-1}(t) := -i\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = {2k-1}\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2k-3\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) \\zeta _{j_1} \\overline{\\zeta _{j_2}} \\zeta _{j_3} (t^{\\prime }) dt^{\\prime }.$ For $k = 2, 3, 4$ , we have (with appropriate restrictions on the range of $s$ ) $\\zeta _3 = z_3 \\in X^{2s-}_\\text{loc},\\qquad \\zeta _5 = z_5 \\in X^{\\frac{5}{2}s-}_\\text{loc},\\qquad \\text{and}\\qquad \\zeta _7 \\in X^{\\frac{11}{4}s-}_\\text{loc}.$ See Proposition REF and Lemmas REF and REF .","In general, we have the following proposition.","Proposition 1.7 Define a sequence $\\lbrace \\alpha _k\\rbrace _{k \\in \\mathbb {N}}$ of positive real numbers by the following recursive relation: $\\alpha _k = \\frac{\\alpha _{k-1} + 3}{2}$ with $\\alpha _1 = 1$ .","Given $0< s < \\alpha _{k-1}^{-1}$ , let $\\phi ^\\omega $ be the Wiener randomization of $\\phi \\in H^s(\\mathbb {R}^3)$ defined in (REF ).","Then, we have $\\zeta _{2k -1} \\in X^\\sigma _\\textup {loc}$ for $\\sigma < \\alpha _k \\cdot s $ , almost surely.","By solving the recursive relation (REF ), we have $\\alpha _k = 2\\bigg \\lbrace 1 - \\bigg (\\frac{1}{2}\\bigg )^{k-1}\\bigg \\rbrace + 1$ and thus we have $\\alpha _2 = 2$ , $\\alpha _3 = \\frac{5}{2}$ , and $\\alpha _4 = \\frac{11}{4}$ .","In particular, Proposition REF agrees with (REF ).","Moreover, since $\\alpha _k$ is increasing and $\\lim _{k \\rightarrow \\infty } \\alpha _k =3$ , the regularity restriction $s < \\alpha _{k-1}^{-1}$ in Proposition REF does not cause any issue since our main focus is to study the probabilistic local well-posedness of (REF ) in the range of $s$ that is not covered by Theorem REF .","Namely, we may assume $s \\le \\frac{1}{5}\\, (< \\frac{1}{3})$ in the following.","In view of Propositions REF and REF , one obvious lower bound for the critical regularity $s_*$ for this iterative procedure is given by $s_0: = 0$ since there is no gain of regularity when $s = 0$ (even in moving from $z_1$ to $z_3$ ).","On the other hand, in order to prove almost sure local well-posedness of (REF ) by carrying out the case-by-case analysis (REF ) for the equation (REF ), we need to show that the $(k+1)$ st order term $\\zeta _{2k+1}$ belongs to $X^\\frac{1}{2}([0, T])$ .","This gives rise to a regularity restriction $s_k := \\frac{1}{2\\alpha _{k+1}}$ at the $k$ th step of the iterative procedure.","By taking $k \\rightarrow \\infty $ , we obtain another “lower” boundThis “lower” bound is based on the upper bounds obtained in Propositions REF and REF .","In other words, if one can improve the bounds in Propositions REF and REF , then one can lower the value of $s_\\infty $ .","See Remark REF .","$s_\\infty := \\frac{1}{6}$ on this critical regularity $s_*$ .","As mentioned above, the case-by-case analysis (REF ) for general $k \\in \\mathbb {N}$ may be a combinatorially overwhelming task due to (i) the number of the increasing combinations in (REF ) and (ii) the random multilinear terms $z_j,$ $j \\in \\lbrace 3, 5, \\dots , 2k-1\\rbrace $ , themselves having non-trivial combinatorial structures which makes it difficult to establish nonlinear estimates; see (REF ).","In the following, we instead consider an alternative iterative procedure based on the following expansion: $u = \\sum _{\\ell = 1}^k \\zeta _{2\\ell -1} + v$ in place of (REF ).","This expansion allows us to prove the following almost sure local well-posedness of (REF ) for $s > s_\\infty = \\frac{1}{6}$ .","Theorem 1.8 Let $\\frac{1}{6} < s < \\frac{1}{2}$ .","Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).","Then, the cubic NLS (REF ) on $\\mathbb {R}^3$ is almost surely locally well-posed with respect to the random initial data $\\phi ^\\omega $ .","More precisely, there exists a set $\\Sigma = \\Sigma (\\phi ) \\subset \\Omega $ with $P(\\Sigma ) = 1$ such that, for any $\\omega \\in \\Sigma $ , there exists a unique function $u = u^\\omega $ in the class: $\\zeta _1 + \\zeta _3 + \\cdots + \\zeta _{2k-1}+ X^\\frac{1}{2}([0, T])& \\subset \\zeta _1 + \\zeta _3 + \\cdots + \\zeta _{2k-1}+ C([0, T]; H^\\frac{1}{2} (\\mathbb {R}^3))\\\\\\ &\\subset C([0, T];H^s(\\mathbb {R}^3))$ with $T = T(\\phi , \\omega ) >0$ such that $u$ is a solution to (REF ) on $[0, T]$ .","Here, $k \\in \\mathbb {N}$ is a unique positive integer such that $\\frac{1}{2\\alpha _{k+1}} < s \\le \\frac{1}{2\\alpha _{k}}$ .","As before, the uniqueness of $u$ in the class: $\\zeta _1 + \\zeta _3 + \\cdots + \\zeta _{2k-1}+ X^\\frac{1}{2}([0, T])$ is to be interpreted as uniqueness of the residual term $v = u -\\sum _{\\ell = 1}^k \\zeta _{2\\ell -1} $ in $X^\\frac{1}{2}([0, T])$ .","See also Remark REF .","In view of the discussion above, Theorem REF proves almost sure local well-posedness of (REF ) in an almost “optimal”Once again, this is based on the estimates in Propositions REF and REF .","In particular, the “optimality” of the regularity threshold in Theorem REF is with respect to Propositions REF and REF .","If one can improve the bounds in Propositions REF and REF , then one can lower the regularity threshold in Theorem REF .","regularity range $s > s_\\infty = \\frac{1}{6}$ with respect to the original iterative procedure based on the partial power series expansion (REF ).","The proof of Theorem REF is analogous to that of Theorem REF .","Given $ \\frac{1}{6} < s<\\frac{1}{2}$ , fix $k \\in \\mathbb {N}$ such that $\\frac{1}{2\\alpha _{k+1}} < s \\le \\frac{1}{2\\alpha _{k}}$ .In view of Proposition REF , the lower bound on $s$ guarantees that $\\zeta _{2k+1} \\in X^\\frac{1}{2}([0, T])$ almost surely, while the upper bound on $s$ states that we can not use Proposition REF to conclude $\\zeta _{2k-1} \\in X^\\frac{1}{2}([0, T])$ almost surely.","Write a solution $u$ as in (REF ).","Note that as $s$ gets closer and closer to the critical value $s_\\infty = \\frac{1}{6}$ , the expansion (REF ) gets arbitrarily long.","In view of (REF ), the residual term $v := u - \\sum _{\\ell = 1}^{k} \\zeta _{2\\ell -1}$ satisfies the following equation: ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}\\bigg (v +\\sum _{\\ell = 1}^{k} \\zeta _{2\\ell -1}\\bigg )-\\sum _{\\ell = 2}^{k}\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 2\\ell - 1\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2\\ell -3\\rbrace \\end{array}}\\zeta _{j_1} \\overline{\\zeta _{j_2}}\\zeta _{j_3}\\\\v|_{t = 0} = 0.\\end{array}\\right.","}$ Hence, we need to carry out the following case-by-case analysis: $\\begin{split}w_1\\overline{w_2} w_3,\\quad & \\text{for $w_i = v, \\zeta _j,$ $j \\in \\lbrace 1, 3, \\dots , 2k-1\\rbrace $,$i = 1, 2, 3$, such that } \\\\& \\text{it is not of the form $\\zeta _{j_1}\\overline{\\zeta _{j_2}}\\zeta _{j_3} $ with$(j_1, j_2, j_3 )= (1, 1, 2\\ell -3)$}\\\\& \\text{(up to permutations) for $\\ell = 2, 3 \\dots , k$}\\end{split}$ in $N^\\frac{1}{2}([0, T])$ .","While Theorem REF yields almost sure local well-posedness in an almost optimal range with respect to the original iterative procedure based on the partial power series expansion (REF ), the required analysis is much simpler than that required for the original iterative procedure based on the expansion (REF ).","First, note that while the case-by-case analysis (REF ) based on the original iterative procedure involves combinatorially non-trivial $z_{2k-1}$ , the case-by-case analysis (REF ) only involves the unbalanced $k$ th order term $\\zeta _{2k-1}$ which has a much simpler structure than $z_{2k-1}$ .","In particular, Proposition REF shows that $\\zeta _{2k-1}$ , $k \\ge 3$ , has a better regularity property than the second order term $\\zeta _3 = z_3$ in (REF ).","In terms of space-time integrability, we show that $\\zeta _{2k-1}$ also enjoys a gain of integrability by giving up a control on derivatives (Lemma REF ).","Finally, by inspecting the proof of Theorem REF (see Lemma REF and Proposition REF below), we see that, except for $z_{j_1} \\overline{z_{j_2}}z_{j_3}$ with $(j_1, j_2, j_3) = (1, 1, 3)$ up to permutations,Namely, the terms constituting the third order term $\\zeta _5 = z_5$ in (REF ).","we can bound all the terms $w_1 \\overline{w_2} w_3$ appearing in the case-by-case analysis (REF ) in $N^\\frac{1}{2}([0, T])$ .","Hence, we can basically apply the result of the case-by-case analysis (REF ) to our problem at hand.","More precisely, by rewriting the case-by-case analysis (REF ) as $ w_i = v, \\zeta _1 = z_1 , \\text{ or }\\zeta _j, \\, j \\in \\lbrace 3, 5, \\dots , 2k-1\\rbrace $ (with the restriction in (REF )) and using the fact that the terms $\\zeta _{2\\ell - 1}$ , $\\ell = 3, \\dots , k$ , behave better than $\\zeta _3 = z_3$ , the proof of Theorem REF (in particular, Proposition REF ) can be used to control all the terms $w_1 \\overline{w_2} w_3$ in (REF ) except for $\\zeta _{j_1}\\overline{\\zeta _{j_2}}\\zeta _{j_3}\\quad \\text{with }(j_1, j_2, j_3 )= (1, 1, 2 k-1)$ (up to permutations).","Note that the contribution to (REF ) under the Duhamel integral is precisely given by $\\zeta _{2k+1}$ in (REF ).","In particular, Proposition REF with $\\sigma = \\frac{1}{2}$ yields the regularity restriction $\\alpha _{k+1}\\cdot s > \\frac{1}{2}$ , i.e.","the lower bound $s > \\frac{1}{2\\alpha _{k+1}}$ stated in Theorem REF .","This allows us to construct a solution $v \\in X^\\frac{1}{2}([0, T])$ by the standard fixed point argument.","See Section for details.","We previously conjectured that the $(k+1)$ st order term $z_{2k+1}$ in (REF ) would be responsible for a regularity restriction at the $k$ th step of the original iterative procedure.","Combining Proposition REF and Theorem REF , we confirmed this claim; the regularity restriction indeed comes only from the unbalanced $(k+1)$ st order term $\\zeta _{2k+1}$ in (REF ).","We conclude this introduction with several remarks.","Remark 1.9 Let $v_0 \\in H^\\frac{1}{2}(\\mathbb {R}^3)$ .","Then, by slightly modifying the proof of Theorem REF based on the modified iterative approach (REF ), we can prove almost sure local well-posedness of the cubic NLS (REF ) with the random initial data of the form $v_0 + \\phi ^\\omega $ for the same range of $s$ .","See Remark REF Remark 1.10 In this paper, we exploit randomness only at the linear level in estimating the stochastic terms.","See Remarks REF and REF .","It may be possible to lower the regularity thresholds in Theorems REF and REF by exploiting randomness at the multilinear level.","We, however, do not pursue this direction in this paper since (i) our main purpose is to present the iterative procedures in their simplest forms and (ii) estimating the higher order stochastic terms by exploiting randomness at the multilinear level would require a significant amount of additional work, which would blur the main focus of this paper.","Remark 1.11 The ill-posedness result in [12] show that the solution map $\\Phi : u_0 \\in H^s(\\mathbb {R}^3) \\longmapsto u \\in C([-T, T]; H^s(\\mathbb {R}^3))$ is not continuous for (REF ) when $s < \\frac{1}{2}$ .","In proving Theorem REF , we studied the perturbed NLS (REF ) for $v = u-z_1$ .","In particular, the proof shows that we can factorize the solution map for (REF ) asSimilarly, we can factorize the solution map for (REF ) as $u |_{t = 0} = v_0 + \\phi ^\\omega \\in H^s(\\mathbb {R}^3)\\longmapsto (v_0, z_1) \\stackrel{\\Psi }{\\longmapsto } v \\in X^\\frac{1}{2}([0, T]) \\subset C([0, T]; H^\\frac{1}{2}(\\mathbb {R}^3))$ such that the second map $\\Psi $ is continuous in $(v_0, z_1) \\in H^\\frac{1}{2}(\\mathbb {R}^3)\\times S^s_1([0, T])$ .","$\\phi ^\\omega \\in H^s(\\mathbb {R}^3)\\longmapsto z_1 \\stackrel{\\Psi }{\\longmapsto } v \\in X^\\frac{1}{2}([0, T]) \\subset C([0, T]; H^\\frac{1}{2}(\\mathbb {R}^3)),$ where the first map can be viewed as a universal lift map and the second map $\\Psi $ is the solution map to (REF ), which is in fact continuous in $z_1 \\in S^s_1([0, T])$ .Here, $S^s_1([0, T]) \\subset C([0, T]; H^s(\\mathbb {R}^3))$ denotes the intersection of suitable space-time function spaces of differentiability at most $s$ .","In the following, we use $S^{\\sigma }_j([0, T])$ in a similar manner.","In the case of Theorem REF (with $\\sigma = \\frac{1}{2}$ ), we have the following factorization of the solution map for (REF ): $ \\phi ^\\omega \\in H^s(\\mathbb {R}^3)\\longmapsto (z_1, z_3 ) \\stackrel{\\Psi }{\\longmapsto } v \\in X^\\frac{1}{2}([0, T]) \\subset C([0, T]; H^\\frac{1}{2}(\\mathbb {R}^3)),$ where the second map $\\Psi $ is the solution map to (REF ), which is continuous from $(z_1, z_3) \\in S^s_1([0, T])\\times S^{2s-}_3([0, T])$ to $v \\in X^\\frac{1}{2}([0, T])$ .","In the case of Theorem REF , we create $k$ stochastic objects in the first step, where $k = k(s) \\in \\mathbb {N}$ : $\\phi ^\\omega \\in H^s(\\mathbb {R}^3)\\longmapsto (\\zeta _1, \\zeta _3, \\dots , \\zeta _{2k-1} ) \\stackrel{\\Psi }{\\longmapsto } v \\in X^\\frac{1}{2}([0, T]) \\subset C([0, T]; H^\\frac{1}{2}(\\mathbb {R}^3)).$ Once again, the second map $\\Psi $ (which is the solution map to (REF )) is continuous from $(\\zeta _1, \\zeta _3, \\dots , \\zeta _{2k-1} ) \\in S^s_1([0, T]) \\times \\prod _{\\ell = 2}^ k S_\\ell ^{\\alpha _\\ell s-} ([0, T])$ to $v \\in X^\\frac{1}{2}([0, T])$ .","We point out an analogy between these factorizations (REF ), (REF ), and (REF ) of the ill-posed solution maps into (i) the first step, involving stochastic analysis and (ii) the second step, where purely deterministic analysis is performed in constructing a continuous map $\\Psi $ and similar factorizations for studying rough differential equations via the rough path theory [19] and singular stochastic parabolic PDEs [21], [23].","In the proof of Theorem REF , we could consider the following expansion of an infinite order: $u = \\sum _{\\ell = 1}^\\infty \\zeta _{2\\ell -1} + v.$ This would allow us to present a single argument that works for all $\\frac{1}{6} < s < \\frac{1}{3}$ .","In this case, the residual part $v := u - \\sum _{\\ell = 1}^\\infty \\zeta _{2\\ell -1}$ satisfies ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}\\bigg (v +\\sum _{\\ell = 1}^{\\infty } \\zeta _{2\\ell -1}\\bigg )-\\sum _{\\ell = 2}^{\\infty }\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 2\\ell - 1\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2\\ell -3\\rbrace \\end{array}}\\zeta _{j_1} \\overline{\\zeta _{j_2}}\\zeta _{j_3}\\\\v|_{t = 0} = 0.\\end{array}\\right.","}$ In particular, we would need to worry about the convergence issue of infinite series and hence there seems to be no simplification in considering the infinite order expansion (REF ).","Another strategy would be to treat $\\zeta _\\infty : = \\sum _{\\ell = 1}^\\infty \\zeta _{2\\ell -1} $ as one stochastic object and write $u =\\zeta _\\infty + v.$ It follows from (REF ) that $\\zeta _\\infty $ satisfies the following equation: ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _t\\zeta _\\infty + \\Delta \\zeta _\\infty =\\sum _{\\ell = 2}^{\\infty }\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 2\\ell - 1\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2\\ell -3\\rbrace \\end{array}}\\zeta _{j_1} \\overline{\\zeta _{j_2}}\\zeta _{j_3}\\\\\\zeta _\\infty |_{t = 0} = \\phi ^\\omega .\\end{array}\\right.","}$ Noting that $\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 2\\ell - 1\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2\\ell -3\\rbrace \\end{array}}\\zeta _{j_1} \\overline{\\zeta _{j_2}}\\zeta _{j_3}= {\\left\\lbrace \\begin{array}{ll}|z_1|^2 z_1& \\text{when }\\ell = 2,\\\\2|z_1|^2\\zeta _{2\\ell -3} + z_1^2 \\overline{\\zeta _{2\\ell -3}},& \\text{when }\\ell \\ge 3,\\end{array}\\right.","}$ we can rewrite (REF ) as ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _t\\zeta _\\infty + \\Delta \\zeta _\\infty =2|z_1|^2 \\zeta _\\infty + z_1^2 \\overline{\\zeta _\\infty }- 2 |z_1|^2 z_1\\\\\\zeta _\\infty |_{t = 0} = \\phi ^\\omega .\\end{array}\\right.","}$ and thus $v : = u - \\zeta _\\infty $ satisfies ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}(v +\\zeta _\\infty )- 2|z_1|^2 \\zeta _\\infty - z_1^2 \\overline{\\zeta _\\infty }+ 2 |z_1|^2 z_1\\\\v|_{t = 0} = 0.\\end{array}\\right.","}$ The equations (REF ) and (REF ) do not particularly appear to be in such a friendly format.","Namely, studying (REF ) and (REF ) does not seem to provide a simplification over the case-by-case analysis (REF ) for the equation (REF ) for each fixed $k \\in \\mathbb {N}$ .","In a recent work [40], the second author (with Tzvetkov and Y. Wang) proved invariance of the white noise for the (renormalized) cubic fourth order NLS on the circle.","One novelty of this work is that we introduced an infinite sequence $\\lbrace z^\\text{res}_{2\\ell -1}\\rbrace _{\\ell \\in \\mathbb {N}}$ of stochastic $(2\\ell -1)$ -linear objects (depending only on the random initial data) and considered the following expansion of an infinite order: $ u = \\sum _{\\ell = 1}^\\infty z^\\text{res}_{2\\ell -1} + v.$ For this problem, it turned out that $z^\\text{res}_\\infty : = \\sum _{\\ell = 1}^\\infty z^\\text{res}_{2\\ell -1}$ satisfies a particularly simple equation.In fact, the series $z^\\text{res}_\\infty = \\sum _{\\ell = 1}^\\infty z^\\text{res}_{2\\ell -1}$ corresponds to the power series expansion of the resonant cubic fourth order NLS.","We point out that $z^\\text{res}_\\infty $ does not belong to the span of Wiener homogeneous chaoses of any finite order.","Then by treating $z^\\text{res}_\\infty $ as one stochastic object, we wrote a solution $u$ as $u = z^\\text{res}_\\infty +v$ , which led to the following factorization: $\\phi ^\\omega \\in H^s(\\longmapsto z^\\text{res}_\\infty = \\sum _{\\ell = 1}^\\infty z^\\text{res}_{2\\ell -1}\\longmapsto v \\in C(\\mathbb {R}; L^2()$ for $s < -\\frac{1}{2}$ , where $\\phi ^\\omega $ denotes the Gaussian white noise on the circle.","Remark 1.12 In [44], the third author (with Y. Wang) recently studied probabilistic local well-posedness of NLS on $\\mathbb {R}^d$ within the framework of the $L^p$ -based Sobolev spaces, using the dispersive estimate.","In the context of the cubic NLS (REF ) on $\\mathbb {R}^3$ , their result yields almost sure local existence of a unique solution $u$ for the randomized initial data $\\phi ^\\omega \\in H^s(\\mathbb {R}^3)$ , provided $s\\ge 0$ .","In particular, the argument in [44] allows us to consider random initial data of lower regularities than Theorems REF , REF , and REF .","Note that, in [44], it was shown that the solution $u$ only belongs to $C([0, T]; W^{s, 4}(\\mathbb {R}^3))$ , almost surely.","We point out that a slight adaptation of the work [39] by the second and third authors (with Y. Wang) shows that the solution $u$ indeed lies in $C([0, T]; H^s(\\mathbb {R}^3))$ .","As compared to [44], the argument presented in this paper provides extra regularity information, namely the decomposition (REF ) of the solution $u$ with the terms of increasing regularities: $ \\zeta _{2\\ell -1} \\in X^{\\alpha _\\ell \\cdot s-}_\\textup {loc} $ and $v \\in X^\\frac{1}{2}([0, T])$ .","In particular, the residual term $v$ lies in the (sub)critical regularity,By slightly tweaking the argument, we can easily place $v$ in $X^{\\frac{1}{2}+}([0, T])$ .","leaving us a possibility of adapting deterministic techniques to study its further properties.","This paper is organized as follows.","In Section , we recall probabilistic and deterministic lemmas along with the definitions of the basic function spaces.","In Section , we study the regularity properties of the second order term $z_3$ in (REF ).","In Section , we further investigate the regularity properties of the higher order terms $z_5$ and $z_7$ and the unbalanced higher order terms $\\zeta _{2k-1}$ .","Note that the analysis on $z_5$ and $z_7$ contains partNote that $z_7$ defined in (REF ) contains the contribution from $z_{j_1}\\overline{z_{j_2}}z_{j_3}$ with $(j_1, j_2, j_3) = (1, 3, 3)$ up to permutations.","of the case-by-case analysis (REF ) needed for proving Theorem REF .","In Section , we then carry out the rest of the case-by-case analysis (REF ) and prove Theorem REF .","In Section , we briefly describe the proof of Theorem REF by indicating how the analysis in the previous sections can lead to the proof.","In Appendix , we prove the deterministic non-smoothing of the Duhamel integral operator discussed in Remark REF .","Notations: We use $a+$ (and $a-$ ) to denote $a + \\varepsilon $ (and $a - \\varepsilon $ , respectively) for arbitrarily small $\\varepsilon \\ll 1$ , where an implicit constant is allowed to depend on $\\varepsilon > 0$ (and it usually diverges as $\\varepsilon \\rightarrow 0$ ).","Given a Banach space $B$ of temporal functions, we use the following short-hand notation: $B_T := B([0, T])$ .","For example, $L^q_T L^r_x = L^q_t([0, T]; L^r_x(\\mathbb {R}^3))$ .","Let $\\eta : \\mathbb {R}\\rightarrow [0, 1]$ be an even, smooth cutoff function supported on $[-\\frac{8}{5}, \\frac{8}{5}]$ such that $\\eta \\equiv 1$ on $[-\\frac{5}{4}, \\frac{5}{4}]$ .","Given a dyadic number $N \\ge 1$ , we set $\\eta _1(\\xi ) = \\eta (|\\xi |)$ and $\\eta _N(\\xi ) = \\eta \\bigg (\\frac{|\\xi |}{N}\\bigg ) - \\eta \\bigg (\\frac{2|\\xi |}{N}\\bigg )$ for $N \\ge 2$ .","Then, we define the Littlewood-Paley projection operator $\\mathbf {P}_N$ as the Fourier multiplier operator with symbol $\\eta _N$ .","In the following, we use the convention that capital letters denote dyadic numbers.","For example, $N = 2^n$ for some $n \\in \\mathbb {N}_0 : = \\mathbb {N}\\cup \\lbrace 0\\rbrace $ .","Given dyadic numbers $N_1, \\dots , N_4 \\in 2^{\\mathbb {N}_0}$ , we set $N_{\\max } := \\max _{j = 1, \\dots , 4} N_j$ .","We also use the following shorthand notation: $f_N = \\mathbf {P}_N f$ .","For example, we have $z_{1, N_j} = \\mathbf {P}_{N_j} z_1$ ."],["Probabilistic Strichartz estimates","First, we recall the usual Strichartz estimates on $\\mathbb {R}^3$ for readers' convenience.","We say that a pair $(q, r)$ is Schrödinger admissible if it satisfies $\\frac{2}{q} + \\frac{3}{r} = \\frac{3}{2}$ with $2\\le q, r \\le \\infty $ .","Then, the following Strichartz estimates are known to hold [46], [49], [20], [28]: $\\Vert S(t) \\phi \\Vert _{L^q_t L^r_x (\\mathbb {R}\\times \\mathbb {R}^3)} \\lesssim \\Vert \\phi \\Vert _{L^2_x(\\mathbb {R}^3)}.$ It follows from (REF ) and Sobolev's inequality that $\\Vert S(t) \\phi \\Vert _{L^p_{t, x} (\\mathbb {R}\\times \\mathbb {R}^3)}\\lesssim \\big \\Vert |\\nabla |^{\\frac{3}{2} - \\frac{5}{p}}\\phi \\big \\Vert _{L^2_x(\\mathbb {R}^3)}$ for $p \\ge \\frac{10}{3}$ .","We will use the following admissible pairs in this paper: $(\\infty , 2), \\ \\big (5, \\tfrac{30}{11}\\big ), \\ \\big (\\tfrac{10}{3}\\tfrac{10}{3}\\big ),\\ (2+, 6-).$ In particular, by Sobolev's inequality, we have $W^{\\frac{1}{2}, \\frac{30}{11}}(\\mathbb {R}^3) \\hookrightarrow L^5(\\mathbb {R}^3).$ One of the important key ingredients for probabilistic well-posedness is the probabilistic Strichartz estimates.","Such probabilistic estimates were first exploited by McKean [34] and Bourgain [5].","In the following, we state the probabilistic Strichartz estimates under the Wiener randomization (REF ).","See [1] for the proofs.","Lemma 2.1 Given $\\phi $ on $\\mathbb {R}^3$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).","Then, given finite $q \\ge 2$ and $2 \\le r \\le \\infty $ , there exist $C, c>0$ such that $P\\Big ( \\Vert S(t) \\phi ^\\omega \\Vert _{L^q_t L^r_x([0, T]\\times \\mathbb {R}^3)}> \\lambda \\Big )\\le C\\exp \\bigg (-c \\frac{\\lambda ^2}{ T^\\frac{2}{q}\\Vert \\phi \\Vert _{H^s}^{2}}\\bigg )$ for all $ T > 0$ and $\\lambda >0$ with (i) $s = 0$ if $r < \\infty $ and (ii) $s > 0$ if $r = \\infty $ .","A similar estimate holds when $q = \\infty $ (with $s > 0$ ) but we will not need it in this paper.","See [38].","We also need the following lemma on the control of the size of $H^s$ -norm of $\\phi ^\\omega $ .","Lemma 2.2 Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).","Then, we have $P\\Big ( \\Vert \\phi ^\\omega \\Vert _{ H^s( \\mathbb {R}^3)} > \\lambda \\Big )\\le C\\exp \\bigg (-c \\frac{\\lambda ^2}{ \\Vert \\phi \\Vert _{H^s}^{2}}\\bigg )$ for all $\\lambda >0$ ."],["Function spaces and their properties","In this subsection, we go over the basic definitions and properties of the functions spaces used for the Fourier restriction norm method (i.e.","analysis involving the $X^{s, b}$ -spaces introduced in [4]) adapted to the space of functions of bounded $p$ -variation and its pre-dual, introduced and developed by Tataru, Koch, and their collaborators [31], [22], [24].","We refer readers to [22], [24] for the proofs of the basic properties.","See also [2].","Let $\\mathcal {Z}$ be the set of finite partitions $-\\infty 2$ .","Given an interval $I \\subset \\mathbb {R}$ , we define the local-in-time versions $X^s(I)$ and $Y^s(I)$ of these spaces as restriction norms.","For example, we define the $X^s(I)$ -norm by $ \\Vert u \\Vert _{X^s(I)} = \\inf \\big \\lbrace \\Vert v\\Vert _{X^s(\\mathbb {R})}: \\, v|_I = u\\big \\rbrace .$ We also define the norm for the nonhomogeneous term on an interval $I = [t_0, t_1)$ : $\\Vert F\\Vert _{N^s(I)} = \\bigg \\Vert \\int _{t_0}^t S(t - t^{\\prime }) F(t^{\\prime }) dt^{\\prime }\\bigg \\Vert _{X^s(I)}.$ We conclude this section by presenting some basic estimates involving these function spaces.","See [22], [24], [2] for the proofs.","Lemma 2.5 Let $s \\in \\mathbb {R}$ and $T\\in ( 0,\\infty ]$ .","Then, the following linear estimates hold: $\\Vert S(t) \\phi \\Vert _{X^s([0, T])}& \\le \\Vert \\phi \\Vert _{H^s}, \\\\\\Vert F \\Vert _{N^{s}([0, T])}& \\le \\sup _{\\begin{array}{c}w \\in Y^{-s}([0, T])\\\\ \\Vert w \\Vert _{Y^{-s}([0, T])}=1\\end{array}} \\left| \\int _{0}^{T} \\langle F(t), w(t) \\rangle _{L_{x}^{2}} dt \\right|$ for any $\\phi \\in H^s(\\mathbb {R}^3)$ and $F \\in L^1([0,T];H^{s}(\\mathbb {R}^3))$ .","The transference principle [22] and the interpolation lemma [22] applied on the Strichartz estimates (REF ) and (REF ) imply the following estimates.","Lemma 2.6 Given any admissible pair $(q,r)$ with $q>2$ and $p \\ge \\frac{10}{3}$ , we have $\\Vert u \\Vert _{L_t^q L_x^r} & \\lesssim \\Vert u \\Vert _{Y^0}, \\\\\\Vert u \\Vert _{L^p_{t,x}}& \\lesssim \\big \\Vert |\\nabla |^{\\frac{3}{2} - \\frac{5}{p}} u\\big \\Vert _{Y^0}.$ Similarly, the bilinear refinement of the Strichartz estimate [6], [41], [14] implies the following bilinear estimate.","Lemma 2.7 Let $N_1, N_2 \\in 2^{\\mathbb {N}_0}$ with $N_1 \\le N_2$ .","Then, we have $\\Vert \\mathbf {P}_{N_1}u_1 \\mathbf {P}_{N_2}u_2\\Vert _{L^2_{t} ([0, T]; L^2_x)}\\lesssim T^{0+}N_1^{1-} N_2^{-\\frac{1}{2}+}\\Vert \\mathbf {P}_{N_1} u_1 \\Vert _{Y^0([0, T])} \\Vert \\mathbf {P}_{N_2} u_2 \\Vert _{Y^0([0, T])}$ for any $T > 0$ and $u_1, u_2 \\in Y^0([0, T])$ .","From the bilinear refinement of the Strichartz estimate [6], [14] and the transference principle, we have $\\Vert \\mathbf {P}_{N_1}u_1 \\mathbf {P}_{N_2}u_2\\Vert _{L^2_{t,x}}\\lesssim N_1^{1-} N_2^{-\\frac{1}{2}+} \\Vert \\mathbf {P}_{N_1} u_1 \\Vert _{Y^0} \\Vert \\mathbf {P}_{N_2} u_2 \\Vert _{Y^0}$ for all $u_1, u_2 \\in Y^0$ .","See [2] for the proof of (REF ).","On the other hand, it follows from Hölder's and Sobolev's inequalities that $\\Vert \\mathbf {P}_{N_1}u_1 \\mathbf {P}_{N_2}u_2\\Vert _{L^2_TL^2_x}& \\lesssim T^\\frac{1}{2} \\Vert \\mathbf {P}_{N_1} u_1 \\Vert _{L^\\infty _TL^4_x} \\Vert \\mathbf {P}_{N_2} u_2 \\Vert _{L^\\infty _TL^4_x}\\\\& \\lesssim T^\\frac{1}{2} N_1^\\frac{3}{4}N_2^\\frac{3}{4} \\Vert \\mathbf {P}_{N_1} u_1 \\Vert _{Y^0_T} \\Vert \\mathbf {P}_{N_2} u_2 \\Vert _{Y^0_T} .$ Then, the estimate (REF ) follows from interpolating (REF ) and (REF )."],["On the second order term $z_3$","In this and the next sections, we study the regularity properties of the various stochastic terms that appear in the iterative procedures.","Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be the Wiener randomization of $\\phi $ defined in (REF ) and set $z_1 = S(t) \\phi ^\\omega .$ In this section, we study the regularity properties of the second order term: $z_3 = -i \\int _0^t S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime }) dt^{\\prime },$ We first present the proof of Proposition REF .","We follow closely the argument in [2].","(i) By Lemma REF , the estimate (REF ) follows once we prove $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}\\langle \\nabla \\rangle ^\\sigma ( z_1 \\overline{z_1} z_1 )\\overline{w} dx dt\\bigg |\\le T^\\theta C(\\omega , \\Vert \\phi \\Vert _{H^s})$ for some almost surely finite constant $ C(\\omega , \\Vert \\phi \\Vert _{H^s} ) > 0$ and $\\theta > 0$ , where $\\Vert w\\Vert _{Y^{0}_T} \\le 1$ .","In the following, we drop the complex conjugate when it does not play any role.","Define $A^s_3(T)$ by $\\Vert z_1\\Vert _{A^s_3(T)}: = \\max \\Big (\\Vert \\langle \\nabla \\rangle ^s z_{1} \\Vert _{L^{\\frac{30}{7}+}_{T}L^\\frac{30}{7}_x},\\Vert \\langle \\nabla \\rangle ^sz_{1} \\Vert _{L^{4}_{T, x}},\\Vert z_1(0)\\Vert _{H^s} \\Big ).$ Then, by applying the dyadic decomposition, it suffices to prove $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}\\mathbf {P}_{N_1} z_1 \\cdot \\mathbf {P}_{N_2} z_1 \\cdot N_3^\\sigma \\mathbf {P}_{N_3} z_1 \\cdot \\mathbf {P}_{N_4} w \\, dx dt\\bigg |\\le T^\\theta N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_3(T)})$ for all $N_1, \\dots , N_4 \\in 2^{\\mathbb {N}_0}$ with $N_3 \\ge N_2 \\ge N_1$ .","Once we prove (REF ), the desired estimate (REF ) follows from summing (REF ) over dyadic blocks and applying Lemmas REF and REF .","Recall our shorthand notation: $z_{1, N_j} = \\mathbf {P}_{N_j} z_1$ and $w_{N_4} = \\mathbf {P}_{N_4} w$ .","In the following, we assume $\\sigma < 2s\\qquad \\text{and}\\qquad 0 \\le s < 1.$ $\\bullet $ Case (1): $N_2 \\sim N_3$ .","By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {O2})& \\lesssim \\Vert z_{1, N_1} \\Vert _{L^\\frac{30}{7}_{T, x}}\\bigg (\\prod _{j = 2}^3 \\Vert \\langle \\nabla \\rangle ^\\frac{\\sigma }{2} z_{1, N_j} \\Vert _{L^{\\frac{30}{7}}_{T, x}}\\bigg )\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_3(T)}) \\Vert w_{N_4}\\Vert _{Y^0_T},$ provided that (REF ) holds.","$\\bullet $ Case (2): $N_3 \\sim N_4 \\gg N_1, N_2$ .","$\\circ $ Subcase (2.a): $N_1, N_2 \\ll N_3^\\frac{1}{2}$ .","By Cauchy-Schwarz' inequality and Lemma REF followed by Lemma REF ,In the remaining part of this paper, we repeatedly apply this argument when there is a frequency separation.","We shall simply refer to it as the “bilinear Strichartz estimate” argument.","we have $\\text{LHS of } (\\ref {O2})& \\le N_3^\\sigma \\Vert z_{1, N_1} z_{1, N_3} \\Vert _{L^2_{T, x}} \\Vert z_{1, N_2} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1-s-} N_2^{1-s-} N_3^{\\sigma - s - 1 +}\\bigg (\\prod _{j = 1}^3\\Vert \\mathbf {P}_{N_j} \\phi ^\\omega \\Vert _{H^{s}}\\bigg )\\Vert w_{N_4}\\Vert _{Y^0_T}\\\\& \\lesssim T^{0+} N_3^{\\sigma - 2s +}\\prod _{j = 1}^3\\Vert \\mathbf {P}_{N_j} \\phi ^\\omega \\Vert _{H^{s}}\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_3(T)}),$ provided that (REF ) holds.","$\\circ $ Subcase (2.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .","By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {O2})& \\le N_3^\\sigma \\bigg (\\prod _{j = 1}^3 \\Vert z_{1, N_j} \\Vert _{L^\\frac{30}{7}_{T, x}}\\bigg )\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{-s} N_2^{-s}N_3^{\\sigma -s}C(\\Vert z_1\\Vert _{A^s_3(T)})\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_3(T)}),$ provided that (REF ) holds.","$\\circ $ Subcase (2.c): $N_2\\gtrsim N_3^\\frac{1}{2} \\gg N_1$ .","By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {O2})& \\le N_3^\\sigma \\Vert z_{1, N_1} w_{N_4} \\Vert _{L^2_{T, x}}\\prod _{j = 2}^3 \\Vert z_{1, N_j}\\Vert _{L^4_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1-s-} N_2^{-s}N_3^{\\sigma -s-\\frac{1}{2}+}\\Vert \\mathbf {P}_{N_1}\\phi ^\\omega \\Vert _{H^s}\\bigg (\\prod _{j = 2}^3 \\Vert \\langle \\nabla \\rangle ^s z_{1, N_j} \\Vert _{L^4_{T, x}}\\bigg )\\Vert w_{N_4}\\Vert _{Y^0_T}\\\\& \\le T^{0+}N_3^{\\sigma -2s+}C(\\Vert z_1\\Vert _{A^s_3(T)})\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_3(T)}),$ provided that (REF ) holds.","Therefore, putting all the cases together, we obtain (REF ).","(ii) Given $N \\gg 1$ and small $\\ell > 0$ , consider the following deterministic initial condition $\\phi $ whose Fourier transform is given by $\\widehat{\\phi }(\\xi ) = \\mathbf {1}_{ N e_1 + \\ell Q}(\\xi ) + \\mathbf {1}_{(N+100\\ell ) e_1 + \\ell Q}(\\xi )+ \\mathbf {1}_{ N e_1 + 100\\ell e_2 + \\ell Q}(\\xi ),$ where $Q = (-\\frac{1}{2}, \\frac{1}{2}]^3$ , $e_1 = (1, 0, 0)$ , and $e_2 = (0, 1, 0)$ .","By taking $\\ell > 0$ sufficiently small, we have $\\operatornamewithlimits{supp}\\widehat{\\phi }\\subset N e_1 + Q$ and thus we can neglect the effect of the randomization in (REF ) since all the three terms on the right-hand side will be multiplied by a common random number $g_{N e_1}$ .","Without loss of generality, we assume that $g_{N e_1} = 1$ in the following.","We estimate from below the contribution to $\\mathbf {1}_{Q_{N, \\ell }} (\\xi ) \\widehat{z}_3 (t, \\xi ) $ , where $Q_{N, \\ell } = Ne_1 + 100 \\ell (e_1 + e_2) + \\ell Q.$ From (REF ), we have $\\mathbf {1}_{Q_{N, \\ell }} & (\\xi ) \\widehat{z}_3 (t, \\xi ) \\\\& = -i \\mathbf {1}_{Q_{N, \\ell }} (\\xi ) e^{-it |\\xi |^2} \\int _0^t \\operatornamewithlimits{\\int }_{\\xi = \\xi _1 - \\xi _2 + \\xi _3}e^{i t^{\\prime } \\Phi (\\bar{\\xi })} \\widehat{\\phi ^\\omega } (\\xi _1) \\overline{\\widehat{\\phi ^\\omega } (\\xi _2)} \\widehat{\\phi ^\\omega } (\\xi _3) d\\xi _1 d\\xi _2 dt^{\\prime },$ where the phase function $\\Phi (\\bar{\\xi })$ is given by $\\Phi (\\bar{\\xi }) = \\Phi (\\xi , \\xi _1, \\xi _2, \\xi _3)= |\\xi |^2 - |\\xi _1|^2 + |\\xi _2|^2 - |\\xi _3|^2= 2\\langle \\xi - \\xi _1, \\xi - \\xi _3 \\rangle _{\\mathbb {R}^3}.$ Then, it follows that the only non-trivial contribution to (REF ) appears if $\\xi _1 \\in (N+100\\ell ) e_1 + \\ell Q,\\qquad \\xi _2 \\in N e_1 + \\ell Q,\\qquad \\xi _3 \\in N e_1 + 100\\ell e_2 + \\ell Q$ (up to the permutation $\\xi _1 \\leftrightarrow \\xi _3$ ).","In this case, we have $|\\Phi (\\bar{\\xi })| \\ll 1 $ and thus $\\operatornamewithlimits{Re}e^{i t^{\\prime } \\Phi (\\bar{\\xi })} \\ge \\frac{1}{2}$ for all $t^{\\prime } \\in [0, 1]$ .","Now, recall the following lemma on the convolution.","Lemma 3.1 There exists $c>0$ such that $\\mathbf {1}_{a + \\ell Q}* \\mathbf {1}_{b + \\ell Q} (\\xi )\\ge c \\ell ^3 \\mathbf {1}_{a+b+\\ell Q}(\\xi )$ for all $a, b, \\xi \\in \\mathbb {R}^3$ .","By applying Lemma REF to (REF ) with (REF ) and (REF ), we obtain $|\\mathbf {1}_{Q_{N, \\ell }} (\\xi ) \\widehat{z}_3 (t, \\xi ) |\\gtrsim t \\ell ^6 \\mathbf {1}_{Q_{N, \\ell }} (\\xi ).$ Therefore, for any $\\sigma > 0$ , we have $\\Vert z_3\\Vert _{X^\\sigma ([0, 1])} \\gtrsim \\Vert z_3 \\Vert _{L^\\infty _t([0, 1]; H^\\sigma )}\\gtrsim \\ell ^\\frac{15}{2} N^\\sigma \\longrightarrow \\infty $ as $N \\rightarrow \\infty $ , while $\\Vert \\phi \\Vert _{L^2} \\sim \\ell ^\\frac{3}{2}$ remains bounded.","This in particular implies that when $s = 0$ , the estimate (REF ) can not hold for any $\\sigma > 0$ .","This proves Part (ii).","Remark 3.2 It follows from the proof of Proposition REF (i) that $\\big \\Vert I( u_1, u_2, u_3)\\big \\Vert _{X^\\sigma ([0, 1])}\\lesssim \\prod _{j = 1}^3 \\Vert u_j \\Vert _{A^s_3(T)},$ where $I( u_1, u_2, u_3)$ is as in (REF ).","In particular, the left-hand side of (REF ) is finite for $u_1, u_2, u_3 \\in \\mathcal {R}^s$ .","The only probabilistic component in the proof of Proposition REF (i) appears in applying Lemmas REF and REF to control the $A^s_3(T)$ -norm of $z_1$ in terms of the $H^s$ -norm of $\\phi $ .","In this sense, we exploit the randomization only at the linear level.","On the one hand, Proposition REF shows that $z_3$ controls almost $2s$ derivatives.","On the other hand, we need to measure $z_3$ in the $X^{2s-}$ -norm, which controls only the admissible space-time Lebesgue norms (with $2s-$ derivatives) via Lemma REF .","The following lemma breaks this rigidity by giving up a control on derivatives.","In particular, it allows us to control a wider range of space-time Lebesgue norms of $z_3$ .","The main idea is to use the dispersive estimate for the linear Schrödinger operator: $\\Vert S(t) f \\Vert _{L^r_x} \\lesssim |t|^{-\\frac{3}{2}(1 - \\frac{2}{r})} \\Vert f\\Vert _{L^{r^{\\prime }}_x}.$ This allows us to reduce the analysis to a product of the random linear solution $z_1 = S(t) \\phi ^\\omega $ and apply Lemma REF .","Lemma 3.3 Let $s \\ge 0$ .","Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).","Then, for any finite $q, r \\ge 1$ , we have $\\Vert \\mathbf {P}_N z_3\\Vert _{L^q_T L^r_x}\\lesssim {\\left\\lbrace \\begin{array}{ll}\\rule [-6mm]{0pt}{15pt} T^{\\frac{3}{r} - \\frac{1}{2}}\\Vert z_1\\Vert _{L^{3q}_T L^{3r^{\\prime }}_x}^3 & \\text{when } 1 \\le r < 6, \\\\T^{0+} N^{\\frac{1}{2} - \\frac{3}{r}+} \\Vert z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}^3& \\text{when } r \\ge 6, \\\\\\end{array}\\right.","}$ for any $T > 0$ and $N \\in 2^{\\mathbb {N}_0}$ .","Note that the right-hand side of (REF ) is almost surely finite thanks to the probabilistic Strichartz estimate (Lemma REF ).","We first consider the case $r < 6$ .","From (REF ) and (REF ), we have $\\Vert \\mathbf {P}_N z_3\\Vert _{L^q_T L^r_x}& \\le \\bigg \\Vert \\int _0^t \\Vert \\mathbf {P}_N S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime })\\Vert _{L^r_x} dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim \\bigg \\Vert \\int _0^t \\frac{1}{|t - t^{\\prime }|^{\\frac{3}{2} - \\frac{3}{r}}} \\Vert z_1(t^{\\prime })\\Vert _{L^{3r^{\\prime }}_x}^3 dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim T^{\\frac{3}{r} - \\frac{1}{2}}\\Vert z_1\\Vert _{L^{3q}_T L^{3r^{\\prime }}_x}^3.$ When $r \\ge 6$ , we proceed as in (REF ) but we apply Sobolev's inequality before applying (REF ): $\\Vert \\mathbf {P}_N z_3\\Vert _{L^q_T L^r_x}& \\le \\bigg \\Vert \\int _0^t \\Vert \\mathbf {P}_N S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime })\\Vert _{L^r_x} dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim N^{\\frac{1}{2} - \\frac{3}{r}+} \\bigg \\Vert \\int _0^t \\Vert \\mathbf {P}_N S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime })\\Vert _{L^{6-}_x} dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim N^{\\frac{1}{2} - \\frac{3}{r}+}\\bigg \\Vert \\int _0^t \\frac{1}{|t - t^{\\prime }|^{1-}} \\Vert z_1(t^{\\prime })\\Vert _{L^{\\frac{18}{5}+}_x}^3 dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim T^{0+} N^{\\frac{1}{2} - \\frac{3}{r}+} \\Vert z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}^3 .$ This completes the proof of Lemma REF ."],["On the higher order terms","In this section, we study the regularity properties of the higher order terms."],["On the third order term $z_5$","In this subsection, we study the third order term: $z_5 = -i \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 5\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) z_{j_1} \\overline{z_{j_2}}z_{j_3} (t^{\\prime }) dt^{\\prime }.$ The following lemma shows that the third order term $z_5$ enjoys a gain of extra $\\frac{1}{2}s$ derivative as compared to the second order term $z_3$ (Proposition REF ).","Lemma 4.1 Given $0 < s < \\frac{1}{2} $ , let $\\phi ^\\omega $ be the Wiener randomization of $\\phi \\in H^s(\\mathbb {R}^3)$ defined in (REF ).","Then, for any $\\sigma < \\frac{5}{2} s$ , we have $z_5 \\in X^\\sigma _\\textup {loc} ,$ almost surely.","In particular, there exists an almost surely finite constant $ C(\\omega , \\Vert \\phi \\Vert _{H^s} ) > 0$ and $\\theta > 0$ such that $\\Vert z_5 \\Vert _{X^{\\sigma }([0, T])} \\le T^\\theta C(\\omega , \\Vert \\phi \\Vert _{H^s})$ for any $T > 0$ .","First, note that the only possible combination for $(j_1, j_2, j_3)$ in (REF ) is $(1, 1, 3)$ up to permutations.","The complex conjugate does not play any role in the subsequent analysis and hence we drop the complex conjugate sign and simply study $z_5 \\sim \\int _0^t S(t - t^{\\prime }) z_{1} z_1 z_{3} (t^{\\prime }) dt^{\\prime }.$ By Lemma REF , it suffices to prove $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}\\langle \\nabla \\rangle ^\\sigma ( z_1 z_1 z_3 )w dx dt\\bigg |\\le T^\\theta C(\\omega , \\Vert \\phi \\Vert _{H^s})$ for some almost surely finite constant $ C(\\omega , \\Vert \\phi \\Vert _{H^s} ) > 0$ and $\\theta > 0$ , where $\\Vert w\\Vert _{Y^{0}_T} \\le 1$ .","Define $A^s_5(T)$ by $ \\Vert z_1\\Vert _{A^s_5(T)}: = \\max \\Big (\\Vert \\langle \\nabla \\rangle ^s z_{1} \\Vert _{L^{5+}_T L^{5}_x},\\Vert z_1(0)\\Vert _{H^s},\\Big ).","$ Then, by applying the dyadic decomposition, it suffices to prove $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}N_{\\max }^\\sigma \\mathbf {P}_{N_1} z_1 \\cdot \\mathbf {P}_{N_2} z_1& \\cdot \\mathbf {P}_{N_3} z_3 \\cdot \\mathbf {P}_{N_4} w \\, dx dt\\bigg | \\\\& \\le T^\\theta N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})$ for all $N_1, \\dots , N_4 \\in 2^{\\mathbb {N}_0}$ .","Once we prove (REF ), the desired estimate (REF ) follows from summing over dyadic blocks and applying Lemmas REF and REF and Proposition REF .","In the following, we fix $0 < s < \\tfrac{1}{2}.$ Without loss of generality, we assume that $N_1 \\ge N_2$ .","$\\bullet $ Case (1): $N_1 \\sim N_2 \\sim N_{\\max }$ .","$\\circ $ Subcase (1.a): $N_3 \\ll N_1^\\frac{1}{2}$ .","By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{\\sigma - s - \\frac{1}{2}+} N_2^{-s}N_3^{1 - 2s-}\\Vert \\mathbf {P}_{N_1}\\phi ^\\omega \\Vert _{H^s}\\Vert \\langle \\nabla \\rangle ^s z_{1, N_2} \\Vert _{L^5_{T, x}}\\Vert z_{3, N_3}\\Vert _{Y^{2s-}_T}\\\\& \\le T^{0+}N_1^{\\sigma -3s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < 3s$ .","$\\circ $ Subcase (1.b): $N_3\\gtrsim N_1^\\frac{1}{2} $ .","By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{\\frac{10}{3}}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma -s} N_2^{-s}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -3s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < 3s$ .","$\\bullet $ Case (2): $N_1 \\sim N_3 \\sim N_{\\max } \\gg N_2$ .","$\\circ $ Subcase (2.a): $N_2 \\ll N_1^\\frac{1}{2}$ .","By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}} \\Vert z_{1, N_2} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{\\sigma - s } N_2^{1-s-}N_3^{- 2s-\\frac{1}{2} +}\\Vert \\langle \\nabla \\rangle ^s z_{1, N_1} \\Vert _{L^5_{T, x}}\\Vert \\mathbf {P}_{N_2}\\phi ^\\omega \\Vert _{H^s}\\Vert z_{3, N_3}\\Vert _{Y^{2s-}_T}\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{7}{2}s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{7}{2}s$ .","$\\circ $ Subcase (2.b): $N_2\\gtrsim N_1^\\frac{1}{2} $ .","By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{\\frac{10}{3}}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma -s} N_2^{-s}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{7}{2} s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{7}{2}s$ .","$\\bullet $ Case (3): $N_1 \\sim N_4 \\sim N_{\\max } \\gg N_2, N_3$ .","$\\circ $ Subcase (3.a): $N_2, N_3 \\ll N_1^\\frac{1}{2} $ .","By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} z_{1, N_2} \\Vert _{L^2_{T, x}} \\Vert z_{3, N_3} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{\\sigma -s-\\frac{1}{2} + }N_2^{1-s - } N_3^{1- 2s -}N_4^{-\\frac{1}{2}+}\\bigg (\\prod _{j = 1}^2\\Vert \\mathbf {P}_{N_j} \\phi ^\\omega \\Vert _{H^{s}}\\bigg )\\Vert z_{3, N_3}\\Vert _{Y^{2s-}_T}\\Vert w_{N_4}\\Vert _{Y^0_T}\\\\& \\le T^{0+} N_1^{\\sigma - \\frac{5}{2} s +}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{5}{2}s$ .","$\\circ $ Subcase (3.b): $N_2, N_3\\gtrsim N_1^\\frac{1}{2} $ .","By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{\\frac{10}{3}}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma -s} N_2^{-s}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{5}{2} s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{5}{2}s$ .","$\\circ $ Subcase (3.c): $N_2\\gtrsim N_1^\\frac{1}{2} \\gg N_3$ .","By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} z_{3, N_3} \\Vert _{L^2_{T, x}} \\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma - s- \\frac{1}{2} + } N_2^{-s}N_3^{1- 2s-}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{5}{2}s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{5}{2}s$ .","$\\circ $ Subcase (3.d): $N_3\\gtrsim N_1^\\frac{1}{2} \\gg N_2$ .","By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma - s } N_2^{1-s-}N_3^{- 2s+} N_4^{-\\frac{1}{2} +}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{5}{2}s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{5}{2}s$ .","$\\bullet $ Case (4): $N_3 \\sim N_4 \\gg N_1 \\ge N_2$ .","$\\circ $ Subcase (4.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .","By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_3^\\sigma \\Vert z_{1, N_1} z_{3, N_3} \\Vert _{L^2_{T, x}} \\Vert z_{1, N_2} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1-s- }N_2^{1-s - } N_3^{\\sigma - 2s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+} N_1^{\\sigma - 3 s +}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < 3s$ .","$\\circ $ Subcase (4.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .","By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_3^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{\\frac{10}{3}}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{ -s} N_2^{-s}N_3^{\\sigma -2s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -3 s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < 3s$ .","$\\circ $ Subcase (4.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .","By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_3^\\sigma \\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}} \\Vert z_{1, N_2} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{ - s} N_2^{1-s-}N_3^{\\sigma - 2s- \\frac{1}{2}+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma - 3s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < 3s$ .","Putting all the cases together, we conclude that (REF ) holds, provided that $\\sigma < \\frac{5}{2} s$ .","This completes the proof of Lemma REF ."],["On the fourth order term $z_7$","Next, we study the following fourth order term: $z_7 = -i \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 7\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3, 5\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) z_{j_1} \\overline{z_{j_2}}z_{j_3} (t^{\\prime }) dt^{\\prime }.$ In this case, there are two possibilities for $(j_1, j_2, j_3)$ in (REF ): $(1, 3, 3)$ and $(1, 1, 5)$ up to permutations.","We denote by $\\widetilde{z}_7$ the contribution to $z_7$ from $(j_1, j_2, j_3) = (1, 3, 3)$ (up to permutations).","Note that the contribution to $z_7$ from $(j_1, j_2, j_3) = (1, 1, 5)$ (up to permutations) corresponds to $\\zeta _7$ defined in (REF ).","Dropping the complex conjugate, we have $\\widetilde{z}_7 & \\sim \\int _0^t S(t - t^{\\prime }) z_{1} z_3 z_{3} (t^{\\prime }) dt^{\\prime }, \\\\\\zeta _7 & \\sim \\int _0^t S(t - t^{\\prime }) z_{1} z_1 z_{5} (t^{\\prime }) dt^{\\prime }.$ The following lemma shows that $\\widetilde{z}_7$ and $\\zeta _7$ enjoy a further gain of derivatives as compared to $z_1$ , $z_3$ , and $z_5$ .","Lemma 4.2 Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be the Wiener randomization of $\\phi $ defined in (REF ).","(i) Let $ 0 < s < \\frac{1}{2}$ .","Then, given any $\\sigma < 3s$ , we have $\\widetilde{z}_7 \\in X^\\sigma _\\textup {loc} ,$ almost surely.","(ii) Let $ 0 < s < \\frac{2}{5}$ .","Then, given any $\\sigma < \\frac{11}{4}s$ , we have $\\zeta _7 \\in X^\\sigma _\\textup {loc} ,$ almost surely.","(i) We first estimate $\\widetilde{z}_7$ in (REF ).","Fix $0 < s < \\frac{1}{2}$ .","We proceed as in the proofs of Proposition REF and Lemma REF .","In view of Lemmas REF and REF and Proposition REF , it suffices to prove that there exists $\\theta > 0$ such that $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}N_{\\max }^\\sigma \\mathbf {P}_{N_1} z_1 \\cdot \\mathbf {P}_{N_2} z_3& \\cdot \\mathbf {P}_{N_3} z_3 \\cdot \\mathbf {P}_{N_4} w \\, dx dt\\bigg |\\\\& \\le T^\\theta N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}).$ for all $N_1, \\dots , N_4 \\in 2^{\\mathbb {N}_0}$ and $\\Vert w\\Vert _{Y^{0}_T} \\le 1$ , where $A^s_7(T)$ is given by $ \\Vert z_1\\Vert _{A^s_7(T)}: = \\max \\Big (\\Vert \\langle \\nabla \\rangle ^s z_{1} \\Vert _{L^{5}_{T, x}},\\Vert \\langle \\nabla \\rangle ^s z_{1} \\Vert _{L^{10+}_T L^{10}_x},\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x},\\Vert z_1(0)\\Vert _{H^s}\\Big ).","$ Without loss of generality, we assume that $N_2 \\ge N_3$ .","$\\bullet $ Case (1): $N_1 \\sim N_2 \\sim N_{\\max }$ .","$\\circ $ Subcase (1.a): $N_3 \\ll N_1^\\frac{1}{2}$ .","By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}} \\Vert z_{3, N_2} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{\\sigma - s } N_2^{-2s - \\frac{1}{2} + }N_3^{1 - 2s-}\\Vert \\langle \\nabla \\rangle ^s z_{1, N_1} \\Vert _{L^5_{T, x}}\\bigg (\\prod _{j = 2}^3 \\Vert z_{3, N_j}\\Vert _{Y^{2s-}_T}\\bigg )\\\\& \\le T^{0+}N_1^{\\sigma -4s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 4s$ .","$\\circ $ Subcase (1.b): $N_3\\gtrsim N_1^\\frac{1}{2} $ .","By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{10}_{T, x}}\\Vert z_{3, N_2} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{\\frac{10}{3}}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma -s} N_2^{-2s+}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -4s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 4s$ .","$\\bullet $ Case (2): $N_1 \\sim N_4 \\sim N_{\\max } \\gg N_2 \\ge N_3$ .","$\\circ $ Subcase (2.a): $N_2, N_3 \\ll N_1^\\frac{1}{2} $ .","By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} z_{3, N_2} \\Vert _{L^2_{T, x}} \\Vert z_{3, N_3} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{\\sigma -s-\\frac{1}{2} + }N_2^{1-2s - } N_3^{1- 2s -}N_4^{-\\frac{1}{2}+}\\Vert \\mathbf {P}_{N_1} \\phi ^\\omega \\Vert _{H^{s}}\\bigg (\\prod _{j = 2}^3\\Vert z_{3, N_j}\\Vert _{Y^{2s-}_T}\\bigg )\\\\& \\le T^{0+} N_1^{\\sigma - 3 s +}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 3s$ .","$\\circ $ Subcase (2.b): $N_2, N_3\\gtrsim N_1^\\frac{1}{2} $ .","By $L^{10}_{T, x}, L^{\\frac{10}{3}}_{T, x},L^{\\frac{10}{3}}_{T, x}, L^{\\frac{10}{3}}_{T, x}$ -Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\le T^{0+}N_1^{\\sigma -s} N_2^{-2s+}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -3 s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 3s$ .","$\\circ $ Subcase (2.c): $N_2\\gtrsim N_1^\\frac{1}{2} \\gg N_3$ .","By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_2} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{3, N_3} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma - s } N_2^{-2s+}N_3^{1- 2s-}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma - 3s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 3s$ .","$\\bullet $ Case (3): $N_2 \\sim N_3 \\sim N_{\\max } \\gg N_1$ .","$\\circ $ Subcase (3.a): $N_1 \\ll N_2^\\frac{1}{2}$ .","By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_2^\\sigma \\Vert z_{1, N_1} z_{3, N_2} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{ 1- s - } N_2^{\\sigma -2s - \\frac{1}{2} + }N_3^{-s}\\Vert \\mathbf {P}_{N_1} \\phi ^\\omega \\Vert _{H^s}\\Vert z_{3, N_2}\\Vert _{Y^{2s-}_T}\\Vert \\langle \\nabla \\rangle ^s z_{3, N_3} \\Vert _{L^{5}_{T, x}}\\\\\\multicolumn{2}{l}{\\text{By applying Lemma \\ref {LEM:Z3_2}and the fractional Leibniz rule,}}\\\\& \\lesssim T^{\\frac{1}{10}+}N_2^{\\sigma -\\frac{7}{2} s+}\\Vert \\mathbf {P}_{N_1} \\phi ^\\omega \\Vert _{H^s}\\Vert z_{3, N_2}\\Vert _{Y^{2s-}_T}\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\\\& \\le T^{\\frac{1}{10}+}N_2^{\\sigma -\\frac{7}{2}s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < \\frac{7}{2}s$ .","$\\circ $ Subcase (3.b): $N_1\\gtrsim N_2^\\frac{1}{2} $ .","By $L^{10}_{T, x}, L^{\\frac{10}{3}}_{T, x},L^{\\frac{10}{3}}_{T, x}, L^{\\frac{10}{3}}_{T, x}$ -Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\le T^{0+}N_1^{ -s} N_2^{\\sigma -2s+}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{9}{2}s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < \\frac{9}{2}s$ .","$\\bullet $ Case (4): $N_2 \\sim N_4 \\gg N_1, N_3$ .","$\\circ $ Subcase (4.a): $N_1, N_3 \\ll N_2^\\frac{1}{2} $ .","By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_2^\\sigma \\Vert z_{1, N_1} z_{3, N_2} \\Vert _{L^2_{T, x}} \\Vert z_{3, N_3} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\le T^{0+} N_1^{1-s- }N_2^{\\sigma -2s -\\frac{1}{2}+ } N_3^{1- 2s-}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+} N_1^{\\sigma - \\frac{7}{2} s +}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < \\frac{7}{2}s$ .","$\\circ $ Subcase (4.b): $N_1, N_3\\gtrsim N_2^\\frac{1}{2} $ .","By $L^{10}_{T, x}, L^{\\frac{10}{3}}_{T, x},L^{\\frac{10}{3}}_{T, x}, L^{\\frac{10}{3}}_{T, x}$ -Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\le T^{0+}N_1^{ -s} N_2^{\\sigma -2s+}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_2^{\\sigma -\\frac{7}{2} s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < \\frac{7}{2}s$ .","$\\circ $ Subcase (4.c): $N_1\\gtrsim N_2^\\frac{1}{2} \\gg N_3$ .","By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_2^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}} \\Vert z_{3, N_2} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{ - s} N_2^{\\sigma -2s-\\frac{1}{2} + }N_3^{1- 2s-}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{7}{2}s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < \\frac{7}{2}s$ .","$\\circ $ Subcase (4.d): $N_3\\gtrsim N_2^\\frac{1}{2} \\gg N_1$ .","By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_2^\\sigma \\Vert z_{1, N_1} z_{3, N_2} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{ 1- s - } N_2^{\\sigma -2s - \\frac{1}{2} + }N_3^{-s}\\Vert \\mathbf {P}_{N_1} \\phi ^\\omega \\Vert _{H^s}\\Vert z_{3, N_2}\\Vert _{Y^{2s-}_T}\\Vert \\langle \\nabla \\rangle ^s z_{3, N_3} \\Vert _{L^{5}_{T, x}}\\\\\\multicolumn{2}{l}{\\text{Proceeding as in Subcase (3.a) with Lemma \\ref {LEM:Z3_2},}}\\\\& \\le T^{\\frac{1}{10}}N_1^{\\sigma - 3s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 3s$ .","Putting all the cases together, we conclude that (REF ) holds when $ \\sigma < 3s$ .","(ii) Next, we estimate $\\zeta _7$ in ().","This term has a similar structure to $z_5$ in (REF ); the only difference appears in the third factor.","Hence, we can estimate $\\zeta _7$ simply by replacing the regularity $2s-$ (for $z_3$ ; see Proposition REF ) with $\\frac{5}{2} s-$ (for $z_5$ ; see Lemma REF ) in the proof of Lemma REF .","In the following, we only indicate the necessary modifications on the powers of dyadic parameters in the proof of Lemma REF .","$\\bullet $ Case (1): $N_1 \\sim N_2 \\sim N_{\\max }$ .","$\\circ $ Subcase (1.a): $N_3 \\ll N_1^\\frac{1}{2}$ .","It suffices to note that $N_1^{\\sigma - s - \\frac{1}{2}+} N_2^{-s}N_3^{1 - \\frac{5}{2} s-}& \\lesssim N_1^{\\sigma - \\frac{13}{4} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $ \\sigma < \\frac{13}{4} s$ and $0 < s < \\frac{2}{5}$ .","The modification for Subcase (1.b) is straightforward under the same regularity restriction.","$\\bullet $ Case (2): $N_1 \\sim N_3 \\sim N_{\\max } \\gg N_2$ .","$\\circ $ Subcase (2.a): $N_2 \\ll N_1^\\frac{1}{2} $ .","It suffices to note that $N_1^{\\sigma - s } N_2^{1-s-}N_3^{- \\frac{5}{2}s-\\frac{1}{2} +}\\lesssim N_1^{\\sigma - 4 s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < 4s$ and $0 \\le s < 1$ .The lower bound on $s$ is needed only for Subcase (2.b).","Similar comments apply to the following cases and also to the proof of Proposition REF .","The modification for Subcase (2.b) is straightforward under the same regularity restriction.","$\\bullet $ Case (3): $N_1 \\sim N_4 \\sim N_{\\max } \\gg N_2, N_3$ .","$\\circ $ Subcase (3.a): $N_2, N_3 \\ll N_1^\\frac{1}{2} $ .","It suffices to note that $N_1^{\\sigma -s-\\frac{1}{2} + }N_2^{1-s - } N_3^{1- \\frac{5}{2}s -}N_4^{-\\frac{1}{2}+}\\lesssim N_1^{\\sigma - \\frac{11}{4} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < \\frac{11}{4}s\\qquad \\text{and}\\qquad 0 < s < \\frac{2}{5}.$ The modifications for Subcases (3.b), (3.c), and (3.d) are straightforward under the regularity restriction (REF ).","$\\bullet $ Case (4): $N_3 \\sim N_4 \\gg N_1 \\ge N_2$ .","$\\circ $ Subcase (4.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .","It suffices to note that $N_1^{1-s- }N_2^{1-s - } N_3^{\\sigma - \\frac{5}{2} s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\lesssim N_1^{\\sigma - \\frac{7}{2} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < \\frac{7}{2} s$ and $0 \\le s < 1$ .","The modifications for Subcases (4.b) and (4.c) are straightforward under the same regularity restriction.","Putting all the cases together, we conclude that (REF ) holds under the regularity restriction (REF ).","This completes the proof of Lemma REF ."],["On the “unbalanced” higher order terms $\\zeta _{2k-1}$","In this subsection, we study the regularity properties of the unbalanced higher order terms $\\zeta _{2k-1}$ defined in (REF ).","Note that $(j_1, j_2, j_3) = (1, 1, 2k-3)$ up to permutations.","Then, by dropping the complex conjugate, we have $\\zeta _{2k-1}\\sim \\int _0^t S(t - t^{\\prime }) z_{1} z_1 \\zeta _{2k-3} (t^{\\prime }) dt^{\\prime }.$ We first present the proof of Proposition REF .","We proceed by induction.","From (REF ), we see that the recursive relation (REF ) is satisfied when $k = 2, 3, 4$ .","In the following, by assuming $\\zeta _{2k -3} \\in X^\\sigma _\\textup {loc}$ for $\\sigma < \\alpha _{k-1} \\cdot s $ almost surely, where $\\alpha _{k-1}$ satisfies (REF ), we prove (REF ) and (REF ).","As in the proof of Lemma REF (ii), the proof follows from the proof of Lemma REF by replacing $z_3$ and $2s-$ with $\\zeta _{2k-3}$ and $\\alpha _{k-1} \\cdot s-$ , respectively.","Therefore, we only indicate the necessary modifications on the powers of dyadic parameters in the proof of Lemma REF .","$\\bullet $ Case (1): $N_1 \\sim N_2 \\sim N_{\\max }$ .","$\\circ $ Subcase (1.a): $N_3 \\ll N_1^\\frac{1}{2}$ .","It suffices to note that $N_1^{\\sigma - s - \\frac{1}{2}+} N_2^{-s}N_3^{1 - \\alpha _{k-1} s-}& \\lesssim N_1^{\\sigma - \\frac{\\alpha _{k-1}+4}{2} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < \\frac{\\alpha _{k-1}+4}{2} s\\qquad \\text{and}\\qquad 0 < s < \\alpha _{k-1}^{-1}.$ The modification for Subcase (1.b) is straightforward, giving the regularity restriction (REF ).","$\\bullet $ Case (2): $N_1 \\sim N_3 \\sim N_{\\max } \\gg N_2$ .","$\\circ $ Subcase (2.a): $N_2 \\ll N_1^\\frac{1}{2} $ .","It suffices to note that $N_1^{\\sigma - s } N_2^{1-s-}N_3^{- \\alpha _{k-1}s-\\frac{1}{2} +}\\lesssim N_1^{\\sigma - \\frac{2 \\alpha _{k-1} + 3}{2} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < \\frac{2\\alpha _{k-1}+3}{2}s\\qquad \\text{and}\\qquad 0 \\le s < 1.$ The modification for Subcase (2.b) is straightforward, giving the regularity restriction (REF ).","$\\bullet $ Case (3): $N_1 \\sim N_4 \\sim N_{\\max } \\gg N_2, N_3$ .","$\\circ $ Subcase (3.a): $N_2, N_3 \\ll N_1^\\frac{1}{2} $ .","It suffices to note that $N_1^{\\sigma -s-\\frac{1}{2} + }N_2^{1-s - } N_3^{1- \\alpha _{k-1}s -}N_4^{-\\frac{1}{2}+}\\lesssim N_1^{\\sigma - \\frac{\\alpha _{k-1}+3}{2} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < \\frac{\\alpha _{k-1}+3}{2}s\\qquad \\text{and}\\qquad 0 < s < \\alpha _{k-1}^{-1}.$ The modifications for Subcases (3.b), (3.c), and (3.d) are straightforward, giving the regularity restriction (REF ).","$\\bullet $ Case (4): $N_3 \\sim N_4 \\gg N_1 \\ge N_2$ .","$\\circ $ Subcase (4.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .","It suffices to note that $N_1^{1-s- }N_2^{1-s - } N_3^{\\sigma - \\alpha _{k-1} s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\lesssim N_1^{\\sigma - (\\alpha _{k-1}+1) s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < (\\alpha _{k-1}+1)s\\qquad \\text{and}\\qquad 0 \\le s < 1.$ The modifications for Subcases (4.a) and (4.b) are straightforward, giving the regularity restriction (REF ).","Since $\\alpha _{k-1}$ satisfies (REF ), we have $\\alpha _{k-1} \\ge 1$ , which in turn implies $\\frac{\\alpha _{k-1}+3}{2} \\le \\alpha _{k-1} + 1$ .","Therefore, we conclude (REF ) and (REF ).","We conclude this section by proving a gain of space-time integrability for $\\zeta _{2k-1}$ analogous to Lemma REF .","Lemma 4.3 Let $s \\ge 0$ .","Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).","Fix an integer $k \\ge 3$ .","Then, for any finite $q, r > 2$ , there exist $\\theta _k = \\theta _k(q, r) > 0$ , $\\delta _k = \\delta _k(q, r) \\in (0, 1)$ , and $1\\le q_k< \\infty $ such that $\\Vert \\mathbf {P}_N \\zeta _{2k-1}\\Vert _{L^q_T L^r_x}\\lesssim {\\left\\lbrace \\begin{array}{ll}\\rule [-6mm]{0pt}{15pt} T^{\\theta _k} \\Vert z_1\\Vert _{L^{q_k}_T L^{2}_x}^{(2k-1)(1-\\delta _k)}\\Vert z_1\\Vert _{L^{q_k}_T L^{6}_x}^{(2k-1)\\delta _k}& \\text{when } 1 \\le r < 6, \\\\T^{\\theta _k} N^{\\frac{1}{2} - \\frac{3}{r}+} \\Vert z_1\\Vert _{L^{q_k}_T L^{2}_x}^{(2k-1)(1-\\delta _k)}\\Vert z_1\\Vert _{L^{q_k}_T L^{6}_x}^{(2k-1)\\delta _k}& \\text{when } r \\ge 6, \\\\\\end{array}\\right.","}$ for any $T > 0$ and $N \\in 2^{\\mathbb {N}_0}$ .","Note that the right-hand side of (REF ) is almost surely finite thanks to the probabilistic Strichartz estimate (Lemma REF ).","We prove (REF ) by induction.","We first consider the case $r < 6$ .","In Lemma REF , we proved (REF ) for $k = 2$ .","Now, suppose that (REF ) holds for $k - 1$ .","Then, from (REF ) and the dispersive estimate (REF ), we have $\\Vert \\mathbf {P}_N \\zeta _{2k-1}\\Vert _{L^q_T L^r_x}& \\le \\bigg \\Vert \\int _0^t \\Vert \\mathbf {P}_N S(t - t^{\\prime }) z_1^2 \\zeta _{2k-3} (t^{\\prime })\\Vert _{L^r_x} dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim \\bigg \\Vert \\int _0^t \\frac{1}{|t - t^{\\prime }|^{\\frac{3}{2} - \\frac{3}{r}}}\\Vert z_1\\Vert _{L^{3r^{\\prime }}_x}^2 \\Vert \\zeta _{2k-3}\\Vert _{L^{3r^{\\prime }}_x} dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\\\multicolumn{2}{l}{\\text{Noting $3r^{\\prime } < 6$and applying the inductive hypothesiswith $\\theta _{k-1} = \\theta _{k-1}(3q, 3r^{\\prime })$,$\\delta _{k-1} = \\delta _{k-1}(3q, 3r^{\\prime })$,and $q_{k-1} = q_{k-1}(3q, 3r^{\\prime })$,}}\\\\& \\lesssim T^{\\frac{3}{r} - \\frac{1}{2} + \\theta _{k-1} }\\Vert z_1\\Vert _{L^{3q}_T L^{3r^{\\prime }}_x}^2\\Vert z_1\\Vert _{L^{q_{k-1}}_T L^{2}_x}^{(2k-3)(1-\\delta _{k-1})}\\Vert z_1\\Vert _{L^{q_{k-1}}_T L^{6}_x}^{(2k-3)\\delta _{k-1}} \\\\\\multicolumn{2}{l}{\\text{By interpolation in $x$ and Hölder's inequality in $t$,}}\\\\& \\lesssim T^{\\theta _k }\\Vert z_1\\Vert _{L^{q_k}_T L^{2}_x}^{(2k-1)(1-\\delta _{k})}\\Vert z_1\\Vert _{L^{q_k}_T L^{6}_x}^{(2k-1)\\delta _{k}} .$ When $r \\ge 6$ , (REF ) follows from Sobolev's inequality and (REF ) as in the proof of Lemma REF ."],["Proof of Theorem ","In this section, we study the fixed point problem (REF ) around the second order expansion (REF ) and present the proof of Theorem REF .","Given $ \\frac{1}{2} \\le \\sigma \\le 1$ , let $ \\frac{2}{5} \\sigma < s < \\frac{1}{2} $ .","Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ) and let $z_1$ and $z_3$ be as in (REF ) and (REF ).","Define $\\Gamma $ by $\\Gamma v(t) = - i \\int _0^t S(t-t^{\\prime })\\big \\lbrace \\mathcal {N}(v + z_1 + z_3) - \\mathcal {N}(z_1)\\big \\rbrace (t^{\\prime }) dt^{\\prime }.$ Then, we have the following nonlinear estimates.","Proposition 5.1 Given $ \\frac{1}{2} \\le \\sigma \\le 1$ , let $\\frac{2}{5} \\sigma < s < \\frac{1}{2}$ .","Then, there exist $\\theta > 0$ , $C_1, C_2> 0$ , and an almost surely finite constant $R = R(\\omega ) >0$ such that $\\Vert \\Gamma v\\Vert _{X^\\sigma ([0, T])}& \\le C_1\\big (\\Vert v\\Vert _{X^\\sigma ([0, T])} ^3 + T^\\theta R(\\omega )\\big ), \\\\\\Vert \\Gamma v_1 - \\Gamma v_2 \\Vert _{X^\\sigma ([0, T])}& \\le C_2\\Big (\\sum _{j = 1}^2 \\Vert v_j\\Vert _{X^\\sigma ([0, T])} ^2+ T^\\theta R(\\omega )\\Big )\\Vert v_1 -v_2 \\Vert _{X^\\sigma ([0, T])},$ for all $v, v_1, v_2 \\in X^{\\sigma }([0, T])$ and $0 < T\\le 1$ .","Once we prove Proposition REF , Theorem REF immediately follows from a standard argument and thus we omit details.","See Section 5 in [2].","Let $0 0$ .","$\\bullet $ Subcase (C.2): $N_3 \\sim N_4 \\sim N_{\\max } \\gg N_2 \\ge N_1$ .","$\\circ $ Subsubcase (C.2.a): $N_1, N_2 \\ll N_3^\\frac{1}{2}$ .","By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert z_{3, N_1} z_{3, N_3} \\Vert _{L^2_{T, x}} \\Vert z_{3, N_2} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1-2s- } N_2^{1-2s -}N_3^{\\sigma - 2s -\\frac{1}{2}+}N_4^{-\\frac{1}{2}+}\\bigg (\\prod _{j = 1}^3\\Vert z_{3, N_j}\\Vert _{Y^{2s-}_T}\\bigg )\\\\& \\lesssim T^{0+} N_3^{\\sigma - 4s +}C( \\Vert z_3\\Vert _{X^{2s-}_T}).$ This yields (REF ), provided that $\\sigma < 4s$ and $s < \\frac{1}{2}$ .","$\\circ $ Subsubcase (C.2.b): $N_1, N_2 \\gtrsim N_3^\\frac{1}{2} $ .","Proceeding as in Subcase (C.1), we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert z_{3, N_1} \\Vert _{L^{q}_T L^{6+}_x}\\bigg (\\prod _{j = 2}^3 \\Vert z_{3, N_2} \\Vert _{L^{2+}_T L^{6-}_x}\\bigg )\\Vert w_{N_4} \\Vert _{L^\\infty _T L^2_x}\\\\& \\lesssim T^{0+}N_1^{ -s +} N_2^{-2s+}N_3^{\\sigma -2s+}\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}\\Vert z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}^2\\bigg (\\prod _{j = 2}^3\\Vert z_{3, N_j}\\Vert _{Y^{2s-}_T}\\bigg )\\\\& \\le T^{0+}N_3^{\\sigma -\\frac{7}{2} s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ This yields (REF ), provided that $\\sigma < \\frac{7}{2} s$ and $s > 0$ .","$\\circ $ Subsubcase (C.2.c): $N_2\\gtrsim N_3^\\frac{1}{2} \\gg N_1$ .","By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\Vert z_{3, N_1} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_2} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{1 - 2s -} N_2^{-s}N_3^{\\sigma - 2s- \\frac{1}{2}+}C( \\Vert z_{3}\\Vert _{X^{2s-}_T})\\Vert \\langle \\nabla \\rangle ^s z_{3, N_2} \\Vert _{L^5_{T, x}}\\\\\\multicolumn{2}{l}{\\text{By applying Lemma \\ref {LEM:Z3_2},}}\\\\& \\le T^{\\frac{1}{10}+}N_1^{1 - 2s -} N_2^{-s}N_3^{\\sigma - 2s- \\frac{1}{2}+}C( \\Vert z_{3}\\Vert _{X^{2s-}_T})\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\\\& \\le T^{\\frac{1}{10}+}N_3^{\\sigma -\\frac{7}{2} s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ This yields (REF ), provided that $\\sigma < \\frac{7}{2} s$ and $ 0 \\le s < \\frac{1}{2}$ .","Case (D): $v v z_1$ case.","By symmetry, we assume $N_1 \\ge N_2$ .","$\\bullet $ Subcase (D.1): $N_1 \\gtrsim N_3$ .","In this case, we have $N_1 \\sim N_{\\max }$ .","$\\circ $ Subsubcase (D.1.a): $\\max (N_2, N_3) \\ge N_1^\\frac{1}{10}$ .","By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\Vert v_{N_1} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert v_{N_2} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_3} \\Vert _{L^{10}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+} N_2^{-\\sigma } N_3^{-s} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $\\sigma > 0$ and $ s > 0$ .","$\\circ $ Subsubcase (D.1.b): $\\max (N_2, N_3) \\ll N_1^\\frac{1}{10}$ .","In this case, we have $N_1\\sim N_4 \\sim N_{\\max }$ .","By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\Vert v_{N_1}\\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_3}\\Vert _{L^5_{T, x}}\\Vert v_{ N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{0+}N_2^{1-\\sigma -}N_3^{-s}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $ \\sigma \\ge 0$ and $ s \\ge 0$ .","$\\bullet $ Subcase (D.2): $N_3 \\sim N_4 \\gg N_1\\ge N_2$ .","$\\circ $ Subsubcase (D.2.a): $N_1, N_2 \\ll N_3^\\frac{1}{2}$ .","By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} z_{1, N_3} \\Vert _{L^2_{T, x}} \\Vert v_{N_2} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1-\\sigma -} N_2^{1-\\sigma -} N_3^{\\sigma - s - \\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\bigg (\\prod _{j = 1}^2 \\Vert v_{N_j}\\Vert _{Y^\\sigma _T}\\bigg )\\Vert \\mathbf {P}_{N_3} \\phi ^\\omega \\Vert _{H^{s}}\\\\& \\le T^{0+} N_3^{ - s +}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $\\sigma \\le 1$ and $s>0$ .","$\\circ $ Subsubcase (D.2.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .","Proceeding as in Subsubcase (D.1.a) with $L^\\frac{10}{3}_{T, x},L^\\frac{10}{3}_{T, x},L^{10}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality, it suffices to note that $N_1^{-\\sigma } N_2^{-\\sigma }N_3^{\\sigma -s}\\lesssim N_3^{ - s }\\lesssim N_{\\max }^{0-},$ provided that $\\sigma \\ge 0$ and $ s > 0$ .","$\\circ $ Subsubcase (D.2.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .","By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1}\\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_3}\\Vert _{L^5_{T, x}}\\Vert v_{ N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{0+}N_1^{-\\sigma } N_2^{1-\\sigma -}N_3^{\\sigma -s}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $0\\le \\sigma \\le 1$ and $ s > 0$ .","Case (E): $v v z_3$ case.","By symmetry, we assume $N_1 \\ge N_2$ .","$\\bullet $ Subcase (E.1): $N_1 \\gtrsim N_3$ .","In this case, we have $N_1 \\sim N_{\\max }$ .","$\\circ $ Subsubcase (E.1.a): $\\max (N_2, N_3) \\ge N_1^\\frac{1}{10}$ .","By Hölder's inequality with (REF ) as in Subcase (C.1) and Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\bigg (\\prod _{j = 1}^2 \\Vert v_{N_j} \\Vert _{L^{2+}_T L^{6-}_x}\\bigg )\\Vert z_{3, N_3} \\Vert _{L^{q}_T L^{6+}_x}\\Vert w_{N_4} \\Vert _{L^\\infty _T L^2_x}\\\\& \\lesssim T^{0+}N_2^{-\\sigma }N_3^{-s+}\\Vert v\\Vert _{X^\\sigma _T}^2\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}\\Vert z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}^2\\\\\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $\\sigma > 0$ and $s > 0$ .","$\\circ $ Subsubcase (E.1.b): $\\max (N_2, N_3) \\ll N_1^\\frac{1}{10}$ .","In this case, we have $N_1\\sim N_4 \\sim N_{\\max }$ .","By Lemmas REF , REF , and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\Vert v_{N_1}\\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{3, N_3}\\Vert _{L^5_{T, x}}\\Vert v_{ N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{\\frac{1}{10}+}N_2^{1-\\sigma -}N_3^{-s}N_4^{-\\frac{1}{2}+}\\Vert v\\Vert _{X^\\sigma _T}^2\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\\\& \\le T^{\\frac{1}{10}+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $ \\sigma \\ge 0$ and $ s \\ge 0$ .","$\\bullet $ Subcase (E.2): $N_3 \\sim N_4 \\gg N_1\\ge N_2$ .","$\\circ $ Subsubcase (E.2.a): $N_1, N_2 \\ll N_3^\\frac{1}{2}$ .","In this case, we proceed as in Subsubcase (D.2.a).","It suffices to note that $N_1^{1-\\sigma -} N_2^{1-\\sigma -} N_3^{\\sigma - 2s - \\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\lesssim N_3^{ - 2s +}\\lesssim N_{\\max }^{0-},$ provided that $\\sigma \\le 1$ and $s>0$ .","$\\circ $ Subsubcase (E.2.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .","By Hölder's inequality with (REF ) as in Subcase (C.1) and Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\bigg (\\prod _{j = 1}^2 \\Vert v_{N_j} \\Vert _{L^{2+}_T L^{6-}_x}\\bigg )\\Vert z_{3, N_3} \\Vert _{L^{q}_T L^{6+}_x}\\Vert w_{N_4} \\Vert _{L^\\infty _T L^2_x}\\\\& \\lesssim T^{0+}N_1^{-\\sigma }N_2^{-\\sigma } N_3^{\\sigma -s+}\\Vert v\\Vert _{X^\\sigma _T}^2\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}\\Vert z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}^2\\\\\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $\\sigma \\ge 0$ and $s > 0$ .","$\\circ $ Subsubcase (E.2.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .","By Lemmas REF , REF , and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1}\\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{3, N_3}\\Vert _{L^5_{T, x}}\\Vert v_{ N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{\\frac{1}{10}}N_1^{-\\sigma } N_2^{1-\\sigma -}N_3^{\\sigma -s}N_4^{-\\frac{1}{2}+}\\Vert v\\Vert _{X^\\sigma _T}^2\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\\\& \\le T^{\\frac{1}{10}} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $0\\le \\sigma \\le 1$ and $ s > 0$ .","Case (F): $v z_1 z_1$ case.","By symmetry, we assume $N_3 \\ge N_2$ .","$\\bullet $ Subcase (F.1): $N_1 \\gtrsim N_3$ .","In this case, we have $N_1 \\sim N_{\\max }$ .","$\\circ $ Subsubcase (F.1.a): $\\max (N_2, N_3) \\ge N_1^\\frac{1}{10}$ .","By Lemma REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\Vert v_{N_1} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^5_{T, x}}\\Vert z_{1, N_3} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+} N_2^{-s} N_3^{-s} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T},$ provided that $ s > 0$ .","$\\circ $ Subsubcase (F.1.b): $\\max (N_2, N_3) \\ll N_1^\\frac{1}{10}$ .","In this case, we proceed as in Subsubcase (D.1.b).","It suffices to note that $N_2^{1-s-} N_3^{-s}N_4^{-\\frac{1}{2}+}\\lesssim N_{\\max }^{0-},$ provided that $ s \\ge 0$ .","$\\bullet $ Subcase (F.2): $N_2 \\sim N_3 \\gg N_1$ .","$\\circ $ Subsubcase (F.2.a): $N_1\\ll N_3^\\frac{1}{2}$ .","By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{ N_1} z_{1, N_2} \\Vert _{L^2_{T, x}}\\Vert z_{1, N_3}\\Vert _{L^5_{T, x}}\\Vert w_{N_4}\\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{1-\\sigma -} N_2^{-s-\\frac{1}{2} + }N_3^{\\sigma -s}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+}N_3^{\\frac{\\sigma }{2}-2s+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $\\sigma \\le \\min (1, 4s-)$ .","$\\circ $ Subsubcase (F.2.b): $N_1\\gtrsim N_3^\\frac{1}{2} $ .","In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality.","It suffices to note that $N_1^{-\\sigma } N_2^{-s}N_3^{\\sigma -s}\\lesssim N_3^{ \\frac{\\sigma }{2} - 2s }\\lesssim N_{\\max }^{0-},$ provided that $0 \\le \\sigma < 4s$ .","$\\bullet $ Subcase (F.3): $N_3 \\sim N_4 \\sim N_{\\max } \\gg N_1, N_2$ .","$\\circ $ Subsubcase (F.3.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .","By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} z_{1, N_3} \\Vert _{L^2_{T, x}} \\Vert z_{1, N_2} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1- \\sigma -}N_2^{1-s - } N_3^{\\sigma - s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\Vert v\\Vert _{X^\\sigma _T}\\bigg (\\prod _{j = 2}^3\\Vert \\mathbf {P}_{N_j}\\phi ^\\omega \\Vert _{H^s}\\bigg )\\\\& \\lesssim T^{0+} N_3^{\\frac{\\sigma }{2} - \\frac{3}{2} s +}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $ \\sigma \\le \\min (1, 3s-)$ and $ s < 1$ .","$\\circ $ Subcase (F.3.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .","In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality.","It suffices to note that $N_1^{-\\sigma } N_2^{-s}N_3^{\\sigma -s}\\lesssim N_3^{ \\frac{\\sigma }{2} - \\frac{3}{2}s }\\lesssim N_{\\max }^{0-},$ provided that $0 \\le \\sigma < 3s$ .","$\\circ $ Subcase (F.3.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .","By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_3} \\Vert _{L^{5}_{T, x}}\\Vert z_{1, N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{0+}N_1^{-\\sigma } N_2^{1-s-}N_3^{\\sigma - s}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+}N_3^{\\frac{\\sigma }{2}- \\frac{3}{2}s+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $0\\le \\sigma < 3s$ and $s < 1$ .","$\\circ $ Subcase (F.3.d): $N_2\\gtrsim N_3^\\frac{1}{2} \\gg N_1$ .","By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} z_{1, N_3} \\Vert _{L^2_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{1-\\sigma -} N_2^{-s}N_3^{\\sigma - s-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+}N_3^{\\frac{\\sigma }{2}- \\frac{3}{2}s+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $ \\sigma \\le \\min (1, 3s-)$ and $s\\ge 0$ .","Case (G): $v z_3 z_3$ case.","By symmetry, we assume $N_3 \\ge N_2$ .","$\\bullet $ Subcase (G.1): $N_1 \\gtrsim N_3$ .","In this case, we have $N_1 \\sim N_{\\max }$ .","We can proceed as in Subcase (E.1) with $v_{N_2}$ replaced by $z_{3, N_2}$ .","More precisely, when $\\max (N_2, N_3) \\ge N_1^\\frac{1}{10}$ , we apply Hölder's inequality with (REF ) as in Subsubcase (E.1.a).","It suffices to note that $N_2^{-2s+}N_3^{-s+}\\lesssim N_{\\max }^{0-},$ provided that $ s> 0$ .","When $\\max (N_2, N_3) \\ll N_1^\\frac{1}{10}$ , we proceed with Lemmas REF , REF , and REF as in Subsubcase (E.1.b).","In this case, it suffices to note that $N_2^{1-2s-}N_3^{-s}N_4^{-\\frac{1}{2}+}\\lesssim N_{\\max }^{0-},$ provided that $s\\ge 0$ .","$\\bullet $ Subcase (G.2): $N_2 \\sim N_3 \\gg N_1$ .","$\\circ $ Subsubcase (G.2.a): $N_1\\ll N_3^\\frac{1}{2}$ .","By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{ N_1} z_{3, N_2} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_3}\\Vert _{L^5_{T, x}}\\Vert w_{N_4}\\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{\\frac{1}{10}}N_1^{1-\\sigma -} N_2^{-2s-\\frac{1}{2} + }N_3^{\\sigma -s} \\Vert v\\Vert _{X^\\sigma _T}\\Vert z_{3, N_2}\\Vert _{X^{2s-}_T}\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2 \\\\& \\le T^{\\frac{1}{10}}N_3^{\\frac{\\sigma }{2} -3 s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T},$ provided that $\\sigma \\le \\min (1, 6s-)$ .","$\\circ $ Subsubcase (G.2.b): $N_1\\gtrsim N_3^\\frac{1}{2} $ .","In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality with Lemma REF .","It suffices to note that $N_1^{-\\sigma } N_2^{-s}N_3^{\\sigma -s}\\lesssim N_3^{ \\frac{\\sigma }{2} - 2s }\\lesssim N_{\\max }^{0-},$ provided that $0 \\le \\sigma < 4s$ .","$\\bullet $ Subcase (G.3): $N_3 \\sim N_4 \\sim N_{\\max } \\gg N_1, N_2$ .","$\\circ $ Subsubcase (G.3.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .","In this case, we proceed as in Subsubcase (F.3.a).","It suffices to note that $N_1^{1- \\sigma -}N_2^{1-2s - } N_3^{\\sigma - 2 s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\lesssim N_3^{\\frac{\\sigma }{2} - 3s+}\\lesssim N_{\\max }^{0-},$ provided that $ \\sigma \\le \\min (1, 6s-)$ and $ s < \\frac{1}{2}$ .","$\\circ $ Subsubcase (G.3.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .","In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality as in Subsubcase (F.3.b) but we apply Lemma REF to estimate the $L^{5}_{T, x}$ -norm of $z_{3, N_j}$ , $j = 2, 3$ .","The regularity restriction is precisely the same as that in Subsubcase (F.3.b).","$\\circ $ Subsubcase (G.3.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .","By Lemmas REF , REF , and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{\\frac{1}{10}+}N_1^{-\\sigma } N_2^{1-2s-}N_3^{\\sigma - s}N_4^{-\\frac{1}{2}+}\\Vert v\\Vert _{X^\\sigma _T}\\Vert z_{3, N_2}\\Vert _{X^{2s-}_T}\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\\\& \\le T^{\\frac{1}{10}+}N_3^{\\frac{\\sigma }{2}- 2s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $0\\le \\sigma < 4s$ and $s < \\frac{1}{2}$ .","$\\circ $ Subcase (G.3.d): $N_2\\gtrsim N_3^\\frac{1}{2} \\gg N_1$ .","By Lemmas REF , REF , and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_2} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{\\frac{1}{10}+}N_1^{1-\\sigma -} N_2^{-s}N_3^{\\sigma - 2s-\\frac{1}{2}+}\\Vert v\\Vert _{X^\\sigma _T}\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\Vert z_{3, N_3} \\Vert _{X^{2s-}_T}\\\\& \\le T^{\\frac{1}{10}+}N_1^{\\frac{\\sigma }{2}- \\frac{5}{2}s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $ \\sigma \\le \\min (1, 5s-)$ and $s\\ge 0$ .","Case (H): $v z_3 z_1$ case.","$\\bullet $ Subcase (H.1): $N_1 \\sim N_{\\max }$ .","In this case, we can proceed as in Subcase (D.1) by replacing $v_{N_2}$ with $z_{3, N_2}$ .","When $\\max (N_2, N_3) \\ge N_1^\\frac{1}{10}$ , it suffices to note that $N_2^{-2s+}N_3^{-s}\\lesssim N_{\\max }^{0-},$ provided that $s> 0$ .","When $\\max (N_2, N_3) \\ll N_1^\\frac{1}{10}$ , it suffices to note that $N_2^{1-2s-}N_3^{-s}N_4^{-\\frac{1}{2}+}\\lesssim N_{\\max }^{0-},$ provided that $s\\ge 0$ .","$\\bullet $ Subcase (H.2): $N_3 \\sim N_{\\max } \\gg N_1$ .","$\\circ $ Subsubcase (H.2.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .","In this case, we have $N_3 \\sim N_4 \\sim N_{\\max }$ .","Then, proceeding as in Subsubcase (F.3.a), it suffices to note that $N_1^{1- \\sigma -}N_2^{1-2s - } N_3^{\\sigma - s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\lesssim N_3^{\\frac{\\sigma }{2} - 2s}\\lesssim N_{\\max }^{0-},$ provided that $ \\sigma \\le \\min (1, 4s-)$ and $ s < \\frac{1}{2}$ .","$\\circ $ Subsubcase (H.2.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .","In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^\\frac{10}{3}_{T, x},L^{10}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality.","It suffices to note that $N_1^{- \\sigma }N_2^{-2s+} N_3^{\\sigma - s}\\lesssim N_3^{\\frac{\\sigma }{2} - 2s+}\\lesssim N_{\\max }^{0-},$ provided that $0 \\le \\sigma < 4s$ .","$\\circ $ Subsubcase (H.2.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .","In this case, we have $N_3 \\sim N_4 \\sim N_{\\max }$ .","We can proceed as in Subsubcase (G.3.c) with $z_{3, N_3}$ replaced by $z_{1,N_3}$ (without applying Lemma REF ).","The regularity restriction is precisely the same as in Subsubcase (G.3.c).","$\\circ $ Subsubcase (H.2.d): $N_2\\gtrsim N_3^\\frac{1}{2} \\gg N_1$ .","We first consider the case $N_3 \\sim N_4 \\sim N_{\\max }$ .","By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} w_{N_4} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_2} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_3} \\Vert _{L^{5}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{1-\\sigma -} N_2^{-2s+}N_3^{\\sigma - s} N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+}N_3^{\\frac{\\sigma }{2}- 2s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $ \\sigma \\le \\min (1, 4s-)$ and $ s> 0$ .","When $N_4\\ll N_3$ , we have $N_2 \\sim N_3 \\sim N_{\\max }$ .","In this case, we can repeat the computation above with the roles of $z_{3, N_2}$ and $w_{N_4}$ switched.","Noting that $N_1^{1-\\sigma -} N_2^{-2s-\\frac{1}{2}+}N_3^{\\sigma - s}\\lesssim N_3^{\\frac{\\sigma }{2}- 3s+} \\lesssim N_{\\max }^{0-},$ we obtain (REF ), provided that $ \\sigma \\le \\min (1, 6s-)$ .","$\\bullet $ Subcase (H.3): $N_2 \\sim N_4 \\gg N_1, N_3$ .","$\\circ $ Subsubcase (H.3.a): $N_1\\ll N_2^\\frac{1}{2}$ .","In this case, we can proceed as in Subsubcase (H.2.d).","It suffices to note that $N_1^{1-\\sigma -} N_2^{\\sigma -2s + }N_3^{-s} N_4^{-\\frac{1}{2} +}\\lesssim N_2^{\\frac{\\sigma }{2} -2 s+}\\lesssim N_{\\max }^{0-},$ provided that $\\sigma \\le \\min (1, 4s-)$ and $s \\ge 0$ .","$\\circ $ Subsubcase (H.3.b): $N_1\\gtrsim N_2^\\frac{1}{2} $ .","In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^\\frac{10}{3}_{T, x},L^{10}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality.","It suffices to note that $N_1^{- \\sigma }N_2^{\\sigma -2s+} N_3^{-s}\\lesssim N_2^{\\frac{\\sigma }{2} - 2s+}\\lesssim N_{\\max }^{0-},$ provided that $0 \\le \\sigma < 4s$ .","Case ( I ): $v vv$ case.","In this case, there is no need to perform the dyadic decomposition.","By Hölder's inequality, Sobolev's inequality (REF ), and Lemma REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim \\Vert v \\Vert _{L^{5}_{T, x}}^2\\Vert \\langle \\nabla \\rangle ^\\sigma v \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert w \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim \\Vert \\langle \\nabla \\rangle ^\\frac{1}{2} v \\Vert _{L^{5}_{T} L^\\frac{30}{11}_x}^2\\Vert \\langle \\nabla \\rangle ^\\sigma v \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim \\Vert v\\Vert _{X^\\sigma _T}^3,$ provided that $ \\sigma \\ge \\frac{1}{2}$ .","Putting all the cases including Cases (A) and (B) treated in Lemmas REF and REF , we conclude that (REF ) holds, provided that $ \\frac{1}{2} \\le \\sigma \\le 1$ and $\\frac{2}{5} \\sigma < s < \\frac{1}{2}$ .","This completes the proof of Proposition REF ."],["Proof of Theorem ","In this section, we briefly discuss the proof of Theorem REF .","Given $ \\frac{1}{6} < s < \\frac{1}{2}$ , let $k \\in \\mathbb {N}$ such that $\\frac{1}{2\\alpha _{k+1}} < s \\le \\frac{1}{2\\alpha _{k}}$ , where $\\alpha _k$ is defined in (REF ).","Our main goal is to study the fixed point problem (REF ).","Define $\\widetilde{\\Gamma }$ by $\\widetilde{\\Gamma }v(t) = -i \\int _0^t S(t - t^{\\prime })\\bigg \\lbrace \\mathcal {N}\\bigg (v +\\sum _{\\ell = 1}^{k} \\zeta _{2\\ell -1}\\bigg )- \\sum _{\\ell = 2}^{k}\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 2\\ell - 1\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2\\ell -3\\rbrace \\end{array}}\\zeta _{j_1} \\overline{\\zeta _{j_2}}\\zeta _{j_3}\\bigg \\rbrace (t^{\\prime }) dt^{\\prime },$ where $\\zeta _{2\\ell -1}$ , $\\ell = 1, \\dots , k$ , is as in (REF ).","Theorem REF follows from a standard fixed point argument, once we prove the following proposition.","Proposition 6.1 Given $ \\frac{1}{6} < s < \\frac{1}{2}$ , let $k \\in \\mathbb {N}$ such that $\\frac{1}{2\\alpha _{k+1}} < s < \\frac{1}{\\alpha _{k}}$ .","Then, there exist $\\theta > 0$ , $C_1, C_2> 0$ , and an almost surely finite constant $R = R(\\omega ) >0$ such that $\\Vert \\widetilde{\\Gamma }v\\Vert _{X^\\frac{1}{2}([0, T])}& \\le C_1\\big (\\Vert v\\Vert _{X^\\frac{1}{2}([0, T])} ^3 + T^\\theta R(\\omega )\\big ),\\\\\\Vert \\widetilde{\\Gamma }v_1 - \\widetilde{\\Gamma }v_2 \\Vert _{X^\\frac{1}{2}([0, T])}& \\le C_2\\Big (\\sum _{j = 1}^2 \\Vert v_j\\Vert _{X^\\frac{1}{2}([0, T])} ^2+ T^\\theta R(\\omega )\\Big )\\Vert v_1 -v_2 \\Vert _{X^\\frac{1}{2}([0, T])},$ for all $v, v_1, v_2 \\in X^\\frac{1}{2}([0, T])$ and $0 < T\\le 1$ .","On the one hand, the upper bound $\\frac{1}{\\alpha _k}$ on the range of $s$ in Proposition REF comes from the restriction in Proposition REF .","On the other hand, given $\\frac{1}{6} < s < \\frac{1}{2}$ , we fix $k \\in \\mathbb {N}$ such that $\\frac{1}{2\\alpha _{k+1}} < s \\le \\frac{1}{2\\alpha _{k}}$ and apply Proposition REF .","Namely, the upper bound on $s$ in Proposition REF does not cause any problem in proving Theorem REF .","We only discuss the proof of (REF ) since () follows in a similar manner.","As in the proof of Proposition REF , it suffices to perform a case-by-case analysis of expressions of the form: $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}\\langle \\nabla \\rangle ^\\frac{1}{2} ( w_1 w_2 w_3 )w dx dt\\bigg |,$ where $\\Vert w\\Vert _{Y^{0}_T} \\le 1$ and $w_j= v$ , $\\zeta _{2\\ell -1}$ , $\\ell = 1, \\dots , k$ , but not of the form $\\zeta _1 \\zeta _1\\zeta _{2\\ell -3}$ for $\\ell \\in \\lbrace 2, 3, \\dots , k\\rbrace $ .","Here, we have dropped the complex conjugate sign on $w_2$ since it does not play any role in our analysis.","More concretely, we need to consider the following cases: Table: NO_CAPTION where $j_1, j_2, j_3$ can take any value in $\\lbrace 3, 5, \\dots , 2k-1\\rbrace $ .","As we see below, the worst interaction appears in Case (A) and all the other cases can be handled as in the proof of Proposition REF .","We first point out that, in the proof of Proposition REF , the regularity restriction coming from Cases (B) - (I) (with $\\sigma = \\frac{1}{2}$ ) is $s > \\frac{1}{6}$ .","Moreover, by comparing Proposition REF and Lemma REF with Proposition REF and Lemma REF , we see that the unbalanced higher order term $\\zeta _{2\\ell -1}$ for $\\ell \\in \\lbrace 2, 3, \\dots , k\\rbrace $ enjoys (at least) the same regularity properties as $z_3 = \\zeta _3$ both in terms of differentiability and space-time integrability.","Therefore, we can simply apply the estimates in the proofs of Lemma REF (i) for Case (B) and Proposition REF for Cases (C) - (I) and conclude that the contributions from Cases (B) - (I) can be bounded, provided $s > \\frac{1}{6}$ .","It remains to consider Case (A).","In view of (REF ), the contribution from Case (A) is nothing but $\\zeta _{2k+1}$ .","Hence, by applying Proposition REF with $\\sigma = \\frac{1}{2}$ , we conclude that the contribution in Case (A) can be estimated by $T^\\theta R(\\omega )$ , provided that $\\frac{1}{2\\alpha _{k+1}} < s < \\frac{1}{\\alpha _{k}}$ .","This completes the proof of Proposition REF ."],["On the deterministic non-smoothing\nof the Duhamel integral operator","In this appendix, we show that there is no deterministic smoothing on the Duhamel integral operator $I$ defined in (REF ) when $s \\le \\frac{1}{2}$ .","Lemma 1.1 Let $\\sigma > 3s-1$ .","Then, the estimate (REF ) does not hold.","In particular, there is no deterministic nonlinear smoothing when $s \\le \\frac{1}{2}$ .","Given $N \\ge L \\ge \\ell \\gg 1$ , consider $u_j = S(t) \\phi _j$ , $j = 1, 2, 3$ , such that $\\widehat{\\phi }_j( \\xi ) \\sim \\mathbf {1}_{A_j}(\\xi )$ , where $A_1 = L e_1 + \\ell Q, \\qquad A_2 = \\ell Q, \\qquad \\text{and}\\qquad A_3 = N e_2 + \\ell Q.$ Then, we have $\\prod _{j = 1}^3 \\Vert u_j(0) \\Vert _{H^s}\\sim \\ell ^{\\frac{9}{2} + s} L^s N^s.$ Let $A = Le_1 + N e_2 + \\ell Q$ .","By writing $\\mathbf {1}_{A} (\\xi )\\mathcal {F}_x[I(u_1, & u_2, u_3)](t, \\xi )\\\\&= -i \\mathbf {1}_{A} (\\xi ) e^{-it |\\xi |^2} \\int _0^t \\operatornamewithlimits{\\int }_{\\xi = \\xi _1 - \\xi _2 + \\xi _3}e^{i t^{\\prime } \\Phi (\\bar{\\xi })} \\prod _{j = 1}^3 \\mathbf {1}_{A_j} (\\xi _j) d\\xi _1 d\\xi _2 dt^{\\prime },$ we see that the non-trivial contribution to (REF ) comes from $\\xi _j \\in A_j$ , $j = 1, 2, 3$ .","Moreover, in this case, $\\Phi (\\bar{\\xi })$ defined (REF ) satisfies $|\\Phi (\\bar{\\xi }) |\\lesssim N L$ .","In particular, by choosing $t_* \\sim \\frac{1}{NL}$ such that $t_*|\\Phi (\\bar{\\xi }) |\\ll 1 $ , we have (REF ) for all $t^{\\prime } \\in [0, t_*]$ .","Hence, by Lemma REF with (REF ), we obtain $\\Vert I(u_1, u_2, u_3) \\Vert _{X^\\sigma ([0, 1])} \\gtrsim \\Vert I(u_1, u_2, u_3)(t_*) \\Vert _{ H^\\sigma }\\gtrsim t_* \\ell ^\\frac{15}{2} N^\\sigma \\sim \\ell ^\\frac{15}{2} L^{-1}N^{\\sigma -1} .$ Therefore, it follows from (REF ) and (REF ) that by choosing $\\ell \\sim L \\sim N$ , we have $\\frac{\\Vert I(u_1, u_2, u_3) \\Vert _{X^\\sigma ([0, 1])}}{\\prod _{j=1}^3 \\Vert u_j(0) \\Vert _{H^s}}\\gtrsim \\ell ^{3-s}L^{-1-s} N^{\\sigma -1-s}\\sim N^{\\sigma - 3s+1} \\longrightarrow \\infty ,$ as $N \\rightarrow \\infty $ , provided that $\\sigma > 3s-1$ .","This shows that the estimate (REF ) can not hold when $\\sigma > 3s-1$ .","Define the quintilinear operator $I^{(2)}$ by $I^{(2)}(u_1, \\dots , u_5) (t):= -i \\sum _{\\ell = 1}^3\\int _0^t S(t - t^{\\prime })v_{j_1} \\overline{v_{j_2}}v_{j_3} (t^{\\prime }) dt^{\\prime },$ where $v_{j_i} = u_i$ for $i \\ne \\ell $ and $v_{j_\\ell } = I(u_\\ell , u_4, u_5)$ .","This basically corresponds to the second order term appearing in the power series expansion for (REF ).","In particular, we have that $z_5 = I^{(2)}(z_1, \\dots , z_1)$ .","By a computation similar to the proof of Lemma REF , we can prove the following deterministic non-smoothing on the second order term.","Let $\\sigma > 5s - 2$ .","Then, there is no finite constant $C> 0$ such that $\\big \\Vert I^{(2)}( S(t) \\phi _1, \\dots , S(t) \\phi _5)\\big \\Vert _{X^\\sigma ([0, 1])}\\le C \\prod _{j = 1}^5 \\Vert \\phi _j \\Vert _{H^s}$ for all $\\phi _j \\in H^s(\\mathbb {R}^3)$ .","In particular, this shows that when $s \\le \\frac{1}{2}$ , there is no deterministic smoothing on the second order term $I^{(2)}$ .","A similar comment applies to the higher order terms.","Acknowledgments This research was partially supported by Research in Groups at International Centre for Mathematical Sciences, Edinburgh, United Kingdom.","Á.B.","was partially supported by a grant from the Simons Foundation (No. 246024).","T.O.","was supported by the European Research Council (grant no.","637995 “ProbDynDispEq”)."]],"string":"[\n [\n \"Higher order expansions for the probabilistic local Cauchy theory of the\\n cubic nonlinear Schr\\\\\\\"odinger equation on $\\\\mathbb{R}^3$\"\n ],\n [\n \"Abstract We consider the cubic nonlinear Schr\\\\\\\"odinger equation (NLS) on $\\\\mathbb{R}^3$ with randomized initial data.\",\n \"In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data.\",\n \"By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work [2].\",\n \"We further investigate a limitation of this iterative procedure.\",\n \"Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion.\"\n ],\n [\n \"Nonlinear Schrödinger equation\",\n \"We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) on $\\\\mathbb {R}^3$ :Our discussion in this paper can be easily adapted to the cubic NLS on $\\\\mathbb {R}^d$ (and to other nonlinear dispersive PDEs).\",\n \"For the sake of concreteness, however, we only consider the $d = 3$ case in the following.\",\n \"See also the comment after Theorem REF for our particular interest in the three-dimensional problem.\",\n \"${\\\\left\\\\lbrace \\\\begin{array}{ll}i \\\\partial _t u + \\\\Delta u = |u|^2 u \\\\\\\\u|_{t = 0} = u_0 \\\\in H^s(\\\\mathbb {R}^3),\\\\end{array}\\\\right.\",\n \"}\\\\qquad ( t, x) \\\\in \\\\mathbb {R}\\\\times \\\\mathbb {R}^3.$ The cubic NLS (REF ) has been studied extensively from both the theoretical and applied points of view.\",\n \"In this paper, we continue our study in [1], [2] and further investigate the probabilistic well-posedness of (REF ) with random and rough initial data.\",\n \"Recall that the equation (REF ) is scaling critical in $\\\\dot{H}^\\\\frac{1}{2}(\\\\mathbb {R}^3)$ in the sense that the scaling symmetry: $u(t, x) \\\\longmapsto \\\\lambda u (\\\\lambda ^2 t, \\\\lambda x)$ preserves the homogeneous $\\\\dot{H}^{\\\\frac{1}{2} }$ -norm (when it is applied to functions only of $x$ ).\",\n \"It is known that the Cauchy problem (REF ) is locally well-posed in $H^s(\\\\mathbb {R}^3)$ for $s \\\\ge \\\\frac{1}{2}$ [10].\",\n \"See also [13], [26], [29], [18] for partialNamely, either for smoother initial data or under an extra hypothesis.\",\n \"global well-posedness and scattering results.\",\n \"On the other hand, (REF ) is known to be ill-posed in $H^s(\\\\mathbb {R}^3)$ for $s < \\\\frac{1}{2}$ [12].\",\n \"In [2], we studied the probabilistic well-posedness property of (REF ) below the scaling critical regularity $s_\\\\text{crit} = \\\\frac{1}{2}$ under a suitable randomization of initial data; see (REF ) below.\",\n \"In particular, we proved that (REF ) is almost surely locally well-posed in $H^s(\\\\mathbb {R}^3)$ , $s > \\\\frac{1}{4}$ .\",\n \"Our main goal in this paper is to introduce an iterative procedure to improve this regularity threshold for almost sure local well-posedness.\",\n \"Furthermore, we study a critical regularity associated with this iterative procedure.\",\n \"By introducing a modified iterative approach, we then prove almost sure local well-posedness of (REF ) in an almost optimal range with respect to the original iterative procedure (Theorem REF ).\",\n \"Beyond the concrete results in this paper, we believe that the iterative procedures based on (modified) partial power series expansions are themselves of interest for further development in the well-posedness theory of dispersive equations with random initial data and/or random forcing.\",\n \"Such a probabilistic construction of solutions to dispersive PDEs first appeared in the works by McKean [34] and Bourgain [5] in the study of invariant Gibbs measures for the cubic NLS on $d$ , $d = 1, 2$ .\",\n \"In particular, they established almost sure local well-posedness with respect to particular random initial data, basically corresponding to the Brownian motion.\",\n \"(These local-in-time solutions were then extended globally in time by invariance of the Gibbs measures.\",\n \"In the following, however, we restrict our attention to local-in-time solutions.)\",\n \"In [8], Burq-Tzvetkov further elaborated this idea and considered a randomization for any rough initial condition via the Fourier series expansion.\",\n \"More precisely, they studied the cubic nonlinear wave equation (NLW) on a three-dimensional compact Riemannian manifold below the scaling critical regularity.\",\n \"By introducing a randomization via the multiplication of the Fourier coefficients by independent random variables, they established almost sure local well-posedness below the critical regularity.\",\n \"Such randomization via the Fourier series expansion is natural on compact domains [15], [35] and more generally in situations where the associated elliptic operators have discrete spectra [47], [45].\",\n \"See [2], [3] for more references therein.\",\n \"In the following, we go over a randomization suitable for our problem on $\\\\mathbb {R}^3$ .\",\n \"Recall the Wiener decomposition of the frequency space [48]: $\\\\mathbb {R}^3_\\\\xi = \\\\bigcup _{n \\\\in \\\\mathbb {Z}^3} (n+ (-\\\\frac{1}{2}, \\\\frac{1}{2}]^3)$ .\",\n \"We employ a randomization adapted to this Wiener decomposition.\",\n \"Let $\\\\psi \\\\in \\\\mathcal {S}(\\\\mathbb {R}^3)$ satisfy $\\\\operatornamewithlimits{supp}\\\\psi \\\\subset [-1,1]^3\\\\qquad \\\\text{and}\\\\qquad \\\\sum _{n \\\\in \\\\mathbb {Z}^3} \\\\psi (\\\\xi -n) =1 \\\\quad \\\\text{for any $\\\\xi \\\\in \\\\mathbb {R}^3$}.$ Then, given a function $\\\\phi $ on $\\\\mathbb {R}^3$ , we have $ \\\\phi = \\\\sum _{n \\\\in \\\\mathbb {Z}^3} \\\\psi (D-n) \\\\phi .$ We define the Wiener randomization $\\\\phi ^\\\\omega $ of $\\\\phi $ by $ \\\\phi ^{\\\\omega } := \\\\sum _{n \\\\in \\\\mathbb {Z}^3} g_n (\\\\omega ) \\\\psi (D-n) \\\\phi ,$ where $\\\\lbrace g_n \\\\rbrace _{n \\\\in \\\\mathbb {Z}^3} $ is a sequence of independent mean-zero complex-valued random variables on a probability space $(\\\\Omega , \\\\mathcal {F} ,P)$ .\",\n \"The randomization (REF ) based on the Wiener decomposition of the frequency space is natural in view of time-frequency analysis; see [1] for a further discussion.\",\n \"Almost simultaneously with [1], Lührmann-Mendelson [32] also considered a randomization of the form (REF ) (with cubes being substituted by appropriately localized balls) in the study of NLW on $\\\\mathbb {R}^3$ .\",\n \"For a similar randomization used in the study of the Navier-Stokes equations, see also the work of Zhang and Fang [50], preceding [1], [32].\",\n \"In [50], the decomposition of the frequency space is not explicitly provided but their randomization certainly includes randomization based on the decomposition of the frequency space by unit cubes or balls.\",\n \"In the following, we assume that the probability distribution $\\\\mu _n$ of $g_n$ satisfies the following exponential moment bound: $\\\\int _{\\\\mathbb {R}^2} e^{\\\\kappa \\\\cdot x} d \\\\mu _{n} (x) \\\\le e^{c |\\\\kappa | ^2}$ for all $\\\\kappa \\\\in \\\\mathbb {R}^2$ and $n \\\\in \\\\mathbb {Z}^3$ .\",\n \"This condition is satisfied by the standard complex-valued Gaussian random variables and the uniform distribution on the circle in the complex plane.\",\n \"We now recall the almost sure local well-posedness result from [2] which is of interest to us.For simplicity, we only consider positive times.\",\n \"By the time reversibility of the equation, the same analysis applies to negative times.\",\n \"Theorem A Let $\\\\frac{1}{4} < s < \\\\frac{1}{2}$ .\",\n \"Given $\\\\phi \\\\in H^s (\\\\mathbb {R}^3)$ , let $\\\\phi ^{\\\\omega }$ be its Wiener randomization defined in (REF ).\",\n \"Then, the Cauchy problem (REF ) is almost surely locally well-posed with respect to the random initial data $\\\\phi ^\\\\omega $ .\",\n \"More precisely, there exists a set $\\\\Sigma = \\\\Sigma (\\\\phi ) \\\\subset \\\\Omega $ with $P(\\\\Sigma ) = 1$ such that, for any $\\\\omega \\\\in \\\\Sigma $ , there exists a unique function $u = u^\\\\omega $ in the class: $S(t)\\\\phi ^\\\\omega + X^\\\\frac{1}{2}([0, T]) \\\\subset S(t) \\\\phi ^{\\\\omega } + C([0,T] ; H^\\\\frac{1}{2}(\\\\mathbb {R}^3)) \\\\subset C([0,T] ; H^s(\\\\mathbb {R}^3))$ with $T = T(\\\\phi , \\\\omega ) >0$ such that $u$ is a solution to (REF ) on $[0, T]$ .\",\n \"Here, $S(t) := e^{it\\\\Delta }$ denotes the linear Schrödinger operator and the function space $X^\\\\frac{1}{2}([0, T]) \\\\subset C([0, T]; H^\\\\frac{1}{2}(\\\\mathbb {R}^3))$ is defined in Section below.\",\n \"Note that the uniqueness statement of Theorem REF in the class: $S(t)\\\\phi ^\\\\omega + X^\\\\frac{1}{2}([0, T])$ is to be interpreted as follows; by setting $v = u - S(t) \\\\phi ^\\\\omega $ (see (REF ) below), uniqueness for the residual term $v$ holds in $X^\\\\frac{1}{2}([0, T])$ .\",\n \"See also Remark REF below.\",\n \"Recall that while the Wiener randomization (REF ) does not improve differentiability, it improves integrability (see Lemma 4 in [1]).\",\n \"See [42], [27], [8] for the corresponding statements in the context of the random Fourier series.\",\n \"The main idea for proving Theorem REF is to exploit this gain of integrability.\",\n \"More precisely, let $z_1(t) := S(t) \\\\phi ^{\\\\omega }$ denote the random linear solution with $\\\\phi ^\\\\omega $ as initial data and write $u = z_1 + v.$ Then, we see that $v: = u - z_1$ satisfies the following perturbed NLS: ${\\\\left\\\\lbrace \\\\begin{array}{ll}i \\\\partial _tv + \\\\Delta v = |v + z_1|^2(v+z_1)\\\\\\\\v|_{t = 0} = 0,\\\\end{array}\\\\right.\",\n \"}$ where $z_1$ is viewed as a given (random) source term.\",\n \"The main point is that the gain of space-time integrability of the random linear solution $z_1$ (Lemma REF ) makes this problem subcriticalThe scaling critical Sobolev regularity $s_\\\\text{crit} = \\\\frac{1}{2}$ for (REF ) is defined by the fact that the $\\\\dot{H}^\\\\frac{1}{2}$ -norm remains invariant under the scaling symmetry (REF ).\",\n \"Given $1 \\\\le r \\\\le \\\\infty $ , we can also define the scaling critical Sobolev regularity $s_\\\\text{crit}(r)$ in terms of the $L^r$ -based Sobolev space $\\\\dot{W}^{s, r}$ .\",\n \"A direct computation gives $s_\\\\text{crit}(r) = \\\\frac{3}{r} - 1\\\\, (\\\\rightarrow -1$ as $r\\\\rightarrow \\\\infty $ ).\",\n \"In particular, the gain of integrability of the random linear solution $z_1$ stated in Lemma REF implies that $z_1$ in (REF ) gives rise only to a subcritical perturbation.\",\n \"See [3] for a further discussion on this issue.\",\n \"Note, however, that if we consider a non-zero initial condition for $v$ , then the critical nature of the equation comes into play through the initial condition.\",\n \"See Remark REF .\",\n \"This plays an important role in studying the global-in-time behavior of solutions.\",\n \"See, for example, [37].\",\n \"and hence we can solve it by a standard fixed point argument.\",\n \"Over the last several years, there have been many results on probabilistic well-posedness of nonlinear dispersive PDEs, using this change of viewpoint.In the field of stochastic parabolic PDEs, this change of viewpoint and solving the fixed point problem for the residual term $v$ is called the Da Prato-Debussche trick [16], [17].\",\n \"See, for example, [34], [5], [8], [15], [9], [35], [32], [1], [2], [43], [38], [25], [33], [7], [30], [37].\",\n \"In [2], we studied the Duhamel formulation for (REF ): $v(t) = - i \\\\int _0^t S(t-t^{\\\\prime }) |v + z_1|^2(v+z_1)(t^{\\\\prime }) dt^{\\\\prime },$ by carrying out case-by-case analysis on terms of the form $v \\\\overline{v} v$ , $v \\\\overline{v} z_1$ , $v \\\\overline{z}_1 z_1$ , etc.\",\n \"in $X^\\\\frac{1}{2}([0, T])$ .\",\n \"The main tools were (i) the gain of space-time integrability of $z_1$ and (ii) the bilinear refinement of the Strichartz estimate (Lemma REF ).\",\n \"This yields Theorem REF .\",\n \"By examining the case-by-case analysis in [2], we see that the regularity restriction $s > \\\\frac{1}{4}$ in Theorem REF comes from the cubic interaction of the random linear solution: $z_3(t) := -i \\\\int _0^t S(t - t^{\\\\prime }) |z_1|^2 z_1(t^{\\\\prime }) dt^{\\\\prime }.$ See Proposition REF below.\",\n \"In the following, we discuss an iterative approach to lower this regularity threshold by studying further expansions in terms of the random linear solution $z_1$ .\",\n \"We will also discuss the limitation of this iterative procedure.\",\n \"Remark 1.1 (i) The proof of Theorem REF , presented in [2], is based on a standard contraction argument for the residual term $v = u - S(t) \\\\phi ^\\\\omega $ in a ball $B$ of radius $O(1)$ in $X^\\\\frac{1}{2}([0, T])$ .\",\n \"As such, the argument in [2] only yields uniqueness of $v$ in the ball $B$ .\",\n \"Noting that $v \\\\in C([0, T]; H^{\\\\frac{1}{2}}(\\\\mathbb {R}^3)) \\\\cap X^\\\\frac{1}{2}([0, T])$ ,By convention, our definition of the $X^s([0, T])$ -space already assumes that functions in $X^s([0, T])$ belong to $ C([0, T]; H^s(\\\\mathbb {R}^3))$ .\",\n \"See Definition REF below.\",\n \"it follows from Lemma A.8 in [2] that the $X^\\\\frac{1}{2}([0, t])$ -norm of $v$ is continuous in $t \\\\in [0, T]$ .\",\n \"Then, by possibly shrinking the local existence time $T>0$ , we can easily upgrade the uniqueness of $v$ in the ball $B$ to uniqueness of $v$ in the entire $X^\\\\frac{1}{2}([0, T])$ .\",\n \"Namely, uniqueness of $u$ holds in the class (REF ).\",\n \"(ii) With the exponential moment assumption (REF ), the proof of Theorem REF , presented in [2], allows us to conclude that the set $\\\\Sigma $ of full probability in Theorem REF has the following decomposition: $ \\\\Sigma = \\\\bigcup _{0 < T \\\\ll 1} \\\\Sigma _T$ such that (a) there exist $c, C >0$ and $C_0 = C_0 (\\\\Vert \\\\phi \\\\Vert _{H^s})>0$ such that $P(\\\\Sigma _T^c) < C \\\\exp \\\\Big (-\\\\frac{C_0(\\\\Vert \\\\phi \\\\Vert _{H^s})}{T^c }\\\\Big )$ for each $0 < T \\\\ll 1$ , (b) for each $\\\\omega \\\\in \\\\Sigma _T\\\\subset \\\\Sigma $ , $0 < T \\\\ll 1$ , the function $u = u^\\\\omega $ constructed in Theorem REF is a solution to (REF ) on $[0, T]$ .\",\n \"See the statement of Theorem 1.1 in [2].\",\n \"A similar decomposition of the set $\\\\Sigma $ of full probability applies to Theorems REF and REF below.\",\n \"We, however, do not state it in an explicit manner in the following.\"\n ],\n [\n \"Improved almost sure local well-posedness\",\n \"Let us first state the following proposition on the regularity property of the cubic term $z_3$ defined above.\",\n \"Proposition 1.2 Given $0 \\\\le s < 1$ , let $\\\\phi ^\\\\omega $ be the Wiener randomization of $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ defined in (REF ) and set $z_1 = S(t) \\\\phi ^\\\\omega $ .\",\n \"(i) For any $\\\\sigma < 2s$ , we have $z_3 \\\\in X^\\\\sigma _\\\\textup {loc}\\\\subset C(\\\\mathbb {R}; H^\\\\sigma (\\\\mathbb {R}^3))$ almost surely.\",\n \"More precisely, there exists an almost surely finite constant $ C(\\\\omega , \\\\Vert \\\\phi \\\\Vert _{H^s} ) > 0$ and $\\\\theta > 0$ such that $\\\\Vert z_3 \\\\Vert _{X^{\\\\sigma }([0, T])} = \\\\bigg \\\\Vert \\\\int _0^t S(t - t^{\\\\prime }) |z_1|^2 z_1(t^{\\\\prime }) dt^{\\\\prime }\\\\bigg \\\\Vert _{X^\\\\sigma ([0, T])}\\\\le T^\\\\theta C(\\\\omega , \\\\Vert \\\\phi \\\\Vert _{H^s})$ for any $T > 0$ .\",\n \"In particular, by taking $\\\\sigma = 2s-\\\\varepsilon $ for small $\\\\varepsilon > 0$ , (REF ) shows that the second orderHere, we referred to $z_3$ as the second order term since it corresponds to the second order term appearing in the (formal) power series expansion of solutions to (REF ) in terms of the random initial data.\",\n \"For the same reason, we refer to $z_5$ in (REF ) and $z_7$ in (REF ) as the third and fourth order terms in the following.\",\n \"See Subsection REF below.\",\n \"term $z_3$ is smoother (by $s - \\\\varepsilon $ ) than the first order term $z_1$ , provided $s > 0$ .\",\n \"(ii) When $s = 0$ , there is no smoothing in the second order term $z_3$ in general; there exists $\\\\phi \\\\in L^2(\\\\mathbb {R}^3)$ such that the estimate (REF ) with $\\\\sigma = \\\\varepsilon > 0$ fails for any $\\\\varepsilon > 0$ .\",\n \"In [2], we already proved (REF ) when $\\\\sigma = \\\\frac{1}{2}$ and $s > \\\\frac{1}{4}$ , giving the regularity restriction in Theorem REF .\",\n \"See Section for the proof of Proposition REF for a general value of $\\\\sigma $ .\",\n \"In the proof of Theorem REF , in order to carry out the case-by-case analysis for (REF ), we need to have $z_3 \\\\in X^\\\\frac{1}{2}_\\\\text{loc}$ , (where we have a deterministic local well-posedness for (REF )).\",\n \"In view of Proposition REF , this imposes the regularity restriction $s > \\\\frac{1}{4}$ .\",\n \"Note, however, that even when $s \\\\le \\\\frac{1}{4}$ , $z_3$ is still a well defined space-time function of spatial regularity $2s- < \\\\frac{1}{2} $ .\",\n \"This motivates us to consider the following second order expansion: $u = z_1 + z_3 + v$ and remove the second order interaction $z_1 \\\\overline{z}_1 z_1$ .\",\n \"Indeed, the residual term $v := u - z_1 - z_3$ satisfies the following equation: ${\\\\left\\\\lbrace \\\\begin{array}{ll}i \\\\partial _tv + \\\\Delta v = \\\\mathcal {N}(v + z_1+z_3) - \\\\mathcal {N}(z_1)\\\\\\\\v|_{t = 0} = 0,\\\\end{array}\\\\right.\",\n \"}$ where $\\\\mathcal {N}(u) = |u|^2 u$ .\",\n \"In terms of the Duhamel formulation, we have $v(t) = - i \\\\int _0^t S(t-t^{\\\\prime })\\\\big \\\\lbrace \\\\mathcal {N}(v + z_1 + z_3) - \\\\mathcal {N}(z_1)\\\\big \\\\rbrace (t^{\\\\prime }) dt^{\\\\prime }.$ Then, by studying the fixed point problem (REF ) for $v$ , we have the following improved almost sure local well-posedness of (REF ).\",\n \"Theorem 1.3 Given $ \\\\frac{1}{2} \\\\le \\\\sigma \\\\le 1$ , let $ \\\\frac{2}{5} \\\\sigma < s < \\\\frac{1}{2}$ .\",\n \"Given $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ , let $\\\\phi ^\\\\omega $ be its Wiener randomization defined in (REF ).\",\n \"Then, the cubic NLS (REF ) on $\\\\mathbb {R}^3$ is almost surely locally well-posed with respect to the random initial data $\\\\phi ^\\\\omega $ .\",\n \"More precisely, there exists a set $\\\\Sigma = \\\\Sigma (\\\\phi , \\\\sigma ) \\\\subset \\\\Omega $ with $P(\\\\Sigma ) = 1$ such that, for any $\\\\omega \\\\in \\\\Sigma $ , there exists a unique function $u = u^\\\\omega $ in the class: $z_1 + z_3+ X^\\\\sigma ([0, T])& \\\\subset z_1 + z_3+ C([0, T]; H^{\\\\sigma } (\\\\mathbb {R}^3))\\\\\\\\\\\\ &\\\\subset C([0, T];H^s(\\\\mathbb {R}^3))$ with $T = T(\\\\phi , \\\\omega ) >0$ such that $u$ is a solution to (REF ) on $[0, T]$ .\",\n \"As in Theorem REF , the uniqueness of $u$ in the class $z_1 + z_3+ X^\\\\sigma ([0, T])$ is to be interpreted as uniqueness of the residual term $v = u - z_1 - z_3$ in $X^\\\\sigma ([0, T])$ .\",\n \"See also Remark REF .\",\n \"By taking $\\\\sigma = \\\\frac{1}{2}$ , Theorem REF states that (REF ) is almost surely locally well-posed in $H^s(\\\\mathbb {R}^3)$ , provided $s > \\\\frac{1}{5}$ .\",\n \"This in particular improves the almost sure local well-posedness in Theorem REF .\",\n \"On the other hand, by taking $\\\\sigma = 1$ , Theorem REF allows us to construct a solution $v = u - z_1 - z_3\\\\in X^1([0, T]) \\\\subset C([0,T] ; H^1(\\\\mathbb {R}^3))$ , provided that $s > \\\\frac{2}{5}$ .\",\n \"In particular, we can take random initial data below the scaling critical regularity $s_\\\\text{crit} = \\\\frac{1}{2}$ , while we construct the residual part $v$ in $X^1([0, T])$ .\",\n \"This opens up a possibility of studying the global-in-time behavior of $v$ , using the (non-conserved) energy of $v$ : $E(v)(t) = \\\\frac{1}{2} \\\\int _{\\\\mathbb {R}^3} |\\\\nabla v(t, x)|^2 dx+\\\\frac{1}{4} \\\\int _{\\\\mathbb {R}^3} |v(t, x)|^4 dx$ with random initial data below the scaling critical regularity.\",\n \"We remark that by inspecting the argument in [2], a modification of (the proof of) Theorem REF yields almost sure local well-posedness of (REF ) in the class: $ S(t)\\\\phi ^\\\\omega + X^1([0, T]) \\\\subset S(t) \\\\phi ^{\\\\omega } + C([0,T] ; H^1(\\\\mathbb {R}^3))$ only for $s > \\\\frac{1}{2}$ .\",\n \"(This restriction on $s$ can be easily seen by setting $\\\\sigma = 1$ in Proposition REF .)\",\n \"In particular, Theorem REF does not allow us to take random initial data below the scaling critical regularity $s_\\\\text{crit} = \\\\frac{1}{2}$ in studying the global-in-time behavior of $v \\\\in X^1([0, T])$ , namely at the level of the energy $E(v)$ .\",\n \"As our focus in this paper is the local-in-time analysis, we do not pursue further this issue on almost sure global well-posedness of (REF ) below the scaling critical regularity in this paper.\",\n \"We, however, point out two recent results [30], [37] on almost sure global well-posedness below the energy space for the defocusing energy-critical NLS in higher dimensions.\",\n \"The main strategy for proving Theorem REF is to study the fixed point problem (REF ) by carrying out case-by-case analysis on $w_1\\\\overline{w_2} w_3,\\\\quad \\\\text{for $w_i = v,$ $z_1$, or $z_3$, $i = 1, 2, 3$, but not all $w_i$ equal to $z_1$}$ in $N^\\\\sigma ([0, T])$ , where the dual norm is defined by $\\\\Vert F\\\\Vert _{N^\\\\sigma ([0, T])} = \\\\bigg \\\\Vert \\\\int _{0}^t S(t - t^{\\\\prime }) F(t^{\\\\prime }) dt^{\\\\prime }\\\\bigg \\\\Vert _{X^\\\\sigma ([0, T])}.$ Note that the number of cases has increased from the case-by-case analysis in the proof of Theorem REF , where we had $w_i = v$ or $z_1$ .\",\n \"One of the main ingredients is the smoothing on $z_3$ stated in Proposition REF above.\",\n \"Note, however, that in order to exploit this smoothing, we need to measure $z_3$ in the $X^{2s-}([0, T])$ -norm, which imposes a certain rigidity on the space-time integrability.Namely, we need to measure $z_3$ in $L^q_t([0, T]; W^{2s-, r}(\\\\mathbb {R}^3))$ for admissible pairs $(q, r)$ .\",\n \"See Lemma REF .\",\n \"In order to prove Theorem REF , we also need to exploit a gain of integrability on $z_3$ .\",\n \"In Lemma REF , we use the dispersive estimate (see (REF ) below) and the gain of integrability on each $z_1$ of the three factors in (REF ) and show that $z_3$ also enjoys a gain of integrability by giving up some differentiability.\",\n \"Remark 1.4 Given $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ , let $\\\\phi ^\\\\omega $ be its Wiener randomization defined in (REF ).\",\n \"Given $v_0 \\\\in H^\\\\frac{1}{2}(\\\\mathbb {R}^3)$ , we can also consider (REF ) with the random initial data of the form $u_0^\\\\omega = v_0 + \\\\phi ^\\\\omega $ : ${\\\\left\\\\lbrace \\\\begin{array}{ll}i \\\\partial _tu + \\\\Delta u = |u|^2 u\\\\\\\\u|_{t = 0} = v_0+ \\\\phi ^\\\\omega .\\\\end{array}\\\\right.\",\n \"}$ Then, by slightly modifying the proofs, we see that the analogues of Theorems A and REF (with $\\\\sigma = \\\\frac{1}{2}$ ) also hold for (REF ).\",\n \"Namely, (REF ) is almost surely locally well-posed, provided $s > \\\\frac{1}{5}$ .\",\n \"This amounts to considering the following Cauchy problems: ${\\\\left\\\\lbrace \\\\begin{array}{ll}i \\\\partial _tv + \\\\Delta v = |v + z_1|^2(v+z_1)\\\\\\\\v|_{t = 0} = v_0,\\\\end{array}\\\\right.\",\n \"}$ when $\\\\frac{1}{4} < s < \\\\frac{1}{2}$ and ${\\\\left\\\\lbrace \\\\begin{array}{ll}i \\\\partial _tv + \\\\Delta v = \\\\mathcal {N}(v + z_1+z_3) - \\\\mathcal {N}(z_1)\\\\\\\\v|_{t = 0} = v_0,\\\\end{array}\\\\right.\",\n \"}$ when $\\\\frac{1}{5} < s \\\\le \\\\frac{1}{4}$ .\",\n \"In this case, the critical nature of the problem appears through the $v \\\\overline{v} v$ interaction in the case-by-case analysis (REF ) due to the deterministic (non-zero) initial data $v_0$ at the critical regularity.\",\n \"The required modification is straightforward and thus we omit details.\",\n \"See Proposition 6.3 in [2] and Lemma 6.2 in [37].\",\n \"We point out that the discussion in the next subsection also applies to (REF ).\",\n \"Remark 1.5 (i) Let $I(u_1, u_2, u_3)$ denote the trilinear operator defined by $I(u_1, u_2, u_3) (t):= -i \\\\int _0^t S(t - t^{\\\\prime }) u_1 \\\\overline{u_2} u_3 (t^{\\\\prime }) dt^{\\\\prime }.$ Then, for $\\\\sigma > 3s-1$ , there is no finite constant $C > 0$ such that $\\\\big \\\\Vert I( u_1, u_2, u_3)\\\\big \\\\Vert _{X^\\\\sigma ([0, 1])}\\\\le C \\\\prod _{j = 1}^3 \\\\Vert \\\\phi _j \\\\Vert _{H^s}$ for all $\\\\phi _j \\\\in H^s(\\\\mathbb {R}^3)$ , where $u_j = S(t) \\\\phi _j$ .\",\n \"See Appendix .\",\n \"In particular, this shows that when $s \\\\le \\\\frac{1}{2}$ , there is no deterministic smoothing for $I$ , i.e.\",\n \"(REF ) does not hold for $\\\\sigma > s$ .\",\n \"(ii) The proof of Proposition REF only exploits “the randomization at the linear level”.\",\n \"Given $s \\\\in \\\\mathbb {R}$ , let $\\\\mathcal {R}^s$ denote the class of functions defined by $\\\\mathcal {R}^s = \\\\big \\\\lbrace u \\\\text{ on } \\\\mathbb {R}\\\\times \\\\mathbb {R}^3:\\\\ & i \\\\partial _tu + \\\\Delta u = 0 \\\\text{ and } \\\\\\\\&u \\\\in L^q_{t, \\\\text{loc}} W^{s, r}_x(\\\\mathbb {R}^3)\\\\text{ for any } 2 \\\\le q, r < \\\\infty \\\\big \\\\rbrace .$ Under the regularity assumption: $\\\\sigma < 2s$ and $0 \\\\le s < 1$ , it follows from the proof of Proposition REF that the left-hand side of (REF ) is finite for any $u_1, u_2, u_3 \\\\in \\\\mathcal {R}^s$ .Here, we only need finiteness of the $A_3^s$ -norm defined in (REF ) for each $u_j$ , $j = 1, 2, 3$ .\",\n \"See Remark REF .\",\n \"In other words, the proof of Proposition REF only uses the fact that the random linear solution $z_1 = S(t) \\\\phi ^\\\\omega $ belongs to $\\\\mathcal {R}^s$ almost surely; see the probabilistic Strichartz estimate (Lemma REF ).\",\n \"The multilinear random structure of $z_3$ in terms of the random linear solution $z_1$ yields further cancellation.\",\n \"See, for example, Lemma 3.6 in [15].\",\n \"Such extra cancellation seems to improve only space-time integrability and we do not know how to use it to improve the regularity threshold (i.e.\",\n \"differentiability) at this point.\",\n \"A similar comment applies to the unbalanced higher order terms $\\\\zeta _{2k-1}$ (including $z_5$ below) studied in Proposition REF below.\"\n ],\n [\n \"Partial power series expansion\\nand the associated critical regularity\",\n \"In this subsection, we discuss possible improvements over Theorem REF by considering further expansions.\",\n \"For this purpose, we fix $ \\\\sigma = \\\\frac{1}{2}$ in the following.\",\n \"By examining the proof of Theorem REF , we see that the regularity restriction $s > \\\\frac{1}{5}$ comes from the following third order term: $z_5 (t) := -i \\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = 5\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 3\\\\rbrace \\\\end{array}}\\\\int _0^t S(t - t^{\\\\prime })z_{j_1} \\\\overline{z_{j_2}}z_{j_3} (t^{\\\\prime }) dt^{\\\\prime }.$ Namely, we have $(j_1, j_2, j_3) = (1, 1, 3)$ up to permutations.\",\n \"In Lemma REF , we show that given $0< s< \\\\frac{1}{2}$ , we have $ z_5 \\\\in X^{\\\\frac{5}{2} s -}_\\\\textup {loc}$ .\",\n \"In particular, we have $ z_5 \\\\in X^\\\\frac{1}{2}_\\\\textup {loc}$ , provided $s > \\\\frac{1}{5} $ , yielding the regularity threshold in Theorem REF .\",\n \"A natural next step is to remove this non-desirable third order interaction: $ \\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = 5\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 3\\\\rbrace \\\\end{array}}z_{j_1} \\\\overline{z_{j_2}}z_{j_3}$ in the case-by-case analysis in (REF ) by considering the following third order expansion: $u = z_1 + z_3 + z_5+ v.$ In this case, the residual term $v := u - z_1 - z_3- z_5$ satisfies the following equation: ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\displaystyle i \\\\partial _tv + \\\\Delta v = \\\\mathcal {N}(v + z_1+z_3+z_5)- \\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 \\\\in \\\\lbrace 3, 5\\\\rbrace \\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 3\\\\rbrace \\\\end{array}}z_{j_1} \\\\overline{z_{j_2}}z_{j_3}\\\\\\\\v|_{t = 0} = 0.\\\\end{array}\\\\right.\",\n \"}$ We expect that (REF ) is almost surely locally well-posed for $s > \\\\frac{2}{11}$ , which would be an improvement over Theorem REF .\",\n \"The proof will be once again based on case-by-case analysis: $w_1\\\\overline{w_2} w_3\\\\quad & \\\\text{for $w_i = v, z_1, z_3$, or $z_5$, $i = 1, 2, 3$, such that } \\\\\\\\& \\\\text{it is not of the form $z_{j_1}\\\\overline{z_{j_2}}z_{j_3} $ with $j_1 + j_2 + j_3 \\\\in \\\\lbrace 3, 5\\\\rbrace $}$ in $N^\\\\frac{1}{2}([0, T])$ .\",\n \"Note the increasing number of combinations.\",\n \"In the following, however, we do not discuss details of this particular improvement over Theorem REF .\",\n \"Instead, we consider further iterative steps and discuss a possible limitation of this procedure.\",\n \"Remark 1.6 We point out that the expansions (REF ), (REF ), and (REF ) correspond to partial power series expansions of the first, second, and third orders,A (partial) power series expansion of a solution to (REF ) in terms of the random initial data can be expressed as a summation of certain multilinear operators over ternary trees.\",\n \"See, for example, [11], [36].\",\n \"Here, the term “order” in our context corresponds to “generation (of the associated trees) $+1$ ” with the convention that the trivial tree consisting only of the root node is of the zeroth generation.\",\n \"For example, the third order term $z_5$ in (REF ) appears as the summation over all the multilinear operators associated to the ternary trees of the third generation.\",\n \"In terms of the graphical representation in [36], we have $z_5 \\\\,\\\\text{``}\\\\!=\\\\!\\\\text{''} \\\\,[baseline=-3,scale=0.15]{(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (0.9, -1) node[dot] {};(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (-0.9, -1) node[dot] {};(0,0)node[ddot] {} -- (1.3,1) node[ddot] {}-- (1.3, 0) node[dot] {};(1.3,1) node[ddot] {}-- (2.6, 0) node[dot] {};}\\\\,+ \\\\,[baseline=-3,scale=0.15]{(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (0.9, -1) node[dot] {};(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (-0.9, -1) node[dot] {};(0,0)node[ddot] {} -- (0,1) node[ddot] {}-- (0.9, 0) node[dot] {};(0,0)node[ddot] {} -- (0,1) node[ddot] {}-- (-0.9, 0) node[dot] {};}\\\\,+\\\\, [baseline=-3,scale=0.15]{(2.6,-1)node[dot] {} -- (2.6,0) node[ddot] {}-- (3.5, -1) node[dot] {};(2.6,-1)node[dot] {} -- (2.6,0) node[ddot] {}-- (1.7, -1) node[dot] {};(0,0)node[dot] {} -- (1.3,1) node[ddot] {}-- (1.3, 0) node[dot] {};(1.3,1) node[ddot] {}-- (2.6, 0) node[ddot] {};} \\\\ ,$ where “ $[baseline=-2.7,scale=0.15]{(0,0) node[dot]{};}$ ” denotes the random linear solution $z_1 = S(t) \\\\phi ^\\\\omega $ and “ $[baseline=-2.7,scale=0.15]{(0,0) node[ddot]{};}$ ” denotes the trilinear Duhamel integral operator $ I(u_1, u_2, u_3)$ defined in (REF ) with its three children as its arguments $u_1, u_2$ , and $u_3$ .\",\n \"respectively, of a solution to (REF ) in terms of the random initial data.\",\n \"Then, by considering the associated equations (REF ), (REF ), and (REF ) for the residual term $v$ , we are recasting the original problem (REF ) as a fixed point problem centered at the partial power series expansions of the first, second, and third orders, respectively.\",\n \"By drawing an analogy to the previous steps, we expect that the worst contribution comes from the following fourth order terms: $z_7 (t): = -i \\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = 7\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 3, 5\\\\rbrace \\\\end{array}}\\\\int _0^t S(t - t^{\\\\prime }) z_{j_1} \\\\overline{z_{j_2}}z_{j_3} (t^{\\\\prime }) dt^{\\\\prime }.$ There are basically two contributions to (REF ): $(j_1, j_2, j_3) = (1, 3, 3)$ or $(1, 1, 5)$ up to permutations.\",\n \"In Lemma REF , we show that the contribution from $(j_1, j_2, j_3) = (1, 1, 5)$ is worse, being responsible for the expected regularity restriction $s > \\\\frac{2}{11}$ .\",\n \"In order to remove this term, we can consider the following fourth order expansion: $u = z_1 + z_3 + z_5+z_7+ v$ as in the previous steps and try to solve the following equation for the residual term $v = u - z_1 - z_3- z_5-z_7$ : ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\displaystyle i \\\\partial _tv + \\\\Delta v = \\\\mathcal {N}(v + z_1+z_3+z_5+z_7)- \\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 \\\\in \\\\lbrace 3, 5, 7\\\\rbrace \\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 3, 5\\\\rbrace \\\\end{array}}z_{j_1} \\\\overline{z_{j_2}}z_{j_3}\\\\\\\\v|_{t = 0} = 0.\\\\end{array}\\\\right.\",\n \"}$ We can obviously iterate this argument and consider the following $k$ th order expansion: $u = \\\\sum _{\\\\ell = 1}^{k} z_{2\\\\ell -1}+ v.$ In this case, we need to consider the following equation for the residual term $v := u - \\\\sum _{\\\\ell = 1}^{k} z_{2\\\\ell -1}$ : ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\displaystyle i \\\\partial _tv + \\\\Delta v = \\\\mathcal {N}\\\\bigg (v +\\\\sum _{\\\\ell = 1}^{k} z_{2\\\\ell -1}\\\\bigg )-\\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 \\\\in \\\\lbrace 3, 5, \\\\dots , 2k-1\\\\rbrace \\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 3, \\\\dots , 2k-3\\\\rbrace \\\\end{array}}z_{j_1} \\\\overline{z_{j_2}}z_{j_3}\\\\\\\\v|_{t = 0} = 0\\\\end{array}\\\\right.\",\n \"}$ and hope to construct a solution $v \\\\in X^\\\\frac{1}{2}([0, T])$ by carrying out the following case-by-case analysis: $\\\\begin{split}w_1\\\\overline{w_2} w_3,\\\\quad & \\\\text{for $w_i = v, z_j,$ $j \\\\in \\\\lbrace 1, 3, \\\\dots , 2k-1\\\\rbrace $,$i = 1, 2, 3$, such that } \\\\\\\\& \\\\text{it is not of the form $z_{j_1}\\\\overline{z_{j_2}}z_{j_3} $ with$j_1 + j_2 + j_3\\\\in \\\\lbrace 3, 5, \\\\dots , 2k-1\\\\rbrace $}\\\\end{split}$ in $N^\\\\frac{1}{2}([0, T])$ .\",\n \"Then, a natural question to ask is “Does this iterative procedure work indefinitely, allowing us to arbitrarily lower the regularity threshold for almost sure local well-posedness for (REF )?\",\n \"Or is there any limitation to it?\",\n \"We now consider a “critical” regularity $s_* < \\\\frac{1}{2}$ with respect to this iterative procedure for proving almost sure local well-posedness of (REF ).\",\n \"We simply define the critical regularity $s_* <\\\\frac{1}{2} $ for this problem to be the infimum of the values of $s< \\\\frac{1}{2}$ such that given any $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ , the above iterative procedureNamely, we solve (REF ) for $v \\\\in X^\\\\frac{1}{2}([0, T])$ with some finite (or infinite) number of steps.\",\n \"shows that (REF ) is almost surely locally well-posed with respect to the Wiener randomization $\\\\phi ^\\\\omega $ of $\\\\phi $ .\",\n \"This is an empirical notion of criticality; unlike the scaling criticality, we can not a priori compute this critical regularity $s_*$ .\",\n \"Moreover, our discussion will be based on the estimates on the stochastic multilinear terms (Proposition REF and Proposition REF below).\",\n \"In the following, we discuss a (possible) lower bound on $s_*$ , presenting a limitation to our iterative procedure based on partial power series expansions.\",\n \"Within the framework of the iterative procedure discussed above, a necessary condition for carrying out the case-by-case analysis (REF ) to study (REF ) in $X^\\\\frac{1}{2}([0, T])$ is that the $(k+1)$ st order term $ z_{2k+1} = z_{2(k+1)-1}$ defined in a recursive manner: $z_{2k+1}(t) : = -i \\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = {2k+1}\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 3, \\\\dots , 2k-1\\\\rbrace \\\\end{array}}\\\\int _0^t S(t - t^{\\\\prime }) z_{j_1} \\\\overline{z_{j_2}}z_{j_3} (t^{\\\\prime }) dt^{\\\\prime }$ belongs to $X^\\\\frac{1}{2}([0, T])$ .\",\n \"By the nature of this iterative procedure, we may assume that the lower order terms $z_{2\\\\ell -1}$ , $\\\\ell \\\\in \\\\lbrace 1, 3, \\\\dots , k\\\\rbrace $ , belong to $X^{ s_\\\\ell }([0, T])$ for some $ s_\\\\ell < \\\\frac{1}{2}$ but not in $X^\\\\frac{1}{2}([0, T])$ , since if any of the lower order terms, say $z_{2\\\\ell -1}$ for some $\\\\ell \\\\in \\\\lbrace 1, 3, \\\\dots , k\\\\rbrace $ , were in $X^\\\\frac{1}{2}([0, T])$ , then we would have stopped the iterative procedure at the $(\\\\ell -1)$ th step.\",\n \"As in the previous steps, we expect that $z_{2k+1}$ is responsible for a regularity restriction at the $k$ th step of this iterative approach.\",\n \"It could be a cumbersome task to study the regularity property of $z_{2k+1}$ due to the increasing number of combinations for $z_{j_1}\\\\overline{z_{j_2}} z_{j_3}$ , satisfying $j_1 + j_2 + j_3 = {2k+1}$ , $j_1, j_2, j_3 \\\\in \\\\lbrace 1, 3, \\\\dots , 2k-1\\\\rbrace $ .\",\n \"In the following, we instead study the regularity property of the $(k+1)$ st order termIn fact, we study the $k$ th order term $\\\\zeta _{2k-1}$ in Proposition REF .\",\n \"of a particular form, corresponding to $(j_1, j_2, j_3) = (1, 1, 2k - 1)$ up to permutations in (REF ); given an integer $k \\\\ge 0$ , define $\\\\zeta _{2k+1}$ by setting $\\\\zeta _1 := z_1 = S(t) \\\\phi ^\\\\omega $ and $\\\\zeta _{2k+1}(t) := -i\\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = {2k+1}\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 2k-1\\\\rbrace \\\\end{array}}\\\\int _0^t S(t - t^{\\\\prime }) \\\\zeta _{j_1} \\\\overline{\\\\zeta _{j_2}} \\\\zeta _{j_3} (t^{\\\\prime }) dt^{\\\\prime }.$ As mentioned above, there are only three terms in this sum: $(j_1, j_2, j_3) = (1, 1, 2k - 1)$ up to permutations.\",\n \"Hence, $\\\\zeta _{2k+1}$ consists of the “unbalanced”By associating the $(2k+1)$ -linear terms appearing in the summation in (REF ) with ternary trees of the $k$ th generation as in [36], the summands in (REF ) correspond to the “unbalanced” trees of the $k$ th generation, where two of the three children of the root node are terminal.\",\n \"$(2k+1)$ -linear terms appearing in the definition (REF ) of $z_{2k+1}$ .\",\n \"We claim that the $(k+1)$ st order term $\\\\zeta _{2k + 1}$ is responsible for the regularity restriction at the $k$ th step of the iterative procedure.\",\n \"See Theorem REF below.\",\n \"In anticipating the alternative expansion (REF ) below, let us study the regularity property of the unbalanced $k$ th order term $\\\\zeta _{2k-1}$ : $\\\\zeta _{2k-1}(t) := -i\\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = {2k-1}\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 2k-3\\\\rbrace \\\\end{array}}\\\\int _0^t S(t - t^{\\\\prime }) \\\\zeta _{j_1} \\\\overline{\\\\zeta _{j_2}} \\\\zeta _{j_3} (t^{\\\\prime }) dt^{\\\\prime }.$ For $k = 2, 3, 4$ , we have (with appropriate restrictions on the range of $s$ ) $\\\\zeta _3 = z_3 \\\\in X^{2s-}_\\\\text{loc},\\\\qquad \\\\zeta _5 = z_5 \\\\in X^{\\\\frac{5}{2}s-}_\\\\text{loc},\\\\qquad \\\\text{and}\\\\qquad \\\\zeta _7 \\\\in X^{\\\\frac{11}{4}s-}_\\\\text{loc}.$ See Proposition REF and Lemmas REF and REF .\",\n \"In general, we have the following proposition.\",\n \"Proposition 1.7 Define a sequence $\\\\lbrace \\\\alpha _k\\\\rbrace _{k \\\\in \\\\mathbb {N}}$ of positive real numbers by the following recursive relation: $\\\\alpha _k = \\\\frac{\\\\alpha _{k-1} + 3}{2}$ with $\\\\alpha _1 = 1$ .\",\n \"Given $0< s < \\\\alpha _{k-1}^{-1}$ , let $\\\\phi ^\\\\omega $ be the Wiener randomization of $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ defined in (REF ).\",\n \"Then, we have $\\\\zeta _{2k -1} \\\\in X^\\\\sigma _\\\\textup {loc}$ for $\\\\sigma < \\\\alpha _k \\\\cdot s $ , almost surely.\",\n \"By solving the recursive relation (REF ), we have $\\\\alpha _k = 2\\\\bigg \\\\lbrace 1 - \\\\bigg (\\\\frac{1}{2}\\\\bigg )^{k-1}\\\\bigg \\\\rbrace + 1$ and thus we have $\\\\alpha _2 = 2$ , $\\\\alpha _3 = \\\\frac{5}{2}$ , and $\\\\alpha _4 = \\\\frac{11}{4}$ .\",\n \"In particular, Proposition REF agrees with (REF ).\",\n \"Moreover, since $\\\\alpha _k$ is increasing and $\\\\lim _{k \\\\rightarrow \\\\infty } \\\\alpha _k =3$ , the regularity restriction $s < \\\\alpha _{k-1}^{-1}$ in Proposition REF does not cause any issue since our main focus is to study the probabilistic local well-posedness of (REF ) in the range of $s$ that is not covered by Theorem REF .\",\n \"Namely, we may assume $s \\\\le \\\\frac{1}{5}\\\\, (< \\\\frac{1}{3})$ in the following.\",\n \"In view of Propositions REF and REF , one obvious lower bound for the critical regularity $s_*$ for this iterative procedure is given by $s_0: = 0$ since there is no gain of regularity when $s = 0$ (even in moving from $z_1$ to $z_3$ ).\",\n \"On the other hand, in order to prove almost sure local well-posedness of (REF ) by carrying out the case-by-case analysis (REF ) for the equation (REF ), we need to show that the $(k+1)$ st order term $\\\\zeta _{2k+1}$ belongs to $X^\\\\frac{1}{2}([0, T])$ .\",\n \"This gives rise to a regularity restriction $s_k := \\\\frac{1}{2\\\\alpha _{k+1}}$ at the $k$ th step of the iterative procedure.\",\n \"By taking $k \\\\rightarrow \\\\infty $ , we obtain another “lower” boundThis “lower” bound is based on the upper bounds obtained in Propositions REF and REF .\",\n \"In other words, if one can improve the bounds in Propositions REF and REF , then one can lower the value of $s_\\\\infty $ .\",\n \"See Remark REF .\",\n \"$s_\\\\infty := \\\\frac{1}{6}$ on this critical regularity $s_*$ .\",\n \"As mentioned above, the case-by-case analysis (REF ) for general $k \\\\in \\\\mathbb {N}$ may be a combinatorially overwhelming task due to (i) the number of the increasing combinations in (REF ) and (ii) the random multilinear terms $z_j,$ $j \\\\in \\\\lbrace 3, 5, \\\\dots , 2k-1\\\\rbrace $ , themselves having non-trivial combinatorial structures which makes it difficult to establish nonlinear estimates; see (REF ).\",\n \"In the following, we instead consider an alternative iterative procedure based on the following expansion: $u = \\\\sum _{\\\\ell = 1}^k \\\\zeta _{2\\\\ell -1} + v$ in place of (REF ).\",\n \"This expansion allows us to prove the following almost sure local well-posedness of (REF ) for $s > s_\\\\infty = \\\\frac{1}{6}$ .\",\n \"Theorem 1.8 Let $\\\\frac{1}{6} < s < \\\\frac{1}{2}$ .\",\n \"Given $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ , let $\\\\phi ^\\\\omega $ be its Wiener randomization defined in (REF ).\",\n \"Then, the cubic NLS (REF ) on $\\\\mathbb {R}^3$ is almost surely locally well-posed with respect to the random initial data $\\\\phi ^\\\\omega $ .\",\n \"More precisely, there exists a set $\\\\Sigma = \\\\Sigma (\\\\phi ) \\\\subset \\\\Omega $ with $P(\\\\Sigma ) = 1$ such that, for any $\\\\omega \\\\in \\\\Sigma $ , there exists a unique function $u = u^\\\\omega $ in the class: $\\\\zeta _1 + \\\\zeta _3 + \\\\cdots + \\\\zeta _{2k-1}+ X^\\\\frac{1}{2}([0, T])& \\\\subset \\\\zeta _1 + \\\\zeta _3 + \\\\cdots + \\\\zeta _{2k-1}+ C([0, T]; H^\\\\frac{1}{2} (\\\\mathbb {R}^3))\\\\\\\\\\\\ &\\\\subset C([0, T];H^s(\\\\mathbb {R}^3))$ with $T = T(\\\\phi , \\\\omega ) >0$ such that $u$ is a solution to (REF ) on $[0, T]$ .\",\n \"Here, $k \\\\in \\\\mathbb {N}$ is a unique positive integer such that $\\\\frac{1}{2\\\\alpha _{k+1}} < s \\\\le \\\\frac{1}{2\\\\alpha _{k}}$ .\",\n \"As before, the uniqueness of $u$ in the class: $\\\\zeta _1 + \\\\zeta _3 + \\\\cdots + \\\\zeta _{2k-1}+ X^\\\\frac{1}{2}([0, T])$ is to be interpreted as uniqueness of the residual term $v = u -\\\\sum _{\\\\ell = 1}^k \\\\zeta _{2\\\\ell -1} $ in $X^\\\\frac{1}{2}([0, T])$ .\",\n \"See also Remark REF .\",\n \"In view of the discussion above, Theorem REF proves almost sure local well-posedness of (REF ) in an almost “optimal”Once again, this is based on the estimates in Propositions REF and REF .\",\n \"In particular, the “optimality” of the regularity threshold in Theorem REF is with respect to Propositions REF and REF .\",\n \"If one can improve the bounds in Propositions REF and REF , then one can lower the regularity threshold in Theorem REF .\",\n \"regularity range $s > s_\\\\infty = \\\\frac{1}{6}$ with respect to the original iterative procedure based on the partial power series expansion (REF ).\",\n \"The proof of Theorem REF is analogous to that of Theorem REF .\",\n \"Given $ \\\\frac{1}{6} < s<\\\\frac{1}{2}$ , fix $k \\\\in \\\\mathbb {N}$ such that $\\\\frac{1}{2\\\\alpha _{k+1}} < s \\\\le \\\\frac{1}{2\\\\alpha _{k}}$ .In view of Proposition REF , the lower bound on $s$ guarantees that $\\\\zeta _{2k+1} \\\\in X^\\\\frac{1}{2}([0, T])$ almost surely, while the upper bound on $s$ states that we can not use Proposition REF to conclude $\\\\zeta _{2k-1} \\\\in X^\\\\frac{1}{2}([0, T])$ almost surely.\",\n \"Write a solution $u$ as in (REF ).\",\n \"Note that as $s$ gets closer and closer to the critical value $s_\\\\infty = \\\\frac{1}{6}$ , the expansion (REF ) gets arbitrarily long.\",\n \"In view of (REF ), the residual term $v := u - \\\\sum _{\\\\ell = 1}^{k} \\\\zeta _{2\\\\ell -1}$ satisfies the following equation: ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\displaystyle i \\\\partial _tv + \\\\Delta v = \\\\mathcal {N}\\\\bigg (v +\\\\sum _{\\\\ell = 1}^{k} \\\\zeta _{2\\\\ell -1}\\\\bigg )-\\\\sum _{\\\\ell = 2}^{k}\\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = 2\\\\ell - 1\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 2\\\\ell -3\\\\rbrace \\\\end{array}}\\\\zeta _{j_1} \\\\overline{\\\\zeta _{j_2}}\\\\zeta _{j_3}\\\\\\\\v|_{t = 0} = 0.\\\\end{array}\\\\right.\",\n \"}$ Hence, we need to carry out the following case-by-case analysis: $\\\\begin{split}w_1\\\\overline{w_2} w_3,\\\\quad & \\\\text{for $w_i = v, \\\\zeta _j,$ $j \\\\in \\\\lbrace 1, 3, \\\\dots , 2k-1\\\\rbrace $,$i = 1, 2, 3$, such that } \\\\\\\\& \\\\text{it is not of the form $\\\\zeta _{j_1}\\\\overline{\\\\zeta _{j_2}}\\\\zeta _{j_3} $ with$(j_1, j_2, j_3 )= (1, 1, 2\\\\ell -3)$}\\\\\\\\& \\\\text{(up to permutations) for $\\\\ell = 2, 3 \\\\dots , k$}\\\\end{split}$ in $N^\\\\frac{1}{2}([0, T])$ .\",\n \"While Theorem REF yields almost sure local well-posedness in an almost optimal range with respect to the original iterative procedure based on the partial power series expansion (REF ), the required analysis is much simpler than that required for the original iterative procedure based on the expansion (REF ).\",\n \"First, note that while the case-by-case analysis (REF ) based on the original iterative procedure involves combinatorially non-trivial $z_{2k-1}$ , the case-by-case analysis (REF ) only involves the unbalanced $k$ th order term $\\\\zeta _{2k-1}$ which has a much simpler structure than $z_{2k-1}$ .\",\n \"In particular, Proposition REF shows that $\\\\zeta _{2k-1}$ , $k \\\\ge 3$ , has a better regularity property than the second order term $\\\\zeta _3 = z_3$ in (REF ).\",\n \"In terms of space-time integrability, we show that $\\\\zeta _{2k-1}$ also enjoys a gain of integrability by giving up a control on derivatives (Lemma REF ).\",\n \"Finally, by inspecting the proof of Theorem REF (see Lemma REF and Proposition REF below), we see that, except for $z_{j_1} \\\\overline{z_{j_2}}z_{j_3}$ with $(j_1, j_2, j_3) = (1, 1, 3)$ up to permutations,Namely, the terms constituting the third order term $\\\\zeta _5 = z_5$ in (REF ).\",\n \"we can bound all the terms $w_1 \\\\overline{w_2} w_3$ appearing in the case-by-case analysis (REF ) in $N^\\\\frac{1}{2}([0, T])$ .\",\n \"Hence, we can basically apply the result of the case-by-case analysis (REF ) to our problem at hand.\",\n \"More precisely, by rewriting the case-by-case analysis (REF ) as $ w_i = v, \\\\zeta _1 = z_1 , \\\\text{ or }\\\\zeta _j, \\\\, j \\\\in \\\\lbrace 3, 5, \\\\dots , 2k-1\\\\rbrace $ (with the restriction in (REF )) and using the fact that the terms $\\\\zeta _{2\\\\ell - 1}$ , $\\\\ell = 3, \\\\dots , k$ , behave better than $\\\\zeta _3 = z_3$ , the proof of Theorem REF (in particular, Proposition REF ) can be used to control all the terms $w_1 \\\\overline{w_2} w_3$ in (REF ) except for $\\\\zeta _{j_1}\\\\overline{\\\\zeta _{j_2}}\\\\zeta _{j_3}\\\\quad \\\\text{with }(j_1, j_2, j_3 )= (1, 1, 2 k-1)$ (up to permutations).\",\n \"Note that the contribution to (REF ) under the Duhamel integral is precisely given by $\\\\zeta _{2k+1}$ in (REF ).\",\n \"In particular, Proposition REF with $\\\\sigma = \\\\frac{1}{2}$ yields the regularity restriction $\\\\alpha _{k+1}\\\\cdot s > \\\\frac{1}{2}$ , i.e.\",\n \"the lower bound $s > \\\\frac{1}{2\\\\alpha _{k+1}}$ stated in Theorem REF .\",\n \"This allows us to construct a solution $v \\\\in X^\\\\frac{1}{2}([0, T])$ by the standard fixed point argument.\",\n \"See Section for details.\",\n \"We previously conjectured that the $(k+1)$ st order term $z_{2k+1}$ in (REF ) would be responsible for a regularity restriction at the $k$ th step of the original iterative procedure.\",\n \"Combining Proposition REF and Theorem REF , we confirmed this claim; the regularity restriction indeed comes only from the unbalanced $(k+1)$ st order term $\\\\zeta _{2k+1}$ in (REF ).\",\n \"We conclude this introduction with several remarks.\",\n \"Remark 1.9 Let $v_0 \\\\in H^\\\\frac{1}{2}(\\\\mathbb {R}^3)$ .\",\n \"Then, by slightly modifying the proof of Theorem REF based on the modified iterative approach (REF ), we can prove almost sure local well-posedness of the cubic NLS (REF ) with the random initial data of the form $v_0 + \\\\phi ^\\\\omega $ for the same range of $s$ .\",\n \"See Remark REF Remark 1.10 In this paper, we exploit randomness only at the linear level in estimating the stochastic terms.\",\n \"See Remarks REF and REF .\",\n \"It may be possible to lower the regularity thresholds in Theorems REF and REF by exploiting randomness at the multilinear level.\",\n \"We, however, do not pursue this direction in this paper since (i) our main purpose is to present the iterative procedures in their simplest forms and (ii) estimating the higher order stochastic terms by exploiting randomness at the multilinear level would require a significant amount of additional work, which would blur the main focus of this paper.\",\n \"Remark 1.11 The ill-posedness result in [12] show that the solution map $\\\\Phi : u_0 \\\\in H^s(\\\\mathbb {R}^3) \\\\longmapsto u \\\\in C([-T, T]; H^s(\\\\mathbb {R}^3))$ is not continuous for (REF ) when $s < \\\\frac{1}{2}$ .\",\n \"In proving Theorem REF , we studied the perturbed NLS (REF ) for $v = u-z_1$ .\",\n \"In particular, the proof shows that we can factorize the solution map for (REF ) asSimilarly, we can factorize the solution map for (REF ) as $u |_{t = 0} = v_0 + \\\\phi ^\\\\omega \\\\in H^s(\\\\mathbb {R}^3)\\\\longmapsto (v_0, z_1) \\\\stackrel{\\\\Psi }{\\\\longmapsto } v \\\\in X^\\\\frac{1}{2}([0, T]) \\\\subset C([0, T]; H^\\\\frac{1}{2}(\\\\mathbb {R}^3))$ such that the second map $\\\\Psi $ is continuous in $(v_0, z_1) \\\\in H^\\\\frac{1}{2}(\\\\mathbb {R}^3)\\\\times S^s_1([0, T])$ .\",\n \"$\\\\phi ^\\\\omega \\\\in H^s(\\\\mathbb {R}^3)\\\\longmapsto z_1 \\\\stackrel{\\\\Psi }{\\\\longmapsto } v \\\\in X^\\\\frac{1}{2}([0, T]) \\\\subset C([0, T]; H^\\\\frac{1}{2}(\\\\mathbb {R}^3)),$ where the first map can be viewed as a universal lift map and the second map $\\\\Psi $ is the solution map to (REF ), which is in fact continuous in $z_1 \\\\in S^s_1([0, T])$ .Here, $S^s_1([0, T]) \\\\subset C([0, T]; H^s(\\\\mathbb {R}^3))$ denotes the intersection of suitable space-time function spaces of differentiability at most $s$ .\",\n \"In the following, we use $S^{\\\\sigma }_j([0, T])$ in a similar manner.\",\n \"In the case of Theorem REF (with $\\\\sigma = \\\\frac{1}{2}$ ), we have the following factorization of the solution map for (REF ): $ \\\\phi ^\\\\omega \\\\in H^s(\\\\mathbb {R}^3)\\\\longmapsto (z_1, z_3 ) \\\\stackrel{\\\\Psi }{\\\\longmapsto } v \\\\in X^\\\\frac{1}{2}([0, T]) \\\\subset C([0, T]; H^\\\\frac{1}{2}(\\\\mathbb {R}^3)),$ where the second map $\\\\Psi $ is the solution map to (REF ), which is continuous from $(z_1, z_3) \\\\in S^s_1([0, T])\\\\times S^{2s-}_3([0, T])$ to $v \\\\in X^\\\\frac{1}{2}([0, T])$ .\",\n \"In the case of Theorem REF , we create $k$ stochastic objects in the first step, where $k = k(s) \\\\in \\\\mathbb {N}$ : $\\\\phi ^\\\\omega \\\\in H^s(\\\\mathbb {R}^3)\\\\longmapsto (\\\\zeta _1, \\\\zeta _3, \\\\dots , \\\\zeta _{2k-1} ) \\\\stackrel{\\\\Psi }{\\\\longmapsto } v \\\\in X^\\\\frac{1}{2}([0, T]) \\\\subset C([0, T]; H^\\\\frac{1}{2}(\\\\mathbb {R}^3)).$ Once again, the second map $\\\\Psi $ (which is the solution map to (REF )) is continuous from $(\\\\zeta _1, \\\\zeta _3, \\\\dots , \\\\zeta _{2k-1} ) \\\\in S^s_1([0, T]) \\\\times \\\\prod _{\\\\ell = 2}^ k S_\\\\ell ^{\\\\alpha _\\\\ell s-} ([0, T])$ to $v \\\\in X^\\\\frac{1}{2}([0, T])$ .\",\n \"We point out an analogy between these factorizations (REF ), (REF ), and (REF ) of the ill-posed solution maps into (i) the first step, involving stochastic analysis and (ii) the second step, where purely deterministic analysis is performed in constructing a continuous map $\\\\Psi $ and similar factorizations for studying rough differential equations via the rough path theory [19] and singular stochastic parabolic PDEs [21], [23].\",\n \"In the proof of Theorem REF , we could consider the following expansion of an infinite order: $u = \\\\sum _{\\\\ell = 1}^\\\\infty \\\\zeta _{2\\\\ell -1} + v.$ This would allow us to present a single argument that works for all $\\\\frac{1}{6} < s < \\\\frac{1}{3}$ .\",\n \"In this case, the residual part $v := u - \\\\sum _{\\\\ell = 1}^\\\\infty \\\\zeta _{2\\\\ell -1}$ satisfies ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\displaystyle i \\\\partial _tv + \\\\Delta v = \\\\mathcal {N}\\\\bigg (v +\\\\sum _{\\\\ell = 1}^{\\\\infty } \\\\zeta _{2\\\\ell -1}\\\\bigg )-\\\\sum _{\\\\ell = 2}^{\\\\infty }\\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = 2\\\\ell - 1\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 2\\\\ell -3\\\\rbrace \\\\end{array}}\\\\zeta _{j_1} \\\\overline{\\\\zeta _{j_2}}\\\\zeta _{j_3}\\\\\\\\v|_{t = 0} = 0.\\\\end{array}\\\\right.\",\n \"}$ In particular, we would need to worry about the convergence issue of infinite series and hence there seems to be no simplification in considering the infinite order expansion (REF ).\",\n \"Another strategy would be to treat $\\\\zeta _\\\\infty : = \\\\sum _{\\\\ell = 1}^\\\\infty \\\\zeta _{2\\\\ell -1} $ as one stochastic object and write $u =\\\\zeta _\\\\infty + v.$ It follows from (REF ) that $\\\\zeta _\\\\infty $ satisfies the following equation: ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\displaystyle i \\\\partial _t\\\\zeta _\\\\infty + \\\\Delta \\\\zeta _\\\\infty =\\\\sum _{\\\\ell = 2}^{\\\\infty }\\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = 2\\\\ell - 1\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 2\\\\ell -3\\\\rbrace \\\\end{array}}\\\\zeta _{j_1} \\\\overline{\\\\zeta _{j_2}}\\\\zeta _{j_3}\\\\\\\\\\\\zeta _\\\\infty |_{t = 0} = \\\\phi ^\\\\omega .\\\\end{array}\\\\right.\",\n \"}$ Noting that $\\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = 2\\\\ell - 1\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 2\\\\ell -3\\\\rbrace \\\\end{array}}\\\\zeta _{j_1} \\\\overline{\\\\zeta _{j_2}}\\\\zeta _{j_3}= {\\\\left\\\\lbrace \\\\begin{array}{ll}|z_1|^2 z_1& \\\\text{when }\\\\ell = 2,\\\\\\\\2|z_1|^2\\\\zeta _{2\\\\ell -3} + z_1^2 \\\\overline{\\\\zeta _{2\\\\ell -3}},& \\\\text{when }\\\\ell \\\\ge 3,\\\\end{array}\\\\right.\",\n \"}$ we can rewrite (REF ) as ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\displaystyle i \\\\partial _t\\\\zeta _\\\\infty + \\\\Delta \\\\zeta _\\\\infty =2|z_1|^2 \\\\zeta _\\\\infty + z_1^2 \\\\overline{\\\\zeta _\\\\infty }- 2 |z_1|^2 z_1\\\\\\\\\\\\zeta _\\\\infty |_{t = 0} = \\\\phi ^\\\\omega .\\\\end{array}\\\\right.\",\n \"}$ and thus $v : = u - \\\\zeta _\\\\infty $ satisfies ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\displaystyle i \\\\partial _tv + \\\\Delta v = \\\\mathcal {N}(v +\\\\zeta _\\\\infty )- 2|z_1|^2 \\\\zeta _\\\\infty - z_1^2 \\\\overline{\\\\zeta _\\\\infty }+ 2 |z_1|^2 z_1\\\\\\\\v|_{t = 0} = 0.\\\\end{array}\\\\right.\",\n \"}$ The equations (REF ) and (REF ) do not particularly appear to be in such a friendly format.\",\n \"Namely, studying (REF ) and (REF ) does not seem to provide a simplification over the case-by-case analysis (REF ) for the equation (REF ) for each fixed $k \\\\in \\\\mathbb {N}$ .\",\n \"In a recent work [40], the second author (with Tzvetkov and Y. Wang) proved invariance of the white noise for the (renormalized) cubic fourth order NLS on the circle.\",\n \"One novelty of this work is that we introduced an infinite sequence $\\\\lbrace z^\\\\text{res}_{2\\\\ell -1}\\\\rbrace _{\\\\ell \\\\in \\\\mathbb {N}}$ of stochastic $(2\\\\ell -1)$ -linear objects (depending only on the random initial data) and considered the following expansion of an infinite order: $ u = \\\\sum _{\\\\ell = 1}^\\\\infty z^\\\\text{res}_{2\\\\ell -1} + v.$ For this problem, it turned out that $z^\\\\text{res}_\\\\infty : = \\\\sum _{\\\\ell = 1}^\\\\infty z^\\\\text{res}_{2\\\\ell -1}$ satisfies a particularly simple equation.In fact, the series $z^\\\\text{res}_\\\\infty = \\\\sum _{\\\\ell = 1}^\\\\infty z^\\\\text{res}_{2\\\\ell -1}$ corresponds to the power series expansion of the resonant cubic fourth order NLS.\",\n \"We point out that $z^\\\\text{res}_\\\\infty $ does not belong to the span of Wiener homogeneous chaoses of any finite order.\",\n \"Then by treating $z^\\\\text{res}_\\\\infty $ as one stochastic object, we wrote a solution $u$ as $u = z^\\\\text{res}_\\\\infty +v$ , which led to the following factorization: $\\\\phi ^\\\\omega \\\\in H^s(\\\\longmapsto z^\\\\text{res}_\\\\infty = \\\\sum _{\\\\ell = 1}^\\\\infty z^\\\\text{res}_{2\\\\ell -1}\\\\longmapsto v \\\\in C(\\\\mathbb {R}; L^2()$ for $s < -\\\\frac{1}{2}$ , where $\\\\phi ^\\\\omega $ denotes the Gaussian white noise on the circle.\",\n \"Remark 1.12 In [44], the third author (with Y. Wang) recently studied probabilistic local well-posedness of NLS on $\\\\mathbb {R}^d$ within the framework of the $L^p$ -based Sobolev spaces, using the dispersive estimate.\",\n \"In the context of the cubic NLS (REF ) on $\\\\mathbb {R}^3$ , their result yields almost sure local existence of a unique solution $u$ for the randomized initial data $\\\\phi ^\\\\omega \\\\in H^s(\\\\mathbb {R}^3)$ , provided $s\\\\ge 0$ .\",\n \"In particular, the argument in [44] allows us to consider random initial data of lower regularities than Theorems REF , REF , and REF .\",\n \"Note that, in [44], it was shown that the solution $u$ only belongs to $C([0, T]; W^{s, 4}(\\\\mathbb {R}^3))$ , almost surely.\",\n \"We point out that a slight adaptation of the work [39] by the second and third authors (with Y. Wang) shows that the solution $u$ indeed lies in $C([0, T]; H^s(\\\\mathbb {R}^3))$ .\",\n \"As compared to [44], the argument presented in this paper provides extra regularity information, namely the decomposition (REF ) of the solution $u$ with the terms of increasing regularities: $ \\\\zeta _{2\\\\ell -1} \\\\in X^{\\\\alpha _\\\\ell \\\\cdot s-}_\\\\textup {loc} $ and $v \\\\in X^\\\\frac{1}{2}([0, T])$ .\",\n \"In particular, the residual term $v$ lies in the (sub)critical regularity,By slightly tweaking the argument, we can easily place $v$ in $X^{\\\\frac{1}{2}+}([0, T])$ .\",\n \"leaving us a possibility of adapting deterministic techniques to study its further properties.\",\n \"This paper is organized as follows.\",\n \"In Section , we recall probabilistic and deterministic lemmas along with the definitions of the basic function spaces.\",\n \"In Section , we study the regularity properties of the second order term $z_3$ in (REF ).\",\n \"In Section , we further investigate the regularity properties of the higher order terms $z_5$ and $z_7$ and the unbalanced higher order terms $\\\\zeta _{2k-1}$ .\",\n \"Note that the analysis on $z_5$ and $z_7$ contains partNote that $z_7$ defined in (REF ) contains the contribution from $z_{j_1}\\\\overline{z_{j_2}}z_{j_3}$ with $(j_1, j_2, j_3) = (1, 3, 3)$ up to permutations.\",\n \"of the case-by-case analysis (REF ) needed for proving Theorem REF .\",\n \"In Section , we then carry out the rest of the case-by-case analysis (REF ) and prove Theorem REF .\",\n \"In Section , we briefly describe the proof of Theorem REF by indicating how the analysis in the previous sections can lead to the proof.\",\n \"In Appendix , we prove the deterministic non-smoothing of the Duhamel integral operator discussed in Remark REF .\",\n \"Notations: We use $a+$ (and $a-$ ) to denote $a + \\\\varepsilon $ (and $a - \\\\varepsilon $ , respectively) for arbitrarily small $\\\\varepsilon \\\\ll 1$ , where an implicit constant is allowed to depend on $\\\\varepsilon > 0$ (and it usually diverges as $\\\\varepsilon \\\\rightarrow 0$ ).\",\n \"Given a Banach space $B$ of temporal functions, we use the following short-hand notation: $B_T := B([0, T])$ .\",\n \"For example, $L^q_T L^r_x = L^q_t([0, T]; L^r_x(\\\\mathbb {R}^3))$ .\",\n \"Let $\\\\eta : \\\\mathbb {R}\\\\rightarrow [0, 1]$ be an even, smooth cutoff function supported on $[-\\\\frac{8}{5}, \\\\frac{8}{5}]$ such that $\\\\eta \\\\equiv 1$ on $[-\\\\frac{5}{4}, \\\\frac{5}{4}]$ .\",\n \"Given a dyadic number $N \\\\ge 1$ , we set $\\\\eta _1(\\\\xi ) = \\\\eta (|\\\\xi |)$ and $\\\\eta _N(\\\\xi ) = \\\\eta \\\\bigg (\\\\frac{|\\\\xi |}{N}\\\\bigg ) - \\\\eta \\\\bigg (\\\\frac{2|\\\\xi |}{N}\\\\bigg )$ for $N \\\\ge 2$ .\",\n \"Then, we define the Littlewood-Paley projection operator $\\\\mathbf {P}_N$ as the Fourier multiplier operator with symbol $\\\\eta _N$ .\",\n \"In the following, we use the convention that capital letters denote dyadic numbers.\",\n \"For example, $N = 2^n$ for some $n \\\\in \\\\mathbb {N}_0 : = \\\\mathbb {N}\\\\cup \\\\lbrace 0\\\\rbrace $ .\",\n \"Given dyadic numbers $N_1, \\\\dots , N_4 \\\\in 2^{\\\\mathbb {N}_0}$ , we set $N_{\\\\max } := \\\\max _{j = 1, \\\\dots , 4} N_j$ .\",\n \"We also use the following shorthand notation: $f_N = \\\\mathbf {P}_N f$ .\",\n \"For example, we have $z_{1, N_j} = \\\\mathbf {P}_{N_j} z_1$ .\"\n ],\n [\n \"Probabilistic Strichartz estimates\",\n \"First, we recall the usual Strichartz estimates on $\\\\mathbb {R}^3$ for readers' convenience.\",\n \"We say that a pair $(q, r)$ is Schrödinger admissible if it satisfies $\\\\frac{2}{q} + \\\\frac{3}{r} = \\\\frac{3}{2}$ with $2\\\\le q, r \\\\le \\\\infty $ .\",\n \"Then, the following Strichartz estimates are known to hold [46], [49], [20], [28]: $\\\\Vert S(t) \\\\phi \\\\Vert _{L^q_t L^r_x (\\\\mathbb {R}\\\\times \\\\mathbb {R}^3)} \\\\lesssim \\\\Vert \\\\phi \\\\Vert _{L^2_x(\\\\mathbb {R}^3)}.$ It follows from (REF ) and Sobolev's inequality that $\\\\Vert S(t) \\\\phi \\\\Vert _{L^p_{t, x} (\\\\mathbb {R}\\\\times \\\\mathbb {R}^3)}\\\\lesssim \\\\big \\\\Vert |\\\\nabla |^{\\\\frac{3}{2} - \\\\frac{5}{p}}\\\\phi \\\\big \\\\Vert _{L^2_x(\\\\mathbb {R}^3)}$ for $p \\\\ge \\\\frac{10}{3}$ .\",\n \"We will use the following admissible pairs in this paper: $(\\\\infty , 2), \\\\ \\\\big (5, \\\\tfrac{30}{11}\\\\big ), \\\\ \\\\big (\\\\tfrac{10}{3}\\\\tfrac{10}{3}\\\\big ),\\\\ (2+, 6-).$ In particular, by Sobolev's inequality, we have $W^{\\\\frac{1}{2}, \\\\frac{30}{11}}(\\\\mathbb {R}^3) \\\\hookrightarrow L^5(\\\\mathbb {R}^3).$ One of the important key ingredients for probabilistic well-posedness is the probabilistic Strichartz estimates.\",\n \"Such probabilistic estimates were first exploited by McKean [34] and Bourgain [5].\",\n \"In the following, we state the probabilistic Strichartz estimates under the Wiener randomization (REF ).\",\n \"See [1] for the proofs.\",\n \"Lemma 2.1 Given $\\\\phi $ on $\\\\mathbb {R}^3$ , let $\\\\phi ^\\\\omega $ be its Wiener randomization defined in (REF ).\",\n \"Then, given finite $q \\\\ge 2$ and $2 \\\\le r \\\\le \\\\infty $ , there exist $C, c>0$ such that $P\\\\Big ( \\\\Vert S(t) \\\\phi ^\\\\omega \\\\Vert _{L^q_t L^r_x([0, T]\\\\times \\\\mathbb {R}^3)}> \\\\lambda \\\\Big )\\\\le C\\\\exp \\\\bigg (-c \\\\frac{\\\\lambda ^2}{ T^\\\\frac{2}{q}\\\\Vert \\\\phi \\\\Vert _{H^s}^{2}}\\\\bigg )$ for all $ T > 0$ and $\\\\lambda >0$ with (i) $s = 0$ if $r < \\\\infty $ and (ii) $s > 0$ if $r = \\\\infty $ .\",\n \"A similar estimate holds when $q = \\\\infty $ (with $s > 0$ ) but we will not need it in this paper.\",\n \"See [38].\",\n \"We also need the following lemma on the control of the size of $H^s$ -norm of $\\\\phi ^\\\\omega $ .\",\n \"Lemma 2.2 Given $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ , let $\\\\phi ^\\\\omega $ be its Wiener randomization defined in (REF ).\",\n \"Then, we have $P\\\\Big ( \\\\Vert \\\\phi ^\\\\omega \\\\Vert _{ H^s( \\\\mathbb {R}^3)} > \\\\lambda \\\\Big )\\\\le C\\\\exp \\\\bigg (-c \\\\frac{\\\\lambda ^2}{ \\\\Vert \\\\phi \\\\Vert _{H^s}^{2}}\\\\bigg )$ for all $\\\\lambda >0$ .\"\n ],\n [\n \"Function spaces and their properties\",\n \"In this subsection, we go over the basic definitions and properties of the functions spaces used for the Fourier restriction norm method (i.e.\",\n \"analysis involving the $X^{s, b}$ -spaces introduced in [4]) adapted to the space of functions of bounded $p$ -variation and its pre-dual, introduced and developed by Tataru, Koch, and their collaborators [31], [22], [24].\",\n \"We refer readers to [22], [24] for the proofs of the basic properties.\",\n \"See also [2].\",\n \"Let $\\\\mathcal {Z}$ be the set of finite partitions $-\\\\infty 2$ .\",\n \"Given an interval $I \\\\subset \\\\mathbb {R}$ , we define the local-in-time versions $X^s(I)$ and $Y^s(I)$ of these spaces as restriction norms.\",\n \"For example, we define the $X^s(I)$ -norm by $ \\\\Vert u \\\\Vert _{X^s(I)} = \\\\inf \\\\big \\\\lbrace \\\\Vert v\\\\Vert _{X^s(\\\\mathbb {R})}: \\\\, v|_I = u\\\\big \\\\rbrace .$ We also define the norm for the nonhomogeneous term on an interval $I = [t_0, t_1)$ : $\\\\Vert F\\\\Vert _{N^s(I)} = \\\\bigg \\\\Vert \\\\int _{t_0}^t S(t - t^{\\\\prime }) F(t^{\\\\prime }) dt^{\\\\prime }\\\\bigg \\\\Vert _{X^s(I)}.$ We conclude this section by presenting some basic estimates involving these function spaces.\",\n \"See [22], [24], [2] for the proofs.\",\n \"Lemma 2.5 Let $s \\\\in \\\\mathbb {R}$ and $T\\\\in ( 0,\\\\infty ]$ .\",\n \"Then, the following linear estimates hold: $\\\\Vert S(t) \\\\phi \\\\Vert _{X^s([0, T])}& \\\\le \\\\Vert \\\\phi \\\\Vert _{H^s}, \\\\\\\\\\\\Vert F \\\\Vert _{N^{s}([0, T])}& \\\\le \\\\sup _{\\\\begin{array}{c}w \\\\in Y^{-s}([0, T])\\\\\\\\ \\\\Vert w \\\\Vert _{Y^{-s}([0, T])}=1\\\\end{array}} \\\\left| \\\\int _{0}^{T} \\\\langle F(t), w(t) \\\\rangle _{L_{x}^{2}} dt \\\\right|$ for any $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ and $F \\\\in L^1([0,T];H^{s}(\\\\mathbb {R}^3))$ .\",\n \"The transference principle [22] and the interpolation lemma [22] applied on the Strichartz estimates (REF ) and (REF ) imply the following estimates.\",\n \"Lemma 2.6 Given any admissible pair $(q,r)$ with $q>2$ and $p \\\\ge \\\\frac{10}{3}$ , we have $\\\\Vert u \\\\Vert _{L_t^q L_x^r} & \\\\lesssim \\\\Vert u \\\\Vert _{Y^0}, \\\\\\\\\\\\Vert u \\\\Vert _{L^p_{t,x}}& \\\\lesssim \\\\big \\\\Vert |\\\\nabla |^{\\\\frac{3}{2} - \\\\frac{5}{p}} u\\\\big \\\\Vert _{Y^0}.$ Similarly, the bilinear refinement of the Strichartz estimate [6], [41], [14] implies the following bilinear estimate.\",\n \"Lemma 2.7 Let $N_1, N_2 \\\\in 2^{\\\\mathbb {N}_0}$ with $N_1 \\\\le N_2$ .\",\n \"Then, we have $\\\\Vert \\\\mathbf {P}_{N_1}u_1 \\\\mathbf {P}_{N_2}u_2\\\\Vert _{L^2_{t} ([0, T]; L^2_x)}\\\\lesssim T^{0+}N_1^{1-} N_2^{-\\\\frac{1}{2}+}\\\\Vert \\\\mathbf {P}_{N_1} u_1 \\\\Vert _{Y^0([0, T])} \\\\Vert \\\\mathbf {P}_{N_2} u_2 \\\\Vert _{Y^0([0, T])}$ for any $T > 0$ and $u_1, u_2 \\\\in Y^0([0, T])$ .\",\n \"From the bilinear refinement of the Strichartz estimate [6], [14] and the transference principle, we have $\\\\Vert \\\\mathbf {P}_{N_1}u_1 \\\\mathbf {P}_{N_2}u_2\\\\Vert _{L^2_{t,x}}\\\\lesssim N_1^{1-} N_2^{-\\\\frac{1}{2}+} \\\\Vert \\\\mathbf {P}_{N_1} u_1 \\\\Vert _{Y^0} \\\\Vert \\\\mathbf {P}_{N_2} u_2 \\\\Vert _{Y^0}$ for all $u_1, u_2 \\\\in Y^0$ .\",\n \"See [2] for the proof of (REF ).\",\n \"On the other hand, it follows from Hölder's and Sobolev's inequalities that $\\\\Vert \\\\mathbf {P}_{N_1}u_1 \\\\mathbf {P}_{N_2}u_2\\\\Vert _{L^2_TL^2_x}& \\\\lesssim T^\\\\frac{1}{2} \\\\Vert \\\\mathbf {P}_{N_1} u_1 \\\\Vert _{L^\\\\infty _TL^4_x} \\\\Vert \\\\mathbf {P}_{N_2} u_2 \\\\Vert _{L^\\\\infty _TL^4_x}\\\\\\\\& \\\\lesssim T^\\\\frac{1}{2} N_1^\\\\frac{3}{4}N_2^\\\\frac{3}{4} \\\\Vert \\\\mathbf {P}_{N_1} u_1 \\\\Vert _{Y^0_T} \\\\Vert \\\\mathbf {P}_{N_2} u_2 \\\\Vert _{Y^0_T} .$ Then, the estimate (REF ) follows from interpolating (REF ) and (REF ).\"\n ],\n [\n \"On the second order term $z_3$\",\n \"In this and the next sections, we study the regularity properties of the various stochastic terms that appear in the iterative procedures.\",\n \"Given $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ , let $\\\\phi ^\\\\omega $ be the Wiener randomization of $\\\\phi $ defined in (REF ) and set $z_1 = S(t) \\\\phi ^\\\\omega .$ In this section, we study the regularity properties of the second order term: $z_3 = -i \\\\int _0^t S(t - t^{\\\\prime }) |z_1|^2 z_1(t^{\\\\prime }) dt^{\\\\prime },$ We first present the proof of Proposition REF .\",\n \"We follow closely the argument in [2].\",\n \"(i) By Lemma REF , the estimate (REF ) follows once we prove $\\\\bigg | \\\\int _0^T \\\\int _{\\\\mathbb {R}^3}\\\\langle \\\\nabla \\\\rangle ^\\\\sigma ( z_1 \\\\overline{z_1} z_1 )\\\\overline{w} dx dt\\\\bigg |\\\\le T^\\\\theta C(\\\\omega , \\\\Vert \\\\phi \\\\Vert _{H^s})$ for some almost surely finite constant $ C(\\\\omega , \\\\Vert \\\\phi \\\\Vert _{H^s} ) > 0$ and $\\\\theta > 0$ , where $\\\\Vert w\\\\Vert _{Y^{0}_T} \\\\le 1$ .\",\n \"In the following, we drop the complex conjugate when it does not play any role.\",\n \"Define $A^s_3(T)$ by $\\\\Vert z_1\\\\Vert _{A^s_3(T)}: = \\\\max \\\\Big (\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_{1} \\\\Vert _{L^{\\\\frac{30}{7}+}_{T}L^\\\\frac{30}{7}_x},\\\\Vert \\\\langle \\\\nabla \\\\rangle ^sz_{1} \\\\Vert _{L^{4}_{T, x}},\\\\Vert z_1(0)\\\\Vert _{H^s} \\\\Big ).$ Then, by applying the dyadic decomposition, it suffices to prove $\\\\bigg | \\\\int _0^T \\\\int _{\\\\mathbb {R}^3}\\\\mathbf {P}_{N_1} z_1 \\\\cdot \\\\mathbf {P}_{N_2} z_1 \\\\cdot N_3^\\\\sigma \\\\mathbf {P}_{N_3} z_1 \\\\cdot \\\\mathbf {P}_{N_4} w \\\\, dx dt\\\\bigg |\\\\le T^\\\\theta N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{A^s_3(T)})$ for all $N_1, \\\\dots , N_4 \\\\in 2^{\\\\mathbb {N}_0}$ with $N_3 \\\\ge N_2 \\\\ge N_1$ .\",\n \"Once we prove (REF ), the desired estimate (REF ) follows from summing (REF ) over dyadic blocks and applying Lemmas REF and REF .\",\n \"Recall our shorthand notation: $z_{1, N_j} = \\\\mathbf {P}_{N_j} z_1$ and $w_{N_4} = \\\\mathbf {P}_{N_4} w$ .\",\n \"In the following, we assume $\\\\sigma < 2s\\\\qquad \\\\text{and}\\\\qquad 0 \\\\le s < 1.$ $\\\\bullet $ Case (1): $N_2 \\\\sim N_3$ .\",\n \"By Hölder's inequality and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {O2})& \\\\lesssim \\\\Vert z_{1, N_1} \\\\Vert _{L^\\\\frac{30}{7}_{T, x}}\\\\bigg (\\\\prod _{j = 2}^3 \\\\Vert \\\\langle \\\\nabla \\\\rangle ^\\\\frac{\\\\sigma }{2} z_{1, N_j} \\\\Vert _{L^{\\\\frac{30}{7}}_{T, x}}\\\\bigg )\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{A^s_3(T)}) \\\\Vert w_{N_4}\\\\Vert _{Y^0_T},$ provided that (REF ) holds.\",\n \"$\\\\bullet $ Case (2): $N_3 \\\\sim N_4 \\\\gg N_1, N_2$ .\",\n \"$\\\\circ $ Subcase (2.a): $N_1, N_2 \\\\ll N_3^\\\\frac{1}{2}$ .\",\n \"By Cauchy-Schwarz' inequality and Lemma REF followed by Lemma REF ,In the remaining part of this paper, we repeatedly apply this argument when there is a frequency separation.\",\n \"We shall simply refer to it as the “bilinear Strichartz estimate” argument.\",\n \"we have $\\\\text{LHS of } (\\\\ref {O2})& \\\\le N_3^\\\\sigma \\\\Vert z_{1, N_1} z_{1, N_3} \\\\Vert _{L^2_{T, x}} \\\\Vert z_{1, N_2} w_{N_4}\\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\lesssim T^{0+} N_1^{1-s-} N_2^{1-s-} N_3^{\\\\sigma - s - 1 +}\\\\bigg (\\\\prod _{j = 1}^3\\\\Vert \\\\mathbf {P}_{N_j} \\\\phi ^\\\\omega \\\\Vert _{H^{s}}\\\\bigg )\\\\Vert w_{N_4}\\\\Vert _{Y^0_T}\\\\\\\\& \\\\lesssim T^{0+} N_3^{\\\\sigma - 2s +}\\\\prod _{j = 1}^3\\\\Vert \\\\mathbf {P}_{N_j} \\\\phi ^\\\\omega \\\\Vert _{H^{s}}\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{A^s_3(T)}),$ provided that (REF ) holds.\",\n \"$\\\\circ $ Subcase (2.b): $N_1, N_2\\\\gtrsim N_3^\\\\frac{1}{2} $ .\",\n \"By Hölder's inequality and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {O2})& \\\\le N_3^\\\\sigma \\\\bigg (\\\\prod _{j = 1}^3 \\\\Vert z_{1, N_j} \\\\Vert _{L^\\\\frac{30}{7}_{T, x}}\\\\bigg )\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{-s} N_2^{-s}N_3^{\\\\sigma -s}C(\\\\Vert z_1\\\\Vert _{A^s_3(T)})\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{A^s_3(T)}),$ provided that (REF ) holds.\",\n \"$\\\\circ $ Subcase (2.c): $N_2\\\\gtrsim N_3^\\\\frac{1}{2} \\\\gg N_1$ .\",\n \"By the bilinear Strichartz estimate, we have $\\\\text{LHS of } (\\\\ref {O2})& \\\\le N_3^\\\\sigma \\\\Vert z_{1, N_1} w_{N_4} \\\\Vert _{L^2_{T, x}}\\\\prod _{j = 2}^3 \\\\Vert z_{1, N_j}\\\\Vert _{L^4_{T, x}}\\\\\\\\& \\\\lesssim T^{0+} N_1^{1-s-} N_2^{-s}N_3^{\\\\sigma -s-\\\\frac{1}{2}+}\\\\Vert \\\\mathbf {P}_{N_1}\\\\phi ^\\\\omega \\\\Vert _{H^s}\\\\bigg (\\\\prod _{j = 2}^3 \\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_{1, N_j} \\\\Vert _{L^4_{T, x}}\\\\bigg )\\\\Vert w_{N_4}\\\\Vert _{Y^0_T}\\\\\\\\& \\\\le T^{0+}N_3^{\\\\sigma -2s+}C(\\\\Vert z_1\\\\Vert _{A^s_3(T)})\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{A^s_3(T)}),$ provided that (REF ) holds.\",\n \"Therefore, putting all the cases together, we obtain (REF ).\",\n \"(ii) Given $N \\\\gg 1$ and small $\\\\ell > 0$ , consider the following deterministic initial condition $\\\\phi $ whose Fourier transform is given by $\\\\widehat{\\\\phi }(\\\\xi ) = \\\\mathbf {1}_{ N e_1 + \\\\ell Q}(\\\\xi ) + \\\\mathbf {1}_{(N+100\\\\ell ) e_1 + \\\\ell Q}(\\\\xi )+ \\\\mathbf {1}_{ N e_1 + 100\\\\ell e_2 + \\\\ell Q}(\\\\xi ),$ where $Q = (-\\\\frac{1}{2}, \\\\frac{1}{2}]^3$ , $e_1 = (1, 0, 0)$ , and $e_2 = (0, 1, 0)$ .\",\n \"By taking $\\\\ell > 0$ sufficiently small, we have $\\\\operatornamewithlimits{supp}\\\\widehat{\\\\phi }\\\\subset N e_1 + Q$ and thus we can neglect the effect of the randomization in (REF ) since all the three terms on the right-hand side will be multiplied by a common random number $g_{N e_1}$ .\",\n \"Without loss of generality, we assume that $g_{N e_1} = 1$ in the following.\",\n \"We estimate from below the contribution to $\\\\mathbf {1}_{Q_{N, \\\\ell }} (\\\\xi ) \\\\widehat{z}_3 (t, \\\\xi ) $ , where $Q_{N, \\\\ell } = Ne_1 + 100 \\\\ell (e_1 + e_2) + \\\\ell Q.$ From (REF ), we have $\\\\mathbf {1}_{Q_{N, \\\\ell }} & (\\\\xi ) \\\\widehat{z}_3 (t, \\\\xi ) \\\\\\\\& = -i \\\\mathbf {1}_{Q_{N, \\\\ell }} (\\\\xi ) e^{-it |\\\\xi |^2} \\\\int _0^t \\\\operatornamewithlimits{\\\\int }_{\\\\xi = \\\\xi _1 - \\\\xi _2 + \\\\xi _3}e^{i t^{\\\\prime } \\\\Phi (\\\\bar{\\\\xi })} \\\\widehat{\\\\phi ^\\\\omega } (\\\\xi _1) \\\\overline{\\\\widehat{\\\\phi ^\\\\omega } (\\\\xi _2)} \\\\widehat{\\\\phi ^\\\\omega } (\\\\xi _3) d\\\\xi _1 d\\\\xi _2 dt^{\\\\prime },$ where the phase function $\\\\Phi (\\\\bar{\\\\xi })$ is given by $\\\\Phi (\\\\bar{\\\\xi }) = \\\\Phi (\\\\xi , \\\\xi _1, \\\\xi _2, \\\\xi _3)= |\\\\xi |^2 - |\\\\xi _1|^2 + |\\\\xi _2|^2 - |\\\\xi _3|^2= 2\\\\langle \\\\xi - \\\\xi _1, \\\\xi - \\\\xi _3 \\\\rangle _{\\\\mathbb {R}^3}.$ Then, it follows that the only non-trivial contribution to (REF ) appears if $\\\\xi _1 \\\\in (N+100\\\\ell ) e_1 + \\\\ell Q,\\\\qquad \\\\xi _2 \\\\in N e_1 + \\\\ell Q,\\\\qquad \\\\xi _3 \\\\in N e_1 + 100\\\\ell e_2 + \\\\ell Q$ (up to the permutation $\\\\xi _1 \\\\leftrightarrow \\\\xi _3$ ).\",\n \"In this case, we have $|\\\\Phi (\\\\bar{\\\\xi })| \\\\ll 1 $ and thus $\\\\operatornamewithlimits{Re}e^{i t^{\\\\prime } \\\\Phi (\\\\bar{\\\\xi })} \\\\ge \\\\frac{1}{2}$ for all $t^{\\\\prime } \\\\in [0, 1]$ .\",\n \"Now, recall the following lemma on the convolution.\",\n \"Lemma 3.1 There exists $c>0$ such that $\\\\mathbf {1}_{a + \\\\ell Q}* \\\\mathbf {1}_{b + \\\\ell Q} (\\\\xi )\\\\ge c \\\\ell ^3 \\\\mathbf {1}_{a+b+\\\\ell Q}(\\\\xi )$ for all $a, b, \\\\xi \\\\in \\\\mathbb {R}^3$ .\",\n \"By applying Lemma REF to (REF ) with (REF ) and (REF ), we obtain $|\\\\mathbf {1}_{Q_{N, \\\\ell }} (\\\\xi ) \\\\widehat{z}_3 (t, \\\\xi ) |\\\\gtrsim t \\\\ell ^6 \\\\mathbf {1}_{Q_{N, \\\\ell }} (\\\\xi ).$ Therefore, for any $\\\\sigma > 0$ , we have $\\\\Vert z_3\\\\Vert _{X^\\\\sigma ([0, 1])} \\\\gtrsim \\\\Vert z_3 \\\\Vert _{L^\\\\infty _t([0, 1]; H^\\\\sigma )}\\\\gtrsim \\\\ell ^\\\\frac{15}{2} N^\\\\sigma \\\\longrightarrow \\\\infty $ as $N \\\\rightarrow \\\\infty $ , while $\\\\Vert \\\\phi \\\\Vert _{L^2} \\\\sim \\\\ell ^\\\\frac{3}{2}$ remains bounded.\",\n \"This in particular implies that when $s = 0$ , the estimate (REF ) can not hold for any $\\\\sigma > 0$ .\",\n \"This proves Part (ii).\",\n \"Remark 3.2 It follows from the proof of Proposition REF (i) that $\\\\big \\\\Vert I( u_1, u_2, u_3)\\\\big \\\\Vert _{X^\\\\sigma ([0, 1])}\\\\lesssim \\\\prod _{j = 1}^3 \\\\Vert u_j \\\\Vert _{A^s_3(T)},$ where $I( u_1, u_2, u_3)$ is as in (REF ).\",\n \"In particular, the left-hand side of (REF ) is finite for $u_1, u_2, u_3 \\\\in \\\\mathcal {R}^s$ .\",\n \"The only probabilistic component in the proof of Proposition REF (i) appears in applying Lemmas REF and REF to control the $A^s_3(T)$ -norm of $z_1$ in terms of the $H^s$ -norm of $\\\\phi $ .\",\n \"In this sense, we exploit the randomization only at the linear level.\",\n \"On the one hand, Proposition REF shows that $z_3$ controls almost $2s$ derivatives.\",\n \"On the other hand, we need to measure $z_3$ in the $X^{2s-}$ -norm, which controls only the admissible space-time Lebesgue norms (with $2s-$ derivatives) via Lemma REF .\",\n \"The following lemma breaks this rigidity by giving up a control on derivatives.\",\n \"In particular, it allows us to control a wider range of space-time Lebesgue norms of $z_3$ .\",\n \"The main idea is to use the dispersive estimate for the linear Schrödinger operator: $\\\\Vert S(t) f \\\\Vert _{L^r_x} \\\\lesssim |t|^{-\\\\frac{3}{2}(1 - \\\\frac{2}{r})} \\\\Vert f\\\\Vert _{L^{r^{\\\\prime }}_x}.$ This allows us to reduce the analysis to a product of the random linear solution $z_1 = S(t) \\\\phi ^\\\\omega $ and apply Lemma REF .\",\n \"Lemma 3.3 Let $s \\\\ge 0$ .\",\n \"Given $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ , let $\\\\phi ^\\\\omega $ be its Wiener randomization defined in (REF ).\",\n \"Then, for any finite $q, r \\\\ge 1$ , we have $\\\\Vert \\\\mathbf {P}_N z_3\\\\Vert _{L^q_T L^r_x}\\\\lesssim {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\rule [-6mm]{0pt}{15pt} T^{\\\\frac{3}{r} - \\\\frac{1}{2}}\\\\Vert z_1\\\\Vert _{L^{3q}_T L^{3r^{\\\\prime }}_x}^3 & \\\\text{when } 1 \\\\le r < 6, \\\\\\\\T^{0+} N^{\\\\frac{1}{2} - \\\\frac{3}{r}+} \\\\Vert z_1\\\\Vert _{L^{3q}_T L^{\\\\frac{18}{5}+}_x}^3& \\\\text{when } r \\\\ge 6, \\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ for any $T > 0$ and $N \\\\in 2^{\\\\mathbb {N}_0}$ .\",\n \"Note that the right-hand side of (REF ) is almost surely finite thanks to the probabilistic Strichartz estimate (Lemma REF ).\",\n \"We first consider the case $r < 6$ .\",\n \"From (REF ) and (REF ), we have $\\\\Vert \\\\mathbf {P}_N z_3\\\\Vert _{L^q_T L^r_x}& \\\\le \\\\bigg \\\\Vert \\\\int _0^t \\\\Vert \\\\mathbf {P}_N S(t - t^{\\\\prime }) |z_1|^2 z_1(t^{\\\\prime })\\\\Vert _{L^r_x} dt^{\\\\prime }\\\\bigg \\\\Vert _{L^q_T}\\\\\\\\& \\\\lesssim \\\\bigg \\\\Vert \\\\int _0^t \\\\frac{1}{|t - t^{\\\\prime }|^{\\\\frac{3}{2} - \\\\frac{3}{r}}} \\\\Vert z_1(t^{\\\\prime })\\\\Vert _{L^{3r^{\\\\prime }}_x}^3 dt^{\\\\prime }\\\\bigg \\\\Vert _{L^q_T}\\\\\\\\& \\\\lesssim T^{\\\\frac{3}{r} - \\\\frac{1}{2}}\\\\Vert z_1\\\\Vert _{L^{3q}_T L^{3r^{\\\\prime }}_x}^3.$ When $r \\\\ge 6$ , we proceed as in (REF ) but we apply Sobolev's inequality before applying (REF ): $\\\\Vert \\\\mathbf {P}_N z_3\\\\Vert _{L^q_T L^r_x}& \\\\le \\\\bigg \\\\Vert \\\\int _0^t \\\\Vert \\\\mathbf {P}_N S(t - t^{\\\\prime }) |z_1|^2 z_1(t^{\\\\prime })\\\\Vert _{L^r_x} dt^{\\\\prime }\\\\bigg \\\\Vert _{L^q_T}\\\\\\\\& \\\\lesssim N^{\\\\frac{1}{2} - \\\\frac{3}{r}+} \\\\bigg \\\\Vert \\\\int _0^t \\\\Vert \\\\mathbf {P}_N S(t - t^{\\\\prime }) |z_1|^2 z_1(t^{\\\\prime })\\\\Vert _{L^{6-}_x} dt^{\\\\prime }\\\\bigg \\\\Vert _{L^q_T}\\\\\\\\& \\\\lesssim N^{\\\\frac{1}{2} - \\\\frac{3}{r}+}\\\\bigg \\\\Vert \\\\int _0^t \\\\frac{1}{|t - t^{\\\\prime }|^{1-}} \\\\Vert z_1(t^{\\\\prime })\\\\Vert _{L^{\\\\frac{18}{5}+}_x}^3 dt^{\\\\prime }\\\\bigg \\\\Vert _{L^q_T}\\\\\\\\& \\\\lesssim T^{0+} N^{\\\\frac{1}{2} - \\\\frac{3}{r}+} \\\\Vert z_1\\\\Vert _{L^{3q}_T L^{\\\\frac{18}{5}+}_x}^3 .$ This completes the proof of Lemma REF .\"\n ],\n [\n \"On the higher order terms\",\n \"In this section, we study the regularity properties of the higher order terms.\"\n ],\n [\n \"On the third order term $z_5$\",\n \"In this subsection, we study the third order term: $z_5 = -i \\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = 5\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 3\\\\rbrace \\\\end{array}}\\\\int _0^t S(t - t^{\\\\prime }) z_{j_1} \\\\overline{z_{j_2}}z_{j_3} (t^{\\\\prime }) dt^{\\\\prime }.$ The following lemma shows that the third order term $z_5$ enjoys a gain of extra $\\\\frac{1}{2}s$ derivative as compared to the second order term $z_3$ (Proposition REF ).\",\n \"Lemma 4.1 Given $0 < s < \\\\frac{1}{2} $ , let $\\\\phi ^\\\\omega $ be the Wiener randomization of $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ defined in (REF ).\",\n \"Then, for any $\\\\sigma < \\\\frac{5}{2} s$ , we have $z_5 \\\\in X^\\\\sigma _\\\\textup {loc} ,$ almost surely.\",\n \"In particular, there exists an almost surely finite constant $ C(\\\\omega , \\\\Vert \\\\phi \\\\Vert _{H^s} ) > 0$ and $\\\\theta > 0$ such that $\\\\Vert z_5 \\\\Vert _{X^{\\\\sigma }([0, T])} \\\\le T^\\\\theta C(\\\\omega , \\\\Vert \\\\phi \\\\Vert _{H^s})$ for any $T > 0$ .\",\n \"First, note that the only possible combination for $(j_1, j_2, j_3)$ in (REF ) is $(1, 1, 3)$ up to permutations.\",\n \"The complex conjugate does not play any role in the subsequent analysis and hence we drop the complex conjugate sign and simply study $z_5 \\\\sim \\\\int _0^t S(t - t^{\\\\prime }) z_{1} z_1 z_{3} (t^{\\\\prime }) dt^{\\\\prime }.$ By Lemma REF , it suffices to prove $\\\\bigg | \\\\int _0^T \\\\int _{\\\\mathbb {R}^3}\\\\langle \\\\nabla \\\\rangle ^\\\\sigma ( z_1 z_1 z_3 )w dx dt\\\\bigg |\\\\le T^\\\\theta C(\\\\omega , \\\\Vert \\\\phi \\\\Vert _{H^s})$ for some almost surely finite constant $ C(\\\\omega , \\\\Vert \\\\phi \\\\Vert _{H^s} ) > 0$ and $\\\\theta > 0$ , where $\\\\Vert w\\\\Vert _{Y^{0}_T} \\\\le 1$ .\",\n \"Define $A^s_5(T)$ by $ \\\\Vert z_1\\\\Vert _{A^s_5(T)}: = \\\\max \\\\Big (\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_{1} \\\\Vert _{L^{5+}_T L^{5}_x},\\\\Vert z_1(0)\\\\Vert _{H^s},\\\\Big ).\",\n \"$ Then, by applying the dyadic decomposition, it suffices to prove $\\\\bigg | \\\\int _0^T \\\\int _{\\\\mathbb {R}^3}N_{\\\\max }^\\\\sigma \\\\mathbf {P}_{N_1} z_1 \\\\cdot \\\\mathbf {P}_{N_2} z_1& \\\\cdot \\\\mathbf {P}_{N_3} z_3 \\\\cdot \\\\mathbf {P}_{N_4} w \\\\, dx dt\\\\bigg | \\\\\\\\& \\\\le T^\\\\theta N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T})$ for all $N_1, \\\\dots , N_4 \\\\in 2^{\\\\mathbb {N}_0}$ .\",\n \"Once we prove (REF ), the desired estimate (REF ) follows from summing over dyadic blocks and applying Lemmas REF and REF and Proposition REF .\",\n \"In the following, we fix $0 < s < \\\\tfrac{1}{2}.$ Without loss of generality, we assume that $N_1 \\\\ge N_2$ .\",\n \"$\\\\bullet $ Case (1): $N_1 \\\\sim N_2 \\\\sim N_{\\\\max }$ .\",\n \"$\\\\circ $ Subcase (1.a): $N_3 \\\\ll N_1^\\\\frac{1}{2}$ .\",\n \"By the bilinear Strichartz estimate and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {P5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} z_{3, N_3} \\\\Vert _{L^2_{T, x}}\\\\Vert z_{1, N_2} \\\\Vert _{L^{5}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\lesssim T^{0+}N_1^{\\\\sigma - s - \\\\frac{1}{2}+} N_2^{-s}N_3^{1 - 2s-}\\\\Vert \\\\mathbf {P}_{N_1}\\\\phi ^\\\\omega \\\\Vert _{H^s}\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_{1, N_2} \\\\Vert _{L^5_{T, x}}\\\\Vert z_{3, N_3}\\\\Vert _{Y^{2s-}_T}\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -3s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\\\sigma < 3s$ .\",\n \"$\\\\circ $ Subcase (1.b): $N_3\\\\gtrsim N_1^\\\\frac{1}{2} $ .\",\n \"By Hölder's inequality and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {P5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{1, N_2} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{3, N_3} \\\\Vert _{L^{\\\\frac{10}{3}}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -s} N_2^{-s}N_3^{-2s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -3s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\\\sigma < 3s$ .\",\n \"$\\\\bullet $ Case (2): $N_1 \\\\sim N_3 \\\\sim N_{\\\\max } \\\\gg N_2$ .\",\n \"$\\\\circ $ Subcase (2.a): $N_2 \\\\ll N_1^\\\\frac{1}{2}$ .\",\n \"By the bilinear Strichartz estimate and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {P5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} \\\\Vert _{L^{5}_{T, x}} \\\\Vert z_{1, N_2} z_{3, N_3} \\\\Vert _{L^2_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\lesssim T^{0+}N_1^{\\\\sigma - s } N_2^{1-s-}N_3^{- 2s-\\\\frac{1}{2} +}\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_{1, N_1} \\\\Vert _{L^5_{T, x}}\\\\Vert \\\\mathbf {P}_{N_2}\\\\phi ^\\\\omega \\\\Vert _{H^s}\\\\Vert z_{3, N_3}\\\\Vert _{Y^{2s-}_T}\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -\\\\frac{7}{2}s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\\\sigma < \\\\frac{7}{2}s$ .\",\n \"$\\\\circ $ Subcase (2.b): $N_2\\\\gtrsim N_1^\\\\frac{1}{2} $ .\",\n \"By Hölder's inequality and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {P5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{1, N_2} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{3, N_3} \\\\Vert _{L^{\\\\frac{10}{3}}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -s} N_2^{-s}N_3^{-2s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -\\\\frac{7}{2} s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\\\sigma < \\\\frac{7}{2}s$ .\",\n \"$\\\\bullet $ Case (3): $N_1 \\\\sim N_4 \\\\sim N_{\\\\max } \\\\gg N_2, N_3$ .\",\n \"$\\\\circ $ Subcase (3.a): $N_2, N_3 \\\\ll N_1^\\\\frac{1}{2} $ .\",\n \"By the bilinear Strichartz estimate, we have $\\\\text{LHS of } (\\\\ref {P5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} z_{1, N_2} \\\\Vert _{L^2_{T, x}} \\\\Vert z_{3, N_3} w_{N_4}\\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\lesssim T^{0+} N_1^{\\\\sigma -s-\\\\frac{1}{2} + }N_2^{1-s - } N_3^{1- 2s -}N_4^{-\\\\frac{1}{2}+}\\\\bigg (\\\\prod _{j = 1}^2\\\\Vert \\\\mathbf {P}_{N_j} \\\\phi ^\\\\omega \\\\Vert _{H^{s}}\\\\bigg )\\\\Vert z_{3, N_3}\\\\Vert _{Y^{2s-}_T}\\\\Vert w_{N_4}\\\\Vert _{Y^0_T}\\\\\\\\& \\\\le T^{0+} N_1^{\\\\sigma - \\\\frac{5}{2} s +}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\\\sigma < \\\\frac{5}{2}s$ .\",\n \"$\\\\circ $ Subcase (3.b): $N_2, N_3\\\\gtrsim N_1^\\\\frac{1}{2} $ .\",\n \"By Hölder's inequality and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {P5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{1, N_2} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{3, N_3} \\\\Vert _{L^{\\\\frac{10}{3}}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -s} N_2^{-s}N_3^{-2s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -\\\\frac{5}{2} s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\\\sigma < \\\\frac{5}{2}s$ .\",\n \"$\\\\circ $ Subcase (3.c): $N_2\\\\gtrsim N_1^\\\\frac{1}{2} \\\\gg N_3$ .\",\n \"By the bilinear Strichartz estimate and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {P5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} z_{3, N_3} \\\\Vert _{L^2_{T, x}} \\\\Vert z_{1, N_2} \\\\Vert _{L^{5}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma - s- \\\\frac{1}{2} + } N_2^{-s}N_3^{1- 2s-}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -\\\\frac{5}{2}s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\\\sigma < \\\\frac{5}{2}s$ .\",\n \"$\\\\circ $ Subcase (3.d): $N_3\\\\gtrsim N_1^\\\\frac{1}{2} \\\\gg N_2$ .\",\n \"By the bilinear Strichartz estimate and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {P5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{3, N_3} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{1, N_2} w_{N_4} \\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma - s } N_2^{1-s-}N_3^{- 2s+} N_4^{-\\\\frac{1}{2} +}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -\\\\frac{5}{2}s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\\\sigma < \\\\frac{5}{2}s$ .\",\n \"$\\\\bullet $ Case (4): $N_3 \\\\sim N_4 \\\\gg N_1 \\\\ge N_2$ .\",\n \"$\\\\circ $ Subcase (4.a): $N_1, N_2 \\\\ll N_3^\\\\frac{1}{2} $ .\",\n \"By the bilinear Strichartz estimate, we have $\\\\text{LHS of } (\\\\ref {P5})& \\\\lesssim N_3^\\\\sigma \\\\Vert z_{1, N_1} z_{3, N_3} \\\\Vert _{L^2_{T, x}} \\\\Vert z_{1, N_2} w_{N_4}\\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\lesssim T^{0+} N_1^{1-s- }N_2^{1-s - } N_3^{\\\\sigma - 2s -\\\\frac{1}{2} +}N_4^{-\\\\frac{1}{2}+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+} N_1^{\\\\sigma - 3 s +}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\\\sigma < 3s$ .\",\n \"$\\\\circ $ Subcase (4.b): $N_1, N_2\\\\gtrsim N_3^\\\\frac{1}{2} $ .\",\n \"By Hölder's inequality and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {P5})& \\\\lesssim N_3^\\\\sigma \\\\Vert z_{1, N_1} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{1, N_2} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{3, N_3} \\\\Vert _{L^{\\\\frac{10}{3}}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{ -s} N_2^{-s}N_3^{\\\\sigma -2s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -3 s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\\\sigma < 3s$ .\",\n \"$\\\\circ $ Subcase (4.c): $N_1\\\\gtrsim N_3^\\\\frac{1}{2} \\\\gg N_2$ .\",\n \"By the bilinear Strichartz estimate and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {P5})& \\\\lesssim N_3^\\\\sigma \\\\Vert z_{1, N_2} \\\\Vert _{L^{5}_{T, x}} \\\\Vert z_{1, N_2} z_{3, N_3} \\\\Vert _{L^2_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\lesssim T^{0+}N_1^{ - s} N_2^{1-s-}N_3^{\\\\sigma - 2s- \\\\frac{1}{2}+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma - 3s+}C(\\\\Vert z_1\\\\Vert _{A^s_5(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\\\sigma < 3s$ .\",\n \"Putting all the cases together, we conclude that (REF ) holds, provided that $\\\\sigma < \\\\frac{5}{2} s$ .\",\n \"This completes the proof of Lemma REF .\"\n ],\n [\n \"On the fourth order term $z_7$\",\n \"Next, we study the following fourth order term: $z_7 = -i \\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = 7\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 3, 5\\\\rbrace \\\\end{array}}\\\\int _0^t S(t - t^{\\\\prime }) z_{j_1} \\\\overline{z_{j_2}}z_{j_3} (t^{\\\\prime }) dt^{\\\\prime }.$ In this case, there are two possibilities for $(j_1, j_2, j_3)$ in (REF ): $(1, 3, 3)$ and $(1, 1, 5)$ up to permutations.\",\n \"We denote by $\\\\widetilde{z}_7$ the contribution to $z_7$ from $(j_1, j_2, j_3) = (1, 3, 3)$ (up to permutations).\",\n \"Note that the contribution to $z_7$ from $(j_1, j_2, j_3) = (1, 1, 5)$ (up to permutations) corresponds to $\\\\zeta _7$ defined in (REF ).\",\n \"Dropping the complex conjugate, we have $\\\\widetilde{z}_7 & \\\\sim \\\\int _0^t S(t - t^{\\\\prime }) z_{1} z_3 z_{3} (t^{\\\\prime }) dt^{\\\\prime }, \\\\\\\\\\\\zeta _7 & \\\\sim \\\\int _0^t S(t - t^{\\\\prime }) z_{1} z_1 z_{5} (t^{\\\\prime }) dt^{\\\\prime }.$ The following lemma shows that $\\\\widetilde{z}_7$ and $\\\\zeta _7$ enjoy a further gain of derivatives as compared to $z_1$ , $z_3$ , and $z_5$ .\",\n \"Lemma 4.2 Given $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ , let $\\\\phi ^\\\\omega $ be the Wiener randomization of $\\\\phi $ defined in (REF ).\",\n \"(i) Let $ 0 < s < \\\\frac{1}{2}$ .\",\n \"Then, given any $\\\\sigma < 3s$ , we have $\\\\widetilde{z}_7 \\\\in X^\\\\sigma _\\\\textup {loc} ,$ almost surely.\",\n \"(ii) Let $ 0 < s < \\\\frac{2}{5}$ .\",\n \"Then, given any $\\\\sigma < \\\\frac{11}{4}s$ , we have $\\\\zeta _7 \\\\in X^\\\\sigma _\\\\textup {loc} ,$ almost surely.\",\n \"(i) We first estimate $\\\\widetilde{z}_7$ in (REF ).\",\n \"Fix $0 < s < \\\\frac{1}{2}$ .\",\n \"We proceed as in the proofs of Proposition REF and Lemma REF .\",\n \"In view of Lemmas REF and REF and Proposition REF , it suffices to prove that there exists $\\\\theta > 0$ such that $\\\\bigg | \\\\int _0^T \\\\int _{\\\\mathbb {R}^3}N_{\\\\max }^\\\\sigma \\\\mathbf {P}_{N_1} z_1 \\\\cdot \\\\mathbf {P}_{N_2} z_3& \\\\cdot \\\\mathbf {P}_{N_3} z_3 \\\\cdot \\\\mathbf {P}_{N_4} w \\\\, dx dt\\\\bigg |\\\\\\\\& \\\\le T^\\\\theta N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T}).$ for all $N_1, \\\\dots , N_4 \\\\in 2^{\\\\mathbb {N}_0}$ and $\\\\Vert w\\\\Vert _{Y^{0}_T} \\\\le 1$ , where $A^s_7(T)$ is given by $ \\\\Vert z_1\\\\Vert _{A^s_7(T)}: = \\\\max \\\\Big (\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_{1} \\\\Vert _{L^{5}_{T, x}},\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_{1} \\\\Vert _{L^{10+}_T L^{10}_x},\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x},\\\\Vert z_1(0)\\\\Vert _{H^s}\\\\Big ).\",\n \"$ Without loss of generality, we assume that $N_2 \\\\ge N_3$ .\",\n \"$\\\\bullet $ Case (1): $N_1 \\\\sim N_2 \\\\sim N_{\\\\max }$ .\",\n \"$\\\\circ $ Subcase (1.a): $N_3 \\\\ll N_1^\\\\frac{1}{2}$ .\",\n \"By the bilinear Strichartz estimate and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {Q5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} \\\\Vert _{L^{5}_{T, x}} \\\\Vert z_{3, N_2} z_{3, N_3} \\\\Vert _{L^2_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\lesssim T^{0+}N_1^{\\\\sigma - s } N_2^{-2s - \\\\frac{1}{2} + }N_3^{1 - 2s-}\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_{1, N_1} \\\\Vert _{L^5_{T, x}}\\\\bigg (\\\\prod _{j = 2}^3 \\\\Vert z_{3, N_j}\\\\Vert _{Y^{2s-}_T}\\\\bigg )\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -4s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\\\sigma < 4s$ .\",\n \"$\\\\circ $ Subcase (1.b): $N_3\\\\gtrsim N_1^\\\\frac{1}{2} $ .\",\n \"By Hölder's inequality and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {Q5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} \\\\Vert _{L^{10}_{T, x}}\\\\Vert z_{3, N_2} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{3, N_3} \\\\Vert _{L^{\\\\frac{10}{3}}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -s} N_2^{-2s+}N_3^{-2s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -4s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\\\sigma < 4s$ .\",\n \"$\\\\bullet $ Case (2): $N_1 \\\\sim N_4 \\\\sim N_{\\\\max } \\\\gg N_2 \\\\ge N_3$ .\",\n \"$\\\\circ $ Subcase (2.a): $N_2, N_3 \\\\ll N_1^\\\\frac{1}{2} $ .\",\n \"By the bilinear Strichartz estimate, we have $\\\\text{LHS of } (\\\\ref {Q5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} z_{3, N_2} \\\\Vert _{L^2_{T, x}} \\\\Vert z_{3, N_3} w_{N_4}\\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\lesssim T^{0+} N_1^{\\\\sigma -s-\\\\frac{1}{2} + }N_2^{1-2s - } N_3^{1- 2s -}N_4^{-\\\\frac{1}{2}+}\\\\Vert \\\\mathbf {P}_{N_1} \\\\phi ^\\\\omega \\\\Vert _{H^{s}}\\\\bigg (\\\\prod _{j = 2}^3\\\\Vert z_{3, N_j}\\\\Vert _{Y^{2s-}_T}\\\\bigg )\\\\\\\\& \\\\le T^{0+} N_1^{\\\\sigma - 3 s +}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\\\sigma < 3s$ .\",\n \"$\\\\circ $ Subcase (2.b): $N_2, N_3\\\\gtrsim N_1^\\\\frac{1}{2} $ .\",\n \"By $L^{10}_{T, x}, L^{\\\\frac{10}{3}}_{T, x},L^{\\\\frac{10}{3}}_{T, x}, L^{\\\\frac{10}{3}}_{T, x}$ -Hölder's inequality and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {Q5})& \\\\le T^{0+}N_1^{\\\\sigma -s} N_2^{-2s+}N_3^{-2s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -3 s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\\\sigma < 3s$ .\",\n \"$\\\\circ $ Subcase (2.c): $N_2\\\\gtrsim N_1^\\\\frac{1}{2} \\\\gg N_3$ .\",\n \"By the bilinear Strichartz estimate and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {Q5})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{1, N_1} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{3, N_2} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{3, N_3} w_{N_4} \\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma - s } N_2^{-2s+}N_3^{1- 2s-}N_4^{-\\\\frac{1}{2}+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma - 3s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\\\sigma < 3s$ .\",\n \"$\\\\bullet $ Case (3): $N_2 \\\\sim N_3 \\\\sim N_{\\\\max } \\\\gg N_1$ .\",\n \"$\\\\circ $ Subcase (3.a): $N_1 \\\\ll N_2^\\\\frac{1}{2}$ .\",\n \"By the bilinear Strichartz estimate and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {Q5})& \\\\lesssim N_2^\\\\sigma \\\\Vert z_{1, N_1} z_{3, N_2} \\\\Vert _{L^2_{T, x}}\\\\Vert z_{3, N_3} \\\\Vert _{L^{5}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\lesssim T^{0+}N_1^{ 1- s - } N_2^{\\\\sigma -2s - \\\\frac{1}{2} + }N_3^{-s}\\\\Vert \\\\mathbf {P}_{N_1} \\\\phi ^\\\\omega \\\\Vert _{H^s}\\\\Vert z_{3, N_2}\\\\Vert _{Y^{2s-}_T}\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_{3, N_3} \\\\Vert _{L^{5}_{T, x}}\\\\\\\\\\\\multicolumn{2}{l}{\\\\text{By applying Lemma \\\\ref {LEM:Z3_2}and the fractional Leibniz rule,}}\\\\\\\\& \\\\lesssim T^{\\\\frac{1}{10}+}N_2^{\\\\sigma -\\\\frac{7}{2} s+}\\\\Vert \\\\mathbf {P}_{N_1} \\\\phi ^\\\\omega \\\\Vert _{H^s}\\\\Vert z_{3, N_2}\\\\Vert _{Y^{2s-}_T}\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}\\\\Vert z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}^2\\\\\\\\& \\\\le T^{\\\\frac{1}{10}+}N_2^{\\\\sigma -\\\\frac{7}{2}s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\\\sigma < \\\\frac{7}{2}s$ .\",\n \"$\\\\circ $ Subcase (3.b): $N_1\\\\gtrsim N_2^\\\\frac{1}{2} $ .\",\n \"By $L^{10}_{T, x}, L^{\\\\frac{10}{3}}_{T, x},L^{\\\\frac{10}{3}}_{T, x}, L^{\\\\frac{10}{3}}_{T, x}$ -Hölder's inequality and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {Q5})& \\\\le T^{0+}N_1^{ -s} N_2^{\\\\sigma -2s+}N_3^{-2s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -\\\\frac{9}{2}s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\\\sigma < \\\\frac{9}{2}s$ .\",\n \"$\\\\bullet $ Case (4): $N_2 \\\\sim N_4 \\\\gg N_1, N_3$ .\",\n \"$\\\\circ $ Subcase (4.a): $N_1, N_3 \\\\ll N_2^\\\\frac{1}{2} $ .\",\n \"By the bilinear Strichartz estimate, we have $\\\\text{LHS of } (\\\\ref {Q5})& \\\\lesssim N_2^\\\\sigma \\\\Vert z_{1, N_1} z_{3, N_2} \\\\Vert _{L^2_{T, x}} \\\\Vert z_{3, N_3} w_{N_4}\\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\le T^{0+} N_1^{1-s- }N_2^{\\\\sigma -2s -\\\\frac{1}{2}+ } N_3^{1- 2s-}N_4^{-\\\\frac{1}{2}+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+} N_1^{\\\\sigma - \\\\frac{7}{2} s +}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\\\sigma < \\\\frac{7}{2}s$ .\",\n \"$\\\\circ $ Subcase (4.b): $N_1, N_3\\\\gtrsim N_2^\\\\frac{1}{2} $ .\",\n \"By $L^{10}_{T, x}, L^{\\\\frac{10}{3}}_{T, x},L^{\\\\frac{10}{3}}_{T, x}, L^{\\\\frac{10}{3}}_{T, x}$ -Hölder's inequality and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {Q5})& \\\\le T^{0+}N_1^{ -s} N_2^{\\\\sigma -2s+}N_3^{-2s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_2^{\\\\sigma -\\\\frac{7}{2} s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\\\sigma < \\\\frac{7}{2}s$ .\",\n \"$\\\\circ $ Subcase (4.c): $N_1\\\\gtrsim N_2^\\\\frac{1}{2} \\\\gg N_3$ .\",\n \"By the bilinear Strichartz estimate and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {Q5})& \\\\lesssim N_2^\\\\sigma \\\\Vert z_{1, N_1} \\\\Vert _{L^{5}_{T, x}} \\\\Vert z_{3, N_2} z_{3, N_3} \\\\Vert _{L^2_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{ - s} N_2^{\\\\sigma -2s-\\\\frac{1}{2} + }N_3^{1- 2s-}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T})\\\\\\\\& \\\\le T^{0+}N_1^{\\\\sigma -\\\\frac{7}{2}s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\\\sigma < \\\\frac{7}{2}s$ .\",\n \"$\\\\circ $ Subcase (4.d): $N_3\\\\gtrsim N_2^\\\\frac{1}{2} \\\\gg N_1$ .\",\n \"By the bilinear Strichartz estimate and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {Q5})& \\\\lesssim N_2^\\\\sigma \\\\Vert z_{1, N_1} z_{3, N_2} \\\\Vert _{L^2_{T, x}}\\\\Vert z_{3, N_3} \\\\Vert _{L^{5}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\lesssim T^{0+}N_1^{ 1- s - } N_2^{\\\\sigma -2s - \\\\frac{1}{2} + }N_3^{-s}\\\\Vert \\\\mathbf {P}_{N_1} \\\\phi ^\\\\omega \\\\Vert _{H^s}\\\\Vert z_{3, N_2}\\\\Vert _{Y^{2s-}_T}\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_{3, N_3} \\\\Vert _{L^{5}_{T, x}}\\\\\\\\\\\\multicolumn{2}{l}{\\\\text{Proceeding as in Subcase (3.a) with Lemma \\\\ref {LEM:Z3_2},}}\\\\\\\\& \\\\le T^{\\\\frac{1}{10}}N_1^{\\\\sigma - 3s+}C(\\\\Vert z_1\\\\Vert _{A^s_7(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\\\sigma < 3s$ .\",\n \"Putting all the cases together, we conclude that (REF ) holds when $ \\\\sigma < 3s$ .\",\n \"(ii) Next, we estimate $\\\\zeta _7$ in ().\",\n \"This term has a similar structure to $z_5$ in (REF ); the only difference appears in the third factor.\",\n \"Hence, we can estimate $\\\\zeta _7$ simply by replacing the regularity $2s-$ (for $z_3$ ; see Proposition REF ) with $\\\\frac{5}{2} s-$ (for $z_5$ ; see Lemma REF ) in the proof of Lemma REF .\",\n \"In the following, we only indicate the necessary modifications on the powers of dyadic parameters in the proof of Lemma REF .\",\n \"$\\\\bullet $ Case (1): $N_1 \\\\sim N_2 \\\\sim N_{\\\\max }$ .\",\n \"$\\\\circ $ Subcase (1.a): $N_3 \\\\ll N_1^\\\\frac{1}{2}$ .\",\n \"It suffices to note that $N_1^{\\\\sigma - s - \\\\frac{1}{2}+} N_2^{-s}N_3^{1 - \\\\frac{5}{2} s-}& \\\\lesssim N_1^{\\\\sigma - \\\\frac{13}{4} s +}\\\\lesssim N_{\\\\max }^{0-} ,$ provided that $ \\\\sigma < \\\\frac{13}{4} s$ and $0 < s < \\\\frac{2}{5}$ .\",\n \"The modification for Subcase (1.b) is straightforward under the same regularity restriction.\",\n \"$\\\\bullet $ Case (2): $N_1 \\\\sim N_3 \\\\sim N_{\\\\max } \\\\gg N_2$ .\",\n \"$\\\\circ $ Subcase (2.a): $N_2 \\\\ll N_1^\\\\frac{1}{2} $ .\",\n \"It suffices to note that $N_1^{\\\\sigma - s } N_2^{1-s-}N_3^{- \\\\frac{5}{2}s-\\\\frac{1}{2} +}\\\\lesssim N_1^{\\\\sigma - 4 s +}\\\\lesssim N_{\\\\max }^{0-} ,$ provided that $\\\\sigma < 4s$ and $0 \\\\le s < 1$ .The lower bound on $s$ is needed only for Subcase (2.b).\",\n \"Similar comments apply to the following cases and also to the proof of Proposition REF .\",\n \"The modification for Subcase (2.b) is straightforward under the same regularity restriction.\",\n \"$\\\\bullet $ Case (3): $N_1 \\\\sim N_4 \\\\sim N_{\\\\max } \\\\gg N_2, N_3$ .\",\n \"$\\\\circ $ Subcase (3.a): $N_2, N_3 \\\\ll N_1^\\\\frac{1}{2} $ .\",\n \"It suffices to note that $N_1^{\\\\sigma -s-\\\\frac{1}{2} + }N_2^{1-s - } N_3^{1- \\\\frac{5}{2}s -}N_4^{-\\\\frac{1}{2}+}\\\\lesssim N_1^{\\\\sigma - \\\\frac{11}{4} s +}\\\\lesssim N_{\\\\max }^{0-} ,$ provided that $\\\\sigma < \\\\frac{11}{4}s\\\\qquad \\\\text{and}\\\\qquad 0 < s < \\\\frac{2}{5}.$ The modifications for Subcases (3.b), (3.c), and (3.d) are straightforward under the regularity restriction (REF ).\",\n \"$\\\\bullet $ Case (4): $N_3 \\\\sim N_4 \\\\gg N_1 \\\\ge N_2$ .\",\n \"$\\\\circ $ Subcase (4.a): $N_1, N_2 \\\\ll N_3^\\\\frac{1}{2} $ .\",\n \"It suffices to note that $N_1^{1-s- }N_2^{1-s - } N_3^{\\\\sigma - \\\\frac{5}{2} s -\\\\frac{1}{2} +}N_4^{-\\\\frac{1}{2}+}\\\\lesssim N_1^{\\\\sigma - \\\\frac{7}{2} s +}\\\\lesssim N_{\\\\max }^{0-} ,$ provided that $\\\\sigma < \\\\frac{7}{2} s$ and $0 \\\\le s < 1$ .\",\n \"The modifications for Subcases (4.b) and (4.c) are straightforward under the same regularity restriction.\",\n \"Putting all the cases together, we conclude that (REF ) holds under the regularity restriction (REF ).\",\n \"This completes the proof of Lemma REF .\"\n ],\n [\n \"On the “unbalanced” higher order terms $\\\\zeta _{2k-1}$\",\n \"In this subsection, we study the regularity properties of the unbalanced higher order terms $\\\\zeta _{2k-1}$ defined in (REF ).\",\n \"Note that $(j_1, j_2, j_3) = (1, 1, 2k-3)$ up to permutations.\",\n \"Then, by dropping the complex conjugate, we have $\\\\zeta _{2k-1}\\\\sim \\\\int _0^t S(t - t^{\\\\prime }) z_{1} z_1 \\\\zeta _{2k-3} (t^{\\\\prime }) dt^{\\\\prime }.$ We first present the proof of Proposition REF .\",\n \"We proceed by induction.\",\n \"From (REF ), we see that the recursive relation (REF ) is satisfied when $k = 2, 3, 4$ .\",\n \"In the following, by assuming $\\\\zeta _{2k -3} \\\\in X^\\\\sigma _\\\\textup {loc}$ for $\\\\sigma < \\\\alpha _{k-1} \\\\cdot s $ almost surely, where $\\\\alpha _{k-1}$ satisfies (REF ), we prove (REF ) and (REF ).\",\n \"As in the proof of Lemma REF (ii), the proof follows from the proof of Lemma REF by replacing $z_3$ and $2s-$ with $\\\\zeta _{2k-3}$ and $\\\\alpha _{k-1} \\\\cdot s-$ , respectively.\",\n \"Therefore, we only indicate the necessary modifications on the powers of dyadic parameters in the proof of Lemma REF .\",\n \"$\\\\bullet $ Case (1): $N_1 \\\\sim N_2 \\\\sim N_{\\\\max }$ .\",\n \"$\\\\circ $ Subcase (1.a): $N_3 \\\\ll N_1^\\\\frac{1}{2}$ .\",\n \"It suffices to note that $N_1^{\\\\sigma - s - \\\\frac{1}{2}+} N_2^{-s}N_3^{1 - \\\\alpha _{k-1} s-}& \\\\lesssim N_1^{\\\\sigma - \\\\frac{\\\\alpha _{k-1}+4}{2} s +}\\\\lesssim N_{\\\\max }^{0-} ,$ provided that $\\\\sigma < \\\\frac{\\\\alpha _{k-1}+4}{2} s\\\\qquad \\\\text{and}\\\\qquad 0 < s < \\\\alpha _{k-1}^{-1}.$ The modification for Subcase (1.b) is straightforward, giving the regularity restriction (REF ).\",\n \"$\\\\bullet $ Case (2): $N_1 \\\\sim N_3 \\\\sim N_{\\\\max } \\\\gg N_2$ .\",\n \"$\\\\circ $ Subcase (2.a): $N_2 \\\\ll N_1^\\\\frac{1}{2} $ .\",\n \"It suffices to note that $N_1^{\\\\sigma - s } N_2^{1-s-}N_3^{- \\\\alpha _{k-1}s-\\\\frac{1}{2} +}\\\\lesssim N_1^{\\\\sigma - \\\\frac{2 \\\\alpha _{k-1} + 3}{2} s +}\\\\lesssim N_{\\\\max }^{0-} ,$ provided that $\\\\sigma < \\\\frac{2\\\\alpha _{k-1}+3}{2}s\\\\qquad \\\\text{and}\\\\qquad 0 \\\\le s < 1.$ The modification for Subcase (2.b) is straightforward, giving the regularity restriction (REF ).\",\n \"$\\\\bullet $ Case (3): $N_1 \\\\sim N_4 \\\\sim N_{\\\\max } \\\\gg N_2, N_3$ .\",\n \"$\\\\circ $ Subcase (3.a): $N_2, N_3 \\\\ll N_1^\\\\frac{1}{2} $ .\",\n \"It suffices to note that $N_1^{\\\\sigma -s-\\\\frac{1}{2} + }N_2^{1-s - } N_3^{1- \\\\alpha _{k-1}s -}N_4^{-\\\\frac{1}{2}+}\\\\lesssim N_1^{\\\\sigma - \\\\frac{\\\\alpha _{k-1}+3}{2} s +}\\\\lesssim N_{\\\\max }^{0-} ,$ provided that $\\\\sigma < \\\\frac{\\\\alpha _{k-1}+3}{2}s\\\\qquad \\\\text{and}\\\\qquad 0 < s < \\\\alpha _{k-1}^{-1}.$ The modifications for Subcases (3.b), (3.c), and (3.d) are straightforward, giving the regularity restriction (REF ).\",\n \"$\\\\bullet $ Case (4): $N_3 \\\\sim N_4 \\\\gg N_1 \\\\ge N_2$ .\",\n \"$\\\\circ $ Subcase (4.a): $N_1, N_2 \\\\ll N_3^\\\\frac{1}{2} $ .\",\n \"It suffices to note that $N_1^{1-s- }N_2^{1-s - } N_3^{\\\\sigma - \\\\alpha _{k-1} s -\\\\frac{1}{2} +}N_4^{-\\\\frac{1}{2}+}\\\\lesssim N_1^{\\\\sigma - (\\\\alpha _{k-1}+1) s +}\\\\lesssim N_{\\\\max }^{0-} ,$ provided that $\\\\sigma < (\\\\alpha _{k-1}+1)s\\\\qquad \\\\text{and}\\\\qquad 0 \\\\le s < 1.$ The modifications for Subcases (4.a) and (4.b) are straightforward, giving the regularity restriction (REF ).\",\n \"Since $\\\\alpha _{k-1}$ satisfies (REF ), we have $\\\\alpha _{k-1} \\\\ge 1$ , which in turn implies $\\\\frac{\\\\alpha _{k-1}+3}{2} \\\\le \\\\alpha _{k-1} + 1$ .\",\n \"Therefore, we conclude (REF ) and (REF ).\",\n \"We conclude this section by proving a gain of space-time integrability for $\\\\zeta _{2k-1}$ analogous to Lemma REF .\",\n \"Lemma 4.3 Let $s \\\\ge 0$ .\",\n \"Given $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ , let $\\\\phi ^\\\\omega $ be its Wiener randomization defined in (REF ).\",\n \"Fix an integer $k \\\\ge 3$ .\",\n \"Then, for any finite $q, r > 2$ , there exist $\\\\theta _k = \\\\theta _k(q, r) > 0$ , $\\\\delta _k = \\\\delta _k(q, r) \\\\in (0, 1)$ , and $1\\\\le q_k< \\\\infty $ such that $\\\\Vert \\\\mathbf {P}_N \\\\zeta _{2k-1}\\\\Vert _{L^q_T L^r_x}\\\\lesssim {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\rule [-6mm]{0pt}{15pt} T^{\\\\theta _k} \\\\Vert z_1\\\\Vert _{L^{q_k}_T L^{2}_x}^{(2k-1)(1-\\\\delta _k)}\\\\Vert z_1\\\\Vert _{L^{q_k}_T L^{6}_x}^{(2k-1)\\\\delta _k}& \\\\text{when } 1 \\\\le r < 6, \\\\\\\\T^{\\\\theta _k} N^{\\\\frac{1}{2} - \\\\frac{3}{r}+} \\\\Vert z_1\\\\Vert _{L^{q_k}_T L^{2}_x}^{(2k-1)(1-\\\\delta _k)}\\\\Vert z_1\\\\Vert _{L^{q_k}_T L^{6}_x}^{(2k-1)\\\\delta _k}& \\\\text{when } r \\\\ge 6, \\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ for any $T > 0$ and $N \\\\in 2^{\\\\mathbb {N}_0}$ .\",\n \"Note that the right-hand side of (REF ) is almost surely finite thanks to the probabilistic Strichartz estimate (Lemma REF ).\",\n \"We prove (REF ) by induction.\",\n \"We first consider the case $r < 6$ .\",\n \"In Lemma REF , we proved (REF ) for $k = 2$ .\",\n \"Now, suppose that (REF ) holds for $k - 1$ .\",\n \"Then, from (REF ) and the dispersive estimate (REF ), we have $\\\\Vert \\\\mathbf {P}_N \\\\zeta _{2k-1}\\\\Vert _{L^q_T L^r_x}& \\\\le \\\\bigg \\\\Vert \\\\int _0^t \\\\Vert \\\\mathbf {P}_N S(t - t^{\\\\prime }) z_1^2 \\\\zeta _{2k-3} (t^{\\\\prime })\\\\Vert _{L^r_x} dt^{\\\\prime }\\\\bigg \\\\Vert _{L^q_T}\\\\\\\\& \\\\lesssim \\\\bigg \\\\Vert \\\\int _0^t \\\\frac{1}{|t - t^{\\\\prime }|^{\\\\frac{3}{2} - \\\\frac{3}{r}}}\\\\Vert z_1\\\\Vert _{L^{3r^{\\\\prime }}_x}^2 \\\\Vert \\\\zeta _{2k-3}\\\\Vert _{L^{3r^{\\\\prime }}_x} dt^{\\\\prime }\\\\bigg \\\\Vert _{L^q_T}\\\\\\\\\\\\multicolumn{2}{l}{\\\\text{Noting $3r^{\\\\prime } < 6$and applying the inductive hypothesiswith $\\\\theta _{k-1} = \\\\theta _{k-1}(3q, 3r^{\\\\prime })$,$\\\\delta _{k-1} = \\\\delta _{k-1}(3q, 3r^{\\\\prime })$,and $q_{k-1} = q_{k-1}(3q, 3r^{\\\\prime })$,}}\\\\\\\\& \\\\lesssim T^{\\\\frac{3}{r} - \\\\frac{1}{2} + \\\\theta _{k-1} }\\\\Vert z_1\\\\Vert _{L^{3q}_T L^{3r^{\\\\prime }}_x}^2\\\\Vert z_1\\\\Vert _{L^{q_{k-1}}_T L^{2}_x}^{(2k-3)(1-\\\\delta _{k-1})}\\\\Vert z_1\\\\Vert _{L^{q_{k-1}}_T L^{6}_x}^{(2k-3)\\\\delta _{k-1}} \\\\\\\\\\\\multicolumn{2}{l}{\\\\text{By interpolation in $x$ and Hölder's inequality in $t$,}}\\\\\\\\& \\\\lesssim T^{\\\\theta _k }\\\\Vert z_1\\\\Vert _{L^{q_k}_T L^{2}_x}^{(2k-1)(1-\\\\delta _{k})}\\\\Vert z_1\\\\Vert _{L^{q_k}_T L^{6}_x}^{(2k-1)\\\\delta _{k}} .$ When $r \\\\ge 6$ , (REF ) follows from Sobolev's inequality and (REF ) as in the proof of Lemma REF .\"\n ],\n [\n \"Proof of Theorem \",\n \"In this section, we study the fixed point problem (REF ) around the second order expansion (REF ) and present the proof of Theorem REF .\",\n \"Given $ \\\\frac{1}{2} \\\\le \\\\sigma \\\\le 1$ , let $ \\\\frac{2}{5} \\\\sigma < s < \\\\frac{1}{2} $ .\",\n \"Given $\\\\phi \\\\in H^s(\\\\mathbb {R}^3)$ , let $\\\\phi ^\\\\omega $ be its Wiener randomization defined in (REF ) and let $z_1$ and $z_3$ be as in (REF ) and (REF ).\",\n \"Define $\\\\Gamma $ by $\\\\Gamma v(t) = - i \\\\int _0^t S(t-t^{\\\\prime })\\\\big \\\\lbrace \\\\mathcal {N}(v + z_1 + z_3) - \\\\mathcal {N}(z_1)\\\\big \\\\rbrace (t^{\\\\prime }) dt^{\\\\prime }.$ Then, we have the following nonlinear estimates.\",\n \"Proposition 5.1 Given $ \\\\frac{1}{2} \\\\le \\\\sigma \\\\le 1$ , let $\\\\frac{2}{5} \\\\sigma < s < \\\\frac{1}{2}$ .\",\n \"Then, there exist $\\\\theta > 0$ , $C_1, C_2> 0$ , and an almost surely finite constant $R = R(\\\\omega ) >0$ such that $\\\\Vert \\\\Gamma v\\\\Vert _{X^\\\\sigma ([0, T])}& \\\\le C_1\\\\big (\\\\Vert v\\\\Vert _{X^\\\\sigma ([0, T])} ^3 + T^\\\\theta R(\\\\omega )\\\\big ), \\\\\\\\\\\\Vert \\\\Gamma v_1 - \\\\Gamma v_2 \\\\Vert _{X^\\\\sigma ([0, T])}& \\\\le C_2\\\\Big (\\\\sum _{j = 1}^2 \\\\Vert v_j\\\\Vert _{X^\\\\sigma ([0, T])} ^2+ T^\\\\theta R(\\\\omega )\\\\Big )\\\\Vert v_1 -v_2 \\\\Vert _{X^\\\\sigma ([0, T])},$ for all $v, v_1, v_2 \\\\in X^{\\\\sigma }([0, T])$ and $0 < T\\\\le 1$ .\",\n \"Once we prove Proposition REF , Theorem REF immediately follows from a standard argument and thus we omit details.\",\n \"See Section 5 in [2].\",\n \"Let $0 0$ .\",\n \"$\\\\bullet $ Subcase (C.2): $N_3 \\\\sim N_4 \\\\sim N_{\\\\max } \\\\gg N_2 \\\\ge N_1$ .\",\n \"$\\\\circ $ Subsubcase (C.2.a): $N_1, N_2 \\\\ll N_3^\\\\frac{1}{2}$ .\",\n \"By the bilinear Strichartz estimate, we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert z_{3, N_1} z_{3, N_3} \\\\Vert _{L^2_{T, x}} \\\\Vert z_{3, N_2} w_{N_4}\\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\lesssim T^{0+} N_1^{1-2s- } N_2^{1-2s -}N_3^{\\\\sigma - 2s -\\\\frac{1}{2}+}N_4^{-\\\\frac{1}{2}+}\\\\bigg (\\\\prod _{j = 1}^3\\\\Vert z_{3, N_j}\\\\Vert _{Y^{2s-}_T}\\\\bigg )\\\\\\\\& \\\\lesssim T^{0+} N_3^{\\\\sigma - 4s +}C( \\\\Vert z_3\\\\Vert _{X^{2s-}_T}).$ This yields (REF ), provided that $\\\\sigma < 4s$ and $s < \\\\frac{1}{2}$ .\",\n \"$\\\\circ $ Subsubcase (C.2.b): $N_1, N_2 \\\\gtrsim N_3^\\\\frac{1}{2} $ .\",\n \"Proceeding as in Subcase (C.1), we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert z_{3, N_1} \\\\Vert _{L^{q}_T L^{6+}_x}\\\\bigg (\\\\prod _{j = 2}^3 \\\\Vert z_{3, N_2} \\\\Vert _{L^{2+}_T L^{6-}_x}\\\\bigg )\\\\Vert w_{N_4} \\\\Vert _{L^\\\\infty _T L^2_x}\\\\\\\\& \\\\lesssim T^{0+}N_1^{ -s +} N_2^{-2s+}N_3^{\\\\sigma -2s+}\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_1\\\\Vert _{L^{3q}_T L^{\\\\frac{18}{5}+}_x}\\\\Vert z_1\\\\Vert _{L^{3q}_T L^{\\\\frac{18}{5}+}_x}^2\\\\bigg (\\\\prod _{j = 2}^3\\\\Vert z_{3, N_j}\\\\Vert _{Y^{2s-}_T}\\\\bigg )\\\\\\\\& \\\\le T^{0+}N_3^{\\\\sigma -\\\\frac{7}{2} s+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ This yields (REF ), provided that $\\\\sigma < \\\\frac{7}{2} s$ and $s > 0$ .\",\n \"$\\\\circ $ Subsubcase (C.2.c): $N_2\\\\gtrsim N_3^\\\\frac{1}{2} \\\\gg N_1$ .\",\n \"By Lemmas REF and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_1^\\\\sigma \\\\Vert z_{3, N_1} z_{3, N_3} \\\\Vert _{L^2_{T, x}}\\\\Vert z_{3, N_2} \\\\Vert _{L^{5}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{1 - 2s -} N_2^{-s}N_3^{\\\\sigma - 2s- \\\\frac{1}{2}+}C( \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T})\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_{3, N_2} \\\\Vert _{L^5_{T, x}}\\\\\\\\\\\\multicolumn{2}{l}{\\\\text{By applying Lemma \\\\ref {LEM:Z3_2},}}\\\\\\\\& \\\\le T^{\\\\frac{1}{10}+}N_1^{1 - 2s -} N_2^{-s}N_3^{\\\\sigma - 2s- \\\\frac{1}{2}+}C( \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T})\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}\\\\Vert z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}^2\\\\\\\\& \\\\le T^{\\\\frac{1}{10}+}N_3^{\\\\sigma -\\\\frac{7}{2} s+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}).$ This yields (REF ), provided that $\\\\sigma < \\\\frac{7}{2} s$ and $ 0 \\\\le s < \\\\frac{1}{2}$ .\",\n \"Case (D): $v v z_1$ case.\",\n \"By symmetry, we assume $N_1 \\\\ge N_2$ .\",\n \"$\\\\bullet $ Subcase (D.1): $N_1 \\\\gtrsim N_3$ .\",\n \"In this case, we have $N_1 \\\\sim N_{\\\\max }$ .\",\n \"$\\\\circ $ Subsubcase (D.1.a): $\\\\max (N_2, N_3) \\\\ge N_1^\\\\frac{1}{10}$ .\",\n \"By Hölder's inequality and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_1^\\\\sigma \\\\Vert v_{N_1} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert v_{N_2} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{1, N_3} \\\\Vert _{L^{10}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+} N_2^{-\\\\sigma } N_3^{-s} C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2,$ provided that $\\\\sigma > 0$ and $ s > 0$ .\",\n \"$\\\\circ $ Subsubcase (D.1.b): $\\\\max (N_2, N_3) \\\\ll N_1^\\\\frac{1}{10}$ .\",\n \"In this case, we have $N_1\\\\sim N_4 \\\\sim N_{\\\\max }$ .\",\n \"By Lemmas REF and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_1^\\\\sigma \\\\Vert v_{N_1}\\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{1, N_3}\\\\Vert _{L^5_{T, x}}\\\\Vert v_{ N_2} w_{N_4} \\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\le T^{0+}N_2^{1-\\\\sigma -}N_3^{-s}N_4^{-\\\\frac{1}{2}+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2,$ provided that $ \\\\sigma \\\\ge 0$ and $ s \\\\ge 0$ .\",\n \"$\\\\bullet $ Subcase (D.2): $N_3 \\\\sim N_4 \\\\gg N_1\\\\ge N_2$ .\",\n \"$\\\\circ $ Subsubcase (D.2.a): $N_1, N_2 \\\\ll N_3^\\\\frac{1}{2}$ .\",\n \"By the bilinear Strichartz estimate, we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert v_{N_1} z_{1, N_3} \\\\Vert _{L^2_{T, x}} \\\\Vert v_{N_2} w_{N_4}\\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\lesssim T^{0+} N_1^{1-\\\\sigma -} N_2^{1-\\\\sigma -} N_3^{\\\\sigma - s - \\\\frac{1}{2} +}N_4^{-\\\\frac{1}{2}+}\\\\bigg (\\\\prod _{j = 1}^2 \\\\Vert v_{N_j}\\\\Vert _{Y^\\\\sigma _T}\\\\bigg )\\\\Vert \\\\mathbf {P}_{N_3} \\\\phi ^\\\\omega \\\\Vert _{H^{s}}\\\\\\\\& \\\\le T^{0+} N_3^{ - s +}C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2,$ provided that $\\\\sigma \\\\le 1$ and $s>0$ .\",\n \"$\\\\circ $ Subsubcase (D.2.b): $N_1, N_2\\\\gtrsim N_3^\\\\frac{1}{2} $ .\",\n \"Proceeding as in Subsubcase (D.1.a) with $L^\\\\frac{10}{3}_{T, x},L^\\\\frac{10}{3}_{T, x},L^{10}_{T, x}, L^\\\\frac{10}{3}_{T, x}$ -Hölder's inequality, it suffices to note that $N_1^{-\\\\sigma } N_2^{-\\\\sigma }N_3^{\\\\sigma -s}\\\\lesssim N_3^{ - s }\\\\lesssim N_{\\\\max }^{0-},$ provided that $\\\\sigma \\\\ge 0$ and $ s > 0$ .\",\n \"$\\\\circ $ Subsubcase (D.2.c): $N_1\\\\gtrsim N_3^\\\\frac{1}{2} \\\\gg N_2$ .\",\n \"By Lemmas REF and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert v_{N_1}\\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{1, N_3}\\\\Vert _{L^5_{T, x}}\\\\Vert v_{ N_2} w_{N_4} \\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{-\\\\sigma } N_2^{1-\\\\sigma -}N_3^{\\\\sigma -s}N_4^{-\\\\frac{1}{2}+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2,$ provided that $0\\\\le \\\\sigma \\\\le 1$ and $ s > 0$ .\",\n \"Case (E): $v v z_3$ case.\",\n \"By symmetry, we assume $N_1 \\\\ge N_2$ .\",\n \"$\\\\bullet $ Subcase (E.1): $N_1 \\\\gtrsim N_3$ .\",\n \"In this case, we have $N_1 \\\\sim N_{\\\\max }$ .\",\n \"$\\\\circ $ Subsubcase (E.1.a): $\\\\max (N_2, N_3) \\\\ge N_1^\\\\frac{1}{10}$ .\",\n \"By Hölder's inequality with (REF ) as in Subcase (C.1) and Lemmas REF and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_1^\\\\sigma \\\\bigg (\\\\prod _{j = 1}^2 \\\\Vert v_{N_j} \\\\Vert _{L^{2+}_T L^{6-}_x}\\\\bigg )\\\\Vert z_{3, N_3} \\\\Vert _{L^{q}_T L^{6+}_x}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\infty _T L^2_x}\\\\\\\\& \\\\lesssim T^{0+}N_2^{-\\\\sigma }N_3^{-s+}\\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_1\\\\Vert _{L^{3q}_T L^{\\\\frac{18}{5}+}_x}\\\\Vert z_1\\\\Vert _{L^{3q}_T L^{\\\\frac{18}{5}+}_x}^2\\\\\\\\\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2,$ provided that $\\\\sigma > 0$ and $s > 0$ .\",\n \"$\\\\circ $ Subsubcase (E.1.b): $\\\\max (N_2, N_3) \\\\ll N_1^\\\\frac{1}{10}$ .\",\n \"In this case, we have $N_1\\\\sim N_4 \\\\sim N_{\\\\max }$ .\",\n \"By Lemmas REF , REF , and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_1^\\\\sigma \\\\Vert v_{N_1}\\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{3, N_3}\\\\Vert _{L^5_{T, x}}\\\\Vert v_{ N_2} w_{N_4} \\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\lesssim T^{\\\\frac{1}{10}+}N_2^{1-\\\\sigma -}N_3^{-s}N_4^{-\\\\frac{1}{2}+}\\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}\\\\Vert z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}^2\\\\\\\\& \\\\le T^{\\\\frac{1}{10}+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2,$ provided that $ \\\\sigma \\\\ge 0$ and $ s \\\\ge 0$ .\",\n \"$\\\\bullet $ Subcase (E.2): $N_3 \\\\sim N_4 \\\\gg N_1\\\\ge N_2$ .\",\n \"$\\\\circ $ Subsubcase (E.2.a): $N_1, N_2 \\\\ll N_3^\\\\frac{1}{2}$ .\",\n \"In this case, we proceed as in Subsubcase (D.2.a).\",\n \"It suffices to note that $N_1^{1-\\\\sigma -} N_2^{1-\\\\sigma -} N_3^{\\\\sigma - 2s - \\\\frac{1}{2} +}N_4^{-\\\\frac{1}{2}+}\\\\lesssim N_3^{ - 2s +}\\\\lesssim N_{\\\\max }^{0-},$ provided that $\\\\sigma \\\\le 1$ and $s>0$ .\",\n \"$\\\\circ $ Subsubcase (E.2.b): $N_1, N_2\\\\gtrsim N_3^\\\\frac{1}{2} $ .\",\n \"By Hölder's inequality with (REF ) as in Subcase (C.1) and Lemmas REF and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\bigg (\\\\prod _{j = 1}^2 \\\\Vert v_{N_j} \\\\Vert _{L^{2+}_T L^{6-}_x}\\\\bigg )\\\\Vert z_{3, N_3} \\\\Vert _{L^{q}_T L^{6+}_x}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\infty _T L^2_x}\\\\\\\\& \\\\lesssim T^{0+}N_1^{-\\\\sigma }N_2^{-\\\\sigma } N_3^{\\\\sigma -s+}\\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_1\\\\Vert _{L^{3q}_T L^{\\\\frac{18}{5}+}_x}\\\\Vert z_1\\\\Vert _{L^{3q}_T L^{\\\\frac{18}{5}+}_x}^2\\\\\\\\\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2,$ provided that $\\\\sigma \\\\ge 0$ and $s > 0$ .\",\n \"$\\\\circ $ Subsubcase (E.2.c): $N_1\\\\gtrsim N_3^\\\\frac{1}{2} \\\\gg N_2$ .\",\n \"By Lemmas REF , REF , and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert v_{N_1}\\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{3, N_3}\\\\Vert _{L^5_{T, x}}\\\\Vert v_{ N_2} w_{N_4} \\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\le T^{\\\\frac{1}{10}}N_1^{-\\\\sigma } N_2^{1-\\\\sigma -}N_3^{\\\\sigma -s}N_4^{-\\\\frac{1}{2}+}\\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}\\\\Vert z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}^2\\\\\\\\& \\\\le T^{\\\\frac{1}{10}} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^2,$ provided that $0\\\\le \\\\sigma \\\\le 1$ and $ s > 0$ .\",\n \"Case (F): $v z_1 z_1$ case.\",\n \"By symmetry, we assume $N_3 \\\\ge N_2$ .\",\n \"$\\\\bullet $ Subcase (F.1): $N_1 \\\\gtrsim N_3$ .\",\n \"In this case, we have $N_1 \\\\sim N_{\\\\max }$ .\",\n \"$\\\\circ $ Subsubcase (F.1.a): $\\\\max (N_2, N_3) \\\\ge N_1^\\\\frac{1}{10}$ .\",\n \"By Lemma REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_1^\\\\sigma \\\\Vert v_{N_1} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{1, N_2} \\\\Vert _{L^5_{T, x}}\\\\Vert z_{1, N_3} \\\\Vert _{L^{5}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+} N_2^{-s} N_3^{-s} C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T},$ provided that $ s > 0$ .\",\n \"$\\\\circ $ Subsubcase (F.1.b): $\\\\max (N_2, N_3) \\\\ll N_1^\\\\frac{1}{10}$ .\",\n \"In this case, we proceed as in Subsubcase (D.1.b).\",\n \"It suffices to note that $N_2^{1-s-} N_3^{-s}N_4^{-\\\\frac{1}{2}+}\\\\lesssim N_{\\\\max }^{0-},$ provided that $ s \\\\ge 0$ .\",\n \"$\\\\bullet $ Subcase (F.2): $N_2 \\\\sim N_3 \\\\gg N_1$ .\",\n \"$\\\\circ $ Subsubcase (F.2.a): $N_1\\\\ll N_3^\\\\frac{1}{2}$ .\",\n \"By Lemmas REF and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert v_{ N_1} z_{1, N_2} \\\\Vert _{L^2_{T, x}}\\\\Vert z_{1, N_3}\\\\Vert _{L^5_{T, x}}\\\\Vert w_{N_4}\\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{1-\\\\sigma -} N_2^{-s-\\\\frac{1}{2} + }N_3^{\\\\sigma -s}C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}\\\\\\\\& \\\\le T^{0+}N_3^{\\\\frac{\\\\sigma }{2}-2s+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}.$ This yields (REF ), provided that $\\\\sigma \\\\le \\\\min (1, 4s-)$ .\",\n \"$\\\\circ $ Subsubcase (F.2.b): $N_1\\\\gtrsim N_3^\\\\frac{1}{2} $ .\",\n \"In this case, we proceed with $L^\\\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\\\frac{10}{3}_{T, x}$ -Hölder's inequality.\",\n \"It suffices to note that $N_1^{-\\\\sigma } N_2^{-s}N_3^{\\\\sigma -s}\\\\lesssim N_3^{ \\\\frac{\\\\sigma }{2} - 2s }\\\\lesssim N_{\\\\max }^{0-},$ provided that $0 \\\\le \\\\sigma < 4s$ .\",\n \"$\\\\bullet $ Subcase (F.3): $N_3 \\\\sim N_4 \\\\sim N_{\\\\max } \\\\gg N_1, N_2$ .\",\n \"$\\\\circ $ Subsubcase (F.3.a): $N_1, N_2 \\\\ll N_3^\\\\frac{1}{2} $ .\",\n \"By the bilinear Strichartz estimate, we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert v_{N_1} z_{1, N_3} \\\\Vert _{L^2_{T, x}} \\\\Vert z_{1, N_2} w_{N_4}\\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\lesssim T^{0+} N_1^{1- \\\\sigma -}N_2^{1-s - } N_3^{\\\\sigma - s -\\\\frac{1}{2} +}N_4^{-\\\\frac{1}{2}+}\\\\Vert v\\\\Vert _{X^\\\\sigma _T}\\\\bigg (\\\\prod _{j = 2}^3\\\\Vert \\\\mathbf {P}_{N_j}\\\\phi ^\\\\omega \\\\Vert _{H^s}\\\\bigg )\\\\\\\\& \\\\lesssim T^{0+} N_3^{\\\\frac{\\\\sigma }{2} - \\\\frac{3}{2} s +}C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}.$ This yields (REF ), provided that $ \\\\sigma \\\\le \\\\min (1, 3s-)$ and $ s < 1$ .\",\n \"$\\\\circ $ Subcase (F.3.b): $N_1, N_2\\\\gtrsim N_3^\\\\frac{1}{2} $ .\",\n \"In this case, we proceed with $L^\\\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\\\frac{10}{3}_{T, x}$ -Hölder's inequality.\",\n \"It suffices to note that $N_1^{-\\\\sigma } N_2^{-s}N_3^{\\\\sigma -s}\\\\lesssim N_3^{ \\\\frac{\\\\sigma }{2} - \\\\frac{3}{2}s }\\\\lesssim N_{\\\\max }^{0-},$ provided that $0 \\\\le \\\\sigma < 3s$ .\",\n \"$\\\\circ $ Subcase (F.3.c): $N_1\\\\gtrsim N_3^\\\\frac{1}{2} \\\\gg N_2$ .\",\n \"By Lemmas REF and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert v_{N_1} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{1, N_3} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{1, N_2} w_{N_4} \\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\le T^{0+}N_1^{-\\\\sigma } N_2^{1-s-}N_3^{\\\\sigma - s}N_4^{-\\\\frac{1}{2}+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}\\\\\\\\& \\\\le T^{0+}N_3^{\\\\frac{\\\\sigma }{2}- \\\\frac{3}{2}s+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}.$ This yields (REF ), provided that $0\\\\le \\\\sigma < 3s$ and $s < 1$ .\",\n \"$\\\\circ $ Subcase (F.3.d): $N_2\\\\gtrsim N_3^\\\\frac{1}{2} \\\\gg N_1$ .\",\n \"By Lemmas REF and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert v_{N_1} z_{1, N_3} \\\\Vert _{L^2_{T, x}}\\\\Vert z_{1, N_2} \\\\Vert _{L^{5}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\lesssim T^{0+}N_1^{1-\\\\sigma -} N_2^{-s}N_3^{\\\\sigma - s-\\\\frac{1}{2}+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}\\\\\\\\& \\\\le T^{0+}N_3^{\\\\frac{\\\\sigma }{2}- \\\\frac{3}{2}s+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}.$ This yields (REF ), provided that $ \\\\sigma \\\\le \\\\min (1, 3s-)$ and $s\\\\ge 0$ .\",\n \"Case (G): $v z_3 z_3$ case.\",\n \"By symmetry, we assume $N_3 \\\\ge N_2$ .\",\n \"$\\\\bullet $ Subcase (G.1): $N_1 \\\\gtrsim N_3$ .\",\n \"In this case, we have $N_1 \\\\sim N_{\\\\max }$ .\",\n \"We can proceed as in Subcase (E.1) with $v_{N_2}$ replaced by $z_{3, N_2}$ .\",\n \"More precisely, when $\\\\max (N_2, N_3) \\\\ge N_1^\\\\frac{1}{10}$ , we apply Hölder's inequality with (REF ) as in Subsubcase (E.1.a).\",\n \"It suffices to note that $N_2^{-2s+}N_3^{-s+}\\\\lesssim N_{\\\\max }^{0-},$ provided that $ s> 0$ .\",\n \"When $\\\\max (N_2, N_3) \\\\ll N_1^\\\\frac{1}{10}$ , we proceed with Lemmas REF , REF , and REF as in Subsubcase (E.1.b).\",\n \"In this case, it suffices to note that $N_2^{1-2s-}N_3^{-s}N_4^{-\\\\frac{1}{2}+}\\\\lesssim N_{\\\\max }^{0-},$ provided that $s\\\\ge 0$ .\",\n \"$\\\\bullet $ Subcase (G.2): $N_2 \\\\sim N_3 \\\\gg N_1$ .\",\n \"$\\\\circ $ Subsubcase (G.2.a): $N_1\\\\ll N_3^\\\\frac{1}{2}$ .\",\n \"By Lemmas REF and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert v_{ N_1} z_{3, N_2} \\\\Vert _{L^2_{T, x}}\\\\Vert z_{3, N_3}\\\\Vert _{L^5_{T, x}}\\\\Vert w_{N_4}\\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\le T^{\\\\frac{1}{10}}N_1^{1-\\\\sigma -} N_2^{-2s-\\\\frac{1}{2} + }N_3^{\\\\sigma -s} \\\\Vert v\\\\Vert _{X^\\\\sigma _T}\\\\Vert z_{3, N_2}\\\\Vert _{X^{2s-}_T}\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}\\\\Vert z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}^2 \\\\\\\\& \\\\le T^{\\\\frac{1}{10}}N_3^{\\\\frac{\\\\sigma }{2} -3 s+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}\\\\\\\\& \\\\le T^{0+} N_{\\\\max }^{0-} C(\\\\Vert z_1\\\\Vert _{B^s(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T},$ provided that $\\\\sigma \\\\le \\\\min (1, 6s-)$ .\",\n \"$\\\\circ $ Subsubcase (G.2.b): $N_1\\\\gtrsim N_3^\\\\frac{1}{2} $ .\",\n \"In this case, we proceed with $L^\\\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\\\frac{10}{3}_{T, x}$ -Hölder's inequality with Lemma REF .\",\n \"It suffices to note that $N_1^{-\\\\sigma } N_2^{-s}N_3^{\\\\sigma -s}\\\\lesssim N_3^{ \\\\frac{\\\\sigma }{2} - 2s }\\\\lesssim N_{\\\\max }^{0-},$ provided that $0 \\\\le \\\\sigma < 4s$ .\",\n \"$\\\\bullet $ Subcase (G.3): $N_3 \\\\sim N_4 \\\\sim N_{\\\\max } \\\\gg N_1, N_2$ .\",\n \"$\\\\circ $ Subsubcase (G.3.a): $N_1, N_2 \\\\ll N_3^\\\\frac{1}{2} $ .\",\n \"In this case, we proceed as in Subsubcase (F.3.a).\",\n \"It suffices to note that $N_1^{1- \\\\sigma -}N_2^{1-2s - } N_3^{\\\\sigma - 2 s -\\\\frac{1}{2} +}N_4^{-\\\\frac{1}{2}+}\\\\lesssim N_3^{\\\\frac{\\\\sigma }{2} - 3s+}\\\\lesssim N_{\\\\max }^{0-},$ provided that $ \\\\sigma \\\\le \\\\min (1, 6s-)$ and $ s < \\\\frac{1}{2}$ .\",\n \"$\\\\circ $ Subsubcase (G.3.b): $N_1, N_2\\\\gtrsim N_3^\\\\frac{1}{2} $ .\",\n \"In this case, we proceed with $L^\\\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\\\frac{10}{3}_{T, x}$ -Hölder's inequality as in Subsubcase (F.3.b) but we apply Lemma REF to estimate the $L^{5}_{T, x}$ -norm of $z_{3, N_j}$ , $j = 2, 3$ .\",\n \"The regularity restriction is precisely the same as that in Subsubcase (F.3.b).\",\n \"$\\\\circ $ Subsubcase (G.3.c): $N_1\\\\gtrsim N_3^\\\\frac{1}{2} \\\\gg N_2$ .\",\n \"By Lemmas REF , REF , and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert v_{N_1} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{3, N_3} \\\\Vert _{L^{5}_{T, x}}\\\\Vert z_{3, N_2} w_{N_4} \\\\Vert _{L^2_{T, x}}\\\\\\\\& \\\\lesssim T^{\\\\frac{1}{10}+}N_1^{-\\\\sigma } N_2^{1-2s-}N_3^{\\\\sigma - s}N_4^{-\\\\frac{1}{2}+}\\\\Vert v\\\\Vert _{X^\\\\sigma _T}\\\\Vert z_{3, N_2}\\\\Vert _{X^{2s-}_T}\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}\\\\Vert z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}^2\\\\\\\\& \\\\le T^{\\\\frac{1}{10}+}N_3^{\\\\frac{\\\\sigma }{2}- 2s+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}.$ This yields (REF ), provided that $0\\\\le \\\\sigma < 4s$ and $s < \\\\frac{1}{2}$ .\",\n \"$\\\\circ $ Subcase (G.3.d): $N_2\\\\gtrsim N_3^\\\\frac{1}{2} \\\\gg N_1$ .\",\n \"By Lemmas REF , REF , and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert v_{N_1} z_{3, N_3} \\\\Vert _{L^2_{T, x}}\\\\Vert z_{3, N_2} \\\\Vert _{L^{5}_{T, x}}\\\\Vert w_{N_4} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\lesssim T^{\\\\frac{1}{10}+}N_1^{1-\\\\sigma -} N_2^{-s}N_3^{\\\\sigma - 2s-\\\\frac{1}{2}+}\\\\Vert v\\\\Vert _{X^\\\\sigma _T}\\\\Vert \\\\langle \\\\nabla \\\\rangle ^s z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}\\\\Vert z_1\\\\Vert _{L^{15}_T L^{\\\\frac{15}{4}}_x}^2\\\\Vert z_{3, N_3} \\\\Vert _{X^{2s-}_T}\\\\\\\\& \\\\le T^{\\\\frac{1}{10}+}N_1^{\\\\frac{\\\\sigma }{2}- \\\\frac{5}{2}s+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}, \\\\Vert z_3\\\\Vert _{X^{2s-}_T}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}.$ This yields (REF ), provided that $ \\\\sigma \\\\le \\\\min (1, 5s-)$ and $s\\\\ge 0$ .\",\n \"Case (H): $v z_3 z_1$ case.\",\n \"$\\\\bullet $ Subcase (H.1): $N_1 \\\\sim N_{\\\\max }$ .\",\n \"In this case, we can proceed as in Subcase (D.1) by replacing $v_{N_2}$ with $z_{3, N_2}$ .\",\n \"When $\\\\max (N_2, N_3) \\\\ge N_1^\\\\frac{1}{10}$ , it suffices to note that $N_2^{-2s+}N_3^{-s}\\\\lesssim N_{\\\\max }^{0-},$ provided that $s> 0$ .\",\n \"When $\\\\max (N_2, N_3) \\\\ll N_1^\\\\frac{1}{10}$ , it suffices to note that $N_2^{1-2s-}N_3^{-s}N_4^{-\\\\frac{1}{2}+}\\\\lesssim N_{\\\\max }^{0-},$ provided that $s\\\\ge 0$ .\",\n \"$\\\\bullet $ Subcase (H.2): $N_3 \\\\sim N_{\\\\max } \\\\gg N_1$ .\",\n \"$\\\\circ $ Subsubcase (H.2.a): $N_1, N_2 \\\\ll N_3^\\\\frac{1}{2} $ .\",\n \"In this case, we have $N_3 \\\\sim N_4 \\\\sim N_{\\\\max }$ .\",\n \"Then, proceeding as in Subsubcase (F.3.a), it suffices to note that $N_1^{1- \\\\sigma -}N_2^{1-2s - } N_3^{\\\\sigma - s -\\\\frac{1}{2} +}N_4^{-\\\\frac{1}{2}+}\\\\lesssim N_3^{\\\\frac{\\\\sigma }{2} - 2s}\\\\lesssim N_{\\\\max }^{0-},$ provided that $ \\\\sigma \\\\le \\\\min (1, 4s-)$ and $ s < \\\\frac{1}{2}$ .\",\n \"$\\\\circ $ Subsubcase (H.2.b): $N_1, N_2\\\\gtrsim N_3^\\\\frac{1}{2} $ .\",\n \"In this case, we proceed with $L^\\\\frac{10}{3}_{T, x},L^\\\\frac{10}{3}_{T, x},L^{10}_{T, x}, L^\\\\frac{10}{3}_{T, x}$ -Hölder's inequality.\",\n \"It suffices to note that $N_1^{- \\\\sigma }N_2^{-2s+} N_3^{\\\\sigma - s}\\\\lesssim N_3^{\\\\frac{\\\\sigma }{2} - 2s+}\\\\lesssim N_{\\\\max }^{0-},$ provided that $0 \\\\le \\\\sigma < 4s$ .\",\n \"$\\\\circ $ Subsubcase (H.2.c): $N_1\\\\gtrsim N_3^\\\\frac{1}{2} \\\\gg N_2$ .\",\n \"In this case, we have $N_3 \\\\sim N_4 \\\\sim N_{\\\\max }$ .\",\n \"We can proceed as in Subsubcase (G.3.c) with $z_{3, N_3}$ replaced by $z_{1,N_3}$ (without applying Lemma REF ).\",\n \"The regularity restriction is precisely the same as in Subsubcase (G.3.c).\",\n \"$\\\\circ $ Subsubcase (H.2.d): $N_2\\\\gtrsim N_3^\\\\frac{1}{2} \\\\gg N_1$ .\",\n \"We first consider the case $N_3 \\\\sim N_4 \\\\sim N_{\\\\max }$ .\",\n \"By Lemmas REF and REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim N_3^\\\\sigma \\\\Vert v_{N_1} w_{N_4} \\\\Vert _{L^2_{T, x}}\\\\Vert z_{3, N_2} \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert z_{1, N_3} \\\\Vert _{L^{5}_{T, x}}\\\\\\\\& \\\\lesssim T^{0+}N_1^{1-\\\\sigma -} N_2^{-2s+}N_3^{\\\\sigma - s} N_4^{-\\\\frac{1}{2}+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}\\\\\\\\& \\\\le T^{0+}N_3^{\\\\frac{\\\\sigma }{2}- 2s+}C(\\\\Vert z_1\\\\Vert _{B^s(T)}, \\\\Vert z_{3}\\\\Vert _{X^{2s-}_T}) \\\\Vert v\\\\Vert _{X^\\\\sigma _T}.$ This yields (REF ), provided that $ \\\\sigma \\\\le \\\\min (1, 4s-)$ and $ s> 0$ .\",\n \"When $N_4\\\\ll N_3$ , we have $N_2 \\\\sim N_3 \\\\sim N_{\\\\max }$ .\",\n \"In this case, we can repeat the computation above with the roles of $z_{3, N_2}$ and $w_{N_4}$ switched.\",\n \"Noting that $N_1^{1-\\\\sigma -} N_2^{-2s-\\\\frac{1}{2}+}N_3^{\\\\sigma - s}\\\\lesssim N_3^{\\\\frac{\\\\sigma }{2}- 3s+} \\\\lesssim N_{\\\\max }^{0-},$ we obtain (REF ), provided that $ \\\\sigma \\\\le \\\\min (1, 6s-)$ .\",\n \"$\\\\bullet $ Subcase (H.3): $N_2 \\\\sim N_4 \\\\gg N_1, N_3$ .\",\n \"$\\\\circ $ Subsubcase (H.3.a): $N_1\\\\ll N_2^\\\\frac{1}{2}$ .\",\n \"In this case, we can proceed as in Subsubcase (H.2.d).\",\n \"It suffices to note that $N_1^{1-\\\\sigma -} N_2^{\\\\sigma -2s + }N_3^{-s} N_4^{-\\\\frac{1}{2} +}\\\\lesssim N_2^{\\\\frac{\\\\sigma }{2} -2 s+}\\\\lesssim N_{\\\\max }^{0-},$ provided that $\\\\sigma \\\\le \\\\min (1, 4s-)$ and $s \\\\ge 0$ .\",\n \"$\\\\circ $ Subsubcase (H.3.b): $N_1\\\\gtrsim N_2^\\\\frac{1}{2} $ .\",\n \"In this case, we proceed with $L^\\\\frac{10}{3}_{T, x},L^\\\\frac{10}{3}_{T, x},L^{10}_{T, x}, L^\\\\frac{10}{3}_{T, x}$ -Hölder's inequality.\",\n \"It suffices to note that $N_1^{- \\\\sigma }N_2^{\\\\sigma -2s+} N_3^{-s}\\\\lesssim N_2^{\\\\frac{\\\\sigma }{2} - 2s+}\\\\lesssim N_{\\\\max }^{0-},$ provided that $0 \\\\le \\\\sigma < 4s$ .\",\n \"Case ( I ): $v vv$ case.\",\n \"In this case, there is no need to perform the dyadic decomposition.\",\n \"By Hölder's inequality, Sobolev's inequality (REF ), and Lemma REF , we have $\\\\text{LHS of } (\\\\ref {XX1})& \\\\lesssim \\\\Vert v \\\\Vert _{L^{5}_{T, x}}^2\\\\Vert \\\\langle \\\\nabla \\\\rangle ^\\\\sigma v \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\Vert w \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\lesssim \\\\Vert \\\\langle \\\\nabla \\\\rangle ^\\\\frac{1}{2} v \\\\Vert _{L^{5}_{T} L^\\\\frac{30}{11}_x}^2\\\\Vert \\\\langle \\\\nabla \\\\rangle ^\\\\sigma v \\\\Vert _{L^\\\\frac{10}{3}_{T, x}}\\\\\\\\& \\\\lesssim \\\\Vert v\\\\Vert _{X^\\\\sigma _T}^3,$ provided that $ \\\\sigma \\\\ge \\\\frac{1}{2}$ .\",\n \"Putting all the cases including Cases (A) and (B) treated in Lemmas REF and REF , we conclude that (REF ) holds, provided that $ \\\\frac{1}{2} \\\\le \\\\sigma \\\\le 1$ and $\\\\frac{2}{5} \\\\sigma < s < \\\\frac{1}{2}$ .\",\n \"This completes the proof of Proposition REF .\"\n ],\n [\n \"Proof of Theorem \",\n \"In this section, we briefly discuss the proof of Theorem REF .\",\n \"Given $ \\\\frac{1}{6} < s < \\\\frac{1}{2}$ , let $k \\\\in \\\\mathbb {N}$ such that $\\\\frac{1}{2\\\\alpha _{k+1}} < s \\\\le \\\\frac{1}{2\\\\alpha _{k}}$ , where $\\\\alpha _k$ is defined in (REF ).\",\n \"Our main goal is to study the fixed point problem (REF ).\",\n \"Define $\\\\widetilde{\\\\Gamma }$ by $\\\\widetilde{\\\\Gamma }v(t) = -i \\\\int _0^t S(t - t^{\\\\prime })\\\\bigg \\\\lbrace \\\\mathcal {N}\\\\bigg (v +\\\\sum _{\\\\ell = 1}^{k} \\\\zeta _{2\\\\ell -1}\\\\bigg )- \\\\sum _{\\\\ell = 2}^{k}\\\\sum _{\\\\begin{array}{c}j_1 + j_2 + j_3 = 2\\\\ell - 1\\\\\\\\j_1, j_2, j_3 \\\\in \\\\lbrace 1, 2\\\\ell -3\\\\rbrace \\\\end{array}}\\\\zeta _{j_1} \\\\overline{\\\\zeta _{j_2}}\\\\zeta _{j_3}\\\\bigg \\\\rbrace (t^{\\\\prime }) dt^{\\\\prime },$ where $\\\\zeta _{2\\\\ell -1}$ , $\\\\ell = 1, \\\\dots , k$ , is as in (REF ).\",\n \"Theorem REF follows from a standard fixed point argument, once we prove the following proposition.\",\n \"Proposition 6.1 Given $ \\\\frac{1}{6} < s < \\\\frac{1}{2}$ , let $k \\\\in \\\\mathbb {N}$ such that $\\\\frac{1}{2\\\\alpha _{k+1}} < s < \\\\frac{1}{\\\\alpha _{k}}$ .\",\n \"Then, there exist $\\\\theta > 0$ , $C_1, C_2> 0$ , and an almost surely finite constant $R = R(\\\\omega ) >0$ such that $\\\\Vert \\\\widetilde{\\\\Gamma }v\\\\Vert _{X^\\\\frac{1}{2}([0, T])}& \\\\le C_1\\\\big (\\\\Vert v\\\\Vert _{X^\\\\frac{1}{2}([0, T])} ^3 + T^\\\\theta R(\\\\omega )\\\\big ),\\\\\\\\\\\\Vert \\\\widetilde{\\\\Gamma }v_1 - \\\\widetilde{\\\\Gamma }v_2 \\\\Vert _{X^\\\\frac{1}{2}([0, T])}& \\\\le C_2\\\\Big (\\\\sum _{j = 1}^2 \\\\Vert v_j\\\\Vert _{X^\\\\frac{1}{2}([0, T])} ^2+ T^\\\\theta R(\\\\omega )\\\\Big )\\\\Vert v_1 -v_2 \\\\Vert _{X^\\\\frac{1}{2}([0, T])},$ for all $v, v_1, v_2 \\\\in X^\\\\frac{1}{2}([0, T])$ and $0 < T\\\\le 1$ .\",\n \"On the one hand, the upper bound $\\\\frac{1}{\\\\alpha _k}$ on the range of $s$ in Proposition REF comes from the restriction in Proposition REF .\",\n \"On the other hand, given $\\\\frac{1}{6} < s < \\\\frac{1}{2}$ , we fix $k \\\\in \\\\mathbb {N}$ such that $\\\\frac{1}{2\\\\alpha _{k+1}} < s \\\\le \\\\frac{1}{2\\\\alpha _{k}}$ and apply Proposition REF .\",\n \"Namely, the upper bound on $s$ in Proposition REF does not cause any problem in proving Theorem REF .\",\n \"We only discuss the proof of (REF ) since () follows in a similar manner.\",\n \"As in the proof of Proposition REF , it suffices to perform a case-by-case analysis of expressions of the form: $\\\\bigg | \\\\int _0^T \\\\int _{\\\\mathbb {R}^3}\\\\langle \\\\nabla \\\\rangle ^\\\\frac{1}{2} ( w_1 w_2 w_3 )w dx dt\\\\bigg |,$ where $\\\\Vert w\\\\Vert _{Y^{0}_T} \\\\le 1$ and $w_j= v$ , $\\\\zeta _{2\\\\ell -1}$ , $\\\\ell = 1, \\\\dots , k$ , but not of the form $\\\\zeta _1 \\\\zeta _1\\\\zeta _{2\\\\ell -3}$ for $\\\\ell \\\\in \\\\lbrace 2, 3, \\\\dots , k\\\\rbrace $ .\",\n \"Here, we have dropped the complex conjugate sign on $w_2$ since it does not play any role in our analysis.\",\n \"More concretely, we need to consider the following cases: Table: NO_CAPTION where $j_1, j_2, j_3$ can take any value in $\\\\lbrace 3, 5, \\\\dots , 2k-1\\\\rbrace $ .\",\n \"As we see below, the worst interaction appears in Case (A) and all the other cases can be handled as in the proof of Proposition REF .\",\n \"We first point out that, in the proof of Proposition REF , the regularity restriction coming from Cases (B) - (I) (with $\\\\sigma = \\\\frac{1}{2}$ ) is $s > \\\\frac{1}{6}$ .\",\n \"Moreover, by comparing Proposition REF and Lemma REF with Proposition REF and Lemma REF , we see that the unbalanced higher order term $\\\\zeta _{2\\\\ell -1}$ for $\\\\ell \\\\in \\\\lbrace 2, 3, \\\\dots , k\\\\rbrace $ enjoys (at least) the same regularity properties as $z_3 = \\\\zeta _3$ both in terms of differentiability and space-time integrability.\",\n \"Therefore, we can simply apply the estimates in the proofs of Lemma REF (i) for Case (B) and Proposition REF for Cases (C) - (I) and conclude that the contributions from Cases (B) - (I) can be bounded, provided $s > \\\\frac{1}{6}$ .\",\n \"It remains to consider Case (A).\",\n \"In view of (REF ), the contribution from Case (A) is nothing but $\\\\zeta _{2k+1}$ .\",\n \"Hence, by applying Proposition REF with $\\\\sigma = \\\\frac{1}{2}$ , we conclude that the contribution in Case (A) can be estimated by $T^\\\\theta R(\\\\omega )$ , provided that $\\\\frac{1}{2\\\\alpha _{k+1}} < s < \\\\frac{1}{\\\\alpha _{k}}$ .\",\n \"This completes the proof of Proposition REF .\"\n ],\n [\n \"On the deterministic non-smoothing\\nof the Duhamel integral operator\",\n \"In this appendix, we show that there is no deterministic smoothing on the Duhamel integral operator $I$ defined in (REF ) when $s \\\\le \\\\frac{1}{2}$ .\",\n \"Lemma 1.1 Let $\\\\sigma > 3s-1$ .\",\n \"Then, the estimate (REF ) does not hold.\",\n \"In particular, there is no deterministic nonlinear smoothing when $s \\\\le \\\\frac{1}{2}$ .\",\n \"Given $N \\\\ge L \\\\ge \\\\ell \\\\gg 1$ , consider $u_j = S(t) \\\\phi _j$ , $j = 1, 2, 3$ , such that $\\\\widehat{\\\\phi }_j( \\\\xi ) \\\\sim \\\\mathbf {1}_{A_j}(\\\\xi )$ , where $A_1 = L e_1 + \\\\ell Q, \\\\qquad A_2 = \\\\ell Q, \\\\qquad \\\\text{and}\\\\qquad A_3 = N e_2 + \\\\ell Q.$ Then, we have $\\\\prod _{j = 1}^3 \\\\Vert u_j(0) \\\\Vert _{H^s}\\\\sim \\\\ell ^{\\\\frac{9}{2} + s} L^s N^s.$ Let $A = Le_1 + N e_2 + \\\\ell Q$ .\",\n \"By writing $\\\\mathbf {1}_{A} (\\\\xi )\\\\mathcal {F}_x[I(u_1, & u_2, u_3)](t, \\\\xi )\\\\\\\\&= -i \\\\mathbf {1}_{A} (\\\\xi ) e^{-it |\\\\xi |^2} \\\\int _0^t \\\\operatornamewithlimits{\\\\int }_{\\\\xi = \\\\xi _1 - \\\\xi _2 + \\\\xi _3}e^{i t^{\\\\prime } \\\\Phi (\\\\bar{\\\\xi })} \\\\prod _{j = 1}^3 \\\\mathbf {1}_{A_j} (\\\\xi _j) d\\\\xi _1 d\\\\xi _2 dt^{\\\\prime },$ we see that the non-trivial contribution to (REF ) comes from $\\\\xi _j \\\\in A_j$ , $j = 1, 2, 3$ .\",\n \"Moreover, in this case, $\\\\Phi (\\\\bar{\\\\xi })$ defined (REF ) satisfies $|\\\\Phi (\\\\bar{\\\\xi }) |\\\\lesssim N L$ .\",\n \"In particular, by choosing $t_* \\\\sim \\\\frac{1}{NL}$ such that $t_*|\\\\Phi (\\\\bar{\\\\xi }) |\\\\ll 1 $ , we have (REF ) for all $t^{\\\\prime } \\\\in [0, t_*]$ .\",\n \"Hence, by Lemma REF with (REF ), we obtain $\\\\Vert I(u_1, u_2, u_3) \\\\Vert _{X^\\\\sigma ([0, 1])} \\\\gtrsim \\\\Vert I(u_1, u_2, u_3)(t_*) \\\\Vert _{ H^\\\\sigma }\\\\gtrsim t_* \\\\ell ^\\\\frac{15}{2} N^\\\\sigma \\\\sim \\\\ell ^\\\\frac{15}{2} L^{-1}N^{\\\\sigma -1} .$ Therefore, it follows from (REF ) and (REF ) that by choosing $\\\\ell \\\\sim L \\\\sim N$ , we have $\\\\frac{\\\\Vert I(u_1, u_2, u_3) \\\\Vert _{X^\\\\sigma ([0, 1])}}{\\\\prod _{j=1}^3 \\\\Vert u_j(0) \\\\Vert _{H^s}}\\\\gtrsim \\\\ell ^{3-s}L^{-1-s} N^{\\\\sigma -1-s}\\\\sim N^{\\\\sigma - 3s+1} \\\\longrightarrow \\\\infty ,$ as $N \\\\rightarrow \\\\infty $ , provided that $\\\\sigma > 3s-1$ .\",\n \"This shows that the estimate (REF ) can not hold when $\\\\sigma > 3s-1$ .\",\n \"Define the quintilinear operator $I^{(2)}$ by $I^{(2)}(u_1, \\\\dots , u_5) (t):= -i \\\\sum _{\\\\ell = 1}^3\\\\int _0^t S(t - t^{\\\\prime })v_{j_1} \\\\overline{v_{j_2}}v_{j_3} (t^{\\\\prime }) dt^{\\\\prime },$ where $v_{j_i} = u_i$ for $i \\\\ne \\\\ell $ and $v_{j_\\\\ell } = I(u_\\\\ell , u_4, u_5)$ .\",\n \"This basically corresponds to the second order term appearing in the power series expansion for (REF ).\",\n \"In particular, we have that $z_5 = I^{(2)}(z_1, \\\\dots , z_1)$ .\",\n \"By a computation similar to the proof of Lemma REF , we can prove the following deterministic non-smoothing on the second order term.\",\n \"Let $\\\\sigma > 5s - 2$ .\",\n \"Then, there is no finite constant $C> 0$ such that $\\\\big \\\\Vert I^{(2)}( S(t) \\\\phi _1, \\\\dots , S(t) \\\\phi _5)\\\\big \\\\Vert _{X^\\\\sigma ([0, 1])}\\\\le C \\\\prod _{j = 1}^5 \\\\Vert \\\\phi _j \\\\Vert _{H^s}$ for all $\\\\phi _j \\\\in H^s(\\\\mathbb {R}^3)$ .\",\n \"In particular, this shows that when $s \\\\le \\\\frac{1}{2}$ , there is no deterministic smoothing on the second order term $I^{(2)}$ .\",\n \"A similar comment applies to the higher order terms.\",\n \"Acknowledgments This research was partially supported by Research in Groups at International Centre for Mathematical Sciences, Edinburgh, United Kingdom.\",\n \"Á.B.\",\n \"was partially supported by a grant from the Simons Foundation (No. 246024).\",\n \"T.O.\",\n \"was supported by the European Research Council (grant no.\",\n \"637995 “ProbDynDispEq”).\"\n ]\n]"},"id":{"kind":"string","value":"1709.01910"}}},{"rowIdx":793,"cells":{"text":{"kind":"list like","value":[["Some Sufficient Conditions for Finding a Nesting of the Normalized\n Matching Posets of Rank 3"],["Abstract Given a graded poset $P$, consider a chain decomposition $\\mathcal{C}$ of $P$.","If $|C_1|\\le |C_2|$ implies that the set of the ranks of elements in $C_1$ is a subset of the ranks of elements in $C_2$ for any chains $C_1,C_2\\in \\mathcal{C}$, then we say $\\mathcal{C}$ is a nested chain decomposition (or nesting, for short) of $P$, and $P$ is said to be nested.","In 1970s, Griggs conjectured that every normalized matching rank-unimodal poset is nested.","This conjecture is proved to be true only for all posets of rank 2 [W:05], some posets of rank 3 [HLS:09,ENSST:11], and the very special cases for higher ranks.","For general cases, it is still widely open.","In this paper, we provide some sufficient conditions on the rank numbers of posets of rank 3 to satisfies the Griggs's conjecuture."],["Introduction","We start with the necessary terminology of poset theory.","A poset $P=(P,\\le )$ is a set $P$ equipped with a partially order relation $\\le $ .","Through out the paper, all posets are finite.","Let $P$ be a poset.","We say a subposet $C$ of $P$ is a chain of length $\\ell $ if $C=\\lbrace x_i\\mid x_i< x_{i+1}\\mbox{ for }0\\le i \\le \\ell -1\\rbrace $ .","A chain decomposition $\\mathcal {C}$ of $P$ is a collection of disjoint chains of $P$ with $\\cup _{C\\in \\mathcal {C}}C=P$ .","We are looking for decompositions with as few number of chains as possible.","The most significant theorem in the literature was given by Dilworth [3].","Here an antichain is a subposet of $P$ such that neither $x\\le y $ nor $y\\le x$ holds for any $x\\ne y$ in $Q$ .","Theorem 1.1 [3] For a poset $P$ , the minimum number of chains in a chain decomposition is equal to the maximum size of an antichain of $P$ .","In the following, we study the chain decompositions of a special class of posets, which includes example such as Boolean lattices, linear lattices, and divisor lattices, etc.","A graded poset is a poset such that every maximal chain has the same length.","For a graded poset $P$ , we define the rank function $r:P\\longrightarrow \\mathbb {N}$ such that $r(x)=i$ , if there exactly $i$ elements $y< x$ in a maximal chain.","Moreover, an element $x$ is of rank $i$ if $r(x)=i$ and the rank of $P$ is $\\max _{x\\in P}r(x)$ .","The $i$ th level of a graded poset $P$ is the collection of all elements of rank $i$ , that is, $L_i=\\lbrace x\\mid r(x)=i, x\\in P\\rbrace $ .","By the definitions, every level is an antichain.","Therefore, $\\max _{i}|L_i|\\le |\\mathcal {C}|$ for every chain decomposition $\\mathcal {C}$ of $P$ .","For graded posets, we define a special chain decomposition: Definition 1.2 (nested chain decomposition) Let $\\mathcal {C}$ be a chain decomposition of a graded poset $P$ .","For any chains $C_i,C_j\\in \\mathcal {C}$ , if $|C_i|\\le |C_j|$ implies $\\lbrace r(x)\\mid x\\in C_i\\rbrace \\subseteq \\lbrace r(x)\\mid x\\in C_j\\rbrace $ , then $\\mathcal {C}$ is called a nested chain decomposition of $P$ .","We say $P$ is nested, or it has a nesting, if such a decomposition of $P$ exists.","Observe that a nesting $\\mathcal {C}$ is a chain decomposition with minimum number of chains.","Because from the inclusion relation, $\\cap _{C\\in \\mathcal {C}}\\lbrace r(x)\\mid x\\in C\\rbrace $ is not empty, and there exists some $m$ and a level $L_m$ such that $m\\in \\cap _{C\\in \\mathcal {C}}\\lbrace r(x)\\mid x\\in C\\rbrace $ .","Thus, every chain in $\\mathcal {C}$ contains an element of rank $m$ , and hence $|L_m|=|\\mathcal {C}|$ .","Since $\\max _i|L_i|\\le |\\mathcal {C}|=|L_m|\\le \\max _i|L_i|$ , we have $\\max _i|L_i|\\le |\\mathcal {C}|$ .","We refer the reader to see more properties of graded posets in [2], [4].","Anderson[1] and Griggs[6] independently gave the same sufficient condition for the existence of a nesting in the graded posets.","Let $r_i$ denote the cardinality of level $i$ .","The rank numbers or Whitney numbers of a graded poset of rank $n$ is the sequence $(r_0,r_1,\\ldots , r_n)$ .","If $r_i=r_{n-i}$ for all $0\\le i\\le n$ , then $P$ is said to be rank-symmetric.","Suppose there exists some $i$ such that $r_0\\le \\cdots \\le r_i\\ge \\cdots \\ge r_n$ , then $P$ is rank-unimodal.","For any levels $L_i$ and $L_j$ of $P$ , consider any subset $S$ of $L_i$ and denote the set $\\Gamma _j(S)=\\lbrace x\\in L_j\\mid x \\le y\\mbox{ or }y\\le x \\mbox{ for some }y\\in S\\rbrace $ .","If the inequality $\\frac{|S|}{r_i} \\le \\frac{|\\Gamma _j(S)|}{r_j}$ holds, then we say $P$ has the normalized matching property from $i$ to $j$ .","By simple calculation, one can see if $P$ has the normalized matching property from $i$ to $j$ , then it also has the property from $j$ to $i$ .","Moreover, if $P$ has the normalized matching property from $i$ to $j$ and $j$ to $k$ for $ir_1>r_0=r_3$ are nested.","For example, if a poset $P\\in (r_0,r_1,r_2,r_3)$ with $r_2>r_1>r_0>r_3$ , then we add $r_0-r_3$ ghosts to the third level of $P$ to get a new poset $P^{\\prime }\\in (r_0,r_1,r_2,r_0)$ .","Now suppose we already have a nesting of $P^{\\prime }$ .","We then remove the ghosts in all the longest chains to get a nesting of $P$ .","See [10] for the details of all the reduction methods.","With this assumption on the rank numbers, Hsu et al.","[10] and Escamilla et al.","[5] proved the following Theorem REF and Theorem REF , respectively.","Theorem 2.3 [10] Let $r_0, r_1$ and $r_2$ be positive integers with $r_0 r_{0}r_{1}$ ; (d) $r_{1}$ divides $r_{2}$ .","Then every $P\\in NM(r_0,r_1,r_2,r_0)$ is nested.","Theorem 2.4 [5] Let $r_0, r_1$ and $r_2$ be positive integers with $r_0 r_{0}r_{1}-r_{0}\\gcd (r_{1},r_{2})$ .","Then every $P\\in NM(r_0,r_1,r_2,r_0)$ is nested.","In [5], the authors also examined the posets of rank 3 with $r_2\\le 13$ .","Using Theorem REF , one can verify that if $r_0r_1>r_0=r_3$ are nested.\",\n \"For example, if a poset $P\\\\in (r_0,r_1,r_2,r_3)$ with $r_2>r_1>r_0>r_3$ , then we add $r_0-r_3$ ghosts to the third level of $P$ to get a new poset $P^{\\\\prime }\\\\in (r_0,r_1,r_2,r_0)$ .\",\n \"Now suppose we already have a nesting of $P^{\\\\prime }$ .\",\n \"We then remove the ghosts in all the longest chains to get a nesting of $P$ .\",\n \"See [10] for the details of all the reduction methods.\",\n \"With this assumption on the rank numbers, Hsu et al.\",\n \"[10] and Escamilla et al.\",\n \"[5] proved the following Theorem REF and Theorem REF , respectively.\",\n \"Theorem 2.3 [10] Let $r_0, r_1$ and $r_2$ be positive integers with $r_0 r_{0}r_{1}$ ; (d) $r_{1}$ divides $r_{2}$ .\",\n \"Then every $P\\\\in NM(r_0,r_1,r_2,r_0)$ is nested.\",\n \"Theorem 2.4 [5] Let $r_0, r_1$ and $r_2$ be positive integers with $r_0 r_{0}r_{1}-r_{0}\\\\gcd (r_{1},r_{2})$ .\",\n \"Then every $P\\\\in NM(r_0,r_1,r_2,r_0)$ is nested.\",\n \"In [5], the authors also examined the posets of rank 3 with $r_2\\\\le 13$ .\",\n \"Using Theorem REF , one can verify that if $r_0T_0$ .","Experimentally, providing a sudden increase in temperature of the bottom plate is challenging and we rely on a third thermostated bath that is set at $\\theta _{h}$ prior to the experiment.","Switching the fluid flow in the bottom plate to this thermostated bath leads to a progressive increase of the temperature of the bottom copper plate, $T_h$ , while the upper copper plate remains at $T_c=T_0=15\\,^{\\rm o}$ C. The temperature $T_h$ increases continuously and, when it reaches a critical value, we observe the sudden destabilization and resuspension of the granular bed."],["Phenomenology","A typical experiment is shown in Fig.","REF (a) where a time lapse shows the evolution of the granular bed.","Initially, the temperature profile is constant in the granular and the fluid layers, equal to $T_0=15^{\\rm o}{\\rm C}$ , and the resulting density profile of the liquid is constant in both regions and is equal to $\\rho _{l}^0$ .","The density of the polystyrene beads at $T_0$ , $\\rho _p^0$ , is larger that the density of the liquid, $\\rho _{l}^0$ , and therefore the granular bed is stable.","At time $t=0$ , the bottom copper plate is connected to the hot thermostated bath at the temperature $\\theta _{h}$ , and the temperature $T_h$ in the bottom plate then starts to increase while the temperature $T_c$ of the top plate remains constant as shown in Fig.","REF (b).","The progressive increase of the temperature of the hot source, $T_h$ , leads to the increase of the temperature in the bottom region of the cell and therefore a decrease of the liquid and particle densities in the granular bed.","Because of the thermal loss between the thermostated bath, whose temperature $\\theta _h$ is fixed, and the bottom copper plate, the final temperature $T_h^{f}$ is smaller than $\\theta _h$ .","Indeed, the thermostated bath is connected to the bottom copper plate through tubings, which, even insulated, lead to heat transfer with the ambient atmosphere and lead to a temperature $T_h$ smaller than $\\theta _h$ .","Nevertheless, the temperature probe measures $T_h$ during the entire duration of an experiment and we therefore refer to this temperature in the following.","During the first phase, no motion of the granular bed occurs (corresponding to region 1 in Fig.","REF ).","Then, at a temperature threshold $T_h^*$ , the granular bed starts to destabilize with the formation of a small corrugation that later grows in time (regions 2 and 3).","The transition from a flat granular bed to the rise of a particle-laden plume occurs suddenly over a time scale that is small compared to the duration of the entire experiment and the evolution of the temperature at the bottom of the cell.","We can therefore accurately estimate the time and the corresponding temperature $T_h$ at the bottom plate at which the granular bed becomes unstable, which in this example corresponds to $t=135\\,{\\rm s}$ and $T_h^* \\simeq 38^{\\rm o}{\\rm C}$ .","In the following, we rationalize quantitatively the threshold temperature $T_h^*$ and the destabilization time at which the granular bed starts to fluidize.","We systematically investigate the influence of the temperature of the hot source, $T_h$ , the initial density contrast between the fluid and the particles, $\\Delta \\rho ^0$ , and the thickness of the granular bed, $h$ ."],["Time-evolution of $T_h$","We observe that a threshold temperature $T_h^*$ needs to be reached to resuspend locally the granular bed.","Because our experimental approach involves a transient increase of the temperature of the bottom copper plate, we characterize the influence of this transient dynamic on the threshold temperature.","To do so, we consider a granular bed of fixed height $h=10\\,{\\rm mm}$ and a fixed initial density contrast $\\Delta \\rho ^0=\\rho _p^0-\\rho _l^0=2.6 \\,\\rm {kg\\,m^{-3}}$ .","We then impose different temperatures at the thermostated bath connected to the bottom hot source, $\\theta _h=[60,\\,65,\\,70,\\,75,\\,80]\\,^{\\rm o}{\\rm C}$ , keeping the temperature at the top of the cell constant and equal to $T_c=15\\,^{\\rm o}C$ during the entire experiment.","As mentioned previously, the thermal loss between the thermostated bath and the bottom copper plate leads to a temperature $T_h^{f}$ reached at the bottom of the cell significantly smaller than the temperature $\\theta _h$ of the thermostated bath.","Experimentally, we determine that $T_h^{f}$ is in the range $42\\,^{\\rm o}$ C to $64\\,^{\\rm o}{\\rm C}$ .","The time evolution of the temperature at the bottom plate, as recorded by the temperature probe, is shown in Fig.","REF for varying values of the temperature $\\theta _h$ of the hot thermalized bath.","The threshold temperature $T_h^*$ at which the granular bed starts to resuspend is also reported.","We observe that the temperature of the bottom plate increases quickly at the beginning and then the increase in temperature becomes slower.","In all the different situations considered here where $T_h^f > T_h^*$ , we observe the resuspension of the granular bed at sufficiently long time.","In addition, the threshold temperature remains approximatively constant, $T_h^*=41.6\\pm 0.8\\,^{\\rm o}{\\rm C}$ , which suggests that, for a constant thickness of granular layer sufficiently large, here $h=10$ mm, and given density contrast, the threshold temperature $T_h^*$ does not depend significantly on the dynamics of the system.","However, we are going to see in the following that the transient effects are required to describe the system when varying the granular bed thickness $h$ .","Although the threshold temperature for the destabilization of the granular bed does not seem to depend significantly on the imposed value of the temperature of the hot thermostated bath $\\theta _h$ , the time needed to reach the destabilization threshold, $T_h^*$ , increases sharply when $\\theta _h$ is decreased.","This observation is mainly explained by the thermodynamics of the system: the time needed to reach the threshold temperature increases when decreasing the temperature of the hot source, leading to a longer waiting time.","In addition, if the temperature of the bottom plate remains smaller than $T_h^*$ , no destabilization of the granular bed is observed at long time.","In the following, we consider a heat flux at the hot source that leads to a maximum steady value of the bottom plate of $T_h^f \\simeq 50\\,^{\\rm o}{\\rm C}$ ."],["Density contrast between the fluid and the particles $\\Delta \\rho ^0$","Another relevant parameter in the destabilization process is the density contrast between the fluid and the particles, $\\Delta \\rho ^0$ .","We thus vary the density of the fluid, $\\rho _l^0$ , by tuning the concentration of CaCl$_2$ salt so that $\\Delta \\rho ^0 = \\rho _p^0-\\rho _l^0$ ranges from $1.6\\,{\\rm kg\\,m^{-3}}$ to $4.1\\,{\\rm kg\\,m^{-3}}$ (see Appendix).","No resuspension of the granular bed has been observed with further increase of the density difference $\\Delta \\rho ^0$ within the range of temperatures that we have access to in our experiments.","For a constant bed thickness $h=10\\,{\\rm mm}$ and $\\theta _h = 65\\,^{\\rm o}{\\rm C}$ , we report in Fig.","REF that the resuspension of the granular bed occurs at different temperature thresholds.","Indeed, the smaller the density contrast $\\Delta \\rho ^0$ is, the sooner the resuspension takes place during the temperature ramp, which corresponds to smaller temperature threshold of the bottom plate $T_h^*$ .","The inset of Fig.","REF highlights that the temperature threshold $T_h^*$ increases linearly with the density contrast $\\Delta \\rho ^0$ in the range of parameters that we have considered.","We come back to this point in the next section when we rationalize our results with dimensionless parameters."],["Granular bed thickness $h$","Finally, we investigate the influence of the granular bed thickness on the temperature threshold and report the results in Fig.","REF .","We keep the density of the liquid and the particles constant, and using the same temperature of the hot thermostated bath $\\theta _h$ , we measure the temperature threshold at which the resuspension occurs as well as the time elapsed before the destabilization, $t_{des}$ .","We observe that the threshold temperature increases with the thickness of the granular layer following a trend close to $T_h^*\\propto h^{1/2}$ .","In addition, the time $t_{des}$ follows a slope $t_{des} \\propto h^2$ .","This scaling suggests that the mechanism responsible for the resuspension of the granular bed is diffusive.","Figure: Temperature threshold T h * T_h^* for increasing thickness of the granular bed, hh.","Red circles are the experimental results and the black dotted line shows T h * ∝h 1/2 T_h^* \\propto h^{1/2}.","Inset: Time elapsed, t des t_{des}, prior to the resuspension threshold for increasing thickness of the granular bed hh.","The temperature of the hot thermostated bath is set at θ h =65 o C\\theta _h=65^{\\rm o}\\rm {C} and Δρ 0 =1.98 kg m -3 \\Delta \\rho ^0=1.98\\,{\\rm kg\\,m^{-3}}.The experimental results highlight that an increase in the density contrast $\\Delta \\rho ^0$ or the granular bed thickness $h$ leads to a larger temperature threshold $T_h^*$ for destabilization.","In this section, we show that these results can be rationalized by considering buoyancy effects in the granular bed."],["Buoyancy number","Buoyancy-driven flows in a Hele-Shaw cell are commonly described using a modified Rayleigh number defined as [24]: $Ra=\\frac{g\\,\\beta \\,\\Delta T\\,H\\,b^2}{12\\,\\nu \\,D},$ where $g$ is the acceleration due to gravity, $b$ and $H$ the width and height of the cell, respectively, $\\beta $ is the thermal expansion coefficient, $\\nu $ is the kinematic viscosity and $D$ is the thermal diffusivity.","For large enough Rayleigh number, the system is unstable and natural convection is observed.","In the present study, the Rayleigh number lies in the range $[10^3,\\,10^6]$ , and we observe large recirculation cells in the top layer (liquid) even below the resuspension threshold.","We therefore need to consider an additional dimensionless number to explain the global destabilization of the granular bed.","Figure: Evolution of the critical buoyancy number B c B_c varying the temperature T h f T_h^f, hence ΔT\\Delta T (open black circles) or the density contrast Δρ 0 \\Delta \\rho ^0 (solid red circles) for a constant granular bed thickness h=10h=10 mm.Here, the presence of a granular bed at the bottom of the cell leads to a more complex situation as we now observe the possible destabilization of the granular layer beyond a temperature threshold.","This situation can be related to past studies that have considered a density stratification in two-layer Newtonian fluids when the two fluids have different densities and viscosities and there is no surface tension between the two fluid layers [25], [26], [27].","In this situation, it has been shown that the onset of convection can be either stationary or oscillatory depending on a dimensionless number, the buoyancy number $B$ , defined as the ratio of the stabilizing density anomaly to the destabilizing thermal density anomaly: $B=\\frac{\\rho _p^0-\\rho _l^0}{\\rho _l^0\\,\\beta \\,\\Delta T}.$ where $\\Delta T=T_h-T_c$ .","Depending on the buoyancy number $B$ , two main regimes have been identified.","Typically, when $B$ is large enough, i.e., $B$ larger than 0.5-1, convective flows develop above and/or below the flat interface to obtain the so-called stratified regime.","When the buoyancy number $B$ is small enough, typically smaller than 0.3-0.5, the interface can become unstable and spontaneous flow occurs in the whole tank, leading to the mixing of the two initial fluid layers in the bed and above.","Using this approach, we rescale the experimental results presented in Secs.","REF and REF , obtained for a given thickness $h$ of the immersed granular bed and varying the density contrast and the temperature of the thermostated bath.","The results reported in Fig.","REF highlight that the destabilization of the granular bed occurs for a constant value of the buoyancy number $B=B_c \\simeq 0.34$ at $h=10$ mm.","In Fig.","REF , we also report the evolution of the critical buoyancy number $B_c$ for varying $h$ while the other parameters are kept constant.","These results show that $B_c$ decreases when increasing the rescaled granular bed thickness $h/H$ as observed for two Newtonian fluids by Le Bars & Davaille [27].","However, such an approach does not explain the evolution of $t_{des}$ with the bed thickness.","Figure: Evolution of the critical buoyancy number B c B_c varying the thickness of the granular bed, hh.","The temperature of the thermostated bath is set at θ h =65 o C\\theta _h=65^{\\rm o}\\rm {C} and Δρ 0 =1.98 kg m -3 \\Delta \\rho ^0=1.98\\,{\\rm kg\\,m^{-3}}.","The black solid line is the best polynomial fit and is a guide for the eye.","In the light yellow region, the granular bed is stable whereas resuspension occurs in the dark blue region."],["Destabilization threshold","We consider a granular bed of compacity $\\phi $ made of polystyrene particles of density $\\rho _{p}^0$ .","Initially, the system is at temperature $T_c=15^{\\rm o}{\\rm C}$ .","We consider an infinitesimal element of length ${\\rm {d}}L$ and width $b$ , which is the gap of the Hele-Shaw cell.","The density of the granular bed averaged over the thickness of the granular bed is written $ \\langle \\rho (T) \\rangle _h = \\frac{1}{h}\\int _{0}^h\\,\\phi \\,\\rho _{p}(T)+({1-\\phi })\\,\\rho _l(T)\\,{\\rm d}z,$ where the temperature $T(z)$ depends on the vertical coordinate.","The evolution of the density with the temperature is given by : $ \\rho _l(z)=\\rho _l^0\\,\\left[1-\\alpha _l(T)\\,\\Delta T\\right],$ for the liquid and by $ \\rho _{p}(z)=\\rho _p^0\\,\\left[1-\\alpha _{p}(T)\\,\\Delta T\\right],$ for the polystyrene beads.","In these equation, $\\alpha _l$ and $\\alpha _p$ are the coefficient of thermal expansion of the liquid and polystyrene beads, respectively (see Apppendix).","We first consider the steady state where the temperature profile only depends on the vertical coordinate $z$ .","Experi- mentally, we observe, even below the resuspension threshold, large-scale flow in the fluid and we therefore assume that at the top of the granular bed the local temperature is comparable to the temperature of the cold source, $T_c$ .","Therefore, the temperature profile in the layer can be approximated as $T(z)=T_c+(T_h-T_c)\\,z/h$ .","In this configuration, the granular bed becomes unstable when the density averaged over the entire thickness h becomes smaller than the density of the fluid on top of it, assumed to be at the temperature $T_c=T_0$ , which means if ${\\langle \\rho (T) \\rangle _h}< {\\rho _l^0} $ .","We can solve this condition numerically for varying density contrast $\\Delta \\rho ^0$ as illustrated in Fig.","REF (a).","We observe that qualitatively the threshold temperature $T_h^*$ increases linearly with the density contrast $\\Delta \\rho ^0$ as observed experimentally.","This observation is consistent with the definition of the Buoyancy number $B$ .","For a constant granular bed thickness, the density contrast has a stabilizing effect, whereas an increase in temperature contributes to the destabilization of the granular bed through a local modification of the density of the granular bed.","We should also emphasize that this description is qualitative but not quantitative because of the experimental limitations.","Indeed, the increase in temperature at the bottom copper plate is not instantaneous and therefore $T_h$ increases as a function of time.","Nevertheless, this approach allows us to highlight the influence of the buoyancy number on the resuspension threshold of an immersed granular bed and the destabilizing effect of the temperature.","Figure: (a) Threshold temperature T h * T_h^* obtained in the steady state regime as a function of the density contrast Δρ 0 \\Delta \\rho ^0.","The thickness of the granular bed is h=10 mm h=10\\,{\\rm mm}.","The dotted line shows the linear scaling T h * ∝Δρ 0 T_h^* \\propto \\Delta \\rho ^0.","(b) Time elapsed, t des t_{des}, prior to the resuspension threshold obtained by solving the diffusion equation for increasing thickness hh of the granular bed.","The numerical parameters are T h =55 o CT_h=55^{\\rm o}{\\rm C}, Δρ=5 kg m -3 \\Delta \\rho = 5\\,{\\rm kg\\,m^{-3}}.","The dotted line is a slope t∝h 2 t\\propto h^2.The influence of the transient state is clearly observed when considering the influence of the granular bed thickness $h$ .","Indeed, if we consider the steady state only, the temperature threshold $T_h^*$ should not depend on $h$ .","However, our experimental results show that $T_h^*$ increases with $h$ (see Fig.","REF ).","Although in our experiments the time variation of the temperature of the hot source is intrinsically related to the thermal properties of the system, we can consider the effect of the time variation to explain the scaling of the time to destabilize the granular bed, $t_{des}$ .","We modified the model developed previously and now consider that, at $t<0$ , the temperature is equal to $T_c$ everywhere in the fluid and in the granular bed.","At time $t=0$ , the lower part at $z=0$ is suddenly put at $T=T_h$ .","In the experiment, this temperature is increasing when connecting the hot thermostated bath but here, for the sake of simplicity, we assume that the temperature of the bottom plate is reached instantaneously.","As a result, the time dependence of the temperature at the position $z$ is the solution of the classical one-dimensional diffusion problem $ T(z)=T_c+\\left(T_h-T_c\\right) \\left[1-{\\rm erf}\\left(\\frac{z}{2\\,\\sqrt{D\\,t}}\\right)\\right],$ with $D \\simeq 1.8 \\times 10^{-7}\\,{\\rm m^2.s^{-1}}$ , the effective diffusion coefficient in the granular bed.","We know that the granular bed becomes unstable when ${\\langle \\rho (T) \\rangle _h}< {\\rho _l^0}$ , and we can solve this condition numerically using equations (REF )-(REF ).","The corresponding results are reported in Fig.","REF (b): the scaling observed experimentally $t_{des} \\propto h^2$ is captured by the diffusion equation in the granular bed.","Therefore, transient effects appear to be important to explain the destabilization of an immersed granular bed.","The qualitative model presented here captures the key physical effects and provides scaling laws to describe the phenomenon.","To take into account the increase in temperature of the hot source a full numerical model will be needed."],["Conclusion","In this paper, we have explored experimentally the resus- pension of an immersed granular bed by a localized heat source.","The granular bed is made of particles that have a density slightly smaller than the density of the surrounding fluid.","Our experiments illustrate that, beyond a temperature threshold, the averaged density of the granular bed can become smaller than the density of the top fluid and produce an overturning instability as observed for two Newtonian fluid [25], [26], [27].","This flow results in the production of thermal plume and the dispersion of the granular particles in the entire container.","We have shown that this mechanism can be described through the buoyancy number $B={(\\rho _p^0-\\rho _l^0)}/({\\rho _l^0\\,\\beta \\,\\Delta T})$ .","The threshold value $B_c$ is observed to be dependent on the granular bed thickness in our experiments owing to the transient regime.","We rationalize our experimental findings with scaling argu- ments that capture the main features of this destabilization process.","Such resuspension of a granular bed could be important in geophysical and environmental processes in which localized heating can induce the transport of deposited particles and the later contamination of the environment.","The dynamics of particle-laden plumes is important to describe the subsequent dispersion of the particles.","The authors are grateful to P. Gondret and A. Davaille for fruitful discussions and to E. Dressaire for a careful reading of the manuscript.","We acknowledge J. Amarni, A. Aubertin, L. Auffray, C. Borget, and R. Pidoux for experimental help.","This work was supported by the French ANR (project Stabingram ANR 2010-BLAN-0927-01).","C.M.","also acknowledges support from the AAP “Attractivité Jeune chercheur” from Paris-Sud University.","*"],["Aqueous solution","The liquid used in our experiments is a mixture of distilled water and CaCl$_2$ (between 0% and 5.5% w/w).","We measured the density of different aqueous mixture using a densimeter (Anton Paar DMA 35) in a range of temperature between $15^{\\rm o}$ C and $30^{\\rm o}$ C as reported in Fig.","REF .","Figure: Density of aqueous mixture for different salt concentration: 4.85%4.85\\% w/w (blue circles), 4.9%4.9\\% w/w (cyan crosses), 4.95%4.95\\% w/w (red squares), 5.05%5.05\\% w/w (green diamonds) and 5.15%5.15\\% w/w (magenta crosses).","The dotted lines are the best fits from the equation () for varying ρ l 0 (c)\\rho ^0_{l}(c).The evolution of the density with the temperature and the salt concentration $\\rho _l(c,T)$ is fitted using the expression $ \\rho _l(c,T)=\\rho ^0_{l}(c)\\,\\left[1-\\alpha (T)\\,(T-T_0)\\right],$ where $\\rho ^0_{l}(c)$ is the density of the aqueous mixture having a concentration $c$ of CaCl$_2$ (w/w), taken at the temperature $T_0=20^{\\rm o}$ C. The coefficient of thermal expansion, $\\alpha (t)=a\\,T+b$ , is fitted from the experimental data ($a=9.6\\times 10^{-6}\\,(^{\\rm o}{\\rm C})^{-2}$ , $b=1.2\\times 10^{-4}\\,(^{\\rm o}{\\rm C})^{-1}$ and $T$ is expressed in $^{\\rm o}{\\rm C}$ )."],["Polystyrene beads","The coefficient of thermal expansion (volume) is taken equal to $\\alpha _{p}=1.9\\,\\times 10^{-4}\\,(^{\\rm o}{\\rm C})^{-1}$ [28], such that the density of the polystyrene beads can be expressed as $\\rho _{p}(T)=\\rho _{p}^0\\,\\left[1-\\alpha _{p}\\,(T-T_0)\\right],$ with $\\rho _{p}^0=1049 \\,{\\rm kg.m^{-3}}$ taken at $T_0= 20^{\\rm o}$ C."]],"string":"[\n [\n \"Resuspension threshold of a granular bed by localized heating\"\n ],\n [\n \"Abstract The resuspension and dispersion of particles occur in industrial fluid dynamic processes as well as environmental and geophysical situations.\",\n \"In this paper, we experimentally investigate the ability to fluidize a granular bed with a vertical gradient of temperature.\",\n \"Using laboratory experiments with a localized heat source, we observe a large entrainment of particles into the fluid volume beyond a threshold temperature.\",\n \"The buoyancy-driven fluidized bed then leads to the transport of solid particles through the generation of particle-laden plumes.\",\n \"We show that the destabilization process is driven by the thermal conductivity inside the granular bed and demonstrate that the threshold temperature depends on the thickness of the granular bed and the buoyancy number, i.e., the ratio of the stabilizing density contrast to the destabilizing thermal density contrast.\"\n ],\n [\n \"Introduction\",\n \"The transport of solid particles induced by shearing particulate beds with water or air flow occurs in many geophysical events, such as in river beds and landscape evolution, wind-blown sand, and dust emission [1], [2], but also in industrial processes: filtration systems or the food industry, for instance [3], [4], [5], [6], [7].\",\n \"Depending on the nature, geometry, and regime of the fluid flow, different types of particle movements can be observed.\",\n \"Conventionally, rolling and sliding motion, saltation, and suspension are distinguished depending on the Reynolds number.\",\n \"In other configurations, when the granular bed is subject to an ascending flow of liquid or gas, the normal stress can fluidize the particulate medium and maintain grains in suspension, which leads to particular properties and characteristics of the medium [8].\",\n \"This situation is observed for instance during the rise of air in an immersed granular bed [9], [10].\",\n \"In addition of these mechanical mechanisms of resuspen- sion, a heat source inside or below the sediment may destabi- lize a loose random packing of granular matter.\",\n \"Resuspension due to thermal effects is of great importance to understand, for example, volcanic ash clouds [11] or seafloor hydrothermal systems such as black smokers.\",\n \"Whereas the resuspension and fluidization of an immersed granular bed by fluid flows such as vortices [12], [13], [14], [15], impacting jets [16], [17], shear flows [18], [19], or gas crossing a liquid-saturated granular bed have been the focus of many studies, the ability of thermal convection to resuspend particles remains poorly understood.\",\n \"Indeed, in recent decades particular attention has been focused on the settling of particles, initially in suspension, in steady cellular convection but only few studies [20] have investigated the destabilization process of an initially loose randomly packed granular bed driven by thermal convection.\",\n \"Several scenarios are possible to induce the reentrainment of solid particles from the bottom: erosion by the bottom shear stress induced by convection current or fluidization by emergence of particle-laden plumes.\",\n \"To explore the mechanisms of the resuspension process by thermal convection, Solomatov et al.\",\n \"[20], [21] used a three-dimensional experimental setup with aqueous solution and polystyrene particles which are initially sedimented to form a loose random packing of granular matter.\",\n \"The bottom wall is uniformly heated from below to ensure the destabilization process of the sedimented particles.\",\n \"The authors defined a Shields number, Sh, which compares the ratio of the destabilizing hydrodynamic stress $\\\\tau $ exerted on a grain to its stabilizing apparent weight $\\\\Delta \\\\rho \\\\,g \\\\,d$ , where $\\\\Delta \\\\rho = \\\\rho _p-\\\\rho _l$ is the density difference between the grains and the ambient fluid, $g$ is the acceleration of the gravity and $d$ is the mean particle diameter.\",\n \"The resuspension of particles from the bottom is driven by the tangential buoyancy stress $\\\\tau = \\\\beta \\\\, \\\\rho _l \\\\,g \\\\,\\\\Delta T \\\\,\\\\delta _T$ (where $\\\\beta $ is the thermal expansion coefficient, $\\\\Delta T$ is the temperature difference between the bottom and the top of the cell and $\\\\delta _T$ is the thermal boundary layer thickness) at the top of the granular bed induced by convection current: particles roll and slide horizontally and can form dunes.\",\n \"At the crest of a dune, the tangential buoyancy stress is vertical and can lead to the resuspension of particles.\",\n \"The buoyancy Shields number, separating a regime without bed motion from a regime with particle entrainment is approximatively constant and of the order of 0.1 in the experiments.\",\n \"On the other hand, Martin and Nokes [22] used a similar setup but a different initial state where particles are in suspension.\",\n \"The reentrainment of particles occurred by the emergence of plumes from the bottom but the authors were unable to extract a criterion for reentrainment.\",\n \"The aim of the present work is to investigate the resus- pension of a granular bed of particles by fluidization from a localized heat source.\",\n \"This situation is different from the uniform heating used in previous studies [20] where the gran- ular bed was eroded over a long time by shear flows induced by the thermal effects and the particles were susceptible to rolling, sliding, and being carried away at the bed surface by saltation [21].\",\n \"Here, we demonstrate that at larger temperature difference the fluidization of the granular bed can also occur from the bottom and actively participate in the resuspension and reentrainment of particles into the bulk through the generation of particle-laden plumes.\",\n \"We focus on the scaling of this fluidization threshold with the relevant dimensionless parameters that describe this destabilization process.\"\n ],\n [\n \"Experimental methods\",\n \"We study the resuspension of spherical polystyrene particles (Dynoseeds purchased from MicroBeads) of diameter $2\\\\,a=250\\\\,\\\\pm 10 \\\\mu {\\\\rm m}$ and a density $\\\\rho _p^0=1.049 \\\\pm 0.003 \\\\,{\\\\rm g\\\\,cm^{-3}}$ [23] induced by a localized heating at the bottom of the granular bed.\",\n \"The setup used in the resuspension experiments is shown in Fig.\",\n \"REF and consists of a rectangular Poly(methyl methacrylate) tank of internal dimensions 20 cm in length, $b=1.2$ cm in width and $H=20$ cm in height.\",\n \"Given the aspect ratio of the tank, no significant motion of the fluid takes place in the short direction.\",\n \"The temperature is imposed at the top and the bottom of the tank through two copper plates whose temperatures are imposed by two circulating thermostated baths and measured by platinum thermocouples located inside the plates.\",\n \"The top copper plate is 20 cm wide and covers the tank, whereas the bottom copper plate is centered and has a width of 4 cm, resulting in a localized heating.\",\n \"Figure: Schematic of the experimentalsetup.The working fluid is composed of water with different concentrations of calcium chloride, CaCl$_2$ .\",\n \"The quantity of salt is varied to increase the density of the working fluid in the range $\\\\rho _l=[1,\\\\,1.049]\\\\,{\\\\rm g.cm^{-3}}$ (measured at 20$^{\\\\rm o}$ C with a densimeter Anton Paar DMA 35; see Appendix).\",\n \"In addition, a small amount of sodium dodecyl sulfate surfactant is added to the water to initially disperse the dry particles into water and avoid the trapping of air bubbles.\",\n \"Each experiment is performed with a new suspension of particles that is allowed to settle and form a loose randomly packed granular bed at the bottom of the tank.\",\n \"The resulting compacity of the granular bed, i.e., the solid volume fraction $\\\\phi =V_p/V_{tot}$ (where $V_p$ is the volume of the particles and $V_{tot}$ is the volume of the liquid and the particles) is equal to $\\\\phi =0.57 \\\\pm 0.01$ and its controlled thickness $h$ lies in the range 1 to $30\\\\,{\\\\rm mm}$ .\",\n \"A vertical laser sheet is set through the short side of the tank so that the fluid and the granular bed are clearly visible.\",\n \"The evolution of the granular bed is recorded using a Phantom MIRO M110 camera (resolution $1200\\\\times 800$ pixels) at 1 frame/s.\",\n \"At the beginning of each experiment, the thermostated baths are first allowed to reach their assigned temperature, leading to a top temperature $T_c$ and a bottom temperature in the copper plate, $T_h$ .\",\n \"Initially, we set $T_c=T_h=T_0=15^{\\\\rm o}$ C. We then wait for a sufficiently long time, typically 30 min, to ensure that the initial temperature in the system is homogeneous and equal to $T_0=15^{\\\\rm o}$ C. At time $t=0$ , we suddenly increase the temperature of the bottom copper plate to the reference temperature $T_{h}>T_0$ .\",\n \"Experimentally, providing a sudden increase in temperature of the bottom plate is challenging and we rely on a third thermostated bath that is set at $\\\\theta _{h}$ prior to the experiment.\",\n \"Switching the fluid flow in the bottom plate to this thermostated bath leads to a progressive increase of the temperature of the bottom copper plate, $T_h$ , while the upper copper plate remains at $T_c=T_0=15\\\\,^{\\\\rm o}$ C. The temperature $T_h$ increases continuously and, when it reaches a critical value, we observe the sudden destabilization and resuspension of the granular bed.\"\n ],\n [\n \"Phenomenology\",\n \"A typical experiment is shown in Fig.\",\n \"REF (a) where a time lapse shows the evolution of the granular bed.\",\n \"Initially, the temperature profile is constant in the granular and the fluid layers, equal to $T_0=15^{\\\\rm o}{\\\\rm C}$ , and the resulting density profile of the liquid is constant in both regions and is equal to $\\\\rho _{l}^0$ .\",\n \"The density of the polystyrene beads at $T_0$ , $\\\\rho _p^0$ , is larger that the density of the liquid, $\\\\rho _{l}^0$ , and therefore the granular bed is stable.\",\n \"At time $t=0$ , the bottom copper plate is connected to the hot thermostated bath at the temperature $\\\\theta _{h}$ , and the temperature $T_h$ in the bottom plate then starts to increase while the temperature $T_c$ of the top plate remains constant as shown in Fig.\",\n \"REF (b).\",\n \"The progressive increase of the temperature of the hot source, $T_h$ , leads to the increase of the temperature in the bottom region of the cell and therefore a decrease of the liquid and particle densities in the granular bed.\",\n \"Because of the thermal loss between the thermostated bath, whose temperature $\\\\theta _h$ is fixed, and the bottom copper plate, the final temperature $T_h^{f}$ is smaller than $\\\\theta _h$ .\",\n \"Indeed, the thermostated bath is connected to the bottom copper plate through tubings, which, even insulated, lead to heat transfer with the ambient atmosphere and lead to a temperature $T_h$ smaller than $\\\\theta _h$ .\",\n \"Nevertheless, the temperature probe measures $T_h$ during the entire duration of an experiment and we therefore refer to this temperature in the following.\",\n \"During the first phase, no motion of the granular bed occurs (corresponding to region 1 in Fig.\",\n \"REF ).\",\n \"Then, at a temperature threshold $T_h^*$ , the granular bed starts to destabilize with the formation of a small corrugation that later grows in time (regions 2 and 3).\",\n \"The transition from a flat granular bed to the rise of a particle-laden plume occurs suddenly over a time scale that is small compared to the duration of the entire experiment and the evolution of the temperature at the bottom of the cell.\",\n \"We can therefore accurately estimate the time and the corresponding temperature $T_h$ at the bottom plate at which the granular bed becomes unstable, which in this example corresponds to $t=135\\\\,{\\\\rm s}$ and $T_h^* \\\\simeq 38^{\\\\rm o}{\\\\rm C}$ .\",\n \"In the following, we rationalize quantitatively the threshold temperature $T_h^*$ and the destabilization time at which the granular bed starts to fluidize.\",\n \"We systematically investigate the influence of the temperature of the hot source, $T_h$ , the initial density contrast between the fluid and the particles, $\\\\Delta \\\\rho ^0$ , and the thickness of the granular bed, $h$ .\"\n ],\n [\n \"Time-evolution of $T_h$\",\n \"We observe that a threshold temperature $T_h^*$ needs to be reached to resuspend locally the granular bed.\",\n \"Because our experimental approach involves a transient increase of the temperature of the bottom copper plate, we characterize the influence of this transient dynamic on the threshold temperature.\",\n \"To do so, we consider a granular bed of fixed height $h=10\\\\,{\\\\rm mm}$ and a fixed initial density contrast $\\\\Delta \\\\rho ^0=\\\\rho _p^0-\\\\rho _l^0=2.6 \\\\,\\\\rm {kg\\\\,m^{-3}}$ .\",\n \"We then impose different temperatures at the thermostated bath connected to the bottom hot source, $\\\\theta _h=[60,\\\\,65,\\\\,70,\\\\,75,\\\\,80]\\\\,^{\\\\rm o}{\\\\rm C}$ , keeping the temperature at the top of the cell constant and equal to $T_c=15\\\\,^{\\\\rm o}C$ during the entire experiment.\",\n \"As mentioned previously, the thermal loss between the thermostated bath and the bottom copper plate leads to a temperature $T_h^{f}$ reached at the bottom of the cell significantly smaller than the temperature $\\\\theta _h$ of the thermostated bath.\",\n \"Experimentally, we determine that $T_h^{f}$ is in the range $42\\\\,^{\\\\rm o}$ C to $64\\\\,^{\\\\rm o}{\\\\rm C}$ .\",\n \"The time evolution of the temperature at the bottom plate, as recorded by the temperature probe, is shown in Fig.\",\n \"REF for varying values of the temperature $\\\\theta _h$ of the hot thermalized bath.\",\n \"The threshold temperature $T_h^*$ at which the granular bed starts to resuspend is also reported.\",\n \"We observe that the temperature of the bottom plate increases quickly at the beginning and then the increase in temperature becomes slower.\",\n \"In all the different situations considered here where $T_h^f > T_h^*$ , we observe the resuspension of the granular bed at sufficiently long time.\",\n \"In addition, the threshold temperature remains approximatively constant, $T_h^*=41.6\\\\pm 0.8\\\\,^{\\\\rm o}{\\\\rm C}$ , which suggests that, for a constant thickness of granular layer sufficiently large, here $h=10$ mm, and given density contrast, the threshold temperature $T_h^*$ does not depend significantly on the dynamics of the system.\",\n \"However, we are going to see in the following that the transient effects are required to describe the system when varying the granular bed thickness $h$ .\",\n \"Although the threshold temperature for the destabilization of the granular bed does not seem to depend significantly on the imposed value of the temperature of the hot thermostated bath $\\\\theta _h$ , the time needed to reach the destabilization threshold, $T_h^*$ , increases sharply when $\\\\theta _h$ is decreased.\",\n \"This observation is mainly explained by the thermodynamics of the system: the time needed to reach the threshold temperature increases when decreasing the temperature of the hot source, leading to a longer waiting time.\",\n \"In addition, if the temperature of the bottom plate remains smaller than $T_h^*$ , no destabilization of the granular bed is observed at long time.\",\n \"In the following, we consider a heat flux at the hot source that leads to a maximum steady value of the bottom plate of $T_h^f \\\\simeq 50\\\\,^{\\\\rm o}{\\\\rm C}$ .\"\n ],\n [\n \"Density contrast between the fluid and the particles $\\\\Delta \\\\rho ^0$\",\n \"Another relevant parameter in the destabilization process is the density contrast between the fluid and the particles, $\\\\Delta \\\\rho ^0$ .\",\n \"We thus vary the density of the fluid, $\\\\rho _l^0$ , by tuning the concentration of CaCl$_2$ salt so that $\\\\Delta \\\\rho ^0 = \\\\rho _p^0-\\\\rho _l^0$ ranges from $1.6\\\\,{\\\\rm kg\\\\,m^{-3}}$ to $4.1\\\\,{\\\\rm kg\\\\,m^{-3}}$ (see Appendix).\",\n \"No resuspension of the granular bed has been observed with further increase of the density difference $\\\\Delta \\\\rho ^0$ within the range of temperatures that we have access to in our experiments.\",\n \"For a constant bed thickness $h=10\\\\,{\\\\rm mm}$ and $\\\\theta _h = 65\\\\,^{\\\\rm o}{\\\\rm C}$ , we report in Fig.\",\n \"REF that the resuspension of the granular bed occurs at different temperature thresholds.\",\n \"Indeed, the smaller the density contrast $\\\\Delta \\\\rho ^0$ is, the sooner the resuspension takes place during the temperature ramp, which corresponds to smaller temperature threshold of the bottom plate $T_h^*$ .\",\n \"The inset of Fig.\",\n \"REF highlights that the temperature threshold $T_h^*$ increases linearly with the density contrast $\\\\Delta \\\\rho ^0$ in the range of parameters that we have considered.\",\n \"We come back to this point in the next section when we rationalize our results with dimensionless parameters.\"\n ],\n [\n \"Granular bed thickness $h$\",\n \"Finally, we investigate the influence of the granular bed thickness on the temperature threshold and report the results in Fig.\",\n \"REF .\",\n \"We keep the density of the liquid and the particles constant, and using the same temperature of the hot thermostated bath $\\\\theta _h$ , we measure the temperature threshold at which the resuspension occurs as well as the time elapsed before the destabilization, $t_{des}$ .\",\n \"We observe that the threshold temperature increases with the thickness of the granular layer following a trend close to $T_h^*\\\\propto h^{1/2}$ .\",\n \"In addition, the time $t_{des}$ follows a slope $t_{des} \\\\propto h^2$ .\",\n \"This scaling suggests that the mechanism responsible for the resuspension of the granular bed is diffusive.\",\n \"Figure: Temperature threshold T h * T_h^* for increasing thickness of the granular bed, hh.\",\n \"Red circles are the experimental results and the black dotted line shows T h * ∝h 1/2 T_h^* \\\\propto h^{1/2}.\",\n \"Inset: Time elapsed, t des t_{des}, prior to the resuspension threshold for increasing thickness of the granular bed hh.\",\n \"The temperature of the hot thermostated bath is set at θ h =65 o C\\\\theta _h=65^{\\\\rm o}\\\\rm {C} and Δρ 0 =1.98 kg m -3 \\\\Delta \\\\rho ^0=1.98\\\\,{\\\\rm kg\\\\,m^{-3}}.The experimental results highlight that an increase in the density contrast $\\\\Delta \\\\rho ^0$ or the granular bed thickness $h$ leads to a larger temperature threshold $T_h^*$ for destabilization.\",\n \"In this section, we show that these results can be rationalized by considering buoyancy effects in the granular bed.\"\n ],\n [\n \"Buoyancy number\",\n \"Buoyancy-driven flows in a Hele-Shaw cell are commonly described using a modified Rayleigh number defined as [24]: $Ra=\\\\frac{g\\\\,\\\\beta \\\\,\\\\Delta T\\\\,H\\\\,b^2}{12\\\\,\\\\nu \\\\,D},$ where $g$ is the acceleration due to gravity, $b$ and $H$ the width and height of the cell, respectively, $\\\\beta $ is the thermal expansion coefficient, $\\\\nu $ is the kinematic viscosity and $D$ is the thermal diffusivity.\",\n \"For large enough Rayleigh number, the system is unstable and natural convection is observed.\",\n \"In the present study, the Rayleigh number lies in the range $[10^3,\\\\,10^6]$ , and we observe large recirculation cells in the top layer (liquid) even below the resuspension threshold.\",\n \"We therefore need to consider an additional dimensionless number to explain the global destabilization of the granular bed.\",\n \"Figure: Evolution of the critical buoyancy number B c B_c varying the temperature T h f T_h^f, hence ΔT\\\\Delta T (open black circles) or the density contrast Δρ 0 \\\\Delta \\\\rho ^0 (solid red circles) for a constant granular bed thickness h=10h=10 mm.Here, the presence of a granular bed at the bottom of the cell leads to a more complex situation as we now observe the possible destabilization of the granular layer beyond a temperature threshold.\",\n \"This situation can be related to past studies that have considered a density stratification in two-layer Newtonian fluids when the two fluids have different densities and viscosities and there is no surface tension between the two fluid layers [25], [26], [27].\",\n \"In this situation, it has been shown that the onset of convection can be either stationary or oscillatory depending on a dimensionless number, the buoyancy number $B$ , defined as the ratio of the stabilizing density anomaly to the destabilizing thermal density anomaly: $B=\\\\frac{\\\\rho _p^0-\\\\rho _l^0}{\\\\rho _l^0\\\\,\\\\beta \\\\,\\\\Delta T}.$ where $\\\\Delta T=T_h-T_c$ .\",\n \"Depending on the buoyancy number $B$ , two main regimes have been identified.\",\n \"Typically, when $B$ is large enough, i.e., $B$ larger than 0.5-1, convective flows develop above and/or below the flat interface to obtain the so-called stratified regime.\",\n \"When the buoyancy number $B$ is small enough, typically smaller than 0.3-0.5, the interface can become unstable and spontaneous flow occurs in the whole tank, leading to the mixing of the two initial fluid layers in the bed and above.\",\n \"Using this approach, we rescale the experimental results presented in Secs.\",\n \"REF and REF , obtained for a given thickness $h$ of the immersed granular bed and varying the density contrast and the temperature of the thermostated bath.\",\n \"The results reported in Fig.\",\n \"REF highlight that the destabilization of the granular bed occurs for a constant value of the buoyancy number $B=B_c \\\\simeq 0.34$ at $h=10$ mm.\",\n \"In Fig.\",\n \"REF , we also report the evolution of the critical buoyancy number $B_c$ for varying $h$ while the other parameters are kept constant.\",\n \"These results show that $B_c$ decreases when increasing the rescaled granular bed thickness $h/H$ as observed for two Newtonian fluids by Le Bars & Davaille [27].\",\n \"However, such an approach does not explain the evolution of $t_{des}$ with the bed thickness.\",\n \"Figure: Evolution of the critical buoyancy number B c B_c varying the thickness of the granular bed, hh.\",\n \"The temperature of the thermostated bath is set at θ h =65 o C\\\\theta _h=65^{\\\\rm o}\\\\rm {C} and Δρ 0 =1.98 kg m -3 \\\\Delta \\\\rho ^0=1.98\\\\,{\\\\rm kg\\\\,m^{-3}}.\",\n \"The black solid line is the best polynomial fit and is a guide for the eye.\",\n \"In the light yellow region, the granular bed is stable whereas resuspension occurs in the dark blue region.\"\n ],\n [\n \"Destabilization threshold\",\n \"We consider a granular bed of compacity $\\\\phi $ made of polystyrene particles of density $\\\\rho _{p}^0$ .\",\n \"Initially, the system is at temperature $T_c=15^{\\\\rm o}{\\\\rm C}$ .\",\n \"We consider an infinitesimal element of length ${\\\\rm {d}}L$ and width $b$ , which is the gap of the Hele-Shaw cell.\",\n \"The density of the granular bed averaged over the thickness of the granular bed is written $ \\\\langle \\\\rho (T) \\\\rangle _h = \\\\frac{1}{h}\\\\int _{0}^h\\\\,\\\\phi \\\\,\\\\rho _{p}(T)+({1-\\\\phi })\\\\,\\\\rho _l(T)\\\\,{\\\\rm d}z,$ where the temperature $T(z)$ depends on the vertical coordinate.\",\n \"The evolution of the density with the temperature is given by : $ \\\\rho _l(z)=\\\\rho _l^0\\\\,\\\\left[1-\\\\alpha _l(T)\\\\,\\\\Delta T\\\\right],$ for the liquid and by $ \\\\rho _{p}(z)=\\\\rho _p^0\\\\,\\\\left[1-\\\\alpha _{p}(T)\\\\,\\\\Delta T\\\\right],$ for the polystyrene beads.\",\n \"In these equation, $\\\\alpha _l$ and $\\\\alpha _p$ are the coefficient of thermal expansion of the liquid and polystyrene beads, respectively (see Apppendix).\",\n \"We first consider the steady state where the temperature profile only depends on the vertical coordinate $z$ .\",\n \"Experi- mentally, we observe, even below the resuspension threshold, large-scale flow in the fluid and we therefore assume that at the top of the granular bed the local temperature is comparable to the temperature of the cold source, $T_c$ .\",\n \"Therefore, the temperature profile in the layer can be approximated as $T(z)=T_c+(T_h-T_c)\\\\,z/h$ .\",\n \"In this configuration, the granular bed becomes unstable when the density averaged over the entire thickness h becomes smaller than the density of the fluid on top of it, assumed to be at the temperature $T_c=T_0$ , which means if ${\\\\langle \\\\rho (T) \\\\rangle _h}< {\\\\rho _l^0} $ .\",\n \"We can solve this condition numerically for varying density contrast $\\\\Delta \\\\rho ^0$ as illustrated in Fig.\",\n \"REF (a).\",\n \"We observe that qualitatively the threshold temperature $T_h^*$ increases linearly with the density contrast $\\\\Delta \\\\rho ^0$ as observed experimentally.\",\n \"This observation is consistent with the definition of the Buoyancy number $B$ .\",\n \"For a constant granular bed thickness, the density contrast has a stabilizing effect, whereas an increase in temperature contributes to the destabilization of the granular bed through a local modification of the density of the granular bed.\",\n \"We should also emphasize that this description is qualitative but not quantitative because of the experimental limitations.\",\n \"Indeed, the increase in temperature at the bottom copper plate is not instantaneous and therefore $T_h$ increases as a function of time.\",\n \"Nevertheless, this approach allows us to highlight the influence of the buoyancy number on the resuspension threshold of an immersed granular bed and the destabilizing effect of the temperature.\",\n \"Figure: (a) Threshold temperature T h * T_h^* obtained in the steady state regime as a function of the density contrast Δρ 0 \\\\Delta \\\\rho ^0.\",\n \"The thickness of the granular bed is h=10 mm h=10\\\\,{\\\\rm mm}.\",\n \"The dotted line shows the linear scaling T h * ∝Δρ 0 T_h^* \\\\propto \\\\Delta \\\\rho ^0.\",\n \"(b) Time elapsed, t des t_{des}, prior to the resuspension threshold obtained by solving the diffusion equation for increasing thickness hh of the granular bed.\",\n \"The numerical parameters are T h =55 o CT_h=55^{\\\\rm o}{\\\\rm C}, Δρ=5 kg m -3 \\\\Delta \\\\rho = 5\\\\,{\\\\rm kg\\\\,m^{-3}}.\",\n \"The dotted line is a slope t∝h 2 t\\\\propto h^2.The influence of the transient state is clearly observed when considering the influence of the granular bed thickness $h$ .\",\n \"Indeed, if we consider the steady state only, the temperature threshold $T_h^*$ should not depend on $h$ .\",\n \"However, our experimental results show that $T_h^*$ increases with $h$ (see Fig.\",\n \"REF ).\",\n \"Although in our experiments the time variation of the temperature of the hot source is intrinsically related to the thermal properties of the system, we can consider the effect of the time variation to explain the scaling of the time to destabilize the granular bed, $t_{des}$ .\",\n \"We modified the model developed previously and now consider that, at $t<0$ , the temperature is equal to $T_c$ everywhere in the fluid and in the granular bed.\",\n \"At time $t=0$ , the lower part at $z=0$ is suddenly put at $T=T_h$ .\",\n \"In the experiment, this temperature is increasing when connecting the hot thermostated bath but here, for the sake of simplicity, we assume that the temperature of the bottom plate is reached instantaneously.\",\n \"As a result, the time dependence of the temperature at the position $z$ is the solution of the classical one-dimensional diffusion problem $ T(z)=T_c+\\\\left(T_h-T_c\\\\right) \\\\left[1-{\\\\rm erf}\\\\left(\\\\frac{z}{2\\\\,\\\\sqrt{D\\\\,t}}\\\\right)\\\\right],$ with $D \\\\simeq 1.8 \\\\times 10^{-7}\\\\,{\\\\rm m^2.s^{-1}}$ , the effective diffusion coefficient in the granular bed.\",\n \"We know that the granular bed becomes unstable when ${\\\\langle \\\\rho (T) \\\\rangle _h}< {\\\\rho _l^0}$ , and we can solve this condition numerically using equations (REF )-(REF ).\",\n \"The corresponding results are reported in Fig.\",\n \"REF (b): the scaling observed experimentally $t_{des} \\\\propto h^2$ is captured by the diffusion equation in the granular bed.\",\n \"Therefore, transient effects appear to be important to explain the destabilization of an immersed granular bed.\",\n \"The qualitative model presented here captures the key physical effects and provides scaling laws to describe the phenomenon.\",\n \"To take into account the increase in temperature of the hot source a full numerical model will be needed.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we have explored experimentally the resus- pension of an immersed granular bed by a localized heat source.\",\n \"The granular bed is made of particles that have a density slightly smaller than the density of the surrounding fluid.\",\n \"Our experiments illustrate that, beyond a temperature threshold, the averaged density of the granular bed can become smaller than the density of the top fluid and produce an overturning instability as observed for two Newtonian fluid [25], [26], [27].\",\n \"This flow results in the production of thermal plume and the dispersion of the granular particles in the entire container.\",\n \"We have shown that this mechanism can be described through the buoyancy number $B={(\\\\rho _p^0-\\\\rho _l^0)}/({\\\\rho _l^0\\\\,\\\\beta \\\\,\\\\Delta T})$ .\",\n \"The threshold value $B_c$ is observed to be dependent on the granular bed thickness in our experiments owing to the transient regime.\",\n \"We rationalize our experimental findings with scaling argu- ments that capture the main features of this destabilization process.\",\n \"Such resuspension of a granular bed could be important in geophysical and environmental processes in which localized heating can induce the transport of deposited particles and the later contamination of the environment.\",\n \"The dynamics of particle-laden plumes is important to describe the subsequent dispersion of the particles.\",\n \"The authors are grateful to P. Gondret and A. Davaille for fruitful discussions and to E. Dressaire for a careful reading of the manuscript.\",\n \"We acknowledge J. Amarni, A. Aubertin, L. Auffray, C. Borget, and R. Pidoux for experimental help.\",\n \"This work was supported by the French ANR (project Stabingram ANR 2010-BLAN-0927-01).\",\n \"C.M.\",\n \"also acknowledges support from the AAP “Attractivité Jeune chercheur” from Paris-Sud University.\",\n \"*\"\n ],\n [\n \"Aqueous solution\",\n \"The liquid used in our experiments is a mixture of distilled water and CaCl$_2$ (between 0% and 5.5% w/w).\",\n \"We measured the density of different aqueous mixture using a densimeter (Anton Paar DMA 35) in a range of temperature between $15^{\\\\rm o}$ C and $30^{\\\\rm o}$ C as reported in Fig.\",\n \"REF .\",\n \"Figure: Density of aqueous mixture for different salt concentration: 4.85%4.85\\\\% w/w (blue circles), 4.9%4.9\\\\% w/w (cyan crosses), 4.95%4.95\\\\% w/w (red squares), 5.05%5.05\\\\% w/w (green diamonds) and 5.15%5.15\\\\% w/w (magenta crosses).\",\n \"The dotted lines are the best fits from the equation () for varying ρ l 0 (c)\\\\rho ^0_{l}(c).The evolution of the density with the temperature and the salt concentration $\\\\rho _l(c,T)$ is fitted using the expression $ \\\\rho _l(c,T)=\\\\rho ^0_{l}(c)\\\\,\\\\left[1-\\\\alpha (T)\\\\,(T-T_0)\\\\right],$ where $\\\\rho ^0_{l}(c)$ is the density of the aqueous mixture having a concentration $c$ of CaCl$_2$ (w/w), taken at the temperature $T_0=20^{\\\\rm o}$ C. The coefficient of thermal expansion, $\\\\alpha (t)=a\\\\,T+b$ , is fitted from the experimental data ($a=9.6\\\\times 10^{-6}\\\\,(^{\\\\rm o}{\\\\rm C})^{-2}$ , $b=1.2\\\\times 10^{-4}\\\\,(^{\\\\rm o}{\\\\rm C})^{-1}$ and $T$ is expressed in $^{\\\\rm o}{\\\\rm C}$ ).\"\n ],\n [\n \"Polystyrene beads\",\n \"The coefficient of thermal expansion (volume) is taken equal to $\\\\alpha _{p}=1.9\\\\,\\\\times 10^{-4}\\\\,(^{\\\\rm o}{\\\\rm C})^{-1}$ [28], such that the density of the polystyrene beads can be expressed as $\\\\rho _{p}(T)=\\\\rho _{p}^0\\\\,\\\\left[1-\\\\alpha _{p}\\\\,(T-T_0)\\\\right],$ with $\\\\rho _{p}^0=1049 \\\\,{\\\\rm kg.m^{-3}}$ taken at $T_0= 20^{\\\\rm o}$ C.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01733"}}},{"rowIdx":795,"cells":{"text":{"kind":"list like","value":[["Counting non-commensurable hyperbolic manifolds and a bound on\n homological torsion"],["Abstract We prove that the cardinality of the torsion subgroups in homology of a closed hyperbolic manifold of any dimension can be bounded by a doubly exponential function of its diameter.","It would follow from a conjecture by Bergeron and Venkatesh that the order of growth in our bound is sharp.","We also determine how the number of non-commensurable closed hyperbolic manifolds of dimension at least 3 and bounded diameter grows.","The lower bound implies that the fraction of arithmetic manifolds tends to zero as the diameter goes up."],["Introduction","Recently there has been a lot of progress on understanding the relation between the volume of a negatively curved manifold and its toplogical complexity.","In this note, we will instead consider the relation between complexity and diameter.","We will restrict to closed hyperbolic (constant sectional curvature $-1$ ) manifolds.","The main upshot of considering the diameter instead of the volume is that we obtain bounds in dimension 3."],["New results","Recall that two manifolds are called commensurable if they have a common finite cover.","The diameter of a commensurability class of manifolds is the minimal diameter realized by a manifold in that class.","Given $d\\in \\mathbb {R}_+$ and $n\\ge 3$ , let $NC^{\\operatorname{diam}}_n(d)$ denote the number of commensurability classes of closed hyperbolic $n$ -manifolds of diameter $\\le d$ .","We will prove Theorem 1 For all $n \\ge 3$ there exist $00$ so that $\\log \\log \\left(\\left|H_i(M,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) \\le C\\cdot \\operatorname{diam}(M) $ for all $i=0,\\ldots ,n$ for any closed hyperbolic $n$ -manifold $M$ .","Let us first note that for $n=2$ our theorem is automatic, since all torsion subgroups in the homology of a closed hyperbolic surface are trivial.","Moreover, in dimension at least 4 results by Bader, Gelander and Sauer [7] (see also Equation (REF )) together with a comparison between volume and diameter (Lemma REF ) also prove Theorem REF .","However, it is known that in dimension 3, this method cannot work.","We also note that to prove that our theorem is sharp, the most likely examples would be sequences hyperbolic manifolds with exponentially growing torsion and logarithmically growing diameter.","Examples of such sequences are abelian covers [37], [32], [1], Liu's construction [25] and random Heegaard splittings [24], [1].","However, these manifolds are not known to have small diameters and in some cases are even known not to have small diameters.","On the other hand, certain sequences of covers of arithmetic manifolds are conjectured to have exponential torsion growth by Bergeron and Venkatesh [14].","It is known that the corresponding lattice has Property $(\\tau )$ with respect to this sequence of covers [38], from which it follows that their diameter is small by a result of Brooks [11] (see Sections REF and REF ).","So assuming the conjecture by Bergeron and Venkatesh, these manifolds would saturate the bound in Theorem REF up to a multiplicative constant.","The proof of Theorem REF consists of two steps.","First we use Young's method from [41] to build a simplicial complex that models our manifold and has a bounded number of cells of any dimension.","We then use a lemma due to Bader, Gelander and Sauer [7] (based on a lemma of Gabber) that bounds the homological torsion in terms of the number of cells (Lemma REF ) to derive our bound."],["Volume and complexity","The main motivation for our work is formed by results that bound the homological complexity of a complete negatively curved $n$ -manifold $M$ in terms of its volume.","A classical result due to Gromov and worked out by Ballmann, Gromov and Schröder [21], [6], states that there exists a constant $C >0$ , depending only on $n$ , so that the Betti numbers $b_i(M)$ satisfy $b_i(M) \\le C \\cdot \\operatorname{vol}(M),$ for $i=0,\\ldots ,n$ , where $\\operatorname{vol}(M)$ denotes the volume of $M$ .","More recently, Bader, Gelander and Sauer [7] have shown that, when the dimension $n$ is at least 4, the cardinality of the torsion subgroups $H_i(M,\\mathbb {Z})_{\\mathrm {tors}}$ in homology can also be bounded in terms of the volume.","They show that for all $n\\ge 4$ , there exists a constant $C>0$ , depending only on $n$ , so that $\\log \\left(\\left|H_i(M,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) \\le C\\cdot \\operatorname{vol}(M).$"],["Counting manifolds by volume","We will again restrict ourselves to closed hyperbolic manifolds.","A classical result due to Wang [39] states that in dimension $n\\ge 4$ the number of closed hyperbolic manifolds of volume $\\le v$ is finite for any $v\\in \\mathbb {R}_+$ .","Let $N^{\\operatorname{vol}}_n(v)$ denote the number such manifolds.","The first bounds on the growth of $N^{\\operatorname{vol}}_n(v)$ are due to Gromov [20].","Burger, Gelander, Lubotzky and Mozes [4] showed that for all $n\\ge 4$ there exist $00$ is a constant depending only on $n$ (see Lemma REF ).","Moreover, because there are hyperbolic surfaces with linearly growing genus and logarithmically growing diameter (for instance random surfaces [8], [28]), a bound of this generality is necessarily exponential in diameter."],["Acknowledgement","The author's research was supported by the ERC Advanced Grant “Moduli”.","The author also thanks the organizers of the Borel Seminar 2017, during which the research for this article was done.","Finally, the author thanks Filippo Cerocchi, Jean Raimbault and Roman Sauer for useful conversations."],["Background material","In what follows, $n$ will be a natural number, $M$ a closed oriented hyperbolic $n$ -manifold.","We will use $\\operatorname{vol}(M)$ , $\\operatorname{diam}(M)$ , $\\operatorname{inj}(M)$ and $\\lambda _1(M)$ to denote the volume, the diameter, the injectivity radius and the first non-zero eigenvalue of the Laplace-Beltrami operator of $M$ respectively.","Moreover, $\\operatorname{\\mathrm {d}}:M\\times M\\rightarrow \\mathbb {R}_+$ will denote the distance function on $M$ .","Finally, $\\mathbb {H}^n$ will denote hyperbolic $n$ -space and $\\operatorname{Isom}^+(\\mathbb {H}^n)$ will denote its group of orientation preserving isometries."],["Bounds involving the diameter","Many of our bounds are based on the following well known fact.","Lemma 2.1 Let $n\\ge 2$ .","There exists a constant $C>0$ , depending only on $n$ , so that for every closed hyperbolic $n$ -manifold $M$ we have $C\\cdot \\log (\\operatorname{vol}(M)) \\le \\operatorname{diam}(M).","$ The crucial observation is that the ball $B_M(p,\\operatorname{diam}(M))$ of radius $\\operatorname{diam}(M)$ around any point $p\\in M$ , by definition of the diameter, covers $M$ .","This implies that $\\operatorname{vol}(M)\\le \\operatorname{vol}(B_{M}(p,\\operatorname{diam}(M))).$ On the other hand, the volume of $B_{M}(p,\\operatorname{diam}(M))$ is at most the volume of a ball of the same radius in $\\mathbb {H}^n$ .","The volume of a ball $B_{\\mathbb {H}^n}(p,R)$ of radius $R$ around $p\\in \\mathbb {H}^n$ is equal to $\\operatorname{vol}(B_{\\mathbb {H}^n}(p,R))=\\operatorname{vol}(\\mathbb {S}^{n-1})\\int _0^R \\sinh ^{n-1}(t)dt,$ where $\\mathbb {S}^{n-1}$ denotes the $(n-1)$ -sphere equipped with the round metric (see for instance [34]).","Putting this together with the inequality above gives the lemma.","Moreover, we shall need a bound on the injectivity radius in terms of the diameter.","The following was proved by Young [41], based on work by Reznikov [35]: Lemma 2.2 Let $n\\ge 2$ .","There exists a constant $C>0$ , depending only on $n$ , so that $\\operatorname{inj}(M) \\ge \\exp (-\\operatorname{diam}(M)/C)$ for all closed hyperbolic $n$ -manifolds $M$ .","In order to control the diameter of a sequence of congruence covers later on, we will use the eigenvalue of their Laplacian in combination with the following theorem due to Brooks [11]: Theorem 2.3 Let $M$ be a closed hyperbolic manifold and Let $\\lbrace M_i\\rbrace _{i\\in \\mathcal {I}}$ be a family of finite covers of $M$ .","If there exists a constant $C>0$ so that $\\lambda _1(M_i)> C$ for all $i\\in \\mathcal {I}$ , then there exist constants $a,b,c>0$ such that $a < \\frac{\\log (\\operatorname{vol}(M_i))+c}{\\operatorname{diam}(M_i)} < b$ for all $i\\in \\mathcal {I}$ ."],["Gelander and Levit's construction","The lower bound in Theorem REF will come from a construction due to Gelander and Levit, which is inspired by a classical construction due to Gromov and Piatetski-Shapiro [19] (see also [33]).","We will briefly describe some, but not all, of the details of their construction.","For more information we refer to [18].","Assume we are given six compact hyperbolic $n$ -manifolds with boundary $V_0$ , $V_1$ , $A_+$ , $A_-$ , $B_+$ and $B_-$ so that - $V_0$ and $V_1$ both have four boundary components and $A_+$ , $A_-$ , $B_+$ and $B_-$ all have two boundary components.","- All the boundary components of these manifolds are isometric to a fixed closed hyperbolic $(n-1)$ -manifold.","- Each of these six manifolds is embedded in an arithmetic manifolds without boundary that are pairwise non-commensurable (see Section REF for a definition of an arithmetic group).","In [18], Gelander and Levit explain how to construct these manifolds.","We will glue these manifolds according to Schreier graphs for finite index subgroups of the free group $\\operatorname{\\mathbb {F}}_2=\\langle a,b\\rangle $ .","Let $\\operatorname{Cay}(\\operatorname{\\mathbb {F}}_2,\\lbrace a,b\\rbrace )$ denote the Cayley graph of $\\operatorname{\\mathbb {F}}_2$ with respect to the generating set $\\lbrace a,b\\rbrace $ .","Given $H < \\operatorname{\\mathbb {F}}_2$ , the Schreier graph $\\Gamma _H$ is the graph $\\Gamma _H = \\operatorname{Cay}(\\operatorname{\\mathbb {F}}_2,\\lbrace a,b\\rbrace ) / H.$ Since the edges in $\\operatorname{Cay}(\\operatorname{\\mathbb {F}}_2,\\lbrace a,b\\rbrace )$ come with a natural labeling with the symbols $\\lbrace a^\\pm ,b^\\pm \\rbrace $ , the edges $\\Gamma _H$ come with such a labeling as well.","Furthermore, note that the number of vertices of $\\Gamma _H$ is equal to the index $[\\operatorname{\\mathbb {F}}_2:H]$ .","Let us denote the vertex and edge set of $\\Gamma _H$ by $V(\\Gamma _H)$ and $E(\\Gamma _H)$ respectively.","Definition 2.4 Given a finite index subgroup $H < \\operatorname{\\mathbb {F}}_2$ and a map $\\tau :V(\\Gamma _H)\\rightarrow \\lbrace 0,1\\rbrace $ , we construct the closed hyperbolic $n$ -manifold $M(H,\\tau )$ as follows: - To each vertex $v\\in V(\\Gamma _H)$ , associate a copy of $V_{\\tau (v)}$ - and to each edge $e\\in E(\\Gamma _H)$ , associate a copy of the pair $A^+,A^-$ or $B^+,B^-$ , according to whether it is labeled with an $a^\\pm $ or a $b^\\pm $ .","- Glue the manifolds together according to the incidence relations in $\\Gamma _H$ .","In particular, the order in which to glue the two blocks associated to an edge depends on whether or not the edge is labeled with an inverse.","Note that there is some ambiguity in the construction above: there is for instance a choice which boundary component to glue to which.","Since we are using the construction for a lower bound, this won't make a difference to us.","We will from now on assume some choice of gluing is given for every pair $(H,\\tau )$ .","Figure REF shows a cartoon of what the local picture of $M(H,\\tau )$ might look like: Figure: A local picture of Γ H \\Gamma _H and M(H,τ)M(H,\\tau ), where τ(v 1 )=τ(v 3 )=0\\tau (v_1)=\\tau (v_3)=0 and τ(v 2 )=1\\tau (v_2)=1In [18], Gelander and Levit show: Proposition 2.5 Let $H,H^{\\prime }<\\operatorname{\\mathbb {F}}_2$ be distinct finite index subgroups and let $\\tau :V(\\Gamma _H)\\rightarrow \\lbrace 0,1\\rbrace $ and $\\tau ^{\\prime }:V(\\Gamma _{H^{\\prime }})\\rightarrow \\lbrace 0,1\\rbrace $ be so that $\\left|\\tau ^{-1}(1)\\right| = \\left|(\\tau ^{\\prime })^{-1}(1)\\right| = 1.$ Then $M(H,\\tau )$ and $M(H^{\\prime },\\tau ^{\\prime })$ are not commensurable.","The upshot of this proposition is that the construction of Gelander and Levit gives rise to at least $a_N(\\operatorname{\\mathbb {F}}_2)$ non-commensurable manifolds on built out of graphs with $n$ vertices, where $a_N(\\operatorname{\\mathbb {F}}_2)$ denotes the number of index $N$ subgroups of $\\operatorname{\\mathbb {F}}_2$ ."],["Graphs and groups","To work with Gelander and Levit's construction, we will need two bounds.","We need a lower bound on $a_N(\\operatorname{\\mathbb {F}}_2)$ and we need to know what the typical diameter of a Schreier graph of an index $N$ subgroup of $\\operatorname{\\mathbb {F}}_2$ is.","For details on the subgroup growth of $\\operatorname{\\mathbb {F}}_2$ , we refer to Chapter 2 in the monograph by Lubotzky and Segal [26].","We will use the following bound, that can be found as a special case of [26]: Theorem 2.6 We have $ a_N(\\operatorname{\\mathbb {F}}_2)\\sim N \\cdot N!$ as $N\\rightarrow \\infty $ .","In the theorem above, we write $f(N)\\sim g(N)$ as $N\\rightarrow \\infty $ to mean that $\\lim _{N\\rightarrow \\infty } f(N)/g(N) =1.$ The distance between two vertices in a connected graph is the minimal number of edges in a path between these two vertices.","The diameter $\\operatorname{diam}(\\Gamma )$ of a finite graph $\\Gamma $ is the maximal distance realized by two vertices in $\\Gamma $ .","To control the diameter of a typical Schreier graph we will use results from random graphs.","The fact that the diameter of a random regular graph is bounded by a logarithmic function of the number of vertices can for instance be derived from the fact that a random regular graph has a large spectral gap with probability tending to 1 as the number of vertices tends to infinity (see [16], [31], [23], [12]).","The sharpest result however does not use this method and is due to Bollobás and Fernandez de la Vega [3].","We will state their result only in the case of 4-regular graphs.","Theorem 2.7 Let $\\operatorname{\\mathbb {P}}_N$ denote the uniform probability measure on the set of isomorphism classes of 4-regular graphs on $N$ .","There exists a function $E:\\mathbb {N}\\rightarrow \\mathbb {R}$ so that $E(N) = o(\\log (N))$ as $N\\rightarrow \\infty $ and $\\operatorname{\\mathbb {P}}_N[\\text{The graph has diameter }\\le \\log _3(N) + E(N)]\\rightarrow 1$ as $N\\rightarrow \\infty $ .","We note that the bound is basically as low as one could possibly expect.","Indeed, with an argument very similar to that in Lemma REF it can be shown that the diameter of a 4-regular graph is at least of the order $\\log _3(N)$ .","We won't go into random regular graphs in this note and refer the reader to [3], [9], [40] for the details.","We do however note that uniformly picking an index $N$ Schreier graph is a slightly different model for random 4-valent graphs than the uniform probability measure on isomorphism classes of 4-valent graphs.","It turns out that the two models are what is called contiguous: they have the same asymptotic 0-sets, which is enough for our purposes.","More details on this can be found in [40] and [17]."],["Arithmetic manifolds","Let $G$ be a semisimple Lie group of noncompact type that is defined over $\\mathbb {Q}$ (in our case $G$ will always be $\\operatorname{Isom}^+(\\mathbb {H}^n)$ ).","A discrete subgroup $\\Gamma < G(\\mathbb {Q})$ will be called arithmetic if there is a $\\mathbb {Q}$ -embedding $\\rho :G\\rightarrow \\operatorname{GL}_m(\\mathbb {R})$ such that $\\rho (\\Gamma )$ is commensurable with $G(\\mathbb {Z}) = \\operatorname{GL}_m(\\mathbb {Z})\\cap \\rho (G)$ .","Arithmetic groups come with a sequence of finite index subgroups called congruence groups.","For general background on arithmetic groups, we refer to [30], [29].","Recall that a lattice $\\Gamma < \\operatorname{Isom}^+(\\mathbb {H}^n)$ is called uniform if $\\Gamma \\backslash \\operatorname{Isom}^+(\\mathbb {H}^n)$ is compact.","We will call $\\Gamma $ maximal if it is not properly contained in another lattice.","The result we will need is a bound on the number of maximal uniform arithmetic lattices up to a given covolume in $\\operatorname{Isom}^+(\\mathbb {H}^n)$ for $n\\ge 3$ .","To this end, let $\\mathrm {MAL}^u_n(v)$ denote the number of maximal uniform arithmetic lattices of covolume $\\le v$ in $\\operatorname{Isom}^+(\\mathbb {H}^n)$ .","The following theorem is due to Belolipetsky [2] in dimension $n\\ge 4$ and Belolipetsky, Gelander, Lubotzky and Shalev [5] in dimensions 2 and 3.","Theorem 2.8 Let $n\\ge 2$ and $\\varepsilon >0$ .","There exist constants $\\alpha =\\alpha (n) \\in \\mathbb {R}_+$ and $\\beta =\\beta (n,\\varepsilon )\\in \\mathbb {R}_+$ so that $v^\\alpha \\le \\mathrm {MAL}^u_n(v) \\le v^{\\beta \\cdot (\\log v)^\\varepsilon }$ for all $v\\in \\mathbb {R}^+$ large enough.","Let us now restrict to hyperbolic 3-manifolds.","To get bounds on the diameters of congruence covers of arithmetic manifolds, we will use the following theorem due to Sarnak and Xue [38]: Theorem 2.9 Let $\\Gamma <\\operatorname{Isom}^+(\\mathbb {H}^3)$ be a uniform arithmetic lattice.","Then there exists a constant $C=C(\\Gamma )>0$ so that for all congruence subgroups $\\Gamma ^{\\prime }<\\Gamma $ we have $\\lambda _1(\\Gamma ^{\\prime }\\backslash \\mathbb {H}^3) \\ge C.$ If a lattice $\\Gamma <\\operatorname{SL}_2(\\mathbb {C})$ has a sequence $\\lbrace \\Gamma _N\\rbrace _{N}$ of finite index subgroups so that $\\lambda _1(\\Gamma _N\\backslash \\mathbb {H}^3)$ is uniformly bounded from below for all $N\\in \\mathbb {N}$ , then $\\Gamma $ is said to have Property ($\\tau $ ) with respect to this sequence.","In the special case where $\\Gamma $ is arithmetic and the sequence consists of congruence subgroups, Property ($\\tau $ ) is sometimes also called the Selberg property.","Congruence covers are also believed to have large torsion subgroups in their homology.","Specifically, there is the following conjecture, due to Bergeron and Venkatesh [14], which we state in the special case of hyperbolic 3-manifolds: Conjecture 2.10 Let $\\Gamma <\\operatorname{SL}_2(\\mathbb {C})$ be a uniform arithmetic lattice and $\\ldots <\\Gamma _N<\\Gamma _{N-1}<\\ldots <\\Gamma _1<\\Gamma $ a sequence of congruence subgroups of $\\Gamma $ so that $\\cap _N \\Gamma _N = \\lbrace 1\\rbrace $ .","Then $\\lim _{N\\rightarrow \\infty } \\frac{\\log \\left(\\left|H_1(\\Gamma _N \\backslash \\mathbb {H}^3,\\mathbb {Z})_\\mathrm {tors}\\right|\\right)}{\\operatorname{vol}(\\Gamma _N \\backslash \\mathbb {H}^3)} = \\frac{1}{6\\pi }.$ The reason for the constant $1/6\\pi $ above is that it is the $\\ell ^2$ -torsion of $\\mathbb {H}^3$ ."],["Torsion and the nerve lemma","To bound torsion in our manifolds we will use a lemma by Bader, Gelander and Sauer [7], that they derived from a lemma due to Gabber (which can for instance be found in [36]).","In this lemma, the degree of a vertex (0-cell) in a simplicial complex is the degree of that vertex in the 1-skeleton of the given complex.","Lemma 2.11 For all $D,p \\in \\mathbb {N}$ there exists a constant $C=C(D,p)>0$ so that for any simplicial complex $X$ with $\\le V$ vertices that all have degree $\\le D$ we have: $ \\log \\left(\\left|H_p(X,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) \\le C\\cdot V.$ The simplicial complex we will use will be the nerve of an open cover of our manifold.","Recall that an open cover of a space $X$ is a collection $\\mathcal {U}=\\lbrace U_i\\rbrace _{i\\in \\mathcal {I}}$ of open subsets of $X$ so that $ X = \\bigcup _{i\\in \\mathcal {I}}U_i.$ The nerve $\\mathcal {N}(\\mathcal {U})$ of this cover is the simplicial complex that has the sets $U_i$ as vertices and contains a $k$ -simplex for every $k$ -tuple of elements in $\\mathcal {U}$ that have a non-trivial intersection.","See [22] for more details.","The following statement is known as the nerve lemma and can for instance be found as [22]: Lemma 2.12 If $\\mathcal {U}$ is an open cover of a paracompact space $X$ such that every nonempty intersection of finitely many sets in $\\mathcal {U}$ is contractible, then $X$ is homotopy equivalent to the nerve $\\mathcal {N}(\\mathcal {U})$ ."],["Counting","We are now ready to prove Theorem REF : thmdiamTheorem REF For all $n \\ge 3$ there exist $00$ so that the index $N$ subgroups of $\\operatorname{\\mathbb {F}}_2$ for $N$ large enough produce at least $C\\cdot N^{CN}$ non-isomorphic graphs of diameter $\\le \\log _3(N) + o(\\log (N))$ .","If we now define maps $\\tau :V(\\Gamma )\\rightarrow \\lbrace 0,1\\rbrace $ that assign the value 1 to only one vertex per graph $\\Gamma $ , we obtain $C\\cdot N^{CN}$ manifolds of diameter $d \\le 2D\\log _3(N) + o(\\log (N))$ .","Working this out, we see that this number of manifolds is at least $\\exp (C^{\\prime }\\cdot d \\cdot \\exp (C^{\\prime }\\cdot d)),$ for some $C^{\\prime }>0$ .","Proposition REF tells us that none of the resulting manifolds will be commensurable.","Corollary REF now also easily follows.","corarithmCorollary REF Let $n\\ge 3$ .","We have $\\lim _{d\\rightarrow \\infty }\\operatorname{\\mathbb {P}}_{n,d}[\\text{The manifold is arithmetic}] = 0$ The only thing we need to control is the number of maximal uniform arithmetic lattices of diameter $\\le d$ in $\\operatorname{Isom}^+(\\mathbb {H}^n)$ .","Let us call this number $\\mathrm {MALD}^u_{n}(d)$ .","By Lemma REF we have $\\mathrm {MALD}^u_{n}(d) \\le \\mathrm {MAL}^u_n(e^{d/C_n})$ for some $C_n>0$ independent of $d$ .","As such, Theorem REF implies that for every $\\varepsilon >0$ there exists a $\\beta ^{\\prime }>0$ so that $\\mathrm {MALD}^u_{n}(d) \\le \\exp (\\beta ^{\\prime }\\cdot d^{1+\\varepsilon }).$ Comparing this to Theorem REF gives the result."],["Torsion","In this section we prove Theorem REF and explain how a positive answer to Conjecture REF would lead to a sequence of manifolds that saturates the bound in that theorem up to a multiplicative constant."],["An upper bound for torsion in homology","We will start with: thmtorsionTheorem REF For every $n\\ge 2$ there exists a constant $C >0$ so that $\\log \\log \\left(\\left|H_i(M,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) \\le C\\cdot \\operatorname{diam}(M) $ for all $i=0,\\ldots ,n$ for any closed hyperbolic $n$ -manifold $M$ .","Our final goal is to employ Lemma REF .","In the language of [7]: we need to show that a closed hyperbolic manifold $M$ is homotopy equivalent to a $(D,C\\cdot \\operatorname{diam}(M))$ -simplicial complex, where $C,D>0$ are constants depending only its dimension.","The simplicial complex we build is the same as that used for the upper bound in Young's result (Equation (REF )).","Set $r=\\operatorname{inj}(M)$ and let $S\\subset M$ be a maximal set of points so that $\\operatorname{\\mathrm {d}}(s,s^{\\prime }) \\ge r/4$ for all $s,s^{\\prime }\\in S$ .","Now consider the collection $\\mathcal {U}=\\lbrace B_M(s,r/2)\\rbrace _{s\\in S},$ where $B_M(s,r/2)$ denotes the open ball in $M$ of radius $r/2$ around $s$ .","It follows from maximality of $S$ that these balls form an open cover.","Moreover, because their radius is half the injectivity radius they are isometrically embedded $n$ -dimensional hyperbolic balls.","As such they are convex, which means that their intersections are convex and thus contractible.","Hence, the nerve lemma (Lemma REF ) applies.","This means that the homology groups we are after are those of $\\mathcal {N}(\\mathcal {U})$ .","So we need to find bounds on the number of vertices and their degrees in $\\mathcal {N}(\\mathcal {U})$ in order to apply Lemma REF .","The number of vertices, or equivalently the number of points in $S$ , can be bounded by $\\left|S\\right| \\le \\frac{\\operatorname{vol}(M)}{\\operatorname{vol}(B_{\\mathbb {H}^n}(p,r/4))} \\le D\\cdot \\frac{\\operatorname{vol}(M)}{(r/4)^n}, $ where $B_{\\mathbb {H}^n}(p,r/4)$ is the ball of radius $r/4$ around some point $p\\in \\mathbb {H}^n$ and $D>0$ is some constant depending only on the dimension.","The second of these bounds again follows from the closed formula for the volume of a ball in $\\mathbb {H}^n$ .","Now we use Lemma REF and Lemma REF to tell us that $(r/4)^n \\ge \\exp (-C\\operatorname{diam}(M)) \\;\\;\\text{and}\\;\\; \\operatorname{vol}(M) \\le \\exp (C\\operatorname{diam}(M))$ for some $C>0$ depending only on $n$ .","So we obtain $\\left|S\\right| \\le A\\cdot \\exp (B\\operatorname{diam}(M)).","$ for some $A,B>0$ depending only on $n$ .","All that remains is to show that each vertex has a bounded number of neighbors.","First of all note that all the neighbors of a point $s\\in S$ lie in $B_M(s,r)\\subset M$ .","By definition of $S$ , the balls of radius $r/8$ around the neighbors of $s$ are all disjoint and all lie in $B_{M}(s,9r/8)$ .","This means that the number of neighbors is at most $\\frac{\\operatorname{vol}(B_M(s,9r/8))}{\\operatorname{vol}(B_M(s,r/8))} \\le \\frac{\\operatorname{vol}(B_{\\mathbb {H}^n}(s,9r/8))}{\\operatorname{vol}(B_{\\mathbb {H}^n}(s,r/8))},$ which, for $r$ small enough, is uniformly bounded in each fixed dimension."],["Sharpness of the bound","Like we said in the introduction, if Conjecture REF holds then we would get a sequence of closed hyperbolic 3-manifolds $\\lbrace M_N\\rbrace _{N\\in \\mathbb {N}}$ such that $\\operatorname{diam}(M_N)\\rightarrow \\infty $ as $N\\rightarrow \\infty $ and $ \\log \\log \\left(\\left| H_1(M_N,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) \\ge C \\operatorname{diam}(M_N).$ for some $C>0$ independent of $N$ .","Indeed, it follows from Theorem REF together with Theorem REF that for a uniform arithmetic group $\\Gamma < \\operatorname{SL}_2(\\mathbb {C})$ and a sequence of congruence subgroups $\\lbrace \\Gamma _N\\rbrace _N$ , the manifolds $M_N = \\Gamma _N \\backslash \\mathbb {H}^3$ satisfy $\\operatorname{diam}(M_N) \\le A \\cdot \\log (\\operatorname{vol}(M_N)).$ It would follow from Conjecture REF that $\\operatorname{vol}(M_N)\\le B\\cdot \\log \\left(\\left| H_1(M_N,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) $ for all $N$ and some $B>0$ independent of $N$ .","Putting these two together would give the desired result."]],"string":"[\n [\n \"Counting non-commensurable hyperbolic manifolds and a bound on\\n homological torsion\"\n ],\n [\n \"Abstract We prove that the cardinality of the torsion subgroups in homology of a closed hyperbolic manifold of any dimension can be bounded by a doubly exponential function of its diameter.\",\n \"It would follow from a conjecture by Bergeron and Venkatesh that the order of growth in our bound is sharp.\",\n \"We also determine how the number of non-commensurable closed hyperbolic manifolds of dimension at least 3 and bounded diameter grows.\",\n \"The lower bound implies that the fraction of arithmetic manifolds tends to zero as the diameter goes up.\"\n ],\n [\n \"Introduction\",\n \"Recently there has been a lot of progress on understanding the relation between the volume of a negatively curved manifold and its toplogical complexity.\",\n \"In this note, we will instead consider the relation between complexity and diameter.\",\n \"We will restrict to closed hyperbolic (constant sectional curvature $-1$ ) manifolds.\",\n \"The main upshot of considering the diameter instead of the volume is that we obtain bounds in dimension 3.\"\n ],\n [\n \"New results\",\n \"Recall that two manifolds are called commensurable if they have a common finite cover.\",\n \"The diameter of a commensurability class of manifolds is the minimal diameter realized by a manifold in that class.\",\n \"Given $d\\\\in \\\\mathbb {R}_+$ and $n\\\\ge 3$ , let $NC^{\\\\operatorname{diam}}_n(d)$ denote the number of commensurability classes of closed hyperbolic $n$ -manifolds of diameter $\\\\le d$ .\",\n \"We will prove Theorem 1 For all $n \\\\ge 3$ there exist $00$ so that $\\\\log \\\\log \\\\left(\\\\left|H_i(M,\\\\mathbb {Z})_{\\\\mathrm {tors}}\\\\right|\\\\right) \\\\le C\\\\cdot \\\\operatorname{diam}(M) $ for all $i=0,\\\\ldots ,n$ for any closed hyperbolic $n$ -manifold $M$ .\",\n \"Let us first note that for $n=2$ our theorem is automatic, since all torsion subgroups in the homology of a closed hyperbolic surface are trivial.\",\n \"Moreover, in dimension at least 4 results by Bader, Gelander and Sauer [7] (see also Equation (REF )) together with a comparison between volume and diameter (Lemma REF ) also prove Theorem REF .\",\n \"However, it is known that in dimension 3, this method cannot work.\",\n \"We also note that to prove that our theorem is sharp, the most likely examples would be sequences hyperbolic manifolds with exponentially growing torsion and logarithmically growing diameter.\",\n \"Examples of such sequences are abelian covers [37], [32], [1], Liu's construction [25] and random Heegaard splittings [24], [1].\",\n \"However, these manifolds are not known to have small diameters and in some cases are even known not to have small diameters.\",\n \"On the other hand, certain sequences of covers of arithmetic manifolds are conjectured to have exponential torsion growth by Bergeron and Venkatesh [14].\",\n \"It is known that the corresponding lattice has Property $(\\\\tau )$ with respect to this sequence of covers [38], from which it follows that their diameter is small by a result of Brooks [11] (see Sections REF and REF ).\",\n \"So assuming the conjecture by Bergeron and Venkatesh, these manifolds would saturate the bound in Theorem REF up to a multiplicative constant.\",\n \"The proof of Theorem REF consists of two steps.\",\n \"First we use Young's method from [41] to build a simplicial complex that models our manifold and has a bounded number of cells of any dimension.\",\n \"We then use a lemma due to Bader, Gelander and Sauer [7] (based on a lemma of Gabber) that bounds the homological torsion in terms of the number of cells (Lemma REF ) to derive our bound.\"\n ],\n [\n \"Volume and complexity\",\n \"The main motivation for our work is formed by results that bound the homological complexity of a complete negatively curved $n$ -manifold $M$ in terms of its volume.\",\n \"A classical result due to Gromov and worked out by Ballmann, Gromov and Schröder [21], [6], states that there exists a constant $C >0$ , depending only on $n$ , so that the Betti numbers $b_i(M)$ satisfy $b_i(M) \\\\le C \\\\cdot \\\\operatorname{vol}(M),$ for $i=0,\\\\ldots ,n$ , where $\\\\operatorname{vol}(M)$ denotes the volume of $M$ .\",\n \"More recently, Bader, Gelander and Sauer [7] have shown that, when the dimension $n$ is at least 4, the cardinality of the torsion subgroups $H_i(M,\\\\mathbb {Z})_{\\\\mathrm {tors}}$ in homology can also be bounded in terms of the volume.\",\n \"They show that for all $n\\\\ge 4$ , there exists a constant $C>0$ , depending only on $n$ , so that $\\\\log \\\\left(\\\\left|H_i(M,\\\\mathbb {Z})_{\\\\mathrm {tors}}\\\\right|\\\\right) \\\\le C\\\\cdot \\\\operatorname{vol}(M).$\"\n ],\n [\n \"Counting manifolds by volume\",\n \"We will again restrict ourselves to closed hyperbolic manifolds.\",\n \"A classical result due to Wang [39] states that in dimension $n\\\\ge 4$ the number of closed hyperbolic manifolds of volume $\\\\le v$ is finite for any $v\\\\in \\\\mathbb {R}_+$ .\",\n \"Let $N^{\\\\operatorname{vol}}_n(v)$ denote the number such manifolds.\",\n \"The first bounds on the growth of $N^{\\\\operatorname{vol}}_n(v)$ are due to Gromov [20].\",\n \"Burger, Gelander, Lubotzky and Mozes [4] showed that for all $n\\\\ge 4$ there exist $00$ is a constant depending only on $n$ (see Lemma REF ).\",\n \"Moreover, because there are hyperbolic surfaces with linearly growing genus and logarithmically growing diameter (for instance random surfaces [8], [28]), a bound of this generality is necessarily exponential in diameter.\"\n ],\n [\n \"Acknowledgement\",\n \"The author's research was supported by the ERC Advanced Grant “Moduli”.\",\n \"The author also thanks the organizers of the Borel Seminar 2017, during which the research for this article was done.\",\n \"Finally, the author thanks Filippo Cerocchi, Jean Raimbault and Roman Sauer for useful conversations.\"\n ],\n [\n \"Background material\",\n \"In what follows, $n$ will be a natural number, $M$ a closed oriented hyperbolic $n$ -manifold.\",\n \"We will use $\\\\operatorname{vol}(M)$ , $\\\\operatorname{diam}(M)$ , $\\\\operatorname{inj}(M)$ and $\\\\lambda _1(M)$ to denote the volume, the diameter, the injectivity radius and the first non-zero eigenvalue of the Laplace-Beltrami operator of $M$ respectively.\",\n \"Moreover, $\\\\operatorname{\\\\mathrm {d}}:M\\\\times M\\\\rightarrow \\\\mathbb {R}_+$ will denote the distance function on $M$ .\",\n \"Finally, $\\\\mathbb {H}^n$ will denote hyperbolic $n$ -space and $\\\\operatorname{Isom}^+(\\\\mathbb {H}^n)$ will denote its group of orientation preserving isometries.\"\n ],\n [\n \"Bounds involving the diameter\",\n \"Many of our bounds are based on the following well known fact.\",\n \"Lemma 2.1 Let $n\\\\ge 2$ .\",\n \"There exists a constant $C>0$ , depending only on $n$ , so that for every closed hyperbolic $n$ -manifold $M$ we have $C\\\\cdot \\\\log (\\\\operatorname{vol}(M)) \\\\le \\\\operatorname{diam}(M).\",\n \"$ The crucial observation is that the ball $B_M(p,\\\\operatorname{diam}(M))$ of radius $\\\\operatorname{diam}(M)$ around any point $p\\\\in M$ , by definition of the diameter, covers $M$ .\",\n \"This implies that $\\\\operatorname{vol}(M)\\\\le \\\\operatorname{vol}(B_{M}(p,\\\\operatorname{diam}(M))).$ On the other hand, the volume of $B_{M}(p,\\\\operatorname{diam}(M))$ is at most the volume of a ball of the same radius in $\\\\mathbb {H}^n$ .\",\n \"The volume of a ball $B_{\\\\mathbb {H}^n}(p,R)$ of radius $R$ around $p\\\\in \\\\mathbb {H}^n$ is equal to $\\\\operatorname{vol}(B_{\\\\mathbb {H}^n}(p,R))=\\\\operatorname{vol}(\\\\mathbb {S}^{n-1})\\\\int _0^R \\\\sinh ^{n-1}(t)dt,$ where $\\\\mathbb {S}^{n-1}$ denotes the $(n-1)$ -sphere equipped with the round metric (see for instance [34]).\",\n \"Putting this together with the inequality above gives the lemma.\",\n \"Moreover, we shall need a bound on the injectivity radius in terms of the diameter.\",\n \"The following was proved by Young [41], based on work by Reznikov [35]: Lemma 2.2 Let $n\\\\ge 2$ .\",\n \"There exists a constant $C>0$ , depending only on $n$ , so that $\\\\operatorname{inj}(M) \\\\ge \\\\exp (-\\\\operatorname{diam}(M)/C)$ for all closed hyperbolic $n$ -manifolds $M$ .\",\n \"In order to control the diameter of a sequence of congruence covers later on, we will use the eigenvalue of their Laplacian in combination with the following theorem due to Brooks [11]: Theorem 2.3 Let $M$ be a closed hyperbolic manifold and Let $\\\\lbrace M_i\\\\rbrace _{i\\\\in \\\\mathcal {I}}$ be a family of finite covers of $M$ .\",\n \"If there exists a constant $C>0$ so that $\\\\lambda _1(M_i)> C$ for all $i\\\\in \\\\mathcal {I}$ , then there exist constants $a,b,c>0$ such that $a < \\\\frac{\\\\log (\\\\operatorname{vol}(M_i))+c}{\\\\operatorname{diam}(M_i)} < b$ for all $i\\\\in \\\\mathcal {I}$ .\"\n ],\n [\n \"Gelander and Levit's construction\",\n \"The lower bound in Theorem REF will come from a construction due to Gelander and Levit, which is inspired by a classical construction due to Gromov and Piatetski-Shapiro [19] (see also [33]).\",\n \"We will briefly describe some, but not all, of the details of their construction.\",\n \"For more information we refer to [18].\",\n \"Assume we are given six compact hyperbolic $n$ -manifolds with boundary $V_0$ , $V_1$ , $A_+$ , $A_-$ , $B_+$ and $B_-$ so that - $V_0$ and $V_1$ both have four boundary components and $A_+$ , $A_-$ , $B_+$ and $B_-$ all have two boundary components.\",\n \"- All the boundary components of these manifolds are isometric to a fixed closed hyperbolic $(n-1)$ -manifold.\",\n \"- Each of these six manifolds is embedded in an arithmetic manifolds without boundary that are pairwise non-commensurable (see Section REF for a definition of an arithmetic group).\",\n \"In [18], Gelander and Levit explain how to construct these manifolds.\",\n \"We will glue these manifolds according to Schreier graphs for finite index subgroups of the free group $\\\\operatorname{\\\\mathbb {F}}_2=\\\\langle a,b\\\\rangle $ .\",\n \"Let $\\\\operatorname{Cay}(\\\\operatorname{\\\\mathbb {F}}_2,\\\\lbrace a,b\\\\rbrace )$ denote the Cayley graph of $\\\\operatorname{\\\\mathbb {F}}_2$ with respect to the generating set $\\\\lbrace a,b\\\\rbrace $ .\",\n \"Given $H < \\\\operatorname{\\\\mathbb {F}}_2$ , the Schreier graph $\\\\Gamma _H$ is the graph $\\\\Gamma _H = \\\\operatorname{Cay}(\\\\operatorname{\\\\mathbb {F}}_2,\\\\lbrace a,b\\\\rbrace ) / H.$ Since the edges in $\\\\operatorname{Cay}(\\\\operatorname{\\\\mathbb {F}}_2,\\\\lbrace a,b\\\\rbrace )$ come with a natural labeling with the symbols $\\\\lbrace a^\\\\pm ,b^\\\\pm \\\\rbrace $ , the edges $\\\\Gamma _H$ come with such a labeling as well.\",\n \"Furthermore, note that the number of vertices of $\\\\Gamma _H$ is equal to the index $[\\\\operatorname{\\\\mathbb {F}}_2:H]$ .\",\n \"Let us denote the vertex and edge set of $\\\\Gamma _H$ by $V(\\\\Gamma _H)$ and $E(\\\\Gamma _H)$ respectively.\",\n \"Definition 2.4 Given a finite index subgroup $H < \\\\operatorname{\\\\mathbb {F}}_2$ and a map $\\\\tau :V(\\\\Gamma _H)\\\\rightarrow \\\\lbrace 0,1\\\\rbrace $ , we construct the closed hyperbolic $n$ -manifold $M(H,\\\\tau )$ as follows: - To each vertex $v\\\\in V(\\\\Gamma _H)$ , associate a copy of $V_{\\\\tau (v)}$ - and to each edge $e\\\\in E(\\\\Gamma _H)$ , associate a copy of the pair $A^+,A^-$ or $B^+,B^-$ , according to whether it is labeled with an $a^\\\\pm $ or a $b^\\\\pm $ .\",\n \"- Glue the manifolds together according to the incidence relations in $\\\\Gamma _H$ .\",\n \"In particular, the order in which to glue the two blocks associated to an edge depends on whether or not the edge is labeled with an inverse.\",\n \"Note that there is some ambiguity in the construction above: there is for instance a choice which boundary component to glue to which.\",\n \"Since we are using the construction for a lower bound, this won't make a difference to us.\",\n \"We will from now on assume some choice of gluing is given for every pair $(H,\\\\tau )$ .\",\n \"Figure REF shows a cartoon of what the local picture of $M(H,\\\\tau )$ might look like: Figure: A local picture of Γ H \\\\Gamma _H and M(H,τ)M(H,\\\\tau ), where τ(v 1 )=τ(v 3 )=0\\\\tau (v_1)=\\\\tau (v_3)=0 and τ(v 2 )=1\\\\tau (v_2)=1In [18], Gelander and Levit show: Proposition 2.5 Let $H,H^{\\\\prime }<\\\\operatorname{\\\\mathbb {F}}_2$ be distinct finite index subgroups and let $\\\\tau :V(\\\\Gamma _H)\\\\rightarrow \\\\lbrace 0,1\\\\rbrace $ and $\\\\tau ^{\\\\prime }:V(\\\\Gamma _{H^{\\\\prime }})\\\\rightarrow \\\\lbrace 0,1\\\\rbrace $ be so that $\\\\left|\\\\tau ^{-1}(1)\\\\right| = \\\\left|(\\\\tau ^{\\\\prime })^{-1}(1)\\\\right| = 1.$ Then $M(H,\\\\tau )$ and $M(H^{\\\\prime },\\\\tau ^{\\\\prime })$ are not commensurable.\",\n \"The upshot of this proposition is that the construction of Gelander and Levit gives rise to at least $a_N(\\\\operatorname{\\\\mathbb {F}}_2)$ non-commensurable manifolds on built out of graphs with $n$ vertices, where $a_N(\\\\operatorname{\\\\mathbb {F}}_2)$ denotes the number of index $N$ subgroups of $\\\\operatorname{\\\\mathbb {F}}_2$ .\"\n ],\n [\n \"Graphs and groups\",\n \"To work with Gelander and Levit's construction, we will need two bounds.\",\n \"We need a lower bound on $a_N(\\\\operatorname{\\\\mathbb {F}}_2)$ and we need to know what the typical diameter of a Schreier graph of an index $N$ subgroup of $\\\\operatorname{\\\\mathbb {F}}_2$ is.\",\n \"For details on the subgroup growth of $\\\\operatorname{\\\\mathbb {F}}_2$ , we refer to Chapter 2 in the monograph by Lubotzky and Segal [26].\",\n \"We will use the following bound, that can be found as a special case of [26]: Theorem 2.6 We have $ a_N(\\\\operatorname{\\\\mathbb {F}}_2)\\\\sim N \\\\cdot N!$ as $N\\\\rightarrow \\\\infty $ .\",\n \"In the theorem above, we write $f(N)\\\\sim g(N)$ as $N\\\\rightarrow \\\\infty $ to mean that $\\\\lim _{N\\\\rightarrow \\\\infty } f(N)/g(N) =1.$ The distance between two vertices in a connected graph is the minimal number of edges in a path between these two vertices.\",\n \"The diameter $\\\\operatorname{diam}(\\\\Gamma )$ of a finite graph $\\\\Gamma $ is the maximal distance realized by two vertices in $\\\\Gamma $ .\",\n \"To control the diameter of a typical Schreier graph we will use results from random graphs.\",\n \"The fact that the diameter of a random regular graph is bounded by a logarithmic function of the number of vertices can for instance be derived from the fact that a random regular graph has a large spectral gap with probability tending to 1 as the number of vertices tends to infinity (see [16], [31], [23], [12]).\",\n \"The sharpest result however does not use this method and is due to Bollobás and Fernandez de la Vega [3].\",\n \"We will state their result only in the case of 4-regular graphs.\",\n \"Theorem 2.7 Let $\\\\operatorname{\\\\mathbb {P}}_N$ denote the uniform probability measure on the set of isomorphism classes of 4-regular graphs on $N$ .\",\n \"There exists a function $E:\\\\mathbb {N}\\\\rightarrow \\\\mathbb {R}$ so that $E(N) = o(\\\\log (N))$ as $N\\\\rightarrow \\\\infty $ and $\\\\operatorname{\\\\mathbb {P}}_N[\\\\text{The graph has diameter }\\\\le \\\\log _3(N) + E(N)]\\\\rightarrow 1$ as $N\\\\rightarrow \\\\infty $ .\",\n \"We note that the bound is basically as low as one could possibly expect.\",\n \"Indeed, with an argument very similar to that in Lemma REF it can be shown that the diameter of a 4-regular graph is at least of the order $\\\\log _3(N)$ .\",\n \"We won't go into random regular graphs in this note and refer the reader to [3], [9], [40] for the details.\",\n \"We do however note that uniformly picking an index $N$ Schreier graph is a slightly different model for random 4-valent graphs than the uniform probability measure on isomorphism classes of 4-valent graphs.\",\n \"It turns out that the two models are what is called contiguous: they have the same asymptotic 0-sets, which is enough for our purposes.\",\n \"More details on this can be found in [40] and [17].\"\n ],\n [\n \"Arithmetic manifolds\",\n \"Let $G$ be a semisimple Lie group of noncompact type that is defined over $\\\\mathbb {Q}$ (in our case $G$ will always be $\\\\operatorname{Isom}^+(\\\\mathbb {H}^n)$ ).\",\n \"A discrete subgroup $\\\\Gamma < G(\\\\mathbb {Q})$ will be called arithmetic if there is a $\\\\mathbb {Q}$ -embedding $\\\\rho :G\\\\rightarrow \\\\operatorname{GL}_m(\\\\mathbb {R})$ such that $\\\\rho (\\\\Gamma )$ is commensurable with $G(\\\\mathbb {Z}) = \\\\operatorname{GL}_m(\\\\mathbb {Z})\\\\cap \\\\rho (G)$ .\",\n \"Arithmetic groups come with a sequence of finite index subgroups called congruence groups.\",\n \"For general background on arithmetic groups, we refer to [30], [29].\",\n \"Recall that a lattice $\\\\Gamma < \\\\operatorname{Isom}^+(\\\\mathbb {H}^n)$ is called uniform if $\\\\Gamma \\\\backslash \\\\operatorname{Isom}^+(\\\\mathbb {H}^n)$ is compact.\",\n \"We will call $\\\\Gamma $ maximal if it is not properly contained in another lattice.\",\n \"The result we will need is a bound on the number of maximal uniform arithmetic lattices up to a given covolume in $\\\\operatorname{Isom}^+(\\\\mathbb {H}^n)$ for $n\\\\ge 3$ .\",\n \"To this end, let $\\\\mathrm {MAL}^u_n(v)$ denote the number of maximal uniform arithmetic lattices of covolume $\\\\le v$ in $\\\\operatorname{Isom}^+(\\\\mathbb {H}^n)$ .\",\n \"The following theorem is due to Belolipetsky [2] in dimension $n\\\\ge 4$ and Belolipetsky, Gelander, Lubotzky and Shalev [5] in dimensions 2 and 3.\",\n \"Theorem 2.8 Let $n\\\\ge 2$ and $\\\\varepsilon >0$ .\",\n \"There exist constants $\\\\alpha =\\\\alpha (n) \\\\in \\\\mathbb {R}_+$ and $\\\\beta =\\\\beta (n,\\\\varepsilon )\\\\in \\\\mathbb {R}_+$ so that $v^\\\\alpha \\\\le \\\\mathrm {MAL}^u_n(v) \\\\le v^{\\\\beta \\\\cdot (\\\\log v)^\\\\varepsilon }$ for all $v\\\\in \\\\mathbb {R}^+$ large enough.\",\n \"Let us now restrict to hyperbolic 3-manifolds.\",\n \"To get bounds on the diameters of congruence covers of arithmetic manifolds, we will use the following theorem due to Sarnak and Xue [38]: Theorem 2.9 Let $\\\\Gamma <\\\\operatorname{Isom}^+(\\\\mathbb {H}^3)$ be a uniform arithmetic lattice.\",\n \"Then there exists a constant $C=C(\\\\Gamma )>0$ so that for all congruence subgroups $\\\\Gamma ^{\\\\prime }<\\\\Gamma $ we have $\\\\lambda _1(\\\\Gamma ^{\\\\prime }\\\\backslash \\\\mathbb {H}^3) \\\\ge C.$ If a lattice $\\\\Gamma <\\\\operatorname{SL}_2(\\\\mathbb {C})$ has a sequence $\\\\lbrace \\\\Gamma _N\\\\rbrace _{N}$ of finite index subgroups so that $\\\\lambda _1(\\\\Gamma _N\\\\backslash \\\\mathbb {H}^3)$ is uniformly bounded from below for all $N\\\\in \\\\mathbb {N}$ , then $\\\\Gamma $ is said to have Property ($\\\\tau $ ) with respect to this sequence.\",\n \"In the special case where $\\\\Gamma $ is arithmetic and the sequence consists of congruence subgroups, Property ($\\\\tau $ ) is sometimes also called the Selberg property.\",\n \"Congruence covers are also believed to have large torsion subgroups in their homology.\",\n \"Specifically, there is the following conjecture, due to Bergeron and Venkatesh [14], which we state in the special case of hyperbolic 3-manifolds: Conjecture 2.10 Let $\\\\Gamma <\\\\operatorname{SL}_2(\\\\mathbb {C})$ be a uniform arithmetic lattice and $\\\\ldots <\\\\Gamma _N<\\\\Gamma _{N-1}<\\\\ldots <\\\\Gamma _1<\\\\Gamma $ a sequence of congruence subgroups of $\\\\Gamma $ so that $\\\\cap _N \\\\Gamma _N = \\\\lbrace 1\\\\rbrace $ .\",\n \"Then $\\\\lim _{N\\\\rightarrow \\\\infty } \\\\frac{\\\\log \\\\left(\\\\left|H_1(\\\\Gamma _N \\\\backslash \\\\mathbb {H}^3,\\\\mathbb {Z})_\\\\mathrm {tors}\\\\right|\\\\right)}{\\\\operatorname{vol}(\\\\Gamma _N \\\\backslash \\\\mathbb {H}^3)} = \\\\frac{1}{6\\\\pi }.$ The reason for the constant $1/6\\\\pi $ above is that it is the $\\\\ell ^2$ -torsion of $\\\\mathbb {H}^3$ .\"\n ],\n [\n \"Torsion and the nerve lemma\",\n \"To bound torsion in our manifolds we will use a lemma by Bader, Gelander and Sauer [7], that they derived from a lemma due to Gabber (which can for instance be found in [36]).\",\n \"In this lemma, the degree of a vertex (0-cell) in a simplicial complex is the degree of that vertex in the 1-skeleton of the given complex.\",\n \"Lemma 2.11 For all $D,p \\\\in \\\\mathbb {N}$ there exists a constant $C=C(D,p)>0$ so that for any simplicial complex $X$ with $\\\\le V$ vertices that all have degree $\\\\le D$ we have: $ \\\\log \\\\left(\\\\left|H_p(X,\\\\mathbb {Z})_{\\\\mathrm {tors}}\\\\right|\\\\right) \\\\le C\\\\cdot V.$ The simplicial complex we will use will be the nerve of an open cover of our manifold.\",\n \"Recall that an open cover of a space $X$ is a collection $\\\\mathcal {U}=\\\\lbrace U_i\\\\rbrace _{i\\\\in \\\\mathcal {I}}$ of open subsets of $X$ so that $ X = \\\\bigcup _{i\\\\in \\\\mathcal {I}}U_i.$ The nerve $\\\\mathcal {N}(\\\\mathcal {U})$ of this cover is the simplicial complex that has the sets $U_i$ as vertices and contains a $k$ -simplex for every $k$ -tuple of elements in $\\\\mathcal {U}$ that have a non-trivial intersection.\",\n \"See [22] for more details.\",\n \"The following statement is known as the nerve lemma and can for instance be found as [22]: Lemma 2.12 If $\\\\mathcal {U}$ is an open cover of a paracompact space $X$ such that every nonempty intersection of finitely many sets in $\\\\mathcal {U}$ is contractible, then $X$ is homotopy equivalent to the nerve $\\\\mathcal {N}(\\\\mathcal {U})$ .\"\n ],\n [\n \"Counting\",\n \"We are now ready to prove Theorem REF : thmdiamTheorem REF For all $n \\\\ge 3$ there exist $00$ so that the index $N$ subgroups of $\\\\operatorname{\\\\mathbb {F}}_2$ for $N$ large enough produce at least $C\\\\cdot N^{CN}$ non-isomorphic graphs of diameter $\\\\le \\\\log _3(N) + o(\\\\log (N))$ .\",\n \"If we now define maps $\\\\tau :V(\\\\Gamma )\\\\rightarrow \\\\lbrace 0,1\\\\rbrace $ that assign the value 1 to only one vertex per graph $\\\\Gamma $ , we obtain $C\\\\cdot N^{CN}$ manifolds of diameter $d \\\\le 2D\\\\log _3(N) + o(\\\\log (N))$ .\",\n \"Working this out, we see that this number of manifolds is at least $\\\\exp (C^{\\\\prime }\\\\cdot d \\\\cdot \\\\exp (C^{\\\\prime }\\\\cdot d)),$ for some $C^{\\\\prime }>0$ .\",\n \"Proposition REF tells us that none of the resulting manifolds will be commensurable.\",\n \"Corollary REF now also easily follows.\",\n \"corarithmCorollary REF Let $n\\\\ge 3$ .\",\n \"We have $\\\\lim _{d\\\\rightarrow \\\\infty }\\\\operatorname{\\\\mathbb {P}}_{n,d}[\\\\text{The manifold is arithmetic}] = 0$ The only thing we need to control is the number of maximal uniform arithmetic lattices of diameter $\\\\le d$ in $\\\\operatorname{Isom}^+(\\\\mathbb {H}^n)$ .\",\n \"Let us call this number $\\\\mathrm {MALD}^u_{n}(d)$ .\",\n \"By Lemma REF we have $\\\\mathrm {MALD}^u_{n}(d) \\\\le \\\\mathrm {MAL}^u_n(e^{d/C_n})$ for some $C_n>0$ independent of $d$ .\",\n \"As such, Theorem REF implies that for every $\\\\varepsilon >0$ there exists a $\\\\beta ^{\\\\prime }>0$ so that $\\\\mathrm {MALD}^u_{n}(d) \\\\le \\\\exp (\\\\beta ^{\\\\prime }\\\\cdot d^{1+\\\\varepsilon }).$ Comparing this to Theorem REF gives the result.\"\n ],\n [\n \"Torsion\",\n \"In this section we prove Theorem REF and explain how a positive answer to Conjecture REF would lead to a sequence of manifolds that saturates the bound in that theorem up to a multiplicative constant.\"\n ],\n [\n \"An upper bound for torsion in homology\",\n \"We will start with: thmtorsionTheorem REF For every $n\\\\ge 2$ there exists a constant $C >0$ so that $\\\\log \\\\log \\\\left(\\\\left|H_i(M,\\\\mathbb {Z})_{\\\\mathrm {tors}}\\\\right|\\\\right) \\\\le C\\\\cdot \\\\operatorname{diam}(M) $ for all $i=0,\\\\ldots ,n$ for any closed hyperbolic $n$ -manifold $M$ .\",\n \"Our final goal is to employ Lemma REF .\",\n \"In the language of [7]: we need to show that a closed hyperbolic manifold $M$ is homotopy equivalent to a $(D,C\\\\cdot \\\\operatorname{diam}(M))$ -simplicial complex, where $C,D>0$ are constants depending only its dimension.\",\n \"The simplicial complex we build is the same as that used for the upper bound in Young's result (Equation (REF )).\",\n \"Set $r=\\\\operatorname{inj}(M)$ and let $S\\\\subset M$ be a maximal set of points so that $\\\\operatorname{\\\\mathrm {d}}(s,s^{\\\\prime }) \\\\ge r/4$ for all $s,s^{\\\\prime }\\\\in S$ .\",\n \"Now consider the collection $\\\\mathcal {U}=\\\\lbrace B_M(s,r/2)\\\\rbrace _{s\\\\in S},$ where $B_M(s,r/2)$ denotes the open ball in $M$ of radius $r/2$ around $s$ .\",\n \"It follows from maximality of $S$ that these balls form an open cover.\",\n \"Moreover, because their radius is half the injectivity radius they are isometrically embedded $n$ -dimensional hyperbolic balls.\",\n \"As such they are convex, which means that their intersections are convex and thus contractible.\",\n \"Hence, the nerve lemma (Lemma REF ) applies.\",\n \"This means that the homology groups we are after are those of $\\\\mathcal {N}(\\\\mathcal {U})$ .\",\n \"So we need to find bounds on the number of vertices and their degrees in $\\\\mathcal {N}(\\\\mathcal {U})$ in order to apply Lemma REF .\",\n \"The number of vertices, or equivalently the number of points in $S$ , can be bounded by $\\\\left|S\\\\right| \\\\le \\\\frac{\\\\operatorname{vol}(M)}{\\\\operatorname{vol}(B_{\\\\mathbb {H}^n}(p,r/4))} \\\\le D\\\\cdot \\\\frac{\\\\operatorname{vol}(M)}{(r/4)^n}, $ where $B_{\\\\mathbb {H}^n}(p,r/4)$ is the ball of radius $r/4$ around some point $p\\\\in \\\\mathbb {H}^n$ and $D>0$ is some constant depending only on the dimension.\",\n \"The second of these bounds again follows from the closed formula for the volume of a ball in $\\\\mathbb {H}^n$ .\",\n \"Now we use Lemma REF and Lemma REF to tell us that $(r/4)^n \\\\ge \\\\exp (-C\\\\operatorname{diam}(M)) \\\\;\\\\;\\\\text{and}\\\\;\\\\; \\\\operatorname{vol}(M) \\\\le \\\\exp (C\\\\operatorname{diam}(M))$ for some $C>0$ depending only on $n$ .\",\n \"So we obtain $\\\\left|S\\\\right| \\\\le A\\\\cdot \\\\exp (B\\\\operatorname{diam}(M)).\",\n \"$ for some $A,B>0$ depending only on $n$ .\",\n \"All that remains is to show that each vertex has a bounded number of neighbors.\",\n \"First of all note that all the neighbors of a point $s\\\\in S$ lie in $B_M(s,r)\\\\subset M$ .\",\n \"By definition of $S$ , the balls of radius $r/8$ around the neighbors of $s$ are all disjoint and all lie in $B_{M}(s,9r/8)$ .\",\n \"This means that the number of neighbors is at most $\\\\frac{\\\\operatorname{vol}(B_M(s,9r/8))}{\\\\operatorname{vol}(B_M(s,r/8))} \\\\le \\\\frac{\\\\operatorname{vol}(B_{\\\\mathbb {H}^n}(s,9r/8))}{\\\\operatorname{vol}(B_{\\\\mathbb {H}^n}(s,r/8))},$ which, for $r$ small enough, is uniformly bounded in each fixed dimension.\"\n ],\n [\n \"Sharpness of the bound\",\n \"Like we said in the introduction, if Conjecture REF holds then we would get a sequence of closed hyperbolic 3-manifolds $\\\\lbrace M_N\\\\rbrace _{N\\\\in \\\\mathbb {N}}$ such that $\\\\operatorname{diam}(M_N)\\\\rightarrow \\\\infty $ as $N\\\\rightarrow \\\\infty $ and $ \\\\log \\\\log \\\\left(\\\\left| H_1(M_N,\\\\mathbb {Z})_{\\\\mathrm {tors}}\\\\right|\\\\right) \\\\ge C \\\\operatorname{diam}(M_N).$ for some $C>0$ independent of $N$ .\",\n \"Indeed, it follows from Theorem REF together with Theorem REF that for a uniform arithmetic group $\\\\Gamma < \\\\operatorname{SL}_2(\\\\mathbb {C})$ and a sequence of congruence subgroups $\\\\lbrace \\\\Gamma _N\\\\rbrace _N$ , the manifolds $M_N = \\\\Gamma _N \\\\backslash \\\\mathbb {H}^3$ satisfy $\\\\operatorname{diam}(M_N) \\\\le A \\\\cdot \\\\log (\\\\operatorname{vol}(M_N)).$ It would follow from Conjecture REF that $\\\\operatorname{vol}(M_N)\\\\le B\\\\cdot \\\\log \\\\left(\\\\left| H_1(M_N,\\\\mathbb {Z})_{\\\\mathrm {tors}}\\\\right|\\\\right) $ for all $N$ and some $B>0$ independent of $N$ .\",\n \"Putting these two together would give the desired result.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01873"}}},{"rowIdx":796,"cells":{"text":{"kind":"list like","value":[["Shocks in the relativistic transonic accretion with low angular momentum"],["Abstract We perform 1D/2D/3D relativistic hydrodynamical simulations of accretion flows with low angular momentum, filling the gap between spherically symmetric Bondi accretion and disc-like accretion flows.","Scenarios with different directional distributions of angular momentum of falling matter and varying values of key parameters such as spin of central black hole, energy and angular momentum of matter are considered.","In some of the scenarios the shock front is formed.","We identify ranges of parameters for which the shock after formation moves towards or outwards the central black hole or the long lasting oscillating shock is observed.","The frequencies of oscillations of shock positions which can cause flaring in mass accretion rate are extracted.","The results are scalable with mass of central black hole and can be compared to the quasi-periodic oscillations of selected microquasars (such as GRS 1915+105, XTE J1550-564 or IGR J17091-3624), as well as to the supermassive black holes in the centres of weakly active galaxies, such as Sgr $A^{*}$."],["Introduction","The weakly active galaxies, where the central nucleus emits the radiation at a moderate level with respect to the most luminous quasars, are frequenlty described by the so called hot accretion flows model.","In such flows, the plasma is virially hot and optically thin, and due to the advection of energy onto black hole, the flow is radiativelly inefficient.","The prototype of an object which can be well described by such model, is the low luminosity black hole in the centre of our Galaxy, the source Sgr $A^{*}$ [64].","Also, the black hole X-ray binaries in their hard and quiescent states can be good representatives for the hot mode of accretion.","These states are frequenlty associated with the ejection of relativistic streams of plasma, which form the jets, responsible for the radio emission of the sources [20].","The matter essentially flows into black hole with the speed of light, while the sound speed at maximum can reach $c/\\sqrt{3}$ .","Therefore, the accretion flow must have transonic nature.","The viscous accretion with transonic solution based on the alpha-disk model was first studied by [46] and [42].","After that, e.g.","[2] examined the stability and structure of transonic disks.","The possibility of collimation of jets by thick accretion tori was proposed by, e.g., [54].","In respect of the value of angular momentum there are two main regimes of accretion, the Bondi accretion, which refers to spherical accretion of gas without any angular momentum, and the disk-like accretion with Keplerian distribution of angular momentum.","In the case of the former, the sonic point is located farther away from the compact object (depending on the energy of the flow) and the flow is supersonic downstream of it.","In the latter case, the flow becomes supersonic quite close to the compact object.","For gas with a low value of constant angular momentum, hence belonging in between these two regimes, the equations allow for the existence of two sonic points of both types.","The possible existence of shocks in low angular momentum flows connected with the presence of multiple critical points in the phase space has been studied from different points of view during the last thirty years.","Quasi-spherical distribution of the gas endowed by constant specific angular momentum $\\lambda $ and the arisen bistability was studied already by [3].","[21] studied the existence of critical points for realistic equation of state of the gas and showed the corresponding Rankine-Hugoniot conditions for standing shocks.","Later, the significance of this phenomenon related to the variability of some X-ray sources has been pointed out by [32] and soon after, the possibility of the shock existence together with shock conditions in different types of geometries was discussed also by [1].","The transonic solution required by the aforementioned boundary conditions is the solution having low subsonic velocity far away from the compact object ($\\mathfrak {M}<1$ , where $\\mathfrak {M}=v/c_{\\rm s}$ is the Mach number, $v=-u^r_{\\rm BL}$ is the inward radial velocity in Boyer-Lindquist coordinates, and $c_{\\rm s}$ is the local sound speed of the gas), and supersonic velocity very near to the horizon ($\\mathfrak {M}>1$ ).","Thus the flow can locally pass only through the outer or also through the inner sonic point.","The latter is globally achieved due to the shock formation between the two critical points.","More recently, the shock existence was found also in the disc-like structure with low angular momentum in hydrostatic equilibrium both in pseudo-Newtonian potential [13] and in full relativistic approach [14].","Regarding the sequence of steady solutions with different values of specific angular momentum, the hysteresis-like behavior of the shock front was proposed in the latter work.","Different geometrical configurations with polytropic or isothermal equation of state were studied in the post Newtonian approach with pseudo-Kerr potential [52].","In the general relativistic description the dependence of the flow properties (Mach number, density, temperature and pressure) in the close vicinity of the horizon was studied by [15], and the asymmetry of prograde versus retrograde accretion was shown.","The complete picture of the accreting microquasar consisting of the Keplerian cold disc and the low angular momentum hot corona, the so called two component advective flow (TCAF), was described by [7].","This model combined with the propagating oscillatory shock (POS) model was later used to explain the evolution of the low frequency QPO during the outburst in several microquasars [10], [11], [45], [19] and it was also implemented into the XSPEC software package [17].","The presence of the low angular momentum component in the accretion flow during the outbursts of microquasars seems to be essential for explaining different timing in the hard and soft bands, especially during the rising phase [63].","[18] showed with the spectral fitting, that the flux of the power-law component is higher than the flux of the disc black body in the time period, when the QPOs are seen in H 1743-322.","[17] showed similar results with the TCAF model, which gives the mass accretion rate of the disc component comparable to the accretion rate of the sub-Keplerian component, when the QPO is seen in three different sources.","Further development in this topic includes numerical simulations of low angular momentum flows in different kinds of geometrical setup.","Hydrodynamical models of the low angular momentum accretion flows have been studied already in two and three dimensions, e.g.","by [48], [30] and [31].","In those simulations, a single, constant value of the specific angular momentum was assumed, while the variability of the flows occurred due to e.g.","non-spherical or non-axisymmetric distribution of the matter.","The level of this variability was also dependent on the adiabatic index.","However, these studies have not concentrated on the existence of the standing shocks as predicted by the theoretical works mentioned above.","The consideration was also put on the problem of viscosity in such flows, especially on the influence of the viscosity on the position of the shock and the shape of the solution [8], [9], [43].","The possible consequences of viscous mechanisms in the shocked accretion flow for the QPOs' evolution was studied by [40].","Later on, [44] added the phenomenon of outflows to the picture.","[53] showed the joint role of viscosity and magnetic field considered in heating of the accretion flow combined with the cooling by Comptonization on the dynamical structure of the global accretion flow.","Such models were also studied by hydrodynamical numerical simulations in the pseudo-Newtonian description of gravity e.g.","by [24], [23], [16].","Our aim is to provide numerical simulations of low angular momentum flows, which would support or correct the semi-analytical findings about the shock existence and behavior mentioned above.","In our previous work, we performed 1D pseudo-Newtonian computations [59], where we studied the dependence of the shock solution of the parameters and the response of the shock front to the change of angular momentum in the incoming matter.","The hysteresis behavior was observed in our simulations and we have seen the repeated creation and disappearance of the shock front due to the oscillations of angular momentum in the flow.","Here we aim to provide more advanced numerical study of the flow using the full relativistic treatment of the gravity with the fixed background metric given by the Schwarzschild/Kerr solution, which is performed in one, two and three dimensions.","The organization of the paper is as follows.","In Section we briefly recall the semi-analytical treatment of the shock existence in the pseudo-Newtonian approach, which is described in details in [59].","The numerical setup of our simulations is given in Section , the different initial conditions are described in Subsections REF , REF and REF .","In Section we present our results, in particular in REF we show the 1D simulations with standing or oscillating shock location and we run models with time-dependent outer boundary condition corresponding to periodic change of angular momentum in the incoming matter.","The major part of the results is presented in REF , where the outcomes of the 2D simulations with different kind of initial conditions are presented.","In REF we confirm the reliability of the 2D results with two full three dimensional tests.","The findings of our study are discussed in Section ."],["Appearance of shocks in 1D low angular momentum flows","In this paper we follow up our previous study of the flow with constant low angular momentum $\\lambda $ , which was held in the pseudo-Newtonian framework.","Here we briefly recall the semi-analytical results in such setup, which we use as an initial setting for our GR computations (for further details see [59]).","For the analytical study we consider a non-viscous quasi-spherical polytropic flow with the equation of state $p=K \\rho ^\\gamma $ , where $p$ is the pressure, and $\\rho $ is the density in the gas.","Our EOS holds for the isentropic flow, hence the specific entropy $K$ is constant.","Using the continuity equation and energy conservation, we can find the position of the critical point $r_c$ as the root of the equation $\\mathcal {E} - \\frac{\\lambda ^2}{2r_c^2} + \\frac{1}{2(r_c-1)}-\\\\\\frac{\\gamma +1}{2(\\gamma -1)} \\left( \\frac{r_c}{4(r_c-1)^2} - \\frac{\\lambda ^2}{2r_c^2} \\right) = 0, $ where $\\mathcal {E}$ stands for energy and where we assume the Paczynski-Wiita gravitational potential in the form of $\\Phi (r)=-\\frac{1}{2(r-1)}$ , so that $r$ is given in the units of $r_g=2GM/c^2$ , and where $\\lambda $ is the value of specific angular momentum.","For a subset of the parameter space ($\\mathcal {E}, \\lambda , \\gamma $ ) there exists more than one solution of this equation (three actually), hence there are more critical points located at different positions.","It can be shown, that only two of the critical points are of a saddle type, so that the solution can pass through it.","We will call them the inner, $r_c^{\\tt in}$ , and the outer, $r_c^{\\tt out}>r_c^{\\tt in}$ , critical points, respectively.","We will call this subset as “multicritical region”.","For changing $\\lambda $ with other parameters kept constant, this region is projected onto one interval of $[\\lambda _{\\tt min}^{cr},\\lambda _{\\tt max}^{cr}]$ .","For decreasing $\\mathcal {E}$ , the interval is shifting up to a higher angular momentum.","Together with equation (REF ) determining the values of all variables at the two critical points, the relation for the derivative ${\\rm d}v/{\\rm d}r$ can be obtained from the continuity equation and the energy conservation, so that the solution can be found by integrating the equations from the critical point downwards and upwards.","The resulting two branches of solutions of course have the same parameters ($\\mathcal {E}, \\lambda , \\gamma $ ), but they differ in value of the constant specific entropy $K$ , which is given by $K = \\left( v r^2 \\frac{c_s^{\\frac{2}{\\gamma - 1}}}{\\gamma ^{\\frac{1}{\\gamma -1}} \\dot{M}} \\right)^{\\gamma - 1}.","$ This is evaluated at the critical point position ($K^{\\rm in/out}=K(r^{\\rm in/out}, v^{\\rm in/out},c_s^{\\rm in/out})$ , where $\\dot{M}$ is the adopted constant accretion rate.","Because in our model we study the motion of test matter, which does not contribute to the gravitational field, and we use a simple polytropic equation of state, the accretion rate can be given in arbitrary units and it does not affect the solution.","The only possible production of entropy is at the shock front, where jumps in radial velocity, density, and other quantities in the flow occur.","Because we are interested in the solution describing the accretion flow, and not winds, the physical requirement for the shock to occur is that the specific entropy at the inner branch is higher than at the outer one ($K^{\\tt in} > K^{\\tt out}$ ).","Moreover, the shock will appear only if the Rankine-Hugoniot conditions, expressing the conservation of mass, energy and angular momentum at the shock position are satisfied, that means that the equation $\\frac{ \\left(\\frac{1}{\\mathfrak {M}_{\\tt in}} + \\gamma \\mathfrak {M}_{\\tt in} \\right)^2 }{ \\mathfrak {M}_{\\tt in}^2(\\gamma - 1) + 2 } = \\frac{ \\left(\\frac{1}{\\mathfrak {M}_{\\tt out}} + \\gamma \\mathfrak {M}_{\\tt out}\\right)^2 }{ \\mathfrak {M}_{\\tt out}^2(\\gamma - 1) + 2 } $ holds at some radius $r_s$ ."],["Numerical setup","We performed 1D to 3D hydrodynamical simulations of the non-magnetized accreting gas on the fixed background using the HARMPI e.g., see https://github.com/atchekho/harmpi computational code [50], [51], [37], [62], [38], [29], [27] based on the original HARM code [22], [35].","The background spacetime is given by the stationary Kerr solution.","The initial conditions are set using Boyer-Lindquist coordinates, and they are transformed into the code coordinates, which are the Kerr-Shild ones.","In order to cover the whole accretion structure with sufficiently fine resolution near the black hole we use logarithmic grid in radius with superexponential grid spacing in the outermost region, so that the outer region is covered with low resolution grid and provide the reservoir of gas for accretion.","In the innermost region, the grid spans below the horizon thanks to the regularity of Kerr-Shild coordinates, having several zones inside the black hole and the free outflow boundary.","The outer boundary condition is mostly given as a free boundary, because the outer boundary is placed far enough from the central region.","In the 1D case, when long-term evolution is studied ( $t_f$ up to $5\\cdot 10^7 {\\rm M}$ )), we prescribe the inflow of matter through the outer boundary according to the PN solution.","Because the outer boundary is very far away from the centre (typically at $\\sim 10^5$ M), the radial inflowing speed is very small and the deviations between GR and PN solution are negligible.","The prescribed properties of the inflowing matter can be also time dependent, when the temporal change of the angular momentum of the matter is considered.","For 2D computations, the resolution is in the range between 256 x 128, up to 384 x 256 and 576 x 192, while in 3D case we use 256 x 128 x 96 cells.","Figure: a) Profiles of Mach number for four values of energy in the converged stationary state for γ=4/3\\gamma =4/3 and λ=3.6M\\lambda = 3.6M at the end of the simulation, t f =10 6 t_f = 10^6M.","Sonic points and shock fronts are located at the points, where the curve crosses the 𝔐=1\\mathfrak {M}=1 line (purple horizontal line).","b) The corresponding profiles of density in arbitrary units.","c) Radial velocity profiles of the flow v=-u BL 𝚛 v=- u_{\\rm BL}^{\\tt r} in the units of the speed of light, cc.In our previous work, [59], we studied the behavior of the shock solution and also its evolution with 1D simulations using the code ZEUS-MP [55], [26] supplied by the pseudo-Newtonian Paczynski-Wiita potential [47], which mimics the strong gravity effects near the black hole.","We will refer to the results presented in that paper as the PW simulations.","The parameters which we used in that work, corresponded either to the evolving frequency of quasi-periodic oscillations seen in some microquasars (e.g.","GRS1915+105 [36], XTE J1550-564 [12], GRO J1655-40 [49], or GX 339-4 [45]), or were estimated from the values holding for Sgr A* [41], [14].","However, such parameters led to an extended accretion structure, meaning especially that $r_c^{\\rm out} \\sim 10^4M$ or more.","Here, we consider different values of parameters, and we perform higher dimensional simulations.","We require that the outer critical point is located inside the computational domain in order to retain the flow subsonic at the outer boundary.","However, the total number of grid cells together with the sufficient resolution near the horizon restrict the maximal radius of the outer boundary, even though we use the logarithmic grid in radial direction.","Because of a sufficiently large value of the energy in the flow, $\\mathcal {E}$ , the radius $r_c^{\\rm out}$ is located quite close to the centre.","The typical values of parameters used in this study are $\\mathcal {E} = 0.0005, \\gamma = 4/3, \\lambda = 3.6M$ .","The shape of the corresponding solution is given in Fig.","REF .","The evolution of initially non-magnetized gas is simulated with the HARMPI package supplied with our own modifications.","The code conserves the vanishing magnetic field and there is no spurious magnetic field generated during the evolution.","We evolve two different types of initial conditions.","In the first case we prescribe $\\rho $ , $\\epsilon $ and $u^r_{\\tt BL}$ (radial component of the four-velocity) according to the Bondi solution, and we modify this solution by adding non-zero $u^\\phi _{\\tt BL}$ component of the four-velocity, where the $u^\\alpha _{\\tt BL}$ is the four-velocity in the Boyer-Lindquist coordinates.","The second option is to prescribe $\\rho $ , $\\epsilon $ and $u^\\alpha _{\\tt BL}$ in accordance to the 1D shock solution (computed with Paczynski-Wiita potential in section ), and then follow the evolution.","Such solution obtained with the simplifying assumptions is of course not the true stationary solution.","However, because of the fact, that for the given initial conditions and for chosen global parameters of the system, the gas evolves towards the true solution, the discrepancies between the GR and PN are not expected to be substantial.","Within the range of parameter space which allows for a shock solution, this 1D approximation can be used for prescribing the initial conditions.","We are interested in the evolution of such initial state towards the correct stationary state.","The EOS of the form $p = (\\gamma - 1) \\rho \\epsilon $ is used to close the system of equations, so that the isentropy assumption is not imposed.","Hence, the specific entropy $K=p/\\rho ^\\gamma $ is not constant and can evolve during the simulation.","The fiducial value of the polytropic index used in our computations is $\\gamma =4/3$ ."],["Initial conditions – Bondi solution equipped with angular momentum","In our first set of computations we adopted initially $\\rho $ , $\\epsilon $ and $u^r_{\\tt BL} = -v$ according to the Bondi solution with $ \\epsilon = K \\rho ^{\\gamma -1} /(\\gamma -1)$ , where $K$ is given by Eq.","(REF ) evaluated for the Bondi critical point (solution of Eq.","(REF ) with $\\lambda = 0$ ).","The solution is parametrized by the value of the polytropic exponent $\\gamma $ and the energy $\\mathcal {E}$ , which fix the position of the critical point $r_c^{\\rm out}$ .","We modify the initial conditions by prescribing the rotation according to relations $\\lambda = \\lambda ^{\\rm eq} \\sin ^2{\\theta }, \\qquad r&>&r_{\\rm b}, \\\\\\lambda = 0 ,\\qquad r&<&r_{\\rm a}, $ in the Boyer-Lindquist coordinates.","Between $r_{\\rm a}$ and $r_{\\rm b}$ the values are smoothened by a cubic spline.","The time component of four-velocity is set from the normalization condition $g_{\\mu \\nu }u^\\mu u^\\nu = -1$ assuming $u^\\theta _{\\tt BL}=0$ .","The factor $\\sin ^2 \\theta $ in Eq.","(REF ) ensures that angular momentum vanishes smoothly at the axis, hence the maximal value of angular momentum $\\lambda ^{\\rm eq}$ is achieved only in the equatorial plane.","One example of such initial conditions is plotted in Fig.","REF ."],["Initial conditions – Shock solution","In the second type of simulations we modify the initial data procedure such that we find the solution with the shock in the same way as in [59].","The values of $\\rho $ , $\\epsilon $ and $u^r_{\\tt BL}$ are set accordingly, with $ \\epsilon = K^{\\rm in/out} \\rho ^{\\gamma -1} /(\\gamma -1)$ , where $K^{\\rm in/out}$ is given by Eq.","(REF ) evaluated at the corresponding critical point $r_c^{\\rm in/out}$ for the two branches of solution.","Figure: Model C4: Initial data with a shock with ℰ=0.0005,λ eq =3.72\\mathcal {E}=0.0005, \\lambda ^{\\rm eq}=3.72M.However, the 1D analysis provided in that paper and here in Section was based on pseudo-Newtonian approximation, while now we use GR MHD code in order to examine the differences between the two approaches.","Moreover, the 1D analysis was held under the assumption of the quasi-spherical shape of the flow and the constant value of angular momentum.","That in particular means, that the dependence of the mass accretion rate, which is constant along the flow, scales with $r^2$ .","We cannot straightforwardly extend this model into 3D, in other words we can not set the spherical distribution of matter with the angular momentum which would be constant everywhere, because we need to avoid a non-zero angular momentum near the vertical axis.","However, the choice of the angular momentum profile in $\\theta $ direction is arbitrary, if it drops to zero near the axis.","Therefore, we choose two different profiles of angular momentum for the initial and boundary conditions, to see how much the results depend on this distribution."],["Angular momentum scaled by $\\sin ^2 \\theta $","The first choice is the same as in Section REF , given by Equations (REF ) and ().","In this case, the maximal value of angular momentum $\\lambda ^{\\rm eq}$ is obtained only in the equatorial plane and it is lower elsewhere.","This type of initial conditions is shown on Fig.","REF ."],["Constant angular momentum in a cone","In the second case we prescribe constant angular momentum in a cone with the half-angle $\\theta _c$ centered along the equatorial plane.","The values of $\\lambda $ are smoothed down from the cone towards the axis.","For this smoothening we choose a cubic spline given by the relations: $\\lambda = \\lambda ^{\\rm eq} f \\qquad \\qquad \\quad \\qquad && \\\\f = \\frac{\\theta ^2(3\\theta _c - 2\\theta )}{\\theta _c^3}, \\quad \\theta <\\frac{\\pi }{2}-\\theta _c \\\\f = 1, \\quad \\theta \\in [\\frac{\\pi }{2}-\\theta _c,\\frac{\\pi }{2}+\\theta _c] \\\\f= \\frac{(\\theta -\\pi )^2(\\frac{\\pi }{2} + 3\\theta _c - 2\\theta )}{(\\theta _c-\\frac{\\pi }{2})^3}, \\quad \\theta >\\frac{\\pi }{2}+\\theta _c$ Thus, all gas within the cone has the maximum angular momentum of $\\lambda ^{\\rm eq}$ , which resembles more the assumptions of the 1D model.","We show the initial conditions for this case in Fig.","REF .","Because of these modifications of angular momentum distribution, such initial conditions are not expected to be the stationary solution in higher dimensions.","However, our experience with 1D simulations shows, that only the presence of the inner sonic point is essential for the creation of the shock in the flow, and the exact stationary solution is not needed.","If the inner sonic point is present, then the shock bubble shape adjusts itself after a short transient time into the appropriate form.","Figure: Top panel: Position of the shock front depending on angular momentum for different values of energy.","In case of oscillations, the minimal and maximal shock position is shown.","Comparison with the shock front position obtained in the semi-analytical approach using Paczynski-Wiita potential is shown with small dots.","Bottom panel: The amplitude 𝒜\\mathcal {A} of the oscillations of the mass accretion rate and its frequency for the oscillating models."],["Initial conditions for spinning black hole","For a spinning black hole we use the initial data described in REF (angular momentum is scaled by $\\sin ^2\\theta $ ).","Our semi-analytical 1D shock solution is obtained for the Schwarzschild black hole only.","In the case of spinning black hole, the relevant values of angular momentum for the shocks can vary significantly and can be outside the possible existence of shocks in the Schwarzschild space time.","Hence, we use two different values of angular momentum: (i) $\\lambda ^s$ , to find the semi-analytic shock solution in non-spinning space time, according to which $\\rho $ , $\\epsilon $ , $u^r_{\\tt BL}$ and $K$ are set, and (ii) $\\lambda ^{\\rm eq}_g$ , which is a different value, according to which the rotation is prescribed using Equations (REF ) and (), and the normalization condition for the four-velocity.","The value of $\\lambda ^s$ is not important for the long term evolution; it only enables us to find a configuration with shock which is used for the initial conditions, so that there exists an inner sonic point at the initial time.","However, the gas is rotating with $\\lambda ^{\\rm eq}_g$ , which determines the evolution of the flow.","After a short transition time, the other variables distribution adjusts to the actual angular momentum."],["1D computations","In the general relativistic framework, the local sound speed relative to the fluid is given by $c_s ^{2} = \\gamma p \\left(\\rho +\\frac{ \\gamma p}{\\gamma -1}\\right)^{-1},$ and the radial Mach number is obtained as $\\mathfrak {M} = - u_{\\rm BL}^{\\tt r}/c_{\\rm s}$ .","The sonic points are the points, where the flow smoothly passes from subsonic to supersonic motion, that is $\\mathfrak {M} = 1$ , and its value increases inwards.","The shock front is at the place, where $\\mathfrak {M}$ discontinuously changes from $\\mathfrak {M}>1$ to $\\mathfrak {M}<1$ along the flow (with decreasing $r$ ).","The resulting profile of the Mach number $\\mathfrak {M}$ for the converged stationary state is shown in Fig.","REF for four different values of $\\mathcal {E}$ .","The inferred shock and outer sonic point locations are given in Table REF .","Table: Location of shock front and the outer sonic point of stationary solutions for different values of ℰ\\mathcal {E}.","The values in geometrized units ([M]) are inferred from the solution at the end of the simulation, at t f =10 6 t_f=10^6M.For higher energy, the outer sonic point is closer to the black hole unlike the shock position, which is located farther, and the outer supersonic region of the flow is thus shrinking.","The strength of the shock is anti-correlated with the energy (and with the location of the shock), so that for increasing energy the ratio of the post-shock density to the preshock density ($\\mathcal {R}$ ) is decreasing (see Table REF and panel b) in Figure REF ).","In comparison with the pseudo-Newtonian analytical estimate, which put the shock position for $\\mathcal {E}=0.0001$ at $r_s^{\\rm PW}(0.0001) = 21M$ , the GR computation tends to put the shock farther from the black hole.","On the other hand, the minimal stable shock position is very similar, because in GR computations the shock exists for slightly lower values of $\\lambda $ , which can be seen on Fig.","REF .","Here the dependence of the shock front position on angular momentum is shown for both the pseudo-Newtonian and GR computations.","The radial extend of possible shock existence agrees very well between PW and GR results, however for the same value of angular momentum the GR shock is located farther from the black hole.","Similarly like in the case of PW simulations, which we presented in [59], also in the GR case computed with HARMPI we have found oscillations of the shock front for higher angular momentum, which causes also the oscillation of the mass accretion rate through the inner boundary.","In Fig.","REF in the top panel we show the minimal and maximal shock positions during the simulation for the oscillating cases.","In the bottom panel, we show the amplitude of the oscillations of the mass accretion rate, which is computed as the ratio of the difference between the maximal and minimal accretion rate to its mean value $\\mathcal {A} = ({\\rm max}(\\dot{M}) - {\\rm min}(\\dot{M}))/\\bar{\\dot{M}}$ .","This ampplitude and the corresponding frequency is presented for the oscillating models.","Table: Location of shock front, if existent, with a=0.3,γ=4/3a=0.3,\\gamma =4/3 for different values of λ\\lambda for ℰ 1 =0.002\\mathcal {E}_1=0.002 and ℰ 2 =0.0000033\\mathcal {E}_2=0.0000033.","The values in geometrized units ([M]) are inferred from the solution at the end of the simulation.The general relativistic semi-analytical study of the shock solutions in Kerr metric was given in [14], however those authors considered the disc in hydrostatic equilibrium with vertically averaged quantities.","The shape of the different regions in the parameter space is given in [14], Fig.","1 for $a=0.3$ .","On Fig.","3 in that paper, the authors show the parameter space of possible shock existence for $\\tilde{\\mathcal {E}}=1.0000033, \\gamma =4/3, a=0.3$ , where $\\tilde{\\mathcal {E}}$ is the total specific energy and corresponds to $\\tilde{\\mathcal {E}} = \\mathcal {E}+1$ .","To compare their solutions with our results, we performed a set of simulations with $a=0.3$ for $\\mathcal {E}_1=0.002$ and $\\mathcal {E}_2=0.0000033$ .","The results are summarised in Table REF .","For $\\mathcal {E}_1$ [14] predicts that the shock exists in the subset of the region A (in particular in the region A$_{\\rm S}$ , which is not shown in the figure), which gives approximately the range 2.8 M $< \\lambda <$ 3.08 M. Our simulations show the shock existence for higher values of angular momentum, in particular the shock front is accreted up to $\\lambda =3.2$ M and the stationary shock exists for $\\lambda \\in [3.21 \\rm {M},3.42 \\rm {M}]$ .","For $\\mathcal {E}_2$ , those authors predict the shock existence for $\\lambda > 3.239$ M. We have found the stable shock solution for $\\lambda =3.25$ M, while for $\\lambda =3.245$ M the shock is accreted.","When the angular momentum is increased, the oscillations develop, for $\\lambda =3.4$ M their frequency $f\\sim 2.9\\cdot 10^{-4}$ M$^{-1}$ with the amplitude $\\mathcal {A} = 1.45$ .","Because a different configuration is considered in the two papers (disc in vertical hydrostatic equilibrium versus quasi-spherical flow), some differences are expected and they are more prominent for higher energy of the flow.","Quite recently, [61] studied the low angular momentum flows with shocks in the Kerr spacetime in several different geometries with different physical conditions at the shock front.","They included also the case of the conical flow with energy-preserving shock, which is close to our scenario.","Figure 2 of [61] shows, that indeed the multicritical region for the quasi-spherical flow exists for higher values of angular momentum than for the flow in vertical hydrostatic equilibrium.","The quantitative comparison of our results with this paper will be given in the future work."],["Properties of the flow for a time dependent angular momentum","Further, we repeated the PW computations done previously with the code ZEUS [59], regarding the hysteresis loop connected with the shock existence with changing angular momentum of the incoming matter.","In this case we prescribe the angular momentum of the matter coming through the outer boundary according to the time dependent relation: $\\lambda ^{\\rm eq} (t) = \\lambda ^{\\rm eq} (0) - A\\sin (t/P),$ where $A$ is the amplitude and $P$ is the period of the perturbation.","If we choose $A$ and $P$ such that the angular momentum crosses the boundary of the multicritical region from below and from above, we observe the creation and disappearance of the shock front, similarly like it was in the PW case.","The comparison of the shock front movement for three different models is given in Figure REF .","Model A1 with $\\lambda ^{\\rm eq} (0)=3.67$ M, $A=0.16$ M, $P=2\\cdot 10^6$ M does not exceed the boundary of the shock existence interval from neither side.","The shock is moving in accordance with the changing angular momentum to and from the black hole, spanning the region (16.6 M, 672.5 M).","Both sonic points exist persistently during the evolution.","Model A2 with $\\lambda ^{\\rm eq} (0)=3.655$ M, $A=0.2$ M, $P=2\\cdot 10^6$ M crosses the shock existence boundary on both sides.","As a consequence, the shock front is being accreted very quickly after it reaches the minimal stable shock position.","The time scale of this event is given by the advection time from the minimal stable shock position and does not depend on the perturbation parameters $A$ and $P$ .","From that moment the inner sonic point vanishes and the flow follows the outer Bondi-like branch of solution (for which only the outer sonic point exists) until the time, when the angular momentum increases such that the outer solution no longer exists.","At that point, the shock is formed at the position of the inner sonic point and it is moving very fast outward, where it merges with the outer sonic point, so that the type of accretion flow with the inner sonic point only is established.","Later, when the angular momentum of the flow is decreasing again, the shock forms at the outer sonic point and it moves slowly towards the black hole.","The third example (model A3 with $\\lambda ^{\\rm eq} (0)=3.68$ M, $A=0.18$ M, $P=2\\cdot 10^6$ M) shows the case, where only the upper boundary is crossed.","Here we have the shock moving slowly towards and away from the centre, where it merges with the outer sonic point.","For some time only the accretion with the inner sonic point exists and then the shock appears again.","In this case, the timescale of such changes and the velocity of the shock front is uniquely given by the parameters $A$ and $P$ .","Figure: Model B2: Bondi initial data with ℰ=0.0025,λ eq =3.8\\mathcal {E}=0.0025, \\lambda ^{\\rm eq}=3.8M.","The shock is expanding (the first snapshot at t=10 3 t=10^3M) until it merges with the outer sonic surface (second snapshot at t=2·10 4 t=2 \\cdot 10^4M).","Note the different spatial range of the second snapshot.The case, in which only the lower boundary is crossed, is not shown, because in this case the shock front is accreted during the first cycle and never appears again.","Hence, under certain circumstances, the standing shock can appear in the low angular momentum flow during accretion.","Then it can either (i) stay at certain position, (ii) move slowly downwards or upwards in the accretion flow with velocity given by the rate of change of angular momentum given by parameters $A$ and $P$ in our model, (iii) be accreted quickly from the the minimal stable shock position (close to the black hole), or (iv) be formed close to the black hole and move quickly towards the outer sonic point.","In the last two cases, the velocity of the shock front does not depend on the perturbation parameters $A$ and $P$ , but it is given by the properties of the medium.","We will study this issue in the next section with 2D simulations; in general, the velocity of the shock front is a few times lower than the sound speed in the preshock medium.","Figure: Model B2: expanding shock with Bondi initial data.","We track the position of the transonic points (i.e.","(𝔐[i]-1)(𝔐[i+1]-1)<1(\\mathfrak {M}[i] -1)(\\mathfrak {M}[i+1] -1) <1) in the equatorial plane (ii is the index of the radial coordinate on the grid) and we labeled the sonic points ( 𝔐[i]>𝔐[i+1]\\mathfrak {M}[i]>\\mathfrak {M}[i+1])by yellow points and the shock position ( 𝔐[i]<𝔐[i+1]\\mathfrak {M}[i]<\\mathfrak {M}[i+1]) by brown points.Figure: Behaviour of the shock front for different values of λ eq \\lambda ^{\\rm eq} with ℰ=0.0025\\mathcal {E}=0.0025.","Panel a) shows the position of the shock front, panel b) shows its velocity v s v_s and panel c) displays the ratio of the preshock medium sound speed c s out c^{\\rm out}_s to the shock front velocity v s v_s during the evolution.After confirming the general outcomes of our previous PW study with full GR 1D simulations, we now continue with simulations in higher dimensions.","Because our initial data are rotationally symmetric (does not depend on $\\phi $ ), is it possible to evolve only a slice spanning ($r,\\theta $ ), and with constant $\\phi $ , thus assuming that the system will remain rotationally symmetric during the evolution.","In order to justify to what extend and under which conditions this assumption is valid the comparison with 3D computation was needed.","We will comment on that in Section REF .","However, within the 2.5-D approach, the $\\phi $ component of the four-velocity is considered.","We prescribe this component in Boyer-Lindquist coordinates according to relations given by Eqs.","(REF ) and ().","We chose several exemplary simulations and we show their results in the form of snapshots.","Every snapshot contains four panels with the slices of $\\mathfrak {M}$ and its equatorial profile, $\\rho $ in arbitrary units, and $\\lambda $ in geometrized units, and it is labelled by the time $t$ in geometrized units, where $M$ is the mass of central black hole.","The axes show the position in geometrized units.","The red colour in the slice of radial Mach number corresponds to the supersonic motion, blue regions indicate the subsonic accretion.","The shock front is located at the place, where the abrupt change from supersonic to subsonic motion occurs, hence it is represented with the white curve separating red region farther from the centre and blue region closer to the centre.","The sonic curves are also white, but they lie between the blue region farther and red region closer to the centre.","On top of the radial Mach number map the velocity streamlines are plotted in blue.","These streamlines are computed from the velocity field at the given instant of time, hence they do not represent the actual history of a certain fluid parcel.","Figure: The same as in Fig.","for different values of ℰ\\mathcal {E} with λ eq =3.85\\lambda ^{\\rm eq}=3.85 M."],["Bondi solution equipped with angular momentum","At the beginning we check, that for pure Bondi solution with $\\lambda =0$ we get the stationary solution of the flow with the properties consistent with the analytical solution.","After that we start to study the slowly rotating flows.","For the first simulation, we pick the parameters such that $\\lambda ^{\\rm eq}$ belongs into the multicritical region (the region in the parameter space, where both the inner and outer sonic points exist in the 1D model), but it is very close to its upper bound, in particular $\\mathcal {E}=0.0025~M$ , $\\lambda ^{\\rm eq} = 3.6~M$ , and $\\gamma =4/3$ .","In that case, even when the shock solution is possible, it is not expected to appear, because the initial configuration is closer to the outer branch (there is no inner sonic point in the Bondi initial data).","In other words, we expect, that the rotation of the gas affects the profile of the Mach number in the innermost region in such a way, that it gets very close to 1, but does not touch it.","In Fig.","REF , the initial conditions at $t=0M$ are shown and the final state at $t_f=10^4M$ of the gas is depicted in Fig REF .","At the later time, the simulation is already relaxed to the stationary state, which resembles the outer branch.","It is interesting to note, that for $t=t_f$ the Mach number actually crosses the $\\mathfrak {M}=1$ line, however only on a very short radial range.","We would expect, that at the moment, when the inner sonic point appears, the shock creates and expands.","The reason, why it is not so in this simulation, is that the angular momentum is very close to the upper bound of the multicritical region, which is a boundary between the two distinct types of evolution.","In such case, the numerical evolution depends also on the resolution of the grid and other numerical settings.","In other words, if the parameters are close to such boundaries, then the simulations with different resolution can lead to a different type of evolution (i.e., the shock creates or not), hence it is difficult to find the exact critical value of the parameter.","This confirms our previous observation [59], that in the multicritical region the evolution of the flow can tend to either the outer “Bondi-like” branch or to the shock solution, and the choice between these two possibilities is given by the initial conditions.","In particular, the profile of the Mach number in the innermost region and the presence of the inner sonic point is crucial for the shock development.","Figure: Model C4: Initial data with a shock with ℰ=0.0005,λ eq =3.72\\mathcal {E}=0.0005, \\lambda ^{\\rm eq}=3.72M.","The shock bubble develops oscillations and changes size quasiperiodically, but does not acretes, nor does it expand outside r out r_{\\rm out}.In [56] we showed a similar computation with $\\lambda ^{\\rm eq}=3.79M, \\mathcal {E}=0.001$ , which is also close to the upper bound of the multicritical region.","Those computations were done in 3D with a different GR code, Einstein toolkithttps://einsteintoolkit.org/ [33].","The qualitative and quantitative agreement with the 1D solution is discussed there.","However the fact, that two very different codes working on different kind of grids (spherical grid logarithmic in radius versus Cartesian-like grid with the mesh refinement) show similar results is a good support for our results.","Last example is the case, when $\\lambda ^{\\rm eq}$ is above the multicritical region, hence the outer type of solution is not possible.","Therefore, the prescribed initial conditions are not close to a physical solution and the shock bubble has to appear.","However, the position of the shock front is not stable and the shock is expanding until it meets the outer sonic point and the distant supersonic region dissolves, yielding only the inner type of accretion.","Two snapshots of the evolution in times $10^3M$ and $2\\cdot 10^4M$ are given in Fig.","REF .","On the first set of panels, the emergent shock bubble is seen, which is the expanding subsonic and more dense region.","Even though the shock front is thought to be very thin, in the numerical simulation it spans across several zones, which can be seen on the equatorial profile of $\\mathfrak {M}$ .","In the last snapshot, the outer supersonic region is already dissolved and only a small supersonic region very close to the black hole can be found.","The time dependence of the shock and sonic point positions in the equatorial plane is shown in Fig.","REF , where it can be seen, that the shock meets the outer sonic point at about $t\\sim 15000~M$ , which corresponds to about 0.75 s in case of a typical microquasar with $M=10~M_\\odot $ .","This is way too short in comparison with the observational data concerning the quasiperiodic oscillations, which change frequency during few weeks.","Hence, if we want to use an oscillatory shock front model to explain the observed low-frequency QPOs, we need to obtain a solution with long lasting shock front oscillating around the slowly changing mean position, and not the fast expanding shock bubble like was observed in this case.","The first step is to find an existing stationary solution with a standing shock.","As we already mentioned, such solution is not expected to occur, if we start the evolution with initial conditions close to the Bondi solution.","We can be, however, interested in the dependence of the shock propagation on the properties of the gas.","On that account, we performed two sets of simulations, where we changed the angular momentum and the energy of the flow such that they are above the multicritical solution, and we followed the expansion of the shock front until it met the outer sonic point.","In Fig.","REF we show the behaviour of the shock front during the evolution for several different values of $\\lambda ^{\\rm eq}$ .","For $\\lambda ^{\\rm eq}=3.65$ M, just above the multicritical region, the growth of the shock bubble is slow with a long transient period, during which it is unclear, if the bubble converges to the stationary state, or it rather expands outwards.","With increasing $\\lambda ^{\\rm eq}$ the bubble expands faster.","The velocity of the shock front ranges between $0.005c$ up to $0.4c$ for the highest angular momentum.","Panel c) shows the ratio of the preshock medium sound speed $c^{\\rm out}_s$ to the shock front velocity $v_s$ .","Except for the highest $\\lambda ^{\\rm eq}=10$ M, the shock front expands by a velocity, which is a few times lower than the corresponding preshock sound speed, for the lowest $\\lambda ^{\\rm eq}=3.65$ M the ratio $c^{\\rm out}_s/v_s$ reaches the value up to 6.","In Fig.","REF we present similar results obtained for different values of $\\mathcal {E}$ .","Here, the position of the outer sonic point differs significantly for different $\\mathcal {E}$ , hence also the time for the shock front to reach the outer sonic point varies considerably.","The plots are therefore given in logarithmic scale of time.","Figure: Model C4: The shock and sonic point position and the corresponding mass accretion rate through the inner boundary.Figure: Model H3: Initial data with a shock and constant angular momentum distribution (ℰ=0.0005,λ=3.65\\mathcal {E}=0.0005, \\lambda =3.65M) in the cone with the half-angle θ c =π/4\\theta _c=\\pi /4.The velocity of the shock front decreases with the decreasing energy, and the same holds for its ratio to the sound speed of the preshock medium.","The trend is similar as for decreasing angular momentum.","The reason is, that for decreasing energy, the multicritical region exists for higher value of angular momentum, as we have seen earlier.","Hence, when we decrease the energy and keep the angular momentum constant, we are approaching the multicritical region in the parameter space.","That is well documented on the case with the lowest energy $\\mathcal {E}=0.0005$ (the purple lines in Fig.","REF ), which lies very close the multicritical region boundary and so the shock bubble waits for a very long time before it starts to expand.","Therefore, we conclude, that the shock front velocity depends on the distance of our chosen parameters ($\\lambda , \\mathcal {E}$ ) from the multicritical region in the parameter space and that it is typically a few times lower than the sound speed in the preshock medium."],["Shock solution","Because we want to find out, if there are any stationary or oscillating solutions with shocks also in 2D, similarly as in 1D, we have to prescribe the initial conditions, which are closer to the shock solution branch than the Bondi-like solution.","These initial conditions are described in Section REF .","Inspired by the range of shock solution in the PW case, we performed a set of simulations with changing angular momentum $\\lambda ^{\\rm eq} \\in [3.52M,3.7M]$ , while keeping $\\mathcal {E}=0.0025$ and $\\gamma =4/3$ .","(Note that the range in 1D paper was defined for $\\mathcal {E}=0.0001$ , not $0.0025$ .","Here, the range is such, that for the lowest $\\lambda $ the shock bubble accretes and for the higher values it expands, so it covers the whole interesting interval.)","These simulations showed, that for lower angular momentum the stationary state is obtained, while for higher angular momentum the shock bubble again expands and merges with the outer sonic surface, which is located around 280M.","The details of these computations are given in [57].","When we choose lower value of energy, the outer sonic surface is located further, so that there is more space, where the shock existence is possible.","We performed several simulations with $\\mathcal {E}=0.0005$ and $\\mathcal {E}=0.0001$ , for which $r_{\\rm out}=1475~M$ and $r_{\\rm out}=7486~M$ , respectively.","Figure: Model H3: The shock and sonic point position and the corresponding mass accretion rate through the inner boundary.For higher values of angular momentum we found the shock bubble unstable - the eddies emerge in the flow, the bubble is growing for some time, after which a quick accretion occurs accompanied by shrinking of the bubble, which is also oscillating in vertical direction.","However, it does not expand or accrete completely.","Several snapshots of the evolution are given in Fig.","REF , where the shock bubble has different shape and size at different times and the asymmetry with respect to equator can be seen.","The time dependence of the shock front position in the equatorial plane is given in Fig.","REF .","Table: Summary of computed 2D models.","All models have γ=4/3\\gamma =4/3.","Initial conditions (IC) are described in Sections (“Bondi”), (“1D+sph”), (“1D+cone”) or (“1D+spin”).","In case of 1D+cone simulations θ c =π 4\\theta _c=\\frac{\\pi }{4}.","The shock behaviour is labeled as “no” for solution without a shock, “EX” for solution with expanding shock, which merges with the outer sonic point, “OS” for solutions with oscillating shock front and “AC” for solutions, in which the shock front is accreted almost immediately.","The next three columns show the range, in which the shock is moving, the average position of the shock and the frequency of the oscillation and flares in the mass accretion rate through the inner boundary, if there are any.","The last column contains the reference to the corresponding Figures.Figure: Model F2: moderately spining black hole (ℰ=0.0005,a=0.8,λ eq =2.8\\mathcal {E}=0.0005, a=0.8, \\lambda ^{\\rm eq}=2.8M) with oscillating shock.This repetitive process causes also flaring in the mass accretion rate through the inner boundary, see Fig REF .","For $10~M_\\odot $ black hole, $t=10^6~M$ corresponds to approx.","50 s, hence the flares occur on a similar time scale as in some of the flaring states of microquasars, e.g.","the heartbeat state of GRS 1915+105 [5] or IGR J17091-3624 [4].","Moreover, we have shown, that the sources in this state show the evidence of nonlinear mechanism behind the emission of the hard component, which corresponds to the low angular momentum flow [58], [60].","However, this fact is only an indirect evidence for the presence of the shock, because there are other possible explanations of the flares, e.g.","they can be a result of the radiation pressure instability in the disc [28], [25].","Another type of simulations are those with different distribution of angular momentum (see Section REF ).","Three snapshots of such evolution are given in Fig.","REF and the corresponding shock position and mass accretion rate is given in Fig.","REF .","We found qualitatively similar behavior for different values of parameters.","The results are summarized in Table REF .","For $\\mathcal {E}=0.0005$ we identified the interval of angular momentum values for which there is oscillating shock to be $[3.59{\\rm M},\\,3.78{\\rm M}]$ (models C2-C5).","For $\\mathcal {E}=0.0001$ the interval is $\\lambda \\in [3.72{\\rm M},\\,3.86{\\rm M}]$ (models D2-D5).","In both series of simulations it is clearly visible, that increasing the value of angular momentum within the interval causes increase of the amplitude of shock oscillations.","So oscillations appear sharpest just before the upper limit of the angular momentum range and the oscillation frequency peaks in the spectrum are clearly visible in those cases.","We can compare the series of simulations C1-C5 with H1-H4 which have the same values of parameters, but differ in distribution of angular momentum (modulation by $\\sin ^2\\theta $ versus constant angular momentum in a cone).","It is clear that the results are very similar: in both cases there is a range of angular momentum in which the shock oscillating with comparable frequencies is present, with amplitude of oscillations increasing with angular momentum value.","This shows, that the main conclusions from our simulations are resistant to changes in spatial distribution of rotation profile."],["Rotating black hole","We chose three values of the spin $a$ , which can represent quite well the estimated range of spins of the know microquasars, in particular $a=0.3$ (which could be a representative value for XTE J1550-564, H1743-322, LMC X-3, A0620-00), $a=0.8$ (M33 X-7, 4U 1543-47, GRO J1655-40) and $a=0.95$ (Cyg X-1, LMC X-1, GRS 1915+105).","The estimated values of the spin are taken from [34].","For $a=0.3$ and $\\lambda ^{\\rm eq}_g \\in [3.35~\\mathrm {M},3.48~\\mathrm {M}]$ we observed a long lasting oscillating shock front.","For $\\lambda ^{\\rm eq}_g=3.34~\\mathrm {M}$ (and lower values) the shock is accreted, and for $\\lambda ^{\\rm eq}_g=3.49~\\mathrm {M}$ (and higher values) the shock surface expands.","The Table REF contains details about five selected simulations of $a=0.3$ scenarios: values on both sides of left and right boundary of interval and one in the middle of the interval of $\\lambda ^{\\rm eq}_g$ values corresponding to stable shock existence.","Figure: Model F2: moderately spining black hole (ℰ=0.0005,a=0.8,λ eq =2.8\\mathcal {E}=0.0005, a=0.8, \\lambda ^{\\rm eq}=2.8M) with oscillating shock.Next in the Table REF the results of several simulations with $a=0.8$ are shown.","In this case the interval of $\\lambda ^{\\rm eq}_g$ for which there are solutions with oscillating shock appears in the range of lower values than for $a=0.3$ .","For $\\lambda ^{\\rm eq}_g = 2.7$ M (model F1), the shock is accreted, for $\\lambda ^{\\rm eq}_g = 2.8$ M (model F2) the accretion flow contains a long lasting oscillating shock front, and for $\\lambda ^{\\rm eq}_g = 2.9$ M (model F3) the shock bubble is expanding.","The qualitative behavior of the shock front is very similar like in the Schwarzschild case, only the exact values of parameters (mainly the angular momentum of the flow) differs.","Again increase of the value of angular momentum within the oscillating shock interval corresponds to increase of the amplitude of oscillations.","Series of models E2-E4 is very similar in this regard to series C2-C5 and D2-D5.","For higher values of spin we found the computations to be more sensitive on the resolution, mainly in the radial direction close to the black hole.","If the resolution is not sufficient, then the the shock bubble can accrete even after a long oscillatory evolution.","However when we increase the resolution for the initial conditions, the shock persists in the evolution, even when it comes very close to the black hole.","Therefore we increased the radial resolution for $a=0.95$ to $N_{\\rm r}=576$ .","Even with this high resolution, the shock bubbles in simulations with $\\lambda ^{\\rm eq}=2.42$ M and $\\lambda ^{\\rm eq}=2.45$ M are accreted after long evolution with repeated shock front oscillations.","We conclude that the qualitative results obtained in the non-spinning case can be generalized for spinning black holes.","What can be clearly seen from the simulations, is that the behavior similar to the case of nonrotating BH is observed for much lower values of the angular momentum, so the spin of the black hole “adds” to the rotation of the gas.","Increasing the value of the spin of black hole decreases both limiting values of angular momentum of accreted gas and leads to a narrower interval in which the oscillating shock solutions are observed.","Figure: Model G2: rapidly spinning black hole (ℰ=0.0005,a=0.95,λ eq =2.42\\mathcal {E}=0.0005, a=0.95, \\lambda ^{\\rm eq}=2.42M) with oscillating shock.We performed two test runs with the initial conditions described in Section REF with the parameters $\\epsilon =0.0005, \\lambda ^{\\rm eq}=3.65, a=0$ in full 3D, and with the resolution $N_{\\rm r}$ x $N_{\\theta }$ x $N_{\\phi }$ equal to 384 x 192 x 64, and 256 x 128 x 96.","The simulations were performed on the Cray supercomupter cluster, typically using 16 nodes, and the message-passing interface supplemented with the hyperthreading technique was used, similarly as in [27].","The code is supposed to conserve the axi-symmetry of the initial state, which we confirm.","Because no non-axisymmetric modes appeared, the solution is the same in each $\\phi $ -slice during the evolution (see Fig.","REF ).","Figure: Model I: Density distribution in the three-dimensional simulation at time 28000M, within the innermost 50 gravitational radii from the black hole.","Resolution of this run is 384 x 192 x 64.","The colour scale and threshold is adopted to show only the densest parts of the flow.We were able evolved the system only up to $t_f = 34100$ M, and $t_f=24400$ M, for these two runs respectively, which is significantly shorter then in the 2D case.","Hence, we cannot directly compare the long term evolution of the flow.","However, the main features of the shock bubble evolution, which we observed in 2D case, appears also in 3D simulation.","Mainly, the shock bubble has similar shape, as can be seen in Fig REF , and we observe similar oscillations of the bubble and of the mass accretion rate (Fig.","REF )."],["Conclusions","In this paper we presented an extensive numerical study of the pseudo-spherical accretion flows with low angular momentum.","The simulations were performed in the general relativistic framework on the Kerr background metric with the GR MHD code HARMPI in one, two and three dimensions.","As a first step, we provided set of 1D computations, which we compared to our earlier results obtained within the pseudo-Newtonian approach with the code ZEUS and published in [59].","We confirm the qualitative properties of those simulations, which are especially the oscillation of the shock front for higher values of angular momentum and the possibility of the hysteresis effect, when the parameters of the flow are changing in time.","The oscillations of the shock front are connected with small oscillations of the value of angular momentum downward from the shock front, hence, they may be triggered by numerical errors at the shock front.","The amplitude of the oscillations reaches the maximum for intermediate shock positions and ceases again for very high shock position/angular momentum.","Figure: Model I: 3D simulation (ϵ=0.0005,λ eq =3.65,a=0.0\\epsilon =0.0005,\\lambda ^{\\rm eq}=3.65,a=0.0) with oscillating shock.If the oscillations are physical, their frequency is in good agreement with the observed low frequency quasi-periodic oscillations seen in several black hole binary systems.","Those are shifting in the range from hundreds of mHz up to few tens of Hz on the time scale of weeks (e.g.","GRS1915+105 [36], XTE J1550-564 [12], GRO J1655-40 [49] or GX 339-4 [45]).","For a fiducial mass of the microquasar equal to 10M$_\\odot $ this corresponds to the frequencies between $10^{-6}$ M$^{-1}$ up to $10^{-3}$ M$^{-1}$ in geometrized unitsThe time unit $1M$ equals to the time, which light needs to travel the half of the Schwarzschild radius of a black hole of mass M, hence for one solar mass $[t]=1M_\\odot =\\frac{GM_\\odot }{c^3}=4.9255\\cdot 10^{-6}s$ ., which is the same range as our observed values (see Fig.","REF , bottom panel for 1D results and Table REF for 2D simulations).","Hence, the change of the observed frequency during the onset and decline of the outburst is possibly connected with the change of angular momentum of the incoming matter (alternative explanation gives e.g.","[40], who consider rather the change of viscosity parameter $\\alpha $ ).","Such change can be either periodic and connected with the orbital motion of the companion, or caused by different conditions during the release of the matter.","[63] proposed the scenario of the outburst of XTE J1550-564, in which the low angular momentum component of the accretion flow is released from the magnetic trap inside the Roche lobe of the black hole, kept by the magnetic field of the active companion.","Depending on the distribution of angular momentum inside the trap and on the mechanism of breaking the magnetic confinement the angular momentum of the incoming matter from this region can slightly vary with time.","Later the angular momentum of the low angular component could be affected by the interaction with the slowly inward propagating Keplerian disc.","Figure: Model I: 3D simulation (ϵ=0.0005,λ eq =3.65,a=0.0\\epsilon =0.0005,\\lambda ^{\\rm eq}=3.65,a=0.0) with oscillating shock.To simulate such scenario, we have studied the case, when the angular momentum of the incoming matter changes periodically with time.","As was shown on Fig.","REF , the behaviour of the shock front depends on whether the value of angular momentum crosses one or both of the values $\\lambda _{\\tt min}^{cr},\\lambda _{\\tt max}^{cr}$ .","When the respective boundary is not crossed, the shock is moving slowly through the permitted region and its speed is determined by the parameters of the perturbation of the angular momentum.","When the angular momentum is in the corresponding range, the oscillations of the shock front develops.","When one of the the critical values is reached, the abrupt change of the flow geometry happens, while the flow transforms from one type of solution into another.","This transformation is achieved via the shock propagation, which is either accreted or expanding.","The velocity of the shock front agrees within the order of magnitude with the sound speed of the preshock medium, when the angular momentum is just slightly higher than $\\lambda _{\\tt max}^{cr}$ .","If the angular momentum is considerably higher, the formation and propagation of the shock can be even higher than the sound speed in the postshock medium (see Figs.","REF and REF ).","However, in nature we expect the first case to realize, because such situation can appear only if the angular momentum is increasing in the flow, which is in the Bondi-like configuration, hence it crosses smoothly through $\\lambda _{\\tt max}^{cr}$ .","When extending our results to higher dimensions, the freedom of the dependence of angular momentum on the angle $\\theta $ arises.","We have chosen two different configurations, described in Section REF .","The comparison of the corresponding models 1D+sph and 1D+cone in Table REF shows, that in the latter case, the shock is placed at larger distances.","That can be understood as the consequence of the fact, that thanks to the relations given by Eqs.","(REF ) and (REF ) larger amount of matter posses the maximal value of angular momentum.","However, the main features of the solution, including the existence of the range of angular momentum enabling the long term shock presence, the shape of the resulting shock bubble and its oscillations, are similar in both cases.","The biggest difference between the two configurations is that in case of constant angular momentum in a cone, the peaks in mass accretion rate are not so prominent as in the case of angular momentum scaled by $\\sin ^2\\theta $ .","Hence, we conclude, that the physical processes in the low angular momentum accretion flows do not qualitatively depend on the exact distribution of angular momentum, but the observable consequences (e.g.","the presence of prominent peaks and their amplitude) may be influenced by the geometry.","Similar conclusion holds also for the change of spin of the black hole.","For all three considered values of spin of the black hole ($0.3$ , $0.8$ and $0.95$ ) we have found a range of values of angular momentum of falling matter in which the long lasting oscillations of shock surface was observed.","We have also observed, that the shock existence interval of angular momentum depends on the spin of the black hole: the higher the spin value, the lower both limiting angular momentum values between which the oscillating shock exists and the narrower is the corresponding interval.","Therefore, for rapidly spinning black holes, even a small change of the angular momentum of the incoming matter leads to significant changes in the flow itself (the existence, position and oscillations of the shock front) and also in the timing properties of the outgoing radiation (which we assume to be related to the accretion rate).","Abrupt emergence, expansion or accretion of the shock which is connected with the crossing of the boundary of the shock existence region in parameter space and which leads to significant changes in the accretion rate, are more likely in the accretion flows around rapidly spinning black holes.","Simulations which we performed cover large variety of configurations: we considered different values of the spin of the central black hole, energy and angular momentum of the accreted gas, and even the distribution of the angular momentum of the matter.","Keeping all the other variables constant we have found the range of angular momentum in which there exist oscillating shock solutions in all scenarios under consideration.","Hence the oscillating regime seems to be intrinsic to the low angular momentum accretion flows.","This finding is supported by [24] who also found an oscillating shock front in their hydrodynamical simulations in the pseudo-Newtonian framework.","Our simulations in two and three dimensions show, that for such parameters the oscillating shock front is long-lasting.","That is an important ingredient for the POS model to be able to explain the QPO frequency change during outbursts of microquasars.","However, the duration of our simulations corresponds to several tens of second for typical microquasar with $M=10 M_\\odot $ , which is still short in comparison with the time scale of the QPOs frequency change (weeks).","Moreover, we did not address this question from the point of view of an analytical stability analysis.","Such analysis was provided by [39] for the spherical accretion onto non-rotating black hole and quite recently by [6] for low angular momentum flow with standing shocks, who also report the stability of the solution.","The dependence of the shock existence interval and consequently the position of the shock on the rotation of the black hole could be the probe of the black hole spin.","However, there is a degeneracy between the spin and the angular momentum of the accreting matter, which itself is mostly unknown and hard to measure.","Hence, the constraints on the spin from the oscillating shock front model explaining QPOs can be posed only when there will be available better observations of the innermost accretion region or better models predicting the angular momentum of the LAF component."],["Acknowledgements","We acknowledge support from Interdisciplinary Center for Computational Modeling of the Warsaw University (grant GB66-3) and Polish National Science Center (2012/05/E/ST9/03914).","PS is supported from Grant No.","GACR-17-06962Y."]],"string":"[\n [\n \"Shocks in the relativistic transonic accretion with low angular momentum\"\n ],\n [\n \"Abstract We perform 1D/2D/3D relativistic hydrodynamical simulations of accretion flows with low angular momentum, filling the gap between spherically symmetric Bondi accretion and disc-like accretion flows.\",\n \"Scenarios with different directional distributions of angular momentum of falling matter and varying values of key parameters such as spin of central black hole, energy and angular momentum of matter are considered.\",\n \"In some of the scenarios the shock front is formed.\",\n \"We identify ranges of parameters for which the shock after formation moves towards or outwards the central black hole or the long lasting oscillating shock is observed.\",\n \"The frequencies of oscillations of shock positions which can cause flaring in mass accretion rate are extracted.\",\n \"The results are scalable with mass of central black hole and can be compared to the quasi-periodic oscillations of selected microquasars (such as GRS 1915+105, XTE J1550-564 or IGR J17091-3624), as well as to the supermassive black holes in the centres of weakly active galaxies, such as Sgr $A^{*}$.\"\n ],\n [\n \"Introduction\",\n \"The weakly active galaxies, where the central nucleus emits the radiation at a moderate level with respect to the most luminous quasars, are frequenlty described by the so called hot accretion flows model.\",\n \"In such flows, the plasma is virially hot and optically thin, and due to the advection of energy onto black hole, the flow is radiativelly inefficient.\",\n \"The prototype of an object which can be well described by such model, is the low luminosity black hole in the centre of our Galaxy, the source Sgr $A^{*}$ [64].\",\n \"Also, the black hole X-ray binaries in their hard and quiescent states can be good representatives for the hot mode of accretion.\",\n \"These states are frequenlty associated with the ejection of relativistic streams of plasma, which form the jets, responsible for the radio emission of the sources [20].\",\n \"The matter essentially flows into black hole with the speed of light, while the sound speed at maximum can reach $c/\\\\sqrt{3}$ .\",\n \"Therefore, the accretion flow must have transonic nature.\",\n \"The viscous accretion with transonic solution based on the alpha-disk model was first studied by [46] and [42].\",\n \"After that, e.g.\",\n \"[2] examined the stability and structure of transonic disks.\",\n \"The possibility of collimation of jets by thick accretion tori was proposed by, e.g., [54].\",\n \"In respect of the value of angular momentum there are two main regimes of accretion, the Bondi accretion, which refers to spherical accretion of gas without any angular momentum, and the disk-like accretion with Keplerian distribution of angular momentum.\",\n \"In the case of the former, the sonic point is located farther away from the compact object (depending on the energy of the flow) and the flow is supersonic downstream of it.\",\n \"In the latter case, the flow becomes supersonic quite close to the compact object.\",\n \"For gas with a low value of constant angular momentum, hence belonging in between these two regimes, the equations allow for the existence of two sonic points of both types.\",\n \"The possible existence of shocks in low angular momentum flows connected with the presence of multiple critical points in the phase space has been studied from different points of view during the last thirty years.\",\n \"Quasi-spherical distribution of the gas endowed by constant specific angular momentum $\\\\lambda $ and the arisen bistability was studied already by [3].\",\n \"[21] studied the existence of critical points for realistic equation of state of the gas and showed the corresponding Rankine-Hugoniot conditions for standing shocks.\",\n \"Later, the significance of this phenomenon related to the variability of some X-ray sources has been pointed out by [32] and soon after, the possibility of the shock existence together with shock conditions in different types of geometries was discussed also by [1].\",\n \"The transonic solution required by the aforementioned boundary conditions is the solution having low subsonic velocity far away from the compact object ($\\\\mathfrak {M}<1$ , where $\\\\mathfrak {M}=v/c_{\\\\rm s}$ is the Mach number, $v=-u^r_{\\\\rm BL}$ is the inward radial velocity in Boyer-Lindquist coordinates, and $c_{\\\\rm s}$ is the local sound speed of the gas), and supersonic velocity very near to the horizon ($\\\\mathfrak {M}>1$ ).\",\n \"Thus the flow can locally pass only through the outer or also through the inner sonic point.\",\n \"The latter is globally achieved due to the shock formation between the two critical points.\",\n \"More recently, the shock existence was found also in the disc-like structure with low angular momentum in hydrostatic equilibrium both in pseudo-Newtonian potential [13] and in full relativistic approach [14].\",\n \"Regarding the sequence of steady solutions with different values of specific angular momentum, the hysteresis-like behavior of the shock front was proposed in the latter work.\",\n \"Different geometrical configurations with polytropic or isothermal equation of state were studied in the post Newtonian approach with pseudo-Kerr potential [52].\",\n \"In the general relativistic description the dependence of the flow properties (Mach number, density, temperature and pressure) in the close vicinity of the horizon was studied by [15], and the asymmetry of prograde versus retrograde accretion was shown.\",\n \"The complete picture of the accreting microquasar consisting of the Keplerian cold disc and the low angular momentum hot corona, the so called two component advective flow (TCAF), was described by [7].\",\n \"This model combined with the propagating oscillatory shock (POS) model was later used to explain the evolution of the low frequency QPO during the outburst in several microquasars [10], [11], [45], [19] and it was also implemented into the XSPEC software package [17].\",\n \"The presence of the low angular momentum component in the accretion flow during the outbursts of microquasars seems to be essential for explaining different timing in the hard and soft bands, especially during the rising phase [63].\",\n \"[18] showed with the spectral fitting, that the flux of the power-law component is higher than the flux of the disc black body in the time period, when the QPOs are seen in H 1743-322.\",\n \"[17] showed similar results with the TCAF model, which gives the mass accretion rate of the disc component comparable to the accretion rate of the sub-Keplerian component, when the QPO is seen in three different sources.\",\n \"Further development in this topic includes numerical simulations of low angular momentum flows in different kinds of geometrical setup.\",\n \"Hydrodynamical models of the low angular momentum accretion flows have been studied already in two and three dimensions, e.g.\",\n \"by [48], [30] and [31].\",\n \"In those simulations, a single, constant value of the specific angular momentum was assumed, while the variability of the flows occurred due to e.g.\",\n \"non-spherical or non-axisymmetric distribution of the matter.\",\n \"The level of this variability was also dependent on the adiabatic index.\",\n \"However, these studies have not concentrated on the existence of the standing shocks as predicted by the theoretical works mentioned above.\",\n \"The consideration was also put on the problem of viscosity in such flows, especially on the influence of the viscosity on the position of the shock and the shape of the solution [8], [9], [43].\",\n \"The possible consequences of viscous mechanisms in the shocked accretion flow for the QPOs' evolution was studied by [40].\",\n \"Later on, [44] added the phenomenon of outflows to the picture.\",\n \"[53] showed the joint role of viscosity and magnetic field considered in heating of the accretion flow combined with the cooling by Comptonization on the dynamical structure of the global accretion flow.\",\n \"Such models were also studied by hydrodynamical numerical simulations in the pseudo-Newtonian description of gravity e.g.\",\n \"by [24], [23], [16].\",\n \"Our aim is to provide numerical simulations of low angular momentum flows, which would support or correct the semi-analytical findings about the shock existence and behavior mentioned above.\",\n \"In our previous work, we performed 1D pseudo-Newtonian computations [59], where we studied the dependence of the shock solution of the parameters and the response of the shock front to the change of angular momentum in the incoming matter.\",\n \"The hysteresis behavior was observed in our simulations and we have seen the repeated creation and disappearance of the shock front due to the oscillations of angular momentum in the flow.\",\n \"Here we aim to provide more advanced numerical study of the flow using the full relativistic treatment of the gravity with the fixed background metric given by the Schwarzschild/Kerr solution, which is performed in one, two and three dimensions.\",\n \"The organization of the paper is as follows.\",\n \"In Section we briefly recall the semi-analytical treatment of the shock existence in the pseudo-Newtonian approach, which is described in details in [59].\",\n \"The numerical setup of our simulations is given in Section , the different initial conditions are described in Subsections REF , REF and REF .\",\n \"In Section we present our results, in particular in REF we show the 1D simulations with standing or oscillating shock location and we run models with time-dependent outer boundary condition corresponding to periodic change of angular momentum in the incoming matter.\",\n \"The major part of the results is presented in REF , where the outcomes of the 2D simulations with different kind of initial conditions are presented.\",\n \"In REF we confirm the reliability of the 2D results with two full three dimensional tests.\",\n \"The findings of our study are discussed in Section .\"\n ],\n [\n \"Appearance of shocks in 1D low angular momentum flows\",\n \"In this paper we follow up our previous study of the flow with constant low angular momentum $\\\\lambda $ , which was held in the pseudo-Newtonian framework.\",\n \"Here we briefly recall the semi-analytical results in such setup, which we use as an initial setting for our GR computations (for further details see [59]).\",\n \"For the analytical study we consider a non-viscous quasi-spherical polytropic flow with the equation of state $p=K \\\\rho ^\\\\gamma $ , where $p$ is the pressure, and $\\\\rho $ is the density in the gas.\",\n \"Our EOS holds for the isentropic flow, hence the specific entropy $K$ is constant.\",\n \"Using the continuity equation and energy conservation, we can find the position of the critical point $r_c$ as the root of the equation $\\\\mathcal {E} - \\\\frac{\\\\lambda ^2}{2r_c^2} + \\\\frac{1}{2(r_c-1)}-\\\\\\\\\\\\frac{\\\\gamma +1}{2(\\\\gamma -1)} \\\\left( \\\\frac{r_c}{4(r_c-1)^2} - \\\\frac{\\\\lambda ^2}{2r_c^2} \\\\right) = 0, $ where $\\\\mathcal {E}$ stands for energy and where we assume the Paczynski-Wiita gravitational potential in the form of $\\\\Phi (r)=-\\\\frac{1}{2(r-1)}$ , so that $r$ is given in the units of $r_g=2GM/c^2$ , and where $\\\\lambda $ is the value of specific angular momentum.\",\n \"For a subset of the parameter space ($\\\\mathcal {E}, \\\\lambda , \\\\gamma $ ) there exists more than one solution of this equation (three actually), hence there are more critical points located at different positions.\",\n \"It can be shown, that only two of the critical points are of a saddle type, so that the solution can pass through it.\",\n \"We will call them the inner, $r_c^{\\\\tt in}$ , and the outer, $r_c^{\\\\tt out}>r_c^{\\\\tt in}$ , critical points, respectively.\",\n \"We will call this subset as “multicritical region”.\",\n \"For changing $\\\\lambda $ with other parameters kept constant, this region is projected onto one interval of $[\\\\lambda _{\\\\tt min}^{cr},\\\\lambda _{\\\\tt max}^{cr}]$ .\",\n \"For decreasing $\\\\mathcal {E}$ , the interval is shifting up to a higher angular momentum.\",\n \"Together with equation (REF ) determining the values of all variables at the two critical points, the relation for the derivative ${\\\\rm d}v/{\\\\rm d}r$ can be obtained from the continuity equation and the energy conservation, so that the solution can be found by integrating the equations from the critical point downwards and upwards.\",\n \"The resulting two branches of solutions of course have the same parameters ($\\\\mathcal {E}, \\\\lambda , \\\\gamma $ ), but they differ in value of the constant specific entropy $K$ , which is given by $K = \\\\left( v r^2 \\\\frac{c_s^{\\\\frac{2}{\\\\gamma - 1}}}{\\\\gamma ^{\\\\frac{1}{\\\\gamma -1}} \\\\dot{M}} \\\\right)^{\\\\gamma - 1}.\",\n \"$ This is evaluated at the critical point position ($K^{\\\\rm in/out}=K(r^{\\\\rm in/out}, v^{\\\\rm in/out},c_s^{\\\\rm in/out})$ , where $\\\\dot{M}$ is the adopted constant accretion rate.\",\n \"Because in our model we study the motion of test matter, which does not contribute to the gravitational field, and we use a simple polytropic equation of state, the accretion rate can be given in arbitrary units and it does not affect the solution.\",\n \"The only possible production of entropy is at the shock front, where jumps in radial velocity, density, and other quantities in the flow occur.\",\n \"Because we are interested in the solution describing the accretion flow, and not winds, the physical requirement for the shock to occur is that the specific entropy at the inner branch is higher than at the outer one ($K^{\\\\tt in} > K^{\\\\tt out}$ ).\",\n \"Moreover, the shock will appear only if the Rankine-Hugoniot conditions, expressing the conservation of mass, energy and angular momentum at the shock position are satisfied, that means that the equation $\\\\frac{ \\\\left(\\\\frac{1}{\\\\mathfrak {M}_{\\\\tt in}} + \\\\gamma \\\\mathfrak {M}_{\\\\tt in} \\\\right)^2 }{ \\\\mathfrak {M}_{\\\\tt in}^2(\\\\gamma - 1) + 2 } = \\\\frac{ \\\\left(\\\\frac{1}{\\\\mathfrak {M}_{\\\\tt out}} + \\\\gamma \\\\mathfrak {M}_{\\\\tt out}\\\\right)^2 }{ \\\\mathfrak {M}_{\\\\tt out}^2(\\\\gamma - 1) + 2 } $ holds at some radius $r_s$ .\"\n ],\n [\n \"Numerical setup\",\n \"We performed 1D to 3D hydrodynamical simulations of the non-magnetized accreting gas on the fixed background using the HARMPI e.g., see https://github.com/atchekho/harmpi computational code [50], [51], [37], [62], [38], [29], [27] based on the original HARM code [22], [35].\",\n \"The background spacetime is given by the stationary Kerr solution.\",\n \"The initial conditions are set using Boyer-Lindquist coordinates, and they are transformed into the code coordinates, which are the Kerr-Shild ones.\",\n \"In order to cover the whole accretion structure with sufficiently fine resolution near the black hole we use logarithmic grid in radius with superexponential grid spacing in the outermost region, so that the outer region is covered with low resolution grid and provide the reservoir of gas for accretion.\",\n \"In the innermost region, the grid spans below the horizon thanks to the regularity of Kerr-Shild coordinates, having several zones inside the black hole and the free outflow boundary.\",\n \"The outer boundary condition is mostly given as a free boundary, because the outer boundary is placed far enough from the central region.\",\n \"In the 1D case, when long-term evolution is studied ( $t_f$ up to $5\\\\cdot 10^7 {\\\\rm M}$ )), we prescribe the inflow of matter through the outer boundary according to the PN solution.\",\n \"Because the outer boundary is very far away from the centre (typically at $\\\\sim 10^5$ M), the radial inflowing speed is very small and the deviations between GR and PN solution are negligible.\",\n \"The prescribed properties of the inflowing matter can be also time dependent, when the temporal change of the angular momentum of the matter is considered.\",\n \"For 2D computations, the resolution is in the range between 256 x 128, up to 384 x 256 and 576 x 192, while in 3D case we use 256 x 128 x 96 cells.\",\n \"Figure: a) Profiles of Mach number for four values of energy in the converged stationary state for γ=4/3\\\\gamma =4/3 and λ=3.6M\\\\lambda = 3.6M at the end of the simulation, t f =10 6 t_f = 10^6M.\",\n \"Sonic points and shock fronts are located at the points, where the curve crosses the 𝔐=1\\\\mathfrak {M}=1 line (purple horizontal line).\",\n \"b) The corresponding profiles of density in arbitrary units.\",\n \"c) Radial velocity profiles of the flow v=-u BL 𝚛 v=- u_{\\\\rm BL}^{\\\\tt r} in the units of the speed of light, cc.In our previous work, [59], we studied the behavior of the shock solution and also its evolution with 1D simulations using the code ZEUS-MP [55], [26] supplied by the pseudo-Newtonian Paczynski-Wiita potential [47], which mimics the strong gravity effects near the black hole.\",\n \"We will refer to the results presented in that paper as the PW simulations.\",\n \"The parameters which we used in that work, corresponded either to the evolving frequency of quasi-periodic oscillations seen in some microquasars (e.g.\",\n \"GRS1915+105 [36], XTE J1550-564 [12], GRO J1655-40 [49], or GX 339-4 [45]), or were estimated from the values holding for Sgr A* [41], [14].\",\n \"However, such parameters led to an extended accretion structure, meaning especially that $r_c^{\\\\rm out} \\\\sim 10^4M$ or more.\",\n \"Here, we consider different values of parameters, and we perform higher dimensional simulations.\",\n \"We require that the outer critical point is located inside the computational domain in order to retain the flow subsonic at the outer boundary.\",\n \"However, the total number of grid cells together with the sufficient resolution near the horizon restrict the maximal radius of the outer boundary, even though we use the logarithmic grid in radial direction.\",\n \"Because of a sufficiently large value of the energy in the flow, $\\\\mathcal {E}$ , the radius $r_c^{\\\\rm out}$ is located quite close to the centre.\",\n \"The typical values of parameters used in this study are $\\\\mathcal {E} = 0.0005, \\\\gamma = 4/3, \\\\lambda = 3.6M$ .\",\n \"The shape of the corresponding solution is given in Fig.\",\n \"REF .\",\n \"The evolution of initially non-magnetized gas is simulated with the HARMPI package supplied with our own modifications.\",\n \"The code conserves the vanishing magnetic field and there is no spurious magnetic field generated during the evolution.\",\n \"We evolve two different types of initial conditions.\",\n \"In the first case we prescribe $\\\\rho $ , $\\\\epsilon $ and $u^r_{\\\\tt BL}$ (radial component of the four-velocity) according to the Bondi solution, and we modify this solution by adding non-zero $u^\\\\phi _{\\\\tt BL}$ component of the four-velocity, where the $u^\\\\alpha _{\\\\tt BL}$ is the four-velocity in the Boyer-Lindquist coordinates.\",\n \"The second option is to prescribe $\\\\rho $ , $\\\\epsilon $ and $u^\\\\alpha _{\\\\tt BL}$ in accordance to the 1D shock solution (computed with Paczynski-Wiita potential in section ), and then follow the evolution.\",\n \"Such solution obtained with the simplifying assumptions is of course not the true stationary solution.\",\n \"However, because of the fact, that for the given initial conditions and for chosen global parameters of the system, the gas evolves towards the true solution, the discrepancies between the GR and PN are not expected to be substantial.\",\n \"Within the range of parameter space which allows for a shock solution, this 1D approximation can be used for prescribing the initial conditions.\",\n \"We are interested in the evolution of such initial state towards the correct stationary state.\",\n \"The EOS of the form $p = (\\\\gamma - 1) \\\\rho \\\\epsilon $ is used to close the system of equations, so that the isentropy assumption is not imposed.\",\n \"Hence, the specific entropy $K=p/\\\\rho ^\\\\gamma $ is not constant and can evolve during the simulation.\",\n \"The fiducial value of the polytropic index used in our computations is $\\\\gamma =4/3$ .\"\n ],\n [\n \"Initial conditions – Bondi solution equipped with angular momentum\",\n \"In our first set of computations we adopted initially $\\\\rho $ , $\\\\epsilon $ and $u^r_{\\\\tt BL} = -v$ according to the Bondi solution with $ \\\\epsilon = K \\\\rho ^{\\\\gamma -1} /(\\\\gamma -1)$ , where $K$ is given by Eq.\",\n \"(REF ) evaluated for the Bondi critical point (solution of Eq.\",\n \"(REF ) with $\\\\lambda = 0$ ).\",\n \"The solution is parametrized by the value of the polytropic exponent $\\\\gamma $ and the energy $\\\\mathcal {E}$ , which fix the position of the critical point $r_c^{\\\\rm out}$ .\",\n \"We modify the initial conditions by prescribing the rotation according to relations $\\\\lambda = \\\\lambda ^{\\\\rm eq} \\\\sin ^2{\\\\theta }, \\\\qquad r&>&r_{\\\\rm b}, \\\\\\\\\\\\lambda = 0 ,\\\\qquad r&<&r_{\\\\rm a}, $ in the Boyer-Lindquist coordinates.\",\n \"Between $r_{\\\\rm a}$ and $r_{\\\\rm b}$ the values are smoothened by a cubic spline.\",\n \"The time component of four-velocity is set from the normalization condition $g_{\\\\mu \\\\nu }u^\\\\mu u^\\\\nu = -1$ assuming $u^\\\\theta _{\\\\tt BL}=0$ .\",\n \"The factor $\\\\sin ^2 \\\\theta $ in Eq.\",\n \"(REF ) ensures that angular momentum vanishes smoothly at the axis, hence the maximal value of angular momentum $\\\\lambda ^{\\\\rm eq}$ is achieved only in the equatorial plane.\",\n \"One example of such initial conditions is plotted in Fig.\",\n \"REF .\"\n ],\n [\n \"Initial conditions – Shock solution\",\n \"In the second type of simulations we modify the initial data procedure such that we find the solution with the shock in the same way as in [59].\",\n \"The values of $\\\\rho $ , $\\\\epsilon $ and $u^r_{\\\\tt BL}$ are set accordingly, with $ \\\\epsilon = K^{\\\\rm in/out} \\\\rho ^{\\\\gamma -1} /(\\\\gamma -1)$ , where $K^{\\\\rm in/out}$ is given by Eq.\",\n \"(REF ) evaluated at the corresponding critical point $r_c^{\\\\rm in/out}$ for the two branches of solution.\",\n \"Figure: Model C4: Initial data with a shock with ℰ=0.0005,λ eq =3.72\\\\mathcal {E}=0.0005, \\\\lambda ^{\\\\rm eq}=3.72M.However, the 1D analysis provided in that paper and here in Section was based on pseudo-Newtonian approximation, while now we use GR MHD code in order to examine the differences between the two approaches.\",\n \"Moreover, the 1D analysis was held under the assumption of the quasi-spherical shape of the flow and the constant value of angular momentum.\",\n \"That in particular means, that the dependence of the mass accretion rate, which is constant along the flow, scales with $r^2$ .\",\n \"We cannot straightforwardly extend this model into 3D, in other words we can not set the spherical distribution of matter with the angular momentum which would be constant everywhere, because we need to avoid a non-zero angular momentum near the vertical axis.\",\n \"However, the choice of the angular momentum profile in $\\\\theta $ direction is arbitrary, if it drops to zero near the axis.\",\n \"Therefore, we choose two different profiles of angular momentum for the initial and boundary conditions, to see how much the results depend on this distribution.\"\n ],\n [\n \"Angular momentum scaled by $\\\\sin ^2 \\\\theta $\",\n \"The first choice is the same as in Section REF , given by Equations (REF ) and ().\",\n \"In this case, the maximal value of angular momentum $\\\\lambda ^{\\\\rm eq}$ is obtained only in the equatorial plane and it is lower elsewhere.\",\n \"This type of initial conditions is shown on Fig.\",\n \"REF .\"\n ],\n [\n \"Constant angular momentum in a cone\",\n \"In the second case we prescribe constant angular momentum in a cone with the half-angle $\\\\theta _c$ centered along the equatorial plane.\",\n \"The values of $\\\\lambda $ are smoothed down from the cone towards the axis.\",\n \"For this smoothening we choose a cubic spline given by the relations: $\\\\lambda = \\\\lambda ^{\\\\rm eq} f \\\\qquad \\\\qquad \\\\quad \\\\qquad && \\\\\\\\f = \\\\frac{\\\\theta ^2(3\\\\theta _c - 2\\\\theta )}{\\\\theta _c^3}, \\\\quad \\\\theta <\\\\frac{\\\\pi }{2}-\\\\theta _c \\\\\\\\f = 1, \\\\quad \\\\theta \\\\in [\\\\frac{\\\\pi }{2}-\\\\theta _c,\\\\frac{\\\\pi }{2}+\\\\theta _c] \\\\\\\\f= \\\\frac{(\\\\theta -\\\\pi )^2(\\\\frac{\\\\pi }{2} + 3\\\\theta _c - 2\\\\theta )}{(\\\\theta _c-\\\\frac{\\\\pi }{2})^3}, \\\\quad \\\\theta >\\\\frac{\\\\pi }{2}+\\\\theta _c$ Thus, all gas within the cone has the maximum angular momentum of $\\\\lambda ^{\\\\rm eq}$ , which resembles more the assumptions of the 1D model.\",\n \"We show the initial conditions for this case in Fig.\",\n \"REF .\",\n \"Because of these modifications of angular momentum distribution, such initial conditions are not expected to be the stationary solution in higher dimensions.\",\n \"However, our experience with 1D simulations shows, that only the presence of the inner sonic point is essential for the creation of the shock in the flow, and the exact stationary solution is not needed.\",\n \"If the inner sonic point is present, then the shock bubble shape adjusts itself after a short transient time into the appropriate form.\",\n \"Figure: Top panel: Position of the shock front depending on angular momentum for different values of energy.\",\n \"In case of oscillations, the minimal and maximal shock position is shown.\",\n \"Comparison with the shock front position obtained in the semi-analytical approach using Paczynski-Wiita potential is shown with small dots.\",\n \"Bottom panel: The amplitude 𝒜\\\\mathcal {A} of the oscillations of the mass accretion rate and its frequency for the oscillating models.\"\n ],\n [\n \"Initial conditions for spinning black hole\",\n \"For a spinning black hole we use the initial data described in REF (angular momentum is scaled by $\\\\sin ^2\\\\theta $ ).\",\n \"Our semi-analytical 1D shock solution is obtained for the Schwarzschild black hole only.\",\n \"In the case of spinning black hole, the relevant values of angular momentum for the shocks can vary significantly and can be outside the possible existence of shocks in the Schwarzschild space time.\",\n \"Hence, we use two different values of angular momentum: (i) $\\\\lambda ^s$ , to find the semi-analytic shock solution in non-spinning space time, according to which $\\\\rho $ , $\\\\epsilon $ , $u^r_{\\\\tt BL}$ and $K$ are set, and (ii) $\\\\lambda ^{\\\\rm eq}_g$ , which is a different value, according to which the rotation is prescribed using Equations (REF ) and (), and the normalization condition for the four-velocity.\",\n \"The value of $\\\\lambda ^s$ is not important for the long term evolution; it only enables us to find a configuration with shock which is used for the initial conditions, so that there exists an inner sonic point at the initial time.\",\n \"However, the gas is rotating with $\\\\lambda ^{\\\\rm eq}_g$ , which determines the evolution of the flow.\",\n \"After a short transition time, the other variables distribution adjusts to the actual angular momentum.\"\n ],\n [\n \"1D computations\",\n \"In the general relativistic framework, the local sound speed relative to the fluid is given by $c_s ^{2} = \\\\gamma p \\\\left(\\\\rho +\\\\frac{ \\\\gamma p}{\\\\gamma -1}\\\\right)^{-1},$ and the radial Mach number is obtained as $\\\\mathfrak {M} = - u_{\\\\rm BL}^{\\\\tt r}/c_{\\\\rm s}$ .\",\n \"The sonic points are the points, where the flow smoothly passes from subsonic to supersonic motion, that is $\\\\mathfrak {M} = 1$ , and its value increases inwards.\",\n \"The shock front is at the place, where $\\\\mathfrak {M}$ discontinuously changes from $\\\\mathfrak {M}>1$ to $\\\\mathfrak {M}<1$ along the flow (with decreasing $r$ ).\",\n \"The resulting profile of the Mach number $\\\\mathfrak {M}$ for the converged stationary state is shown in Fig.\",\n \"REF for four different values of $\\\\mathcal {E}$ .\",\n \"The inferred shock and outer sonic point locations are given in Table REF .\",\n \"Table: Location of shock front and the outer sonic point of stationary solutions for different values of ℰ\\\\mathcal {E}.\",\n \"The values in geometrized units ([M]) are inferred from the solution at the end of the simulation, at t f =10 6 t_f=10^6M.For higher energy, the outer sonic point is closer to the black hole unlike the shock position, which is located farther, and the outer supersonic region of the flow is thus shrinking.\",\n \"The strength of the shock is anti-correlated with the energy (and with the location of the shock), so that for increasing energy the ratio of the post-shock density to the preshock density ($\\\\mathcal {R}$ ) is decreasing (see Table REF and panel b) in Figure REF ).\",\n \"In comparison with the pseudo-Newtonian analytical estimate, which put the shock position for $\\\\mathcal {E}=0.0001$ at $r_s^{\\\\rm PW}(0.0001) = 21M$ , the GR computation tends to put the shock farther from the black hole.\",\n \"On the other hand, the minimal stable shock position is very similar, because in GR computations the shock exists for slightly lower values of $\\\\lambda $ , which can be seen on Fig.\",\n \"REF .\",\n \"Here the dependence of the shock front position on angular momentum is shown for both the pseudo-Newtonian and GR computations.\",\n \"The radial extend of possible shock existence agrees very well between PW and GR results, however for the same value of angular momentum the GR shock is located farther from the black hole.\",\n \"Similarly like in the case of PW simulations, which we presented in [59], also in the GR case computed with HARMPI we have found oscillations of the shock front for higher angular momentum, which causes also the oscillation of the mass accretion rate through the inner boundary.\",\n \"In Fig.\",\n \"REF in the top panel we show the minimal and maximal shock positions during the simulation for the oscillating cases.\",\n \"In the bottom panel, we show the amplitude of the oscillations of the mass accretion rate, which is computed as the ratio of the difference between the maximal and minimal accretion rate to its mean value $\\\\mathcal {A} = ({\\\\rm max}(\\\\dot{M}) - {\\\\rm min}(\\\\dot{M}))/\\\\bar{\\\\dot{M}}$ .\",\n \"This ampplitude and the corresponding frequency is presented for the oscillating models.\",\n \"Table: Location of shock front, if existent, with a=0.3,γ=4/3a=0.3,\\\\gamma =4/3 for different values of λ\\\\lambda for ℰ 1 =0.002\\\\mathcal {E}_1=0.002 and ℰ 2 =0.0000033\\\\mathcal {E}_2=0.0000033.\",\n \"The values in geometrized units ([M]) are inferred from the solution at the end of the simulation.The general relativistic semi-analytical study of the shock solutions in Kerr metric was given in [14], however those authors considered the disc in hydrostatic equilibrium with vertically averaged quantities.\",\n \"The shape of the different regions in the parameter space is given in [14], Fig.\",\n \"1 for $a=0.3$ .\",\n \"On Fig.\",\n \"3 in that paper, the authors show the parameter space of possible shock existence for $\\\\tilde{\\\\mathcal {E}}=1.0000033, \\\\gamma =4/3, a=0.3$ , where $\\\\tilde{\\\\mathcal {E}}$ is the total specific energy and corresponds to $\\\\tilde{\\\\mathcal {E}} = \\\\mathcal {E}+1$ .\",\n \"To compare their solutions with our results, we performed a set of simulations with $a=0.3$ for $\\\\mathcal {E}_1=0.002$ and $\\\\mathcal {E}_2=0.0000033$ .\",\n \"The results are summarised in Table REF .\",\n \"For $\\\\mathcal {E}_1$ [14] predicts that the shock exists in the subset of the region A (in particular in the region A$_{\\\\rm S}$ , which is not shown in the figure), which gives approximately the range 2.8 M $< \\\\lambda <$ 3.08 M. Our simulations show the shock existence for higher values of angular momentum, in particular the shock front is accreted up to $\\\\lambda =3.2$ M and the stationary shock exists for $\\\\lambda \\\\in [3.21 \\\\rm {M},3.42 \\\\rm {M}]$ .\",\n \"For $\\\\mathcal {E}_2$ , those authors predict the shock existence for $\\\\lambda > 3.239$ M. We have found the stable shock solution for $\\\\lambda =3.25$ M, while for $\\\\lambda =3.245$ M the shock is accreted.\",\n \"When the angular momentum is increased, the oscillations develop, for $\\\\lambda =3.4$ M their frequency $f\\\\sim 2.9\\\\cdot 10^{-4}$ M$^{-1}$ with the amplitude $\\\\mathcal {A} = 1.45$ .\",\n \"Because a different configuration is considered in the two papers (disc in vertical hydrostatic equilibrium versus quasi-spherical flow), some differences are expected and they are more prominent for higher energy of the flow.\",\n \"Quite recently, [61] studied the low angular momentum flows with shocks in the Kerr spacetime in several different geometries with different physical conditions at the shock front.\",\n \"They included also the case of the conical flow with energy-preserving shock, which is close to our scenario.\",\n \"Figure 2 of [61] shows, that indeed the multicritical region for the quasi-spherical flow exists for higher values of angular momentum than for the flow in vertical hydrostatic equilibrium.\",\n \"The quantitative comparison of our results with this paper will be given in the future work.\"\n ],\n [\n \"Properties of the flow for a time dependent angular momentum\",\n \"Further, we repeated the PW computations done previously with the code ZEUS [59], regarding the hysteresis loop connected with the shock existence with changing angular momentum of the incoming matter.\",\n \"In this case we prescribe the angular momentum of the matter coming through the outer boundary according to the time dependent relation: $\\\\lambda ^{\\\\rm eq} (t) = \\\\lambda ^{\\\\rm eq} (0) - A\\\\sin (t/P),$ where $A$ is the amplitude and $P$ is the period of the perturbation.\",\n \"If we choose $A$ and $P$ such that the angular momentum crosses the boundary of the multicritical region from below and from above, we observe the creation and disappearance of the shock front, similarly like it was in the PW case.\",\n \"The comparison of the shock front movement for three different models is given in Figure REF .\",\n \"Model A1 with $\\\\lambda ^{\\\\rm eq} (0)=3.67$ M, $A=0.16$ M, $P=2\\\\cdot 10^6$ M does not exceed the boundary of the shock existence interval from neither side.\",\n \"The shock is moving in accordance with the changing angular momentum to and from the black hole, spanning the region (16.6 M, 672.5 M).\",\n \"Both sonic points exist persistently during the evolution.\",\n \"Model A2 with $\\\\lambda ^{\\\\rm eq} (0)=3.655$ M, $A=0.2$ M, $P=2\\\\cdot 10^6$ M crosses the shock existence boundary on both sides.\",\n \"As a consequence, the shock front is being accreted very quickly after it reaches the minimal stable shock position.\",\n \"The time scale of this event is given by the advection time from the minimal stable shock position and does not depend on the perturbation parameters $A$ and $P$ .\",\n \"From that moment the inner sonic point vanishes and the flow follows the outer Bondi-like branch of solution (for which only the outer sonic point exists) until the time, when the angular momentum increases such that the outer solution no longer exists.\",\n \"At that point, the shock is formed at the position of the inner sonic point and it is moving very fast outward, where it merges with the outer sonic point, so that the type of accretion flow with the inner sonic point only is established.\",\n \"Later, when the angular momentum of the flow is decreasing again, the shock forms at the outer sonic point and it moves slowly towards the black hole.\",\n \"The third example (model A3 with $\\\\lambda ^{\\\\rm eq} (0)=3.68$ M, $A=0.18$ M, $P=2\\\\cdot 10^6$ M) shows the case, where only the upper boundary is crossed.\",\n \"Here we have the shock moving slowly towards and away from the centre, where it merges with the outer sonic point.\",\n \"For some time only the accretion with the inner sonic point exists and then the shock appears again.\",\n \"In this case, the timescale of such changes and the velocity of the shock front is uniquely given by the parameters $A$ and $P$ .\",\n \"Figure: Model B2: Bondi initial data with ℰ=0.0025,λ eq =3.8\\\\mathcal {E}=0.0025, \\\\lambda ^{\\\\rm eq}=3.8M.\",\n \"The shock is expanding (the first snapshot at t=10 3 t=10^3M) until it merges with the outer sonic surface (second snapshot at t=2·10 4 t=2 \\\\cdot 10^4M).\",\n \"Note the different spatial range of the second snapshot.The case, in which only the lower boundary is crossed, is not shown, because in this case the shock front is accreted during the first cycle and never appears again.\",\n \"Hence, under certain circumstances, the standing shock can appear in the low angular momentum flow during accretion.\",\n \"Then it can either (i) stay at certain position, (ii) move slowly downwards or upwards in the accretion flow with velocity given by the rate of change of angular momentum given by parameters $A$ and $P$ in our model, (iii) be accreted quickly from the the minimal stable shock position (close to the black hole), or (iv) be formed close to the black hole and move quickly towards the outer sonic point.\",\n \"In the last two cases, the velocity of the shock front does not depend on the perturbation parameters $A$ and $P$ , but it is given by the properties of the medium.\",\n \"We will study this issue in the next section with 2D simulations; in general, the velocity of the shock front is a few times lower than the sound speed in the preshock medium.\",\n \"Figure: Model B2: expanding shock with Bondi initial data.\",\n \"We track the position of the transonic points (i.e.\",\n \"(𝔐[i]-1)(𝔐[i+1]-1)<1(\\\\mathfrak {M}[i] -1)(\\\\mathfrak {M}[i+1] -1) <1) in the equatorial plane (ii is the index of the radial coordinate on the grid) and we labeled the sonic points ( 𝔐[i]>𝔐[i+1]\\\\mathfrak {M}[i]>\\\\mathfrak {M}[i+1])by yellow points and the shock position ( 𝔐[i]<𝔐[i+1]\\\\mathfrak {M}[i]<\\\\mathfrak {M}[i+1]) by brown points.Figure: Behaviour of the shock front for different values of λ eq \\\\lambda ^{\\\\rm eq} with ℰ=0.0025\\\\mathcal {E}=0.0025.\",\n \"Panel a) shows the position of the shock front, panel b) shows its velocity v s v_s and panel c) displays the ratio of the preshock medium sound speed c s out c^{\\\\rm out}_s to the shock front velocity v s v_s during the evolution.After confirming the general outcomes of our previous PW study with full GR 1D simulations, we now continue with simulations in higher dimensions.\",\n \"Because our initial data are rotationally symmetric (does not depend on $\\\\phi $ ), is it possible to evolve only a slice spanning ($r,\\\\theta $ ), and with constant $\\\\phi $ , thus assuming that the system will remain rotationally symmetric during the evolution.\",\n \"In order to justify to what extend and under which conditions this assumption is valid the comparison with 3D computation was needed.\",\n \"We will comment on that in Section REF .\",\n \"However, within the 2.5-D approach, the $\\\\phi $ component of the four-velocity is considered.\",\n \"We prescribe this component in Boyer-Lindquist coordinates according to relations given by Eqs.\",\n \"(REF ) and ().\",\n \"We chose several exemplary simulations and we show their results in the form of snapshots.\",\n \"Every snapshot contains four panels with the slices of $\\\\mathfrak {M}$ and its equatorial profile, $\\\\rho $ in arbitrary units, and $\\\\lambda $ in geometrized units, and it is labelled by the time $t$ in geometrized units, where $M$ is the mass of central black hole.\",\n \"The axes show the position in geometrized units.\",\n \"The red colour in the slice of radial Mach number corresponds to the supersonic motion, blue regions indicate the subsonic accretion.\",\n \"The shock front is located at the place, where the abrupt change from supersonic to subsonic motion occurs, hence it is represented with the white curve separating red region farther from the centre and blue region closer to the centre.\",\n \"The sonic curves are also white, but they lie between the blue region farther and red region closer to the centre.\",\n \"On top of the radial Mach number map the velocity streamlines are plotted in blue.\",\n \"These streamlines are computed from the velocity field at the given instant of time, hence they do not represent the actual history of a certain fluid parcel.\",\n \"Figure: The same as in Fig.\",\n \"for different values of ℰ\\\\mathcal {E} with λ eq =3.85\\\\lambda ^{\\\\rm eq}=3.85 M.\"\n ],\n [\n \"Bondi solution equipped with angular momentum\",\n \"At the beginning we check, that for pure Bondi solution with $\\\\lambda =0$ we get the stationary solution of the flow with the properties consistent with the analytical solution.\",\n \"After that we start to study the slowly rotating flows.\",\n \"For the first simulation, we pick the parameters such that $\\\\lambda ^{\\\\rm eq}$ belongs into the multicritical region (the region in the parameter space, where both the inner and outer sonic points exist in the 1D model), but it is very close to its upper bound, in particular $\\\\mathcal {E}=0.0025~M$ , $\\\\lambda ^{\\\\rm eq} = 3.6~M$ , and $\\\\gamma =4/3$ .\",\n \"In that case, even when the shock solution is possible, it is not expected to appear, because the initial configuration is closer to the outer branch (there is no inner sonic point in the Bondi initial data).\",\n \"In other words, we expect, that the rotation of the gas affects the profile of the Mach number in the innermost region in such a way, that it gets very close to 1, but does not touch it.\",\n \"In Fig.\",\n \"REF , the initial conditions at $t=0M$ are shown and the final state at $t_f=10^4M$ of the gas is depicted in Fig REF .\",\n \"At the later time, the simulation is already relaxed to the stationary state, which resembles the outer branch.\",\n \"It is interesting to note, that for $t=t_f$ the Mach number actually crosses the $\\\\mathfrak {M}=1$ line, however only on a very short radial range.\",\n \"We would expect, that at the moment, when the inner sonic point appears, the shock creates and expands.\",\n \"The reason, why it is not so in this simulation, is that the angular momentum is very close to the upper bound of the multicritical region, which is a boundary between the two distinct types of evolution.\",\n \"In such case, the numerical evolution depends also on the resolution of the grid and other numerical settings.\",\n \"In other words, if the parameters are close to such boundaries, then the simulations with different resolution can lead to a different type of evolution (i.e., the shock creates or not), hence it is difficult to find the exact critical value of the parameter.\",\n \"This confirms our previous observation [59], that in the multicritical region the evolution of the flow can tend to either the outer “Bondi-like” branch or to the shock solution, and the choice between these two possibilities is given by the initial conditions.\",\n \"In particular, the profile of the Mach number in the innermost region and the presence of the inner sonic point is crucial for the shock development.\",\n \"Figure: Model C4: Initial data with a shock with ℰ=0.0005,λ eq =3.72\\\\mathcal {E}=0.0005, \\\\lambda ^{\\\\rm eq}=3.72M.\",\n \"The shock bubble develops oscillations and changes size quasiperiodically, but does not acretes, nor does it expand outside r out r_{\\\\rm out}.In [56] we showed a similar computation with $\\\\lambda ^{\\\\rm eq}=3.79M, \\\\mathcal {E}=0.001$ , which is also close to the upper bound of the multicritical region.\",\n \"Those computations were done in 3D with a different GR code, Einstein toolkithttps://einsteintoolkit.org/ [33].\",\n \"The qualitative and quantitative agreement with the 1D solution is discussed there.\",\n \"However the fact, that two very different codes working on different kind of grids (spherical grid logarithmic in radius versus Cartesian-like grid with the mesh refinement) show similar results is a good support for our results.\",\n \"Last example is the case, when $\\\\lambda ^{\\\\rm eq}$ is above the multicritical region, hence the outer type of solution is not possible.\",\n \"Therefore, the prescribed initial conditions are not close to a physical solution and the shock bubble has to appear.\",\n \"However, the position of the shock front is not stable and the shock is expanding until it meets the outer sonic point and the distant supersonic region dissolves, yielding only the inner type of accretion.\",\n \"Two snapshots of the evolution in times $10^3M$ and $2\\\\cdot 10^4M$ are given in Fig.\",\n \"REF .\",\n \"On the first set of panels, the emergent shock bubble is seen, which is the expanding subsonic and more dense region.\",\n \"Even though the shock front is thought to be very thin, in the numerical simulation it spans across several zones, which can be seen on the equatorial profile of $\\\\mathfrak {M}$ .\",\n \"In the last snapshot, the outer supersonic region is already dissolved and only a small supersonic region very close to the black hole can be found.\",\n \"The time dependence of the shock and sonic point positions in the equatorial plane is shown in Fig.\",\n \"REF , where it can be seen, that the shock meets the outer sonic point at about $t\\\\sim 15000~M$ , which corresponds to about 0.75 s in case of a typical microquasar with $M=10~M_\\\\odot $ .\",\n \"This is way too short in comparison with the observational data concerning the quasiperiodic oscillations, which change frequency during few weeks.\",\n \"Hence, if we want to use an oscillatory shock front model to explain the observed low-frequency QPOs, we need to obtain a solution with long lasting shock front oscillating around the slowly changing mean position, and not the fast expanding shock bubble like was observed in this case.\",\n \"The first step is to find an existing stationary solution with a standing shock.\",\n \"As we already mentioned, such solution is not expected to occur, if we start the evolution with initial conditions close to the Bondi solution.\",\n \"We can be, however, interested in the dependence of the shock propagation on the properties of the gas.\",\n \"On that account, we performed two sets of simulations, where we changed the angular momentum and the energy of the flow such that they are above the multicritical solution, and we followed the expansion of the shock front until it met the outer sonic point.\",\n \"In Fig.\",\n \"REF we show the behaviour of the shock front during the evolution for several different values of $\\\\lambda ^{\\\\rm eq}$ .\",\n \"For $\\\\lambda ^{\\\\rm eq}=3.65$ M, just above the multicritical region, the growth of the shock bubble is slow with a long transient period, during which it is unclear, if the bubble converges to the stationary state, or it rather expands outwards.\",\n \"With increasing $\\\\lambda ^{\\\\rm eq}$ the bubble expands faster.\",\n \"The velocity of the shock front ranges between $0.005c$ up to $0.4c$ for the highest angular momentum.\",\n \"Panel c) shows the ratio of the preshock medium sound speed $c^{\\\\rm out}_s$ to the shock front velocity $v_s$ .\",\n \"Except for the highest $\\\\lambda ^{\\\\rm eq}=10$ M, the shock front expands by a velocity, which is a few times lower than the corresponding preshock sound speed, for the lowest $\\\\lambda ^{\\\\rm eq}=3.65$ M the ratio $c^{\\\\rm out}_s/v_s$ reaches the value up to 6.\",\n \"In Fig.\",\n \"REF we present similar results obtained for different values of $\\\\mathcal {E}$ .\",\n \"Here, the position of the outer sonic point differs significantly for different $\\\\mathcal {E}$ , hence also the time for the shock front to reach the outer sonic point varies considerably.\",\n \"The plots are therefore given in logarithmic scale of time.\",\n \"Figure: Model C4: The shock and sonic point position and the corresponding mass accretion rate through the inner boundary.Figure: Model H3: Initial data with a shock and constant angular momentum distribution (ℰ=0.0005,λ=3.65\\\\mathcal {E}=0.0005, \\\\lambda =3.65M) in the cone with the half-angle θ c =π/4\\\\theta _c=\\\\pi /4.The velocity of the shock front decreases with the decreasing energy, and the same holds for its ratio to the sound speed of the preshock medium.\",\n \"The trend is similar as for decreasing angular momentum.\",\n \"The reason is, that for decreasing energy, the multicritical region exists for higher value of angular momentum, as we have seen earlier.\",\n \"Hence, when we decrease the energy and keep the angular momentum constant, we are approaching the multicritical region in the parameter space.\",\n \"That is well documented on the case with the lowest energy $\\\\mathcal {E}=0.0005$ (the purple lines in Fig.\",\n \"REF ), which lies very close the multicritical region boundary and so the shock bubble waits for a very long time before it starts to expand.\",\n \"Therefore, we conclude, that the shock front velocity depends on the distance of our chosen parameters ($\\\\lambda , \\\\mathcal {E}$ ) from the multicritical region in the parameter space and that it is typically a few times lower than the sound speed in the preshock medium.\"\n ],\n [\n \"Shock solution\",\n \"Because we want to find out, if there are any stationary or oscillating solutions with shocks also in 2D, similarly as in 1D, we have to prescribe the initial conditions, which are closer to the shock solution branch than the Bondi-like solution.\",\n \"These initial conditions are described in Section REF .\",\n \"Inspired by the range of shock solution in the PW case, we performed a set of simulations with changing angular momentum $\\\\lambda ^{\\\\rm eq} \\\\in [3.52M,3.7M]$ , while keeping $\\\\mathcal {E}=0.0025$ and $\\\\gamma =4/3$ .\",\n \"(Note that the range in 1D paper was defined for $\\\\mathcal {E}=0.0001$ , not $0.0025$ .\",\n \"Here, the range is such, that for the lowest $\\\\lambda $ the shock bubble accretes and for the higher values it expands, so it covers the whole interesting interval.)\",\n \"These simulations showed, that for lower angular momentum the stationary state is obtained, while for higher angular momentum the shock bubble again expands and merges with the outer sonic surface, which is located around 280M.\",\n \"The details of these computations are given in [57].\",\n \"When we choose lower value of energy, the outer sonic surface is located further, so that there is more space, where the shock existence is possible.\",\n \"We performed several simulations with $\\\\mathcal {E}=0.0005$ and $\\\\mathcal {E}=0.0001$ , for which $r_{\\\\rm out}=1475~M$ and $r_{\\\\rm out}=7486~M$ , respectively.\",\n \"Figure: Model H3: The shock and sonic point position and the corresponding mass accretion rate through the inner boundary.For higher values of angular momentum we found the shock bubble unstable - the eddies emerge in the flow, the bubble is growing for some time, after which a quick accretion occurs accompanied by shrinking of the bubble, which is also oscillating in vertical direction.\",\n \"However, it does not expand or accrete completely.\",\n \"Several snapshots of the evolution are given in Fig.\",\n \"REF , where the shock bubble has different shape and size at different times and the asymmetry with respect to equator can be seen.\",\n \"The time dependence of the shock front position in the equatorial plane is given in Fig.\",\n \"REF .\",\n \"Table: Summary of computed 2D models.\",\n \"All models have γ=4/3\\\\gamma =4/3.\",\n \"Initial conditions (IC) are described in Sections (“Bondi”), (“1D+sph”), (“1D+cone”) or (“1D+spin”).\",\n \"In case of 1D+cone simulations θ c =π 4\\\\theta _c=\\\\frac{\\\\pi }{4}.\",\n \"The shock behaviour is labeled as “no” for solution without a shock, “EX” for solution with expanding shock, which merges with the outer sonic point, “OS” for solutions with oscillating shock front and “AC” for solutions, in which the shock front is accreted almost immediately.\",\n \"The next three columns show the range, in which the shock is moving, the average position of the shock and the frequency of the oscillation and flares in the mass accretion rate through the inner boundary, if there are any.\",\n \"The last column contains the reference to the corresponding Figures.Figure: Model F2: moderately spining black hole (ℰ=0.0005,a=0.8,λ eq =2.8\\\\mathcal {E}=0.0005, a=0.8, \\\\lambda ^{\\\\rm eq}=2.8M) with oscillating shock.This repetitive process causes also flaring in the mass accretion rate through the inner boundary, see Fig REF .\",\n \"For $10~M_\\\\odot $ black hole, $t=10^6~M$ corresponds to approx.\",\n \"50 s, hence the flares occur on a similar time scale as in some of the flaring states of microquasars, e.g.\",\n \"the heartbeat state of GRS 1915+105 [5] or IGR J17091-3624 [4].\",\n \"Moreover, we have shown, that the sources in this state show the evidence of nonlinear mechanism behind the emission of the hard component, which corresponds to the low angular momentum flow [58], [60].\",\n \"However, this fact is only an indirect evidence for the presence of the shock, because there are other possible explanations of the flares, e.g.\",\n \"they can be a result of the radiation pressure instability in the disc [28], [25].\",\n \"Another type of simulations are those with different distribution of angular momentum (see Section REF ).\",\n \"Three snapshots of such evolution are given in Fig.\",\n \"REF and the corresponding shock position and mass accretion rate is given in Fig.\",\n \"REF .\",\n \"We found qualitatively similar behavior for different values of parameters.\",\n \"The results are summarized in Table REF .\",\n \"For $\\\\mathcal {E}=0.0005$ we identified the interval of angular momentum values for which there is oscillating shock to be $[3.59{\\\\rm M},\\\\,3.78{\\\\rm M}]$ (models C2-C5).\",\n \"For $\\\\mathcal {E}=0.0001$ the interval is $\\\\lambda \\\\in [3.72{\\\\rm M},\\\\,3.86{\\\\rm M}]$ (models D2-D5).\",\n \"In both series of simulations it is clearly visible, that increasing the value of angular momentum within the interval causes increase of the amplitude of shock oscillations.\",\n \"So oscillations appear sharpest just before the upper limit of the angular momentum range and the oscillation frequency peaks in the spectrum are clearly visible in those cases.\",\n \"We can compare the series of simulations C1-C5 with H1-H4 which have the same values of parameters, but differ in distribution of angular momentum (modulation by $\\\\sin ^2\\\\theta $ versus constant angular momentum in a cone).\",\n \"It is clear that the results are very similar: in both cases there is a range of angular momentum in which the shock oscillating with comparable frequencies is present, with amplitude of oscillations increasing with angular momentum value.\",\n \"This shows, that the main conclusions from our simulations are resistant to changes in spatial distribution of rotation profile.\"\n ],\n [\n \"Rotating black hole\",\n \"We chose three values of the spin $a$ , which can represent quite well the estimated range of spins of the know microquasars, in particular $a=0.3$ (which could be a representative value for XTE J1550-564, H1743-322, LMC X-3, A0620-00), $a=0.8$ (M33 X-7, 4U 1543-47, GRO J1655-40) and $a=0.95$ (Cyg X-1, LMC X-1, GRS 1915+105).\",\n \"The estimated values of the spin are taken from [34].\",\n \"For $a=0.3$ and $\\\\lambda ^{\\\\rm eq}_g \\\\in [3.35~\\\\mathrm {M},3.48~\\\\mathrm {M}]$ we observed a long lasting oscillating shock front.\",\n \"For $\\\\lambda ^{\\\\rm eq}_g=3.34~\\\\mathrm {M}$ (and lower values) the shock is accreted, and for $\\\\lambda ^{\\\\rm eq}_g=3.49~\\\\mathrm {M}$ (and higher values) the shock surface expands.\",\n \"The Table REF contains details about five selected simulations of $a=0.3$ scenarios: values on both sides of left and right boundary of interval and one in the middle of the interval of $\\\\lambda ^{\\\\rm eq}_g$ values corresponding to stable shock existence.\",\n \"Figure: Model F2: moderately spining black hole (ℰ=0.0005,a=0.8,λ eq =2.8\\\\mathcal {E}=0.0005, a=0.8, \\\\lambda ^{\\\\rm eq}=2.8M) with oscillating shock.Next in the Table REF the results of several simulations with $a=0.8$ are shown.\",\n \"In this case the interval of $\\\\lambda ^{\\\\rm eq}_g$ for which there are solutions with oscillating shock appears in the range of lower values than for $a=0.3$ .\",\n \"For $\\\\lambda ^{\\\\rm eq}_g = 2.7$ M (model F1), the shock is accreted, for $\\\\lambda ^{\\\\rm eq}_g = 2.8$ M (model F2) the accretion flow contains a long lasting oscillating shock front, and for $\\\\lambda ^{\\\\rm eq}_g = 2.9$ M (model F3) the shock bubble is expanding.\",\n \"The qualitative behavior of the shock front is very similar like in the Schwarzschild case, only the exact values of parameters (mainly the angular momentum of the flow) differs.\",\n \"Again increase of the value of angular momentum within the oscillating shock interval corresponds to increase of the amplitude of oscillations.\",\n \"Series of models E2-E4 is very similar in this regard to series C2-C5 and D2-D5.\",\n \"For higher values of spin we found the computations to be more sensitive on the resolution, mainly in the radial direction close to the black hole.\",\n \"If the resolution is not sufficient, then the the shock bubble can accrete even after a long oscillatory evolution.\",\n \"However when we increase the resolution for the initial conditions, the shock persists in the evolution, even when it comes very close to the black hole.\",\n \"Therefore we increased the radial resolution for $a=0.95$ to $N_{\\\\rm r}=576$ .\",\n \"Even with this high resolution, the shock bubbles in simulations with $\\\\lambda ^{\\\\rm eq}=2.42$ M and $\\\\lambda ^{\\\\rm eq}=2.45$ M are accreted after long evolution with repeated shock front oscillations.\",\n \"We conclude that the qualitative results obtained in the non-spinning case can be generalized for spinning black holes.\",\n \"What can be clearly seen from the simulations, is that the behavior similar to the case of nonrotating BH is observed for much lower values of the angular momentum, so the spin of the black hole “adds” to the rotation of the gas.\",\n \"Increasing the value of the spin of black hole decreases both limiting values of angular momentum of accreted gas and leads to a narrower interval in which the oscillating shock solutions are observed.\",\n \"Figure: Model G2: rapidly spinning black hole (ℰ=0.0005,a=0.95,λ eq =2.42\\\\mathcal {E}=0.0005, a=0.95, \\\\lambda ^{\\\\rm eq}=2.42M) with oscillating shock.We performed two test runs with the initial conditions described in Section REF with the parameters $\\\\epsilon =0.0005, \\\\lambda ^{\\\\rm eq}=3.65, a=0$ in full 3D, and with the resolution $N_{\\\\rm r}$ x $N_{\\\\theta }$ x $N_{\\\\phi }$ equal to 384 x 192 x 64, and 256 x 128 x 96.\",\n \"The simulations were performed on the Cray supercomupter cluster, typically using 16 nodes, and the message-passing interface supplemented with the hyperthreading technique was used, similarly as in [27].\",\n \"The code is supposed to conserve the axi-symmetry of the initial state, which we confirm.\",\n \"Because no non-axisymmetric modes appeared, the solution is the same in each $\\\\phi $ -slice during the evolution (see Fig.\",\n \"REF ).\",\n \"Figure: Model I: Density distribution in the three-dimensional simulation at time 28000M, within the innermost 50 gravitational radii from the black hole.\",\n \"Resolution of this run is 384 x 192 x 64.\",\n \"The colour scale and threshold is adopted to show only the densest parts of the flow.We were able evolved the system only up to $t_f = 34100$ M, and $t_f=24400$ M, for these two runs respectively, which is significantly shorter then in the 2D case.\",\n \"Hence, we cannot directly compare the long term evolution of the flow.\",\n \"However, the main features of the shock bubble evolution, which we observed in 2D case, appears also in 3D simulation.\",\n \"Mainly, the shock bubble has similar shape, as can be seen in Fig REF , and we observe similar oscillations of the bubble and of the mass accretion rate (Fig.\",\n \"REF ).\"\n ],\n [\n \"Conclusions\",\n \"In this paper we presented an extensive numerical study of the pseudo-spherical accretion flows with low angular momentum.\",\n \"The simulations were performed in the general relativistic framework on the Kerr background metric with the GR MHD code HARMPI in one, two and three dimensions.\",\n \"As a first step, we provided set of 1D computations, which we compared to our earlier results obtained within the pseudo-Newtonian approach with the code ZEUS and published in [59].\",\n \"We confirm the qualitative properties of those simulations, which are especially the oscillation of the shock front for higher values of angular momentum and the possibility of the hysteresis effect, when the parameters of the flow are changing in time.\",\n \"The oscillations of the shock front are connected with small oscillations of the value of angular momentum downward from the shock front, hence, they may be triggered by numerical errors at the shock front.\",\n \"The amplitude of the oscillations reaches the maximum for intermediate shock positions and ceases again for very high shock position/angular momentum.\",\n \"Figure: Model I: 3D simulation (ϵ=0.0005,λ eq =3.65,a=0.0\\\\epsilon =0.0005,\\\\lambda ^{\\\\rm eq}=3.65,a=0.0) with oscillating shock.If the oscillations are physical, their frequency is in good agreement with the observed low frequency quasi-periodic oscillations seen in several black hole binary systems.\",\n \"Those are shifting in the range from hundreds of mHz up to few tens of Hz on the time scale of weeks (e.g.\",\n \"GRS1915+105 [36], XTE J1550-564 [12], GRO J1655-40 [49] or GX 339-4 [45]).\",\n \"For a fiducial mass of the microquasar equal to 10M$_\\\\odot $ this corresponds to the frequencies between $10^{-6}$ M$^{-1}$ up to $10^{-3}$ M$^{-1}$ in geometrized unitsThe time unit $1M$ equals to the time, which light needs to travel the half of the Schwarzschild radius of a black hole of mass M, hence for one solar mass $[t]=1M_\\\\odot =\\\\frac{GM_\\\\odot }{c^3}=4.9255\\\\cdot 10^{-6}s$ ., which is the same range as our observed values (see Fig.\",\n \"REF , bottom panel for 1D results and Table REF for 2D simulations).\",\n \"Hence, the change of the observed frequency during the onset and decline of the outburst is possibly connected with the change of angular momentum of the incoming matter (alternative explanation gives e.g.\",\n \"[40], who consider rather the change of viscosity parameter $\\\\alpha $ ).\",\n \"Such change can be either periodic and connected with the orbital motion of the companion, or caused by different conditions during the release of the matter.\",\n \"[63] proposed the scenario of the outburst of XTE J1550-564, in which the low angular momentum component of the accretion flow is released from the magnetic trap inside the Roche lobe of the black hole, kept by the magnetic field of the active companion.\",\n \"Depending on the distribution of angular momentum inside the trap and on the mechanism of breaking the magnetic confinement the angular momentum of the incoming matter from this region can slightly vary with time.\",\n \"Later the angular momentum of the low angular component could be affected by the interaction with the slowly inward propagating Keplerian disc.\",\n \"Figure: Model I: 3D simulation (ϵ=0.0005,λ eq =3.65,a=0.0\\\\epsilon =0.0005,\\\\lambda ^{\\\\rm eq}=3.65,a=0.0) with oscillating shock.To simulate such scenario, we have studied the case, when the angular momentum of the incoming matter changes periodically with time.\",\n \"As was shown on Fig.\",\n \"REF , the behaviour of the shock front depends on whether the value of angular momentum crosses one or both of the values $\\\\lambda _{\\\\tt min}^{cr},\\\\lambda _{\\\\tt max}^{cr}$ .\",\n \"When the respective boundary is not crossed, the shock is moving slowly through the permitted region and its speed is determined by the parameters of the perturbation of the angular momentum.\",\n \"When the angular momentum is in the corresponding range, the oscillations of the shock front develops.\",\n \"When one of the the critical values is reached, the abrupt change of the flow geometry happens, while the flow transforms from one type of solution into another.\",\n \"This transformation is achieved via the shock propagation, which is either accreted or expanding.\",\n \"The velocity of the shock front agrees within the order of magnitude with the sound speed of the preshock medium, when the angular momentum is just slightly higher than $\\\\lambda _{\\\\tt max}^{cr}$ .\",\n \"If the angular momentum is considerably higher, the formation and propagation of the shock can be even higher than the sound speed in the postshock medium (see Figs.\",\n \"REF and REF ).\",\n \"However, in nature we expect the first case to realize, because such situation can appear only if the angular momentum is increasing in the flow, which is in the Bondi-like configuration, hence it crosses smoothly through $\\\\lambda _{\\\\tt max}^{cr}$ .\",\n \"When extending our results to higher dimensions, the freedom of the dependence of angular momentum on the angle $\\\\theta $ arises.\",\n \"We have chosen two different configurations, described in Section REF .\",\n \"The comparison of the corresponding models 1D+sph and 1D+cone in Table REF shows, that in the latter case, the shock is placed at larger distances.\",\n \"That can be understood as the consequence of the fact, that thanks to the relations given by Eqs.\",\n \"(REF ) and (REF ) larger amount of matter posses the maximal value of angular momentum.\",\n \"However, the main features of the solution, including the existence of the range of angular momentum enabling the long term shock presence, the shape of the resulting shock bubble and its oscillations, are similar in both cases.\",\n \"The biggest difference between the two configurations is that in case of constant angular momentum in a cone, the peaks in mass accretion rate are not so prominent as in the case of angular momentum scaled by $\\\\sin ^2\\\\theta $ .\",\n \"Hence, we conclude, that the physical processes in the low angular momentum accretion flows do not qualitatively depend on the exact distribution of angular momentum, but the observable consequences (e.g.\",\n \"the presence of prominent peaks and their amplitude) may be influenced by the geometry.\",\n \"Similar conclusion holds also for the change of spin of the black hole.\",\n \"For all three considered values of spin of the black hole ($0.3$ , $0.8$ and $0.95$ ) we have found a range of values of angular momentum of falling matter in which the long lasting oscillations of shock surface was observed.\",\n \"We have also observed, that the shock existence interval of angular momentum depends on the spin of the black hole: the higher the spin value, the lower both limiting angular momentum values between which the oscillating shock exists and the narrower is the corresponding interval.\",\n \"Therefore, for rapidly spinning black holes, even a small change of the angular momentum of the incoming matter leads to significant changes in the flow itself (the existence, position and oscillations of the shock front) and also in the timing properties of the outgoing radiation (which we assume to be related to the accretion rate).\",\n \"Abrupt emergence, expansion or accretion of the shock which is connected with the crossing of the boundary of the shock existence region in parameter space and which leads to significant changes in the accretion rate, are more likely in the accretion flows around rapidly spinning black holes.\",\n \"Simulations which we performed cover large variety of configurations: we considered different values of the spin of the central black hole, energy and angular momentum of the accreted gas, and even the distribution of the angular momentum of the matter.\",\n \"Keeping all the other variables constant we have found the range of angular momentum in which there exist oscillating shock solutions in all scenarios under consideration.\",\n \"Hence the oscillating regime seems to be intrinsic to the low angular momentum accretion flows.\",\n \"This finding is supported by [24] who also found an oscillating shock front in their hydrodynamical simulations in the pseudo-Newtonian framework.\",\n \"Our simulations in two and three dimensions show, that for such parameters the oscillating shock front is long-lasting.\",\n \"That is an important ingredient for the POS model to be able to explain the QPO frequency change during outbursts of microquasars.\",\n \"However, the duration of our simulations corresponds to several tens of second for typical microquasar with $M=10 M_\\\\odot $ , which is still short in comparison with the time scale of the QPOs frequency change (weeks).\",\n \"Moreover, we did not address this question from the point of view of an analytical stability analysis.\",\n \"Such analysis was provided by [39] for the spherical accretion onto non-rotating black hole and quite recently by [6] for low angular momentum flow with standing shocks, who also report the stability of the solution.\",\n \"The dependence of the shock existence interval and consequently the position of the shock on the rotation of the black hole could be the probe of the black hole spin.\",\n \"However, there is a degeneracy between the spin and the angular momentum of the accreting matter, which itself is mostly unknown and hard to measure.\",\n \"Hence, the constraints on the spin from the oscillating shock front model explaining QPOs can be posed only when there will be available better observations of the innermost accretion region or better models predicting the angular momentum of the LAF component.\"\n ],\n [\n \"Acknowledgements\",\n \"We acknowledge support from Interdisciplinary Center for Computational Modeling of the Warsaw University (grant GB66-3) and Polish National Science Center (2012/05/E/ST9/03914).\",\n \"PS is supported from Grant No.\",\n \"GACR-17-06962Y.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01824"}}},{"rowIdx":797,"cells":{"text":{"kind":"list like","value":[["Theory of excitation of Rydberg polarons in an atomic quantum gas"],["Abstract We present a quantum many-body description of the excitation spectrum of Rydberg polarons in a Bose gas.","The many-body Hamiltonian is solved with functional determinant theory, and we extend this technique to describe Rydberg polarons of finite mass.","Mean-field and classical descriptions of the spectrum are derived as approximations of the many-body theory.","The various approaches are applied to experimental observations of polarons created by excitation of Rydberg atoms in a strontium Bose-Einstein condensate."],["Introduction","When an impurity is immersed in a polarizable medium, the collective response of the medium can form quasi-particles, labelled as polarons, which describe the dressing of the impurity by excitations of the background medium.","Polarons play important roles in the conduction in ionic crystals and polar semiconductors [1], spin-current transport in organic semiconductors [2], dynamics of molecules in superfluid helium nanodroplets [3], [4], [5], and collective excitations in strongly interacting fermionic and bosonic ultracold gases [6], [7], [8].","In this paper, which accompanies the publication heralding the observation of Rydberg Bose polarons [9], we present details of calculations and the interpretation of this observation as a new class of Bose polarons, formed through excitation of Sr($5sns$ $^3S_1$ ) Rydberg atoms in a strontium Bose-Einstein condensate (BEC).","We begin with a general outline of different polaron Hamiltonians, and construct the Rydberg polaron Hamiltonian used in this work.","The spectral response function in the linear response limit is derived and the many-body mean field shift of the spectral response is obtained.","The mean field theory particularly fails to describe the limits of small and large detuning, where the detailed description of quantum few- and many-body processes is particularly relevant.","These processes are fully accounted for by the bosonic functional determinant approach (FDA) [10] that solves an extended Fröhlich Hamiltonian for an impurity in a Bose gas.","In the frequency domain, the FDA predicts a gaussian shape for the intrinsic spectrum, which is a hallmark of Rydberg polarons.","A classical Monte Carlo simulation [11] which reproduces the background spectral shape is shown to miss spectral features arising from quantization of bound states.","Agreement between experimental results and FDA theory for both the observed few-body molecular spectra and the many-body polaronic states is excellent.","In the companion paper [9], we provide experimental evidence for the observation of polarons created by excitation of Rydberg atoms in a strontium Bose-Einstein condensate, with an emphasis on the determination of the excitation spectrum in the absence of density inhomogeneity.","Here we provide a detailed discussion of the theoretical methods and experimental analysis."],["Bose Polaron Hamiltonians","With ultracold atomic systems, various quantum impurity models can be studied in which an impurity interacts with a bosonic bath.","Here we focus on models that follow from the general Hamiltonian describing an impurity of mass $M$ interacting with a gas of weakly interacting bosons of mass $m$ : $\\hat{H}&=&\\sum _\\mathbf {p}\\epsilon _\\mathbf {p}^I\\hat{d}^\\dagger _\\mathbf {p}\\hat{d}_\\mathbf {p}+\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}+\\frac{g_{bb}}{2\\cal {V}}\\sum _{\\mathbf {k}\\mathbf {k}^{\\prime }\\mathbf {q}}\\hat{a}^\\dagger _{\\mathbf {k}^{\\prime }+\\mathbf {q}}\\hat{a}^\\dagger _{\\mathbf {k}-\\mathbf {q}}\\hat{a}_{\\mathbf {k}^{\\prime }}\\hat{a}_{\\mathbf {k}}\\nonumber \\\\&+&{\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {k}^{\\prime }\\mathbf {q}}V(\\mathbf {q})\\hat{d}^\\dagger _{\\mathbf {k}^{\\prime }-\\mathbf {q}}\\hat{d}_{\\mathbf {k}^{\\prime }}\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}}}_{ \\hat{H}_{IB}}.$ Here, the first two terms describe the kinetic energy of the impurities ($\\hat{d}_\\mathbf {p}$ ) and bosons ($\\hat{a}_\\mathbf {k}$ ) with dispersion relations $\\epsilon _\\mathbf {p}^I=\\frac{\\mathbf {p}^2}{2M}$ and $\\epsilon _\\mathbf {k}=\\frac{\\mathbf {k}^2}{2m}$ , respectively (unless explicitly stated we set $\\hbar =1$ ).","The third term accounts for the interaction between the bosons.","Assuming weak coupling between bosons, the microscopic coupling constant is given by the relation $g_{bb}=\\pi a_{bb}/m$ , with $a_{bb}$ the s-wave scattering length describing the low-energy boson-boson interactions.","The last term, $\\hat{H}_{IB}$ describes the impurity-boson interaction in momentum space, which, in the real space, reads $\\hat{H}_{IB}=\\int d^3r d^3r^{\\prime } \\,\\,\\hat{n}_I(\\mathbf {r}^{\\prime })\\,V(\\mathbf {r}^{\\prime }-\\mathbf {r})\\,\\hat{n}_B(\\mathbf {r}).$ Here $\\hat{n}_I(\\mathbf {r})=\\hat{\\psi }^\\dagger _I(\\mathbf {r})\\hat{\\psi }_I(\\mathbf {r})=\\frac{1}{\\cal {V}}\\sum _{\\mathbf {k}\\mathbf {q}} \\hat{d}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{d}_\\mathbf {k}e^{-i\\mathbf {q}\\mathbf {r}}$ and $\\hat{n}_B(\\mathbf {r})=\\hat{\\psi }^\\dagger _B(\\mathbf {r})\\hat{\\psi }_B(\\mathbf {r})=\\frac{1}{\\cal {V}}\\sum _{\\mathbf {k}\\mathbf {q}} \\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_\\mathbf {k}e^{-i\\mathbf {q}\\mathbf {r}}$ are the impurity and boson density, respectively.","As we consider the limit of a single impurity it is convenient to switch to the first quantized description of the impurity, which is characterized by its position and momentum operator $\\hat{\\mathbf {R}}$ and $\\hat{\\mathbf {p}}$ .","The density becomes $\\hat{n}_I(\\mathbf {r})\\rightarrow \\delta ^{(3)}(\\mathbf {r}- \\hat{\\mathbf {R}})$ and Eq.","(REF ) takes the form $\\hat{H}&=& \\frac{\\hat{\\mathbf {p}}^2}{2M}+\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}+\\frac{g_{bb}}{2\\cal {V}}\\sum _{\\mathbf {k}\\mathbf {k}^{\\prime }\\mathbf {q}}\\hat{a}^\\dagger _{\\mathbf {k}^{\\prime }+\\mathbf {q}}\\hat{a}^\\dagger _{\\mathbf {k}-\\mathbf {q}}\\hat{a}_{\\mathbf {k}^{\\prime }}\\hat{a}_{\\mathbf {k}} \\\\ \\nonumber &+&\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}}.$ From this Hamiltonian various polaron models can be derived, and we briefly explain their relevance and regimes of validity.","At $T=0$ a large fraction of bosons is condensed in a BEC.","This condensate can be regarded as a coherent state such that the zero-momentum mode $\\langle \\hat{a}_\\mathbf {k}\\rangle =\\sqrt{N_0}\\delta _{\\mathbf {k},\\mathbf {0}}$ takes on a macroscopic expectation value.","Within the Bogoliubov approximation the bosonic creation and annihilation operators in Eq.","(REF ) are expanded in fluctuations $\\hat{B}_\\mathbf {p}$ around this expectation value $\\sqrt{N_0}$ and terms of higher than quadratic order are neglected in the resulting Hamiltonian.","The purely bosonic part of the Hamiltonian can then be diagonalized by the Bogoliubov rotation $\\hat{B}_\\mathbf {p}=u_\\mathbf {p}\\hat{b}_\\mathbf {p}+ v_{-\\mathbf {p}}^* \\hat{b}^\\dagger _{-\\mathbf {p}}\\,\\,,\\,\\,\\hat{B}^\\dagger _\\mathbf {p}=u_\\mathbf {p}^* \\hat{b}^\\dagger _\\mathbf {p}+ v_{-\\mathbf {p}} \\hat{b}_{-\\mathbf {p}},$ where $\\omega _\\mathbf {k}=\\sqrt{\\epsilon _\\mathbf {p}(\\epsilon _\\mathbf {p}+2 g_{bb}\\rho )}$ is the Bogoliubov dispersion relation and $u_\\mathbf {p},v_{-\\mathbf {p}}=\\pm \\sqrt{\\frac{\\epsilon _\\mathbf {p}+g_{bb} \\rho }{2\\omega _\\mathbf {p}}\\pm \\frac{1}{2}}$ .","The density of the homogeneous condensate is given by $\\rho $ .","The transformation (REF ) yields the Bose-impurity Hamiltonian $\\hat{H}&=& \\frac{\\hat{\\mathbf {p}}^2}{2M}+\\sum _\\mathbf {k}\\omega _\\mathbf {k}\\hat{b}^\\dagger _\\mathbf {k}\\hat{b}_\\mathbf {k}\\nonumber \\\\&+&\\rho V(\\mathbf {0})+{\\frac{1}{\\sqrt{\\cal {V}}}\\sum _{ \\mathbf {q}}g(\\mathbf {q}) e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\left(\\hat{b}^\\dagger _{\\mathbf {q}}+\\hat{b}_{-\\mathbf {q}}\\right)}_{\\text{Fröhlich interaction}}\\nonumber \\\\&+&{\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\hat{B}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{B}_{\\mathbf {k}}}_{\\text{extended Fröhlich interaction}}$ where we introduced the `Fröhlich coupling' $g(\\mathbf {q})=\\sqrt{\\frac{\\rho \\epsilon _\\mathbf {q}}{\\omega _\\mathbf {q}}}V(\\mathbf {q})$ and in the last term we kept the untransformed expression for notational brevity.","In Eq.","(REF ) the term $\\rho V(\\mathbf {0})= \\rho \\int d^3r V(\\mathbf {r})$ describes the mean-field energy shift of the polaron in the Born approximation.","Neglecting the constant mean-field shift and the last term in Eq.","(REF ) one arrives at the celebrated Fröhlich model [12]: $\\hat{H}&= \\frac{\\hat{\\mathbf {p}}^2}{2M}+\\sum _\\mathbf {k}\\omega _\\mathbf {k}\\hat{b}^\\dagger _\\mathbf {k}\\hat{b}_\\mathbf {k}+\\frac{1}{\\sqrt{\\cal {V}}}\\sum _{ \\mathbf {q}}g(\\mathbf {q}) e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\left(\\hat{b}^\\dagger _{\\mathbf {q}}+\\hat{b}_{-\\mathbf {q}}\\right)$ In initial attempts to describe impurities in weakly interacting Bose gases, this Hamiltonian was used for systems close to Feshbach resonances [13], [14].","However, as shown in [15], the Fröhlich Hamiltonian alone is insufficient to describe impurities that interact strongly with atomic quantum gases (for an explicit third-order perturbation theory analysis see Ref. [16]).","Indeed the major theoretical shortcoming of Eq.","(REF ) is that from this model one cannot recover the Lippmann-Schwinger equation for two-body impurity-bose scattering.","Hence it fails to describe the underlying two-body scattering physics between the impurity and bosons including molecule formation.","Similarly the Fröhlich Hamiltonian cannot account for the intricate dynamics leading to the formation of Rydberg polarons.","For such strongly interacting systems the inclusion of the last term in Eq.","(REF ) becomes crucial.","This term accounts for pairing of the impurity with the atoms in the environment and accounts for the detailed `short-distance' physics of the problem, which is completely neglected in the Fröhlich model that is tailored for the description of long-wave length (low-energy) physics.","So far it has been experimentally verified that the inclusion of this term is relevant for the observation of Bose polarons [7], [8] where the impurity-Bose interaction can be modeled by a potential that supports only a single, weakly bound two-body molecular state.","In the present work we encounter a new type of impurity problem where the impurity is dressed by large sets of molecular states that have ultra-long-range character.","This yields the novel physics of Rydberg polarons that is beyond the physics of Bose polarons so far observed in experiments and discussed in the literature.","To account for the relevant Rydberg molecular physics theoretically, we analyze the full Hamiltonian Eq.","(REF ).","We note that the physics of the Rydberg oligomer states is different from that of Efimov states in the Bose polaron problem [15], [17], [18], [19].","While Efimov states are also multi-body bound states, they arise due a quantum anomaly of the underlying quantum field theory [20], [21], [22].","Indeed, the Efimov effect [23], [24], [25] gives rise to an infinite series of three-body states (and also states of larger atom number [26], [27]) that respect a discrete scaling symmetry.","The Efimov effect arises for resonant short-range two-body interactions in three dimensions.","In previous experiments studying Bose polarons close to a Feshbach resonance it was found that these Efimov states do not strongly influence the polaron physics [7], [8].","In our case the multi-molecular states are not related to the Efimov effect and dimer, trimer, etc.","states are rather comprised of nearly independently bound two-body molecular states that feature only very weak three- and higher-body correlations.","In contrast, in Efimov physics, a single impurity potentially mediates strong correlations between two particles in the Bose gas.","Due to large binding energies of Rydberg molecular states, this induced interaction is expected to be small for Rydberg polarons [28]."],["Polaron formation","The formation of a polaron can be understood as dressing of the impurity by excitations of the bosonic bath, which entangles the momentum of the impurity with that of the bath excitations.","This can be illustrated with a simple wave function expansion for a polaron of zero momentum $\\mathinner {|{\\Psi }\\rangle }&=\\sqrt{Z}\\mathinner {|{BEC}\\rangle }\\mathinner {|{\\mathbf {p}=0}\\rangle }_I+\\sum _\\mathbf {k}\\alpha _\\mathbf {k}\\hat{b}^\\dagger _{-\\mathbf {k}} \\hat{b}_\\mathbf {0}\\mathinner {|{BEC}\\rangle } \\mathinner {|{\\mathbf {k}}\\rangle }_I\\nonumber \\\\&+\\sum _{\\mathbf {k}\\mathbf {q}} \\alpha _{\\mathbf {k}\\mathbf {q}} \\hat{b}^\\dagger _{-\\mathbf {k}} \\hat{b}^\\dagger _{\\mathbf {k}-\\mathbf {q}} \\hat{b}_\\mathbf {0}\\hat{b}_\\mathbf {0}\\mathinner {|{BEC}\\rangle } \\mathinner {|{\\mathbf {q}}\\rangle }_I+\\ldots $ where $\\mathinner {|{\\mathbf {p}}\\rangle }_I$ denotes impurity momentum states.","The unperturbed impurity-bath state in the first terms becomes supplemented by `particle-hole' fluctuations of the BEC state given by the second, etc.","terms.","These terms describe the polaronic dressing of the impurity by bath excitations that leads to the formation of the polaron.","A Fourier transformation of the parameter $\\alpha _\\mathbf {k}$ to real space reveals the creation of density modulations in the medium, which represent the formation of a dressing cloud that is build around the impurity particle.","In fact, wave functions of the type in Eq.","(REF ) were found to yield a rather accurate description of impurities coupled to a fermionic bath of cold atoms via Feshbach resonances [29], [30], [31], [32], [33] and also describe exciton-impurities in two-dimensional semiconductors [34].","When truncated at the one-excitation level, such an ansatz leads to limited agreement with experimental observations for impurities in a Bose gas [35], [36].","Dressing can also be understood from a perturbative expansion in terms of Feynman diagrams as shown in Fig.","REF .","Here solid lines represent the propagating impurity, and the dashed lines denote Bosons that are excited from the BEC.","The Fröhlich model considers only processes where bosons are excited out of the condensate and then re-enter the BEC in their next scattering event with the impurity.","However, these `Fröhlich scattering processes' can neither account for molecular bound-state formation, nor for intricate strong-coupling physics found close to Feshbach resonances.","To describe these phenomena, the last term in Eq.","(REF ) has to be considered.","This term allows for scattering of bosons where an excited boson does not directly reenter the BEC but can scatter off the impurity arbitrarily many times.","In this process, illustrated as gray disk in Fig.","REF , the boson changes its momentum.","The infinite repetition of such scattering processes represents the Lippmann-Schwinger equation in terms of Feynman diagrams and accounts for molecular bound state formation."],["Rydberg Polaron Hamiltonian","We now specialize to the case of Rydberg impurities.","When a bath atom is excited into a Rydberg state of principal quantum number $n$ , it interacts with the surrounding ground-state atoms through a pseudopotential, first proposed by Fermi [37], in which molecular binding occurs through frequent scattering of the nearly free and zero-energy Rydberg electron from the ground-state atom.","In this picture, the Born-Oppenheimer potential for a ground-state atom at distance $\\mathbf {r}$ from the Rydberg impurity, retaining $s$ -wave and $p$ -wave scattering partial waves, is given as [37], [38], [39], [40], [41] $V_\\text{Ryd}(\\mathbf {r})&=&\\frac{2\\pi \\hbar ^2}{m_e} a_s|\\Psi (\\mathbf {r})|^2+\\frac{6\\pi \\hbar ^2 }{m_e} a_p^3|\\overrightarrow{\\nabla }\\Psi (\\mathbf {r})|^2, $ where we kept explicit factors of $\\hbar $ and $\\Psi (\\mathbf {r})$ is the Rydberg electron wave function, $a_s$ and $a_p$ are the momentum-dependent $s$ -wave and $p$ -wave scattering lengths, and $m_e$ is the electron mass.","When $a_s<0$ , $V(\\mathbf {r})$ can support molecular states with one or more ground-state atoms bound to the impurity [38], [42], [43].","A Rydberg polaron is formed when the Rydberg impurity is dressed by the occupation of a large number of these bound states in addition to finite momentum states.","This process gives rise to an absorption spectrum of a distribution of molecular peaks with a Gaussian envelope, which is the key spectral signature of Rydberg polaron formation.","The large energy scale of the Rydberg molecular states involved in the formation of Rydberg polarons allow us to simplify the extended Fröhlich Hamiltonian Eq.","(REF ): the typical energy range of Rydberg molecules is $0.1-10$ MHz for high quantum number $n$ , while the typical energy scale for Bose-Bose interactions is 1-10 KHz.","Therefore bosons that are bound to a Rydberg impurity probe momentum scales deep in the particle branch of the Bogoliubov dispersion relation, in a regime where the Bogoliubov factors $u_\\mathbf {p}=1$ and $v_\\mathbf {p}=0$ .","Hence we can neglect the Bose-Bose interactions and Eq.","(REF ) reduces to the simplified, extended Fröhlich model [11] $\\hat{H}&=& \\frac{\\hat{\\mathbf {p}}^2}{2M}+\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}+\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}}\\nonumber \\\\&=& \\frac{\\hat{\\mathbf {p}}^2}{2M}+\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\hat{b}^\\dagger _\\mathbf {k}\\hat{b}_\\mathbf {k}+\\rho V(\\mathbf {0}) \\nonumber \\\\&+&{\\frac{1}{\\sqrt{\\cal {V}}}\\sum _{ \\mathbf {q}}\\sqrt{\\rho }V(\\mathbf {q})e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\left(\\hat{b}^\\dagger _{\\mathbf {q}}+\\hat{b}_{-\\mathbf {q}}\\right)}_{\\text{Fröhlich interaction}}\\nonumber \\\\&+&{\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\hat{b}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{b}_{\\mathbf {k}}}_{\\text{extended Fröhlich interaction}}.$ where in the second equivalent expression we expanded the boson operators $\\hat{a}^\\dagger _\\mathbf {k}$ around their expectation value $\\sqrt{N_0}$ which highlights the connection to the Fröhlich model.","In order to describe Rydberg impurities the interaction $V(\\mathbf {q})$ is given as the Fourier transform of the Rydberg molecular potential $V(\\mathbf {q})=\\int d^3r V_\\text{Ryd}(\\mathbf {r})e^{i\\mathbf {q}\\cdot \\mathbf {r}}$ .","The molecular potentials and interacting single-particle wave functions ($\\mathinner {|{\\beta _i}\\rangle }$ ) are calculated as described in [43], [10].","We note that the depletion of a BEC by a single electron in a BEC has been studied in [44] including only the linear Fröhlich term in Eq.","(REF ).","While such an approach can approximately account for the rate of depletion of the condensate it fails to describe the formation of molecules that is essential for the formation of Rydberg polarons."],["Linear-response absorption from quench dynamics","One way to probe polaron structure and dynamics is absorption spectroscopy.","Here one utilizes the fact that before being excited to a Rydberg state, an atom is in its ground state $\\mathinner {|{5s}\\rangle }$ , and its interaction with the surrounding Bosons is negligible.","The system is described by the Hamiltonian $\\hat{H}_0$ given by the first two terms in Eq.","(REF ).","In contrast, when the atom is in its Rydberg state $\\mathinner {|{ns}\\rangle }$ the potential $V(\\mathbf {q})$ is switched on.","In experiments [9], transitions between both states are driven by a two-photon excitation.","Within linear response, the corresponding absorption of laser light at frequency $\\nu $ is given by Fermi's Golden rule $\\mathcal {A}(\\nu )&= 2\\pi \\sum _{if}w_i |\\mathinner {\\langle {f}|}\\hat{V}_\\text{L}\\mathinner {|{i}\\rangle }|^2\\times \\,\\delta (\\nu -(E_i-E_f)).$ Here the sum extends over all initial and final states of the impurity plus bosonic bath that fulfill $\\hat{H}_0 \\mathinner {|{i}\\rangle }=E_i\\mathinner {|{i}\\rangle }$ and $\\hat{H} \\mathinner {|{f}\\rangle }=E_f\\mathinner {|{f}\\rangle }$ , where $\\hat{H}$ is given by Eq.","(REF ) so that the final states $\\mathinner {|{f}\\rangle }$ are eigenstates in the presence of the strong perturbation from Rydberg-boson interactions.","In the states $\\mathinner {|{i}\\rangle }$ , all atoms (including the impurity atom) are in the $\\mathinner {|{5s}\\rangle }$ state, while in final states $\\mathinner {|{f}\\rangle }$ the impurity is in the atomic $\\mathinner {|{ns}\\rangle }$ state.","The laser operator that drives the two-photon transition between these two atomic states $\\mathinner {|{5s}\\rangle }$ and $\\mathinner {|{ns}\\rangle }$ , is given by $\\hat{V}_L\\sim \\mathinner {|{ns}\\rangle }\\mathinner {\\langle {5s}|}+h.c.$ .","Using the Fourier-representation of the delta function in Eq.","(REF ), one can show that the absorption spectrum follows from [10] (for details see [45]) $\\mathcal {A}(\\nu )=2\\,\\text{Re}\\,\\int _{0}^\\infty dt\\, e^{i \\nu t}\\, S(t)$ with the many-body overlap (also called the Loschmidt echo) $S(t)$ .","In the general case of finite temperature, the $w_i$ in Eq.","(REF ) are the thermodynamic weights of each initial state given by the diagonal elements of the density matrix $\\hat{\\rho }_\\text{ini}=e^{-\\beta H_0}/Z_p$ , for sample temperature $k_B T=1/\\beta $ and partition function $Z_p$ .","For finite temperature the Loschmidt echo is then given by $S(t)&=&\\text{Tr}[\\hat{\\rho }_\\text{ini}\\, e^{i \\hat{H}_0 t }e^{-i \\hat{H} t}]$ where $\\text{Tr}$ denotes the trace over the complete many-body Fock space.","This expression shows that the overlap function $S(t)$ encodes the non-equilibrium time evolution of the system following a quantum quench represented by the introduction of the Rydberg impurity at time $t=0$ .","From this time evolution then follows the absorption spectrum by Fourier transformation.","The relevant time scales for the dynamics of Rydberg impurities in a BEC is given by the Rydberg molecular energies which exceed both the temperature scale $k_B T$ and the energy scale associated with boson-boson interactions.","Thus temperature and boson interaction effects can be neglected in the calculation of $S(t)$ for Rydberg polarons.","In this limit the initial state of the system is given by $\\mathinner {|{i}\\rangle }=\\mathinner {|{\\text{BEC}}\\rangle }\\otimes \\mathinner {|{\\mathbf {p}=0}\\rangle }_I \\otimes \\mathinner {|{5s}\\rangle }_I$ and Rydberg polarons are well described by the T=0 limit of Eq.","(REF ), $S(t)={_I\\mathinner {\\langle {\\mathbf {p}=0}|}}\\mathinner {\\langle {\\Psi _\\text{BEC}}|}e^{i \\hat{H}_0 t}e^{-i \\hat{H} t}\\mathinner {|{\\Psi _\\text{BEC}}\\rangle }\\mathinner {|{\\mathbf {p}=0}\\rangle }_I,$ where $\\mathinner {|{\\Psi _\\text{BEC}}\\rangle }$ represents an ideal BEC of atoms.","The accurate calculation of the absorption response presents a formidable theoretical challenge.","In the following we will present three different approaches of different degrees of sophistication.","In a simple approximation one may employ mean-field theory (Section ) where the impurity-boson interactions are solely described by the mean-field result $\\rho V(\\mathbf {0})$ in Eq.","(REF ).","In this approach all quantum effects and fluctuations are neglected.","Furthermore one may employ a classical stochastic model (Section ), which at least can treat fluctuations to a certain degree.","Finally, we employ a functional determinant approach, which was developed in [10] and which we review in Section VI.","This approach treats all interaction terms in Eq.","(REF ) fully on a quantum level and allows us to explain the intricate Rydberg polaron formation dynamics observed in our experiments."],["Mean Field Approach","The simplest treatment of the polaron excitation spectrum is to neglect all interaction terms other than $\\rho {V}(\\mathbf {0})$ in Eq.","(REF ), which yields the mean-field approximation.","Hence at a given constant density $\\rho (\\mathbf {r})$ in the trap the absorption spectrum is obtained by Fourier transformation of (cf.","Eq.","(REF )) $S(t,\\vec{r})=e^{-i \\rho (\\mathbf {r}) V(\\mathbf {0}) t} = e^{-i \\Delta (\\mathbf {r})t},$ which yields a delta function Rydberg-absorption response $\\mathcal {A}(\\nu ,\\mathbf {r})=\\delta (\\nu -\\Delta (\\mathbf {r}))$ for a given local, homogeneous density $\\rho (\\mathbf {r})$ at a detuning from the atomic transition $\\Delta (\\mathbf {r})&=& \\rho (\\mathbf {r}) V(\\mathbf {q}=\\mathbf {0})=\\rho (\\mathbf {r})\\int d^3\\mathbf {r}^{\\prime } V(\\mathbf {r}-\\mathbf {r}^{\\prime }) .$ In the experiment, the density $\\rho (\\mathbf {r})$ varies with position $\\mathbf {r}$ .","Since the excitation laser illuminates the entire atomic cloud it excites Rydberg atoms in regions of varying density.","To theoretically model the resulting average over various contributions from the atomic cloud, we perform a local density approximation (LDA), which assumes the density variation is negligible over the range of the Rydberg interaction $V(\\mathbf {r})$ given by Eq.","REF .","The spectrum for creation of a Rydberg impurity is then given by $A(\\nu )\\propto \\int d\\mathbf {r}^3 \\rho (\\mathbf {r}) \\mathcal {A}(\\nu ,\\mathbf {r}).$ In the mean-field approximation this yields the response $A(\\nu )\\propto \\int d^3r\\, \\rho (\\mathbf {r}) \\delta (\\nu -\\Delta (\\mathbf {r}))$ for laser detuning $\\nu $ from unperturbed atomic resonance.","Note that the mean-field treatment is similar to the description of $1S-2S$ spectroscopy of a quantum degenerate hydrogen gas given in [46].","An intuitive way to express this shift is in terms of an effective $s$ -wave electron-atom scattering length that reflects the average of the interactions over the Rydberg wave function for the remainder of this section we keep factors of $\\hbar $, $a_{s,\\textup {eff}}=\\int d^3r^{\\prime } \\frac{{2\\pi m_e}}{h^2}V(\\mathbf {r^{\\prime }}),$ yielding $h\\Delta (\\mathbf {r})&=& \\frac{h^2 a_{s,\\textup {eff}}}{2\\pi m_e} \\rho (\\mathbf {r}).$ $a_{s,\\textup {eff}}$ varies with principal quantum number through the variation in the Rydberg electron wavefunction and the classical dependence of the electron momentum on position.","Figure REF shows calculated values of $a_{s,\\textup {eff}}$ for the interaction of Sr($5sns$ $^3S_1$ ) Rydberg atoms with background strontium atoms [43].","To obtain this result, Eq.","(REF ) is only integrated over $|\\mathbf {r}^{\\prime }|>0.06 n^2 a_0$ because the approximation breaks down near the Rydberg core, where, among other modifications, the ion-atom polarization potential becomes important.","Figure: Calculated values of a s,eff a_{s,\\textup {eff}} from Eq.","() versus n * =n-δn^*=n-\\delta , where δ=3.371\\delta =3.371 is the quantum defect.Figure: Mean-field description of the excitation spectrum of Sr(5sns5sns 3 S 1 ^3S_1) Rydberg atoms in a strontium Bose-Einstein condensate (BEC) for (Left) n=49, (Middle) n=60, (Right) n=72.","Symbols are the experimental data.","Blue bands represent the confidence interval for the mean-field fit of the BEC contribution to the spectrum corresponding to the uncertainties in parameters given in Tab.",".","The central black line indicates the best fit.","The dashed red line is the mean-field prediction for thecontribution from the low-density region of the atomic cloud formed by thermal atoms.","This contribution is calculated using the parametersgiven in Tab.","including a sample temperature adjusted to reproduce the observed BEC fraction.Table: Parameters for data and fits shown in Fig.",": Peak detuning of the mean-field fit (Δ max \\Delta _{\\textup {max}}) and BEC fraction (η\\eta ) are determined from the spectra.","Number of atoms in the condensate (N BEC N_{BEC}) and mean trap frequency ω ¯\\bar{\\omega } are determined from time-of-flight-absorption images and measurements of collective mode frequencies for trapped atoms respectively, and they determine the peak condensate density (ρ max \\rho _{\\textup {max}}) and chemical potential (μ\\mu ).","The second-to-last column provides the ratio of Δ max \\Delta _{\\textup {max}} to Δ ˜ max =ω ¯ 4π15Na bb a ho 2/5 m m e a s,eff a bb \\tilde{{\\Delta }}_{\\textup {max}} = \\frac{\\overline{\\omega }}{4\\pi }\\left(\\frac{15 N a_{bb}}{a_{ho}}\\right)^{2/5}\\frac{m}{m_e}\\frac{a_{s,eff}}{a_{bb}}, the peak shift predicted using information independent of the mean-field fits.","The sample temperature (TT) is the result of a fit of the observed BEC fraction using a numerical calculation of the number of non-condensed atoms in the trap fixing all the BEC parameters at values given in the table.Figure REF shows the mean-field description of experimental results for excitation of Sr($5sns$ $^3S_1$ ) Rydberg atoms in a strontium Bose-Einstein condensate (BEC).","For the prediction of the absorption response the effect of temperature enters by determining the local density of the cloud.","In fact, in the local-density-approximation model of the impurity excitation spectrum (Eq.","(REF )), the density of atoms in the trap, $\\rho (\\mathbf {r})=\\rho _{BEC}(\\mathbf {r})+\\rho _{\\textup {th}}(\\mathbf {r})$ , has contributions from thermal, non-condensed atoms and condensed atoms.","The contribution from the thermal gas is restricted to the sharp peak near zero detuning and a very small contribution towards the red in each data set in Fig.","REF .","Thus the BEC and thermal contributions can be calculated separately in the mean-field approximation.","The profile of the condensate density is well-approximated for our conditions with a Thomas-Fermi (TF) distribution for a harmonic trap [48].","If we neglect the contribution of thermal atoms to the density, the condensate contribution to the spectrum is $A_{\\textup {MF}}(\\nu )\\propto N_{BEC}\\frac{-\\nu }{\\Delta _{\\textup {max}}^2}\\sqrt{1-\\nu /\\Delta _{\\textup {max}}},$ for $\\Delta _{\\textup {max}}\\le \\nu \\le 0$ and zero otherwise [46].","Here, $\\Delta _{\\textup {max}} =\\frac{ h a_{s,\\textup {eff}}}{2\\pi m_e} \\rho _{\\textup {max}}$ , and the peak BEC density is $\\rho _{\\textup {max}}=\\mu _{TF}/g$ , where the chemical potential is $\\mu _{TF}=(\\hbar \\overline{\\omega }/2)\\left( 15 N_{BEC} a_{bb}/ a_{\\textup {ho}} \\right)^{2/5}$ and $g=4\\pi \\hbar ^2 a_{bb}/m$ for harmonic oscillator length $a_{\\textup {ho}}=(\\hbar /m\\overline{\\omega })^{1/2}$ and mean trap radial frequency $\\overline{\\omega }=(\\omega _1 \\omega _2 \\omega _3)^{1/3}$ .","This functional form is used to fit several data sets in Fig.","REF with $\\Delta _{\\textup {max}}$ and the overall signal amplitude as the only fit parameters.","When frequency is scaled by $\\Delta _{\\textup {max}}$ , the mean-field prediction for a BEC in a harmonic trap is universal, making $\\Delta _{\\textup {max}}$ an important parameter for describing the spectrum.","The resulting fit parameters are given in table REF .","We ascribe the difference between the mean-field fit and the observed spectra at small detuning to the contribution from the low-density region of the atomic cloud formed by thermal atoms.","From the ratio of the areas of these two signal components, we extract the condensate fraction.","Given the clean spectral separation between the contributions from thermal and condensed atoms, this is a promising technique for measuring very small thermal fractions and thus temperature of very cold Bose gases.","Once the BEC parameters are set, for a consistency check, we then calculate the contribution to the spectrum from thermal atoms using the mean-field approximation and adjusting the sample temperature to match the BEC fraction.","Additional details of the fitting process are described in App.",".","For $n=72$ , the mean-field approach is quite accurate in describing the overall shape of the response across the entire spectrum.","However, at lower principal quantum numbers, the data and fit deviate significantly, especially at larger detunings.","Generally, mean-field fails both at small detuning (i.e.","for response from low-density regions of the cloud), where the deviation arises from the formation of few-body molecular states, and at large detuning, where fluctuations in the macroscopic occupation of molecular states become important.","Fluctuations correspond to a spread in binding energies of the polaron states excited for a given average density.","All of these effects are neglected in the mean-field description."],["Functional determinant approach","The mean-field approach is insufficient to describe the main features found in the absorption spectrum both at small and large detuning from the atomic transition.","For instance its failure at low detuning (low densities) has its origin in the formation of Rydberg molecular states, which is not described in mean-field theory.","The low density regime can be most conveniently studied in detail by working at a low principal quantum number (here $n=38$ ) where only few atoms are in the Rydberg orbit, and the signal is thus dominated by the low-density response.","To calculate $S(t)$ efficiently we evaluate it within the FDA.","In [10] the FDA was developed to include (for impurities of infinite mass) also the finite-temperature corrections that are however irrelevant for our experiment.","Thus we can restrict ourselves to the description of the zero-temperature response given by Eq.","(REF ), where the bosonic ground state can be expressed as a state of fixed particle number $\\mathinner {|{\\text{BEC}}\\rangle }=(\\hat{a}_\\mathbf {0}^\\dagger )^N_0/\\sqrt{N_0!","}\\mathinner {|{0}\\rangle }$ .","For sufficiently large particle number we may equally use its representation as a coherent state $\\mathinner {|{\\text{BEC}}\\rangle }=\\exp [{\\sqrt{N_0}(\\hat{a}_\\mathbf {0}^\\dagger -\\hat{a}_\\mathbf {0})}]\\mathinner {|{0}\\rangle }$ where $\\mathinner {|{0}\\rangle }$ is the boson vacuum.","The FDA, as developed in Ref.","[10], did not include the description of impurity recoil which is, however, essential for an accurate prediction of Rydberg molecular spectra.","In the following subsection we develop a extension of the FDA approach that overcomes this limitation."],["Canonical transformation: mobile Rydberg impurity","The FDA can be applied to time evolutions described by a Hamiltonian bilinear in creation and destruction operators.","This is fulfilled for the Hamiltonian Eq.","(REF ) in the case of an infinitely heavy impurity $M=\\infty $ .","However, for a mobile Rydberg impurity, the presence of the non-commuting operators $\\hat{\\mathbf {p}}$ and $\\hat{\\mathbf {r}}$ effectively gives rise to non-bilinear terms.","Figure: Exemplary Ramsey signal S(t)=|S(t)|e iϕ(t) S(t)= |S(t)| e^{i \\varphi (t)} including contrast |S||S| and phase ϕ\\varphi underlying the calculation of the Rydberg polaron absorption spectrum for n=49n=49 at peak density in absence of a finite Rydberg lifetime.","A fast decay visible in the contrast accompanies fast oscillations of the full complex system.","The combination of both gives rise to the distinct Rydberg polaron features of molecular peaks that are distributed according to a gaussian envelope.","The Ramsey signal thus provides an alternative pathway for observing Rydberg polaron formation dynamics in real-time.","The time scales of this coherent dressing dynamics are ultrafast compared to the typical time scales of collective low-energy excitations of ultracold quantum gases allowing study of ultrafast dynamics in a new setting.To remedy this challenge we combine the FDA with a canonical transformation.","Here we focus on the zero-temperature case.","In this case the response is obtained from the time evolution using Eq.","(REF ), where the Hamiltonian includes bosonic and impurity operators.","To deal with the impurity motion we perform a canonical transformation first proposed by Lee, Low, and Pines [49] which effectively transforms into the system comoving with the impurity.","To this end we define the translation operator [49] $U=\\exp \\left\\lbrace i \\hat{\\mathbf {R}}\\sum _\\mathbf {k}\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\right\\rbrace $ which is inserted in the time evolution $S(t)&=\\mathinner {\\langle {\\mathbf {p}=0}|}_I\\mathinner {\\langle {\\Psi _\\text{BEC}}|}U U^{-1}e^{-i \\hat{H} t}U U^{-1}\\mathinner {|{\\Psi _\\text{BEC}}\\rangle }\\mathinner {|{\\mathbf {p}=0}\\rangle }_I\\nonumber \\\\&=\\mathinner {\\langle {\\mathbf {p}=0}|}_I\\mathinner {\\langle {\\Psi _\\text{BEC}}|}e^{-i \\hat{\\mathcal {H}} t} \\mathinner {|{\\Psi _\\text{BEC}}\\rangle }\\mathinner {|{\\mathbf {p}=0}\\rangle }_I.$ In the first line we used that the term $e^{i\\hat{H}_0 t}$ can be dropped as the initial state is a zero energy state.","Furthermore, in the second line we made use of the fact that the total boson momentum of the BEC state is zero and we defined the transformed Hamiltonian $\\hat{\\mathcal {H}}=U^{-1}\\hat{H} U&= \\frac{\\left(\\hat{\\mathbf {p}}-\\sum _\\mathbf {k}\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\right)^2}{2M}+\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\nonumber \\\\&+\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}}$ By virtue of the transformation $U$ the impurity coordinate is eliminated in the interaction and only the impurity momentum operator $\\hat{\\mathbf {p}}$ remains.","It thus commutes with the many-body Hamiltonian and is replaced by a c-number, $\\hat{\\mathbf {p}}\\rightarrow \\mathbf {p}$ .","Since in our case we are interested in the polaron at zero momentum, $\\mathbf {p}\\rightarrow 0$ .","After normal ordering, we arrive at $\\hat{\\mathcal {H}}&=& \\sum _{\\mathbf {k}\\mathbf {k}^{\\prime }} \\frac{\\mathbf {k}\\mathbf {k}^{\\prime }}{2M} \\hat{a}^\\dagger _{\\mathbf {k}^{\\prime }}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\hat{a}_{\\mathbf {k}^{\\prime }}+\\sum _\\mathbf {k}\\frac{\\mathbf {k}^2}{2\\mu _\\text{red}} \\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\nonumber \\\\&+&\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}}$ The above transformation has the effect that the boson dispersion relation $\\epsilon _\\mathbf {k}\\rightarrow \\mathbf {k}^2/2\\mu _\\text{red}$ now has the reduced mass $\\mu _\\text{red}=mM/(M+m)$ of boson-impurity partners, and the Hamiltonian includes an induced interaction of bosons described by the first term in Eq.","(REF ).","Due to spherical symmetry and the large energy scale of the Rydberg impurity-Bose gas interaction, we may neglect this term.","The reliability of this approximation has been demonstrated in recent work of some of the authors [36] where a time-dependent variational principle has been applied to the evaluation of the time-evolution of Eq.","(REF ).","In the approximation where the time-dependent wave function $\\mathinner {|{\\Psi (t)}\\rangle }$ evolved by $\\exp (-i\\hat{\\mathcal {H}}t)$ is taken to be a product of coherent states of the form $\\mathinner {|{\\Psi (t)}\\rangle }=e^{\\sum _\\mathbf {k}(\\gamma _\\mathbf {k}(t) \\hat{a}_\\mathbf {k}^\\dagger +\\text{h.c.})}\\mathinner {|{0}\\rangle }$ , one finds from the equation of motions of the variational parameters $\\gamma _\\mathbf {q}$ that the expectation value $\\mathinner {\\langle {\\Psi (t)}|}\\sum _{\\mathbf {k}\\mathbf {k}^{\\prime }} \\frac{\\mathbf {k}\\mathbf {k}^{\\prime }}{2M} \\hat{a}^\\dagger _{\\mathbf {k}^{\\prime }}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\hat{a}_{\\mathbf {k}^{\\prime }}\\mathinner {|{\\Psi (t)}\\rangle }$ always remains zero and this term hence does not contribute to the dynamics.","For our case of Rydberg impurities the FDA solution presented here reproduces the variational result of Ref.","[36] and again shows excellent agreement with experimental data attesting a postiori to the accuracy of the method and the neglect of the first term in Eq.","(REF ).","To which extent this remains valid for other Bose polaron scenarios is an open question and subject to ongoing theoretical studies [50].","In summary, we finally arrive at the Hamiltonian $\\hat{ \\mathcal {H}}= \\sum _\\mathbf {k}\\frac{\\mathbf {k}^2}{2\\mu _\\text{red}} \\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}+\\frac{1}{\\mathcal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}} \\mathinner {|{ns}\\rangle }\\mathinner {\\langle {ns}|}$ whose dynamics we simulate to obtain Rydberg polaron excitation spectra.","Note, in Eq.","(REF ), we make explicit that the impurity-boson interaction is only present when the impurity is in its excited atomic Rydberg state $\\mathinner {|{n s}\\rangle }$ .","In this state the Rydberg electron scatters off ground state atoms in the environment, leading to the strong Rydberg Born-Oppenheimer potential $V(\\mathbf {q})$ .","For simplified notation we will now switch again to the symbol $\\mathcal {\\hat{H}} \\rightarrow \\hat{H}$ ."],["Time-domain $S(t)$","The strength of the FDA is that it allows one to express overlaps of many-body states in terms of single-particle eigenstates [51], [52], [53], [54], [10].","In our case this implies that we must calculate single particle eigenstates and energies of the free and `interacting' Hamiltonian, i.e.","$\\hat{h}_0\\mathinner {|{n}\\rangle }=\\epsilon _n \\mathinner {|{n}\\rangle }$ and $\\hat{h}\\mathinner {|{\\beta }\\rangle } = \\omega _\\beta \\mathinner {|{\\beta }\\rangle }$ , respectively ($\\hat{h}_0$ and $\\hat{h}$ are the single-particle representatives of the many-body operators $\\hat{H}_0$ and $\\hat{H}$ , respectively).","The single-particle eigenstates and energies $\\mathinner {|{\\beta }\\rangle }$ and $\\omega _\\beta $ are calculated using exact diagonalization.","Here we solve the radial Schrödinger equation for a spherical box of radius $R$ discretized in real-space.","Since the BEC is initially in a zero angular state and the Rydberg molecular potential is spherically symmetric we can restrict the analysis to single-particle states with zero-angular momentum.","We include the 300 energetically lowest eigenstates, which leads to convergent results.","In terms of these states and energies the overlap $S(t)$ becomes [10] $S(t)=\\left(\\sum _{\\beta } |\\langle \\beta | s \\rangle |^2 e^{i(\\epsilon _s-\\omega _\\beta )t}\\right)^{N},$ where $\\mathinner {|{s}\\rangle }$ denotes the lowest single particle eigenstate [55] of $\\hat{h}_0$ , and $N$ is the number of atoms in the spherical box of radius $R$ chosen to reproduce a given local experimental density of a Bose gas.","We choose $R=10^6 a_0$ with $a_0$ the Bohr radius so that we find negligible finite size corrections.","We emphasize again that the temperature enters the calculation only in determining the local density of the atomic cloud.","For the dynamics and thus the prediction of the absorption spectra at a given local density the temperature $T$ is irrelevant.","This is due to the fact that the energy scales of the dynamics of the system is determined by the binding energy of Rydberg molecular states, which exceed $k_BT$ .","We obtain absorption spectra by predicting the many-body overlap, or Loschmidt echo, $S(t)$ .","This calculation corresponds to solving the full time evolution of the system following a quantum quench, where at time $t=0$ the Rydberg impurity is suddenly introduced into the BEC.","The overlap $S(t)$ describes the dephasing dynamics of the many-body system and thus the evolution of the dressing of the Rydberg impurity by bath excitations.","It is one of the virtues of cold atomic systems that the signal $S(t)$ can be directly measured experimentally by Ramsey spectroscopy where via a $\\pi /2$ rotation at time $t=0$ , the impurity is prepared in a superposition state of $\\mathinner {|{5s}\\rangle }$ and $\\mathinner {|{ns}\\rangle }$ .","Following a time evolution of duration $t$ , a further $\\pi $ rotation is performed and $\\sigma _z$ is measured.","This gives the Ramsey contrast $|S(t)|$ .","Changing the phase of the final $\\pi $ rotation, the full signal $S(t)= |S(t)| e^{i \\varphi (t)}$ can be measured [56], [57], [45].","In Fig.","REF we show the predicted Ramsey signal that underlies the calculation of the Rydberg polaron absorption spectrum for $n=49$ at peak density, here shown in absence of a finite Rydberg lifetime.","We observe a fast decay in the contrast that indicates strong dephasing and thus efficient creation of polaron dressing by particle-hole excitations.","This decay is accompanied by fast oscillations of the complex signal $S(t)$ as visible in the shown evolution of the Ramsey phase $\\varphi (t)$ restricted to the branch $(-\\pi ,\\pi )$ .","The combination of the oscillations at the Rydberg molecular binding energies and the decay of the Ramsey signal $S(t)$ gives rise to the distinct Rydberg polaron features of molecular peaks that are distributed according to a gaussian envelope, to be discussed below.","The Ramsey signal thus provides an alternative pathway to observing Rydberg polaron formation dynamics in real-time.","The time scales of this coherent dressing dynamics are ultrafast compared to the typical time scales of collective low-energy excitations of ultracold quantum gases.","This again highlights that effects arising from Bose-Bose interaction as well as finite temperature will play only a minor role in the prediction of the absorption response as those start to influence dynamics only on much longer time scales.","Figure: Density profile of bosonic atoms in a harmonic trap for parameters T=180T=180 nK, N BEC =3.7·10 5 N_\\text{BEC}=3.7 \\cdot 10^5, N tot =5.2·10 5 N_\\text{tot}=5.2\\cdot 10^5, ω r =104\\omega _r = 104 Hz, and ω a =111\\omega _a=111 Hz, used for the prediction of the spectrum of the n=60n=60 Rydberg excitation shown in Fig. .","In blue is shown the contribution from the condensed atoms, while the red area represents the contribution from thermal atoms.Figure: (a) LDA absorption spectrum for n=49n=49 as calculated by FDA (solid blue) in comparison with experimental data (symbols).Using the data of the non-equilibrium quench dynamics, the FDA can accurately capture the formation of Rydberg molecular dimers, trimers, tetramers, etc.","This is demonstrated by comparing the FDA spectrum and experimental measurement for low principal quantum number, as shown in [9].","Indeed, from a many-body wave-function perspective capturing the trimer, tetrameter and higher-order oligamer state requires the inclusion of the corresponding higher order terms in Eq.","(REF ).","This attests to the challenge of describing Rydberg polarons which are formed by the dressing with many deeply bound atoms, rather than just a small number of bath excitations.","In fact due to the magnitude of the binding energies involved, coherent formation of Rydberg polarons takes place on a $\\mu $ s timescale, which is short compared to typical ultracold-atom time scales."],["FDA absorption spectra","The accurate description of molecular formation underlies the precise analysis of Rydberg-polaron absorption response.","As the principal quantum number is increased to $n=49$ , more atoms are situated on average within the Rydberg orbit and many-body effects become relevant.","The density-averaged prediction using FDA is shown in Fig.","REF .","For all density-averaged spectra, we rely on density profiles such as shown in Fig.","REF for parameters used for the FDA prediction of the $n=60$ Rydberg-absorption spectrum.","We include Hartree-Fock corrections [48], [58] as discussed in Appendix .","Furthermore, the absorption spectrum is calculated from the Fourier transformation according to Eq.","(REF ), where in the time-evolution of $S(t)$ we choose a temporal cutoff determined by the experimental laser-pulse duration, and in the Fourier transform we account for the finite laser line width (400 kHz).","Figure: LDA absorption spectrum for n=49n=49 as calculated from the classical Monte Carlo sampling model (black) compared to the FDA prediction.","The shaded area, which corresponds to the prediction of absorption response from the cloud center at peak density, reveals that the Gaussian line shape of the response is due to discrete, Gaussian-distributed molecular peaks and not a continuous response as described by a simple classical model.In Fig.","REF , we show again the $n=49$ absorption spectrum obtained theoretically, but with a simulated laser linewidth of 100 kHz (solid blue line).","The shaded region shows the result of an FDA calculation of the spectrum for a central region of the atomic cloud with roughly constant density.","We find that the signal from this region shows a series of molecular peaks whose weights follow a Gaussian envelope.","This distribution of molecular peaks is one of the key signatures of Rydberg polarons as predicted theoretically in the accompanying work [10].","The comparison of experiment and theory in Fig.","6 in turn shows that the response is indeed composed of many individual molecular lines.","While due to the finite lifetimes of Rydberg molecules and the fact that their binding energies decrease for increasing principal number, these individual molecular peaks cannot be resolved experimentally for high principal numbers, their Gaussian envelope is a robust signature of the formation of Rydberg polarons.","As our simulation shows, the broad tail to the red and the characteristic gaussian profile of the signal are both clear signatures of Rydberg polarons.","The excellent agreement between many-body theory and experiment confirms the presence of Rydberg polarons in the Sr experiment.","Figure: Experimentally observed absorption spectrum (symbols) for n=60n=60 in comparison with the theoretical prediction from FDA (lines).","The experimental input parameters such as the trap frequencies are varied within the experimental uncertainty, giving rise to the gray band around the solid red line.","The specific set of values is given in Table .","The inset shows the signal on a logarithmic scale.Table: Parameters used for the prediction of the theoretical spectrum shown in Fig.","(b) for the Rydberg n=60n=60 excitation.","Inferred quantities are the condensate fraction η=N BEC /N tot \\eta =N_\\text{BEC}/N_\\text{tot}, peak BEC density ρ peak,BEC \\rho _\\text{peak,BEC}, and chemical potential μ\\mu .","Densities are given in cm -3 \\text{cm}^{-3}.The Gaussian signature of the Rydberg-polaron spectrum becomes more pronounced when exciting $n=60$ states (Fig.","REF ).","In the accompanying work [10] we observe this Gaussian response experimentally for Rydberg polarons of large principal number $n=60$ and $n=72$ .","While due to the finite lifetimes of Rydberg molecules and the fact that their binding energies decrease for increasing principal number, these individual molecular peaks cannot be resolved experimentally for high principal numbers, their Gaussian envelope is a robust signature of the formation of Rydberg polarons.","The comparison of experiment and theory in Fig.","6 in turn shows that the response is indeed composed of many individual molecular lines.","The FDA calculation of the absorption spectra takes as input the experimentally determined number of atoms in the condensate $N_\\text{BEC}$ and the trap frequencies $\\omega _i$ .","The temperature $T$ and the total number of atoms $N_\\text{tot}$ are taken as fit parameters, and results are consistent with their determination using the mean-field model and absorption imaging described in Section .","For the $n=60$ spectrum, we also show a range of FDA predictions resulting from varying the trap frequencies within experimental uncertainty, which illustrates the impact of these uncertainties.","Table REF lists the parameters used in the theoretical simulation.","Note that the requirement to explain the spectrum over the whole frequency regime places tight constraints on the fit parameters $T$ , and $N_\\text{tot}$ .","(For Rydberg polaron response at $n=72$ , refer to Ref.","[9]).","As discussed in Ref.","[10], the emergence of the Gaussian response can be understood from a direct, analytical calculation of $S(t)$ in terms of single particle eigenstates and energies.","Indeed expanding the sum in Eq.","(REF ) explicitly in a multinominal form leads, after Fourier transform, to the expression $\\mathcal {A}(\\nu )&=&N!","\\sum _{\\Sigma n_i = N}\\frac{|\\langle \\alpha _1 | s \\rangle |^{2 n_1}\\cdot \\ldots \\cdot |\\langle \\alpha _M | s \\rangle |^{2 n_M}\\cdot \\ldots }{n_1!\\ldots n_M!","\\ldots } \\nonumber \\\\&&\\times \\delta (-\\nu +n_1 \\omega _1+\\ldots +n_M \\omega _M+\\ldots ).$ This expression captures not only the bound states but also continuum states.","Expressing the spectrum in this explicit form again highlights the fact that the actual spectrum is indeed built by $\\delta $ -peaks corresponding to the many configurations in which bound molecules can be formed and thus dress the impurity.","It also explains the emergence of the gaussian lineshape as a limit of the multinominal distribution of delta-function peaks for large particle number $N$ .","We note that the fact that the spectrum is built by molecular excitation peaks (of up to hundreds of atoms bound to a single impurity) is missed by the classical description of the Rydberg polarons discussed in Section .","We note that while Eq.","(REF ) is appealing as it qualitatively explains the observed spectral features, it is less useful for quantitative calculations due to the exponential growth of the number of terms in the sum (reflecting the exponential growth of Hilbert space for increasing particle number).","In contrast, the calculation of the time-evolution in Eq.","(REF ) is numerically efficient (when limited to finite evolution times) and only requires a final Fourier transformation."],["From quantum to classical description of Rydberg absorption response","For a sufficiently large average number of atoms within the Rydberg electron orbit, the overall line shape of Rydberg spectra can be described with classical statistical arguments.","In Fig.","REF , we show the experimental spectrum obtained for $n=60$ in comparison with the prediction from the FDA (dashed red line) and a classical statistical approach (solid black).","In the latter approach, using a classical Monte Carlo (CMC) algorithm [11], we randomly distribute atoms in three-dimensional space around the Rydberg ion such that the correct density profile is obtained.","Due to statistical fluctuations in the random sampling of coordinates (sampled from a uniform distribution), the local density within the Rydberg-electron radius fluctuates.","For each of the random configurations of atoms $\\mathcal {C}=(\\mathbf {r}_1, \\mathbf {r}_2,\\ldots )$ we calculate the classical energy of the configuration $E_{\\mathcal {C}}=\\sum _i V(\\mathbf {r}_i)$ , where $\\mathbf {r}_i$ are the atom coordinates.","The resulting energies $E_{\\mathcal {C}}$ are collected in an energy histogram and shown as the black solid line in Fig.","REF .","The validity of the classical description reflects the fact that the macroscopic occupation of the molecular bound and scattering states probes the Rydberg molecular potential uniformly over its entire range.","This process is also well described by a repeated sampling of different classical atom configurations drawn from a homogeneous density distribution.","Figure: LDA absorption spectrum for n=60n=60 as calculated by the FDA (dashed red) and classical statistical Monte Carlo sampling method (black).","The inset shows the signal on a logarithmic scale.The quantum-classical correspondence also becomes evident when considering the specific approximation that underlies the classical statistical approach.","The Rydberg absorption response is given as a Fourier transform of the full quantum quench dynamics as given by Eq.","(REF ).","The classical statistical model arises as an approximation of the time evolution of $S(t)$ where the kinetic energy of both the impurity and the Bose gas is completely neglected.","Hence both bosons and the impurity are treated as effectively infinitely heavy objects; their motion is `frozen' in space.","The disregard of the kinetic terms has the consequence that the Hamiltonian now commutes with the position operators $\\hat{\\mathbf {r}}_j$ of impurity and bath atoms (non-commuting $\\hat{\\mathbf {p}}_j$ operators are now absent), and hence dynamics becomes completely classical.","In this approximation the Hamiltonian becomes (without loss of generality, we assume that the impurity is at the center of the coordinate system) $\\hat{H}= \\int d^3r V(\\mathbf {r})\\hat{a}^\\dagger (\\mathbf {r})\\hat{a}(\\mathbf {r})$ Furthermore, $\\hat{H}_0=0$ , and hence $\\hat{\\rho }_\\text{ini} = 1/Z$ in Eq.","(REF ).","Since all boson coordinates $\\hat{\\mathbf {r}}_j$ now commute with the Hamiltonian, the many-body eigenstates are given by $\\mathinner {|{\\phi ^{(j)}}\\rangle }=\\mathinner {|{\\mathbf {r}^{(j)}_1,\\mathbf {r}^{(j)}_2,\\ldots , \\mathbf {r}^{(j)}_N}\\rangle }$ .","The trace in Eq.","(REF ) reduces to the sum over the set of all basis states $\\lbrace \\mathinner {|{\\phi ^{(j)}}\\rangle }\\rbrace $ .","The $\\mathinner {|{\\phi ^{(j)}}\\rangle }$ are eigenstates of $\\hat{H}$ with eigenvalues $\\hat{H} \\mathinner {|{\\phi ^{(j)}}\\rangle }= \\sum _{\\lbrace \\mathbf {r}_i^{(j)}\\rbrace } V(r_i^{(j)}) \\mathinner {|{\\phi ^{(j)}}\\rangle }$ and $S(t)$ becomes $S_\\text{cl}(t)=\\mathcal {N} \\sum _{\\lbrace \\mathbf {r}_i^{(j)}\\rbrace }e^{i\\left(\\nu - \\sum _{i} V(r_i^{(j)})\\right)t},$ where the sum extends over all possible atomic configurations in real-space, and $\\mathcal {N}$ is a normalization factor (representing the partition function $Z$ in the density operator $\\rho _\\text{ini}$ ).","Finally, performing the Fourier transform of $S(t)$ one arrives at $\\mathcal {A}(\\nu )=\\mathcal {N} \\sum _{\\lbrace \\mathbf {r}_i^{(j)}\\rbrace }\\delta \\left(\\nu - \\sum _{i} V(r_i^{(j)})\\right).$ A finite lifetime of the Rydberg or laser excitation, as well as a finite linewidth of the laser leads to a broadening of the delta function in Eq.","(REF ).","The sum in Eq.","(REF ) is exactly the object sampled in the classical Monte Carlo (CMC) approach derived from first principles.","Moreover we note that for impurities interacting with an ideal Bose gas with contact interactions, the gaussian spectral signature of polarons [10] would remain, while in the classical model, a sharp excitation at the atomic transition frequency is expected.","Indications of a Gaussian response have been seen in a recent experiment performed independently at Aarhus [8] and JILA [7], in agreement with theory [36].","The classical statistical approach fails to describe the Bose polarons close to a Feshbach resonance [7], [8].","Here the impurity indeed interacts with the bath via a contact interaction, and, due to a diverging scattering length, the single bound state present in the problem is highly delocalized and extends in size far beyond the range of the potential.","This effect cannot be captured by a classical approach.","While Rydberg absorption spectra find an effective, yet approximate description in terms of classical statistics, such an approach does not reveal the quantum mechanical origin of Rydberg polarons.","The classical sampling model treats both the Rydberg and the ground state atoms as infinitely heavy objects.","Zero-point motion and the discrete nature of the bound energy levels are absent in this treatment, so that it cannot describe the formation of Rydberg molecular states.","This short-coming of the classical approach is evident in Fig.","REF , where we show the absorption spectrum for $n=49$ .","The FDA (solid blue) fully describes the quantum mechanical formation of molecules (dimers, trimers, tetrameters, excited molecular states, etc.","), which gives rise to discrete molecular peaks visible in the spectrum.","In contrast, and as evident from Fig.","REF , the existence of molecular states, which are the quantum mechanical building block of Rydberg polarons, is not described by the classical approach (solid black).","The fact that the classical model can only describe the envelope of the Rydberg polaron response but not the underlying distribution of molecular peaks, is further emphasized by the comparison of FDA and CMC simulation from the center of the atomic cloud shown as shaded region in Fig.","7."],["Conclusion","In this work, we have detailed the descriptions of polarons as encountered in condensed matter and in ultracold atomic systems.","We time-evolve an extended Fröhlich Hamiltonian as relevant for Rydberg excitations, Eq.","(REF ), unitarily to obtain the overlap function, Eq.","(REF ), whose Fourier transform leads to the spectral function for Rydberg impurity excitation in a Bose gas, Eq.","(REF ).","We show how different approximations to the many-body quantum description, such as mean field and classical Monte Carlo treatments can be derived.","Here we extend the bosonic functional determinant approach to the Rydberg polaron problem to account for recoil of the impurity.","We show that the FDA approach can correctly and accurately account for the few-body molecular bound-state formation, as well as the macroscopic occupation of many-body bound and continuum states.","The various treatments are compared with experimental data for Rydberg excitation in a $^{84}$ Sr BEC [9] with the FDA results reproducing the observed data over a wide range of Rydberg line spectral intensities.","Rydberg polarons are exemplified on the one hand by their large energy scales, 1-10 MHz, that allow for coherent polaronic dressing on correspondingly short time scales, and on the other hand large spatial extent, $~1\\,\\mu $ m, leading to large-scale variations of the density in the many-body medium.","Another key feature of Rydberg polarons is the coherent dressing of the quantum impurity not by collective low-energy excitations, but by a large number of molecular bound states.","This is a new dressing mechanism, distinguishing them from polarons encountered so far in the solid state physics context.","Our FDA time domain analysis suggests a way to investigate the dynamical formation of the Rydberg polarons.","A natural extension will be to explore the feasibility of observing Pauli blocking in a degenerate Fermi atomic gas with non-trivial spatial correlations.","Note that one can study polarons with nonzero momentum by keeping all terms in Eq.","(REF ) and allowing $\\mathbf {p}$ to be finite.","The effective polaron mass can then be analyzed as previously demonstrated [59].","Acknowledgements: Research supported by the AFOSR (FA9550-14-1-0007), the NSF (1301773, 1600059, and 1205946), the Robert A, Welch Foundation (C-0734 and C-1844), the FWF(Austria) (P23359-N16, and FWF-SFB049 NextLite).","The Vienna scientific cluster was used for the calculations.","H. R. S. was supported by a grant to ITAMP from the NSF.","R. S. and H. R. S. were supported by the NSF through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and the Smithsonian Astrophysical Observatory.","R. S. acknowledges support from the ETH Pauli Center for Theoretical Studies.","E. D. acknowledges support from Harvard-MIT CUA, NSF Grant (DMR-1308435), AFOSR Quantum Simulation MURI, the ARO-MURI on Atomtronics, and support from Dr. Max Rössler, the Walter Haefner Foundation and the ETH Foundation.","T. C. K acknowledges support from Yale University during writing of this manuscript."],["Determination of BEC parameters and Mean-Field description of the Impurity Excitation Spectrum in a Strontium BEC","The mean-field approximation neglects fluctuations in the density around the Rydberg impurity, which correspond to a spread in binding energies of the polaron states excited for a given average density.","This explains the discrepancy between data and fit at large detuning.","But as shown in [9], the broadening resulting from these fluctuations is proportional to $\\sqrt{\\rho }$ and vanishes as detuning approaches zero.","Thus, we assume that the difference between data and the mean-field fit in Fig.","REF for small detuning ($\\nu /\\Delta _{\\textup {max}}<0.5$ ) arises from non-condensed atoms.","We adjust the area of the mean-field BEC contribution so the sum of the non-condensed and mean-field-BEC signal matches the total experimental spectral area.","Because the deviations between the mean-field fit and the BEC spectrum are significant, the fit of the peak shift $\\Delta _{\\textup {max}}$ is less rigorous.","We adjust it to qualitatively match the data and so that the mean-field fit and the data have approximately equal area for ($\\nu /\\Delta _{\\textup {max}}>0.5$ ).","This uncertainty in the fitting procedure decreases with increasing principal quantum number as the experimental data converges towards the mean-field form.","In the mean-field description we neglect the laser linewidth of 400 kHz in this analysis.","The fit parameters are given in Table REF .","The condensate fraction $\\eta $ is directly determined from the spectrum.","The uncertainties correspond to values calculated for the extremes of the confidence intervals shown as bands in Fig.","REF , except for uncertainties in $\\omega _i$ and total atom number, which reflect uncertainties from the independent procedures for measuring these quantities.","The fit value of the peak shift, $\\Delta _{\\textup {max}}$ , can be compared to the predicted peak shift, $\\tilde{\\Delta }_\\textup {max}$ , calculated from independent, a priori information: the number of atoms in the condensate $N_{BEC}$ determined from time-of-flight-absorption images, the trap oscillation frequencies $\\omega _i$ determined from measurements of collective mode frequencies for trapped atoms, and the value of $a_{s,\\textup {eff}}$ found theoretically from $V_{Ryd}(\\mathbf {r})$ .","The values of $\\Delta _{\\textup {max}}$ are all about 10% below $\\tilde{\\Delta }_\\textup {max}$ .","This may point to something systematic in our analysis procedure, but it is also a reasonable agreement given our uncertainty in determination of $\\omega _i$ and total atom number.","The calculation of $a_{s,\\textup {eff}}$ from $V_{Ryd}(\\mathbf {r})$ (Eq.","(REF )) could also account for some of this discrepancy through approximations made to describe the Rydberg-atom potential at short range.","Residual deviations in the description of the short-range details of the Rydberg molecular potential lead, for instance, also to the minor discrepancies between the FDA and CMC simulation of the Rydberg response from the center of the atomic cloud, shown in Fig.","REF .","For a non-interacting gas in a harmonic trap, the temperature is easily found from $\\eta $ , and $\\bar{\\omega }$ [60], and the standard expressions imply a temperature of approximately 240 nK for the data presented in Fig.","REF .","However, the situation is much more complicated than this for an interacting gas.","There are corrections to the condensate fraction that are independent of the trapping potential, and these are often discussed in terms of the shift of the critical temperature for condensation [48], [61], [62], but these are all small in our case, reflecting the smallness of relevant expansion parameters: $\\rho _{\\textup {max}}a_{bb}^3=10^{-4}, a_{bb}/\\lambda _{\\textup {th}}=10^{-2}$ at 100 nK, and $a_{\\textup {ho}}/R_{TF}=10^{-1}$ , where $\\lambda _{\\textup {th}}$ is the thermal de Broglie wavelength and $a_{\\textup {ho}}$ is the trap harmonic oscillator length.","For a gas trapped in an inhomogeneous external potential $V_{ext}(\\textbf {r})$ , the Hartree-Fock approximation (as described in detail in Refs.","[63], [60]) yields a mean-field interaction between thermal and BEC atoms that creates an effective potential for thermal atoms, $V_{\\textup {eff}}(\\textbf {r})= V_{\\text{ext}}(\\textbf {r})+2g[\\rho _{BEC}(\\textbf {r})+\\rho _{\\textup {th}}(\\textbf {r})]$ that is of a “Mexican hat\" shape rather than parabolic [60], [63], [64].","This effect is particularly important when the sample temperature is close to or lower than the chemical potential, which is the case here.","It increases the volume available to non-condensed atoms, and implies a lower sample temperature for a given value of $\\eta $ compared to the non-interacting case.","In other words, the number of thermal atoms can significantly exceed the critical number for condensation predicted for an ideal gas [64], [58].","240 nK is thus an upper limit of the sample temperature.","Sample temperatures extracted from bimodal fits to the time-of-flight absorption images are 100-190 nK.","Both the mean-field and the FDA fits use a local density approximation for the spectrum.","Here the temperature and interaction between bosons enter by determining the precise form of the overall density distribution of the atomic gas.","As discussed above, the density profile has contributions from both the condensed and thermal, non-condensed atoms $\\rho (\\mathbf {r})= \\rho _\\text{BEC}(\\mathbf {r}) +\\rho _\\text{th}(\\mathbf {r}).$ The BEC density $\\rho _\\text{BEC}$ and resulting chemical potential are assumed to be given by the Thomas-Fermi expressions [63] $\\rho _{BEC}(\\mathbf {r})= \\frac{m}{4\\pi \\hbar ^2 a_{bb}}[\\mu _{TF}-V_{ext}(\\mathbf {r})],$ with $\\mu _\\text{TF}=\\hbar \\overline{\\omega }/2(15 N_\\text{BEC} a_{bb}/a_{ho})^{2/5}$ .","$N_\\text{BEC}$ is the total number of condensed atoms, $a_{bb}=123 a_0$ for $^{84}$ Sr, and $a_{ho}=(\\hbar /m\\overline{\\omega })^{1/2}$ is the harmonic oscillator length.","(In this section we make explicit factors of $\\hbar $ and $k_B$ .)","The density distribution for atoms in the thermal gas can be calculated from known trap and BEC parameters using a Bose-distribution $\\rho _\\text{th}(\\mathbf {r})= \\frac{1}{\\lambda _T^3}g_{3/2}\\left\\lbrace e^{-\\frac{1}{k_BT}\\left[V_\\textup {eff}(\\mathbf {r})-\\mu _{TF}\\right]}\\right\\rbrace ,$ where $\\lambda _T=\\sqrt{\\frac{2\\pi \\hbar ^2}{m k_BT}}$ is the thermal wavelength and $g_{3/2}(z)$ the polylogarithm $\\text{PolyLog}[\\frac{3}{2};z]$ [63], and the chemical potential is set to $\\mu _{TF}$ .","The parameters $T$ and $\\mu _{TF}$ are varied self-consistently to fit that spectrum, which can be seen as matching the experimentally observed total particle number $N_\\text{tot}=N_\\text{BEC}+N_\\text{th}$ and BEC fraction $\\eta =N_\\text{BEC}/N_\\text{tot}$ .","The mean-field calculations use the full geometry of the external trapping potential as determined by the optical-dipole-trap lasers.","Integrals involving $V_\\text{ext}(\\mathbf {r})$ are evaluated numerically.","This procedure yields temperatures of $155-170\\,nK$ , in good agreement with other determinations.","The modification of the potential seen by thermal atoms due to Hartree-Fock corrections has an important consequence for our analysis.","The peak shift in the Rydberg excitation spectrum should in principle reflect the peak condensate density plus the density of thermal atoms at the center of the trapping potential.","The contribution from thermal atoms would be a significant correction if the mean-field repulsion of thermal atoms from the center of the trap were ignored.","But because the sample temperature is close to the chemical potential, the density of thermal atoms is suppressed at trap center and we neglect it in discussions of the peak mean-field shift in the spectrum.","For the FDA simulation, the dipole trap potential is approximated by $V_\\text{ext}(\\mathbf {r})=m(\\omega _r^2 r^2+\\omega _z^2 z^2)/2$ .","In Fig.","REF (a) we show the density profile along the radial direction for parameters as used for the FDA prediction of the $n=60$ Rydberg absorption spectrum including Hartree-Fock corrections.","We emphasize again that the temperature and interaction between bosons are relevant only for the determination of the density profile of atoms.","In the simulation of the absorption spectrum, which is obtained within a local density approximation, this density distribution of atoms enters as input.","In this simulation, then locally performed for constant density, both temperature effects and boson-interaction effects can be neglected due to the short time scales associated with Rydberg polaron dressing dynamics (that are given by the Rydberg molecular binding energies)."]],"string":"[\n [\n \"Theory of excitation of Rydberg polarons in an atomic quantum gas\"\n ],\n [\n \"Abstract We present a quantum many-body description of the excitation spectrum of Rydberg polarons in a Bose gas.\",\n \"The many-body Hamiltonian is solved with functional determinant theory, and we extend this technique to describe Rydberg polarons of finite mass.\",\n \"Mean-field and classical descriptions of the spectrum are derived as approximations of the many-body theory.\",\n \"The various approaches are applied to experimental observations of polarons created by excitation of Rydberg atoms in a strontium Bose-Einstein condensate.\"\n ],\n [\n \"Introduction\",\n \"When an impurity is immersed in a polarizable medium, the collective response of the medium can form quasi-particles, labelled as polarons, which describe the dressing of the impurity by excitations of the background medium.\",\n \"Polarons play important roles in the conduction in ionic crystals and polar semiconductors [1], spin-current transport in organic semiconductors [2], dynamics of molecules in superfluid helium nanodroplets [3], [4], [5], and collective excitations in strongly interacting fermionic and bosonic ultracold gases [6], [7], [8].\",\n \"In this paper, which accompanies the publication heralding the observation of Rydberg Bose polarons [9], we present details of calculations and the interpretation of this observation as a new class of Bose polarons, formed through excitation of Sr($5sns$ $^3S_1$ ) Rydberg atoms in a strontium Bose-Einstein condensate (BEC).\",\n \"We begin with a general outline of different polaron Hamiltonians, and construct the Rydberg polaron Hamiltonian used in this work.\",\n \"The spectral response function in the linear response limit is derived and the many-body mean field shift of the spectral response is obtained.\",\n \"The mean field theory particularly fails to describe the limits of small and large detuning, where the detailed description of quantum few- and many-body processes is particularly relevant.\",\n \"These processes are fully accounted for by the bosonic functional determinant approach (FDA) [10] that solves an extended Fröhlich Hamiltonian for an impurity in a Bose gas.\",\n \"In the frequency domain, the FDA predicts a gaussian shape for the intrinsic spectrum, which is a hallmark of Rydberg polarons.\",\n \"A classical Monte Carlo simulation [11] which reproduces the background spectral shape is shown to miss spectral features arising from quantization of bound states.\",\n \"Agreement between experimental results and FDA theory for both the observed few-body molecular spectra and the many-body polaronic states is excellent.\",\n \"In the companion paper [9], we provide experimental evidence for the observation of polarons created by excitation of Rydberg atoms in a strontium Bose-Einstein condensate, with an emphasis on the determination of the excitation spectrum in the absence of density inhomogeneity.\",\n \"Here we provide a detailed discussion of the theoretical methods and experimental analysis.\"\n ],\n [\n \"Bose Polaron Hamiltonians\",\n \"With ultracold atomic systems, various quantum impurity models can be studied in which an impurity interacts with a bosonic bath.\",\n \"Here we focus on models that follow from the general Hamiltonian describing an impurity of mass $M$ interacting with a gas of weakly interacting bosons of mass $m$ : $\\\\hat{H}&=&\\\\sum _\\\\mathbf {p}\\\\epsilon _\\\\mathbf {p}^I\\\\hat{d}^\\\\dagger _\\\\mathbf {p}\\\\hat{d}_\\\\mathbf {p}+\\\\sum _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {k}\\\\hat{a}^\\\\dagger _\\\\mathbf {k}\\\\hat{a}_\\\\mathbf {k}+\\\\frac{g_{bb}}{2\\\\cal {V}}\\\\sum _{\\\\mathbf {k}\\\\mathbf {k}^{\\\\prime }\\\\mathbf {q}}\\\\hat{a}^\\\\dagger _{\\\\mathbf {k}^{\\\\prime }+\\\\mathbf {q}}\\\\hat{a}^\\\\dagger _{\\\\mathbf {k}-\\\\mathbf {q}}\\\\hat{a}_{\\\\mathbf {k}^{\\\\prime }}\\\\hat{a}_{\\\\mathbf {k}}\\\\nonumber \\\\\\\\&+&{\\\\frac{1}{\\\\cal {V}}\\\\sum _{ \\\\mathbf {k}\\\\mathbf {k}^{\\\\prime }\\\\mathbf {q}}V(\\\\mathbf {q})\\\\hat{d}^\\\\dagger _{\\\\mathbf {k}^{\\\\prime }-\\\\mathbf {q}}\\\\hat{d}_{\\\\mathbf {k}^{\\\\prime }}\\\\hat{a}^\\\\dagger _{\\\\mathbf {k}+\\\\mathbf {q}}\\\\hat{a}_{\\\\mathbf {k}}}_{ \\\\hat{H}_{IB}}.$ Here, the first two terms describe the kinetic energy of the impurities ($\\\\hat{d}_\\\\mathbf {p}$ ) and bosons ($\\\\hat{a}_\\\\mathbf {k}$ ) with dispersion relations $\\\\epsilon _\\\\mathbf {p}^I=\\\\frac{\\\\mathbf {p}^2}{2M}$ and $\\\\epsilon _\\\\mathbf {k}=\\\\frac{\\\\mathbf {k}^2}{2m}$ , respectively (unless explicitly stated we set $\\\\hbar =1$ ).\",\n \"The third term accounts for the interaction between the bosons.\",\n \"Assuming weak coupling between bosons, the microscopic coupling constant is given by the relation $g_{bb}=\\\\pi a_{bb}/m$ , with $a_{bb}$ the s-wave scattering length describing the low-energy boson-boson interactions.\",\n \"The last term, $\\\\hat{H}_{IB}$ describes the impurity-boson interaction in momentum space, which, in the real space, reads $\\\\hat{H}_{IB}=\\\\int d^3r d^3r^{\\\\prime } \\\\,\\\\,\\\\hat{n}_I(\\\\mathbf {r}^{\\\\prime })\\\\,V(\\\\mathbf {r}^{\\\\prime }-\\\\mathbf {r})\\\\,\\\\hat{n}_B(\\\\mathbf {r}).$ Here $\\\\hat{n}_I(\\\\mathbf {r})=\\\\hat{\\\\psi }^\\\\dagger _I(\\\\mathbf {r})\\\\hat{\\\\psi }_I(\\\\mathbf {r})=\\\\frac{1}{\\\\cal {V}}\\\\sum _{\\\\mathbf {k}\\\\mathbf {q}} \\\\hat{d}^\\\\dagger _{\\\\mathbf {k}+\\\\mathbf {q}}\\\\hat{d}_\\\\mathbf {k}e^{-i\\\\mathbf {q}\\\\mathbf {r}}$ and $\\\\hat{n}_B(\\\\mathbf {r})=\\\\hat{\\\\psi }^\\\\dagger _B(\\\\mathbf {r})\\\\hat{\\\\psi }_B(\\\\mathbf {r})=\\\\frac{1}{\\\\cal {V}}\\\\sum _{\\\\mathbf {k}\\\\mathbf {q}} \\\\hat{a}^\\\\dagger _{\\\\mathbf {k}+\\\\mathbf {q}}\\\\hat{a}_\\\\mathbf {k}e^{-i\\\\mathbf {q}\\\\mathbf {r}}$ are the impurity and boson density, respectively.\",\n \"As we consider the limit of a single impurity it is convenient to switch to the first quantized description of the impurity, which is characterized by its position and momentum operator $\\\\hat{\\\\mathbf {R}}$ and $\\\\hat{\\\\mathbf {p}}$ .\",\n \"The density becomes $\\\\hat{n}_I(\\\\mathbf {r})\\\\rightarrow \\\\delta ^{(3)}(\\\\mathbf {r}- \\\\hat{\\\\mathbf {R}})$ and Eq.\",\n \"(REF ) takes the form $\\\\hat{H}&=& \\\\frac{\\\\hat{\\\\mathbf {p}}^2}{2M}+\\\\sum _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {k}\\\\hat{a}^\\\\dagger _\\\\mathbf {k}\\\\hat{a}_\\\\mathbf {k}+\\\\frac{g_{bb}}{2\\\\cal {V}}\\\\sum _{\\\\mathbf {k}\\\\mathbf {k}^{\\\\prime }\\\\mathbf {q}}\\\\hat{a}^\\\\dagger _{\\\\mathbf {k}^{\\\\prime }+\\\\mathbf {q}}\\\\hat{a}^\\\\dagger _{\\\\mathbf {k}-\\\\mathbf {q}}\\\\hat{a}_{\\\\mathbf {k}^{\\\\prime }}\\\\hat{a}_{\\\\mathbf {k}} \\\\\\\\ \\\\nonumber &+&\\\\frac{1}{\\\\cal {V}}\\\\sum _{ \\\\mathbf {k}\\\\mathbf {q}}V(\\\\mathbf {q})e^{-i\\\\mathbf {q}\\\\hat{\\\\mathbf {R}}}\\\\hat{a}^\\\\dagger _{\\\\mathbf {k}+\\\\mathbf {q}}\\\\hat{a}_{\\\\mathbf {k}}.$ From this Hamiltonian various polaron models can be derived, and we briefly explain their relevance and regimes of validity.\",\n \"At $T=0$ a large fraction of bosons is condensed in a BEC.\",\n \"This condensate can be regarded as a coherent state such that the zero-momentum mode $\\\\langle \\\\hat{a}_\\\\mathbf {k}\\\\rangle =\\\\sqrt{N_0}\\\\delta _{\\\\mathbf {k},\\\\mathbf {0}}$ takes on a macroscopic expectation value.\",\n \"Within the Bogoliubov approximation the bosonic creation and annihilation operators in Eq.\",\n \"(REF ) are expanded in fluctuations $\\\\hat{B}_\\\\mathbf {p}$ around this expectation value $\\\\sqrt{N_0}$ and terms of higher than quadratic order are neglected in the resulting Hamiltonian.\",\n \"The purely bosonic part of the Hamiltonian can then be diagonalized by the Bogoliubov rotation $\\\\hat{B}_\\\\mathbf {p}=u_\\\\mathbf {p}\\\\hat{b}_\\\\mathbf {p}+ v_{-\\\\mathbf {p}}^* \\\\hat{b}^\\\\dagger _{-\\\\mathbf {p}}\\\\,\\\\,,\\\\,\\\\,\\\\hat{B}^\\\\dagger _\\\\mathbf {p}=u_\\\\mathbf {p}^* \\\\hat{b}^\\\\dagger _\\\\mathbf {p}+ v_{-\\\\mathbf {p}} \\\\hat{b}_{-\\\\mathbf {p}},$ where $\\\\omega _\\\\mathbf {k}=\\\\sqrt{\\\\epsilon _\\\\mathbf {p}(\\\\epsilon _\\\\mathbf {p}+2 g_{bb}\\\\rho )}$ is the Bogoliubov dispersion relation and $u_\\\\mathbf {p},v_{-\\\\mathbf {p}}=\\\\pm \\\\sqrt{\\\\frac{\\\\epsilon _\\\\mathbf {p}+g_{bb} \\\\rho }{2\\\\omega _\\\\mathbf {p}}\\\\pm \\\\frac{1}{2}}$ .\",\n \"The density of the homogeneous condensate is given by $\\\\rho $ .\",\n \"The transformation (REF ) yields the Bose-impurity Hamiltonian $\\\\hat{H}&=& \\\\frac{\\\\hat{\\\\mathbf {p}}^2}{2M}+\\\\sum _\\\\mathbf {k}\\\\omega _\\\\mathbf {k}\\\\hat{b}^\\\\dagger _\\\\mathbf {k}\\\\hat{b}_\\\\mathbf {k}\\\\nonumber \\\\\\\\&+&\\\\rho V(\\\\mathbf {0})+{\\\\frac{1}{\\\\sqrt{\\\\cal {V}}}\\\\sum _{ \\\\mathbf {q}}g(\\\\mathbf {q}) e^{-i\\\\mathbf {q}\\\\hat{\\\\mathbf {R}}}\\\\left(\\\\hat{b}^\\\\dagger _{\\\\mathbf {q}}+\\\\hat{b}_{-\\\\mathbf {q}}\\\\right)}_{\\\\text{Fröhlich interaction}}\\\\nonumber \\\\\\\\&+&{\\\\frac{1}{\\\\cal {V}}\\\\sum _{ \\\\mathbf {k}\\\\mathbf {q}}V(\\\\mathbf {q})e^{-i\\\\mathbf {q}\\\\hat{\\\\mathbf {R}}}\\\\hat{B}^\\\\dagger _{\\\\mathbf {k}+\\\\mathbf {q}}\\\\hat{B}_{\\\\mathbf {k}}}_{\\\\text{extended Fröhlich interaction}}$ where we introduced the `Fröhlich coupling' $g(\\\\mathbf {q})=\\\\sqrt{\\\\frac{\\\\rho \\\\epsilon _\\\\mathbf {q}}{\\\\omega _\\\\mathbf {q}}}V(\\\\mathbf {q})$ and in the last term we kept the untransformed expression for notational brevity.\",\n \"In Eq.\",\n \"(REF ) the term $\\\\rho V(\\\\mathbf {0})= \\\\rho \\\\int d^3r V(\\\\mathbf {r})$ describes the mean-field energy shift of the polaron in the Born approximation.\",\n \"Neglecting the constant mean-field shift and the last term in Eq.\",\n \"(REF ) one arrives at the celebrated Fröhlich model [12]: $\\\\hat{H}&= \\\\frac{\\\\hat{\\\\mathbf {p}}^2}{2M}+\\\\sum _\\\\mathbf {k}\\\\omega _\\\\mathbf {k}\\\\hat{b}^\\\\dagger _\\\\mathbf {k}\\\\hat{b}_\\\\mathbf {k}+\\\\frac{1}{\\\\sqrt{\\\\cal {V}}}\\\\sum _{ \\\\mathbf {q}}g(\\\\mathbf {q}) e^{-i\\\\mathbf {q}\\\\hat{\\\\mathbf {R}}}\\\\left(\\\\hat{b}^\\\\dagger _{\\\\mathbf {q}}+\\\\hat{b}_{-\\\\mathbf {q}}\\\\right)$ In initial attempts to describe impurities in weakly interacting Bose gases, this Hamiltonian was used for systems close to Feshbach resonances [13], [14].\",\n \"However, as shown in [15], the Fröhlich Hamiltonian alone is insufficient to describe impurities that interact strongly with atomic quantum gases (for an explicit third-order perturbation theory analysis see Ref. [16]).\",\n \"Indeed the major theoretical shortcoming of Eq.\",\n \"(REF ) is that from this model one cannot recover the Lippmann-Schwinger equation for two-body impurity-bose scattering.\",\n \"Hence it fails to describe the underlying two-body scattering physics between the impurity and bosons including molecule formation.\",\n \"Similarly the Fröhlich Hamiltonian cannot account for the intricate dynamics leading to the formation of Rydberg polarons.\",\n \"For such strongly interacting systems the inclusion of the last term in Eq.\",\n \"(REF ) becomes crucial.\",\n \"This term accounts for pairing of the impurity with the atoms in the environment and accounts for the detailed `short-distance' physics of the problem, which is completely neglected in the Fröhlich model that is tailored for the description of long-wave length (low-energy) physics.\",\n \"So far it has been experimentally verified that the inclusion of this term is relevant for the observation of Bose polarons [7], [8] where the impurity-Bose interaction can be modeled by a potential that supports only a single, weakly bound two-body molecular state.\",\n \"In the present work we encounter a new type of impurity problem where the impurity is dressed by large sets of molecular states that have ultra-long-range character.\",\n \"This yields the novel physics of Rydberg polarons that is beyond the physics of Bose polarons so far observed in experiments and discussed in the literature.\",\n \"To account for the relevant Rydberg molecular physics theoretically, we analyze the full Hamiltonian Eq.\",\n \"(REF ).\",\n \"We note that the physics of the Rydberg oligomer states is different from that of Efimov states in the Bose polaron problem [15], [17], [18], [19].\",\n \"While Efimov states are also multi-body bound states, they arise due a quantum anomaly of the underlying quantum field theory [20], [21], [22].\",\n \"Indeed, the Efimov effect [23], [24], [25] gives rise to an infinite series of three-body states (and also states of larger atom number [26], [27]) that respect a discrete scaling symmetry.\",\n \"The Efimov effect arises for resonant short-range two-body interactions in three dimensions.\",\n \"In previous experiments studying Bose polarons close to a Feshbach resonance it was found that these Efimov states do not strongly influence the polaron physics [7], [8].\",\n \"In our case the multi-molecular states are not related to the Efimov effect and dimer, trimer, etc.\",\n \"states are rather comprised of nearly independently bound two-body molecular states that feature only very weak three- and higher-body correlations.\",\n \"In contrast, in Efimov physics, a single impurity potentially mediates strong correlations between two particles in the Bose gas.\",\n \"Due to large binding energies of Rydberg molecular states, this induced interaction is expected to be small for Rydberg polarons [28].\"\n ],\n [\n \"Polaron formation\",\n \"The formation of a polaron can be understood as dressing of the impurity by excitations of the bosonic bath, which entangles the momentum of the impurity with that of the bath excitations.\",\n \"This can be illustrated with a simple wave function expansion for a polaron of zero momentum $\\\\mathinner {|{\\\\Psi }\\\\rangle }&=\\\\sqrt{Z}\\\\mathinner {|{BEC}\\\\rangle }\\\\mathinner {|{\\\\mathbf {p}=0}\\\\rangle }_I+\\\\sum _\\\\mathbf {k}\\\\alpha _\\\\mathbf {k}\\\\hat{b}^\\\\dagger _{-\\\\mathbf {k}} \\\\hat{b}_\\\\mathbf {0}\\\\mathinner {|{BEC}\\\\rangle } \\\\mathinner {|{\\\\mathbf {k}}\\\\rangle }_I\\\\nonumber \\\\\\\\&+\\\\sum _{\\\\mathbf {k}\\\\mathbf {q}} \\\\alpha _{\\\\mathbf {k}\\\\mathbf {q}} \\\\hat{b}^\\\\dagger _{-\\\\mathbf {k}} \\\\hat{b}^\\\\dagger _{\\\\mathbf {k}-\\\\mathbf {q}} \\\\hat{b}_\\\\mathbf {0}\\\\hat{b}_\\\\mathbf {0}\\\\mathinner {|{BEC}\\\\rangle } \\\\mathinner {|{\\\\mathbf {q}}\\\\rangle }_I+\\\\ldots $ where $\\\\mathinner {|{\\\\mathbf {p}}\\\\rangle }_I$ denotes impurity momentum states.\",\n \"The unperturbed impurity-bath state in the first terms becomes supplemented by `particle-hole' fluctuations of the BEC state given by the second, etc.\",\n \"terms.\",\n \"These terms describe the polaronic dressing of the impurity by bath excitations that leads to the formation of the polaron.\",\n \"A Fourier transformation of the parameter $\\\\alpha _\\\\mathbf {k}$ to real space reveals the creation of density modulations in the medium, which represent the formation of a dressing cloud that is build around the impurity particle.\",\n \"In fact, wave functions of the type in Eq.\",\n \"(REF ) were found to yield a rather accurate description of impurities coupled to a fermionic bath of cold atoms via Feshbach resonances [29], [30], [31], [32], [33] and also describe exciton-impurities in two-dimensional semiconductors [34].\",\n \"When truncated at the one-excitation level, such an ansatz leads to limited agreement with experimental observations for impurities in a Bose gas [35], [36].\",\n \"Dressing can also be understood from a perturbative expansion in terms of Feynman diagrams as shown in Fig.\",\n \"REF .\",\n \"Here solid lines represent the propagating impurity, and the dashed lines denote Bosons that are excited from the BEC.\",\n \"The Fröhlich model considers only processes where bosons are excited out of the condensate and then re-enter the BEC in their next scattering event with the impurity.\",\n \"However, these `Fröhlich scattering processes' can neither account for molecular bound-state formation, nor for intricate strong-coupling physics found close to Feshbach resonances.\",\n \"To describe these phenomena, the last term in Eq.\",\n \"(REF ) has to be considered.\",\n \"This term allows for scattering of bosons where an excited boson does not directly reenter the BEC but can scatter off the impurity arbitrarily many times.\",\n \"In this process, illustrated as gray disk in Fig.\",\n \"REF , the boson changes its momentum.\",\n \"The infinite repetition of such scattering processes represents the Lippmann-Schwinger equation in terms of Feynman diagrams and accounts for molecular bound state formation.\"\n ],\n [\n \"Rydberg Polaron Hamiltonian\",\n \"We now specialize to the case of Rydberg impurities.\",\n \"When a bath atom is excited into a Rydberg state of principal quantum number $n$ , it interacts with the surrounding ground-state atoms through a pseudopotential, first proposed by Fermi [37], in which molecular binding occurs through frequent scattering of the nearly free and zero-energy Rydberg electron from the ground-state atom.\",\n \"In this picture, the Born-Oppenheimer potential for a ground-state atom at distance $\\\\mathbf {r}$ from the Rydberg impurity, retaining $s$ -wave and $p$ -wave scattering partial waves, is given as [37], [38], [39], [40], [41] $V_\\\\text{Ryd}(\\\\mathbf {r})&=&\\\\frac{2\\\\pi \\\\hbar ^2}{m_e} a_s|\\\\Psi (\\\\mathbf {r})|^2+\\\\frac{6\\\\pi \\\\hbar ^2 }{m_e} a_p^3|\\\\overrightarrow{\\\\nabla }\\\\Psi (\\\\mathbf {r})|^2, $ where we kept explicit factors of $\\\\hbar $ and $\\\\Psi (\\\\mathbf {r})$ is the Rydberg electron wave function, $a_s$ and $a_p$ are the momentum-dependent $s$ -wave and $p$ -wave scattering lengths, and $m_e$ is the electron mass.\",\n \"When $a_s<0$ , $V(\\\\mathbf {r})$ can support molecular states with one or more ground-state atoms bound to the impurity [38], [42], [43].\",\n \"A Rydberg polaron is formed when the Rydberg impurity is dressed by the occupation of a large number of these bound states in addition to finite momentum states.\",\n \"This process gives rise to an absorption spectrum of a distribution of molecular peaks with a Gaussian envelope, which is the key spectral signature of Rydberg polaron formation.\",\n \"The large energy scale of the Rydberg molecular states involved in the formation of Rydberg polarons allow us to simplify the extended Fröhlich Hamiltonian Eq.\",\n \"(REF ): the typical energy range of Rydberg molecules is $0.1-10$ MHz for high quantum number $n$ , while the typical energy scale for Bose-Bose interactions is 1-10 KHz.\",\n \"Therefore bosons that are bound to a Rydberg impurity probe momentum scales deep in the particle branch of the Bogoliubov dispersion relation, in a regime where the Bogoliubov factors $u_\\\\mathbf {p}=1$ and $v_\\\\mathbf {p}=0$ .\",\n \"Hence we can neglect the Bose-Bose interactions and Eq.\",\n \"(REF ) reduces to the simplified, extended Fröhlich model [11] $\\\\hat{H}&=& \\\\frac{\\\\hat{\\\\mathbf {p}}^2}{2M}+\\\\sum _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {k}\\\\hat{a}^\\\\dagger _\\\\mathbf {k}\\\\hat{a}_\\\\mathbf {k}+\\\\frac{1}{\\\\cal {V}}\\\\sum _{ \\\\mathbf {k}\\\\mathbf {q}}V(\\\\mathbf {q})e^{-i\\\\mathbf {q}\\\\hat{\\\\mathbf {R}}}\\\\hat{a}^\\\\dagger _{\\\\mathbf {k}+\\\\mathbf {q}}\\\\hat{a}_{\\\\mathbf {k}}\\\\nonumber \\\\\\\\&=& \\\\frac{\\\\hat{\\\\mathbf {p}}^2}{2M}+\\\\sum _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {k}\\\\hat{b}^\\\\dagger _\\\\mathbf {k}\\\\hat{b}_\\\\mathbf {k}+\\\\rho V(\\\\mathbf {0}) \\\\nonumber \\\\\\\\&+&{\\\\frac{1}{\\\\sqrt{\\\\cal {V}}}\\\\sum _{ \\\\mathbf {q}}\\\\sqrt{\\\\rho }V(\\\\mathbf {q})e^{-i\\\\mathbf {q}\\\\hat{\\\\mathbf {R}}}\\\\left(\\\\hat{b}^\\\\dagger _{\\\\mathbf {q}}+\\\\hat{b}_{-\\\\mathbf {q}}\\\\right)}_{\\\\text{Fröhlich interaction}}\\\\nonumber \\\\\\\\&+&{\\\\frac{1}{\\\\cal {V}}\\\\sum _{ \\\\mathbf {k}\\\\mathbf {q}}V(\\\\mathbf {q})e^{-i\\\\mathbf {q}\\\\hat{\\\\mathbf {R}}}\\\\hat{b}^\\\\dagger _{\\\\mathbf {k}+\\\\mathbf {q}}\\\\hat{b}_{\\\\mathbf {k}}}_{\\\\text{extended Fröhlich interaction}}.$ where in the second equivalent expression we expanded the boson operators $\\\\hat{a}^\\\\dagger _\\\\mathbf {k}$ around their expectation value $\\\\sqrt{N_0}$ which highlights the connection to the Fröhlich model.\",\n \"In order to describe Rydberg impurities the interaction $V(\\\\mathbf {q})$ is given as the Fourier transform of the Rydberg molecular potential $V(\\\\mathbf {q})=\\\\int d^3r V_\\\\text{Ryd}(\\\\mathbf {r})e^{i\\\\mathbf {q}\\\\cdot \\\\mathbf {r}}$ .\",\n \"The molecular potentials and interacting single-particle wave functions ($\\\\mathinner {|{\\\\beta _i}\\\\rangle }$ ) are calculated as described in [43], [10].\",\n \"We note that the depletion of a BEC by a single electron in a BEC has been studied in [44] including only the linear Fröhlich term in Eq.\",\n \"(REF ).\",\n \"While such an approach can approximately account for the rate of depletion of the condensate it fails to describe the formation of molecules that is essential for the formation of Rydberg polarons.\"\n ],\n [\n \"Linear-response absorption from quench dynamics\",\n \"One way to probe polaron structure and dynamics is absorption spectroscopy.\",\n \"Here one utilizes the fact that before being excited to a Rydberg state, an atom is in its ground state $\\\\mathinner {|{5s}\\\\rangle }$ , and its interaction with the surrounding Bosons is negligible.\",\n \"The system is described by the Hamiltonian $\\\\hat{H}_0$ given by the first two terms in Eq.\",\n \"(REF ).\",\n \"In contrast, when the atom is in its Rydberg state $\\\\mathinner {|{ns}\\\\rangle }$ the potential $V(\\\\mathbf {q})$ is switched on.\",\n \"In experiments [9], transitions between both states are driven by a two-photon excitation.\",\n \"Within linear response, the corresponding absorption of laser light at frequency $\\\\nu $ is given by Fermi's Golden rule $\\\\mathcal {A}(\\\\nu )&= 2\\\\pi \\\\sum _{if}w_i |\\\\mathinner {\\\\langle {f}|}\\\\hat{V}_\\\\text{L}\\\\mathinner {|{i}\\\\rangle }|^2\\\\times \\\\,\\\\delta (\\\\nu -(E_i-E_f)).$ Here the sum extends over all initial and final states of the impurity plus bosonic bath that fulfill $\\\\hat{H}_0 \\\\mathinner {|{i}\\\\rangle }=E_i\\\\mathinner {|{i}\\\\rangle }$ and $\\\\hat{H} \\\\mathinner {|{f}\\\\rangle }=E_f\\\\mathinner {|{f}\\\\rangle }$ , where $\\\\hat{H}$ is given by Eq.\",\n \"(REF ) so that the final states $\\\\mathinner {|{f}\\\\rangle }$ are eigenstates in the presence of the strong perturbation from Rydberg-boson interactions.\",\n \"In the states $\\\\mathinner {|{i}\\\\rangle }$ , all atoms (including the impurity atom) are in the $\\\\mathinner {|{5s}\\\\rangle }$ state, while in final states $\\\\mathinner {|{f}\\\\rangle }$ the impurity is in the atomic $\\\\mathinner {|{ns}\\\\rangle }$ state.\",\n \"The laser operator that drives the two-photon transition between these two atomic states $\\\\mathinner {|{5s}\\\\rangle }$ and $\\\\mathinner {|{ns}\\\\rangle }$ , is given by $\\\\hat{V}_L\\\\sim \\\\mathinner {|{ns}\\\\rangle }\\\\mathinner {\\\\langle {5s}|}+h.c.$ .\",\n \"Using the Fourier-representation of the delta function in Eq.\",\n \"(REF ), one can show that the absorption spectrum follows from [10] (for details see [45]) $\\\\mathcal {A}(\\\\nu )=2\\\\,\\\\text{Re}\\\\,\\\\int _{0}^\\\\infty dt\\\\, e^{i \\\\nu t}\\\\, S(t)$ with the many-body overlap (also called the Loschmidt echo) $S(t)$ .\",\n \"In the general case of finite temperature, the $w_i$ in Eq.\",\n \"(REF ) are the thermodynamic weights of each initial state given by the diagonal elements of the density matrix $\\\\hat{\\\\rho }_\\\\text{ini}=e^{-\\\\beta H_0}/Z_p$ , for sample temperature $k_B T=1/\\\\beta $ and partition function $Z_p$ .\",\n \"For finite temperature the Loschmidt echo is then given by $S(t)&=&\\\\text{Tr}[\\\\hat{\\\\rho }_\\\\text{ini}\\\\, e^{i \\\\hat{H}_0 t }e^{-i \\\\hat{H} t}]$ where $\\\\text{Tr}$ denotes the trace over the complete many-body Fock space.\",\n \"This expression shows that the overlap function $S(t)$ encodes the non-equilibrium time evolution of the system following a quantum quench represented by the introduction of the Rydberg impurity at time $t=0$ .\",\n \"From this time evolution then follows the absorption spectrum by Fourier transformation.\",\n \"The relevant time scales for the dynamics of Rydberg impurities in a BEC is given by the Rydberg molecular energies which exceed both the temperature scale $k_B T$ and the energy scale associated with boson-boson interactions.\",\n \"Thus temperature and boson interaction effects can be neglected in the calculation of $S(t)$ for Rydberg polarons.\",\n \"In this limit the initial state of the system is given by $\\\\mathinner {|{i}\\\\rangle }=\\\\mathinner {|{\\\\text{BEC}}\\\\rangle }\\\\otimes \\\\mathinner {|{\\\\mathbf {p}=0}\\\\rangle }_I \\\\otimes \\\\mathinner {|{5s}\\\\rangle }_I$ and Rydberg polarons are well described by the T=0 limit of Eq.\",\n \"(REF ), $S(t)={_I\\\\mathinner {\\\\langle {\\\\mathbf {p}=0}|}}\\\\mathinner {\\\\langle {\\\\Psi _\\\\text{BEC}}|}e^{i \\\\hat{H}_0 t}e^{-i \\\\hat{H} t}\\\\mathinner {|{\\\\Psi _\\\\text{BEC}}\\\\rangle }\\\\mathinner {|{\\\\mathbf {p}=0}\\\\rangle }_I,$ where $\\\\mathinner {|{\\\\Psi _\\\\text{BEC}}\\\\rangle }$ represents an ideal BEC of atoms.\",\n \"The accurate calculation of the absorption response presents a formidable theoretical challenge.\",\n \"In the following we will present three different approaches of different degrees of sophistication.\",\n \"In a simple approximation one may employ mean-field theory (Section ) where the impurity-boson interactions are solely described by the mean-field result $\\\\rho V(\\\\mathbf {0})$ in Eq.\",\n \"(REF ).\",\n \"In this approach all quantum effects and fluctuations are neglected.\",\n \"Furthermore one may employ a classical stochastic model (Section ), which at least can treat fluctuations to a certain degree.\",\n \"Finally, we employ a functional determinant approach, which was developed in [10] and which we review in Section VI.\",\n \"This approach treats all interaction terms in Eq.\",\n \"(REF ) fully on a quantum level and allows us to explain the intricate Rydberg polaron formation dynamics observed in our experiments.\"\n ],\n [\n \"Mean Field Approach\",\n \"The simplest treatment of the polaron excitation spectrum is to neglect all interaction terms other than $\\\\rho {V}(\\\\mathbf {0})$ in Eq.\",\n \"(REF ), which yields the mean-field approximation.\",\n \"Hence at a given constant density $\\\\rho (\\\\mathbf {r})$ in the trap the absorption spectrum is obtained by Fourier transformation of (cf.\",\n \"Eq.\",\n \"(REF )) $S(t,\\\\vec{r})=e^{-i \\\\rho (\\\\mathbf {r}) V(\\\\mathbf {0}) t} = e^{-i \\\\Delta (\\\\mathbf {r})t},$ which yields a delta function Rydberg-absorption response $\\\\mathcal {A}(\\\\nu ,\\\\mathbf {r})=\\\\delta (\\\\nu -\\\\Delta (\\\\mathbf {r}))$ for a given local, homogeneous density $\\\\rho (\\\\mathbf {r})$ at a detuning from the atomic transition $\\\\Delta (\\\\mathbf {r})&=& \\\\rho (\\\\mathbf {r}) V(\\\\mathbf {q}=\\\\mathbf {0})=\\\\rho (\\\\mathbf {r})\\\\int d^3\\\\mathbf {r}^{\\\\prime } V(\\\\mathbf {r}-\\\\mathbf {r}^{\\\\prime }) .$ In the experiment, the density $\\\\rho (\\\\mathbf {r})$ varies with position $\\\\mathbf {r}$ .\",\n \"Since the excitation laser illuminates the entire atomic cloud it excites Rydberg atoms in regions of varying density.\",\n \"To theoretically model the resulting average over various contributions from the atomic cloud, we perform a local density approximation (LDA), which assumes the density variation is negligible over the range of the Rydberg interaction $V(\\\\mathbf {r})$ given by Eq.\",\n \"REF .\",\n \"The spectrum for creation of a Rydberg impurity is then given by $A(\\\\nu )\\\\propto \\\\int d\\\\mathbf {r}^3 \\\\rho (\\\\mathbf {r}) \\\\mathcal {A}(\\\\nu ,\\\\mathbf {r}).$ In the mean-field approximation this yields the response $A(\\\\nu )\\\\propto \\\\int d^3r\\\\, \\\\rho (\\\\mathbf {r}) \\\\delta (\\\\nu -\\\\Delta (\\\\mathbf {r}))$ for laser detuning $\\\\nu $ from unperturbed atomic resonance.\",\n \"Note that the mean-field treatment is similar to the description of $1S-2S$ spectroscopy of a quantum degenerate hydrogen gas given in [46].\",\n \"An intuitive way to express this shift is in terms of an effective $s$ -wave electron-atom scattering length that reflects the average of the interactions over the Rydberg wave function for the remainder of this section we keep factors of $\\\\hbar $, $a_{s,\\\\textup {eff}}=\\\\int d^3r^{\\\\prime } \\\\frac{{2\\\\pi m_e}}{h^2}V(\\\\mathbf {r^{\\\\prime }}),$ yielding $h\\\\Delta (\\\\mathbf {r})&=& \\\\frac{h^2 a_{s,\\\\textup {eff}}}{2\\\\pi m_e} \\\\rho (\\\\mathbf {r}).$ $a_{s,\\\\textup {eff}}$ varies with principal quantum number through the variation in the Rydberg electron wavefunction and the classical dependence of the electron momentum on position.\",\n \"Figure REF shows calculated values of $a_{s,\\\\textup {eff}}$ for the interaction of Sr($5sns$ $^3S_1$ ) Rydberg atoms with background strontium atoms [43].\",\n \"To obtain this result, Eq.\",\n \"(REF ) is only integrated over $|\\\\mathbf {r}^{\\\\prime }|>0.06 n^2 a_0$ because the approximation breaks down near the Rydberg core, where, among other modifications, the ion-atom polarization potential becomes important.\",\n \"Figure: Calculated values of a s,eff a_{s,\\\\textup {eff}} from Eq.\",\n \"() versus n * =n-δn^*=n-\\\\delta , where δ=3.371\\\\delta =3.371 is the quantum defect.Figure: Mean-field description of the excitation spectrum of Sr(5sns5sns 3 S 1 ^3S_1) Rydberg atoms in a strontium Bose-Einstein condensate (BEC) for (Left) n=49, (Middle) n=60, (Right) n=72.\",\n \"Symbols are the experimental data.\",\n \"Blue bands represent the confidence interval for the mean-field fit of the BEC contribution to the spectrum corresponding to the uncertainties in parameters given in Tab.\",\n \".\",\n \"The central black line indicates the best fit.\",\n \"The dashed red line is the mean-field prediction for thecontribution from the low-density region of the atomic cloud formed by thermal atoms.\",\n \"This contribution is calculated using the parametersgiven in Tab.\",\n \"including a sample temperature adjusted to reproduce the observed BEC fraction.Table: Parameters for data and fits shown in Fig.\",\n \": Peak detuning of the mean-field fit (Δ max \\\\Delta _{\\\\textup {max}}) and BEC fraction (η\\\\eta ) are determined from the spectra.\",\n \"Number of atoms in the condensate (N BEC N_{BEC}) and mean trap frequency ω ¯\\\\bar{\\\\omega } are determined from time-of-flight-absorption images and measurements of collective mode frequencies for trapped atoms respectively, and they determine the peak condensate density (ρ max \\\\rho _{\\\\textup {max}}) and chemical potential (μ\\\\mu ).\",\n \"The second-to-last column provides the ratio of Δ max \\\\Delta _{\\\\textup {max}} to Δ ˜ max =ω ¯ 4π15Na bb a ho 2/5 m m e a s,eff a bb \\\\tilde{{\\\\Delta }}_{\\\\textup {max}} = \\\\frac{\\\\overline{\\\\omega }}{4\\\\pi }\\\\left(\\\\frac{15 N a_{bb}}{a_{ho}}\\\\right)^{2/5}\\\\frac{m}{m_e}\\\\frac{a_{s,eff}}{a_{bb}}, the peak shift predicted using information independent of the mean-field fits.\",\n \"The sample temperature (TT) is the result of a fit of the observed BEC fraction using a numerical calculation of the number of non-condensed atoms in the trap fixing all the BEC parameters at values given in the table.Figure REF shows the mean-field description of experimental results for excitation of Sr($5sns$ $^3S_1$ ) Rydberg atoms in a strontium Bose-Einstein condensate (BEC).\",\n \"For the prediction of the absorption response the effect of temperature enters by determining the local density of the cloud.\",\n \"In fact, in the local-density-approximation model of the impurity excitation spectrum (Eq.\",\n \"(REF )), the density of atoms in the trap, $\\\\rho (\\\\mathbf {r})=\\\\rho _{BEC}(\\\\mathbf {r})+\\\\rho _{\\\\textup {th}}(\\\\mathbf {r})$ , has contributions from thermal, non-condensed atoms and condensed atoms.\",\n \"The contribution from the thermal gas is restricted to the sharp peak near zero detuning and a very small contribution towards the red in each data set in Fig.\",\n \"REF .\",\n \"Thus the BEC and thermal contributions can be calculated separately in the mean-field approximation.\",\n \"The profile of the condensate density is well-approximated for our conditions with a Thomas-Fermi (TF) distribution for a harmonic trap [48].\",\n \"If we neglect the contribution of thermal atoms to the density, the condensate contribution to the spectrum is $A_{\\\\textup {MF}}(\\\\nu )\\\\propto N_{BEC}\\\\frac{-\\\\nu }{\\\\Delta _{\\\\textup {max}}^2}\\\\sqrt{1-\\\\nu /\\\\Delta _{\\\\textup {max}}},$ for $\\\\Delta _{\\\\textup {max}}\\\\le \\\\nu \\\\le 0$ and zero otherwise [46].\",\n \"Here, $\\\\Delta _{\\\\textup {max}} =\\\\frac{ h a_{s,\\\\textup {eff}}}{2\\\\pi m_e} \\\\rho _{\\\\textup {max}}$ , and the peak BEC density is $\\\\rho _{\\\\textup {max}}=\\\\mu _{TF}/g$ , where the chemical potential is $\\\\mu _{TF}=(\\\\hbar \\\\overline{\\\\omega }/2)\\\\left( 15 N_{BEC} a_{bb}/ a_{\\\\textup {ho}} \\\\right)^{2/5}$ and $g=4\\\\pi \\\\hbar ^2 a_{bb}/m$ for harmonic oscillator length $a_{\\\\textup {ho}}=(\\\\hbar /m\\\\overline{\\\\omega })^{1/2}$ and mean trap radial frequency $\\\\overline{\\\\omega }=(\\\\omega _1 \\\\omega _2 \\\\omega _3)^{1/3}$ .\",\n \"This functional form is used to fit several data sets in Fig.\",\n \"REF with $\\\\Delta _{\\\\textup {max}}$ and the overall signal amplitude as the only fit parameters.\",\n \"When frequency is scaled by $\\\\Delta _{\\\\textup {max}}$ , the mean-field prediction for a BEC in a harmonic trap is universal, making $\\\\Delta _{\\\\textup {max}}$ an important parameter for describing the spectrum.\",\n \"The resulting fit parameters are given in table REF .\",\n \"We ascribe the difference between the mean-field fit and the observed spectra at small detuning to the contribution from the low-density region of the atomic cloud formed by thermal atoms.\",\n \"From the ratio of the areas of these two signal components, we extract the condensate fraction.\",\n \"Given the clean spectral separation between the contributions from thermal and condensed atoms, this is a promising technique for measuring very small thermal fractions and thus temperature of very cold Bose gases.\",\n \"Once the BEC parameters are set, for a consistency check, we then calculate the contribution to the spectrum from thermal atoms using the mean-field approximation and adjusting the sample temperature to match the BEC fraction.\",\n \"Additional details of the fitting process are described in App.\",\n \".\",\n \"For $n=72$ , the mean-field approach is quite accurate in describing the overall shape of the response across the entire spectrum.\",\n \"However, at lower principal quantum numbers, the data and fit deviate significantly, especially at larger detunings.\",\n \"Generally, mean-field fails both at small detuning (i.e.\",\n \"for response from low-density regions of the cloud), where the deviation arises from the formation of few-body molecular states, and at large detuning, where fluctuations in the macroscopic occupation of molecular states become important.\",\n \"Fluctuations correspond to a spread in binding energies of the polaron states excited for a given average density.\",\n \"All of these effects are neglected in the mean-field description.\"\n ],\n [\n \"Functional determinant approach\",\n \"The mean-field approach is insufficient to describe the main features found in the absorption spectrum both at small and large detuning from the atomic transition.\",\n \"For instance its failure at low detuning (low densities) has its origin in the formation of Rydberg molecular states, which is not described in mean-field theory.\",\n \"The low density regime can be most conveniently studied in detail by working at a low principal quantum number (here $n=38$ ) where only few atoms are in the Rydberg orbit, and the signal is thus dominated by the low-density response.\",\n \"To calculate $S(t)$ efficiently we evaluate it within the FDA.\",\n \"In [10] the FDA was developed to include (for impurities of infinite mass) also the finite-temperature corrections that are however irrelevant for our experiment.\",\n \"Thus we can restrict ourselves to the description of the zero-temperature response given by Eq.\",\n \"(REF ), where the bosonic ground state can be expressed as a state of fixed particle number $\\\\mathinner {|{\\\\text{BEC}}\\\\rangle }=(\\\\hat{a}_\\\\mathbf {0}^\\\\dagger )^N_0/\\\\sqrt{N_0!\",\n \"}\\\\mathinner {|{0}\\\\rangle }$ .\",\n \"For sufficiently large particle number we may equally use its representation as a coherent state $\\\\mathinner {|{\\\\text{BEC}}\\\\rangle }=\\\\exp [{\\\\sqrt{N_0}(\\\\hat{a}_\\\\mathbf {0}^\\\\dagger -\\\\hat{a}_\\\\mathbf {0})}]\\\\mathinner {|{0}\\\\rangle }$ where $\\\\mathinner {|{0}\\\\rangle }$ is the boson vacuum.\",\n \"The FDA, as developed in Ref.\",\n \"[10], did not include the description of impurity recoil which is, however, essential for an accurate prediction of Rydberg molecular spectra.\",\n \"In the following subsection we develop a extension of the FDA approach that overcomes this limitation.\"\n ],\n [\n \"Canonical transformation: mobile Rydberg impurity\",\n \"The FDA can be applied to time evolutions described by a Hamiltonian bilinear in creation and destruction operators.\",\n \"This is fulfilled for the Hamiltonian Eq.\",\n \"(REF ) in the case of an infinitely heavy impurity $M=\\\\infty $ .\",\n \"However, for a mobile Rydberg impurity, the presence of the non-commuting operators $\\\\hat{\\\\mathbf {p}}$ and $\\\\hat{\\\\mathbf {r}}$ effectively gives rise to non-bilinear terms.\",\n \"Figure: Exemplary Ramsey signal S(t)=|S(t)|e iϕ(t) S(t)= |S(t)| e^{i \\\\varphi (t)} including contrast |S||S| and phase ϕ\\\\varphi underlying the calculation of the Rydberg polaron absorption spectrum for n=49n=49 at peak density in absence of a finite Rydberg lifetime.\",\n \"A fast decay visible in the contrast accompanies fast oscillations of the full complex system.\",\n \"The combination of both gives rise to the distinct Rydberg polaron features of molecular peaks that are distributed according to a gaussian envelope.\",\n \"The Ramsey signal thus provides an alternative pathway for observing Rydberg polaron formation dynamics in real-time.\",\n \"The time scales of this coherent dressing dynamics are ultrafast compared to the typical time scales of collective low-energy excitations of ultracold quantum gases allowing study of ultrafast dynamics in a new setting.To remedy this challenge we combine the FDA with a canonical transformation.\",\n \"Here we focus on the zero-temperature case.\",\n \"In this case the response is obtained from the time evolution using Eq.\",\n \"(REF ), where the Hamiltonian includes bosonic and impurity operators.\",\n \"To deal with the impurity motion we perform a canonical transformation first proposed by Lee, Low, and Pines [49] which effectively transforms into the system comoving with the impurity.\",\n \"To this end we define the translation operator [49] $U=\\\\exp \\\\left\\\\lbrace i \\\\hat{\\\\mathbf {R}}\\\\sum _\\\\mathbf {k}\\\\mathbf {k}\\\\hat{a}^\\\\dagger _\\\\mathbf {k}\\\\hat{a}_\\\\mathbf {k}\\\\right\\\\rbrace $ which is inserted in the time evolution $S(t)&=\\\\mathinner {\\\\langle {\\\\mathbf {p}=0}|}_I\\\\mathinner {\\\\langle {\\\\Psi _\\\\text{BEC}}|}U U^{-1}e^{-i \\\\hat{H} t}U U^{-1}\\\\mathinner {|{\\\\Psi _\\\\text{BEC}}\\\\rangle }\\\\mathinner {|{\\\\mathbf {p}=0}\\\\rangle }_I\\\\nonumber \\\\\\\\&=\\\\mathinner {\\\\langle {\\\\mathbf {p}=0}|}_I\\\\mathinner {\\\\langle {\\\\Psi _\\\\text{BEC}}|}e^{-i \\\\hat{\\\\mathcal {H}} t} \\\\mathinner {|{\\\\Psi _\\\\text{BEC}}\\\\rangle }\\\\mathinner {|{\\\\mathbf {p}=0}\\\\rangle }_I.$ In the first line we used that the term $e^{i\\\\hat{H}_0 t}$ can be dropped as the initial state is a zero energy state.\",\n \"Furthermore, in the second line we made use of the fact that the total boson momentum of the BEC state is zero and we defined the transformed Hamiltonian $\\\\hat{\\\\mathcal {H}}=U^{-1}\\\\hat{H} U&= \\\\frac{\\\\left(\\\\hat{\\\\mathbf {p}}-\\\\sum _\\\\mathbf {k}\\\\mathbf {k}\\\\hat{a}^\\\\dagger _\\\\mathbf {k}\\\\hat{a}_\\\\mathbf {k}\\\\right)^2}{2M}+\\\\sum _\\\\mathbf {k}\\\\epsilon _\\\\mathbf {k}\\\\hat{a}^\\\\dagger _\\\\mathbf {k}\\\\hat{a}_\\\\mathbf {k}\\\\nonumber \\\\\\\\&+\\\\frac{1}{\\\\cal {V}}\\\\sum _{ \\\\mathbf {k}\\\\mathbf {q}}V(\\\\mathbf {q})\\\\hat{a}^\\\\dagger _{\\\\mathbf {k}+\\\\mathbf {q}}\\\\hat{a}_{\\\\mathbf {k}}$ By virtue of the transformation $U$ the impurity coordinate is eliminated in the interaction and only the impurity momentum operator $\\\\hat{\\\\mathbf {p}}$ remains.\",\n \"It thus commutes with the many-body Hamiltonian and is replaced by a c-number, $\\\\hat{\\\\mathbf {p}}\\\\rightarrow \\\\mathbf {p}$ .\",\n \"Since in our case we are interested in the polaron at zero momentum, $\\\\mathbf {p}\\\\rightarrow 0$ .\",\n \"After normal ordering, we arrive at $\\\\hat{\\\\mathcal {H}}&=& \\\\sum _{\\\\mathbf {k}\\\\mathbf {k}^{\\\\prime }} \\\\frac{\\\\mathbf {k}\\\\mathbf {k}^{\\\\prime }}{2M} \\\\hat{a}^\\\\dagger _{\\\\mathbf {k}^{\\\\prime }}\\\\hat{a}^\\\\dagger _\\\\mathbf {k}\\\\hat{a}_\\\\mathbf {k}\\\\hat{a}_{\\\\mathbf {k}^{\\\\prime }}+\\\\sum _\\\\mathbf {k}\\\\frac{\\\\mathbf {k}^2}{2\\\\mu _\\\\text{red}} \\\\hat{a}^\\\\dagger _\\\\mathbf {k}\\\\hat{a}_\\\\mathbf {k}\\\\nonumber \\\\\\\\&+&\\\\frac{1}{\\\\cal {V}}\\\\sum _{ \\\\mathbf {k}\\\\mathbf {q}}V(\\\\mathbf {q})\\\\hat{a}^\\\\dagger _{\\\\mathbf {k}+\\\\mathbf {q}}\\\\hat{a}_{\\\\mathbf {k}}$ The above transformation has the effect that the boson dispersion relation $\\\\epsilon _\\\\mathbf {k}\\\\rightarrow \\\\mathbf {k}^2/2\\\\mu _\\\\text{red}$ now has the reduced mass $\\\\mu _\\\\text{red}=mM/(M+m)$ of boson-impurity partners, and the Hamiltonian includes an induced interaction of bosons described by the first term in Eq.\",\n \"(REF ).\",\n \"Due to spherical symmetry and the large energy scale of the Rydberg impurity-Bose gas interaction, we may neglect this term.\",\n \"The reliability of this approximation has been demonstrated in recent work of some of the authors [36] where a time-dependent variational principle has been applied to the evaluation of the time-evolution of Eq.\",\n \"(REF ).\",\n \"In the approximation where the time-dependent wave function $\\\\mathinner {|{\\\\Psi (t)}\\\\rangle }$ evolved by $\\\\exp (-i\\\\hat{\\\\mathcal {H}}t)$ is taken to be a product of coherent states of the form $\\\\mathinner {|{\\\\Psi (t)}\\\\rangle }=e^{\\\\sum _\\\\mathbf {k}(\\\\gamma _\\\\mathbf {k}(t) \\\\hat{a}_\\\\mathbf {k}^\\\\dagger +\\\\text{h.c.})}\\\\mathinner {|{0}\\\\rangle }$ , one finds from the equation of motions of the variational parameters $\\\\gamma _\\\\mathbf {q}$ that the expectation value $\\\\mathinner {\\\\langle {\\\\Psi (t)}|}\\\\sum _{\\\\mathbf {k}\\\\mathbf {k}^{\\\\prime }} \\\\frac{\\\\mathbf {k}\\\\mathbf {k}^{\\\\prime }}{2M} \\\\hat{a}^\\\\dagger _{\\\\mathbf {k}^{\\\\prime }}\\\\hat{a}^\\\\dagger _\\\\mathbf {k}\\\\hat{a}_\\\\mathbf {k}\\\\hat{a}_{\\\\mathbf {k}^{\\\\prime }}\\\\mathinner {|{\\\\Psi (t)}\\\\rangle }$ always remains zero and this term hence does not contribute to the dynamics.\",\n \"For our case of Rydberg impurities the FDA solution presented here reproduces the variational result of Ref.\",\n \"[36] and again shows excellent agreement with experimental data attesting a postiori to the accuracy of the method and the neglect of the first term in Eq.\",\n \"(REF ).\",\n \"To which extent this remains valid for other Bose polaron scenarios is an open question and subject to ongoing theoretical studies [50].\",\n \"In summary, we finally arrive at the Hamiltonian $\\\\hat{ \\\\mathcal {H}}= \\\\sum _\\\\mathbf {k}\\\\frac{\\\\mathbf {k}^2}{2\\\\mu _\\\\text{red}} \\\\hat{a}^\\\\dagger _\\\\mathbf {k}\\\\hat{a}_\\\\mathbf {k}+\\\\frac{1}{\\\\mathcal {V}}\\\\sum _{ \\\\mathbf {k}\\\\mathbf {q}}V(\\\\mathbf {q})\\\\hat{a}^\\\\dagger _{\\\\mathbf {k}+\\\\mathbf {q}}\\\\hat{a}_{\\\\mathbf {k}} \\\\mathinner {|{ns}\\\\rangle }\\\\mathinner {\\\\langle {ns}|}$ whose dynamics we simulate to obtain Rydberg polaron excitation spectra.\",\n \"Note, in Eq.\",\n \"(REF ), we make explicit that the impurity-boson interaction is only present when the impurity is in its excited atomic Rydberg state $\\\\mathinner {|{n s}\\\\rangle }$ .\",\n \"In this state the Rydberg electron scatters off ground state atoms in the environment, leading to the strong Rydberg Born-Oppenheimer potential $V(\\\\mathbf {q})$ .\",\n \"For simplified notation we will now switch again to the symbol $\\\\mathcal {\\\\hat{H}} \\\\rightarrow \\\\hat{H}$ .\"\n ],\n [\n \"Time-domain $S(t)$\",\n \"The strength of the FDA is that it allows one to express overlaps of many-body states in terms of single-particle eigenstates [51], [52], [53], [54], [10].\",\n \"In our case this implies that we must calculate single particle eigenstates and energies of the free and `interacting' Hamiltonian, i.e.\",\n \"$\\\\hat{h}_0\\\\mathinner {|{n}\\\\rangle }=\\\\epsilon _n \\\\mathinner {|{n}\\\\rangle }$ and $\\\\hat{h}\\\\mathinner {|{\\\\beta }\\\\rangle } = \\\\omega _\\\\beta \\\\mathinner {|{\\\\beta }\\\\rangle }$ , respectively ($\\\\hat{h}_0$ and $\\\\hat{h}$ are the single-particle representatives of the many-body operators $\\\\hat{H}_0$ and $\\\\hat{H}$ , respectively).\",\n \"The single-particle eigenstates and energies $\\\\mathinner {|{\\\\beta }\\\\rangle }$ and $\\\\omega _\\\\beta $ are calculated using exact diagonalization.\",\n \"Here we solve the radial Schrödinger equation for a spherical box of radius $R$ discretized in real-space.\",\n \"Since the BEC is initially in a zero angular state and the Rydberg molecular potential is spherically symmetric we can restrict the analysis to single-particle states with zero-angular momentum.\",\n \"We include the 300 energetically lowest eigenstates, which leads to convergent results.\",\n \"In terms of these states and energies the overlap $S(t)$ becomes [10] $S(t)=\\\\left(\\\\sum _{\\\\beta } |\\\\langle \\\\beta | s \\\\rangle |^2 e^{i(\\\\epsilon _s-\\\\omega _\\\\beta )t}\\\\right)^{N},$ where $\\\\mathinner {|{s}\\\\rangle }$ denotes the lowest single particle eigenstate [55] of $\\\\hat{h}_0$ , and $N$ is the number of atoms in the spherical box of radius $R$ chosen to reproduce a given local experimental density of a Bose gas.\",\n \"We choose $R=10^6 a_0$ with $a_0$ the Bohr radius so that we find negligible finite size corrections.\",\n \"We emphasize again that the temperature enters the calculation only in determining the local density of the atomic cloud.\",\n \"For the dynamics and thus the prediction of the absorption spectra at a given local density the temperature $T$ is irrelevant.\",\n \"This is due to the fact that the energy scales of the dynamics of the system is determined by the binding energy of Rydberg molecular states, which exceed $k_BT$ .\",\n \"We obtain absorption spectra by predicting the many-body overlap, or Loschmidt echo, $S(t)$ .\",\n \"This calculation corresponds to solving the full time evolution of the system following a quantum quench, where at time $t=0$ the Rydberg impurity is suddenly introduced into the BEC.\",\n \"The overlap $S(t)$ describes the dephasing dynamics of the many-body system and thus the evolution of the dressing of the Rydberg impurity by bath excitations.\",\n \"It is one of the virtues of cold atomic systems that the signal $S(t)$ can be directly measured experimentally by Ramsey spectroscopy where via a $\\\\pi /2$ rotation at time $t=0$ , the impurity is prepared in a superposition state of $\\\\mathinner {|{5s}\\\\rangle }$ and $\\\\mathinner {|{ns}\\\\rangle }$ .\",\n \"Following a time evolution of duration $t$ , a further $\\\\pi $ rotation is performed and $\\\\sigma _z$ is measured.\",\n \"This gives the Ramsey contrast $|S(t)|$ .\",\n \"Changing the phase of the final $\\\\pi $ rotation, the full signal $S(t)= |S(t)| e^{i \\\\varphi (t)}$ can be measured [56], [57], [45].\",\n \"In Fig.\",\n \"REF we show the predicted Ramsey signal that underlies the calculation of the Rydberg polaron absorption spectrum for $n=49$ at peak density, here shown in absence of a finite Rydberg lifetime.\",\n \"We observe a fast decay in the contrast that indicates strong dephasing and thus efficient creation of polaron dressing by particle-hole excitations.\",\n \"This decay is accompanied by fast oscillations of the complex signal $S(t)$ as visible in the shown evolution of the Ramsey phase $\\\\varphi (t)$ restricted to the branch $(-\\\\pi ,\\\\pi )$ .\",\n \"The combination of the oscillations at the Rydberg molecular binding energies and the decay of the Ramsey signal $S(t)$ gives rise to the distinct Rydberg polaron features of molecular peaks that are distributed according to a gaussian envelope, to be discussed below.\",\n \"The Ramsey signal thus provides an alternative pathway to observing Rydberg polaron formation dynamics in real-time.\",\n \"The time scales of this coherent dressing dynamics are ultrafast compared to the typical time scales of collective low-energy excitations of ultracold quantum gases.\",\n \"This again highlights that effects arising from Bose-Bose interaction as well as finite temperature will play only a minor role in the prediction of the absorption response as those start to influence dynamics only on much longer time scales.\",\n \"Figure: Density profile of bosonic atoms in a harmonic trap for parameters T=180T=180 nK, N BEC =3.7·10 5 N_\\\\text{BEC}=3.7 \\\\cdot 10^5, N tot =5.2·10 5 N_\\\\text{tot}=5.2\\\\cdot 10^5, ω r =104\\\\omega _r = 104 Hz, and ω a =111\\\\omega _a=111 Hz, used for the prediction of the spectrum of the n=60n=60 Rydberg excitation shown in Fig. .\",\n \"In blue is shown the contribution from the condensed atoms, while the red area represents the contribution from thermal atoms.Figure: (a) LDA absorption spectrum for n=49n=49 as calculated by FDA (solid blue) in comparison with experimental data (symbols).Using the data of the non-equilibrium quench dynamics, the FDA can accurately capture the formation of Rydberg molecular dimers, trimers, tetramers, etc.\",\n \"This is demonstrated by comparing the FDA spectrum and experimental measurement for low principal quantum number, as shown in [9].\",\n \"Indeed, from a many-body wave-function perspective capturing the trimer, tetrameter and higher-order oligamer state requires the inclusion of the corresponding higher order terms in Eq.\",\n \"(REF ).\",\n \"This attests to the challenge of describing Rydberg polarons which are formed by the dressing with many deeply bound atoms, rather than just a small number of bath excitations.\",\n \"In fact due to the magnitude of the binding energies involved, coherent formation of Rydberg polarons takes place on a $\\\\mu $ s timescale, which is short compared to typical ultracold-atom time scales.\"\n ],\n [\n \"FDA absorption spectra\",\n \"The accurate description of molecular formation underlies the precise analysis of Rydberg-polaron absorption response.\",\n \"As the principal quantum number is increased to $n=49$ , more atoms are situated on average within the Rydberg orbit and many-body effects become relevant.\",\n \"The density-averaged prediction using FDA is shown in Fig.\",\n \"REF .\",\n \"For all density-averaged spectra, we rely on density profiles such as shown in Fig.\",\n \"REF for parameters used for the FDA prediction of the $n=60$ Rydberg-absorption spectrum.\",\n \"We include Hartree-Fock corrections [48], [58] as discussed in Appendix .\",\n \"Furthermore, the absorption spectrum is calculated from the Fourier transformation according to Eq.\",\n \"(REF ), where in the time-evolution of $S(t)$ we choose a temporal cutoff determined by the experimental laser-pulse duration, and in the Fourier transform we account for the finite laser line width (400 kHz).\",\n \"Figure: LDA absorption spectrum for n=49n=49 as calculated from the classical Monte Carlo sampling model (black) compared to the FDA prediction.\",\n \"The shaded area, which corresponds to the prediction of absorption response from the cloud center at peak density, reveals that the Gaussian line shape of the response is due to discrete, Gaussian-distributed molecular peaks and not a continuous response as described by a simple classical model.In Fig.\",\n \"REF , we show again the $n=49$ absorption spectrum obtained theoretically, but with a simulated laser linewidth of 100 kHz (solid blue line).\",\n \"The shaded region shows the result of an FDA calculation of the spectrum for a central region of the atomic cloud with roughly constant density.\",\n \"We find that the signal from this region shows a series of molecular peaks whose weights follow a Gaussian envelope.\",\n \"This distribution of molecular peaks is one of the key signatures of Rydberg polarons as predicted theoretically in the accompanying work [10].\",\n \"The comparison of experiment and theory in Fig.\",\n \"6 in turn shows that the response is indeed composed of many individual molecular lines.\",\n \"While due to the finite lifetimes of Rydberg molecules and the fact that their binding energies decrease for increasing principal number, these individual molecular peaks cannot be resolved experimentally for high principal numbers, their Gaussian envelope is a robust signature of the formation of Rydberg polarons.\",\n \"As our simulation shows, the broad tail to the red and the characteristic gaussian profile of the signal are both clear signatures of Rydberg polarons.\",\n \"The excellent agreement between many-body theory and experiment confirms the presence of Rydberg polarons in the Sr experiment.\",\n \"Figure: Experimentally observed absorption spectrum (symbols) for n=60n=60 in comparison with the theoretical prediction from FDA (lines).\",\n \"The experimental input parameters such as the trap frequencies are varied within the experimental uncertainty, giving rise to the gray band around the solid red line.\",\n \"The specific set of values is given in Table .\",\n \"The inset shows the signal on a logarithmic scale.Table: Parameters used for the prediction of the theoretical spectrum shown in Fig.\",\n \"(b) for the Rydberg n=60n=60 excitation.\",\n \"Inferred quantities are the condensate fraction η=N BEC /N tot \\\\eta =N_\\\\text{BEC}/N_\\\\text{tot}, peak BEC density ρ peak,BEC \\\\rho _\\\\text{peak,BEC}, and chemical potential μ\\\\mu .\",\n \"Densities are given in cm -3 \\\\text{cm}^{-3}.The Gaussian signature of the Rydberg-polaron spectrum becomes more pronounced when exciting $n=60$ states (Fig.\",\n \"REF ).\",\n \"In the accompanying work [10] we observe this Gaussian response experimentally for Rydberg polarons of large principal number $n=60$ and $n=72$ .\",\n \"While due to the finite lifetimes of Rydberg molecules and the fact that their binding energies decrease for increasing principal number, these individual molecular peaks cannot be resolved experimentally for high principal numbers, their Gaussian envelope is a robust signature of the formation of Rydberg polarons.\",\n \"The comparison of experiment and theory in Fig.\",\n \"6 in turn shows that the response is indeed composed of many individual molecular lines.\",\n \"The FDA calculation of the absorption spectra takes as input the experimentally determined number of atoms in the condensate $N_\\\\text{BEC}$ and the trap frequencies $\\\\omega _i$ .\",\n \"The temperature $T$ and the total number of atoms $N_\\\\text{tot}$ are taken as fit parameters, and results are consistent with their determination using the mean-field model and absorption imaging described in Section .\",\n \"For the $n=60$ spectrum, we also show a range of FDA predictions resulting from varying the trap frequencies within experimental uncertainty, which illustrates the impact of these uncertainties.\",\n \"Table REF lists the parameters used in the theoretical simulation.\",\n \"Note that the requirement to explain the spectrum over the whole frequency regime places tight constraints on the fit parameters $T$ , and $N_\\\\text{tot}$ .\",\n \"(For Rydberg polaron response at $n=72$ , refer to Ref.\",\n \"[9]).\",\n \"As discussed in Ref.\",\n \"[10], the emergence of the Gaussian response can be understood from a direct, analytical calculation of $S(t)$ in terms of single particle eigenstates and energies.\",\n \"Indeed expanding the sum in Eq.\",\n \"(REF ) explicitly in a multinominal form leads, after Fourier transform, to the expression $\\\\mathcal {A}(\\\\nu )&=&N!\",\n \"\\\\sum _{\\\\Sigma n_i = N}\\\\frac{|\\\\langle \\\\alpha _1 | s \\\\rangle |^{2 n_1}\\\\cdot \\\\ldots \\\\cdot |\\\\langle \\\\alpha _M | s \\\\rangle |^{2 n_M}\\\\cdot \\\\ldots }{n_1!\\\\ldots n_M!\",\n \"\\\\ldots } \\\\nonumber \\\\\\\\&&\\\\times \\\\delta (-\\\\nu +n_1 \\\\omega _1+\\\\ldots +n_M \\\\omega _M+\\\\ldots ).$ This expression captures not only the bound states but also continuum states.\",\n \"Expressing the spectrum in this explicit form again highlights the fact that the actual spectrum is indeed built by $\\\\delta $ -peaks corresponding to the many configurations in which bound molecules can be formed and thus dress the impurity.\",\n \"It also explains the emergence of the gaussian lineshape as a limit of the multinominal distribution of delta-function peaks for large particle number $N$ .\",\n \"We note that the fact that the spectrum is built by molecular excitation peaks (of up to hundreds of atoms bound to a single impurity) is missed by the classical description of the Rydberg polarons discussed in Section .\",\n \"We note that while Eq.\",\n \"(REF ) is appealing as it qualitatively explains the observed spectral features, it is less useful for quantitative calculations due to the exponential growth of the number of terms in the sum (reflecting the exponential growth of Hilbert space for increasing particle number).\",\n \"In contrast, the calculation of the time-evolution in Eq.\",\n \"(REF ) is numerically efficient (when limited to finite evolution times) and only requires a final Fourier transformation.\"\n ],\n [\n \"From quantum to classical description of Rydberg absorption response\",\n \"For a sufficiently large average number of atoms within the Rydberg electron orbit, the overall line shape of Rydberg spectra can be described with classical statistical arguments.\",\n \"In Fig.\",\n \"REF , we show the experimental spectrum obtained for $n=60$ in comparison with the prediction from the FDA (dashed red line) and a classical statistical approach (solid black).\",\n \"In the latter approach, using a classical Monte Carlo (CMC) algorithm [11], we randomly distribute atoms in three-dimensional space around the Rydberg ion such that the correct density profile is obtained.\",\n \"Due to statistical fluctuations in the random sampling of coordinates (sampled from a uniform distribution), the local density within the Rydberg-electron radius fluctuates.\",\n \"For each of the random configurations of atoms $\\\\mathcal {C}=(\\\\mathbf {r}_1, \\\\mathbf {r}_2,\\\\ldots )$ we calculate the classical energy of the configuration $E_{\\\\mathcal {C}}=\\\\sum _i V(\\\\mathbf {r}_i)$ , where $\\\\mathbf {r}_i$ are the atom coordinates.\",\n \"The resulting energies $E_{\\\\mathcal {C}}$ are collected in an energy histogram and shown as the black solid line in Fig.\",\n \"REF .\",\n \"The validity of the classical description reflects the fact that the macroscopic occupation of the molecular bound and scattering states probes the Rydberg molecular potential uniformly over its entire range.\",\n \"This process is also well described by a repeated sampling of different classical atom configurations drawn from a homogeneous density distribution.\",\n \"Figure: LDA absorption spectrum for n=60n=60 as calculated by the FDA (dashed red) and classical statistical Monte Carlo sampling method (black).\",\n \"The inset shows the signal on a logarithmic scale.The quantum-classical correspondence also becomes evident when considering the specific approximation that underlies the classical statistical approach.\",\n \"The Rydberg absorption response is given as a Fourier transform of the full quantum quench dynamics as given by Eq.\",\n \"(REF ).\",\n \"The classical statistical model arises as an approximation of the time evolution of $S(t)$ where the kinetic energy of both the impurity and the Bose gas is completely neglected.\",\n \"Hence both bosons and the impurity are treated as effectively infinitely heavy objects; their motion is `frozen' in space.\",\n \"The disregard of the kinetic terms has the consequence that the Hamiltonian now commutes with the position operators $\\\\hat{\\\\mathbf {r}}_j$ of impurity and bath atoms (non-commuting $\\\\hat{\\\\mathbf {p}}_j$ operators are now absent), and hence dynamics becomes completely classical.\",\n \"In this approximation the Hamiltonian becomes (without loss of generality, we assume that the impurity is at the center of the coordinate system) $\\\\hat{H}= \\\\int d^3r V(\\\\mathbf {r})\\\\hat{a}^\\\\dagger (\\\\mathbf {r})\\\\hat{a}(\\\\mathbf {r})$ Furthermore, $\\\\hat{H}_0=0$ , and hence $\\\\hat{\\\\rho }_\\\\text{ini} = 1/Z$ in Eq.\",\n \"(REF ).\",\n \"Since all boson coordinates $\\\\hat{\\\\mathbf {r}}_j$ now commute with the Hamiltonian, the many-body eigenstates are given by $\\\\mathinner {|{\\\\phi ^{(j)}}\\\\rangle }=\\\\mathinner {|{\\\\mathbf {r}^{(j)}_1,\\\\mathbf {r}^{(j)}_2,\\\\ldots , \\\\mathbf {r}^{(j)}_N}\\\\rangle }$ .\",\n \"The trace in Eq.\",\n \"(REF ) reduces to the sum over the set of all basis states $\\\\lbrace \\\\mathinner {|{\\\\phi ^{(j)}}\\\\rangle }\\\\rbrace $ .\",\n \"The $\\\\mathinner {|{\\\\phi ^{(j)}}\\\\rangle }$ are eigenstates of $\\\\hat{H}$ with eigenvalues $\\\\hat{H} \\\\mathinner {|{\\\\phi ^{(j)}}\\\\rangle }= \\\\sum _{\\\\lbrace \\\\mathbf {r}_i^{(j)}\\\\rbrace } V(r_i^{(j)}) \\\\mathinner {|{\\\\phi ^{(j)}}\\\\rangle }$ and $S(t)$ becomes $S_\\\\text{cl}(t)=\\\\mathcal {N} \\\\sum _{\\\\lbrace \\\\mathbf {r}_i^{(j)}\\\\rbrace }e^{i\\\\left(\\\\nu - \\\\sum _{i} V(r_i^{(j)})\\\\right)t},$ where the sum extends over all possible atomic configurations in real-space, and $\\\\mathcal {N}$ is a normalization factor (representing the partition function $Z$ in the density operator $\\\\rho _\\\\text{ini}$ ).\",\n \"Finally, performing the Fourier transform of $S(t)$ one arrives at $\\\\mathcal {A}(\\\\nu )=\\\\mathcal {N} \\\\sum _{\\\\lbrace \\\\mathbf {r}_i^{(j)}\\\\rbrace }\\\\delta \\\\left(\\\\nu - \\\\sum _{i} V(r_i^{(j)})\\\\right).$ A finite lifetime of the Rydberg or laser excitation, as well as a finite linewidth of the laser leads to a broadening of the delta function in Eq.\",\n \"(REF ).\",\n \"The sum in Eq.\",\n \"(REF ) is exactly the object sampled in the classical Monte Carlo (CMC) approach derived from first principles.\",\n \"Moreover we note that for impurities interacting with an ideal Bose gas with contact interactions, the gaussian spectral signature of polarons [10] would remain, while in the classical model, a sharp excitation at the atomic transition frequency is expected.\",\n \"Indications of a Gaussian response have been seen in a recent experiment performed independently at Aarhus [8] and JILA [7], in agreement with theory [36].\",\n \"The classical statistical approach fails to describe the Bose polarons close to a Feshbach resonance [7], [8].\",\n \"Here the impurity indeed interacts with the bath via a contact interaction, and, due to a diverging scattering length, the single bound state present in the problem is highly delocalized and extends in size far beyond the range of the potential.\",\n \"This effect cannot be captured by a classical approach.\",\n \"While Rydberg absorption spectra find an effective, yet approximate description in terms of classical statistics, such an approach does not reveal the quantum mechanical origin of Rydberg polarons.\",\n \"The classical sampling model treats both the Rydberg and the ground state atoms as infinitely heavy objects.\",\n \"Zero-point motion and the discrete nature of the bound energy levels are absent in this treatment, so that it cannot describe the formation of Rydberg molecular states.\",\n \"This short-coming of the classical approach is evident in Fig.\",\n \"REF , where we show the absorption spectrum for $n=49$ .\",\n \"The FDA (solid blue) fully describes the quantum mechanical formation of molecules (dimers, trimers, tetrameters, excited molecular states, etc.\",\n \"), which gives rise to discrete molecular peaks visible in the spectrum.\",\n \"In contrast, and as evident from Fig.\",\n \"REF , the existence of molecular states, which are the quantum mechanical building block of Rydberg polarons, is not described by the classical approach (solid black).\",\n \"The fact that the classical model can only describe the envelope of the Rydberg polaron response but not the underlying distribution of molecular peaks, is further emphasized by the comparison of FDA and CMC simulation from the center of the atomic cloud shown as shaded region in Fig.\",\n \"7.\"\n ],\n [\n \"Conclusion\",\n \"In this work, we have detailed the descriptions of polarons as encountered in condensed matter and in ultracold atomic systems.\",\n \"We time-evolve an extended Fröhlich Hamiltonian as relevant for Rydberg excitations, Eq.\",\n \"(REF ), unitarily to obtain the overlap function, Eq.\",\n \"(REF ), whose Fourier transform leads to the spectral function for Rydberg impurity excitation in a Bose gas, Eq.\",\n \"(REF ).\",\n \"We show how different approximations to the many-body quantum description, such as mean field and classical Monte Carlo treatments can be derived.\",\n \"Here we extend the bosonic functional determinant approach to the Rydberg polaron problem to account for recoil of the impurity.\",\n \"We show that the FDA approach can correctly and accurately account for the few-body molecular bound-state formation, as well as the macroscopic occupation of many-body bound and continuum states.\",\n \"The various treatments are compared with experimental data for Rydberg excitation in a $^{84}$ Sr BEC [9] with the FDA results reproducing the observed data over a wide range of Rydberg line spectral intensities.\",\n \"Rydberg polarons are exemplified on the one hand by their large energy scales, 1-10 MHz, that allow for coherent polaronic dressing on correspondingly short time scales, and on the other hand large spatial extent, $~1\\\\,\\\\mu $ m, leading to large-scale variations of the density in the many-body medium.\",\n \"Another key feature of Rydberg polarons is the coherent dressing of the quantum impurity not by collective low-energy excitations, but by a large number of molecular bound states.\",\n \"This is a new dressing mechanism, distinguishing them from polarons encountered so far in the solid state physics context.\",\n \"Our FDA time domain analysis suggests a way to investigate the dynamical formation of the Rydberg polarons.\",\n \"A natural extension will be to explore the feasibility of observing Pauli blocking in a degenerate Fermi atomic gas with non-trivial spatial correlations.\",\n \"Note that one can study polarons with nonzero momentum by keeping all terms in Eq.\",\n \"(REF ) and allowing $\\\\mathbf {p}$ to be finite.\",\n \"The effective polaron mass can then be analyzed as previously demonstrated [59].\",\n \"Acknowledgements: Research supported by the AFOSR (FA9550-14-1-0007), the NSF (1301773, 1600059, and 1205946), the Robert A, Welch Foundation (C-0734 and C-1844), the FWF(Austria) (P23359-N16, and FWF-SFB049 NextLite).\",\n \"The Vienna scientific cluster was used for the calculations.\",\n \"H. R. S. was supported by a grant to ITAMP from the NSF.\",\n \"R. S. and H. R. S. were supported by the NSF through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and the Smithsonian Astrophysical Observatory.\",\n \"R. S. acknowledges support from the ETH Pauli Center for Theoretical Studies.\",\n \"E. D. acknowledges support from Harvard-MIT CUA, NSF Grant (DMR-1308435), AFOSR Quantum Simulation MURI, the ARO-MURI on Atomtronics, and support from Dr. Max Rössler, the Walter Haefner Foundation and the ETH Foundation.\",\n \"T. C. K acknowledges support from Yale University during writing of this manuscript.\"\n ],\n [\n \"Determination of BEC parameters and Mean-Field description of the Impurity Excitation Spectrum in a Strontium BEC\",\n \"The mean-field approximation neglects fluctuations in the density around the Rydberg impurity, which correspond to a spread in binding energies of the polaron states excited for a given average density.\",\n \"This explains the discrepancy between data and fit at large detuning.\",\n \"But as shown in [9], the broadening resulting from these fluctuations is proportional to $\\\\sqrt{\\\\rho }$ and vanishes as detuning approaches zero.\",\n \"Thus, we assume that the difference between data and the mean-field fit in Fig.\",\n \"REF for small detuning ($\\\\nu /\\\\Delta _{\\\\textup {max}}<0.5$ ) arises from non-condensed atoms.\",\n \"We adjust the area of the mean-field BEC contribution so the sum of the non-condensed and mean-field-BEC signal matches the total experimental spectral area.\",\n \"Because the deviations between the mean-field fit and the BEC spectrum are significant, the fit of the peak shift $\\\\Delta _{\\\\textup {max}}$ is less rigorous.\",\n \"We adjust it to qualitatively match the data and so that the mean-field fit and the data have approximately equal area for ($\\\\nu /\\\\Delta _{\\\\textup {max}}>0.5$ ).\",\n \"This uncertainty in the fitting procedure decreases with increasing principal quantum number as the experimental data converges towards the mean-field form.\",\n \"In the mean-field description we neglect the laser linewidth of 400 kHz in this analysis.\",\n \"The fit parameters are given in Table REF .\",\n \"The condensate fraction $\\\\eta $ is directly determined from the spectrum.\",\n \"The uncertainties correspond to values calculated for the extremes of the confidence intervals shown as bands in Fig.\",\n \"REF , except for uncertainties in $\\\\omega _i$ and total atom number, which reflect uncertainties from the independent procedures for measuring these quantities.\",\n \"The fit value of the peak shift, $\\\\Delta _{\\\\textup {max}}$ , can be compared to the predicted peak shift, $\\\\tilde{\\\\Delta }_\\\\textup {max}$ , calculated from independent, a priori information: the number of atoms in the condensate $N_{BEC}$ determined from time-of-flight-absorption images, the trap oscillation frequencies $\\\\omega _i$ determined from measurements of collective mode frequencies for trapped atoms, and the value of $a_{s,\\\\textup {eff}}$ found theoretically from $V_{Ryd}(\\\\mathbf {r})$ .\",\n \"The values of $\\\\Delta _{\\\\textup {max}}$ are all about 10% below $\\\\tilde{\\\\Delta }_\\\\textup {max}$ .\",\n \"This may point to something systematic in our analysis procedure, but it is also a reasonable agreement given our uncertainty in determination of $\\\\omega _i$ and total atom number.\",\n \"The calculation of $a_{s,\\\\textup {eff}}$ from $V_{Ryd}(\\\\mathbf {r})$ (Eq.\",\n \"(REF )) could also account for some of this discrepancy through approximations made to describe the Rydberg-atom potential at short range.\",\n \"Residual deviations in the description of the short-range details of the Rydberg molecular potential lead, for instance, also to the minor discrepancies between the FDA and CMC simulation of the Rydberg response from the center of the atomic cloud, shown in Fig.\",\n \"REF .\",\n \"For a non-interacting gas in a harmonic trap, the temperature is easily found from $\\\\eta $ , and $\\\\bar{\\\\omega }$ [60], and the standard expressions imply a temperature of approximately 240 nK for the data presented in Fig.\",\n \"REF .\",\n \"However, the situation is much more complicated than this for an interacting gas.\",\n \"There are corrections to the condensate fraction that are independent of the trapping potential, and these are often discussed in terms of the shift of the critical temperature for condensation [48], [61], [62], but these are all small in our case, reflecting the smallness of relevant expansion parameters: $\\\\rho _{\\\\textup {max}}a_{bb}^3=10^{-4}, a_{bb}/\\\\lambda _{\\\\textup {th}}=10^{-2}$ at 100 nK, and $a_{\\\\textup {ho}}/R_{TF}=10^{-1}$ , where $\\\\lambda _{\\\\textup {th}}$ is the thermal de Broglie wavelength and $a_{\\\\textup {ho}}$ is the trap harmonic oscillator length.\",\n \"For a gas trapped in an inhomogeneous external potential $V_{ext}(\\\\textbf {r})$ , the Hartree-Fock approximation (as described in detail in Refs.\",\n \"[63], [60]) yields a mean-field interaction between thermal and BEC atoms that creates an effective potential for thermal atoms, $V_{\\\\textup {eff}}(\\\\textbf {r})= V_{\\\\text{ext}}(\\\\textbf {r})+2g[\\\\rho _{BEC}(\\\\textbf {r})+\\\\rho _{\\\\textup {th}}(\\\\textbf {r})]$ that is of a “Mexican hat\\\" shape rather than parabolic [60], [63], [64].\",\n \"This effect is particularly important when the sample temperature is close to or lower than the chemical potential, which is the case here.\",\n \"It increases the volume available to non-condensed atoms, and implies a lower sample temperature for a given value of $\\\\eta $ compared to the non-interacting case.\",\n \"In other words, the number of thermal atoms can significantly exceed the critical number for condensation predicted for an ideal gas [64], [58].\",\n \"240 nK is thus an upper limit of the sample temperature.\",\n \"Sample temperatures extracted from bimodal fits to the time-of-flight absorption images are 100-190 nK.\",\n \"Both the mean-field and the FDA fits use a local density approximation for the spectrum.\",\n \"Here the temperature and interaction between bosons enter by determining the precise form of the overall density distribution of the atomic gas.\",\n \"As discussed above, the density profile has contributions from both the condensed and thermal, non-condensed atoms $\\\\rho (\\\\mathbf {r})= \\\\rho _\\\\text{BEC}(\\\\mathbf {r}) +\\\\rho _\\\\text{th}(\\\\mathbf {r}).$ The BEC density $\\\\rho _\\\\text{BEC}$ and resulting chemical potential are assumed to be given by the Thomas-Fermi expressions [63] $\\\\rho _{BEC}(\\\\mathbf {r})= \\\\frac{m}{4\\\\pi \\\\hbar ^2 a_{bb}}[\\\\mu _{TF}-V_{ext}(\\\\mathbf {r})],$ with $\\\\mu _\\\\text{TF}=\\\\hbar \\\\overline{\\\\omega }/2(15 N_\\\\text{BEC} a_{bb}/a_{ho})^{2/5}$ .\",\n \"$N_\\\\text{BEC}$ is the total number of condensed atoms, $a_{bb}=123 a_0$ for $^{84}$ Sr, and $a_{ho}=(\\\\hbar /m\\\\overline{\\\\omega })^{1/2}$ is the harmonic oscillator length.\",\n \"(In this section we make explicit factors of $\\\\hbar $ and $k_B$ .)\",\n \"The density distribution for atoms in the thermal gas can be calculated from known trap and BEC parameters using a Bose-distribution $\\\\rho _\\\\text{th}(\\\\mathbf {r})= \\\\frac{1}{\\\\lambda _T^3}g_{3/2}\\\\left\\\\lbrace e^{-\\\\frac{1}{k_BT}\\\\left[V_\\\\textup {eff}(\\\\mathbf {r})-\\\\mu _{TF}\\\\right]}\\\\right\\\\rbrace ,$ where $\\\\lambda _T=\\\\sqrt{\\\\frac{2\\\\pi \\\\hbar ^2}{m k_BT}}$ is the thermal wavelength and $g_{3/2}(z)$ the polylogarithm $\\\\text{PolyLog}[\\\\frac{3}{2};z]$ [63], and the chemical potential is set to $\\\\mu _{TF}$ .\",\n \"The parameters $T$ and $\\\\mu _{TF}$ are varied self-consistently to fit that spectrum, which can be seen as matching the experimentally observed total particle number $N_\\\\text{tot}=N_\\\\text{BEC}+N_\\\\text{th}$ and BEC fraction $\\\\eta =N_\\\\text{BEC}/N_\\\\text{tot}$ .\",\n \"The mean-field calculations use the full geometry of the external trapping potential as determined by the optical-dipole-trap lasers.\",\n \"Integrals involving $V_\\\\text{ext}(\\\\mathbf {r})$ are evaluated numerically.\",\n \"This procedure yields temperatures of $155-170\\\\,nK$ , in good agreement with other determinations.\",\n \"The modification of the potential seen by thermal atoms due to Hartree-Fock corrections has an important consequence for our analysis.\",\n \"The peak shift in the Rydberg excitation spectrum should in principle reflect the peak condensate density plus the density of thermal atoms at the center of the trapping potential.\",\n \"The contribution from thermal atoms would be a significant correction if the mean-field repulsion of thermal atoms from the center of the trap were ignored.\",\n \"But because the sample temperature is close to the chemical potential, the density of thermal atoms is suppressed at trap center and we neglect it in discussions of the peak mean-field shift in the spectrum.\",\n \"For the FDA simulation, the dipole trap potential is approximated by $V_\\\\text{ext}(\\\\mathbf {r})=m(\\\\omega _r^2 r^2+\\\\omega _z^2 z^2)/2$ .\",\n \"In Fig.\",\n \"REF (a) we show the density profile along the radial direction for parameters as used for the FDA prediction of the $n=60$ Rydberg absorption spectrum including Hartree-Fock corrections.\",\n \"We emphasize again that the temperature and interaction between bosons are relevant only for the determination of the density profile of atoms.\",\n \"In the simulation of the absorption spectrum, which is obtained within a local density approximation, this density distribution of atoms enters as input.\",\n \"In this simulation, then locally performed for constant density, both temperature effects and boson-interaction effects can be neglected due to the short time scales associated with Rydberg polaron dressing dynamics (that are given by the Rydberg molecular binding energies).\"\n ]\n]"},"id":{"kind":"string","value":"1709.01838"}}},{"rowIdx":798,"cells":{"text":{"kind":"list like","value":[["On the geometry pervading One Particle States"],["Abstract In this paper, a way is given to obtain explicitly the representations of the Poincar\\'e group as can be prescribed by Geometric Quantization.","Thus one obtains some forms of the Space of Quantum States of the different relativistic free particles, and I give explicitly these spaces and the corresponding operators for the usually accepted as realistic physical particles.","The general description of the massless particles I obtain, is given in terms of solutions of Penrose equations.","In the case of Photon, I also give other descriptions, one in terms of the Electromagnetic Field.","Since the results are derived from Geometric Quantization, they are related to certain Contact and Symplectic manifols, that I study in detail.","The symplectic manifold must be interpreted, according with Souriau, as the Movement Space of the corresponding classical particle, and that leads to propose one of the spaces I use as the State Space of the corresponding classical particle.","These spaces are also described in each case."],["Introduction.","The Wave Equations of relativistic elementary particles, Klein-Gordon, Dirac, Weil etc.., have been originally derived each one in a independent way.","An unification was the discovery of the relation of these equations with the representations of Poincaré group (inhomogeneous Lorenz group).","The classification of the representations of Poincaré group made by Wigner [17] with important contributions of Majorana, Dirac and Proca [12], [9], [14] leads to a group theoretical study of Wave Equations by Bargmann and Wigner [3].","On the other hand, in Kirillov-Kostant-Souriau theory (Geometric Quantization), a description of Quantum Systems is given in terms of elements of the dual of the Lie algebra of the Lie group under consideration.","Of course, this way of seing Quantum Mechanics is not completely independent of the preceding one, since it has its origin in a method to obtain representations [10], [1].","The correspondence of Quantum States in the sense of Geometric Quantization with Wave Functions in the Quantum Mechanical sense is not clear in all cases.","In Souriau's book [15], a very general way to do the passage is given, but it doesn't work in all cases.","In this paper we give a geometrical construction that stablishes a one to one correspondence from Quantum States in the sense of Geometric Quantization to Wave Functions in the Quantum Mechanical sense, that is valid for all kinds of relativistic elementary particles.","The idea is as follows.","In Geometric Quantization, one begins with a regular contact manifold or its associated hermitian line bundle.","Quantum states ( in the sense of Geometric Quantification ) can be considered as being the collection of those sections of the hermitian line bundle which satisfies \" Planck's condition\" (cf.","Souriau's book).","In this paper we see that these sections are in a one to one correspondence with the (unrestricted) sections of another hermitian line bundle.","Thus, this fibre bundle is a good setting to describe the quantum processes under consideration.","The main idea to pass from this description to the usual one in terms of Wave Functions, can be intuitively explained as follows.","Quantum States in G.Q.","attribute an “amplitude of probability” to each movement of the particle.","To obtain the corresponding wave function one must proceed as follows: for each event, the corresponding amplitude of probability is obtained by taking all movements passing through the given event, and then “adding up” ( in a suitable sense) the corresponding amplitudes of probability.","Of course the concept of “movement passing through an event” is only obvious in the case of the ordinary massive spinless particle, and it is defined in section .","The Contact Manifold under consideration, fibers on a Symplectic Manifold that, following Souriau, must be considered as being composed by the \"movements of the corresponding classical particle\".","Then, with the constructions made in this paper, it becomes clear what space must be considered as composed by the \"states of the corresponding classical particle\", for each kind of relativistic particle.","The spaces of Wave Functions and many relevant isomorphic vector spaces we obtain, are the spaces of the representations of Poincaré group that I describe in section .","The preceding constructions are made in a explicit way for the relativistic particles usually considered as having physical sense.","A section is devoted to massive particles and other to massless particles.","In section a explicit application of the preceeding constructions is done for massive particles.","One obtains solutions of Klein-Gordon and Dirac equations, and also a description of the wave functions for massive particles of higher spin.","Massless particles are studied in section .","For massless particles of spin 1/2 one obtains solutions of Weyl equations and for general spin this method leads in a natural way to the description of massless particles by means of solutions of Penrose wave equations [13].","This is related to the fact that the Contact and Symplectic Manifolds corresponding to Massless Particles are expresed in a natural way in Twistor Space and its Projective Space, as explained in section REF .","In the particular case of Photon, in section REF ,I give also other forms for the Wave Functions.","One of them is in terms of the Electromagnetic Field,and are similar to the Wave Functions proposed by Bialynicki-Birula.","In section I describe for each particle the concrete space that must be considered to be the \"Space of states of the corresponding classical particle\"."],["Notation ","All differentiable manifolds appearing in this paper are assumed to be $C^{\\infty }$ , finite dimensional, Hausdorff and second countable.","The set composed by the differentiable vector fields on a differential manifold, $M$ , is denoted by ${\\cal D}(M)$ .","The set of differential $k$ -forms on $M$ is denoted by $\\Omega ^{k}(M)$ .","Let $X\\in {\\cal D}(M),\\ \\omega \\in \\Omega ^{k}(M)$ .","We denote by $i(X)\\omega $ the interior product of X by $\\omega $ and by $L(X)\\omega $ the Lie derivative of $\\omega $ with respect to $X$ .","Let $G$ be a Lie group.","The Lie algebra of $G$ is the set consisting of the left invariant vector fields on $G$ , provided with its canonical structure of Lie algebra.","It is denoted in this paper by $\\underline{G}$ .","I denote by $\\underline{G}^{\\ast }$ the dual of $\\underline{G}$ .","$\\underline{G}^{\\ast }$ is canonically identified with the set composed by the left invariant 1-forms on $G$ .","By identification of each element of $\\underline{G}$ or $\\underline{G}^{\\ast }$ with its value at the neutral element, $e$ , the lie algebra of $G$ is identified to the tangent space at $e$ , and $\\underline{G}^{\\ast }$ to its dual.","There exist a map, the exponential map, $Exp: G \\rightarrow \\underline{G}$ , such that: if $X\\in \\underline{G}$ , the integral curve of $X$ with initial value $e$ is $t\\in {\\rightarrow } Exp \\,tX$ .","This integral curve is a one parameter subgroup of $G$ .","The Lie algebra of the circle Lie group, $S^1$ , is identified to $R$ in such a way that for all $ t\\in R,\\ Exp\\,t=e^{2 \\pi i t}$ .","If we have an homomorphism of $G$ in $S^1$ , its differential is a linear map from the Lie algebra of $G$ into the Lie algebra of $S^1$ , which is $R$ .","Thus the differential of an homomorphism of $G$ into $S^1,$ can be considered as an element of $\\underline{G}^{\\ast }$ .","In the same way, the Lie algebra of the Lie group $R$ is identified to $R$ in such a way that the exponential map becomes the identity.","Thus, also in this case, the differential of an homomorphism of $G$ into $R$ , can be considered as an element of $\\underline{G}^{\\ast }$ .","The coadjoint representation is the homomorphism $Ad^{\\ast }:g\\in G\\rightarrow Ad_{g}^{\\ast }\\in $ Aut $(\\underline{G}^{\\ast })$ , given by $(Ad_{g}^{\\ast }(\\alpha ))(X)=\\alpha (Ad_{g^{-1}}(X))$ for all $\\alpha \\in \\underline{G}^{\\ast },X\\in \\underline{G}$ , where $Ad_{g^{-1}}$ is the differential of the automorphism of $G$ that sends each $h\\in G$ to $g^{-1}hg.$ Let $M$ be a differentiable manifold and $G$ a Lie group acting on $M$ on the left (resp.","on the right).","Given an element, $X$ , of $\\underline{G}$ , we denote by $X_{M}$ the vector field on $M$ whose flow is given by the diffeomorphisms associated by the action to $\\lbrace $ Exp$(-tX):t\\in {\\bf R}\\rbrace $ (resp.","$\\lbrace $ Exp$tX:t\\in {\\bf R}\\rbrace $ ).","$X_{M}$ is called the infinitesimal generator of the action associated to $X$ .","A principal fibre bundle having $M$ as total space, $B$ as base and $G$ as structural group, will be denoted by $M(B,G)$ .","In all that concerns fibre bundles, we use the notation of [11]."],["A Particular Classical State Space. ","In this section I motivate the physical interpretation of the geometrical constructions that are to be made in this paper.","I do not enter in the details of the computations.","Most part of this section is an easy consecuence of Souriau's book [15], which has inspirated this paper.","Let us consider a classical relativistic free particle without spin in Minkowski space-time, with rest mass $ m\\ne 0$ .","Space-time is interpreted as being an abstract four dimensional manifold, M, each inertial observer, R, providing us with a global chart, $\\phi _{R}=(x_{R}^1,...,x_{R}^{4})$ .","Changes of these global charts are given by transformations corresponding to elements of the Poincaré (i.e.","inhomogeneous Lorenz) group, ${\\cal P}$ .","We consider a family of inertial frames such that changes are all given by ortochronous proper Poincaré transformations, i.e.","elements of the connected component of the identity, ${\\cal P}_{+}^{\\uparrow }$ .","A geometrical object in $\\bf R^4\\rm $ invariant under the action on $\\bf R^4 \\rm $, provides us with a well defined object in M , whose local expression is the same for all inertial frames.","An example is the Minkowski metric, $g$ .","When provided with this metric, M becomes Minkowski space.","The objects defined in this way would be well defined even if the charts were not global.","Charts $\\phi _R $ give rise in the canonical way to charts of TM, $\\dot{\\phi }_R=(x_R^i,\\dot{x}_R^i)$ , where $\\dot{x}_R^i(v)=v(x_R^i)$ , for all $ v \\in TM.$ The charts $\\overline{\\phi }_R=(x_R^i,P_R^i) $ , $P_R^i=m\\,\\dot{x}_R^i$ , are more natural in Physics.","Changes of these global charts are given by the transformations corresponding to elements of the Poincaré group , in its canonical action on $\\bf R^8 \\rm \\ $$i.e.$ the action given by $(L,C)*\\left(\\begin{array}{c}z^1\\\\ \\vdots \\\\z^8\\end{array}\\right)=\\left(\\begin{array}{l} L\\left(\\begin{array}{c}z^1\\\\\\vdots \\\\z^4\\end{array}\\right)+C\\\\ L\\left(\\begin{array}{c}z^5\\\\\\vdots \\\\z^8\\end{array}\\right)\\end{array}\\right)$ Geometrical objects on $\\bf R^8 \\rm $ invariant under this action, give us well defined geometrical objects on TM, whose local expressions are equal in the different inertial frames.","For example, if $(z^1,\\ldots ,z^8)$ is the canonical coordinate system of $\\bf R^8 \\rm $ , the 1-form: ${\\mbox{$\\omega $}}_0 \\equiv \\ z^8 dz^4-\\sum _{i=1}^3\\ z^{i+4}dz^i$ the vector field: $X_0 \\equiv \\ \\frac{1}{m}\\ \\sum _{i=1}^4\\ z^{i+4}\\frac{\\partial }{\\partial z^i}$ and the submanifold, ${\\cal E}_0$, given by: $z^8 \\equiv \\ \\sqrt{m^2+(z^5)^2+(z^6)^2+(z^7)^2}$ are invariant by the action on $\\bf R^8 \\rm $.","Thus the following 1-form, vectorfield, and submanifold are well defined on TM: $\\widetilde{\\mbox{$\\omega $}} = P_R^4\\,dx_R^4 -\\sum _{i=1}^3\\ P_R^i\\,dx_R^i$ $\\widetilde{X}={ \\frac{1}{m}}\\ {\\sum _{i=1}^4}P_R^i\\,\\frac{\\partial }{\\partial x_R^i}$ ${\\cal E} & = & \\left\\lbrace \\, v\\in TM: P_R^4(v)=\\left( m^2+\\sum _{i=1}^3(P_R^i(v))^2\\right)^{\\frac{1}{2}} \\right\\rbrace \\\\& = & \\left\\lbrace \\,v\\in TM: g(v,v)=1,\\dot{x}_R^4(v) > 0 \\right\\rbrace $ where R is an arbitrary inertial frame.","The restriction of $ \\widetilde{\\mbox{$\\omega $}} $ to $\\cal E,$ $\\omega ,$ is a contact form.","$\\widetilde{X}$ is tangent to ${\\cal E}$ .","The restriction of $\\widetilde{X}$ to $\\cal E,$ $X$ , is the unique vectorfield such that $ i_X {\\mbox{$\\omega $}} = m,\\ i_Xd{\\mbox{$\\omega $}} = 0.","$ Each inertial observer associates to each element, v, of $\\cal E$ eight numbers, $\\overline{\\phi }_R(v)=(x_R^1(v), ...,\\ x_R^4(v), P_R^1(v),...,\\ P_R^4(v))$ which are interpreted as giving position, time and momentum-energy ( we take c=1).","Thus $\\cal E$ must be interpreted as being state space.","The movements of the free particle under consideration are curves in $\\cal E,$ $\\gamma $ , having, for each inertial observer, a constant four velocity $i.e.$ $\\dot{\\phi }_R\\circ \\gamma (s)=(a^i+s(\\dot{x}_R^i)_0, (\\dot{x}_R^i)_0) $ where the $(\\dot{x}_R^i)_0$ are constant, $(\\dot{x}_R^4)_0 > 0, (\\dot{x}_R^4)_0^2-\\sum _{i=1}^3 (\\dot{x}_R^i)_0^2 = 1.$ Since the preceeding relation is equivalent to $\\overline{\\phi }_R\\circ \\gamma (s) = ( a^i+s\\frac{{(P_R^i)}_0}{m},{(P_R^i)}_0),$ where ${(P_R^i)}_0=m(\\dot{x}_R^i)_0,$ one sees that the movements are the integral curves of X.","The parameter $s$ coincides, since we take c=1, with proper time (of the trajectory in space time).","The action on $\\bf R^8 \\rm $ gives, by means of any of the inertial observers, $R,$ an action on M by means of $(L,C)\\odot v=(\\overline{\\phi }_R)^{-1}((L,C)*(\\overline{\\phi }_R(v))),$ for all $v\\in {\\cal E},\\ (L,C)\\in {\\cal P}_{+}^{\\uparrow }$ .","This action obviously preserves $\\cal E,$ and $ {\\mbox{$\\omega $}} $ in such a way that $({\\mbox{${\\cal E}$}},{\\mbox{$\\omega $}})$ is a homogeneous contact manifold for the given action, but different inertial observers lead to different actions.","Now, we shall describe ${\\mbox{${\\cal E}_0$}}$ in a different way, thus obtaining a picture of state space more suitable for its generalisation to other kind of particles.","Let $Y_8$ be the infinitesimal generator of the action on $\\bf R^8 \\rm $ associated to the element Y of the Lie algebra of ${\\cal P}_{+}^{\\uparrow },\\ \\underline{\\cal P}$ .","Since the action on $\\bf R^8 \\rm $ preserves ${\\mbox{$\\omega $}}_0$ , we have $L_{Y_8} {\\mbox{$\\omega $}}_0=0,$ so that $i_{Y_8}\\,d {\\mbox{$\\omega $}}_0=-d\\,(i_{Y_8}{\\mbox{$\\omega $}}_0)=-d({\\mbox{$\\omega $}}_0(Y_8)).$ Thus we define a map, $ {\\mu }_0$ , called the momentum map, from $\\bf R^8 \\rm $ into $\\underline{\\cal P}^*$ , by means of $ {\\mu _0}(z)\\cdot Y=-( {\\mbox{$\\omega $}}_0(Y_8))(z),$ for all $Y \\in \\underline{\\cal P}, z \\in \\bf R^8 \\rm .$ Let $\\lbrace Y_{\\alpha }^i, Y_\\beta ^i, Y_{\\gamma }^i, Y_\\delta :\\ i=1, 2, 3 \\rbrace $ be the basis of $\\underline{\\cal P}$ , composed by the usual generators of Lorenz rotations and space-time translations.","It can be proved that $\\mu _0=z^8\\,Y_\\delta ^* - \\sum _{i=1}^3\\ \\lbrace [(z^1, z^2,z^3)\\times (z^5, z^6, z^7)]^i\\, Y_\\alpha ^{i*}+$ $+ ( z^4z^{i+4}\\,-z^8z^i)\\,Y_\\beta ^{i*} +z^{i+4}\\, Y_{\\gamma }^{i*}\\,\\rbrace $ where $\\lbrace Y_\\alpha ^{1*},..., Y_\\delta ^*\\rbrace $ is the dual basis of $\\lbrace Y_\\alpha ^1,..., Y_\\delta \\rbrace $ .","The map $\\mu _0$ is equivariant for the given action on $\\bf R^8 \\rm $ and coadjoint action.","Thus, since $(0, ...,0,m)\\in {\\mbox{${\\cal E}_0$}}$ and ${\\mu _0}(0, ...,0,m)=m\\, Y_\\delta ^*,\\ {\\mu _0}({\\mbox{${\\cal E}_0$}})$ is the coadjoint orbit of $m\\,Y_\\delta ^*$ .","We also have ${\\mu _0^*}\\, {\\mbox{$\\Omega $}} =d {\\mbox{$\\omega $}} _o$ , where $ {\\mbox{$\\Omega $}} $ is the Kirillov symplectic form on the coadjoint orbit.","For each $\\alpha $ in the coadjoint orbit, $\\mu _0^{-1}\\,\\lbrace {\\mbox{$\\alpha $}}\\rbrace $ is the image of a integral curve of $X_0$ , that we know to be local expresion of movements.","Then $\\mu _0$ gives a one to one correspondence between points of the coadjoint orbit and movements of the particle under consideration.","Now we consider the map $f:\\ (a,b)\\in {\\mbox{${\\cal E}_0$}}\\longrightarrow (a, \\mu _0\\,(a,b))\\in {{\\bf R^4}\\rm }\\times \\underline{\\cal P}^*.$ By using the above expression for $\\mu _0$ , one sees that $f$ is an imbedding.","Also, $f$ is equivariant when one considers the given action in ${\\mbox{${\\cal E}_0$}}$ and the “product action” in ${{\\bf R^4}\\rm }\\times \\underline{\\cal P}^*$ given by $(L,C)*(a, {\\mbox{$\\alpha $} } )=(La+C,Ad_{(L,C)}^*\\,{\\mbox{$\\alpha $} } ),\\ \\ \\ \\ \\forall (L,C)\\in {\\cal P}_{+}^{\\uparrow }, (a, {\\mbox{$\\alpha $} } )\\in f({\\mbox{${\\cal E}_0$}}).$ Observe that, as a consequence, $f({\\mbox{${\\cal E}_0$}})$ is an orbit of that action.","Each inertial frame, R, enables us to “see” state space, $\\cal E$ , by means of $f\\circ \\overline{\\phi }_R$ , as being $f({\\mbox{${\\cal E}_0$}})\\subset { {\\bf R}^4}\\times \\underline{\\cal P}^*$ .","Changes of inertial frames are now given by transformations coming from the product action.","Since $f\\circ \\overline{\\phi }_R(e)=(x_R^1(e),...,x_R^4(e),\\ \\mu _0(\\overline{\\phi }_R(e)))$ , if we denote $\\mu _R \\equiv \\mu _0\\circ \\overline{\\phi }_R$ we have $f\\circ \\overline{\\phi }_R =(\\phi _R, \\mu _R),$ and $\\mu _R$ stablishes a one to one correspondence of trajectories of X ($i.e.$ movements of the particle parametrized by proper time) with points of the coadjoint orbit.","Thus, coadjoint orbit is to be identified to movement space for all inertial frames, althought the identification is different for each $R$ .","The picture of a state, $e\\in \\cal E$ , obtained by R is now $(\\phi _R(e),\\mu _R(e)) \\ i.e.$ an event, $\\phi _R(e)$ , and a movement, $\\mu _R(e)$ , “passing through” the event.","Each $Y\\in \\underline{\\cal P}$ defines a function on $\\underline{\\cal P}^*$ , denoted by the same symbol.","Thus $Y\\circ \\pi _2\\circ f\\circ \\overline{\\phi }_R$ is a function on $\\cal E$ ($i.e.$ a dynamical variable) where $\\pi _2$ is the canonical projection of $ { {\\bf R}^4}\\times \\underline{\\cal P}^*$ onto $\\underline{\\cal P}^*$ .","We have: $- {Y_{\\gamma }^i}\\circ \\mu _R &=&P_R^i \\\\{Y_\\delta }\\circ \\mu _R&=& P_R^4 \\\\- {Y_{\\alpha }}^i\\circ \\mu _R&=&({\\overrightarrow{x}_R} \\times \\overrightarrow{P}_R)^i \\\\{Y_\\beta }^i\\circ \\mu _R&=&(P_R^4\\overrightarrow{x}_R -x_R^4\\overrightarrow{P}_R)^i$ where $i=1,2,3,\\overrightarrow{x}_R= (x_R^1, x_R^2,x_R^3),$ and $\\overrightarrow{P}_R= (P_R^1,P_R^2,P_R^3)$ .","Thus the function on ${ {\\bf R}^4}\\times \\underline{\\cal P}^*$ given by $P=(P^1,P^2,P^3,P^4) \\equiv (- {Y_{\\gamma }^1}\\circ \\pi _2,- {Y_{\\gamma }^2}\\circ \\pi _2, - {Y_{\\gamma }^3}\\circ \\pi _2,{Y_\\delta }\\circ \\pi _2)$ ,can be considered as an abstraction of linear momentum: each inertial observer obtains $f\\circ \\overline{\\phi }_R(e)$ as the picture of $e\\in {\\mbox{$\\cal E$}}$ , and the linear momentum he measures is $P(f\\circ \\overline{\\phi }_R(e))$ .","In the same way the components of $\\overrightarrow{l}= ( - Y_\\alpha ^1\\circ \\pi _2, - {Y_\\alpha ^2}\\circ \\pi _2 ,- {Y_\\alpha ^3}\\circ \\pi _2)$ and $\\overrightarrow{g}= ( Y_\\beta ^1\\circ \\pi _2,Y_\\beta ^2\\circ \\pi _2 , Y_\\beta ^3\\circ \\pi _2)$ must be interpreted as being the components of (relativistic) angular momentum.","If we denote ${\\cal P}_{+}^{\\uparrow }$ by G, we can summarize what has been said as follows State space of our free particle is such that each inertial observer establishes a diffeomorphism from it to an orbit of G in ${ {\\bf R}^4}\\times \\underline{G}^*,$ for the canonical action .","Changes of inertial observer are given by the transformations given by the same action.","An inertial observer, R, thus sees any state as a pair, the first component is an event and the second a movement containig the event.","The values at that point of $P,\\overrightarrow{l},\\overrightarrow{g}$ , are the values measured by R of linear momentum and angular momentum.","In this paper I accept that this is also valid for all relativistic free particles, with or without mass or spin.","More precisely, it is assumed that the possible state space of the relativistic free particles are the orbits of G in ${ {\\bf R}^4}\\times \\underline{G}^*$ , where now G is the universal (two fold) covering group of ${\\cal P}_{+}^{\\uparrow }$ .","The preceeding interpretations of movements, states, linear and angular momentum are also considered as valid.","In what follows, we assume that an inertial observer, R, has been fixed.","If state space is the orbit $\\cal O$ and $( x, \\alpha )\\in \\cal O$ , we have seen how $\\alpha $ can be considered as a movement.","In fact, this movement is composed by the elements of $\\cal O$ of the form $( y, \\alpha ).$ The set composed by such $y$ , $M_{\\alpha },$ compose the ordinary portrait of the movement $\\alpha $ in ${\\bf R}^4,$ obtained by the inertial observer R. Obviously $M_{\\alpha }=\\lbrace g*x: g\\in G_\\alpha \\rbrace $ where $G_\\alpha $ is the isotropy subgroup at $\\alpha $ of the coadjoint representation.","This way of looking at states also enables us to determine all movements containing a given event: if $( x, \\alpha )\\in \\cal O$ , the movements containing the event $x$ compose the set $N_x=\\lbrace Ad^*_g\\alpha :g\\in G_x\\rbrace $ where $G_x$ is the isotropy group at $x$ of the action of $G$ on ${\\bf R}^4.$"],["Universal covering group of the Poincaré Group. ","It is a well known fact that the universal covering group of Poincaré group is a semidirect product of $\\bf {SL}(2,\\bf {C})$ by a four dimensional real vector space.","In this section, I recall some general facts about this group and stablishes the notation.","Let $(x^1,x^2,x^3,x^4)$ be the canonical coordinates in $\\bf {R}^4$ , I the $2\\times 2$ unit matrix and ${\\sigma }_1, {\\sigma }_2, {\\sigma }_3$ the Pauli matrices, $i.e.$ $\\left(\\begin{array}{cc} 0 & 1\\\\1 & 0\\end{array}\\right),\\ \\left(\\begin{array}{cc}0 & -i\\\\i & 0\\end{array}\\right), \\,\\,\\left(\\begin{array}{cc}1 & 0\\\\0 & -1\\end{array}\\right)$ respectively.","A generic point of $\\bf {R}^4$ will be denoted $x=(x^1,x^2,x^3,x^4)$ .","We define an isomorphism $h$ , from $\\bf {R}^4$ onto the real vector espace, $\\bf {H}(2)$ , of the hermitian $2\\times 2$ matrices, by means of $h(x)=x^4I+\\sum _{i=1}^{3}x^i{\\sigma }_i=\\left(\\begin{array}{cc}x^4+x^3 , & x^1-ix^2\\\\x^1+ix^2 , &x^4-x^3\\end{array}\\right)$ We have $Det \\, h(x)=_{ m}$ , where $<\\, ,\\, >_m $ is Minkowski pseudo-scalar product $_m=x^4y^4-\\sum _{i=1}^3 x^iy^i.$ If $x=(x^1,x^2,x^3,x^4)$ we denote $\\vec{x}=(x^1,x^2,x^3)$ and $h(\\vec{x},x^4)=h(x).$ The following formulae are useful for many computations along this paper $[h(\\vec{k},k_4),h(\\vec{x},x^4)]\\stackrel{def}{=}h(\\vec{k},k_4)h(\\vec{x},x^4)-h(\\vec{x},x^4)h(\\vec{k},k_4)=2ih(\\vec{k} \\times \\vec{x},0),$ where $\\times $ means ordinary vector product, $\\lbrace h(\\vec{k},k_4),h(\\vec{x},x^4)\\rbrace & \\stackrel{def}{=}&h(\\vec{k},k_4)h(\\vec{x},x^4)+h(\\vec{x},x^4)h(\\vec{k},k_4)=\\\\ &=& 2h(k_4\\vec{x} +x^4 \\vec{k},k_4x^4+\\langle \\vec{k},\\vec{x} \\rangle ), $ where $\\langle \\, .","\\, ,\\, .","\\, \\rangle $ is the usual scalar product in $\\mathbb {R}^3,$ $h(\\vec{k},k_4)\\varepsilon \\overline{ h(\\vec{x},x^4)} \\varepsilon -h(\\vec{x},x^4)\\varepsilon \\overline{ h(\\vec{k},k_4)} \\varepsilon =\\\\\\nonumber =2[h(k_4 \\vec{x}-x^4 \\vec{k},0)+ih(\\vec{k} \\times \\vec{x},0)]$ where the bar means complex conjugation, and we denote by ${\\varepsilon }$ the matrix $i\\sigma _2,\\ i.e.$ ${\\varepsilon }=\\left(\\begin{array}{cc}0 & 1\\\\-1 & 0\\end{array}\\right).$ Notice that ${ }^tA\\,{\\varepsilon }\\,A=(Det \\,A)\\,{\\varepsilon },$ so that $A\\,{\\varepsilon }={\\varepsilon }\\,({}^tA)^{-1}$ if $A\\in \\bf {SL}(2,\\bf {C})$ .","Also we have $\\frac{1}{2}Tr(h(x){\\varepsilon }\\overline{h(y)}{\\varepsilon })=-_m$ for all $x, y\\in \\bf {R}^4.$ We define an action on the left of the Lie group $\\bf {SL}(2,\\bf {C})$ on the abelian Lie group $\\bf {H}(2)$ by means of $A*H=AHA^*$ for all $A\\in \\bf {SL}(2,\\bf {C})$ , $H\\in \\bf {H}(2)$ , where $A^*$ is the transposed of the complex conjugate of $A$ .","To this action by automorphisms of $\\bf {H}(2)$ then corresponds a semidirect product, $\\bf {SL}(2,\\bf {C})\\oplus \\bf {H}(2)$ , whose group law is given by $(A,H)*(B,K)=(AB,AKA^*+H)$ The identity element is $(I,0)$ and $(A,H)^{-1}=(A^{-1},-A^{-1}HA^{*^{-1}})$ .","This semidirect product acts on the left on $\\bf {R}^4$ by means of $(A,H)* x=h^{-1}(Ah(x)A^*+H).$ Poincaré group, ${\\cal {P}}$ , is identified to the closed subgroup of $GL(5; \\bf {R})$ composed by the matrices $\\left(\\begin{array}{cc}L & C\\\\0 & 1\\end{array}\\right)$ where $C\\in \\bf {R}^4$ and $L\\in {\\cal {O}}(3,1)$ (such a matrix is denoted in the following simply by $(L,C)$ ).","For all $(A,H)\\in \\bf {SL}\\oplus \\bf {H}(2)$ (where $\\bf {SL}$ stands for $\\bf {SL}(2;\\bf {C})$ ), there exists a unique $(L,C)\\in {\\cal {P}}$ such that $(A,H)*x=Lx+C$ for all $x\\in \\bf {R}^4$ .","The map, $\\rho $ , from $\\bf {SL}\\oplus \\bf {H}(2)$ into ${\\cal {P}}$ defined by sending such a $(A,H)$ to the corresponding $(L,C)$ , is a homomorphism of Lie groups, whose kernel consists of $(I,0)$ and $(-I,0)$ .","In fact, $\\rho $ is a two fold covering map of the identity component in ${\\cal {P}}$ , ${\\cal {P}}_+^{\\uparrow }$ .","Since $\\bf {SL}$ and $\\bf {H}(2)$ are connected and simply connected it follows that $\\bf {SL} \\oplus \\bf {H}(2)$ is the universal covering group of ${\\cal {P}}_+^{\\uparrow }$ .","The differential of $\\rho $ is an isomorphism from the Lie algebra of $\\bf {SL}\\oplus \\bf {H}(2)$ onto the Lie algebra of ${\\cal {P}}$ .","The standard method to handle semidirect products enable us to identify the Lie algebra of $\\bf {SL}\\oplus \\bf {H}(2)$ with $\\bf {sl(2,{\\bf C})}\\times \\bf {H}(2)$ , the Lie bracket being $[(a,k),(a^{\\prime },k^{\\prime })]=([a,a^{\\prime }], ak^{\\prime }+ k^{\\prime }a^*-(a^{\\prime }k+ka^{\\prime *})).$ If $(a,h)\\in \\bf {sl(2,{\\bf C})}\\times \\bf {H}(2),\\ t\\in {\\bf R},$ we have $Exp[t(a,h)] =\\left(e^{ta},\\int _0^t e^{sa}\\,h\\, e^{sa^*}\\,ds \\right).$ In this paper, we use the basis of $\\bf {sl}\\times \\bf {H}(2)$ corresponding by $d\\rho $ to the basis of ${\\cal {P}}$ associated to linear momentum and angular momentum in section .","This basis is composed by the following elements $P^k &=&(0,-\\sigma _k),\\ \\ \\ k=1,2,3, \\nonumber \\\\P^4 &=&(0,\\sigma _4)=(0,I), \\nonumber \\\\l^k &=&(i\\frac{\\sigma _k}{2},0), \\\\g^k &=&(\\frac{\\sigma _k}{2},0).","\\nonumber $ An element, $X$ , of $\\bf {sl(2,{\\bf C})}\\times \\bf {H}(2)$ defines a (linear) function on $(\\bf {sl(2,{\\bf C})}\\times \\bf {H}(2))^*.$ Its restriction to movement space ( a coadjoint orbit) must be considered as a dynamical variable.","But also, if we have a representation on a vector space (resp.","an action on a manifold) of $\\bf {SL}\\oplus \\bf {H}(2),$ $X$ gives rise to a infinitesimal generator of the representation (resp.","the action), i.e., an endomorphism of the vector space (resp.","a vector field on the manifold).","We define vector valued functions on the dual of the Lie algebra as follows $P&=&(P^1,\\,P^2,\\,P^3,\\,P^4) \\\\\\overrightarrow{l}&=&(l^1,\\,l^2,\\,l^3)\\\\\\overrightarrow{g}&=&(g^1,\\,g^2,\\,g^3)$ what will be considered as being linear momentum and angular momentum.","We define a non degenerate scalar product in $\\bf {sl}\\times \\bf {H}(2)$ by means of $<(a,k),(b,l)> & =&2Re Tr (\\frac{1}{4}k{\\varepsilon }\\overline{l}{\\varepsilon }-ab)= \\nonumber \\\\& =&\\frac{1}{2}Tr(k{\\varepsilon }\\overline{l}{\\varepsilon })-2 Re Tr\\, ab.","\\nonumber $ This scalar product defines in the standard way an isomorphism from the Lie algebra of $\\bf {SL}\\oplus \\bf {H}(2)$ onto its dual.","The image of $(a,k)\\in \\bf {sl}\\times \\bf {H}(2)$ by this isomorphism, will be denoted by $\\lbrace a,k\\rbrace \\in \\left(\\bf {sl}\\times \\bf {H}(2)\\right)^{*}$ , and is given by $\\lbrace a,k\\rbrace \\left((b,m)\\right)=<(a,k),(b,m)>.$ With this notation, the values of $P,\\ \\overrightarrow{l}$ and $\\overrightarrow{g}$ at $\\lbrace a,k\\rbrace $ are, when written in terms of its hermitian form $h(P( \\lbrace a,\\ k \\rbrace ))&=&-k, \\\\h(\\vec{l}(\\lbrace a,\\ k \\rbrace ),\\ 0)&=& i\\,(a^*-a), \\\\h(\\vec{g}(\\lbrace a,\\ k \\rbrace ),\\ 0)&=& -(a+a^*).$ so that $\\lbrace a,k\\rbrace =\\lbrace -\\frac{1}{2}h(\\vec{g}(\\lbrace a,\\ k \\rbrace ),0)+\\frac{i}{2}h(\\vec{l}(\\lbrace a,\\ k \\rbrace ),0),-h(P( \\lbrace a,\\ k \\rbrace ))\\rbrace .$ i.e.","$-(1/2)h(\\vec{g}(\\lbrace a,\\ k \\rbrace ),0)$ is the hermitian real part of $a$ and the matrix $(1/2)h(\\vec{l}(\\lbrace a,\\ k \\rbrace ),0)$ its hermitian imaginary part.","A straightforward computation, leads to the following formula for the coadjoint representation $Ad^*_{(A,H)}\\lbrace a,k\\rbrace =\\lbrace AaA^{-1}+\\frac{1}{4}(AkA^*{\\varepsilon }\\overline{H}{\\varepsilon }-H{\\varepsilon }\\overline{AkA^*}{\\varepsilon }),AkA^*\\rbrace .$ To end this section, we define other functions on the dual of the Lie algebra of $\\bf {SL}\\oplus \\bf {H}(2).$ One of these is $\\vert P \\vert $ , defined by $\\left| P \\right| ( \\lbrace a, k\\rbrace )= Det\\,\\left( h(P( \\lbrace a,\\ k \\rbrace )) \\right)=Det\\,\\left( k \\right),$ whose physical meaning is mass square.","Then, $\\left| P \\right| ( Ad^*_{(A,H)}\\lbrace a, k\\rbrace )=Det (-AkA^*)=Det(k),$ so that the value of $\\vert P \\vert $ , is constant along any coadjoint orbit.","The other is defined in terms of the Pauli-Lubanski fourvector, given by $W&=&(\\overrightarrow{ W},W^4)\\\\\\overrightarrow{ W}&=&P^4 \\vec{l}+\\vec{P} \\times \\vec{g},\\\\W^4&=&\\langle \\vec{P},\\vec{l} \\rangle .$ Using (REF ), (REF ) and (REF ), the hermitian form of $W$ is found to be $h(W( \\lbrace a, k\\rbrace ))= i(a\\ k\\ -k\\ a^*).","$ One can prove that $h(W(Ad^*_{(A,H)}\\lbrace a,\\ k\\rbrace ))=A\\ h( W(\\lbrace a,\\ k\\rbrace ))\\ A^*,$ so that the function $\\vert W \\vert ( \\lbrace a, k\\rbrace )= Det(h(W( \\lbrace a, k\\rbrace ))),$ is also constant along each coadjoint orbit."],["Quantizable forms","In this section I recall some of the geometric constructions done in [16], [4], [5], [6], [7].","I shall give a way to construct homogeneous contact manifolds that fibers on the coadjoint orbits of Lie groups.","This is not possible on an arbitrary orbit, but only on the so called quantizable ones.","The idea is to consider the quantizable coadjoint orbits of $\\bf {SL}(2;{\\bf C}) \\oplus \\bf {H}(2)$ as the classical movement space of relativistic free particles.","The corresponding homogeneous contact manifolds are geometrical objects enabling us to construct the Wave Functions, and representations of this group.","Let $G$ be a Lie group and $\\alpha $ an element of the dual of the Lie algebra of $G$ , $\\underline{G}^*$ , where we consider the coadjoint action.","We say that , $\\alpha $ is quantizable if there exists a surjective homomorphism, $C_\\alpha $ , from the isotropy subgroup at $\\alpha $ , $G_\\alpha $ , onto the unit circle, ${\\bf S}^1 \\rm $, whose differential is $\\alpha $ (cf.","section where the way in which the differential is identified to an element of $\\underline{G}^*$ , is detailed).","In a more explicit way, this means that $\\alpha $ is quantizable if there exists a surjective homomorphism, $C_\\alpha ,$ from the isotropy subgroup at $\\alpha $ , $G_\\alpha ,$ onto the unit circle, such that, for all $X$ in the Lie algebra of $G_\\alpha $ and t in R $C_{\\alpha }(Exp\\,tX)=e^{2 \\pi i t {\\alpha }(X)}.$ If a form is quantizable, all the elements of its coadjoint orbit are quantizable.","These orbits are called quantizable orbits .","The form $\\alpha $ is said to be R-quantizable if there exists a surjective homomorphism from $G_\\alpha $ onto the usual additive Lie group of reals, R , whose differential is $\\alpha $ .","In other words, $\\alpha $ is R-quantizable if there exists a surjective homomorphism from $G_\\alpha $ onto R , $H_{\\alpha }$ , such that, for all $X$ in the Lie algebra of $G_\\alpha $ , and $t$ in R $H_{\\alpha }(Exp\\,tX)=t\\,{\\alpha }(X).$ To such a $H_{\\alpha }$ , one can associate an homomorphism, $C_{\\alpha }$ , onto ${\\bf S}^1$ by means of $C_{\\alpha }:g \\in {G_\\alpha } \\rightarrow e^{2 \\pi i H_{\\alpha }(g)} \\in {\\mbox{${\\bf S}^1 \\rm $}}.$ The differential of $C_{\\alpha }$ is $\\alpha $ .","Thus when $\\alpha $ is R-quantizable , it is quantizable .","All elements of a coadjoint orbit containing a R-quantizable form, are R-quantizable.","In this case, the orbit is said to be a R-quantizable orbit.","In [7] a slightly more general concept of quantizability is used, but it is unnecesary for the purposes of the present paper.","In what follows we assume that $\\alpha $ is quantizable and $C_\\alpha $ is a homomorphism from $G_\\alpha $ onto the unit circle, whose differential is $\\alpha $ .","We identify the coadjoint orbit, ${\\cal O}_{\\alpha },$ with $G/G_{\\alpha }$ by means of the diffeomorphism $g G_{\\alpha } \\in \\frac{G}{G_{\\alpha }} \\rightarrow Ad^*_g \\alpha \\in {\\cal O}_{\\alpha }.$ We define an action of ${\\bf S}^1$ on $G/Ker\\,C_\\alpha $ by means of $(g\\,Ker\\,C _\\alpha )*s=g\\,h\\,Ker\\,C_\\alpha $ where $h$ is any element of $G_{\\mbox{$ \\alpha $}}$ such that $C_\\alpha (h)=s$ .","Actually $(G/Ker\\,C_\\alpha )$ $(G/G_{\\mbox{$\\alpha $}},\\bf S\\rm ^1)$ is a principal fibre bundle , the bundle action is given by (REF ) and the bundle projection, by the canonical map, $\\pi : g{KerC_\\alpha } \\longrightarrow g {G_\\alpha }$ This action of ${\\bf S}^1$ , commutes with the canonical action of $G$ on $G/Ker\\,C_\\alpha $ .","We denote by $\\pi ^c$ and $\\pi ^s$ the canonical maps $\\pi ^c&:& g\\in G \\longrightarrow g {KerC_\\alpha } \\in \\frac{G}{{KerC_\\alpha }}\\\\\\pi ^s &:& g\\in G \\longrightarrow g {G_\\alpha } \\in \\frac{G}{{G_\\alpha }}.$ There exist an unique 1-form, $\\Omega $ , on the homogeneous space $G/Ker\\,C_\\alpha $ such that $(\\pi ^c)^*\\Omega =\\alpha .$ The 1-form $\\Omega $ is a contact form, invariant under the action of $G.$ Let $Z(\\Omega )$ be the vectorfield defined by $i_{Z (\\Omega )} \\Omega = 1,\\ \\ i_{Z(\\Omega )}\\,d\\Omega =0.$ All the integral curves of $Z(\\Omega )$ have the same period.","If we denote by ${T(\\Omega )}$ this period, then $\\Omega /{T(\\Omega )}$ is a connexion form.","Since the structural group is abelian, the curvature form is $d\\Omega /{{T(\\Omega )}} $ .","There exist an unique 2-form, $\\omega ,$ on $G/G_{\\alpha }$ , such that $\\pi _*\\omega =\\frac{d\\Omega }{{T(\\Omega )}}.$ Thus $(\\pi ^s)^*\\omega =\\frac{d\\alpha }{T(\\Omega )}.$ The form $\\omega $ is an invariant symplectic form, and its cohomology class is integral.","Each $m\\in \\underline{G},$ define a function on $\\underline{G}^*$ so that it does also on the coadjoint orbit.","Since we think in the coadjoint orbit as being the movement space of a particle, we can say that m defines a dynamical variable, $D_m.$ Such a $m\\in \\underline{G},$ also define a infinitesimal generator of the canonical action of $G$ on $G/Ker\\,C_\\alpha $ , denoted by $X_m^c,$ and a infinitesimal generator of the canonical action of $G$ on $G/G_\\alpha $ , denoted by $X_m^s.$ Since $\\Omega $ is left invariant by the action of $G,$ we have $L_{X_m^c}\\Omega =0,$ so that the relation $L_X=d\\,i_X+i_X\\,d$ leads to $i_{X_m^c}\\,d\\Omega =-d[\\Omega (X_m^c)].$ A computation gives ${\\begin{array}{c}(\\Omega (X_m^c) \\circ \\pi ^c)(g)= Ad^*_g \\alpha _0 (-m)=-D_m(Ad^*_g \\alpha _0)=\\\\=- D_m(g\\, G_\\alpha )=-D_m\\circ \\pi (g\\, Ker\\,C_\\alpha ),\\end{array}}$ so that $\\Omega (X_m^c)=-D_m\\circ \\pi .$ Thus (REF ), (REF ), and the fact that $\\pi _* X_m^c=X_m^s,$ lead to $i_{X_m^s}\\omega =\\frac{1}{T\\left( \\Omega \\right)}d D_m.$ Since the flow of the vector field $X_m^s$ (resp.","$X_m^c$ ) preserves $\\omega $ (resp.","$\\Omega $ ), it is an infinitesimal automorphism of the symplectic (resp.","contact) structure, i.e.","a locally hamiltonian vector field.","Equation (REF ) tell us that in fact $X_m^s$ is globally hamiltonian and $ D_m/{T\\left( \\Omega \\right)}$ is the corresponding hamiltonian.","As usual in the Theory of Connections, a map, $f$ , into $G/{KerC_\\alpha }$ is called horizontal, if $f^{*}\\Omega =0.$ Let $f_0$ be a map into $G/{G_\\alpha }$ .","An horizontal lift of $f_0$ is an horizontal map into $G/{KerC_\\alpha }$ , such that $\\pi \\circ f =f_0$ .","The horizontal lift of curves can be described as follows.","Given a curve, $\\gamma $ in $G/G_{\\mbox{$\\alpha $}}$ , the horizontal lift of $\\gamma $ to $g\\,KerC_\\alpha $ is $\\widetilde{\\gamma }\\left( t \\right)=\\left( \\overline{\\gamma }\\left( t \\right) KerC_\\alpha \\right) * e^{-2 \\pi i \\int _{\\overline{\\gamma }|_{[0,\\ t]}}\\alpha } $ where $\\overline{\\gamma }$ is any lifting of $\\gamma $ to G, such that $\\overline{\\gamma }(0)=g$ , and the vertical bar means restriction.","Associated to this principal fibre bundle and the canonical action of ${\\bf S}^1$ on ${\\mathbb {C}}$ , one can consider the 1- dimensional vector bundle whose total space is $\\left( \\frac{G}{KerC_\\alpha }\\right) \\times _{{\\bf S}^1} {\\mathbb {C}}.$ Let us recall its definition.","Consider in $\\left( G/KerC_\\alpha \\right) \\times {\\mathbb {C}}$ , an action of ${\\bf S}^1$ defined by $(g\\, KerC_\\alpha , t)*z=\\left(\\left(g\\, KerC_\\alpha \\right)*z ,z^{-1} t\\right),$ for all $z\\in {\\bf S}^1$ .","The elements of $\\left( G/KerC_\\alpha \\right)\\times _{{\\bf S}^1}{\\mathbb {C}}$ are the orbits of this action.","Let us denote by $[g\\, KerC_\\alpha , t]$ the orbit of $(g\\, KerC_\\alpha , t).$ We have $\\nonumber [g\\, KerC_\\alpha , t]&=&\\left\\lbrace \\left(\\left(g\\,KerC_\\alpha \\right)*s, s^{-1} t\\right):\\ s\\in S^{\\rm 1}\\right\\rbrace = \\\\ &=&\\nonumber \\left\\lbrace \\left(gh\\,KerC_\\alpha , C_\\alpha \\left(h^{\\rm -1}\\right) t\\right):\\ h\\in G_\\alpha \\right\\rbrace $ We define a map, $\\overline{\\pi }$ , from $ \\frac{G}{KerC_\\alpha } \\times _{{\\bf S }^1} {\\mathbb {C}}$ onto the coadjoint orbit by means of $\\overline{\\pi }([g\\, KerC_\\alpha , t])=g\\,G_\\alpha $ .","Let $m\\in G/G_\\alpha $ , and $g\\in G$ one of its representatives.","Since the action of ${\\bf S}^1$ on $\\pi ^{-1}(m)$ is transitive, we have ${\\overline{\\pi }}^{-1}(m)=\\lbrace [g\\,Ker\\,C_{\\alpha },t]:t\\in \\bf C\\rm \\rbrace .$ Thus, for each $g$ we obtain a bijection of ${\\overline{\\pi }}^{-1}(m)$ onto $\\bf C$ .","We consider in ${\\overline{\\pi }}^{-1}(m)$ the structure of hermitian one dimensional complex vector space such that this bijection is a unitary isomorphism.","This structure is independent of the representative $g$ .","If $g$ and $g^{\\prime }$ are representatives of $m$ , we have for all $t,\\ t^\\prime ,\\ a \\in \\bf C$ $ \\left[g\\,KerC_\\alpha ,\\ t\\right] + \\left[g^\\prime \\,KerC_\\alpha ,\\ t^\\prime \\right]=\\left[g\\,KerC_\\alpha ,\\ t+C_\\alpha (g^{-1}\\,g^\\prime )\\,t^\\prime \\right]$ , $ a \\cdot \\left[g\\,KerC _\\alpha ,\\ t\\right] =\\left[g\\,KerC_\\alpha ,\\ a\\,t\\right],$ $\\langle \\left[g\\,KerC_\\alpha ,\\ t\\right] ,\\left[g^\\prime \\, KerC_\\alpha ,\\ t^\\prime \\right] \\rangle ={\\overline{t}}\\,{C_\\alpha (g^{-1}g^\\prime )\\,t^\\prime } .$ With these operations, $\\overline{\\pi }$ becomes a complex vector bundle of dimension one with a hermitian product in each fiber i.e.","a hermitian line bundle.","The sections of the hermitian line bundle are in a one to one correspondence with the functions on $G/{KerC_\\alpha } $ , f, such that f((g ${\\mbox{$Ker\\,C_\\alpha $}} )\\,*\\,s)= s^{-1} $ f(g ${\\mbox{$Ker\\,C_\\alpha $}} $ ), for all $s\\in {\\mbox{$\\bf S \\rm ^1$}}$ .","These functions will be called from now on pseudotensorial functions .","This correspondence is as follows.","If f is a pseudotensorial function, the corresponding section sends $m \\in G/G_\\alpha $ to $\\left[ r,f(r) \\right]$ where r is arbitrary in $\\pi ^{-1}(m)$ .","If $\\sigma $ is a given section of the hermitian line bundle, the corresponding pseudotensorial function, f, is defined by $\\sigma (\\pi (r)) = \\left[ r,f(r) \\right]$ for all $ r \\in G/Ker C_\\alpha $ .","The canonical action of $G$ on $\\left( G/KerC_\\alpha \\right)$ , leads to an action on $\\left( G/KerC_\\alpha \\right)\\times _{\\mbox{$\\bf S \\rm ^1$}}{\\mbox{{$C$}} \\rm },$ given by $g*[h\\, Ker C_\\alpha ,t]=[gh\\, Ker C_\\alpha ,t].$ This action is well defined, as a consecuence of the fact that the action of $\\bf S \\rm ^1$ conmutes with the canonical action of $G$ .","In the following, I use the same construction of a hermitian line bundle, for each principal bundle, $ G/(Ker\\,C)(G/H,S),$ where $C$ is an homomorphism of $H$ into $\\bf S \\rm ^1$, whose image is $S$ ."],["Geometric Quantum States .","In this section I recall some definitions and results from [8].","Let $G=SL(2,{\\bf C}) \\oplus H(2)$ and $\\alpha $ a quantizable form of $G$ .","We use the notation of section .","The sections of the hermitian line bundle whose total space is $ \\left( G/KerC_\\alpha \\right)\\times _{\\mbox{$\\bf S \\rm ^1$}}{\\mbox{{$C$}} \\rm } $ are called Prequantum States .","We use the same denomination for the corresponding pseudotensorial functions.","Now, let us consider the actions of the abelian subgroup $\\lbrace I\\rbrace \\times H(2)$ on $G/Ker\\,C_\\alpha $ and $G/G_\\alpha $ , induced by the canonical action.","There exist an unique action of $\\lbrace I\\rbrace \\times H(2)$ on $G/Ker C_\\alpha $ whose orbits are horizontal and such that $\\pi $ becomes equivariant.","This action is called horizontal action and is given by $(I,\\ K)*((A,\\ H)\\,KerC_\\alpha )=((A,\\ H+K)\\,KerC_\\alpha )*e^{-i \\pi Tr\\left(AkA^* \\varepsilon \\overline{K} \\varepsilon \\right)}$ for all $ K \\in H(2),\\ (A,\\ H) \\in G $ , where $*$ in the left hand side stands for the new action and in the right hand one, corresponds to the bundle action.","k is given by $\\alpha = \\lbrace a,\\ k\\rbrace $ .","We define Quantum States as being the Prequantum States that correspond to pseudotensorial functions left invariant by the horizontal action.","Let $\\pi _1$ and $\\pi _2$ be the canonical projections of $G$ on ${ SL(2,\\bf C)}$ and $ H(2) $ respectively.","We denote $\\pi _1(G_\\alpha )$ by ${\\mbox{$(G_\\alpha )_{SL}$}}$ and $\\pi _2(G_\\alpha )$ by ${\\mbox{$(G_\\alpha )_{H}$}}$ .","In section 4 of [8] it is proved that the map $(C_{{\\mbox{$\\alpha $}}})_{SL} :(G_{{\\mbox{$\\alpha $}}})_{SL} \\longmapsto S^1 ,$ defined by $(C_{{\\mbox{$\\alpha $}}})_{SL}(g)\\ =\\ C_{{\\mbox{$\\alpha $}}}(g,h)\\ e^{-i\\pi Tr\\left(k{\\mbox{$\\varepsilon $}} \\overline{h}{\\mbox{$\\varepsilon $}}\\right) },$ for all $(g,h)\\in {\\mbox{$G_\\alpha $}},$ is well defined and a homomorphism.","As a consecuence, we can define $\\widetilde{C}_\\alpha :(G_\\alpha )_{SL}\\ \\oplus \\ H(2)\\longmapsto S^1 ,$ by means of $ \\widetilde{C}_{{\\mbox{$\\alpha $}}}(g,r)\\ =\\ (C_{{\\mbox{$\\alpha $}}})_{SL}(g)\\ e^{i\\pi Tr\\left(k{\\mbox{$\\varepsilon $}} \\overline{r}{\\mbox{$\\varepsilon $}}\\right)}.$ $ \\widetilde{C}_{{\\mbox{$\\alpha $}}}$ is an extension of $C_\\alpha $ to $(G_{{\\mbox{$\\alpha $}}})_{SL}\\ \\oplus \\ {{\\mbox{$ H(2)\\rm $}}},$ and a homomorphism .","Its differential coincides with the restriction of $\\alpha $ to the Lie algebra of this group.","The canonical action of $G$ on $G/KerC_\\alpha $ maps horizontal orbits to horizontal orbits, thus defining a transitive action on the space of horizontal orbits, ${\\cal W}_{\\alpha }.$ Let us consider the canonical map $\\tau :\\frac{G}{KerC_\\alpha } \\rightarrow {\\cal W}_{\\alpha }$ defined by sending each element of $G/KerC_\\alpha $ onto its horizontal orbit.","The isotropy subgroup at $\\tau (KerC_\\alpha )$ is $Ker\\widetilde{C}_\\alpha $ .","As a consequence, we can identify ${\\cal W}_{\\alpha }$ to $G/Ker\\,\\widetilde{C}_{\\alpha },$ by means of the bijective map $(A,H)\\,Ker\\,\\widetilde{C}_{\\alpha }\\in \\frac{G}{Ker\\,\\widetilde{C}_{\\alpha }}\\rightarrow \\tau ((A,H)Ker\\,C_{\\alpha }) \\in {\\cal W}_{\\alpha }.$ With the definitions given in section , we have seen that $\\frac{G}{Ker\\,C_{\\alpha }}\\left(\\frac{G}{G_{\\alpha }},{\\bf S}^1\\right)$ becomes a principal fibre bundle.","Similar definitions provides $\\frac{G}{Ker\\,\\widetilde{C}_{\\alpha }}\\left(\\frac{G}{(G_{\\alpha })_{SL}\\oplus H(2)},{\\bf S}^1\\right)$ with a structure of principal fibre bundle.","The bundle projection is the canonical map from ${G}/{Ker\\,\\widetilde{C}_{\\alpha }}$ onto ${G}/({(G_{\\alpha })_{SL}\\oplus H(2)}).$ The bundle action is given by $((A,H){Ker\\,\\widetilde{C}_{\\alpha }})*s=(A,H)(B,K){Ker\\,\\widetilde{C}_{\\alpha }},$ where $(B,K)$ is such that $\\widetilde{C}_{\\alpha }(B,K)=s.$ But the map $(A,H)((G_\\alpha )_{SL} \\oplus H(2))\\in \\frac{G}{(G_{\\mbox{$\\alpha $}})_{SL} \\oplus H(2)}\\rightarrow A\\,(G_{\\alpha })_{SL} \\in \\frac{SL}{(G_{\\alpha })_{SL}}$ is a diffeomorphism.","We identify these homogeneous spaces by means of this map, so that the principal fibre bundle becomes $\\frac{G}{Ker\\,\\widetilde{C}_{\\alpha }}\\left(\\frac{SL}{(G_{\\alpha })_{SL}},{\\bf S}^1\\right),$ where the bundle projection is $\\tau _2:(A,H){Ker\\,\\widetilde{C}_{\\alpha }}\\rightarrow A\\,(G_{\\alpha })_{SL}.$ The canonical maps define the homomorphism of principal ${\\bf S}^1 \\rm $-bundles given in Figure REF Figure: Fibre Bundles for Quantum StatesSince Quantum States correspond to pseudotensorial functions left invariant by the horizontal action, Quantum States are the pull back by $\\iota _1$ of unrestricted pseudotensorial functions on $G/Ker\\widetilde{C}_\\alpha $ ."],["Wave Functions","Now, let us associate, to each Quantum State in the sense of section , a Wave Function in the ordinary sense of Quantum Mechanics.","From now on, we assume, unless the contrary is explicitly stated, that the State Space is the orbit of $(0,\\alpha )$ in $H(2)\\times \\underline{G}^*.$ This choice do not carry very important consequences for the study of the free particle at the quantum level, in the sense that the other choices lead to isomorphic spaces of Quantum States (in all of its forms), and the same Wave Functions.","These facts are proved in Remark 5.2 of [8].","The isotropy subgroup at $(0,\\alpha ),$ $G_{(0,\\alpha )},$ is composed by the elements of $G_\\alpha $ of the form $(A,0)$ i.e.","$G_{(0,\\alpha )}=G_\\alpha \\cap (SL \\oplus \\lbrace 0\\rbrace ).$ Let $\\alpha =\\lbrace a,k\\rbrace $ and denote $SL_1=\\lbrace A\\in SL: AaA^{-1}=a\\rbrace ,$ $SL_2=\\lbrace A\\in SL: AkA^*=k\\rbrace .$ Then $G_{(0,\\alpha )}=(SL_1 \\cap SL_2)\\oplus \\lbrace 0\\rbrace ,$ Notice that $(G_\\alpha )_{SL}\\subset SL_2$ and $SL_1 \\cap SL_2\\subset (G_\\alpha )_{SL}$ so that $SL_1 \\cap SL_2=(G_\\alpha )_{SL}\\cap SL_1.$ But the map $(A,H)((G_\\alpha )_{SL}\\cap SL_1)\\oplus \\lbrace 0\\rbrace \\in \\frac{G}{((G_\\alpha )_{SL}\\cap SL_1)\\oplus \\lbrace 0\\rbrace }\\rightarrow $ $ \\rightarrow (H,A((G_\\alpha )_{SL}\\cap SL_1))\\in H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1}$ is a diffeomorphism.","Thus, State Space is the image of the injective map $i:(H,A(G_\\alpha )_{SL}\\cap SL_1))\\in H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1} \\rightarrow $ $\\rightarrow (H, Ad^*_{(A,H)}\\alpha )\\in H(2)\\times \\underline{G}^*.$ This image need not, a priori, be a proper submanifold of $H(2)\\times \\underline{G}^*,$ and we consider it provided with the topology and differentiable structure such that $i$ becomes a diffeomorphism.","In the following we identify each $i(X)$ with $X,$ so that we can say that $H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1}$ is State Space.","The canonical map from State Space onto Movement Space, can be generalised to all the homogeneous spaces appearing in the commutative diagram of Figure 1.","This will be done with the following geometrical construction.","Let $ \\cal L $ be a closed subgroup of G, and $ {\\cal S}=\\lbrace s\\in SL : (s,0) \\in {\\cal L} \\rbrace ,$ so that ${\\cal S} \\oplus \\lbrace 0\\rbrace = {\\cal L} \\cap (SL \\oplus \\lbrace 0\\rbrace ).$ Thus we define the map $(H,A {\\cal S})\\in {H(2) \\times \\frac{SL}{\\cal S} } \\stackrel{\\nu }{\\longrightarrow } (A,H) {\\cal L} \\in \\frac{G}{\\cal L}.$ This map is well defined and, if $H$ is a fixed element in $H(2)$ , its restriction to $ \\lbrace H\\rbrace \\times \\frac{SL}{\\cal S} $ is injective.","Now we apply this to the cases in Figure 1.","When ${\\cal L}=Ker\\,C_\\alpha $ we have ${\\cal S}=Ker(C_\\alpha )_{SL} \\cap SL_1$ , so that we have a map $(H,A ( Ker(C_\\alpha )_{SL} \\cap SL_1))\\in {H(2) \\times \\frac{SL}{ Ker(C_\\alpha )_{SL} \\cap SL_1 }} \\stackrel{\\nu _1}{\\longrightarrow }$ $\\stackrel{\\nu _1}{\\longrightarrow } (A,H) {Ker\\,C_\\alpha } \\in \\frac{G}{Ker\\,C_\\alpha }.","$ If ${\\cal L}=G_\\alpha $ we have ${\\cal S}= SL_1 \\cap SL_2=(G_{\\alpha })_{SL} \\cap SL_1 $ , and we thus obtain the map $(H,A ((G_{\\alpha })_{SL} \\cap SL_1 ))\\in {H(2) \\times \\frac{SL}{(G_{\\alpha })_{SL} \\cap SL_1 }} \\stackrel{\\nu _2}{\\longrightarrow }$ $\\stackrel{\\nu _2}{\\longrightarrow } (A,H) {G_\\alpha } \\in \\frac{G}{G_\\alpha }.","$ When ${\\cal L}=Ker{\\widetilde{C}}_\\alpha $ we have ${\\cal S}= Ker(C_\\alpha )_{SL}, $ so that we have a map $(H,A Ker(C_\\alpha )_{SL})\\in {H(2) \\times \\frac{SL}{ Ker(C_\\alpha )_{SL}}} \\stackrel{\\nu _3}{\\longrightarrow }(A,H) {Ker\\,\\widetilde{C}_\\alpha } \\in \\frac{G}{Ker\\,\\widetilde{C}_\\alpha }.","$ If ${\\cal L}=(G_\\alpha )_{SL} \\oplus H(2)$ we have ${\\cal S}= (G_\\alpha )_{SL} $ , and we thus have a map ${(H,A (G_\\alpha )_{SL}))\\in {H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}}} } \\stackrel{\\nu _4}{\\longrightarrow }$ $\\stackrel{\\nu _4}{\\longrightarrow } (A,H) {((G_\\alpha )}_{SL}\\oplus H(2)) \\in \\frac{G}{(G_\\alpha )_{SL}\\oplus H(2)}$ which, using the identification (REF ), can be written ${(H,A (G_\\alpha )_{SL}))\\in {H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}}} } \\stackrel{\\nu _4}{\\longrightarrow }$ $\\stackrel{\\nu _4}{\\longrightarrow } A (G_\\alpha )_{SL} \\in \\frac{SL}{(G_\\alpha )_{SL}}.$ When we denote by $\\iota _3,\\ \\iota _4,\\ \\tau _3,\\ \\tau _4,$ the canonical maps of homogeneous spaces that appears in Figure REF , we obtain the commutative diagram in that Figure.","Figure: Fibre Bundles for Wave FunctionsThe maps $\\tau _i$ are the bundle maps of principal fibre bundles whose structural groups are identified by means of $(C_\\alpha )_{SL},$ $\\widetilde{C}_\\alpha $ , or $C_\\alpha $ to subgroups of $S^1$ .","We already know the bundle actions of ${\\bf S}^1$ on $G/KerC_\\alpha $ and $G/Ker\\widetilde{C}_\\alpha .$ The other are defined in a similar way as follows.","The action for the principal bundle corresponding to $\\tau _4,$ $H(2) \\times \\frac{SL}{Ker(C_\\alpha )_{SL}}\\left(H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}},(C_\\alpha )_{SL}((G_\\alpha )_{SL})\\right),$ is given by $(H,A{Ker(C_\\alpha )_{SL}})*s=(H,AB{Ker(C_\\alpha )_{SL}})$ if $s\\in (C_\\alpha )_{SL}((G_\\alpha )_{SL}$ and $B\\in (G_\\alpha )_{SL}$ is such that $(C_\\alpha )_{SL}(B)=s.$ In the case of the principal bundle corresponding to $\\tau _3,$ $H(2) \\times \\frac{SL}{Ker(C_\\alpha )_{SL} \\cap SL_1}\\left(H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1},(C_\\alpha )_{SL}((G_\\alpha )_{SL}\\cap SL_1)\\right),$ the bundle action is defined in the same way but using the restriction of $(C_\\alpha )_{SL}$ to $(G_\\alpha )_{SL}\\cap SL_1.$ The pairs $(\\nu _1,\\nu _2),(\\nu _3,\\nu _4),\\ (\\iota _1,\\iota _2),\\ (\\iota _3,\\iota _4)$ define homomorphisms of principal fibre bundles.","In what concerns the structural groups, we have $(C_\\alpha )_{SL}((G_\\alpha )_{SL}\\cap SL_1) \\subset (C_\\alpha )_{SL}((G_\\alpha )_{SL})\\subset {\\mathbf {S}}^1$ and the homomorphism of structural groups is, in all cases, the canonical injection of the group of the first bundle into the group of the second.","As a consecuence of (REF ), the linear momentum, is given on the coadjoint orbit of $\\alpha $ by $P(Ad^{*}_{(A, H)} \\alpha )= -AkA^{*}$ , so that, with the identification of the coadjoint orbit with $G/G_\\alpha $ , we can write $P((A, H)G_\\alpha )=-AkA^{*}$ .","On the other hand, since $ (G_\\alpha )_{SL} \\subset SL_2,$ we can define in $ {SL}/{(G_\\alpha )_{SL}} $ a function, $P_0$ , by means of $P_0(A(G_\\alpha )_{SL})=-AkA^{*}$ .","But then, $P_0$ is the projection of $P$ by the canonical map $(A,\\ H)G_\\alpha \\in \\frac{G}{G_\\alpha } \\stackrel{\\iota _0}{\\longrightarrow } A (G_\\alpha )_{SL}\\in \\frac{SL}{(G_\\alpha )_{SL}},$ and will be denoted in the following simply by $P$ and also called linear momentum, thus we can write $P(A (G_\\alpha )_{SL})=-AkA^{*}$ where $k$ is given by $\\alpha =\\lbrace a,k\\rbrace $ .","Let us denote $(C_\\alpha )_{SL}(G_\\alpha )_{SL})$ by $S$ .","In [8] I prove that The pull back by $\\nu _3$ , maps in a one to one way the set of quantum states (considered as pseudotensorial functions on $G/Ker\\widetilde{C}_\\alpha $ onto the set composed by the functions on $H(2) \\times (SL/Ker(C_\\alpha )_{SL})$ having the form $ \\phi _f(H,A\\,Ker(C_\\alpha )_{SL})\\,=\\,f(A\\,Ker(C_\\alpha )_{SL})\\,e^{i\\pi Tr(P(A (G_\\alpha )_{SL})\\varepsilon \\overline{H}\\varepsilon )}$ where f is a pseudotensorial function on the principal fibre bundle $\\frac{SL}{Ker(C_\\alpha )_{SL}} \\left(\\frac{SL}{(G_\\alpha )_{SL}},S\\right) .$ In order to be more precise we will use the following definitions.","Let ${\\cal C}$ be the complex vector space composed by the pseudotensorial functions of the bundle $\\frac{SL}{\\mbox{$Ker\\,(C_\\alpha )_{SL}$}} \\left(\\frac{SL}{\\mbox{$(G_\\alpha )_{SL}$}},S\\right),$ and $\\widetilde{\\cal V}$ the complex vector space composed by the pseudotensorial functions on $G/Ker\\widetilde{C}_\\alpha .$ For each $f\\in {\\cal C}$ we define a pseudotensorial function on $G/Ker\\widetilde{C}_\\alpha ,$ $\\Psi _f,$ by $\\Psi _f \\circ \\nu _3 \\left(H,A\\,Ker(C_\\alpha )_{SL}\\right)= \\phi _f\\left(H,A\\,Ker(C_\\alpha )_{SL}\\right)=$ $=f(A\\,Ker(C_\\alpha )_{SL})\\,e^{i\\pi \\,Tr(P(A\\,(G_\\alpha )_{SL} )\\,\\varepsilon \\, \\overline{H}\\,\\varepsilon )}.$ The map $\\Psi :f\\in {\\cal C} \\rightarrow \\Psi _f \\in \\widetilde{\\cal V}$ is an isomorphism.","Also we define $\\Phi _f=\\Psi _f \\circ \\iota _1.$ These functions are pseudotensorial on $G/KerC_\\alpha $ and invariant by the horizontal action, so that they represent Quantum States in the most primitive sense adopted in this paper.","We denote by ${\\cal V}$ the complex vector space composed by these $\\Phi _f,$ so that the map $\\Phi :f\\in {\\cal C} \\rightarrow \\Phi _f \\in {\\cal V}$ is an isomorphism.","Thus, we can consider Quantum States as being elements of ${\\cal C}$ or elements of $\\widetilde{\\cal V}$ or elements of ${\\cal V}.$ Now we shall regard Quantum States under another form: Wave Functions.","The differentiable manifold $W \\stackrel{\\mathrm {d}ef}{=}( H(2) \\times (SL/ Ker\\,(C_\\alpha )_{SL} ) \\times _S \\,{\\bf C}$ is defined in the same way as we have defined $\\left( G/KerC_\\alpha \\right)\\times _{\\mbox{$\\bf S \\rm ^1$}}{\\mbox{{$C$}} \\rm }$ in section .","$W$ is the total space of the hermitian line bundle associated to the principal fibre bundle $\\left(H(2) \\times \\left(SL/Ker \\left(C_\\alpha \\right)_{SL}\\right)\\right)\\left(H(2) \\times \\left(SL/\\left(G_\\alpha \\right)_{SL} \\right),\\,S \\right)$ and the canonical action of $S$ on C .","The bundle projection is $\\eta :[(H,A Ker\\,(C_\\alpha )_{SL} ), c]_S \\in W \\rightarrow (H,A (G_\\alpha )_{SL} ) \\in H(2)\\times ( SL/(G_\\alpha )_{SL}),$ where $[(H,A Ker\\,(C_\\alpha )_{SL} ), c]_S$ is the orbit of $((H,A Ker\\,(C_\\alpha )_{SL} ), c)$ under the action of $S$ .","Since for each $f\\in {\\cal C}$ the function $\\Psi _f \\circ \\nu _3$ is pseudotensorial in $H(2) \\times \\left(SL/Ker \\left(C_\\alpha \\right)_{SL}\\right),$ it defines a section of $\\eta .$ This section can be considered as another description of the Quantum State given by $f$ .","To complete our way towards Wave Functions, we need to “represent” Quantum States as functions with values in a fixed complex vector space, not in a different vector space for each point in the base space, as does the sections of $\\eta $ .","If $(C_\\alpha )_{SL}(g)=1,\\ \\forall g\\in (G_\\alpha )_{SL},$ , we have $S=\\lbrace 1\\rbrace ,$ and $Ker\\,(C_\\alpha )_{SL}=(G_\\alpha )_{SL}$ so that $W \\approx ( H(2) \\times (SL/ (G_\\alpha )_{SL} ) \\times \\,{\\bf C},$ and $\\eta $ is the canonical projection onto the first two factors.","The section of $\\eta $ corresponding to the function $\\phi _f $ in (REF ) is $(H,\\,A\\,(G_\\alpha )_{SL})\\, \\longrightarrow (H,\\,A\\,(G_\\alpha )_{SL},\\phi _f(\\,H,\\,A\\,\\,(G_\\alpha )_{SL})).$ Thus, if $(C_\\alpha )_{SL}={\\bf 1 },$ our task is accomplished by $\\phi _f(\\,H,\\,A\\,Ker\\,{\\mbox{$(C_\\alpha )_{SL}$}})\\,=\\,f(\\,A\\,Ker\\,{\\mbox{$(C_\\alpha )_{SL}$}})\\,e^{i\\pi \\,Tr(P(A\\,(G_\\alpha )_{SL} )\\,\\varepsilon \\, \\overline{H}\\,\\varepsilon }$ itself as a complex valued function on the base space.","In this case, $\\phi _f$ is called the Prewave Function associated to $f,$ and denoted by $\\psi _f.$ In the case where ${\\mbox{$(C_\\alpha )_{SL}$}}$ is not trivial, our goal will be attained by imbedding the hermitian fibre bundle in a trivial one.","We do this in a direct way, but a more geometrical view of the method is exposed in remark 5.1 of [8].","A key concept in our construction of Wave Functions is the following: A Trivialization of $ C_\\alpha $ is a triple $(\\rho ,\\ {L},\\ z_0)$ , where $ { L}$ is a finite dimensional complex vector space, $z_0\\in { L}$ and $\\rho $ is a representation of $SL$(2,C) in ${ L}$ such that $ &1)&\\ \\rho (A)(z_0)\\,=\\,(C_\\alpha )_{SL}(A)\\ z_0,\\ \\ \\ \\forall A\\in (G_\\alpha )_{SL}.","\\nonumber \\\\&2)&\\ {\\rm The \\ isotropy \\ subgroup\\ at}\\ z_0\\ {\\rm is}\\ Ker\\,(C_\\alpha )_{SL}.$ In what follows, we assume that a trivialization of $C_\\alpha $ is given.","The homogeneous space $ { SL}/{\\mbox{$Ker\\,(C_\\alpha )_{SL}$}} $ is identified to the orbit of $z_0$ , $\\cal B$ , by means of $A Ker\\,(C_\\alpha )_{SL} \\in \\frac{ SL}{Ker\\,(C_\\alpha )_{SL}} \\longrightarrow \\rho (A)(z_0)\\in {\\cal B}.$ The action of $S$ on $ { SL}/{\\mbox{$Ker\\,(C_\\alpha )_{SL}$}} $ becomes, with this identification, multiplication in $L$ of elements of $S,$ as complex numbers, by elements of ${\\cal B},$ as elements of $L$ .","The canonical map from $ { SL}/{\\mbox{$Ker\\,(C_\\alpha )_{SL}$}} $ onto $ {SL}/{\\mbox{$(G_\\alpha )_{SL}$}} $ will be denoted by r, and is given by $r\\left(\\rho (A)(z_0)\\right)=A {\\mbox{$(G_\\alpha )_{SL}$}} .$ Each $f\\in {\\cal D}$ thus becomes a function on ${\\cal B},$ homogeneous of degree $-1$ under ordinary multiplication by elements of $S$ .","The functions having these characteristics, will be called S-homogeneous of degree -1.","The $S$ -homogeneous of degree T functions are defined in a similar way.","Let us denote by ${\\cal C}$ the complex vector space composed by the S-homogeneous of degree -1 functions on ${\\cal B}$ .","Now, $W$ is $( H(2) \\times {\\cal B} ) \\times _S \\,{\\bf C}$ , and $\\eta $ maps $[(H,z),c]$ onto $(H,r(z)).$ The sections of $\\eta $ corresponding to the functions having the form of $\\phi _f$ in (REF ) are as follows.","Let $f$ be a $S$ -homogeneous of degree -1 function.","The corresponding section, $\\sigma ,$ maps $(H,m)$ to $\\sigma (H,m)=[(H,z),\\phi _f(H,z)]$ where $z$ is arbitrary in $r^{-1}(m).$ We define a map, $\\chi $ , from W into $ H(2) \\times ( SL/(G_\\alpha )_{SL}) \\times \\ L $ , by sending $[(H,z ), c]_S \\in ( H(2) \\times {\\cal B}) ) \\times _S \\,{\\bf C}$ to $ (H,\\ r(z), cz)$ .","The map $\\chi $ is injective .","In fact the relation $\\chi ([(H,z),c]_S)=\\chi ([(H^\\prime ,z^\\prime ),c^\\prime ]_S )$ is equivalent to $(H,r(z),cz)=(H^\\prime ,r(z^\\prime ),c^\\prime z^\\prime )$ so that there exist $e^{i\\gamma }\\in S$ such that $H^\\prime =H,$ $z^\\prime =e^{i\\gamma } z,$ and $c^\\prime z^\\prime =cz$ .","Thus $[(H^\\prime ,z^\\prime ),c^\\prime ]_S=[(H,ze^{i\\gamma } ),ce^{-i\\gamma } ]_S=[(H,z),c]_S.$ The fiber of $W$ on $(H,m)$ is $\\lbrace [(H,z ),c]: c\\in {\\bf C}\\rbrace $ , where $z$ is any fixed element in $r^{-1}(m).$ Its image under $\\chi $ is composed by the $(H,m,y)$ such that $y$ is in the one dimensional subspace of $L$ generated by $z.$ We have thus inmersed our, in general, non trivial bundle, in a trivial one with fiber $L$ .","This enable us to identify sections of $\\eta $ with functions with values in $L,$ as follows.","The section $\\sigma $ of $\\eta $ , that corresponds to $f$ can be identifed with $\\chi \\circ \\sigma (H,m) = \\left(H,m,\\phi _f(z)z\\right),$ where $z$ is arbitrary in $r^{-1}(m).$ The right hand side in the preceeding equation is completely determined by its third component: $\\psi _f(H,m)= \\phi _f(z)z=f(z)\\,e^{i\\pi \\,Tr(P(m )\\,\\varepsilon \\, \\overline{H}\\, \\varepsilon )}z.$ The function $\\psi _f,$ with $f$ $S$ -homogeneous of degree -1, will be called Prewave Function associated to $f$ .","The complex vector space composed by the Prewave Functions is denoted by PW, so that the map $\\psi :f\\in {\\cal C}\\ \\rightarrow \\psi _f\\in {\\cal PW}$ is an isomorphism.","The composition of any Prewave Function with $\\iota _4$ is a function in State Space that can be considered as giving an amplitude of probability for each state.","As said in the Introduction, we obtain from this Prewave Function a Wave Function, defined on space-time points ($i.e.$ hermitian matrices), by adding up, for each $H \\in H(2),$ the amplitudes of probability corresponding to the states $(H,\\beta )$ where $\\beta $ is a movement whose portrait in space-time contains $H$ .","According to (REF ), we see that these states are the elements of the set $\\lbrace (H,Ad^*_{(B,H)}\\alpha : B\\in SL\\rbrace ,$ that is identified by $i$ ($c.f.$ (REF )) with $\\lbrace H\\rbrace \\times {\\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1}},$ whose image by $\\iota _4$ is $\\lbrace H\\rbrace \\times \\frac{SL}{(G_\\alpha )_{SL}}.$ Then to any given Prewave Function, $\\psi _f$ , we associate a Wave Function, $\\widetilde{ \\psi _f}$ , as follows $\\widetilde{\\psi _f}(H)=\\int _{{SL}/(G_\\alpha )_{SL}} {\\mbox{$\\psi $}}_f(H,\\,m)\\ {\\mbox{$\\omega $}}_m$ where $m$ is a generic element in $ { SL}/({\\mbox{$G_\\alpha $}})_{SL},$ and $\\omega $ is a volume element on $ { SL}/({\\mbox{$G_\\alpha $}})_{SL},$ left invariant by the canonical action of $SL$ on $ { SL}/({\\mbox{$G_\\alpha $}})_{SL}.$ This means that, if we denote by $d^{\\prime }_A$ the diffeomorphism of $SL/(G_\\alpha )_{SL}$ defined by sending $B\\,(G_\\alpha )_{SL}$ to $AB\\,(G_\\alpha )_{SL},$ then $(d^{\\prime }_A)^*\\omega =\\omega .$ In the following, we assume that such a invariant volume element is given.","This definition of Wave Functions, forces us to do a restriction on the class of the functions to be considered: it is necessary that the integral exists.","In what follows I consider only Quantum States corresponding to the functions in ${\\cal C}$ that are continuous with compact support.","The complex vector space composed by these Wave Functions is denoted by ${\\cal WF}.$ Of course, there are other possible conditions that can be imposed on $f$ in order to assure integrability.","Also, if one obtains for some choice a Prehilbert space, one can consider its completed, in order to have a Hilbert Space.","But in this paper I prefer to maintain the \"continuous with compact support\" condition.","The Wave Functions are another form of description of Quantum States, in fact it is the most usual in Quantum Mechanics.","In the following I change the notation in such a way that, ${\\cal C}$ stands for the complex vector space composed by the $S$ -homogeneous of degree -1 functions on ${\\cal B}$ that also are continuous with compact support, and ${\\cal PW},\\ {\\cal V},\\ \\widetilde{\\cal V},$ the corresponding isomorphic spaces.","Remark 7.1 Now, let us assume that we know a section of the map $r,$ defined an open set, $D,$ in ${SL/(G_\\alpha )_{SL}}$ , $\\sigma : D \\longrightarrow {\\cal B}.$ In this case, we can associate a prewave function, and thus a wave function, to each $C^\\infty $ function, with compact support contained in $D,$ as follows.","Let $f_0$ be a $C^\\infty $ function, with compact support contained in $D$ .","We define a function on $\\cal B$ , $f$ , by means of $f(z)={\\left\\lbrace \\begin{array}{ll}t^{-1} f_0(r(z))& {\\rm if}\\ z\\in r^{-1}(U)\\ {\\rm where}\\ t \\ {\\rm is\\ such\\ that}\\ z= t \\sigma (r(z))\\\\0& {\\rm if}\\ z\\notin r^{-1}(U)\\end{array}\\right.","}$ This function is $S$ -homogeneous of degree -1 since, for all $e^{i a}\\in S,\\ z\\in {r^{-1}(U)},$ we have $f(e^{i a}z)=t^{\\prime -1} f_0(r(e^{i a}z)),$ where $t^{\\prime }$ is such that $e^{i a}z=t^{\\prime } \\sigma (r(e^{i a}z))=t^{\\prime } \\sigma (r(z))=t^{\\prime } t^{-1} z,$ so that $e^{i a}=t^{\\prime } t^{-1}$ and $f(e^{i a}z)=e^{-i a}t^{-1} f_0(r(e^{i a}z))=e^{-i a}t^{-1} f_0(r(z))=e^{-i a}f(z).$ If $z\\notin {r^{-1}(U)}$ , $f(e^{i a}z)=0=f(z)=e^{-i a}f(z)$ .","The Prewave Function $\\psi _f$ then is $\\psi _f(H,\\ m)={\\left\\lbrace \\begin{array}{ll}f_0(m)\\,e^{i\\pi Tr(P(m)\\,\\varepsilon \\overline{H} \\varepsilon )}\\ \\sigma (m) & \\mathrm {if}\\ m\\in U \\\\0& \\mathrm {if}\\ m\\notin U\\end{array}\\right.","}$ and we have an associated Wave Function, $\\tilde{\\psi }_f.$ This expression contains no reference to any homogeneous function and suggest directly the usual form of Wave Functions."],[" Representation on $S$ -homogeneous functions of degree -1 ","If $(C_\\alpha )_{SL}={\\bf 1},$ , the $S$ -homogeneous functions of degree -1 are simply functions on $SL/(G_\\alpha )_{SL}.$ In this case we denote by $\\cal C$ the complex vector space composed by the continuous functions on $SL/(G_\\alpha )_{SL}$ with compact support.","For all $B\\in SL$ we denote by $d^\\prime _B$ the canonical diffeomorphism of $SL/(G_\\alpha )_{SL}$ given by $d^\\prime _B(A (G_\\alpha )_{SL})=BA (G_\\alpha )_{SL}).$ Then, we define a representation, $\\delta ,$ of $SL$ on $\\cal C$ by $\\delta (f)=f\\circ d^\\prime _{A^{-1}}.$ In $\\cal C$ we also define an hermitian product by $\\langle f,\\, f^\\prime \\rangle = \\int _{{SL}/(G_\\alpha )_{SL}} \\overline{f} f^\\prime \\ {\\mbox{$\\omega $}} ,$ where $\\omega $ is an invariant volume element on $SL/(G_\\alpha )_{SL}.$ With this inner product, $\\cal C$ becomes a prehilbert space.","The invariance of $\\omega $ enable us to write $\\langle \\delta (A)(f),\\, \\delta (A)(f^\\prime ) \\rangle =\\langle f,\\, f^\\prime \\rangle $ so that the representation $\\delta $ is unitary.","In case $(C_\\alpha )_{SL}\\ne {\\bf 1},$ , we assume that we have a trivialization, $(\\rho ,\\ {L},\\ z_0),$ and we define ${\\cal B}$ and ${\\cal C}$ as in the preceeding section.","Of course, also in case $(C_\\alpha )_{SL}={\\bf 1},$ we can have a trvialization and thus apply all that follows.","In ${\\cal C}$ we define a hermitian product: $\\langle f,\\, f^\\prime \\rangle = \\int _{{SL}/(G_\\alpha )_{SL}} \\overline{f} f^\\prime \\ {\\mbox{$\\omega $}} , $ where by $ \\overline{f} f^\\prime $ we means the function defined on $SL/(G_\\alpha )_{SL},$ by $\\overline{f} f^\\prime (m)=\\overline{f}(z) f^\\prime (z)$ for all $m\\in SL/(G_\\alpha )_{SL},$ where $z$ is arbitrary in $r^{-1}(m),$ and $\\omega $ is an invariant volume element on $SL/(G_\\alpha )_{SL}.$ Also in this case, $\\cal C$ becomes a prehilbert space.","A representation, $\\delta ,$ of $SL$ on $\\cal C,$ is defined by $\\delta (A): f\\in {\\cal C} \\longrightarrow f\\circ \\rho (A^{-1})\\in {\\cal C}.$ for all $A\\in SL.$ If $A\\in SL$ we have $\\langle \\delta (A)(f),\\, \\delta (A)(f^\\prime ) \\rangle = \\int _{{ SL}/(G_\\alpha )_{SL}} \\left((\\overline{f} f^\\prime )\\circ d^{\\prime }_{A^{-1}} \\right)\\ {\\mbox{$\\omega $}}$ so that the invariance of $\\omega $ enable us to write $\\langle \\delta (A)(f),\\, \\delta (A)(f^\\prime ) \\rangle =\\langle f,\\, f^\\prime \\rangle $ We see that the representation $\\delta $ is, also in this case, unitary.","Also we have a representation, $\\delta ^\\prime ,$ of $SL \\oplus H(2)$ on $ {\\cal C}$ defined by: $(\\delta ^\\prime (A,H)\\cdot f)(z)= f( \\rho (A^{-1})\\cdot z) Exp\\left(-i\\pi Tr\\left(\\,P( r(z)) \\varepsilon \\overline{H} \\varepsilon \\right)\\right),$ where $(A,H)\\in SL \\oplus H(2),\\ f\\in {\\cal C},$ and $z\\in {\\cal B}.$ To prove that $\\delta ^\\prime $ is a representation , notice that $r$ is equivariant i.e.","$r(\\rho (A)\\cdot z)=d^{\\prime }_A(r(z))$ for all $A\\in SL.$ Formula (REF ), leads to $P(d^{\\prime }_B(A(G_\\alpha )_{SL}))=P(BA(G_\\alpha )_{SL})=-BAkA^*B^*=BP(A(G_\\alpha )_{SL})B^*$ so that $P(r(\\rho (A)\\cdot z))=P(d^{\\prime }_A(r(z)))=AP(r(z))A^*.$ For all $A\\in SL,$ $A\\, \\varepsilon = \\varepsilon \\,{}^t\\hspace{0.0pt}A^{-1}$ Now, we can prove that $\\delta ^\\prime $ is a representation as follows.","If $(B,K),(A,H)\\in SL \\oplus H(2),\\ z\\in {\\cal B},$ then ${\\begin{array}{c}\\delta ^\\prime (B,K)\\cdot \\left(\\left(\\delta ^\\prime (A,H)\\cdot f\\right)\\right)(z)=\\left(\\delta ^\\prime (A,H)\\cdot f\\right)(\\rho (B^{-1})\\cdot z) \\\\Exp\\left(-i\\pi Tr[P(r( z))\\varepsilon \\overline{K} \\varepsilon ]\\right)=f(\\rho (A^{-1})\\cdot (\\rho (B^{-1})\\cdot z))\\\\Exp\\left(-i\\pi Tr[P(r(\\rho (B^{-1})\\cdot z))\\varepsilon \\overline{H} \\varepsilon + P(r( z))\\varepsilon \\overline{K} \\varepsilon ] \\right)=\\\\=f(\\rho ((BA)^{-1})\\cdot z) Exp\\left(-i\\pi Tr[P(r(z))\\varepsilon \\overline{(B HB^* + K) }\\varepsilon ]\\right)=\\\\=\\left(\\delta ^\\prime \\left(\\left(B,K\\right)\\left(A,H\\right)\\right)\\cdot f\\right)\\left( z\\right)\\end{array}}$ The infinitesimal generator of $\\delta $ associated to $a\\in sl(2,\\bf C)$ is the linear map from $\\cal C$ into itself, $d\\,\\delta (a)$ given by $(d\\,\\delta (a) \\cdot f) (z)=\\left(\\frac{d}{dt}\\right)_{t=0} \\left(\\delta \\left(e^{t\\,a}\\right)\\cdot f \\right) \\left(z\\right).$ Then $(d\\,\\delta (a) \\cdot f) (z)=\\left(\\frac{d}{dt}\\right)_{t=0} \\left(f \\left(\\rho \\left(e^{-t\\,a}\\right) \\cdot z\\right) \\right),$ and thus we see that $d\\,\\delta (a)$ acts on $f$ as the vector field, $X_a,$ infinitesimal generator of the action on $\\cal B,$ defined by $A*z=\\rho (A) \\cdot z,$ associated to $a$ .","We can give a more explicit form of $X_a$ as follows.","Let us fix a basis, $\\beta =\\lbrace e_1, \\dots ,e_q\\rbrace ,$ of $L.$ By means of $\\beta $ we identify $L$ with ${\\bf C}^q$ .","Let us denote by ${}^t(z^1,\\dots ,z^q)$ the matrix of $z\\in L$ in the basis $\\beta $ and $(d\\rho (a))_i^j,$ the element in the file $j$ column $i$ of the matrix of $(d\\rho (a))$ in the basis $\\beta .$ We denote by $\\beta ^*=\\lbrace w^1, \\dots ,w^q\\rbrace $ the system of (complex) coordinates associated to $\\beta $ (i.e.","$\\beta ^*$ is the dual basis of $\\beta .$ ) If $w^j_x$ (resp $w^j_y$ ) is the real (resp.","imaginary) part of $w^j,$ it is usually writen $\\frac{\\partial }{\\partial w^j}=\\frac{1}{2}\\left(\\frac{\\partial }{\\partial w^j_x}-i\\ \\frac{\\partial }{\\partial w^j_y}\\right),$ $\\frac{\\partial }{\\partial \\overline{w}^j}=\\frac{1}{2}\\left(\\frac{\\partial }{\\partial w^j_x}+i\\ \\frac{\\partial }{\\partial w^j_y}\\right).$ Thus, when $f$ is a complex valued differentiable function on an open subset, M, of ${\\bf C}^q$ , and $v(t)$ a differentiable map from an open neigbourhood of 0 in ${\\bf R}$ into M, we have, using summation convention ${\\begin{array}{c}\\left( \\frac{d}{dt}\\right)_0 (f\\circ v) =\\left(\\frac{\\partial \\, f}{\\partial w^j_x}\\right)_{v(0)}\\, (w_x^j(v(t)))^{\\prime }(0)+\\left(\\frac{\\partial \\, f}{\\partial w^j_y}\\right)_{v(0)}\\, (w_y^j(v(t)))^{\\prime }(0)=\\\\=\\left(\\frac{\\partial \\, f}{\\partial w^j}\\right)_{v(0)}\\, (w^j(v(t)))^{\\prime }(0)+\\left(\\frac{\\partial \\, f}{\\partial \\overline{w}^j}\\right)_{v(0)}\\, (\\overline{w}^j(v(t)))^{\\prime }(0).\\end{array}}$ Since $\\left(\\frac{d}{dt}\\right)_{t=0} \\left(f \\left(\\rho \\left(e^{-t\\,a}\\right) \\cdot z\\right) \\right)=\\left(\\frac{d}{dt}\\right)_{t=0} \\left(f \\left(\\left(e^{-t\\,d\\rho \\left(a\\right)}\\right) \\cdot z\\right) \\right)$ it follows that ${\\begin{array}{c}\\left(X_a)_z(f) \\right)=\\left(\\frac{\\partial \\, f}{\\partial w^j}\\right)_{z}\\, (-(d\\rho (a))^j_i\\,z^i)+\\left(\\frac{\\partial \\, f}{\\partial \\overline{w}^j}\\right)_{z}\\, ({-(\\overline{d\\rho (a)})} ^j_i\\,\\overline{z}^i)\\end{array}}$ so that an extension of $X_a$ from $\\cal B$ to $\\mathbb {C}^q$ is $X_a=-\\left(w^i \\,(d\\rho (a))^j_i\\,\\frac{\\partial }{\\partial w^j}+\\overline{w}^i \\,(\\overline{d\\rho (a)})^j_i\\,\\frac{\\partial }{\\partial \\overline{w}^j} \\right).$ If $(C_\\alpha )_{SL}={\\bf 1},$ , and do not use a trivialization, we have similar results, but $X_a$ is the infinitesimal generator of the canonical action of $SL$ on $SL/(G_\\alpha )_{SL},$ associated to $a.$ Thus, in all cases $d\\,\\delta (a) \\cdot f=X_{a}(f),$ or simply, as linear maps on $\\cal C$ , $d\\,\\delta (a) =X_{a}.$ On the other hand, the infinitesimal generator of $\\delta ^{\\prime }$ associated to $(a,h)\\ \\ \\in sl(2,\\bf C)\\oplus H(2)$ is the endomorphism, $d\\,\\delta ^{\\prime }(a,h)$ of $\\cal C$ given by $(d\\,\\delta ^{\\prime }(a,h) \\cdot f) (z)=\\left(\\frac{d}{dt}\\right)_{t=0} \\left(\\delta ^{\\prime }\\left(Exp({t\\,(a,h))}\\right)\\cdot f \\right) \\left(z\\right),$ but the exponential map in $SL\\oplus H(2)$ is given by (REF ), so that $(d\\,\\delta ^{\\prime }(a,h) \\cdot f) (z)=\\left(\\frac{d}{dt}\\right)_{t=0} \\left(\\delta ^{\\prime }\\left(\\left(e^{ta},\\int _0^t e^{sa}\\,h\\, e^{sa^*}\\,ds \\right)\\right)\\cdot f \\right) \\left(z\\right),$ and a short computation thus leads to ${\\begin{array}{c}d\\,\\delta ^{\\prime }(a,h) \\cdot f= X_a(f)-i\\pi Tr[P(r(\\cdot ))\\varepsilon \\overline{h}\\varepsilon ]\\ f=\\\\=X_a(f)+2\\pi i \\langle P(r(\\cdot )),h \\rangle \\ f,\\end{array}}$ where I have used the same symbol for any hermitian matrix and the corresponding element in ${\\bf R}^4,$ and $\\langle \\ ,\\ \\rangle $ is Minkowski product.","Also, we can write $d\\,\\delta ^{\\prime }(a,h)=X_a+2\\pi i \\langle P(r(\\cdot )),h \\rangle .$ where $X_a$ is the endomorphism given by the vectorfield (REF ) , and $2\\pi i\\langle P(r(\\cdot )),h \\rangle $ is the endomorphism given by ordinary multiplication by this function.","Instead of the infinitesimal generators $d\\,\\delta ^{\\prime }(a,h),$ we can use $(a,h)^{\\theta }\\stackrel{\\mathrm {d}ef}{ =}\\frac{1}{2 \\pi i}\\ d\\,\\delta ^{\\prime }(a,h) ,$ so that $(a,h)^{\\theta } =\\frac{1}{2 \\pi i}\\,X_a+ \\langle P(r(\\cdot )),h \\rangle .$ The endomorphism $(a,h)^{\\theta },$ is hermitian for our hermitian product and is the Quantum Operator associated to the Dynamical Variable (a,h), when Quantum States are represented by $S$ -homogeneous functions.","When Quantum States are represented by Wave Functions, the Quantum Operators acquires its usual form in Quantum Mechanics (c.f.","(REF )).","In the particular case of the Linear Momentum (c.f.","(REF )), one sees that $(P^k)^{\\theta }\\cdot f=(P^k \\circ r)\\,f,\\ \\ k=1,\\,2,\\,3,\\,4.$ where the $P^k$ in the left hand side are the components of Linear momentum as a dynamical variable, and the ones in the right hand side, the components of Linear momentum considered as a function on $SL/(G_\\alpha )_{SL},$ given by (REF ).","In what concerns to the Quantum Operators corresponding to Angular Momentum we have $(l^j)^{\\theta }= \\frac{1}{4\\pi i} \\,X_{i\\,\\sigma _j}, \\ \\ j=1,\\,2,\\,3.$ $(g^j)^{\\theta }=\\frac{1}{4\\pi i} \\,X_{\\sigma _j}, \\ \\ j=1,\\,2,\\,3.$ Of course, the representations on ${\\cal C}$ lead to equivalent representations on the isomorphic vector spaces $ {\\cal V},\\ \\widetilde{\\cal V},$ and ${\\cal PW}$ .","We do that in detail in ${\\cal PW}$ and ${\\cal V}$ as follows:"],["Representation on Prewave Functions.","In ${\\cal PW}$ we define an hermitian product by $\\langle \\psi _f ,\\psi _{f^{\\prime }} \\rangle \\stackrel{def}{=}\\langle f ,{f^{\\prime }} \\rangle ,$ and thus becomes a prehilbert space, isomorphic as inner product spaces to ${\\cal C.}$ The hermitian product can be given in terms of the prewave functions themselves, as follows.","Let $\\beta $ be a sesquilinear form on $L,$ nonvanishing on ${\\cal B}$ .","We define $ \\psi _f\\, \\beta \\,\\psi _{f^\\prime } \\,: m \\in SL/(G_\\alpha )_{SL} \\mapsto \\frac{\\beta ( \\psi _f ( H,\\,m),\\psi _{f^\\prime } ( H,\\,m ))}{\\beta (z,\\ z )} $ where $z$ is arbitrary in ${\\bf r}^{-1}(m)$ and $H$ is arbitrary in $H(2)$ .","Thus $\\langle \\psi _f,\\ \\psi _{f^\\prime } \\rangle =\\int _{{SL}/(G_\\alpha )_{SL}}\\psi _f\\, \\beta \\,\\psi _{f^\\prime }\\ \\omega .$ The representation of $G$ on ${\\cal PW}$ equivalent to $\\delta ^\\prime $ under $\\psi $ is given by $\\delta ^{pw}(A,H)\\cdot \\psi _f=\\psi _{\\delta ^\\prime (A,H) \\cdot f}.$ The infinitesimal generator associated to $(a,h)\\ \\ \\in sl(2,\\bf C)\\oplus H(2)$ is $d\\delta ^{pw}(a,h)\\cdot \\psi _f=\\psi _{d\\delta ^{\\prime }(a,h)\\cdot f}.$ The Quantum Operators for Prewave Functions must be defined as $(a,h)^{\\upsilon }\\stackrel{\\mathrm {d}ef}{=}\\ \\frac{1}{2\\pi i}\\,d\\delta ^{pw}(a,h),$ and we have $(a,h)^{\\upsilon }\\cdot \\psi _f{=}\\ \\psi _{(a,h)^{\\theta }\\cdot f}.$ On the other hand, we have a natural action on $H(2)\\times SL/(G_\\alpha )_{SL}$ defined by $(B,K)* (H,A\\,(G_\\alpha )_{SL})=(BHB^*+K,BA (G_\\alpha )_{SL}).$ Thus we define an, a priori different, representation, $\\delta _{pw},$ on ${\\cal PW}$ by means of $\\delta _{pw}(A,H) \\cdot \\psi _f =\\rho (A) \\circ \\psi _f \\circ \\left((A,H)*\\right)^{-1},$ i.e., for all $(K,m)\\in H(2)\\times SL/(G_\\alpha )_{SL}$ $\\left(\\delta _{pw}(A,H) \\cdot \\psi _f \\right)(K,m) =\\rho (A) \\cdot \\psi _f \\left((A,H)^{-1}*(K,m)\\right).$ Using (REF ) and (REF ) one can see that $\\delta _{pw}=\\delta ^{pw}$ as follows ${\\begin{array}{c}\\left(\\delta ^{pw}(A,H) \\cdot \\psi _f \\right)(K,m)=\\psi _{\\delta ^\\prime \\left( A,H\\right)\\cdot f}(K,m)= f(\\rho (A^{-1})\\cdot z) \\\\Exp\\left(- i\\pi Tr [ P( m))\\varepsilon \\overline{H} \\varepsilon ]\\right)Exp\\left( i\\pi Tr [ P( m)\\varepsilon \\overline{K} \\varepsilon ]\\right) z=(*)\\end{array}}$ where $z\\in r^{-1}(m).$ If we denote $ z^{\\prime }=\\rho (A^{-1})\\cdot z ,$ we have ${\\begin{array}{c}(*)= f( z^{\\prime }) Exp\\left( i\\pi Tr [ P( r(\\rho (A)\\cdot z^{\\prime }))\\varepsilon \\overline{(K-H)} \\varepsilon ]\\right)( \\rho (A)\\cdot z^{\\prime }) =\\\\=f( z^{\\prime }) Exp\\left( i\\pi Tr [ P( r( z^{\\prime }))\\varepsilon \\overline{(A^{-1}(K-H)(A^{-1})^*)} \\varepsilon ]\\right) (\\rho (A)\\cdot z^{\\prime })=\\\\=\\rho (A) \\psi _f((A,H)^{-1}*(K,m) =\\left(\\delta _{pw}(A,H) \\cdot \\psi _f \\right)(K,m).\\end{array}}$"],["Representation on Wave Functions.","Now, let us consider ${\\cal WF},$ the complex vector space composed by the Wave Functions $\\widetilde{ \\psi _f}$ such that $f\\in {\\cal C}.$ A \"translation\" of the preceeding representations of $SL\\oplus H(2)$ to ${\\cal WF},$ is given by $\\delta _{w}(A,H) \\cdot \\widetilde{\\psi _f} = \\lbrace \\delta _{pw}(A,H) \\cdot \\psi _{ f}\\rbrace {\\widetilde{}}= \\lbrace \\psi _{\\delta ^{\\prime }(A,H)\\cdot f}\\rbrace {\\widetilde{}},$ where $\\lbrace \\psi \\rbrace {\\widetilde{}}$ means $\\widetilde{\\psi }.$ Then $\\delta _{w}(A,H) \\cdot \\widetilde{\\psi _f}(K)&=& \\lbrace \\delta _{pw}(A,H) \\cdot \\psi _{ f}\\rbrace {\\widetilde{}}(K)\\\\&=& \\int _{SL/(G_\\alpha )_{SL}}\\rho (A) \\cdot \\psi _f((A,H)^{-1}*(K,m))\\omega _m=\\\\&=& \\rho (A) \\cdot \\int _{SL/(G_\\alpha )_{SL}} \\psi _f((A,H)^{-1}*K,d^{\\prime }_{A^{-1}}m))\\omega _m=\\\\&=& \\rho (A) \\cdot \\int _{SL/(G_\\alpha )_{SL}} \\psi _f((A,H)^{-1}*K,m))\\omega _m$ because of the invariance of $\\omega .$ Thus $\\left(\\delta _{w}(A,H) \\cdot \\widetilde{\\psi _f}\\right)(K)=\\rho (A) \\cdot \\widetilde{\\psi _f } \\left((A,H)^{-1}*K)\\right).$ Compare to (REF ).","Let us denote by $d\\delta _{w}\\cdot (a,h)$ the infinitesimal generator of the representation $\\delta _w$ associated to $ (a,h) \\in sl(2, C)\\oplus { H(2)},$ and $\\widehat{ (a,h)} \\stackrel{def}{=}\\frac{1}{ 2 \\pi i } (d\\delta _{w}\\cdot (a,h) ).$ The endomorphism $\\widehat{ (a,h)}$ of ${\\cal WF}$ is the Quantum Operator corresponding to the Dynamical Variable $(a,h)$ when the Quantum States are represented by Wave Functions.","A straightforward computation leads to the following expresions for the operators corresponding to Linear and Angular Momentum $\\widehat{ P^k} \\cdot \\widetilde{ \\psi _f }&=&\\frac{1}{2\\pi i}\\frac{\\partial }{\\partial x^k} \\ \\widetilde{ \\psi _f } \\nonumber \\\\\\widehat{ P^4} \\cdot \\widetilde{ \\psi _f } &=&\\frac{i}{2\\pi }\\frac{\\partial }{\\partial x^4} \\ \\widetilde{ \\psi _f } \\nonumber \\\\\\widehat{ l^k} \\cdot \\widetilde{ \\psi _f }&=&\\frac{1}{2\\pi i}\\left( d\\rho \\left( {\\frac{i\\sigma _k}{2}}\\right)+\\sum _{j,r=1}^3\\varepsilon _{kjr} x^j\\frac{\\partial }{\\partial x^r} \\right) \\ \\widetilde{ \\psi _f } \\\\\\widehat{ g^k} \\cdot \\widetilde{ \\psi _f } &=&\\frac{1}{2\\pi i}\\left( d\\rho \\left(\\frac{\\sigma _k}{2}\\right)- \\left( x^4\\frac{\\partial }{\\partial x^k}+ x^k\\frac{\\partial }{\\partial x^4}\\right)\\right)\\ \\widetilde{ \\psi _f } \\nonumber $ where $\\varepsilon _{ijk}$ are the components of an antisymmetric tensor such that $\\varepsilon _{123}=1$ ."],["Representation on Pseudotensorial Functions in the contact manifold.","Recall the isomorphism (REF ) $ \\nonumber \\Phi :f\\in {\\cal C} \\rightarrow \\Phi _f \\in {\\cal V}$ If $f\\in {\\cal C},$ we have $\\Phi _f((A,H)KerC_\\alpha )=\\,f(A\\,Ker(C_\\alpha )_{SL}) \\,e^{i\\pi \\,Tr(P (A\\,(G_\\alpha )_{SL}) \\,\\varepsilon \\, \\overline{H}\\,\\varepsilon )}.$ When $(C_{\\alpha })_{SL}={\\bf 1},$ we have $Ker(C_\\alpha )_{SL}=(G_\\alpha )_{SL},$ so that $\\Phi _f((A,H)KerC_\\alpha )=\\,f(A\\,(G_\\alpha )_{SL}) \\,e^{i\\pi \\,Tr(P (A\\,(G_\\alpha )_{SL}) \\,\\varepsilon \\, \\overline{H}\\, \\varepsilon )}.$ In the case $(C_{\\alpha })_{SL}\\ne {\\bf 1},$ we assume the existence of a trivialization, $(L,\\,\\rho ,\\,z_0),$ and we identify $SL/Ker\\,(C_\\alpha )_{SL}$ with ${\\cal B}.$ Then the pseudotensorial function $\\Phi _f$ becomes $\\Phi _f\\left(\\left(A,H\\right)\\,Ker C_\\alpha \\right)=\\,f(\\rho (A)\\cdot z_0) \\,e^{i\\pi \\,Tr( P(r (\\rho (A)\\cdot z_0) )\\,\\varepsilon \\, \\overline{H}\\,\\varepsilon )}.$ A natural representation of $G$ on $V$ is the $\\delta ^c$ given by $\\delta ^c(A,H) \\cdot \\Phi _f=\\Phi _f \\circ ((A,H)^{-1}*)$ where $*$ is the canonical action on $G/Ker\\, C_\\alpha .$ Let us prove that $\\delta ^c(A,H) \\cdot \\Phi _f=\\Phi _{\\delta ^{\\prime }(A,H) \\cdot f}$ $i.e.$ that $\\Phi $ is equivariant for the representations $\\delta ^{\\prime }$ in ${\\cal C}$ and $\\delta ^c$ in ${\\cal V}$ .","In fact, we have ${\\begin{array}{c}\\left( \\delta ^c(A,H) \\cdot \\Phi _f\\right) \\left((B,K)Ker\\, C_\\alpha \\right)=\\Phi _f\\left( A^{-1} B, A^{-1}(K-H)(A^{-1})^* \\right)=\\\\=f\\left( \\rho (A^{-1})\\cdot (\\rho (B)\\cdot z_0)\\right)\\,e^{i\\pi \\,Tr(P(r (\\rho (A^{-1})\\cdot (\\rho (B)\\cdot z_0)) )\\,\\varepsilon \\, \\overline{ ( A^{-1}(K-H)(A^{-1})^*)}\\,\\varepsilon )}=\\\\=f\\left( \\rho (A^{-1})\\cdot (\\rho (B)\\cdot z_0)\\right)\\,e^{i\\pi \\,Tr(A^{-1}\\,P(\\rho (B)\\cdot z_0)(A^{-1})^* A^*\\varepsilon \\, \\overline{ (K-H)}\\,\\varepsilon \\,A)}=\\\\= f\\left( \\rho (A^{-1})\\cdot (\\rho (B)\\cdot z_0)\\right)\\,e^{i\\pi \\,Tr(P(\\rho (B)\\cdot z_0)\\,\\varepsilon \\, \\overline{ (K-H)}\\,\\varepsilon ) }=\\\\=\\left( \\delta ^{\\prime }(A,H)\\cdot f\\right)\\left(\\rho (B) \\cdot z_0 \\right)\\,e^{i\\pi \\,Tr(P(\\rho (B)\\cdot z_0)\\,\\varepsilon \\, \\overline{ K}\\,\\varepsilon }=\\\\=\\Phi _{\\delta ^{\\prime }(A,H)\\cdot f}\\left((B,K)Ker\\, C_\\alpha \\right).\\end{array}}$ As a particular consecuence, for all $(a,h)\\in \\underline{G},$ we have for the infinitesimal generators of $\\delta ^c$ $d\\delta ^c(a,h) \\cdot \\Phi _f=\\Phi _{d\\delta ^\\prime (a,h)\\cdot f}.$ Equation (REF ), tell us that the infinitesimal generator of the representation $\\delta ^c$ associated to $(a,h)\\in \\underline{G},$ acts on $\\Phi _f$ as the (vector field)infinitesimal generator of the action on $G/Ker\\,C_\\alpha ,$ what will be denoted by $X_{(a,h)}^c.$ If we denote by $(a,h)^c,$ the Quantum Operator $(a,h)^c \\stackrel{def}{=}\\frac{1}{2\\pi i}\\,d\\delta ^c(a,h)$ we have $(a,h)^c \\cdot \\Phi _f=\\Phi _{(a,h)^\\theta \\cdot f}.$ Thus, our results in subsection REF , on Quantum Operators in that case, gives us results on Quantum Operators in our present case."],["Quantizable forms in $ SL(2,\\mathbb {C}) \\oplus H(2)$ .","In [7] I give a classification of the coadjoint orbits of the group under consideration.","The orbits are divided into 9 Types, and a canonical representative of each orbit is given, according with its type.","Let $\\alpha =\\lbrace a,k\\rbrace $ be a nonzero element of the dual of the Lie algebra of $ SL \\oplus H(2),$ and denote by $W$ and $P$ the Pauli- Lubanski and Linear momentum respectively at $\\alpha $ (cf.","section ).","Figure 3 gives what type of coadjoint orbit is the one of $\\alpha ,$ according with the values of $W$ and $P.$ Figure: Types of coadjoint orbitsWhen one knows the type of $\\alpha ,$ Figure 4 gives us the canonical representative of the orbit of $\\alpha .$ Figure: Canonical Representatives.The $\\mathbb {R} $ -quantizable orbits are all of the types 3, 6, 8, 9 and these of type 5 corresponding to the case $|W|=0$ Quantizable but not $\\bf R\\rm $-quantizable are the orbits whose canonical representatives are: $\\left\\lbrace \\frac{iT}{8 \\pi } \\left(\\begin{array}{cc} 1 &0 \\\\ 0 &-1 \\end{array} \\right) ,0\\right\\rbrace \\ \\ (type\\ 2),$ $\\left\\lbrace \\frac{i\\chi T}{8 \\pi }\\ \\left( \\begin{array}{cc}1 &0 \\\\0 &-1\\end{array} \\right),-sign (Tr\\left( P \\right) )\\left( \\begin{array}{cc}1 &0 \\\\0 &0\\end{array} \\right)\\right\\rbrace \\ \\ (type\\ 4), $ $\\left\\lbrace \\frac{iT}{8 \\pi }\\left( \\begin{array}{cc}1 &0 \\\\0 &-1\\end{array} \\right),-sign (Tr\\left( P \\right) )\\sqrt{|P|}\\ I \\right\\rbrace \\ \\ (type\\ 5), $ $ \\left\\lbrace \\frac{i\\chi T}{8 \\pi }\\left( \\begin{array}{cc}1 &0 \\\\0 &-1\\end{array} \\right),\\sqrt{-|P|}\\ \\left( \\begin{array}{cc}1 &0 \\\\0 &-1\\end{array} \\right)\\right\\rbrace \\ \\ (type\\ 7).$ where, in all cases, $T \\in \\bf Z^+\\rm , \\chi =1,-1.$ The concrete particles we study in this paper are those corresponding to Type 5 (massive particles) and Type 4 (massless particles such that the Pauly-Lubanski fourvector is proportional to Impulsion-Energy).","Results concerning other types of particles will be published elsewhere."],["Guide to the explicit construction of Wave Functions.","Let $\\alpha $ be a quantizable element of $\\underline{G}^*.$ To determine the explicit form of the Wave Functions of the corresponding free particles, one can proceed as follows 1.","Evaluate the isotropy subgroup at $\\alpha ,$ of the coadjoint representation, $G_{\\alpha }$ .","The coadjoint orbit of $\\alpha ,$ that is the main symplectic manifold associated to $\\alpha ,$ is identified to the homogeneous space $G/G_{\\alpha }.$ 2.","Find a surjective homomorphism, $C_{\\alpha },$ from $G_{\\alpha }$ onto the unit circle S$^1,$ whose differential is $\\alpha .$ 3.","Evaluate $(G_{\\alpha })_{SL},$ that is composed by the $g\\in SL$ such that $(g,h)\\in G_{\\alpha }$ for some $h\\in H(2).$ 4.","Determine $(C_{\\alpha })_{SL},$ defined in (REF ), and $S\\equiv (C_{\\alpha })_{SL}\\left((G_{\\alpha })_{SL}\\right).$ 5.","Find an action of $SL(2,{\\mathbb {C}})$ on a manifold, such that the isotropy subgroup at some point, $p$ , is $(G_{\\alpha })_{SL}.$ Identify $SL/(G_{\\alpha })_{SL}$ with the orbit of $p,$ ${\\cal K}$ .","6.","Determine a volume element on ${\\cal K},$ $\\omega ,$ invariant under the action of $SL$ .","7.","If $S=\\lbrace 1\\rbrace ,$ the Prewave Functions are the $\\psi _f(H,K)= f(K)\\,e^{i\\pi Tr(P(K )\\varepsilon \\, \\overline{H}\\, \\varepsilon ) }$ where $(H,K)\\in H(2)\\times {\\cal K},$ $f$ is a function on ${\\cal K},$ and $P$ is the dynamical variable linear momentum, on ${\\cal K},$ given by (REF ).","8.","If $S\\ne \\lbrace 1\\rbrace ,$ find a Trivialization of $ C_\\alpha $ ($c.f.$ (REF )).","Let ${\\cal B}$ the orbit of $z_0$ , identified to $SL/Ker(C_{\\alpha })_{SL},$ $r:{\\cal B} \\rightarrow {\\cal K}$ the canonical map from $SL/Ker(C_{\\alpha })_{SL})$ onto $SL/(G_{\\alpha })_{SL}$ .","The Prewave Functions are given by ($c.f.$ (REF )) $\\psi _f(H,K)= f(z)\\,e^{i\\pi Tr(P(K )\\varepsilon \\, \\overline{H}\\, \\varepsilon ) }z$ where $(H,K)\\in H(2)\\times {\\cal K},$ $z\\in r^{-1}(K),$ and $f$ is a function on ${\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers.","9.","The Prewave Functions corresponding to continuous with compact support $f$ define Wave Functions by means of ($c.f.$ (REF )) $\\widetilde{\\psi _f}(H)=\\int _{\\cal K} {\\mbox{$\\psi $}}_f(H,\\cdot )\\ {\\mbox{$\\omega $}} .$"],["Wave Functions. Klein-Gordon equation.","Let us consider a particle whose movement space is the coadjoint orbit of ${\\alpha _o}=\\left\\lbrace 0,\\ \\eta \\,m\\,I\\right\\rbrace ,\\ \\ \\ m \\in {\\mathbb {R}}^+, \\eta =\\pm 1.$ This orbit is a $\\mathbb {R} \\rm $ -quantizable orbit of the type 5, in the notation of section .","The number $(-\\eta )$ is the sign of energy i.e.","the sign of the value of the dynamical variable $P^4$ ($c.f.$ section ) at any point of the orbit.","According with the usual interpretation of the determinant of momentum-energy, $\\vert P \\vert ,$ as mass square, the number $m$ must be interpreted as being the mass of the particle.","By direct computation, one sees that ${\\mbox{$G_{\\alpha _o}$}} = \\lbrace \\ (A, \\ hI ): \\ A\\in SU(2),\\ h\\in \\mathbb {R} \\rbrace .$ The unique homomorphism onto ${\\mathbb {R}}$ whose differential is ${\\alpha _o}$ is given by $ C^\\prime _{\\alpha _o} (A ,\\ hI)=-\\eta m h .$ The unique homomorphism onto ${\\bf S}^1$ whose differential is $\\alpha _o$ is given by $C_{\\alpha _o} (A ,\\ hI)=e^{-2\\pi i\\eta m h}.$ Then we have ${\\begin{array}{c}Ker\\,C^\\prime _{\\alpha _o}=\\lbrace (A,0):A\\in SU(2)\\rbrace =SU(2)\\oplus \\lbrace 0\\rbrace ,\\\\Ker\\,C_{\\alpha _o}=\\lbrace (A,\\,\\frac{\\eta N}{m}\\, I):A\\in SU(2),\\ N \\in \\mathbb {Z}\\rbrace ,\\\\{\\mbox{$\\widetilde{C}_{\\alpha _o}$}} (A ,\\ H)=e^{-\\pi i\\eta m TrH} ,\\\\{\\mbox{$(G_{\\alpha _o})_{SL}$}}=SU(2) ,\\\\{\\mbox{$(C_{\\alpha _o})_{SL}$}}={\\bf 1 } ,\\\\SL_1 =SL_2= {\\mbox{$Ker\\,(C_{\\alpha _o})_{SL}$}}={\\mbox{$(G_{\\alpha _o} ) _{SL}$}},\\end{array}} $ Thus, in the commutative diagram of figure REF , section , the four spaces on the left are the same.","All of them represent State Space, $H(2)\\times \\frac{SL}{(G_{\\alpha _o})_{SL}\\cap SL_1}$ what in our present case becomes $H(2) \\times \\frac{SL}{SU(2)},$ and can be obviously identified to $\\frac{G}{SU(2) \\oplus \\lbrace 0\\rbrace },$ by means of the diffeomorphism $(H,A\\,SU(2))\\rightarrow (A,H)(SU(2) \\oplus \\lbrace 0\\rbrace ).$ Then, in this case, besides the canonical map from State Space onto Movement Space (c.f.","figure REF ), $\\nu _2: (H,ASU(2))\\in H(2) \\times \\frac{SL}{SU(2)} \\rightarrow (A,H)G_\\alpha \\in \\frac{G}{G_{\\alpha _o}}$ we have a natural map from State Space onto $G/Ker\\,C_{\\alpha _o}$ $\\nu _1:(A,H)(SU(2) \\oplus \\lbrace 0\\rbrace )\\in \\frac{G}{SU(2) \\oplus \\lbrace 0\\rbrace } \\rightarrow (A,H){Ker\\,C_{\\alpha _o}} \\in \\frac{G}{Ker\\,C_{\\alpha _o}}.$ The diagram of figure REF , becomes that of figure REF .","Figure: Diagram for Klein-Gordon particles.Let ${\\cal H}^m$ be the positive mass hyperboloid ${\\cal H}^m=\\lbrace H \\in H(2):\\ det\\, H\\,=\\,m^2,\\ Tr\\,H\\,>\\,0\\,\\rbrace .$ In ${\\cal H}^m$ we consider the action of $SL(2,C)$ given by $ A* H=A\\,H\\,A^*,$ for all $H\\in {\\cal H}^m$ and $A \\in SL(2,C).$ In these conditions we have $Tr(A\\,H\\,A^*)>0$ as a consecuence of the fact that $SL(2,C)$ is connected.","Since the group of restricted homogeneous Lorenz transformations is transitive on ${\\cal H}^m,$ so does $SL(2,\\mathbb {C}).$ The isotropy subgroup at $mI$ is $SU(2).$ Thus $SL/(G_{\\alpha _o})_{SL}$ is identified to ${\\cal H}^m,$ by means of the diffeomorphism $A\\, SU(2)\\in \\frac{SL}{SU(2)} \\longrightarrow m\\,A\\,A^* \\in {\\cal H}^m.$ If $K= m\\,A\\,A^* \\in {\\cal H}^m,$ then $K$ is identified to $A\\, SU(2)\\in \\frac{SL}{SU(2)},$ and the function P thus becomes $P(K)=P(ASU(2))=-A(\\eta \\,m\\,I)A^*=-\\eta K.$ The manifold ${\\cal H}^m$ has the global parametrization $\\varphi :(p^1,\\ p^2,\\ p^3)\\in {{\\mathbb {R}} ^3}\\ \\ \\mapsto \\ \\ h(p^1,\\ p^2,\\ p^3,\\ (m^2+ p^2)^{\\frac{1}{2}} )\\ \\in \\ {\\cal H}^m$ where $p^2= \\sum _{i=1}^3 (p^i)^2.$ In particular it is orientable.","On the other hand, the restriction of the pseudoriemannian metric defined on ${\\mathbb {R}}^4$ by Minkowski metric, to ${\\cal H}^m,$ is negative definite so that its opposite is a riemannian metric, that provides us with a canonical volume element, $\\nu $ .","A computation of the matrix of the riemannian metric in the chart $\\varphi $ leads to $\\nu =\\frac{dp^1 \\wedge dp^2 \\wedge dp^3}{\\sqrt{m^2+ p^2}}$ where the $p^i$ are the coordinates in the chart $\\varphi .$ The action on $H(2)$ defined as in (REF ) preserves Minkowski metric, so that the action on ${\\cal H}^m$ preserves $\\nu $ .","As a consequence, $\\nu $ is an invariant volume element under the action (REF ).","Since $(C_{\\alpha _o})_{SL}={\\bf 1}$ , $\\cal C$ is composed by functions on ${\\cal H}^m.$ The prewave functions have the form $\\psi _f:(H,\\ K) \\in H(2) \\times {\\cal H}^m\\ \\mapsto \\ \\ f\\left(K\\right) e^{-i\\pi \\eta Tr(K \\varepsilon \\overline{H} \\varepsilon ) }.$ where f is in $\\cal C.$ The corresponding wave function is $\\tilde{\\psi }_f(X)=\\int _{{\\cal H}^m}\\psi _f(h(X),\\,\\cdot \\,)\\ \\nu $ By direct computation one sees that the prewave and wave functions satisfy Klein-Gordon equation.","If $f^\\prime $ is another continuous with compact support function on ${\\cal H}^m$ , the hermitian product of the quantum states corresponding to $\\psi _f$ and $\\psi _{f^\\prime },$ defined in section REF , can be writen $\\langle \\psi _f,\\ \\psi _{f^\\prime } \\rangle =\\int _{{\\cal H}^m }\\psi _f^*\\,\\psi _{f^\\prime }\\,\\nu .$"],["The Homogeneous Contact and Symplectic Manifolds for Klein-Gordon particles","Let us consider the action of $G$ on ${\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1$ given by $(A,H)*(K,\\overrightarrow{y},z)=(AKA^*,\\overrightarrow{x}(A,H,\\overrightarrow{y},K), e^{2\\pi i \\eta m \\ell (A,H,\\overrightarrow{y},K)}z)$ where $\\overrightarrow{x}(A,H,\\overrightarrow{y},K)$ and $\\ell (A,H,\\overrightarrow{y},K)$ are given by $h(\\overrightarrow{x}(A,H,\\overrightarrow{y},K),0)=A\\,h(\\overrightarrow{y},0)\\,A^*+H+\\ell (A,H,\\overrightarrow{y},K) A\\,\\frac{K}{m}\\,A^*.$ Obviously we have $\\ell (A,H,\\overrightarrow{y},K)=-m\\frac{Tr(A\\,h(\\overrightarrow{y},0)\\,A^*+H)}{Tr(AKA^*)}.$ This is a transitive action, as can be proved by direct computation.","The isotropy subgroup at $(mI,\\,\\overrightarrow{0},1),$ is $Ker\\,C_{\\alpha _0},$ so that the homogeneous space $G/Ker\\,C_{\\alpha _0},$ can be identified to ${\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1$ by means of the map $(A,H) Ker\\,C_{\\alpha _0} \\longrightarrow (A,H)*(mI,\\overrightarrow{0},1).$ We know (c.f.","section ) that the left-invariant 1-form on $G$ whose value at $(I,0)$ is $\\alpha _o$ , $\\tilde{\\alpha }_o,$ projects in a 1-form , $\\Omega ,$ in $G/Ker\\,C_{\\alpha _o}$ which is a contact form, and is homogeneous in the sense that it is preserved by the diffeomorphisms corresponding to the canonical action of $G$ on $G/Ker\\,C_{\\alpha _o}.$ One way to give explicitly $\\Omega ,$ is to use coordinate domains in $G/Ker\\,C_{\\alpha _0}$ that are also domains of sections of the canonical map from the group $G$ onto $G/Ker\\,C_{\\alpha _0}.$ The pull back of $\\tilde{\\alpha }_0$ by that section, is the restriction of $\\Omega $ to the domain, and we can give its local expresion in the coordinate system.","With our identification , the just cited canonical map becomes $\\mu :(A,H)\\in G \\longrightarrow (A,H) * (mI,\\overrightarrow{0},1)\\in {\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1.$ In ${\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1$ be define a coordinate system for each $\\tau \\in \\mathbb {R}$ as the inverse of the parametrization ${\\begin{array}{c}\\phi _\\tau : (k_1,k_2,k_3,x^1,x^2,x^3,t)\\in \\mathbb {R}^6 \\times \\left(-\\frac{1}{2},\\frac{1}{2}\\right) \\rightarrow \\\\\\rightarrow \\left( m\\ h(k_1,k_2,k_3,k_4),x^1,x^2,x^3,e^{2\\pi i(t+\\tau )}\\right)\\in \\\\\\in {\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1\\end{array}}$ where $k_4=\\sqrt{1+k_1^2+k_2^2+k_3^2}.$ Now, let us consider the map ${\\begin{array}{c}\\sigma _\\tau :\\phi _\\tau (k_1,k_2,k_3,x^1,x^2,x^3,t) \\rightarrow \\\\ \\rightarrow \\left( \\left( \\begin{array}{cc}\\sqrt{\\frac{k_4+k_3}{1+k_1^2+k_2^2}}&\\sqrt{\\frac{k_4+k_3}{1+k_1^2+k_2^2}}(k_1-ik_2)\\\\0&\\sqrt{\\frac{1+k_1^2+k_2^2 }{ k_4+k_3}}\\end{array}\\right),h(\\overrightarrow{x},0)-\\frac{\\eta }{m}(t+\\tau )) h(\\overrightarrow{k},k_4)\\right) \\\\ \\in G.\\hspace{341.43306pt}\\end{array}}$ We have $\\mu \\circ \\sigma _\\tau (\\phi _\\tau (R))=\\sigma _\\tau (\\phi _\\tau (R))*(mI,\\overrightarrow{0},1)=\\phi _\\tau (R),$ for all $R\\in \\mathbb {R}^6 \\times (-1/2,\\ 1/2),$ so that $\\sigma _\\tau $ is a section of the canonical map $\\mu $ .","Then, on the image of $\\phi _\\tau $ we have $\\Omega =\\sigma _\\tau ^* \\tilde{\\alpha }_0.$ In order to obtain, for exemple, the value $(\\Omega )_{\\left( \\phi _\\tau \\left( R \\right) \\right)}\\cdot \\left( \\frac{\\partial }{\\partial k_1} \\right)_{\\left( \\phi _\\tau \\left( R \\right) \\right)},$ we must find the value of $\\alpha _0$ on the tangent vector to the curve $\\gamma (s)= \\left( \\sigma _\\tau ( \\phi _\\tau (R) )\\right)^{-1}\\left( \\sigma _\\tau ( \\phi _\\tau (R+(s,0,0,0,0,0,0))\\right)$ at $0.$ Since $\\alpha _0=\\left\\lbrace 0,\\ \\eta \\,m\\,I\\right\\rbrace ,$ the first component of the tangent vector at 0 of $\\gamma $ has no incidence in the value of $\\alpha _0$ on it.","The preceding computation gives us $\\langle (o,\\ \\eta m I),\\stackrel{.","}{\\gamma }_0 \\rangle =0.$ A similar computation for each of the other coordinates, or a common reasoning with a curve $\\gamma (s)=\\phi _\\tau (c(s)),$ leads to $\\Omega =d\\,t+\\eta \\,m\\,k_i\\,dx^i.$ Obviously $d\\Omega =\\eta \\,m \\,dk_i\\wedge \\,dx^i,$ so that $\\Omega \\wedge \\left(d\\Omega \\right)^3=\\frac{\\eta m^3}{16}\\ dt\\wedge dk_1\\wedge dk_2\\wedge dk_3\\wedge dx^1\\wedge dx^2\\wedge dx^3$ what confirms the fact that $\\Omega $ is a contact form.","The characteristic vectorfield $Z(\\Omega )$ defined by $i(Z(\\Omega ))\\Omega =1,\\ \\ i(Z(\\Omega ))\\,d\\Omega =0,$ has flow, ${\\phi _t},$ given by $\\phi _t(K,\\overrightarrow{x},z)=(K,\\overrightarrow{x}, e^{2\\pi i t}z).$ As a consecuence, the period of all its integral curves is $T(\\Omega )=1,$ so that $\\Omega $ is a connection form, and $d\\Omega $ the corresponding curvature form.","Now, let us consider the action of $G$ on ${\\cal H}^m \\times \\mathbb {R}^3 $ given by $(A,H)*(K,\\overrightarrow{y})=(AKA^*,\\overrightarrow{x}(A,H,\\overrightarrow{y},K))$ where $h(\\overrightarrow{x}(A,H,\\overrightarrow{y},K),0)=A\\,h(\\overrightarrow{y},0)\\,A^*+H+\\ell (A,H,\\overrightarrow{y},K) A\\,\\frac{K}{m}\\,A^*.$ This is also a transitive action.The isotropy subgroup at $(mI,\\,\\overrightarrow{0}),$ is $G_{\\alpha _0},$ so that $G/G_{\\alpha _0},$ can be identified to ${\\cal H}^m \\times \\mathbb {R}^3 $ by means of the map $(A,H) G_{\\alpha _0} \\longrightarrow (A,H)*(mI,\\overrightarrow{0}).$ The left-invariant 2-form on $G$ whose value at $(I,0)$ is $d\\alpha _0$ , $d\\tilde{\\alpha }_0,$ projects in a 2-form , $\\omega ,$ in $G/G_{\\alpha _0}$ which is a simplectic form, and is homogeneous in the sense that it is preserved by the diffeomorphisms corresponding to the canonical action of $G$ on $G/G_{\\alpha _0}.$ With our identifications, the canonical map $(A,H) Ker\\,C_\\alpha \\rightarrow (A,H) G_\\alpha $ becomes $(K,\\overrightarrow{x},z )\\in {\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1 \\rightarrow (K,\\overrightarrow{x})\\in {\\cal H}^m \\times {\\mathbb {R}}^3$ and, since $T(\\Omega )=1,$ $\\omega $ must be a projection of $d\\Omega ,$ so that $\\omega = {\\eta \\,m} \\,d{k}_i \\wedge \\,d{x}^i,$ where now $\\lbrace k_1,\\dots ,x^3\\rbrace $ are the coordinates corresponding to the following global parametrization of ${\\cal H}^m \\times {\\mathbb {R}}^3 $ ${\\begin{array}{c}\\underline{\\phi }: ({k}_1,{k}_2,{k}_3,{x}^1,{x}^2,{x}^3)\\in {\\mathbb {R}}^6 \\rightarrow \\\\\\rightarrow \\left( m\\ h({k}_1,{k}_2,{k}_3,{k}_4),{x}_1,{x}_2,{x}_3\\right)\\\\\\in {\\cal H}^m \\times {\\mathbb {R}}^3\\end{array}}$ where ${k}_4=\\sqrt{1+{k}_1^2+{k}_2^2+{k}_3^2}.$ In this case we have a global section of the canonical map ${\\begin{array}{c}\\underline{\\sigma }(\\underline{\\phi }({k}_1,{k}_2,{k}_3,{x}^1,{x}^2,{x}^3))=\\\\=\\left( \\left( \\begin{array}{cc}\\sqrt{\\frac{k_4+k_3}{1+k_1^2+k_2^2}}&\\sqrt{\\frac{k_4+k_3}{1+k_1^2+k_2^2}}(k_1-ik_2)\\\\0&\\sqrt{\\frac{1+k_1^2+k_2^2 }{ k_4+k_3}}\\end{array}\\right),h(\\overrightarrow{x},0) \\right).\\end{array}}$ The map $\\underline{\\sigma }$ is a section since $\\underline{\\sigma }(\\phi (R))*(mI,\\overrightarrow{0})=\\phi (R).$ The coadjoint orbit of $\\alpha _0,$ becomes identified to $ {\\cal H}^m \\times {\\mathbb {R}}^3 $ by means of $Ad^*_{(A,H)}\\cdot \\alpha _0 \\rightarrow (A,H)*(mI,\\overrightarrow{0}),$ whose inverse is given by $(K,\\overrightarrow{x}) \\rightarrow Ad^*_{\\underline{\\sigma }(K,\\overrightarrow{x})} \\cdot \\alpha _0.$ Thus, the $(e,g)\\in \\underline{G},$ what are dynamical variables on the coadjoint orbit, becomes functions on $ {\\cal H}^m \\times \\mathbb {R}^3 , $ denoted by $D_{(e,g)}$ in section , defined as follows $D_(e,g)(\\underline{\\phi }(k_1,\\dots ,x_3))=(Ad^*_{\\underline{\\sigma }(\\underline{\\phi }(k_1,\\dots ,x_3))}\\alpha _0)( e,g).$ In particular, the Linear and Angular Momentum, given in (REF ), are, as functions on $ {\\cal H}^m \\times \\mathbb {R}^3, $ $\\nonumber P(m\\,k,\\overrightarrow{x})&=&- \\eta \\,m\\ k, \\\\ \\overrightarrow{l}(m\\,k,\\overrightarrow{x})&=&\\eta \\,m\\,\\overrightarrow{k} \\times \\overrightarrow{x}=\\overrightarrow{x} \\times \\left(\\overrightarrow{P}(m\\,k,\\overrightarrow{x}) \\right) \\\\ \\nonumber \\overrightarrow{g}(m\\,k,\\overrightarrow{x})&=&-\\eta \\,m\\,k^4 \\overrightarrow{x}=\\left( P^4(m\\,k,\\overrightarrow{x}) \\right) \\ \\overrightarrow{x} ,$ where we have denoted $D_{P^k},\\ D_{l^i},$ and $D_{g^j}$ by $P^k, l^i$ and $g^j$ respectively.","Also, each $a\\in \\underline{G}, $ define an infinitesimal generator, $X^s_{a},$ of the action (REF ).","These infinitesimal generators are defined by means of its flow, and its local expresions can be obtained directly from the definition, but it is also possible to use formula (REF ) to obtain the following local expresions in the chart $\\underline{\\Phi }^{-1}$ $X^s_{P^j}&=&\\frac{\\partial }{\\partial x^j}\\\\X^s_{P^4}&=&\\sum _{j=1}^3\\frac{k_j}{k_4}\\ \\frac{\\partial }{\\partial x^j}\\\\X_{l^k}^s &=&\\sum _{j,r=1}^3 \\varepsilon _{kjr} k_j \\frac{\\partial }{\\partial k_r} +\\sum _{j,r=1}^3 \\varepsilon _{kjr} x^j\\frac{\\partial }{\\partial x^r} \\\\X^s_{g^j}&=&x^j\\, X_{P^4}^s-k_4\\frac{\\partial }{\\partial k_j}$ where $ \\varepsilon _{kjr}$ is as in (REF ).","Equation (REF ) proves that these vector fields are globally hamiltonian, and the corresponding hamiltonian is, with our actual notation, the function appearing in the subindex in each case.","The infinitesimal generator corresponding to $a\\in \\underline{G}$ for the action in the contact manifold is denoted by $X_a^c.$ We have in the charts $\\Phi _\\tau ^{-1}$ $X^c_{P^j}&=&\\frac{\\partial }{\\partial x^j}\\nonumber \\\\X^c_{P^4}&=&\\frac{1}{k_4}\\left(\\sum _{j=1}^3\\,k_j\\, \\frac{\\partial }{\\partial x^j}+\\eta m \\frac{\\partial }{\\partial t} \\right)\\nonumber \\\\X_{l^k}^c &=&\\sum _{j,r=1}^3 \\varepsilon _{kjr} k_j \\frac{\\partial }{\\partial k_r} +\\sum _{j,r=1}^3 \\varepsilon _{kjr} x^j\\frac{\\partial }{\\partial x^r} \\\\X^c_{g^j}&=&x^j\\, X_{P^4}^c-k_4\\frac{\\partial }{\\partial k_j}.\\nonumber $ The Quantum Operators representing Linear and Angular Momentum for Quantum States on the Contact Manifold are $\\frac{1}{2 \\pi i}$ times the vectorfiels in (REF ).","If $f$ is a $C^\\infty $ function on ${\\cal H}^m$ the corresponding Quantum State in the contact manifold is $\\Phi _f((A,H)\\,Ker\\,C_\\alpha )=f(A\\,(G_\\alpha )_{SL})Exp[i \\pi Tr(P(A\\,(G_\\alpha )_{SL})\\varepsilon \\overline{H} \\varepsilon )]$ or, with the identifications we have made $\\Phi _f((A,H)* (mI,\\overrightarrow{0},1))=f(mAA^*) Exp[i \\pi Tr((-\\eta mAA^*)\\varepsilon \\overline{H} \\varepsilon )]$ but $\\Phi _f((A,H)* (mI,\\overrightarrow{0},1))=\\Phi _f(mAA^*,h^{-1}( H+\\ell AA^*), Exp[2 \\pi i \\eta m \\ell ]),$ where $ \\ell $ is such that $Tr(H+\\ell AA^*)=0.$ Thus $\\Phi _f(K,\\overrightarrow{x},z)=f(K) Exp[i \\pi Tr((-\\eta K)\\varepsilon \\overline{(h(\\overrightarrow{x},0)-\\ell \\frac{K}{m})} \\varepsilon )]$ where $\\ell $ is such that $z= Exp[2 \\pi i \\eta m \\ell ].$ Then $\\Phi _f(K,\\overrightarrow{x},z)=f(K) Exp[-i \\eta \\pi Tr( K \\varepsilon \\overline{h(\\overrightarrow{x},0)}\\varepsilon ]Exp[i \\ell \\eta \\pi Tr( K \\varepsilon \\overline{\\frac{K}{m}}\\varepsilon ]$ and one sees that, if $K=m h(\\overrightarrow{k},k_4),$ $\\Phi _f(K,\\overrightarrow{x},z)=f(K)\\, Exp[-2 \\pi i \\eta m \\langle \\overrightarrow{k} ,\\overrightarrow{x} \\rangle ] \\,\\overline{z} .$ In the coordinate system associated to $\\phi _\\tau ,$ we have $\\Phi _f \\circ \\phi _\\tau (\\overrightarrow{k} ,\\overrightarrow{x} ,t)=f(mh(\\overrightarrow{k},k_4))\\, \\, Exp[-2 \\pi i ( \\eta m \\langle \\overrightarrow{k} ,\\overrightarrow{x} \\rangle +t+\\tau )].$"],["Wave Functions for $T=1.$ Dirac equation.","Let us consider a particle whose movement space is the coadjoint orbit of $\\alpha _1=\\left\\lbrace \\ \\frac{i}{8\\pi }\\left( \\begin{array}{cc} 1&0 \\\\ 0&-1\\end{array} \\right) \\ ,\\ \\eta \\,m\\,I\\right\\rbrace ,$ where $m \\in {\\bf R}^+, \\eta =\\pm 1.$ This orbit is a quantizable, not $\\bf R \\rm $ -quantizable orbit, of the type 5, in the notation of section .","In this case we have ${\\mbox{$G_{\\alpha _1}$}} = \\lbrace \\ (\\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right), \\ hI ): \\ z\\in S^1,\\ h\\in {\\mbox{$\\bf R \\rm $}} \\rbrace .$ The unique homomorphism from $G_{\\alpha _1}$ onto ${\\bf S}^1$ whose differential is ${\\alpha _1}$ is given by ${\\mbox{$C_{\\alpha _1}$}} (\\ \\left( \\begin{array}{cc}e^{2\\pi i\\phi }&0 \\\\0&e^{-2\\pi i\\phi }\\end{array} \\right) ,\\ hI)=e^{2\\pi i(\\phi -\\eta m h)}.$ Then $\\begin{array}{l}Ker\\, C_{\\alpha _1}=\\lbrace (\\ \\left( \\begin{array}{cc}e^{2\\pi i\\eta m h}&0 \\\\0&e^{-2\\pi i\\eta m h}\\end{array} \\right) ,\\ hI):h\\in \\mathbb {R} \\rbrace ,\\\\\\vspace{7.22743pt}(G_{\\alpha _1} ) _{SL}=\\lbrace \\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right):\\ z\\in S^1\\rbrace , \\\\\\vspace{7.22743pt}{{\\mbox{$(C_{\\alpha _1})_{SL}$}}(\\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right))=z}, \\\\\\vspace{7.22743pt}SL_1 \\cap SL_2 ={\\mbox{$(G_{\\alpha _1} ) _{SL}$}}, \\\\\\vspace{7.22743pt}SL_1 \\cap {\\mbox{$Ker\\,(C_{\\alpha _1})_{SL}$}}={\\mbox{$Ker\\,(C_{\\alpha _1})_{SL}$}}=\\lbrace I \\rbrace .","\\\\\\end{array}$ Let us denote $(G_{\\alpha _1})_{SL}$ by $[S^1],$ $G_{\\alpha _1}$ by $[S^1]\\oplus \\mathbb {R},$ and $Ker\\, C_{\\alpha _1}$ by $[R].$ Then, in the conmutative diagram of Figure REF , $\\iota _3$ and $\\iota _4$ become identical maps, $\\tau _3=\\tau _4$ and the diagram becomes, with obvious conventions, that of Figure REF .","Figure: Fibre Bundles for Dirac ParticlesThe homogeneous space $SL/[S^1]$ can be characterised as follows.","Let ${\\bf P}_1(\\mathbb { C})$ be the complex projective space corresponding to $ \\mathbb { C}^2$ ( $i.e.$ the space $ {\\bf P}_1(\\mathbb { C}) $ consists of the one dimensional complex subspaces of $ {\\mathbb { C}}^2$ ).","In ${\\cal H}^m\\times {\\bf P}_1(\\mathbb { C})$ we can consider the action of $ SL(2, C)$ given by $A\\,*\\,(H,[z])=(AHA^*,\\,[Az])$ where $[z]\\in {\\bf P}_1(\\mathbb { C})$ is the vector subspace generated by $z \\in \\ {\\mathbb { C}}^2-\\lbrace (0, 0) \\rbrace $ .","We know that the partial action on ${\\cal H}^m$ is transitive.","Now let us see that the complete action on ${\\cal H}^m\\times {\\bf P}_1(\\mathbb { C})$ is also transitive.","Let $(K,[w])\\in {\\cal H}^m\\times {\\bf P}_1(\\mathbb {C}).$ Then, there exist $A\\in SL$ such that $A*(K,[w])=(mI,[w^{\\prime }]),$ for some $w^{\\prime }\\in {\\mathbb {C}}^2-\\lbrace (0,0)\\rbrace ,$ but there exist obviously an element $B$ of $SU(2)$ such that $\\left[B \\left(\\begin{array}{c}1\\\\0 \\end{array}\\right) \\right]=[w^{\\prime }],$ and we see that $(B^{-1}A)*(K,[w])=\\left( mI,\\left[ \\left(\\begin{array}{c}1\\\\0 \\end{array}\\right) \\right] \\right).$ Transitivity follows.","On the other hand, the isotropy subgroup at $(mI,\\ [ \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array}\\right) ])$ is $[S^1] $ , so that, since the action is transitive, one can identify $SL/[S^1]$ to ${\\cal H}^m\\times {\\bf P}_1(\\mathbb { C})$ .","It is a well known fact that ${\\bf P}_1(\\mathbb {C})$ is diffeomorphic to the sphere $S^2.$ A diffeomorphism can be described as follows.","Let $C^+$ be the future lightcone, composed by the hermitian matrices having 0 determinant and positive trace.","The subset, $S_2,$ of $C^+$ composed by the elements whose trace is 2 is $S_2=\\lbrace h(x,y,z,1):1-x^2+y^2+z^2=0 \\rbrace $ that can be identified to the sphere $S^2,$ by means of $\\delta :(x,y,z)\\in S^2\\subset \\mathbb {R}^3 \\longleftrightarrow h(x,y,z,1)\\in S_2\\subset C^+.$ The map $\\beta _0:[z]\\in {\\bf P}_1(\\mathbb { C}) \\longrightarrow \\frac{2}{z^*\\,z}\\,z\\,z^* \\in S_2,$ is bijective.","Its inverse is $\\beta _0^{-1}: C \\in S_2\\subset C^+ \\longrightarrow [w]\\in {\\bf P}_1(\\mathbb { C}),$ where $w$ is an eigenvector of $C$ corresponding to the eigenvalue 2.","This is a consecuence of the fact that $\\begin{split} &\\left(\\frac{2}{z^*\\,z}\\,z\\,z^*\\right)z=2z\\\\&\\left(\\frac{2}{z^*\\,z}\\,z\\,z^*\\right)\\varepsilon \\overline{z}=0\\end{split} $ Thus $\\beta \\stackrel{def}{=} \\delta ^{-1}\\circ \\beta _0: {\\bf P}_1(\\mathbb { C}) \\rightarrow S^2,$ can be described by $\\beta ([z])=\\vec{u} \\ \\Longleftrightarrow \\ \\frac{2}{z^*\\,z}\\,z\\,z^*=h(\\vec{u},1).$ that is practical in order to use $\\beta .$ More practical in order to use $\\beta ^{-1}$ is $\\beta ([z])=\\vec{u} \\ \\Longleftrightarrow \\ \\ h(\\vec{u},-1)z=0.$ If $p_s$ and $p_n$ denote the stereographical projections from poles $s=(0,0,-1)$ and $n=(0,0,1)$ respectively, we have $p_s\\left(\\beta \\left[ \\left( \\begin{array}{c} z^1\\\\z^2 \\end{array}\\right)\\right]\\right)= \\frac{z^2}{z^1}\\ \\ \\ \\mathrm {if}\\ z^1\\ne 0,\\\\p_n\\left(\\beta \\left[\\left( \\begin{array}{c} z^1\\\\z^2 \\end{array}\\right)\\right]\\right)= \\frac{\\overline{z}^1}{\\overline{z}^2}\\ \\ \\ \\mathrm {if}\\ z^2\\ne 0.","\\nonumber $ This can be used to prove easily the differentiability of $\\beta $ and $\\beta ^{-1}.","$ Thus, it is possible to change in all that follows ${\\bf P}_1(\\mathbb {C})$ by $S^2,$ but, at this moment, I prefer to use ${\\bf P}_1( \\mathbb {C}).$ In order to describe Prewave Functions in this case , one can consider the trivialisation $(\\rho ,\\ {\\bf C}^4,z_0)$ , where $z_0={}^t\\hspace{0.0pt}(1, 0, 1, 0)$ and $\\rho $ is given by $\\rho (A)=\\left(\\begin{array}{cc}A&0 \\\\0&(A^*)^{-1}\\end{array}\\right)$ The orbit of $z_0$ is ${\\cal B}=\\left\\lbrace \\left( \\!\\begin{array}{c}w \\\\z\\end{array}\\!","\\right)\\ :\\ w,\\ z \\in \\mathbb { C }^2,\\ z^*\\,w=1 \\right\\rbrace .","$ This can be seen by consideration of the map $\\phi _0:\\left(\\begin{array}{c}w \\\\z\\end{array}\\right) \\in {\\cal B} \\longrightarrow \\left(w \\vert -\\varepsilon \\overline{z} \\right)\\in SL(2,\\bf C)$ (it is a diffeomorphism), and the fact that $\\rho (w|-\\varepsilon \\overline{z} )\\left(\\begin{array}{c}1 \\\\0\\\\1\\\\0\\end{array}\\right)=\\left(\\begin{array}{c}w \\\\z\\end{array}\\right).$ When one identifies $SL / Ker(C_{\\alpha _1})_{SL}$ to $\\cal B$ , and $SL / (G_{\\alpha _1} ) _{SL}$ to ${\\cal H}^m \\times {\\bf P}_1( \\mathbb {C})$ , by means of the preceeding actions, the canonical map between these homogeneous spaces becomes a map, ${\\mbox{$ r$}}$ , from $\\cal B$ onto ${\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})$ .","As a consecuence of (REF ), we have $(w \\ |\\,-\\varepsilon \\overline{z})* (mI,\\left[\\left( \\begin{array}{c}1 \\\\0\\end{array}\\right)\\right])={\\mathbf {r}}\\left(\\begin{array}{c}w \\\\z\\end{array}\\right)$ and this leads easily to ${{\\mbox{$ r$}}} \\left(\\!\\begin{array}{c} w\\\\ z\\end{array}\\!","\\right)=(m(ww^*\\,-\\,\\varepsilon \\overline{zz^*} \\varepsilon ),\\, [w]).$ On the other hand, if we define $\\sigma (K,w)=\\frac{1}{\\sqrt{mw^*K^{-1}w}}\\left(w\\ \\ -\\frac{1}{m}K \\varepsilon \\overline{w}\\right),$ where the vertical bar separates the two columns of the $2\\times 2$ matrix, we have $\\sigma (K,w)*(mI,\\left[\\left( \\begin{array}{c}1 \\\\0\\end{array}\\right)\\right])=(K,[w]).$ Notice that if $z\\ \\mathrm {and}\\ z^{\\prime }$ are representatives of the same element of $ P^1(\\mathbb {C}),$ $\\sigma \\left(K, z \\right)$ and $\\sigma \\left(K, z^{\\prime } \\right)$ are in general different.","Thus, since ${\\mathbf {r}}^{-1}(mI,\\left[\\left( \\begin{array}{c}1 \\\\0\\end{array}\\right)\\right])=\\lbrace s {\\left(\\begin{array}{c}1 \\\\0\\\\1\\\\0\\end{array}\\right)}:s\\in S^1 \\rbrace $ we have ${\\mathbf {r}}^{-1}(K,[w])=\\rho (\\sigma (K,w)) \\lbrace s \\left( {\\begin{array}{c}1 \\\\0\\\\1\\\\0\\end{array} }\\right):s\\in S^1 \\rbrace $ which leads to $ {{\\mbox{$ r$}}}^{-1}(K,\\ [a])=\\left\\lbrace s \\left(\\!\\begin{array}{c} I\\\\m\\,K^{-1}\\end{array}\\!","\\right)\\frac{a}{\\sqrt{ma^*K^{-1}a}}:\\ s \\in {\\bf S}^1 \\right\\rbrace $ Also we have $P(K,\\ [w])\\,=P(\\sigma (K,w)G_{(\\alpha _1)_{SL}})=\\,-\\eta K.$ If $f$ is a continuous function with compact support in $\\cal B,$ and is $S^1$ -homogeneous of degree -1 (i.e.","$f\\in {\\cal C}$ in the notation of section REF ) , the corresponding prewave function is $\\psi _f:(H,\\ K,\\ [a]) \\in H(2) \\times {\\cal H}^m \\times {{\\mbox{$ P_1$}}}( {\\bf C})\\ \\ \\mapsto \\ \\ f\\left(\\!\\begin{array}{c} w \\\\ z \\end{array} \\!","\\right) e^{-i\\pi \\eta \\ Tr(K\\,\\varepsilon \\overline{H} \\varepsilon ) }\\ \\left(\\!\\begin{array}{c} w\\\\ z \\end{array} \\!","\\right)$ where $\\left(\\!\\begin{array}{c} w\\\\ z \\end{array} \\!","\\right) $ is arbitrary in ${\\mbox{$r$}}^{-1}(K,\\ [a])$ .","When one considers the Dirac matrices in the representation $\\gamma ^4=\\left( \\begin{array}{cc}0&I \\\\I&0\\end{array} \\right)\\ \\ \\ ,\\gamma ^k=\\left( \\begin{array}{cc}0&-\\sigma _k \\\\\\sigma _k&0\\end{array} \\right)\\ \\ \\ ,k=1,\\ 2,\\ 3,$ a straightforward computation proves that that these prewave functions satisfy Dirac Equation $(\\gamma ^\\nu \\ \\partial _\\nu \\ -2 \\pi i \\eta m)\\ \\psi _f \\ =\\ 0.$ A sesquilinear form on ${\\bf C}^4$ whose value on ${\\cal B}$ is 1, is defined by $ \\Phi \\left( Z,\\ Z^\\prime \\right)=\\frac{1}{2} Z^* \\gamma ^4 Z.$ Thus, if $f,\\ f^{\\prime } \\in {\\cal C},$ the hermitian product of $\\psi _f$ and $\\psi _{f^\\prime }$ can be writen $\\langle \\psi _f,\\ \\psi _{f^\\prime } \\rangle =\\frac{1}{2} \\int _{{\\cal H}^m \\times P_1(C)}\\psi _f^*\\,\\gamma ^4\\,\\psi _{f^\\prime }\\,\\mu .$ To describe Wave Functions we need an invariant volume element on ${\\cal H}^m\\times {\\bf P}_1 (\\mathbb {C}).$ Let us consider the 5-form in ${\\cal H}^m \\times ( {\\mathbb {C}}^2-\\lbrace (0,0)\\rbrace )$ , given by $(\\mu _0)_{(K,\\ z)}=\\nu \\ \\wedge \\frac{(z^1\\,dz^2-z^2\\,dz^1)\\,\\wedge \\,{(\\overline{z^1}\\,d\\overline{z^2}-\\overline{z^2}\\,d\\overline{z^1})}}{(z^* {\\mbox{$\\varepsilon $}}\\overline{K} {\\mbox{$\\varepsilon $}} z)^2},$ where the $z^k$ are the two canonical projections of ${\\mathbb {C} \\rm }^2$ onto ${\\mathbb {C}}$ .","This form is well defined since, for all $K\\in {\\cal H}^m, $ the hermitian product defined in ${\\mathbb {C}}^2$ by $\\langle x,y \\rangle =x^* \\overline{K} y$ is positive definite.","This differential form projects to an invariant volume element, $\\mu $ , in ${\\cal H}^m \\times {\\mbox{$ P_1$}}(\\mathbb {C})$ .To prove this fact, one must proceed in many steps.","Let us denote by $\\tau $ the canonical map $\\tau :(K,z)\\in {\\cal H}^m \\times ( {\\mathbb {C}}^2-\\lbrace (0,0)\\rbrace ) \\rightarrow (K,[z])\\in {\\cal H}^m \\times {\\mbox{$ P_1$}}(\\mathbb {C}).$ The triple ${\\cal H}^m \\times ( {\\mathbb {C}}^2-\\lbrace (0,0)\\rbrace )({\\cal H}^m \\times {\\mbox{$ P_1$}}(\\mathbb {C}), {\\mathbb {C}}-\\lbrace 0\\rbrace )$ is a principal fibre bundle with projection $\\tau .$ To prove that there exist a form, $\\mu ,$ such that $\\tau ^*\\mu =\\mu _0$ it is enought to see that $\\mu _0$ is invariant under the bundle action, what is obvious, and that $\\mu _0$ vanishes on vertical vectors, what can be proved as follows.","The vertical vectors at $(K,z)$ are tangent at $t=0$ to the curves $\\gamma (t)=(K,\\lambda (t)z)$ with $\\lambda $ a $C^\\infty $ map from $\\mathbb {R}$ into $\\mathbb {C}$ such that $\\lambda (0)=1.$ Let $X_{(K,z)}$ be the tangent vector to $\\gamma $ at $0.$ We have $X_{(K,z)}=\\lambda ^\\prime (0)\\left(z^1 \\frac{\\partial }{\\partial z^1}+z^2 \\frac{\\partial }{\\partial z^2}\\right)+\\overline{\\lambda ^\\prime (0)}\\left(\\overline{z^1} \\frac{\\partial }{\\partial \\overline{z^1}}+\\overline{z^2} \\frac{\\partial }{\\partial \\overline{z^2}}\\right),$ and it is easily seen that $i_{X_{(K,z)}}{(\\mu _0)_{(K,z)}}=0.$ This finishes the proof of the existence of $\\mu .$ Let us denote $\\Vert p \\Vert ^2=\\sum _{i=1}^3 (p^i)^2$ and $q_1(p_1,p_2,p_3,z)&=&(h(p_1,p_2,p_3,(m^2+ \\Vert p \\Vert ^2 )^{\\frac{1}{2}} ),\\left[ \\left(\\begin{array}{c}1\\\\z \\end{array}\\right) \\right]) \\\\q_2(p_1,p_2,p_3,z)&=&(h(p_1,p_2,p_3,(m^2+ \\Vert p \\Vert ^2)^{\\frac{1}{2}} ),\\left[ \\left(\\begin{array}{c}z\\\\1 \\end{array}\\right) \\right]) \\nonumber $ for all $p_1,p_2,p_3,z\\in {\\mathbb {R}}^3 \\times \\mathbb {C}.$ The maps $q_1$ and $q_2$ are parametrizations, whose inverses are local charts that compose an atlas of ${\\cal H}^m \\times {\\mbox{$ P_1$}}(\\mathbb {C}).$ The local expressions of $\\mu $ in these charts can be obtained as the reciprocal image of $\\mu _0$ by the maps $\\sigma _1(p_1,p_2,p_3,z)=(h(p_1,p_2,p_3,(m^2+ {\\mbox{$\\sum _{i=1}^3$}} \\ (p^i)^2)^{\\frac{1}{2}} ),\\left(\\begin{array}{c}1\\\\z \\end{array}\\right) ),$ and $\\sigma _(p_1,p_2,p_3,z)=(h(p_1,p_2,p_3,(m^2+ {\\mbox{$\\sum _{i=1}^3$}} \\ (p^i)^2)^{\\frac{1}{2}} ),\\left(\\begin{array}{c}z\\\\1 \\end{array}\\right) ).$ These local expressions are given by $\\sigma _1^*\\mu _0=\\frac{\\nu \\wedge dz \\wedge d \\overline{z}}{2\\Re (z(p_1-ip_2))+(1-\\vert z \\vert ^2)p_3-(1+\\vert z \\vert ^2)(m^2+ \\Vert p \\Vert ^2 )^{\\frac{1}{2}}},$ $\\sigma _2^*\\mu _0=\\frac{\\nu \\wedge dz \\wedge d \\overline{z}}{2\\Re (z(p_1+ip_2))+(\\vert z \\vert ^2-1)p_3-(1+\\vert z \\vert ^2)(m^2+ \\Vert p \\Vert ^2 )^{\\frac{1}{2}}}.$ As a particular consequence, $\\mu $ is a volume element.","Now, let us consider the action of $SL$ on ${\\cal H}^m \\times ( {\\mathbb {C}}^2-\\lbrace (0,0)\\rbrace )$ given by $A*(K,z)=(AKA^*,Az).$ We already know that $\\nu $ is invariant under the action on ${\\cal H}^m ,$ and it is not a difficult matter to see that $z^1\\,dz^2-z^2\\,dz^1,\\ {\\overline{z^1}\\,d\\overline{z^2}-\\overline{z^2}\\,d\\overline{z^1} } $ and $z^* {\\mbox{$\\varepsilon $}}\\overline{K} {\\mbox{$\\varepsilon $}} z$ are invariant under this action.","As a consecuence, $\\mu _0$ is invariant under the same action.","It follows that $\\mu $ is an invariant volume element on ${\\cal H}^m \\times {\\mbox{$ P_1$}}(\\mathbb {C}).$ The wave functions have the form $\\tilde{\\psi }_f(X)=\\int _{{\\cal H}^m \\times P_1(C)}\\psi _f(h(X),\\cdot ,\\cdot )\\ \\mu $ and also satisfies Dirac Equation $(\\gamma ^\\nu \\ \\partial _\\nu \\ -2 \\pi i \\eta m)\\ \\tilde{\\psi }_f \\ =\\ 0.$"],["Wave functions for $T>1$ .","Now, let us consider a particle whose movement space is the coadjoint orbit of $\\alpha _T=\\left\\lbrace \\ \\frac{iT}{8\\pi }\\left( \\begin{array}{cc} 1&0 \\\\ 0&-1\\end{array} \\right) \\ ,\\ \\eta \\,m\\,I\\right\\rbrace ,$ where $T\\in {\\bf Z}^+,\\ m \\in {\\bf R}^+, \\eta =\\pm 1.$ The case of the preceeding section is the particular one given by $T=1.$ For all $T$ the orbit is a quantizable, not $\\bf R \\rm $ -quantizable orbit, of the type 5.","Now we have ${\\mbox{$G_{\\alpha _T}$}} = \\lbrace \\ (\\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right), \\ hI ): \\ z\\in S^1,\\ h\\in {\\mbox{$\\bf R \\rm $}} \\rbrace .$ The unique homomorphism from $G_{\\alpha _T}$ onto ${\\bf S}^1$ whose differential is $\\alpha _T$ is given by ${\\mbox{$C_{\\alpha _T}$}} (\\ \\left( \\begin{array}{cc}e^{2\\pi i\\phi }&0 \\\\0&e^{-2\\pi i\\phi }\\end{array} \\right) ,\\ hI)=e^{2\\pi i(\\phi T-\\eta m h)}.$ Then $\\begin{array}{l}\\vspace{7.22743pt}{{\\mbox{$(G_{\\alpha _T } ) _{SL}$}}=\\lbrace \\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right):\\ z\\in S^1\\rbrace }, \\\\\\vspace{7.22743pt}{{\\mbox{$(C_{\\alpha _T})_{SL}$}}(\\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right))=z^T}, \\\\\\vspace{7.22743pt}SL_1 \\cap SL_2 ={\\mbox{$(G_{\\alpha _T} ) _{SL}$}}, \\\\\\vspace{7.22743pt}SL_1 \\cap Ker\\,(C_{\\alpha _T})_{SL}=Ker\\,(C_{\\alpha _T})_{SL}=\\lbrace \\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right):\\ z\\in \\@root T \\of {\\mathbf {1}}\\rbrace .","\\\\\\end{array}$ where $\\@root T \\of {\\mathbf {1}}$ is the subgroup of ${\\mathbb {C}}^*$ composed by the roots of order $T$ of 1.","The homogeneous space SL/${\\mbox{$(G_{\\alpha _T} ) _{SL}$}}$ , is the same as in the preceeding section so that it can be identified to ${\\cal H}^m\\times {\\bf P}_1(\\mathbb {C})$ , and consider on it the invariant volume element $\\mu .$ In order to give the wave functions in the case of arbitrary ${T}$ , the following results are useful.","Let ${\\beta },\\ \\beta ^\\prime $ be quantizable elements of $\\underline{G}^*$ with ${\\beta ^\\prime }$ not R-quantizable, $(G_{\\beta })_{SL}=(G_{{\\beta ^\\prime }})_{SL}$ , $(C_{\\beta })_{SL} =\\left((C_{{\\beta ^\\prime }})_{SL}\\right)^T,$ where $T \\in {\\bf Z}^+$ .","Notice that, as a consecuence of the fact that $\\beta ^{\\prime }$ is not ${\\mathbb {R} }$ -quantizable, we must have $(C_{\\beta ^\\prime })_{SL}((G_{{\\beta ^\\prime }})_{SL})=S^1.$ If $(\\rho , L, z_0)$ is a trivialization of $C_{\\beta ^\\prime }$ , we consider the triple $(\\rho ^{\\otimes T},\\ L^ {\\otimes T},\\ z_0^ {\\otimes T}),$ where $L^ {\\otimes T}=L \\otimes \\stackrel{(T}{\\cdots } \\nolinebreak \\otimes L,\\ z_0^{\\otimes T}=z_0 \\otimes \\stackrel{(T}{\\cdots } \\otimes z_0$ , and $\\rho ^{\\otimes T}$ is the representation such that $\\rho ^{\\otimes T}(A)(z_1 \\otimes \\cdots \\otimes z_T)=\\rho (A)(z_1) \\otimes \\cdots \\otimes \\rho (A)(z_T).$ In lemma 6.1 of [8] I have proved that, under these circunstances, if $(G_{\\beta })_{SL}$ is connected, $(\\rho ^{\\otimes T},\\ L^ {\\otimes T},\\ z_0^ {\\otimes T})$ is a trivialization of $C_\\beta $ .","Let us assume that $(\\rho ^{\\otimes T},\\ L^ {\\otimes T},\\ z_0^ {\\otimes T})$ is a trivialization of $C_\\beta $ and let ${\\cal B}_T$ be the orbit of $ z_0^{\\otimes T}$ .","The pullback by $z \\in {\\cal B} \\mapsto z^{\\otimes T} \\in {\\cal B}_T$ , establishes a one to one map from the set of the $S^1$ -homogeneous functions of degree -1 on $ {\\cal B}_T$ , onto the set of the ${S^1}$ -homogeneous functions of degree -T on ${\\cal B}$ .","If $f$ is one of these functions, the corresponding prewave function of particles corresponding to $\\beta $ has the form ${\\mbox{$\\psi $}}_f(H,\\ m)\\,=\\,f(z)\\,e^{i\\pi Tr(P(m)\\,\\varepsilon \\overline{H} \\varepsilon )}z^{\\otimes T}$ where $z \\in { \\bf r}^{-1}(m).$ Now we apply these results to the particles of type 5 with $T> 1.$ Let $\\beta ^{\\prime }=\\alpha _1$ and $\\beta =\\alpha _T.$ The remark just made leads to the following Prewave Function $\\psi _f:(H,\\ K,\\ [a]) \\in H(2) \\times {\\cal H}^m \\times {{\\mbox{$ P_1$}}}( {\\bf C})\\ \\ \\mapsto \\ \\ f\\left(\\!\\!\\begin{array}{c} w \\\\ z \\end{array} \\!","\\!","\\right) e^{-i\\pi \\eta \\ Tr(K\\,\\varepsilon \\overline{H} \\varepsilon ) }\\ \\left(\\!\\!\\begin{array}{c} w\\\\ z \\end{array} \\!\\!","\\right)^{\\otimes T}$ where $\\left(\\!\\!\\begin{array}{c} w\\\\ z \\end{array} \\!\\!","\\right) $ is arbitrary in ${\\mbox{$r$}}^{-1}(K,\\ [a])$ , and f is a function on ${\\cal B}$ , $C^\\infty ,$ with compact support and homogeneous of degree -T under multiplication by complex numbers of modulus one.","The Wave Functions are obtained by integration as usual."],["Movement Space for massive particles with $T\\ge 1$ .","We consider the action of $G$ on ${\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3$ given by $(A,H)*(K,[z],\\overrightarrow{y})=(AKA^*,[Az], \\overrightarrow{x}(A,H,\\overrightarrow{y},K))$ where $\\overrightarrow{x}(A,H,\\overrightarrow{y},K)$ is given by $h(\\overrightarrow{x}(A,H,\\overrightarrow{y},K),0)=Ah(\\overrightarrow{y},0)A^*+H+\\ell (A,H,\\overrightarrow{y},K) A\\frac{K}{m}A^* \\\\\\ell (A,H,\\overrightarrow{y},K)=\\frac{-mTr(Ah(\\overrightarrow{y},0)A^*+H)}{Tr( AKA^*)}.$ This is a transitive action.","The isotropy subgroup at $(mI,[{}^t(1,0)], \\overrightarrow{0}),$ is found to be $G_{\\alpha _T}.$ Thus, the map $\\lambda :(A,H) G_{\\alpha _T}\\in G/G_{\\alpha _T} \\leftrightarrow (A,H)*\\left(mI,\\left[\\left(\\begin{array}{c}1 \\\\0\\end{array}\\right)\\right], \\overrightarrow{0}\\right)\\in {\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3$ enables us to identify the coadjoint orbit with ${\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3.$ A parametrization whose image is a neighborhood of $(mI,[{}^t\\hspace{0.0pt}(1,0)], \\overrightarrow{0}),$ is $\\Gamma _s:(\\overrightarrow{k},\\overrightarrow{x},z)\\in \\mathbb {R}^3 \\times \\mathbb {R}^3\\times \\mathbb {C}\\rightarrow &&\\left(mh(\\overrightarrow{k},k_4),\\left[\\left(\\begin{array}{c}1 \\\\z\\end{array}\\right)\\right], \\overrightarrow{x}\\right)\\\\&&\\in {\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3.\\nonumber $ where $k_4=\\sqrt{1+k_1^2+k_2^2+k_3^2}.$ The corresponding local chart is given by $\\Gamma _s^{-1}:\\left(K,\\left[\\left(\\begin{array}{c}z^1 \\\\z^2\\end{array}\\right)\\right],\\overrightarrow{x}\\right)\\in \\lbrace z^1\\ne 0 \\rbrace \\rightarrow \\left(\\frac{1}{m}h^{-1}(K-Tr(K)I),\\overrightarrow{x},\\frac{z^2}{z^1}\\right)$ Another parametrization is $\\Gamma _n:(\\overrightarrow{k},\\overrightarrow{x},z)\\in \\mathbb {R}^3 \\times \\mathbb {R}^3\\times \\mathbb {C}\\rightarrow &&\\left(mh(\\overrightarrow{k},k_4),\\left[\\left(\\begin{array}{c}z \\\\1\\end{array}\\right)\\right], \\overrightarrow{x}\\right)\\\\&&\\in {\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3.\\nonumber $ with $k_4=\\sqrt{1+k_1^2+k_2^2+k_3^2},$ whose inverse is given by $\\Gamma _n^{-1}:\\left(K,\\left[\\left(\\begin{array}{c}z^1 \\\\z^2\\end{array}\\right)\\right],\\overrightarrow{x}\\right)\\in \\lbrace z^2\\ne 0 \\rbrace \\rightarrow \\left(\\frac{1}{m}h^{-1}(K-Tr(K)I),\\overrightarrow{x},\\frac{z^1}{z^2}\\right) \\nonumber $ Obviously, these two charts compose an atlas.","The identification of ${\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3$ with the coadjoint orbit is given by $(A,H)*((mI,[{}^t(1,0)], \\overrightarrow{0})\\longleftrightarrow Ad^*_{(A,H)}\\alpha _T.$ Now, let $\\left(K,\\left[ z\\right],\\overrightarrow{x}\\right)\\in {\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3.$ If $\\sigma \\left(K, w \\right)$ is given by (REF ), then $\\left(\\sigma \\left(K, z \\right),h(\\overrightarrow{x},0)\\right)*(mI,[{}^t(1,0)], \\overrightarrow{0})=\\left(K,\\left[ z\\right],\\overrightarrow{x}\\right),$ so that $\\left(K,\\left[ z\\right],\\overrightarrow{x}\\right)$ must be identified to $Ad^*_{\\left(\\sigma \\left(K, z \\right),h(\\overrightarrow{x},0)\\right)}\\cdot \\alpha _T.$ Now let $F$ be a dynamical variable on the coadjoint orbit.","$F$ becomes a function on ${\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3$ .","To give explicit expresions of these functions, is now preferable to use the sphere $S^2$ instead of ${\\ bf P}_1(\\mathbb {C}) $ thus writing $F(K,[z],\\vec{x}) =F( Ad^*_{\\left(\\sigma \\left(K, z \\right),h(\\overrightarrow{x},0)\\right)}\\cdot \\alpha _T)= F(K,\\vec{u},\\vec{x}),$ where $\\vec{u}\\in S^2$ is related to $[z]\\in \\mathbf {P_1(\\mathbb {C})}$ by (c.f.","(REF )) $\\frac{2}{z^*z}zz^*=h(\\vec{u},1).$ Let us evaluate $Ad^*_{\\left(\\sigma \\left(K, z \\right),h(\\vec{x},0)\\right)}\\cdot \\alpha _T.$ To do that computation we need some of the equations (REF ) to (REF ), and $(\\sigma (K,z))^{-1}=\\frac{1}{\\sqrt{mz^*K^{-1}z}} \\left( \\frac{ mz^*K^{-1}}{ z \\varepsilon }\\right),$ where the horizontal line separates the two files of the $2\\times 2$ matrix.","This leads to $Ad^*_{\\left(\\sigma \\left(K, z \\right),h(\\vec{x},0)\\right)}\\cdot \\alpha _T&=&\\left\\lbrace \\left[ -\\frac{T}{8\\pi }\\frac{1}{\\langle k_4\\vec{ u}-\\vec{ k},\\vec{u} \\rangle }h(\\vec{k}\\times \\vec{u},0)+\\frac{\\eta m}{2}k_4h(\\vec{x},0) \\right]+ \\right.","\\\\ \\nonumber &+&i \\left[\\frac{T}{8 \\pi } \\frac{1}{\\langle k_4\\vec{ u}-\\vec{ k},\\vec{u} \\rangle }h( k_4\\vec{ u}-\\vec{ k},0)+ \\frac{\\eta m}{2} h(\\vec{k}\\times \\vec{x},0)\\right], \\\\ \\nonumber && \\left.","\\eta m h(\\vec{k},k_4) \\right\\rbrace ,$ where $h(\\vec{k},k_4)=\\frac{1}{m}K.$ Thus, using (REF ), we obtain $\\begin{split}P(mh(\\vec{k},k_4),\\vec{u},\\vec{x})&=-\\eta m h(\\vec{k},k_4),\\\\\\vec{ l}(mh(\\vec{k},k_4),\\vec{u},\\vec{x})&=\\frac{T}{4\\pi }\\frac{k_4\\vec{u}-\\vec{k}}{k_4-\\langle \\vec{ k},\\overrightarrow{u} \\rangle }+\\eta m \\vec{k}\\times \\vec{x},\\\\\\vec{g}(mh(\\vec{k},k_4),\\vec{u},\\vec{x})&=\\frac{T}{4\\pi }\\frac{\\vec{k}\\times \\vec{u}}{k_4-\\langle \\vec{ k},\\overrightarrow{u} \\rangle }-\\eta m k_4 \\vec{x}.\\end{split}$ Notice that $k_4-\\langle \\vec{ k},\\overrightarrow{u} \\rangle =\\langle k_4 \\overrightarrow{u} - \\vec{ k},\\overrightarrow{u} \\rangle .$ If we denote $P(K,\\vec{u},\\vec{x}),\\vec{l}(K,\\vec{u},\\vec{x})$ and $\\vec{g}(K,\\vec{u},\\vec{x})$ simply by $P,\\ \\vec{l}$ and $\\vec{g}$ respectively when no danger of confusion exist, and $\\vec{P}=(P^1,P^2,P^3),$ we have the following relations between these dynamical variables $\\vec{ l}&=&\\frac{T}{4\\pi }\\frac{P^4\\vec{u}-\\vec{P}}{P^4-\\langle \\vec{ P},\\vec{u} \\rangle }+ \\vec{x}\\times \\vec{P},\\\\ \\vec{g}&=&\\frac{T}{4\\pi }\\frac{\\vec{P}\\times \\vec{u}}{ P^4-\\langle \\vec{ P},\\vec{u} \\rangle }+P^4\\vec{x}$ The value of the Pauli-Lubanski fourvector at $(mh(\\vec{k},k_4),\\vec{u},\\vec{x})$ can be calculated using (REF ) , (),(REF ),...,() and is found to be $ \\begin{split}\\overrightarrow{W}&=P^4\\frac{T}{4\\pi }\\frac{P^4\\vec{u}-\\vec{P}}{\\langle P^4\\vec{u}-\\vec{P},\\vec{u} \\rangle }=-\\eta m k_4\\frac{T}{4\\pi } \\frac{k_4\\vec{u}-\\vec{k}}{\\langle k_4\\vec{u}- \\vec{ k},\\vec{u} \\rangle } \\\\W^4&=\\frac{T}{4\\pi }\\frac{\\langle P^4\\vec{u}-\\vec{P},\\vec{P} \\rangle }{\\langle P^4\\vec{u}-\\vec{P},\\vec{u} \\rangle }= -\\eta m\\frac{T}{4\\pi } \\frac{\\langle k_4\\vec{u}-\\vec{k},\\vec{k} \\rangle }{\\langle k_4\\vec{u}- \\vec{ k},\\vec{u} \\rangle },\\end{split}$ so that we can also write $\\vec{ l}=\\frac{1}{P^4}\\overrightarrow{W}+ \\vec{x}\\times \\vec{P} .$"],["Contact manifold for Dirac particles","Now we pay attention to the contact manifold in the case $T=1$ .","We consider the action of $G$ on ${\\cal B}\\times {\\mathbb {R}}^3$ given by $(A,H)*(w,z,\\overrightarrow{y})=\\left( e^{2\\pi i\\eta m \\ell }Aw,e^{2\\pi i\\eta m \\ell }(A^*)^{-1}z,\\overrightarrow{x}(A,H,w,z,\\overrightarrow{y})\\right)$ where $h(\\overrightarrow{x}(A,H,w,z,\\overrightarrow{y}),0)= Ah(\\overrightarrow{y},0)A^*+H+\\ell A(ww^*-\\varepsilon \\overline{zz^*} \\varepsilon )A^*.$ Obviously, we have $\\ell =-\\frac{Tr(Ah(\\vec{x},0)A^*+H)}{Tr( A(ww^*-\\varepsilon \\overline{zz^*} \\varepsilon )A^*)}.$ This is a transitive action and the isotropy subgroup at $\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0}\\right)$ is $Ker\\,C_{\\alpha _1},$ so that we can identify $G/Ker\\,C_{\\alpha _1}$ with ${\\cal B}\\times {\\mathbb {R}}^3$ by means of $(A,H)Ker\\,C_{\\alpha _1} \\longleftrightarrow (A,H)* \\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0}\\right).$ When $w\\in {\\mathbb {C}}^2-\\lbrace 0 \\rbrace ,$ the $z$ such that $(w,z)\\in {\\cal B}$ compose the set $\\left\\lbrace \\frac{1}{w^*w}(w+y\\varepsilon \\overline{w}): y\\in {\\mathbb {C}} \\right\\rbrace .$ Thus, the map $\\Delta _0:(w,y)\\in ({\\mathbb {C}}^2-\\lbrace 0 \\rbrace )\\times {\\mathbb {C}} \\longrightarrow \\left( w , z(w,y)\\right)\\in {\\cal B},$ where $z(w,y)=\\frac{1}{w^*w}(w+y\\varepsilon \\overline{w}),$ is a bijection, and in fact a diffeomorphism.","Its inverse is given by $(\\Delta _0)^{-1}:(w,z)\\in {\\cal B} \\longrightarrow (w, {}^tw \\varepsilon z)\\in ({\\mathbb {C}}^2-\\lbrace 0 \\rbrace )\\times {\\mathbb {C}}.$ Then, $(\\Delta _0)^{-1}$ is a global complex coordinate system of ${\\cal B},$ and $\\Delta :(w,y,\\vec{x})\\in ({\\mathbb {C}}^2-\\lbrace 0 \\rbrace )\\times {\\mathbb {C}}\\times {\\mathbb {R}}^3 \\longrightarrow \\left( w , z(w,y),\\vec{x}\\right)\\in {\\cal B}\\times {\\mathbb {R}}^3,$ is a parametrization of the Contact Manifold defined in an an open subset of ${\\mathbb {C}}^3 \\times {\\mathbb {R}}^3.$ Notice that, since we know a diffeomorphism, $\\phi _0,$ from ${\\cal B}$ onto $SL(2,\\mathbb {C})$ (c.f.","(REF )), we obtain also a complex parametrization of this group by means of $\\phi _0 \\circ \\Delta _0:(w,y)\\in ({\\mathbb {C}}^2-\\lbrace 0 \\rbrace )\\times {\\mathbb {C}} \\longrightarrow \\left( w\\ \\ \\frac{1}{w^*w}({\\overline{y}}w-\\varepsilon \\overline{w}\\right)\\in SL(2,\\mathbb {C}).$ The inverse of $\\phi _0 \\circ \\Delta _0$ is the global complex chart of $SL(2,\\mathbb {C})$ given by $\\phi :(w|v)\\in SL(2,\\mathbb {C}) \\longrightarrow (w, {}^tw \\overline{v})\\in ({\\mathbb {C}}^2-\\lbrace 0 \\rbrace )\\times {\\mathbb {C}}.$ The canonical map of $G$ onto $G/Ker\\,C_{\\alpha _1}$ becomes $(A,H)\\in G \\longleftrightarrow (A,H)* \\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0}\\right),$ and the map $\\Sigma :(w,z,\\vec{x})\\in {\\cal B}\\times {\\mathbb {R}}^3 \\longrightarrow \\left( \\left( w\\ \\ -\\varepsilon \\overline{z}\\right),h(\\vec{x},0)\\right)\\in G,$ is a section of this canonical map , as a consecuence of the fact that $\\Sigma (w,z,\\vec{x})*\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0} \\right)=(w,z,\\vec{x}).$ The canonical map from $G/Ker\\,C_\\alpha $ onto $G/G_\\alpha $ becomes $\\tilde{\\mathbf {r}}&:&(w,z,\\vec{x})\\in {\\cal B}\\times {\\mathbb {R}}^3 \\longrightarrow \\Sigma (w,z,\\vec{x})* (mI,[\\left(\\begin{array}{c}1\\\\0 \\end{array} \\right)],\\vec{0})=\\\\&=&(m(ww^*-\\varepsilon \\overline{zz^*}\\varepsilon ),[w],\\vec{x})\\in {\\cal H}^m\\times {\\mathbf {P}_1(\\mathbb {C})}\\times {\\mathbb {R}}^3,$ i.e.","coincides with ${\\bf r}\\times Id_{{\\mathbb {R}}^3}.$ The Linear Momentum, Angular Momentum and Pauli-Lubanski fourvector, whose descriptions in the symplectic manifold are given by (REF ) and (REF ), when composed with $\\tilde{\\mathbf {r}},$ give functions $P_c,\\ \\vec{l}_c,\\vec{g}_c,W_c,$ on the Contact Manifold, that are the translation of these dynamical variables to this homogeneous space.","In order to give explicit expressions for these functions ($c.f.$ (REF )), let us denote $\\tilde{\\mathbf {r}}(w,z,\\vec{x})=(m\\,h(\\vec{k},k_4),\\vec{u}, \\vec{x})\\in {\\cal H}^m \\times S^2 \\times {\\mathbb {R}}^3$ we have $h(\\vec{u},1)= \\frac{2}{w^*w}w\\,w^*,$ and $h(\\vec{k},k_4)=ww^*-\\varepsilon \\overline{zz^*}\\varepsilon ,$ but $-h(\\vec{u},1)\\varepsilon \\overline{h(\\vec{k},k_4)}\\varepsilon =h(k_4\\vec{u}-\\vec{k},k_4-\\langle \\vec{k},\\vec{u} \\rangle ) + i h(\\vec{k} \\times \\vec{u},0)$ and $-h(\\vec{u},1)\\varepsilon \\overline{h(\\vec{k},k_4)}\\varepsilon =- \\frac{2}{w^*w}ww^*\\varepsilon \\overline{(ww^*-\\varepsilon \\overline{zz^*}\\varepsilon )}\\varepsilon =\\frac{2}{w^*w}w\\,z^*,$ leads to $h(k_4\\vec{u}-\\vec{k},k_4-\\langle \\vec{k},\\vec{u} \\rangle ) + i h(\\vec{k} \\times \\vec{u},0)=\\frac{2}{w^*w}w\\,z^*,$ so that $\\begin{split}&h(k_4\\vec{u}-\\vec{k},k_4-\\langle \\vec{k},\\vec{u} \\rangle ) =\\frac{1}{w^*w}(w\\,z^*+zw^*),\\\\&h(\\vec{k} \\times \\vec{u},0)=i\\frac{1}{w^*w}(zw^*-w\\,z^*)\\end{split}$ and $\\begin{split}&\\langle k_4\\vec{u}-\\vec{k},\\vec{u} \\rangle =k_4-\\langle \\vec{k},\\vec{u} \\rangle =\\frac{1}{w^*w}\\\\&h\\left(\\frac{k_4\\vec{u}-\\vec{k}}{k_4-\\langle \\vec{k},\\vec{u} \\rangle },1\\right)=w\\,z^*+zw^*\\\\&h\\left(\\frac{\\vec{k} \\times \\vec{u}}{k_4-\\langle \\vec{k},\\vec{u} \\rangle },0\\right)=i(zw^*-w\\,z^*)\\end{split}$ Thus $\\begin{split}&P_c(w,z,\\vec{x})=-\\eta m\\,(ww^*-\\varepsilon \\overline{zz^*}\\varepsilon )\\\\&h(\\vec{l}_c(w,z,\\vec{x}),0)=\\frac{T}{4\\pi }(wz^*+zw^*-I)+h(\\vec{x} \\times \\vec{P}_c,0)\\\\&h(\\vec{g}_c(w,z,\\vec{x}),0)=\\frac{iT}{4\\pi } (zw^*-wz^*)+P^4 h(\\vec{x},0)\\\\&h(\\vec{W}_c(w,z,\\vec{x}),0)=P^4\\frac{T}{4\\pi }(wz^*+zw^*-I)\\\\&W^4_c(w,z,\\vec{x})=\\frac{-\\eta m T}{8\\pi }(w^*w-z^*z)\\end{split}$ The formula for $W_c^4$ can be obtained as follows $\\begin{split}&\\langle \\vec{l},\\vec{P} \\rangle \\circ \\mathbf {\\tilde{r}}=\\left\\langle \\frac{T}{4\\pi }\\frac{k_4\\vec{u}-\\vec{k}}{\\langle k_4\\vec{u}-\\vec{k},\\vec{u} \\rangle },\\vec{P} \\right\\rangle \\circ \\mathbf {\\tilde{r}}= \\\\&=\\frac{1}{2}Tr\\left(h\\left(\\frac{T}{4\\pi }\\frac{k_4\\vec{u}-\\vec{k}}{\\langle k_4\\vec{u}-\\vec{k},\\vec{u} \\rangle },0\\right)P \\right) \\circ \\mathbf {\\tilde{r}}=\\\\ &=\\frac{1}{2}Tr\\left(\\left(\\frac{T}{4\\pi }(wz^*+zw^*-I)\\right)(-\\eta m)(ww^*-\\varepsilon \\overline{zz^*}\\varepsilon )\\right)=\\\\ &=\\frac{-\\eta mT}{8\\pi }(w^*w-z^*z).\\end{split}$ The local expresions of these functions in the chart $\\Delta ^{-1}$ are obtained simply by changing in the above expresions $z$ by the $z(w,y)$ given in (REF ).","Let $\\Omega _1$ be the contact form and $\\tilde{\\alpha }_1$ the left invariant differential form on $G$ whose value at $(I,0)$ is $\\alpha _1.$ We have $\\Omega _1=\\Sigma ^* \\tilde{\\alpha }_1,$ and this formula enables us, by similar procedures to those of section REF , to see that $\\Omega _1=\\frac{iT}{4\\pi }\\left(z^1 d\\overline{w^1}-\\overline{ z^1} d w^1+z^2 d\\overline{ w^2}-\\overline{ z^2} d w^2\\right)-P^1dx^1-P^2 dx^2-P^3 d x^3,$ where (c.f.","(REF )) $\\left(\\begin{array}{c}z^1\\\\z^2\\end{array}\\right)=z(w,y).$ This equation must be interpreted as follows: the right hand side of (REF ) is a differential 1-form in $\\mathbb {C}^4 \\times \\mathbb {R}^3$ refered to coordinates $w^1,\\dots ,x^3,$ and $\\Omega _1$ is the restriction of this form to the submanifold ${\\cal B}\\times \\mathbb {R}^3.$ Obviously, under the same conditions $\\begin{split}d\\Omega _1=\\frac{iT}{4\\pi }&\\left(dz^1\\wedge d\\overline{w^1}-d\\overline{ z^1}\\wedge d w^1+dz^2 \\wedge d\\overline{ w^2}-d\\overline{ z^2} \\wedge d w^2\\right)\\\\&-dP^1\\wedge dx^1-dP^2 \\wedge dx^2-dP^3 \\wedge d x^3.","\\end{split}$ Now, to obtain the symplectic form we only need sections of the map $\\tilde{\\mathbf {r}}.$ On the domain, $U_s,$ of the local chart $\\Gamma _s^{-1},$ the map ${\\tilde{\\sigma }}_s: \\Gamma _s(\\vec{k},\\vec{x},t) \\rightarrow \\left( \\sigma \\left(mh(\\vec{k},k_4),\\left( \\begin{array}{c} 1\\\\ t\\end{array}\\right)\\right),h(\\vec{x},0)\\right)*\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0} \\right), $ where $\\sigma (K,w)$ is given by (REF ), is a section.","Then ${\\tilde{\\sigma }}_s \\circ \\Gamma _s(\\vec{k},\\vec{x},t) =\\left(\\frac{1}{\\sqrt{(1,\\overline{t})(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}1\\\\t \\end{array}\\right)}} \\left(\\begin{array}{c}1\\\\t \\end{array}\\right),\\right.$ $\\left.","\\frac{1}{\\sqrt{(1,\\overline{t})(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}1\\\\t \\end{array}\\right)}}(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}1\\\\t \\end{array}\\right), \\vec{x} \\right).","$ If $\\omega _1$ is the symplectic form, we have on $U_s$ $ \\omega _1={\\tilde{\\sigma }}_s ^* d\\Omega _1.$ In the same way, on the domain, $U_n,$ of the chart $\\Gamma _n,$ we define a section, $\\tilde{\\sigma }_n,$ by ${\\tilde{\\sigma }}_n \\circ \\Gamma _n(\\vec{k},\\vec{x},t)=\\left( \\sigma \\left(mh(\\vec{k},k_4),\\left(\\begin{array}{c} t\\\\ 1\\end{array}\\right)\\right),h(\\vec{x},0)\\right)*\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0} \\right).$ Then ${\\tilde{\\sigma }}_n \\circ \\Gamma _n(\\vec{k},\\vec{x},t) =\\left(\\frac{1}{\\sqrt{(\\overline{t},1)(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}t\\\\1 \\end{array}\\right)}} \\left(\\begin{array}{c}t\\\\1 \\end{array}\\right),\\right.$ $\\left.","\\frac{1}{\\sqrt{(\\overline{t},1)(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}t\\\\1 \\end{array}\\right)}}(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}t\\\\1 \\end{array}\\right), \\vec{x} \\right).","$ On $U_n$ we have $\\omega _1={\\tilde{\\sigma }}_n ^* d\\Omega _1.$ Since $\\lbrace U_s,U_n\\rbrace $ is an open covering of the symplectic manifold, (REF ) and (REF ), determine $\\omega $ everywhere."],["Contact manifold for $T > 1.$","Let us denote by $r_1,\\dots , r_T$ the elements of $\\@root T \\of {\\mathbf {1}}.$ We define a properly discontinuous free action, $\\cdot ,$ of $\\@root T \\of {\\mathbf {1}}$ on ${\\cal B}\\times {\\mathbb {R}}^3 $ by means of $r_j\\cdot (w,z,\\overrightarrow{x})=(r_j w,r_j z,\\overrightarrow{x}).$ The quotient space is denoted by $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}},$ and the canonical map from ${\\cal B}\\times {\\mathbb {R}}^3$ onto $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}},$ is denoted by $\\pi _T.$ Thus $\\pi _T(w,z,\\overrightarrow{x})$ is the orbit of $(w,z,\\overrightarrow{x})$ , considered as an element of $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}}.$ The map $\\pi _T$ is a T-fold covering map and, since ${\\cal B}\\times {\\mathbb {R}}^3$ is simply connected, the Universal covering map of the quotient space.","The action of $G$ on ${\\cal B}\\times {\\mathbb {R}}^3$ conmutes with that of $\\@root T \\of {\\mathbf {1}}.$ As a consequence, the action on $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}}$ given by $(A,H)*((w,z,\\overrightarrow{x})\\@root T \\of {\\mathbf {1}})=((A,H)*(w,z,\\overrightarrow{x}))\\@root T \\of {\\mathbf {1}}$ is well defined, and obviously makes the covering map $\\pi _T$ equivariant.","The isotropy subgroup at $\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0}\\right)\\@root T \\of {\\mathbf {1}}$ is $Ker\\,C_{\\alpha _T},$ so that we can identify $G/Ker\\,C_{\\alpha _T}$ with $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}}$ by means of $(A,H)Ker\\,C_{\\alpha _T} \\longleftrightarrow (A,H)* \\left(\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0}\\right)\\@root T \\of {\\mathbf {1}}\\right).$ The contact form on ${\\cal B}\\times {\\mathbb {R}}^3,\\ \\Omega _1,$ is invariant under the action $``\\cdot \",$ so that there exist an unique contact form, $\\Omega _T,$ on $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}},$ such that $\\pi _T^*\\Omega _T=\\Omega _1.$ The action of $G$ on $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}}$ is transitive and preserves $\\Omega _T.$ The manifold $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}},$ when provided with the contact form $\\Omega _T$ and the cited action of $G,$ is the homogeneous contact manifold that correspond to massive particles with $T\\ge 1.$"],["Massless Type 4 particles. ","In this section we consider particles whose movement space is the coadjoint orbit of $\\alpha =\\left\\lbrace \\frac{i \\chi T}{8 \\pi }\\left(\\begin{array}{cc} 1 & 0\\\\0 & -1\\end{array}\\right),\\eta \\,\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)\\right\\rbrace \\ \\ \\ \\ ,\\ \\chi ,\\ \\eta \\in \\lbrace \\pm 1\\rbrace ,\\ T \\in {\\bf Z}^+ ,$ where $\\eta =-sign(Tr(P)),$ (see Table 2 of section ).","So, at least at the classical level, $\\eta $ must be interpreted as the opposite of the sign of energy.","We will see in the following subsections that this interpretation is also exact at the quantum level.","Since $|P|$ is mass square and $Det\\left( \\eta \\,\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right) \\right)=0$ these orbits correspond to particles with zero mass.","These are quantizable, not $\\bf R\\rm $ -quantizable orbits of the type 4.","The group $G_\\alpha $ is connected, so that there exists at most one homomorphism from $G_\\alpha $ onto ${\\bf S}^1$ whose differential is $\\alpha $ .","In fact we have $\\begin{array}{l}\\vspace{7.22743pt}G_\\alpha = \\left\\lbrace \\left(\\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&i \\chi \\eta T a z/2 \\pi \\\\\\overline{i \\chi \\eta T a z / 2 \\pi }&0\\end{array}\\right)\\right):\\ z \\in {\\bf S^1},\\right.","\\\\\\left.","\\ \\ \\ \\ \\ \\ \\ \\ \\ b\\in {\\bf R},\\ a \\in {\\bf C}\\right\\rbrace \\\\\\ \\\\\\mathrm {and\\ the\\ homomorphism} \\\\\\ \\\\C_\\alpha \\left( \\left( \\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&{i \\chi \\eta T a z/2 \\pi } \\\\{\\overline{i \\chi \\eta T a z/2 \\pi }}&0\\end{array}\\right)\\right)\\right)=z^{\\chi T}\\end{array}$ has differential $\\alpha .$ Other computations give us $\\begin{array}{l}({\\mbox{$G_\\alpha $}} )_{SL}=\\left\\lbrace \\left(\\begin{array}{cc} z&a \\\\0&{\\overline{z}}\\end{array}\\right): z \\in {\\bf S^1},\\ a \\in {\\bf C} \\right\\rbrace \\\\\\vspace{7.22743pt}(C_\\alpha )_{SL}\\left(\\ \\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right)\\right)=\\ z^{\\chi T} \\\\\\vspace{7.22743pt}SL_1=\\left\\lbrace \\left(\\begin{array}{cc}a&0 \\\\0&{1/a}\\end{array}\\right):\\ a \\in {\\bf C}\\right\\rbrace \\\\\\vspace{7.22743pt}SL_2=(G_\\alpha )_{SL} \\\\\\vspace{7.22743pt}SL_1 \\cap SL_2=\\left\\lbrace \\left(\\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array}\\right):\\ z \\in {\\bf S}^1\\right\\rbrace \\end{array}.$ The isotropy subgroup at $\\left(\\begin{array}{cc}1&0 \\\\0&0\\end{array}\\right.$ , for the usual action of $SL$(2,$C$) on H(2) (that is $A\\ast m=AmA^*$ ) is ${\\mbox{$(G_\\alpha )$}}_{SL}$ .","Thus SL/${\\mbox{$(G_\\alpha )$}}_{SL}$ will be identified to the orbit of $\\left(\\begin{array}{cc}1&0 \\\\0&0\\end{array}\\right),$ which is ${\\mbox{$\\bf C^+$}}=\\lbrace H \\in {H(2)}:\\ Det\\,H=0,\\ Tr\\,H>0\\rbrace .$ This set is composed by the hermitian matrices that represent space-time points in the future lightcone, so that it will be called itself, future lightcone.","When this identification is made, the function $P$ on SL/${\\mbox{$(G_\\alpha )$}}_{SL}$ becomes $P(H)=-\\eta \\,H$ .","An invariant volume element in ${\\mbox{$\\bf C^+$}}$ is $\\omega =\\frac{1}{\\Vert \\vec{ p}\\Vert } \\ dp^1\\wedge \\,dp^2\\wedge \\,dp^3,$ where $(p^1,\\ p^2,\\ p^3)$ is the coordinate system corresponding to the parametrization $\\phi :(p^1,\\ p^2,\\ p^3) \\in {\\bf R^3}-\\lbrace \\,0\\,\\rbrace \\mapsto h(\\vec{ p},\\Vert {\\vec{ p}\\Vert }) \\in {\\mbox{$\\bf C^+$}}$ where $\\vec{p}= (p^1,\\ p^2\\ p^3),$ and $\\Vert {\\vec{ p}}\\Vert =+ \\sqrt{ \\sum _{i=1}^3 (p^i)^2}.$ In this parametrization, the linear momentum is given by $P(\\phi (\\vec{ p}))=-\\eta \\ h(\\vec{ p},\\Vert {\\vec{ p}\\Vert }).$ Before to proceed to the study of the general case, we shall consider two particular ones."],["Massless antineutrino","Let us consider the case $ {T=1,\\ \\chi =1,\\ \\eta =\\,-\\,1}$ .","A trivialization is given by $\\left(\\rho _+ ,\\ {\\bf C}^2,\\ \\left(\\begin{array}{c}1 \\\\0\\end{array}\\right) \\right),$ where $\\rho _+$ (A) is multiplication by A.","The orbit of $\\left(\\begin{array}{c} 1\\\\0\\end{array}\\right)$ is ${\\bf C}^2-\\lbrace 0\\rbrace $ .","This space can thus be identified to SL/${\\mbox{$Ker\\,(C_\\alpha )$}}_{SL}$ so that the canonical map $SL/{\\mbox{$Ker\\,(C_\\alpha )$}}_{SL} \\longrightarrow SL/(G_\\alpha ) $ becomes a map $r_+ :{\\bf C}^2-\\lbrace 0\\rbrace \\mapsto {\\bf C}^+ .$ This map must be equivariant, so that, for all $A\\in SL$ $r_+\\left(A \\left(\\begin{array}{c}1 \\\\0\\end{array} \\right) \\right) =A\\left(\\begin{array}{cc}1&0 \\\\0&0\\end{array}\\right.","A^*.$ Thus $r_+$ is explicitly given by $r_+(z)=z\\,z^* \\in {\\mbox{$\\bf C^+$}} .$ Since $z$ and $\\varepsilon \\overline{z}$ are eigenvectors of $z\\,z^*$ corresponding to the eigenvalues $\\Vert z\\Vert ^2$ and 0 respectively, $r_+^{-1}(H)$ is composed by the eigenvectors of $H$ corresponding to the positive eigenvalue, whose norm is the square root of that eigenvalue.","The principal $ {\\bf S}^1$ -bundle whose projection is $r_+$ is related to the Hopf fibration as follows.","The image of the restriction of $r_+$ to the sphere $S^3(R)=\\lbrace z \\in { {\\bf C}}^2\\,:\\,\\Vert z\\Vert ^2=R^2\\rbrace $ is composed by the elements of ${\\mbox{$\\bf C^+$}}$ whose trace is $R^2$ .","Since the image of this subset by the preceeding chart is the sphere of radius $R^2$ /2, we obtain maps from spheres $S^3$ onto spheres $S^2$ .","Each one of these mappings is, up to the radius and a reflection, the Hopf fibration.","For each $S$ -homogeneous of degree -1 function on ${\\bf C}^2-{0},\\ f, $ we have the following prewave function ${\\psi } _f^+:\\,(N,\\ H) \\in {\\mbox{$\\bf C^+$}}\\times {H(2)}\\mapsto f(z)\\ e^{i\\pi \\,Tr(\\,N \\varepsilon \\overline{H} \\varepsilon ) } z \\in { {\\bf C}}^2,$ where z is an arbitrary element of $r_+^{-1}(N)$ .","Here we use the fact that, since $\\eta =-1$ , the linear momentum is given in $C^+$ by $P(N)=N.$ If $X$ and $Q$ are the elements of ${\\bf R}^4$ corresponding to $H$ and $N$ , one can write ${\\psi } _f^+(Q,X) = f(z)\\ e^{-2\\pi i\\langle Q,X \\rangle } z ,$ where $z$ is arbitrary in $r_+^{-1}(Q).$ The corresponding wave function , if $f$ is continuous with compact support, is ${\\widetilde{\\psi }} _f^+(H)= \\int _{C^+} {\\psi } _f^+(\\cdot ,\\ H) \\omega .$ By direct computation one can see that $\\left(\\ \\sigma _1 \\ \\frac{\\partial }{\\partial x^1}+\\ \\sigma _2 \\ \\frac{\\partial }{\\partial x^2}+\\ \\sigma _3 \\ \\frac{\\partial }{\\partial x^3}+\\ \\sigma _4 \\ \\frac{\\partial }{\\partial x^4} \\right)\\ {\\psi }_f^+(N,\\ h(x))\\,=\\,0$ The corresponding wave function thus satisfy the same equation, which is the Weyl equation (positive energy), usually admised as corresponding to the antineutrino.","If $f$ and $f^\\prime $ are pseudotensorial functions, we have $({\\psi } _f^+(N,H))^*\\ {\\psi }_{f^\\prime }^+(N,H)=\\overline{f(z)}\\, f^\\prime (z)\\ Tr\\,N.$ Thus, the hermitian product of the corresponding quantum states ($cf.$ section ) can be written as follows ${\\frac{1}{2}}\\ \\int _{\\bf C ^+} \\frac{({\\psi }^+ _f)^*\\ {\\psi }^+_{f^\\prime }}{\\sum _{i=1}^3 (p^i)^2}\\ dp^1\\,dp^2\\,dp^3.$ To obtain prewave functions directly from functions on $C^+,$ we use Remark REF , as follows.","Let $U= \\phi ({\\bf R}^3-\\lbrace p^1=p^2=0, p^3<0\\rbrace ),$ $V= \\phi ({\\bf R}^3-\\lbrace p^1=p^2=0, p^3>0\\rbrace ).$ Then the maps $\\sigma _U : \\phi (p^1,p^2,p^3)\\in U \\mapsto \\left({\\begin{array}{c} \\sqrt{\\Vert \\vec{ p}\\Vert +p^3} \\\\\\frac{p^1+i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert +p^3}} \\end{array}}\\right) \\in {\\bf C}^2-{0}$ $\\sigma _V : \\phi (p^1,p^2,p^3)\\in V \\mapsto \\left( {\\begin{array}{c} \\frac{p^1-i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert -p^3}} \\\\\\sqrt{\\Vert \\vec{ p}\\Vert -p^3} \\end{array}}\\right) \\in {\\bf C}^2-{0}$ are sections of $r_+,$ so that we obtain prewave functions having the form $\\psi (\\phi (p^1,p^2,p^3), X)=F(p^1,p^2,p^3) \\left({\\begin{array}{c} \\sqrt{\\Vert \\vec{ p}\\Vert +p^3} \\\\\\frac{p^1+i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert +p^3}} \\end{array}}\\right)$ $ \\displaystyle e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ where $F$ is a complex valued function continuous on ${\\bf R}^3-\\lbrace 0\\rbrace ,$ with compact support in ${\\bf R}^3-\\lbrace p^1=p^2=0, p^3<0\\rbrace ,$ and prewave functions having the form $\\theta (\\phi (p^1,p^2,p^3), X)=J(p^1,p^2,p^3) \\left( {\\begin{array}{c} \\frac{p^1-i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert -p^3}} \\\\\\sqrt{\\Vert \\vec{ p}\\Vert -p^3} \\end{array}}\\right)$ $ \\displaystyle e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ where $J$ is a complex valued function continuous on ${\\bf R}^3-\\lbrace 0\\rbrace ,$ with compact support in ${\\bf R}^3-\\lbrace p^1=p^2=0, p^3>0\\rbrace .$ If one is interested in the case $T=1,\\ \\chi =1,\\ \\eta =+1$ , everything is as in the case $\\eta =-1$ , but the exponent of $e$ in the prewave functions changes its sign, thus giving wave functions for quantum states of negative energy.","Remenber that $\\eta $ is the opposite of the sign of energy at the clasical level."],["Massless neutrino","Now let us consider the case in which $T=1,\\ \\chi =-1,\\ \\eta =-1$ .","A trivialisation in this case is $\\left(\\rho _- ,\\ {\\bf C}^2,\\ \\left(\\begin{array}{c}0 \\\\1\\end{array}\\right) \\right),$ $ \\rho _- $ (A) being multiplication by $(A^*)^{-1}.$ Thus we identify $SL/Ker\\,C_{\\alpha }$ with the orbit of $ \\left(\\begin{array}{c}0 \\\\1\\end{array}\\right),$ which, also in this case, is ${\\bf C}^2-\\lbrace 0\\rbrace .$ Thus,the canonical map $SL/{\\mbox{$Ker\\,(C_\\alpha )$}}_{SL} \\longrightarrow SL/(G_\\alpha ) $ becomes a map $r_- :{\\bf C}^2-\\lbrace 0\\rbrace \\mapsto {\\bf C}^+ .$ Now, using equivariance as in the case of antineutrino, we obtain $r_-(z)\\,=\\,-\\varepsilon \\overline{z\\,z^* }\\varepsilon .","$ The prewave functions one obtains have exactly the same form that (REF ) and (REF ), but now $z$ is in $(r_-)^{-1}(N)$ or $(r_-)^{-1}(h(Q))$ respectively.","The wave functions one obtains in this case satisfies the Weyl equation that, according to Feynman, corresponds to the neutrino.","The map from the real vector space $\\bf C \\rm ^2$ onto itself defined by sending $z$ to $ \\varepsilon \\overline{z}$ , is a complex structure and its restriction to $\\bf C\\rm ^2-\\lbrace 0 \\rbrace $ , gives us an isomorphism of the principal circle bundle corresponding to ${\\bf r}_{-}$ ( resp.${\\bf r}_+$ ) onto the principal circle bundle corresponding to ${\\bf r}_+$ ( resp.","${\\bf r}_-$ ).","The isomorphism of the structural group is defined by sending each element to its inverse.","Thus if $z\\in r_+^{-1}(H)$ then $\\varepsilon \\overline{z}\\in r_-^{-1}(H)$ and conversely.","As a consequence, the sections of the map $r_+$ defined in REF give rise to the following sections of $r_-$ $\\sigma ^\\prime _U(u)=\\varepsilon \\overline{\\sigma _U(u)},\\ \\ \\sigma ^\\prime _V(v)=\\varepsilon \\overline{\\sigma _V(v)},$ for all $u\\in U$ and $v\\in V,$ and the prewave functions of antineutrino $\\psi $ and $\\theta $ give rise to prewave functions of neutrino, $\\psi ^\\prime $ and $\\theta ^\\prime $ , by means of $\\psi ^\\prime (\\phi (p^1,p^2,p^3), X)=F(p^1,p^2,p^3)\\varepsilon \\overline{\\left({\\begin{array}{c} \\sqrt{\\Vert \\vec{ p}\\Vert +p^3} \\\\\\frac{p^1+i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert +p^3}} \\end{array}}\\right) }$ $e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ $\\theta ^\\prime (\\phi (p^1,p^2,p^3), X)=J(p^1,p^2,p^3) \\varepsilon \\overline{\\left( {\\begin{array}{c} \\frac{p^1-i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert -p^3}} \\\\\\sqrt{\\Vert \\vec{ p}\\Vert -p^3} \\end{array}}\\right) }$ $e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ that leads to $\\psi ^{\\prime }(\\phi (p^1,p^2,p^3), X)&=& F(p^1,p^2,p^3){\\left({{\\begin{array}{c} \\frac{p^1-i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert +p^3}}\\\\-\\sqrt{\\Vert \\vec{ p}\\Vert +p^3} \\end{array}}}\\right) }\\\\ &&{e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}} \\\\\\ \\theta ^{\\prime }(\\phi (p^1,p^2,p^3), X)&=&J(p^1,p^2,p^3){\\left( {{\\begin{array}{c}\\sqrt{\\Vert \\vec{ p}\\Vert -p^3}\\\\{-\\frac{p^1+i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert -p^3}}} \\end{array}}}\\right)}\\\\ && {e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}}.$ If one consider the case $T=1,\\ \\chi =-1,\\ \\eta =+1$ one obtain similar wave functions but corresponding to negative energy."],["General case","We proceed as in section REF .","In the case $\\chi =+1$ a trivialization is given by $\\left((\\rho _+)^{\\otimes T},\\ \\left({\\bf C}^2\\right)^ {\\otimes T},\\ \\left(\\begin{array}{c} 1\\\\0\\end{array}\\right)^{\\otimes T}\\right),$ and if $\\chi =-1$ a trivialization is given by $\\left((\\rho _-)^{\\otimes T},\\ \\left({\\bf C}^2\\right)^{\\otimes T},\\ \\left(\\begin{array}{c} 0\\\\1\\end{array}\\right)^{\\otimes T}\\right),$ The prewave functions are given by functions on ${\\bf C} \\rm ^2-\\lbrace 0\\rbrace $ , which are $C^\\infty $ with compact support and homogeneous of degree -T under multiplication by modulus one complex numbers.","Let $f_T$ be one of these functions.","If $\\chi =1$ , the corresponding prewave function is given by $\\psi ^+_{f_T}:\\,(N,\\ H) \\in {\\mbox{$\\bf C^+$}}\\times {H(2)}\\mapsto f_T(z)\\ e^{-i \\pi \\eta \\,Tr(N \\varepsilon \\overline{H} \\varepsilon ) } z^{\\otimes T}\\in ({\\bf C}^2)^{\\otimes T}$ where $z$ is an arbitrary element of $r_+^{-1}(N)$ .","In the case $\\chi =-1$ , the corresponding prewave function is $\\psi ^-_{f_T}:\\,(N,\\ H) \\in {\\mbox{$\\bf C^+$}}\\times {H(2)}\\mapsto f_T(z)\\ e^{-i \\pi \\eta \\,Tr(N \\varepsilon \\overline{H} \\varepsilon )} z^{\\otimes T}\\in ({\\bf C}^2)^{\\otimes T}$ but now, z is an arbitrary element of $r_-^{-1}(N)$ .","The associated wave functions, ${\\widetilde{\\psi }}^+_{f_T}(H)&=&\\int _{C^+} {\\psi } ^+_{f_T}(\\cdot ,\\ H) \\omega ,\\\\{\\widetilde{\\psi }}^-_{f_T}(H)&=&\\int _{C^+} {\\psi } ^-_{f_T}(\\cdot ,\\ H) \\omega $ satisfies Penrose's wave equations,that we will describe here for the sake of completeness.","Let us consider in $({\\bf C}^2)^{\\otimes T}$ the basis $\\lbrace e_A \\otimes e_B \\otimes \\stackrel{(T}{\\cdots }\\ :\\ A,\\ B,\\dots \\in \\lbrace 1,\\ 2\\rbrace \\rbrace $ , where $\\lbrace e_1,\\ e_2\\rbrace $ is the canonical basis of $\\bf C \\rm ^2$ .","The prewave functions $\\psi ^\\pm _{f_T}$ and the wave functions $\\tilde{\\psi }^\\pm _{f_T}$ , have components in this basis which will be denoted by $\\lbrace \\psi _\\pm ^{A\\,B\\dots } \\rbrace $ and $\\lbrace \\tilde{\\psi }_\\pm ^{A\\,B\\dots } \\rbrace $ , respectively.","Let us consider the vector fields in $\\bf R \\rm ^4$ given by $\\nabla _{11}&=&\\frac{1}{2} \\left( \\frac{\\partial }{\\partial x^3}+\\frac{\\partial }{\\partial x^4} \\right)\\\\\\nabla _{12}&=&\\frac{1}{2} \\left( \\frac{\\partial }{\\partial x^1}-i\\frac{\\partial }{\\partial x^2} \\right)\\\\\\nabla _{21}&=&\\frac{1}{2} \\left( \\frac{\\partial }{\\partial x^1}+i\\frac{\\partial }{\\partial x^2} \\right)\\\\\\nabla _{22}&=&\\frac{1}{2} \\left( \\frac{\\partial }{\\partial x^4}-\\frac{\\partial }{\\partial x^3} \\right)$ and, for all $A, A^\\prime \\in \\lbrace 1,\\ 2\\rbrace ,\\ \\nabla ^{A \\ A^\\prime }=\\varepsilon ^{A\\ B}\\ \\varepsilon ^{A^\\prime \\ B^\\prime }\\ \\nabla _{B\\ B^\\prime }$ (summation convention), where $\\lbrace \\varepsilon ^{A\\ B}\\rbrace $ are the elements of $-\\varepsilon $ .","We also define $\\psi ^\\pm _{A\\,B\\dots }&=&\\varepsilon _{A\\ A^\\prime }\\ \\varepsilon _{B\\ B^\\prime }\\dots \\psi _\\pm ^{A^\\prime \\,B^\\prime \\dots } \\\\\\tilde{\\psi }^\\pm _{A\\,B\\dots }&=&\\varepsilon _{A\\ A^\\prime }\\ \\varepsilon _{B\\ B^\\prime }\\dots \\tilde{\\psi }_\\pm ^{A^\\prime \\,B^\\prime \\dots }$ where $\\lbrace \\varepsilon _{A\\ B}\\rbrace $ are the elements of $\\varepsilon $ .","Thus we have for all $h(x) \\in C^+$ $\\nabla ^{A\\ A^\\prime }\\ \\psi ^+_{A^\\prime \\,B\\ C\\dots }(h(x),\\,\\cdot \\, )&=&0\\\\\\nabla ^{ A^\\prime \\ A}\\ \\psi ^-_{A^\\prime \\,B\\ C\\dots }(h(x),\\,\\cdot \\,)&=&0$ so that, by derivation under the integral sign, we see that Penrose wave equations: $\\nabla ^{A\\ A^\\prime }\\ \\tilde{\\psi }^+_{A^\\prime \\,B\\ C\\dots }&=&0\\\\\\nabla ^{ A^\\prime \\ A}\\ \\tilde{\\psi }^-_{A^\\prime \\,B\\ C\\dots }&=&0$ are satisfied."],["Helicity","In formula (REF ) one sees that, if $(\\rho ,L,z_0)$ is the trivialization we use, the components of spin operator are given by $\\hat{s^k}:\\widetilde{\\psi }_f \\longrightarrow s^k \\circ \\widetilde{\\psi }_f,$ $s^k$ being the following endomophism of $L$ $s^k=\\frac{1}{2\\pi i}\\frac{\\widetilde{i\\sigma _k}}{2}=\\frac{1}{4\\pi i}\\widetilde{i\\sigma _k},$ where, $\\widetilde{i\\sigma _k}=d\\rho ({i\\sigma _k}).$ Thus, in the case $\\chi =1,$ $\\widetilde{i\\sigma _k}=d\\left((\\rho _+)^{\\otimes T}\\right)({i\\sigma _k}),$ and in the case $\\chi =-1,$ $\\widetilde{i\\sigma _k}=d\\left((\\rho _-)^{\\otimes T}\\right)({i\\sigma _k}).$ As a consecuence, in both cases, $\\chi =+ 1$ or $\\chi = 1,$ when acting on monomials we have $\\widetilde{i\\sigma _k}\\cdot z_1\\otimes \\dots \\otimes z_T=\\sum _{j=1}^T\\ z_1\\otimes \\dots \\otimes z_{j-1}\\otimes i\\, \\sigma _k\\,z_j\\otimes z_{j+1}\\otimes \\dots \\otimes z_T.$ We consider the operators $\\hat{s^k}$ as also acting on prewave functions by the same formula that in the case of wave functions, without the tilde.","On the other hand, Linear Momentum is given on $C^+$ by $P(K)=-\\eta K,\\ \\text{\\rm for all}\\ K\\in C^+.$ Then, with the notation $K=h(K^1,K^2,K^3,K^4)$ and $P=h(P^1,P^2,P^3,P^4),$ the $P^k$ are functions on $C^+,$ given by $P^k(K)=-\\eta K^k.$ We also denote $\\overrightarrow{K}&=&(K^1,K^2,K^3)\\\\\\Vert \\overrightarrow{K}\\Vert &=&+\\left(\\sum _{k=1}^3({K^k})^2\\right)^{1/2}\\\\\\overrightarrow{P}&=&(P^1,P^2,P^3)\\\\ \\Vert \\overrightarrow{P}\\Vert &=&+\\left(\\sum _{k=1}^3({P^k})^2\\right)^{1/2}.$ The Helicity Operator is defined by $\\mathfrak {h}=\\frac{1}{\\Vert \\overrightarrow{P}\\Vert } {\\sum _{k=1}^3}{P^k}\\hat{s^k},$ which means $\\left(\\mathfrak {h}\\cdot \\psi _{f_T}^{\\pm }\\right)(K,H)=\\frac{1}{\\Vert \\overrightarrow{P}\\Vert (K)} {\\sum _{k=1}^3}{P^k(K)}{s^k}\\left(\\psi _{f_T}^{\\pm }(K,H)\\right).$ Then, if $z\\in r^{-1}_{\\pm }(K)$ and $z_1=\\dots =z_T=z,$ we have $&&\\left(\\mathfrak {h}\\cdot \\psi _{f_T}^{\\pm }\\right)(K,H)=\\frac{1}{K^4} \\sum _{k=1}^3{P^k(K)}{s^k}\\left(f_T(z)e^{-i\\eta \\pi TrK\\epsilon \\overline{H}\\epsilon } z^{\\otimes T}\\right)=\\\\&&=\\frac{1}{4 \\pi K^4}\\ f_T(z)e^{-i\\eta \\pi TrK\\epsilon \\overline{H}\\epsilon }\\\\&&\\left( \\sum _{j=1}^T\\ z_1\\otimes \\dots \\otimes z_{j-1}\\otimes (\\sum _{k=1}^3{P^k(K)}\\, \\sigma _k\\,z_j)\\otimes z_{j+1}\\otimes \\dots \\otimes z_T\\right)=\\\\&&=\\frac{1}{4 \\pi K^4} \\ f_T(z)e^{-i\\eta \\pi TrK\\epsilon \\overline{H}\\epsilon } \\\\&& \\left( \\sum _{j=1}^T\\ z_1\\otimes \\dots \\otimes z_{j-1}\\otimes (-\\eta )((K-K^4\\,I)z_j)\\otimes z_{j+1}\\otimes \\dots \\otimes z_T\\right).$ In the case $\\chi =+1,$ we have $K= r_+(z)=zz^*,$ so that $ (K-K^4\\,I)z=(zz^*-\\frac{\\Vert z \\Vert }{2}\\,I)z=K^4\\,z,$ and then $\\left(\\mathfrak {h}\\cdot \\psi _{f_T}^{+}\\right)(K,H)=\\frac{-\\eta T}{4\\pi }\\psi _{f_T}^{+}(K,H).$ On the other hand, in the case $\\chi =-1,$ we have $ K=-\\epsilon \\overline{zz^*}\\epsilon .$ Thus $(K-K^4\\,I)z=(-\\epsilon \\overline{zz^*}\\epsilon -K^4\\,I)z=-K^4\\,z,$ so that $\\left(\\mathfrak {h}\\cdot \\psi _{f_T}^{-}\\right)(K,H)=\\frac{\\eta T}{4\\pi }\\psi _{f_T}^{-}(K,H).$ Thus $\\psi _{f_T}^{\\pm }$ is an eigenvector of the helicity operator, corresponding to the eigenvalue $-\\eta \\chi T/4 \\pi .$ In particular, the sign of helicity is $-\\eta \\chi .$"],["The Homogeneous Contact and Symplectic Manifolds for Massless particles of type 4. Twistors.","Let us denote by $\\@root T \\of {\\mathbf {1}}=\\lbrace r_1, \\dots ,r_T\\rbrace ,$ the group composed by the roots of order $T$ of 1, and $\\nu =\\frac{- \\eta \\chi T}{4\\pi } .$ In section REF we have seen that all wave functions obtained in section REF are eigenvectors of helicity with eigenvalue $\\nu .$ The group $Ker\\,C_\\alpha ,$ has $T$ connected components $(Ker\\,C_\\alpha )_j =\\left\\lbrace \\left(\\left(\\begin{array}{cc}r_j&a \\\\0&{\\overline{r}_j}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&-2i\\nu \\,a\\, r_j \\\\\\overline{-2i\\nu \\,a\\, r_j } &0\\end{array}\\right)\\right):b\\in {\\bf R},\\ a \\in {\\bf C}\\right\\rbrace $ The component of the identity is $(Ker\\,C_\\alpha )_o =\\left\\lbrace \\left(\\left(\\begin{array}{cc}1&a \\\\0&{1}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&-2i\\nu \\,a \\\\\\overline{\\-2i\\nu \\,a}&0\\end{array}\\right)\\right):b\\in {\\bf R},\\ a \\in {\\bf C}\\right\\rbrace $ and $(Ker\\,C_\\alpha )_j $ is the left translation by $\\tilde{\\bf r}_j \\stackrel{def}{=}\\left( \\left( \\begin{array}{cc}r_j&0 \\\\0&{\\overline{r}_j}\\end{array}\\right),\\ 0 \\right)$ of $(Ker\\,C_\\alpha )_o .$ As a consequence, the group $Ker\\,C_\\alpha /(Ker\\,C_\\alpha )_o$ is isomorphic to $\\@root T \\of {\\mathbf {1}}$ .","The canonical map ${\\begin{array}{c}\\tilde{\\bf c} : (A,H)\\,(Ker\\,C_\\alpha )_o\\in \\frac{G}{(Ker\\,C_\\alpha )_o} \\longrightarrow \\\\\\longrightarrow (A,H)\\,Ker\\,C_\\alpha \\in \\frac{G}{Ker\\,C_\\alpha }.\\end{array}}$ is a $T$ -fold covering map, whose structural group is $Ker\\,C_\\alpha /(Ker\\,C_\\alpha )_o$ .","Thus, $G/Ker\\,C_\\alpha $ is the quotient of $G/(Ker\\,C_\\alpha )_o$ by a properly discontinuous with no fixed point action of $Ker\\,C_\\alpha /(Ker\\,C_\\alpha )_o$ .","But, as we have seen, this group can be changed to $\\@root T \\of {\\mathbf {1}}$ .","The action of $\\@root T \\of {\\mathbf {1}}$ on $G/(Ker\\,C_\\alpha )_o$ corresponding to this construction is: $((A,H)\\,(Ker\\,C_\\alpha )_o)* r_j= ((A,H)\\,\\tilde{\\bf r}_j)\\,(Ker\\,C_\\alpha )_o.$ In what follows, we identify $G/Ker\\,C_\\alpha $ with the quotient space of $G/(Ker\\,C_\\alpha )_o$ by this action.","We have the following conmutative diagram ${\\begin{matrix}\\frac{G}{(Ker\\,C_\\alpha )_o } &\\xrightarrow{}& \\frac{G}{Ker\\,C_\\alpha } \\\\\\downarrow && \\downarrow && \\\\\\frac{G}{G_\\alpha } &=& \\frac{G}{G_\\alpha }\\end{matrix}}$ where the vertical arrow on the right is the bundle map of the contact manifold onto the symplectic manifold, for general $T.$ If $T=1$ both vertical arrows are the same.","The homomorphism $\\mu _1:(A, H) \\in SL\\oplus H(2) \\longrightarrow \\left( \\begin{array}{cc}A&{-iH{A^*}^{-1}} \\\\0&{A^*}^{-1}\\end{array} \\right)\\in GL(4,\\ \\text{\\bf C}).$ is a representation in ${\\bf C}^4.$ The isotropy subgroup at $q \\stackrel{def}{=}\\left( \\begin{array}{c}0\\\\ 2\\nu \\\\0\\\\ 1\\end{array}\\right)$ is $(Ker\\,C_\\alpha )_o .$ Let us denote by $\\mathrm {O}_q$ the orbit of $q$ by this representation.","In order to describe $\\mathrm {O}_q$ in another way, we consider in $\\text{\\bf C}^4$ the hermitian product $<\\binom{\\hat{w}}{\\hat{z}},\\ \\binom{w}{z}>=\\frac{1}{2} (\\hat{w}^*z+\\hat{z}^*w)\\ \\ \\ \\ \\ \\forall \\hat{w},\\,\\hat{z},\\,w,\\,z \\in \\text{\\bf C}^2$ whose signature and quadratic form are respectively (+, +, -, -) and $\\Phi \\binom{w}{z} =Re\\,z^*\\,w.$ The complex vector space $\\text{\\bf C}^4$ provided with this hermitian product is Penrose's Twistor Space.","The representation $\\mu _1$ preserves $\\Phi $ , so that any orbit must be contained in a subset of the form $\\Phi =\\mathnormal {constant}.$ It follows that the orbit of $q$ is contained in the seven dimensional submanifold, $\\mathcal {O}_\\nu $ , given by $\\Phi =2\\nu .$ Let $\\binom{w}{z}\\in \\text{\\bf C}^4$ be such that $sign\\left(\\Phi \\binom{w}{z} \\right)=sign(\\nu ),$ and define $A(w,z)&=&\\left(\\frac{2\\nu }{ \\Phi \\binom{w}{z}}\\right)^{1/2}\\left(\\begin{array}{c} {}\\\\\\epsilon \\overline{z}\\\\{} \\end{array}\\ \\Bigg | \\ \\frac{1}{2\\nu }\\left(i\\frac{Im(z^*w)}{\\Vert z \\Vert ^2}z-w\\right)\\right)\\\\ H(w,z) &=& -\\frac{Im(z^*w)}{\\Vert z \\Vert ^2}\\ I.$ Then we have $\\mu _1(A(w,z),H(w,z)) q=\\left(\\frac{2\\nu }{ \\Phi \\binom{w}{z}}\\right)^{1/2}\\ \\binom{w}{z}.$ This enables us to prove that each element of $\\mathcal {O}_\\nu $ is in the orbit, so that $\\mathrm {O}_q=\\mathcal {O}_\\nu $ .","Then, the map from $G/(Ker\\,C_\\alpha )_o$ onto $ \\mathcal {O}_\\nu ,$ given by $\\tau _T:(A,H)\\,(Ker\\,C_\\alpha )_o\\in \\frac{G}{(Ker\\,C_\\alpha )_o} \\longrightarrow \\mu _1(A,H)\\,q\\in \\mathcal {O}_\\nu ,$ is a diffeomorphism.","If we identify these spaces by $\\tau _T,$ the action of $\\@root T \\of {\\mathbf {1}}$ on $G/(Ker\\,C_\\alpha )_o$ given by (REF ), traslates to an action on $\\mathcal {O}_\\nu .$ Since $\\tau _T((A,H)\\,(Ker\\,C_\\alpha )_o) *r_j)&=&\\mu _1 ((A,H)\\,\\tilde{\\bf r}_j)q=\\mu _1 (A,H)\\mu _1(\\tilde{\\bf r}_j)q\\\\=\\mu _1 (A,H)(\\overline{r_j}q)&=&\\overline{r_j}\\ \\tau _T((A,H)\\,(Ker\\,C_\\alpha )_o) ,$ the action of $\\@root T \\of {\\mathbf {1}}$ on $\\mathcal {O}_\\nu $ is given by ordinary product by the conjugated: $V*r_j=\\overline{r_j}\\ V$ Let us denote by $\\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}}$ the quotient space of $\\mathcal {O}_\\nu $ by this action.","It follows from the preceeding construction that the contact manifold, $G/Ker\\,C_\\alpha ,$ is diffeomorphic, and will be identified, to $\\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}}$ .","The contact form will be determinated later on.","Let ${}^t(w^1,w^2,z^1,z^2)\\in {\\mathbb {C}}^4-\\lbrace 0\\rbrace ,$ and let us denote by $[{}^t(w^1,w^2,z^1,z^2)]$ the complex vector subspace of ${\\mathbb {C}}^4$ generated by ${}^t(w^1,w^2,z^1,z^2).$ The set composed by these subspaces is the complex projective space of ${\\mathbb {C}}^4,$ ${\\bf P}_3(\\mathbb {C}),$ and we consider it as provided with its canonical differentiable structure.","Penrose also considers the subsets of twistor space, ${\\bf T}^+,\\ {\\bf T}^-,\\ {\\bf T}^0,$ given by $\\Phi \\,>\\,0,\\ \\Phi \\,<\\,0,\\ \\Phi \\,=\\,0$ , respectively, and the subsets of projective space $ {\\bf P}_3^+=\\pi ({\\bf T}^+),\\ {\\bf P}_3^-=\\pi ({\\bf T}^-),\\ {\\bf P}_3^0=\\pi ({\\bf T}^0)$ , where $\\pi :{\\mathbb {C}}^4 -\\lbrace 0\\rbrace \\longrightarrow {\\bf P}_3({\\mathbb {C}}) $ is the canonical map.","Since $\\Phi $ is preserved by the representation, the subsets ${\\bf T}^+,\\ {\\bf T}^-,\\ {\\bf T}^0,$ are stable under $\\mu _1$ .","The representation $\\mu _1$ also defines an action of $G$ on ${\\bf P}_3(\\text{\\bf C}) $ such that $\\pi $ is equivariant.","Explicitly $(A,H)*\\left[ \\left( \\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\ z^2\\end{array}\\right)\\right]=\\left[\\left( \\begin{array}{cc}A&{-iH{A^*}^{-1}} \\\\0&{A^*}^{-1}\\end{array} \\right)\\left( \\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\ z^2\\end{array}\\right)\\right].$ As a consecuence of formula (REF ), the open subsets of ${\\bf P}_3(\\text{\\bf C}) $ denoted by ${\\bf P}_3^+$ and ${\\bf P}_3^-$ are orbits of this action.","The point $[q]$ of ${\\bf P}_3(\\text{\\bf C}) $ is obviously in ${\\bf P}_3^{sign(\\nu )}$ so that its orbit is this open subset.","The isotropy subgroup at $[q]$ is $G_\\alpha .$ As a consecuence we have a diffeomorphism from ${\\bf P}_3^{sign(\\nu )}$ onto the coadjoint orbit of $\\alpha ,$ given by $\\Pi : [\\mu _1(A,H)\\,q]\\in {\\bf P}_3^{sign(\\nu )} \\longrightarrow Ad^*_{(A,H)}\\alpha \\in {\\mathcal {M}S} .$ were I have denoted the coadjoint orbit by ${\\mathcal {M}S},$ because of its interpretation as Movement Space.","We have $\\Pi \\left(\\left[ \\binom{ w}{ z} \\right]\\right)= Ad^*_{(A(w,z),H(w,z))}\\alpha $ so that, the formula for coadjoint representation in section ,(REF ), thus leads, after some computation, to $\\Pi \\left( \\left[ \\binom{ w}{ z} \\right] \\right)=\\frac{2\\nu \\eta }{ \\Phi \\binom{ w}{ z}} \\biggl \\lbrace \\frac{i}{4}(wz^*+\\epsilon \\overline{zw^*}\\epsilon ),\\ - \\epsilon \\overline{zz^*}\\epsilon \\biggr \\rbrace .$ We identify $P_3^{sign(\\nu )}$ to ${\\mathcal {M}S},$ by means of $\\Pi .$ Section provides us with well defined expresions for linear and angular momentum in $\\mathcal {M}S,$ $c.f.$ (REF ), which, when composed with $\\Pi ,$ give expressions for linear and angular momentum in $P^{sign(\\nu )}_3.$ In its hermitian form, these expressions are $h(P\\left( \\left[ \\binom{ w}{ z} \\right] \\right))&=& \\frac{2\\eta \\nu }{\\Phi \\binom{ w}{ z}} \\epsilon \\overline{zz^*}\\epsilon \\\\ h(\\vec{l} \\left( \\left[ \\binom{ w}{ z} \\right] \\right),0)&=&\\frac{\\eta \\nu }{2\\Phi \\binom{ w}{ z}} (zw^*+wz^*+\\epsilon \\overline{(zw^*+wz^*)}\\epsilon )\\\\ h(\\vec{g} \\left( \\left[ \\binom{ w}{ z} \\right] \\right),0)&=&\\frac{i\\,\\eta \\nu }{2\\Phi \\binom{ w}{ z}}(zw^*-wz^*-\\epsilon \\overline{(zw^*-wz^*)}\\epsilon )$ The Pauli-Lubanski four vector, when evaluated according with (REF ), is found to be $h(W\\left( \\left[ \\binom{ w}{ z} \\right] \\right)) =-\\eta \\nu \\,h(P\\left( \\left[ \\binom{ w}{ z}\\right] \\right)).","$ In case T=1, the bundle map of the contact manifold onto the coadjoint orbit becomes the canonical map $\\pi _1:\\binom{ w}{ z} \\in \\mathcal {O}_1\\ \\longrightarrow \\left[ \\binom{ w}{ z} \\right]\\in P^{sign(\\nu )}_3,$ and in the general case, the bundle map is $\\pi _\\nu :\\binom{ w}{z} \\@root T \\of {\\mathbf {1}}\\in \\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}} \\longrightarrow \\left[ \\binom{ w}{ z} \\right]\\in P^{sign(\\nu )}_3.$ Then, the composition with $\\pi _\\nu $ of the canonical dynamical variables, are also given by the right hand sides of (), () and (), but taking $\\Phi =2\\nu .$ We obtain the following formulae for the components of these dynamical variables $P^1\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right) &=\\frac{\\eta }{2}(z^1\\overline{z}^2\\,+\\,z^2\\overline{z}^1) \\\\P^2\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{i\\eta }{2}(z^1\\overline{z}^2\\,-\\,z^2\\overline{z}^1)\\\\P^3\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{\\eta }{2}(\\vert z^1\\vert ^2\\,-\\,\\vert z^2\\vert ^2) \\\\P^4\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=-\\frac{\\eta }{2}(\\vert z^1\\vert ^2\\,+\\,\\vert z^2\\vert ^2)$ $l^1\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{\\eta }{4} (z^1\\overline{w}^2+w^1\\overline{z}^2+z^2\\overline{w}^1+w^2\\overline{z}^1) \\\\l^2\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{i\\eta }{4} (z^1\\overline{w}^2+w^1\\overline{z}^2-z^2\\overline{w}^1-w^2\\overline{z}^1) \\\\l^3\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{\\eta }{4} (z^1\\overline{w}^1+w^1\\overline{z}^1-z^2\\overline{w}^2-w^2\\overline{z}^2)$ $g^1\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{i\\eta }{4} (z^1\\overline{w}^2-w^1\\overline{z}^2+z^2\\overline{w}^1-w^2\\overline{z}^1) \\\\g^2\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=-\\frac{\\eta }{4} (z^1\\overline{w}^2-w^1\\overline{z}^2-z^2\\overline{w}^1+w^2\\overline{z}^1) \\\\g^3\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{i\\eta }{4} (z^1\\overline{w}^1-w^1\\overline{z}^1-z^2\\overline{w}^2+w^2\\overline{z}^2).$ We can also consider the functions on ${\\bf T}^{sgn(\\nu )}$ obtained by composing the dynamical variables in $\\mathcal {M}S$ with the canonical projection, $\\pi ,$ from ${\\bf T}^{sgn(\\nu )}$ onto ${\\bf P}_3^{sgn(\\nu )},$ thus obtaining functions whose expresion is as the right hand sides of the preceeding formulae multiplied by $\\frac{2\\nu }{ \\Phi \\binom{ w}{ z}}$ These expressions coincide, up to notational conventions, with the expressions that R.Penrose gives for its energy - momentum and angular momentum in twistor space.","We denote these functions by $\\widetilde{P}^k,\\ \\widetilde{l}^k,\\ \\widetilde{g}^k.$ In general, for all $(a,h)$ in the Lie algebra of $G,$ the function it defines on the coadjoint orbit, identified to ${\\bf P}_3^{sgn(\\nu )},$ is denoted by the same symbol, $(a,h),$ and its composition with $\\pi ,$ $\\widetilde{(a,h)}.$ In a similar way, the infinitesimal generator of the action on ${\\bf T}^{sgn(\\nu )}$ defined by the representation $\\mu _1,$ associated to $(a,h),\\ i.e.$ the vector field whose flow is given by $\\mu _1(Exp(-t(a,h))),$ is denoted by $\\widetilde{X_{(a,h)} }.$ Let us consider the following one form on ${\\bf T}^{sgn(\\nu )}$ $\\omega _0=\\frac{i\\eta \\nu }{2\\Phi } (z^1d\\overline{w}^1+w^1d\\overline{z}^1+z^2d\\overline{w}^2+ w^2d\\overline{z}^2 -$ $- \\overline{z}^1 d w^1- \\overline{w}^1d z^1-\\overline{z}^2d w^2-\\overline{w}^2dz^2)$ A computation lead us to $\\omega _0\\left( \\widetilde{X_{(a,h)} }\\right)= -\\widetilde{(a,h)}.$ Let us denote by $\\omega $ the restriction of $\\omega _0$ to $\\mathcal {O}_\\nu $ .","The one form $\\omega $ is invariant by the action of $\\@root T \\of {\\mathbf {1}},$ so that it projects to a well defined one form on $\\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}},$ that we denote by $\\omega _\\nu .$ We know that $\\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}},$ represent the homogeneous contact manifold corresponding to the kind of particle under consideration.","As a consecuence of (REF ) and (REF ) is not difficult to prove that $\\omega _\\nu $ is the contact form.","As a consecuence of the fact that the map of $\\mathcal {O}_\\nu $ on $\\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}}$ is a covering map, $\\omega $ is also a contact form on $\\mathcal {O}_\\nu ,$ that becomes itself an homogeneous contact manifold.","The two form $d(\\omega _\\nu )$ thus projects under $\\pi _\\nu $ on the symplectic form on $P_3^{sign(\\nu )},\\ \\Omega _\\nu ,$ that corresponds to Kirillov form in the identification of $P_3^{sign(\\nu )}$ with the coadjoint orbit.","Then, $d\\omega $ also projects on $\\Omega _\\nu ,$ under the restriction of the canonical map $\\pi $ to $\\mathcal {O}_\\nu .$ But $d\\omega $ is the restriction to $\\mathcal {O}_\\nu $ of $d\\omega _0$ and we have $d\\omega _0=\\frac{i\\eta \\nu }{ \\Phi } (dz^1\\wedge d\\overline{w}^1+dw^1\\wedge d\\overline{z}^1+ dz^2\\wedge d\\overline{w}^2+dw^2 \\wedge d\\overline{z}^2 )+(d\\Phi )\\wedge \\delta ,$ where $\\delta $ is a one form.","Since $d\\Phi $ vanishes on $\\mathcal {O}_\\nu ,$ it follows that $d\\omega $ is also the restriction to $\\mathcal {O}_\\nu $ of $\\Omega =\\frac{i\\eta \\nu }{ \\Phi } (dz^1\\wedge d\\overline{w}^1+dw^1\\wedge d\\overline{z}^1+ dz^2\\wedge d\\overline{w}^2+dw^2 \\wedge d\\overline{z}^2).$ Thus, we can obtain explicit expressions of $\\Omega _\\nu $ as follows: for each differentiable section of $\\pi $ with values in $\\mathcal {O}_\\nu $ $\\sigma : U \\rightarrow {\\cal O}_\\nu \\subset {\\bf T}^{sgn(\\nu )},$ we have on the open set $U$ $\\Omega _\\nu =\\sigma ^*\\Omega _0,$ Also we have $\\Omega _\\nu =d\\,\\left(\\sigma ^*\\omega _0 \\right).$"],["Local expression of the symplectic form.","In ${{\\mathbb {C}}}^{3}$ we define ${\\cal D}=\\lbrace (t,u,v):\\,sign(\\nu )\\,\\Re (\\phi (t,u,v)) > 0\\rbrace ,$ where $\\phi (t,u,v)=u+\\overline{t} v$ and $\\Re {}$ stands for real part.","In ${\\bf P }^{sgn(\\nu )}_3$ we define ${\\cal D}_1=\\left\\lbrace \\left[\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array} \\right]:w^1\\ne 0,\\ sign(\\nu )\\,\\Re (\\Phi (\\left(\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array}\\right))) > 0\\right\\rbrace $ ${\\cal D}_2=\\left\\lbrace \\left[\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array} \\right]:w^2\\ne 0,\\ sign(\\nu )\\,\\Re (\\Phi (\\left(\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array}\\right))) > 0\\right\\rbrace $ ${\\cal D}_3=\\left\\lbrace \\left[\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array} \\right]:z^1\\ne 0,\\ sign(\\nu )\\,\\Re (\\Phi (\\left(\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array}\\right))) > 0\\right\\rbrace $ ${\\cal D}_4=\\left\\lbrace \\left[\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array} \\right]:z^2\\ne 0,\\ sign(\\nu )\\,\\Re (\\Phi (\\left(\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array}\\right))) > 0\\right\\rbrace .$ The ${\\cal D}_k$ compose an open cover of ${\\bf P }^{sgn(\\nu )}_3.$ The maps $\\psi _1: (t,u,v)\\in {\\cal D} \\rightarrow \\left[\\begin{array}{c}1\\\\t\\\\u\\\\v \\end{array} \\right] \\in {\\cal D}_1$ $\\psi _2: (t,u,v)\\in {\\cal D} \\rightarrow \\left[\\begin{array}{c}v\\\\1\\\\t\\\\ u \\end{array} \\right] \\in {\\cal D}_2$ $\\psi _3: (t,u,v)\\in {\\cal D} \\rightarrow \\left[\\begin{array}{c}u\\\\v\\\\1\\\\t \\end{array} \\right] \\in {\\cal D}_3$ $\\psi _4: (t,u,v)\\in {\\cal D} \\rightarrow \\left[\\begin{array}{c}t\\\\ u \\\\v\\\\1 \\end{array} \\right] \\in {\\cal D}_4$ are such that the $({\\cal D}_k,(\\psi _k)^{-1})$ compose an atlas of ${\\bf P }^{sgn(\\nu )}_3.$ For all of these charts the coordinates will be denoted by $(t,u,v).$ We also define sections of $\\pi _\\nu ,$ $\\sigma _k:{\\cal D}_k \\rightarrow {\\cal O}_\\nu ,\\ \\ \\ \\ k=1,\\dots ,4,$ by means of $\\sigma _1\\circ \\psi _1: (t,u,v)\\in {\\cal D} \\rightarrow F(t,u,v)\\left(\\begin{array}{c}1\\\\t\\\\u\\\\v \\end{array} \\right) \\in {\\cal D}_1$ $\\sigma _2\\circ \\psi _2: (t,u,v)\\in {\\cal D} \\rightarrow F(t,u,v)\\left(\\begin{array}{c}v\\\\1\\\\t\\\\ u \\end{array} \\right) \\in {\\cal D}_2$ $\\sigma _3\\circ \\psi _3: (t,u,v)\\in {\\cal D} \\rightarrow F(t,u,v)\\left(\\begin{array}{c}u\\\\v\\\\1\\\\t \\end{array} \\right) \\in {\\cal D}_3$ $\\sigma _4\\circ \\psi _4: (t,u,v)\\in {\\cal D} \\rightarrow F(t,u,v)\\left(\\begin{array}{c}t\\\\ u \\\\v\\\\1 \\end{array} \\right) \\in {\\cal D}_4$ where $F(t,u,v)=\\sqrt{\\frac{2\\nu }{\\Re (\\phi (t,u,v))}}.$ For all $k$ we have $(\\sigma _k\\circ \\psi _k)^*\\omega = \\frac{\\eta \\nu }{\\Re (\\phi )}(d(\\Im (\\phi ))+i(v\\,d\\overline{t}-\\overline{v}\\,dt)),$ Where $\\Im (\\phi )$ is the imaginary part of $\\phi .$ Then, the local expression of $\\Omega _\\nu $ in all of these coordinate systems can be evaluated by means of $\\Omega _\\nu \\stackrel{loc}{=}d\\,((\\sigma _k\\circ \\psi _k)^*\\omega ).$ for all $k.$ Notice that $\\omega $ is not projectable on ${\\bf P }^{sgn(\\nu )}_3.$ Thus (REF ) need not be local expressions of a well defined 1-form on all of ${\\bf P }^{sgn(\\nu )}_3.$"],["Symmetric Wave Functions of the Photon.","There are many proposals in the literature for Wave Functions of the Photon.","Including someones that asserts that such a thing does not exist.","In this paper I have described the Wave Functions of the Massless Type 4 Particles, where the case $T=2$ corresponds to the Photon.","These Wave Functions satisfies the Penrose Wave Equations.","In this section I describe other forms of the Wave Functions of Photon.","Equivalent representations.These forms enables us to relate directly the Wave Functions with the Electromagnetic Potential and the Electromagnetic Field.","Photon is the name of four kinds of particles: the massless Type 4 particles with $T=2,$ $i.e.$ those corresponding to $\\gamma =\\left\\lbrace \\frac{i \\chi }{4 \\pi }\\left(\\begin{array}{cc} 1 & 0\\\\0 & -1\\end{array}\\right),\\eta \\,\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)\\right\\rbrace \\ \\ \\ \\ ,\\ \\chi ,\\ \\eta \\in \\lbrace \\pm 1\\rbrace ,$ We already know that the value of $-\\eta $ is the sign of energy and that $-\\eta \\chi $ is the sign of helicity ($c.f.$ section REF ).","We denote $\\ell =-\\eta \\chi $ .","From section we obtain $\\begin{array}{l}\\vspace{7.22743pt}G_\\gamma = \\left\\lbrace \\left(\\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&i \\chi \\eta a z/ \\pi \\\\\\overline{i \\chi \\eta a z / \\pi }&0\\end{array}\\right)\\right):\\ z \\in {\\bf S^1},\\right.","\\\\\\left.","\\ \\ \\ \\ \\ \\ \\ \\ \\ b\\in {\\bf R},\\ a \\in {\\bf C}\\right\\rbrace \\\\\\ \\\\\\mathrm {and\\ the\\ homomorphism} \\\\\\ \\\\C_\\gamma \\left( \\left( \\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&{i \\chi \\eta a z/ \\pi } \\\\{\\overline{i \\chi \\eta a z/ \\pi }}&0\\end{array}\\right)\\right)\\right)=z^{2\\chi }\\end{array}$ has differential $\\gamma .$ $\\begin{array}{l}({\\mbox{$G_\\gamma $}} )_{SL}=\\left\\lbrace \\left(\\begin{array}{cc} z&a \\\\0&{\\overline{z}}\\end{array}\\right): z \\in {\\bf S^1},\\ a \\in {\\mathbb {C}} \\right\\rbrace \\\\\\vspace{7.22743pt}(C_\\gamma )_{SL}\\left(\\ \\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right)\\right)=\\ z^{2\\chi } \\\\\\vspace{7.22743pt}SL_1=\\left\\lbrace \\left(\\begin{array}{cc}a&0 \\\\0&{1/a}\\end{array}\\right):\\ a \\in {\\mathbb {C}}\\right\\rbrace \\\\\\vspace{7.22743pt}SL_2=(G_\\gamma )_{SL} \\\\\\vspace{7.22743pt}SL_1 \\cap SL_2=\\left\\lbrace \\left(\\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array}\\right):\\ z \\in {\\bf S}^1\\right\\rbrace \\end{array}.$ In section we have identified the space $SL/(G_\\gamma )_{SL}$ to the future lightcone, ${\\mbox{${\\bf C}^+$}}=\\lbrace H \\in {H(2)}:\\ Det\\,H=0,\\ Tr\\,H>0\\rbrace ,$ by means of the diffeomorphism $A\\, (G_\\gamma )_{SL} \\in SL/(G_\\gamma )_{SL} \\mapsto A \\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right) A^* \\in {\\bf C}^+.$ In this section I give another description of the wave functions of a photon, by means of different trivialisations than the ones I have used in section .","Of course these trivialisations are “isomorphic”, in an obvious sense.","Let us consider first the cases $\\chi =+1$ $i.e.$ $\\ell =-\\eta .$ In these cases, a trivialisation ($c.\\,f.$ section ) is $(\\mu _{+}, { \\cal {S}},s_0^+),$ where $\\cal S$ is the vector subspace of $gl(2,\\bf C),$ composed by the symmetric matrices, $s_0^+=\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)$ and $\\mu _+(A)\\cdot s=A \\,s\\, {}^tA,$ for all $A \\in SL(2,{\\bf C}),\\ s\\in \\cal S$ .","Since $\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)=\\left(\\begin{array}{c} 1 \\\\0\\end{array}\\right)\\left(\\begin{array}{cc} 1 & 0\\end{array}\\right) $ the orbit of $\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)$ by $\\mu _+$ is composed by the elements of ${\\cal S}^2$ of the form $(A \\left(\\begin{array}{c} 1 \\\\0\\end{array}\\right) )\\ {}^t (A \\left(\\begin{array}{c} 1 \\\\0\\end{array}\\right) ),$ for some $A \\in SL(2,\\bf C)$ .","Then, one sees that the orbit is contained in ${\\cal B}=\\lbrace z \\ {}^t z: z\\in {\\bf C}^2-\\lbrace 0\\rbrace \\rbrace .$ But every $z\\in {\\bf C}^2-\\lbrace 0\\rbrace $ has the form $ A\\, \\left(\\begin{array}{c} 1 \\\\0\\end{array}\\right),$ for some $A \\in SL(2,{\\bf C})$ , so that the orbit coincides with ${\\cal B}$ .","For each $ s\\in {\\cal S}$ such that $s\\ne 0$ and $Det\\, s=0,$ there exist exactly two $z\\in {\\bf C}^2-{0}$ such that $s=z \\ {}^t z.$ In fact, if $s=\\left(\\begin{array}{cc} a & b\\\\b & d\\end{array}\\right)\\in B$ the $z$ such that $s=z \\ {}^t z$ are $\\pm \\left(\\begin{array}{c} \\alpha \\\\\\sigma \\delta \\end{array}\\right)$ where $\\alpha $ is a square root of $a$ , $\\delta $ is a square root of $d$ , and $\\sigma \\in \\lbrace \\pm 1\\rbrace $ such that $b=\\sigma \\alpha \\delta .$ As a consequence, we also have ${\\cal B}=\\lbrace s\\in {\\cal S}: s\\ne 0,Det\\, s=0\\rbrace .$ The homogeneous space $SL/Ker(C_\\gamma )_{SL}$ is identified to ${\\cal B}$ by means of $ A\\,Ker(C_\\gamma )_{SL} \\in SL/Ker(C_\\gamma )_{SL}\\longrightarrow \\mu _+(A)\\cdot s_0^+ \\in {\\cal B}$ Then, the canonical map $SL/Ker(C_\\gamma )_{SL} \\longrightarrow SL/(G_\\gamma )_{SL},$ denoted in the following by $r_+,$ becomes $r_+:(A \\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right){}^tA)\\in {\\cal B} \\longrightarrow A \\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)A^*\\in C^+,$ for all $A\\in SL,$ so that $r_+(z{}^t z)= zz^*$ for all $z \\in {\\bf C}^2-{0}$ .","By elementary operations with matrices, one can prove that the relation $r_+(s)=C $ is equivalent to $C=(+(Tr\\,s\\overline{s})^{-1/2})\\,s\\overline{s}$ and also to $s=(+(Tr\\,C {}^t C)^{-1/2})e^{i\\phi }(C {}^t C)$ for some $\\phi \\in {\\bf R}.$ To obtain Wave Functions, we need functions on ${\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers.","If $f$ is one of such functions the corresponding Prewave Function is $ \\psi _f^{(-\\eta ,-\\eta )}(H,K)=f(s)\\,s\\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m},$ where $s$ is arbitrary in $r_+^{-1}(K).$ In ${(-\\eta ,-\\eta )}$ the first $-\\eta $ stands for the sign of energy and the second by $\\ell ,$ helicity.","Now, let us consider the cases $\\chi =-1$ $i.e.$ $\\ell =\\eta .$ In these cases, a trivialisation is $(\\mu _{-}, { \\cal {S}},s_0^-),$ where $\\cal S$ is as in the $\\chi =+1$ case, $s_0^-=\\left(\\begin{array}{cc} 0 & 0\\\\0 & 1\\end{array}\\right)$ and $\\mu _-(A)\\cdot s=(A^*)^{-1} \\,s\\, (\\overline{A})^{-1},$ for all $A \\in SL(2,{\\bf C}),\\ s\\in \\cal S$ .","The orbit of $\\left(\\begin{array}{cc} 0 & 0\\\\0 & 1\\end{array}\\right)$ by $\\mu _-$ is, as in case $\\chi =+1$ , ${\\cal B}$ .","The canonical map $r_-: SL/Ker(C_{\\gamma })_{SL} \\longrightarrow SL/(G_{\\gamma })_{SL}$ becomes a map from ${\\cal B}$ onto ${\\bf C}^+,$ given by $r_-( \\mu _-(A)\\cdot s_0^-)=A \\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)A^*.$ The maps $r_+$ and $r_-$ are geometrically related as follows.","Let us consider the map $J:s\\in {\\cal S} \\longrightarrow -\\epsilon \\overline{s}\\epsilon \\in {\\cal S} .$ This is an antilinear map with $J^2=I.$ Since $J(z{}^tz)=(-\\epsilon \\overline{z}){}^t(-\\epsilon \\overline{z}),$ we have $J({\\cal B})={\\cal B}.$ On the other hand $r_-\\circ J(\\mu _+(A)\\cdot s_0^+)=r_-(-\\epsilon \\, \\overline{A}\\,\\overline{s_0^+}\\,A^*\\,\\epsilon )=r_-(-(A^*)^{-1}\\,\\epsilon \\,\\overline{s_0^+} \\,\\epsilon \\, (\\overline{A})^{-1})=\\\\=r_-(\\mu _-(A)\\cdot s_0^-)=A\\ ({\\mbox{$G_\\gamma $}} )_{SL}=r_+(\\mu _+(A)\\cdot s_0^+) , $ so that $r_+=r_-\\circ J.$ Since $J^2=I,$ we also have $r_-=r_+\\circ J.$ As a consequence, $J$ stablishes a principal fibre bundle isomorphism over the identical map of $C^+$ , the isomorphism of structural groups being the map defined by sending each element of $S^1$ to its inverse.","Another consequence is $r_-(z {}^tz)=-\\epsilon \\overline{zz^*}\\epsilon .$ If $f$ is a function on ${\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers, it also defines a Prewave Function for this kind of photon by means of $ \\psi _f^{(-\\eta ,\\eta )}(H,K)=f(s)\\,s\\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m},$ where $s,$ now, is arbitrary in $r_-^{-1}(K).$ Here, in ${(-\\eta ,\\eta )},$ $-\\eta $ stands for the sign of energy and the second $\\eta ,$ which in the present case coincides with $\\ell ,$ stands for helicity.","Thus, the Prewave Functions corresponding to the four kinds of Photon are different.","The sign of energy, $-\\eta ,$ and the sign of helicity, $\\ell ,$ determines the type of Prewave Functions to be used, $\\psi _f^{(-\\eta , \\ell )},$ that are given by (REF ) or (REF ).","When $f$ is continuous with compact support, the corresponding Wave Function is $\\widetilde{\\psi }_f^{(-\\eta , \\ell )}(x)=\\int _{C^+} \\psi _f^{(-\\eta , \\ell )}(h(x),K) \\omega _K,$ where $\\omega $ is the invariant volume element on $C^+$ defined in (REF ).","These Wave Functions represent states whose energy has sign $-\\eta $ and are eigenvectors of helicity ($c.f.","$ section REF ) corresponding to the eigenvalues $\\frac{\\ell }{2\\pi }.$"],["Prehilbert Space estructure.","Let us denote by ${\\cal {F}}$ the vector subspace composed by the functions on ${\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers, and by ${\\cal {F}}_c$ the subspace of ${\\cal {F}}$ composed by the continuous elements with compact support.","The space ${\\cal {F}}_c$ can be provided with a prehilbert space structure, by means of the general method described in section , for each kind of Photon.","In more detail we proceed as follows.","We separate the cases ${-\\eta \\ell }=\\pm 1.$ If $\\ f , f^\\prime \\in {\\cal F}_c,$ its hermitian product is in each case $\\langle f,\\, f^\\prime \\rangle _{(-\\eta , \\ell )}= \\int _{C^+} (\\overline{f} f^\\prime )_{(-\\eta , \\ell )}\\ {\\mbox{$\\omega $}} , $ where $ (\\overline{f} f^\\prime )_{(-\\eta , \\ell )}$ is the function defined on $C^+$ by $(\\overline{f} f^\\prime )_{(-\\eta , \\ell )} (K)=\\overline{f(s)} f^\\prime (s)$ for all $K\\in C^+,$ where $\\ s\\in r_{-\\eta \\ell }^{-1}(K).$ With each of these inner products, ${\\cal F}_c$ becomes a prehilbert space.","For prewave functions we define $\\langle \\psi _f^{(-\\eta , \\ell )},\\, \\psi _{f^\\prime }^{(-\\eta , \\ell )} \\rangle _{(-\\eta , \\ell )}= \\langle f,\\, f^\\prime \\rangle _{(-\\eta , \\ell )}.$ A sexquilinear form on S is given by $\\Phi (s,s^\\prime )=Tr(\\overline{s}\\,{s^\\prime }).$ Thus ($c.f.$ section ), the hermitian product of Prewave Functions can be given in terms of the Prewave Functions themselves instead of the functions $f,$ by $\\langle \\psi ^{(-\\eta , \\ell )}_f,\\, \\psi ^{(-\\eta , \\ell )}_{f^\\prime } \\rangle _{(-\\eta , \\ell )}=\\int _{C^+} \\psi _f^{(-\\eta , \\ell )}\\Phi _{(-\\eta , \\ell )} \\psi _{f^\\prime }^{(-\\eta , \\ell )} \\ \\omega ,$ where $\\psi _f^{(-\\eta , \\ell )} \\Phi _{(-\\eta , \\ell )} \\psi _{f^\\prime }^{(-\\eta , \\ell )}$ is the function on $C^+$ given by $\\psi _f^{(-\\eta , \\ell )} \\Phi _{(-\\eta , \\ell )} \\psi _{f^\\prime }^{(-\\eta , \\ell )} (K)=\\frac{Tr(\\overline{\\psi _f^{(-\\eta , \\ell )}(H,K)}{ \\psi _{f^\\prime }^{(-\\eta , \\ell )}(H,K))}}{Tr(\\overline{s}\\,{s})}$ for all $K\\in C^+,$ $H \\in H(2)$ and $s\\in r_{-\\eta \\ell }^{-1}(K).$ The hermitian product of Wave Functions is defined as being the hermitian product of the corresponding Prewave Functions."],["Electromagnetic Potential. ","If $K\\in C^+$ , the tangent space to $C^+$ at $K$ can be identified to the subspace of $H(2)$ given by $T_KC^+=\\lbrace M\\in H(2): Tr\\,M\\epsilon \\overline{K}\\epsilon =0 \\rbrace .$ Then, the complexified tangent space can be identified to ${}^{\\bf C}T_KC^+=\\lbrace M\\in gl(2,\\mathbb { C}\\rm ): Tr\\,M\\epsilon \\overline{K}\\epsilon =0 \\rbrace .$ A real vector field on $C^+$ is thus a map $A:C^+ \\longrightarrow H(2),$ such that $Tr\\,A(K)\\epsilon \\overline{K}\\epsilon =0,$ for all $K\\in C^+$ .","A complex vector field on $C^+$ is thus given by a function $A:C^+ \\longrightarrow gl(2,\\bf C),$ whose real and imaginary hermitian parts are real vectorfields on $C^+$ .","Since $Tr M\\epsilon \\overline{K}\\epsilon $ is real for $M$ and $K$ hermitian, we see that $A$ is a complex vector field on $C^+$ if and only if $Tr\\,A(K)\\epsilon \\overline{K}\\epsilon =0,$ for all $K\\in C^+$ .","Equation (REF ) is equivalent to say that $A(K)\\epsilon \\overline{K}$ is a symmetric matrix.","Let $A$ be a complex vector field on $C^+.$ We denote by $A_R$ and $A_I$ the real and imaginary hermitian parts of $A$ and $A_R(K)&=&h(A_R^1(K),.., A_R^4(K))\\\\A_I(K)&=&h(A_I^1(K),.., A_I^4(K))\\\\A^\\mu (K)&=&A_R^\\mu +i\\, A_I^\\mu ,\\ \\ \\mu =1,..,4\\\\\\overrightarrow{A_R}(K)&=&(A_R^1(K),.., A_R^3(K))\\\\\\overrightarrow{A_I}(K)&=&(A_I^1(K),.., A_I^3(K))\\\\\\overrightarrow{A}(K)&=&(A^1(K),.., A^3(K)).$ If $A_i^j(K)$ is the element of $A(K)$ in the row $i$ column $j,$ we also have $A^1 (K)&=&\\frac{1}{2}(A_1^2(K)+A_2^1(K))\\\\A^2 (K)&=&\\frac{i}{2}(A_1^2(K)-A_2^1(K))\\\\A^3 (K)&=&\\frac{1}{2}(A_1^1(K)-A_2^2(K))\\\\A^4 (K)&=&\\frac{1}{2}(A_1^1(K)+A_2^2(K))$ If $A(K)$ is , for exemple, continuous with compact support, for $\\mu =1,..,4,\\ x\\in \\mathbb {R}^4$ we define $\\widetilde{ A^\\mu }(x)=\\int _{C^+} A^\\mu (K) Exp(2\\pi i \\eta \\langle h^{-1}( K),x \\rangle ) \\omega .$ Then $\\square \\widetilde{A^\\mu }=0,$ for all $\\mu ,$ and $\\sum _{\\mu =1}^4 \\frac{\\partial \\widetilde{A^\\mu }}{\\partial x^\\mu }=0.$ We thus see that the $\\widetilde{A^\\mu }(x)$ define a complex electromagnetic potential in the Lorenz gauje.","The corresponding electric field is given by $\\overrightarrow{E}(x)=-\\nabla \\widetilde{A^4}(x)-\\frac{\\partial \\overrightarrow{\\widetilde{A}}(x)}{\\partial x^4 },$ where $\\overrightarrow{\\widetilde{A}}(x)=(\\widetilde{ A^1}(x),\\widetilde{ A^2}(x),\\widetilde{ A^3}(x)),$ that can be written $\\overrightarrow{E}(x)=2\\pi i \\eta \\int _{C^+}(A^4(K)\\overrightarrow{K}-K^4 \\overrightarrow{A}(K)) Exp(2\\pi i \\eta \\langle h^{-1}( K),x \\rangle ) \\omega .$ The magnetic field is given by $\\overrightarrow{B}(x)=\\nabla \\times \\overrightarrow{\\widetilde{A}}(x),$ that can be written $\\overrightarrow{B}(x)=2\\pi i \\eta \\int _{C^+}(\\overrightarrow{A}(K)\\times \\overrightarrow{K} ) Exp(2\\pi i \\eta \\langle h^{-1}( K),x \\rangle ) \\omega .$ This electromagnetic field take in general complex values, the real parts of $\\overrightarrow{E}$ and $\\overrightarrow{B}$ are real electric and magnetic fields, corresponding to the electromagnetic potential given by the real parts of the $A^\\mu (x).$"],["Wave Functions and the Electromagnetic Potential. ","In section REF we have seen that a complex vector field on $C^+$ , define a complex electromagnetic potential in the Lorenz gauje.","Let $A$ be a complex vector field on $C^+$ .","Then, the matrix $A(K)\\epsilon \\overline{K}$ is a symmetric matrix, for all $K\\in C^+$ .","If $ s\\in r_+^{-1}(K)$ there exist an unique complex number, $f_A(s),$ such that $A(K)\\epsilon \\overline{K}=f_A(s)\\, s .$ In fact, if $z\\in {\\bf C}^2-{0}$ , is such that $s= z {}^t z$ , we have $K=zz^*$ , and, since $0=Tr A(K)\\epsilon \\overline{z}{}^t z\\epsilon ={}^tz\\epsilon A(K)\\epsilon \\overline{z},$ there exist a number, $\\lambda $ , such that $A(K)\\epsilon \\overline{z}=\\lambda \\, z,$ so that $A(K)\\epsilon \\overline{K}=\\lambda \\, s .$ Since $s\\ne 0$ such a $\\lambda $ is unique and is denoted by $f_A(s).$ Thus the complex vector field $A$ defines a function, $f_A,$ on ${\\cal B}.$ If we take $e^{i\\phi }\\,s$ instead of $s$ in $ r_+^{-1}(K)$ , we can take in the preceeding reasoning $e^{i\\phi /2}z$ instead of $z$ , and thus $A(K)\\epsilon \\overline{e^{i\\phi /2}z}=\\lambda ^\\prime \\, e^{i\\phi /2}z$ leads to $\\lambda ^\\prime = e^{-i\\phi }\\,\\lambda .$ We thus see that $f_A(e^{i\\phi }\\,s)=e^{-i\\phi }\\,f_A(s)$ which proves that $f_A $ is $S^1$ -homogeneous of degree -1.","The Prewave Function corresponding to $f_A$ for $-\\eta \\ell =1$ can be written as $\\psi _A^{(-\\eta , -\\eta )}(H,K)=f_A(s)\\,s\\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m},$ where $s$ is arbitrary in $r_+^{-1}(K).$ As a consecuence of (REF ) we have $\\psi _A^{(-\\eta , -\\eta )}(H,K)=A(K)\\epsilon \\overline{K}e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m},$ that gives Prewave Functions directly in terms of vectorfields.","If $A(K)$ is continuous with compact support, the corresponding Wave Function is $ \\widetilde{\\psi }_A^{(-\\eta , -\\eta )}(x)=\\int _{C^+}A(K)\\epsilon \\overline{K}\\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,x\\rangle _m}\\omega _K.$ Now we define $\\widehat{f}_A=\\overline{f_A\\circ J},$ where $J$ is given by (REF ).","I shall prove that $\\widehat{f}_A$ is $S^1$ -homogeneous of degree -1 on ${\\cal B}$ and that, for all $ s\\in r_-^{-1}(K),$ we have $-\\epsilon \\overline{A(K)}\\epsilon K \\epsilon =\\widehat{f}_A(s)\\, s .$ This will enable us to give a Prewave Function of Photons with $\\ell =\\eta ,$ directly in terms of $A$ .","For all $ s\\in B,\\ a\\in S^1$ we have $\\widehat{f}_A(as)=\\overline{f_A( J(as))}=\\overline{f_A(\\overline{a} J(s)}=\\overline{af_A( J(s)}=\\overline{a}\\,\\widehat{f}_A(s),$ so that $\\widehat{f}_A$ has the appropiate homogeneity to define a Prewave Function.","On the other hand, if $s \\in r_-^{-1}(K)$ , we have $\\widehat{f}_A(s)\\,s=\\overline{f_A(J(s))}\\,J(J(s))=J(f_A(J(s))\\,J({s}))=$ $=J(A(r_+(J(s)))\\epsilon \\overline{r_+(J(s))})=J(A(r_-(s))\\epsilon \\overline{r_-(s)}=$ $=J(A(K)\\epsilon \\overline{K}))=-\\epsilon \\overline{A(K)}\\epsilon K \\epsilon .$ Then, the Prewave Function for Photons with $\\ell =\\eta $ corresponding to $\\widehat{f}_A$ is $\\psi _A^{(-\\eta , \\eta )}(H,K)=-\\epsilon \\overline{A(K)}\\epsilon K \\epsilon \\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m}.$ Obviously $\\psi _A^{(-\\eta , \\eta )}=-\\epsilon \\overline{\\psi _A^{(\\eta , \\eta )}}\\epsilon , \\ \\ \\psi _A^{(\\eta , \\eta )}=-\\epsilon \\overline{\\psi _A^{(-\\eta , \\eta )}}\\epsilon .$ If $A(K)$ is continuous with compact support, the corresponding Wave Function is $ \\widetilde{\\psi }_A^{(-\\eta , \\eta )}(x)=-\\int _{C^+}\\epsilon \\overline{A(K)}\\epsilon K \\epsilon \\,\\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,x\\rangle _m}\\omega _K.$ We have $\\widetilde{\\psi }_A^{(-\\eta , \\eta )}=-\\epsilon \\overline{\\widetilde{\\psi }_A^{(\\eta , \\eta )}}\\epsilon ,\\ \\ \\widetilde{\\psi }_A^{(\\eta , \\eta )}=-\\epsilon \\overline{\\widetilde{\\psi }_A^{(-\\eta , \\eta )}}\\epsilon .$ With equations (REF ) and (REF ) we see that a single complex vectorfied on $C^+$ gives rise to Wave Functions of the four kinds of Photons: those with positive and negative energy and helicity.","The inner products defined in section REF lead to the following results.","If $A$ and $A^\\prime $ are vector fields on $C^+$ we have $\\langle \\psi ^{(-\\eta , -\\eta )}_A,\\, \\psi ^{(-\\eta , -\\eta )}_{A^\\prime } \\rangle _{(-\\eta , -\\eta )}= \\langle f_A,\\, f_{A^\\prime } \\rangle _{(-\\eta , -\\eta )},$ and $\\langle \\psi ^{(-\\eta , \\eta )}_A,\\, \\psi ^{(-\\eta , \\eta )}_{A^\\prime } \\rangle _{(-\\eta , \\eta )}=\\langle \\widehat{f}_A,\\, \\widehat{f}_{A^\\prime } \\rangle _{(-\\eta , \\eta )}.$ Then, from (REF ) and (REF ) one can obtain $\\langle \\psi ^{(-\\eta , -\\eta )}_A,\\, \\psi ^{(-\\eta , -\\eta )}_{A^\\prime } \\rangle _{(-\\eta , -\\eta )}=\\int _{C^+} A\\Phi A^\\prime =$ $=\\langle \\psi ^{(-\\eta , \\eta )}_{A^\\prime },\\, \\psi ^{(-\\eta , \\eta )}_{A} \\rangle _{(-\\eta , \\eta )}$ where $A\\Phi A^\\prime (K)= \\frac{Tr(\\overline{A(K)}\\epsilon K A^\\prime (K)\\epsilon \\overline{K})}{Tr(\\overline{s}{s})}$ for all $K\\in C^+,$ and $s\\in r_{\\pm }^{-1}(K).$"],["Wave Functions in terms of the Electromagnetic Field.","Let $A$ a complex vectorfield on $C^+$ and denote $\\overrightarrow{E}(K)&=& A^4(K)\\overrightarrow{K}-K^4 \\overrightarrow{A}(K) \\\\\\overrightarrow{B}(K)&=& \\overrightarrow{A}(K)\\times \\overrightarrow{K}$ Thus, the Electric and Magnetic Fields can be given by $\\overrightarrow{E}(x)=2\\pi i \\eta \\int _{C^+}\\overrightarrow{E}(K) Exp(2\\pi i \\eta \\langle h^{-1}( K),x \\rangle ) \\omega _K$ $\\overrightarrow{B}(x)=2\\pi i \\eta \\int _{C^+}\\overrightarrow{B}(K) Exp(2\\pi i \\eta \\langle h^{-1}( K),x \\rangle ) \\omega _K.$ Let us prove that $A(K) \\epsilon \\overline{K} \\epsilon =\\left(\\overrightarrow{E}(K)+i\\overrightarrow{B}(K)\\right)\\cdot \\overrightarrow{\\sigma },º$ which means $A(K) \\epsilon \\overline{K} \\epsilon =\\left(\\begin{array}{cc}(\\overrightarrow{E}+i\\overrightarrow{B})^3 & (\\overrightarrow{E}+i\\overrightarrow{B})^1-i(\\overrightarrow{E}+i\\overrightarrow{B})^2\\\\(\\overrightarrow{E}+i\\overrightarrow{B})^1+i(\\overrightarrow{E}+i\\overrightarrow{B})^2 &-(\\overrightarrow{E}+i\\overrightarrow{B})^3\\end{array}\\right)$ In fact, we have $A(K) \\epsilon \\overline{K} \\epsilon =\\frac{1}{2} \\left( A(K) \\epsilon \\overline{K}+ {}^{t}(A(K) \\epsilon \\overline{K}) \\right)\\epsilon =$ $=\\frac{1}{2}\\left((A_R+iA_I)\\epsilon \\overline{K}\\epsilon -K \\epsilon ({}^{t}A_R+i {}^{t}A_I)\\epsilon \\right)=$ $=\\frac{1}{2}(A_R\\epsilon \\overline{K} \\epsilon -K\\epsilon \\overline{A_R} \\epsilon )+\\frac{i}{2}(A_I\\epsilon \\overline{K} \\epsilon -K\\epsilon \\overline{A_I} \\epsilon )$ and thus (REF ) follows from (REF ).","As a consequence, the Prewave Functions corresponding to $A$ are $\\psi _A^{(-\\eta , -\\eta )}(H,K)=-\\left(\\left(\\overrightarrow{E}(K)+i\\overrightarrow{B}(K)\\right)\\cdot \\overrightarrow{ \\sigma } \\right)\\epsilon \\ e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m},$ $\\psi _A^{(-\\eta , \\eta )}(H,K)=-\\epsilon \\ \\left(\\overline{\\left(\\overrightarrow{E}(K)+i\\overrightarrow{B}(K)\\right)\\cdot \\overrightarrow{ \\sigma }}\\right) \\ e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m}.$ If $A$ is continuous with compact support, we have $\\widetilde{\\psi }_A^{(-\\eta , -\\eta )}(x)=-\\int _{C^+}((\\overrightarrow{E}(K)+i\\overrightarrow{B}(K))\\cdot \\overrightarrow{\\sigma })\\ \\epsilon \\ e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m}\\omega _K,$ $\\widetilde{\\psi }_A^{(-\\eta , \\eta )}(x)=-\\int _{C^+}\\ \\epsilon \\ (\\overline{(\\overrightarrow{E}(K)+i\\overrightarrow{B}(K))\\cdot \\overrightarrow{\\sigma })} \\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m}\\omega _K.$ so that the Wave Functions are $\\widetilde{\\psi }_A^{(-\\eta , -\\eta )}(x)=\\frac{i\\eta }{2\\pi }((\\overrightarrow{E}(x)+i\\overrightarrow{B}(x))\\cdot \\overrightarrow{\\sigma })\\ \\epsilon ,$ or $\\widetilde{\\psi }_A^{(-\\eta , \\eta )}(x)=\\frac{i\\eta }{2\\pi }\\epsilon \\ (\\overline{(\\overrightarrow{E}(x)+i\\overrightarrow{B}(x))\\cdot \\overrightarrow{\\sigma })} .$ These Wave Functions are given in terms of the components of $\\overrightarrow{E}(x)+i\\overrightarrow{B}(x)$ as (up to constants) the Wave Functions of Bialynicki-Birula [2]."],["Gauje invariance for Photons.","Let us denote by $\\cal {D}$ the complex vector space of complex vector fields on $C^+.$ The map $A\\in {\\cal D} \\longrightarrow f_A \\in {\\cal F}$ is linear and its kernel is composed by the vector fields on $C^+$ having the form $A(K)=\\left( \\begin{array}{c}\\lambda (z)\\\\ \\mu (z) \\end{array}\\right)\\,z^*$ where $z\\in {\\bf C}^2-{0}$ is such that $K=zz^*$ and $\\lambda ,\\ \\mu $ are funtions on ${\\bf C}^2-{0}$ $S^1$ -homogeneous of degree +1.","The kernel of the map $A\\in {\\cal D} \\longrightarrow \\widehat{f}_A \\in {\\cal F}$ is the same and will be denoted by $\\cal {N}.$ In particular, for each map $L:K\\in C^+ \\rightarrow L(K)\\in gl(2,{\\bf C}),$ the vectorfield $A_L(K)=L(K)K$ is in $\\cal {N}.$ If $A\\in \\cal {D}$ , $N \\in \\cal {N}$ we have $f_{A+N}=f_A$ $\\widehat{f}_{A+N}= \\widehat{f}_{A},$ so that the Prewave and Wave Functions are invariant under the changes $A\\rightarrow A+N$ : $\\widetilde{\\psi }_{A+N}^{(-\\eta , \\ell )}=\\widetilde{\\psi }_{A}^{(-\\eta , \\ell )}.$ Because of (REF ) and (REF ) we see that also the Electric and Magnetic Fiels corresponding to Photon are invariant under the changes $A\\rightarrow A+N$ .","In the particular case $N=A_L$ with $L(K)=g(K)\\,I,$ where $g$ is a complex valued function on $C^+,$ the change $A\\rightarrow A+N$ lead to the following change in the Electromagnetic Potential $\\widetilde{(A+N)}^\\mu (x)=\\widetilde{A}^\\mu (x)+\\partial ^\\mu \\phi (x),$ where $\\phi (x)=-\\frac{i\\eta }{2\\pi }\\int _{C^+}g(K) e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,x\\rangle _m}\\omega _K$ and $\\partial ^\\mu \\phi =g^{\\mu \\nu }\\frac{\\partial }{\\partial x^\\nu }\\,\\phi $ where the $g^{\\mu \\nu }$ are the components of Minkowski metric.","Thus, the invariance of the Electromagnetic Field of Photons under the change $A(K) \\rightarrow A(K)+g(K)K$ is a particular case of the well known gauje invariance of general Electromagnetic Fields."],["General remarks","In this paper, State Space for a particle defined by $\\alpha \\in \\underline{G}^*$ has been stated as the homogeneous space that correspond to the orbit of $(0,\\alpha )$ in $H(2)\\times \\underline{G}^*.$ Thus, as we have seen in section , it is identified to $\\frac{G}{G_{(0,\\alpha )}}$ where $G_{(0,\\alpha )}=G_\\alpha \\cap (SL \\oplus \\lbrace 0\\rbrace ),$ or, equivalently, to $H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1} ,$ that is the form adopted in Figure REF of that section.","The equivariant maps $\\iota _4:H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1} \\rightarrow H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}}$ and $\\nu _2:H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1} \\rightarrow \\frac{G}{G_\\alpha }.$ in that Figure, will be used in this section.","When we characterize $H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1}$ in terms of other concrete manifolds or of some parametrization, we need to physically interpret the geometric objects that appear, and, in order to do that, the more effective way is , in general, to know the values of the Canonical Dynamical Variables in terms of these objects.","But the value of these Dynamical Variables is well known on the coadjoint orbit i.e.","on $G/G_\\alpha .$ Thus, its values in State Space can be obtained by composition with $\\nu _2.$ Since Prewave Functions are defined on $H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}}$ its composition with $\\iota _4$ gives another version of Quantum States, now defined on the Classical State Space.","In the definition of Prewave Functions REF , space-time only appears in the exponential, whose exponent is $-2\\pi i\\langle h^{-1}(P),h^{-1}(H)\\rangle _m.$ where $P$ is the “traslation\" of momentum-energy to $SL/(G_\\alpha )_{SL}.$ The exponent in the composition of Prewave Functions with $\\iota _4$ must have the same expression, but changing $P$ by $P\\circ \\iota _4.$ The conmutativity of the diagramm in Figure REF imply that $P\\circ \\iota _4$ can be obtained as the impulsion-energy in $G/G_\\alpha $ composite with $\\nu _2.$ In the next sections I give the expressions of the canonical dynamical variables on State Space, for different kind of particles."],["Klein-Gordon particles","We have seen in section REF that State Space for Klein-Gordon particles can be identified to $H(2) \\oplus \\frac{SL}{SU(2)},$ and this homogeneous space to ${\\cal H}^m\\times H(2).$ If $(K,H)\\in {\\cal H}^m\\times H(2) $ and $A\\in SL$ is such that $K=mAA^*,$ we denote $H&=&h(\\overrightarrow{x},x^4)\\\\\\overrightarrow{x}&=& (x^1,x^2,x^3)\\\\K&=&m h(\\overrightarrow{k}, k_4) \\\\\\overrightarrow{k}&=& (k^1,k^2,k^3)\\\\k^4&=&\\sqrt{1+\\Vert \\overrightarrow{k} \\Vert ^2}.$ Thus, the map (REF ) becomes, $\\nu _1(K,H)=(A,H)*(mI,\\overrightarrow{0},1)=$ $=(K,\\overrightarrow{x}-\\frac{x^4}{k^4} \\overrightarrow{k},e^{-2\\pi i \\eta m x^4/k^4})$ and the map (REF ), $\\nu _2(K,H))=(A,H)*(mI,\\overrightarrow{0})=(K,\\overrightarrow{x}-\\frac{x^4}{k^4} \\overrightarrow{k}).$ The Canonical Dynamical Variables on State Space are obtained as composition of (REF ) with $\\nu _2.$ If we denote $V_{KG}=V\\circ \\nu _2$ for each dynamical variable, $V,$ we have $P_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4))=- \\eta \\,mh(\\overrightarrow{k},k_4),$ $\\overrightarrow{l}_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4))=\\overrightarrow{x} \\times \\left(\\overrightarrow{P}_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4)\\right) ,$ $\\overrightarrow{g}_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4))=$ $ = P^4_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4)) \\overrightarrow{x}-x^4 \\overrightarrow{P}_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4)).$ where $ k_4=\\sqrt{1+\\Vert \\overrightarrow{k} \\Vert ^2}.$ The Prewave functions have been defined on a manifold that, in this case, coincides with State Space.","The map $\\iota _4$ is the identical map.","Another fact is that, in this case, State Space is the Universal Covering of the homogeneous contact manifold.","Let us prove that $\\nu _1$ is a covering map.","The local expresion of $\\nu _1$ in the charts corresponding to $\\phi _\\tau $ and $\\phi ^\\prime :(k_1,k_2,k_3,x^1,x^2,x^3,x^4)\\in {\\mathbb {R}}^7 \\longrightarrow $ $ \\longrightarrow (h(m\\overrightarrow{k},m k_4),h(x^1,x^2,x^3,x^4))\\in {\\cal H}^m\\times H(2) ,$ is given by $(\\phi _\\tau )^{-1}\\circ \\nu _1 \\circ \\phi ^{\\prime }(k_1,\\dots ,x^4)=$ $=(k_1,k_2,k_3,x^1-\\frac{x^4}{k_4}k_1,x^2-\\frac{x^4}{k_4}k_2,x^3-\\frac{x^4}{k_4}k_3,-\\tau -\\eta m \\frac{x^4}{k_4}+N)$ where the domain of definition is defined by the condition that $\\tau +\\eta m \\frac{x^4}{k_4} \\notin {\\mathbb {Z}}+\\frac{1}{2} $ and $N$ is defined by the condition that $-\\tau -\\eta m \\frac{x^4}{k_4}+N\\in (-\\frac{1}{2},\\frac{1}{2}).$ We have $((\\phi _\\tau )^{-1}\\circ \\nu _1 \\circ \\phi ^{\\prime })^{-1}\\lbrace (\\overrightarrow{n},\\overrightarrow{y},t)\\rbrace =$ $=\\lbrace (\\overrightarrow{n},\\overrightarrow{y}-\\frac{\\eta }{m}(t+\\tau +N)\\overrightarrow{n}, -\\frac{\\eta }{m}(t+\\tau +N)\\sqrt{1+\\Vert \\overrightarrow{n} \\Vert ^2}): N\\in {\\mathbb {Z}}\\rbrace .$ Let $d\\in {\\cal H}^m \\times {\\mathbb {R}}^3\\times {\\mathbb {S}}^1,$ and let $\\tau $ be such that $d$ is in the image of $(\\phi _\\tau )^{-1}.$ This image, $U_o,$ is an open neighborhood of $d$ , and we shall prove that its antiimage by $\\nu _1$ is union of disjoint open sets, each of them diffeomorphic to $U_o$ under the restriction of $\\nu _1.$ We denote by ${\\cal U}$ the domain of $\\phi _\\tau $ .","The domain of $(\\phi _\\tau )^{-1}\\circ \\nu _1 \\circ \\phi ^{\\prime },$ ${\\cal W}=((\\phi _\\tau )^{-1}\\circ \\nu _1 \\circ \\phi ^{\\prime })^{-1}({\\cal U})$ is the union of the open sets ${\\cal W}_N=\\lbrace &(&\\overrightarrow{n},\\overrightarrow{y}-\\frac{\\eta }{m}(t+\\tau +N)\\overrightarrow{n}, -\\frac{\\eta }{m}(t+\\tau +N))\\sqrt{1+\\Vert \\overrightarrow{n} \\Vert ^2}: \\\\&(&\\overrightarrow{n},\\overrightarrow{y},t)\\in {\\cal U}\\rbrace .$ where $N\\in \\mathbb {Z}.$ But the ${\\cal W}_N$ are disjoint.","In fact, if $(\\overrightarrow{n},\\overrightarrow{y}-\\frac{\\eta }{m}(t+\\tau +N)\\overrightarrow{n}, -\\frac{\\eta }{m}(t+\\tau +N)\\sqrt{1+\\Vert \\overrightarrow{n} \\Vert ^2})=$ $=(\\overrightarrow{n}^{\\prime },\\overrightarrow{y}^{\\prime }-\\frac{\\eta }{m}(t^{\\prime }+\\tau +N^{\\prime })\\overrightarrow{n}^{\\prime }, -\\frac{\\eta }{m}(t^{\\prime }+\\tau +N^{\\prime })\\sqrt{1+\\Vert \\overrightarrow{n}^{\\prime } \\Vert ^2})$ where $(\\overrightarrow{n},\\overrightarrow{y},t),\\ (\\overrightarrow{n}^{\\prime },\\overrightarrow{y}^{\\prime },t^{\\prime })\\in \\cal U,$ we obtain first $\\overrightarrow{n}=\\overrightarrow{n}^{\\prime }$ and then $t+N=t^{\\prime }+N^{\\prime }.$ Since $-1/20$ , the values of the canonical dynamical variables in this state are obtained by composition of (REF ) with the map $\\nu _2$ .","When for each dynamical variable $V$ we denote $V\\circ \\nu _2$ by $V_{massive}$ we have $P_{massive}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&-\\eta m h(\\vec{k},k_4),\\\\\\overrightarrow{l}_{massive}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&\\frac{T}{4\\pi }\\frac{k_4\\overrightarrow{v}-\\vec{k}}{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle }+\\overrightarrow{x}\\times \\overrightarrow{P}_{massive},\\\\\\overrightarrow{g}_{massive}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&\\frac{T}{4\\pi }\\frac{\\vec{k}\\times \\overrightarrow{v}}{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle }+ \\\\ &+& P^4_{massive}\\overrightarrow{x}-x^4\\overrightarrow{P}_{massive},$ In what concerns the Pauli-Lubanski fourvector, we have $\\begin{split}\\overrightarrow{W}_{massive}&=P^4_{massive}\\frac{T}{4\\pi } \\frac{k_4\\overrightarrow{v}-\\vec{k}}{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle } \\\\W^4_{massive}&= -\\eta m\\frac{T}{4\\pi } \\frac{\\langle k_4\\overrightarrow{v}-\\vec{k},\\vec{k} \\rangle }{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle },\\end{split}$ As in $T=0$ case, the prewave functions have been defined directly on State Space, the map $\\iota _4$ being the identical map.","Thus they are exactly as in section REF ."],["Massless particles of type 4","Since $(G_\\alpha )_{SL}\\cap SL_1=[S^1],$ State Space for massless particles, $H(2)\\times (SL/((G_\\alpha )_{SL}\\cap SL_1),$ can be identified to State Space for massive particles, ${\\cal H}^m \\times P_1(C) \\times H(2)$ .","Here $m$ is an arbitrary positive number, with no physical significance.","On the other hand, Movement Space, $G/G_\\alpha ,$ is identified to $P^{sign(\\nu )}_3$ by means of the action (REF ).","The canonical map $\\nu _2$ from State Space onto Movement Space thus becomes for this kind of particle $\\nu _2((A,H)*\\left(mI,\\left[ \\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right) \\right],0 \\right)=(A,H)*[q]$ where $q=\\left( \\begin{array}{c} 0\\\\2\\nu \\\\0\\\\1\\end{array} \\right).$ Then $\\nu _2\\left(mAA^*,\\left[A \\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right) \\right],H\\right)=\\left[\\begin{array}{c} 2\\nu A \\left(\\begin{array}{c} 0\\\\1 \\end{array}\\right)-iH(A^*)^{-1}\\left(\\begin{array}{c} 0\\\\1 \\end{array}\\right)\\\\ (A^*)^{-1}\\left(\\begin{array}{c} 0\\\\1 \\end{array}\\right)\\end{array} \\right].$ But $(A^*)^{-1}=-\\varepsilon \\overline{A} \\varepsilon ,$ so that $\\nu _2\\left(mAA^*,\\left[A \\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right) \\right],H\\right)=\\left[\\begin{array}{c} (2\\nu AA^* -iH)(A^*)^{-1}\\left(\\begin{array}{c} 0\\\\1 \\end{array}\\right)\\\\ (A^*)^{-1}\\left(\\begin{array}{c} 0\\\\1 \\end{array}\\right)\\end{array} \\right]=$ $=\\left[\\begin{array}{c} (2\\nu AA^* -iH)\\varepsilon \\overline{A}\\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right)\\\\\\varepsilon \\overline{A}\\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right)\\end{array} \\right].$ Then $\\nu _2:(K,[u],H)\\in {\\cal H}^m \\times P_1(C) \\times H(2) \\rightarrow \\left[\\begin{array}{c} \\left(\\frac{2\\nu }{m} K -iH \\right)\\varepsilon \\overline{u}\\\\ \\varepsilon \\overline{u}\\end{array} \\right] \\in P_3^{sign(\\nu )}.$ If we fix, for exemple, $m=1,$ then for all $\\left[\\left(\\begin{array}{c} \\omega \\\\\\pi \\end{array}\\right) \\right]\\in P_3^{sign(\\nu )},$ we have $(\\nu _2)^{-1}\\left\\lbrace \\left[\\left(\\begin{array}{c} \\omega \\\\\\pi \\end{array}\\right) \\right]\\right\\rbrace =$ $=\\lbrace (K,[\\varepsilon \\overline{\\pi }],H)\\in {\\cal H}^1 \\times P_1(C) \\times H(2): (\\frac{2\\nu }{m} K-iH)\\pi =\\omega \\rbrace .$ This set represent the “portrait\" of the movement $\\left[\\left(\\begin{array}{c} \\omega \\\\\\pi \\end{array}\\right)\\right]$ in ${\\cal H}^1 \\times P_1(C) \\times H(2)$ considered as State Space of massless particles.","The values of dynamical variables $P,\\ \\overrightarrow{l} ,\\ \\overrightarrow{g},$ on ${\\cal H}^m \\times P_1(C) \\times H(2)$ for the massless Type 4 particles can be obtained by composition of its expressions (REF ), (), (), with the map $\\nu _2$ given in (REF ).","If $(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x_4))\\in {\\cal H}^m \\times S^2 \\times H(2),$ is interpreted as a state of a massless Type 4 particle, where now $m$ is an arbitrary positive number with no physical significance, the values of the canonical dynamical variables in this state are $\\overrightarrow{P}_{massless}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&P_{massless}^4 \\overrightarrow{v},\\\\P_{massless}^4(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&\\frac{-\\eta }{ 2 (k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle ) } \\\\\\overrightarrow{l}_{massless}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&\\frac{\\chi T}{4\\pi }\\frac{k_4\\overrightarrow{v}-\\vec{k}}{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle }+ \\\\ &+&\\overrightarrow{x}\\times \\overrightarrow{P}_{massless},\\\\\\overrightarrow{g}_{massless}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&\\frac{\\chi T}{4\\pi }\\frac{\\vec{k}\\times \\overrightarrow{v}}{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle }+ \\\\ &+& P^4_{massless}\\overrightarrow{x}-x^4\\overrightarrow{P}_{massless}.$ The Pauly-Lubanski four vector for type 4 particles is (c.f.","section ) $W_{massless}=\\frac{\\chi T}{4\\pi } P_{massless}.$ We thus see that the expression of $\\overline{l}$ and $\\overline{g}$ are formally almost identical for massive and massless particles.","But this is not the case for $P.$ The point $(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))$ can be interpreted as being the state of a massive particle or the state of a massless particle, but the value in this state of linear momentum, energy and angular momentum is different in the massive or in the massless case.","The map $\\iota _4,$ what in the case of massive particles whas the identical map, in the case of massless particles can be found to be $\\iota _4:(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))\\in {\\cal H}^m \\times S^2 \\times H(2) \\rightarrow $ $ \\rightarrow \\left( \\frac{h(\\overrightarrow{v},1)}{2(k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle )}, h(\\overrightarrow{x},x^4))\\right)\\in C^+ \\times H(2)$ The $P_{KG},\\ P_{massive}$ and $P_{massless}$ appears in the exponent of the Prewave Functions in State Space of the corresponding particles."]],"string":"[\n [\n \"On the geometry pervading One Particle States\"\n ],\n [\n \"Abstract In this paper, a way is given to obtain explicitly the representations of the Poincar\\\\'e group as can be prescribed by Geometric Quantization.\",\n \"Thus one obtains some forms of the Space of Quantum States of the different relativistic free particles, and I give explicitly these spaces and the corresponding operators for the usually accepted as realistic physical particles.\",\n \"The general description of the massless particles I obtain, is given in terms of solutions of Penrose equations.\",\n \"In the case of Photon, I also give other descriptions, one in terms of the Electromagnetic Field.\",\n \"Since the results are derived from Geometric Quantization, they are related to certain Contact and Symplectic manifols, that I study in detail.\",\n \"The symplectic manifold must be interpreted, according with Souriau, as the Movement Space of the corresponding classical particle, and that leads to propose one of the spaces I use as the State Space of the corresponding classical particle.\",\n \"These spaces are also described in each case.\"\n ],\n [\n \"Introduction.\",\n \"The Wave Equations of relativistic elementary particles, Klein-Gordon, Dirac, Weil etc.., have been originally derived each one in a independent way.\",\n \"An unification was the discovery of the relation of these equations with the representations of Poincaré group (inhomogeneous Lorenz group).\",\n \"The classification of the representations of Poincaré group made by Wigner [17] with important contributions of Majorana, Dirac and Proca [12], [9], [14] leads to a group theoretical study of Wave Equations by Bargmann and Wigner [3].\",\n \"On the other hand, in Kirillov-Kostant-Souriau theory (Geometric Quantization), a description of Quantum Systems is given in terms of elements of the dual of the Lie algebra of the Lie group under consideration.\",\n \"Of course, this way of seing Quantum Mechanics is not completely independent of the preceding one, since it has its origin in a method to obtain representations [10], [1].\",\n \"The correspondence of Quantum States in the sense of Geometric Quantization with Wave Functions in the Quantum Mechanical sense is not clear in all cases.\",\n \"In Souriau's book [15], a very general way to do the passage is given, but it doesn't work in all cases.\",\n \"In this paper we give a geometrical construction that stablishes a one to one correspondence from Quantum States in the sense of Geometric Quantization to Wave Functions in the Quantum Mechanical sense, that is valid for all kinds of relativistic elementary particles.\",\n \"The idea is as follows.\",\n \"In Geometric Quantization, one begins with a regular contact manifold or its associated hermitian line bundle.\",\n \"Quantum states ( in the sense of Geometric Quantification ) can be considered as being the collection of those sections of the hermitian line bundle which satisfies \\\" Planck's condition\\\" (cf.\",\n \"Souriau's book).\",\n \"In this paper we see that these sections are in a one to one correspondence with the (unrestricted) sections of another hermitian line bundle.\",\n \"Thus, this fibre bundle is a good setting to describe the quantum processes under consideration.\",\n \"The main idea to pass from this description to the usual one in terms of Wave Functions, can be intuitively explained as follows.\",\n \"Quantum States in G.Q.\",\n \"attribute an “amplitude of probability” to each movement of the particle.\",\n \"To obtain the corresponding wave function one must proceed as follows: for each event, the corresponding amplitude of probability is obtained by taking all movements passing through the given event, and then “adding up” ( in a suitable sense) the corresponding amplitudes of probability.\",\n \"Of course the concept of “movement passing through an event” is only obvious in the case of the ordinary massive spinless particle, and it is defined in section .\",\n \"The Contact Manifold under consideration, fibers on a Symplectic Manifold that, following Souriau, must be considered as being composed by the \\\"movements of the corresponding classical particle\\\".\",\n \"Then, with the constructions made in this paper, it becomes clear what space must be considered as composed by the \\\"states of the corresponding classical particle\\\", for each kind of relativistic particle.\",\n \"The spaces of Wave Functions and many relevant isomorphic vector spaces we obtain, are the spaces of the representations of Poincaré group that I describe in section .\",\n \"The preceding constructions are made in a explicit way for the relativistic particles usually considered as having physical sense.\",\n \"A section is devoted to massive particles and other to massless particles.\",\n \"In section a explicit application of the preceeding constructions is done for massive particles.\",\n \"One obtains solutions of Klein-Gordon and Dirac equations, and also a description of the wave functions for massive particles of higher spin.\",\n \"Massless particles are studied in section .\",\n \"For massless particles of spin 1/2 one obtains solutions of Weyl equations and for general spin this method leads in a natural way to the description of massless particles by means of solutions of Penrose wave equations [13].\",\n \"This is related to the fact that the Contact and Symplectic Manifolds corresponding to Massless Particles are expresed in a natural way in Twistor Space and its Projective Space, as explained in section REF .\",\n \"In the particular case of Photon, in section REF ,I give also other forms for the Wave Functions.\",\n \"One of them is in terms of the Electromagnetic Field,and are similar to the Wave Functions proposed by Bialynicki-Birula.\",\n \"In section I describe for each particle the concrete space that must be considered to be the \\\"Space of states of the corresponding classical particle\\\".\"\n ],\n [\n \"Notation \",\n \"All differentiable manifolds appearing in this paper are assumed to be $C^{\\\\infty }$ , finite dimensional, Hausdorff and second countable.\",\n \"The set composed by the differentiable vector fields on a differential manifold, $M$ , is denoted by ${\\\\cal D}(M)$ .\",\n \"The set of differential $k$ -forms on $M$ is denoted by $\\\\Omega ^{k}(M)$ .\",\n \"Let $X\\\\in {\\\\cal D}(M),\\\\ \\\\omega \\\\in \\\\Omega ^{k}(M)$ .\",\n \"We denote by $i(X)\\\\omega $ the interior product of X by $\\\\omega $ and by $L(X)\\\\omega $ the Lie derivative of $\\\\omega $ with respect to $X$ .\",\n \"Let $G$ be a Lie group.\",\n \"The Lie algebra of $G$ is the set consisting of the left invariant vector fields on $G$ , provided with its canonical structure of Lie algebra.\",\n \"It is denoted in this paper by $\\\\underline{G}$ .\",\n \"I denote by $\\\\underline{G}^{\\\\ast }$ the dual of $\\\\underline{G}$ .\",\n \"$\\\\underline{G}^{\\\\ast }$ is canonically identified with the set composed by the left invariant 1-forms on $G$ .\",\n \"By identification of each element of $\\\\underline{G}$ or $\\\\underline{G}^{\\\\ast }$ with its value at the neutral element, $e$ , the lie algebra of $G$ is identified to the tangent space at $e$ , and $\\\\underline{G}^{\\\\ast }$ to its dual.\",\n \"There exist a map, the exponential map, $Exp: G \\\\rightarrow \\\\underline{G}$ , such that: if $X\\\\in \\\\underline{G}$ , the integral curve of $X$ with initial value $e$ is $t\\\\in {\\\\rightarrow } Exp \\\\,tX$ .\",\n \"This integral curve is a one parameter subgroup of $G$ .\",\n \"The Lie algebra of the circle Lie group, $S^1$ , is identified to $R$ in such a way that for all $ t\\\\in R,\\\\ Exp\\\\,t=e^{2 \\\\pi i t}$ .\",\n \"If we have an homomorphism of $G$ in $S^1$ , its differential is a linear map from the Lie algebra of $G$ into the Lie algebra of $S^1$ , which is $R$ .\",\n \"Thus the differential of an homomorphism of $G$ into $S^1,$ can be considered as an element of $\\\\underline{G}^{\\\\ast }$ .\",\n \"In the same way, the Lie algebra of the Lie group $R$ is identified to $R$ in such a way that the exponential map becomes the identity.\",\n \"Thus, also in this case, the differential of an homomorphism of $G$ into $R$ , can be considered as an element of $\\\\underline{G}^{\\\\ast }$ .\",\n \"The coadjoint representation is the homomorphism $Ad^{\\\\ast }:g\\\\in G\\\\rightarrow Ad_{g}^{\\\\ast }\\\\in $ Aut $(\\\\underline{G}^{\\\\ast })$ , given by $(Ad_{g}^{\\\\ast }(\\\\alpha ))(X)=\\\\alpha (Ad_{g^{-1}}(X))$ for all $\\\\alpha \\\\in \\\\underline{G}^{\\\\ast },X\\\\in \\\\underline{G}$ , where $Ad_{g^{-1}}$ is the differential of the automorphism of $G$ that sends each $h\\\\in G$ to $g^{-1}hg.$ Let $M$ be a differentiable manifold and $G$ a Lie group acting on $M$ on the left (resp.\",\n \"on the right).\",\n \"Given an element, $X$ , of $\\\\underline{G}$ , we denote by $X_{M}$ the vector field on $M$ whose flow is given by the diffeomorphisms associated by the action to $\\\\lbrace $ Exp$(-tX):t\\\\in {\\\\bf R}\\\\rbrace $ (resp.\",\n \"$\\\\lbrace $ Exp$tX:t\\\\in {\\\\bf R}\\\\rbrace $ ).\",\n \"$X_{M}$ is called the infinitesimal generator of the action associated to $X$ .\",\n \"A principal fibre bundle having $M$ as total space, $B$ as base and $G$ as structural group, will be denoted by $M(B,G)$ .\",\n \"In all that concerns fibre bundles, we use the notation of [11].\"\n ],\n [\n \"A Particular Classical State Space. \",\n \"In this section I motivate the physical interpretation of the geometrical constructions that are to be made in this paper.\",\n \"I do not enter in the details of the computations.\",\n \"Most part of this section is an easy consecuence of Souriau's book [15], which has inspirated this paper.\",\n \"Let us consider a classical relativistic free particle without spin in Minkowski space-time, with rest mass $ m\\\\ne 0$ .\",\n \"Space-time is interpreted as being an abstract four dimensional manifold, M, each inertial observer, R, providing us with a global chart, $\\\\phi _{R}=(x_{R}^1,...,x_{R}^{4})$ .\",\n \"Changes of these global charts are given by transformations corresponding to elements of the Poincaré (i.e.\",\n \"inhomogeneous Lorenz) group, ${\\\\cal P}$ .\",\n \"We consider a family of inertial frames such that changes are all given by ortochronous proper Poincaré transformations, i.e.\",\n \"elements of the connected component of the identity, ${\\\\cal P}_{+}^{\\\\uparrow }$ .\",\n \"A geometrical object in $\\\\bf R^4\\\\rm $ invariant under the action on $\\\\bf R^4 \\\\rm $, provides us with a well defined object in M , whose local expression is the same for all inertial frames.\",\n \"An example is the Minkowski metric, $g$ .\",\n \"When provided with this metric, M becomes Minkowski space.\",\n \"The objects defined in this way would be well defined even if the charts were not global.\",\n \"Charts $\\\\phi _R $ give rise in the canonical way to charts of TM, $\\\\dot{\\\\phi }_R=(x_R^i,\\\\dot{x}_R^i)$ , where $\\\\dot{x}_R^i(v)=v(x_R^i)$ , for all $ v \\\\in TM.$ The charts $\\\\overline{\\\\phi }_R=(x_R^i,P_R^i) $ , $P_R^i=m\\\\,\\\\dot{x}_R^i$ , are more natural in Physics.\",\n \"Changes of these global charts are given by the transformations corresponding to elements of the Poincaré group , in its canonical action on $\\\\bf R^8 \\\\rm \\\\ $$i.e.$ the action given by $(L,C)*\\\\left(\\\\begin{array}{c}z^1\\\\\\\\ \\\\vdots \\\\\\\\z^8\\\\end{array}\\\\right)=\\\\left(\\\\begin{array}{l} L\\\\left(\\\\begin{array}{c}z^1\\\\\\\\\\\\vdots \\\\\\\\z^4\\\\end{array}\\\\right)+C\\\\\\\\ L\\\\left(\\\\begin{array}{c}z^5\\\\\\\\\\\\vdots \\\\\\\\z^8\\\\end{array}\\\\right)\\\\end{array}\\\\right)$ Geometrical objects on $\\\\bf R^8 \\\\rm $ invariant under this action, give us well defined geometrical objects on TM, whose local expressions are equal in the different inertial frames.\",\n \"For example, if $(z^1,\\\\ldots ,z^8)$ is the canonical coordinate system of $\\\\bf R^8 \\\\rm $ , the 1-form: ${\\\\mbox{$\\\\omega $}}_0 \\\\equiv \\\\ z^8 dz^4-\\\\sum _{i=1}^3\\\\ z^{i+4}dz^i$ the vector field: $X_0 \\\\equiv \\\\ \\\\frac{1}{m}\\\\ \\\\sum _{i=1}^4\\\\ z^{i+4}\\\\frac{\\\\partial }{\\\\partial z^i}$ and the submanifold, ${\\\\cal E}_0$, given by: $z^8 \\\\equiv \\\\ \\\\sqrt{m^2+(z^5)^2+(z^6)^2+(z^7)^2}$ are invariant by the action on $\\\\bf R^8 \\\\rm $.\",\n \"Thus the following 1-form, vectorfield, and submanifold are well defined on TM: $\\\\widetilde{\\\\mbox{$\\\\omega $}} = P_R^4\\\\,dx_R^4 -\\\\sum _{i=1}^3\\\\ P_R^i\\\\,dx_R^i$ $\\\\widetilde{X}={ \\\\frac{1}{m}}\\\\ {\\\\sum _{i=1}^4}P_R^i\\\\,\\\\frac{\\\\partial }{\\\\partial x_R^i}$ ${\\\\cal E} & = & \\\\left\\\\lbrace \\\\, v\\\\in TM: P_R^4(v)=\\\\left( m^2+\\\\sum _{i=1}^3(P_R^i(v))^2\\\\right)^{\\\\frac{1}{2}} \\\\right\\\\rbrace \\\\\\\\& = & \\\\left\\\\lbrace \\\\,v\\\\in TM: g(v,v)=1,\\\\dot{x}_R^4(v) > 0 \\\\right\\\\rbrace $ where R is an arbitrary inertial frame.\",\n \"The restriction of $ \\\\widetilde{\\\\mbox{$\\\\omega $}} $ to $\\\\cal E,$ $\\\\omega ,$ is a contact form.\",\n \"$\\\\widetilde{X}$ is tangent to ${\\\\cal E}$ .\",\n \"The restriction of $\\\\widetilde{X}$ to $\\\\cal E,$ $X$ , is the unique vectorfield such that $ i_X {\\\\mbox{$\\\\omega $}} = m,\\\\ i_Xd{\\\\mbox{$\\\\omega $}} = 0.\",\n \"$ Each inertial observer associates to each element, v, of $\\\\cal E$ eight numbers, $\\\\overline{\\\\phi }_R(v)=(x_R^1(v), ...,\\\\ x_R^4(v), P_R^1(v),...,\\\\ P_R^4(v))$ which are interpreted as giving position, time and momentum-energy ( we take c=1).\",\n \"Thus $\\\\cal E$ must be interpreted as being state space.\",\n \"The movements of the free particle under consideration are curves in $\\\\cal E,$ $\\\\gamma $ , having, for each inertial observer, a constant four velocity $i.e.$ $\\\\dot{\\\\phi }_R\\\\circ \\\\gamma (s)=(a^i+s(\\\\dot{x}_R^i)_0, (\\\\dot{x}_R^i)_0) $ where the $(\\\\dot{x}_R^i)_0$ are constant, $(\\\\dot{x}_R^4)_0 > 0, (\\\\dot{x}_R^4)_0^2-\\\\sum _{i=1}^3 (\\\\dot{x}_R^i)_0^2 = 1.$ Since the preceeding relation is equivalent to $\\\\overline{\\\\phi }_R\\\\circ \\\\gamma (s) = ( a^i+s\\\\frac{{(P_R^i)}_0}{m},{(P_R^i)}_0),$ where ${(P_R^i)}_0=m(\\\\dot{x}_R^i)_0,$ one sees that the movements are the integral curves of X.\",\n \"The parameter $s$ coincides, since we take c=1, with proper time (of the trajectory in space time).\",\n \"The action on $\\\\bf R^8 \\\\rm $ gives, by means of any of the inertial observers, $R,$ an action on M by means of $(L,C)\\\\odot v=(\\\\overline{\\\\phi }_R)^{-1}((L,C)*(\\\\overline{\\\\phi }_R(v))),$ for all $v\\\\in {\\\\cal E},\\\\ (L,C)\\\\in {\\\\cal P}_{+}^{\\\\uparrow }$ .\",\n \"This action obviously preserves $\\\\cal E,$ and $ {\\\\mbox{$\\\\omega $}} $ in such a way that $({\\\\mbox{${\\\\cal E}$}},{\\\\mbox{$\\\\omega $}})$ is a homogeneous contact manifold for the given action, but different inertial observers lead to different actions.\",\n \"Now, we shall describe ${\\\\mbox{${\\\\cal E}_0$}}$ in a different way, thus obtaining a picture of state space more suitable for its generalisation to other kind of particles.\",\n \"Let $Y_8$ be the infinitesimal generator of the action on $\\\\bf R^8 \\\\rm $ associated to the element Y of the Lie algebra of ${\\\\cal P}_{+}^{\\\\uparrow },\\\\ \\\\underline{\\\\cal P}$ .\",\n \"Since the action on $\\\\bf R^8 \\\\rm $ preserves ${\\\\mbox{$\\\\omega $}}_0$ , we have $L_{Y_8} {\\\\mbox{$\\\\omega $}}_0=0,$ so that $i_{Y_8}\\\\,d {\\\\mbox{$\\\\omega $}}_0=-d\\\\,(i_{Y_8}{\\\\mbox{$\\\\omega $}}_0)=-d({\\\\mbox{$\\\\omega $}}_0(Y_8)).$ Thus we define a map, $ {\\\\mu }_0$ , called the momentum map, from $\\\\bf R^8 \\\\rm $ into $\\\\underline{\\\\cal P}^*$ , by means of $ {\\\\mu _0}(z)\\\\cdot Y=-( {\\\\mbox{$\\\\omega $}}_0(Y_8))(z),$ for all $Y \\\\in \\\\underline{\\\\cal P}, z \\\\in \\\\bf R^8 \\\\rm .$ Let $\\\\lbrace Y_{\\\\alpha }^i, Y_\\\\beta ^i, Y_{\\\\gamma }^i, Y_\\\\delta :\\\\ i=1, 2, 3 \\\\rbrace $ be the basis of $\\\\underline{\\\\cal P}$ , composed by the usual generators of Lorenz rotations and space-time translations.\",\n \"It can be proved that $\\\\mu _0=z^8\\\\,Y_\\\\delta ^* - \\\\sum _{i=1}^3\\\\ \\\\lbrace [(z^1, z^2,z^3)\\\\times (z^5, z^6, z^7)]^i\\\\, Y_\\\\alpha ^{i*}+$ $+ ( z^4z^{i+4}\\\\,-z^8z^i)\\\\,Y_\\\\beta ^{i*} +z^{i+4}\\\\, Y_{\\\\gamma }^{i*}\\\\,\\\\rbrace $ where $\\\\lbrace Y_\\\\alpha ^{1*},..., Y_\\\\delta ^*\\\\rbrace $ is the dual basis of $\\\\lbrace Y_\\\\alpha ^1,..., Y_\\\\delta \\\\rbrace $ .\",\n \"The map $\\\\mu _0$ is equivariant for the given action on $\\\\bf R^8 \\\\rm $ and coadjoint action.\",\n \"Thus, since $(0, ...,0,m)\\\\in {\\\\mbox{${\\\\cal E}_0$}}$ and ${\\\\mu _0}(0, ...,0,m)=m\\\\, Y_\\\\delta ^*,\\\\ {\\\\mu _0}({\\\\mbox{${\\\\cal E}_0$}})$ is the coadjoint orbit of $m\\\\,Y_\\\\delta ^*$ .\",\n \"We also have ${\\\\mu _0^*}\\\\, {\\\\mbox{$\\\\Omega $}} =d {\\\\mbox{$\\\\omega $}} _o$ , where $ {\\\\mbox{$\\\\Omega $}} $ is the Kirillov symplectic form on the coadjoint orbit.\",\n \"For each $\\\\alpha $ in the coadjoint orbit, $\\\\mu _0^{-1}\\\\,\\\\lbrace {\\\\mbox{$\\\\alpha $}}\\\\rbrace $ is the image of a integral curve of $X_0$ , that we know to be local expresion of movements.\",\n \"Then $\\\\mu _0$ gives a one to one correspondence between points of the coadjoint orbit and movements of the particle under consideration.\",\n \"Now we consider the map $f:\\\\ (a,b)\\\\in {\\\\mbox{${\\\\cal E}_0$}}\\\\longrightarrow (a, \\\\mu _0\\\\,(a,b))\\\\in {{\\\\bf R^4}\\\\rm }\\\\times \\\\underline{\\\\cal P}^*.$ By using the above expression for $\\\\mu _0$ , one sees that $f$ is an imbedding.\",\n \"Also, $f$ is equivariant when one considers the given action in ${\\\\mbox{${\\\\cal E}_0$}}$ and the “product action” in ${{\\\\bf R^4}\\\\rm }\\\\times \\\\underline{\\\\cal P}^*$ given by $(L,C)*(a, {\\\\mbox{$\\\\alpha $} } )=(La+C,Ad_{(L,C)}^*\\\\,{\\\\mbox{$\\\\alpha $} } ),\\\\ \\\\ \\\\ \\\\ \\\\forall (L,C)\\\\in {\\\\cal P}_{+}^{\\\\uparrow }, (a, {\\\\mbox{$\\\\alpha $} } )\\\\in f({\\\\mbox{${\\\\cal E}_0$}}).$ Observe that, as a consequence, $f({\\\\mbox{${\\\\cal E}_0$}})$ is an orbit of that action.\",\n \"Each inertial frame, R, enables us to “see” state space, $\\\\cal E$ , by means of $f\\\\circ \\\\overline{\\\\phi }_R$ , as being $f({\\\\mbox{${\\\\cal E}_0$}})\\\\subset { {\\\\bf R}^4}\\\\times \\\\underline{\\\\cal P}^*$ .\",\n \"Changes of inertial frames are now given by transformations coming from the product action.\",\n \"Since $f\\\\circ \\\\overline{\\\\phi }_R(e)=(x_R^1(e),...,x_R^4(e),\\\\ \\\\mu _0(\\\\overline{\\\\phi }_R(e)))$ , if we denote $\\\\mu _R \\\\equiv \\\\mu _0\\\\circ \\\\overline{\\\\phi }_R$ we have $f\\\\circ \\\\overline{\\\\phi }_R =(\\\\phi _R, \\\\mu _R),$ and $\\\\mu _R$ stablishes a one to one correspondence of trajectories of X ($i.e.$ movements of the particle parametrized by proper time) with points of the coadjoint orbit.\",\n \"Thus, coadjoint orbit is to be identified to movement space for all inertial frames, althought the identification is different for each $R$ .\",\n \"The picture of a state, $e\\\\in \\\\cal E$ , obtained by R is now $(\\\\phi _R(e),\\\\mu _R(e)) \\\\ i.e.$ an event, $\\\\phi _R(e)$ , and a movement, $\\\\mu _R(e)$ , “passing through” the event.\",\n \"Each $Y\\\\in \\\\underline{\\\\cal P}$ defines a function on $\\\\underline{\\\\cal P}^*$ , denoted by the same symbol.\",\n \"Thus $Y\\\\circ \\\\pi _2\\\\circ f\\\\circ \\\\overline{\\\\phi }_R$ is a function on $\\\\cal E$ ($i.e.$ a dynamical variable) where $\\\\pi _2$ is the canonical projection of $ { {\\\\bf R}^4}\\\\times \\\\underline{\\\\cal P}^*$ onto $\\\\underline{\\\\cal P}^*$ .\",\n \"We have: $- {Y_{\\\\gamma }^i}\\\\circ \\\\mu _R &=&P_R^i \\\\\\\\{Y_\\\\delta }\\\\circ \\\\mu _R&=& P_R^4 \\\\\\\\- {Y_{\\\\alpha }}^i\\\\circ \\\\mu _R&=&({\\\\overrightarrow{x}_R} \\\\times \\\\overrightarrow{P}_R)^i \\\\\\\\{Y_\\\\beta }^i\\\\circ \\\\mu _R&=&(P_R^4\\\\overrightarrow{x}_R -x_R^4\\\\overrightarrow{P}_R)^i$ where $i=1,2,3,\\\\overrightarrow{x}_R= (x_R^1, x_R^2,x_R^3),$ and $\\\\overrightarrow{P}_R= (P_R^1,P_R^2,P_R^3)$ .\",\n \"Thus the function on ${ {\\\\bf R}^4}\\\\times \\\\underline{\\\\cal P}^*$ given by $P=(P^1,P^2,P^3,P^4) \\\\equiv (- {Y_{\\\\gamma }^1}\\\\circ \\\\pi _2,- {Y_{\\\\gamma }^2}\\\\circ \\\\pi _2, - {Y_{\\\\gamma }^3}\\\\circ \\\\pi _2,{Y_\\\\delta }\\\\circ \\\\pi _2)$ ,can be considered as an abstraction of linear momentum: each inertial observer obtains $f\\\\circ \\\\overline{\\\\phi }_R(e)$ as the picture of $e\\\\in {\\\\mbox{$\\\\cal E$}}$ , and the linear momentum he measures is $P(f\\\\circ \\\\overline{\\\\phi }_R(e))$ .\",\n \"In the same way the components of $\\\\overrightarrow{l}= ( - Y_\\\\alpha ^1\\\\circ \\\\pi _2, - {Y_\\\\alpha ^2}\\\\circ \\\\pi _2 ,- {Y_\\\\alpha ^3}\\\\circ \\\\pi _2)$ and $\\\\overrightarrow{g}= ( Y_\\\\beta ^1\\\\circ \\\\pi _2,Y_\\\\beta ^2\\\\circ \\\\pi _2 , Y_\\\\beta ^3\\\\circ \\\\pi _2)$ must be interpreted as being the components of (relativistic) angular momentum.\",\n \"If we denote ${\\\\cal P}_{+}^{\\\\uparrow }$ by G, we can summarize what has been said as follows State space of our free particle is such that each inertial observer establishes a diffeomorphism from it to an orbit of G in ${ {\\\\bf R}^4}\\\\times \\\\underline{G}^*,$ for the canonical action .\",\n \"Changes of inertial observer are given by the transformations given by the same action.\",\n \"An inertial observer, R, thus sees any state as a pair, the first component is an event and the second a movement containig the event.\",\n \"The values at that point of $P,\\\\overrightarrow{l},\\\\overrightarrow{g}$ , are the values measured by R of linear momentum and angular momentum.\",\n \"In this paper I accept that this is also valid for all relativistic free particles, with or without mass or spin.\",\n \"More precisely, it is assumed that the possible state space of the relativistic free particles are the orbits of G in ${ {\\\\bf R}^4}\\\\times \\\\underline{G}^*$ , where now G is the universal (two fold) covering group of ${\\\\cal P}_{+}^{\\\\uparrow }$ .\",\n \"The preceeding interpretations of movements, states, linear and angular momentum are also considered as valid.\",\n \"In what follows, we assume that an inertial observer, R, has been fixed.\",\n \"If state space is the orbit $\\\\cal O$ and $( x, \\\\alpha )\\\\in \\\\cal O$ , we have seen how $\\\\alpha $ can be considered as a movement.\",\n \"In fact, this movement is composed by the elements of $\\\\cal O$ of the form $( y, \\\\alpha ).$ The set composed by such $y$ , $M_{\\\\alpha },$ compose the ordinary portrait of the movement $\\\\alpha $ in ${\\\\bf R}^4,$ obtained by the inertial observer R. Obviously $M_{\\\\alpha }=\\\\lbrace g*x: g\\\\in G_\\\\alpha \\\\rbrace $ where $G_\\\\alpha $ is the isotropy subgroup at $\\\\alpha $ of the coadjoint representation.\",\n \"This way of looking at states also enables us to determine all movements containing a given event: if $( x, \\\\alpha )\\\\in \\\\cal O$ , the movements containing the event $x$ compose the set $N_x=\\\\lbrace Ad^*_g\\\\alpha :g\\\\in G_x\\\\rbrace $ where $G_x$ is the isotropy group at $x$ of the action of $G$ on ${\\\\bf R}^4.$\"\n ],\n [\n \"Universal covering group of the Poincaré Group. \",\n \"It is a well known fact that the universal covering group of Poincaré group is a semidirect product of $\\\\bf {SL}(2,\\\\bf {C})$ by a four dimensional real vector space.\",\n \"In this section, I recall some general facts about this group and stablishes the notation.\",\n \"Let $(x^1,x^2,x^3,x^4)$ be the canonical coordinates in $\\\\bf {R}^4$ , I the $2\\\\times 2$ unit matrix and ${\\\\sigma }_1, {\\\\sigma }_2, {\\\\sigma }_3$ the Pauli matrices, $i.e.$ $\\\\left(\\\\begin{array}{cc} 0 & 1\\\\\\\\1 & 0\\\\end{array}\\\\right),\\\\ \\\\left(\\\\begin{array}{cc}0 & -i\\\\\\\\i & 0\\\\end{array}\\\\right), \\\\,\\\\,\\\\left(\\\\begin{array}{cc}1 & 0\\\\\\\\0 & -1\\\\end{array}\\\\right)$ respectively.\",\n \"A generic point of $\\\\bf {R}^4$ will be denoted $x=(x^1,x^2,x^3,x^4)$ .\",\n \"We define an isomorphism $h$ , from $\\\\bf {R}^4$ onto the real vector espace, $\\\\bf {H}(2)$ , of the hermitian $2\\\\times 2$ matrices, by means of $h(x)=x^4I+\\\\sum _{i=1}^{3}x^i{\\\\sigma }_i=\\\\left(\\\\begin{array}{cc}x^4+x^3 , & x^1-ix^2\\\\\\\\x^1+ix^2 , &x^4-x^3\\\\end{array}\\\\right)$ We have $Det \\\\, h(x)=_{ m}$ , where $<\\\\, ,\\\\, >_m $ is Minkowski pseudo-scalar product $_m=x^4y^4-\\\\sum _{i=1}^3 x^iy^i.$ If $x=(x^1,x^2,x^3,x^4)$ we denote $\\\\vec{x}=(x^1,x^2,x^3)$ and $h(\\\\vec{x},x^4)=h(x).$ The following formulae are useful for many computations along this paper $[h(\\\\vec{k},k_4),h(\\\\vec{x},x^4)]\\\\stackrel{def}{=}h(\\\\vec{k},k_4)h(\\\\vec{x},x^4)-h(\\\\vec{x},x^4)h(\\\\vec{k},k_4)=2ih(\\\\vec{k} \\\\times \\\\vec{x},0),$ where $\\\\times $ means ordinary vector product, $\\\\lbrace h(\\\\vec{k},k_4),h(\\\\vec{x},x^4)\\\\rbrace & \\\\stackrel{def}{=}&h(\\\\vec{k},k_4)h(\\\\vec{x},x^4)+h(\\\\vec{x},x^4)h(\\\\vec{k},k_4)=\\\\\\\\ &=& 2h(k_4\\\\vec{x} +x^4 \\\\vec{k},k_4x^4+\\\\langle \\\\vec{k},\\\\vec{x} \\\\rangle ), $ where $\\\\langle \\\\, .\",\n \"\\\\, ,\\\\, .\",\n \"\\\\, \\\\rangle $ is the usual scalar product in $\\\\mathbb {R}^3,$ $h(\\\\vec{k},k_4)\\\\varepsilon \\\\overline{ h(\\\\vec{x},x^4)} \\\\varepsilon -h(\\\\vec{x},x^4)\\\\varepsilon \\\\overline{ h(\\\\vec{k},k_4)} \\\\varepsilon =\\\\\\\\\\\\nonumber =2[h(k_4 \\\\vec{x}-x^4 \\\\vec{k},0)+ih(\\\\vec{k} \\\\times \\\\vec{x},0)]$ where the bar means complex conjugation, and we denote by ${\\\\varepsilon }$ the matrix $i\\\\sigma _2,\\\\ i.e.$ ${\\\\varepsilon }=\\\\left(\\\\begin{array}{cc}0 & 1\\\\\\\\-1 & 0\\\\end{array}\\\\right).$ Notice that ${ }^tA\\\\,{\\\\varepsilon }\\\\,A=(Det \\\\,A)\\\\,{\\\\varepsilon },$ so that $A\\\\,{\\\\varepsilon }={\\\\varepsilon }\\\\,({}^tA)^{-1}$ if $A\\\\in \\\\bf {SL}(2,\\\\bf {C})$ .\",\n \"Also we have $\\\\frac{1}{2}Tr(h(x){\\\\varepsilon }\\\\overline{h(y)}{\\\\varepsilon })=-_m$ for all $x, y\\\\in \\\\bf {R}^4.$ We define an action on the left of the Lie group $\\\\bf {SL}(2,\\\\bf {C})$ on the abelian Lie group $\\\\bf {H}(2)$ by means of $A*H=AHA^*$ for all $A\\\\in \\\\bf {SL}(2,\\\\bf {C})$ , $H\\\\in \\\\bf {H}(2)$ , where $A^*$ is the transposed of the complex conjugate of $A$ .\",\n \"To this action by automorphisms of $\\\\bf {H}(2)$ then corresponds a semidirect product, $\\\\bf {SL}(2,\\\\bf {C})\\\\oplus \\\\bf {H}(2)$ , whose group law is given by $(A,H)*(B,K)=(AB,AKA^*+H)$ The identity element is $(I,0)$ and $(A,H)^{-1}=(A^{-1},-A^{-1}HA^{*^{-1}})$ .\",\n \"This semidirect product acts on the left on $\\\\bf {R}^4$ by means of $(A,H)* x=h^{-1}(Ah(x)A^*+H).$ Poincaré group, ${\\\\cal {P}}$ , is identified to the closed subgroup of $GL(5; \\\\bf {R})$ composed by the matrices $\\\\left(\\\\begin{array}{cc}L & C\\\\\\\\0 & 1\\\\end{array}\\\\right)$ where $C\\\\in \\\\bf {R}^4$ and $L\\\\in {\\\\cal {O}}(3,1)$ (such a matrix is denoted in the following simply by $(L,C)$ ).\",\n \"For all $(A,H)\\\\in \\\\bf {SL}\\\\oplus \\\\bf {H}(2)$ (where $\\\\bf {SL}$ stands for $\\\\bf {SL}(2;\\\\bf {C})$ ), there exists a unique $(L,C)\\\\in {\\\\cal {P}}$ such that $(A,H)*x=Lx+C$ for all $x\\\\in \\\\bf {R}^4$ .\",\n \"The map, $\\\\rho $ , from $\\\\bf {SL}\\\\oplus \\\\bf {H}(2)$ into ${\\\\cal {P}}$ defined by sending such a $(A,H)$ to the corresponding $(L,C)$ , is a homomorphism of Lie groups, whose kernel consists of $(I,0)$ and $(-I,0)$ .\",\n \"In fact, $\\\\rho $ is a two fold covering map of the identity component in ${\\\\cal {P}}$ , ${\\\\cal {P}}_+^{\\\\uparrow }$ .\",\n \"Since $\\\\bf {SL}$ and $\\\\bf {H}(2)$ are connected and simply connected it follows that $\\\\bf {SL} \\\\oplus \\\\bf {H}(2)$ is the universal covering group of ${\\\\cal {P}}_+^{\\\\uparrow }$ .\",\n \"The differential of $\\\\rho $ is an isomorphism from the Lie algebra of $\\\\bf {SL}\\\\oplus \\\\bf {H}(2)$ onto the Lie algebra of ${\\\\cal {P}}$ .\",\n \"The standard method to handle semidirect products enable us to identify the Lie algebra of $\\\\bf {SL}\\\\oplus \\\\bf {H}(2)$ with $\\\\bf {sl(2,{\\\\bf C})}\\\\times \\\\bf {H}(2)$ , the Lie bracket being $[(a,k),(a^{\\\\prime },k^{\\\\prime })]=([a,a^{\\\\prime }], ak^{\\\\prime }+ k^{\\\\prime }a^*-(a^{\\\\prime }k+ka^{\\\\prime *})).$ If $(a,h)\\\\in \\\\bf {sl(2,{\\\\bf C})}\\\\times \\\\bf {H}(2),\\\\ t\\\\in {\\\\bf R},$ we have $Exp[t(a,h)] =\\\\left(e^{ta},\\\\int _0^t e^{sa}\\\\,h\\\\, e^{sa^*}\\\\,ds \\\\right).$ In this paper, we use the basis of $\\\\bf {sl}\\\\times \\\\bf {H}(2)$ corresponding by $d\\\\rho $ to the basis of ${\\\\cal {P}}$ associated to linear momentum and angular momentum in section .\",\n \"This basis is composed by the following elements $P^k &=&(0,-\\\\sigma _k),\\\\ \\\\ \\\\ k=1,2,3, \\\\nonumber \\\\\\\\P^4 &=&(0,\\\\sigma _4)=(0,I), \\\\nonumber \\\\\\\\l^k &=&(i\\\\frac{\\\\sigma _k}{2},0), \\\\\\\\g^k &=&(\\\\frac{\\\\sigma _k}{2},0).\",\n \"\\\\nonumber $ An element, $X$ , of $\\\\bf {sl(2,{\\\\bf C})}\\\\times \\\\bf {H}(2)$ defines a (linear) function on $(\\\\bf {sl(2,{\\\\bf C})}\\\\times \\\\bf {H}(2))^*.$ Its restriction to movement space ( a coadjoint orbit) must be considered as a dynamical variable.\",\n \"But also, if we have a representation on a vector space (resp.\",\n \"an action on a manifold) of $\\\\bf {SL}\\\\oplus \\\\bf {H}(2),$ $X$ gives rise to a infinitesimal generator of the representation (resp.\",\n \"the action), i.e., an endomorphism of the vector space (resp.\",\n \"a vector field on the manifold).\",\n \"We define vector valued functions on the dual of the Lie algebra as follows $P&=&(P^1,\\\\,P^2,\\\\,P^3,\\\\,P^4) \\\\\\\\\\\\overrightarrow{l}&=&(l^1,\\\\,l^2,\\\\,l^3)\\\\\\\\\\\\overrightarrow{g}&=&(g^1,\\\\,g^2,\\\\,g^3)$ what will be considered as being linear momentum and angular momentum.\",\n \"We define a non degenerate scalar product in $\\\\bf {sl}\\\\times \\\\bf {H}(2)$ by means of $<(a,k),(b,l)> & =&2Re Tr (\\\\frac{1}{4}k{\\\\varepsilon }\\\\overline{l}{\\\\varepsilon }-ab)= \\\\nonumber \\\\\\\\& =&\\\\frac{1}{2}Tr(k{\\\\varepsilon }\\\\overline{l}{\\\\varepsilon })-2 Re Tr\\\\, ab.\",\n \"\\\\nonumber $ This scalar product defines in the standard way an isomorphism from the Lie algebra of $\\\\bf {SL}\\\\oplus \\\\bf {H}(2)$ onto its dual.\",\n \"The image of $(a,k)\\\\in \\\\bf {sl}\\\\times \\\\bf {H}(2)$ by this isomorphism, will be denoted by $\\\\lbrace a,k\\\\rbrace \\\\in \\\\left(\\\\bf {sl}\\\\times \\\\bf {H}(2)\\\\right)^{*}$ , and is given by $\\\\lbrace a,k\\\\rbrace \\\\left((b,m)\\\\right)=<(a,k),(b,m)>.$ With this notation, the values of $P,\\\\ \\\\overrightarrow{l}$ and $\\\\overrightarrow{g}$ at $\\\\lbrace a,k\\\\rbrace $ are, when written in terms of its hermitian form $h(P( \\\\lbrace a,\\\\ k \\\\rbrace ))&=&-k, \\\\\\\\h(\\\\vec{l}(\\\\lbrace a,\\\\ k \\\\rbrace ),\\\\ 0)&=& i\\\\,(a^*-a), \\\\\\\\h(\\\\vec{g}(\\\\lbrace a,\\\\ k \\\\rbrace ),\\\\ 0)&=& -(a+a^*).$ so that $\\\\lbrace a,k\\\\rbrace =\\\\lbrace -\\\\frac{1}{2}h(\\\\vec{g}(\\\\lbrace a,\\\\ k \\\\rbrace ),0)+\\\\frac{i}{2}h(\\\\vec{l}(\\\\lbrace a,\\\\ k \\\\rbrace ),0),-h(P( \\\\lbrace a,\\\\ k \\\\rbrace ))\\\\rbrace .$ i.e.\",\n \"$-(1/2)h(\\\\vec{g}(\\\\lbrace a,\\\\ k \\\\rbrace ),0)$ is the hermitian real part of $a$ and the matrix $(1/2)h(\\\\vec{l}(\\\\lbrace a,\\\\ k \\\\rbrace ),0)$ its hermitian imaginary part.\",\n \"A straightforward computation, leads to the following formula for the coadjoint representation $Ad^*_{(A,H)}\\\\lbrace a,k\\\\rbrace =\\\\lbrace AaA^{-1}+\\\\frac{1}{4}(AkA^*{\\\\varepsilon }\\\\overline{H}{\\\\varepsilon }-H{\\\\varepsilon }\\\\overline{AkA^*}{\\\\varepsilon }),AkA^*\\\\rbrace .$ To end this section, we define other functions on the dual of the Lie algebra of $\\\\bf {SL}\\\\oplus \\\\bf {H}(2).$ One of these is $\\\\vert P \\\\vert $ , defined by $\\\\left| P \\\\right| ( \\\\lbrace a, k\\\\rbrace )= Det\\\\,\\\\left( h(P( \\\\lbrace a,\\\\ k \\\\rbrace )) \\\\right)=Det\\\\,\\\\left( k \\\\right),$ whose physical meaning is mass square.\",\n \"Then, $\\\\left| P \\\\right| ( Ad^*_{(A,H)}\\\\lbrace a, k\\\\rbrace )=Det (-AkA^*)=Det(k),$ so that the value of $\\\\vert P \\\\vert $ , is constant along any coadjoint orbit.\",\n \"The other is defined in terms of the Pauli-Lubanski fourvector, given by $W&=&(\\\\overrightarrow{ W},W^4)\\\\\\\\\\\\overrightarrow{ W}&=&P^4 \\\\vec{l}+\\\\vec{P} \\\\times \\\\vec{g},\\\\\\\\W^4&=&\\\\langle \\\\vec{P},\\\\vec{l} \\\\rangle .$ Using (REF ), (REF ) and (REF ), the hermitian form of $W$ is found to be $h(W( \\\\lbrace a, k\\\\rbrace ))= i(a\\\\ k\\\\ -k\\\\ a^*).\",\n \"$ One can prove that $h(W(Ad^*_{(A,H)}\\\\lbrace a,\\\\ k\\\\rbrace ))=A\\\\ h( W(\\\\lbrace a,\\\\ k\\\\rbrace ))\\\\ A^*,$ so that the function $\\\\vert W \\\\vert ( \\\\lbrace a, k\\\\rbrace )= Det(h(W( \\\\lbrace a, k\\\\rbrace ))),$ is also constant along each coadjoint orbit.\"\n ],\n [\n \"Quantizable forms\",\n \"In this section I recall some of the geometric constructions done in [16], [4], [5], [6], [7].\",\n \"I shall give a way to construct homogeneous contact manifolds that fibers on the coadjoint orbits of Lie groups.\",\n \"This is not possible on an arbitrary orbit, but only on the so called quantizable ones.\",\n \"The idea is to consider the quantizable coadjoint orbits of $\\\\bf {SL}(2;{\\\\bf C}) \\\\oplus \\\\bf {H}(2)$ as the classical movement space of relativistic free particles.\",\n \"The corresponding homogeneous contact manifolds are geometrical objects enabling us to construct the Wave Functions, and representations of this group.\",\n \"Let $G$ be a Lie group and $\\\\alpha $ an element of the dual of the Lie algebra of $G$ , $\\\\underline{G}^*$ , where we consider the coadjoint action.\",\n \"We say that , $\\\\alpha $ is quantizable if there exists a surjective homomorphism, $C_\\\\alpha $ , from the isotropy subgroup at $\\\\alpha $ , $G_\\\\alpha $ , onto the unit circle, ${\\\\bf S}^1 \\\\rm $, whose differential is $\\\\alpha $ (cf.\",\n \"section where the way in which the differential is identified to an element of $\\\\underline{G}^*$ , is detailed).\",\n \"In a more explicit way, this means that $\\\\alpha $ is quantizable if there exists a surjective homomorphism, $C_\\\\alpha ,$ from the isotropy subgroup at $\\\\alpha $ , $G_\\\\alpha ,$ onto the unit circle, such that, for all $X$ in the Lie algebra of $G_\\\\alpha $ and t in R $C_{\\\\alpha }(Exp\\\\,tX)=e^{2 \\\\pi i t {\\\\alpha }(X)}.$ If a form is quantizable, all the elements of its coadjoint orbit are quantizable.\",\n \"These orbits are called quantizable orbits .\",\n \"The form $\\\\alpha $ is said to be R-quantizable if there exists a surjective homomorphism from $G_\\\\alpha $ onto the usual additive Lie group of reals, R , whose differential is $\\\\alpha $ .\",\n \"In other words, $\\\\alpha $ is R-quantizable if there exists a surjective homomorphism from $G_\\\\alpha $ onto R , $H_{\\\\alpha }$ , such that, for all $X$ in the Lie algebra of $G_\\\\alpha $ , and $t$ in R $H_{\\\\alpha }(Exp\\\\,tX)=t\\\\,{\\\\alpha }(X).$ To such a $H_{\\\\alpha }$ , one can associate an homomorphism, $C_{\\\\alpha }$ , onto ${\\\\bf S}^1$ by means of $C_{\\\\alpha }:g \\\\in {G_\\\\alpha } \\\\rightarrow e^{2 \\\\pi i H_{\\\\alpha }(g)} \\\\in {\\\\mbox{${\\\\bf S}^1 \\\\rm $}}.$ The differential of $C_{\\\\alpha }$ is $\\\\alpha $ .\",\n \"Thus when $\\\\alpha $ is R-quantizable , it is quantizable .\",\n \"All elements of a coadjoint orbit containing a R-quantizable form, are R-quantizable.\",\n \"In this case, the orbit is said to be a R-quantizable orbit.\",\n \"In [7] a slightly more general concept of quantizability is used, but it is unnecesary for the purposes of the present paper.\",\n \"In what follows we assume that $\\\\alpha $ is quantizable and $C_\\\\alpha $ is a homomorphism from $G_\\\\alpha $ onto the unit circle, whose differential is $\\\\alpha $ .\",\n \"We identify the coadjoint orbit, ${\\\\cal O}_{\\\\alpha },$ with $G/G_{\\\\alpha }$ by means of the diffeomorphism $g G_{\\\\alpha } \\\\in \\\\frac{G}{G_{\\\\alpha }} \\\\rightarrow Ad^*_g \\\\alpha \\\\in {\\\\cal O}_{\\\\alpha }.$ We define an action of ${\\\\bf S}^1$ on $G/Ker\\\\,C_\\\\alpha $ by means of $(g\\\\,Ker\\\\,C _\\\\alpha )*s=g\\\\,h\\\\,Ker\\\\,C_\\\\alpha $ where $h$ is any element of $G_{\\\\mbox{$ \\\\alpha $}}$ such that $C_\\\\alpha (h)=s$ .\",\n \"Actually $(G/Ker\\\\,C_\\\\alpha )$ $(G/G_{\\\\mbox{$\\\\alpha $}},\\\\bf S\\\\rm ^1)$ is a principal fibre bundle , the bundle action is given by (REF ) and the bundle projection, by the canonical map, $\\\\pi : g{KerC_\\\\alpha } \\\\longrightarrow g {G_\\\\alpha }$ This action of ${\\\\bf S}^1$ , commutes with the canonical action of $G$ on $G/Ker\\\\,C_\\\\alpha $ .\",\n \"We denote by $\\\\pi ^c$ and $\\\\pi ^s$ the canonical maps $\\\\pi ^c&:& g\\\\in G \\\\longrightarrow g {KerC_\\\\alpha } \\\\in \\\\frac{G}{{KerC_\\\\alpha }}\\\\\\\\\\\\pi ^s &:& g\\\\in G \\\\longrightarrow g {G_\\\\alpha } \\\\in \\\\frac{G}{{G_\\\\alpha }}.$ There exist an unique 1-form, $\\\\Omega $ , on the homogeneous space $G/Ker\\\\,C_\\\\alpha $ such that $(\\\\pi ^c)^*\\\\Omega =\\\\alpha .$ The 1-form $\\\\Omega $ is a contact form, invariant under the action of $G.$ Let $Z(\\\\Omega )$ be the vectorfield defined by $i_{Z (\\\\Omega )} \\\\Omega = 1,\\\\ \\\\ i_{Z(\\\\Omega )}\\\\,d\\\\Omega =0.$ All the integral curves of $Z(\\\\Omega )$ have the same period.\",\n \"If we denote by ${T(\\\\Omega )}$ this period, then $\\\\Omega /{T(\\\\Omega )}$ is a connexion form.\",\n \"Since the structural group is abelian, the curvature form is $d\\\\Omega /{{T(\\\\Omega )}} $ .\",\n \"There exist an unique 2-form, $\\\\omega ,$ on $G/G_{\\\\alpha }$ , such that $\\\\pi _*\\\\omega =\\\\frac{d\\\\Omega }{{T(\\\\Omega )}}.$ Thus $(\\\\pi ^s)^*\\\\omega =\\\\frac{d\\\\alpha }{T(\\\\Omega )}.$ The form $\\\\omega $ is an invariant symplectic form, and its cohomology class is integral.\",\n \"Each $m\\\\in \\\\underline{G},$ define a function on $\\\\underline{G}^*$ so that it does also on the coadjoint orbit.\",\n \"Since we think in the coadjoint orbit as being the movement space of a particle, we can say that m defines a dynamical variable, $D_m.$ Such a $m\\\\in \\\\underline{G},$ also define a infinitesimal generator of the canonical action of $G$ on $G/Ker\\\\,C_\\\\alpha $ , denoted by $X_m^c,$ and a infinitesimal generator of the canonical action of $G$ on $G/G_\\\\alpha $ , denoted by $X_m^s.$ Since $\\\\Omega $ is left invariant by the action of $G,$ we have $L_{X_m^c}\\\\Omega =0,$ so that the relation $L_X=d\\\\,i_X+i_X\\\\,d$ leads to $i_{X_m^c}\\\\,d\\\\Omega =-d[\\\\Omega (X_m^c)].$ A computation gives ${\\\\begin{array}{c}(\\\\Omega (X_m^c) \\\\circ \\\\pi ^c)(g)= Ad^*_g \\\\alpha _0 (-m)=-D_m(Ad^*_g \\\\alpha _0)=\\\\\\\\=- D_m(g\\\\, G_\\\\alpha )=-D_m\\\\circ \\\\pi (g\\\\, Ker\\\\,C_\\\\alpha ),\\\\end{array}}$ so that $\\\\Omega (X_m^c)=-D_m\\\\circ \\\\pi .$ Thus (REF ), (REF ), and the fact that $\\\\pi _* X_m^c=X_m^s,$ lead to $i_{X_m^s}\\\\omega =\\\\frac{1}{T\\\\left( \\\\Omega \\\\right)}d D_m.$ Since the flow of the vector field $X_m^s$ (resp.\",\n \"$X_m^c$ ) preserves $\\\\omega $ (resp.\",\n \"$\\\\Omega $ ), it is an infinitesimal automorphism of the symplectic (resp.\",\n \"contact) structure, i.e.\",\n \"a locally hamiltonian vector field.\",\n \"Equation (REF ) tell us that in fact $X_m^s$ is globally hamiltonian and $ D_m/{T\\\\left( \\\\Omega \\\\right)}$ is the corresponding hamiltonian.\",\n \"As usual in the Theory of Connections, a map, $f$ , into $G/{KerC_\\\\alpha }$ is called horizontal, if $f^{*}\\\\Omega =0.$ Let $f_0$ be a map into $G/{G_\\\\alpha }$ .\",\n \"An horizontal lift of $f_0$ is an horizontal map into $G/{KerC_\\\\alpha }$ , such that $\\\\pi \\\\circ f =f_0$ .\",\n \"The horizontal lift of curves can be described as follows.\",\n \"Given a curve, $\\\\gamma $ in $G/G_{\\\\mbox{$\\\\alpha $}}$ , the horizontal lift of $\\\\gamma $ to $g\\\\,KerC_\\\\alpha $ is $\\\\widetilde{\\\\gamma }\\\\left( t \\\\right)=\\\\left( \\\\overline{\\\\gamma }\\\\left( t \\\\right) KerC_\\\\alpha \\\\right) * e^{-2 \\\\pi i \\\\int _{\\\\overline{\\\\gamma }|_{[0,\\\\ t]}}\\\\alpha } $ where $\\\\overline{\\\\gamma }$ is any lifting of $\\\\gamma $ to G, such that $\\\\overline{\\\\gamma }(0)=g$ , and the vertical bar means restriction.\",\n \"Associated to this principal fibre bundle and the canonical action of ${\\\\bf S}^1$ on ${\\\\mathbb {C}}$ , one can consider the 1- dimensional vector bundle whose total space is $\\\\left( \\\\frac{G}{KerC_\\\\alpha }\\\\right) \\\\times _{{\\\\bf S}^1} {\\\\mathbb {C}}.$ Let us recall its definition.\",\n \"Consider in $\\\\left( G/KerC_\\\\alpha \\\\right) \\\\times {\\\\mathbb {C}}$ , an action of ${\\\\bf S}^1$ defined by $(g\\\\, KerC_\\\\alpha , t)*z=\\\\left(\\\\left(g\\\\, KerC_\\\\alpha \\\\right)*z ,z^{-1} t\\\\right),$ for all $z\\\\in {\\\\bf S}^1$ .\",\n \"The elements of $\\\\left( G/KerC_\\\\alpha \\\\right)\\\\times _{{\\\\bf S}^1}{\\\\mathbb {C}}$ are the orbits of this action.\",\n \"Let us denote by $[g\\\\, KerC_\\\\alpha , t]$ the orbit of $(g\\\\, KerC_\\\\alpha , t).$ We have $\\\\nonumber [g\\\\, KerC_\\\\alpha , t]&=&\\\\left\\\\lbrace \\\\left(\\\\left(g\\\\,KerC_\\\\alpha \\\\right)*s, s^{-1} t\\\\right):\\\\ s\\\\in S^{\\\\rm 1}\\\\right\\\\rbrace = \\\\\\\\ &=&\\\\nonumber \\\\left\\\\lbrace \\\\left(gh\\\\,KerC_\\\\alpha , C_\\\\alpha \\\\left(h^{\\\\rm -1}\\\\right) t\\\\right):\\\\ h\\\\in G_\\\\alpha \\\\right\\\\rbrace $ We define a map, $\\\\overline{\\\\pi }$ , from $ \\\\frac{G}{KerC_\\\\alpha } \\\\times _{{\\\\bf S }^1} {\\\\mathbb {C}}$ onto the coadjoint orbit by means of $\\\\overline{\\\\pi }([g\\\\, KerC_\\\\alpha , t])=g\\\\,G_\\\\alpha $ .\",\n \"Let $m\\\\in G/G_\\\\alpha $ , and $g\\\\in G$ one of its representatives.\",\n \"Since the action of ${\\\\bf S}^1$ on $\\\\pi ^{-1}(m)$ is transitive, we have ${\\\\overline{\\\\pi }}^{-1}(m)=\\\\lbrace [g\\\\,Ker\\\\,C_{\\\\alpha },t]:t\\\\in \\\\bf C\\\\rm \\\\rbrace .$ Thus, for each $g$ we obtain a bijection of ${\\\\overline{\\\\pi }}^{-1}(m)$ onto $\\\\bf C$ .\",\n \"We consider in ${\\\\overline{\\\\pi }}^{-1}(m)$ the structure of hermitian one dimensional complex vector space such that this bijection is a unitary isomorphism.\",\n \"This structure is independent of the representative $g$ .\",\n \"If $g$ and $g^{\\\\prime }$ are representatives of $m$ , we have for all $t,\\\\ t^\\\\prime ,\\\\ a \\\\in \\\\bf C$ $ \\\\left[g\\\\,KerC_\\\\alpha ,\\\\ t\\\\right] + \\\\left[g^\\\\prime \\\\,KerC_\\\\alpha ,\\\\ t^\\\\prime \\\\right]=\\\\left[g\\\\,KerC_\\\\alpha ,\\\\ t+C_\\\\alpha (g^{-1}\\\\,g^\\\\prime )\\\\,t^\\\\prime \\\\right]$ , $ a \\\\cdot \\\\left[g\\\\,KerC _\\\\alpha ,\\\\ t\\\\right] =\\\\left[g\\\\,KerC_\\\\alpha ,\\\\ a\\\\,t\\\\right],$ $\\\\langle \\\\left[g\\\\,KerC_\\\\alpha ,\\\\ t\\\\right] ,\\\\left[g^\\\\prime \\\\, KerC_\\\\alpha ,\\\\ t^\\\\prime \\\\right] \\\\rangle ={\\\\overline{t}}\\\\,{C_\\\\alpha (g^{-1}g^\\\\prime )\\\\,t^\\\\prime } .$ With these operations, $\\\\overline{\\\\pi }$ becomes a complex vector bundle of dimension one with a hermitian product in each fiber i.e.\",\n \"a hermitian line bundle.\",\n \"The sections of the hermitian line bundle are in a one to one correspondence with the functions on $G/{KerC_\\\\alpha } $ , f, such that f((g ${\\\\mbox{$Ker\\\\,C_\\\\alpha $}} )\\\\,*\\\\,s)= s^{-1} $ f(g ${\\\\mbox{$Ker\\\\,C_\\\\alpha $}} $ ), for all $s\\\\in {\\\\mbox{$\\\\bf S \\\\rm ^1$}}$ .\",\n \"These functions will be called from now on pseudotensorial functions .\",\n \"This correspondence is as follows.\",\n \"If f is a pseudotensorial function, the corresponding section sends $m \\\\in G/G_\\\\alpha $ to $\\\\left[ r,f(r) \\\\right]$ where r is arbitrary in $\\\\pi ^{-1}(m)$ .\",\n \"If $\\\\sigma $ is a given section of the hermitian line bundle, the corresponding pseudotensorial function, f, is defined by $\\\\sigma (\\\\pi (r)) = \\\\left[ r,f(r) \\\\right]$ for all $ r \\\\in G/Ker C_\\\\alpha $ .\",\n \"The canonical action of $G$ on $\\\\left( G/KerC_\\\\alpha \\\\right)$ , leads to an action on $\\\\left( G/KerC_\\\\alpha \\\\right)\\\\times _{\\\\mbox{$\\\\bf S \\\\rm ^1$}}{\\\\mbox{{$C$}} \\\\rm },$ given by $g*[h\\\\, Ker C_\\\\alpha ,t]=[gh\\\\, Ker C_\\\\alpha ,t].$ This action is well defined, as a consecuence of the fact that the action of $\\\\bf S \\\\rm ^1$ conmutes with the canonical action of $G$ .\",\n \"In the following, I use the same construction of a hermitian line bundle, for each principal bundle, $ G/(Ker\\\\,C)(G/H,S),$ where $C$ is an homomorphism of $H$ into $\\\\bf S \\\\rm ^1$, whose image is $S$ .\"\n ],\n [\n \"Geometric Quantum States .\",\n \"In this section I recall some definitions and results from [8].\",\n \"Let $G=SL(2,{\\\\bf C}) \\\\oplus H(2)$ and $\\\\alpha $ a quantizable form of $G$ .\",\n \"We use the notation of section .\",\n \"The sections of the hermitian line bundle whose total space is $ \\\\left( G/KerC_\\\\alpha \\\\right)\\\\times _{\\\\mbox{$\\\\bf S \\\\rm ^1$}}{\\\\mbox{{$C$}} \\\\rm } $ are called Prequantum States .\",\n \"We use the same denomination for the corresponding pseudotensorial functions.\",\n \"Now, let us consider the actions of the abelian subgroup $\\\\lbrace I\\\\rbrace \\\\times H(2)$ on $G/Ker\\\\,C_\\\\alpha $ and $G/G_\\\\alpha $ , induced by the canonical action.\",\n \"There exist an unique action of $\\\\lbrace I\\\\rbrace \\\\times H(2)$ on $G/Ker C_\\\\alpha $ whose orbits are horizontal and such that $\\\\pi $ becomes equivariant.\",\n \"This action is called horizontal action and is given by $(I,\\\\ K)*((A,\\\\ H)\\\\,KerC_\\\\alpha )=((A,\\\\ H+K)\\\\,KerC_\\\\alpha )*e^{-i \\\\pi Tr\\\\left(AkA^* \\\\varepsilon \\\\overline{K} \\\\varepsilon \\\\right)}$ for all $ K \\\\in H(2),\\\\ (A,\\\\ H) \\\\in G $ , where $*$ in the left hand side stands for the new action and in the right hand one, corresponds to the bundle action.\",\n \"k is given by $\\\\alpha = \\\\lbrace a,\\\\ k\\\\rbrace $ .\",\n \"We define Quantum States as being the Prequantum States that correspond to pseudotensorial functions left invariant by the horizontal action.\",\n \"Let $\\\\pi _1$ and $\\\\pi _2$ be the canonical projections of $G$ on ${ SL(2,\\\\bf C)}$ and $ H(2) $ respectively.\",\n \"We denote $\\\\pi _1(G_\\\\alpha )$ by ${\\\\mbox{$(G_\\\\alpha )_{SL}$}}$ and $\\\\pi _2(G_\\\\alpha )$ by ${\\\\mbox{$(G_\\\\alpha )_{H}$}}$ .\",\n \"In section 4 of [8] it is proved that the map $(C_{{\\\\mbox{$\\\\alpha $}}})_{SL} :(G_{{\\\\mbox{$\\\\alpha $}}})_{SL} \\\\longmapsto S^1 ,$ defined by $(C_{{\\\\mbox{$\\\\alpha $}}})_{SL}(g)\\\\ =\\\\ C_{{\\\\mbox{$\\\\alpha $}}}(g,h)\\\\ e^{-i\\\\pi Tr\\\\left(k{\\\\mbox{$\\\\varepsilon $}} \\\\overline{h}{\\\\mbox{$\\\\varepsilon $}}\\\\right) },$ for all $(g,h)\\\\in {\\\\mbox{$G_\\\\alpha $}},$ is well defined and a homomorphism.\",\n \"As a consecuence, we can define $\\\\widetilde{C}_\\\\alpha :(G_\\\\alpha )_{SL}\\\\ \\\\oplus \\\\ H(2)\\\\longmapsto S^1 ,$ by means of $ \\\\widetilde{C}_{{\\\\mbox{$\\\\alpha $}}}(g,r)\\\\ =\\\\ (C_{{\\\\mbox{$\\\\alpha $}}})_{SL}(g)\\\\ e^{i\\\\pi Tr\\\\left(k{\\\\mbox{$\\\\varepsilon $}} \\\\overline{r}{\\\\mbox{$\\\\varepsilon $}}\\\\right)}.$ $ \\\\widetilde{C}_{{\\\\mbox{$\\\\alpha $}}}$ is an extension of $C_\\\\alpha $ to $(G_{{\\\\mbox{$\\\\alpha $}}})_{SL}\\\\ \\\\oplus \\\\ {{\\\\mbox{$ H(2)\\\\rm $}}},$ and a homomorphism .\",\n \"Its differential coincides with the restriction of $\\\\alpha $ to the Lie algebra of this group.\",\n \"The canonical action of $G$ on $G/KerC_\\\\alpha $ maps horizontal orbits to horizontal orbits, thus defining a transitive action on the space of horizontal orbits, ${\\\\cal W}_{\\\\alpha }.$ Let us consider the canonical map $\\\\tau :\\\\frac{G}{KerC_\\\\alpha } \\\\rightarrow {\\\\cal W}_{\\\\alpha }$ defined by sending each element of $G/KerC_\\\\alpha $ onto its horizontal orbit.\",\n \"The isotropy subgroup at $\\\\tau (KerC_\\\\alpha )$ is $Ker\\\\widetilde{C}_\\\\alpha $ .\",\n \"As a consequence, we can identify ${\\\\cal W}_{\\\\alpha }$ to $G/Ker\\\\,\\\\widetilde{C}_{\\\\alpha },$ by means of the bijective map $(A,H)\\\\,Ker\\\\,\\\\widetilde{C}_{\\\\alpha }\\\\in \\\\frac{G}{Ker\\\\,\\\\widetilde{C}_{\\\\alpha }}\\\\rightarrow \\\\tau ((A,H)Ker\\\\,C_{\\\\alpha }) \\\\in {\\\\cal W}_{\\\\alpha }.$ With the definitions given in section , we have seen that $\\\\frac{G}{Ker\\\\,C_{\\\\alpha }}\\\\left(\\\\frac{G}{G_{\\\\alpha }},{\\\\bf S}^1\\\\right)$ becomes a principal fibre bundle.\",\n \"Similar definitions provides $\\\\frac{G}{Ker\\\\,\\\\widetilde{C}_{\\\\alpha }}\\\\left(\\\\frac{G}{(G_{\\\\alpha })_{SL}\\\\oplus H(2)},{\\\\bf S}^1\\\\right)$ with a structure of principal fibre bundle.\",\n \"The bundle projection is the canonical map from ${G}/{Ker\\\\,\\\\widetilde{C}_{\\\\alpha }}$ onto ${G}/({(G_{\\\\alpha })_{SL}\\\\oplus H(2)}).$ The bundle action is given by $((A,H){Ker\\\\,\\\\widetilde{C}_{\\\\alpha }})*s=(A,H)(B,K){Ker\\\\,\\\\widetilde{C}_{\\\\alpha }},$ where $(B,K)$ is such that $\\\\widetilde{C}_{\\\\alpha }(B,K)=s.$ But the map $(A,H)((G_\\\\alpha )_{SL} \\\\oplus H(2))\\\\in \\\\frac{G}{(G_{\\\\mbox{$\\\\alpha $}})_{SL} \\\\oplus H(2)}\\\\rightarrow A\\\\,(G_{\\\\alpha })_{SL} \\\\in \\\\frac{SL}{(G_{\\\\alpha })_{SL}}$ is a diffeomorphism.\",\n \"We identify these homogeneous spaces by means of this map, so that the principal fibre bundle becomes $\\\\frac{G}{Ker\\\\,\\\\widetilde{C}_{\\\\alpha }}\\\\left(\\\\frac{SL}{(G_{\\\\alpha })_{SL}},{\\\\bf S}^1\\\\right),$ where the bundle projection is $\\\\tau _2:(A,H){Ker\\\\,\\\\widetilde{C}_{\\\\alpha }}\\\\rightarrow A\\\\,(G_{\\\\alpha })_{SL}.$ The canonical maps define the homomorphism of principal ${\\\\bf S}^1 \\\\rm $-bundles given in Figure REF Figure: Fibre Bundles for Quantum StatesSince Quantum States correspond to pseudotensorial functions left invariant by the horizontal action, Quantum States are the pull back by $\\\\iota _1$ of unrestricted pseudotensorial functions on $G/Ker\\\\widetilde{C}_\\\\alpha $ .\"\n ],\n [\n \"Wave Functions\",\n \"Now, let us associate, to each Quantum State in the sense of section , a Wave Function in the ordinary sense of Quantum Mechanics.\",\n \"From now on, we assume, unless the contrary is explicitly stated, that the State Space is the orbit of $(0,\\\\alpha )$ in $H(2)\\\\times \\\\underline{G}^*.$ This choice do not carry very important consequences for the study of the free particle at the quantum level, in the sense that the other choices lead to isomorphic spaces of Quantum States (in all of its forms), and the same Wave Functions.\",\n \"These facts are proved in Remark 5.2 of [8].\",\n \"The isotropy subgroup at $(0,\\\\alpha ),$ $G_{(0,\\\\alpha )},$ is composed by the elements of $G_\\\\alpha $ of the form $(A,0)$ i.e.\",\n \"$G_{(0,\\\\alpha )}=G_\\\\alpha \\\\cap (SL \\\\oplus \\\\lbrace 0\\\\rbrace ).$ Let $\\\\alpha =\\\\lbrace a,k\\\\rbrace $ and denote $SL_1=\\\\lbrace A\\\\in SL: AaA^{-1}=a\\\\rbrace ,$ $SL_2=\\\\lbrace A\\\\in SL: AkA^*=k\\\\rbrace .$ Then $G_{(0,\\\\alpha )}=(SL_1 \\\\cap SL_2)\\\\oplus \\\\lbrace 0\\\\rbrace ,$ Notice that $(G_\\\\alpha )_{SL}\\\\subset SL_2$ and $SL_1 \\\\cap SL_2\\\\subset (G_\\\\alpha )_{SL}$ so that $SL_1 \\\\cap SL_2=(G_\\\\alpha )_{SL}\\\\cap SL_1.$ But the map $(A,H)((G_\\\\alpha )_{SL}\\\\cap SL_1)\\\\oplus \\\\lbrace 0\\\\rbrace \\\\in \\\\frac{G}{((G_\\\\alpha )_{SL}\\\\cap SL_1)\\\\oplus \\\\lbrace 0\\\\rbrace }\\\\rightarrow $ $ \\\\rightarrow (H,A((G_\\\\alpha )_{SL}\\\\cap SL_1))\\\\in H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}\\\\cap SL_1}$ is a diffeomorphism.\",\n \"Thus, State Space is the image of the injective map $i:(H,A(G_\\\\alpha )_{SL}\\\\cap SL_1))\\\\in H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}\\\\cap SL_1} \\\\rightarrow $ $\\\\rightarrow (H, Ad^*_{(A,H)}\\\\alpha )\\\\in H(2)\\\\times \\\\underline{G}^*.$ This image need not, a priori, be a proper submanifold of $H(2)\\\\times \\\\underline{G}^*,$ and we consider it provided with the topology and differentiable structure such that $i$ becomes a diffeomorphism.\",\n \"In the following we identify each $i(X)$ with $X,$ so that we can say that $H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}\\\\cap SL_1}$ is State Space.\",\n \"The canonical map from State Space onto Movement Space, can be generalised to all the homogeneous spaces appearing in the commutative diagram of Figure 1.\",\n \"This will be done with the following geometrical construction.\",\n \"Let $ \\\\cal L $ be a closed subgroup of G, and $ {\\\\cal S}=\\\\lbrace s\\\\in SL : (s,0) \\\\in {\\\\cal L} \\\\rbrace ,$ so that ${\\\\cal S} \\\\oplus \\\\lbrace 0\\\\rbrace = {\\\\cal L} \\\\cap (SL \\\\oplus \\\\lbrace 0\\\\rbrace ).$ Thus we define the map $(H,A {\\\\cal S})\\\\in {H(2) \\\\times \\\\frac{SL}{\\\\cal S} } \\\\stackrel{\\\\nu }{\\\\longrightarrow } (A,H) {\\\\cal L} \\\\in \\\\frac{G}{\\\\cal L}.$ This map is well defined and, if $H$ is a fixed element in $H(2)$ , its restriction to $ \\\\lbrace H\\\\rbrace \\\\times \\\\frac{SL}{\\\\cal S} $ is injective.\",\n \"Now we apply this to the cases in Figure 1.\",\n \"When ${\\\\cal L}=Ker\\\\,C_\\\\alpha $ we have ${\\\\cal S}=Ker(C_\\\\alpha )_{SL} \\\\cap SL_1$ , so that we have a map $(H,A ( Ker(C_\\\\alpha )_{SL} \\\\cap SL_1))\\\\in {H(2) \\\\times \\\\frac{SL}{ Ker(C_\\\\alpha )_{SL} \\\\cap SL_1 }} \\\\stackrel{\\\\nu _1}{\\\\longrightarrow }$ $\\\\stackrel{\\\\nu _1}{\\\\longrightarrow } (A,H) {Ker\\\\,C_\\\\alpha } \\\\in \\\\frac{G}{Ker\\\\,C_\\\\alpha }.\",\n \"$ If ${\\\\cal L}=G_\\\\alpha $ we have ${\\\\cal S}= SL_1 \\\\cap SL_2=(G_{\\\\alpha })_{SL} \\\\cap SL_1 $ , and we thus obtain the map $(H,A ((G_{\\\\alpha })_{SL} \\\\cap SL_1 ))\\\\in {H(2) \\\\times \\\\frac{SL}{(G_{\\\\alpha })_{SL} \\\\cap SL_1 }} \\\\stackrel{\\\\nu _2}{\\\\longrightarrow }$ $\\\\stackrel{\\\\nu _2}{\\\\longrightarrow } (A,H) {G_\\\\alpha } \\\\in \\\\frac{G}{G_\\\\alpha }.\",\n \"$ When ${\\\\cal L}=Ker{\\\\widetilde{C}}_\\\\alpha $ we have ${\\\\cal S}= Ker(C_\\\\alpha )_{SL}, $ so that we have a map $(H,A Ker(C_\\\\alpha )_{SL})\\\\in {H(2) \\\\times \\\\frac{SL}{ Ker(C_\\\\alpha )_{SL}}} \\\\stackrel{\\\\nu _3}{\\\\longrightarrow }(A,H) {Ker\\\\,\\\\widetilde{C}_\\\\alpha } \\\\in \\\\frac{G}{Ker\\\\,\\\\widetilde{C}_\\\\alpha }.\",\n \"$ If ${\\\\cal L}=(G_\\\\alpha )_{SL} \\\\oplus H(2)$ we have ${\\\\cal S}= (G_\\\\alpha )_{SL} $ , and we thus have a map ${(H,A (G_\\\\alpha )_{SL}))\\\\in {H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}}} } \\\\stackrel{\\\\nu _4}{\\\\longrightarrow }$ $\\\\stackrel{\\\\nu _4}{\\\\longrightarrow } (A,H) {((G_\\\\alpha )}_{SL}\\\\oplus H(2)) \\\\in \\\\frac{G}{(G_\\\\alpha )_{SL}\\\\oplus H(2)}$ which, using the identification (REF ), can be written ${(H,A (G_\\\\alpha )_{SL}))\\\\in {H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}}} } \\\\stackrel{\\\\nu _4}{\\\\longrightarrow }$ $\\\\stackrel{\\\\nu _4}{\\\\longrightarrow } A (G_\\\\alpha )_{SL} \\\\in \\\\frac{SL}{(G_\\\\alpha )_{SL}}.$ When we denote by $\\\\iota _3,\\\\ \\\\iota _4,\\\\ \\\\tau _3,\\\\ \\\\tau _4,$ the canonical maps of homogeneous spaces that appears in Figure REF , we obtain the commutative diagram in that Figure.\",\n \"Figure: Fibre Bundles for Wave FunctionsThe maps $\\\\tau _i$ are the bundle maps of principal fibre bundles whose structural groups are identified by means of $(C_\\\\alpha )_{SL},$ $\\\\widetilde{C}_\\\\alpha $ , or $C_\\\\alpha $ to subgroups of $S^1$ .\",\n \"We already know the bundle actions of ${\\\\bf S}^1$ on $G/KerC_\\\\alpha $ and $G/Ker\\\\widetilde{C}_\\\\alpha .$ The other are defined in a similar way as follows.\",\n \"The action for the principal bundle corresponding to $\\\\tau _4,$ $H(2) \\\\times \\\\frac{SL}{Ker(C_\\\\alpha )_{SL}}\\\\left(H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}},(C_\\\\alpha )_{SL}((G_\\\\alpha )_{SL})\\\\right),$ is given by $(H,A{Ker(C_\\\\alpha )_{SL}})*s=(H,AB{Ker(C_\\\\alpha )_{SL}})$ if $s\\\\in (C_\\\\alpha )_{SL}((G_\\\\alpha )_{SL}$ and $B\\\\in (G_\\\\alpha )_{SL}$ is such that $(C_\\\\alpha )_{SL}(B)=s.$ In the case of the principal bundle corresponding to $\\\\tau _3,$ $H(2) \\\\times \\\\frac{SL}{Ker(C_\\\\alpha )_{SL} \\\\cap SL_1}\\\\left(H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}\\\\cap SL_1},(C_\\\\alpha )_{SL}((G_\\\\alpha )_{SL}\\\\cap SL_1)\\\\right),$ the bundle action is defined in the same way but using the restriction of $(C_\\\\alpha )_{SL}$ to $(G_\\\\alpha )_{SL}\\\\cap SL_1.$ The pairs $(\\\\nu _1,\\\\nu _2),(\\\\nu _3,\\\\nu _4),\\\\ (\\\\iota _1,\\\\iota _2),\\\\ (\\\\iota _3,\\\\iota _4)$ define homomorphisms of principal fibre bundles.\",\n \"In what concerns the structural groups, we have $(C_\\\\alpha )_{SL}((G_\\\\alpha )_{SL}\\\\cap SL_1) \\\\subset (C_\\\\alpha )_{SL}((G_\\\\alpha )_{SL})\\\\subset {\\\\mathbf {S}}^1$ and the homomorphism of structural groups is, in all cases, the canonical injection of the group of the first bundle into the group of the second.\",\n \"As a consecuence of (REF ), the linear momentum, is given on the coadjoint orbit of $\\\\alpha $ by $P(Ad^{*}_{(A, H)} \\\\alpha )= -AkA^{*}$ , so that, with the identification of the coadjoint orbit with $G/G_\\\\alpha $ , we can write $P((A, H)G_\\\\alpha )=-AkA^{*}$ .\",\n \"On the other hand, since $ (G_\\\\alpha )_{SL} \\\\subset SL_2,$ we can define in $ {SL}/{(G_\\\\alpha )_{SL}} $ a function, $P_0$ , by means of $P_0(A(G_\\\\alpha )_{SL})=-AkA^{*}$ .\",\n \"But then, $P_0$ is the projection of $P$ by the canonical map $(A,\\\\ H)G_\\\\alpha \\\\in \\\\frac{G}{G_\\\\alpha } \\\\stackrel{\\\\iota _0}{\\\\longrightarrow } A (G_\\\\alpha )_{SL}\\\\in \\\\frac{SL}{(G_\\\\alpha )_{SL}},$ and will be denoted in the following simply by $P$ and also called linear momentum, thus we can write $P(A (G_\\\\alpha )_{SL})=-AkA^{*}$ where $k$ is given by $\\\\alpha =\\\\lbrace a,k\\\\rbrace $ .\",\n \"Let us denote $(C_\\\\alpha )_{SL}(G_\\\\alpha )_{SL})$ by $S$ .\",\n \"In [8] I prove that The pull back by $\\\\nu _3$ , maps in a one to one way the set of quantum states (considered as pseudotensorial functions on $G/Ker\\\\widetilde{C}_\\\\alpha $ onto the set composed by the functions on $H(2) \\\\times (SL/Ker(C_\\\\alpha )_{SL})$ having the form $ \\\\phi _f(H,A\\\\,Ker(C_\\\\alpha )_{SL})\\\\,=\\\\,f(A\\\\,Ker(C_\\\\alpha )_{SL})\\\\,e^{i\\\\pi Tr(P(A (G_\\\\alpha )_{SL})\\\\varepsilon \\\\overline{H}\\\\varepsilon )}$ where f is a pseudotensorial function on the principal fibre bundle $\\\\frac{SL}{Ker(C_\\\\alpha )_{SL}} \\\\left(\\\\frac{SL}{(G_\\\\alpha )_{SL}},S\\\\right) .$ In order to be more precise we will use the following definitions.\",\n \"Let ${\\\\cal C}$ be the complex vector space composed by the pseudotensorial functions of the bundle $\\\\frac{SL}{\\\\mbox{$Ker\\\\,(C_\\\\alpha )_{SL}$}} \\\\left(\\\\frac{SL}{\\\\mbox{$(G_\\\\alpha )_{SL}$}},S\\\\right),$ and $\\\\widetilde{\\\\cal V}$ the complex vector space composed by the pseudotensorial functions on $G/Ker\\\\widetilde{C}_\\\\alpha .$ For each $f\\\\in {\\\\cal C}$ we define a pseudotensorial function on $G/Ker\\\\widetilde{C}_\\\\alpha ,$ $\\\\Psi _f,$ by $\\\\Psi _f \\\\circ \\\\nu _3 \\\\left(H,A\\\\,Ker(C_\\\\alpha )_{SL}\\\\right)= \\\\phi _f\\\\left(H,A\\\\,Ker(C_\\\\alpha )_{SL}\\\\right)=$ $=f(A\\\\,Ker(C_\\\\alpha )_{SL})\\\\,e^{i\\\\pi \\\\,Tr(P(A\\\\,(G_\\\\alpha )_{SL} )\\\\,\\\\varepsilon \\\\, \\\\overline{H}\\\\,\\\\varepsilon )}.$ The map $\\\\Psi :f\\\\in {\\\\cal C} \\\\rightarrow \\\\Psi _f \\\\in \\\\widetilde{\\\\cal V}$ is an isomorphism.\",\n \"Also we define $\\\\Phi _f=\\\\Psi _f \\\\circ \\\\iota _1.$ These functions are pseudotensorial on $G/KerC_\\\\alpha $ and invariant by the horizontal action, so that they represent Quantum States in the most primitive sense adopted in this paper.\",\n \"We denote by ${\\\\cal V}$ the complex vector space composed by these $\\\\Phi _f,$ so that the map $\\\\Phi :f\\\\in {\\\\cal C} \\\\rightarrow \\\\Phi _f \\\\in {\\\\cal V}$ is an isomorphism.\",\n \"Thus, we can consider Quantum States as being elements of ${\\\\cal C}$ or elements of $\\\\widetilde{\\\\cal V}$ or elements of ${\\\\cal V}.$ Now we shall regard Quantum States under another form: Wave Functions.\",\n \"The differentiable manifold $W \\\\stackrel{\\\\mathrm {d}ef}{=}( H(2) \\\\times (SL/ Ker\\\\,(C_\\\\alpha )_{SL} ) \\\\times _S \\\\,{\\\\bf C}$ is defined in the same way as we have defined $\\\\left( G/KerC_\\\\alpha \\\\right)\\\\times _{\\\\mbox{$\\\\bf S \\\\rm ^1$}}{\\\\mbox{{$C$}} \\\\rm }$ in section .\",\n \"$W$ is the total space of the hermitian line bundle associated to the principal fibre bundle $\\\\left(H(2) \\\\times \\\\left(SL/Ker \\\\left(C_\\\\alpha \\\\right)_{SL}\\\\right)\\\\right)\\\\left(H(2) \\\\times \\\\left(SL/\\\\left(G_\\\\alpha \\\\right)_{SL} \\\\right),\\\\,S \\\\right)$ and the canonical action of $S$ on C .\",\n \"The bundle projection is $\\\\eta :[(H,A Ker\\\\,(C_\\\\alpha )_{SL} ), c]_S \\\\in W \\\\rightarrow (H,A (G_\\\\alpha )_{SL} ) \\\\in H(2)\\\\times ( SL/(G_\\\\alpha )_{SL}),$ where $[(H,A Ker\\\\,(C_\\\\alpha )_{SL} ), c]_S$ is the orbit of $((H,A Ker\\\\,(C_\\\\alpha )_{SL} ), c)$ under the action of $S$ .\",\n \"Since for each $f\\\\in {\\\\cal C}$ the function $\\\\Psi _f \\\\circ \\\\nu _3$ is pseudotensorial in $H(2) \\\\times \\\\left(SL/Ker \\\\left(C_\\\\alpha \\\\right)_{SL}\\\\right),$ it defines a section of $\\\\eta .$ This section can be considered as another description of the Quantum State given by $f$ .\",\n \"To complete our way towards Wave Functions, we need to “represent” Quantum States as functions with values in a fixed complex vector space, not in a different vector space for each point in the base space, as does the sections of $\\\\eta $ .\",\n \"If $(C_\\\\alpha )_{SL}(g)=1,\\\\ \\\\forall g\\\\in (G_\\\\alpha )_{SL},$ , we have $S=\\\\lbrace 1\\\\rbrace ,$ and $Ker\\\\,(C_\\\\alpha )_{SL}=(G_\\\\alpha )_{SL}$ so that $W \\\\approx ( H(2) \\\\times (SL/ (G_\\\\alpha )_{SL} ) \\\\times \\\\,{\\\\bf C},$ and $\\\\eta $ is the canonical projection onto the first two factors.\",\n \"The section of $\\\\eta $ corresponding to the function $\\\\phi _f $ in (REF ) is $(H,\\\\,A\\\\,(G_\\\\alpha )_{SL})\\\\, \\\\longrightarrow (H,\\\\,A\\\\,(G_\\\\alpha )_{SL},\\\\phi _f(\\\\,H,\\\\,A\\\\,\\\\,(G_\\\\alpha )_{SL})).$ Thus, if $(C_\\\\alpha )_{SL}={\\\\bf 1 },$ our task is accomplished by $\\\\phi _f(\\\\,H,\\\\,A\\\\,Ker\\\\,{\\\\mbox{$(C_\\\\alpha )_{SL}$}})\\\\,=\\\\,f(\\\\,A\\\\,Ker\\\\,{\\\\mbox{$(C_\\\\alpha )_{SL}$}})\\\\,e^{i\\\\pi \\\\,Tr(P(A\\\\,(G_\\\\alpha )_{SL} )\\\\,\\\\varepsilon \\\\, \\\\overline{H}\\\\,\\\\varepsilon }$ itself as a complex valued function on the base space.\",\n \"In this case, $\\\\phi _f$ is called the Prewave Function associated to $f,$ and denoted by $\\\\psi _f.$ In the case where ${\\\\mbox{$(C_\\\\alpha )_{SL}$}}$ is not trivial, our goal will be attained by imbedding the hermitian fibre bundle in a trivial one.\",\n \"We do this in a direct way, but a more geometrical view of the method is exposed in remark 5.1 of [8].\",\n \"A key concept in our construction of Wave Functions is the following: A Trivialization of $ C_\\\\alpha $ is a triple $(\\\\rho ,\\\\ {L},\\\\ z_0)$ , where $ { L}$ is a finite dimensional complex vector space, $z_0\\\\in { L}$ and $\\\\rho $ is a representation of $SL$(2,C) in ${ L}$ such that $ &1)&\\\\ \\\\rho (A)(z_0)\\\\,=\\\\,(C_\\\\alpha )_{SL}(A)\\\\ z_0,\\\\ \\\\ \\\\ \\\\forall A\\\\in (G_\\\\alpha )_{SL}.\",\n \"\\\\nonumber \\\\\\\\&2)&\\\\ {\\\\rm The \\\\ isotropy \\\\ subgroup\\\\ at}\\\\ z_0\\\\ {\\\\rm is}\\\\ Ker\\\\,(C_\\\\alpha )_{SL}.$ In what follows, we assume that a trivialization of $C_\\\\alpha $ is given.\",\n \"The homogeneous space $ { SL}/{\\\\mbox{$Ker\\\\,(C_\\\\alpha )_{SL}$}} $ is identified to the orbit of $z_0$ , $\\\\cal B$ , by means of $A Ker\\\\,(C_\\\\alpha )_{SL} \\\\in \\\\frac{ SL}{Ker\\\\,(C_\\\\alpha )_{SL}} \\\\longrightarrow \\\\rho (A)(z_0)\\\\in {\\\\cal B}.$ The action of $S$ on $ { SL}/{\\\\mbox{$Ker\\\\,(C_\\\\alpha )_{SL}$}} $ becomes, with this identification, multiplication in $L$ of elements of $S,$ as complex numbers, by elements of ${\\\\cal B},$ as elements of $L$ .\",\n \"The canonical map from $ { SL}/{\\\\mbox{$Ker\\\\,(C_\\\\alpha )_{SL}$}} $ onto $ {SL}/{\\\\mbox{$(G_\\\\alpha )_{SL}$}} $ will be denoted by r, and is given by $r\\\\left(\\\\rho (A)(z_0)\\\\right)=A {\\\\mbox{$(G_\\\\alpha )_{SL}$}} .$ Each $f\\\\in {\\\\cal D}$ thus becomes a function on ${\\\\cal B},$ homogeneous of degree $-1$ under ordinary multiplication by elements of $S$ .\",\n \"The functions having these characteristics, will be called S-homogeneous of degree -1.\",\n \"The $S$ -homogeneous of degree T functions are defined in a similar way.\",\n \"Let us denote by ${\\\\cal C}$ the complex vector space composed by the S-homogeneous of degree -1 functions on ${\\\\cal B}$ .\",\n \"Now, $W$ is $( H(2) \\\\times {\\\\cal B} ) \\\\times _S \\\\,{\\\\bf C}$ , and $\\\\eta $ maps $[(H,z),c]$ onto $(H,r(z)).$ The sections of $\\\\eta $ corresponding to the functions having the form of $\\\\phi _f$ in (REF ) are as follows.\",\n \"Let $f$ be a $S$ -homogeneous of degree -1 function.\",\n \"The corresponding section, $\\\\sigma ,$ maps $(H,m)$ to $\\\\sigma (H,m)=[(H,z),\\\\phi _f(H,z)]$ where $z$ is arbitrary in $r^{-1}(m).$ We define a map, $\\\\chi $ , from W into $ H(2) \\\\times ( SL/(G_\\\\alpha )_{SL}) \\\\times \\\\ L $ , by sending $[(H,z ), c]_S \\\\in ( H(2) \\\\times {\\\\cal B}) ) \\\\times _S \\\\,{\\\\bf C}$ to $ (H,\\\\ r(z), cz)$ .\",\n \"The map $\\\\chi $ is injective .\",\n \"In fact the relation $\\\\chi ([(H,z),c]_S)=\\\\chi ([(H^\\\\prime ,z^\\\\prime ),c^\\\\prime ]_S )$ is equivalent to $(H,r(z),cz)=(H^\\\\prime ,r(z^\\\\prime ),c^\\\\prime z^\\\\prime )$ so that there exist $e^{i\\\\gamma }\\\\in S$ such that $H^\\\\prime =H,$ $z^\\\\prime =e^{i\\\\gamma } z,$ and $c^\\\\prime z^\\\\prime =cz$ .\",\n \"Thus $[(H^\\\\prime ,z^\\\\prime ),c^\\\\prime ]_S=[(H,ze^{i\\\\gamma } ),ce^{-i\\\\gamma } ]_S=[(H,z),c]_S.$ The fiber of $W$ on $(H,m)$ is $\\\\lbrace [(H,z ),c]: c\\\\in {\\\\bf C}\\\\rbrace $ , where $z$ is any fixed element in $r^{-1}(m).$ Its image under $\\\\chi $ is composed by the $(H,m,y)$ such that $y$ is in the one dimensional subspace of $L$ generated by $z.$ We have thus inmersed our, in general, non trivial bundle, in a trivial one with fiber $L$ .\",\n \"This enable us to identify sections of $\\\\eta $ with functions with values in $L,$ as follows.\",\n \"The section $\\\\sigma $ of $\\\\eta $ , that corresponds to $f$ can be identifed with $\\\\chi \\\\circ \\\\sigma (H,m) = \\\\left(H,m,\\\\phi _f(z)z\\\\right),$ where $z$ is arbitrary in $r^{-1}(m).$ The right hand side in the preceeding equation is completely determined by its third component: $\\\\psi _f(H,m)= \\\\phi _f(z)z=f(z)\\\\,e^{i\\\\pi \\\\,Tr(P(m )\\\\,\\\\varepsilon \\\\, \\\\overline{H}\\\\, \\\\varepsilon )}z.$ The function $\\\\psi _f,$ with $f$ $S$ -homogeneous of degree -1, will be called Prewave Function associated to $f$ .\",\n \"The complex vector space composed by the Prewave Functions is denoted by PW, so that the map $\\\\psi :f\\\\in {\\\\cal C}\\\\ \\\\rightarrow \\\\psi _f\\\\in {\\\\cal PW}$ is an isomorphism.\",\n \"The composition of any Prewave Function with $\\\\iota _4$ is a function in State Space that can be considered as giving an amplitude of probability for each state.\",\n \"As said in the Introduction, we obtain from this Prewave Function a Wave Function, defined on space-time points ($i.e.$ hermitian matrices), by adding up, for each $H \\\\in H(2),$ the amplitudes of probability corresponding to the states $(H,\\\\beta )$ where $\\\\beta $ is a movement whose portrait in space-time contains $H$ .\",\n \"According to (REF ), we see that these states are the elements of the set $\\\\lbrace (H,Ad^*_{(B,H)}\\\\alpha : B\\\\in SL\\\\rbrace ,$ that is identified by $i$ ($c.f.$ (REF )) with $\\\\lbrace H\\\\rbrace \\\\times {\\\\frac{SL}{(G_\\\\alpha )_{SL}\\\\cap SL_1}},$ whose image by $\\\\iota _4$ is $\\\\lbrace H\\\\rbrace \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}}.$ Then to any given Prewave Function, $\\\\psi _f$ , we associate a Wave Function, $\\\\widetilde{ \\\\psi _f}$ , as follows $\\\\widetilde{\\\\psi _f}(H)=\\\\int _{{SL}/(G_\\\\alpha )_{SL}} {\\\\mbox{$\\\\psi $}}_f(H,\\\\,m)\\\\ {\\\\mbox{$\\\\omega $}}_m$ where $m$ is a generic element in $ { SL}/({\\\\mbox{$G_\\\\alpha $}})_{SL},$ and $\\\\omega $ is a volume element on $ { SL}/({\\\\mbox{$G_\\\\alpha $}})_{SL},$ left invariant by the canonical action of $SL$ on $ { SL}/({\\\\mbox{$G_\\\\alpha $}})_{SL}.$ This means that, if we denote by $d^{\\\\prime }_A$ the diffeomorphism of $SL/(G_\\\\alpha )_{SL}$ defined by sending $B\\\\,(G_\\\\alpha )_{SL}$ to $AB\\\\,(G_\\\\alpha )_{SL},$ then $(d^{\\\\prime }_A)^*\\\\omega =\\\\omega .$ In the following, we assume that such a invariant volume element is given.\",\n \"This definition of Wave Functions, forces us to do a restriction on the class of the functions to be considered: it is necessary that the integral exists.\",\n \"In what follows I consider only Quantum States corresponding to the functions in ${\\\\cal C}$ that are continuous with compact support.\",\n \"The complex vector space composed by these Wave Functions is denoted by ${\\\\cal WF}.$ Of course, there are other possible conditions that can be imposed on $f$ in order to assure integrability.\",\n \"Also, if one obtains for some choice a Prehilbert space, one can consider its completed, in order to have a Hilbert Space.\",\n \"But in this paper I prefer to maintain the \\\"continuous with compact support\\\" condition.\",\n \"The Wave Functions are another form of description of Quantum States, in fact it is the most usual in Quantum Mechanics.\",\n \"In the following I change the notation in such a way that, ${\\\\cal C}$ stands for the complex vector space composed by the $S$ -homogeneous of degree -1 functions on ${\\\\cal B}$ that also are continuous with compact support, and ${\\\\cal PW},\\\\ {\\\\cal V},\\\\ \\\\widetilde{\\\\cal V},$ the corresponding isomorphic spaces.\",\n \"Remark 7.1 Now, let us assume that we know a section of the map $r,$ defined an open set, $D,$ in ${SL/(G_\\\\alpha )_{SL}}$ , $\\\\sigma : D \\\\longrightarrow {\\\\cal B}.$ In this case, we can associate a prewave function, and thus a wave function, to each $C^\\\\infty $ function, with compact support contained in $D,$ as follows.\",\n \"Let $f_0$ be a $C^\\\\infty $ function, with compact support contained in $D$ .\",\n \"We define a function on $\\\\cal B$ , $f$ , by means of $f(z)={\\\\left\\\\lbrace \\\\begin{array}{ll}t^{-1} f_0(r(z))& {\\\\rm if}\\\\ z\\\\in r^{-1}(U)\\\\ {\\\\rm where}\\\\ t \\\\ {\\\\rm is\\\\ such\\\\ that}\\\\ z= t \\\\sigma (r(z))\\\\\\\\0& {\\\\rm if}\\\\ z\\\\notin r^{-1}(U)\\\\end{array}\\\\right.\",\n \"}$ This function is $S$ -homogeneous of degree -1 since, for all $e^{i a}\\\\in S,\\\\ z\\\\in {r^{-1}(U)},$ we have $f(e^{i a}z)=t^{\\\\prime -1} f_0(r(e^{i a}z)),$ where $t^{\\\\prime }$ is such that $e^{i a}z=t^{\\\\prime } \\\\sigma (r(e^{i a}z))=t^{\\\\prime } \\\\sigma (r(z))=t^{\\\\prime } t^{-1} z,$ so that $e^{i a}=t^{\\\\prime } t^{-1}$ and $f(e^{i a}z)=e^{-i a}t^{-1} f_0(r(e^{i a}z))=e^{-i a}t^{-1} f_0(r(z))=e^{-i a}f(z).$ If $z\\\\notin {r^{-1}(U)}$ , $f(e^{i a}z)=0=f(z)=e^{-i a}f(z)$ .\",\n \"The Prewave Function $\\\\psi _f$ then is $\\\\psi _f(H,\\\\ m)={\\\\left\\\\lbrace \\\\begin{array}{ll}f_0(m)\\\\,e^{i\\\\pi Tr(P(m)\\\\,\\\\varepsilon \\\\overline{H} \\\\varepsilon )}\\\\ \\\\sigma (m) & \\\\mathrm {if}\\\\ m\\\\in U \\\\\\\\0& \\\\mathrm {if}\\\\ m\\\\notin U\\\\end{array}\\\\right.\",\n \"}$ and we have an associated Wave Function, $\\\\tilde{\\\\psi }_f.$ This expression contains no reference to any homogeneous function and suggest directly the usual form of Wave Functions.\"\n ],\n [\n \" Representation on $S$ -homogeneous functions of degree -1 \",\n \"If $(C_\\\\alpha )_{SL}={\\\\bf 1},$ , the $S$ -homogeneous functions of degree -1 are simply functions on $SL/(G_\\\\alpha )_{SL}.$ In this case we denote by $\\\\cal C$ the complex vector space composed by the continuous functions on $SL/(G_\\\\alpha )_{SL}$ with compact support.\",\n \"For all $B\\\\in SL$ we denote by $d^\\\\prime _B$ the canonical diffeomorphism of $SL/(G_\\\\alpha )_{SL}$ given by $d^\\\\prime _B(A (G_\\\\alpha )_{SL})=BA (G_\\\\alpha )_{SL}).$ Then, we define a representation, $\\\\delta ,$ of $SL$ on $\\\\cal C$ by $\\\\delta (f)=f\\\\circ d^\\\\prime _{A^{-1}}.$ In $\\\\cal C$ we also define an hermitian product by $\\\\langle f,\\\\, f^\\\\prime \\\\rangle = \\\\int _{{SL}/(G_\\\\alpha )_{SL}} \\\\overline{f} f^\\\\prime \\\\ {\\\\mbox{$\\\\omega $}} ,$ where $\\\\omega $ is an invariant volume element on $SL/(G_\\\\alpha )_{SL}.$ With this inner product, $\\\\cal C$ becomes a prehilbert space.\",\n \"The invariance of $\\\\omega $ enable us to write $\\\\langle \\\\delta (A)(f),\\\\, \\\\delta (A)(f^\\\\prime ) \\\\rangle =\\\\langle f,\\\\, f^\\\\prime \\\\rangle $ so that the representation $\\\\delta $ is unitary.\",\n \"In case $(C_\\\\alpha )_{SL}\\\\ne {\\\\bf 1},$ , we assume that we have a trivialization, $(\\\\rho ,\\\\ {L},\\\\ z_0),$ and we define ${\\\\cal B}$ and ${\\\\cal C}$ as in the preceeding section.\",\n \"Of course, also in case $(C_\\\\alpha )_{SL}={\\\\bf 1},$ we can have a trvialization and thus apply all that follows.\",\n \"In ${\\\\cal C}$ we define a hermitian product: $\\\\langle f,\\\\, f^\\\\prime \\\\rangle = \\\\int _{{SL}/(G_\\\\alpha )_{SL}} \\\\overline{f} f^\\\\prime \\\\ {\\\\mbox{$\\\\omega $}} , $ where by $ \\\\overline{f} f^\\\\prime $ we means the function defined on $SL/(G_\\\\alpha )_{SL},$ by $\\\\overline{f} f^\\\\prime (m)=\\\\overline{f}(z) f^\\\\prime (z)$ for all $m\\\\in SL/(G_\\\\alpha )_{SL},$ where $z$ is arbitrary in $r^{-1}(m),$ and $\\\\omega $ is an invariant volume element on $SL/(G_\\\\alpha )_{SL}.$ Also in this case, $\\\\cal C$ becomes a prehilbert space.\",\n \"A representation, $\\\\delta ,$ of $SL$ on $\\\\cal C,$ is defined by $\\\\delta (A): f\\\\in {\\\\cal C} \\\\longrightarrow f\\\\circ \\\\rho (A^{-1})\\\\in {\\\\cal C}.$ for all $A\\\\in SL.$ If $A\\\\in SL$ we have $\\\\langle \\\\delta (A)(f),\\\\, \\\\delta (A)(f^\\\\prime ) \\\\rangle = \\\\int _{{ SL}/(G_\\\\alpha )_{SL}} \\\\left((\\\\overline{f} f^\\\\prime )\\\\circ d^{\\\\prime }_{A^{-1}} \\\\right)\\\\ {\\\\mbox{$\\\\omega $}}$ so that the invariance of $\\\\omega $ enable us to write $\\\\langle \\\\delta (A)(f),\\\\, \\\\delta (A)(f^\\\\prime ) \\\\rangle =\\\\langle f,\\\\, f^\\\\prime \\\\rangle $ We see that the representation $\\\\delta $ is, also in this case, unitary.\",\n \"Also we have a representation, $\\\\delta ^\\\\prime ,$ of $SL \\\\oplus H(2)$ on $ {\\\\cal C}$ defined by: $(\\\\delta ^\\\\prime (A,H)\\\\cdot f)(z)= f( \\\\rho (A^{-1})\\\\cdot z) Exp\\\\left(-i\\\\pi Tr\\\\left(\\\\,P( r(z)) \\\\varepsilon \\\\overline{H} \\\\varepsilon \\\\right)\\\\right),$ where $(A,H)\\\\in SL \\\\oplus H(2),\\\\ f\\\\in {\\\\cal C},$ and $z\\\\in {\\\\cal B}.$ To prove that $\\\\delta ^\\\\prime $ is a representation , notice that $r$ is equivariant i.e.\",\n \"$r(\\\\rho (A)\\\\cdot z)=d^{\\\\prime }_A(r(z))$ for all $A\\\\in SL.$ Formula (REF ), leads to $P(d^{\\\\prime }_B(A(G_\\\\alpha )_{SL}))=P(BA(G_\\\\alpha )_{SL})=-BAkA^*B^*=BP(A(G_\\\\alpha )_{SL})B^*$ so that $P(r(\\\\rho (A)\\\\cdot z))=P(d^{\\\\prime }_A(r(z)))=AP(r(z))A^*.$ For all $A\\\\in SL,$ $A\\\\, \\\\varepsilon = \\\\varepsilon \\\\,{}^t\\\\hspace{0.0pt}A^{-1}$ Now, we can prove that $\\\\delta ^\\\\prime $ is a representation as follows.\",\n \"If $(B,K),(A,H)\\\\in SL \\\\oplus H(2),\\\\ z\\\\in {\\\\cal B},$ then ${\\\\begin{array}{c}\\\\delta ^\\\\prime (B,K)\\\\cdot \\\\left(\\\\left(\\\\delta ^\\\\prime (A,H)\\\\cdot f\\\\right)\\\\right)(z)=\\\\left(\\\\delta ^\\\\prime (A,H)\\\\cdot f\\\\right)(\\\\rho (B^{-1})\\\\cdot z) \\\\\\\\Exp\\\\left(-i\\\\pi Tr[P(r( z))\\\\varepsilon \\\\overline{K} \\\\varepsilon ]\\\\right)=f(\\\\rho (A^{-1})\\\\cdot (\\\\rho (B^{-1})\\\\cdot z))\\\\\\\\Exp\\\\left(-i\\\\pi Tr[P(r(\\\\rho (B^{-1})\\\\cdot z))\\\\varepsilon \\\\overline{H} \\\\varepsilon + P(r( z))\\\\varepsilon \\\\overline{K} \\\\varepsilon ] \\\\right)=\\\\\\\\=f(\\\\rho ((BA)^{-1})\\\\cdot z) Exp\\\\left(-i\\\\pi Tr[P(r(z))\\\\varepsilon \\\\overline{(B HB^* + K) }\\\\varepsilon ]\\\\right)=\\\\\\\\=\\\\left(\\\\delta ^\\\\prime \\\\left(\\\\left(B,K\\\\right)\\\\left(A,H\\\\right)\\\\right)\\\\cdot f\\\\right)\\\\left( z\\\\right)\\\\end{array}}$ The infinitesimal generator of $\\\\delta $ associated to $a\\\\in sl(2,\\\\bf C)$ is the linear map from $\\\\cal C$ into itself, $d\\\\,\\\\delta (a)$ given by $(d\\\\,\\\\delta (a) \\\\cdot f) (z)=\\\\left(\\\\frac{d}{dt}\\\\right)_{t=0} \\\\left(\\\\delta \\\\left(e^{t\\\\,a}\\\\right)\\\\cdot f \\\\right) \\\\left(z\\\\right).$ Then $(d\\\\,\\\\delta (a) \\\\cdot f) (z)=\\\\left(\\\\frac{d}{dt}\\\\right)_{t=0} \\\\left(f \\\\left(\\\\rho \\\\left(e^{-t\\\\,a}\\\\right) \\\\cdot z\\\\right) \\\\right),$ and thus we see that $d\\\\,\\\\delta (a)$ acts on $f$ as the vector field, $X_a,$ infinitesimal generator of the action on $\\\\cal B,$ defined by $A*z=\\\\rho (A) \\\\cdot z,$ associated to $a$ .\",\n \"We can give a more explicit form of $X_a$ as follows.\",\n \"Let us fix a basis, $\\\\beta =\\\\lbrace e_1, \\\\dots ,e_q\\\\rbrace ,$ of $L.$ By means of $\\\\beta $ we identify $L$ with ${\\\\bf C}^q$ .\",\n \"Let us denote by ${}^t(z^1,\\\\dots ,z^q)$ the matrix of $z\\\\in L$ in the basis $\\\\beta $ and $(d\\\\rho (a))_i^j,$ the element in the file $j$ column $i$ of the matrix of $(d\\\\rho (a))$ in the basis $\\\\beta .$ We denote by $\\\\beta ^*=\\\\lbrace w^1, \\\\dots ,w^q\\\\rbrace $ the system of (complex) coordinates associated to $\\\\beta $ (i.e.\",\n \"$\\\\beta ^*$ is the dual basis of $\\\\beta .$ ) If $w^j_x$ (resp $w^j_y$ ) is the real (resp.\",\n \"imaginary) part of $w^j,$ it is usually writen $\\\\frac{\\\\partial }{\\\\partial w^j}=\\\\frac{1}{2}\\\\left(\\\\frac{\\\\partial }{\\\\partial w^j_x}-i\\\\ \\\\frac{\\\\partial }{\\\\partial w^j_y}\\\\right),$ $\\\\frac{\\\\partial }{\\\\partial \\\\overline{w}^j}=\\\\frac{1}{2}\\\\left(\\\\frac{\\\\partial }{\\\\partial w^j_x}+i\\\\ \\\\frac{\\\\partial }{\\\\partial w^j_y}\\\\right).$ Thus, when $f$ is a complex valued differentiable function on an open subset, M, of ${\\\\bf C}^q$ , and $v(t)$ a differentiable map from an open neigbourhood of 0 in ${\\\\bf R}$ into M, we have, using summation convention ${\\\\begin{array}{c}\\\\left( \\\\frac{d}{dt}\\\\right)_0 (f\\\\circ v) =\\\\left(\\\\frac{\\\\partial \\\\, f}{\\\\partial w^j_x}\\\\right)_{v(0)}\\\\, (w_x^j(v(t)))^{\\\\prime }(0)+\\\\left(\\\\frac{\\\\partial \\\\, f}{\\\\partial w^j_y}\\\\right)_{v(0)}\\\\, (w_y^j(v(t)))^{\\\\prime }(0)=\\\\\\\\=\\\\left(\\\\frac{\\\\partial \\\\, f}{\\\\partial w^j}\\\\right)_{v(0)}\\\\, (w^j(v(t)))^{\\\\prime }(0)+\\\\left(\\\\frac{\\\\partial \\\\, f}{\\\\partial \\\\overline{w}^j}\\\\right)_{v(0)}\\\\, (\\\\overline{w}^j(v(t)))^{\\\\prime }(0).\\\\end{array}}$ Since $\\\\left(\\\\frac{d}{dt}\\\\right)_{t=0} \\\\left(f \\\\left(\\\\rho \\\\left(e^{-t\\\\,a}\\\\right) \\\\cdot z\\\\right) \\\\right)=\\\\left(\\\\frac{d}{dt}\\\\right)_{t=0} \\\\left(f \\\\left(\\\\left(e^{-t\\\\,d\\\\rho \\\\left(a\\\\right)}\\\\right) \\\\cdot z\\\\right) \\\\right)$ it follows that ${\\\\begin{array}{c}\\\\left(X_a)_z(f) \\\\right)=\\\\left(\\\\frac{\\\\partial \\\\, f}{\\\\partial w^j}\\\\right)_{z}\\\\, (-(d\\\\rho (a))^j_i\\\\,z^i)+\\\\left(\\\\frac{\\\\partial \\\\, f}{\\\\partial \\\\overline{w}^j}\\\\right)_{z}\\\\, ({-(\\\\overline{d\\\\rho (a)})} ^j_i\\\\,\\\\overline{z}^i)\\\\end{array}}$ so that an extension of $X_a$ from $\\\\cal B$ to $\\\\mathbb {C}^q$ is $X_a=-\\\\left(w^i \\\\,(d\\\\rho (a))^j_i\\\\,\\\\frac{\\\\partial }{\\\\partial w^j}+\\\\overline{w}^i \\\\,(\\\\overline{d\\\\rho (a)})^j_i\\\\,\\\\frac{\\\\partial }{\\\\partial \\\\overline{w}^j} \\\\right).$ If $(C_\\\\alpha )_{SL}={\\\\bf 1},$ , and do not use a trivialization, we have similar results, but $X_a$ is the infinitesimal generator of the canonical action of $SL$ on $SL/(G_\\\\alpha )_{SL},$ associated to $a.$ Thus, in all cases $d\\\\,\\\\delta (a) \\\\cdot f=X_{a}(f),$ or simply, as linear maps on $\\\\cal C$ , $d\\\\,\\\\delta (a) =X_{a}.$ On the other hand, the infinitesimal generator of $\\\\delta ^{\\\\prime }$ associated to $(a,h)\\\\ \\\\ \\\\in sl(2,\\\\bf C)\\\\oplus H(2)$ is the endomorphism, $d\\\\,\\\\delta ^{\\\\prime }(a,h)$ of $\\\\cal C$ given by $(d\\\\,\\\\delta ^{\\\\prime }(a,h) \\\\cdot f) (z)=\\\\left(\\\\frac{d}{dt}\\\\right)_{t=0} \\\\left(\\\\delta ^{\\\\prime }\\\\left(Exp({t\\\\,(a,h))}\\\\right)\\\\cdot f \\\\right) \\\\left(z\\\\right),$ but the exponential map in $SL\\\\oplus H(2)$ is given by (REF ), so that $(d\\\\,\\\\delta ^{\\\\prime }(a,h) \\\\cdot f) (z)=\\\\left(\\\\frac{d}{dt}\\\\right)_{t=0} \\\\left(\\\\delta ^{\\\\prime }\\\\left(\\\\left(e^{ta},\\\\int _0^t e^{sa}\\\\,h\\\\, e^{sa^*}\\\\,ds \\\\right)\\\\right)\\\\cdot f \\\\right) \\\\left(z\\\\right),$ and a short computation thus leads to ${\\\\begin{array}{c}d\\\\,\\\\delta ^{\\\\prime }(a,h) \\\\cdot f= X_a(f)-i\\\\pi Tr[P(r(\\\\cdot ))\\\\varepsilon \\\\overline{h}\\\\varepsilon ]\\\\ f=\\\\\\\\=X_a(f)+2\\\\pi i \\\\langle P(r(\\\\cdot )),h \\\\rangle \\\\ f,\\\\end{array}}$ where I have used the same symbol for any hermitian matrix and the corresponding element in ${\\\\bf R}^4,$ and $\\\\langle \\\\ ,\\\\ \\\\rangle $ is Minkowski product.\",\n \"Also, we can write $d\\\\,\\\\delta ^{\\\\prime }(a,h)=X_a+2\\\\pi i \\\\langle P(r(\\\\cdot )),h \\\\rangle .$ where $X_a$ is the endomorphism given by the vectorfield (REF ) , and $2\\\\pi i\\\\langle P(r(\\\\cdot )),h \\\\rangle $ is the endomorphism given by ordinary multiplication by this function.\",\n \"Instead of the infinitesimal generators $d\\\\,\\\\delta ^{\\\\prime }(a,h),$ we can use $(a,h)^{\\\\theta }\\\\stackrel{\\\\mathrm {d}ef}{ =}\\\\frac{1}{2 \\\\pi i}\\\\ d\\\\,\\\\delta ^{\\\\prime }(a,h) ,$ so that $(a,h)^{\\\\theta } =\\\\frac{1}{2 \\\\pi i}\\\\,X_a+ \\\\langle P(r(\\\\cdot )),h \\\\rangle .$ The endomorphism $(a,h)^{\\\\theta },$ is hermitian for our hermitian product and is the Quantum Operator associated to the Dynamical Variable (a,h), when Quantum States are represented by $S$ -homogeneous functions.\",\n \"When Quantum States are represented by Wave Functions, the Quantum Operators acquires its usual form in Quantum Mechanics (c.f.\",\n \"(REF )).\",\n \"In the particular case of the Linear Momentum (c.f.\",\n \"(REF )), one sees that $(P^k)^{\\\\theta }\\\\cdot f=(P^k \\\\circ r)\\\\,f,\\\\ \\\\ k=1,\\\\,2,\\\\,3,\\\\,4.$ where the $P^k$ in the left hand side are the components of Linear momentum as a dynamical variable, and the ones in the right hand side, the components of Linear momentum considered as a function on $SL/(G_\\\\alpha )_{SL},$ given by (REF ).\",\n \"In what concerns to the Quantum Operators corresponding to Angular Momentum we have $(l^j)^{\\\\theta }= \\\\frac{1}{4\\\\pi i} \\\\,X_{i\\\\,\\\\sigma _j}, \\\\ \\\\ j=1,\\\\,2,\\\\,3.$ $(g^j)^{\\\\theta }=\\\\frac{1}{4\\\\pi i} \\\\,X_{\\\\sigma _j}, \\\\ \\\\ j=1,\\\\,2,\\\\,3.$ Of course, the representations on ${\\\\cal C}$ lead to equivalent representations on the isomorphic vector spaces $ {\\\\cal V},\\\\ \\\\widetilde{\\\\cal V},$ and ${\\\\cal PW}$ .\",\n \"We do that in detail in ${\\\\cal PW}$ and ${\\\\cal V}$ as follows:\"\n ],\n [\n \"Representation on Prewave Functions.\",\n \"In ${\\\\cal PW}$ we define an hermitian product by $\\\\langle \\\\psi _f ,\\\\psi _{f^{\\\\prime }} \\\\rangle \\\\stackrel{def}{=}\\\\langle f ,{f^{\\\\prime }} \\\\rangle ,$ and thus becomes a prehilbert space, isomorphic as inner product spaces to ${\\\\cal C.}$ The hermitian product can be given in terms of the prewave functions themselves, as follows.\",\n \"Let $\\\\beta $ be a sesquilinear form on $L,$ nonvanishing on ${\\\\cal B}$ .\",\n \"We define $ \\\\psi _f\\\\, \\\\beta \\\\,\\\\psi _{f^\\\\prime } \\\\,: m \\\\in SL/(G_\\\\alpha )_{SL} \\\\mapsto \\\\frac{\\\\beta ( \\\\psi _f ( H,\\\\,m),\\\\psi _{f^\\\\prime } ( H,\\\\,m ))}{\\\\beta (z,\\\\ z )} $ where $z$ is arbitrary in ${\\\\bf r}^{-1}(m)$ and $H$ is arbitrary in $H(2)$ .\",\n \"Thus $\\\\langle \\\\psi _f,\\\\ \\\\psi _{f^\\\\prime } \\\\rangle =\\\\int _{{SL}/(G_\\\\alpha )_{SL}}\\\\psi _f\\\\, \\\\beta \\\\,\\\\psi _{f^\\\\prime }\\\\ \\\\omega .$ The representation of $G$ on ${\\\\cal PW}$ equivalent to $\\\\delta ^\\\\prime $ under $\\\\psi $ is given by $\\\\delta ^{pw}(A,H)\\\\cdot \\\\psi _f=\\\\psi _{\\\\delta ^\\\\prime (A,H) \\\\cdot f}.$ The infinitesimal generator associated to $(a,h)\\\\ \\\\ \\\\in sl(2,\\\\bf C)\\\\oplus H(2)$ is $d\\\\delta ^{pw}(a,h)\\\\cdot \\\\psi _f=\\\\psi _{d\\\\delta ^{\\\\prime }(a,h)\\\\cdot f}.$ The Quantum Operators for Prewave Functions must be defined as $(a,h)^{\\\\upsilon }\\\\stackrel{\\\\mathrm {d}ef}{=}\\\\ \\\\frac{1}{2\\\\pi i}\\\\,d\\\\delta ^{pw}(a,h),$ and we have $(a,h)^{\\\\upsilon }\\\\cdot \\\\psi _f{=}\\\\ \\\\psi _{(a,h)^{\\\\theta }\\\\cdot f}.$ On the other hand, we have a natural action on $H(2)\\\\times SL/(G_\\\\alpha )_{SL}$ defined by $(B,K)* (H,A\\\\,(G_\\\\alpha )_{SL})=(BHB^*+K,BA (G_\\\\alpha )_{SL}).$ Thus we define an, a priori different, representation, $\\\\delta _{pw},$ on ${\\\\cal PW}$ by means of $\\\\delta _{pw}(A,H) \\\\cdot \\\\psi _f =\\\\rho (A) \\\\circ \\\\psi _f \\\\circ \\\\left((A,H)*\\\\right)^{-1},$ i.e., for all $(K,m)\\\\in H(2)\\\\times SL/(G_\\\\alpha )_{SL}$ $\\\\left(\\\\delta _{pw}(A,H) \\\\cdot \\\\psi _f \\\\right)(K,m) =\\\\rho (A) \\\\cdot \\\\psi _f \\\\left((A,H)^{-1}*(K,m)\\\\right).$ Using (REF ) and (REF ) one can see that $\\\\delta _{pw}=\\\\delta ^{pw}$ as follows ${\\\\begin{array}{c}\\\\left(\\\\delta ^{pw}(A,H) \\\\cdot \\\\psi _f \\\\right)(K,m)=\\\\psi _{\\\\delta ^\\\\prime \\\\left( A,H\\\\right)\\\\cdot f}(K,m)= f(\\\\rho (A^{-1})\\\\cdot z) \\\\\\\\Exp\\\\left(- i\\\\pi Tr [ P( m))\\\\varepsilon \\\\overline{H} \\\\varepsilon ]\\\\right)Exp\\\\left( i\\\\pi Tr [ P( m)\\\\varepsilon \\\\overline{K} \\\\varepsilon ]\\\\right) z=(*)\\\\end{array}}$ where $z\\\\in r^{-1}(m).$ If we denote $ z^{\\\\prime }=\\\\rho (A^{-1})\\\\cdot z ,$ we have ${\\\\begin{array}{c}(*)= f( z^{\\\\prime }) Exp\\\\left( i\\\\pi Tr [ P( r(\\\\rho (A)\\\\cdot z^{\\\\prime }))\\\\varepsilon \\\\overline{(K-H)} \\\\varepsilon ]\\\\right)( \\\\rho (A)\\\\cdot z^{\\\\prime }) =\\\\\\\\=f( z^{\\\\prime }) Exp\\\\left( i\\\\pi Tr [ P( r( z^{\\\\prime }))\\\\varepsilon \\\\overline{(A^{-1}(K-H)(A^{-1})^*)} \\\\varepsilon ]\\\\right) (\\\\rho (A)\\\\cdot z^{\\\\prime })=\\\\\\\\=\\\\rho (A) \\\\psi _f((A,H)^{-1}*(K,m) =\\\\left(\\\\delta _{pw}(A,H) \\\\cdot \\\\psi _f \\\\right)(K,m).\\\\end{array}}$\"\n ],\n [\n \"Representation on Wave Functions.\",\n \"Now, let us consider ${\\\\cal WF},$ the complex vector space composed by the Wave Functions $\\\\widetilde{ \\\\psi _f}$ such that $f\\\\in {\\\\cal C}.$ A \\\"translation\\\" of the preceeding representations of $SL\\\\oplus H(2)$ to ${\\\\cal WF},$ is given by $\\\\delta _{w}(A,H) \\\\cdot \\\\widetilde{\\\\psi _f} = \\\\lbrace \\\\delta _{pw}(A,H) \\\\cdot \\\\psi _{ f}\\\\rbrace {\\\\widetilde{}}= \\\\lbrace \\\\psi _{\\\\delta ^{\\\\prime }(A,H)\\\\cdot f}\\\\rbrace {\\\\widetilde{}},$ where $\\\\lbrace \\\\psi \\\\rbrace {\\\\widetilde{}}$ means $\\\\widetilde{\\\\psi }.$ Then $\\\\delta _{w}(A,H) \\\\cdot \\\\widetilde{\\\\psi _f}(K)&=& \\\\lbrace \\\\delta _{pw}(A,H) \\\\cdot \\\\psi _{ f}\\\\rbrace {\\\\widetilde{}}(K)\\\\\\\\&=& \\\\int _{SL/(G_\\\\alpha )_{SL}}\\\\rho (A) \\\\cdot \\\\psi _f((A,H)^{-1}*(K,m))\\\\omega _m=\\\\\\\\&=& \\\\rho (A) \\\\cdot \\\\int _{SL/(G_\\\\alpha )_{SL}} \\\\psi _f((A,H)^{-1}*K,d^{\\\\prime }_{A^{-1}}m))\\\\omega _m=\\\\\\\\&=& \\\\rho (A) \\\\cdot \\\\int _{SL/(G_\\\\alpha )_{SL}} \\\\psi _f((A,H)^{-1}*K,m))\\\\omega _m$ because of the invariance of $\\\\omega .$ Thus $\\\\left(\\\\delta _{w}(A,H) \\\\cdot \\\\widetilde{\\\\psi _f}\\\\right)(K)=\\\\rho (A) \\\\cdot \\\\widetilde{\\\\psi _f } \\\\left((A,H)^{-1}*K)\\\\right).$ Compare to (REF ).\",\n \"Let us denote by $d\\\\delta _{w}\\\\cdot (a,h)$ the infinitesimal generator of the representation $\\\\delta _w$ associated to $ (a,h) \\\\in sl(2, C)\\\\oplus { H(2)},$ and $\\\\widehat{ (a,h)} \\\\stackrel{def}{=}\\\\frac{1}{ 2 \\\\pi i } (d\\\\delta _{w}\\\\cdot (a,h) ).$ The endomorphism $\\\\widehat{ (a,h)}$ of ${\\\\cal WF}$ is the Quantum Operator corresponding to the Dynamical Variable $(a,h)$ when the Quantum States are represented by Wave Functions.\",\n \"A straightforward computation leads to the following expresions for the operators corresponding to Linear and Angular Momentum $\\\\widehat{ P^k} \\\\cdot \\\\widetilde{ \\\\psi _f }&=&\\\\frac{1}{2\\\\pi i}\\\\frac{\\\\partial }{\\\\partial x^k} \\\\ \\\\widetilde{ \\\\psi _f } \\\\nonumber \\\\\\\\\\\\widehat{ P^4} \\\\cdot \\\\widetilde{ \\\\psi _f } &=&\\\\frac{i}{2\\\\pi }\\\\frac{\\\\partial }{\\\\partial x^4} \\\\ \\\\widetilde{ \\\\psi _f } \\\\nonumber \\\\\\\\\\\\widehat{ l^k} \\\\cdot \\\\widetilde{ \\\\psi _f }&=&\\\\frac{1}{2\\\\pi i}\\\\left( d\\\\rho \\\\left( {\\\\frac{i\\\\sigma _k}{2}}\\\\right)+\\\\sum _{j,r=1}^3\\\\varepsilon _{kjr} x^j\\\\frac{\\\\partial }{\\\\partial x^r} \\\\right) \\\\ \\\\widetilde{ \\\\psi _f } \\\\\\\\\\\\widehat{ g^k} \\\\cdot \\\\widetilde{ \\\\psi _f } &=&\\\\frac{1}{2\\\\pi i}\\\\left( d\\\\rho \\\\left(\\\\frac{\\\\sigma _k}{2}\\\\right)- \\\\left( x^4\\\\frac{\\\\partial }{\\\\partial x^k}+ x^k\\\\frac{\\\\partial }{\\\\partial x^4}\\\\right)\\\\right)\\\\ \\\\widetilde{ \\\\psi _f } \\\\nonumber $ where $\\\\varepsilon _{ijk}$ are the components of an antisymmetric tensor such that $\\\\varepsilon _{123}=1$ .\"\n ],\n [\n \"Representation on Pseudotensorial Functions in the contact manifold.\",\n \"Recall the isomorphism (REF ) $ \\\\nonumber \\\\Phi :f\\\\in {\\\\cal C} \\\\rightarrow \\\\Phi _f \\\\in {\\\\cal V}$ If $f\\\\in {\\\\cal C},$ we have $\\\\Phi _f((A,H)KerC_\\\\alpha )=\\\\,f(A\\\\,Ker(C_\\\\alpha )_{SL}) \\\\,e^{i\\\\pi \\\\,Tr(P (A\\\\,(G_\\\\alpha )_{SL}) \\\\,\\\\varepsilon \\\\, \\\\overline{H}\\\\,\\\\varepsilon )}.$ When $(C_{\\\\alpha })_{SL}={\\\\bf 1},$ we have $Ker(C_\\\\alpha )_{SL}=(G_\\\\alpha )_{SL},$ so that $\\\\Phi _f((A,H)KerC_\\\\alpha )=\\\\,f(A\\\\,(G_\\\\alpha )_{SL}) \\\\,e^{i\\\\pi \\\\,Tr(P (A\\\\,(G_\\\\alpha )_{SL}) \\\\,\\\\varepsilon \\\\, \\\\overline{H}\\\\, \\\\varepsilon )}.$ In the case $(C_{\\\\alpha })_{SL}\\\\ne {\\\\bf 1},$ we assume the existence of a trivialization, $(L,\\\\,\\\\rho ,\\\\,z_0),$ and we identify $SL/Ker\\\\,(C_\\\\alpha )_{SL}$ with ${\\\\cal B}.$ Then the pseudotensorial function $\\\\Phi _f$ becomes $\\\\Phi _f\\\\left(\\\\left(A,H\\\\right)\\\\,Ker C_\\\\alpha \\\\right)=\\\\,f(\\\\rho (A)\\\\cdot z_0) \\\\,e^{i\\\\pi \\\\,Tr( P(r (\\\\rho (A)\\\\cdot z_0) )\\\\,\\\\varepsilon \\\\, \\\\overline{H}\\\\,\\\\varepsilon )}.$ A natural representation of $G$ on $V$ is the $\\\\delta ^c$ given by $\\\\delta ^c(A,H) \\\\cdot \\\\Phi _f=\\\\Phi _f \\\\circ ((A,H)^{-1}*)$ where $*$ is the canonical action on $G/Ker\\\\, C_\\\\alpha .$ Let us prove that $\\\\delta ^c(A,H) \\\\cdot \\\\Phi _f=\\\\Phi _{\\\\delta ^{\\\\prime }(A,H) \\\\cdot f}$ $i.e.$ that $\\\\Phi $ is equivariant for the representations $\\\\delta ^{\\\\prime }$ in ${\\\\cal C}$ and $\\\\delta ^c$ in ${\\\\cal V}$ .\",\n \"In fact, we have ${\\\\begin{array}{c}\\\\left( \\\\delta ^c(A,H) \\\\cdot \\\\Phi _f\\\\right) \\\\left((B,K)Ker\\\\, C_\\\\alpha \\\\right)=\\\\Phi _f\\\\left( A^{-1} B, A^{-1}(K-H)(A^{-1})^* \\\\right)=\\\\\\\\=f\\\\left( \\\\rho (A^{-1})\\\\cdot (\\\\rho (B)\\\\cdot z_0)\\\\right)\\\\,e^{i\\\\pi \\\\,Tr(P(r (\\\\rho (A^{-1})\\\\cdot (\\\\rho (B)\\\\cdot z_0)) )\\\\,\\\\varepsilon \\\\, \\\\overline{ ( A^{-1}(K-H)(A^{-1})^*)}\\\\,\\\\varepsilon )}=\\\\\\\\=f\\\\left( \\\\rho (A^{-1})\\\\cdot (\\\\rho (B)\\\\cdot z_0)\\\\right)\\\\,e^{i\\\\pi \\\\,Tr(A^{-1}\\\\,P(\\\\rho (B)\\\\cdot z_0)(A^{-1})^* A^*\\\\varepsilon \\\\, \\\\overline{ (K-H)}\\\\,\\\\varepsilon \\\\,A)}=\\\\\\\\= f\\\\left( \\\\rho (A^{-1})\\\\cdot (\\\\rho (B)\\\\cdot z_0)\\\\right)\\\\,e^{i\\\\pi \\\\,Tr(P(\\\\rho (B)\\\\cdot z_0)\\\\,\\\\varepsilon \\\\, \\\\overline{ (K-H)}\\\\,\\\\varepsilon ) }=\\\\\\\\=\\\\left( \\\\delta ^{\\\\prime }(A,H)\\\\cdot f\\\\right)\\\\left(\\\\rho (B) \\\\cdot z_0 \\\\right)\\\\,e^{i\\\\pi \\\\,Tr(P(\\\\rho (B)\\\\cdot z_0)\\\\,\\\\varepsilon \\\\, \\\\overline{ K}\\\\,\\\\varepsilon }=\\\\\\\\=\\\\Phi _{\\\\delta ^{\\\\prime }(A,H)\\\\cdot f}\\\\left((B,K)Ker\\\\, C_\\\\alpha \\\\right).\\\\end{array}}$ As a particular consecuence, for all $(a,h)\\\\in \\\\underline{G},$ we have for the infinitesimal generators of $\\\\delta ^c$ $d\\\\delta ^c(a,h) \\\\cdot \\\\Phi _f=\\\\Phi _{d\\\\delta ^\\\\prime (a,h)\\\\cdot f}.$ Equation (REF ), tell us that the infinitesimal generator of the representation $\\\\delta ^c$ associated to $(a,h)\\\\in \\\\underline{G},$ acts on $\\\\Phi _f$ as the (vector field)infinitesimal generator of the action on $G/Ker\\\\,C_\\\\alpha ,$ what will be denoted by $X_{(a,h)}^c.$ If we denote by $(a,h)^c,$ the Quantum Operator $(a,h)^c \\\\stackrel{def}{=}\\\\frac{1}{2\\\\pi i}\\\\,d\\\\delta ^c(a,h)$ we have $(a,h)^c \\\\cdot \\\\Phi _f=\\\\Phi _{(a,h)^\\\\theta \\\\cdot f}.$ Thus, our results in subsection REF , on Quantum Operators in that case, gives us results on Quantum Operators in our present case.\"\n ],\n [\n \"Quantizable forms in $ SL(2,\\\\mathbb {C}) \\\\oplus H(2)$ .\",\n \"In [7] I give a classification of the coadjoint orbits of the group under consideration.\",\n \"The orbits are divided into 9 Types, and a canonical representative of each orbit is given, according with its type.\",\n \"Let $\\\\alpha =\\\\lbrace a,k\\\\rbrace $ be a nonzero element of the dual of the Lie algebra of $ SL \\\\oplus H(2),$ and denote by $W$ and $P$ the Pauli- Lubanski and Linear momentum respectively at $\\\\alpha $ (cf.\",\n \"section ).\",\n \"Figure 3 gives what type of coadjoint orbit is the one of $\\\\alpha ,$ according with the values of $W$ and $P.$ Figure: Types of coadjoint orbitsWhen one knows the type of $\\\\alpha ,$ Figure 4 gives us the canonical representative of the orbit of $\\\\alpha .$ Figure: Canonical Representatives.The $\\\\mathbb {R} $ -quantizable orbits are all of the types 3, 6, 8, 9 and these of type 5 corresponding to the case $|W|=0$ Quantizable but not $\\\\bf R\\\\rm $-quantizable are the orbits whose canonical representatives are: $\\\\left\\\\lbrace \\\\frac{iT}{8 \\\\pi } \\\\left(\\\\begin{array}{cc} 1 &0 \\\\\\\\ 0 &-1 \\\\end{array} \\\\right) ,0\\\\right\\\\rbrace \\\\ \\\\ (type\\\\ 2),$ $\\\\left\\\\lbrace \\\\frac{i\\\\chi T}{8 \\\\pi }\\\\ \\\\left( \\\\begin{array}{cc}1 &0 \\\\\\\\0 &-1\\\\end{array} \\\\right),-sign (Tr\\\\left( P \\\\right) )\\\\left( \\\\begin{array}{cc}1 &0 \\\\\\\\0 &0\\\\end{array} \\\\right)\\\\right\\\\rbrace \\\\ \\\\ (type\\\\ 4), $ $\\\\left\\\\lbrace \\\\frac{iT}{8 \\\\pi }\\\\left( \\\\begin{array}{cc}1 &0 \\\\\\\\0 &-1\\\\end{array} \\\\right),-sign (Tr\\\\left( P \\\\right) )\\\\sqrt{|P|}\\\\ I \\\\right\\\\rbrace \\\\ \\\\ (type\\\\ 5), $ $ \\\\left\\\\lbrace \\\\frac{i\\\\chi T}{8 \\\\pi }\\\\left( \\\\begin{array}{cc}1 &0 \\\\\\\\0 &-1\\\\end{array} \\\\right),\\\\sqrt{-|P|}\\\\ \\\\left( \\\\begin{array}{cc}1 &0 \\\\\\\\0 &-1\\\\end{array} \\\\right)\\\\right\\\\rbrace \\\\ \\\\ (type\\\\ 7).$ where, in all cases, $T \\\\in \\\\bf Z^+\\\\rm , \\\\chi =1,-1.$ The concrete particles we study in this paper are those corresponding to Type 5 (massive particles) and Type 4 (massless particles such that the Pauly-Lubanski fourvector is proportional to Impulsion-Energy).\",\n \"Results concerning other types of particles will be published elsewhere.\"\n ],\n [\n \"Guide to the explicit construction of Wave Functions.\",\n \"Let $\\\\alpha $ be a quantizable element of $\\\\underline{G}^*.$ To determine the explicit form of the Wave Functions of the corresponding free particles, one can proceed as follows 1.\",\n \"Evaluate the isotropy subgroup at $\\\\alpha ,$ of the coadjoint representation, $G_{\\\\alpha }$ .\",\n \"The coadjoint orbit of $\\\\alpha ,$ that is the main symplectic manifold associated to $\\\\alpha ,$ is identified to the homogeneous space $G/G_{\\\\alpha }.$ 2.\",\n \"Find a surjective homomorphism, $C_{\\\\alpha },$ from $G_{\\\\alpha }$ onto the unit circle S$^1,$ whose differential is $\\\\alpha .$ 3.\",\n \"Evaluate $(G_{\\\\alpha })_{SL},$ that is composed by the $g\\\\in SL$ such that $(g,h)\\\\in G_{\\\\alpha }$ for some $h\\\\in H(2).$ 4.\",\n \"Determine $(C_{\\\\alpha })_{SL},$ defined in (REF ), and $S\\\\equiv (C_{\\\\alpha })_{SL}\\\\left((G_{\\\\alpha })_{SL}\\\\right).$ 5.\",\n \"Find an action of $SL(2,{\\\\mathbb {C}})$ on a manifold, such that the isotropy subgroup at some point, $p$ , is $(G_{\\\\alpha })_{SL}.$ Identify $SL/(G_{\\\\alpha })_{SL}$ with the orbit of $p,$ ${\\\\cal K}$ .\",\n \"6.\",\n \"Determine a volume element on ${\\\\cal K},$ $\\\\omega ,$ invariant under the action of $SL$ .\",\n \"7.\",\n \"If $S=\\\\lbrace 1\\\\rbrace ,$ the Prewave Functions are the $\\\\psi _f(H,K)= f(K)\\\\,e^{i\\\\pi Tr(P(K )\\\\varepsilon \\\\, \\\\overline{H}\\\\, \\\\varepsilon ) }$ where $(H,K)\\\\in H(2)\\\\times {\\\\cal K},$ $f$ is a function on ${\\\\cal K},$ and $P$ is the dynamical variable linear momentum, on ${\\\\cal K},$ given by (REF ).\",\n \"8.\",\n \"If $S\\\\ne \\\\lbrace 1\\\\rbrace ,$ find a Trivialization of $ C_\\\\alpha $ ($c.f.$ (REF )).\",\n \"Let ${\\\\cal B}$ the orbit of $z_0$ , identified to $SL/Ker(C_{\\\\alpha })_{SL},$ $r:{\\\\cal B} \\\\rightarrow {\\\\cal K}$ the canonical map from $SL/Ker(C_{\\\\alpha })_{SL})$ onto $SL/(G_{\\\\alpha })_{SL}$ .\",\n \"The Prewave Functions are given by ($c.f.$ (REF )) $\\\\psi _f(H,K)= f(z)\\\\,e^{i\\\\pi Tr(P(K )\\\\varepsilon \\\\, \\\\overline{H}\\\\, \\\\varepsilon ) }z$ where $(H,K)\\\\in H(2)\\\\times {\\\\cal K},$ $z\\\\in r^{-1}(K),$ and $f$ is a function on ${\\\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers.\",\n \"9.\",\n \"The Prewave Functions corresponding to continuous with compact support $f$ define Wave Functions by means of ($c.f.$ (REF )) $\\\\widetilde{\\\\psi _f}(H)=\\\\int _{\\\\cal K} {\\\\mbox{$\\\\psi $}}_f(H,\\\\cdot )\\\\ {\\\\mbox{$\\\\omega $}} .$\"\n ],\n [\n \"Wave Functions. Klein-Gordon equation.\",\n \"Let us consider a particle whose movement space is the coadjoint orbit of ${\\\\alpha _o}=\\\\left\\\\lbrace 0,\\\\ \\\\eta \\\\,m\\\\,I\\\\right\\\\rbrace ,\\\\ \\\\ \\\\ m \\\\in {\\\\mathbb {R}}^+, \\\\eta =\\\\pm 1.$ This orbit is a $\\\\mathbb {R} \\\\rm $ -quantizable orbit of the type 5, in the notation of section .\",\n \"The number $(-\\\\eta )$ is the sign of energy i.e.\",\n \"the sign of the value of the dynamical variable $P^4$ ($c.f.$ section ) at any point of the orbit.\",\n \"According with the usual interpretation of the determinant of momentum-energy, $\\\\vert P \\\\vert ,$ as mass square, the number $m$ must be interpreted as being the mass of the particle.\",\n \"By direct computation, one sees that ${\\\\mbox{$G_{\\\\alpha _o}$}} = \\\\lbrace \\\\ (A, \\\\ hI ): \\\\ A\\\\in SU(2),\\\\ h\\\\in \\\\mathbb {R} \\\\rbrace .$ The unique homomorphism onto ${\\\\mathbb {R}}$ whose differential is ${\\\\alpha _o}$ is given by $ C^\\\\prime _{\\\\alpha _o} (A ,\\\\ hI)=-\\\\eta m h .$ The unique homomorphism onto ${\\\\bf S}^1$ whose differential is $\\\\alpha _o$ is given by $C_{\\\\alpha _o} (A ,\\\\ hI)=e^{-2\\\\pi i\\\\eta m h}.$ Then we have ${\\\\begin{array}{c}Ker\\\\,C^\\\\prime _{\\\\alpha _o}=\\\\lbrace (A,0):A\\\\in SU(2)\\\\rbrace =SU(2)\\\\oplus \\\\lbrace 0\\\\rbrace ,\\\\\\\\Ker\\\\,C_{\\\\alpha _o}=\\\\lbrace (A,\\\\,\\\\frac{\\\\eta N}{m}\\\\, I):A\\\\in SU(2),\\\\ N \\\\in \\\\mathbb {Z}\\\\rbrace ,\\\\\\\\{\\\\mbox{$\\\\widetilde{C}_{\\\\alpha _o}$}} (A ,\\\\ H)=e^{-\\\\pi i\\\\eta m TrH} ,\\\\\\\\{\\\\mbox{$(G_{\\\\alpha _o})_{SL}$}}=SU(2) ,\\\\\\\\{\\\\mbox{$(C_{\\\\alpha _o})_{SL}$}}={\\\\bf 1 } ,\\\\\\\\SL_1 =SL_2= {\\\\mbox{$Ker\\\\,(C_{\\\\alpha _o})_{SL}$}}={\\\\mbox{$(G_{\\\\alpha _o} ) _{SL}$}},\\\\end{array}} $ Thus, in the commutative diagram of figure REF , section , the four spaces on the left are the same.\",\n \"All of them represent State Space, $H(2)\\\\times \\\\frac{SL}{(G_{\\\\alpha _o})_{SL}\\\\cap SL_1}$ what in our present case becomes $H(2) \\\\times \\\\frac{SL}{SU(2)},$ and can be obviously identified to $\\\\frac{G}{SU(2) \\\\oplus \\\\lbrace 0\\\\rbrace },$ by means of the diffeomorphism $(H,A\\\\,SU(2))\\\\rightarrow (A,H)(SU(2) \\\\oplus \\\\lbrace 0\\\\rbrace ).$ Then, in this case, besides the canonical map from State Space onto Movement Space (c.f.\",\n \"figure REF ), $\\\\nu _2: (H,ASU(2))\\\\in H(2) \\\\times \\\\frac{SL}{SU(2)} \\\\rightarrow (A,H)G_\\\\alpha \\\\in \\\\frac{G}{G_{\\\\alpha _o}}$ we have a natural map from State Space onto $G/Ker\\\\,C_{\\\\alpha _o}$ $\\\\nu _1:(A,H)(SU(2) \\\\oplus \\\\lbrace 0\\\\rbrace )\\\\in \\\\frac{G}{SU(2) \\\\oplus \\\\lbrace 0\\\\rbrace } \\\\rightarrow (A,H){Ker\\\\,C_{\\\\alpha _o}} \\\\in \\\\frac{G}{Ker\\\\,C_{\\\\alpha _o}}.$ The diagram of figure REF , becomes that of figure REF .\",\n \"Figure: Diagram for Klein-Gordon particles.Let ${\\\\cal H}^m$ be the positive mass hyperboloid ${\\\\cal H}^m=\\\\lbrace H \\\\in H(2):\\\\ det\\\\, H\\\\,=\\\\,m^2,\\\\ Tr\\\\,H\\\\,>\\\\,0\\\\,\\\\rbrace .$ In ${\\\\cal H}^m$ we consider the action of $SL(2,C)$ given by $ A* H=A\\\\,H\\\\,A^*,$ for all $H\\\\in {\\\\cal H}^m$ and $A \\\\in SL(2,C).$ In these conditions we have $Tr(A\\\\,H\\\\,A^*)>0$ as a consecuence of the fact that $SL(2,C)$ is connected.\",\n \"Since the group of restricted homogeneous Lorenz transformations is transitive on ${\\\\cal H}^m,$ so does $SL(2,\\\\mathbb {C}).$ The isotropy subgroup at $mI$ is $SU(2).$ Thus $SL/(G_{\\\\alpha _o})_{SL}$ is identified to ${\\\\cal H}^m,$ by means of the diffeomorphism $A\\\\, SU(2)\\\\in \\\\frac{SL}{SU(2)} \\\\longrightarrow m\\\\,A\\\\,A^* \\\\in {\\\\cal H}^m.$ If $K= m\\\\,A\\\\,A^* \\\\in {\\\\cal H}^m,$ then $K$ is identified to $A\\\\, SU(2)\\\\in \\\\frac{SL}{SU(2)},$ and the function P thus becomes $P(K)=P(ASU(2))=-A(\\\\eta \\\\,m\\\\,I)A^*=-\\\\eta K.$ The manifold ${\\\\cal H}^m$ has the global parametrization $\\\\varphi :(p^1,\\\\ p^2,\\\\ p^3)\\\\in {{\\\\mathbb {R}} ^3}\\\\ \\\\ \\\\mapsto \\\\ \\\\ h(p^1,\\\\ p^2,\\\\ p^3,\\\\ (m^2+ p^2)^{\\\\frac{1}{2}} )\\\\ \\\\in \\\\ {\\\\cal H}^m$ where $p^2= \\\\sum _{i=1}^3 (p^i)^2.$ In particular it is orientable.\",\n \"On the other hand, the restriction of the pseudoriemannian metric defined on ${\\\\mathbb {R}}^4$ by Minkowski metric, to ${\\\\cal H}^m,$ is negative definite so that its opposite is a riemannian metric, that provides us with a canonical volume element, $\\\\nu $ .\",\n \"A computation of the matrix of the riemannian metric in the chart $\\\\varphi $ leads to $\\\\nu =\\\\frac{dp^1 \\\\wedge dp^2 \\\\wedge dp^3}{\\\\sqrt{m^2+ p^2}}$ where the $p^i$ are the coordinates in the chart $\\\\varphi .$ The action on $H(2)$ defined as in (REF ) preserves Minkowski metric, so that the action on ${\\\\cal H}^m$ preserves $\\\\nu $ .\",\n \"As a consequence, $\\\\nu $ is an invariant volume element under the action (REF ).\",\n \"Since $(C_{\\\\alpha _o})_{SL}={\\\\bf 1}$ , $\\\\cal C$ is composed by functions on ${\\\\cal H}^m.$ The prewave functions have the form $\\\\psi _f:(H,\\\\ K) \\\\in H(2) \\\\times {\\\\cal H}^m\\\\ \\\\mapsto \\\\ \\\\ f\\\\left(K\\\\right) e^{-i\\\\pi \\\\eta Tr(K \\\\varepsilon \\\\overline{H} \\\\varepsilon ) }.$ where f is in $\\\\cal C.$ The corresponding wave function is $\\\\tilde{\\\\psi }_f(X)=\\\\int _{{\\\\cal H}^m}\\\\psi _f(h(X),\\\\,\\\\cdot \\\\,)\\\\ \\\\nu $ By direct computation one sees that the prewave and wave functions satisfy Klein-Gordon equation.\",\n \"If $f^\\\\prime $ is another continuous with compact support function on ${\\\\cal H}^m$ , the hermitian product of the quantum states corresponding to $\\\\psi _f$ and $\\\\psi _{f^\\\\prime },$ defined in section REF , can be writen $\\\\langle \\\\psi _f,\\\\ \\\\psi _{f^\\\\prime } \\\\rangle =\\\\int _{{\\\\cal H}^m }\\\\psi _f^*\\\\,\\\\psi _{f^\\\\prime }\\\\,\\\\nu .$\"\n ],\n [\n \"The Homogeneous Contact and Symplectic Manifolds for Klein-Gordon particles\",\n \"Let us consider the action of $G$ on ${\\\\cal H}^m \\\\times \\\\mathbb {R}^3 \\\\times \\\\mathbb {S}^1$ given by $(A,H)*(K,\\\\overrightarrow{y},z)=(AKA^*,\\\\overrightarrow{x}(A,H,\\\\overrightarrow{y},K), e^{2\\\\pi i \\\\eta m \\\\ell (A,H,\\\\overrightarrow{y},K)}z)$ where $\\\\overrightarrow{x}(A,H,\\\\overrightarrow{y},K)$ and $\\\\ell (A,H,\\\\overrightarrow{y},K)$ are given by $h(\\\\overrightarrow{x}(A,H,\\\\overrightarrow{y},K),0)=A\\\\,h(\\\\overrightarrow{y},0)\\\\,A^*+H+\\\\ell (A,H,\\\\overrightarrow{y},K) A\\\\,\\\\frac{K}{m}\\\\,A^*.$ Obviously we have $\\\\ell (A,H,\\\\overrightarrow{y},K)=-m\\\\frac{Tr(A\\\\,h(\\\\overrightarrow{y},0)\\\\,A^*+H)}{Tr(AKA^*)}.$ This is a transitive action, as can be proved by direct computation.\",\n \"The isotropy subgroup at $(mI,\\\\,\\\\overrightarrow{0},1),$ is $Ker\\\\,C_{\\\\alpha _0},$ so that the homogeneous space $G/Ker\\\\,C_{\\\\alpha _0},$ can be identified to ${\\\\cal H}^m \\\\times \\\\mathbb {R}^3 \\\\times \\\\mathbb {S}^1$ by means of the map $(A,H) Ker\\\\,C_{\\\\alpha _0} \\\\longrightarrow (A,H)*(mI,\\\\overrightarrow{0},1).$ We know (c.f.\",\n \"section ) that the left-invariant 1-form on $G$ whose value at $(I,0)$ is $\\\\alpha _o$ , $\\\\tilde{\\\\alpha }_o,$ projects in a 1-form , $\\\\Omega ,$ in $G/Ker\\\\,C_{\\\\alpha _o}$ which is a contact form, and is homogeneous in the sense that it is preserved by the diffeomorphisms corresponding to the canonical action of $G$ on $G/Ker\\\\,C_{\\\\alpha _o}.$ One way to give explicitly $\\\\Omega ,$ is to use coordinate domains in $G/Ker\\\\,C_{\\\\alpha _0}$ that are also domains of sections of the canonical map from the group $G$ onto $G/Ker\\\\,C_{\\\\alpha _0}.$ The pull back of $\\\\tilde{\\\\alpha }_0$ by that section, is the restriction of $\\\\Omega $ to the domain, and we can give its local expresion in the coordinate system.\",\n \"With our identification , the just cited canonical map becomes $\\\\mu :(A,H)\\\\in G \\\\longrightarrow (A,H) * (mI,\\\\overrightarrow{0},1)\\\\in {\\\\cal H}^m \\\\times \\\\mathbb {R}^3 \\\\times \\\\mathbb {S}^1.$ In ${\\\\cal H}^m \\\\times \\\\mathbb {R}^3 \\\\times \\\\mathbb {S}^1$ be define a coordinate system for each $\\\\tau \\\\in \\\\mathbb {R}$ as the inverse of the parametrization ${\\\\begin{array}{c}\\\\phi _\\\\tau : (k_1,k_2,k_3,x^1,x^2,x^3,t)\\\\in \\\\mathbb {R}^6 \\\\times \\\\left(-\\\\frac{1}{2},\\\\frac{1}{2}\\\\right) \\\\rightarrow \\\\\\\\\\\\rightarrow \\\\left( m\\\\ h(k_1,k_2,k_3,k_4),x^1,x^2,x^3,e^{2\\\\pi i(t+\\\\tau )}\\\\right)\\\\in \\\\\\\\\\\\in {\\\\cal H}^m \\\\times \\\\mathbb {R}^3 \\\\times \\\\mathbb {S}^1\\\\end{array}}$ where $k_4=\\\\sqrt{1+k_1^2+k_2^2+k_3^2}.$ Now, let us consider the map ${\\\\begin{array}{c}\\\\sigma _\\\\tau :\\\\phi _\\\\tau (k_1,k_2,k_3,x^1,x^2,x^3,t) \\\\rightarrow \\\\\\\\ \\\\rightarrow \\\\left( \\\\left( \\\\begin{array}{cc}\\\\sqrt{\\\\frac{k_4+k_3}{1+k_1^2+k_2^2}}&\\\\sqrt{\\\\frac{k_4+k_3}{1+k_1^2+k_2^2}}(k_1-ik_2)\\\\\\\\0&\\\\sqrt{\\\\frac{1+k_1^2+k_2^2 }{ k_4+k_3}}\\\\end{array}\\\\right),h(\\\\overrightarrow{x},0)-\\\\frac{\\\\eta }{m}(t+\\\\tau )) h(\\\\overrightarrow{k},k_4)\\\\right) \\\\\\\\ \\\\in G.\\\\hspace{341.43306pt}\\\\end{array}}$ We have $\\\\mu \\\\circ \\\\sigma _\\\\tau (\\\\phi _\\\\tau (R))=\\\\sigma _\\\\tau (\\\\phi _\\\\tau (R))*(mI,\\\\overrightarrow{0},1)=\\\\phi _\\\\tau (R),$ for all $R\\\\in \\\\mathbb {R}^6 \\\\times (-1/2,\\\\ 1/2),$ so that $\\\\sigma _\\\\tau $ is a section of the canonical map $\\\\mu $ .\",\n \"Then, on the image of $\\\\phi _\\\\tau $ we have $\\\\Omega =\\\\sigma _\\\\tau ^* \\\\tilde{\\\\alpha }_0.$ In order to obtain, for exemple, the value $(\\\\Omega )_{\\\\left( \\\\phi _\\\\tau \\\\left( R \\\\right) \\\\right)}\\\\cdot \\\\left( \\\\frac{\\\\partial }{\\\\partial k_1} \\\\right)_{\\\\left( \\\\phi _\\\\tau \\\\left( R \\\\right) \\\\right)},$ we must find the value of $\\\\alpha _0$ on the tangent vector to the curve $\\\\gamma (s)= \\\\left( \\\\sigma _\\\\tau ( \\\\phi _\\\\tau (R) )\\\\right)^{-1}\\\\left( \\\\sigma _\\\\tau ( \\\\phi _\\\\tau (R+(s,0,0,0,0,0,0))\\\\right)$ at $0.$ Since $\\\\alpha _0=\\\\left\\\\lbrace 0,\\\\ \\\\eta \\\\,m\\\\,I\\\\right\\\\rbrace ,$ the first component of the tangent vector at 0 of $\\\\gamma $ has no incidence in the value of $\\\\alpha _0$ on it.\",\n \"The preceding computation gives us $\\\\langle (o,\\\\ \\\\eta m I),\\\\stackrel{.\",\n \"}{\\\\gamma }_0 \\\\rangle =0.$ A similar computation for each of the other coordinates, or a common reasoning with a curve $\\\\gamma (s)=\\\\phi _\\\\tau (c(s)),$ leads to $\\\\Omega =d\\\\,t+\\\\eta \\\\,m\\\\,k_i\\\\,dx^i.$ Obviously $d\\\\Omega =\\\\eta \\\\,m \\\\,dk_i\\\\wedge \\\\,dx^i,$ so that $\\\\Omega \\\\wedge \\\\left(d\\\\Omega \\\\right)^3=\\\\frac{\\\\eta m^3}{16}\\\\ dt\\\\wedge dk_1\\\\wedge dk_2\\\\wedge dk_3\\\\wedge dx^1\\\\wedge dx^2\\\\wedge dx^3$ what confirms the fact that $\\\\Omega $ is a contact form.\",\n \"The characteristic vectorfield $Z(\\\\Omega )$ defined by $i(Z(\\\\Omega ))\\\\Omega =1,\\\\ \\\\ i(Z(\\\\Omega ))\\\\,d\\\\Omega =0,$ has flow, ${\\\\phi _t},$ given by $\\\\phi _t(K,\\\\overrightarrow{x},z)=(K,\\\\overrightarrow{x}, e^{2\\\\pi i t}z).$ As a consecuence, the period of all its integral curves is $T(\\\\Omega )=1,$ so that $\\\\Omega $ is a connection form, and $d\\\\Omega $ the corresponding curvature form.\",\n \"Now, let us consider the action of $G$ on ${\\\\cal H}^m \\\\times \\\\mathbb {R}^3 $ given by $(A,H)*(K,\\\\overrightarrow{y})=(AKA^*,\\\\overrightarrow{x}(A,H,\\\\overrightarrow{y},K))$ where $h(\\\\overrightarrow{x}(A,H,\\\\overrightarrow{y},K),0)=A\\\\,h(\\\\overrightarrow{y},0)\\\\,A^*+H+\\\\ell (A,H,\\\\overrightarrow{y},K) A\\\\,\\\\frac{K}{m}\\\\,A^*.$ This is also a transitive action.The isotropy subgroup at $(mI,\\\\,\\\\overrightarrow{0}),$ is $G_{\\\\alpha _0},$ so that $G/G_{\\\\alpha _0},$ can be identified to ${\\\\cal H}^m \\\\times \\\\mathbb {R}^3 $ by means of the map $(A,H) G_{\\\\alpha _0} \\\\longrightarrow (A,H)*(mI,\\\\overrightarrow{0}).$ The left-invariant 2-form on $G$ whose value at $(I,0)$ is $d\\\\alpha _0$ , $d\\\\tilde{\\\\alpha }_0,$ projects in a 2-form , $\\\\omega ,$ in $G/G_{\\\\alpha _0}$ which is a simplectic form, and is homogeneous in the sense that it is preserved by the diffeomorphisms corresponding to the canonical action of $G$ on $G/G_{\\\\alpha _0}.$ With our identifications, the canonical map $(A,H) Ker\\\\,C_\\\\alpha \\\\rightarrow (A,H) G_\\\\alpha $ becomes $(K,\\\\overrightarrow{x},z )\\\\in {\\\\cal H}^m \\\\times \\\\mathbb {R}^3 \\\\times \\\\mathbb {S}^1 \\\\rightarrow (K,\\\\overrightarrow{x})\\\\in {\\\\cal H}^m \\\\times {\\\\mathbb {R}}^3$ and, since $T(\\\\Omega )=1,$ $\\\\omega $ must be a projection of $d\\\\Omega ,$ so that $\\\\omega = {\\\\eta \\\\,m} \\\\,d{k}_i \\\\wedge \\\\,d{x}^i,$ where now $\\\\lbrace k_1,\\\\dots ,x^3\\\\rbrace $ are the coordinates corresponding to the following global parametrization of ${\\\\cal H}^m \\\\times {\\\\mathbb {R}}^3 $ ${\\\\begin{array}{c}\\\\underline{\\\\phi }: ({k}_1,{k}_2,{k}_3,{x}^1,{x}^2,{x}^3)\\\\in {\\\\mathbb {R}}^6 \\\\rightarrow \\\\\\\\\\\\rightarrow \\\\left( m\\\\ h({k}_1,{k}_2,{k}_3,{k}_4),{x}_1,{x}_2,{x}_3\\\\right)\\\\\\\\\\\\in {\\\\cal H}^m \\\\times {\\\\mathbb {R}}^3\\\\end{array}}$ where ${k}_4=\\\\sqrt{1+{k}_1^2+{k}_2^2+{k}_3^2}.$ In this case we have a global section of the canonical map ${\\\\begin{array}{c}\\\\underline{\\\\sigma }(\\\\underline{\\\\phi }({k}_1,{k}_2,{k}_3,{x}^1,{x}^2,{x}^3))=\\\\\\\\=\\\\left( \\\\left( \\\\begin{array}{cc}\\\\sqrt{\\\\frac{k_4+k_3}{1+k_1^2+k_2^2}}&\\\\sqrt{\\\\frac{k_4+k_3}{1+k_1^2+k_2^2}}(k_1-ik_2)\\\\\\\\0&\\\\sqrt{\\\\frac{1+k_1^2+k_2^2 }{ k_4+k_3}}\\\\end{array}\\\\right),h(\\\\overrightarrow{x},0) \\\\right).\\\\end{array}}$ The map $\\\\underline{\\\\sigma }$ is a section since $\\\\underline{\\\\sigma }(\\\\phi (R))*(mI,\\\\overrightarrow{0})=\\\\phi (R).$ The coadjoint orbit of $\\\\alpha _0,$ becomes identified to $ {\\\\cal H}^m \\\\times {\\\\mathbb {R}}^3 $ by means of $Ad^*_{(A,H)}\\\\cdot \\\\alpha _0 \\\\rightarrow (A,H)*(mI,\\\\overrightarrow{0}),$ whose inverse is given by $(K,\\\\overrightarrow{x}) \\\\rightarrow Ad^*_{\\\\underline{\\\\sigma }(K,\\\\overrightarrow{x})} \\\\cdot \\\\alpha _0.$ Thus, the $(e,g)\\\\in \\\\underline{G},$ what are dynamical variables on the coadjoint orbit, becomes functions on $ {\\\\cal H}^m \\\\times \\\\mathbb {R}^3 , $ denoted by $D_{(e,g)}$ in section , defined as follows $D_(e,g)(\\\\underline{\\\\phi }(k_1,\\\\dots ,x_3))=(Ad^*_{\\\\underline{\\\\sigma }(\\\\underline{\\\\phi }(k_1,\\\\dots ,x_3))}\\\\alpha _0)( e,g).$ In particular, the Linear and Angular Momentum, given in (REF ), are, as functions on $ {\\\\cal H}^m \\\\times \\\\mathbb {R}^3, $ $\\\\nonumber P(m\\\\,k,\\\\overrightarrow{x})&=&- \\\\eta \\\\,m\\\\ k, \\\\\\\\ \\\\overrightarrow{l}(m\\\\,k,\\\\overrightarrow{x})&=&\\\\eta \\\\,m\\\\,\\\\overrightarrow{k} \\\\times \\\\overrightarrow{x}=\\\\overrightarrow{x} \\\\times \\\\left(\\\\overrightarrow{P}(m\\\\,k,\\\\overrightarrow{x}) \\\\right) \\\\\\\\ \\\\nonumber \\\\overrightarrow{g}(m\\\\,k,\\\\overrightarrow{x})&=&-\\\\eta \\\\,m\\\\,k^4 \\\\overrightarrow{x}=\\\\left( P^4(m\\\\,k,\\\\overrightarrow{x}) \\\\right) \\\\ \\\\overrightarrow{x} ,$ where we have denoted $D_{P^k},\\\\ D_{l^i},$ and $D_{g^j}$ by $P^k, l^i$ and $g^j$ respectively.\",\n \"Also, each $a\\\\in \\\\underline{G}, $ define an infinitesimal generator, $X^s_{a},$ of the action (REF ).\",\n \"These infinitesimal generators are defined by means of its flow, and its local expresions can be obtained directly from the definition, but it is also possible to use formula (REF ) to obtain the following local expresions in the chart $\\\\underline{\\\\Phi }^{-1}$ $X^s_{P^j}&=&\\\\frac{\\\\partial }{\\\\partial x^j}\\\\\\\\X^s_{P^4}&=&\\\\sum _{j=1}^3\\\\frac{k_j}{k_4}\\\\ \\\\frac{\\\\partial }{\\\\partial x^j}\\\\\\\\X_{l^k}^s &=&\\\\sum _{j,r=1}^3 \\\\varepsilon _{kjr} k_j \\\\frac{\\\\partial }{\\\\partial k_r} +\\\\sum _{j,r=1}^3 \\\\varepsilon _{kjr} x^j\\\\frac{\\\\partial }{\\\\partial x^r} \\\\\\\\X^s_{g^j}&=&x^j\\\\, X_{P^4}^s-k_4\\\\frac{\\\\partial }{\\\\partial k_j}$ where $ \\\\varepsilon _{kjr}$ is as in (REF ).\",\n \"Equation (REF ) proves that these vector fields are globally hamiltonian, and the corresponding hamiltonian is, with our actual notation, the function appearing in the subindex in each case.\",\n \"The infinitesimal generator corresponding to $a\\\\in \\\\underline{G}$ for the action in the contact manifold is denoted by $X_a^c.$ We have in the charts $\\\\Phi _\\\\tau ^{-1}$ $X^c_{P^j}&=&\\\\frac{\\\\partial }{\\\\partial x^j}\\\\nonumber \\\\\\\\X^c_{P^4}&=&\\\\frac{1}{k_4}\\\\left(\\\\sum _{j=1}^3\\\\,k_j\\\\, \\\\frac{\\\\partial }{\\\\partial x^j}+\\\\eta m \\\\frac{\\\\partial }{\\\\partial t} \\\\right)\\\\nonumber \\\\\\\\X_{l^k}^c &=&\\\\sum _{j,r=1}^3 \\\\varepsilon _{kjr} k_j \\\\frac{\\\\partial }{\\\\partial k_r} +\\\\sum _{j,r=1}^3 \\\\varepsilon _{kjr} x^j\\\\frac{\\\\partial }{\\\\partial x^r} \\\\\\\\X^c_{g^j}&=&x^j\\\\, X_{P^4}^c-k_4\\\\frac{\\\\partial }{\\\\partial k_j}.\\\\nonumber $ The Quantum Operators representing Linear and Angular Momentum for Quantum States on the Contact Manifold are $\\\\frac{1}{2 \\\\pi i}$ times the vectorfiels in (REF ).\",\n \"If $f$ is a $C^\\\\infty $ function on ${\\\\cal H}^m$ the corresponding Quantum State in the contact manifold is $\\\\Phi _f((A,H)\\\\,Ker\\\\,C_\\\\alpha )=f(A\\\\,(G_\\\\alpha )_{SL})Exp[i \\\\pi Tr(P(A\\\\,(G_\\\\alpha )_{SL})\\\\varepsilon \\\\overline{H} \\\\varepsilon )]$ or, with the identifications we have made $\\\\Phi _f((A,H)* (mI,\\\\overrightarrow{0},1))=f(mAA^*) Exp[i \\\\pi Tr((-\\\\eta mAA^*)\\\\varepsilon \\\\overline{H} \\\\varepsilon )]$ but $\\\\Phi _f((A,H)* (mI,\\\\overrightarrow{0},1))=\\\\Phi _f(mAA^*,h^{-1}( H+\\\\ell AA^*), Exp[2 \\\\pi i \\\\eta m \\\\ell ]),$ where $ \\\\ell $ is such that $Tr(H+\\\\ell AA^*)=0.$ Thus $\\\\Phi _f(K,\\\\overrightarrow{x},z)=f(K) Exp[i \\\\pi Tr((-\\\\eta K)\\\\varepsilon \\\\overline{(h(\\\\overrightarrow{x},0)-\\\\ell \\\\frac{K}{m})} \\\\varepsilon )]$ where $\\\\ell $ is such that $z= Exp[2 \\\\pi i \\\\eta m \\\\ell ].$ Then $\\\\Phi _f(K,\\\\overrightarrow{x},z)=f(K) Exp[-i \\\\eta \\\\pi Tr( K \\\\varepsilon \\\\overline{h(\\\\overrightarrow{x},0)}\\\\varepsilon ]Exp[i \\\\ell \\\\eta \\\\pi Tr( K \\\\varepsilon \\\\overline{\\\\frac{K}{m}}\\\\varepsilon ]$ and one sees that, if $K=m h(\\\\overrightarrow{k},k_4),$ $\\\\Phi _f(K,\\\\overrightarrow{x},z)=f(K)\\\\, Exp[-2 \\\\pi i \\\\eta m \\\\langle \\\\overrightarrow{k} ,\\\\overrightarrow{x} \\\\rangle ] \\\\,\\\\overline{z} .$ In the coordinate system associated to $\\\\phi _\\\\tau ,$ we have $\\\\Phi _f \\\\circ \\\\phi _\\\\tau (\\\\overrightarrow{k} ,\\\\overrightarrow{x} ,t)=f(mh(\\\\overrightarrow{k},k_4))\\\\, \\\\, Exp[-2 \\\\pi i ( \\\\eta m \\\\langle \\\\overrightarrow{k} ,\\\\overrightarrow{x} \\\\rangle +t+\\\\tau )].$\"\n ],\n [\n \"Wave Functions for $T=1.$ Dirac equation.\",\n \"Let us consider a particle whose movement space is the coadjoint orbit of $\\\\alpha _1=\\\\left\\\\lbrace \\\\ \\\\frac{i}{8\\\\pi }\\\\left( \\\\begin{array}{cc} 1&0 \\\\\\\\ 0&-1\\\\end{array} \\\\right) \\\\ ,\\\\ \\\\eta \\\\,m\\\\,I\\\\right\\\\rbrace ,$ where $m \\\\in {\\\\bf R}^+, \\\\eta =\\\\pm 1.$ This orbit is a quantizable, not $\\\\bf R \\\\rm $ -quantizable orbit, of the type 5, in the notation of section .\",\n \"In this case we have ${\\\\mbox{$G_{\\\\alpha _1}$}} = \\\\lbrace \\\\ (\\\\ \\\\left( \\\\begin{array}{cc}z&0 \\\\\\\\0&\\\\overline{z}\\\\end{array} \\\\right), \\\\ hI ): \\\\ z\\\\in S^1,\\\\ h\\\\in {\\\\mbox{$\\\\bf R \\\\rm $}} \\\\rbrace .$ The unique homomorphism from $G_{\\\\alpha _1}$ onto ${\\\\bf S}^1$ whose differential is ${\\\\alpha _1}$ is given by ${\\\\mbox{$C_{\\\\alpha _1}$}} (\\\\ \\\\left( \\\\begin{array}{cc}e^{2\\\\pi i\\\\phi }&0 \\\\\\\\0&e^{-2\\\\pi i\\\\phi }\\\\end{array} \\\\right) ,\\\\ hI)=e^{2\\\\pi i(\\\\phi -\\\\eta m h)}.$ Then $\\\\begin{array}{l}Ker\\\\, C_{\\\\alpha _1}=\\\\lbrace (\\\\ \\\\left( \\\\begin{array}{cc}e^{2\\\\pi i\\\\eta m h}&0 \\\\\\\\0&e^{-2\\\\pi i\\\\eta m h}\\\\end{array} \\\\right) ,\\\\ hI):h\\\\in \\\\mathbb {R} \\\\rbrace ,\\\\\\\\\\\\vspace{7.22743pt}(G_{\\\\alpha _1} ) _{SL}=\\\\lbrace \\\\ \\\\left( \\\\begin{array}{cc}z&0 \\\\\\\\0&\\\\overline{z}\\\\end{array} \\\\right):\\\\ z\\\\in S^1\\\\rbrace , \\\\\\\\\\\\vspace{7.22743pt}{{\\\\mbox{$(C_{\\\\alpha _1})_{SL}$}}(\\\\ \\\\left( \\\\begin{array}{cc}z&0 \\\\\\\\0&\\\\overline{z}\\\\end{array} \\\\right))=z}, \\\\\\\\\\\\vspace{7.22743pt}SL_1 \\\\cap SL_2 ={\\\\mbox{$(G_{\\\\alpha _1} ) _{SL}$}}, \\\\\\\\\\\\vspace{7.22743pt}SL_1 \\\\cap {\\\\mbox{$Ker\\\\,(C_{\\\\alpha _1})_{SL}$}}={\\\\mbox{$Ker\\\\,(C_{\\\\alpha _1})_{SL}$}}=\\\\lbrace I \\\\rbrace .\",\n \"\\\\\\\\\\\\end{array}$ Let us denote $(G_{\\\\alpha _1})_{SL}$ by $[S^1],$ $G_{\\\\alpha _1}$ by $[S^1]\\\\oplus \\\\mathbb {R},$ and $Ker\\\\, C_{\\\\alpha _1}$ by $[R].$ Then, in the conmutative diagram of Figure REF , $\\\\iota _3$ and $\\\\iota _4$ become identical maps, $\\\\tau _3=\\\\tau _4$ and the diagram becomes, with obvious conventions, that of Figure REF .\",\n \"Figure: Fibre Bundles for Dirac ParticlesThe homogeneous space $SL/[S^1]$ can be characterised as follows.\",\n \"Let ${\\\\bf P}_1(\\\\mathbb { C})$ be the complex projective space corresponding to $ \\\\mathbb { C}^2$ ( $i.e.$ the space $ {\\\\bf P}_1(\\\\mathbb { C}) $ consists of the one dimensional complex subspaces of $ {\\\\mathbb { C}}^2$ ).\",\n \"In ${\\\\cal H}^m\\\\times {\\\\bf P}_1(\\\\mathbb { C})$ we can consider the action of $ SL(2, C)$ given by $A\\\\,*\\\\,(H,[z])=(AHA^*,\\\\,[Az])$ where $[z]\\\\in {\\\\bf P}_1(\\\\mathbb { C})$ is the vector subspace generated by $z \\\\in \\\\ {\\\\mathbb { C}}^2-\\\\lbrace (0, 0) \\\\rbrace $ .\",\n \"We know that the partial action on ${\\\\cal H}^m$ is transitive.\",\n \"Now let us see that the complete action on ${\\\\cal H}^m\\\\times {\\\\bf P}_1(\\\\mathbb { C})$ is also transitive.\",\n \"Let $(K,[w])\\\\in {\\\\cal H}^m\\\\times {\\\\bf P}_1(\\\\mathbb {C}).$ Then, there exist $A\\\\in SL$ such that $A*(K,[w])=(mI,[w^{\\\\prime }]),$ for some $w^{\\\\prime }\\\\in {\\\\mathbb {C}}^2-\\\\lbrace (0,0)\\\\rbrace ,$ but there exist obviously an element $B$ of $SU(2)$ such that $\\\\left[B \\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right) \\\\right]=[w^{\\\\prime }],$ and we see that $(B^{-1}A)*(K,[w])=\\\\left( mI,\\\\left[ \\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right) \\\\right] \\\\right).$ Transitivity follows.\",\n \"On the other hand, the isotropy subgroup at $(mI,\\\\ [ \\\\left( \\\\begin{array}{c} 1 \\\\\\\\ 0 \\\\end{array}\\\\right) ])$ is $[S^1] $ , so that, since the action is transitive, one can identify $SL/[S^1]$ to ${\\\\cal H}^m\\\\times {\\\\bf P}_1(\\\\mathbb { C})$ .\",\n \"It is a well known fact that ${\\\\bf P}_1(\\\\mathbb {C})$ is diffeomorphic to the sphere $S^2.$ A diffeomorphism can be described as follows.\",\n \"Let $C^+$ be the future lightcone, composed by the hermitian matrices having 0 determinant and positive trace.\",\n \"The subset, $S_2,$ of $C^+$ composed by the elements whose trace is 2 is $S_2=\\\\lbrace h(x,y,z,1):1-x^2+y^2+z^2=0 \\\\rbrace $ that can be identified to the sphere $S^2,$ by means of $\\\\delta :(x,y,z)\\\\in S^2\\\\subset \\\\mathbb {R}^3 \\\\longleftrightarrow h(x,y,z,1)\\\\in S_2\\\\subset C^+.$ The map $\\\\beta _0:[z]\\\\in {\\\\bf P}_1(\\\\mathbb { C}) \\\\longrightarrow \\\\frac{2}{z^*\\\\,z}\\\\,z\\\\,z^* \\\\in S_2,$ is bijective.\",\n \"Its inverse is $\\\\beta _0^{-1}: C \\\\in S_2\\\\subset C^+ \\\\longrightarrow [w]\\\\in {\\\\bf P}_1(\\\\mathbb { C}),$ where $w$ is an eigenvector of $C$ corresponding to the eigenvalue 2.\",\n \"This is a consecuence of the fact that $\\\\begin{split} &\\\\left(\\\\frac{2}{z^*\\\\,z}\\\\,z\\\\,z^*\\\\right)z=2z\\\\\\\\&\\\\left(\\\\frac{2}{z^*\\\\,z}\\\\,z\\\\,z^*\\\\right)\\\\varepsilon \\\\overline{z}=0\\\\end{split} $ Thus $\\\\beta \\\\stackrel{def}{=} \\\\delta ^{-1}\\\\circ \\\\beta _0: {\\\\bf P}_1(\\\\mathbb { C}) \\\\rightarrow S^2,$ can be described by $\\\\beta ([z])=\\\\vec{u} \\\\ \\\\Longleftrightarrow \\\\ \\\\frac{2}{z^*\\\\,z}\\\\,z\\\\,z^*=h(\\\\vec{u},1).$ that is practical in order to use $\\\\beta .$ More practical in order to use $\\\\beta ^{-1}$ is $\\\\beta ([z])=\\\\vec{u} \\\\ \\\\Longleftrightarrow \\\\ \\\\ h(\\\\vec{u},-1)z=0.$ If $p_s$ and $p_n$ denote the stereographical projections from poles $s=(0,0,-1)$ and $n=(0,0,1)$ respectively, we have $p_s\\\\left(\\\\beta \\\\left[ \\\\left( \\\\begin{array}{c} z^1\\\\\\\\z^2 \\\\end{array}\\\\right)\\\\right]\\\\right)= \\\\frac{z^2}{z^1}\\\\ \\\\ \\\\ \\\\mathrm {if}\\\\ z^1\\\\ne 0,\\\\\\\\p_n\\\\left(\\\\beta \\\\left[\\\\left( \\\\begin{array}{c} z^1\\\\\\\\z^2 \\\\end{array}\\\\right)\\\\right]\\\\right)= \\\\frac{\\\\overline{z}^1}{\\\\overline{z}^2}\\\\ \\\\ \\\\ \\\\mathrm {if}\\\\ z^2\\\\ne 0.\",\n \"\\\\nonumber $ This can be used to prove easily the differentiability of $\\\\beta $ and $\\\\beta ^{-1}.\",\n \"$ Thus, it is possible to change in all that follows ${\\\\bf P}_1(\\\\mathbb {C})$ by $S^2,$ but, at this moment, I prefer to use ${\\\\bf P}_1( \\\\mathbb {C}).$ In order to describe Prewave Functions in this case , one can consider the trivialisation $(\\\\rho ,\\\\ {\\\\bf C}^4,z_0)$ , where $z_0={}^t\\\\hspace{0.0pt}(1, 0, 1, 0)$ and $\\\\rho $ is given by $\\\\rho (A)=\\\\left(\\\\begin{array}{cc}A&0 \\\\\\\\0&(A^*)^{-1}\\\\end{array}\\\\right)$ The orbit of $z_0$ is ${\\\\cal B}=\\\\left\\\\lbrace \\\\left( \\\\!\\\\begin{array}{c}w \\\\\\\\z\\\\end{array}\\\\!\",\n \"\\\\right)\\\\ :\\\\ w,\\\\ z \\\\in \\\\mathbb { C }^2,\\\\ z^*\\\\,w=1 \\\\right\\\\rbrace .\",\n \"$ This can be seen by consideration of the map $\\\\phi _0:\\\\left(\\\\begin{array}{c}w \\\\\\\\z\\\\end{array}\\\\right) \\\\in {\\\\cal B} \\\\longrightarrow \\\\left(w \\\\vert -\\\\varepsilon \\\\overline{z} \\\\right)\\\\in SL(2,\\\\bf C)$ (it is a diffeomorphism), and the fact that $\\\\rho (w|-\\\\varepsilon \\\\overline{z} )\\\\left(\\\\begin{array}{c}1 \\\\\\\\0\\\\\\\\1\\\\\\\\0\\\\end{array}\\\\right)=\\\\left(\\\\begin{array}{c}w \\\\\\\\z\\\\end{array}\\\\right).$ When one identifies $SL / Ker(C_{\\\\alpha _1})_{SL}$ to $\\\\cal B$ , and $SL / (G_{\\\\alpha _1} ) _{SL}$ to ${\\\\cal H}^m \\\\times {\\\\bf P}_1( \\\\mathbb {C})$ , by means of the preceeding actions, the canonical map between these homogeneous spaces becomes a map, ${\\\\mbox{$ r$}}$ , from $\\\\cal B$ onto ${\\\\cal H}^m \\\\times {\\\\bf P}_1(\\\\mathbb {C})$ .\",\n \"As a consecuence of (REF ), we have $(w \\\\ |\\\\,-\\\\varepsilon \\\\overline{z})* (mI,\\\\left[\\\\left( \\\\begin{array}{c}1 \\\\\\\\0\\\\end{array}\\\\right)\\\\right])={\\\\mathbf {r}}\\\\left(\\\\begin{array}{c}w \\\\\\\\z\\\\end{array}\\\\right)$ and this leads easily to ${{\\\\mbox{$ r$}}} \\\\left(\\\\!\\\\begin{array}{c} w\\\\\\\\ z\\\\end{array}\\\\!\",\n \"\\\\right)=(m(ww^*\\\\,-\\\\,\\\\varepsilon \\\\overline{zz^*} \\\\varepsilon ),\\\\, [w]).$ On the other hand, if we define $\\\\sigma (K,w)=\\\\frac{1}{\\\\sqrt{mw^*K^{-1}w}}\\\\left(w\\\\ \\\\ -\\\\frac{1}{m}K \\\\varepsilon \\\\overline{w}\\\\right),$ where the vertical bar separates the two columns of the $2\\\\times 2$ matrix, we have $\\\\sigma (K,w)*(mI,\\\\left[\\\\left( \\\\begin{array}{c}1 \\\\\\\\0\\\\end{array}\\\\right)\\\\right])=(K,[w]).$ Notice that if $z\\\\ \\\\mathrm {and}\\\\ z^{\\\\prime }$ are representatives of the same element of $ P^1(\\\\mathbb {C}),$ $\\\\sigma \\\\left(K, z \\\\right)$ and $\\\\sigma \\\\left(K, z^{\\\\prime } \\\\right)$ are in general different.\",\n \"Thus, since ${\\\\mathbf {r}}^{-1}(mI,\\\\left[\\\\left( \\\\begin{array}{c}1 \\\\\\\\0\\\\end{array}\\\\right)\\\\right])=\\\\lbrace s {\\\\left(\\\\begin{array}{c}1 \\\\\\\\0\\\\\\\\1\\\\\\\\0\\\\end{array}\\\\right)}:s\\\\in S^1 \\\\rbrace $ we have ${\\\\mathbf {r}}^{-1}(K,[w])=\\\\rho (\\\\sigma (K,w)) \\\\lbrace s \\\\left( {\\\\begin{array}{c}1 \\\\\\\\0\\\\\\\\1\\\\\\\\0\\\\end{array} }\\\\right):s\\\\in S^1 \\\\rbrace $ which leads to $ {{\\\\mbox{$ r$}}}^{-1}(K,\\\\ [a])=\\\\left\\\\lbrace s \\\\left(\\\\!\\\\begin{array}{c} I\\\\\\\\m\\\\,K^{-1}\\\\end{array}\\\\!\",\n \"\\\\right)\\\\frac{a}{\\\\sqrt{ma^*K^{-1}a}}:\\\\ s \\\\in {\\\\bf S}^1 \\\\right\\\\rbrace $ Also we have $P(K,\\\\ [w])\\\\,=P(\\\\sigma (K,w)G_{(\\\\alpha _1)_{SL}})=\\\\,-\\\\eta K.$ If $f$ is a continuous function with compact support in $\\\\cal B,$ and is $S^1$ -homogeneous of degree -1 (i.e.\",\n \"$f\\\\in {\\\\cal C}$ in the notation of section REF ) , the corresponding prewave function is $\\\\psi _f:(H,\\\\ K,\\\\ [a]) \\\\in H(2) \\\\times {\\\\cal H}^m \\\\times {{\\\\mbox{$ P_1$}}}( {\\\\bf C})\\\\ \\\\ \\\\mapsto \\\\ \\\\ f\\\\left(\\\\!\\\\begin{array}{c} w \\\\\\\\ z \\\\end{array} \\\\!\",\n \"\\\\right) e^{-i\\\\pi \\\\eta \\\\ Tr(K\\\\,\\\\varepsilon \\\\overline{H} \\\\varepsilon ) }\\\\ \\\\left(\\\\!\\\\begin{array}{c} w\\\\\\\\ z \\\\end{array} \\\\!\",\n \"\\\\right)$ where $\\\\left(\\\\!\\\\begin{array}{c} w\\\\\\\\ z \\\\end{array} \\\\!\",\n \"\\\\right) $ is arbitrary in ${\\\\mbox{$r$}}^{-1}(K,\\\\ [a])$ .\",\n \"When one considers the Dirac matrices in the representation $\\\\gamma ^4=\\\\left( \\\\begin{array}{cc}0&I \\\\\\\\I&0\\\\end{array} \\\\right)\\\\ \\\\ \\\\ ,\\\\gamma ^k=\\\\left( \\\\begin{array}{cc}0&-\\\\sigma _k \\\\\\\\\\\\sigma _k&0\\\\end{array} \\\\right)\\\\ \\\\ \\\\ ,k=1,\\\\ 2,\\\\ 3,$ a straightforward computation proves that that these prewave functions satisfy Dirac Equation $(\\\\gamma ^\\\\nu \\\\ \\\\partial _\\\\nu \\\\ -2 \\\\pi i \\\\eta m)\\\\ \\\\psi _f \\\\ =\\\\ 0.$ A sesquilinear form on ${\\\\bf C}^4$ whose value on ${\\\\cal B}$ is 1, is defined by $ \\\\Phi \\\\left( Z,\\\\ Z^\\\\prime \\\\right)=\\\\frac{1}{2} Z^* \\\\gamma ^4 Z.$ Thus, if $f,\\\\ f^{\\\\prime } \\\\in {\\\\cal C},$ the hermitian product of $\\\\psi _f$ and $\\\\psi _{f^\\\\prime }$ can be writen $\\\\langle \\\\psi _f,\\\\ \\\\psi _{f^\\\\prime } \\\\rangle =\\\\frac{1}{2} \\\\int _{{\\\\cal H}^m \\\\times P_1(C)}\\\\psi _f^*\\\\,\\\\gamma ^4\\\\,\\\\psi _{f^\\\\prime }\\\\,\\\\mu .$ To describe Wave Functions we need an invariant volume element on ${\\\\cal H}^m\\\\times {\\\\bf P}_1 (\\\\mathbb {C}).$ Let us consider the 5-form in ${\\\\cal H}^m \\\\times ( {\\\\mathbb {C}}^2-\\\\lbrace (0,0)\\\\rbrace )$ , given by $(\\\\mu _0)_{(K,\\\\ z)}=\\\\nu \\\\ \\\\wedge \\\\frac{(z^1\\\\,dz^2-z^2\\\\,dz^1)\\\\,\\\\wedge \\\\,{(\\\\overline{z^1}\\\\,d\\\\overline{z^2}-\\\\overline{z^2}\\\\,d\\\\overline{z^1})}}{(z^* {\\\\mbox{$\\\\varepsilon $}}\\\\overline{K} {\\\\mbox{$\\\\varepsilon $}} z)^2},$ where the $z^k$ are the two canonical projections of ${\\\\mathbb {C} \\\\rm }^2$ onto ${\\\\mathbb {C}}$ .\",\n \"This form is well defined since, for all $K\\\\in {\\\\cal H}^m, $ the hermitian product defined in ${\\\\mathbb {C}}^2$ by $\\\\langle x,y \\\\rangle =x^* \\\\overline{K} y$ is positive definite.\",\n \"This differential form projects to an invariant volume element, $\\\\mu $ , in ${\\\\cal H}^m \\\\times {\\\\mbox{$ P_1$}}(\\\\mathbb {C})$ .To prove this fact, one must proceed in many steps.\",\n \"Let us denote by $\\\\tau $ the canonical map $\\\\tau :(K,z)\\\\in {\\\\cal H}^m \\\\times ( {\\\\mathbb {C}}^2-\\\\lbrace (0,0)\\\\rbrace ) \\\\rightarrow (K,[z])\\\\in {\\\\cal H}^m \\\\times {\\\\mbox{$ P_1$}}(\\\\mathbb {C}).$ The triple ${\\\\cal H}^m \\\\times ( {\\\\mathbb {C}}^2-\\\\lbrace (0,0)\\\\rbrace )({\\\\cal H}^m \\\\times {\\\\mbox{$ P_1$}}(\\\\mathbb {C}), {\\\\mathbb {C}}-\\\\lbrace 0\\\\rbrace )$ is a principal fibre bundle with projection $\\\\tau .$ To prove that there exist a form, $\\\\mu ,$ such that $\\\\tau ^*\\\\mu =\\\\mu _0$ it is enought to see that $\\\\mu _0$ is invariant under the bundle action, what is obvious, and that $\\\\mu _0$ vanishes on vertical vectors, what can be proved as follows.\",\n \"The vertical vectors at $(K,z)$ are tangent at $t=0$ to the curves $\\\\gamma (t)=(K,\\\\lambda (t)z)$ with $\\\\lambda $ a $C^\\\\infty $ map from $\\\\mathbb {R}$ into $\\\\mathbb {C}$ such that $\\\\lambda (0)=1.$ Let $X_{(K,z)}$ be the tangent vector to $\\\\gamma $ at $0.$ We have $X_{(K,z)}=\\\\lambda ^\\\\prime (0)\\\\left(z^1 \\\\frac{\\\\partial }{\\\\partial z^1}+z^2 \\\\frac{\\\\partial }{\\\\partial z^2}\\\\right)+\\\\overline{\\\\lambda ^\\\\prime (0)}\\\\left(\\\\overline{z^1} \\\\frac{\\\\partial }{\\\\partial \\\\overline{z^1}}+\\\\overline{z^2} \\\\frac{\\\\partial }{\\\\partial \\\\overline{z^2}}\\\\right),$ and it is easily seen that $i_{X_{(K,z)}}{(\\\\mu _0)_{(K,z)}}=0.$ This finishes the proof of the existence of $\\\\mu .$ Let us denote $\\\\Vert p \\\\Vert ^2=\\\\sum _{i=1}^3 (p^i)^2$ and $q_1(p_1,p_2,p_3,z)&=&(h(p_1,p_2,p_3,(m^2+ \\\\Vert p \\\\Vert ^2 )^{\\\\frac{1}{2}} ),\\\\left[ \\\\left(\\\\begin{array}{c}1\\\\\\\\z \\\\end{array}\\\\right) \\\\right]) \\\\\\\\q_2(p_1,p_2,p_3,z)&=&(h(p_1,p_2,p_3,(m^2+ \\\\Vert p \\\\Vert ^2)^{\\\\frac{1}{2}} ),\\\\left[ \\\\left(\\\\begin{array}{c}z\\\\\\\\1 \\\\end{array}\\\\right) \\\\right]) \\\\nonumber $ for all $p_1,p_2,p_3,z\\\\in {\\\\mathbb {R}}^3 \\\\times \\\\mathbb {C}.$ The maps $q_1$ and $q_2$ are parametrizations, whose inverses are local charts that compose an atlas of ${\\\\cal H}^m \\\\times {\\\\mbox{$ P_1$}}(\\\\mathbb {C}).$ The local expressions of $\\\\mu $ in these charts can be obtained as the reciprocal image of $\\\\mu _0$ by the maps $\\\\sigma _1(p_1,p_2,p_3,z)=(h(p_1,p_2,p_3,(m^2+ {\\\\mbox{$\\\\sum _{i=1}^3$}} \\\\ (p^i)^2)^{\\\\frac{1}{2}} ),\\\\left(\\\\begin{array}{c}1\\\\\\\\z \\\\end{array}\\\\right) ),$ and $\\\\sigma _(p_1,p_2,p_3,z)=(h(p_1,p_2,p_3,(m^2+ {\\\\mbox{$\\\\sum _{i=1}^3$}} \\\\ (p^i)^2)^{\\\\frac{1}{2}} ),\\\\left(\\\\begin{array}{c}z\\\\\\\\1 \\\\end{array}\\\\right) ).$ These local expressions are given by $\\\\sigma _1^*\\\\mu _0=\\\\frac{\\\\nu \\\\wedge dz \\\\wedge d \\\\overline{z}}{2\\\\Re (z(p_1-ip_2))+(1-\\\\vert z \\\\vert ^2)p_3-(1+\\\\vert z \\\\vert ^2)(m^2+ \\\\Vert p \\\\Vert ^2 )^{\\\\frac{1}{2}}},$ $\\\\sigma _2^*\\\\mu _0=\\\\frac{\\\\nu \\\\wedge dz \\\\wedge d \\\\overline{z}}{2\\\\Re (z(p_1+ip_2))+(\\\\vert z \\\\vert ^2-1)p_3-(1+\\\\vert z \\\\vert ^2)(m^2+ \\\\Vert p \\\\Vert ^2 )^{\\\\frac{1}{2}}}.$ As a particular consequence, $\\\\mu $ is a volume element.\",\n \"Now, let us consider the action of $SL$ on ${\\\\cal H}^m \\\\times ( {\\\\mathbb {C}}^2-\\\\lbrace (0,0)\\\\rbrace )$ given by $A*(K,z)=(AKA^*,Az).$ We already know that $\\\\nu $ is invariant under the action on ${\\\\cal H}^m ,$ and it is not a difficult matter to see that $z^1\\\\,dz^2-z^2\\\\,dz^1,\\\\ {\\\\overline{z^1}\\\\,d\\\\overline{z^2}-\\\\overline{z^2}\\\\,d\\\\overline{z^1} } $ and $z^* {\\\\mbox{$\\\\varepsilon $}}\\\\overline{K} {\\\\mbox{$\\\\varepsilon $}} z$ are invariant under this action.\",\n \"As a consecuence, $\\\\mu _0$ is invariant under the same action.\",\n \"It follows that $\\\\mu $ is an invariant volume element on ${\\\\cal H}^m \\\\times {\\\\mbox{$ P_1$}}(\\\\mathbb {C}).$ The wave functions have the form $\\\\tilde{\\\\psi }_f(X)=\\\\int _{{\\\\cal H}^m \\\\times P_1(C)}\\\\psi _f(h(X),\\\\cdot ,\\\\cdot )\\\\ \\\\mu $ and also satisfies Dirac Equation $(\\\\gamma ^\\\\nu \\\\ \\\\partial _\\\\nu \\\\ -2 \\\\pi i \\\\eta m)\\\\ \\\\tilde{\\\\psi }_f \\\\ =\\\\ 0.$\"\n ],\n [\n \"Wave functions for $T>1$ .\",\n \"Now, let us consider a particle whose movement space is the coadjoint orbit of $\\\\alpha _T=\\\\left\\\\lbrace \\\\ \\\\frac{iT}{8\\\\pi }\\\\left( \\\\begin{array}{cc} 1&0 \\\\\\\\ 0&-1\\\\end{array} \\\\right) \\\\ ,\\\\ \\\\eta \\\\,m\\\\,I\\\\right\\\\rbrace ,$ where $T\\\\in {\\\\bf Z}^+,\\\\ m \\\\in {\\\\bf R}^+, \\\\eta =\\\\pm 1.$ The case of the preceeding section is the particular one given by $T=1.$ For all $T$ the orbit is a quantizable, not $\\\\bf R \\\\rm $ -quantizable orbit, of the type 5.\",\n \"Now we have ${\\\\mbox{$G_{\\\\alpha _T}$}} = \\\\lbrace \\\\ (\\\\ \\\\left( \\\\begin{array}{cc}z&0 \\\\\\\\0&\\\\overline{z}\\\\end{array} \\\\right), \\\\ hI ): \\\\ z\\\\in S^1,\\\\ h\\\\in {\\\\mbox{$\\\\bf R \\\\rm $}} \\\\rbrace .$ The unique homomorphism from $G_{\\\\alpha _T}$ onto ${\\\\bf S}^1$ whose differential is $\\\\alpha _T$ is given by ${\\\\mbox{$C_{\\\\alpha _T}$}} (\\\\ \\\\left( \\\\begin{array}{cc}e^{2\\\\pi i\\\\phi }&0 \\\\\\\\0&e^{-2\\\\pi i\\\\phi }\\\\end{array} \\\\right) ,\\\\ hI)=e^{2\\\\pi i(\\\\phi T-\\\\eta m h)}.$ Then $\\\\begin{array}{l}\\\\vspace{7.22743pt}{{\\\\mbox{$(G_{\\\\alpha _T } ) _{SL}$}}=\\\\lbrace \\\\ \\\\left( \\\\begin{array}{cc}z&0 \\\\\\\\0&\\\\overline{z}\\\\end{array} \\\\right):\\\\ z\\\\in S^1\\\\rbrace }, \\\\\\\\\\\\vspace{7.22743pt}{{\\\\mbox{$(C_{\\\\alpha _T})_{SL}$}}(\\\\ \\\\left( \\\\begin{array}{cc}z&0 \\\\\\\\0&\\\\overline{z}\\\\end{array} \\\\right))=z^T}, \\\\\\\\\\\\vspace{7.22743pt}SL_1 \\\\cap SL_2 ={\\\\mbox{$(G_{\\\\alpha _T} ) _{SL}$}}, \\\\\\\\\\\\vspace{7.22743pt}SL_1 \\\\cap Ker\\\\,(C_{\\\\alpha _T})_{SL}=Ker\\\\,(C_{\\\\alpha _T})_{SL}=\\\\lbrace \\\\ \\\\left( \\\\begin{array}{cc}z&0 \\\\\\\\0&\\\\overline{z}\\\\end{array} \\\\right):\\\\ z\\\\in \\\\@root T \\\\of {\\\\mathbf {1}}\\\\rbrace .\",\n \"\\\\\\\\\\\\end{array}$ where $\\\\@root T \\\\of {\\\\mathbf {1}}$ is the subgroup of ${\\\\mathbb {C}}^*$ composed by the roots of order $T$ of 1.\",\n \"The homogeneous space SL/${\\\\mbox{$(G_{\\\\alpha _T} ) _{SL}$}}$ , is the same as in the preceeding section so that it can be identified to ${\\\\cal H}^m\\\\times {\\\\bf P}_1(\\\\mathbb {C})$ , and consider on it the invariant volume element $\\\\mu .$ In order to give the wave functions in the case of arbitrary ${T}$ , the following results are useful.\",\n \"Let ${\\\\beta },\\\\ \\\\beta ^\\\\prime $ be quantizable elements of $\\\\underline{G}^*$ with ${\\\\beta ^\\\\prime }$ not R-quantizable, $(G_{\\\\beta })_{SL}=(G_{{\\\\beta ^\\\\prime }})_{SL}$ , $(C_{\\\\beta })_{SL} =\\\\left((C_{{\\\\beta ^\\\\prime }})_{SL}\\\\right)^T,$ where $T \\\\in {\\\\bf Z}^+$ .\",\n \"Notice that, as a consecuence of the fact that $\\\\beta ^{\\\\prime }$ is not ${\\\\mathbb {R} }$ -quantizable, we must have $(C_{\\\\beta ^\\\\prime })_{SL}((G_{{\\\\beta ^\\\\prime }})_{SL})=S^1.$ If $(\\\\rho , L, z_0)$ is a trivialization of $C_{\\\\beta ^\\\\prime }$ , we consider the triple $(\\\\rho ^{\\\\otimes T},\\\\ L^ {\\\\otimes T},\\\\ z_0^ {\\\\otimes T}),$ where $L^ {\\\\otimes T}=L \\\\otimes \\\\stackrel{(T}{\\\\cdots } \\\\nolinebreak \\\\otimes L,\\\\ z_0^{\\\\otimes T}=z_0 \\\\otimes \\\\stackrel{(T}{\\\\cdots } \\\\otimes z_0$ , and $\\\\rho ^{\\\\otimes T}$ is the representation such that $\\\\rho ^{\\\\otimes T}(A)(z_1 \\\\otimes \\\\cdots \\\\otimes z_T)=\\\\rho (A)(z_1) \\\\otimes \\\\cdots \\\\otimes \\\\rho (A)(z_T).$ In lemma 6.1 of [8] I have proved that, under these circunstances, if $(G_{\\\\beta })_{SL}$ is connected, $(\\\\rho ^{\\\\otimes T},\\\\ L^ {\\\\otimes T},\\\\ z_0^ {\\\\otimes T})$ is a trivialization of $C_\\\\beta $ .\",\n \"Let us assume that $(\\\\rho ^{\\\\otimes T},\\\\ L^ {\\\\otimes T},\\\\ z_0^ {\\\\otimes T})$ is a trivialization of $C_\\\\beta $ and let ${\\\\cal B}_T$ be the orbit of $ z_0^{\\\\otimes T}$ .\",\n \"The pullback by $z \\\\in {\\\\cal B} \\\\mapsto z^{\\\\otimes T} \\\\in {\\\\cal B}_T$ , establishes a one to one map from the set of the $S^1$ -homogeneous functions of degree -1 on $ {\\\\cal B}_T$ , onto the set of the ${S^1}$ -homogeneous functions of degree -T on ${\\\\cal B}$ .\",\n \"If $f$ is one of these functions, the corresponding prewave function of particles corresponding to $\\\\beta $ has the form ${\\\\mbox{$\\\\psi $}}_f(H,\\\\ m)\\\\,=\\\\,f(z)\\\\,e^{i\\\\pi Tr(P(m)\\\\,\\\\varepsilon \\\\overline{H} \\\\varepsilon )}z^{\\\\otimes T}$ where $z \\\\in { \\\\bf r}^{-1}(m).$ Now we apply these results to the particles of type 5 with $T> 1.$ Let $\\\\beta ^{\\\\prime }=\\\\alpha _1$ and $\\\\beta =\\\\alpha _T.$ The remark just made leads to the following Prewave Function $\\\\psi _f:(H,\\\\ K,\\\\ [a]) \\\\in H(2) \\\\times {\\\\cal H}^m \\\\times {{\\\\mbox{$ P_1$}}}( {\\\\bf C})\\\\ \\\\ \\\\mapsto \\\\ \\\\ f\\\\left(\\\\!\\\\!\\\\begin{array}{c} w \\\\\\\\ z \\\\end{array} \\\\!\",\n \"\\\\!\",\n \"\\\\right) e^{-i\\\\pi \\\\eta \\\\ Tr(K\\\\,\\\\varepsilon \\\\overline{H} \\\\varepsilon ) }\\\\ \\\\left(\\\\!\\\\!\\\\begin{array}{c} w\\\\\\\\ z \\\\end{array} \\\\!\\\\!\",\n \"\\\\right)^{\\\\otimes T}$ where $\\\\left(\\\\!\\\\!\\\\begin{array}{c} w\\\\\\\\ z \\\\end{array} \\\\!\\\\!\",\n \"\\\\right) $ is arbitrary in ${\\\\mbox{$r$}}^{-1}(K,\\\\ [a])$ , and f is a function on ${\\\\cal B}$ , $C^\\\\infty ,$ with compact support and homogeneous of degree -T under multiplication by complex numbers of modulus one.\",\n \"The Wave Functions are obtained by integration as usual.\"\n ],\n [\n \"Movement Space for massive particles with $T\\\\ge 1$ .\",\n \"We consider the action of $G$ on ${\\\\cal H}^m \\\\times {\\\\bf P}_1(\\\\mathbb {C})\\\\times \\\\mathbb {R}^3$ given by $(A,H)*(K,[z],\\\\overrightarrow{y})=(AKA^*,[Az], \\\\overrightarrow{x}(A,H,\\\\overrightarrow{y},K))$ where $\\\\overrightarrow{x}(A,H,\\\\overrightarrow{y},K)$ is given by $h(\\\\overrightarrow{x}(A,H,\\\\overrightarrow{y},K),0)=Ah(\\\\overrightarrow{y},0)A^*+H+\\\\ell (A,H,\\\\overrightarrow{y},K) A\\\\frac{K}{m}A^* \\\\\\\\\\\\ell (A,H,\\\\overrightarrow{y},K)=\\\\frac{-mTr(Ah(\\\\overrightarrow{y},0)A^*+H)}{Tr( AKA^*)}.$ This is a transitive action.\",\n \"The isotropy subgroup at $(mI,[{}^t(1,0)], \\\\overrightarrow{0}),$ is found to be $G_{\\\\alpha _T}.$ Thus, the map $\\\\lambda :(A,H) G_{\\\\alpha _T}\\\\in G/G_{\\\\alpha _T} \\\\leftrightarrow (A,H)*\\\\left(mI,\\\\left[\\\\left(\\\\begin{array}{c}1 \\\\\\\\0\\\\end{array}\\\\right)\\\\right], \\\\overrightarrow{0}\\\\right)\\\\in {\\\\cal H}^m \\\\times {\\\\bf P}_1(\\\\mathbb {C})\\\\times \\\\mathbb {R}^3$ enables us to identify the coadjoint orbit with ${\\\\cal H}^m \\\\times {\\\\bf P}_1(\\\\mathbb {C})\\\\times \\\\mathbb {R}^3.$ A parametrization whose image is a neighborhood of $(mI,[{}^t\\\\hspace{0.0pt}(1,0)], \\\\overrightarrow{0}),$ is $\\\\Gamma _s:(\\\\overrightarrow{k},\\\\overrightarrow{x},z)\\\\in \\\\mathbb {R}^3 \\\\times \\\\mathbb {R}^3\\\\times \\\\mathbb {C}\\\\rightarrow &&\\\\left(mh(\\\\overrightarrow{k},k_4),\\\\left[\\\\left(\\\\begin{array}{c}1 \\\\\\\\z\\\\end{array}\\\\right)\\\\right], \\\\overrightarrow{x}\\\\right)\\\\\\\\&&\\\\in {\\\\cal H}^m \\\\times {\\\\bf P}_1(\\\\mathbb {C})\\\\times \\\\mathbb {R}^3.\\\\nonumber $ where $k_4=\\\\sqrt{1+k_1^2+k_2^2+k_3^2}.$ The corresponding local chart is given by $\\\\Gamma _s^{-1}:\\\\left(K,\\\\left[\\\\left(\\\\begin{array}{c}z^1 \\\\\\\\z^2\\\\end{array}\\\\right)\\\\right],\\\\overrightarrow{x}\\\\right)\\\\in \\\\lbrace z^1\\\\ne 0 \\\\rbrace \\\\rightarrow \\\\left(\\\\frac{1}{m}h^{-1}(K-Tr(K)I),\\\\overrightarrow{x},\\\\frac{z^2}{z^1}\\\\right)$ Another parametrization is $\\\\Gamma _n:(\\\\overrightarrow{k},\\\\overrightarrow{x},z)\\\\in \\\\mathbb {R}^3 \\\\times \\\\mathbb {R}^3\\\\times \\\\mathbb {C}\\\\rightarrow &&\\\\left(mh(\\\\overrightarrow{k},k_4),\\\\left[\\\\left(\\\\begin{array}{c}z \\\\\\\\1\\\\end{array}\\\\right)\\\\right], \\\\overrightarrow{x}\\\\right)\\\\\\\\&&\\\\in {\\\\cal H}^m \\\\times {\\\\bf P}_1(\\\\mathbb {C})\\\\times \\\\mathbb {R}^3.\\\\nonumber $ with $k_4=\\\\sqrt{1+k_1^2+k_2^2+k_3^2},$ whose inverse is given by $\\\\Gamma _n^{-1}:\\\\left(K,\\\\left[\\\\left(\\\\begin{array}{c}z^1 \\\\\\\\z^2\\\\end{array}\\\\right)\\\\right],\\\\overrightarrow{x}\\\\right)\\\\in \\\\lbrace z^2\\\\ne 0 \\\\rbrace \\\\rightarrow \\\\left(\\\\frac{1}{m}h^{-1}(K-Tr(K)I),\\\\overrightarrow{x},\\\\frac{z^1}{z^2}\\\\right) \\\\nonumber $ Obviously, these two charts compose an atlas.\",\n \"The identification of ${\\\\cal H}^m \\\\times {\\\\bf P}_1(\\\\mathbb {C})\\\\times \\\\mathbb {R}^3$ with the coadjoint orbit is given by $(A,H)*((mI,[{}^t(1,0)], \\\\overrightarrow{0})\\\\longleftrightarrow Ad^*_{(A,H)}\\\\alpha _T.$ Now, let $\\\\left(K,\\\\left[ z\\\\right],\\\\overrightarrow{x}\\\\right)\\\\in {\\\\cal H}^m \\\\times {\\\\bf P}_1(\\\\mathbb {C})\\\\times \\\\mathbb {R}^3.$ If $\\\\sigma \\\\left(K, w \\\\right)$ is given by (REF ), then $\\\\left(\\\\sigma \\\\left(K, z \\\\right),h(\\\\overrightarrow{x},0)\\\\right)*(mI,[{}^t(1,0)], \\\\overrightarrow{0})=\\\\left(K,\\\\left[ z\\\\right],\\\\overrightarrow{x}\\\\right),$ so that $\\\\left(K,\\\\left[ z\\\\right],\\\\overrightarrow{x}\\\\right)$ must be identified to $Ad^*_{\\\\left(\\\\sigma \\\\left(K, z \\\\right),h(\\\\overrightarrow{x},0)\\\\right)}\\\\cdot \\\\alpha _T.$ Now let $F$ be a dynamical variable on the coadjoint orbit.\",\n \"$F$ becomes a function on ${\\\\cal H}^m \\\\times {\\\\bf P}_1(\\\\mathbb {C})\\\\times \\\\mathbb {R}^3$ .\",\n \"To give explicit expresions of these functions, is now preferable to use the sphere $S^2$ instead of ${\\\\ bf P}_1(\\\\mathbb {C}) $ thus writing $F(K,[z],\\\\vec{x}) =F( Ad^*_{\\\\left(\\\\sigma \\\\left(K, z \\\\right),h(\\\\overrightarrow{x},0)\\\\right)}\\\\cdot \\\\alpha _T)= F(K,\\\\vec{u},\\\\vec{x}),$ where $\\\\vec{u}\\\\in S^2$ is related to $[z]\\\\in \\\\mathbf {P_1(\\\\mathbb {C})}$ by (c.f.\",\n \"(REF )) $\\\\frac{2}{z^*z}zz^*=h(\\\\vec{u},1).$ Let us evaluate $Ad^*_{\\\\left(\\\\sigma \\\\left(K, z \\\\right),h(\\\\vec{x},0)\\\\right)}\\\\cdot \\\\alpha _T.$ To do that computation we need some of the equations (REF ) to (REF ), and $(\\\\sigma (K,z))^{-1}=\\\\frac{1}{\\\\sqrt{mz^*K^{-1}z}} \\\\left( \\\\frac{ mz^*K^{-1}}{ z \\\\varepsilon }\\\\right),$ where the horizontal line separates the two files of the $2\\\\times 2$ matrix.\",\n \"This leads to $Ad^*_{\\\\left(\\\\sigma \\\\left(K, z \\\\right),h(\\\\vec{x},0)\\\\right)}\\\\cdot \\\\alpha _T&=&\\\\left\\\\lbrace \\\\left[ -\\\\frac{T}{8\\\\pi }\\\\frac{1}{\\\\langle k_4\\\\vec{ u}-\\\\vec{ k},\\\\vec{u} \\\\rangle }h(\\\\vec{k}\\\\times \\\\vec{u},0)+\\\\frac{\\\\eta m}{2}k_4h(\\\\vec{x},0) \\\\right]+ \\\\right.\",\n \"\\\\\\\\ \\\\nonumber &+&i \\\\left[\\\\frac{T}{8 \\\\pi } \\\\frac{1}{\\\\langle k_4\\\\vec{ u}-\\\\vec{ k},\\\\vec{u} \\\\rangle }h( k_4\\\\vec{ u}-\\\\vec{ k},0)+ \\\\frac{\\\\eta m}{2} h(\\\\vec{k}\\\\times \\\\vec{x},0)\\\\right], \\\\\\\\ \\\\nonumber && \\\\left.\",\n \"\\\\eta m h(\\\\vec{k},k_4) \\\\right\\\\rbrace ,$ where $h(\\\\vec{k},k_4)=\\\\frac{1}{m}K.$ Thus, using (REF ), we obtain $\\\\begin{split}P(mh(\\\\vec{k},k_4),\\\\vec{u},\\\\vec{x})&=-\\\\eta m h(\\\\vec{k},k_4),\\\\\\\\\\\\vec{ l}(mh(\\\\vec{k},k_4),\\\\vec{u},\\\\vec{x})&=\\\\frac{T}{4\\\\pi }\\\\frac{k_4\\\\vec{u}-\\\\vec{k}}{k_4-\\\\langle \\\\vec{ k},\\\\overrightarrow{u} \\\\rangle }+\\\\eta m \\\\vec{k}\\\\times \\\\vec{x},\\\\\\\\\\\\vec{g}(mh(\\\\vec{k},k_4),\\\\vec{u},\\\\vec{x})&=\\\\frac{T}{4\\\\pi }\\\\frac{\\\\vec{k}\\\\times \\\\vec{u}}{k_4-\\\\langle \\\\vec{ k},\\\\overrightarrow{u} \\\\rangle }-\\\\eta m k_4 \\\\vec{x}.\\\\end{split}$ Notice that $k_4-\\\\langle \\\\vec{ k},\\\\overrightarrow{u} \\\\rangle =\\\\langle k_4 \\\\overrightarrow{u} - \\\\vec{ k},\\\\overrightarrow{u} \\\\rangle .$ If we denote $P(K,\\\\vec{u},\\\\vec{x}),\\\\vec{l}(K,\\\\vec{u},\\\\vec{x})$ and $\\\\vec{g}(K,\\\\vec{u},\\\\vec{x})$ simply by $P,\\\\ \\\\vec{l}$ and $\\\\vec{g}$ respectively when no danger of confusion exist, and $\\\\vec{P}=(P^1,P^2,P^3),$ we have the following relations between these dynamical variables $\\\\vec{ l}&=&\\\\frac{T}{4\\\\pi }\\\\frac{P^4\\\\vec{u}-\\\\vec{P}}{P^4-\\\\langle \\\\vec{ P},\\\\vec{u} \\\\rangle }+ \\\\vec{x}\\\\times \\\\vec{P},\\\\\\\\ \\\\vec{g}&=&\\\\frac{T}{4\\\\pi }\\\\frac{\\\\vec{P}\\\\times \\\\vec{u}}{ P^4-\\\\langle \\\\vec{ P},\\\\vec{u} \\\\rangle }+P^4\\\\vec{x}$ The value of the Pauli-Lubanski fourvector at $(mh(\\\\vec{k},k_4),\\\\vec{u},\\\\vec{x})$ can be calculated using (REF ) , (),(REF ),...,() and is found to be $ \\\\begin{split}\\\\overrightarrow{W}&=P^4\\\\frac{T}{4\\\\pi }\\\\frac{P^4\\\\vec{u}-\\\\vec{P}}{\\\\langle P^4\\\\vec{u}-\\\\vec{P},\\\\vec{u} \\\\rangle }=-\\\\eta m k_4\\\\frac{T}{4\\\\pi } \\\\frac{k_4\\\\vec{u}-\\\\vec{k}}{\\\\langle k_4\\\\vec{u}- \\\\vec{ k},\\\\vec{u} \\\\rangle } \\\\\\\\W^4&=\\\\frac{T}{4\\\\pi }\\\\frac{\\\\langle P^4\\\\vec{u}-\\\\vec{P},\\\\vec{P} \\\\rangle }{\\\\langle P^4\\\\vec{u}-\\\\vec{P},\\\\vec{u} \\\\rangle }= -\\\\eta m\\\\frac{T}{4\\\\pi } \\\\frac{\\\\langle k_4\\\\vec{u}-\\\\vec{k},\\\\vec{k} \\\\rangle }{\\\\langle k_4\\\\vec{u}- \\\\vec{ k},\\\\vec{u} \\\\rangle },\\\\end{split}$ so that we can also write $\\\\vec{ l}=\\\\frac{1}{P^4}\\\\overrightarrow{W}+ \\\\vec{x}\\\\times \\\\vec{P} .$\"\n ],\n [\n \"Contact manifold for Dirac particles\",\n \"Now we pay attention to the contact manifold in the case $T=1$ .\",\n \"We consider the action of $G$ on ${\\\\cal B}\\\\times {\\\\mathbb {R}}^3$ given by $(A,H)*(w,z,\\\\overrightarrow{y})=\\\\left( e^{2\\\\pi i\\\\eta m \\\\ell }Aw,e^{2\\\\pi i\\\\eta m \\\\ell }(A^*)^{-1}z,\\\\overrightarrow{x}(A,H,w,z,\\\\overrightarrow{y})\\\\right)$ where $h(\\\\overrightarrow{x}(A,H,w,z,\\\\overrightarrow{y}),0)= Ah(\\\\overrightarrow{y},0)A^*+H+\\\\ell A(ww^*-\\\\varepsilon \\\\overline{zz^*} \\\\varepsilon )A^*.$ Obviously, we have $\\\\ell =-\\\\frac{Tr(Ah(\\\\vec{x},0)A^*+H)}{Tr( A(ww^*-\\\\varepsilon \\\\overline{zz^*} \\\\varepsilon )A^*)}.$ This is a transitive action and the isotropy subgroup at $\\\\left(\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right),\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right), \\\\vec{0}\\\\right)$ is $Ker\\\\,C_{\\\\alpha _1},$ so that we can identify $G/Ker\\\\,C_{\\\\alpha _1}$ with ${\\\\cal B}\\\\times {\\\\mathbb {R}}^3$ by means of $(A,H)Ker\\\\,C_{\\\\alpha _1} \\\\longleftrightarrow (A,H)* \\\\left(\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right),\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right), \\\\vec{0}\\\\right).$ When $w\\\\in {\\\\mathbb {C}}^2-\\\\lbrace 0 \\\\rbrace ,$ the $z$ such that $(w,z)\\\\in {\\\\cal B}$ compose the set $\\\\left\\\\lbrace \\\\frac{1}{w^*w}(w+y\\\\varepsilon \\\\overline{w}): y\\\\in {\\\\mathbb {C}} \\\\right\\\\rbrace .$ Thus, the map $\\\\Delta _0:(w,y)\\\\in ({\\\\mathbb {C}}^2-\\\\lbrace 0 \\\\rbrace )\\\\times {\\\\mathbb {C}} \\\\longrightarrow \\\\left( w , z(w,y)\\\\right)\\\\in {\\\\cal B},$ where $z(w,y)=\\\\frac{1}{w^*w}(w+y\\\\varepsilon \\\\overline{w}),$ is a bijection, and in fact a diffeomorphism.\",\n \"Its inverse is given by $(\\\\Delta _0)^{-1}:(w,z)\\\\in {\\\\cal B} \\\\longrightarrow (w, {}^tw \\\\varepsilon z)\\\\in ({\\\\mathbb {C}}^2-\\\\lbrace 0 \\\\rbrace )\\\\times {\\\\mathbb {C}}.$ Then, $(\\\\Delta _0)^{-1}$ is a global complex coordinate system of ${\\\\cal B},$ and $\\\\Delta :(w,y,\\\\vec{x})\\\\in ({\\\\mathbb {C}}^2-\\\\lbrace 0 \\\\rbrace )\\\\times {\\\\mathbb {C}}\\\\times {\\\\mathbb {R}}^3 \\\\longrightarrow \\\\left( w , z(w,y),\\\\vec{x}\\\\right)\\\\in {\\\\cal B}\\\\times {\\\\mathbb {R}}^3,$ is a parametrization of the Contact Manifold defined in an an open subset of ${\\\\mathbb {C}}^3 \\\\times {\\\\mathbb {R}}^3.$ Notice that, since we know a diffeomorphism, $\\\\phi _0,$ from ${\\\\cal B}$ onto $SL(2,\\\\mathbb {C})$ (c.f.\",\n \"(REF )), we obtain also a complex parametrization of this group by means of $\\\\phi _0 \\\\circ \\\\Delta _0:(w,y)\\\\in ({\\\\mathbb {C}}^2-\\\\lbrace 0 \\\\rbrace )\\\\times {\\\\mathbb {C}} \\\\longrightarrow \\\\left( w\\\\ \\\\ \\\\frac{1}{w^*w}({\\\\overline{y}}w-\\\\varepsilon \\\\overline{w}\\\\right)\\\\in SL(2,\\\\mathbb {C}).$ The inverse of $\\\\phi _0 \\\\circ \\\\Delta _0$ is the global complex chart of $SL(2,\\\\mathbb {C})$ given by $\\\\phi :(w|v)\\\\in SL(2,\\\\mathbb {C}) \\\\longrightarrow (w, {}^tw \\\\overline{v})\\\\in ({\\\\mathbb {C}}^2-\\\\lbrace 0 \\\\rbrace )\\\\times {\\\\mathbb {C}}.$ The canonical map of $G$ onto $G/Ker\\\\,C_{\\\\alpha _1}$ becomes $(A,H)\\\\in G \\\\longleftrightarrow (A,H)* \\\\left(\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right),\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right), \\\\vec{0}\\\\right),$ and the map $\\\\Sigma :(w,z,\\\\vec{x})\\\\in {\\\\cal B}\\\\times {\\\\mathbb {R}}^3 \\\\longrightarrow \\\\left( \\\\left( w\\\\ \\\\ -\\\\varepsilon \\\\overline{z}\\\\right),h(\\\\vec{x},0)\\\\right)\\\\in G,$ is a section of this canonical map , as a consecuence of the fact that $\\\\Sigma (w,z,\\\\vec{x})*\\\\left(\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right),\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right), \\\\vec{0} \\\\right)=(w,z,\\\\vec{x}).$ The canonical map from $G/Ker\\\\,C_\\\\alpha $ onto $G/G_\\\\alpha $ becomes $\\\\tilde{\\\\mathbf {r}}&:&(w,z,\\\\vec{x})\\\\in {\\\\cal B}\\\\times {\\\\mathbb {R}}^3 \\\\longrightarrow \\\\Sigma (w,z,\\\\vec{x})* (mI,[\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array} \\\\right)],\\\\vec{0})=\\\\\\\\&=&(m(ww^*-\\\\varepsilon \\\\overline{zz^*}\\\\varepsilon ),[w],\\\\vec{x})\\\\in {\\\\cal H}^m\\\\times {\\\\mathbf {P}_1(\\\\mathbb {C})}\\\\times {\\\\mathbb {R}}^3,$ i.e.\",\n \"coincides with ${\\\\bf r}\\\\times Id_{{\\\\mathbb {R}}^3}.$ The Linear Momentum, Angular Momentum and Pauli-Lubanski fourvector, whose descriptions in the symplectic manifold are given by (REF ) and (REF ), when composed with $\\\\tilde{\\\\mathbf {r}},$ give functions $P_c,\\\\ \\\\vec{l}_c,\\\\vec{g}_c,W_c,$ on the Contact Manifold, that are the translation of these dynamical variables to this homogeneous space.\",\n \"In order to give explicit expressions for these functions ($c.f.$ (REF )), let us denote $\\\\tilde{\\\\mathbf {r}}(w,z,\\\\vec{x})=(m\\\\,h(\\\\vec{k},k_4),\\\\vec{u}, \\\\vec{x})\\\\in {\\\\cal H}^m \\\\times S^2 \\\\times {\\\\mathbb {R}}^3$ we have $h(\\\\vec{u},1)= \\\\frac{2}{w^*w}w\\\\,w^*,$ and $h(\\\\vec{k},k_4)=ww^*-\\\\varepsilon \\\\overline{zz^*}\\\\varepsilon ,$ but $-h(\\\\vec{u},1)\\\\varepsilon \\\\overline{h(\\\\vec{k},k_4)}\\\\varepsilon =h(k_4\\\\vec{u}-\\\\vec{k},k_4-\\\\langle \\\\vec{k},\\\\vec{u} \\\\rangle ) + i h(\\\\vec{k} \\\\times \\\\vec{u},0)$ and $-h(\\\\vec{u},1)\\\\varepsilon \\\\overline{h(\\\\vec{k},k_4)}\\\\varepsilon =- \\\\frac{2}{w^*w}ww^*\\\\varepsilon \\\\overline{(ww^*-\\\\varepsilon \\\\overline{zz^*}\\\\varepsilon )}\\\\varepsilon =\\\\frac{2}{w^*w}w\\\\,z^*,$ leads to $h(k_4\\\\vec{u}-\\\\vec{k},k_4-\\\\langle \\\\vec{k},\\\\vec{u} \\\\rangle ) + i h(\\\\vec{k} \\\\times \\\\vec{u},0)=\\\\frac{2}{w^*w}w\\\\,z^*,$ so that $\\\\begin{split}&h(k_4\\\\vec{u}-\\\\vec{k},k_4-\\\\langle \\\\vec{k},\\\\vec{u} \\\\rangle ) =\\\\frac{1}{w^*w}(w\\\\,z^*+zw^*),\\\\\\\\&h(\\\\vec{k} \\\\times \\\\vec{u},0)=i\\\\frac{1}{w^*w}(zw^*-w\\\\,z^*)\\\\end{split}$ and $\\\\begin{split}&\\\\langle k_4\\\\vec{u}-\\\\vec{k},\\\\vec{u} \\\\rangle =k_4-\\\\langle \\\\vec{k},\\\\vec{u} \\\\rangle =\\\\frac{1}{w^*w}\\\\\\\\&h\\\\left(\\\\frac{k_4\\\\vec{u}-\\\\vec{k}}{k_4-\\\\langle \\\\vec{k},\\\\vec{u} \\\\rangle },1\\\\right)=w\\\\,z^*+zw^*\\\\\\\\&h\\\\left(\\\\frac{\\\\vec{k} \\\\times \\\\vec{u}}{k_4-\\\\langle \\\\vec{k},\\\\vec{u} \\\\rangle },0\\\\right)=i(zw^*-w\\\\,z^*)\\\\end{split}$ Thus $\\\\begin{split}&P_c(w,z,\\\\vec{x})=-\\\\eta m\\\\,(ww^*-\\\\varepsilon \\\\overline{zz^*}\\\\varepsilon )\\\\\\\\&h(\\\\vec{l}_c(w,z,\\\\vec{x}),0)=\\\\frac{T}{4\\\\pi }(wz^*+zw^*-I)+h(\\\\vec{x} \\\\times \\\\vec{P}_c,0)\\\\\\\\&h(\\\\vec{g}_c(w,z,\\\\vec{x}),0)=\\\\frac{iT}{4\\\\pi } (zw^*-wz^*)+P^4 h(\\\\vec{x},0)\\\\\\\\&h(\\\\vec{W}_c(w,z,\\\\vec{x}),0)=P^4\\\\frac{T}{4\\\\pi }(wz^*+zw^*-I)\\\\\\\\&W^4_c(w,z,\\\\vec{x})=\\\\frac{-\\\\eta m T}{8\\\\pi }(w^*w-z^*z)\\\\end{split}$ The formula for $W_c^4$ can be obtained as follows $\\\\begin{split}&\\\\langle \\\\vec{l},\\\\vec{P} \\\\rangle \\\\circ \\\\mathbf {\\\\tilde{r}}=\\\\left\\\\langle \\\\frac{T}{4\\\\pi }\\\\frac{k_4\\\\vec{u}-\\\\vec{k}}{\\\\langle k_4\\\\vec{u}-\\\\vec{k},\\\\vec{u} \\\\rangle },\\\\vec{P} \\\\right\\\\rangle \\\\circ \\\\mathbf {\\\\tilde{r}}= \\\\\\\\&=\\\\frac{1}{2}Tr\\\\left(h\\\\left(\\\\frac{T}{4\\\\pi }\\\\frac{k_4\\\\vec{u}-\\\\vec{k}}{\\\\langle k_4\\\\vec{u}-\\\\vec{k},\\\\vec{u} \\\\rangle },0\\\\right)P \\\\right) \\\\circ \\\\mathbf {\\\\tilde{r}}=\\\\\\\\ &=\\\\frac{1}{2}Tr\\\\left(\\\\left(\\\\frac{T}{4\\\\pi }(wz^*+zw^*-I)\\\\right)(-\\\\eta m)(ww^*-\\\\varepsilon \\\\overline{zz^*}\\\\varepsilon )\\\\right)=\\\\\\\\ &=\\\\frac{-\\\\eta mT}{8\\\\pi }(w^*w-z^*z).\\\\end{split}$ The local expresions of these functions in the chart $\\\\Delta ^{-1}$ are obtained simply by changing in the above expresions $z$ by the $z(w,y)$ given in (REF ).\",\n \"Let $\\\\Omega _1$ be the contact form and $\\\\tilde{\\\\alpha }_1$ the left invariant differential form on $G$ whose value at $(I,0)$ is $\\\\alpha _1.$ We have $\\\\Omega _1=\\\\Sigma ^* \\\\tilde{\\\\alpha }_1,$ and this formula enables us, by similar procedures to those of section REF , to see that $\\\\Omega _1=\\\\frac{iT}{4\\\\pi }\\\\left(z^1 d\\\\overline{w^1}-\\\\overline{ z^1} d w^1+z^2 d\\\\overline{ w^2}-\\\\overline{ z^2} d w^2\\\\right)-P^1dx^1-P^2 dx^2-P^3 d x^3,$ where (c.f.\",\n \"(REF )) $\\\\left(\\\\begin{array}{c}z^1\\\\\\\\z^2\\\\end{array}\\\\right)=z(w,y).$ This equation must be interpreted as follows: the right hand side of (REF ) is a differential 1-form in $\\\\mathbb {C}^4 \\\\times \\\\mathbb {R}^3$ refered to coordinates $w^1,\\\\dots ,x^3,$ and $\\\\Omega _1$ is the restriction of this form to the submanifold ${\\\\cal B}\\\\times \\\\mathbb {R}^3.$ Obviously, under the same conditions $\\\\begin{split}d\\\\Omega _1=\\\\frac{iT}{4\\\\pi }&\\\\left(dz^1\\\\wedge d\\\\overline{w^1}-d\\\\overline{ z^1}\\\\wedge d w^1+dz^2 \\\\wedge d\\\\overline{ w^2}-d\\\\overline{ z^2} \\\\wedge d w^2\\\\right)\\\\\\\\&-dP^1\\\\wedge dx^1-dP^2 \\\\wedge dx^2-dP^3 \\\\wedge d x^3.\",\n \"\\\\end{split}$ Now, to obtain the symplectic form we only need sections of the map $\\\\tilde{\\\\mathbf {r}}.$ On the domain, $U_s,$ of the local chart $\\\\Gamma _s^{-1},$ the map ${\\\\tilde{\\\\sigma }}_s: \\\\Gamma _s(\\\\vec{k},\\\\vec{x},t) \\\\rightarrow \\\\left( \\\\sigma \\\\left(mh(\\\\vec{k},k_4),\\\\left( \\\\begin{array}{c} 1\\\\\\\\ t\\\\end{array}\\\\right)\\\\right),h(\\\\vec{x},0)\\\\right)*\\\\left(\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right),\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right), \\\\vec{0} \\\\right), $ where $\\\\sigma (K,w)$ is given by (REF ), is a section.\",\n \"Then ${\\\\tilde{\\\\sigma }}_s \\\\circ \\\\Gamma _s(\\\\vec{k},\\\\vec{x},t) =\\\\left(\\\\frac{1}{\\\\sqrt{(1,\\\\overline{t})(h(\\\\vec{k},k_4))^{-1}\\\\left(\\\\begin{array}{c}1\\\\\\\\t \\\\end{array}\\\\right)}} \\\\left(\\\\begin{array}{c}1\\\\\\\\t \\\\end{array}\\\\right),\\\\right.$ $\\\\left.\",\n \"\\\\frac{1}{\\\\sqrt{(1,\\\\overline{t})(h(\\\\vec{k},k_4))^{-1}\\\\left(\\\\begin{array}{c}1\\\\\\\\t \\\\end{array}\\\\right)}}(h(\\\\vec{k},k_4))^{-1}\\\\left(\\\\begin{array}{c}1\\\\\\\\t \\\\end{array}\\\\right), \\\\vec{x} \\\\right).\",\n \"$ If $\\\\omega _1$ is the symplectic form, we have on $U_s$ $ \\\\omega _1={\\\\tilde{\\\\sigma }}_s ^* d\\\\Omega _1.$ In the same way, on the domain, $U_n,$ of the chart $\\\\Gamma _n,$ we define a section, $\\\\tilde{\\\\sigma }_n,$ by ${\\\\tilde{\\\\sigma }}_n \\\\circ \\\\Gamma _n(\\\\vec{k},\\\\vec{x},t)=\\\\left( \\\\sigma \\\\left(mh(\\\\vec{k},k_4),\\\\left(\\\\begin{array}{c} t\\\\\\\\ 1\\\\end{array}\\\\right)\\\\right),h(\\\\vec{x},0)\\\\right)*\\\\left(\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right),\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right), \\\\vec{0} \\\\right).$ Then ${\\\\tilde{\\\\sigma }}_n \\\\circ \\\\Gamma _n(\\\\vec{k},\\\\vec{x},t) =\\\\left(\\\\frac{1}{\\\\sqrt{(\\\\overline{t},1)(h(\\\\vec{k},k_4))^{-1}\\\\left(\\\\begin{array}{c}t\\\\\\\\1 \\\\end{array}\\\\right)}} \\\\left(\\\\begin{array}{c}t\\\\\\\\1 \\\\end{array}\\\\right),\\\\right.$ $\\\\left.\",\n \"\\\\frac{1}{\\\\sqrt{(\\\\overline{t},1)(h(\\\\vec{k},k_4))^{-1}\\\\left(\\\\begin{array}{c}t\\\\\\\\1 \\\\end{array}\\\\right)}}(h(\\\\vec{k},k_4))^{-1}\\\\left(\\\\begin{array}{c}t\\\\\\\\1 \\\\end{array}\\\\right), \\\\vec{x} \\\\right).\",\n \"$ On $U_n$ we have $\\\\omega _1={\\\\tilde{\\\\sigma }}_n ^* d\\\\Omega _1.$ Since $\\\\lbrace U_s,U_n\\\\rbrace $ is an open covering of the symplectic manifold, (REF ) and (REF ), determine $\\\\omega $ everywhere.\"\n ],\n [\n \"Contact manifold for $T > 1.$\",\n \"Let us denote by $r_1,\\\\dots , r_T$ the elements of $\\\\@root T \\\\of {\\\\mathbf {1}}.$ We define a properly discontinuous free action, $\\\\cdot ,$ of $\\\\@root T \\\\of {\\\\mathbf {1}}$ on ${\\\\cal B}\\\\times {\\\\mathbb {R}}^3 $ by means of $r_j\\\\cdot (w,z,\\\\overrightarrow{x})=(r_j w,r_j z,\\\\overrightarrow{x}).$ The quotient space is denoted by $({\\\\cal B}\\\\times {\\\\mathbb {R}}^3)/\\\\@root T \\\\of {\\\\mathbf {1}},$ and the canonical map from ${\\\\cal B}\\\\times {\\\\mathbb {R}}^3$ onto $({\\\\cal B}\\\\times {\\\\mathbb {R}}^3)/\\\\@root T \\\\of {\\\\mathbf {1}},$ is denoted by $\\\\pi _T.$ Thus $\\\\pi _T(w,z,\\\\overrightarrow{x})$ is the orbit of $(w,z,\\\\overrightarrow{x})$ , considered as an element of $({\\\\cal B}\\\\times {\\\\mathbb {R}}^3)/\\\\@root T \\\\of {\\\\mathbf {1}}.$ The map $\\\\pi _T$ is a T-fold covering map and, since ${\\\\cal B}\\\\times {\\\\mathbb {R}}^3$ is simply connected, the Universal covering map of the quotient space.\",\n \"The action of $G$ on ${\\\\cal B}\\\\times {\\\\mathbb {R}}^3$ conmutes with that of $\\\\@root T \\\\of {\\\\mathbf {1}}.$ As a consequence, the action on $({\\\\cal B}\\\\times {\\\\mathbb {R}}^3)/\\\\@root T \\\\of {\\\\mathbf {1}}$ given by $(A,H)*((w,z,\\\\overrightarrow{x})\\\\@root T \\\\of {\\\\mathbf {1}})=((A,H)*(w,z,\\\\overrightarrow{x}))\\\\@root T \\\\of {\\\\mathbf {1}}$ is well defined, and obviously makes the covering map $\\\\pi _T$ equivariant.\",\n \"The isotropy subgroup at $\\\\left(\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right),\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right), \\\\vec{0}\\\\right)\\\\@root T \\\\of {\\\\mathbf {1}}$ is $Ker\\\\,C_{\\\\alpha _T},$ so that we can identify $G/Ker\\\\,C_{\\\\alpha _T}$ with $({\\\\cal B}\\\\times {\\\\mathbb {R}}^3)/\\\\@root T \\\\of {\\\\mathbf {1}}$ by means of $(A,H)Ker\\\\,C_{\\\\alpha _T} \\\\longleftrightarrow (A,H)* \\\\left(\\\\left(\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right),\\\\left(\\\\begin{array}{c}1\\\\\\\\0 \\\\end{array}\\\\right), \\\\vec{0}\\\\right)\\\\@root T \\\\of {\\\\mathbf {1}}\\\\right).$ The contact form on ${\\\\cal B}\\\\times {\\\\mathbb {R}}^3,\\\\ \\\\Omega _1,$ is invariant under the action $``\\\\cdot \\\",$ so that there exist an unique contact form, $\\\\Omega _T,$ on $({\\\\cal B}\\\\times {\\\\mathbb {R}}^3)/\\\\@root T \\\\of {\\\\mathbf {1}},$ such that $\\\\pi _T^*\\\\Omega _T=\\\\Omega _1.$ The action of $G$ on $({\\\\cal B}\\\\times {\\\\mathbb {R}}^3)/\\\\@root T \\\\of {\\\\mathbf {1}}$ is transitive and preserves $\\\\Omega _T.$ The manifold $({\\\\cal B}\\\\times {\\\\mathbb {R}}^3)/\\\\@root T \\\\of {\\\\mathbf {1}},$ when provided with the contact form $\\\\Omega _T$ and the cited action of $G,$ is the homogeneous contact manifold that correspond to massive particles with $T\\\\ge 1.$\"\n ],\n [\n \"Massless Type 4 particles. \",\n \"In this section we consider particles whose movement space is the coadjoint orbit of $\\\\alpha =\\\\left\\\\lbrace \\\\frac{i \\\\chi T}{8 \\\\pi }\\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & -1\\\\end{array}\\\\right),\\\\eta \\\\,\\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & 0\\\\end{array}\\\\right)\\\\right\\\\rbrace \\\\ \\\\ \\\\ \\\\ ,\\\\ \\\\chi ,\\\\ \\\\eta \\\\in \\\\lbrace \\\\pm 1\\\\rbrace ,\\\\ T \\\\in {\\\\bf Z}^+ ,$ where $\\\\eta =-sign(Tr(P)),$ (see Table 2 of section ).\",\n \"So, at least at the classical level, $\\\\eta $ must be interpreted as the opposite of the sign of energy.\",\n \"We will see in the following subsections that this interpretation is also exact at the quantum level.\",\n \"Since $|P|$ is mass square and $Det\\\\left( \\\\eta \\\\,\\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & 0\\\\end{array}\\\\right) \\\\right)=0$ these orbits correspond to particles with zero mass.\",\n \"These are quantizable, not $\\\\bf R\\\\rm $ -quantizable orbits of the type 4.\",\n \"The group $G_\\\\alpha $ is connected, so that there exists at most one homomorphism from $G_\\\\alpha $ onto ${\\\\bf S}^1$ whose differential is $\\\\alpha $ .\",\n \"In fact we have $\\\\begin{array}{l}\\\\vspace{7.22743pt}G_\\\\alpha = \\\\left\\\\lbrace \\\\left(\\\\left(\\\\begin{array}{cc}z&a \\\\\\\\0&{\\\\overline{z}}\\\\end{array}\\\\right),\\\\ \\\\left(\\\\begin{array}{cc}b&i \\\\chi \\\\eta T a z/2 \\\\pi \\\\\\\\\\\\overline{i \\\\chi \\\\eta T a z / 2 \\\\pi }&0\\\\end{array}\\\\right)\\\\right):\\\\ z \\\\in {\\\\bf S^1},\\\\right.\",\n \"\\\\\\\\\\\\left.\",\n \"\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ b\\\\in {\\\\bf R},\\\\ a \\\\in {\\\\bf C}\\\\right\\\\rbrace \\\\\\\\\\\\ \\\\\\\\\\\\mathrm {and\\\\ the\\\\ homomorphism} \\\\\\\\\\\\ \\\\\\\\C_\\\\alpha \\\\left( \\\\left( \\\\left(\\\\begin{array}{cc}z&a \\\\\\\\0&{\\\\overline{z}}\\\\end{array}\\\\right),\\\\ \\\\left(\\\\begin{array}{cc}b&{i \\\\chi \\\\eta T a z/2 \\\\pi } \\\\\\\\{\\\\overline{i \\\\chi \\\\eta T a z/2 \\\\pi }}&0\\\\end{array}\\\\right)\\\\right)\\\\right)=z^{\\\\chi T}\\\\end{array}$ has differential $\\\\alpha .$ Other computations give us $\\\\begin{array}{l}({\\\\mbox{$G_\\\\alpha $}} )_{SL}=\\\\left\\\\lbrace \\\\left(\\\\begin{array}{cc} z&a \\\\\\\\0&{\\\\overline{z}}\\\\end{array}\\\\right): z \\\\in {\\\\bf S^1},\\\\ a \\\\in {\\\\bf C} \\\\right\\\\rbrace \\\\\\\\\\\\vspace{7.22743pt}(C_\\\\alpha )_{SL}\\\\left(\\\\ \\\\left(\\\\begin{array}{cc}z&a \\\\\\\\0&{\\\\overline{z}}\\\\end{array}\\\\right)\\\\right)=\\\\ z^{\\\\chi T} \\\\\\\\\\\\vspace{7.22743pt}SL_1=\\\\left\\\\lbrace \\\\left(\\\\begin{array}{cc}a&0 \\\\\\\\0&{1/a}\\\\end{array}\\\\right):\\\\ a \\\\in {\\\\bf C}\\\\right\\\\rbrace \\\\\\\\\\\\vspace{7.22743pt}SL_2=(G_\\\\alpha )_{SL} \\\\\\\\\\\\vspace{7.22743pt}SL_1 \\\\cap SL_2=\\\\left\\\\lbrace \\\\left(\\\\begin{array}{cc}z&0 \\\\\\\\0&\\\\overline{z}\\\\end{array}\\\\right):\\\\ z \\\\in {\\\\bf S}^1\\\\right\\\\rbrace \\\\end{array}.$ The isotropy subgroup at $\\\\left(\\\\begin{array}{cc}1&0 \\\\\\\\0&0\\\\end{array}\\\\right.$ , for the usual action of $SL$(2,$C$) on H(2) (that is $A\\\\ast m=AmA^*$ ) is ${\\\\mbox{$(G_\\\\alpha )$}}_{SL}$ .\",\n \"Thus SL/${\\\\mbox{$(G_\\\\alpha )$}}_{SL}$ will be identified to the orbit of $\\\\left(\\\\begin{array}{cc}1&0 \\\\\\\\0&0\\\\end{array}\\\\right),$ which is ${\\\\mbox{$\\\\bf C^+$}}=\\\\lbrace H \\\\in {H(2)}:\\\\ Det\\\\,H=0,\\\\ Tr\\\\,H>0\\\\rbrace .$ This set is composed by the hermitian matrices that represent space-time points in the future lightcone, so that it will be called itself, future lightcone.\",\n \"When this identification is made, the function $P$ on SL/${\\\\mbox{$(G_\\\\alpha )$}}_{SL}$ becomes $P(H)=-\\\\eta \\\\,H$ .\",\n \"An invariant volume element in ${\\\\mbox{$\\\\bf C^+$}}$ is $\\\\omega =\\\\frac{1}{\\\\Vert \\\\vec{ p}\\\\Vert } \\\\ dp^1\\\\wedge \\\\,dp^2\\\\wedge \\\\,dp^3,$ where $(p^1,\\\\ p^2,\\\\ p^3)$ is the coordinate system corresponding to the parametrization $\\\\phi :(p^1,\\\\ p^2,\\\\ p^3) \\\\in {\\\\bf R^3}-\\\\lbrace \\\\,0\\\\,\\\\rbrace \\\\mapsto h(\\\\vec{ p},\\\\Vert {\\\\vec{ p}\\\\Vert }) \\\\in {\\\\mbox{$\\\\bf C^+$}}$ where $\\\\vec{p}= (p^1,\\\\ p^2\\\\ p^3),$ and $\\\\Vert {\\\\vec{ p}}\\\\Vert =+ \\\\sqrt{ \\\\sum _{i=1}^3 (p^i)^2}.$ In this parametrization, the linear momentum is given by $P(\\\\phi (\\\\vec{ p}))=-\\\\eta \\\\ h(\\\\vec{ p},\\\\Vert {\\\\vec{ p}\\\\Vert }).$ Before to proceed to the study of the general case, we shall consider two particular ones.\"\n ],\n [\n \"Massless antineutrino\",\n \"Let us consider the case $ {T=1,\\\\ \\\\chi =1,\\\\ \\\\eta =\\\\,-\\\\,1}$ .\",\n \"A trivialization is given by $\\\\left(\\\\rho _+ ,\\\\ {\\\\bf C}^2,\\\\ \\\\left(\\\\begin{array}{c}1 \\\\\\\\0\\\\end{array}\\\\right) \\\\right),$ where $\\\\rho _+$ (A) is multiplication by A.\",\n \"The orbit of $\\\\left(\\\\begin{array}{c} 1\\\\\\\\0\\\\end{array}\\\\right)$ is ${\\\\bf C}^2-\\\\lbrace 0\\\\rbrace $ .\",\n \"This space can thus be identified to SL/${\\\\mbox{$Ker\\\\,(C_\\\\alpha )$}}_{SL}$ so that the canonical map $SL/{\\\\mbox{$Ker\\\\,(C_\\\\alpha )$}}_{SL} \\\\longrightarrow SL/(G_\\\\alpha ) $ becomes a map $r_+ :{\\\\bf C}^2-\\\\lbrace 0\\\\rbrace \\\\mapsto {\\\\bf C}^+ .$ This map must be equivariant, so that, for all $A\\\\in SL$ $r_+\\\\left(A \\\\left(\\\\begin{array}{c}1 \\\\\\\\0\\\\end{array} \\\\right) \\\\right) =A\\\\left(\\\\begin{array}{cc}1&0 \\\\\\\\0&0\\\\end{array}\\\\right.\",\n \"A^*.$ Thus $r_+$ is explicitly given by $r_+(z)=z\\\\,z^* \\\\in {\\\\mbox{$\\\\bf C^+$}} .$ Since $z$ and $\\\\varepsilon \\\\overline{z}$ are eigenvectors of $z\\\\,z^*$ corresponding to the eigenvalues $\\\\Vert z\\\\Vert ^2$ and 0 respectively, $r_+^{-1}(H)$ is composed by the eigenvectors of $H$ corresponding to the positive eigenvalue, whose norm is the square root of that eigenvalue.\",\n \"The principal $ {\\\\bf S}^1$ -bundle whose projection is $r_+$ is related to the Hopf fibration as follows.\",\n \"The image of the restriction of $r_+$ to the sphere $S^3(R)=\\\\lbrace z \\\\in { {\\\\bf C}}^2\\\\,:\\\\,\\\\Vert z\\\\Vert ^2=R^2\\\\rbrace $ is composed by the elements of ${\\\\mbox{$\\\\bf C^+$}}$ whose trace is $R^2$ .\",\n \"Since the image of this subset by the preceeding chart is the sphere of radius $R^2$ /2, we obtain maps from spheres $S^3$ onto spheres $S^2$ .\",\n \"Each one of these mappings is, up to the radius and a reflection, the Hopf fibration.\",\n \"For each $S$ -homogeneous of degree -1 function on ${\\\\bf C}^2-{0},\\\\ f, $ we have the following prewave function ${\\\\psi } _f^+:\\\\,(N,\\\\ H) \\\\in {\\\\mbox{$\\\\bf C^+$}}\\\\times {H(2)}\\\\mapsto f(z)\\\\ e^{i\\\\pi \\\\,Tr(\\\\,N \\\\varepsilon \\\\overline{H} \\\\varepsilon ) } z \\\\in { {\\\\bf C}}^2,$ where z is an arbitrary element of $r_+^{-1}(N)$ .\",\n \"Here we use the fact that, since $\\\\eta =-1$ , the linear momentum is given in $C^+$ by $P(N)=N.$ If $X$ and $Q$ are the elements of ${\\\\bf R}^4$ corresponding to $H$ and $N$ , one can write ${\\\\psi } _f^+(Q,X) = f(z)\\\\ e^{-2\\\\pi i\\\\langle Q,X \\\\rangle } z ,$ where $z$ is arbitrary in $r_+^{-1}(Q).$ The corresponding wave function , if $f$ is continuous with compact support, is ${\\\\widetilde{\\\\psi }} _f^+(H)= \\\\int _{C^+} {\\\\psi } _f^+(\\\\cdot ,\\\\ H) \\\\omega .$ By direct computation one can see that $\\\\left(\\\\ \\\\sigma _1 \\\\ \\\\frac{\\\\partial }{\\\\partial x^1}+\\\\ \\\\sigma _2 \\\\ \\\\frac{\\\\partial }{\\\\partial x^2}+\\\\ \\\\sigma _3 \\\\ \\\\frac{\\\\partial }{\\\\partial x^3}+\\\\ \\\\sigma _4 \\\\ \\\\frac{\\\\partial }{\\\\partial x^4} \\\\right)\\\\ {\\\\psi }_f^+(N,\\\\ h(x))\\\\,=\\\\,0$ The corresponding wave function thus satisfy the same equation, which is the Weyl equation (positive energy), usually admised as corresponding to the antineutrino.\",\n \"If $f$ and $f^\\\\prime $ are pseudotensorial functions, we have $({\\\\psi } _f^+(N,H))^*\\\\ {\\\\psi }_{f^\\\\prime }^+(N,H)=\\\\overline{f(z)}\\\\, f^\\\\prime (z)\\\\ Tr\\\\,N.$ Thus, the hermitian product of the corresponding quantum states ($cf.$ section ) can be written as follows ${\\\\frac{1}{2}}\\\\ \\\\int _{\\\\bf C ^+} \\\\frac{({\\\\psi }^+ _f)^*\\\\ {\\\\psi }^+_{f^\\\\prime }}{\\\\sum _{i=1}^3 (p^i)^2}\\\\ dp^1\\\\,dp^2\\\\,dp^3.$ To obtain prewave functions directly from functions on $C^+,$ we use Remark REF , as follows.\",\n \"Let $U= \\\\phi ({\\\\bf R}^3-\\\\lbrace p^1=p^2=0, p^3<0\\\\rbrace ),$ $V= \\\\phi ({\\\\bf R}^3-\\\\lbrace p^1=p^2=0, p^3>0\\\\rbrace ).$ Then the maps $\\\\sigma _U : \\\\phi (p^1,p^2,p^3)\\\\in U \\\\mapsto \\\\left({\\\\begin{array}{c} \\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert +p^3} \\\\\\\\\\\\frac{p^1+i p^2}{ \\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert +p^3}} \\\\end{array}}\\\\right) \\\\in {\\\\bf C}^2-{0}$ $\\\\sigma _V : \\\\phi (p^1,p^2,p^3)\\\\in V \\\\mapsto \\\\left( {\\\\begin{array}{c} \\\\frac{p^1-i p^2}{ \\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert -p^3}} \\\\\\\\\\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert -p^3} \\\\end{array}}\\\\right) \\\\in {\\\\bf C}^2-{0}$ are sections of $r_+,$ so that we obtain prewave functions having the form $\\\\psi (\\\\phi (p^1,p^2,p^3), X)=F(p^1,p^2,p^3) \\\\left({\\\\begin{array}{c} \\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert +p^3} \\\\\\\\\\\\frac{p^1+i p^2}{ \\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert +p^3}} \\\\end{array}}\\\\right)$ $ \\\\displaystyle e^{-2\\\\pi i (\\\\Vert \\\\vec{ p}\\\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ where $F$ is a complex valued function continuous on ${\\\\bf R}^3-\\\\lbrace 0\\\\rbrace ,$ with compact support in ${\\\\bf R}^3-\\\\lbrace p^1=p^2=0, p^3<0\\\\rbrace ,$ and prewave functions having the form $\\\\theta (\\\\phi (p^1,p^2,p^3), X)=J(p^1,p^2,p^3) \\\\left( {\\\\begin{array}{c} \\\\frac{p^1-i p^2}{ \\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert -p^3}} \\\\\\\\\\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert -p^3} \\\\end{array}}\\\\right)$ $ \\\\displaystyle e^{-2\\\\pi i (\\\\Vert \\\\vec{ p}\\\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ where $J$ is a complex valued function continuous on ${\\\\bf R}^3-\\\\lbrace 0\\\\rbrace ,$ with compact support in ${\\\\bf R}^3-\\\\lbrace p^1=p^2=0, p^3>0\\\\rbrace .$ If one is interested in the case $T=1,\\\\ \\\\chi =1,\\\\ \\\\eta =+1$ , everything is as in the case $\\\\eta =-1$ , but the exponent of $e$ in the prewave functions changes its sign, thus giving wave functions for quantum states of negative energy.\",\n \"Remenber that $\\\\eta $ is the opposite of the sign of energy at the clasical level.\"\n ],\n [\n \"Massless neutrino\",\n \"Now let us consider the case in which $T=1,\\\\ \\\\chi =-1,\\\\ \\\\eta =-1$ .\",\n \"A trivialisation in this case is $\\\\left(\\\\rho _- ,\\\\ {\\\\bf C}^2,\\\\ \\\\left(\\\\begin{array}{c}0 \\\\\\\\1\\\\end{array}\\\\right) \\\\right),$ $ \\\\rho _- $ (A) being multiplication by $(A^*)^{-1}.$ Thus we identify $SL/Ker\\\\,C_{\\\\alpha }$ with the orbit of $ \\\\left(\\\\begin{array}{c}0 \\\\\\\\1\\\\end{array}\\\\right),$ which, also in this case, is ${\\\\bf C}^2-\\\\lbrace 0\\\\rbrace .$ Thus,the canonical map $SL/{\\\\mbox{$Ker\\\\,(C_\\\\alpha )$}}_{SL} \\\\longrightarrow SL/(G_\\\\alpha ) $ becomes a map $r_- :{\\\\bf C}^2-\\\\lbrace 0\\\\rbrace \\\\mapsto {\\\\bf C}^+ .$ Now, using equivariance as in the case of antineutrino, we obtain $r_-(z)\\\\,=\\\\,-\\\\varepsilon \\\\overline{z\\\\,z^* }\\\\varepsilon .\",\n \"$ The prewave functions one obtains have exactly the same form that (REF ) and (REF ), but now $z$ is in $(r_-)^{-1}(N)$ or $(r_-)^{-1}(h(Q))$ respectively.\",\n \"The wave functions one obtains in this case satisfies the Weyl equation that, according to Feynman, corresponds to the neutrino.\",\n \"The map from the real vector space $\\\\bf C \\\\rm ^2$ onto itself defined by sending $z$ to $ \\\\varepsilon \\\\overline{z}$ , is a complex structure and its restriction to $\\\\bf C\\\\rm ^2-\\\\lbrace 0 \\\\rbrace $ , gives us an isomorphism of the principal circle bundle corresponding to ${\\\\bf r}_{-}$ ( resp.${\\\\bf r}_+$ ) onto the principal circle bundle corresponding to ${\\\\bf r}_+$ ( resp.\",\n \"${\\\\bf r}_-$ ).\",\n \"The isomorphism of the structural group is defined by sending each element to its inverse.\",\n \"Thus if $z\\\\in r_+^{-1}(H)$ then $\\\\varepsilon \\\\overline{z}\\\\in r_-^{-1}(H)$ and conversely.\",\n \"As a consequence, the sections of the map $r_+$ defined in REF give rise to the following sections of $r_-$ $\\\\sigma ^\\\\prime _U(u)=\\\\varepsilon \\\\overline{\\\\sigma _U(u)},\\\\ \\\\ \\\\sigma ^\\\\prime _V(v)=\\\\varepsilon \\\\overline{\\\\sigma _V(v)},$ for all $u\\\\in U$ and $v\\\\in V,$ and the prewave functions of antineutrino $\\\\psi $ and $\\\\theta $ give rise to prewave functions of neutrino, $\\\\psi ^\\\\prime $ and $\\\\theta ^\\\\prime $ , by means of $\\\\psi ^\\\\prime (\\\\phi (p^1,p^2,p^3), X)=F(p^1,p^2,p^3)\\\\varepsilon \\\\overline{\\\\left({\\\\begin{array}{c} \\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert +p^3} \\\\\\\\\\\\frac{p^1+i p^2}{ \\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert +p^3}} \\\\end{array}}\\\\right) }$ $e^{-2\\\\pi i (\\\\Vert \\\\vec{ p}\\\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ $\\\\theta ^\\\\prime (\\\\phi (p^1,p^2,p^3), X)=J(p^1,p^2,p^3) \\\\varepsilon \\\\overline{\\\\left( {\\\\begin{array}{c} \\\\frac{p^1-i p^2}{ \\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert -p^3}} \\\\\\\\\\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert -p^3} \\\\end{array}}\\\\right) }$ $e^{-2\\\\pi i (\\\\Vert \\\\vec{ p}\\\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ that leads to $\\\\psi ^{\\\\prime }(\\\\phi (p^1,p^2,p^3), X)&=& F(p^1,p^2,p^3){\\\\left({{\\\\begin{array}{c} \\\\frac{p^1-i p^2}{ \\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert +p^3}}\\\\\\\\-\\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert +p^3} \\\\end{array}}}\\\\right) }\\\\\\\\ &&{e^{-2\\\\pi i (\\\\Vert \\\\vec{ p}\\\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}} \\\\\\\\\\\\ \\\\theta ^{\\\\prime }(\\\\phi (p^1,p^2,p^3), X)&=&J(p^1,p^2,p^3){\\\\left( {{\\\\begin{array}{c}\\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert -p^3}\\\\\\\\{-\\\\frac{p^1+i p^2}{ \\\\sqrt{\\\\Vert \\\\vec{ p}\\\\Vert -p^3}}} \\\\end{array}}}\\\\right)}\\\\\\\\ && {e^{-2\\\\pi i (\\\\Vert \\\\vec{ p}\\\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}}.$ If one consider the case $T=1,\\\\ \\\\chi =-1,\\\\ \\\\eta =+1$ one obtain similar wave functions but corresponding to negative energy.\"\n ],\n [\n \"General case\",\n \"We proceed as in section REF .\",\n \"In the case $\\\\chi =+1$ a trivialization is given by $\\\\left((\\\\rho _+)^{\\\\otimes T},\\\\ \\\\left({\\\\bf C}^2\\\\right)^ {\\\\otimes T},\\\\ \\\\left(\\\\begin{array}{c} 1\\\\\\\\0\\\\end{array}\\\\right)^{\\\\otimes T}\\\\right),$ and if $\\\\chi =-1$ a trivialization is given by $\\\\left((\\\\rho _-)^{\\\\otimes T},\\\\ \\\\left({\\\\bf C}^2\\\\right)^{\\\\otimes T},\\\\ \\\\left(\\\\begin{array}{c} 0\\\\\\\\1\\\\end{array}\\\\right)^{\\\\otimes T}\\\\right),$ The prewave functions are given by functions on ${\\\\bf C} \\\\rm ^2-\\\\lbrace 0\\\\rbrace $ , which are $C^\\\\infty $ with compact support and homogeneous of degree -T under multiplication by modulus one complex numbers.\",\n \"Let $f_T$ be one of these functions.\",\n \"If $\\\\chi =1$ , the corresponding prewave function is given by $\\\\psi ^+_{f_T}:\\\\,(N,\\\\ H) \\\\in {\\\\mbox{$\\\\bf C^+$}}\\\\times {H(2)}\\\\mapsto f_T(z)\\\\ e^{-i \\\\pi \\\\eta \\\\,Tr(N \\\\varepsilon \\\\overline{H} \\\\varepsilon ) } z^{\\\\otimes T}\\\\in ({\\\\bf C}^2)^{\\\\otimes T}$ where $z$ is an arbitrary element of $r_+^{-1}(N)$ .\",\n \"In the case $\\\\chi =-1$ , the corresponding prewave function is $\\\\psi ^-_{f_T}:\\\\,(N,\\\\ H) \\\\in {\\\\mbox{$\\\\bf C^+$}}\\\\times {H(2)}\\\\mapsto f_T(z)\\\\ e^{-i \\\\pi \\\\eta \\\\,Tr(N \\\\varepsilon \\\\overline{H} \\\\varepsilon )} z^{\\\\otimes T}\\\\in ({\\\\bf C}^2)^{\\\\otimes T}$ but now, z is an arbitrary element of $r_-^{-1}(N)$ .\",\n \"The associated wave functions, ${\\\\widetilde{\\\\psi }}^+_{f_T}(H)&=&\\\\int _{C^+} {\\\\psi } ^+_{f_T}(\\\\cdot ,\\\\ H) \\\\omega ,\\\\\\\\{\\\\widetilde{\\\\psi }}^-_{f_T}(H)&=&\\\\int _{C^+} {\\\\psi } ^-_{f_T}(\\\\cdot ,\\\\ H) \\\\omega $ satisfies Penrose's wave equations,that we will describe here for the sake of completeness.\",\n \"Let us consider in $({\\\\bf C}^2)^{\\\\otimes T}$ the basis $\\\\lbrace e_A \\\\otimes e_B \\\\otimes \\\\stackrel{(T}{\\\\cdots }\\\\ :\\\\ A,\\\\ B,\\\\dots \\\\in \\\\lbrace 1,\\\\ 2\\\\rbrace \\\\rbrace $ , where $\\\\lbrace e_1,\\\\ e_2\\\\rbrace $ is the canonical basis of $\\\\bf C \\\\rm ^2$ .\",\n \"The prewave functions $\\\\psi ^\\\\pm _{f_T}$ and the wave functions $\\\\tilde{\\\\psi }^\\\\pm _{f_T}$ , have components in this basis which will be denoted by $\\\\lbrace \\\\psi _\\\\pm ^{A\\\\,B\\\\dots } \\\\rbrace $ and $\\\\lbrace \\\\tilde{\\\\psi }_\\\\pm ^{A\\\\,B\\\\dots } \\\\rbrace $ , respectively.\",\n \"Let us consider the vector fields in $\\\\bf R \\\\rm ^4$ given by $\\\\nabla _{11}&=&\\\\frac{1}{2} \\\\left( \\\\frac{\\\\partial }{\\\\partial x^3}+\\\\frac{\\\\partial }{\\\\partial x^4} \\\\right)\\\\\\\\\\\\nabla _{12}&=&\\\\frac{1}{2} \\\\left( \\\\frac{\\\\partial }{\\\\partial x^1}-i\\\\frac{\\\\partial }{\\\\partial x^2} \\\\right)\\\\\\\\\\\\nabla _{21}&=&\\\\frac{1}{2} \\\\left( \\\\frac{\\\\partial }{\\\\partial x^1}+i\\\\frac{\\\\partial }{\\\\partial x^2} \\\\right)\\\\\\\\\\\\nabla _{22}&=&\\\\frac{1}{2} \\\\left( \\\\frac{\\\\partial }{\\\\partial x^4}-\\\\frac{\\\\partial }{\\\\partial x^3} \\\\right)$ and, for all $A, A^\\\\prime \\\\in \\\\lbrace 1,\\\\ 2\\\\rbrace ,\\\\ \\\\nabla ^{A \\\\ A^\\\\prime }=\\\\varepsilon ^{A\\\\ B}\\\\ \\\\varepsilon ^{A^\\\\prime \\\\ B^\\\\prime }\\\\ \\\\nabla _{B\\\\ B^\\\\prime }$ (summation convention), where $\\\\lbrace \\\\varepsilon ^{A\\\\ B}\\\\rbrace $ are the elements of $-\\\\varepsilon $ .\",\n \"We also define $\\\\psi ^\\\\pm _{A\\\\,B\\\\dots }&=&\\\\varepsilon _{A\\\\ A^\\\\prime }\\\\ \\\\varepsilon _{B\\\\ B^\\\\prime }\\\\dots \\\\psi _\\\\pm ^{A^\\\\prime \\\\,B^\\\\prime \\\\dots } \\\\\\\\\\\\tilde{\\\\psi }^\\\\pm _{A\\\\,B\\\\dots }&=&\\\\varepsilon _{A\\\\ A^\\\\prime }\\\\ \\\\varepsilon _{B\\\\ B^\\\\prime }\\\\dots \\\\tilde{\\\\psi }_\\\\pm ^{A^\\\\prime \\\\,B^\\\\prime \\\\dots }$ where $\\\\lbrace \\\\varepsilon _{A\\\\ B}\\\\rbrace $ are the elements of $\\\\varepsilon $ .\",\n \"Thus we have for all $h(x) \\\\in C^+$ $\\\\nabla ^{A\\\\ A^\\\\prime }\\\\ \\\\psi ^+_{A^\\\\prime \\\\,B\\\\ C\\\\dots }(h(x),\\\\,\\\\cdot \\\\, )&=&0\\\\\\\\\\\\nabla ^{ A^\\\\prime \\\\ A}\\\\ \\\\psi ^-_{A^\\\\prime \\\\,B\\\\ C\\\\dots }(h(x),\\\\,\\\\cdot \\\\,)&=&0$ so that, by derivation under the integral sign, we see that Penrose wave equations: $\\\\nabla ^{A\\\\ A^\\\\prime }\\\\ \\\\tilde{\\\\psi }^+_{A^\\\\prime \\\\,B\\\\ C\\\\dots }&=&0\\\\\\\\\\\\nabla ^{ A^\\\\prime \\\\ A}\\\\ \\\\tilde{\\\\psi }^-_{A^\\\\prime \\\\,B\\\\ C\\\\dots }&=&0$ are satisfied.\"\n ],\n [\n \"Helicity\",\n \"In formula (REF ) one sees that, if $(\\\\rho ,L,z_0)$ is the trivialization we use, the components of spin operator are given by $\\\\hat{s^k}:\\\\widetilde{\\\\psi }_f \\\\longrightarrow s^k \\\\circ \\\\widetilde{\\\\psi }_f,$ $s^k$ being the following endomophism of $L$ $s^k=\\\\frac{1}{2\\\\pi i}\\\\frac{\\\\widetilde{i\\\\sigma _k}}{2}=\\\\frac{1}{4\\\\pi i}\\\\widetilde{i\\\\sigma _k},$ where, $\\\\widetilde{i\\\\sigma _k}=d\\\\rho ({i\\\\sigma _k}).$ Thus, in the case $\\\\chi =1,$ $\\\\widetilde{i\\\\sigma _k}=d\\\\left((\\\\rho _+)^{\\\\otimes T}\\\\right)({i\\\\sigma _k}),$ and in the case $\\\\chi =-1,$ $\\\\widetilde{i\\\\sigma _k}=d\\\\left((\\\\rho _-)^{\\\\otimes T}\\\\right)({i\\\\sigma _k}).$ As a consecuence, in both cases, $\\\\chi =+ 1$ or $\\\\chi = 1,$ when acting on monomials we have $\\\\widetilde{i\\\\sigma _k}\\\\cdot z_1\\\\otimes \\\\dots \\\\otimes z_T=\\\\sum _{j=1}^T\\\\ z_1\\\\otimes \\\\dots \\\\otimes z_{j-1}\\\\otimes i\\\\, \\\\sigma _k\\\\,z_j\\\\otimes z_{j+1}\\\\otimes \\\\dots \\\\otimes z_T.$ We consider the operators $\\\\hat{s^k}$ as also acting on prewave functions by the same formula that in the case of wave functions, without the tilde.\",\n \"On the other hand, Linear Momentum is given on $C^+$ by $P(K)=-\\\\eta K,\\\\ \\\\text{\\\\rm for all}\\\\ K\\\\in C^+.$ Then, with the notation $K=h(K^1,K^2,K^3,K^4)$ and $P=h(P^1,P^2,P^3,P^4),$ the $P^k$ are functions on $C^+,$ given by $P^k(K)=-\\\\eta K^k.$ We also denote $\\\\overrightarrow{K}&=&(K^1,K^2,K^3)\\\\\\\\\\\\Vert \\\\overrightarrow{K}\\\\Vert &=&+\\\\left(\\\\sum _{k=1}^3({K^k})^2\\\\right)^{1/2}\\\\\\\\\\\\overrightarrow{P}&=&(P^1,P^2,P^3)\\\\\\\\ \\\\Vert \\\\overrightarrow{P}\\\\Vert &=&+\\\\left(\\\\sum _{k=1}^3({P^k})^2\\\\right)^{1/2}.$ The Helicity Operator is defined by $\\\\mathfrak {h}=\\\\frac{1}{\\\\Vert \\\\overrightarrow{P}\\\\Vert } {\\\\sum _{k=1}^3}{P^k}\\\\hat{s^k},$ which means $\\\\left(\\\\mathfrak {h}\\\\cdot \\\\psi _{f_T}^{\\\\pm }\\\\right)(K,H)=\\\\frac{1}{\\\\Vert \\\\overrightarrow{P}\\\\Vert (K)} {\\\\sum _{k=1}^3}{P^k(K)}{s^k}\\\\left(\\\\psi _{f_T}^{\\\\pm }(K,H)\\\\right).$ Then, if $z\\\\in r^{-1}_{\\\\pm }(K)$ and $z_1=\\\\dots =z_T=z,$ we have $&&\\\\left(\\\\mathfrak {h}\\\\cdot \\\\psi _{f_T}^{\\\\pm }\\\\right)(K,H)=\\\\frac{1}{K^4} \\\\sum _{k=1}^3{P^k(K)}{s^k}\\\\left(f_T(z)e^{-i\\\\eta \\\\pi TrK\\\\epsilon \\\\overline{H}\\\\epsilon } z^{\\\\otimes T}\\\\right)=\\\\\\\\&&=\\\\frac{1}{4 \\\\pi K^4}\\\\ f_T(z)e^{-i\\\\eta \\\\pi TrK\\\\epsilon \\\\overline{H}\\\\epsilon }\\\\\\\\&&\\\\left( \\\\sum _{j=1}^T\\\\ z_1\\\\otimes \\\\dots \\\\otimes z_{j-1}\\\\otimes (\\\\sum _{k=1}^3{P^k(K)}\\\\, \\\\sigma _k\\\\,z_j)\\\\otimes z_{j+1}\\\\otimes \\\\dots \\\\otimes z_T\\\\right)=\\\\\\\\&&=\\\\frac{1}{4 \\\\pi K^4} \\\\ f_T(z)e^{-i\\\\eta \\\\pi TrK\\\\epsilon \\\\overline{H}\\\\epsilon } \\\\\\\\&& \\\\left( \\\\sum _{j=1}^T\\\\ z_1\\\\otimes \\\\dots \\\\otimes z_{j-1}\\\\otimes (-\\\\eta )((K-K^4\\\\,I)z_j)\\\\otimes z_{j+1}\\\\otimes \\\\dots \\\\otimes z_T\\\\right).$ In the case $\\\\chi =+1,$ we have $K= r_+(z)=zz^*,$ so that $ (K-K^4\\\\,I)z=(zz^*-\\\\frac{\\\\Vert z \\\\Vert }{2}\\\\,I)z=K^4\\\\,z,$ and then $\\\\left(\\\\mathfrak {h}\\\\cdot \\\\psi _{f_T}^{+}\\\\right)(K,H)=\\\\frac{-\\\\eta T}{4\\\\pi }\\\\psi _{f_T}^{+}(K,H).$ On the other hand, in the case $\\\\chi =-1,$ we have $ K=-\\\\epsilon \\\\overline{zz^*}\\\\epsilon .$ Thus $(K-K^4\\\\,I)z=(-\\\\epsilon \\\\overline{zz^*}\\\\epsilon -K^4\\\\,I)z=-K^4\\\\,z,$ so that $\\\\left(\\\\mathfrak {h}\\\\cdot \\\\psi _{f_T}^{-}\\\\right)(K,H)=\\\\frac{\\\\eta T}{4\\\\pi }\\\\psi _{f_T}^{-}(K,H).$ Thus $\\\\psi _{f_T}^{\\\\pm }$ is an eigenvector of the helicity operator, corresponding to the eigenvalue $-\\\\eta \\\\chi T/4 \\\\pi .$ In particular, the sign of helicity is $-\\\\eta \\\\chi .$\"\n ],\n [\n \"The Homogeneous Contact and Symplectic Manifolds for Massless particles of type 4. Twistors.\",\n \"Let us denote by $\\\\@root T \\\\of {\\\\mathbf {1}}=\\\\lbrace r_1, \\\\dots ,r_T\\\\rbrace ,$ the group composed by the roots of order $T$ of 1, and $\\\\nu =\\\\frac{- \\\\eta \\\\chi T}{4\\\\pi } .$ In section REF we have seen that all wave functions obtained in section REF are eigenvectors of helicity with eigenvalue $\\\\nu .$ The group $Ker\\\\,C_\\\\alpha ,$ has $T$ connected components $(Ker\\\\,C_\\\\alpha )_j =\\\\left\\\\lbrace \\\\left(\\\\left(\\\\begin{array}{cc}r_j&a \\\\\\\\0&{\\\\overline{r}_j}\\\\end{array}\\\\right),\\\\ \\\\left(\\\\begin{array}{cc}b&-2i\\\\nu \\\\,a\\\\, r_j \\\\\\\\\\\\overline{-2i\\\\nu \\\\,a\\\\, r_j } &0\\\\end{array}\\\\right)\\\\right):b\\\\in {\\\\bf R},\\\\ a \\\\in {\\\\bf C}\\\\right\\\\rbrace $ The component of the identity is $(Ker\\\\,C_\\\\alpha )_o =\\\\left\\\\lbrace \\\\left(\\\\left(\\\\begin{array}{cc}1&a \\\\\\\\0&{1}\\\\end{array}\\\\right),\\\\ \\\\left(\\\\begin{array}{cc}b&-2i\\\\nu \\\\,a \\\\\\\\\\\\overline{\\\\-2i\\\\nu \\\\,a}&0\\\\end{array}\\\\right)\\\\right):b\\\\in {\\\\bf R},\\\\ a \\\\in {\\\\bf C}\\\\right\\\\rbrace $ and $(Ker\\\\,C_\\\\alpha )_j $ is the left translation by $\\\\tilde{\\\\bf r}_j \\\\stackrel{def}{=}\\\\left( \\\\left( \\\\begin{array}{cc}r_j&0 \\\\\\\\0&{\\\\overline{r}_j}\\\\end{array}\\\\right),\\\\ 0 \\\\right)$ of $(Ker\\\\,C_\\\\alpha )_o .$ As a consequence, the group $Ker\\\\,C_\\\\alpha /(Ker\\\\,C_\\\\alpha )_o$ is isomorphic to $\\\\@root T \\\\of {\\\\mathbf {1}}$ .\",\n \"The canonical map ${\\\\begin{array}{c}\\\\tilde{\\\\bf c} : (A,H)\\\\,(Ker\\\\,C_\\\\alpha )_o\\\\in \\\\frac{G}{(Ker\\\\,C_\\\\alpha )_o} \\\\longrightarrow \\\\\\\\\\\\longrightarrow (A,H)\\\\,Ker\\\\,C_\\\\alpha \\\\in \\\\frac{G}{Ker\\\\,C_\\\\alpha }.\\\\end{array}}$ is a $T$ -fold covering map, whose structural group is $Ker\\\\,C_\\\\alpha /(Ker\\\\,C_\\\\alpha )_o$ .\",\n \"Thus, $G/Ker\\\\,C_\\\\alpha $ is the quotient of $G/(Ker\\\\,C_\\\\alpha )_o$ by a properly discontinuous with no fixed point action of $Ker\\\\,C_\\\\alpha /(Ker\\\\,C_\\\\alpha )_o$ .\",\n \"But, as we have seen, this group can be changed to $\\\\@root T \\\\of {\\\\mathbf {1}}$ .\",\n \"The action of $\\\\@root T \\\\of {\\\\mathbf {1}}$ on $G/(Ker\\\\,C_\\\\alpha )_o$ corresponding to this construction is: $((A,H)\\\\,(Ker\\\\,C_\\\\alpha )_o)* r_j= ((A,H)\\\\,\\\\tilde{\\\\bf r}_j)\\\\,(Ker\\\\,C_\\\\alpha )_o.$ In what follows, we identify $G/Ker\\\\,C_\\\\alpha $ with the quotient space of $G/(Ker\\\\,C_\\\\alpha )_o$ by this action.\",\n \"We have the following conmutative diagram ${\\\\begin{matrix}\\\\frac{G}{(Ker\\\\,C_\\\\alpha )_o } &\\\\xrightarrow{}& \\\\frac{G}{Ker\\\\,C_\\\\alpha } \\\\\\\\\\\\downarrow && \\\\downarrow && \\\\\\\\\\\\frac{G}{G_\\\\alpha } &=& \\\\frac{G}{G_\\\\alpha }\\\\end{matrix}}$ where the vertical arrow on the right is the bundle map of the contact manifold onto the symplectic manifold, for general $T.$ If $T=1$ both vertical arrows are the same.\",\n \"The homomorphism $\\\\mu _1:(A, H) \\\\in SL\\\\oplus H(2) \\\\longrightarrow \\\\left( \\\\begin{array}{cc}A&{-iH{A^*}^{-1}} \\\\\\\\0&{A^*}^{-1}\\\\end{array} \\\\right)\\\\in GL(4,\\\\ \\\\text{\\\\bf C}).$ is a representation in ${\\\\bf C}^4.$ The isotropy subgroup at $q \\\\stackrel{def}{=}\\\\left( \\\\begin{array}{c}0\\\\\\\\ 2\\\\nu \\\\\\\\0\\\\\\\\ 1\\\\end{array}\\\\right)$ is $(Ker\\\\,C_\\\\alpha )_o .$ Let us denote by $\\\\mathrm {O}_q$ the orbit of $q$ by this representation.\",\n \"In order to describe $\\\\mathrm {O}_q$ in another way, we consider in $\\\\text{\\\\bf C}^4$ the hermitian product $<\\\\binom{\\\\hat{w}}{\\\\hat{z}},\\\\ \\\\binom{w}{z}>=\\\\frac{1}{2} (\\\\hat{w}^*z+\\\\hat{z}^*w)\\\\ \\\\ \\\\ \\\\ \\\\ \\\\forall \\\\hat{w},\\\\,\\\\hat{z},\\\\,w,\\\\,z \\\\in \\\\text{\\\\bf C}^2$ whose signature and quadratic form are respectively (+, +, -, -) and $\\\\Phi \\\\binom{w}{z} =Re\\\\,z^*\\\\,w.$ The complex vector space $\\\\text{\\\\bf C}^4$ provided with this hermitian product is Penrose's Twistor Space.\",\n \"The representation $\\\\mu _1$ preserves $\\\\Phi $ , so that any orbit must be contained in a subset of the form $\\\\Phi =\\\\mathnormal {constant}.$ It follows that the orbit of $q$ is contained in the seven dimensional submanifold, $\\\\mathcal {O}_\\\\nu $ , given by $\\\\Phi =2\\\\nu .$ Let $\\\\binom{w}{z}\\\\in \\\\text{\\\\bf C}^4$ be such that $sign\\\\left(\\\\Phi \\\\binom{w}{z} \\\\right)=sign(\\\\nu ),$ and define $A(w,z)&=&\\\\left(\\\\frac{2\\\\nu }{ \\\\Phi \\\\binom{w}{z}}\\\\right)^{1/2}\\\\left(\\\\begin{array}{c} {}\\\\\\\\\\\\epsilon \\\\overline{z}\\\\\\\\{} \\\\end{array}\\\\ \\\\Bigg | \\\\ \\\\frac{1}{2\\\\nu }\\\\left(i\\\\frac{Im(z^*w)}{\\\\Vert z \\\\Vert ^2}z-w\\\\right)\\\\right)\\\\\\\\ H(w,z) &=& -\\\\frac{Im(z^*w)}{\\\\Vert z \\\\Vert ^2}\\\\ I.$ Then we have $\\\\mu _1(A(w,z),H(w,z)) q=\\\\left(\\\\frac{2\\\\nu }{ \\\\Phi \\\\binom{w}{z}}\\\\right)^{1/2}\\\\ \\\\binom{w}{z}.$ This enables us to prove that each element of $\\\\mathcal {O}_\\\\nu $ is in the orbit, so that $\\\\mathrm {O}_q=\\\\mathcal {O}_\\\\nu $ .\",\n \"Then, the map from $G/(Ker\\\\,C_\\\\alpha )_o$ onto $ \\\\mathcal {O}_\\\\nu ,$ given by $\\\\tau _T:(A,H)\\\\,(Ker\\\\,C_\\\\alpha )_o\\\\in \\\\frac{G}{(Ker\\\\,C_\\\\alpha )_o} \\\\longrightarrow \\\\mu _1(A,H)\\\\,q\\\\in \\\\mathcal {O}_\\\\nu ,$ is a diffeomorphism.\",\n \"If we identify these spaces by $\\\\tau _T,$ the action of $\\\\@root T \\\\of {\\\\mathbf {1}}$ on $G/(Ker\\\\,C_\\\\alpha )_o$ given by (REF ), traslates to an action on $\\\\mathcal {O}_\\\\nu .$ Since $\\\\tau _T((A,H)\\\\,(Ker\\\\,C_\\\\alpha )_o) *r_j)&=&\\\\mu _1 ((A,H)\\\\,\\\\tilde{\\\\bf r}_j)q=\\\\mu _1 (A,H)\\\\mu _1(\\\\tilde{\\\\bf r}_j)q\\\\\\\\=\\\\mu _1 (A,H)(\\\\overline{r_j}q)&=&\\\\overline{r_j}\\\\ \\\\tau _T((A,H)\\\\,(Ker\\\\,C_\\\\alpha )_o) ,$ the action of $\\\\@root T \\\\of {\\\\mathbf {1}}$ on $\\\\mathcal {O}_\\\\nu $ is given by ordinary product by the conjugated: $V*r_j=\\\\overline{r_j}\\\\ V$ Let us denote by $\\\\mathcal {O}_\\\\nu /\\\\@root T \\\\of {\\\\mathbf {1}}$ the quotient space of $\\\\mathcal {O}_\\\\nu $ by this action.\",\n \"It follows from the preceeding construction that the contact manifold, $G/Ker\\\\,C_\\\\alpha ,$ is diffeomorphic, and will be identified, to $\\\\mathcal {O}_\\\\nu /\\\\@root T \\\\of {\\\\mathbf {1}}$ .\",\n \"The contact form will be determinated later on.\",\n \"Let ${}^t(w^1,w^2,z^1,z^2)\\\\in {\\\\mathbb {C}}^4-\\\\lbrace 0\\\\rbrace ,$ and let us denote by $[{}^t(w^1,w^2,z^1,z^2)]$ the complex vector subspace of ${\\\\mathbb {C}}^4$ generated by ${}^t(w^1,w^2,z^1,z^2).$ The set composed by these subspaces is the complex projective space of ${\\\\mathbb {C}}^4,$ ${\\\\bf P}_3(\\\\mathbb {C}),$ and we consider it as provided with its canonical differentiable structure.\",\n \"Penrose also considers the subsets of twistor space, ${\\\\bf T}^+,\\\\ {\\\\bf T}^-,\\\\ {\\\\bf T}^0,$ given by $\\\\Phi \\\\,>\\\\,0,\\\\ \\\\Phi \\\\,<\\\\,0,\\\\ \\\\Phi \\\\,=\\\\,0$ , respectively, and the subsets of projective space $ {\\\\bf P}_3^+=\\\\pi ({\\\\bf T}^+),\\\\ {\\\\bf P}_3^-=\\\\pi ({\\\\bf T}^-),\\\\ {\\\\bf P}_3^0=\\\\pi ({\\\\bf T}^0)$ , where $\\\\pi :{\\\\mathbb {C}}^4 -\\\\lbrace 0\\\\rbrace \\\\longrightarrow {\\\\bf P}_3({\\\\mathbb {C}}) $ is the canonical map.\",\n \"Since $\\\\Phi $ is preserved by the representation, the subsets ${\\\\bf T}^+,\\\\ {\\\\bf T}^-,\\\\ {\\\\bf T}^0,$ are stable under $\\\\mu _1$ .\",\n \"The representation $\\\\mu _1$ also defines an action of $G$ on ${\\\\bf P}_3(\\\\text{\\\\bf C}) $ such that $\\\\pi $ is equivariant.\",\n \"Explicitly $(A,H)*\\\\left[ \\\\left( \\\\begin{array}{c}w^1\\\\\\\\w^2\\\\\\\\z^1\\\\\\\\ z^2\\\\end{array}\\\\right)\\\\right]=\\\\left[\\\\left( \\\\begin{array}{cc}A&{-iH{A^*}^{-1}} \\\\\\\\0&{A^*}^{-1}\\\\end{array} \\\\right)\\\\left( \\\\begin{array}{c}w^1\\\\\\\\w^2\\\\\\\\z^1\\\\\\\\ z^2\\\\end{array}\\\\right)\\\\right].$ As a consecuence of formula (REF ), the open subsets of ${\\\\bf P}_3(\\\\text{\\\\bf C}) $ denoted by ${\\\\bf P}_3^+$ and ${\\\\bf P}_3^-$ are orbits of this action.\",\n \"The point $[q]$ of ${\\\\bf P}_3(\\\\text{\\\\bf C}) $ is obviously in ${\\\\bf P}_3^{sign(\\\\nu )}$ so that its orbit is this open subset.\",\n \"The isotropy subgroup at $[q]$ is $G_\\\\alpha .$ As a consecuence we have a diffeomorphism from ${\\\\bf P}_3^{sign(\\\\nu )}$ onto the coadjoint orbit of $\\\\alpha ,$ given by $\\\\Pi : [\\\\mu _1(A,H)\\\\,q]\\\\in {\\\\bf P}_3^{sign(\\\\nu )} \\\\longrightarrow Ad^*_{(A,H)}\\\\alpha \\\\in {\\\\mathcal {M}S} .$ were I have denoted the coadjoint orbit by ${\\\\mathcal {M}S},$ because of its interpretation as Movement Space.\",\n \"We have $\\\\Pi \\\\left(\\\\left[ \\\\binom{ w}{ z} \\\\right]\\\\right)= Ad^*_{(A(w,z),H(w,z))}\\\\alpha $ so that, the formula for coadjoint representation in section ,(REF ), thus leads, after some computation, to $\\\\Pi \\\\left( \\\\left[ \\\\binom{ w}{ z} \\\\right] \\\\right)=\\\\frac{2\\\\nu \\\\eta }{ \\\\Phi \\\\binom{ w}{ z}} \\\\biggl \\\\lbrace \\\\frac{i}{4}(wz^*+\\\\epsilon \\\\overline{zw^*}\\\\epsilon ),\\\\ - \\\\epsilon \\\\overline{zz^*}\\\\epsilon \\\\biggr \\\\rbrace .$ We identify $P_3^{sign(\\\\nu )}$ to ${\\\\mathcal {M}S},$ by means of $\\\\Pi .$ Section provides us with well defined expresions for linear and angular momentum in $\\\\mathcal {M}S,$ $c.f.$ (REF ), which, when composed with $\\\\Pi ,$ give expressions for linear and angular momentum in $P^{sign(\\\\nu )}_3.$ In its hermitian form, these expressions are $h(P\\\\left( \\\\left[ \\\\binom{ w}{ z} \\\\right] \\\\right))&=& \\\\frac{2\\\\eta \\\\nu }{\\\\Phi \\\\binom{ w}{ z}} \\\\epsilon \\\\overline{zz^*}\\\\epsilon \\\\\\\\ h(\\\\vec{l} \\\\left( \\\\left[ \\\\binom{ w}{ z} \\\\right] \\\\right),0)&=&\\\\frac{\\\\eta \\\\nu }{2\\\\Phi \\\\binom{ w}{ z}} (zw^*+wz^*+\\\\epsilon \\\\overline{(zw^*+wz^*)}\\\\epsilon )\\\\\\\\ h(\\\\vec{g} \\\\left( \\\\left[ \\\\binom{ w}{ z} \\\\right] \\\\right),0)&=&\\\\frac{i\\\\,\\\\eta \\\\nu }{2\\\\Phi \\\\binom{ w}{ z}}(zw^*-wz^*-\\\\epsilon \\\\overline{(zw^*-wz^*)}\\\\epsilon )$ The Pauli-Lubanski four vector, when evaluated according with (REF ), is found to be $h(W\\\\left( \\\\left[ \\\\binom{ w}{ z} \\\\right] \\\\right)) =-\\\\eta \\\\nu \\\\,h(P\\\\left( \\\\left[ \\\\binom{ w}{ z}\\\\right] \\\\right)).\",\n \"$ In case T=1, the bundle map of the contact manifold onto the coadjoint orbit becomes the canonical map $\\\\pi _1:\\\\binom{ w}{ z} \\\\in \\\\mathcal {O}_1\\\\ \\\\longrightarrow \\\\left[ \\\\binom{ w}{ z} \\\\right]\\\\in P^{sign(\\\\nu )}_3,$ and in the general case, the bundle map is $\\\\pi _\\\\nu :\\\\binom{ w}{z} \\\\@root T \\\\of {\\\\mathbf {1}}\\\\in \\\\mathcal {O}_\\\\nu /\\\\@root T \\\\of {\\\\mathbf {1}} \\\\longrightarrow \\\\left[ \\\\binom{ w}{ z} \\\\right]\\\\in P^{sign(\\\\nu )}_3.$ Then, the composition with $\\\\pi _\\\\nu $ of the canonical dynamical variables, are also given by the right hand sides of (), () and (), but taking $\\\\Phi =2\\\\nu .$ We obtain the following formulae for the components of these dynamical variables $P^1\\\\left(\\\\binom{ w}{ z} \\\\@root T \\\\of {\\\\mathbf {1}}\\\\right) &=\\\\frac{\\\\eta }{2}(z^1\\\\overline{z}^2\\\\,+\\\\,z^2\\\\overline{z}^1) \\\\\\\\P^2\\\\left(\\\\binom{ w}{ z} \\\\@root T \\\\of {\\\\mathbf {1}}\\\\right)&=\\\\frac{i\\\\eta }{2}(z^1\\\\overline{z}^2\\\\,-\\\\,z^2\\\\overline{z}^1)\\\\\\\\P^3\\\\left(\\\\binom{ w}{ z} \\\\@root T \\\\of {\\\\mathbf {1}}\\\\right)&=\\\\frac{\\\\eta }{2}(\\\\vert z^1\\\\vert ^2\\\\,-\\\\,\\\\vert z^2\\\\vert ^2) \\\\\\\\P^4\\\\left(\\\\binom{ w}{ z} \\\\@root T \\\\of {\\\\mathbf {1}}\\\\right)&=-\\\\frac{\\\\eta }{2}(\\\\vert z^1\\\\vert ^2\\\\,+\\\\,\\\\vert z^2\\\\vert ^2)$ $l^1\\\\left(\\\\binom{ w}{ z} \\\\@root T \\\\of {\\\\mathbf {1}}\\\\right)&=\\\\frac{\\\\eta }{4} (z^1\\\\overline{w}^2+w^1\\\\overline{z}^2+z^2\\\\overline{w}^1+w^2\\\\overline{z}^1) \\\\\\\\l^2\\\\left(\\\\binom{ w}{ z} \\\\@root T \\\\of {\\\\mathbf {1}}\\\\right)&=\\\\frac{i\\\\eta }{4} (z^1\\\\overline{w}^2+w^1\\\\overline{z}^2-z^2\\\\overline{w}^1-w^2\\\\overline{z}^1) \\\\\\\\l^3\\\\left(\\\\binom{ w}{ z} \\\\@root T \\\\of {\\\\mathbf {1}}\\\\right)&=\\\\frac{\\\\eta }{4} (z^1\\\\overline{w}^1+w^1\\\\overline{z}^1-z^2\\\\overline{w}^2-w^2\\\\overline{z}^2)$ $g^1\\\\left(\\\\binom{ w}{ z} \\\\@root T \\\\of {\\\\mathbf {1}}\\\\right)&=\\\\frac{i\\\\eta }{4} (z^1\\\\overline{w}^2-w^1\\\\overline{z}^2+z^2\\\\overline{w}^1-w^2\\\\overline{z}^1) \\\\\\\\g^2\\\\left(\\\\binom{ w}{ z} \\\\@root T \\\\of {\\\\mathbf {1}}\\\\right)&=-\\\\frac{\\\\eta }{4} (z^1\\\\overline{w}^2-w^1\\\\overline{z}^2-z^2\\\\overline{w}^1+w^2\\\\overline{z}^1) \\\\\\\\g^3\\\\left(\\\\binom{ w}{ z} \\\\@root T \\\\of {\\\\mathbf {1}}\\\\right)&=\\\\frac{i\\\\eta }{4} (z^1\\\\overline{w}^1-w^1\\\\overline{z}^1-z^2\\\\overline{w}^2+w^2\\\\overline{z}^2).$ We can also consider the functions on ${\\\\bf T}^{sgn(\\\\nu )}$ obtained by composing the dynamical variables in $\\\\mathcal {M}S$ with the canonical projection, $\\\\pi ,$ from ${\\\\bf T}^{sgn(\\\\nu )}$ onto ${\\\\bf P}_3^{sgn(\\\\nu )},$ thus obtaining functions whose expresion is as the right hand sides of the preceeding formulae multiplied by $\\\\frac{2\\\\nu }{ \\\\Phi \\\\binom{ w}{ z}}$ These expressions coincide, up to notational conventions, with the expressions that R.Penrose gives for its energy - momentum and angular momentum in twistor space.\",\n \"We denote these functions by $\\\\widetilde{P}^k,\\\\ \\\\widetilde{l}^k,\\\\ \\\\widetilde{g}^k.$ In general, for all $(a,h)$ in the Lie algebra of $G,$ the function it defines on the coadjoint orbit, identified to ${\\\\bf P}_3^{sgn(\\\\nu )},$ is denoted by the same symbol, $(a,h),$ and its composition with $\\\\pi ,$ $\\\\widetilde{(a,h)}.$ In a similar way, the infinitesimal generator of the action on ${\\\\bf T}^{sgn(\\\\nu )}$ defined by the representation $\\\\mu _1,$ associated to $(a,h),\\\\ i.e.$ the vector field whose flow is given by $\\\\mu _1(Exp(-t(a,h))),$ is denoted by $\\\\widetilde{X_{(a,h)} }.$ Let us consider the following one form on ${\\\\bf T}^{sgn(\\\\nu )}$ $\\\\omega _0=\\\\frac{i\\\\eta \\\\nu }{2\\\\Phi } (z^1d\\\\overline{w}^1+w^1d\\\\overline{z}^1+z^2d\\\\overline{w}^2+ w^2d\\\\overline{z}^2 -$ $- \\\\overline{z}^1 d w^1- \\\\overline{w}^1d z^1-\\\\overline{z}^2d w^2-\\\\overline{w}^2dz^2)$ A computation lead us to $\\\\omega _0\\\\left( \\\\widetilde{X_{(a,h)} }\\\\right)= -\\\\widetilde{(a,h)}.$ Let us denote by $\\\\omega $ the restriction of $\\\\omega _0$ to $\\\\mathcal {O}_\\\\nu $ .\",\n \"The one form $\\\\omega $ is invariant by the action of $\\\\@root T \\\\of {\\\\mathbf {1}},$ so that it projects to a well defined one form on $\\\\mathcal {O}_\\\\nu /\\\\@root T \\\\of {\\\\mathbf {1}},$ that we denote by $\\\\omega _\\\\nu .$ We know that $\\\\mathcal {O}_\\\\nu /\\\\@root T \\\\of {\\\\mathbf {1}},$ represent the homogeneous contact manifold corresponding to the kind of particle under consideration.\",\n \"As a consecuence of (REF ) and (REF ) is not difficult to prove that $\\\\omega _\\\\nu $ is the contact form.\",\n \"As a consecuence of the fact that the map of $\\\\mathcal {O}_\\\\nu $ on $\\\\mathcal {O}_\\\\nu /\\\\@root T \\\\of {\\\\mathbf {1}}$ is a covering map, $\\\\omega $ is also a contact form on $\\\\mathcal {O}_\\\\nu ,$ that becomes itself an homogeneous contact manifold.\",\n \"The two form $d(\\\\omega _\\\\nu )$ thus projects under $\\\\pi _\\\\nu $ on the symplectic form on $P_3^{sign(\\\\nu )},\\\\ \\\\Omega _\\\\nu ,$ that corresponds to Kirillov form in the identification of $P_3^{sign(\\\\nu )}$ with the coadjoint orbit.\",\n \"Then, $d\\\\omega $ also projects on $\\\\Omega _\\\\nu ,$ under the restriction of the canonical map $\\\\pi $ to $\\\\mathcal {O}_\\\\nu .$ But $d\\\\omega $ is the restriction to $\\\\mathcal {O}_\\\\nu $ of $d\\\\omega _0$ and we have $d\\\\omega _0=\\\\frac{i\\\\eta \\\\nu }{ \\\\Phi } (dz^1\\\\wedge d\\\\overline{w}^1+dw^1\\\\wedge d\\\\overline{z}^1+ dz^2\\\\wedge d\\\\overline{w}^2+dw^2 \\\\wedge d\\\\overline{z}^2 )+(d\\\\Phi )\\\\wedge \\\\delta ,$ where $\\\\delta $ is a one form.\",\n \"Since $d\\\\Phi $ vanishes on $\\\\mathcal {O}_\\\\nu ,$ it follows that $d\\\\omega $ is also the restriction to $\\\\mathcal {O}_\\\\nu $ of $\\\\Omega =\\\\frac{i\\\\eta \\\\nu }{ \\\\Phi } (dz^1\\\\wedge d\\\\overline{w}^1+dw^1\\\\wedge d\\\\overline{z}^1+ dz^2\\\\wedge d\\\\overline{w}^2+dw^2 \\\\wedge d\\\\overline{z}^2).$ Thus, we can obtain explicit expressions of $\\\\Omega _\\\\nu $ as follows: for each differentiable section of $\\\\pi $ with values in $\\\\mathcal {O}_\\\\nu $ $\\\\sigma : U \\\\rightarrow {\\\\cal O}_\\\\nu \\\\subset {\\\\bf T}^{sgn(\\\\nu )},$ we have on the open set $U$ $\\\\Omega _\\\\nu =\\\\sigma ^*\\\\Omega _0,$ Also we have $\\\\Omega _\\\\nu =d\\\\,\\\\left(\\\\sigma ^*\\\\omega _0 \\\\right).$\"\n ],\n [\n \"Local expression of the symplectic form.\",\n \"In ${{\\\\mathbb {C}}}^{3}$ we define ${\\\\cal D}=\\\\lbrace (t,u,v):\\\\,sign(\\\\nu )\\\\,\\\\Re (\\\\phi (t,u,v)) > 0\\\\rbrace ,$ where $\\\\phi (t,u,v)=u+\\\\overline{t} v$ and $\\\\Re {}$ stands for real part.\",\n \"In ${\\\\bf P }^{sgn(\\\\nu )}_3$ we define ${\\\\cal D}_1=\\\\left\\\\lbrace \\\\left[\\\\begin{array}{c}w^1\\\\\\\\w^2\\\\\\\\z^1\\\\\\\\z^2 \\\\end{array} \\\\right]:w^1\\\\ne 0,\\\\ sign(\\\\nu )\\\\,\\\\Re (\\\\Phi (\\\\left(\\\\begin{array}{c}w^1\\\\\\\\w^2\\\\\\\\z^1\\\\\\\\z^2 \\\\end{array}\\\\right))) > 0\\\\right\\\\rbrace $ ${\\\\cal D}_2=\\\\left\\\\lbrace \\\\left[\\\\begin{array}{c}w^1\\\\\\\\w^2\\\\\\\\z^1\\\\\\\\z^2 \\\\end{array} \\\\right]:w^2\\\\ne 0,\\\\ sign(\\\\nu )\\\\,\\\\Re (\\\\Phi (\\\\left(\\\\begin{array}{c}w^1\\\\\\\\w^2\\\\\\\\z^1\\\\\\\\z^2 \\\\end{array}\\\\right))) > 0\\\\right\\\\rbrace $ ${\\\\cal D}_3=\\\\left\\\\lbrace \\\\left[\\\\begin{array}{c}w^1\\\\\\\\w^2\\\\\\\\z^1\\\\\\\\z^2 \\\\end{array} \\\\right]:z^1\\\\ne 0,\\\\ sign(\\\\nu )\\\\,\\\\Re (\\\\Phi (\\\\left(\\\\begin{array}{c}w^1\\\\\\\\w^2\\\\\\\\z^1\\\\\\\\z^2 \\\\end{array}\\\\right))) > 0\\\\right\\\\rbrace $ ${\\\\cal D}_4=\\\\left\\\\lbrace \\\\left[\\\\begin{array}{c}w^1\\\\\\\\w^2\\\\\\\\z^1\\\\\\\\z^2 \\\\end{array} \\\\right]:z^2\\\\ne 0,\\\\ sign(\\\\nu )\\\\,\\\\Re (\\\\Phi (\\\\left(\\\\begin{array}{c}w^1\\\\\\\\w^2\\\\\\\\z^1\\\\\\\\z^2 \\\\end{array}\\\\right))) > 0\\\\right\\\\rbrace .$ The ${\\\\cal D}_k$ compose an open cover of ${\\\\bf P }^{sgn(\\\\nu )}_3.$ The maps $\\\\psi _1: (t,u,v)\\\\in {\\\\cal D} \\\\rightarrow \\\\left[\\\\begin{array}{c}1\\\\\\\\t\\\\\\\\u\\\\\\\\v \\\\end{array} \\\\right] \\\\in {\\\\cal D}_1$ $\\\\psi _2: (t,u,v)\\\\in {\\\\cal D} \\\\rightarrow \\\\left[\\\\begin{array}{c}v\\\\\\\\1\\\\\\\\t\\\\\\\\ u \\\\end{array} \\\\right] \\\\in {\\\\cal D}_2$ $\\\\psi _3: (t,u,v)\\\\in {\\\\cal D} \\\\rightarrow \\\\left[\\\\begin{array}{c}u\\\\\\\\v\\\\\\\\1\\\\\\\\t \\\\end{array} \\\\right] \\\\in {\\\\cal D}_3$ $\\\\psi _4: (t,u,v)\\\\in {\\\\cal D} \\\\rightarrow \\\\left[\\\\begin{array}{c}t\\\\\\\\ u \\\\\\\\v\\\\\\\\1 \\\\end{array} \\\\right] \\\\in {\\\\cal D}_4$ are such that the $({\\\\cal D}_k,(\\\\psi _k)^{-1})$ compose an atlas of ${\\\\bf P }^{sgn(\\\\nu )}_3.$ For all of these charts the coordinates will be denoted by $(t,u,v).$ We also define sections of $\\\\pi _\\\\nu ,$ $\\\\sigma _k:{\\\\cal D}_k \\\\rightarrow {\\\\cal O}_\\\\nu ,\\\\ \\\\ \\\\ \\\\ k=1,\\\\dots ,4,$ by means of $\\\\sigma _1\\\\circ \\\\psi _1: (t,u,v)\\\\in {\\\\cal D} \\\\rightarrow F(t,u,v)\\\\left(\\\\begin{array}{c}1\\\\\\\\t\\\\\\\\u\\\\\\\\v \\\\end{array} \\\\right) \\\\in {\\\\cal D}_1$ $\\\\sigma _2\\\\circ \\\\psi _2: (t,u,v)\\\\in {\\\\cal D} \\\\rightarrow F(t,u,v)\\\\left(\\\\begin{array}{c}v\\\\\\\\1\\\\\\\\t\\\\\\\\ u \\\\end{array} \\\\right) \\\\in {\\\\cal D}_2$ $\\\\sigma _3\\\\circ \\\\psi _3: (t,u,v)\\\\in {\\\\cal D} \\\\rightarrow F(t,u,v)\\\\left(\\\\begin{array}{c}u\\\\\\\\v\\\\\\\\1\\\\\\\\t \\\\end{array} \\\\right) \\\\in {\\\\cal D}_3$ $\\\\sigma _4\\\\circ \\\\psi _4: (t,u,v)\\\\in {\\\\cal D} \\\\rightarrow F(t,u,v)\\\\left(\\\\begin{array}{c}t\\\\\\\\ u \\\\\\\\v\\\\\\\\1 \\\\end{array} \\\\right) \\\\in {\\\\cal D}_4$ where $F(t,u,v)=\\\\sqrt{\\\\frac{2\\\\nu }{\\\\Re (\\\\phi (t,u,v))}}.$ For all $k$ we have $(\\\\sigma _k\\\\circ \\\\psi _k)^*\\\\omega = \\\\frac{\\\\eta \\\\nu }{\\\\Re (\\\\phi )}(d(\\\\Im (\\\\phi ))+i(v\\\\,d\\\\overline{t}-\\\\overline{v}\\\\,dt)),$ Where $\\\\Im (\\\\phi )$ is the imaginary part of $\\\\phi .$ Then, the local expression of $\\\\Omega _\\\\nu $ in all of these coordinate systems can be evaluated by means of $\\\\Omega _\\\\nu \\\\stackrel{loc}{=}d\\\\,((\\\\sigma _k\\\\circ \\\\psi _k)^*\\\\omega ).$ for all $k.$ Notice that $\\\\omega $ is not projectable on ${\\\\bf P }^{sgn(\\\\nu )}_3.$ Thus (REF ) need not be local expressions of a well defined 1-form on all of ${\\\\bf P }^{sgn(\\\\nu )}_3.$\"\n ],\n [\n \"Symmetric Wave Functions of the Photon.\",\n \"There are many proposals in the literature for Wave Functions of the Photon.\",\n \"Including someones that asserts that such a thing does not exist.\",\n \"In this paper I have described the Wave Functions of the Massless Type 4 Particles, where the case $T=2$ corresponds to the Photon.\",\n \"These Wave Functions satisfies the Penrose Wave Equations.\",\n \"In this section I describe other forms of the Wave Functions of Photon.\",\n \"Equivalent representations.These forms enables us to relate directly the Wave Functions with the Electromagnetic Potential and the Electromagnetic Field.\",\n \"Photon is the name of four kinds of particles: the massless Type 4 particles with $T=2,$ $i.e.$ those corresponding to $\\\\gamma =\\\\left\\\\lbrace \\\\frac{i \\\\chi }{4 \\\\pi }\\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & -1\\\\end{array}\\\\right),\\\\eta \\\\,\\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & 0\\\\end{array}\\\\right)\\\\right\\\\rbrace \\\\ \\\\ \\\\ \\\\ ,\\\\ \\\\chi ,\\\\ \\\\eta \\\\in \\\\lbrace \\\\pm 1\\\\rbrace ,$ We already know that the value of $-\\\\eta $ is the sign of energy and that $-\\\\eta \\\\chi $ is the sign of helicity ($c.f.$ section REF ).\",\n \"We denote $\\\\ell =-\\\\eta \\\\chi $ .\",\n \"From section we obtain $\\\\begin{array}{l}\\\\vspace{7.22743pt}G_\\\\gamma = \\\\left\\\\lbrace \\\\left(\\\\left(\\\\begin{array}{cc}z&a \\\\\\\\0&{\\\\overline{z}}\\\\end{array}\\\\right),\\\\ \\\\left(\\\\begin{array}{cc}b&i \\\\chi \\\\eta a z/ \\\\pi \\\\\\\\\\\\overline{i \\\\chi \\\\eta a z / \\\\pi }&0\\\\end{array}\\\\right)\\\\right):\\\\ z \\\\in {\\\\bf S^1},\\\\right.\",\n \"\\\\\\\\\\\\left.\",\n \"\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ b\\\\in {\\\\bf R},\\\\ a \\\\in {\\\\bf C}\\\\right\\\\rbrace \\\\\\\\\\\\ \\\\\\\\\\\\mathrm {and\\\\ the\\\\ homomorphism} \\\\\\\\\\\\ \\\\\\\\C_\\\\gamma \\\\left( \\\\left( \\\\left(\\\\begin{array}{cc}z&a \\\\\\\\0&{\\\\overline{z}}\\\\end{array}\\\\right),\\\\ \\\\left(\\\\begin{array}{cc}b&{i \\\\chi \\\\eta a z/ \\\\pi } \\\\\\\\{\\\\overline{i \\\\chi \\\\eta a z/ \\\\pi }}&0\\\\end{array}\\\\right)\\\\right)\\\\right)=z^{2\\\\chi }\\\\end{array}$ has differential $\\\\gamma .$ $\\\\begin{array}{l}({\\\\mbox{$G_\\\\gamma $}} )_{SL}=\\\\left\\\\lbrace \\\\left(\\\\begin{array}{cc} z&a \\\\\\\\0&{\\\\overline{z}}\\\\end{array}\\\\right): z \\\\in {\\\\bf S^1},\\\\ a \\\\in {\\\\mathbb {C}} \\\\right\\\\rbrace \\\\\\\\\\\\vspace{7.22743pt}(C_\\\\gamma )_{SL}\\\\left(\\\\ \\\\left(\\\\begin{array}{cc}z&a \\\\\\\\0&{\\\\overline{z}}\\\\end{array}\\\\right)\\\\right)=\\\\ z^{2\\\\chi } \\\\\\\\\\\\vspace{7.22743pt}SL_1=\\\\left\\\\lbrace \\\\left(\\\\begin{array}{cc}a&0 \\\\\\\\0&{1/a}\\\\end{array}\\\\right):\\\\ a \\\\in {\\\\mathbb {C}}\\\\right\\\\rbrace \\\\\\\\\\\\vspace{7.22743pt}SL_2=(G_\\\\gamma )_{SL} \\\\\\\\\\\\vspace{7.22743pt}SL_1 \\\\cap SL_2=\\\\left\\\\lbrace \\\\left(\\\\begin{array}{cc}z&0 \\\\\\\\0&\\\\overline{z}\\\\end{array}\\\\right):\\\\ z \\\\in {\\\\bf S}^1\\\\right\\\\rbrace \\\\end{array}.$ In section we have identified the space $SL/(G_\\\\gamma )_{SL}$ to the future lightcone, ${\\\\mbox{${\\\\bf C}^+$}}=\\\\lbrace H \\\\in {H(2)}:\\\\ Det\\\\,H=0,\\\\ Tr\\\\,H>0\\\\rbrace ,$ by means of the diffeomorphism $A\\\\, (G_\\\\gamma )_{SL} \\\\in SL/(G_\\\\gamma )_{SL} \\\\mapsto A \\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & 0\\\\end{array}\\\\right) A^* \\\\in {\\\\bf C}^+.$ In this section I give another description of the wave functions of a photon, by means of different trivialisations than the ones I have used in section .\",\n \"Of course these trivialisations are “isomorphic”, in an obvious sense.\",\n \"Let us consider first the cases $\\\\chi =+1$ $i.e.$ $\\\\ell =-\\\\eta .$ In these cases, a trivialisation ($c.\\\\,f.$ section ) is $(\\\\mu _{+}, { \\\\cal {S}},s_0^+),$ where $\\\\cal S$ is the vector subspace of $gl(2,\\\\bf C),$ composed by the symmetric matrices, $s_0^+=\\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & 0\\\\end{array}\\\\right)$ and $\\\\mu _+(A)\\\\cdot s=A \\\\,s\\\\, {}^tA,$ for all $A \\\\in SL(2,{\\\\bf C}),\\\\ s\\\\in \\\\cal S$ .\",\n \"Since $\\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & 0\\\\end{array}\\\\right)=\\\\left(\\\\begin{array}{c} 1 \\\\\\\\0\\\\end{array}\\\\right)\\\\left(\\\\begin{array}{cc} 1 & 0\\\\end{array}\\\\right) $ the orbit of $\\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & 0\\\\end{array}\\\\right)$ by $\\\\mu _+$ is composed by the elements of ${\\\\cal S}^2$ of the form $(A \\\\left(\\\\begin{array}{c} 1 \\\\\\\\0\\\\end{array}\\\\right) )\\\\ {}^t (A \\\\left(\\\\begin{array}{c} 1 \\\\\\\\0\\\\end{array}\\\\right) ),$ for some $A \\\\in SL(2,\\\\bf C)$ .\",\n \"Then, one sees that the orbit is contained in ${\\\\cal B}=\\\\lbrace z \\\\ {}^t z: z\\\\in {\\\\bf C}^2-\\\\lbrace 0\\\\rbrace \\\\rbrace .$ But every $z\\\\in {\\\\bf C}^2-\\\\lbrace 0\\\\rbrace $ has the form $ A\\\\, \\\\left(\\\\begin{array}{c} 1 \\\\\\\\0\\\\end{array}\\\\right),$ for some $A \\\\in SL(2,{\\\\bf C})$ , so that the orbit coincides with ${\\\\cal B}$ .\",\n \"For each $ s\\\\in {\\\\cal S}$ such that $s\\\\ne 0$ and $Det\\\\, s=0,$ there exist exactly two $z\\\\in {\\\\bf C}^2-{0}$ such that $s=z \\\\ {}^t z.$ In fact, if $s=\\\\left(\\\\begin{array}{cc} a & b\\\\\\\\b & d\\\\end{array}\\\\right)\\\\in B$ the $z$ such that $s=z \\\\ {}^t z$ are $\\\\pm \\\\left(\\\\begin{array}{c} \\\\alpha \\\\\\\\\\\\sigma \\\\delta \\\\end{array}\\\\right)$ where $\\\\alpha $ is a square root of $a$ , $\\\\delta $ is a square root of $d$ , and $\\\\sigma \\\\in \\\\lbrace \\\\pm 1\\\\rbrace $ such that $b=\\\\sigma \\\\alpha \\\\delta .$ As a consequence, we also have ${\\\\cal B}=\\\\lbrace s\\\\in {\\\\cal S}: s\\\\ne 0,Det\\\\, s=0\\\\rbrace .$ The homogeneous space $SL/Ker(C_\\\\gamma )_{SL}$ is identified to ${\\\\cal B}$ by means of $ A\\\\,Ker(C_\\\\gamma )_{SL} \\\\in SL/Ker(C_\\\\gamma )_{SL}\\\\longrightarrow \\\\mu _+(A)\\\\cdot s_0^+ \\\\in {\\\\cal B}$ Then, the canonical map $SL/Ker(C_\\\\gamma )_{SL} \\\\longrightarrow SL/(G_\\\\gamma )_{SL},$ denoted in the following by $r_+,$ becomes $r_+:(A \\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & 0\\\\end{array}\\\\right){}^tA)\\\\in {\\\\cal B} \\\\longrightarrow A \\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & 0\\\\end{array}\\\\right)A^*\\\\in C^+,$ for all $A\\\\in SL,$ so that $r_+(z{}^t z)= zz^*$ for all $z \\\\in {\\\\bf C}^2-{0}$ .\",\n \"By elementary operations with matrices, one can prove that the relation $r_+(s)=C $ is equivalent to $C=(+(Tr\\\\,s\\\\overline{s})^{-1/2})\\\\,s\\\\overline{s}$ and also to $s=(+(Tr\\\\,C {}^t C)^{-1/2})e^{i\\\\phi }(C {}^t C)$ for some $\\\\phi \\\\in {\\\\bf R}.$ To obtain Wave Functions, we need functions on ${\\\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers.\",\n \"If $f$ is one of such functions the corresponding Prewave Function is $ \\\\psi _f^{(-\\\\eta ,-\\\\eta )}(H,K)=f(s)\\\\,s\\\\,e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,h^{-1}(H)\\\\rangle _m},$ where $s$ is arbitrary in $r_+^{-1}(K).$ In ${(-\\\\eta ,-\\\\eta )}$ the first $-\\\\eta $ stands for the sign of energy and the second by $\\\\ell ,$ helicity.\",\n \"Now, let us consider the cases $\\\\chi =-1$ $i.e.$ $\\\\ell =\\\\eta .$ In these cases, a trivialisation is $(\\\\mu _{-}, { \\\\cal {S}},s_0^-),$ where $\\\\cal S$ is as in the $\\\\chi =+1$ case, $s_0^-=\\\\left(\\\\begin{array}{cc} 0 & 0\\\\\\\\0 & 1\\\\end{array}\\\\right)$ and $\\\\mu _-(A)\\\\cdot s=(A^*)^{-1} \\\\,s\\\\, (\\\\overline{A})^{-1},$ for all $A \\\\in SL(2,{\\\\bf C}),\\\\ s\\\\in \\\\cal S$ .\",\n \"The orbit of $\\\\left(\\\\begin{array}{cc} 0 & 0\\\\\\\\0 & 1\\\\end{array}\\\\right)$ by $\\\\mu _-$ is, as in case $\\\\chi =+1$ , ${\\\\cal B}$ .\",\n \"The canonical map $r_-: SL/Ker(C_{\\\\gamma })_{SL} \\\\longrightarrow SL/(G_{\\\\gamma })_{SL}$ becomes a map from ${\\\\cal B}$ onto ${\\\\bf C}^+,$ given by $r_-( \\\\mu _-(A)\\\\cdot s_0^-)=A \\\\left(\\\\begin{array}{cc} 1 & 0\\\\\\\\0 & 0\\\\end{array}\\\\right)A^*.$ The maps $r_+$ and $r_-$ are geometrically related as follows.\",\n \"Let us consider the map $J:s\\\\in {\\\\cal S} \\\\longrightarrow -\\\\epsilon \\\\overline{s}\\\\epsilon \\\\in {\\\\cal S} .$ This is an antilinear map with $J^2=I.$ Since $J(z{}^tz)=(-\\\\epsilon \\\\overline{z}){}^t(-\\\\epsilon \\\\overline{z}),$ we have $J({\\\\cal B})={\\\\cal B}.$ On the other hand $r_-\\\\circ J(\\\\mu _+(A)\\\\cdot s_0^+)=r_-(-\\\\epsilon \\\\, \\\\overline{A}\\\\,\\\\overline{s_0^+}\\\\,A^*\\\\,\\\\epsilon )=r_-(-(A^*)^{-1}\\\\,\\\\epsilon \\\\,\\\\overline{s_0^+} \\\\,\\\\epsilon \\\\, (\\\\overline{A})^{-1})=\\\\\\\\=r_-(\\\\mu _-(A)\\\\cdot s_0^-)=A\\\\ ({\\\\mbox{$G_\\\\gamma $}} )_{SL}=r_+(\\\\mu _+(A)\\\\cdot s_0^+) , $ so that $r_+=r_-\\\\circ J.$ Since $J^2=I,$ we also have $r_-=r_+\\\\circ J.$ As a consequence, $J$ stablishes a principal fibre bundle isomorphism over the identical map of $C^+$ , the isomorphism of structural groups being the map defined by sending each element of $S^1$ to its inverse.\",\n \"Another consequence is $r_-(z {}^tz)=-\\\\epsilon \\\\overline{zz^*}\\\\epsilon .$ If $f$ is a function on ${\\\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers, it also defines a Prewave Function for this kind of photon by means of $ \\\\psi _f^{(-\\\\eta ,\\\\eta )}(H,K)=f(s)\\\\,s\\\\,e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,h^{-1}(H)\\\\rangle _m},$ where $s,$ now, is arbitrary in $r_-^{-1}(K).$ Here, in ${(-\\\\eta ,\\\\eta )},$ $-\\\\eta $ stands for the sign of energy and the second $\\\\eta ,$ which in the present case coincides with $\\\\ell ,$ stands for helicity.\",\n \"Thus, the Prewave Functions corresponding to the four kinds of Photon are different.\",\n \"The sign of energy, $-\\\\eta ,$ and the sign of helicity, $\\\\ell ,$ determines the type of Prewave Functions to be used, $\\\\psi _f^{(-\\\\eta , \\\\ell )},$ that are given by (REF ) or (REF ).\",\n \"When $f$ is continuous with compact support, the corresponding Wave Function is $\\\\widetilde{\\\\psi }_f^{(-\\\\eta , \\\\ell )}(x)=\\\\int _{C^+} \\\\psi _f^{(-\\\\eta , \\\\ell )}(h(x),K) \\\\omega _K,$ where $\\\\omega $ is the invariant volume element on $C^+$ defined in (REF ).\",\n \"These Wave Functions represent states whose energy has sign $-\\\\eta $ and are eigenvectors of helicity ($c.f.\",\n \"$ section REF ) corresponding to the eigenvalues $\\\\frac{\\\\ell }{2\\\\pi }.$\"\n ],\n [\n \"Prehilbert Space estructure.\",\n \"Let us denote by ${\\\\cal {F}}$ the vector subspace composed by the functions on ${\\\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers, and by ${\\\\cal {F}}_c$ the subspace of ${\\\\cal {F}}$ composed by the continuous elements with compact support.\",\n \"The space ${\\\\cal {F}}_c$ can be provided with a prehilbert space structure, by means of the general method described in section , for each kind of Photon.\",\n \"In more detail we proceed as follows.\",\n \"We separate the cases ${-\\\\eta \\\\ell }=\\\\pm 1.$ If $\\\\ f , f^\\\\prime \\\\in {\\\\cal F}_c,$ its hermitian product is in each case $\\\\langle f,\\\\, f^\\\\prime \\\\rangle _{(-\\\\eta , \\\\ell )}= \\\\int _{C^+} (\\\\overline{f} f^\\\\prime )_{(-\\\\eta , \\\\ell )}\\\\ {\\\\mbox{$\\\\omega $}} , $ where $ (\\\\overline{f} f^\\\\prime )_{(-\\\\eta , \\\\ell )}$ is the function defined on $C^+$ by $(\\\\overline{f} f^\\\\prime )_{(-\\\\eta , \\\\ell )} (K)=\\\\overline{f(s)} f^\\\\prime (s)$ for all $K\\\\in C^+,$ where $\\\\ s\\\\in r_{-\\\\eta \\\\ell }^{-1}(K).$ With each of these inner products, ${\\\\cal F}_c$ becomes a prehilbert space.\",\n \"For prewave functions we define $\\\\langle \\\\psi _f^{(-\\\\eta , \\\\ell )},\\\\, \\\\psi _{f^\\\\prime }^{(-\\\\eta , \\\\ell )} \\\\rangle _{(-\\\\eta , \\\\ell )}= \\\\langle f,\\\\, f^\\\\prime \\\\rangle _{(-\\\\eta , \\\\ell )}.$ A sexquilinear form on S is given by $\\\\Phi (s,s^\\\\prime )=Tr(\\\\overline{s}\\\\,{s^\\\\prime }).$ Thus ($c.f.$ section ), the hermitian product of Prewave Functions can be given in terms of the Prewave Functions themselves instead of the functions $f,$ by $\\\\langle \\\\psi ^{(-\\\\eta , \\\\ell )}_f,\\\\, \\\\psi ^{(-\\\\eta , \\\\ell )}_{f^\\\\prime } \\\\rangle _{(-\\\\eta , \\\\ell )}=\\\\int _{C^+} \\\\psi _f^{(-\\\\eta , \\\\ell )}\\\\Phi _{(-\\\\eta , \\\\ell )} \\\\psi _{f^\\\\prime }^{(-\\\\eta , \\\\ell )} \\\\ \\\\omega ,$ where $\\\\psi _f^{(-\\\\eta , \\\\ell )} \\\\Phi _{(-\\\\eta , \\\\ell )} \\\\psi _{f^\\\\prime }^{(-\\\\eta , \\\\ell )}$ is the function on $C^+$ given by $\\\\psi _f^{(-\\\\eta , \\\\ell )} \\\\Phi _{(-\\\\eta , \\\\ell )} \\\\psi _{f^\\\\prime }^{(-\\\\eta , \\\\ell )} (K)=\\\\frac{Tr(\\\\overline{\\\\psi _f^{(-\\\\eta , \\\\ell )}(H,K)}{ \\\\psi _{f^\\\\prime }^{(-\\\\eta , \\\\ell )}(H,K))}}{Tr(\\\\overline{s}\\\\,{s})}$ for all $K\\\\in C^+,$ $H \\\\in H(2)$ and $s\\\\in r_{-\\\\eta \\\\ell }^{-1}(K).$ The hermitian product of Wave Functions is defined as being the hermitian product of the corresponding Prewave Functions.\"\n ],\n [\n \"Electromagnetic Potential. \",\n \"If $K\\\\in C^+$ , the tangent space to $C^+$ at $K$ can be identified to the subspace of $H(2)$ given by $T_KC^+=\\\\lbrace M\\\\in H(2): Tr\\\\,M\\\\epsilon \\\\overline{K}\\\\epsilon =0 \\\\rbrace .$ Then, the complexified tangent space can be identified to ${}^{\\\\bf C}T_KC^+=\\\\lbrace M\\\\in gl(2,\\\\mathbb { C}\\\\rm ): Tr\\\\,M\\\\epsilon \\\\overline{K}\\\\epsilon =0 \\\\rbrace .$ A real vector field on $C^+$ is thus a map $A:C^+ \\\\longrightarrow H(2),$ such that $Tr\\\\,A(K)\\\\epsilon \\\\overline{K}\\\\epsilon =0,$ for all $K\\\\in C^+$ .\",\n \"A complex vector field on $C^+$ is thus given by a function $A:C^+ \\\\longrightarrow gl(2,\\\\bf C),$ whose real and imaginary hermitian parts are real vectorfields on $C^+$ .\",\n \"Since $Tr M\\\\epsilon \\\\overline{K}\\\\epsilon $ is real for $M$ and $K$ hermitian, we see that $A$ is a complex vector field on $C^+$ if and only if $Tr\\\\,A(K)\\\\epsilon \\\\overline{K}\\\\epsilon =0,$ for all $K\\\\in C^+$ .\",\n \"Equation (REF ) is equivalent to say that $A(K)\\\\epsilon \\\\overline{K}$ is a symmetric matrix.\",\n \"Let $A$ be a complex vector field on $C^+.$ We denote by $A_R$ and $A_I$ the real and imaginary hermitian parts of $A$ and $A_R(K)&=&h(A_R^1(K),.., A_R^4(K))\\\\\\\\A_I(K)&=&h(A_I^1(K),.., A_I^4(K))\\\\\\\\A^\\\\mu (K)&=&A_R^\\\\mu +i\\\\, A_I^\\\\mu ,\\\\ \\\\ \\\\mu =1,..,4\\\\\\\\\\\\overrightarrow{A_R}(K)&=&(A_R^1(K),.., A_R^3(K))\\\\\\\\\\\\overrightarrow{A_I}(K)&=&(A_I^1(K),.., A_I^3(K))\\\\\\\\\\\\overrightarrow{A}(K)&=&(A^1(K),.., A^3(K)).$ If $A_i^j(K)$ is the element of $A(K)$ in the row $i$ column $j,$ we also have $A^1 (K)&=&\\\\frac{1}{2}(A_1^2(K)+A_2^1(K))\\\\\\\\A^2 (K)&=&\\\\frac{i}{2}(A_1^2(K)-A_2^1(K))\\\\\\\\A^3 (K)&=&\\\\frac{1}{2}(A_1^1(K)-A_2^2(K))\\\\\\\\A^4 (K)&=&\\\\frac{1}{2}(A_1^1(K)+A_2^2(K))$ If $A(K)$ is , for exemple, continuous with compact support, for $\\\\mu =1,..,4,\\\\ x\\\\in \\\\mathbb {R}^4$ we define $\\\\widetilde{ A^\\\\mu }(x)=\\\\int _{C^+} A^\\\\mu (K) Exp(2\\\\pi i \\\\eta \\\\langle h^{-1}( K),x \\\\rangle ) \\\\omega .$ Then $\\\\square \\\\widetilde{A^\\\\mu }=0,$ for all $\\\\mu ,$ and $\\\\sum _{\\\\mu =1}^4 \\\\frac{\\\\partial \\\\widetilde{A^\\\\mu }}{\\\\partial x^\\\\mu }=0.$ We thus see that the $\\\\widetilde{A^\\\\mu }(x)$ define a complex electromagnetic potential in the Lorenz gauje.\",\n \"The corresponding electric field is given by $\\\\overrightarrow{E}(x)=-\\\\nabla \\\\widetilde{A^4}(x)-\\\\frac{\\\\partial \\\\overrightarrow{\\\\widetilde{A}}(x)}{\\\\partial x^4 },$ where $\\\\overrightarrow{\\\\widetilde{A}}(x)=(\\\\widetilde{ A^1}(x),\\\\widetilde{ A^2}(x),\\\\widetilde{ A^3}(x)),$ that can be written $\\\\overrightarrow{E}(x)=2\\\\pi i \\\\eta \\\\int _{C^+}(A^4(K)\\\\overrightarrow{K}-K^4 \\\\overrightarrow{A}(K)) Exp(2\\\\pi i \\\\eta \\\\langle h^{-1}( K),x \\\\rangle ) \\\\omega .$ The magnetic field is given by $\\\\overrightarrow{B}(x)=\\\\nabla \\\\times \\\\overrightarrow{\\\\widetilde{A}}(x),$ that can be written $\\\\overrightarrow{B}(x)=2\\\\pi i \\\\eta \\\\int _{C^+}(\\\\overrightarrow{A}(K)\\\\times \\\\overrightarrow{K} ) Exp(2\\\\pi i \\\\eta \\\\langle h^{-1}( K),x \\\\rangle ) \\\\omega .$ This electromagnetic field take in general complex values, the real parts of $\\\\overrightarrow{E}$ and $\\\\overrightarrow{B}$ are real electric and magnetic fields, corresponding to the electromagnetic potential given by the real parts of the $A^\\\\mu (x).$\"\n ],\n [\n \"Wave Functions and the Electromagnetic Potential. \",\n \"In section REF we have seen that a complex vector field on $C^+$ , define a complex electromagnetic potential in the Lorenz gauje.\",\n \"Let $A$ be a complex vector field on $C^+$ .\",\n \"Then, the matrix $A(K)\\\\epsilon \\\\overline{K}$ is a symmetric matrix, for all $K\\\\in C^+$ .\",\n \"If $ s\\\\in r_+^{-1}(K)$ there exist an unique complex number, $f_A(s),$ such that $A(K)\\\\epsilon \\\\overline{K}=f_A(s)\\\\, s .$ In fact, if $z\\\\in {\\\\bf C}^2-{0}$ , is such that $s= z {}^t z$ , we have $K=zz^*$ , and, since $0=Tr A(K)\\\\epsilon \\\\overline{z}{}^t z\\\\epsilon ={}^tz\\\\epsilon A(K)\\\\epsilon \\\\overline{z},$ there exist a number, $\\\\lambda $ , such that $A(K)\\\\epsilon \\\\overline{z}=\\\\lambda \\\\, z,$ so that $A(K)\\\\epsilon \\\\overline{K}=\\\\lambda \\\\, s .$ Since $s\\\\ne 0$ such a $\\\\lambda $ is unique and is denoted by $f_A(s).$ Thus the complex vector field $A$ defines a function, $f_A,$ on ${\\\\cal B}.$ If we take $e^{i\\\\phi }\\\\,s$ instead of $s$ in $ r_+^{-1}(K)$ , we can take in the preceeding reasoning $e^{i\\\\phi /2}z$ instead of $z$ , and thus $A(K)\\\\epsilon \\\\overline{e^{i\\\\phi /2}z}=\\\\lambda ^\\\\prime \\\\, e^{i\\\\phi /2}z$ leads to $\\\\lambda ^\\\\prime = e^{-i\\\\phi }\\\\,\\\\lambda .$ We thus see that $f_A(e^{i\\\\phi }\\\\,s)=e^{-i\\\\phi }\\\\,f_A(s)$ which proves that $f_A $ is $S^1$ -homogeneous of degree -1.\",\n \"The Prewave Function corresponding to $f_A$ for $-\\\\eta \\\\ell =1$ can be written as $\\\\psi _A^{(-\\\\eta , -\\\\eta )}(H,K)=f_A(s)\\\\,s\\\\,e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,h^{-1}(H)\\\\rangle _m},$ where $s$ is arbitrary in $r_+^{-1}(K).$ As a consecuence of (REF ) we have $\\\\psi _A^{(-\\\\eta , -\\\\eta )}(H,K)=A(K)\\\\epsilon \\\\overline{K}e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,h^{-1}(H)\\\\rangle _m},$ that gives Prewave Functions directly in terms of vectorfields.\",\n \"If $A(K)$ is continuous with compact support, the corresponding Wave Function is $ \\\\widetilde{\\\\psi }_A^{(-\\\\eta , -\\\\eta )}(x)=\\\\int _{C^+}A(K)\\\\epsilon \\\\overline{K}\\\\,e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,x\\\\rangle _m}\\\\omega _K.$ Now we define $\\\\widehat{f}_A=\\\\overline{f_A\\\\circ J},$ where $J$ is given by (REF ).\",\n \"I shall prove that $\\\\widehat{f}_A$ is $S^1$ -homogeneous of degree -1 on ${\\\\cal B}$ and that, for all $ s\\\\in r_-^{-1}(K),$ we have $-\\\\epsilon \\\\overline{A(K)}\\\\epsilon K \\\\epsilon =\\\\widehat{f}_A(s)\\\\, s .$ This will enable us to give a Prewave Function of Photons with $\\\\ell =\\\\eta ,$ directly in terms of $A$ .\",\n \"For all $ s\\\\in B,\\\\ a\\\\in S^1$ we have $\\\\widehat{f}_A(as)=\\\\overline{f_A( J(as))}=\\\\overline{f_A(\\\\overline{a} J(s)}=\\\\overline{af_A( J(s)}=\\\\overline{a}\\\\,\\\\widehat{f}_A(s),$ so that $\\\\widehat{f}_A$ has the appropiate homogeneity to define a Prewave Function.\",\n \"On the other hand, if $s \\\\in r_-^{-1}(K)$ , we have $\\\\widehat{f}_A(s)\\\\,s=\\\\overline{f_A(J(s))}\\\\,J(J(s))=J(f_A(J(s))\\\\,J({s}))=$ $=J(A(r_+(J(s)))\\\\epsilon \\\\overline{r_+(J(s))})=J(A(r_-(s))\\\\epsilon \\\\overline{r_-(s)}=$ $=J(A(K)\\\\epsilon \\\\overline{K}))=-\\\\epsilon \\\\overline{A(K)}\\\\epsilon K \\\\epsilon .$ Then, the Prewave Function for Photons with $\\\\ell =\\\\eta $ corresponding to $\\\\widehat{f}_A$ is $\\\\psi _A^{(-\\\\eta , \\\\eta )}(H,K)=-\\\\epsilon \\\\overline{A(K)}\\\\epsilon K \\\\epsilon \\\\,e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,h^{-1}(H)\\\\rangle _m}.$ Obviously $\\\\psi _A^{(-\\\\eta , \\\\eta )}=-\\\\epsilon \\\\overline{\\\\psi _A^{(\\\\eta , \\\\eta )}}\\\\epsilon , \\\\ \\\\ \\\\psi _A^{(\\\\eta , \\\\eta )}=-\\\\epsilon \\\\overline{\\\\psi _A^{(-\\\\eta , \\\\eta )}}\\\\epsilon .$ If $A(K)$ is continuous with compact support, the corresponding Wave Function is $ \\\\widetilde{\\\\psi }_A^{(-\\\\eta , \\\\eta )}(x)=-\\\\int _{C^+}\\\\epsilon \\\\overline{A(K)}\\\\epsilon K \\\\epsilon \\\\,\\\\,e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,x\\\\rangle _m}\\\\omega _K.$ We have $\\\\widetilde{\\\\psi }_A^{(-\\\\eta , \\\\eta )}=-\\\\epsilon \\\\overline{\\\\widetilde{\\\\psi }_A^{(\\\\eta , \\\\eta )}}\\\\epsilon ,\\\\ \\\\ \\\\widetilde{\\\\psi }_A^{(\\\\eta , \\\\eta )}=-\\\\epsilon \\\\overline{\\\\widetilde{\\\\psi }_A^{(-\\\\eta , \\\\eta )}}\\\\epsilon .$ With equations (REF ) and (REF ) we see that a single complex vectorfied on $C^+$ gives rise to Wave Functions of the four kinds of Photons: those with positive and negative energy and helicity.\",\n \"The inner products defined in section REF lead to the following results.\",\n \"If $A$ and $A^\\\\prime $ are vector fields on $C^+$ we have $\\\\langle \\\\psi ^{(-\\\\eta , -\\\\eta )}_A,\\\\, \\\\psi ^{(-\\\\eta , -\\\\eta )}_{A^\\\\prime } \\\\rangle _{(-\\\\eta , -\\\\eta )}= \\\\langle f_A,\\\\, f_{A^\\\\prime } \\\\rangle _{(-\\\\eta , -\\\\eta )},$ and $\\\\langle \\\\psi ^{(-\\\\eta , \\\\eta )}_A,\\\\, \\\\psi ^{(-\\\\eta , \\\\eta )}_{A^\\\\prime } \\\\rangle _{(-\\\\eta , \\\\eta )}=\\\\langle \\\\widehat{f}_A,\\\\, \\\\widehat{f}_{A^\\\\prime } \\\\rangle _{(-\\\\eta , \\\\eta )}.$ Then, from (REF ) and (REF ) one can obtain $\\\\langle \\\\psi ^{(-\\\\eta , -\\\\eta )}_A,\\\\, \\\\psi ^{(-\\\\eta , -\\\\eta )}_{A^\\\\prime } \\\\rangle _{(-\\\\eta , -\\\\eta )}=\\\\int _{C^+} A\\\\Phi A^\\\\prime =$ $=\\\\langle \\\\psi ^{(-\\\\eta , \\\\eta )}_{A^\\\\prime },\\\\, \\\\psi ^{(-\\\\eta , \\\\eta )}_{A} \\\\rangle _{(-\\\\eta , \\\\eta )}$ where $A\\\\Phi A^\\\\prime (K)= \\\\frac{Tr(\\\\overline{A(K)}\\\\epsilon K A^\\\\prime (K)\\\\epsilon \\\\overline{K})}{Tr(\\\\overline{s}{s})}$ for all $K\\\\in C^+,$ and $s\\\\in r_{\\\\pm }^{-1}(K).$\"\n ],\n [\n \"Wave Functions in terms of the Electromagnetic Field.\",\n \"Let $A$ a complex vectorfield on $C^+$ and denote $\\\\overrightarrow{E}(K)&=& A^4(K)\\\\overrightarrow{K}-K^4 \\\\overrightarrow{A}(K) \\\\\\\\\\\\overrightarrow{B}(K)&=& \\\\overrightarrow{A}(K)\\\\times \\\\overrightarrow{K}$ Thus, the Electric and Magnetic Fields can be given by $\\\\overrightarrow{E}(x)=2\\\\pi i \\\\eta \\\\int _{C^+}\\\\overrightarrow{E}(K) Exp(2\\\\pi i \\\\eta \\\\langle h^{-1}( K),x \\\\rangle ) \\\\omega _K$ $\\\\overrightarrow{B}(x)=2\\\\pi i \\\\eta \\\\int _{C^+}\\\\overrightarrow{B}(K) Exp(2\\\\pi i \\\\eta \\\\langle h^{-1}( K),x \\\\rangle ) \\\\omega _K.$ Let us prove that $A(K) \\\\epsilon \\\\overline{K} \\\\epsilon =\\\\left(\\\\overrightarrow{E}(K)+i\\\\overrightarrow{B}(K)\\\\right)\\\\cdot \\\\overrightarrow{\\\\sigma },º$ which means $A(K) \\\\epsilon \\\\overline{K} \\\\epsilon =\\\\left(\\\\begin{array}{cc}(\\\\overrightarrow{E}+i\\\\overrightarrow{B})^3 & (\\\\overrightarrow{E}+i\\\\overrightarrow{B})^1-i(\\\\overrightarrow{E}+i\\\\overrightarrow{B})^2\\\\\\\\(\\\\overrightarrow{E}+i\\\\overrightarrow{B})^1+i(\\\\overrightarrow{E}+i\\\\overrightarrow{B})^2 &-(\\\\overrightarrow{E}+i\\\\overrightarrow{B})^3\\\\end{array}\\\\right)$ In fact, we have $A(K) \\\\epsilon \\\\overline{K} \\\\epsilon =\\\\frac{1}{2} \\\\left( A(K) \\\\epsilon \\\\overline{K}+ {}^{t}(A(K) \\\\epsilon \\\\overline{K}) \\\\right)\\\\epsilon =$ $=\\\\frac{1}{2}\\\\left((A_R+iA_I)\\\\epsilon \\\\overline{K}\\\\epsilon -K \\\\epsilon ({}^{t}A_R+i {}^{t}A_I)\\\\epsilon \\\\right)=$ $=\\\\frac{1}{2}(A_R\\\\epsilon \\\\overline{K} \\\\epsilon -K\\\\epsilon \\\\overline{A_R} \\\\epsilon )+\\\\frac{i}{2}(A_I\\\\epsilon \\\\overline{K} \\\\epsilon -K\\\\epsilon \\\\overline{A_I} \\\\epsilon )$ and thus (REF ) follows from (REF ).\",\n \"As a consequence, the Prewave Functions corresponding to $A$ are $\\\\psi _A^{(-\\\\eta , -\\\\eta )}(H,K)=-\\\\left(\\\\left(\\\\overrightarrow{E}(K)+i\\\\overrightarrow{B}(K)\\\\right)\\\\cdot \\\\overrightarrow{ \\\\sigma } \\\\right)\\\\epsilon \\\\ e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,h^{-1}(H)\\\\rangle _m},$ $\\\\psi _A^{(-\\\\eta , \\\\eta )}(H,K)=-\\\\epsilon \\\\ \\\\left(\\\\overline{\\\\left(\\\\overrightarrow{E}(K)+i\\\\overrightarrow{B}(K)\\\\right)\\\\cdot \\\\overrightarrow{ \\\\sigma }}\\\\right) \\\\ e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,h^{-1}(H)\\\\rangle _m}.$ If $A$ is continuous with compact support, we have $\\\\widetilde{\\\\psi }_A^{(-\\\\eta , -\\\\eta )}(x)=-\\\\int _{C^+}((\\\\overrightarrow{E}(K)+i\\\\overrightarrow{B}(K))\\\\cdot \\\\overrightarrow{\\\\sigma })\\\\ \\\\epsilon \\\\ e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,h^{-1}(H)\\\\rangle _m}\\\\omega _K,$ $\\\\widetilde{\\\\psi }_A^{(-\\\\eta , \\\\eta )}(x)=-\\\\int _{C^+}\\\\ \\\\epsilon \\\\ (\\\\overline{(\\\\overrightarrow{E}(K)+i\\\\overrightarrow{B}(K))\\\\cdot \\\\overrightarrow{\\\\sigma })} \\\\,e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,h^{-1}(H)\\\\rangle _m}\\\\omega _K.$ so that the Wave Functions are $\\\\widetilde{\\\\psi }_A^{(-\\\\eta , -\\\\eta )}(x)=\\\\frac{i\\\\eta }{2\\\\pi }((\\\\overrightarrow{E}(x)+i\\\\overrightarrow{B}(x))\\\\cdot \\\\overrightarrow{\\\\sigma })\\\\ \\\\epsilon ,$ or $\\\\widetilde{\\\\psi }_A^{(-\\\\eta , \\\\eta )}(x)=\\\\frac{i\\\\eta }{2\\\\pi }\\\\epsilon \\\\ (\\\\overline{(\\\\overrightarrow{E}(x)+i\\\\overrightarrow{B}(x))\\\\cdot \\\\overrightarrow{\\\\sigma })} .$ These Wave Functions are given in terms of the components of $\\\\overrightarrow{E}(x)+i\\\\overrightarrow{B}(x)$ as (up to constants) the Wave Functions of Bialynicki-Birula [2].\"\n ],\n [\n \"Gauje invariance for Photons.\",\n \"Let us denote by $\\\\cal {D}$ the complex vector space of complex vector fields on $C^+.$ The map $A\\\\in {\\\\cal D} \\\\longrightarrow f_A \\\\in {\\\\cal F}$ is linear and its kernel is composed by the vector fields on $C^+$ having the form $A(K)=\\\\left( \\\\begin{array}{c}\\\\lambda (z)\\\\\\\\ \\\\mu (z) \\\\end{array}\\\\right)\\\\,z^*$ where $z\\\\in {\\\\bf C}^2-{0}$ is such that $K=zz^*$ and $\\\\lambda ,\\\\ \\\\mu $ are funtions on ${\\\\bf C}^2-{0}$ $S^1$ -homogeneous of degree +1.\",\n \"The kernel of the map $A\\\\in {\\\\cal D} \\\\longrightarrow \\\\widehat{f}_A \\\\in {\\\\cal F}$ is the same and will be denoted by $\\\\cal {N}.$ In particular, for each map $L:K\\\\in C^+ \\\\rightarrow L(K)\\\\in gl(2,{\\\\bf C}),$ the vectorfield $A_L(K)=L(K)K$ is in $\\\\cal {N}.$ If $A\\\\in \\\\cal {D}$ , $N \\\\in \\\\cal {N}$ we have $f_{A+N}=f_A$ $\\\\widehat{f}_{A+N}= \\\\widehat{f}_{A},$ so that the Prewave and Wave Functions are invariant under the changes $A\\\\rightarrow A+N$ : $\\\\widetilde{\\\\psi }_{A+N}^{(-\\\\eta , \\\\ell )}=\\\\widetilde{\\\\psi }_{A}^{(-\\\\eta , \\\\ell )}.$ Because of (REF ) and (REF ) we see that also the Electric and Magnetic Fiels corresponding to Photon are invariant under the changes $A\\\\rightarrow A+N$ .\",\n \"In the particular case $N=A_L$ with $L(K)=g(K)\\\\,I,$ where $g$ is a complex valued function on $C^+,$ the change $A\\\\rightarrow A+N$ lead to the following change in the Electromagnetic Potential $\\\\widetilde{(A+N)}^\\\\mu (x)=\\\\widetilde{A}^\\\\mu (x)+\\\\partial ^\\\\mu \\\\phi (x),$ where $\\\\phi (x)=-\\\\frac{i\\\\eta }{2\\\\pi }\\\\int _{C^+}g(K) e^{2 \\\\pi i\\\\eta \\\\langle \\\\, h^{-1}(K),\\\\,x\\\\rangle _m}\\\\omega _K$ and $\\\\partial ^\\\\mu \\\\phi =g^{\\\\mu \\\\nu }\\\\frac{\\\\partial }{\\\\partial x^\\\\nu }\\\\,\\\\phi $ where the $g^{\\\\mu \\\\nu }$ are the components of Minkowski metric.\",\n \"Thus, the invariance of the Electromagnetic Field of Photons under the change $A(K) \\\\rightarrow A(K)+g(K)K$ is a particular case of the well known gauje invariance of general Electromagnetic Fields.\"\n ],\n [\n \"General remarks\",\n \"In this paper, State Space for a particle defined by $\\\\alpha \\\\in \\\\underline{G}^*$ has been stated as the homogeneous space that correspond to the orbit of $(0,\\\\alpha )$ in $H(2)\\\\times \\\\underline{G}^*.$ Thus, as we have seen in section , it is identified to $\\\\frac{G}{G_{(0,\\\\alpha )}}$ where $G_{(0,\\\\alpha )}=G_\\\\alpha \\\\cap (SL \\\\oplus \\\\lbrace 0\\\\rbrace ),$ or, equivalently, to $H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}\\\\cap SL_1} ,$ that is the form adopted in Figure REF of that section.\",\n \"The equivariant maps $\\\\iota _4:H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}\\\\cap SL_1} \\\\rightarrow H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}}$ and $\\\\nu _2:H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}\\\\cap SL_1} \\\\rightarrow \\\\frac{G}{G_\\\\alpha }.$ in that Figure, will be used in this section.\",\n \"When we characterize $H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}\\\\cap SL_1}$ in terms of other concrete manifolds or of some parametrization, we need to physically interpret the geometric objects that appear, and, in order to do that, the more effective way is , in general, to know the values of the Canonical Dynamical Variables in terms of these objects.\",\n \"But the value of these Dynamical Variables is well known on the coadjoint orbit i.e.\",\n \"on $G/G_\\\\alpha .$ Thus, its values in State Space can be obtained by composition with $\\\\nu _2.$ Since Prewave Functions are defined on $H(2) \\\\times \\\\frac{SL}{(G_\\\\alpha )_{SL}}$ its composition with $\\\\iota _4$ gives another version of Quantum States, now defined on the Classical State Space.\",\n \"In the definition of Prewave Functions REF , space-time only appears in the exponential, whose exponent is $-2\\\\pi i\\\\langle h^{-1}(P),h^{-1}(H)\\\\rangle _m.$ where $P$ is the “traslation\\\" of momentum-energy to $SL/(G_\\\\alpha )_{SL}.$ The exponent in the composition of Prewave Functions with $\\\\iota _4$ must have the same expression, but changing $P$ by $P\\\\circ \\\\iota _4.$ The conmutativity of the diagramm in Figure REF imply that $P\\\\circ \\\\iota _4$ can be obtained as the impulsion-energy in $G/G_\\\\alpha $ composite with $\\\\nu _2.$ In the next sections I give the expressions of the canonical dynamical variables on State Space, for different kind of particles.\"\n ],\n [\n \"Klein-Gordon particles\",\n \"We have seen in section REF that State Space for Klein-Gordon particles can be identified to $H(2) \\\\oplus \\\\frac{SL}{SU(2)},$ and this homogeneous space to ${\\\\cal H}^m\\\\times H(2).$ If $(K,H)\\\\in {\\\\cal H}^m\\\\times H(2) $ and $A\\\\in SL$ is such that $K=mAA^*,$ we denote $H&=&h(\\\\overrightarrow{x},x^4)\\\\\\\\\\\\overrightarrow{x}&=& (x^1,x^2,x^3)\\\\\\\\K&=&m h(\\\\overrightarrow{k}, k_4) \\\\\\\\\\\\overrightarrow{k}&=& (k^1,k^2,k^3)\\\\\\\\k^4&=&\\\\sqrt{1+\\\\Vert \\\\overrightarrow{k} \\\\Vert ^2}.$ Thus, the map (REF ) becomes, $\\\\nu _1(K,H)=(A,H)*(mI,\\\\overrightarrow{0},1)=$ $=(K,\\\\overrightarrow{x}-\\\\frac{x^4}{k^4} \\\\overrightarrow{k},e^{-2\\\\pi i \\\\eta m x^4/k^4})$ and the map (REF ), $\\\\nu _2(K,H))=(A,H)*(mI,\\\\overrightarrow{0})=(K,\\\\overrightarrow{x}-\\\\frac{x^4}{k^4} \\\\overrightarrow{k}).$ The Canonical Dynamical Variables on State Space are obtained as composition of (REF ) with $\\\\nu _2.$ If we denote $V_{KG}=V\\\\circ \\\\nu _2$ for each dynamical variable, $V,$ we have $P_{KG}(mh(\\\\overrightarrow{k},k_4),h(\\\\overrightarrow{x},x^4))=- \\\\eta \\\\,mh(\\\\overrightarrow{k},k_4),$ $\\\\overrightarrow{l}_{KG}(mh(\\\\overrightarrow{k},k_4),h(\\\\overrightarrow{x},x^4))=\\\\overrightarrow{x} \\\\times \\\\left(\\\\overrightarrow{P}_{KG}(mh(\\\\overrightarrow{k},k_4),h(\\\\overrightarrow{x},x^4)\\\\right) ,$ $\\\\overrightarrow{g}_{KG}(mh(\\\\overrightarrow{k},k_4),h(\\\\overrightarrow{x},x^4))=$ $ = P^4_{KG}(mh(\\\\overrightarrow{k},k_4),h(\\\\overrightarrow{x},x^4)) \\\\overrightarrow{x}-x^4 \\\\overrightarrow{P}_{KG}(mh(\\\\overrightarrow{k},k_4),h(\\\\overrightarrow{x},x^4)).$ where $ k_4=\\\\sqrt{1+\\\\Vert \\\\overrightarrow{k} \\\\Vert ^2}.$ The Prewave functions have been defined on a manifold that, in this case, coincides with State Space.\",\n \"The map $\\\\iota _4$ is the identical map.\",\n \"Another fact is that, in this case, State Space is the Universal Covering of the homogeneous contact manifold.\",\n \"Let us prove that $\\\\nu _1$ is a covering map.\",\n \"The local expresion of $\\\\nu _1$ in the charts corresponding to $\\\\phi _\\\\tau $ and $\\\\phi ^\\\\prime :(k_1,k_2,k_3,x^1,x^2,x^3,x^4)\\\\in {\\\\mathbb {R}}^7 \\\\longrightarrow $ $ \\\\longrightarrow (h(m\\\\overrightarrow{k},m k_4),h(x^1,x^2,x^3,x^4))\\\\in {\\\\cal H}^m\\\\times H(2) ,$ is given by $(\\\\phi _\\\\tau )^{-1}\\\\circ \\\\nu _1 \\\\circ \\\\phi ^{\\\\prime }(k_1,\\\\dots ,x^4)=$ $=(k_1,k_2,k_3,x^1-\\\\frac{x^4}{k_4}k_1,x^2-\\\\frac{x^4}{k_4}k_2,x^3-\\\\frac{x^4}{k_4}k_3,-\\\\tau -\\\\eta m \\\\frac{x^4}{k_4}+N)$ where the domain of definition is defined by the condition that $\\\\tau +\\\\eta m \\\\frac{x^4}{k_4} \\\\notin {\\\\mathbb {Z}}+\\\\frac{1}{2} $ and $N$ is defined by the condition that $-\\\\tau -\\\\eta m \\\\frac{x^4}{k_4}+N\\\\in (-\\\\frac{1}{2},\\\\frac{1}{2}).$ We have $((\\\\phi _\\\\tau )^{-1}\\\\circ \\\\nu _1 \\\\circ \\\\phi ^{\\\\prime })^{-1}\\\\lbrace (\\\\overrightarrow{n},\\\\overrightarrow{y},t)\\\\rbrace =$ $=\\\\lbrace (\\\\overrightarrow{n},\\\\overrightarrow{y}-\\\\frac{\\\\eta }{m}(t+\\\\tau +N)\\\\overrightarrow{n}, -\\\\frac{\\\\eta }{m}(t+\\\\tau +N)\\\\sqrt{1+\\\\Vert \\\\overrightarrow{n} \\\\Vert ^2}): N\\\\in {\\\\mathbb {Z}}\\\\rbrace .$ Let $d\\\\in {\\\\cal H}^m \\\\times {\\\\mathbb {R}}^3\\\\times {\\\\mathbb {S}}^1,$ and let $\\\\tau $ be such that $d$ is in the image of $(\\\\phi _\\\\tau )^{-1}.$ This image, $U_o,$ is an open neighborhood of $d$ , and we shall prove that its antiimage by $\\\\nu _1$ is union of disjoint open sets, each of them diffeomorphic to $U_o$ under the restriction of $\\\\nu _1.$ We denote by ${\\\\cal U}$ the domain of $\\\\phi _\\\\tau $ .\",\n \"The domain of $(\\\\phi _\\\\tau )^{-1}\\\\circ \\\\nu _1 \\\\circ \\\\phi ^{\\\\prime },$ ${\\\\cal W}=((\\\\phi _\\\\tau )^{-1}\\\\circ \\\\nu _1 \\\\circ \\\\phi ^{\\\\prime })^{-1}({\\\\cal U})$ is the union of the open sets ${\\\\cal W}_N=\\\\lbrace &(&\\\\overrightarrow{n},\\\\overrightarrow{y}-\\\\frac{\\\\eta }{m}(t+\\\\tau +N)\\\\overrightarrow{n}, -\\\\frac{\\\\eta }{m}(t+\\\\tau +N))\\\\sqrt{1+\\\\Vert \\\\overrightarrow{n} \\\\Vert ^2}: \\\\\\\\&(&\\\\overrightarrow{n},\\\\overrightarrow{y},t)\\\\in {\\\\cal U}\\\\rbrace .$ where $N\\\\in \\\\mathbb {Z}.$ But the ${\\\\cal W}_N$ are disjoint.\",\n \"In fact, if $(\\\\overrightarrow{n},\\\\overrightarrow{y}-\\\\frac{\\\\eta }{m}(t+\\\\tau +N)\\\\overrightarrow{n}, -\\\\frac{\\\\eta }{m}(t+\\\\tau +N)\\\\sqrt{1+\\\\Vert \\\\overrightarrow{n} \\\\Vert ^2})=$ $=(\\\\overrightarrow{n}^{\\\\prime },\\\\overrightarrow{y}^{\\\\prime }-\\\\frac{\\\\eta }{m}(t^{\\\\prime }+\\\\tau +N^{\\\\prime })\\\\overrightarrow{n}^{\\\\prime }, -\\\\frac{\\\\eta }{m}(t^{\\\\prime }+\\\\tau +N^{\\\\prime })\\\\sqrt{1+\\\\Vert \\\\overrightarrow{n}^{\\\\prime } \\\\Vert ^2})$ where $(\\\\overrightarrow{n},\\\\overrightarrow{y},t),\\\\ (\\\\overrightarrow{n}^{\\\\prime },\\\\overrightarrow{y}^{\\\\prime },t^{\\\\prime })\\\\in \\\\cal U,$ we obtain first $\\\\overrightarrow{n}=\\\\overrightarrow{n}^{\\\\prime }$ and then $t+N=t^{\\\\prime }+N^{\\\\prime }.$ Since $-1/20$ , the values of the canonical dynamical variables in this state are obtained by composition of (REF ) with the map $\\\\nu _2$ .\",\n \"When for each dynamical variable $V$ we denote $V\\\\circ \\\\nu _2$ by $V_{massive}$ we have $P_{massive}(mh(\\\\overrightarrow{k},k_4),\\\\overrightarrow{v},h(\\\\overrightarrow{x},x^4))&=&-\\\\eta m h(\\\\vec{k},k_4),\\\\\\\\\\\\overrightarrow{l}_{massive}(mh(\\\\overrightarrow{k},k_4),\\\\overrightarrow{v},h(\\\\overrightarrow{x},x^4))&=&\\\\frac{T}{4\\\\pi }\\\\frac{k_4\\\\overrightarrow{v}-\\\\vec{k}}{k_4-\\\\langle \\\\vec{ k},\\\\overrightarrow{v} \\\\rangle }+\\\\overrightarrow{x}\\\\times \\\\overrightarrow{P}_{massive},\\\\\\\\\\\\overrightarrow{g}_{massive}(mh(\\\\overrightarrow{k},k_4),\\\\overrightarrow{v},h(\\\\overrightarrow{x},x^4))&=&\\\\frac{T}{4\\\\pi }\\\\frac{\\\\vec{k}\\\\times \\\\overrightarrow{v}}{k_4-\\\\langle \\\\vec{ k},\\\\overrightarrow{v} \\\\rangle }+ \\\\\\\\ &+& P^4_{massive}\\\\overrightarrow{x}-x^4\\\\overrightarrow{P}_{massive},$ In what concerns the Pauli-Lubanski fourvector, we have $\\\\begin{split}\\\\overrightarrow{W}_{massive}&=P^4_{massive}\\\\frac{T}{4\\\\pi } \\\\frac{k_4\\\\overrightarrow{v}-\\\\vec{k}}{k_4-\\\\langle \\\\vec{ k},\\\\overrightarrow{v} \\\\rangle } \\\\\\\\W^4_{massive}&= -\\\\eta m\\\\frac{T}{4\\\\pi } \\\\frac{\\\\langle k_4\\\\overrightarrow{v}-\\\\vec{k},\\\\vec{k} \\\\rangle }{k_4-\\\\langle \\\\vec{ k},\\\\overrightarrow{v} \\\\rangle },\\\\end{split}$ As in $T=0$ case, the prewave functions have been defined directly on State Space, the map $\\\\iota _4$ being the identical map.\",\n \"Thus they are exactly as in section REF .\"\n ],\n [\n \"Massless particles of type 4\",\n \"Since $(G_\\\\alpha )_{SL}\\\\cap SL_1=[S^1],$ State Space for massless particles, $H(2)\\\\times (SL/((G_\\\\alpha )_{SL}\\\\cap SL_1),$ can be identified to State Space for massive particles, ${\\\\cal H}^m \\\\times P_1(C) \\\\times H(2)$ .\",\n \"Here $m$ is an arbitrary positive number, with no physical significance.\",\n \"On the other hand, Movement Space, $G/G_\\\\alpha ,$ is identified to $P^{sign(\\\\nu )}_3$ by means of the action (REF ).\",\n \"The canonical map $\\\\nu _2$ from State Space onto Movement Space thus becomes for this kind of particle $\\\\nu _2((A,H)*\\\\left(mI,\\\\left[ \\\\left(\\\\begin{array}{c} 1\\\\\\\\0 \\\\end{array}\\\\right) \\\\right],0 \\\\right)=(A,H)*[q]$ where $q=\\\\left( \\\\begin{array}{c} 0\\\\\\\\2\\\\nu \\\\\\\\0\\\\\\\\1\\\\end{array} \\\\right).$ Then $\\\\nu _2\\\\left(mAA^*,\\\\left[A \\\\left(\\\\begin{array}{c} 1\\\\\\\\0 \\\\end{array}\\\\right) \\\\right],H\\\\right)=\\\\left[\\\\begin{array}{c} 2\\\\nu A \\\\left(\\\\begin{array}{c} 0\\\\\\\\1 \\\\end{array}\\\\right)-iH(A^*)^{-1}\\\\left(\\\\begin{array}{c} 0\\\\\\\\1 \\\\end{array}\\\\right)\\\\\\\\ (A^*)^{-1}\\\\left(\\\\begin{array}{c} 0\\\\\\\\1 \\\\end{array}\\\\right)\\\\end{array} \\\\right].$ But $(A^*)^{-1}=-\\\\varepsilon \\\\overline{A} \\\\varepsilon ,$ so that $\\\\nu _2\\\\left(mAA^*,\\\\left[A \\\\left(\\\\begin{array}{c} 1\\\\\\\\0 \\\\end{array}\\\\right) \\\\right],H\\\\right)=\\\\left[\\\\begin{array}{c} (2\\\\nu AA^* -iH)(A^*)^{-1}\\\\left(\\\\begin{array}{c} 0\\\\\\\\1 \\\\end{array}\\\\right)\\\\\\\\ (A^*)^{-1}\\\\left(\\\\begin{array}{c} 0\\\\\\\\1 \\\\end{array}\\\\right)\\\\end{array} \\\\right]=$ $=\\\\left[\\\\begin{array}{c} (2\\\\nu AA^* -iH)\\\\varepsilon \\\\overline{A}\\\\left(\\\\begin{array}{c} 1\\\\\\\\0 \\\\end{array}\\\\right)\\\\\\\\\\\\varepsilon \\\\overline{A}\\\\left(\\\\begin{array}{c} 1\\\\\\\\0 \\\\end{array}\\\\right)\\\\end{array} \\\\right].$ Then $\\\\nu _2:(K,[u],H)\\\\in {\\\\cal H}^m \\\\times P_1(C) \\\\times H(2) \\\\rightarrow \\\\left[\\\\begin{array}{c} \\\\left(\\\\frac{2\\\\nu }{m} K -iH \\\\right)\\\\varepsilon \\\\overline{u}\\\\\\\\ \\\\varepsilon \\\\overline{u}\\\\end{array} \\\\right] \\\\in P_3^{sign(\\\\nu )}.$ If we fix, for exemple, $m=1,$ then for all $\\\\left[\\\\left(\\\\begin{array}{c} \\\\omega \\\\\\\\\\\\pi \\\\end{array}\\\\right) \\\\right]\\\\in P_3^{sign(\\\\nu )},$ we have $(\\\\nu _2)^{-1}\\\\left\\\\lbrace \\\\left[\\\\left(\\\\begin{array}{c} \\\\omega \\\\\\\\\\\\pi \\\\end{array}\\\\right) \\\\right]\\\\right\\\\rbrace =$ $=\\\\lbrace (K,[\\\\varepsilon \\\\overline{\\\\pi }],H)\\\\in {\\\\cal H}^1 \\\\times P_1(C) \\\\times H(2): (\\\\frac{2\\\\nu }{m} K-iH)\\\\pi =\\\\omega \\\\rbrace .$ This set represent the “portrait\\\" of the movement $\\\\left[\\\\left(\\\\begin{array}{c} \\\\omega \\\\\\\\\\\\pi \\\\end{array}\\\\right)\\\\right]$ in ${\\\\cal H}^1 \\\\times P_1(C) \\\\times H(2)$ considered as State Space of massless particles.\",\n \"The values of dynamical variables $P,\\\\ \\\\overrightarrow{l} ,\\\\ \\\\overrightarrow{g},$ on ${\\\\cal H}^m \\\\times P_1(C) \\\\times H(2)$ for the massless Type 4 particles can be obtained by composition of its expressions (REF ), (), (), with the map $\\\\nu _2$ given in (REF ).\",\n \"If $(mh(\\\\overrightarrow{k},k_4),\\\\overrightarrow{v},h(\\\\overrightarrow{x},x_4))\\\\in {\\\\cal H}^m \\\\times S^2 \\\\times H(2),$ is interpreted as a state of a massless Type 4 particle, where now $m$ is an arbitrary positive number with no physical significance, the values of the canonical dynamical variables in this state are $\\\\overrightarrow{P}_{massless}(mh(\\\\overrightarrow{k},k_4),\\\\overrightarrow{v},h(\\\\overrightarrow{x},x^4))&=&P_{massless}^4 \\\\overrightarrow{v},\\\\\\\\P_{massless}^4(mh(\\\\overrightarrow{k},k_4),\\\\overrightarrow{v},h(\\\\overrightarrow{x},x^4))&=&\\\\frac{-\\\\eta }{ 2 (k_4-\\\\langle \\\\vec{ k},\\\\overrightarrow{v} \\\\rangle ) } \\\\\\\\\\\\overrightarrow{l}_{massless}(mh(\\\\overrightarrow{k},k_4),\\\\overrightarrow{v},h(\\\\overrightarrow{x},x^4))&=&\\\\frac{\\\\chi T}{4\\\\pi }\\\\frac{k_4\\\\overrightarrow{v}-\\\\vec{k}}{k_4-\\\\langle \\\\vec{ k},\\\\overrightarrow{v} \\\\rangle }+ \\\\\\\\ &+&\\\\overrightarrow{x}\\\\times \\\\overrightarrow{P}_{massless},\\\\\\\\\\\\overrightarrow{g}_{massless}(mh(\\\\overrightarrow{k},k_4),\\\\overrightarrow{v},h(\\\\overrightarrow{x},x^4))&=&\\\\frac{\\\\chi T}{4\\\\pi }\\\\frac{\\\\vec{k}\\\\times \\\\overrightarrow{v}}{k_4-\\\\langle \\\\vec{ k},\\\\overrightarrow{v} \\\\rangle }+ \\\\\\\\ &+& P^4_{massless}\\\\overrightarrow{x}-x^4\\\\overrightarrow{P}_{massless}.$ The Pauly-Lubanski four vector for type 4 particles is (c.f.\",\n \"section ) $W_{massless}=\\\\frac{\\\\chi T}{4\\\\pi } P_{massless}.$ We thus see that the expression of $\\\\overline{l}$ and $\\\\overline{g}$ are formally almost identical for massive and massless particles.\",\n \"But this is not the case for $P.$ The point $(mh(\\\\overrightarrow{k},k_4),\\\\overrightarrow{v},h(\\\\overrightarrow{x},x^4))$ can be interpreted as being the state of a massive particle or the state of a massless particle, but the value in this state of linear momentum, energy and angular momentum is different in the massive or in the massless case.\",\n \"The map $\\\\iota _4,$ what in the case of massive particles whas the identical map, in the case of massless particles can be found to be $\\\\iota _4:(mh(\\\\overrightarrow{k},k_4),\\\\overrightarrow{v},h(\\\\overrightarrow{x},x^4))\\\\in {\\\\cal H}^m \\\\times S^2 \\\\times H(2) \\\\rightarrow $ $ \\\\rightarrow \\\\left( \\\\frac{h(\\\\overrightarrow{v},1)}{2(k_4-\\\\langle \\\\vec{ k},\\\\overrightarrow{v} \\\\rangle )}, h(\\\\overrightarrow{x},x^4))\\\\right)\\\\in C^+ \\\\times H(2)$ The $P_{KG},\\\\ P_{massive}$ and $P_{massless}$ appears in the exponent of the Prewave Functions in State Space of the corresponding particles.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01863"}}},{"rowIdx":799,"cells":{"text":{"kind":"list like","value":[["Dynamic Multiscale Tree Learning Using Ensemble Strong Classifiers for\n Multi-label Segmentation of Medical Images with Lesions"],["Abstract We introduce a dynamic multiscale tree (DMT) architecture that learns how to leverage the strengths of different existing classifiers for supervised multi-label image segmentation.","Unlike previous works that simply aggregate or cascade classifiers for addressing image segmentation and labeling tasks, we propose to embed strong classifiers into a tree structure that allows bi-directional flow of information between its classifier nodes to gradually improve their performances.","Our DMT is a generic classification model that inherently embeds different cascades of classifiers while enhancing learning transfer between them to boost up their classification accuracies.","Specifically, each node in our DMT can nest a Structured Random Forest (SRF) classifier or a Bayesian Network (BN) classifier.","The proposed SRF-BN DMT architecture has several appealing properties.","First, while SRF operates at a patch-level (regular image region), BN operates at the super-pixel level (irregular image region), thereby enabling the DMT to integrate multi-level image knowledge in the learning process.","Second, although BN is powerful in modeling dependencies between image elements (superpixels, edges) and their features, the learning of its structure and parameters is challenging.","On the other hand, SRF may fail to accurately detect very irregular object boundaries.","The proposed DMT robustly overcomes these limitations for both classifiers through the ascending and descending flow of contextual information between each parent node and its children nodes.","Third, we train DMT using different scales, where we progressively decrease the patch and superpixel sizes as we go deeper along the tree edges nearing its leaf nodes.","Last, DMT demonstrates its outperformance in comparison to several state-of-the-art segmentation methods for multi-labeling of brain images with gliomas."],["Introduction","Accurate multi-label image segmentation is one of the top challenges in both computer vision and medical image analysis.","Specifically, in computer-aided healthcare applications, medical image segmentation constitutes a critical step for tracking the evolution of anatomical structures and lesions in the brain using neuroimaging, as well as quantitatively measuring group structural difference between image populations [5], [17], [3], [18], [10].","Multi-label image segmentation is widely addressed as a classification problem.","Previous works [9], [19] used individual classifiers such as support vector machine (SVM) to segment each label class independently, then fuse the different label maps into a multi-label map.","However, prior to the fusion step, the produced label maps may largely overlap one another, which might yield to biased fused label map.","Alternatively, the integration of multiple classifiers within the same segmentation framework would help reduce this bias and improve the overall multi-label classification performance since M heads are better than one as reported in [8].","Broadly, one can categorize the segmentation methods that combine multiple classifiers into two groups:(1) cascaded classifiers, and(2) ensemble classifiers.","In the first group, classifiers are chained such that the output of each classifier is fed into the next classifier in the cascade to generate the final segmentation result at the end of the cascade.","Such architecture can be adopted for two different goals.","First, cascaded classifiers take into account contextual information, encoded in the segmentation map outputted from the previous classifier, thereby enforcing spatial consistency between neighboring image elements (e.g., patches, superpixels) in the spirit of an auto-context model [5], [16], [14], [25], [10].","Second, this allows to combine classifiers hierarchically, where each classifier in the cascade is assigned to a more specific segmentation task (or a sub-task), as it further sub-labels the output label map of its antecedent classifier [4], [17], [3], [18].","Although these methods produced promising results, and clearly outperformed the use of single (non-cascaded) classifiers in different image segmentation applications, cascading classifiers only allows a unidirectional learning transfer, where the learned mapping from the previous classifier is somehow ‘communicated' to the next classifier in the chain for instance through the output segmentation map.","The second group represents ensemble classifiers based methods, which train individual classifiers, then aggregate their segmentation results [15].","Specifically, such frameworks combine a set of independently trained classifiers on the same labeling problem and generates the final segmentation result by fusing the individual segmentation results using a fusion method, which is typically weighted or unweighted voting [6].","Hence, it constructs a strong classifier that outperforms each individual `weak' classifier (or base classifier) [8].","For instance, Random Forest (RF) classification algorithm, independently trains weak decision trees using bootstrap samples generated from the training data to learn a mapping between the feature and the label sets [2].","The segmentation map of a new input image is the aggregation of the trees' decisions by major voting.","RF demonstrated its efficiency in solving different image classification problems [14], [25], which reflects the power of the ensemble classifiers technique.","In addition to significantly improving the segmentation results when compared with single classifiers, ensemble classifiers based methods are powerful in addressing several known classification problems such as imbalanced correlation and over-fitting [21].","However, such combination technique is not enough to fully exploit the training of classifiers and leverage their strengths.","Indeed, the base classifiers perform segmentation independently without any cooperation to solve the target classification problem.","Moreover, the learning of each classifier in the ensemble is performed in one-step, as opposed to multi-step classifier training, where the learning of each classifier gradually improves from one step to the next one.","We note that this differs from cascaded classifiers, where each classifier is ‘visited' or trained once through combining the contextual segmentation map of the previous classifier along with the original input image.","To address the aforementioned limitations of both categories, we propose a Dynamic Multi-scale Tree (DMT) architecture for multi-label image segmentation.","DMT is a binary tree, where each node nests a classifier, and each traversed path from the root node to a leaf node encodes a cascade of classifiers (i.e., nodes on the path).","Our proposed DMT architecture allows a bidirectional information flow between two successive nodes in the tree (from parent node to child node and from child node back to parent node).","Thus, DMT is based on ascending and descending feedbacks between each parent node and its children nodes.","This allows to gradually refine the learning of each node classifier, while benefitting from the learning of its immediate neighboring nodes.","To generate the final segmentation results, we combine the elementary segmentation results produced at the leaf nodes using majority voting strategy.","The proposed architecture integrates different possible combinations of different classifiers, while taking advantage of their strengths and overcoming their limitations through the bidirectional learning transfer between them, which defines the dynamic aspect of the proposed architecture.","Furthermore, the DMT inherently integrates contextual information in the classification task, since each classifier inputs the segmentation result of its parent node or children nodes classifiers.","Additionally, to capture a coarse-to-fine image details for accurate segmentation, the DMT is designed to consider different scale at each level in the tree in a way that the adopted scale decreases as we go deeper along the tree edges nearing its leaf nodes.","In this work, we define our DMT classification model using two strong classifiers: Structured Random Forest (SRF) and Bayesian Network (BN).","SRF is an improved version of Random Forest [7].","In addition of being fast, resistant to over-fitting and having a good performance in classifying high-dimensional data, SRF handles structural informationand integrates spatial information.","It has shown good performance in several classification tasks especially muli-label image segmentation [7], [22].","On the other hand, BN is a learning graphical model that statistically represents the dependencies between the image elements and their features.","It is suitable for multi-label segmentation for its effectiveness in fusing complex relationships between image features of different natures and handling noisy as well as missing signals in images [23], [12], [24], [20].","Embedding SRF and BN within our DMT leverages their strengths and helps overcome their limitations (i.e.","not accurately classifying transitions between label classes for SRF and the problem of parameters learning such as prior probabilities for BN).","Moreover, the SRF-BN bidirectional cooperation during learning and testing stages enables the integration of multi-level image knowledge through the combination of regular and irregular image elements (i.e.","patch-level classification produced by SRF and superpixel-level classification produced by BN).","To sum up, our SRF-BN DMT has promise for multi-label image segmentation as it: Gradually improves the classification accuracy through the bidirectional flow between parents and children nodes, each nesting a BN or SRF classifier Simultaneously integrates multi-level and multi-scale knowledge from training images, thereby examining in depth the different inherent image characteristics Overcomes SRF and BN limitations when used independently through multiple cascades (or tree paths) composed of different combinations of BN and SRF classifiers."],["Base classifiers","In this section we briefly introduce the SRF and BN classifiers, that are embedded as nodes in our DMT classification framework.","Then,we explain in detail how we define our DMT architecture and elaborate on how to perform the training and testing stages on an image dataset for multi-label image segmentation."],["Structured Random Forest","SRF is a variant of the traditional Random Forest classifier, which better handles and preserves the structure of different labels in the image [7].","While, standard RF maps an intensity feature vector extracted from a 2D patch centered at pixel $\\emph {x}$ to the label of its center pixel $\\emph {x}$ (i.e., patch-to-pixel mapping), SRF maps the intensity feature vector to a 2D label patch centered at $\\emph {x}$ (patch-to-patch mapping).","This is achieved at each node in the SRF tree, where the function that splits patch features between right and left children nodes depends on the joint distribution of two labels: a first label at the patch center and a second label selected at a random position within the training patch [7].","We also note that in SRF, both feature space and label space nest patches that might have different dimensions.","Despite its elegant and solid mathematical foundation as well as its improved performance in image segmentation compared with RF, SRF might perform poorly at irregular boundaries between different label classes since it is trained using regularly structured patches [7].","Besides, it does not include contextual information to enforce spatial consistency between neighboring label patches.","To address these limitations, we first propose to embed SRF as a classifier node into our DMT architecture, where the contextual information is provided as a segmentation map by its parent and children nodes.","Second, we improve its training around irregular boundaries through leveraging the strength of one or more its neighboring BN classifiers, which learns to segment the image at the superpixel level, thereby better capturing irregular boundaries in the image."],["Bayesian network","Various BN-based models have been proposed for image segmentation [23], [12], [24].","In our work, we adopt the BN architecture proposed in [24].","As a preprocessing step, we first generate the edge maps from the input MR image modalities FigREF .","This edge map consists of a set of superpixels ${S_{p_{i}}} ; i=1 \\dots N$ (or regional blobs) and edge segments ${E_{j}}; j=1 \\dots L$ .","We define our BN as a four-layer network, where each node in the first layer stores a superpixel.","The second layer is composed of nodes, each storing a single edge from the edge map.","The two remaining layers store the extracted superpixel features and edge features, respectively.","During the training stage, to set BN parameters, we define the prior probability $P(S_{p_{i}})$ of $S_{p_{i}}$ as a uniform distribution and then learn the conditional probability representing the relationship between the superpixels' features and their corresponding labels using a mixture of Gaussians model.","In addition, we empirically define the conditional probability modeling the relationship between each superpixel label and each edge state (i.e., true or false edge) $P(E_{j}|p_a (E_{j}))$ , where $p_a (E_{j})$ denotes the parent superpixel nodes of $E_{j}$ .","During the testing stage, we learn the BN structure through encoding the semantic relationships between superpixels and edge segments.","Specifically, each edge node has for parent nodes the two superpixel nodes that are separated by this edge.","In other words, each superpixel provides contextual information to judge whether the edge is on the object boundary or not.","If two superpixels have different labels, it is more likely that there is a true object boundary between them, i.e.","$E_{j} = 1$ , otherwise $E_{j} = 0$ .","Although automatic segmentation methods based on BN have shown great results in the state-of-the-art, they might perform poorly in segmenting low-contrast image regions and different regions with similar features [24].","To further improve the segmentation accuracy of BN, we propose to include additional information through embedding BN classifier into our proposed DMT learning architecture."],["Proposed Multi-scale Dynamic Tree Learning","In this section, we present the main steps in devising our Multi-scale Dynamic Tree segmentation framework, which aims to boost up the performance of classifiers nested in its nodes.","FigREF illustrates the proposed binary tree architecture composed of classifier nodes, where each classifier ultimately communicates the output of its learning (i.e., semantic context or probability segmentation maps) to its parent and children nodes.","Therefore, the learning of the tree is dynamic as it is based on ascending and descending feedbacks between each parent node and its children nodes.","Specifically, each node output is fed to the children nodes as semantic context, in turn the children nodes transfer their learning (i.e.","probability maps) to their common parent node.","Then, after merging these transferred probability maps from children nodes, the parent node uses the merged maps as a contextual information to generate a new segmentation result that will be subsequently communicated again to its children nodes.","This gradually improves the learning of its classifier nodes at each depth level of the tree.","In the following sections, we further detail the DMT architecture."],["Dynamic Tree Learning","We define a binary tree T(V,E), where V denotes the set of nodes in T and E represents the set of edges in T. Each node $i$ in T represents a classifier $c_i$ and each edge $e_{ij}$ connecting two nodes i and j carries bidirectional contextual information flow between the classifiers $c_i$ and $c_j$ that are always inputting the original image characteristics (i.e.","the features for SRF, superpixel features and input image edge map for BN).","Specifically, we define bidirectional feedbacks between two neighboring classifier nodes $i$ and $j$ , encoding two flows: a descending flow $F_{i\\rightarrow j} $ that represents the transfer of the probability maps generated by parent classifier node $c_i$ to its child classifier node $c_j$ as contextual information and an ascending flow $F_{j\\rightarrow i} $ that models the transfer of the probability maps generated by a child node $c_j$ back to its parent node $c_i$ (Fig.","REF ).","This depth-wise bidirectional learning transfer occurs locally along each edge between a parent node and its child node, thereby defining the dynamics of the tree.","In addition, as our Dynamic Tree (DT) grows exponentially, it integrates various combinations of classifiers.","Thus, each path of the tree implements a unique cascade of classifiers.","To generate the final segmentation result we aggregate the segmentation maps produced at each leaf node in the binary tree by applying the majority voting.","Figure: Implicit and explicitlearning transfer in the proposed dynamic multi-scale tree-based classification architecture..","The dashed gray arrows denote the descending flows from the parent classifier c i c_i to its children c j c_j and c j ' c_j^{\\prime } (iteration kk), while the dashed red arrows denote the ascending flows derived from the children nodes to their parent node.","Both ascending flows are fused for the parent node to integrate them in generating a new segmentation map that will be communicated to its two children classifier nodes (iteration k+1k+1).Inherent implicit and explicit transfer learning between nodes in DT architecture.","We note that the bidirectional flow between parent nodes and their corresponding children nodes defines a new traversing strategy of the tree nodes, that in addition to the dynamic learning aspect, encodes two different types of learning transfer: explicit and implicit.","Indeed, the ascending and descending flows between parent nodes and their children through the direct transfer of their generated probability maps is an explicit learning transfer.","However, in our binary tree, when a parent node $i$ receives the ascending flows ($F^k(j \\rightarrow i)$ and $F^k(j^{\\prime } \\rightarrow i)$ from its left and right children nodes j and j', they are fused before being passed on, in a second round, as contextual information ($F^{k+1}(i \\rightarrow j)$ and $F^{k+1}(i \\rightarrow j^{\\prime }) $ ) to the children nodes (Fig.","REF ).","The probability maps fusion at the parent node level is performed through simple averaging.","In particular, the parent node concatenates the fused probability map with the original input features to generate a new segmentation probability map result that will be communicated to its two children classifier nodes.","Hence, the children nodes of the same parent node explicitly cooperate to improve their parent learning, and implicitly cooperate to improve their own learning while using their parent node as a proxy.","SRF-BN Dynamic Tree.","In this work, each classifier node is assigned a SRF or a BN model, previously described in Section 2, to define our Dynamic Tree architecture.","The transferred information between classifiers through the descending and ascending flows is used in addition to the testing image features as contextual information, while BN classifier uses this information as prior knowledge (i.e prior probability) to perform the multi-label segmentation task.","The combination of SRF and BN classifiers is compelling for the following reasons.","First, it enhances the performance of BN by taking the posterior probability generated by SRF as prior probability.","This justifies our choice of the root node of our DT as a SRF.","Second, it improves SRF performance around irregular between-class boundaries since SRF benefits from BN structure learning, which is based on image over-segmentation that is guided by object boundaries.","Third, as the SRF maps image information at the patch level, while BN models knowledge at the superpixel level, their combination allows the aggregation of regular (i.e.","patch) and irregular (i.e.","superpixel) structures in the image for our target multi-label segmentation task."],["Dynamic Multi-scale Tree Learning","To further boost the performance of our multi-label segmentation framework and enhance the segmentation accuracy, we introduce a multi-scale learning strategy in our dynamic tree architecture by varying the size of the input patches and superpixels used to grow the SRF and construct the BN classifier.","Specifically, we use a different scale at each depth level such as we go deeper along the tree edges nearing its leaf nodes, we progressively decrease the size of both patches and superpixels in the training and testing stages.","In addition to capturing coarse-to-fine details of the image anatomical structure, the application of the multi-scale strategy to the proposed DT allows to capture fine-to-coarse information.","Indeed, DMT learning semantically divides the image into different patterns (e.g., different patches and superpixels at each depth of the tree) in both intensity and label domains at different scales.","However, thanks to the bidirectional dynamic flow, the scale defined at each depth influences the performance of parent nodes (in previous level) and children nodes (in next level), which allows to simultaneously perform coarse-to-fine and fine-to-coarse information integration in the multi-label classification task.","Moreover, a depth-wise multi-scale feature representation adaptively encodes image features at different scales for each image pixel in the image element (superpixel or patch)."],["Statistical superpixel-based and patch-based feature extraction","To train each classifier node in the tree, we extract the following statistical features at the superpixel level (for BN) and 2D patch level (for SRF): first order operators (mean, standard deviation, max, min, median, Sobel, gradient); higher order operators (Laplacian, difference of Gaussian, entropy, curvatures, kurtosis, skewness), texture features (Gabor filter), and spatial context features (symmetry, projection, neighborhoods) [13]."],[" Results and Discussion","Dataset and parameters.","We evaluate our proposed brain tumor segmentation framework on 50 subjects with high-grade gliomas, randomly selected from the Brain Tumor Image Segmentation Challenge (BRATS 2015) dataset [11].","For each patient, we use three MRI modalities (FLAIR, T2-w, T1-c) along with the corresponding manually segmented glioma lesions.","They are rigidly co-registered and resampled to a common resolution to establish patch-to-patch correspondence across modalities.","Then, we apply N4 filter for inhomogeneity correction, and use histogram linear transformation for intensity normalization.","Table: Segmentation results of the proposed framework and comparison methods averaged across 50 patients.","(HT: whole Tumor; CT: Core Tumor; ET: Enhanced Tumor; depth of the tree; * indicates outperformed methods with p-value≺0.05p-value \\prec 0.05).For the baseline methods training we adopt the following parameters:(1) Edgemap generation: we use the SLICE oversegmentation algorithm with a superpixel number fixed to 1000 and compactness fixed to 10 [1].","To establish superpixel-to-superpixel correspondence across modalities for each subject, we first oversegment the FLAIR MRI, then we apply the generated edgemap (i.e., superpixel partition) to the corresponding T1-c and T2-w MR images.","(2) SRF training: we grow 15 trees using intensity feature patches of size 10x10 and label patches of size 7x7.","(3) BN construction: the BN model is built using the generated edgemap as detailed in Section 2; the conditional probabilities modeling the relationships between the superpixel labeling and the edge state are defined as follows: $P(E_{j}=1|p_a (E_{j}))= 0.8$ if the parent region nodes have different labels, and $P(E_{j}=1|p_a (E_{j})) = 0.2$ otherwise.","Evaluation and comparison methods.","For comparison, as baseline methods we use: (1) SRF: the Random Forest version that exploits structural information described in Section 2, (2) BN: the classification algorithm described in Section 2 where the prior probability of superpixels is set as a uniform distribution, (3) SRF-SRF denotes the auto-context Structured Random Forest, (4) BN-BN denotes the auto-context Bayesian Network, where the first BN prior probability is set as a uniform distribution while the second classifier use the posterior probability of its previous as prior probability.","Of note, by conventional auto-context classifier, we mean a uni-directional contextual flow from one classifier to the next one.","The segmentation frameworks were trained using leave-one-patient cross-validation experiments.","For evaluation, we use the Dice score between the ground truth region area $A_{gt}$ and the segmented region area $A_s$ as follows $D = (A_{gt} \\bigcap A_s)/ 2( A_{gt} + A_s)$ .","Next, we investigate the influence of the tree depth as well as the multi-scale tree learning strategy on the performance of the proposed architecture.","Varying tree architectures.","In this experiment, we evaluate two different tree architectures to examine the impact of the tree depth on the framework performance.","Table.","REF shows the segmentation results for 2-level tree (i.e.","depth=2) and 1-level tree (i.e.","depth=1) for tumor lesion multi-label segmentation with and without multiscale variant.","Although the average Dice Score has improved from depth 1 to 2, the improvement wasn't statistically significant.","We did not explore larger depths (d>2), since as the binary tree grows exponentially, its computational time dramatically increases and becomes demanding in terms of resources (especially memory).","Multi-scale tree architecture.","To examine the influence of the multiscale DT learning strategy, we compare the conventional DT architecture (at a fixed-scale) to MDT architecture.","For the fixed-scale architecture, all tree nodes nest either an SRF classifier trained using intensity feature patches of size 10x10 and label patches of size 7x7 or a BN classifier constructed using an edgemap of 1000 superpixels generated with a compactness of 10.","In the multiscale architecture, we keep the same parameters of the fixed-scale architecture at the first level of the tree while the classifiers of the second level are trained with different parameters.","Specifically, we use smaller intensity patches (of size 8x8) and label patches (of size 5x5) for the SRF training, and a smaller number of superpixels for BN construction (1200 superpixels).","Figure: Qualitative segmentation results for all the baseline methods applied on 5 subjects: (a) BN segmentation result.","(b) SRF segmentation result.","(c) Auto-context BN.","(d) Auto-context SRF.","(e) SRF+BN segmentation result.","(f) DMT segmentation result.","(depth=2).","(g) Ground truth label map.Clearly, the quantitative results show the outperformance (improvement of 7%) of both proposed DT and DMT architectures in comparison with several baseline methods for multi-label tumor lesion segmentation with statistical significance (p < 0.05) .","This indicates that a deeper combination of different learning models helps increase the segmentation accuracy.","When comparing the results of the SRF and BN we found that SRF outperforms BN in segmenting the three classes: wHole Tumor (HT), Core Tumor (CT) and Enhancing Tumor (ET) Table.","REF .","This is due to the fact that BN have difficulties in segmenting low-contrast images and identifying different superpixels having similar characteristics, especially with the lack of any prior knowledge on the anatomical structure of the testing image.","Although BN has a low Dice score compared to SRF, in Fig.","REF we can note that it has better performance in detecting the boundaries between different classes.","This shows the impact of the irregular structure of superpixels used during BN training and testing, which gives BN the ability to be more accurate in detecting object boundaries compared to SRF that considers regular image patches.","Notably, BN structure is individualized during the testing stage for each testing subject since it is based on the testing image oversegmentation map.","Thus, SRF and BN classifiers are complementary.","First, they perform segmentation at regular and irregular structures of the image.","Second, one (SRF) learns image knowledge during the training stage, while the other (BN) is structured using the input testing image during the testing stage through modeling the testing image structure.","Further, the results of SRF-SRF and BN-BN models that implement the auto-context approach show an improvement of the segmentation results at both qualitative and quantitative levels when compared with baseline SRF and BN models.","More importantly, we note that BN-BN cascade outperforms SRF-SRF cascade when segmenting the Core Tumor and Enhancing Tumor (ET) lesions.","This can be explained by the fine and irregular anatomical details of these image structures when compared to the whole tumor lesion.","Since BN is trained using irregular superpixels, it produced more accurate segmentations for these classes (e.g., BN-BN:56.14 vs SRF-SRF: 37.12 for ET).","Through further cascading both SRF and BN classifiers, we note that the heterogenoues SRF-BN cascade produced much better results compared to both autocontext SRF and autocontext BN for two main reasons.","First SRF aids in defining BN prior based on the testing image structure, while BN enhances the performance of SRF at the boundaries level.","This further highlights the importance of integrating both regular and irregular image elements for training classifiers that capture different image structures.","The outperformance of the proposed DMT architecture also lays ground for our assumption that embedding SRF and BN into our a unified dynamic architecture where they mutually benefit from their learning boosts up the multi-label segmentation accuracy.","In addition to the previously mentionned advantages of combining SRF and BN, it is important to note that the integration of variant cascades of SRF and BN endows our architecture with a an efficient learning ability, where it incorporates in a deep manner the knowledge of SRF based on modeling the dataset during the training step and the individualized learning of BN based on the testing image during the testing step.","Notably, our DMT architecture has a few limitations.","First, it becomes more demanding in terms of computational and memory resources, as the tree grows exponentially (in the order of $O(2^n)$ ).","The more nodes we add to the binary tree, the slower the algorithm converges.","Second, the patch and superpixel sizes in the multi-scale learning strategy can be further learned, instead of empirically fixing them through inner cross-validation.","Third, the bidirectional flow is currently restricted between neighboring parent and children nodes at a fixed tree depth.","This can be extended to further nodes (e.g., root node), where the semantic context progressively diffuses from each node $i$ along tree paths to far-away nodes."],["Conclusion","We proposed a Dynamic Multi-scale Tree (DMT) learning architecture that both cascades and aggregates classifiers for multi-label medical image segmentation.","Specifically, our DMT embeds classifiers into a binary tree architecture, where each node nests a classifier and each edge encodes a learning transfer between the classifiers.","A new tree traversal strategy is proposed where a depth-wise bidirectional feedbacks are performed along each edge between a parent node and its child node.","This allows explicit learning between parent and children nodes and implicit learning transfer between children of the same parent.","Moreover, we train DMT using different scales for input patches and superpixels to capture a coarse-to-fine image details as well as a fine-to-coarse image structures through the depth-wise bidirectional flow.","To sum up, our DMT integrates compound and complementary aspects: deep learning, cooperative learning, dynamic learning, coarse-to-fine and fine-to-coarse learning.","In our future work, we will devise a more comprehensive tree traversal strategy where the learning transfer starts from the root node, descending all the way down to the leaf nodes and then ascending all the way up to the root node.","We will also evaluate our DMT semantic segmentation architecture on different large datasets."]],"string":"[\n [\n \"Dynamic Multiscale Tree Learning Using Ensemble Strong Classifiers for\\n Multi-label Segmentation of Medical Images with Lesions\"\n ],\n [\n \"Abstract We introduce a dynamic multiscale tree (DMT) architecture that learns how to leverage the strengths of different existing classifiers for supervised multi-label image segmentation.\",\n \"Unlike previous works that simply aggregate or cascade classifiers for addressing image segmentation and labeling tasks, we propose to embed strong classifiers into a tree structure that allows bi-directional flow of information between its classifier nodes to gradually improve their performances.\",\n \"Our DMT is a generic classification model that inherently embeds different cascades of classifiers while enhancing learning transfer between them to boost up their classification accuracies.\",\n \"Specifically, each node in our DMT can nest a Structured Random Forest (SRF) classifier or a Bayesian Network (BN) classifier.\",\n \"The proposed SRF-BN DMT architecture has several appealing properties.\",\n \"First, while SRF operates at a patch-level (regular image region), BN operates at the super-pixel level (irregular image region), thereby enabling the DMT to integrate multi-level image knowledge in the learning process.\",\n \"Second, although BN is powerful in modeling dependencies between image elements (superpixels, edges) and their features, the learning of its structure and parameters is challenging.\",\n \"On the other hand, SRF may fail to accurately detect very irregular object boundaries.\",\n \"The proposed DMT robustly overcomes these limitations for both classifiers through the ascending and descending flow of contextual information between each parent node and its children nodes.\",\n \"Third, we train DMT using different scales, where we progressively decrease the patch and superpixel sizes as we go deeper along the tree edges nearing its leaf nodes.\",\n \"Last, DMT demonstrates its outperformance in comparison to several state-of-the-art segmentation methods for multi-labeling of brain images with gliomas.\"\n ],\n [\n \"Introduction\",\n \"Accurate multi-label image segmentation is one of the top challenges in both computer vision and medical image analysis.\",\n \"Specifically, in computer-aided healthcare applications, medical image segmentation constitutes a critical step for tracking the evolution of anatomical structures and lesions in the brain using neuroimaging, as well as quantitatively measuring group structural difference between image populations [5], [17], [3], [18], [10].\",\n \"Multi-label image segmentation is widely addressed as a classification problem.\",\n \"Previous works [9], [19] used individual classifiers such as support vector machine (SVM) to segment each label class independently, then fuse the different label maps into a multi-label map.\",\n \"However, prior to the fusion step, the produced label maps may largely overlap one another, which might yield to biased fused label map.\",\n \"Alternatively, the integration of multiple classifiers within the same segmentation framework would help reduce this bias and improve the overall multi-label classification performance since M heads are better than one as reported in [8].\",\n \"Broadly, one can categorize the segmentation methods that combine multiple classifiers into two groups:(1) cascaded classifiers, and(2) ensemble classifiers.\",\n \"In the first group, classifiers are chained such that the output of each classifier is fed into the next classifier in the cascade to generate the final segmentation result at the end of the cascade.\",\n \"Such architecture can be adopted for two different goals.\",\n \"First, cascaded classifiers take into account contextual information, encoded in the segmentation map outputted from the previous classifier, thereby enforcing spatial consistency between neighboring image elements (e.g., patches, superpixels) in the spirit of an auto-context model [5], [16], [14], [25], [10].\",\n \"Second, this allows to combine classifiers hierarchically, where each classifier in the cascade is assigned to a more specific segmentation task (or a sub-task), as it further sub-labels the output label map of its antecedent classifier [4], [17], [3], [18].\",\n \"Although these methods produced promising results, and clearly outperformed the use of single (non-cascaded) classifiers in different image segmentation applications, cascading classifiers only allows a unidirectional learning transfer, where the learned mapping from the previous classifier is somehow ‘communicated' to the next classifier in the chain for instance through the output segmentation map.\",\n \"The second group represents ensemble classifiers based methods, which train individual classifiers, then aggregate their segmentation results [15].\",\n \"Specifically, such frameworks combine a set of independently trained classifiers on the same labeling problem and generates the final segmentation result by fusing the individual segmentation results using a fusion method, which is typically weighted or unweighted voting [6].\",\n \"Hence, it constructs a strong classifier that outperforms each individual `weak' classifier (or base classifier) [8].\",\n \"For instance, Random Forest (RF) classification algorithm, independently trains weak decision trees using bootstrap samples generated from the training data to learn a mapping between the feature and the label sets [2].\",\n \"The segmentation map of a new input image is the aggregation of the trees' decisions by major voting.\",\n \"RF demonstrated its efficiency in solving different image classification problems [14], [25], which reflects the power of the ensemble classifiers technique.\",\n \"In addition to significantly improving the segmentation results when compared with single classifiers, ensemble classifiers based methods are powerful in addressing several known classification problems such as imbalanced correlation and over-fitting [21].\",\n \"However, such combination technique is not enough to fully exploit the training of classifiers and leverage their strengths.\",\n \"Indeed, the base classifiers perform segmentation independently without any cooperation to solve the target classification problem.\",\n \"Moreover, the learning of each classifier in the ensemble is performed in one-step, as opposed to multi-step classifier training, where the learning of each classifier gradually improves from one step to the next one.\",\n \"We note that this differs from cascaded classifiers, where each classifier is ‘visited' or trained once through combining the contextual segmentation map of the previous classifier along with the original input image.\",\n \"To address the aforementioned limitations of both categories, we propose a Dynamic Multi-scale Tree (DMT) architecture for multi-label image segmentation.\",\n \"DMT is a binary tree, where each node nests a classifier, and each traversed path from the root node to a leaf node encodes a cascade of classifiers (i.e., nodes on the path).\",\n \"Our proposed DMT architecture allows a bidirectional information flow between two successive nodes in the tree (from parent node to child node and from child node back to parent node).\",\n \"Thus, DMT is based on ascending and descending feedbacks between each parent node and its children nodes.\",\n \"This allows to gradually refine the learning of each node classifier, while benefitting from the learning of its immediate neighboring nodes.\",\n \"To generate the final segmentation results, we combine the elementary segmentation results produced at the leaf nodes using majority voting strategy.\",\n \"The proposed architecture integrates different possible combinations of different classifiers, while taking advantage of their strengths and overcoming their limitations through the bidirectional learning transfer between them, which defines the dynamic aspect of the proposed architecture.\",\n \"Furthermore, the DMT inherently integrates contextual information in the classification task, since each classifier inputs the segmentation result of its parent node or children nodes classifiers.\",\n \"Additionally, to capture a coarse-to-fine image details for accurate segmentation, the DMT is designed to consider different scale at each level in the tree in a way that the adopted scale decreases as we go deeper along the tree edges nearing its leaf nodes.\",\n \"In this work, we define our DMT classification model using two strong classifiers: Structured Random Forest (SRF) and Bayesian Network (BN).\",\n \"SRF is an improved version of Random Forest [7].\",\n \"In addition of being fast, resistant to over-fitting and having a good performance in classifying high-dimensional data, SRF handles structural informationand integrates spatial information.\",\n \"It has shown good performance in several classification tasks especially muli-label image segmentation [7], [22].\",\n \"On the other hand, BN is a learning graphical model that statistically represents the dependencies between the image elements and their features.\",\n \"It is suitable for multi-label segmentation for its effectiveness in fusing complex relationships between image features of different natures and handling noisy as well as missing signals in images [23], [12], [24], [20].\",\n \"Embedding SRF and BN within our DMT leverages their strengths and helps overcome their limitations (i.e.\",\n \"not accurately classifying transitions between label classes for SRF and the problem of parameters learning such as prior probabilities for BN).\",\n \"Moreover, the SRF-BN bidirectional cooperation during learning and testing stages enables the integration of multi-level image knowledge through the combination of regular and irregular image elements (i.e.\",\n \"patch-level classification produced by SRF and superpixel-level classification produced by BN).\",\n \"To sum up, our SRF-BN DMT has promise for multi-label image segmentation as it: Gradually improves the classification accuracy through the bidirectional flow between parents and children nodes, each nesting a BN or SRF classifier Simultaneously integrates multi-level and multi-scale knowledge from training images, thereby examining in depth the different inherent image characteristics Overcomes SRF and BN limitations when used independently through multiple cascades (or tree paths) composed of different combinations of BN and SRF classifiers.\"\n ],\n [\n \"Base classifiers\",\n \"In this section we briefly introduce the SRF and BN classifiers, that are embedded as nodes in our DMT classification framework.\",\n \"Then,we explain in detail how we define our DMT architecture and elaborate on how to perform the training and testing stages on an image dataset for multi-label image segmentation.\"\n ],\n [\n \"Structured Random Forest\",\n \"SRF is a variant of the traditional Random Forest classifier, which better handles and preserves the structure of different labels in the image [7].\",\n \"While, standard RF maps an intensity feature vector extracted from a 2D patch centered at pixel $\\\\emph {x}$ to the label of its center pixel $\\\\emph {x}$ (i.e., patch-to-pixel mapping), SRF maps the intensity feature vector to a 2D label patch centered at $\\\\emph {x}$ (patch-to-patch mapping).\",\n \"This is achieved at each node in the SRF tree, where the function that splits patch features between right and left children nodes depends on the joint distribution of two labels: a first label at the patch center and a second label selected at a random position within the training patch [7].\",\n \"We also note that in SRF, both feature space and label space nest patches that might have different dimensions.\",\n \"Despite its elegant and solid mathematical foundation as well as its improved performance in image segmentation compared with RF, SRF might perform poorly at irregular boundaries between different label classes since it is trained using regularly structured patches [7].\",\n \"Besides, it does not include contextual information to enforce spatial consistency between neighboring label patches.\",\n \"To address these limitations, we first propose to embed SRF as a classifier node into our DMT architecture, where the contextual information is provided as a segmentation map by its parent and children nodes.\",\n \"Second, we improve its training around irregular boundaries through leveraging the strength of one or more its neighboring BN classifiers, which learns to segment the image at the superpixel level, thereby better capturing irregular boundaries in the image.\"\n ],\n [\n \"Bayesian network\",\n \"Various BN-based models have been proposed for image segmentation [23], [12], [24].\",\n \"In our work, we adopt the BN architecture proposed in [24].\",\n \"As a preprocessing step, we first generate the edge maps from the input MR image modalities FigREF .\",\n \"This edge map consists of a set of superpixels ${S_{p_{i}}} ; i=1 \\\\dots N$ (or regional blobs) and edge segments ${E_{j}}; j=1 \\\\dots L$ .\",\n \"We define our BN as a four-layer network, where each node in the first layer stores a superpixel.\",\n \"The second layer is composed of nodes, each storing a single edge from the edge map.\",\n \"The two remaining layers store the extracted superpixel features and edge features, respectively.\",\n \"During the training stage, to set BN parameters, we define the prior probability $P(S_{p_{i}})$ of $S_{p_{i}}$ as a uniform distribution and then learn the conditional probability representing the relationship between the superpixels' features and their corresponding labels using a mixture of Gaussians model.\",\n \"In addition, we empirically define the conditional probability modeling the relationship between each superpixel label and each edge state (i.e., true or false edge) $P(E_{j}|p_a (E_{j}))$ , where $p_a (E_{j})$ denotes the parent superpixel nodes of $E_{j}$ .\",\n \"During the testing stage, we learn the BN structure through encoding the semantic relationships between superpixels and edge segments.\",\n \"Specifically, each edge node has for parent nodes the two superpixel nodes that are separated by this edge.\",\n \"In other words, each superpixel provides contextual information to judge whether the edge is on the object boundary or not.\",\n \"If two superpixels have different labels, it is more likely that there is a true object boundary between them, i.e.\",\n \"$E_{j} = 1$ , otherwise $E_{j} = 0$ .\",\n \"Although automatic segmentation methods based on BN have shown great results in the state-of-the-art, they might perform poorly in segmenting low-contrast image regions and different regions with similar features [24].\",\n \"To further improve the segmentation accuracy of BN, we propose to include additional information through embedding BN classifier into our proposed DMT learning architecture.\"\n ],\n [\n \"Proposed Multi-scale Dynamic Tree Learning\",\n \"In this section, we present the main steps in devising our Multi-scale Dynamic Tree segmentation framework, which aims to boost up the performance of classifiers nested in its nodes.\",\n \"FigREF illustrates the proposed binary tree architecture composed of classifier nodes, where each classifier ultimately communicates the output of its learning (i.e., semantic context or probability segmentation maps) to its parent and children nodes.\",\n \"Therefore, the learning of the tree is dynamic as it is based on ascending and descending feedbacks between each parent node and its children nodes.\",\n \"Specifically, each node output is fed to the children nodes as semantic context, in turn the children nodes transfer their learning (i.e.\",\n \"probability maps) to their common parent node.\",\n \"Then, after merging these transferred probability maps from children nodes, the parent node uses the merged maps as a contextual information to generate a new segmentation result that will be subsequently communicated again to its children nodes.\",\n \"This gradually improves the learning of its classifier nodes at each depth level of the tree.\",\n \"In the following sections, we further detail the DMT architecture.\"\n ],\n [\n \"Dynamic Tree Learning\",\n \"We define a binary tree T(V,E), where V denotes the set of nodes in T and E represents the set of edges in T. Each node $i$ in T represents a classifier $c_i$ and each edge $e_{ij}$ connecting two nodes i and j carries bidirectional contextual information flow between the classifiers $c_i$ and $c_j$ that are always inputting the original image characteristics (i.e.\",\n \"the features for SRF, superpixel features and input image edge map for BN).\",\n \"Specifically, we define bidirectional feedbacks between two neighboring classifier nodes $i$ and $j$ , encoding two flows: a descending flow $F_{i\\\\rightarrow j} $ that represents the transfer of the probability maps generated by parent classifier node $c_i$ to its child classifier node $c_j$ as contextual information and an ascending flow $F_{j\\\\rightarrow i} $ that models the transfer of the probability maps generated by a child node $c_j$ back to its parent node $c_i$ (Fig.\",\n \"REF ).\",\n \"This depth-wise bidirectional learning transfer occurs locally along each edge between a parent node and its child node, thereby defining the dynamics of the tree.\",\n \"In addition, as our Dynamic Tree (DT) grows exponentially, it integrates various combinations of classifiers.\",\n \"Thus, each path of the tree implements a unique cascade of classifiers.\",\n \"To generate the final segmentation result we aggregate the segmentation maps produced at each leaf node in the binary tree by applying the majority voting.\",\n \"Figure: Implicit and explicitlearning transfer in the proposed dynamic multi-scale tree-based classification architecture..\",\n \"The dashed gray arrows denote the descending flows from the parent classifier c i c_i to its children c j c_j and c j ' c_j^{\\\\prime } (iteration kk), while the dashed red arrows denote the ascending flows derived from the children nodes to their parent node.\",\n \"Both ascending flows are fused for the parent node to integrate them in generating a new segmentation map that will be communicated to its two children classifier nodes (iteration k+1k+1).Inherent implicit and explicit transfer learning between nodes in DT architecture.\",\n \"We note that the bidirectional flow between parent nodes and their corresponding children nodes defines a new traversing strategy of the tree nodes, that in addition to the dynamic learning aspect, encodes two different types of learning transfer: explicit and implicit.\",\n \"Indeed, the ascending and descending flows between parent nodes and their children through the direct transfer of their generated probability maps is an explicit learning transfer.\",\n \"However, in our binary tree, when a parent node $i$ receives the ascending flows ($F^k(j \\\\rightarrow i)$ and $F^k(j^{\\\\prime } \\\\rightarrow i)$ from its left and right children nodes j and j', they are fused before being passed on, in a second round, as contextual information ($F^{k+1}(i \\\\rightarrow j)$ and $F^{k+1}(i \\\\rightarrow j^{\\\\prime }) $ ) to the children nodes (Fig.\",\n \"REF ).\",\n \"The probability maps fusion at the parent node level is performed through simple averaging.\",\n \"In particular, the parent node concatenates the fused probability map with the original input features to generate a new segmentation probability map result that will be communicated to its two children classifier nodes.\",\n \"Hence, the children nodes of the same parent node explicitly cooperate to improve their parent learning, and implicitly cooperate to improve their own learning while using their parent node as a proxy.\",\n \"SRF-BN Dynamic Tree.\",\n \"In this work, each classifier node is assigned a SRF or a BN model, previously described in Section 2, to define our Dynamic Tree architecture.\",\n \"The transferred information between classifiers through the descending and ascending flows is used in addition to the testing image features as contextual information, while BN classifier uses this information as prior knowledge (i.e prior probability) to perform the multi-label segmentation task.\",\n \"The combination of SRF and BN classifiers is compelling for the following reasons.\",\n \"First, it enhances the performance of BN by taking the posterior probability generated by SRF as prior probability.\",\n \"This justifies our choice of the root node of our DT as a SRF.\",\n \"Second, it improves SRF performance around irregular between-class boundaries since SRF benefits from BN structure learning, which is based on image over-segmentation that is guided by object boundaries.\",\n \"Third, as the SRF maps image information at the patch level, while BN models knowledge at the superpixel level, their combination allows the aggregation of regular (i.e.\",\n \"patch) and irregular (i.e.\",\n \"superpixel) structures in the image for our target multi-label segmentation task.\"\n ],\n [\n \"Dynamic Multi-scale Tree Learning\",\n \"To further boost the performance of our multi-label segmentation framework and enhance the segmentation accuracy, we introduce a multi-scale learning strategy in our dynamic tree architecture by varying the size of the input patches and superpixels used to grow the SRF and construct the BN classifier.\",\n \"Specifically, we use a different scale at each depth level such as we go deeper along the tree edges nearing its leaf nodes, we progressively decrease the size of both patches and superpixels in the training and testing stages.\",\n \"In addition to capturing coarse-to-fine details of the image anatomical structure, the application of the multi-scale strategy to the proposed DT allows to capture fine-to-coarse information.\",\n \"Indeed, DMT learning semantically divides the image into different patterns (e.g., different patches and superpixels at each depth of the tree) in both intensity and label domains at different scales.\",\n \"However, thanks to the bidirectional dynamic flow, the scale defined at each depth influences the performance of parent nodes (in previous level) and children nodes (in next level), which allows to simultaneously perform coarse-to-fine and fine-to-coarse information integration in the multi-label classification task.\",\n \"Moreover, a depth-wise multi-scale feature representation adaptively encodes image features at different scales for each image pixel in the image element (superpixel or patch).\"\n ],\n [\n \"Statistical superpixel-based and patch-based feature extraction\",\n \"To train each classifier node in the tree, we extract the following statistical features at the superpixel level (for BN) and 2D patch level (for SRF): first order operators (mean, standard deviation, max, min, median, Sobel, gradient); higher order operators (Laplacian, difference of Gaussian, entropy, curvatures, kurtosis, skewness), texture features (Gabor filter), and spatial context features (symmetry, projection, neighborhoods) [13].\"\n ],\n [\n \" Results and Discussion\",\n \"Dataset and parameters.\",\n \"We evaluate our proposed brain tumor segmentation framework on 50 subjects with high-grade gliomas, randomly selected from the Brain Tumor Image Segmentation Challenge (BRATS 2015) dataset [11].\",\n \"For each patient, we use three MRI modalities (FLAIR, T2-w, T1-c) along with the corresponding manually segmented glioma lesions.\",\n \"They are rigidly co-registered and resampled to a common resolution to establish patch-to-patch correspondence across modalities.\",\n \"Then, we apply N4 filter for inhomogeneity correction, and use histogram linear transformation for intensity normalization.\",\n \"Table: Segmentation results of the proposed framework and comparison methods averaged across 50 patients.\",\n \"(HT: whole Tumor; CT: Core Tumor; ET: Enhanced Tumor; depth of the tree; * indicates outperformed methods with p-value≺0.05p-value \\\\prec 0.05).For the baseline methods training we adopt the following parameters:(1) Edgemap generation: we use the SLICE oversegmentation algorithm with a superpixel number fixed to 1000 and compactness fixed to 10 [1].\",\n \"To establish superpixel-to-superpixel correspondence across modalities for each subject, we first oversegment the FLAIR MRI, then we apply the generated edgemap (i.e., superpixel partition) to the corresponding T1-c and T2-w MR images.\",\n \"(2) SRF training: we grow 15 trees using intensity feature patches of size 10x10 and label patches of size 7x7.\",\n \"(3) BN construction: the BN model is built using the generated edgemap as detailed in Section 2; the conditional probabilities modeling the relationships between the superpixel labeling and the edge state are defined as follows: $P(E_{j}=1|p_a (E_{j}))= 0.8$ if the parent region nodes have different labels, and $P(E_{j}=1|p_a (E_{j})) = 0.2$ otherwise.\",\n \"Evaluation and comparison methods.\",\n \"For comparison, as baseline methods we use: (1) SRF: the Random Forest version that exploits structural information described in Section 2, (2) BN: the classification algorithm described in Section 2 where the prior probability of superpixels is set as a uniform distribution, (3) SRF-SRF denotes the auto-context Structured Random Forest, (4) BN-BN denotes the auto-context Bayesian Network, where the first BN prior probability is set as a uniform distribution while the second classifier use the posterior probability of its previous as prior probability.\",\n \"Of note, by conventional auto-context classifier, we mean a uni-directional contextual flow from one classifier to the next one.\",\n \"The segmentation frameworks were trained using leave-one-patient cross-validation experiments.\",\n \"For evaluation, we use the Dice score between the ground truth region area $A_{gt}$ and the segmented region area $A_s$ as follows $D = (A_{gt} \\\\bigcap A_s)/ 2( A_{gt} + A_s)$ .\",\n \"Next, we investigate the influence of the tree depth as well as the multi-scale tree learning strategy on the performance of the proposed architecture.\",\n \"Varying tree architectures.\",\n \"In this experiment, we evaluate two different tree architectures to examine the impact of the tree depth on the framework performance.\",\n \"Table.\",\n \"REF shows the segmentation results for 2-level tree (i.e.\",\n \"depth=2) and 1-level tree (i.e.\",\n \"depth=1) for tumor lesion multi-label segmentation with and without multiscale variant.\",\n \"Although the average Dice Score has improved from depth 1 to 2, the improvement wasn't statistically significant.\",\n \"We did not explore larger depths (d>2), since as the binary tree grows exponentially, its computational time dramatically increases and becomes demanding in terms of resources (especially memory).\",\n \"Multi-scale tree architecture.\",\n \"To examine the influence of the multiscale DT learning strategy, we compare the conventional DT architecture (at a fixed-scale) to MDT architecture.\",\n \"For the fixed-scale architecture, all tree nodes nest either an SRF classifier trained using intensity feature patches of size 10x10 and label patches of size 7x7 or a BN classifier constructed using an edgemap of 1000 superpixels generated with a compactness of 10.\",\n \"In the multiscale architecture, we keep the same parameters of the fixed-scale architecture at the first level of the tree while the classifiers of the second level are trained with different parameters.\",\n \"Specifically, we use smaller intensity patches (of size 8x8) and label patches (of size 5x5) for the SRF training, and a smaller number of superpixels for BN construction (1200 superpixels).\",\n \"Figure: Qualitative segmentation results for all the baseline methods applied on 5 subjects: (a) BN segmentation result.\",\n \"(b) SRF segmentation result.\",\n \"(c) Auto-context BN.\",\n \"(d) Auto-context SRF.\",\n \"(e) SRF+BN segmentation result.\",\n \"(f) DMT segmentation result.\",\n \"(depth=2).\",\n \"(g) Ground truth label map.Clearly, the quantitative results show the outperformance (improvement of 7%) of both proposed DT and DMT architectures in comparison with several baseline methods for multi-label tumor lesion segmentation with statistical significance (p < 0.05) .\",\n \"This indicates that a deeper combination of different learning models helps increase the segmentation accuracy.\",\n \"When comparing the results of the SRF and BN we found that SRF outperforms BN in segmenting the three classes: wHole Tumor (HT), Core Tumor (CT) and Enhancing Tumor (ET) Table.\",\n \"REF .\",\n \"This is due to the fact that BN have difficulties in segmenting low-contrast images and identifying different superpixels having similar characteristics, especially with the lack of any prior knowledge on the anatomical structure of the testing image.\",\n \"Although BN has a low Dice score compared to SRF, in Fig.\",\n \"REF we can note that it has better performance in detecting the boundaries between different classes.\",\n \"This shows the impact of the irregular structure of superpixels used during BN training and testing, which gives BN the ability to be more accurate in detecting object boundaries compared to SRF that considers regular image patches.\",\n \"Notably, BN structure is individualized during the testing stage for each testing subject since it is based on the testing image oversegmentation map.\",\n \"Thus, SRF and BN classifiers are complementary.\",\n \"First, they perform segmentation at regular and irregular structures of the image.\",\n \"Second, one (SRF) learns image knowledge during the training stage, while the other (BN) is structured using the input testing image during the testing stage through modeling the testing image structure.\",\n \"Further, the results of SRF-SRF and BN-BN models that implement the auto-context approach show an improvement of the segmentation results at both qualitative and quantitative levels when compared with baseline SRF and BN models.\",\n \"More importantly, we note that BN-BN cascade outperforms SRF-SRF cascade when segmenting the Core Tumor and Enhancing Tumor (ET) lesions.\",\n \"This can be explained by the fine and irregular anatomical details of these image structures when compared to the whole tumor lesion.\",\n \"Since BN is trained using irregular superpixels, it produced more accurate segmentations for these classes (e.g., BN-BN:56.14 vs SRF-SRF: 37.12 for ET).\",\n \"Through further cascading both SRF and BN classifiers, we note that the heterogenoues SRF-BN cascade produced much better results compared to both autocontext SRF and autocontext BN for two main reasons.\",\n \"First SRF aids in defining BN prior based on the testing image structure, while BN enhances the performance of SRF at the boundaries level.\",\n \"This further highlights the importance of integrating both regular and irregular image elements for training classifiers that capture different image structures.\",\n \"The outperformance of the proposed DMT architecture also lays ground for our assumption that embedding SRF and BN into our a unified dynamic architecture where they mutually benefit from their learning boosts up the multi-label segmentation accuracy.\",\n \"In addition to the previously mentionned advantages of combining SRF and BN, it is important to note that the integration of variant cascades of SRF and BN endows our architecture with a an efficient learning ability, where it incorporates in a deep manner the knowledge of SRF based on modeling the dataset during the training step and the individualized learning of BN based on the testing image during the testing step.\",\n \"Notably, our DMT architecture has a few limitations.\",\n \"First, it becomes more demanding in terms of computational and memory resources, as the tree grows exponentially (in the order of $O(2^n)$ ).\",\n \"The more nodes we add to the binary tree, the slower the algorithm converges.\",\n \"Second, the patch and superpixel sizes in the multi-scale learning strategy can be further learned, instead of empirically fixing them through inner cross-validation.\",\n \"Third, the bidirectional flow is currently restricted between neighboring parent and children nodes at a fixed tree depth.\",\n \"This can be extended to further nodes (e.g., root node), where the semantic context progressively diffuses from each node $i$ along tree paths to far-away nodes.\"\n ],\n [\n \"Conclusion\",\n \"We proposed a Dynamic Multi-scale Tree (DMT) learning architecture that both cascades and aggregates classifiers for multi-label medical image segmentation.\",\n \"Specifically, our DMT embeds classifiers into a binary tree architecture, where each node nests a classifier and each edge encodes a learning transfer between the classifiers.\",\n \"A new tree traversal strategy is proposed where a depth-wise bidirectional feedbacks are performed along each edge between a parent node and its child node.\",\n \"This allows explicit learning between parent and children nodes and implicit learning transfer between children of the same parent.\",\n \"Moreover, we train DMT using different scales for input patches and superpixels to capture a coarse-to-fine image details as well as a fine-to-coarse image structures through the depth-wise bidirectional flow.\",\n \"To sum up, our DMT integrates compound and complementary aspects: deep learning, cooperative learning, dynamic learning, coarse-to-fine and fine-to-coarse learning.\",\n \"In our future work, we will devise a more comprehensive tree traversal strategy where the learning transfer starts from the root node, descending all the way down to the leaf nodes and then ascending all the way up to the root node.\",\n \"We will also evaluate our DMT semantic segmentation architecture on different large datasets.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01602"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":7,"numItemsPerPage":100,"numTotalItems":1867602,"offset":700,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc1MDU0Nzk3NCwic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzUwNTUxNTc0LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.rQ9y87SDuaBaCzeGVRKU0Q0Mx9QziEi1MFC7fLIwi53lQfhRh6iLWY8XSvK8XC2b2K1SHcny6i8evemX6yGgDg","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
text
sequencelengths 2
2.54k
| id
stringlengths 9
16
|
|---|---|
[
[
"Polar Transformer Networks"
],
[
"Abstract Convolutional neural networks (CNNs) are inherently equivariant to translation.",
"Efforts to embed other forms of equivariance have concentrated solely on rotation.",
"We expand the notion of equivariance in CNNs through the Polar Transformer Network (PTN).",
"PTN combines ideas from the Spatial Transformer Network (STN) and canonical coordinate representations.",
"The result is a network invariant to translation and equivariant to both rotation and scale.",
"PTN is trained end-to-end and composed of three distinct stages: a polar origin predictor, the newly introduced polar transformer module and a classifier.",
"PTN achieves state-of-the-art on rotated MNIST and the newly introduced SIM2MNIST dataset, an MNIST variation obtained by adding clutter and perturbing digits with translation, rotation and scaling.",
"The ideas of PTN are extensible to 3D which we demonstrate through the Cylindrical Transformer Network."
],
[
"Introduction",
"Whether at the global pattern or local feature level [9], the quest for (in/equi)variant representations is as old as the field of computer vision and pattern recognition itself.",
"State-of-the-art in “hand-crafted” approaches is typified by SIFT [21].",
"These detector/descriptors identify the intrinsic scale or rotation of a region [20], [2] and produce an equivariant descriptor which is normalized for scale and/or rotation invariance.",
"The burden of these methods is in the computation of the orbit (i.e.",
"a sampling the transformation space) which is necessary to achieve equivariance.",
"This motivated steerable filtering which guarantees transformed filter responses can be interpolated from a finite number of filter responses.",
"Steerability was proved for rotations of Gaussian derivatives [7] and extended to scale and translations in the shiftable pyramid [32].",
"Use of the orbit and SVD to create a filter basis was proposed by [27]and in parallel, [30] proved for certain classes of transformations there exists canonical coordinates where deformation of the input presents as translation of the output.",
"Following this work, [26] and [11], [34] proposed a methodology for computing the bases of equivariant spaces given the Lie generators of a transformation.",
"and most recently, [31] proposed the scattering transform which offers representations invariant to translation, scaling, and rotations.",
"Figure: In the log-polar representation, rotations around the origin become vertical shifts, and dilations around the origin become horizontal shifts.",
"The distance between the yellow and green lines is proportional to the rotation angle/scale factor.",
"Top rows: sequence of rotations, and the corresponding polar images.",
"Bottom rows: sequence of dilations, and the corresponding polar images.The current consensus is representations should be learned not designed.",
"Equivariance to translations by convolution and invariance to local deformations by pooling are now textbook ([18], p.335) but approaches to equivariance of more general deformations are still maturing.",
"The main veins are: Spatial Transformer Network (STN) [14] which similarly to SIFT learn a canonical pose and produce an invariant representation through warping, work which constrains the structure of convolutional filters [37] and work which uses the filter orbit [4] to enforce an equivariance to a specific transformation group.",
"In this paper, we propose the Polar Transformer Network (PTN), which combines the ideas of STN and canonical coordinate representations to achieve equivariance to translations, rotations, and dilations.",
"The three stage network learns to identify the object center then transforms the input into log-polar coordinates.",
"In this coordinate system, planar convolutions correspond to group-convolutions in rotation and scale.",
"PTN produces a representation equivariant to rotations and dilations without the challenging parameter regression of STN.",
"We enlarge the notion of equivariance in CNNs beyond Harmonic Networks [37] and Group Convolutions [4] by capturing both rotations and dilations of arbitrary precision.",
"Similar to STN; however, PTN accommodates only global deformations.",
"We present state-of-the-art performance on rotated MNIST and SIM2MNIST, which we introduce.",
"To summarize our contributions: We develop a CNN architecture capable of learning an image representation invariant to translation and equivariant to rotation and dilation.",
"We propose the polar transformer module, which performs a differentiable log-polar transform, amenable to backpropagation training.",
"The transform origin is a latent variable.",
"We show how the polar transform origin can be learned effectively as the centroid of a single channel heatmap predicted by a fully convolutional network."
],
[
"Related Work",
"One of the first equivariant feature extraction schemes was proposed by [26] who suggested the discrete sampling of 2D-rotations of a complex angle modulated filter.",
"About the same time, the image and optical processing community discovered the Mellin transform as a modification of the Fourier transform [40], [1].",
"The Fourier-Mellin transform is equivariant to rotation and scale while its modulus is invariant.",
"During the 80's and 90's invariances of integral transforms were developed through methods based in the Lie generators of the respective transforms starting from one-parameter transforms [6] and generalizing to Abelian subgroups of the affine group [30].",
"Closely related to the (in/equi)variance work is work in steerability, the interpolation of responses to any group action using the response of a finite filter basis.",
"An exact steerability framework began in [7], where rotational steerability for Gaussian derivatives was explicitly computed.",
"It was extended to the shiftable pyramid [32], which handle rotation and scale.",
"A method of approximating steerability by learning a lower dimensional representation of the image deformation from the transformation orbit and the SVD was proposed by [27].",
"A unification of Lie generator and steerability approaches was introduced by [34] who used SVD to reduce the number of basis functions for a given transformation group.",
"Teo and Hel-Or developed the most extensive framework for steerability [34], [11], and proposed the first approach for non-Abelian groups starting with exact steerability for the largest Abelian subgroup and incrementally steering for the remaining subgroups.",
"[3], [13] recently combined steerability and learnable filters.",
"The most recent “hand-crafted” approach to equivariant representations is the scattering transform [31] which composes rotated and dilated wavelets.",
"Similar to SIFT [21] this approach relies on the equivariance of anchor points (e.g.",
"the maxima of filtered responses in (translation) space).",
"Translation invariance is obtained through the modulus operation which is computed after each convolution.",
"The final scattering coefficient is invariant to translations and equivariant to local rotations and scalings.",
"[16] achieve transformation invariance by pooling feature maps computed over the input orbit, which scales poorly as it requires forward and backward passes for each orbit element.",
"Within the context of CNNs, methods of enforcing equivariance fall to two main veins.",
"In the first, equivariance is obtained by constraining filter structure similarly to Lie generator based approaches [30], [11].",
"Harmonic Networks [37] use filters derived from the complex harmonics achieving both rotational and translational equivariance.",
"The second requires the use of a filter orbit which is itself equivariant to obtain group equivariance.",
"[4] convolve with the orbit of a learned filter and prove the equivariance of group-convolutions and preservation of rotational equivariance in the presence of rectification and pooling.",
"[5] process elements of the image orbit individually and use the set of outputs for classification.",
"[8] produce maps of finite-multiparameter groups, [39] and [22] use a rotational filter orbit to produce oriented feature maps and rotationally invariant features, and [19] propose a transformation layer which acts as a group-convolution by first permuting then transforming by a linear filter.",
"Our approach, PTN, is akin to the second vein.",
"We achieve global rotational equivariance and expand the notion of CNN equivariance to include scaling.",
"PTN employs log-polar coordinates (canonical coordinates in [30]) to achieve rotation-dilation group-convolution through translational convolution subject to the assumption of an image center estimated similarly to the STN.",
"Most related to our method is [12], which achieves equivariance by warping the inputs to a fixed grid, with no learned parameters.",
"When learning features from 3D objects, invariance to transformations is usually achieved through augmenting the training data with transformed versions of the inputs [38], or pooling over transformed versions during training and/or test [23], [28].",
"[29] show that a multi-task approach, i.e.",
"prediction of both the orientation and class, improves classification performance.",
"In our extension to 3D object classification, we explicitly learn representations equivariant to rotations around a family of parallel axes by transforming the input to cylindrical coordinates about a predicted axis."
],
[
"Theoretical Background",
"This section is divided into two parts, the first offers a review of equivariance and group-convolutions.",
"The second offers an explicit example of the equivariance of group-convolutions through the 2D similarity transformations group, SIM(2), comprised of translations, dilations and rotations.",
"Reparameterization of SIM(2) to canonical coordinates allows for the application of the SIM(2) group-convolution using translational convolution."
],
[
"Group Equivariance",
"Equivariant representations are highly sought after as they encode both class and deformation information in a predictable way.",
"Let $G$ be a transformation group and $L_gI$ be the group action applied to an image $I$ .",
"A mapping $\\Phi :E\\rightarrow F$ is said to be equivariant to the group action $L_g$ , $g\\in G$ if $\\Phi (L_gI) = L^{\\prime }_g(\\Phi (I))$ where $L_g$ and $L^{\\prime }_g$ correspond to application of $g$ to $E$ and $F$ respectively and satisfy $L_{gh} = L_{g}L_{h}$ .",
"Invariance is the special case of equivariance where $L^{\\prime }_g$ is the identity.",
"In the context of image classification and CNNs, $g\\in G$ can be thought of as an image deformation and $\\Phi $ a mapping from the image to a feature map.",
"The inherent translational equivariance of CNNs is independent of the convolutional kernel and evident in the corresponding translation of the output in response to translation of the input.",
"Equivariance to other types of deformations can be achieved through application of the group-convolution, a generalization of translational convolution.",
"Letting $f(g)$ and $\\phi (g)$ be real valued functions on $G$ with $L_h f(g) = f(h^{-1}g)$ , the group-convolution is defined [15] $(f \\star _G \\phi )(g) = \\int _{h \\in G} f(h) \\phi (h^{-1}g) \\, dh.$ A slight modification to the definition is necessary in the first CNN layer since the group is acting on the image.",
"The group-convolution reduces to translational convolution when $G$ is translation in $\\mathbb {R}^n$ with addition as the group operator, $\\begin{aligned}(f \\star \\phi )(x) &= \\int _h f(h) \\phi (h^{-1}x) \\, dh\\\\&= \\int _h f(h) \\phi (x-h) \\, dh.\\end{aligned}$ Group-convolution requires integrability over a group and identification of the appropriate measure $dg$ .",
"It can be proved that given the measure $dg$ , group-convolution is always group equivariant: $\\begin{aligned}(L_a f\\star _{G} \\phi )(g)&= \\int _{h \\in G} f(a^{-1}h) \\phi (h^{-1}g) \\, dh\\\\&= \\int _{b \\in G} f(b) \\phi ((ab)^{-1}g) \\, db\\\\&= \\int _{b \\in G} f(b) \\phi (b^{-1}a^{-1}g) \\, db\\\\&= (f\\star _G \\phi )(a^{-1}g) \\\\&= L_a ((f\\star _G \\phi ))(g).\\end{aligned}$ This is depicted in response of an equivariant representation to input deformation (Figure REF (left))."
],
[
"Equivariance in SIM(2)",
"A similarity transformation, $\\rho \\in \\mbox{SIM(2)}$ , acts on a point in $x\\in \\mathbb {R}^2$ by $\\rho x \\rightarrow s\\,R\\,x + t\\quad s\\in \\mathbb {R}^+,\\, R\\in SO(2),\\, t\\in \\mathbb {R}^2,$ where $SO(2)$ is the rotation group.",
"To take advantage of the standard planar convolution in classical CNNs we decompose a $\\rho \\in \\mbox{SIM(2)}$ into a translation, $t$ in $\\mathbb {R}^2$ and a dilated-rotation $r$ in $\\mbox{SO(2)}\\times \\mathbb {R}^+$ .",
"Equivariance to SIM(2) is achieved by learning the center of the dilated rotation, shifting the original image accordingly then transforming the image to canonical coordinates.",
"In this reparameterization the standard translational convolution is equivalent to the dilated-rotation group-convolution.",
"The origin predictor is an application of STN to global translation prediction [14], the centroid of the output is taken as the origin of the input.",
"Transformation of the image $L_t I= I(t-t_0)$ (canonization in [33]) reduces the $\\mbox{SIM(2)}$ deformation to a dilated-rotation if $t_o$ is the true translation.",
"After centering, we perform $\\mbox{SO(2)}\\times \\mathbb {R}^+$ convolutions on the new image $I_o=I(x-t_o)$ : $f(r) = \\int _{x \\in \\mathbb {R}^2} I_o(x) \\phi (r^{-1}x) \\,\\,dx$ and the feature maps $f$ in subsequent layers $h(r) = \\int _{s \\in SO(2)\\times \\mathbb {R}^+} f(s) \\phi (s^{-1}r) \\,\\,ds$ where $r,s\\in $ $\\mbox{SO(2)}\\times \\mathbb {R}^+$ .",
"We compute this convolution through use of canonical coordinates for Abelian Lie-groups [30].",
"The centered image $I_o(x,y)$we abuse the notation here and momentarily we use $x$ as the x-coordinate instead of $x \\in \\mathbb {R}^2$ .",
"is transformed to log-polar coordinates, $I( e^{\\xi } \\cos (\\theta ),e^{\\xi } \\sin (\\theta ))$ hereafter written $\\lambda (\\xi ,\\theta )$ with $(\\xi ,\\theta )\\in $ $\\mbox{SO(2)}\\times \\mathbb {R}^+$ for notational convenience.",
"The shift of the dilated-rotation equivariant representation in response to input deformation is shown in Figure REF (right) using canonical coordinates.",
"Figure: Left: Group-convolutions in SO(2)SO(2).The images in the left most column differ by 90 ∘ 90^{\\circ } rotation, the filters are shown in the top row.Application of the rotational group-convolution with an arbitrary filter results is shown to produce an equivariant representation.The inner-product each of filter orbit (rotated from0-360 ∘ 0-360^\\circ ) and the image is plotted in blue for the top image and red for the bottom image.",
"Observe how the filter response is shifted by 90 ∘ 90^{\\circ }.Right: Group-convolutions in SO(2)×ℝ + \\mbox{SO(2)}\\times \\mathbb {R}^+.Images in the left most column differ by a rotation of π/4\\pi /4 and scaling of 1.21.2.Careful consideration of the resulting heatmaps (shown in canonical coordinates) reveals a shift corresponding to the deformation of the input image.In canonical coordinates $s^{-1}r = \\xi _r -\\xi ,\\theta _r-\\theta $ and the $\\mbox{SO(2)}\\times \\mathbb {R}^+$ group-convolutionabuse of the term, $\\mbox{SO(2)}\\times \\mathbb {R}^+$ is not a group because the dilation $\\xi $ is not compact.",
"can be expressed and efficiently implemented as a planar convolution $\\int _{s } f(s) \\phi (s^{-1}r) \\,\\,ds= \\int _{s } \\lambda (\\xi ,\\theta ) \\phi (\\xi _r -\\xi ,\\theta _r-\\theta )\\,\\, d\\xi d\\theta .$ To summarize, we (1) construct a network of translational convolutions, (2) take the centroid of the last layer, (3) shift the original image to accordingly, (4) convert to log-polar coordinates, and (5) apply a second networkthe network employs rectifier and pooling which have been shown to preserve equivariance [4].",
"of translational convolutions.",
"The result is a feature map equivariant to dilated-rotations around the origin."
],
[
"Architecture",
"PTN is comprised of two main components connected by the polar transformer module.",
"The first part is the polar origin predictor and the second is the classifier (a conventional fully convolutional network).",
"The building block of the network is a $3 \\times 3\\times K$ convolutional layer followed by batch normalization, an ReLU and occasional subsampling through strided convolution.",
"We will refer to this building block simply as block.",
"Figure REF shows the architecture.",
"Figure: Network architecture.",
"The input image passes through a fully convolutional network, the polar origin predictor, which outputs a heatmap.",
"The centroid of the heatmap (two coordinates), together with the input image, goes into the polar transformer module, which performs a polar transform with origin at the input coordinates.",
"The obtained polar representation is invariant with respect to the original object location; and rotations and dilations are now shifts, which are handled equivariantly by a conventional classifier CNN."
],
[
"Polar Origin Predictor",
"The polar origin predictor operates on the original image and comprises a sequence of blocks followed by a $1 \\times 1$ convolution.",
"The output is a single channel feature map, the centroid of which is taken as the origin of the polar transform.",
"There are some difficulties in training a neural network to predict coordinates in images.",
"Some approaches [36] attempt to use fully connected layers to directly regress the coordinates with limited success.",
"A better option is to predict heatmaps [35], [25], and take their argmax.",
"However, this can be problematic since backpropogation gradients are zero in all but one point, which impedes learning.",
"The usual approach to heatmap prediction is evaluation of a loss against some ground truth.",
"In this approach the argmax gradient problem is circumvented by supervision.",
"In PTN the the gradient of the output coordinates must be taken with respect to the heatmap since the polar origin is unknown and must be learned.",
"Use of argmax is avoided by using the centroid of the heatmap as the polar origin.",
"The gradient of the centroid with respect to the heatmap is constant and nonzero for all points, making learning possible."
],
[
"Polar transformer module",
"The polar transformer module takes the origin prediction and image as inputs and outputs the log-polar representation of the input.",
"The module uses the same differentiable image sampling technique as STN [14], which allows output coordinates $V_{i}$ to be expressed in terms of the input $U$ and the source sample point coordinates $(x_i^s, y_i^s)$ .",
"The log-polar transform in terms of the source sample points and target regular grid $(x_i^t, y_i^t)$ is: $x_i^s &= x_0 + r^{{x_i^t}/{W}} \\cos {\\frac{2\\pi y_i^t}{H}} \\\\y_i^s &= y_0 + r^{{x_i^t}/{W}} \\sin {\\frac{2\\pi y_i^t}{H}}$ where $(x_0, y_0)$ is the origin, $W, H$ are the output width and height, and $r$ is the maximum distance from the origin, set to $0.5\\sqrt{H^2 + W^2}$ in our experiments."
],
[
"Wrap-around padding",
"To maintain feature map resolution, most CNN implementations use zero-padding.",
"This is not ideal for the polar representation, as it is periodic about the angular axis.",
"A rotation of the input result in a vertical shift of the output, wrapping at the boundary; hence, identification of the top and bottom most rows is most appropriate.",
"This is achieved with wrap-around padding on the vertical dimension.The top most row of the feature map is padded using the bottom rows and vice versa.",
"Zero-padding is used in the horizontal dimension.",
"tab:ablation shows a performance evaluation."
],
[
"Polar origin augmentation",
"To improve robustness of our method, we augment the polar origin during training time by adding a random shift to the regressed polar origin coordinates.",
"Note that this comes for little computational cost compared to conventional augmentation methods such as rotating the input image.",
"tab:ablation quantifies the performance gains of this kind of augmentation."
],
[
"Architectures",
"We briefly define the architectures in this section, see for details.",
"CCNN is a conventional fully convolutional network; PCNN is the same, but applied to polar images with central origin.",
"STN is our implementation of the spatial transformer networks [14].",
"PTN is our polar transformer networks, and PTN-CNN is a combination of PTN and CCNN.",
"The suffixes S and B indicate small and big networks, according to the number of parameters.",
"The suffixes + and ++ indicate training and training+test rotation augmentation.",
"We perform rotation augmentation for polar-based methods.",
"In theory, the effect of input rotation is just a shift in the corresponding polar image, which should not affect the classifier CNN.",
"In practice, interpolation and angle discretization effects result in slightly different polar images for rotated inputs, so even the polar-based methods benefit from this kind of augmentation."
],
[
"Rotated MNIST {{cite:98e8fed9514d356730fe73835f58bb87eb66f382}}",
"tab:rot shows the results.",
"We divide the analysis in two parts; on the left, we show approaches with smaller networks and no rotation augmentation, on the right there are no restrictions.",
"Between the restricted approaches, the Harmonic Network [37] outperforms the PTN by a small margin, but with almost 4x more training time, because the convolutions on complex variables are more costly.",
"Also worth mentioning is the poor performance of the STN with no augmentation, which shows that learning the transformation parameters is much harder than learning the polar origin coordinates.",
"Between the unrestricted approaches, most variants of PTN-B outperform the current state of the art, with significant improvements when combined with CCNN and/or test time augmentation.",
"Finally, we note that the PCNN achieves a relatively high accuracy in this dataset because the digits are mostly centered, so using the polar transform origin as the image center is reasonable.",
"Our method, however, outperforms it by a high margin, showing that even in this case, it is possible to find an origin away from the image center that results in a more distinctive representation.",
"[h] Performance on rotated MNIST.",
"Errors are averages of several runs, with standard deviations within parenthesis.",
"Times are average training time per epoch.",
"Table: NO_CAPTION1, 2, 3, 4, 5 [37], [4], [39], [16], [22] 6 Test time performance is 8x slower when using test time augmentation Other MNIST variants We also perform experiments in other MNIST variants.",
"MNIST R, RTS are replicated from [14].",
"We introduce SIM2MNIST, with a more challenging set of transformations from SIM(2).",
"See for more details about the datasets.",
"Table REF shows the results.",
"We can see that the PTN performance mostly matches the STN on both MNIST R and RTS.",
"The deformations on these datasets are mild and data is plenty, so the performance may be saturated.",
"On SIM2MNIST, however, the deformations are more challenging and the training set 5x smaller.",
"The PCNN performance is significantly lower, which reiterates the importance of predicting the best polar origin.",
"The HNet outperforms the other methods (except the PTN), thanks to its translation and rotation equivariance properties.",
"Our method is more efficient both in number of parameters and training time, and is also equivariant to dilations, achieving the best performance by a large margin.",
"Performance on MNIST variants.",
"Table: NO_CAPTION 1 No augmentation is used with SIM2MNIST, despite the + suffixes 2 Our modified version, with two extra layers with subsampling to account for larger input Visualization Figure: Left: The rows alternate between samples from SIM2MNIST, where the predicted origin is shown in green, and their learned polar representation.Note how rotations and dilations of the object become shifts.Right: Each row shows a different input and correspondent feature maps on the last convolutional layer.The first and second rows show that the 180 ∘ 180^\\circ rotation results in a half-height vertical shift of the feature maps.The third and fourth rows show that the 2.4×2.4 \\times dilation results in a shift right of the feature maps.The first and third rows show invariance to translation.We visualize network activations to confirm our claims about invariance to translation and equivariance to rotations and dilations.",
"Figure REF (left) shows some of the predicted polar origins and the results of the polar transform.",
"We can see that the network learns to reject clutter and to find a suitable origin for the polar transform, and that the representation after the polar transformer module does present the properties claimed.",
"We proceed to visualize if the properties are preserved in deeper layers.",
"Figure REF (right) shows the activations of selected channels from the last convolutional layer, for different rotations, dilations, and translations of the input.",
"The reader can verify that the equivariance to rotations and dilations, and the invariance to translations are indeed preserved during the sequence of convolutional layers.",
"Extension to 3D object classification We extend our model to perform 3D object classification from voxel occupancy grids.",
"We assume that the inputs are transformed by random rotations around an axis from a family of parallel axes.",
"Then, a rotation around that axis corresponds to a translation in cylindrical coordinates.",
"In order to achieve equivariance to rotations, we predict an axis and use it as the origin to transform to cylindrical coordinates.",
"If the axis is parallel to one of the input grid axes, the cylindrical transform amounts to channel-wise polar transforms, where the origin is the same for all channels and each channel is a 2D slice of the 3D voxel grid.",
"In this setting, we can just apply the polar transformer layer to each slice.",
"We use a technique similar to the anisotropic probing of [28] to predict the axis.",
"Let $z$ denote the input grid axis parallel to the rotation axis.",
"We treat the dimension indexed by $z$ as channels, and run regular 2D convolutional layers, reducing the number of channels on each layer, eventually collapsing to a single 2D heatmap.",
"The heatmap centroid gives one point of the axis, and the direction is parallel to $z$ .",
"In other words, the centroid is the origin of all channel-wise polar transforms.",
"We then proceed with a regular 3D CNN classifier, acting on the cylindrical representation.",
"The 3D convolutions are equivariant to translations; since they act on cylindrical coordinates, the learned representation is equivariant to input rotations around axes parallel to $z$ .",
"We run experiments on ModelNet40 [38], which contains objects rotated around the gravity direction ($z$ ).",
"Figure REF shows examples of input voxel grids and their cylindrical coordinates representation, while table REF shows the classification performance.",
"To the best of our knowledge, our method outperforms all published voxel-based methods, even with no test time augmentation.",
"However, the multi-view based methods generally outperform the voxel-based.",
"[28].",
"Note that we could also achieve equivariance to scale by using log-cylindrical or log-spherical coordinates, but none of these change of coordinates would result in equivariance to arbitrary 3D rotations.",
"Figure: Top: rotated voxel occupancy grids.Bottom: corresponding cylindrical representations.Note how rotations around a vertical axis correspond to translations over a horizontal axis.ModelNet40 classification performance.",
"We compare only with voxel-based methods.",
"Table: NO_CAPTION Conclusion We have proposed a novel network whose output is invariant to translations and equivariant to the group of dilations/rotations.",
"We have combined the idea of learning the translation (similar to the spatial transformer) but providing equivariance for the scaling and rotation, avoiding, thus, fully connected layers required for the pose regression in the spatial transformer.",
"Equivariance with respect to dilated rotations is achieved by convolution in this group.",
"Such a convolution would require the production of multiple group copies, however, we avoid this by transforming into canonical coordinates.",
"We improve the state of the art performance on rotated MNIST by a large margin, and outperform all other tested methods on a new dataset we call SIM2MNIST.",
"We expect our approach to be applicable to other problems, where the presence of different orientations and scales hinder the performance of conventional CNNs.",
"Appendices Architectures details We implement the following architectures for comparison, Conventional CNN (CCNN), a fully convolutional network, composed of a sequence of convolutional layers and some rounds of subsampling .",
"Polar CNN (PCNN), same architecture as CCNN, operating on polar images.",
"The log-polar transform is pre-computed at the image center before training, as in [12].",
"The fundamental difference between our method and this is that we learn the polar origin implicitly, instead of fixing it.",
"Spatial Transformer Network (STN), our implementation of [14], replacing the localization network by four blocks of 20 filters and stride 2, followed by a 20 unit fully connected layer, which we found to perform better.",
"The transformation regressed is in $\\mbox{SIM(2)}$ , and a CCNN comes after the transform.",
"Polar Transformer Network (PTN), our proposed method.",
"The polar origin predictor comprises three blocks of 20 filters each, with stride 2 on the first block (or the first two blocks, when input is $96 \\times 96$ ).",
"The classification network is the CCNN.",
"PTN-CNN, we classify based on the sum of the per class scores of instances of PTN and CCNN trained independently.",
"The following suffixes qualify the architectures described above: S, “small” network, with seven blocks of 20 filters and one round of subsampling (equivalent to the Z2CNN in [4]).",
"B, “big” network, with 8 blocks with the following number of filters: 16, 16, 32, 32, 32, 64, 64, 64.",
"Subsampling by strided convolution is used whenever the number of filters increase.",
"We add up to two 2 extra blocks of 16 filters with stride 2 at the beginning to handle larger input resolutions (one for $42 \\times 42$ and two for $96 \\times 96$ ).",
"+, training time rotation augmentation by continuous angles.",
"++, training and test time rotation augmentation.",
"We input 8 rotated versions the the query image and classify using the sum of the per class scores.",
"Cylindrical transformer network: The axis prediction part of the cylindrical transformer network is composed of four 2D blocks, with $5\\times 5$ kernels and 32, 16, 8, and 4 channels, no subsampling.",
"The classifier is composed of eight 3D convolutional blocks, with $3\\times 3 \\times 3$ kernels, the following number of filters: 32, 32, 32, 64, 64, 64, 128, 128, and subsampling whenever the number of filters increase.",
"Total number of params is approximately 1M.",
"Dataset details Rotated MNIST The rotated MNIST dataset [17] is composed of $28 \\times 28$ , $360^\\circ $ rotated images of handwritten digits.",
"The training, validation and test sets are of sizes 10k, 2k, and 50k, respectively.",
"MNIST R, we replicate it from [14].",
"It has 60k training and 10k testing samples, where the digits of the original MNIST are rotated between $[-90^\\circ , 90^\\circ ]$ .",
"It is also know as half-rotated MNIST [16].",
"MNIST RTS, we replicate it from [14].",
"It has 60k training and 10k testing samples, where the digits of the original MNIST are rotated between $[-45^\\circ , 45^\\circ ]$ , scaled between 0.7 and 1.2, and shifted within a $42 \\times 42$ black canvas.",
"SIM2MNIST, we introduce a more challenging dataset, based on MNIST, perturbed by random transformations from $\\mbox{SIM(2)}$ .",
"The images are $96 \\times 96$ , with $360^\\circ $ rotations; the scale factors range from 1 to $2.4$ , and the digits can appear anywhere in the image.",
"The training, validation and test set have size 10k, 5k, and 50k, respectively.",
"SVHN Experiments In order to demonstrate the efficacy of PTN on real-world RGB images, we run experiments on the Street View House Numbers (SVHN) dataset [24], and a rotated version that we introduce (ROTSVHN) .",
"The dataset contains cropped images of single digits, as well as the slightly larger images from where the digits are cropped.",
"Using the latter, we can extract the rotated digits without introducing artifacts.",
"Figure REF shows some examples from the ROTSVHN.",
"We use a 32 layer Residual Network [10] as a baseline (ResNet32).",
"The PTN-ResNet32 has 8 residual convolutional layers as the origin predictor, followed by a ResNet32.",
"In contrast with handwritten digits, the 6s and 9s in house numbers are usually indistinguishable.",
"To remove this effect from our analysis, we also run experiments removing those classes from the datasets (which is denoted by appending a minus to the dataset name).",
"Table REF shows the results.",
"The reader will note that rotations cause a significant performance loss on the conventional ResNet; the error increases from 2.09% to 5.39%, even when removing 6s and 9s from the dataset.",
"With PTN, on the other hand, the error goes from 2.85% to 3.96%, which shows our method is more robust to the perturbations, although the performance on the unperturbed datasets is slightly worse.",
"We expect the PTN to be even more advantageous when large scale variations are also present.",
"Figure: ROTSVHN samples.Since the digits are cropped from larger images, no artifacts are introduced when rotating.The 6s and 9s are indistinguishable when rotated.Note that there are usually visible digits on the sides, which pose a challenge for classification and PTN origin prediction.Table: SVHN classification performance.The minus suffix indicate removal of 6s and 9s.PTN shows slightly worse performance on the unperturbed dataset, but is clearly superior when rotations are present.",
"Ablation Study We quantify the performance boost obtained with wrap around padding, polar origin augmentation, and training time rotation augmentation.",
"Results are based on the PTN-B variant trained on Rotated MNIST.",
"We remove one operation at a time and verify that the performance consistently drops, which indicates that all operations are indeed helpful.",
"tab:ablation shows the results.",
"Table: Ablation study.Rotation and polar origin augmentation during training time, and wrap around padding all contribute to reduce the error.Results are from PTN-B on the rotated MNIST."
],
[
"Conclusion",
"We have proposed a novel network whose output is invariant to translations and equivariant to the group of dilations/rotations.",
"We have combined the idea of learning the translation (similar to the spatial transformer) but providing equivariance for the scaling and rotation, avoiding, thus, fully connected layers required for the pose regression in the spatial transformer.",
"Equivariance with respect to dilated rotations is achieved by convolution in this group.",
"Such a convolution would require the production of multiple group copies, however, we avoid this by transforming into canonical coordinates.",
"We improve the state of the art performance on rotated MNIST by a large margin, and outperform all other tested methods on a new dataset we call SIM2MNIST.",
"We expect our approach to be applicable to other problems, where the presence of different orientations and scales hinder the performance of conventional CNNs."
],
[
"Architectures details",
"We implement the following architectures for comparison, Conventional CNN (CCNN), a fully convolutional network, composed of a sequence of convolutional layers and some rounds of subsampling .",
"Polar CNN (PCNN), same architecture as CCNN, operating on polar images.",
"The log-polar transform is pre-computed at the image center before training, as in [12].",
"The fundamental difference between our method and this is that we learn the polar origin implicitly, instead of fixing it.",
"Spatial Transformer Network (STN), our implementation of [14], replacing the localization network by four blocks of 20 filters and stride 2, followed by a 20 unit fully connected layer, which we found to perform better.",
"The transformation regressed is in $\\mbox{SIM(2)}$ , and a CCNN comes after the transform.",
"Polar Transformer Network (PTN), our proposed method.",
"The polar origin predictor comprises three blocks of 20 filters each, with stride 2 on the first block (or the first two blocks, when input is $96 \\times 96$ ).",
"The classification network is the CCNN.",
"PTN-CNN, we classify based on the sum of the per class scores of instances of PTN and CCNN trained independently.",
"The following suffixes qualify the architectures described above: S, “small” network, with seven blocks of 20 filters and one round of subsampling (equivalent to the Z2CNN in [4]).",
"B, “big” network, with 8 blocks with the following number of filters: 16, 16, 32, 32, 32, 64, 64, 64.",
"Subsampling by strided convolution is used whenever the number of filters increase.",
"We add up to two 2 extra blocks of 16 filters with stride 2 at the beginning to handle larger input resolutions (one for $42 \\times 42$ and two for $96 \\times 96$ ).",
"+, training time rotation augmentation by continuous angles.",
"++, training and test time rotation augmentation.",
"We input 8 rotated versions the the query image and classify using the sum of the per class scores.",
"Cylindrical transformer network: The axis prediction part of the cylindrical transformer network is composed of four 2D blocks, with $5\\times 5$ kernels and 32, 16, 8, and 4 channels, no subsampling.",
"The classifier is composed of eight 3D convolutional blocks, with $3\\times 3 \\times 3$ kernels, the following number of filters: 32, 32, 32, 64, 64, 64, 128, 128, and subsampling whenever the number of filters increase.",
"Total number of params is approximately 1M."
],
[
"Dataset details",
" Rotated MNIST The rotated MNIST dataset [17] is composed of $28 \\times 28$ , $360^\\circ $ rotated images of handwritten digits.",
"The training, validation and test sets are of sizes 10k, 2k, and 50k, respectively.",
"MNIST R, we replicate it from [14].",
"It has 60k training and 10k testing samples, where the digits of the original MNIST are rotated between $[-90^\\circ , 90^\\circ ]$ .",
"It is also know as half-rotated MNIST [16].",
"MNIST RTS, we replicate it from [14].",
"It has 60k training and 10k testing samples, where the digits of the original MNIST are rotated between $[-45^\\circ , 45^\\circ ]$ , scaled between 0.7 and 1.2, and shifted within a $42 \\times 42$ black canvas.",
"SIM2MNIST, we introduce a more challenging dataset, based on MNIST, perturbed by random transformations from $\\mbox{SIM(2)}$ .",
"The images are $96 \\times 96$ , with $360^\\circ $ rotations; the scale factors range from 1 to $2.4$ , and the digits can appear anywhere in the image.",
"The training, validation and test set have size 10k, 5k, and 50k, respectively."
],
[
"SVHN Experiments",
"In order to demonstrate the efficacy of PTN on real-world RGB images, we run experiments on the Street View House Numbers (SVHN) dataset [24], and a rotated version that we introduce (ROTSVHN) .",
"The dataset contains cropped images of single digits, as well as the slightly larger images from where the digits are cropped.",
"Using the latter, we can extract the rotated digits without introducing artifacts.",
"Figure REF shows some examples from the ROTSVHN.",
"We use a 32 layer Residual Network [10] as a baseline (ResNet32).",
"The PTN-ResNet32 has 8 residual convolutional layers as the origin predictor, followed by a ResNet32.",
"In contrast with handwritten digits, the 6s and 9s in house numbers are usually indistinguishable.",
"To remove this effect from our analysis, we also run experiments removing those classes from the datasets (which is denoted by appending a minus to the dataset name).",
"Table REF shows the results.",
"The reader will note that rotations cause a significant performance loss on the conventional ResNet; the error increases from 2.09% to 5.39%, even when removing 6s and 9s from the dataset.",
"With PTN, on the other hand, the error goes from 2.85% to 3.96%, which shows our method is more robust to the perturbations, although the performance on the unperturbed datasets is slightly worse.",
"We expect the PTN to be even more advantageous when large scale variations are also present.",
"Figure: ROTSVHN samples.Since the digits are cropped from larger images, no artifacts are introduced when rotating.The 6s and 9s are indistinguishable when rotated.Note that there are usually visible digits on the sides, which pose a challenge for classification and PTN origin prediction.Table: SVHN classification performance.The minus suffix indicate removal of 6s and 9s.PTN shows slightly worse performance on the unperturbed dataset, but is clearly superior when rotations are present."
],
[
"Ablation Study",
"We quantify the performance boost obtained with wrap around padding, polar origin augmentation, and training time rotation augmentation.",
"Results are based on the PTN-B variant trained on Rotated MNIST.",
"We remove one operation at a time and verify that the performance consistently drops, which indicates that all operations are indeed helpful.",
"tab:ablation shows the results.",
"Table: Ablation study.Rotation and polar origin augmentation during training time, and wrap around padding all contribute to reduce the error.Results are from PTN-B on the rotated MNIST."
]
] | 1709.01889 |
[
[
"Phase-Encoded Hyperpolarized Nanodiamond for Magnetic Resonance Imaging"
],
[
"Abstract Surface-functionalized nanomaterials can act as theranostic agents that detect disease and track biological processes using hyperpolarized magnetic resonance imaging (MRI).",
"Candidate materials are sparse however, requiring spinful nuclei with long spin-lattice relaxation (T1) and spin-dephasing times (T2), together with a reservoir of electrons to impart hyperpolarization.",
"Here, we demonstrate the versatility of the nanodiamond material system for hyperpolarized 13C MRI, making use of its intrinsic paramagnetic defect centers, hours-long nuclear T1 times, and T2 times suitable for spatially resolving millimeter-scale structures.",
"Combining these properties, we enable a new imaging modality that exploits the phase-contrast between spins encoded with a hyperpolarization that is aligned, or anti-aligned with the external magnetic field.",
"The use of phase-encoded hyperpolarization allows nanodiamonds to be tagged and distinguished in an MRI based on their spin-orientation alone, and could permit the action of specific bio-functionalized complexes to be directly compared and imaged."
],
[
"Methods",
"Diamond Particles: Nanodiamonds used in this work were purchased from Microdiamant (Switzerland).",
"Specific nanodiamond types were monocrystalline, synthetic HPHT and NAT particles.",
"Both types were used in sizes of 210 nm (0-500 nm, median diameter 210 nm) and 2 $$ m (1.5-2.5 $$ m, median diameter 2 $$ m).",
"The 2 $$ m HPHT diamonds used here are well suited to injection as they have a zeta potential of -38$\\pm $ 7 mV in water.",
"This zeta potential shows that they are aggregation-resistant, a characteristic important to benefiting fully from the surface of nanodiamond in biological applications [42].",
"The 2 $$ m particles display sedimentation on the timescale of hours, as is to be expected for diamond particles larger than 500 nm, which is the point at which gravitational forces overcome Brownian motion.",
"210 nm HPHT nanodiamonds have a near identical zeta potential of -39 $\\pm $ 8 mV with limited sedimentation observed over a period of weeks.",
"For details of dynamic light scattering measurements and calculation of particle stability, see Supp.",
"Note .",
"EPR Characterization: EPR spectra were measured using a Bruker EMX EPR Spectrometer operating at 9.735 GHz and room temperature.",
"Resulting spectra were fit using Easyspin to a three spin component model [11], [43].",
"Defect concentrations were calculated relative to the signal from an irradiated quartz EPR standard [44].",
"Hyperpolarization of diamond: Hyperpolarization is performed in a 2.88 T superconducting NMR magnet with a homebuilt DNP probe [45].",
"80-82 GHz microwaves are generated via a frequency multiplier (Virginia Diodes) and 2 W power amplifier (Quinstar) before being directed to the sample via a slotted waveguide antenna.",
"A helium flow cryostat (Janis) was used to cool the sample to 4.5 K. NMR measurements in the polarizer were taken with a custom saddle coil tuned to the $^{13}$ C Larmor frequency of 30.912 MHz.",
"All hyperpolarization events begin with a $^{13}$ C NMR saturation sequence ($64 \\times \\pi /2$ pulses) to zero the initial polarization.",
"Individual points in the frequency sweeps of Fig.",
"REF d correspond to unique hyperpolarization events, showing the $^{13}$ C NMR signal from a $\\pi /2$ pulse after 60 s of microwave saturation at a constant frequency.",
"NMR enhancement plots in Fig.",
"REF e and 1f show the $^{13}$ C signal magnitude after 20 minutes of microwave saturation compared with the signal seen after an identical period of thermal polarization.",
"Absolute polarization values are calculated from the thermally polarized $^{13}$ C signal at equilibrium and the expected Boltzmann polarization [11].",
"Diamond samples were hyperpolarized in teflon tubes, chosen for their transparency to microwaves and robustness to thermal cycling.",
"Transfer and dissolution: All post-transfer NMR signals were acquired in the same 7 T superconducting NMR magnet used for imaging.",
"Adiabatic sample transfer between the DNP probe and the MRI scanner occurs via a series of “magnetic shields” based on permanent magnets and Halbach arrays (see Supp.",
"Fig.",
"and Supp.",
"Note for further detail).",
"Halbach arrays are especially useful for transferring samples between superconducting magnets as they experience no net translational force in external magnetic fields.",
"Depolarization at 7 T was measured via small tip angle for 2 $$ m and 210 nm samples hyperpolarized for 2 hours and 1 hour respectively.",
"Data were divided by $\\left( \\cos {\\alpha } \\right)^{ n-1 }$ , where $\\alpha $ is the tip angle and $n$ the pulse number, to account for RF-induced signal decay.",
"$T_1$ versus magnetic field strength was measured in the stray field of the imaging magnet with samples hyperpolarized for 20 minutes.",
"Small tip angle measurements of the $^{13}$ C signal magnitude were taken at 7 T before and after shuttling the sample to a lower field region for repeated periods of 30 s. The phantom in Fig.",
"REF a was prepared in an approximately 200 mT permanent magnet by rapid thawing of a $^{13}$ C hyperpolarized 2:1 mixture of nanodiamond and water.",
"The hyperpolarized mixture was then mixed with additional water to give a 120 mg mL$^{-1}$ concentration.",
"Transfer to the MRI system occurred via a 110 mT Halbach array.",
"MRI Experiments: All MRI was performed in a 7 T widebore, microimaging system with $^{1}$ H and $^{13}$ C Larmor frequencies of 299.97 MHz and 75.381 MHz respectively.",
"The microimaging gradient set produces gradient fields up to 250 mT m$^{-1}$ .",
"Phantoms were imaged using a 10 mm, dual resonance $^{1}$ H:$^{13}$ C NMR probe.",
"$^{1}$ H phantom images were acquired with a GRE sequence with 60 $$ m $\\times $ 60 $$ m pixel size and 6 mm slice thickness.",
"$^{13}$ C slice thickness was restricted to 20 mm by the active region of the detection coil.",
"Concentration phantoms contained 140 $$ L of diamond and water mixture.",
"$^{13}$ C GRE image data in Fig.",
"REF b-d and REF d were acquired with a 64 $\\times $ 32 matrix and pixel resolution of 0.7 mm $\\times $ 0.6 mm.",
"The refocusing time (TE) of 1.22 ms was minimized by ramping gradients to full strength and using short, 60 $$ s, excitation pulses (see Supp.",
"Fig.",
"for complete timing parameters).",
"Centrically-ordered phase encoding with a constant flip angle of 20 was used to increase SNR by 1.96 times at the expense of limited blurring in the phase encode direction [46].",
"All images displaying $^{13}$ C data only were interpolated to 128 $\\times $ 128 resolution by zero-filling and Gaussian filtering in $k$ -space.",
"Co-registered $^{13}$ C images were interpolated to the resolution of the accompanying $^{1}$ H image and values smaller than 3 times (Fig.",
"REF b) or 5 times (Fig.",
"REF e, Fig.",
"REF d) the SNR made transparent to reveal the underlying structure in the $^{1}$ H image.",
"Animal Imaging: A 150 mg sample of 2 $$ m nanodiamonds was hyperpolarized for 2.5 hours in a 1:1 mixture with water with 0.2 mL of hot water added during dissolution.",
"Small tip angle characterization showed that hyperpolarizing in water or ethanol gave the same $^{13}$ C enhancement seen for dry samples.",
"The resulting 0.5 mL bolus was injected into the thoracic cavity of a three-week old female mouse post mortem (11 g, mus musculus) with a 21G needle and syringe before transfer to the MRI system.",
"Hyperpolarized nanodiamonds were magnetically shielded with custom magnets when not in the polarizer or imager (further details in Supp.",
"Fig. )",
".",
"Total time for dissolution, injection and transfer to the imager was approximately 60 s. A $^{1}$ H:$^{13}$ C double resonance microimaging probe with 18 mm bore size was used for imaging the mouse torso.",
"$^{13}$ C SNR of this probe was measured to be one third of the 10 mm probe used for phantom imaging.",
"A conventional $^{1}$ H spin-echo sequence was used for high-contrast anatomical imaging.",
"$^{1}$ H images were acquired with 1 mm slice thickness with a 256 $\\times $ 256 matrix and pixel resolution of 230 $$ m $\\times $ 176 $$ m. The same $^{13}$ C GRE sequence was used as in Fig.",
"REF b-d but with a gradient echo time of 0.90 ms and pixel size of 2.1 mm $\\times $ 1.9 mm.",
"Slice thickness was limited to less than 18 mm by the sensitive region of the probe.",
"Phase-contrast Imaging: To demonstrate phase-contrast imaging, a 2 $$ m HPHT sample was positively polarized with 80.87 GHz microwaves for 20 minutes and transferred to the MRI scanner.",
"A second, nominally identical sample was then negatively polarized with 80.94 GHz microwaves for 4 minutes and transferred to the MRI scanner.",
"The authors thank M. S. Rosen for useful discussions and M. C. Cassidy for constructing the hyperpolarization probe.",
"We are also grateful to R. Marsh and A. Itkin for the supply of custom hardware for these experiments.",
"For EPR measurements, the authors acknowledge use of facilities in the Mark Wainwright Analytical Centre at the University of New South Wales.",
"This work was supported by the Australian Research Council Centre of Excellence Scheme (Grant No.",
"EQuS CE110001013) and ARC DP1094439.",
"We also acknowledge use of the nanoparticle analysis tools provided by the Bosch Institute Molecular Biology Facility at the University of Sydney.",
"$^*$ Correspondence and requests for materials should be addressed to: D.J.R.",
"(email: [email protected])."
]
] | 1709.01851 |
[
[
"Neither pulled nor pushed: Genetic drift and front wandering uncover a\n new class of reaction-diffusion waves"
],
[
"Abstract Traveling waves describe diverse natural phenomena from crystal growth in physics to range expansions in biology.",
"Two classes of waves exist with very different properties: pulled and pushed.",
"Pulled waves are driven by high growth rates at the expansion edge, where the number of organisms is small and fluctuations are large.",
"In contrast, fluctuations are suppressed in pushed waves because the region of maximal growth is shifted towards the population bulk.",
"Although it is commonly believed that expansions are either pulled or pushed, we found an intermediate class of waves with bulk-driven growth, but exceedingly large fluctuations.",
"These waves are unusual because their properties are controlled by both the leading edge and the bulk of the front."
],
[
"[itemize]noitemsep,topsep=0pt 0.0pt Short Abstract: Traveling waves describe diverse natural phenomena from crystal growth in physics to range expansions in biology.",
"Two classes of waves exist with very different properties: pulled and pushed.",
"Pulled waves are driven by high growth rates at the expansion edge, where the number of organisms is small and fluctuations are large.",
"In contrast, fluctuations are suppressed in pushed waves because the region of maximal growth is shifted towards the population bulk.",
"Although it is commonly believed that expansions are either pulled or pushed, we found an intermediate class of waves with bulk-driven growth, but exceedingly large fluctuations.",
"These waves are unusual because their properties are controlled by both the leading edge and the bulk of the front.",
"Long Abstract: Epidemics, flame propagation, and cardiac rhythms are classic examples of reaction-diffusion waves that describe a switch from one alternative state to another.",
"Only two types of waves are known: pulled, driven by the leading edge, and pushed, driven by the bulk of the wave.",
"Here, we report a distinct class of semi-pushed waves for which both the bulk and the leading edge contribute to the dynamics.",
"These hybrid waves have the kinetics of pushed waves, but exhibit giant fluctuations similar to pulled waves.",
"The transitions between pulled, semi-pushed, and fully-pushed waves occur at universal ratios of the wave velocity to the Fisher velocity.",
"We derive these results in the context of a species invading a new habitat by examining front diffusion, rate of diversity loss, and fluctuation-induced corrections to the expansion velocity.",
"All three quantities decrease as a power law of the population density with the same exponent.",
"We analytically calculate this exponent taking into account the fluctuations in the shape of the wave front.",
"For fully-pushed waves, the exponent is -1 consistent with the central limit theorem.",
"In semi-pushed waves, however, the fluctuations average out much more slowly, and the exponent approaches 0 towards the transition to pulled waves.",
"As a result, a rapid loss of genetic diversity and large fluctuations in the position of the front occur even for populations with cooperative growth and other forms of an Allee effect.",
"The evolutionary outcome of spatial spreading in such populations could therefore be less predictable than previously thought.",
"Introduction ave-like phenomena are ubiquitous in nature and have been extensively studied across many disciplines.",
"In physics, traveling waves describe chemical reactions, kinetics of phase transitions, and fluid flow [1], [2], [3], [4], [5], [6], [7], [8].",
"In biology, traveling waves describe invasions, disease outbreaks, and spatial processes in physiology and development [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].",
"Even non-spatial phenomena such as Darwinian evolution and dynamics on networks can be successfully modeled by waves propagating in more abstract spaces such as fitness [21], [22], [23], [24], [25], [20].",
"The wide range of applications stimulated substantial effort to develop a general theory of traveling waves that is now commonly used to understand, predict, and control spreading phenomena [1], [16], [17], [20], [9], [14], [26], [27].",
"A major achievement of this theory was the division of traveling waves into two classes with very different properties [28], [29], [9], [1], [30], [31], [6], [26], [32], [33].",
"The first class contains waves that are “pulled” forward by the dynamics at the leading edge.",
"Kinetics of pulled waves are independent from the nonlinearities behind the front, but extremely sensitive to noise and external perturbations [1], [34], [29].",
"In contrast, the waves in the second class are resilient to fluctuations and are “pushed” forward by the nonlinear dynamics behind the wave front.",
"Fluctuations in traveling waves arise due to the randomness associated with discrete events such chemical reaction or birth and deaths.",
"This microscopic stochasticity manifests in many macroscopic properties of the wave including its velocity, the diffusive wandering of the front position, and the loss of genetic diversity [33], [35], [36], [37], [38].",
"For pulled waves, these quantities have been intensely studied because they show an apparent violation of the central limit theorem [39], [33], [1], [34], [29], [30], [35], [36], [25], [37], [40].",
"Naively, one might expect that fluctuations self-average, and their variance is, therefore, inversely proportional to the population density.",
"Instead, the strength of fluctuations in pulled waves has only a logarithmic dependence on the population density.",
"This weak dependence is now completely understood and is explained by the extreme sensitivity of pulled waves to the dynamics at the front [33], [1], [37], [40].",
"A complete understanding is however lacking for fluctuations in pushed waves [1], [34], [29], [30], [31], [35], [36], [25].",
"Since pushed waves are driven by the dynamics at the bulk of the wave front, it is reasonable to expect that the central limit theorem holds, and fluctuations decrease as one over the population density $N$ .",
"Consistent with this expectation, the $1/N$ scaling was theoretically derived both for the effective diffusion constant of the front [38] and for the rate of diversity loss [35].",
"Numerical simulations confirmed the $1/N$ scaling for the diffusion constant [41], but showed a much weaker dependence for the rate of diversity loss [35].",
"Ref.",
"[41], however, considered only propagation into a metastable state, while Ref.",
"[35] analyzed only one particular choice of the nonlinear growth function.",
"As a result, it is not clear whether the effective diffusion constant and the rate of diversity loss behave differently or if there are two distinct types of dynamics within the class of pushed waves.",
"The latter possibility was anticipated by the analysis of how the wave velocity changes if one sets the growth rate to zero below a certain population density [31].",
"This study found that the velocity correction scales as a power law of the growth-rate cutoff with a continuously varying exponent.",
"If the cutoff was a faithful approximation of fluctuations at the front, this result would suggest that the central limit theorem does not apply to pushed waves.",
"Stochastic simulations, however, were not carried out in Ref.",
"[31] to test this prediction.",
"Taken together, previous findings highlight the need to characterize the dynamics of pushed waves more thoroughly.",
"Here, we develop a unified theoretical approach to fluctuations in reaction-diffusion waves and show how to handle divergences and cutoffs that typically arise in analytical calculations.",
"Theoretical predictions are tested against extensive numerical simulations.",
"In simulations, we vary the model parameters to tune the propagation dynamics from pulled to pushed and determine how the front diffusion, diversity loss, and wave velocity depend on the population density.",
"Our main result is that the simple pulled vs. pushed classification does not hold.",
"Instead, there are three distinct classes of traveling waves.",
"Only one of these classes shows weak fluctuations consistent with the central limit theorem.",
"The other two classes exhibit large fluctuations because they are very sensitive to the dynamics at the leading edge of the wave front.",
"Model Traveling waves occur when a transport mechanism couples dynamics at different spatial locations.",
"The nature of these wave-generating processes could be very different and ranges from reactions and diffusion in chemistry to growth and dispersal in ecology.",
"The simplest and most widely-used model of a reaction-diffusion waveThroughout the paper we use the term reaction-diffusion wave to describe propagating fronts that connect two states with different population densities.",
"Reaction-diffusion models, especially with several components, also describe more intricate phenomena such as periodic waves, spatio-temporal chaos, and pulse propagation.",
"While some of our results could be useful in these more general settings, our theory and numerical simulations are limited to regular fronts only.",
"is the generalized Fisher-Kolmogorov equation: $\\frac{\\partial n}{\\partial t} = D\\frac{\\partial ^2 n}{\\partial x^2} + r(n) n + \\sqrt{\\gamma _{n}(n)n}\\eta (t,x) ,$ which, in the context of ecology, describes how a species colonizes a new habitat [1], [9], [42], [43], [44].",
"Here, $n(t,x)$ is the population density of the species, $D$ is the dispersal rate, and $r(n)$ is the density-dependent per capita growth rate.",
"The last term accounts for demographic fluctuations: $\\eta (t,x)$ is a Gaussian white noise, and $\\gamma _n(n)$ quantifies the strength of demographic fluctuations.",
"In simple birth-death models, $\\gamma _n$ is a constant, but we allow for an arbitrary dependence on $n$ provided that $\\gamma _n(0)>0$ .",
"The origin of the noise term and its effects on the wave dynamics are further discussed in Sec.",
"IV of the SI.",
"Pulled waves occur when $r(n)$ is maximal at small $n$ ; for example, when the growth is logistic: $r(n)=r_0(1-n/N)$ [1], [9].",
"Here, $r_0$ is the growth rate at low densities, and $N$ is the carrying capacity that sets the population density behind the front.",
"For pulled waves, the expansion dynamics are controlled by the very tip of the front, where the organisms not only grow at the fastest rate, but also have an unhindered access to the uncolonized territories.",
"As a result, the expansion velocity is independent of the functional form of $r(n)$ and is given by the celebrated result due to Fisher, Kolmogorov, and Skellam [42], [43], [44]: $v_{\\mathrm {\\textsc {f}}} = 2\\sqrt{Dr(0)}.$ Expression REF , to which we refer as the Fisher velocity, can be defined for any model with $r(0)>0$ even when the expansion is not pulled.",
"We show below that $v_{\\mathrm {\\textsc {f}}} $ provides a useful baseline for comparing different types of waves.",
"Pushed waves occur when a species grows best at intermediate population densities [1], [9].",
"Such non-monotonic behavior of $r(n)$ arises through a diverse set of mechanisms and is known as an Allee effect in ecology [45], [46].",
"Most common causes of an Allee effect are cooperative feeding, collective defense against predators, and the difficulty in finding mates at low population densities [47], [48], [49], [50].",
"The velocity of pushed waves is always greater than Fisher's prediction $(v>v_{\\mathrm {\\textsc {f}}})$ and depends on all aspects of the functional form of $r(n)$ [1], [9].",
"Allee effects are typically described by adding a cooperative term to the logistic equation: $r(n) = r_0 \\left(1-\\frac{n}{N}\\right)\\left(1+B\\frac{n}{N}\\right),$ where $B$ is the strength of cooperativity.",
"For this model, the exact solutions are known for the expansion velocity and the population density profile; see SI(Sec.",
"II) and Ref.",
"[9], [51], [52].",
"For $B\\le 2$ , expansions are pulled, and the expansion velocity equals $v_{\\mathrm {\\textsc {f}}}$ , which is independent of $B$ .",
"That is cooperativity does not always increase the expansion velocity even though it always increases the growth rates at high densities.",
"For $B>2$ , expansions are pushed, and $v$ increases with $B$ .",
"Figure REF A illustrates this transition from pulled to pushed waves as cooperativity is increased.",
"In Methods and SI(Sec.",
"II), we also present several alternative models of an Allee effect and show that our conclusions do not depend on a particular choice of $r(n)$ .",
"Figure: Waves transition from pulled to pushed as growth becomes more cooperative.",
"(A) shows the expansion velocity as a function of cooperativity for the growth rate specified by Eq. ().",
"For low cooperativity, expansions are pulled and their velocity equals the Fisher velocity.",
"Beyond the critical value of B=2B=2, expansions become pushed and their velocity exceeds v f v_{\\mathrm {\\textsc {f}}}.",
"Note that the region of high growth is at the leading edge of the front in pulled waves (B), but in the interior of the front in pushed waves (C).",
"This difference is due to the dependence of the growth rate on the population density.",
"For low cooperativity, the growth rate is maximal at low population densities, but, for high cooperativity, the growth rate is maximal at intermediate population densities.",
"In all panels, the exact solution of Eq.",
"() is plotted; B=0B=0 in panel B, and B=4B=4 in panel C.Increasing the value of cooperativity beyond $B=2$ not only makes the expansion faster, but also shifts the region of high growth from the tip to the interior of the expansion front (Fig.",
"REF BC).",
"This shift is the most fundamental difference between pulled and pushed waves because it indicates the transition from a wave being “pulled” by its leading edge to a wave being “pushed” by its bulk growth.",
"The edge-dominated dynamics make pulled waves extremely sensitive to the vagaries of reproduction, death, and dispersal [29], [1], [34], [33].",
"Indeed, the number of organisms at the leading edge is always small, so strong number fluctuations are expected even in populations with a large carrying capacity, $N$ .",
"These fluctuation affect both physical properties, such as the shape and position of the wave front, and evolutionary properties, such as the genetic diversity of the expanding population.We refer to genetic drift and genetic diversity as an evolutionary property because they occurs only in systems where agents can be assigned heritable labels.",
"In contrast, front wandering occurs in any physical system and can be quantified even when all agents are indistinguishable as is the case in chemical processes.",
"Consistent with these expectations, experiments with pulled waves reported an unusual roughness of the expansion front [6] and a rapid loss of genetic diversity [53], [54].",
"Figure: Ancestral lineages occupy distinct locations in pulled and pushed waves.",
"(A) illustrates the fixation of a particular genotype.",
"Initially, a unique and heritable color was assigned to every organism to visualize its ancestral lineage.",
"There are no fitness differences in the population, so fixations are caused by genetic drift.",
"(B) and (C) show the probability that the fixed genotype was initially present at a specific position in the reference frame comoving with the expansion.",
"The transition from pulled to pushed waves is marked by a shift in the fixation probability from the tip to the interior of the expansion front.",
"This shift indicates that most ancestral lineages are focused at the leading edge in pulled waves, but near the middle of the front in pushed waves.",
"The fixation probabilities were computed analytically, following Refs.",
", , as described in the SI.",
"We used B=0B=0 in panel B and B=4B=4 in panel C.The transition from “pulled” to “pushed” dynamics is also evident in the number of organisms that trace their ancestry to the leading edge vs. the bulk of the front.",
"The expected number of descendants has been determined for any spatial position along the front for both pulled and pushed waves [55], [35], [36], [56].",
"For pulled waves, only the very tip of the expansion contributes to future generations.",
"On the contrary, the organisms at the leading edge leave few progeny in pushed waves, and the population descends primarily from the organisms in the region of high growth.",
"This shift in the spatial patterns of ancestry has a profound effect on species evolution.",
"In pulled waves, only mutations near the very edge of the expansion have an appreciable fixation probability, but the entire expansion front contributes to evolution in pushed waves (Fig.",
"REF ).",
"Figure: Fluctuations are much stronger in pulled than in pushed waves.",
"The top row compares front wandering between pulled (A) and pushed (B) expansions.",
"Each line shows the position of the front X f (t)X_{\\mathrm {f}}(t) in a single simulation relative to the mean over all simulations in the plot.",
"The bottom row compares the strength of genetic drift between pulled (C) and pushed (D) expansions.",
"We started the simulations with two neutral genotypes equally distributed throughout the front and then tracked how the fraction of one of the genotypes changes with time.",
"This fraction was computed from 300 patches centered on X f X_{\\mathrm {f}} to exclude the fluctuations well behind the expansion front.Fixation probabilities and, more generally, the dynamics of heritable markers provides an important window into the internal dynamics of a reaction-diffusion wave [55].",
"When the markers are neutral, i.e.",
"they do not affect the growth and dispersal of the agents, the relative abundance of the markers changes only stochastically.",
"In population genetics, such random changes in the genotype frequencies are known as genetic drift.",
"To describe genetic drift mathematically, we introduce the relative fraction of one of the genotypes in the population $f(t,x)$ .",
"The dynamics of $f(t,x)$ follow from Eq.",
"(REF ) and are derived in Sec.",
"III of the SI (see also Ref.",
"[57], [58], [55]).",
"The result reads $\\frac{\\partial f}{\\partial t} = D\\frac{\\partial ^2 f}{\\partial x^2} + 2\\frac{\\partial \\ln n}{\\partial x}\\frac{\\partial f}{\\partial x} + \\sqrt{\\frac{\\gamma _f(n)}{n}f(1-f)}\\eta _{f}(t),$ where $\\gamma _{f}(n)>0$ is the strength of genetic drift.",
"Equation (REF ) preserves the expectation value of $f$ , but the variance of $f$ increases with time until one of the absorbing states is reached.",
"The two absorbing states are $f=0$ and $f=1$ , which correspond to the extinction and fixation of a particular genotype respectively.",
"The fluctuations of $f$ and front position are shown in Fig.",
"REF .",
"Both quantities show an order of magnitude differences between pulled and pushed waves even though the corresponding change in cooperativity is quite small.",
"Although the difference between pulled and pushed waves seems well-established, little is known about the transition between the two types of behavior.",
"In particular, it is not clear how increasing the nonlinearity of $r(n)$ transforms the patterns of fluctuations and other properties of a traveling wave.",
"To answer this question, we solved Eqs.",
"(REF ) and (REF ) numerically.",
"Specifically, our simulations described the dynamics of both the population density and the relative abundance of two neutral genotypes.",
"The former was used to estimate the fluctuations in the position of the front, and the latter was used to quantify the decay rate of genetic diversity.",
"In simulations, the species expanded in a one-dimensional array of habitable patches connected by dispersal between the nearest neighbors.",
"Each time step consisted of a deterministic dispersal and growth followed by random sampling to simulate demographic fluctuations and genetic drift (see Methods and Sec.",
"XIII in the SI).",
"By increasing the cooperativity of the growth rate, we observed a clear transition from pulled ($v=v_{\\mathrm {\\textsc {f}}}$ ) to pushed ($v>v_{\\mathrm {\\textsc {f}}}$ ) waves accompanied by a dramatic reduction in fluctuations; see Fig.",
"REF .",
"Results Fluctuations provide an easy readout of the internal dynamics in a traveling wave, so we decided to determine how they change as a function of cooperativity.",
"Because the magnitude of the fluctuations also depends on the population density, we looked for a qualitative change in this dependence while varying $B$ .",
"In particular, we aimed to determine whether population dynamics change gradually or discontinuously at the transition between pulled and pushed waves.",
"Spatial wandering of the front We first examined the fluctuations of the front position in the comoving reference frame.",
"The position of the front $X_{\\mathrm {f}}$ was defined as the total population size in the colonized space normalized by the carrying capacity $X_{\\mathrm {f}}=\\frac{1}{N}\\int _{0}^{+\\infty }n(t,x)dx$ .",
"As expected [1], [34], [33], [38], $X_{\\mathrm {f}}$ performed a random walk due to demographic fluctuations in addition to the average motion with a constant velocity (Fig.",
"REF AB).",
"For both pulled and pushed waves, the variance of $X_{\\mathrm {f}}$ grew linearly in time (Fig.",
"REF A), i.e.",
"the front wandering was diffusive and could be quantified by an effective diffusion constant $D_{\\mathrm {f}}$ .",
"The magnitude of the front wandering is expected to depend strongly on the type of the expansion [1], [34], [33], [38].",
"For pulled waves, Ref.",
"[33] found that $D_{\\mathrm {f}}\\sim \\ln ^{-3}N$ , but a very different scaling $D_{\\mathrm {f}} \\sim N^{-1}$ was predicted for certain pushed waves [38]; see Fig.",
"REF B.",
"Given that pulled and pushed waves belong to distinct universality classes, it is easy to assume that the transition between the two scaling regimens should be discontinuous [1], [34], [33], [38], [35], [36].",
"This assumption, however, has not been carefully investigated, and we hypothesized that there could be an intermediate regime with $D_{\\mathrm {f}}\\sim N^{\\alpha _{\\mathrm {\\textsc {d}}}}$ .",
"From simulations, we computed how $\\alpha _{\\mathrm {\\textsc {d}}}$ changes with $B$ and indeed found that pushed waves have intermediate values of $\\alpha _{\\mathrm {\\textsc {d}}}$ between 0 and $-1$ when $B\\in (2,4)$ (Fig. S3).",
"The dependence of the scaling exponent on the value of cooperativity is shown in Fig.",
"REF C. For large $B$ , we found that $\\alpha _{\\mathrm {\\textsc {d}}}$ is constant and equal to $-1$ , which is consistent with the previous work [38].",
"Below a critical value of cooperativity, however, the exponent $\\alpha _{\\mathrm {\\textsc {d}}}$ continually changes with $B$ towards 0.",
"The critical cooperativity is much larger than the transition point between pulled and pushed waves, so the change in the scaling occurs within the class of pushed waves.",
"This transition divides pushed waves into two subclasses, which we termed fully-pushed and semi-pushed waves.",
"For pulled waves, we found that $\\alpha _{\\mathrm {\\textsc {d}}}$ is independent of $B$ , but our estimate of $\\alpha _{\\mathrm {\\textsc {d}}}$ deviated slightly from the expected value due to the finite range of $N$ in the simulations (compare $N^{\\alpha _{\\mathrm {\\textsc {d}}}}$ and $\\ln ^{-3}N$ fits in Fig.",
"REF B).",
"Figure: Front wandering identifies a new class of pushed waves.",
"(A) Fluctuations in the front position can be described by simple diffusion for both pulled and pushed waves.",
"(B) The front diffusion is caused by the number fluctuations, so the effective diffusion constant, D f D_{\\mathrm {f}}, decreases with the carrying capacity, NN.",
"For pulled waves, D f ∼ln -3 ND_{\\mathrm {f}}\\sim \\ln ^{-3}N , while, for pushed waves, D f D_{\\mathrm {f}} can decrease much faster as N -1 N^{-1} .",
"We quantify the scaling of D f D_{\\mathrm {f}} with NN by the exponent α d \\alpha _{\\mathrm {\\textsc {d}}} equal to the slope on the log-log plot shown.",
"The overlap of the two red lines highlight the fact that, even though α d \\alpha _{\\mathrm {\\textsc {d}}} should equal 0 for pulled waves, the limited range of NN results in a different value of α d ≈-0.33\\alpha _{\\mathrm {\\textsc {d}}}\\approx -0.33.",
"(C) The dependence of the scaling exponent on cooperativity identifies two distinct classes of pushed waves.Loss of genetic diversity Our analysis of the front wandering showed that pushed waves consist of two classes with a very different response to demographic fluctuations.",
"To determine whether this difference extends to other properties of expansions, we turned to genetic drift, a different process that describes fluctuations in the genetic composition of the front.",
"Genetic drift occurs even in the absence of front wandering (see Sec.",
"III in the SI and Ref.",
"[59]), so these two properties are largely independent from each other and capture complementary aspects related to physical and evolutionary dynamics in traveling waves.Front wandering and genetic drift are in general coupled because both arise due to the randomness of birth and death.",
"The two processes are however not identical because the fluctuations in the total population density could differ from the fluctuations in the relative frequency of the genotypes.",
"For example, in the standard Wright-Fisher model, only genetic drift is present since the total population size is fixed; see SI(Sec.",
"III) for further details.",
"We quantified genetic fluctuations by the rate at which genetic diversity is lost during an expansion.",
"The simulations were started in a diverse state with each habitable patch containing an equal number of two neutral genotypes.",
"As the expansion proceeded, the relative fractions of the genotypes fluctuated and eventually one of them was lost from the expansion front (Fig.",
"REF C).",
"To capture the loss of diversity, we computed the average heterozygosity $H$ , defined as the probability to sample two different genotypes at the front.",
"Mathematically, $H$ equals the average of $2f(1-f)$ , where $f$ is the fraction of one of the genotypes in an array of patches comoving with the front, and the averaging is done over independent realizations.",
"Consistent with the previous work [33], [35], we found that the heterozygosity decays exponentially at long times: $H\\sim e^{-\\Lambda t}$ for both pulled and pushed waves (Fig.",
"REF A).",
"Therefore, the rate $\\Lambda $ was used to measure the strength of genetic drift across all values of cooperativity.",
"By analogy with the front wandering, we reasoned that $\\Lambda $ would scales as $N^{\\alpha _{\\mathrm {\\textsc {h}}}}$ for large $N$ , and $\\alpha _{\\mathrm {\\textsc {h}}}$ would serve as an effective “order parameter” that distinguishes different classes of traveling waves.",
"Indeed, Ref.",
"[33] showed that $\\Lambda \\sim \\ln ^{-3}N$ for pulled waves, i.e.",
"the expected $\\alpha _{\\mathrm {\\textsc {h}}}$ is zero.",
"Although no conclusive results have been reported for pushed waves, the work on adaptation waves in fitness space suggests $\\alpha _{\\mathrm {\\textsc {h}}}=-1$ for fully-pushed waves [25].",
"Our simulations confirmed both of these predictions (Fig.",
"REF B) and showed $\\Lambda \\sim N^{\\alpha _{\\mathrm {\\textsc {h}}}}$ scaling for all values of cooperativity.",
"The dependence of $\\alpha _{\\mathrm {\\textsc {h}}}$ on $B$ shows that genetic fluctuations follow exactly the same pattern as the front wandering (Fig.",
"REF C).",
"In particular, both exponents undergo a simultaneous transition from $\\alpha _{\\mathrm {\\textsc {h}}}=\\alpha _{\\mathrm {\\textsc {d}}}=-1$ to a continual dependence on $B$ as cooperativity is decreased.",
"Thus, genetic fluctuations also become large as waves switch from fully-pushed to semi-pushed.",
"In the region of pulled waves, $\\alpha _{\\mathrm {\\textsc {d}}}$ and $\\alpha _{\\mathrm {\\textsc {h}}}$ are independent of $B$ , but their values deviate from the theoretical expectation due to the finite range of $N$ explored in the simulations.",
"Overall, the consistent behavior of the fluctuations in the position and composition of the front strongly suggests the existence of two classes of pushed waves, each with a distinct set of properties.",
"Figure: Genetic diversity is lost at different rates in pulled, semi-pushed waves, and fully-pushed waves.",
"(A) The average heterozygosity, HH, is a measure of diversity equal to the probability to sample two distinct genotypes in the population.",
"For both pulled and pushed expansions, the decay of genetic diversity is exponential in time: H∼e -Λt H\\sim e^{-\\Lambda t}, so we used Λ\\Lambda to measure the strength of genetic drift.",
"(B) Genetic drift decrease with NN.",
"For pulled waves, Λ∼ln -3 N\\Lambda \\sim \\ln ^{-3}N , while, for fully-pushed waves, we predict that Λ∼N -1 \\Lambda \\sim N^{-1}; see Eq. ().",
"To quantify the dependence of Λ\\Lambda on NN, we fit Λ∼N α h \\Lambda \\sim N^{\\alpha _{\\mathrm {\\textsc {h}}}}.",
"The dashed red line shows that even though α h \\alpha _{\\mathrm {\\textsc {h}}} should equal 0 for pulled waves, the limited range of NN results in a different value of α h ≈-0.33\\alpha _{\\mathrm {\\textsc {h}}}\\approx -0.33.",
"(C) The dependence of the scaling exponent on cooperativity identifies the same three classes of waves as in Fig.",
"C; the transitions between the classes occur at the same values of BB.The origin of semi-pushed waves We next sought an analytical argument that can explain the origin of the giant fluctuations in semi-pushed waves.",
"In the SI(Sec.",
"VI and VIII), we explain and extend the approaches from Refs.",
"[35] and [38] to compute $D_{\\mathrm {f}}$ and $\\Lambda $ using a perturbation expansion in $1/N$ .",
"The main results are $\\begin{aligned}D_{\\mathrm {f}} &= \\frac{1}{N} \\frac{\\int _{-\\infty }^{+\\infty }\\gamma _n(\\rho )[\\rho ^{\\prime }(\\zeta )]^2\\rho (\\zeta )e^{\\frac{2v\\zeta }{D}}d\\zeta }{2\\left(\\int ^{+\\infty }_{-\\infty }[\\rho ^{\\prime }(\\zeta )]^2e^{\\frac{v\\zeta }{D}}d\\zeta \\right)^2}, \\\\\\Lambda &= \\frac{1}{N} \\frac{\\int _{-\\infty }^{+\\infty }\\gamma _{f}(\\rho )\\rho ^{3}(\\zeta )e^{\\frac{2v\\zeta }{D}}d\\zeta }{\\left(\\int ^{+\\infty }_{-\\infty }\\rho ^2(\\zeta )e^{\\frac{v\\zeta }{D}}d\\zeta \\right)^2}.",
"\\\\\\end{aligned}$ Here, primes denote derivatives; $\\zeta =x-vt$ is the coordinate in the reference frame comoving with the expansion; $\\rho (\\zeta )=n(\\zeta )/N$ is the normalized population density profile in the steady state; $v$ is the expansion velocity; $D$ is the dispersal rate as in Eq.",
"(REF ); and $\\gamma _n$ and $\\gamma _{f}$ are the strength of demographic fluctuations and genetic drift, which in general could be different (see Sec.",
"III in the SI).",
"The $N^{-1}$ scaling that we observed for fully-pushed waves is readily apparent from Eqs.",
"(REF ).",
"The prefactors of $1/N$ account for the dependence of microscopic fluctuations on the carrying capacity, and the ratios of the integrals describe the relative contribution of the different locations within the wave front.",
"For fully-pushed waves, the integrands in Eqs.",
"(REF ) vanish both in the bulk and at the leading edge, so $\\Lambda $ and $D_{\\mathrm {f}}$ are controlled by the number of organisms within the wave front.",
"Hence, the $N^{-1}$ scaling can be viewed as a manifestation of the central limit theorem, which predicts that the variance in the position and genetic diversity of the front should be inversely proportional to the effective population size of the front.",
"To test this theory, we calculated the integrals in Eqs.",
"(REF ) analytically for the model specified by Eq.",
"(REF ); see Sec.",
"VI and VIII in the SI.",
"These exact results show excellent agreement with our simulations (Fig.",
"S4) and thus confirm the validity of the perturbation approach for fully-pushed waves.",
"Why does the $N^{-1}$ scaling break down in semi-pushed waves?",
"We found that the integrals in the numerators in Eqs.",
"(REF ) become more and more dominated by large $\\zeta $ as cooperativity decreases, and, at a critical value of $B$ , they diverge.",
"To pinpoint this transition, we determined the behavior of $\\rho (\\zeta )$ for large $\\zeta $ by linearizing Eq.",
"(REF ) for small population densities: $D\\frac{d^2\\rho }{d\\zeta ^2} + v\\frac{d\\rho }{d\\zeta } + r(0)\\rho = 0,$ where we replaced $n$ by $\\rho $ and shifted into the reference frame comoving with the front.",
"Equation (REF ) is linear, so the population density decreases exponentially at the front as $\\rho \\sim e^{-k\\zeta }$ .",
"The value of $k$ is obtained by substituting this exponential form into Eq.",
"(REF ) and is given by $k=\\frac{v}{2D}\\left(1+\\sqrt{1-v_{\\mathrm {\\textsc {f}}}^2/v^2}\\right)$ with $v_{\\mathrm {\\textsc {f}}}$ as in Eq.",
"(REF ) (see Sec.",
"II and Sec.",
"IX in the SI).",
"From the asymptotic behavior of $\\rho $ , it is clear that the integrands in the numerators in Eqs.",
"(REF ) scale as $e^{(2v/D-3k)\\zeta }$ , and the integrals diverge when $v/D = 3k/2$ .",
"The integrals in the denominators converge for all pushed waves.",
"The divergence condition can be stated more clearly by expressing $k$ in terms of $v$ and then solving for the critical velocity $v_{\\mathrm {critical}}$ .",
"From this calculation, we found that the transition from fully-pushed to semi-pushed waves occurs at a universal ratio of the expansion velocity $v$ to the linear spreading velocity $v_{\\mathrm {\\textsc {f}}}$ : $v_{\\mathrm {critical}} = \\frac{3}{2\\sqrt{2}}v_{\\mathrm {\\textsc {f}}}.$ This result does not rely on Eq.",
"(REF ) and holds for any model of cooperative growth.",
"The ratio $v/v_{\\mathrm {\\textsc {f}}}$ increases with cooperativity and serves as a model-independent metric of the extent to which a wave is pushed.",
"Equation (REF ) and the results below further show that this metric is universal, i.e.",
"different models with the same $v/v_{\\mathrm {\\textsc {f}}}$ have the same patterns of fluctuations.",
"We can then classify all reaction-diffusion waves using this metric.",
"Pulled waves correspond to the special point of $v/v_{\\mathrm {\\textsc {f}}}=1$ .",
"When $1<v/v_{\\mathrm {\\textsc {f}}}<\\frac{3}{2\\sqrt{2}}$ , waves are semi-pushed, and fully-pushed waves occur when $v/v_{\\mathrm {\\textsc {f}}}\\ge \\frac{3}{2\\sqrt{2}}$ .",
"Fully-pushed waves also occur when $r(0)<0$ ; see Sec.",
"X in the SI.",
"Such situations are called propagation into metastable state in physics [1] and strong Allee effect in ecology [47].",
"Because the growth rate at the front is negative, $v_{\\mathrm {\\textsc {f}}}$ does not exist, and the expansion proceeds only due to the growth in the bulk, where the fluctuations are small.",
"Properties of semi-pushed waves Although the perturbation theory breaks down for $v<v_{\\mathrm {critical}}$ , we can nevertheless estimate the scaling exponents $\\alpha _{\\mathrm {\\textsc {d}}}$ and $\\alpha _{\\mathrm {\\textsc {h}}}$ by imposing an appropriate cutoff in the integrals in Eqs.",
"(REF ).",
"One reasonable choice of the cutoff is $\\rho _{c} \\sim 1/N$ , which ensures that there is no growth in patches that have fewer than one organism.",
"In Sec.",
"IX of the SI, we show that this cutoff is appropriate for deterministic fronts with $\\gamma _n=0$ , but a different cutoff is needed for fluctuating fronts with $\\gamma _n>0$ .",
"The need for a different cutoff had been recognized for a long time both from simulations [31] and theoretical considerations [33].",
"However, a method to compute the cutoff has been developed only recently.",
"For pulled waves, the correct value of the cutoff was obtained in Ref.",
"[37] using a nonstandard moment-closure approximation for Eq.",
"(REF ).",
"We extended this method to pushed waves and found that the integrals should be cut off when $\\rho $ falls below $\\rho _c \\sim (1/N)^{\\frac{1}{v/Dk-1}}$ ; see Sec.",
"IX in the SI.",
"Note that the value of the cutoff depends not only on the absolute number of organisms, but also on the shape and velocity of the front.",
"This dependence arises because population dynamics are much more sensitive to the rare excursions of the front ahead of its deterministic position than to the local fluctuations of the population density; see Sec.",
"IX in the SI and [33].",
"Since front excursions occur into typically unoccupied regions, we find that $\\rho _c<1/N$ and, therefore, genetic drift and front wandering are stronger than one would expect from $\\rho _c=1/N$ .",
"Upon applying the correct cutoff to Eqs.",
"(REF ), we find that the fluctuations in semi-pushed waves have a power-law dependence on $N$ with a nontrivial exponent between 0 and $-1$ .",
"The exponent is the same for both $\\Lambda $ and $D_{\\mathrm {f}}$ and depends only on $v/v_{\\mathrm {\\textsc {f}}}$ .",
"Overall, our theoretical results can be summarized as follows $\\alpha _{\\mathrm {\\textsc {d}}} = \\alpha _{\\mathrm {\\textsc {h}}} = \\left\\lbrace \\begin{aligned}& 0,\\quad \\quad v/v_{\\mathrm {\\textsc {f}}}=1,\\\\& - 2\\frac{\\sqrt{1-v^2_{\\mathrm {\\textsc {f}}}/v^2}}{1-\\sqrt{1-v^2_{\\mathrm {\\textsc {f}}}/v^2}} ,\\quad \\quad v/v_{\\mathrm {\\textsc {f}}}\\in (1,\\frac{3}{2\\sqrt{2}}),\\\\& -1,\\quad \\quad v/v_{\\mathrm {\\textsc {f}}}\\ge \\frac{3}{2\\sqrt{2}}.\\\\\\end{aligned}\\right.$ In the case of pulled waves, our cutoff-based calculation not only predicts the correct values of $\\alpha _{\\mathrm {\\textsc {d}}} = \\alpha _{\\mathrm {\\textsc {h}}}=0$ , but also reproduces the expected $\\ln ^{-3}N$ scaling (Sec.",
"X in the SI).",
"To test the validity of the cutoff approach, we compared its predictions to the simulations of Eq.",
"(REF ) and two other models of cooperative growth; see Fig.",
"REF , Methods, and Fig.",
"S5.",
"The simulations confirm that the values of $\\alpha _{\\mathrm {\\textsc {d}}}$ and $\\alpha _{\\mathrm {\\textsc {h}}}$ are equal to each other and depend only on $v/v_{\\mathrm {\\textsc {f}}}$ .",
"Moreover, there is a reasonable quantitative agreement between the theory and the data, given the errors in $\\alpha _{\\mathrm {\\textsc {d}}}$ and $\\alpha _{\\mathrm {\\textsc {h}}}$ due to the finite range of $N$ in our simulations.",
"The success of the cutoff-based calculation leads to the following conclusion about the dynamics in semi-pushed waves: The fluctuations are controlled only by the very tip of the front while the growth and ancestry are controlled by the front bulk (see Figs.",
"REF C and REF C).",
"Thus, the counter-intuitive behavior of semi-pushed waves originates from the spatial segregation of different processes within a wave front.",
"This segregation is not present in either pulled or fully-pushed waves and signifies a new state of the internal dynamics in a traveling wave.",
"Figure: The universal transition from semi-pushed to fully-pushed waves.",
"For three different models on an Allee effect, the scaling exponents for the heterozygosity and front diffusion collapse on the same curve when plotted as a function of v/v f v/v_{\\mathrm {\\textsc {f}}}.",
"Thus, v/v f v/v_{\\mathrm {\\textsc {f}}} serves as a universal metric that quantifies the effects of cooperativity and separates semi-pushed from fully-pushed waves.",
"We used used squares and plus signs for the model specified by Eq.",
"(), triangles and crosses for the model specified by Eq.",
"(), inverted triangles and stars for the model specified by Eq.",
"(), and the red line for the theoretical prediction from Eq.",
"().Corrections to the expansion velocity due to demographic fluctuations Finally, we examined how the expansion velocity depends on the strength of demographic fluctuations.",
"To quantify this dependence, we computed $\\Delta v$ , the difference between the actual wave velocity $v$ and the deterministic wave velocity $v_{\\mathrm {d}}$ obtained by setting $\\gamma _n=0$ in Eq.",
"(REF ).",
"The perturbation theory in $1/N$ shows that $\\Delta v\\sim N^{\\alpha _{\\mathrm {\\textsc {v}}}}$ with $\\alpha _{\\mathrm {\\textsc {v}}}$ equal to $\\alpha _{\\mathrm {\\textsc {d}}}=\\alpha _{\\mathrm {\\textsc {h}}}$ (see Sec.",
"VII in the SI).",
"Thus, we predict $1/N$ scaling for fully-pushed waves and a weaker power-law dependence for semi-pushed waves with the exponent given by Eq.",
"(REF )Note that, for pulled waves, $v-v_{\\mathrm {d}}\\sim \\ln ^{-2}N$ , which is different from the $\\ln ^{-3}N$ scaling of $D_{\\mathrm {f}}$ and $\\Lambda $ [33], [37].",
"All three quantities, however, scale identically with $N$ for semi- and fully-pushed waves.",
"Our simulations agreed with these results (Fig.",
"S6) and, therefore, provided further support for the existence of two distinct classes of pushed waves.",
"Historically, corrections to wave velocity have been used to test the theories of fluctuating fronts [1], [34], [31].",
"For pulled waves, the scaling $\\Delta v\\sim \\ln ^{-2}N$ was first obtained using the $1/N$ growth-rate cutoff [39].",
"This calculation yielded the right answer because the correct value of the cutoff $\\rho _c \\sim (1/N)^{\\frac{1}{v/Dk-1}}$ reduces to $1/N$ in the limit of pulled waves.For pulled waves, $\\Delta v$ depends on $\\ln \\rho _c$ , so any power-law dependence of $\\rho _c$ on $N$ leads to the same scaling with $N$ .",
"The coefficient of proportionality between $\\Delta v$ and $\\ln ^{-2}N$ is, however, also universal, and the correct value is obtained only for $\\rho _c\\sim 1/N$ .",
"It is then natural to expect that the approach based on the $1/N$ cutoff must fail for pushed waves.",
"Indeed, Kessler et al.",
"[31] extended the cutoff-based approach to pushed wave and obtained results quite different from what we report here.",
"They analyzed deterministic fronts and imposed a fixed growth-rate cutoff.",
"Upon setting the value of this cutoff to $1/N$ , one obtains that $\\alpha _{\\mathrm {\\textsc {v}}}$ changes continuously from 0 to $-2$ as cooperativity increases.",
"Thus, for some values of cooperativity, the decrease with $N$ is faster than would be expected from the central limit theorem.",
"This clearly indicates that fluctuations rather than the modification of the growth rates play the dominant role.",
"In Sec.",
"X of the SI, we show that the approach of Ref.",
"[31] supplemented with the correct value of the cutoff $\\rho _c \\sim (1/N)^{\\frac{1}{v/Dk-1}}$ captures the dependence of $\\Delta v$ on $N$ for semi-pushed waves.",
"We also explain why this approach does not apply to fully-pushed waves, in which $\\Delta v$ is not sensitive to the growth dynamics at the expansion edge, but is instead controlled by the fluctuations throughout the wave front.",
"The SI also provides a detailed comparison of the rate of diversity loss in fluctuating vs. deterministic fronts (Sec.",
"X and Figs.",
"S7, S8 and S9).",
"Discussion Spatially extended systems often change through a wave-like process.",
"In reaction-diffusion systems, two types of waves have been known for a long time: pulled and pushed.",
"Pulled waves are driven by the dynamics at the leading edge, and all their properties can be obtained by linearizing the equations of motion.",
"In contrast, the kinetics of pushed waves are determined by nonlinear reaction processes.",
"The distinction between pulled and pushed waves has been further supported by the recent work on the evolutionary dynamics during range expansions [35], [36].",
"In pulled waves, mutations spread only if they occur at the expansion edge, but the entire front contributes to adaptation in pushed waves.",
"A natural conclusion from the previous work is that all aspects of the wave behavior are determined by whether the wave is pulled or pushed.",
"Here, we challenged this view by reporting how fluctuation patterns change as the growth of a species becomes more nonlinear.",
"Our main finding is that both front wandering (a physical property) and genetic drift (an evolutionary property) show identical behavior with increasing nonlinearity and undergo two phase transitions.",
"The first phase transition is the classic transition between pulled and pushed waves.",
"The second phase transition is novel and separates pushed waves into two distinct subclasses, which we termed fully-pushed and semi-pushed waves.",
"The differences between the three wave classes can be understood from the spatial distribution of population dynamics.",
"The transition from pulled to semi-pushed waves is marked by a shift of growth and ancestry from the edge to the bulk of the front (Fig. S2).",
"In pulled waves, the expansion velocity is determined only by the growth rate at the expansion edge, while the velocity of semi-pushed waves depends on the growth rates throughout the front.",
"Similarly, all organisms descend from the individuals right at the edge of the front in pulled, but not in semi-pushed waves, where any organism at the front has a nonzero probability to become the sole ancestor of the future generations.",
"The transition from semi- to fully-pushed waves is marked by an additional change in the spatial pattern of fluctuations.",
"In fully-pushed waves, the wandering of the front arises due to the fluctuations in the shape of the entire wave front.",
"Similarly, genetic drift at all regions of the wave front contributes to the overall fluctuations in genotype frequencies.",
"The dynamics of semi-pushed wave are different: Both the bulk processes and rare excursions of the leading edge control the rate of diversity loss and front wandering.",
"As a result, semi-pushed waves possess characteristics of both pulled and pushed expansions and require analysis that relies on neither linearization of the reaction-diffusion equation nor on an effective averaging within the wave front.",
"The shift of the fluctuations from the front to the bulk of the wave front explains the different scalings of fluctuations with the population density, $N$ .",
"In fully-pushed waves, fluctuations obey the central limit theorem and decrease with the carrying capacity as $N^{-1}$ .",
"This simple behavior arises because all processes are localized in a region behind the front.",
"The number of organisms in this region grows linearly with $N$ , so the variance of the fluctuations scales as $N^{-1}$ .",
"The central limit theorem seems not to apply to semi-pushed waves, for which we observed a nontrivial power-law scaling with variable exponents.",
"The new scaling reflects the balance between the large fluctuations at the leading edge and the localization of the growth and ancestry processes behind the front.",
"The departure from the $N^{-1}$ scaling is the strongest in pulled waves, where all processes localize at the tip of the front.",
"Since the number of organisms at the leading edge is always close to 1, the fluctuations are very large and decrease only logarithmically with the population size.",
"The different scalings of fluctuations with population density may reflect the different structure of genealogies in pulled, semi-pushed, and fully-pushed waves.",
"Although little is known about the structure of genealogies in the context of range expansions, we can nevertheless propose a conjecture based on an analogy with evolutionary waves in fitness space.",
"Similar to range expansions, evolutionary waves are described by a one-dimensional reaction-diffusion equation, where the role of dispersal is assumed by mutations, which take populations to neighboring regions of the fitness space.",
"The growth rate in evolutionary waves, however, depends not only on the local population density, but also on the location itself because the location of an organisms is its fitness.",
"Despite this important difference, evolutionary waves and range expansions have striking similarities.",
"Some evolutionary waves driven by frequent adaptive mutations are similar to pulled waves because their velocity is controlled by the dynamics at the wave edge, and their rate of diversity loss scales as $\\ln ^{-3}N$ [21], [22], [33], [25], [24], [60], [61].",
"Approximately neutral evolution is in turn similar to pushed waves because its dynamics is controlled by the entire population, and the rate of diversity loss scales as $N^{-1}$ [25], [62].",
"The transition between these two regimes is not fully understood [25], [62], and range expansions might provide a simpler context in which to approach this problem.",
"Based on the above similarity and the known structure of genealogies in evolutionary waves, it has been conjectured that genealogies in pulled waves are described by the Bolthausen-Sznitman coalescent with multiple mergers [63], [24], [62], [25], [60], [61], [64], [65], [66].",
"For fully-pushed waves we conjecture that their genealogies are described by the standard Kingman coalescent with pairwise merges.",
"The Kingman coalescent was rigorously derived for well-mixed populations with arbitrary complex demographic structure [64], so it is natural to expect that it should apply to fully-pushed waves, where all of the dynamics occur in a well-defined region within the wave front.",
"The structure of genealogies in semi-pushed waves is likely to be intermediate and could be similar to that of a $\\Lambda -$ coalescent with multiple mergers [66], [25].",
"Although these conjectures are in line with the results for evolutionary waves [25], [62], [63], [62], their applicability to range expansions requires further study, which we hope to carry out in the near future.",
"Given that genealogies can be readily inferred from population sequencing, they could provide a convenient method to identify the class of a wave and characterize the pattern of fluctuations.",
"Our analysis of diversity loss and front wandering also revealed surprising universality in pushed waves.",
"Because pushed waves are nonlinear, their velocity and front shape depend on all aspects of the growth rate, and it is natural to assume that there are as many types of pushed waves as there are nonlinear growth functions.",
"Contrary to this expectation, we showed that many consequences of nonlinearities can be captured by a single dimensionless parameter $v/v_{\\mathrm {\\textsc {f}}}$ .",
"This ratio was first used to distinguish pulled and pushed waves, but we found that $v/v_{\\mathrm {\\textsc {f}}}$ also determines the transition from semi-pushed to fully-pushed waves and the magnitude of the fluctuations.",
"We therefore suggest that $v/v_{\\mathrm {\\textsc {f}}}$ could be a useful and possibly universal metric of the extent to which an expansion is pushed.",
"Such a metric is needed to compare dynamics in different ecosystems and could play an important role in connecting the theory to empirical studies that can measure $v/v_{\\mathrm {\\textsc {f}}}$ sufficiently accurately.",
"In most ecological studies, however, the measurements of both the observed and the Fisher velocities have substantial uncertainty.",
"Our results caution against the common practice of using the approximate equality of $v$ and $v_{\\mathrm {\\textsc {f}}}$ to conclude that the invasion is pulled.",
"The transition to fully-pushed waves occurs at $v/v_{\\mathrm {\\textsc {f}}}=3/(2\\sqrt{2})\\approx 1.06$ , which is very close to the regime of pulled waves $v/v_{\\mathrm {\\textsc {f}}}=1$ .",
"Therefore, expansions with velocities that are only a few percent greater than $v_{\\mathrm {\\textsc {f}}}$ could behave very differently from pulled waves, e.g., have orders of magnitude lower rates of diversity loss.",
"Given that Allee effects arise via a large number of mechanisms and are usually difficult to detect [47], [48], [67], it is possible that many expansions thought to be pulled based on $v\\approx v_{\\mathrm {\\textsc {f}}}$ are actually semi- or even fully-pushed.",
"The utility of $v/v_{\\mathrm {\\textsc {f}}}$ for distinguishing pulled from semi-pushed waves could, therefore, be limited to systems where accurate measurements are possible such as waves in physical systems or in well-controlled experimental populations.",
"Identifying fully-pushed waves based on the velocity ratio is, however, more straightforward because $v$ substantially grater than $v_{\\mathrm {\\textsc {f}}}$ unambiguously signals that the wave is fully-pushed and that the fluctuations are weak.",
"The somewhat narrow range of velocity ratios for semi-pushed waves, $1<v/v_{\\mathrm {\\textsc {f}}}\\lesssim 1.06$ , does not imply that semi-pushed waves are rare.",
"Indeed, the entire class of pulled waves is mapped to a single point $v/v_{\\mathrm {\\textsc {f}}}=1$ even though a large number of growth functions lead to pulled expansions.",
"For the growth function in Eq.",
"(REF ), pulled and semi-pushed waves occupy equally sized regions in the parameter space: $B\\in [0,2]$ for pulled and $B\\in (2,4)$ for semi-pushed waves.",
"We examined several other models of cooperative growth in the SI(Sec.",
"XII and Fig.",
"S1), including the one that describes the observed transition from pulled to pushed waves in an experimental yeast population [32].",
"For all models, we found that pulled, semi-pushed, and fully-pushed waves occupy regions in the parameter space that have comparable size.",
"Thus, all three classes of waves should be readily observable in cooperatively growing populations.",
"Conclusions Despite the critical role that evolution plays in biological invasions [18], [68], [27], [69], [70], [71], [72], [10], [73], [74], only a handful of studies examined the link between genetic diversity and species ecology in this context [35], [36], [59], [56].",
"The main result of the previous work is that Allee effects reduce genetic drift and preserve diversity.",
"This conclusion, however, was reached without systematically varying the strength on the Allee effect in simulations and was often motivated by the behavior of the fixation probabilities rather than the diversity itself.",
"Our findings not only provide firm analytical and numerical support for the previous results, but also demonstrate that the simple picture of reduced fluctuations in pushed waves does not accurately reflect the entire complexity of the eco-evolutionary feedback in traveling waves.",
"In particular, we showed that the strength of genetic drift varies greatly between semi-pushed and fully-pushed waves.",
"As a result, even a large Allee effect that makes the expansion pushed could be insufficient to substantially slow down the rate of diversity loss.",
"Beyond specific applications in the evolution and ecology of expanding populations, our work provides an important conceptual advance in the theory of fluctuations in reaction-diffusion waves.",
"We showed that there are three distinct classes of traveling waves and developed a unified approach to describe their fluctuations.",
"In fully-pushed waves, fluctuations throughout the entire wave front contribute to the population dynamics.",
"In contrast, the behavior of pulled and semi-pushed waves is largely controlled by rare front excursions, which can be captured by an effective cutoff at low population densities.",
"Both the contribution of the dynamics at the leading edge and the value of the cutoff depend on the ratio of the wave velocity to the Fisher velocity.",
"This dependence explains the transition from giant, $\\ln ^{-3} N$ , fluctuations in pulled waves to regular $1/N$ fluctuations in fully-pushed waves.",
"Extensions of our analytical approach could potentially be useful in other settings, where one needs to describe stochastic dynamics of non-linear waves.",
"Methods The simulations in Figs.",
"REF -REF were carried out for the growth model defined by Eq.",
"(REF ).",
"In Fig.",
"REF , we also used two other growth models to demonstrate that our results do not depend on the choice of $r(n)$ .",
"These growth models are specified by the following equations: $r(n) = g_0\\left(1-\\frac{n}{N}\\right) \\left(\\frac{n}{N} - \\frac{n^*}{N}\\right),$ and $r(n) = g_0\\left[1-\\left(\\frac{n}{N}\\right)^3\\right] \\left[\\left(\\frac{n}{N}\\right)^3-\\left(\\frac{n^*}{N}\\right)^3\\right],$ where $N>0$ is the carrying capacity, $g_0>0$ sets the time scale of growth, and $c^*$ is the Allee threshold, which could assume both positive and negative values; see SI(Sec. II).",
"We simulated range expansions of two neutral genotypes in a one-dimensional habitat modeled by an array of patches separated by distance $a$ ; the time was discretized in steps of duration $\\tau $ .",
"Thus, the abundance of each genotype was represented as $n_i(t,x)$ , where $i\\in \\lbrace 1,2\\rbrace $ is the index of the genotype, and $t$ and $x$ are integer multiples of $\\tau $ and $a$ .",
"Each time step, we updated the abundance of both genotypes simultaneously by drawing from a multinomial distribution with $N$ trials and probability $p_i$ to sample genotype $i$ .",
"The values of $p_i$ reflected the expected abundances of the genotypes following dispersal and growth: $p_i = \\frac{\\frac{m}{2}n_i(t,x-a) + (1-m)n_i(t,x) + \\frac{m}{2}n_i(t,x+a)}{N(1-r(\\tilde{n})\\tau )},$ where $\\tilde{n}=\\frac{m}{2}n_1(t,x-a) + (1-m)n_1(t,x) + \\frac{m}{2}n_1(t,x+a) + \\frac{m}{2}n_2(t,x-a) + (1-m)n_2(t,x) + \\frac{m}{2}n_2(t,x+a)$ is the total population density after dispersal.",
"Note that $p_1+p_2<1$ in patches, where the population density is less than the carrying capacity.",
"In the continuum limit, when $r(n)\\tau \\ll 1$ and $ka\\ll 1$ , our model becomes equivalent to Eq.",
"(REF ) for the population density and to Eq.",
"(REF ) for the relative fraction of the two genotypes with $D=ma^2/2$ , $\\gamma _n = (1-n/N)/\\tau $ , and $\\gamma _f=1/(a\\tau )$ .",
"For simplicity, we set both $a$ and $\\tau $ to 1 in all of our simulations.",
"We used $r0=g0=0.01$ and $m=0.25$ for all simulations, unless noted otherwise.",
"These values were chosen to minimize the effects of discreteness of space and time while preserving computational efficiency.",
"Acknowledgements We thank Jeff Gore and Saurabh Gandhi for useful discussions.",
"This work was supported by a grant from the Simons Foundation (#409704, Kirill S. Korolev; #327934, Oskar Hallatschek), by the startup fund from Boston University to Kirill S. Korolev, and by a National Science Foundation Career Award (#1555330, Oskar Hallatschek).",
"Simulations were carried out on the Boston University Shared Computing Cluster.",
"Supplemental Information Supplemental Information (SI) provides additional results and explanations that further support the classification of reaction-diffusion waves into pulled, semi-pushed, and fully-pushed waves.",
"SI can be roughly divided into two parts.",
"The first part (up to “Cutoffs for deterministic and fluctuating fronts”) mostly reviews previous findings while the second part contains mostly new results.",
"We describe the content of each part in more detail below.",
"The goal of the first half is to introduce common notation, clarify terminology, and state the results in a way that makes it easy to compare theoretical predictions to simulations, experiments, and field studies.",
"The first two sections summarize the standard theory of deterministic reaction-diffusion waves and explain the terms that physicists and ecologists use for cooperative growth.",
"The third section discusses the patterns of ancestry in reaction-diffusion waves.",
"The fourth section introduces demographic fluctuations and genetic drift paying special attention to distinguishing fluctuations in population density from fluctuations in the genetic composition of the population.",
"This distinction is not always drawn in the literature, but is important for applying the theory to specific populations.",
"Sections V-VIII develop a perturbation theory in $1/N$ to compute $\\Delta v$ , $D_{\\mathrm {f}}$ , and $\\Lambda $ .",
"The only new results here are the second order perturbation theory for $\\Delta v$ and the expressions for $D_{\\mathrm {f}}$ and $\\Lambda $ for exactly solvable models.",
"The second half of the SI begins with section IX, which shows how to regularize the perturbation theory by introducing an effective cutoff at low population densities.",
"The value of the cutoff differs between deterministic and fluctuating fronts and is a nontrivial result from our work.",
"The following section contains our main arguments for the existence of the three distinct classes of reaction-diffusion waves.",
"This section combines the results of the perturbation theory with an appropriate cutoff and provides the derivation of the scaling exponents $\\alpha _{\\mathrm {\\textsc {v}}}$ , $\\alpha _{\\mathrm {\\textsc {d}}}$ , and $\\alpha _{\\mathrm {\\textsc {h}}}$ .",
"The separation between foci of growth, ancestry, and diversity is discussed in Section XI.",
"Section XII discusses the parameter range for pulled, semi-pushed and fully-pushed waves in different models.",
"The details of computer simulations and data analysis are given in Section XIII.",
"The final section of the SI contains additional results from simulations.",
"In particular, we show that (i) the perturbation theory agrees with simulations for fully-pushed waves without any adjustable parameters; (ii) the scaling of $\\Delta v$ with $N$ is the same as for $D_{\\mathrm {f}}$ and $\\Lambda $ ; (iii) the predicted scaling exponents match simulation results for both deterministic and fluctuating fronts.",
"In this section, we also show that waves propagating into a metastable state are fully-pushed.",
"Table of Contents I.",
"Classification of the growth rate $r(n)$ in physics and ecology: metastability and Allee effects21 II.",
"Deterministic theory and classification of reaction-diffusion waves21 Reduced equation for traveling wave solutions22 Behavior near the boundaries22 Waves propagating into a metastable state are pushed23 Infinite number of solutions for propagation into an unstable state23 Fisher waves - a simple case of pulled waves23 Transition between pulled and pushed regimes of propagation into an unstable state25 Summary25 Example of pulled and pushed waves in exactly solvable models26 Connection to the model of cooperative growth in the main text26 Other exactly solvable models27 Comments on notation28 III.",
"Dynamics of neutral markers and fixation probabilities28 Forward-in-time dynamics29 Fixation probabilities30 Contribution to neutral evolution by different regions of the frontd31 Spatial distribution of ancestors31 Backward-in-time dynamics and the patterns of ancestry31 Fixation probabilities and ancestry in pulled vs. pushed waves33 Evaluation of integrals34 IV.",
"Demographic fluctuations and genetic drift35 Fluctuations in population size35 Fluctuations in population composition35 Relationship between demographic fluctuations and genetic drift36 Fluctuations in spatial models37 V. Correction to the wave velocity, $v$ , due to a cutoff38 VI.",
"Diffusion constant of the front, $D_\\mathrm {f}$41 Perturbation theory for demographic fluctuation41 Perturbation theory for migration fluctuations44 Results for exactly solvable models45 VII.",
"Correction to velocity due to demographic fluctuations46 VIII.",
"Rate of diversity loss, $\\Lambda $50 Forward-in-time analysis of the decay of heterozygosity50 Backward-in-time analysis of lineage coalescence53 Explicit results for $\\Lambda $ in exactly solvable models and connection54 IX.",
"Cutoffs for deterministic and fluctuating fronts55 Cutoff for deterministic fronts55 Cutoff for fluctuating fronts56 Cutoff for pushed waves expanding into a metastable state57 Cutoff for pushed waves expanding into an unstable state57 Cutoff for pulled waves58 X.",
"Scaling of $\\Delta v$ , $D_{\\mathrm {f}}$ , and $\\Lambda $ in pulled, semi-pushed, and fully-pushed waves58 $1/N$ scaling in fully-pushed waves59 $N^{\\alpha }$ scaling in semi-pushed waves60 Logarithmic scaling in pulled waves60 Scaling with N in deterministic fronts61 Comparison of $\\Delta v$ due to a cutoff in growth and due to demographic fluctuations61 XI.",
"Precise definition of the foci of growth, ancestry, and diversity63 XII.",
"Prevalence of semi-pushed waves 64 XIII.",
"Computer simulations66 Interpretation of the simulations as the Wright-Fisher model with vacancies66 Simulations of deterministic fronts66 Boundary and initial conditions66 Duration of simulations and data collection67 Computing the front velocity67 Computing the diffusion constant of the front67 Computing the heterozygosity and the rate of its decay67 Computing the scaling exponents for $D_{\\mathrm {f}}$ , $\\Lambda $ , and $v-v_{\\mathrm {d}}$68 XIV.",
"Supplemental results and figures68 I.",
"Classification of the growth rate $r(n)$ in physics and ecology: metastability and Allee effects Here, we introduce the terminology used to characterize the growth term in Eq.",
"(5) in physics and ecology.",
"The physics literature typically distinguishes between propagation into an unstable state when $r(0)>0$ and propagation into a metastable state when $r(0)<0$ .",
"The main difference between these two cases is their response to small perturbations.",
"For $r(0)>0$ , the introduction of any number of organisms into an uncolonized habitat results in a successful invasion, so $n=0$ is an unstable state.",
"In contrast, for $r(0)<0$ , invasions fail when the number of introduced organisms is sufficiently small.",
"Large introductions, however, do result in an invasion, so $n=0$ is a metastable state.",
"Since $n=N$ is stable against any perturbation, it is referred to as a stable state.",
"When $n=0$ and $n=N$ are the only states stable against small perturbations, $r(n)$ is often termed bistable.",
"In ecology, populations with a metastable state at $n=0$ ($r(0)<0$ ) are said to exhibit a strong Allee effect.",
"When $r(0)>0$ , the growth dynamics is further classified as exhibiting either a weak Allee effect or no Allee effect.",
"A population exhibits no Allee effect if $r(0)\\ge r(n)$ for all $n$ ; otherwise, it exhibits a weak Allee effect.",
"A common example of $r(n)$ without an Allee effect is the logistic growth, for which $r(n)$ decays monotonically from $r_0$ at $n=0$ to 0 at $n=N$ .",
"II.",
"Deterministic theory and classification of reaction-diffusion waves The goal of this section is to explain the difference between pulled and pushed waves.",
"Despite considerable work on the topic, it is easy to conflate related but distinct properties of traveling waves, and few concise and self-contained accounts are available in the literature.",
"Here, we only provide minimal and mostly intuitive discussion following Ref.",
"[1], which is one of the most lucid and comprehensive reviews on propagating reaction-diffusion fronts.",
"This section contains no new results.",
"The distinction between pulled and pushed waves arises already at the level of deterministic reaction-diffusion equations, so stochastic effects are not considered in this section.",
"Specifically, we are interested in the asymptotic behavior as $t\\rightarrow +\\infty $ of the solutions of the following one-dimensional problem: $\\frac{\\partial n}{\\partial t} = D\\frac{\\partial ^2 n}{\\partial x^2} + r(n) n,$ where $r(n)$ is the per capita growth rate that is negative for large $n$ , but could be either positive or negative at small $n$ .",
"Except for possibly $n=0$ , we assume that there is only one other stable fixed point of $dn/dt=r(n)n$ at $n=N$ , where $N$ is the carrying capacity.",
"We further assume that the initial conditions are sufficiently localized, e.g.",
"$n(0,x)$ is strictly zero outside a finite domain.",
"Under these assumptions, the long time behavior of $n(t,x)$ is that of a traveling wave: $\\begin{aligned}&n(t,x) = n(\\zeta ), \\\\&\\zeta = x - vt, \\\\&n(-\\infty ) = N,\\\\&n(+\\infty ) = 0.\\end{aligned}$ and our main task is to determine the wave velocity $v$ and the shape of the front in the comoving reference frame $n(\\zeta )$ .",
"Throughout this paper, we focus on the right-moving part of the expansion; the behavior of the left-moving expansion is completely analogous.",
"Reduced equation for traveling wave solutions By substituting Eq.",
"(REF ) into Eq.",
"(REF ), we obtain the necessary condition on $v$ and $n(\\zeta )$ : $Dn^{\\prime \\prime } + vn^{\\prime } + r(n)n = 0,$ where primes denote derivatives with respect to $\\zeta $ .",
"The solutions of Eq.",
"(REF ) clearly have a translational degree of freedom, i.e., if $n(\\zeta )$ is a solution, then $n(\\zeta + \\mathrm {const})$ is a solution.",
"We can eliminate this degree of freedom by choosing the references frame such that $n(0)=N/2$ .",
"The general solution of Eq.",
"(REF ) then has only one remaining degree of freedom, which corresponds to the value of $n^{\\prime }(0)$ .",
"For a given $v$ , the existence of the solution depends on whether the value of $n^{\\prime }(0)$ can be adjusted to match the boundary conditions specified by Eq.",
"(REF ).",
"Depending on the behavior of the solution at $\\zeta \\rightarrow \\pm \\infty $ , each boundary condition may or may not provide a constraint on the solution and, therefore, remove either one or zero degrees of freedom.",
"To determine the number constraints, we linearize Eq.",
"(REF ) near each of the boundaries.",
"Behavior near the boundaries For $\\zeta \\rightarrow -\\infty $ , we let $u=N-n$ and obtain: $Du^{\\prime \\prime } + vu^{\\prime } + \\left.\\frac{dr}{dn}\\right|_{n=N}Nu = 0,$ which has the following solution: $u = A_b e^{k_b \\zeta } + C_b e^{q_b \\zeta },$ where $\\begin{aligned}&k_b = \\frac{\\sqrt{v^2 - 4D\\left.\\frac{dr}{dn}\\right|_{n=N}N}-v}{2D},\\\\&q_b = \\frac{-\\sqrt{v^2 - 4D\\left.\\frac{dr}{dn}\\right|_{n=N}N}-v}{2D}.\\\\\\end{aligned}$ Here, we used subscript $b$ to indicate that we refer to the behavior of the population bulk.",
"Because the carrying capacity is an attractive fixed point, $dr/dn$ is negative at $n=N$ , and therefore $k>0$ and $q<0$ .",
"We then conclude that the boundary condition at $\\zeta \\rightarrow -\\infty $ requires that $C_b=0$ and thus selects a unique value of $n^{\\prime }(0)$ .",
"Next, we analyze the behavior at the front and linearize Eq.",
"(REF ) for small $n$ : $Dn^{\\prime \\prime } + vn^{\\prime } + r(0)n = 0,$ which has the following solution: $n = A e^{-k \\zeta } + C e^{-q \\zeta },$ where $\\begin{aligned}&k = \\frac{v+\\sqrt{v^2 - 4Dr(0)}}{2D},\\\\&q = \\frac{v-\\sqrt{v^2 - 4Dr(0)}}{2D}.\\\\\\end{aligned}$ Waves propagating into a metastable state are pushed The implications of Eq.",
"(REF ) depend on the sign of $r(0)$ .",
"For a strong Allee effect ($r(0)<0$ ), that is when the wave propagates into a metastable state, we find that $k>0$ and $q<0$ .",
"In consequence, the boundary condition requires that $C=0$ and imposes an additional constraint.",
"For an arbitrary value of $v$ this constraint cannot be satisfied because the value of $n^{\\prime }(0)$ is already determined by the behavior in the bulk.",
"However, there could be a value of $v$ for which the constraints in the bulk and at the front are satisfied simultaneously.",
"This special $v$ is then the velocity of the wave.",
"Because the value of $v$ depends population dynamics throughout the wave front, the propagation into a metastable state is classified as a pushed wave.",
"Infinite number of solutions for propagation into an unstable state For $r(0)>0$ , i.e.",
"when the wave propagates into an unstable state, the analysis is more subtle because the behavior at the front does not fully constrain the velocity of the wave.",
"To demonstrate this, we draw the following two conclusions from Eq.",
"(REF ).",
"First, $v$ must be greater or equal to $v_{\\mathrm {\\textsc {f}}}=2\\sqrt{Dr(0)}$ ; otherwise, the solutions are oscillating around zero as $\\zeta \\rightarrow +\\infty $ and violate the biological constraint that $n\\ge 0$ .",
"Second, both $k$ and $q$ are positive for $v\\ge v_{\\mathrm {\\textsc {f}}}$ , so the solution decays to 0 for arbitrary $A$ and $C$ .",
"Thus, the boundary condition at the front does not impose an additional constraint, and one can find a solution of Eq.",
"(REF ) satisfying Eq.",
"(REF ) for arbitrary $v\\ge v_{\\mathrm {\\textsc {f}}}$ .",
"Fisher waves—a simple case of pulled waves The multiplicity of solutions for $r(0)>0$ posed a great challenge for applied mathematics, statistical physics, and chemistry, and its resolution greatly stimulated the development of the theory of front propagation.",
"The simplest context in which one can show that the wave velocity is unique is when there is no Allee effect, i.e.",
"$r(0)\\ge r(n)$ for all $n\\in (0,N)$ .",
"This condition guarantees that the expansion velocity for Eq.",
"(REF ) is less or equal than the expansion velocity for the dynamical equation linearized around $n=0$ : $\\frac{\\partial n}{\\partial t} = D\\frac{\\partial ^2 n}{\\partial x^2} + r(0) n.$ Moreover, we show below that the upper bound on $v$ from the linearized dynamics coincides with the lower bound imposed by Eq.",
"(REF ).",
"Thus, the velocity of the wave is $2\\sqrt{Dr(0)}$ .",
"This analysis was first carried out in Ref.",
"[43] for an equation, which is now commonly known as the Fisher, Fisher-Kolmogorov, or Fisher-Kolmogorov-Petrovskii-Piskunov equation [43], [42].",
"Therefore, we refer to waves with $r(0)\\ge r(n)$ as Fisher or Fisher-like waves.",
"Since the expansion of the population is determined by the linearized dynamics, Fisher waves are pulled.",
"The upper bound on the velocity from the linearized equation can be obtained for an arbitrary initial condition via the standard technique of Fourier and Laplace transforms [1].",
"However, it is much simpler to use an initial condition for which a closed form solution is available.",
"The results of such an analysis are completely general because the velocity of the wave should not depend on the exact shape of $n(0,x)$ at least when the population density is zero outside a finite domain.",
"So, we make a convenient choice of $n(0,x)=\\delta (x)$ , where $\\delta (x)$ is the Dirac delta function.",
"Then, the solution of Eq.",
"(REF ) is the product of an exponential growth term and a diffusively widening Gaussian: $n(t,x) = \\frac{1}{\\sqrt{4\\pi D t}}e^{r(0)t}e^{-\\frac{x^2}{4Dt}}.$ To determine the long-time behavior, it is convenient to shift into a comoving reference frame (see Eq.",
"(REF )): $n(t,\\zeta ) = \\frac{1}{\\sqrt{4\\pi D t}}e^{\\left(r(0)-\\frac{v^2}{4D}\\right)t}e^{-\\frac{\\zeta ^2}{4Dt}}e^{-\\frac{v}{4D}\\zeta }.$ From the first exponential term, it immediately clear that the wave velocity must be equal to $2\\sqrt{Dr(0)}$ ; otherwise the population size will either exponentially grow or decline as $t\\rightarrow +\\infty $ .",
"As we said earlier, this result demonstrates that the upper and the lower bounds on $v$ coincide with each other and uniquely specify the expansion velocity.",
"The last exponential term in Eq.",
"(REF ) further shows that the wave profile decays exponentially with $\\zeta $ as $e^{-k_{\\mathrm {\\textsc {f}}}\\zeta }$ , where $k_{\\mathrm {\\textsc {f}}} = \\sqrt{\\frac{r(0)}{D}}$ consistent with Eq.",
"(REF ) for $v=v_{\\mathrm {\\textsc {f}}}$ .",
"Finally, the middle exponential term in Eq.",
"(REF ) describes the transition from a sharp density profile at $t=0$ to the asymptotic exponential decay of $n(\\zeta )$ .",
"For $\\zeta < 2\\sqrt{Dt}$ , this term is order one, and the front is well approximated by the asymptotic exponential profile.",
"For $\\zeta > 2\\sqrt{Dt}$ , the Gaussian term becomes important, and the shape of the front is primarily dictated by the diffusive spreading from the initial conditions.",
"Thus, the linear spreading dynamics builds up a gradual decay of the population density starting from a much sharper population front due to localized initial conditions.",
"The steepness of the front created by the linearized dynamics is very important for the transition from pulled to pushed waves, so we emphasize that Eq.",
"(REF ) specifies the lower bound on the rate of exponential decay of the population density.",
"Indeed, Eq.",
"(REF ) indicates that $n$ decays no slower than $e^{-k_{\\mathrm {\\textsc {f}}}\\zeta }$ at all times.",
"Transition between pulled and pushed regimes of propagation into an unstable state Up to here, we have shown that waves are pushed when the Allee effect is strong, but the waves are pulled when there is no Allee effect.",
"Now, we shift the focus to the remaining case of the weak Allee effect and show that the transition from pulled to pushed waves occurs when the growth at intermediate $n$ is sufficiently high to support an expansion velocity greater than the linear spreading velocity $v_{\\mathrm {\\textsc {f}}}$ .",
"To understand the origin of pushed waves, we need to consider the behavior of the population density near the front for $v>v_{\\mathrm {\\textsc {f}}}$ .",
"Equations (REF ) and (REF ) predict that $n(\\zeta )$ is a sum of two exponentially decaying terms with different decay rates: One term decays with the rate $k>k_{\\mathrm {\\textsc {f}}}$ , but the other with the rate $q<k_{\\mathrm {\\textsc {f}}}$ .",
"This behavior is in general inconsistent with the solution of the linearized dynamical equation (REF ).",
"Indeed, our analysis of a localized initial condition suggests and more rigorous analysis proves [1] that $n(t,x)$ decays to zero at least as fast as $k_{\\mathrm {\\textsc {f}}}$ .",
"Therefore, we must require that $C=0$ just as in the case of propagation into a metastable state.",
"The extra requirement makes the problem of satisfying the boundary conditions overdetermined.",
"As a result, there are two alternatives.",
"First, there could be no solutions for any $v>v_{\\mathrm {\\textsc {f}}}$ .",
"In this case, the expansion must be pulled because it is the only feasible solution.",
"Second, a special value of $v$ exists for which the boundary conditions at $\\zeta \\rightarrow \\pm \\infty $ can be satisfied simultaneously.",
"In this case, the wave is pushed because the pulled expansion at low $n$ is quickly overtaken by the faster expansion from the bulk.",
"For completeness, we also mention that initial conditions that are not localized and decay asymptotically as $e^{-\\tilde{k}x}$ with $\\tilde{k}<k_{\\mathrm {\\textsc {f}}}$ lead to pulled expansions with $v=1/2v_{\\mathrm {\\textsc {f}}}(\\tilde{k}/k_{\\mathrm {\\textsc {f}}}+k_{\\mathrm {\\textsc {f}}}/\\tilde{k})>v_{\\mathrm {\\textsc {f}}}$ .",
"This result immediately follows from the solution of Eq.",
"(REF ) using either Laplace transforms or a substitution of an exponential ansatz $n=e^{-\\tilde{k}(x-vt)}$ .",
"Discreteness of molecules or individuals, however, make such initial conditions impossible, so the wave propagation with $v>v_{\\mathrm {\\textsc {f}}}$ can only describe transient behavior for initial conditions with a slow decay at large $x$ .",
"Summary To summarize, we have shown that all waves propagating into a metastable state are pushed, and their profile decays as $e^{-k\\zeta }$ at the front with $k$ given by Eq.",
"(REF ).",
"Propagation into an unstable state could be either pulled or pushed depending on the relative strength of the growth behind vs. at the front.",
"Pushed waves expand with $v>v_{\\mathrm {\\textsc {f}}}$ , and their profile decays as $e^{-k\\zeta }$ .",
"In contrast, pulled waves expand with $v=v_{\\mathrm {\\textsc {f}}}$ , and their profile decays as $e^{-k_\\mathrm {\\textsc {f}}\\zeta }$ .",
"In general, the class of a wave propagating into an unstable state cannot be determined without solving the full nonlinear problem (Eq.",
"(REF ) or Eq.",
"(REF )).",
"However, there are a few rigorous results that determine the class of the wave from simple properties of $r(n)$ .",
"To the best of our knowledge, the most general result of this type is that waves are always pulled when there is no Allee effect.",
"Example of pulled and pushed waves in exactly solvable models We conclude this section by illustrating the transition from pulled to pushed waves in an exactly solvable model.",
"Our main goal is to provide an concrete example for the abstract concepts discussed so far.",
"In addition, the models presented below are used in simulations and to explicitly calculate the diffusion constant of the front and the rate of diversity loss in the regime of fully-pushed waves.",
"Consider $r(n)$ specified by $r(n) = g_0\\left(1-\\frac{n}{N}\\right) \\left(\\frac{n}{N} - \\frac{n^*}{N}\\right),$ where $g_0$ sets the time scale of growth, $N$ is the carrying capacity, and $n^*$ is a parameter that controls the strength of an Allee effect.",
"For every value of $n^*$ , the growth function $r(n)$ can be classified in one of three types: no Allee effect, weak Allee effect, or strong Allee effect.",
"The growth function does not exhibit an Allee effect when $r(0)\\ge r(n)$ for all $n\\in (0,N)$ .",
"For the model defined above, the region of no Allee effect corresponds to $n^*/N\\le -1$ .",
"When an Allee effect is present, one distinguished between a strong Allee effect, $r(0)<0$ , and a weak Allee effect, $r(0)\\ge 0$ .",
"Thus, the Allee effect is weak for $n^*/N\\in (-1,0]$ and strong for $n^*>0$ .",
"In the latter case, $n^*$ represents the minimal population density required for net growth and is known as the Allee threshold.",
"In the following, we will refer to $n^*$ as the Allee threshold regardless of its sign.",
"With $r(n)$ specified by Eq.",
"(REF ), Eq.",
"(REF ) admits an exact solution [52], [51], [9], [75]: $\\begin{aligned}& n(\\zeta ) = \\frac{N}{1 + e^{\\sqrt{\\frac{g_0}{2D}} \\zeta }}\\\\& v = \\sqrt{\\frac{g_0D}{2}}\\left(1 - 2\\frac{n^*}{N} \\right).\\end{aligned}$ For $n^*>0$ , we expect a unique pushed wave, so Eq.",
"(REF ) must provide the desired solution.",
"For $n^*/N<-1$ , we know that the wave must be pulled with $v=v_{\\mathrm {\\textsc {f}}}=2\\sqrt{-Dg_0n^*/N}$ and $k=k_{\\mathrm {\\textsc {f}}}=\\sqrt{-g_0n^*/(ND)}$ .",
"To identify the transition between pulled and pushed waves, we equate the two expressions for the velocity and obtain that the critical value of the Allee threshold is given by $n^*/N=-1/2$ .",
"Thus, the transition between pulled and pushed waves occurs within the region of a weak Allee effect in agreement with the general theory developed above.",
"The behavior of this exactly solvable model is summarized in Table REF .",
"Table: Comparison of wave type, state stability, and Allee effect for an exactly solvable model of range expansions given by Eq. ().",
"Note that the transition from pulled to pushed waves does not coincide with a change in the type of growth.",
"In particular, the pulled-pushed transition is distinct from the transition between no Allee effect and a weak Allee effect; it is also distinct from the transition between propagation into unstable and metastable state.",
"For c * /N>1/2c^*/N>1/2, the relative stability of the populated and unpopulated states changes, and the expansion wave propagates from n=0n=0 state into n=Nn=N state.Connection to the model of cooperative growth in the main text The model of cooperative growth that we defined in the main text (Eq.",
"(3)) is a simple re-parameterization of Eq.",
"(REF ) with $r_0=-g_0n^*/N$ and $B=-N/n^*$ .",
"In consequence, $r(n)$ exhibits no Allee effect for $B\\le 1$ and a weak Allee effect for $B>1$ .",
"A strong Allee effect is not possible for any $B$ because $r(0)>0$ .",
"Hence, the wave always propagates into an unstable state.",
"This model choice was convenient for us because it ensures that the transition between semi-pushed and fully-pushed waves is unambiguously distinct from the transition between propagation into unstable and metastable states.",
"The transition between pulled and pushed waves occurs at $B=2$ .",
"For pushed waves ($B>2$ ), the velocity and front shape are given by $\\begin{aligned}&n(\\zeta ) = \\frac{N}{1 + e^{\\sqrt{\\frac{r_0B}{2D}} \\zeta }}\\\\&v = \\sqrt{\\frac{r_0DB}{2}}\\left(1 + \\frac{2}{B} \\right),\\end{aligned}$ while for pulled waves ($B\\le 2$ ) the corresponding results are $\\begin{aligned}&n(\\zeta ) \\sim e^{-\\sqrt{\\frac{r_0}{D}}\\zeta },\\quad \\zeta \\rightarrow +\\infty ,\\\\&v = 2\\sqrt{Dr_0}.\\end{aligned}$ Although Eq.",
"(3) is equivalent to Eq.",
"(REF ), our computer simulations of these models reveal complementary information because they explore different cuts through the parameter space.",
"In particular, when we vary $B$ in one model, we change both $g_0$ and $n^*/N$ in the other model.",
"Similarly, changes in $n^*/N$ modify both $r_0$ and $B$ .",
"Concordant results for the two parameterizations indicate that the transitions from pulled to semi-pushed and from semi-pushed to fully-pushed waves are universal and do not depend on the precise definition of cooperativity or the strength of an Allee effect.",
"Other exactly solvable models For completeness, we also mention that exact solutions for pushed waves are known for a slightly more general class of $r(n)$ than the quadratic growth function discussed so far.",
"The following results are from Ref.",
"[76], which is an excellent resource for exactly solvable models of traveling waves.",
"For $r(n)$ defined by $r(n) = g_0\\left[1-\\left(\\frac{n}{N}\\right)^b\\right] \\left[\\left(\\frac{n}{N}\\right)^b-\\left(\\frac{n^*}{N}\\right)^b\\right],$ pushed waves occurs for $n^*/N>-(b+1)^{-1/b}$ , and their velocity and profile shape are given by $\\begin{aligned}& v = \\sqrt{(b + 1)g_{0}D} \\left[ \\frac{1}{b + 1} - \\left(\\frac{n^{*}}{N}\\right)^b \\right],\\\\& n(\\zeta ) = \\frac{N}{\\left(1 + e^{b\\sqrt{\\frac{g_0}{(b+1)D}} \\zeta } \\right)^{1/b}}.\\end{aligned}$ The transition point from semi-pushed to fully-pushed waves follows from these results and the condition that $v=\\sqrt{9/8}v_\\mathrm {\\textsc {f}}$ .",
"The value of this critical Allee threshold is given by $n^*/N=-[2(b+1)]^{-1/b}$ .",
"Finally, the transition from weak to no Allee effect occurs at $n^*=-N$ and from weak to strong Allee effect at $n^*=0$ .",
"Comments on notation For all models of $r(n)$ , it is sometimes convenient to use the normalized population density $\\rho = n/N$ .",
"For the exactly solvable models introduced above, it is also convenient to define $\\rho ^*=n^*/N$ .",
"This notation is used in the main text and in the following sections.",
"We can also now be more precise about the definitions of population bulk, front, interior regions of the front, and the leading edge, which we use throughout the paper.",
"The population bulk is defined the region where $n$ is close to $N$ and Eq.",
"(REF ) holds.",
"Similarly, the leading edge, the tip of the front, front edge, etc.",
"refer to the region of $n\\ll N$ , where Eq.",
"(REF ) holds.",
"The region with intermediate $n$ is termed as the front or more precisely as the interior region of the front or the bulk of the front.",
"We tried to avoid using the generic term front whenever that can cause confusion between the leading edge and the interior region of the front.",
"III.",
"Dynamics of neutral markers and fixation probabilities Heritable neutral markers provide a window in the internal dynamics of an expanding population.",
"These dynamics can be studied either forward in time or backward in time.",
"The former approach describes how the spatial distribution of neutral markers changes over time and provides an easy way to compute the fixation probabilities of neutral mutations.",
"The latter approach describes the patterns of ancestry that emerge during a range expansion and provides a natural way to infer population parameters from genetic data.",
"The main goal of this section is to demonstrate that both forward-in-time and backward-in-time dynamics fundamentally change at the transition from pulled to pushed waves.",
"In pulled waves, all organisms trace their ancestry to the very tip of the expansion front, which is also the only source of successful mutations.",
"In contrast, the entire expansion front contributes to the evolutionary dynamics in pushed waves.",
"This section contains no new results except for the analytical calculation of the fixation probabilities in the exactly solvable models.",
"The discussion largely follows that in Refs.",
"[35] and [36].",
"Our main goal here is to introduce the notation to be used in the following sections and explain how the patterns of ancestry depend on cooperativity.",
"Forward-in-time dynamics Let us consider a subpopulation carrying a neutral marker $i$ and describe how its population density $n_i(t,x)$ changes forward in time.",
"Since the growth and migration rates are the same for all markers, $n_i$ obey the following equation: $\\frac{\\partial n_i}{\\partial t} = D\\frac{\\partial ^2 n_i}{\\partial x^2} + r(n)n_i,$ where, as before, $n$ is the total population density, which is given by $n=\\sum _i n_i$ if all individuals are labeled by some marker.",
"To isolate the behavior of neutral markers from the overall population growth, it is convenient to define their relative frequency in the population: $f_i=n_i/n$ .",
"From Eqs.",
"(REF ) and (REF ), it follows that [55], [35] $\\frac{\\partial f_i}{\\partial t} = D\\frac{\\partial ^2 f_i}{\\partial x^2} + 2D\\frac{\\partial \\ln n}{\\partial x}\\frac{\\partial f_i}{\\partial x}.$ The new advection-like term arises from the nonlinear change of variables and accounts for a larger change in $f_i$ due to immigration from regions with high population density compared to regions with low population density.",
"The main effect of the new term is to establish a “flow” of $f_i$ from the posterior to the anterior of the front.",
"It is also convenient to shift into the comoving reference frame (see Eq.",
"(REF )) in order to focus on the dynamics that occurs at a fixed position within the front region rather than at a fixed position in the stationary reference frame.",
"The result reads $\\frac{\\partial f_i}{\\partial t} = D\\frac{\\partial ^2 f_i}{\\partial \\zeta ^2} + v\\frac{\\partial f_i}{\\partial \\zeta } + 2D\\frac{\\partial \\ln n}{\\partial \\zeta }\\frac{\\partial f_i}{\\partial \\zeta }.$ We now drop the index $i$ because, for the rest of this section, we focus on the frequency of a single marker, which we denote simply by $f(t,\\zeta )$ .",
"In the following, we also assume that $n(t,\\zeta )$ has reached the steady-state given by Eq.",
"(REF ).",
"The evolutionary dynamics are typically much slower than ecological dynamics, so the initial transient in the dynamics of $n$ could be neglected.",
"The analysis of Eq.",
"(REF ) is greatly simplified by the existence of a time-invariant quantity: $\\pi = \\frac{\\int _{-\\infty }^{+\\infty }f(t,\\zeta )n^{2}(\\zeta )e^{v\\zeta /D}d\\zeta }{\\int _{-\\infty }^{+\\infty }n^{2}(\\zeta )e^{v\\zeta /D}d\\zeta },$ which was first demonstrated in Refs.",
"[35] and [36].",
"To show that $\\pi $ is conserved, we evaluate its time derivative: $\\frac{d\\pi }{dt} = \\frac{\\int _{-\\infty }^{+\\infty }\\frac{\\partial f}{\\partial t}n^{2}e^{v\\zeta /D}d\\zeta }{\\int _{-\\infty }^{+\\infty }n^{2}e^{v\\zeta /D}d\\zeta },$ and replace $\\partial f/\\partial t$ by the right hand side of Eq.",
"(REF ).",
"The numerator can then be simplified via the integration by parts to eliminate the derivatives of $f$ with respect to $\\zeta $ in favor of $f$ .",
"This leads to the cancellation of all the terms and thereby proves that $\\pi $ does not depend on time.",
"The conservation on $\\pi $ makes it quite straightforward to determine the fixation probabilities, the spatial distribution of ancestors, and the contribution of different parts of the front to the neutral evolution.",
"We now discuss each of these results separately.",
"Fixation probabilities Since only spatial derivatives of $f$ enter Eq.",
"(REF ), we conclude that $f=\\mathrm {const}$ is a solution that describes the steady state after migration has smoothed out the spatial variations in the initial conditions.",
"The value of the constant is given by $\\pi $ because $\\pi = \\lim _{t\\rightarrow +\\infty } \\frac{\\int _{-\\infty }^{+\\infty }f(t,\\zeta )n^{2}(\\zeta )e^{v\\zeta /D}d\\zeta }{\\int _{-\\infty }^{+\\infty }n^{2}(\\zeta )e^{v\\zeta /D}d\\zeta } = \\frac{\\int _{-\\infty }^{+\\infty }\\left(\\lim _{t\\rightarrow +\\infty }f\\right)n^{2}e^{v\\zeta /D}d\\zeta }{\\int _{-\\infty }^{+\\infty }n^{2}e^{v\\zeta /D}d\\zeta } = \\lim _{t\\rightarrow +\\infty }f(t,\\zeta ).$ At the level of deterministic dynamics, this result captures how the final fraction of a genotype depends on its initial distribution in the population.",
"Genetic drift, however, leads to the extinction of all but one genotype, so $f(t,\\zeta )$ should be interpreted as the average over the stochastic dynamics (one can take the expectation value of both sides of Eq.",
"(REF ) below).",
"Since the expected value of $f$ is 1 times the probability of fixation plus 0 times the probability of extinction, we immediately conclude that the fixation probability equals $\\pi $ .",
"Thus, given the initial distribution of a neutral marker $f(0,\\zeta )$ , we can obtain its fixation probability by evaluating the integrals in Eq.",
"(REF ) at $t=0$ .",
"We can also express this result in terms of the absolute abundance of the neutral marker $n_i$ (here we keep the subscript to distinguish $n_i$ from the total population density).",
"Since $f_i = n_i/n$ , the fixation probability is given by $\\pi = \\frac{\\int _{-\\infty }^{+\\infty }n_i(t,\\zeta )n(\\zeta )e^{v\\zeta /D}d\\zeta }{\\int _{-\\infty }^{+\\infty }n^{2}(\\zeta )e^{v\\zeta /D}d\\zeta }.$ For a single organism present at location $\\zeta _0$ , we can approximate $n_i$ as $\\delta (\\zeta -\\zeta _0)$ and thus obtain the fixation probability of a single mutant: $u(\\zeta _0) = \\frac{n(\\zeta _0)e^{v\\zeta _0/D}}{\\int _{-\\infty }^{+\\infty }n^{2}(\\zeta )e^{v\\zeta /D}d\\zeta }.$ Contribution to neutral evolution by different regions of the front From Eq.",
"(REF ), we can determine how different regions contribute to the neutral evolution during a range expansion.",
"Neutral evolution proceeds through two steps: first a random mutations appears somewhere in the population and second the frequency of the mutation fluctuations until the mutation either reaches fixation or becomes extinct.",
"For a given mutation, the probability that it first occurred at location $\\zeta $ is proportional to $n(\\zeta )$ , and its fixation probability is given by $u(\\zeta )$ .",
"Thus, the fraction of fixed mutations that first originated at $\\zeta $ is given by $n^{2}(\\zeta )e^{v\\zeta /D}/\\int _{-\\infty }^{+\\infty }n^{2}(\\zeta )e^{v\\zeta /D}d\\zeta $ .",
"Spatial distribution of ancestors Finally, we note that the last result also represents the probability that the ancestor of a randomly sampled individual from the population used to live at location $\\zeta $ sufficiently long ago.",
"To demonstrate this, we label all individuals between $\\zeta $ and $\\zeta + d\\zeta $ at a long time in the past and note that the probability of a random individual to have its ancestor at $\\zeta $ equals the expected number of labeled descendants.",
"Since the long time limit of $f$ is given by $\\pi $ , we immediately conclude that the probability distribution of ancestor locations is given by $S(\\zeta ) = \\frac{n^{2}(\\zeta )e^{v\\zeta /D}}{\\int _{-\\infty }^{+\\infty }n^{2}(\\zeta )e^{v\\zeta /D}d\\zeta }.$ Backward-in-time dynamics and the patterns of ancestry To characterize the patterns of ancestry in a population, it is convenient to describe the dynamics of ancestral lineages backward in time.",
"Following the approach of Ref.",
"[35], we show below that the probability $S(\\tau ,\\zeta )$ that an ancestor of a given individual lived at position $\\zeta $ time $\\tau $ ago is governed by the following equation [35]: $\\frac{\\partial S}{\\partial \\tau } = D\\frac{\\partial ^2 S}{\\partial \\zeta ^2} - v\\frac{\\partial S}{\\partial \\zeta } - 2D\\frac{\\partial }{\\partial \\zeta }\\left(\\frac{\\partial \\ln n}{\\partial \\zeta }S\\right).$ Here, the diffusion term randomizes the position of the ancestor; the term proportional to $v$ pushes the ancestor towards the tip of the wave and reflects the change into the comoving reference frame; the last term pushes the ancestor towards the population bulk and reflects the fact that the ancestor is more likely to have emigrated from the region where the population density is higher.",
"For large $\\tau $ , it is easy to check that these forces balance and result in a stationary distribution $S(\\zeta )$ given by Eq.",
"(REF ).",
"Note that, unlike in Eq.",
"(REF ) for the dynamics of $f$ , the right hand side of Eq.",
"(REF ) contains a divergence of a flux and, therefore, preserves the normalization of $S$ , i.e.",
"$\\int S(\\tau ,\\zeta ) d\\zeta = \\mathrm {const}$ .",
"Indeed, the probability that an ancestor was present somewhere in the population must always equal to one.",
"The derivation of Eq.",
"(REF ) from Eq.",
"(REF ) follows the standard procedure for changing from the forward-in-time to the backward-in-time description [77], [78] and consists of three steps.",
"The first step is to define the “propagator” function that can describe both forward-in-time and backward-in-time processes.",
"We denote this function as $G(t_d,\\zeta _d;t_a,\\zeta _a)$ and define it as the probability that a descendant located at $\\zeta _d$ at time $t_d$ originated from an ancestor who lived at time $t_a$ at position $\\zeta _a$ .",
"On the one hand, with $t_d$ and $\\zeta _d$ fixed, $G$ can be viewed as a function of $t_a$ and $\\zeta _a$ that specifies the probability distribution of ancestor location at a specific time.",
"On the other hand, with $t_a$ and $\\zeta _a$ fixed, $G$ can be viewed as a function of $t_d$ and $\\zeta _d$ that describes the spatial and temporal dynamics of the expected frequency of the descendants from all organisms that were present at $\\zeta _a$ at time $t_a$ .",
"This two-way interpretation follows from the labeling thought-experiment that we used to derive $S(\\zeta )$ using the forward-in-time formulation.",
"In the second step, we claim that $G$ obeys the same equation as $f(t,\\zeta )$ , i.e.",
"Eq.",
"(REF ).",
"This statement immediately follows from the forward-in-time interpretation of $G$ .",
"We formally state this result as $\\frac{\\partial G}{\\partial t_d} = \\mathcal {L}_{\\zeta _d}G,$ where $\\mathcal {L}_{\\zeta _d}$ is the linear operator from the right hand side of Eq.",
"(REF ), and we used the subscript $\\zeta _{d}$ to indicate variable on which the operator acts.",
"The third step is to derive an equation for $G$ that involves only the ancestor-related variables.",
"To this purpose, we consider an infinitesimal change in $t_a$ , $\\begin{aligned}G(t_d, \\zeta _d; t_a-dt, \\zeta _a) &= \\int _{-\\infty }^{+\\infty } G(t_d, \\zeta _d; t_a, \\zeta ^{\\prime }) G(t_a, \\zeta ^{\\prime }; t_a-dt, \\zeta _a) d\\zeta ^{\\prime } \\\\ &= \\int _{-\\infty }^{+\\infty } G(t_d, \\zeta _d; t_a, \\zeta ^{\\prime }) [G(t_a-dt, \\zeta ^{\\prime }; t_a-dt, \\zeta _a) + dt \\mathcal {L}_{\\zeta ^{\\prime }}G(t_a-dt, \\zeta ^{\\prime }; t_a-dt, \\zeta _a)] d\\zeta ^{\\prime } \\\\ & = \\int _{-\\infty }^{+\\infty } G(t_d, \\zeta _d; t_a, \\zeta ^{\\prime }) [\\delta (\\zeta ^{\\prime }-\\zeta _a)+ dt \\mathcal {L}_{\\zeta ^{\\prime }}\\delta (\\zeta ^{\\prime }-\\zeta _a)] d\\zeta ^{\\prime }\\end{aligned}$ where we used the Markov property of the forward-in-time dynamics, then Eq.",
"(REF ), and finally the fact that $G(t, \\zeta _a; t, \\zeta _d) = \\delta (\\zeta _a-\\zeta _d)$ , which immediately follows from the definition of $G$ .",
"Equation (REF ) can be further simplified by expanding the left hand side in $dt$ : $-\\frac{\\partial G}{\\partial t_a} = \\int _{-\\infty }^{+\\infty } G(t_d, \\zeta _d; t_a, \\zeta ^{\\prime })\\mathcal {L}_{\\zeta ^{\\prime }}\\delta (\\zeta ^{\\prime }-\\zeta _a) d\\zeta ^{\\prime }.$ Finally, we integrate by parts to transfer the derivatives in $\\mathcal {L}_{\\zeta _a}$ from the delta function to $G(t_d, \\zeta _d; t_a, \\zeta ^{\\prime })$ : $-\\frac{\\partial G}{\\partial t_a} = \\int _{-\\infty }^{+\\infty } \\delta (\\zeta ^{\\prime }-\\zeta _a) \\mathcal {L}^{+}_{\\zeta ^{\\prime }} G(t_d, \\zeta _d; t_a, \\zeta ^{\\prime }) d\\zeta ^{\\prime } = \\mathcal {L}^{+}_{\\zeta _a} G(t_d, \\zeta _d; t_a, \\zeta _a),$ where, $\\mathcal {L}^{+}_{\\zeta _a}$ is the operator that results from the integration by parts and is known as the adjoint operator of $\\mathcal {L}_{\\zeta _a}$ .",
"Equation (REF ) is the desired backward-in-time formulation that involves only the ancestor-related variables.",
"To see that it is equivalent to Eq.",
"(REF ), one needs to explicitly compute $\\mathcal {L}^{+}_{\\zeta _a}$ , substitute the definition of $\\tau = t_d - t_a$ , and change the notation from $G$ to $S$ .",
"Fixation probabilities and ancestry in pulled vs. pushed waves We conclude this section by comparing the fixation probabilities and patterns of ancestry in pulled and pushed waves.",
"We focus on $S(\\zeta )$ as a typical example; other quantities, e.g., $u(\\zeta )$ can be analyzed in the same fashion.",
"Up to a constant factor, $S(\\zeta )$ is given by $n^2(\\zeta )e^{v\\zeta /D}$ .",
"For large negative $\\zeta $ , the exponential factor tends rapidly to zero indicating that the bulk of the wave contributes little to the neutral evolution and is unlikely to contain the ancestor of future generations.",
"This conclusion applies to both pulled and pushed waves.",
"The behavior at the front is more subtle because $n(\\zeta )\\rightarrow 0$ and $e^{v\\zeta /D}\\rightarrow +\\infty $ as $\\zeta \\rightarrow +\\infty $ .",
"To determine the scaling of $S(\\zeta )$ at the front, we replace $n$ by its asymptotic form $e^{-k\\zeta }$ and obtain that $S(\\zeta ) \\propto e^{-\\zeta (2k - v/D)} \\propto e^{-\\zeta (k-q)},$ where the last expression follows from the fact that $v = D(k+q)$ ; see Eq.",
"(REF ).",
"For pushed waves, $k>q$ so the tip of the front makes a vanishing contribution to the neutral evolution.",
"Therefore, the main contribution to $S(\\zeta )$ must come from the interior regions of the front.",
"In fact, for the exactly solvable model specified by Eq.",
"(REF ), one can express $\\zeta $ in terms of $n$ and show that $S(\\zeta ) = \\frac{2\\pi \\rho ^*}{\\sin (2\\pi \\rho ^*)}\\rho ^{1+2\\rho ^*}(1-\\rho )^{1-2\\rho ^*},$ where $\\rho =n/N$ , and $\\rho ^*=n^*/N$ .",
"This result clearly demonstrates that $S(\\zeta )$ is peaked at intermediate population densities (specifically at $\\rho = (1+2\\rho ^*)/2$ ).",
"For pulled waves, $k=q$ , and $n^2(\\zeta )e^{v\\zeta /D}\\rightarrow \\mathrm {const}$ as $\\zeta \\rightarrow +\\infty $ .",
"Thus, every point arbitrarily far ahead of the front contributes equally to the neutral evolution.",
"Since the region ahead of the front is infinite, the relative contribution of the bulk and the interior of the front must be negligible compared to that of the leading edge.",
"A more careful analysis requires one to impose a cutoff on $\\zeta $ at sufficiently low densities so that $S(\\zeta )$ can be normalized.",
"We show how to introduce such a cutoff in section IX.",
"In summary, we determined fixation probabilities and patterns of ancestry, which are plotted in Fig.",
"2 of the main text.",
"We also compared the dynamics in pulled and pushed waves and found that they are driven by distinct spatial regions of the front.",
"In pulled waves, the very tip of the front not only “pulls” the wave forward, but also acts as the focus of ancestry and the sole source of successful mutations.",
"In pushed waves, however, the entire front contributes to both the expansion dynamics and evolutionary processes.",
"We refer the readers to Ref.",
"[35], [36], [37] for the original derivations of these results and further discussion.",
"Evaluation of integrals Let us briefly explain how one can evaluate the integrals that appear in Eq.",
"(REF ) and in similar equations for the diffusion constant of the front and the rate of diversity loss.",
"The main insight is to change the independent variable from $\\zeta $ to $\\rho $ using equation Eq.",
"(REF ).",
"The following formulas are useful for this purpose: $\\begin{aligned}& \\zeta = \\sqrt{\\frac{2D}{g_0}}\\ln \\left(\\frac{1-\\rho }{\\rho }\\right), \\\\& d\\zeta = -\\sqrt{\\frac{2D}{g_0}}\\frac{d\\rho }{\\rho (1-\\rho )}, \\\\& e^{\\zeta v/D} = \\rho ^{-(1-2\\rho ^*)}(1-\\rho )^{1-2\\rho ^*}, \\\\& \\frac{d\\rho }{d\\zeta } = - \\sqrt{\\frac{g_0}{2D}}\\rho (1-\\rho ).\\end{aligned}$ After this change of variable, all integrals become beta functions [79].",
"For example, $\\int ^{+\\infty }_{-\\infty }n^2(\\zeta )e^{\\frac{v\\zeta }{D}}d\\zeta = \\sqrt{\\frac{2D}{g_0}} \\int _{0}^{1} \\rho ^{2\\rho ^*}(1-\\rho )^{-2\\rho ^*} = \\mathrm {Be}(1+2\\rho ^*,1-2\\rho ^*).$ The integrals of this type can be evaluated in the complex plane.",
"Specifically, one can equate the integral around the branch cut from 0 to 1 to the residue at $\\zeta =\\infty $ .",
"For the integral above, this results in $\\sqrt{\\frac{2D}{g_0}} \\int _{0}^{1} \\rho ^{2\\rho ^*}(1-\\rho )^{-2\\rho ^*} = \\frac{2\\pi \\rho ^*}{\\sin (2\\pi \\rho ^*)}.$ An alternative method to evaluate the integrals is to use the following properties of the gamma and beta functions [79]: $\\begin{aligned}& \\mathrm {Be}(x,y) = \\frac{\\Gamma (x)\\Gamma (y)}{\\Gamma (x+y)}, \\\\& \\Gamma (z)\\Gamma (1 - z) = \\frac{\\pi }{\\sin {\\pi z}}.",
"\\\\\\end{aligned}$ Using these formulas, we can evaluate all beta functions of the type $\\mathrm {Be}(n + z, m - z)$ , where $n$ and $m$ are positive integers and $z \\in (0,1)$ .",
"The general expressions are $\\begin{aligned}\\mathrm {Be}(n+z, 1 - z) & = \\frac{\\prod _{k = 0}^{n - 1}(z + k)}{n!",
"}\\frac{\\pi }{\\sin {\\pi z}}, &\\\\\\mathrm {Be}(n+z, m - z) & = \\frac{\\prod _{k = 0}^{n - 1}(z + k)\\prod _{l = 1}^{m - 1}(l - z)}{(n + m - 1)!",
"}\\frac{\\pi }{\\sin {\\pi z}}, & m > 1.\\end{aligned}$ IV.",
"Demographic fluctuations and genetic drift In this section, we describe how to move beyond the deterministic approximation in Eqs.",
"(REF ) and (REF ) and account for the effects of demographic fluctuations and genetic drift.",
"Because the magnitude of the fluctuations depends on the details of the reproductive process, we need to introduce two additional functions of the population density $\\gamma _{n}(n)$ and $\\gamma _{f}(n)$ that describe the strength of fluctuations in the population size and composition respectively.",
"This section contains no new results, and its main purpose is to specify the relevant notation and carefully discuss how stochastic dynamics should be added to deterministic reaction-diffusion equations.",
"For simplicity, we will consider well-mixed populations first and limit the discussion to neutral markers, e.g.",
"genotypes that do not differ in fitness.",
"Fluctuations in population size Demographic fluctuations in the size $n(t)$ of a well-mixed population arise due to the randomness of births and deaths.",
"The simplest and most commonly used assumption is that an independent decision is made for each organism on whether it dies or to reproduces [57], [80].",
"In a short time interval $\\Delta t$ , the number of births and deaths are therefore two independent Poisson random variables with mean and variance equal to $\\mu _b(n)n\\Delta t$ for births and to $\\mu _d(n)n\\Delta t$ for deaths.",
"Here, $\\mu _b(n)$ and $\\mu _d(n)$ are the per capita rates of birth and death respectively.",
"Since the change in the population size is the difference between these two independent random variables, we conclude that the mean change of $n$ is $[\\mu _b(n)-\\mu _d(n)]n\\Delta t$ and the variance is $[\\mu _b(n)+\\mu _d(n)]n\\Delta t$ .",
"In the continuum limit, the dynamics of the population size is then described by the following stochastic differential equation $\\frac{dn}{dt} = [b(n) - d(n)]n + \\sqrt{[b(n) + d(n)]n}\\eta (t),$ where $\\eta (t)$ is the Itô white noise, i.e.",
"$\\langle \\eta (t_1) \\eta (t_2) \\rangle = \\delta (t_1-t_2)$ .",
"In the following, we denote ensemble averages by angular brackets, and use $\\delta (t)$ for the Dirac delta function.",
"The assumption that births and death events are independent random variables is however too restrictive.",
"For example, the number of birth always equals the number of deaths in the classic Wright-Fisher model, which exhibits no fluctuations in $n$ as a result.",
"In addition, the number of births or deaths could deviate from the Poisson distribution and therefore have the variance not equal to the mean.",
"To account for such scenarios, we need to generalize Eq.",
"(REF ) as follows $\\frac{dn}{dt} = r(n) n + \\sqrt{\\gamma _{n}(n)n}\\eta (t),$ where $r(n)$ is the difference between the birth and death rates, and $\\gamma _{n}$ characterizes the strength of the demographic fluctuations.",
"The value of $\\gamma _{n}$ can be easily determined from model parameters because $\\gamma _{n}n\\Delta t$ is the sum of the variances of births and deaths during $\\Delta t$ minus twice their covariance.",
"Fluctuations in population composition Genetic drift arises because the choice of the genotype that is affected by a specific birth or death event is random.",
"This randomness does not lead to a change in the average abundance of neutral genotypes, but induces a random walk in the space of population compositions described by the species fractions $\\lbrace f_i\\rbrace $ .",
"It is easy to show that both births and deaths contribute equally to the increase in the variance of $f_{i}$ in a short time interval $\\Delta t$ [81], [57], [80], so the strength of the genetic drift depends only on the total number of updates due to both births and deaths: $\\gamma _f(n)n\\Delta t$ , where $\\gamma _f=\\mu _b(n)+\\mu _d(n)$ .",
"Since probability to choose genotype $i$ for an update is proportional to its current fraction in the population $f_{i}(t)$ , the number of updates for each genotype will be given by a multinomial distribution with $\\gamma _f(n)n\\Delta t$ trials and outcome probabilities given by $\\lbrace f_i\\rbrace $ .",
"This leads to the following continuum limit [81], [57], [80], [58]: $\\frac{df_{i}}{dt} = \\sqrt{\\frac{\\gamma _f(n)}{n}f_{i}(1-f_{i})}\\eta _{i}(t),$ with the covariance structure of the noises $\\eta _{i}$ specified by $\\langle \\eta _{i}(t_1)\\eta _{j}(t_2) \\rangle = \\delta (t_1-t_2)\\left\\lbrace \\begin{aligned}& 1,\\quad i=j,\\\\& -\\sqrt{\\frac{f_i f_j}{(1-f_i)(1-f_j)}}, \\quad i\\ne j.\\end{aligned}\\right.$ The factor of $1/n$ under the square root in Eq.",
"(REF ) arises because a single birth or death event changes the frequency of the genotype at most by $1/n$ .",
"The dependence on $f_i$ reflects the properties of the multinomial distribution and ensures that the sum of $f_i$ does not fluctuate and remains equal to 1.",
"Since genetic drift and demographic fluctuations are independent from each other, i.e.",
"$\\langle \\eta (t_1)\\eta _{i}(t_2)\\rangle =0$ , and Eqs.",
"(REF ), (REF ), and (REF ) completely specify population dynamics.",
"In particular, one can easily obtain the dynamical equations for the genotype abundances $n_{i}$ by differentiating $n_i=f_in$ .",
"An alternative, but completely equivalent, formulation of Eqs.",
"(REF ) and (REF ) reads $\\frac{df_{i}}{dt} = \\sqrt{\\frac{\\gamma _f(n)}{n}f_{i}}\\left(\\tilde{\\eta }_{i}(t)-\\sqrt{f_i}\\sum _j\\tilde{\\eta }_j\\right),$ where $\\langle \\tilde{\\eta }_i(t_1)\\tilde{\\eta }_{j}(t_2)\\rangle =\\delta _{ij}\\delta (t_1-t_2)$ ; we use $\\delta _{ij}$ to denote the Kronecker delta, i.e.",
"the identity matrix.",
"This alternative definition arises naturally when one derives the equations for $f_i$ starting from the dynamical equations for species abundances $n_i$ and shows that $\\sum _j f_j$ is constant more clearly.",
"We provide this formulation only for completeness and do not use in the following.",
"Relationships between demographic fluctuations and genetic drift For the simple processes of uncorrelated births and deaths described by Eq.",
"(REF ), one can show that $\\gamma _f(n)=\\gamma _n(n)$ [57], [80], but this is equality does not hold in general.",
"For example, in the simulations that we describe below $\\gamma _f$ is independent of $n$ , but $\\gamma _n$ monotonically decreases to zero as the population size approaches the carrying capacity.",
"Nevertheless, in a wide set of models, $\\gamma _f(0)=\\gamma _n(0)$ because the dynamics of different genotypes becomes uncorrelated at low population densities and their fluctuations are determined by $\\gamma _0 = \\mu _b(0) + \\mu _d(0)$ .",
"As a result, the fluctuations in pulled and semi-pushed waves depends only on $\\gamma _0$ when the carrying capacity is large enough to justify the asymptotic limit.",
"Exceptions to $\\gamma _f(0)=\\gamma _n(0)$ are in principle possible, for example, when many cycles of birth and death occur without an appreciable change in the total population.",
"Such dynamics could arise when a slow and quasi-deterministic niche construction is required to increase the current limit on the population size.",
"For completeness, we also mention that the dynamical equations for $n_{i}$ take a particularly simple form when $\\gamma _n(n) = \\gamma _f(n) = \\gamma (n)$ : $\\frac{dn_{i}}{dt} = g(n)n_i + \\sqrt{\\gamma (n)n_{i}}\\tilde{\\eta }_{i}(t),$ where $\\langle \\tilde{\\eta }_i(t_1)\\tilde{\\eta }_{j}(t_2)\\rangle =\\delta _{ij}\\delta (t_1-t_2)$ .",
"Thus, for simple birth-death models, the fluctuations in genotype abundances are independent from each other as expected.",
"Although Eq.",
"(REF ) is often used as a starting point for the analysis [40], it does not capture the full complexity of possible eco-evolutionary dynamics.",
"Fluctuations in spatial models It is straightforward to extend the above discussion to spatial populations where $n$ and $f_i$ depend on both $t$ and $x$ .",
"The net result is that Eqs.",
"(REF ) and (REF ) acquire stochastic terms specified by Eqs.",
"(REF ) and (REF ).",
"The results read $\\frac{\\partial n}{\\partial t} = D\\frac{\\partial ^2 n}{\\partial x^2} + r(n) n + \\sqrt{\\gamma _{n}(n)n}\\eta (t),$ and $\\frac{\\partial f_{i}}{\\partial t} = D\\frac{\\partial ^2 f_{i}}{\\partial x^2} + 2\\frac{\\partial \\ln n}{\\partial x}\\frac{\\partial f_{i}}{\\partial x} + \\sqrt{\\frac{\\gamma _f(n)}{n}f_{i}(1-f_{i})}\\eta _{i}(t).$ The noise-noise correlations are specified by the following equations $\\langle \\eta (t_1,x_1)\\eta _i(t_2,x_2) \\rangle = 0,$ $\\langle \\eta (t_1,x_1)\\eta (t_2,x_2) \\rangle = \\delta (t_1-t_2)\\delta (x_1-x_2),$ and $\\langle \\eta _{i}(t_1,x_1)\\eta _{j}(t_2,x_2) \\rangle = \\delta (t_1-t_2)\\delta (x_1-x_2) \\left[\\delta _{ij}- (1-\\delta _{ij})\\sqrt{\\frac{f_i f_j}{(1-f_i)(1-f_j)}}\\right].$ Note that, in Eqs.",
"(REF ) and (REF ), we omitted a noise term that accounts for the randomness of migration (or diffusion in the context of chemical reactions).",
"Such noise inevitably arises when each organisms makes an independent decision on whether to migrate to a particular nearby site.",
"Because this noise conserves the number of individual it appears as a derivative of the flux in the dynamical equation for $n_{i}$ .",
"The general form of this noise is $\\partial _x[\\sqrt{\\gamma _m n_i} \\chi _i]$ with $\\langle \\chi _i(t_1,x_1)\\chi _{j}(t_2,x_2)\\rangle =\\delta _{ij}\\delta (t_1-t_2)\\delta (x_1-x_2)$ , and $\\chi _i$ are uncorrelated with $\\eta $ and $\\eta _i$ [38].",
"Migration noise does not typically lead to any new qualitative dynamics, and we will show below it leads to the same scaling of the fluctuations with the bulk population density $N$ .",
"Moreover, migration noise is often negligible compared to genetic drift.",
"For example, it can be neglected when the migration rate is small or the number of organisms at the dispersal stage is much larger then the number of reproducing adults (compare the number of seeds vs. the number of trees).",
"We do not consider migration noise further because it is absent in our computer simulation.",
"For the sake of simplicity and greater computational speed, we chose to perform the migration update deterministically.",
"V. Correction to the wave velocity, $v$ , due to a cutoff How do demographic fluctuations modify the dynamics of wave propagation?",
"This question is central to our paper and has generated significant interest in nonequilibrium statistical physics.",
"Most early studies explored how demographic stochasticity modifies the expansion velocity [39], [31], [21], [29], [34].",
"While velocity corrections are small and likely negligible in the context of range expansions, they are essential for the description of evolving populations, which are often modeled as traveling waves in fitness space [21], [22], [37].",
"More importantly, wave velocity serves a salient and easy to measure observable that has been frequently used to test the theories of fluctuating fronts.",
"This section shows how to compute the corrections to wave velocity using perturbation theory.",
"All results in this section have been derived previously in Refs.",
"[82], [29], [83], [84], [38], [1].",
"Our main goal here is to introduce the relevant notation and to explain the perturbation theory in the simplest context.",
"Because non-linear stochastic equations are notoriously difficult to analyze, a direct calculation of $v$ is challenging, and several approximate approaches were developed instead [1].",
"In this section, we describe the simplest of these approaches that imposes a cutoff on the growth rate below a certain population density $n_{c}$ : $r_{\\mathrm {cutoff}}(n) = r(n)\\theta (n-n_c),$ where $\\theta (n)$ is the Heaviside step function, which equals one for positive arguments and zero for negative arguments.",
"Although the value of $n_c$ must reflect the strength of the demographic fluctuations, it is not entirely clear how to determine $n_c$ a priori.",
"A natural guess is to set $n_c$ to one over the size of the patch size in simulations so that no growth occurs in regions where the expected number of individuals is less than one.",
"However, this choice does not capture the full complexity of demographic fluctuations as shown in section IX.",
"For now, we keep $n_c$ as an unspecified parameter and focus on the corrections to $v$ due to the change in the growth rate specified by Eq.",
"(REF ).",
"The position of the cuttoff where the deterministic profile reaches $n_c$ is denoted as $\\zeta _c$ , i.e.",
"$n(\\zeta _c)=n_c$ .",
"The corrections to $v$ can be computed using a perturbation expansion in $\\Delta r(n) = r_{\\mathrm {cutoff}}(n) - r(n)$ .",
"This approach has been developed by different groups either for computing the corrections due to a cutoff or for computing the diffusion constant of the front [82], [29], [83], [84], [38], [1].",
"Let us first introduce a convenient notation for the perturbation expansion that is also used in the following sections, where the perturbation is a stochastic variable rather than a deterministic cutoff.",
"All quantities that solve the deterministic, unperturbed problem (Eq.",
"(REF ) or Eq.",
"(REF )) are denoted with subscript $d$ .",
"All quantities that solve the full, perturbed problem are denoted without a subscript.",
"And, the differences between the two types of quantities are denotes with $\\Delta $ .",
"With this notation, the perturbed equation reads $\\frac{\\partial n}{\\partial t} = D\\frac{\\partial ^2 n}{\\partial x^2} + n r_{\\mathrm {cutoff}}(n),$ or equivalently $\\frac{\\partial n}{\\partial t} = D\\frac{\\partial ^2 n}{\\partial x^2} + n r(n) + n \\Delta r(n).$ We seek the solution correct to the first order in $\\Delta r$ via the following ansatz $n(t,x) = n_d(x-v_dt-\\Delta vt) + \\Delta n (x-v_dt-\\Delta vt),$ where $\\Delta n$ is the correction to the shape of the stationary density profile, and $\\Delta v$ is the correction to the expansion velocity.",
"The zeroth order in perturbation theory yields the unperturbed equation: $Dn^{\\prime \\prime }_d + v_d n^{\\prime }_d + r n_d = 0,$ which is automatically satisfied by our choice of $n_d$ .",
"To obtain the equations for the next order, we expand $\\partial n/\\partial t$ as $\\frac{\\partial n}{\\partial t} \\approx - v_d n^{\\prime }_d - n^{\\prime }_d \\Delta v- v_d \\Delta n^{\\prime },$ the diffusion term as $D\\frac{\\partial ^2 n}{\\partial x^2} \\approx D n^{\\prime \\prime }_d + \\Delta n^{\\prime \\prime },$ and the growth term as $r_{\\mathrm {cutoff}}(n)n \\approx n_dr(n_d) + r(n_d)\\Delta n + r^{\\prime }(n_d)n_d\\Delta n + n_d\\Delta r(n_d).$ As before, we use primes to denote derivatives of functions of a single argument.",
"The resulting equation for the first order in perturbation theory reads $\\mathfrak {L}_p \\Delta n = - n_d\\Delta v - n_d\\Delta r,$ where $\\mathfrak {L}_p = D\\frac{d^2}{d\\zeta ^2} + v_d\\frac{d}{d\\zeta } + r(n_d) + r^{\\prime }(n_d)n_d,$ is the linear operator that acts on the comoving spatial variable $\\zeta =x-vt=x-(v_d+\\Delta v)t$ .",
"Although Eq.",
"(REF ) has two unknowns $\\Delta n$ and $\\Delta v$ , both quantities can be determined simultaneously because the solution for $\\Delta n$ exists only for a specific value of $\\Delta v$ .",
"The constraint on $\\Delta v$ comes from the fact that $\\mathfrak {L}_p$ has an eigenvalue equal to zero and, therefore, its image does not span the entire space of functions possible on the right hand side of Eq.",
"(REF ).",
"As a result, $\\Delta v$ must be chosen to make $-n_d\\Delta v - n_d\\Delta r$ lie in the image of $\\mathfrak {L}_p$ .",
"The zero mode of $\\mathfrak {L}_p$ originates from the translational invariance of the unperturbed problem, for which both $n_d(\\zeta )$ and $n_d(\\zeta +\\mathrm {const})$ are solutions.",
"Therefore, there should be no restoring force from the dynamical equation for $\\Delta n$ that effectively translates the front by an infinitesimal distance $\\delta \\zeta $ .",
"Since $n_d(\\zeta +\\delta \\zeta ) \\approx n_d(\\zeta ) + n^{\\prime }_d(\\zeta ) \\delta \\zeta $ , we expect that $\\Delta n \\propto n_d^{\\prime }(\\zeta )$ should not alter the left hand side of Eq.",
"(REF ).",
"Consistent with reasoning, the differentiation of Eq.",
"(REF ) with respect to $\\zeta $ shows that $\\mathfrak {L}_pn^{\\prime }_d=0$ .",
"Thus, $\\mathfrak {L}_p$ indeed has a zero mode with $n^{\\prime }_d$ being the right eigenvector.",
"The corresponding left eigenvector can be obtained by solving $\\mathfrak {L}^{+}_{p}L(\\zeta ) = 0$ and is given by $L(\\zeta ) = n^{\\prime }_d(\\zeta )e^{v_d\\zeta /D}.$ To compute $\\Delta v$ , we multiply both sides of Eq.",
"(REF ) by $L(\\zeta )$ and integrate over $\\zeta $ .",
"Since $L\\mathfrak {L}_p$ is equivalent to zero, the terms on the left hand side cancel, and we obtain that $\\Delta v =-\\frac{\\int _{-\\infty }^{+\\infty } e^{v_d \\zeta /D} n_d^{\\prime }(\\zeta ) n_d(\\zeta ) \\Delta r(n_d(\\zeta )) d\\zeta }{\\int _{-\\infty }^{+\\infty } e^{v_d \\zeta /D} [n_d^{\\prime }(\\zeta )]^2 d\\zeta },$ which is the same result as in Refs.",
"[29], [85].",
"For the specific form of $\\Delta r$ due to a cutoff, this formula simplifies to $\\Delta v = \\frac{\\int _{\\zeta _c}^{+\\infty } e^{v_d \\zeta /D} n_d^{\\prime }(\\zeta ) n_d(\\zeta ) r(n_d(\\zeta )) d\\zeta }{\\int _{-\\infty }^{+\\infty } e^{v_d \\zeta /D} [n_d^{\\prime }(\\zeta )]^2 d\\zeta },$ which is the main result of this section.",
"The solvability condition that we used to compute $\\Delta v$ has a simple interpretation: All perturbations that act along the zero eigenmode of $\\mathfrak {L}_p$ accumulate unattenuated and contribute to the translation of the front, i.e.",
"to $\\Delta v$ rather than to $\\Delta n$ .",
"This fact can be seen more clearly from the time-dependent perturbation theory that we use in sections VI and VII to compute the diffusion constant of the front and the corrections to the wave velocity due to demographic noise rather than a cutoff.",
"VI.",
"Diffusion constant of the front, $D_{\\mathrm {f}}$ While a cutoff can account for changes in the velocity due to demographic fluctuations, it cannot capture the fluctuations in the front shape and position.",
"In this section, we describe the stochastic properties of the front using an extension of the perturbation theory developed above.",
"Originally developed in Refs.",
"[82] and [84], [38], this approach shows that the position of the front performs a random walk that can be described by an effective diffusion constant.",
"Following Ref.",
"[38], we derive the general formula for $D_{\\mathrm {f}}$ given by Eq.",
"(5) of the main text and evaluate it explicitly for the exactly solvable models introduced in the beginning of the SI.",
"The calculations for the exactly solvable models are the only new results in this section.",
"Perturbation theory for demographic fluctuations The calculation follows exactly the same steps as in section V. We begin by restating Eq.",
"(REF ) in a more convenient form: $\\frac{\\partial \\rho }{\\partial t} = D\\frac{\\partial ^2 \\rho }{\\partial x^2} + r(\\rho )\\rho +\\frac{1}{\\sqrt{N}}\\Gamma (\\rho )\\eta $ where $\\Gamma $ denotes the strength of the noise term $\\Gamma (\\rho ) = \\sqrt{\\gamma _n(\\rho )\\rho }.$ In this section, we use the normalized population density $\\rho = n/N$ instead of $n$ to indicate that the stochastic term is small and scales as $1/\\sqrt{N}$ .",
"We also introduce a more compact notation for the noise strength $\\Gamma $ to avoid taking explicit derivatives of $\\sqrt{\\gamma _n(\\rho )\\rho }$ .",
"We seek the solution of Eq.",
"(REF ) in the following form $\\rho (t,x) = \\rho _d(x-v_dt-\\xi (t)) + \\Delta \\rho (t, x-v_dt-\\xi (t)),$ where $\\rho _d$ is the deterministic stationary solution satisfying Eq.",
"(REF ), $\\xi (t)$ is the shift in the front position due to fluctuations, and $\\Delta \\rho (t,\\zeta )$ accounts for the effect of the perturbation on the front shape.",
"Because the perturbation, $\\Gamma \\eta $ , is time dependent, $\\Delta \\rho $ explicitly depends on time in addition to the dependence on $t$ through the comoving coordinate $\\zeta =x-v_dt-\\xi (t)$ .",
"To the first order in perturbation theory, there are no terms due to the special rules of Itô calculus, and we obtain the following expansions for the deterministic terms in Eq.",
"(REF ) $\\frac{\\partial \\rho }{\\partial t} \\approx - v_d \\rho ^{\\prime }_d - \\rho ^{\\prime }_d \\xi ^{\\prime }- v_d \\frac{\\partial \\Delta \\rho }{\\partial \\zeta } + \\frac{\\partial \\Delta \\rho }{\\partial t},$ the diffusion term as $D\\frac{\\partial ^2 \\rho }{\\partial x^2} \\approx D \\rho ^{\\prime \\prime }_d + D\\frac{\\partial ^2 \\Delta \\rho }{\\partial \\zeta ^2},$ and the growth term as $r(\\rho )\\rho \\approx \\rho _d r(\\rho _d) + r(\\rho _d)\\Delta \\rho + r^{\\prime }(\\rho _d)\\rho _d\\Delta \\rho .$ For functions with a single argument, primes denote derivatives with respect to that argument.",
"As before, the zeroth order of the perturbation theory is automatically satisfied, and the first non-trivial equation arises at the first order: $\\frac{\\partial \\Delta \\rho }{\\partial t} - \\mathfrak {L}_p \\Delta \\rho = \\rho ^{\\prime }_d \\xi ^{\\prime } + \\frac{1}{\\sqrt{N}}\\Gamma (\\rho _d)\\eta ,$ where $\\mathfrak {L}_p$ is the same as in Eq.",
"(REF ).",
"To obtain the equation for $\\xi $ , we multiply both sides by $L(\\zeta )$ , the left eigenvector of $\\mathfrak {L}_p$ with zero eigenvalue, and integrating over $\\zeta $ .",
"The result reads $\\frac{\\partial }{\\partial t}\\int _{-\\infty }^{+\\infty }L\\Delta \\rho d\\zeta = \\int _{-\\infty }^{+\\infty } L\\rho ^{\\prime }_d\\xi ^{\\prime } d\\zeta + \\frac{1}{\\sqrt{N}}\\int _{-\\infty }^{+\\infty }L\\Gamma (\\rho _d)\\eta d\\zeta .$ We now use the fact that $\\int _{-\\infty }^{+\\infty }L(\\zeta ) \\Delta \\rho (\\zeta ) d\\zeta =0$ .",
"The projection of $\\Delta \\rho $ on $L$ vanishes because translations of $\\rho _d$ are excluded from the fluctuations of the front shape and are instead included through $\\xi (t)$ .",
"Imposing this condition is also necessary for the perturbation theory to be self-consistent.",
"Otherwise, according to Eq.",
"(REF ), $\\int _{-\\infty }^{\\infty } L\\Delta \\rho d\\zeta $ would perform an unconstrained random walk and grow arbitrarily large, which would violate the assumption that $\\Delta \\rho $ is small.",
"After imposing $\\int _{-\\infty }^{+\\infty } L\\Delta \\rho d\\zeta =0$ , we obtain $\\xi ^{\\prime }(t) = -\\frac{1}{\\sqrt{N}} \\frac{\\int _{-\\infty }^{+\\infty }L(\\zeta )\\Gamma (\\rho _d(\\zeta ))\\eta (t,\\zeta ) d\\zeta }{\\int _{-\\infty }^{+\\infty }L(\\zeta )\\rho ^{\\prime }_d(\\zeta )d\\zeta },$ From Eq.",
"(REF ), it immediately follows that $\\langle \\xi ^{\\prime }(t) \\rangle = 0.$ Thus, there are no corrections to the wave velocity at this order in the perturbation theory, and the motion of the front position is a random walk.",
"The deviation between the position of the front relative to the deterministic expectation is given by $\\xi $ , which we obtain by integrating Eq.",
"(REF ): $X_{\\mathrm {f}} - v_d t = \\xi = -\\frac{1}{\\sqrt{N}} \\int _0^{t} \\frac{\\int _{-\\infty }^{+\\infty }L(\\zeta )\\Gamma (\\rho _d(\\zeta ))\\eta (t,\\zeta ) d\\zeta }{\\int _{-\\infty }^{+\\infty }L(\\zeta )\\rho ^{\\prime }_d(\\zeta )d\\zeta } dt$ To determine the diffusion constant of front wandering, we evaluate the mean square displacement of the front position: $\\begin{aligned}D_{\\mathrm {f}} &= \\frac{\\mathrm {Var} \\lbrace X_{\\mathrm {f}}^2\\rbrace }{2t} = \\frac{\\langle \\xi ^2\\rangle }{2t} \\\\& = \\frac{1}{2tN} \\left\\langle \\int _0^{t}\\frac{\\int _{-\\infty }^{+\\infty }L(\\zeta _1)\\Gamma (\\rho _d(\\zeta _1))\\eta (t_1,\\zeta _1) d\\zeta _1}{\\int _{-\\infty }^{+\\infty }L(\\zeta )\\rho ^{\\prime }_d(\\zeta )d\\zeta }dt_1 \\int _0^{t} \\frac{\\int _{-\\infty }^{+\\infty }L(\\zeta _2)\\Gamma (\\rho _d(\\zeta _2))\\eta (t_2,\\zeta _2) d\\zeta _2}{\\int _{-\\infty }^{+\\infty }L(\\zeta )\\rho ^{\\prime }_d(\\zeta )d\\zeta }dt_2\\right\\rangle \\\\&= \\frac{1}{2tN} \\frac{ \\int _0^{t}dt_1 \\int _0^{t}dt_2\\int _{-\\infty }^{+\\infty }d\\zeta _1 \\int _{-\\infty }^{+\\infty }d\\zeta _2 L(\\zeta _1)\\Gamma (\\rho _d(\\zeta _1))L(\\zeta _2)\\Gamma (\\rho _d(\\zeta _2)) \\left\\langle \\eta (t_1,\\zeta _1)\\eta (t_2,\\zeta _2) \\right\\rangle }{\\left(\\int _{-\\infty }^{+\\infty }L(\\zeta )\\rho ^{\\prime }_d(\\zeta )d\\zeta \\right)^2} \\\\& = \\frac{1}{2N} \\frac{ \\int _{-\\infty }^{+\\infty }d\\zeta L^2(\\zeta )\\Gamma ^2(\\rho _d(\\zeta ))}{\\left(\\int _{-\\infty }^{+\\infty }L(\\zeta )\\rho ^{\\prime }_d(\\zeta )d\\zeta \\right)^2},\\end{aligned}$ where we used Eq.",
"(REF ) to express $\\xi (t)$ and Eq.",
"(REF ) to average over the noise.",
"Finally, we substitute the expression for $L(\\zeta )$ from Eq.",
"(REF ) and use the explicit form of $\\Gamma $ from Eq.",
"(REF ) to obtain Eq.",
"(5) from the main text: $D_{\\mathrm {f}} = \\frac{1}{2N} \\frac{\\int _{-\\infty }^{+\\infty }[\\rho ^{\\prime }_d(\\zeta )]^2\\rho _d(\\zeta )\\gamma _n(\\rho _d(\\zeta ))e^{\\frac{2v_d\\zeta }{D}}d\\zeta }{\\left(\\int ^{+\\infty }_{-\\infty }[\\rho _d^{\\prime }(\\zeta )]^2e^{\\frac{v_d\\zeta }{D}}d\\zeta \\right)^2},$ which was originally derived in Refs.",
"[82] and [84], [38].",
"Perturbation theory for migration fluctuations The analysis that we performed to compute $D_\\mathrm {f}$ due to demographic noise can be easily generalized to account for the noise in migration; see the discussion below Eq.",
"(REF ).",
"This was first done in Ref.",
"[38] that extend the perturbation theory to the following equation $\\frac{\\partial \\rho }{\\partial t} = D\\frac{\\partial ^2 \\rho }{\\partial x^2} + r(\\rho )\\rho + \\frac{1}{\\sqrt{N}}\\Gamma (\\rho )\\eta + \\frac{1}{\\sqrt{N}}\\frac{\\partial (\\Gamma _m(\\rho )\\chi )}{\\partial \\zeta },$ where $\\Gamma _m$ denotes the strength of the migration fluctuations $\\Gamma _m(\\rho ) = \\sqrt{\\gamma _m(\\rho )\\rho },$ and $\\chi =\\sum _i \\sqrt{n_i}\\chi _i/\\sqrt{n}$ is a unit-strength, delta-correlated, Gaussian noise that enters the equation for the total population density $n = \\sum _i n_i$ of all neutral genotypes.",
"Note that, for the standard diffusion, $\\gamma _m(\\rho ) = \\mathrm {const}$ , but we allow the dependence on $\\rho $ because it does not affect the calculation below.",
"The solution for $\\xi $ acquires an additional term due to migration fluctuations: $\\xi = -\\frac{1}{\\sqrt{N}} \\int _0^{t} \\frac{\\int _{-\\infty }^{+\\infty }L(\\zeta )\\Gamma (\\rho _d(\\zeta ))\\eta (t,\\zeta ) d\\zeta }{\\int _{-\\infty }^{+\\infty }L(\\zeta )\\rho ^{\\prime }_d(\\zeta )d\\zeta } dt -\\frac{1}{\\sqrt{N}} \\int _0^{t} \\frac{\\int _{-\\infty }^{+\\infty }L(\\zeta )\\frac{\\partial \\Gamma _m(\\rho _d(\\zeta ))\\chi (t,\\zeta )}{\\partial \\zeta } d\\zeta }{\\int _{-\\infty }^{+\\infty }L(\\zeta )\\rho ^{\\prime }_d(\\zeta )d\\zeta } dt.$ Because $\\chi $ and $\\eta $ are uncorrelated, their contributions to $D_{\\mathrm {f}}$ simply add: $D_{\\mathrm {f}} = \\frac{1}{2N} \\frac{\\int _{-\\infty }^{+\\infty }[\\rho ^{\\prime }_d(\\zeta )]^2\\rho _d(\\zeta )\\gamma _n(\\rho _d(\\zeta ))e^{\\frac{2v_d\\zeta }{D}}d\\zeta }{\\left(\\int ^{+\\infty }_{-\\infty }[\\rho _d^{\\prime }(\\zeta )]^2e^{\\frac{v_d\\zeta }{D}}d\\zeta \\right)^2} + \\frac{1}{2N} \\frac{\\int _{-\\infty }^{+\\infty }\\gamma _m(\\rho _d(\\zeta ))\\rho _d(\\zeta ) (\\rho ^{\\prime \\prime }_d(\\zeta ) + \\rho ^{\\prime }_d(\\zeta )v_d/D)^2e^{\\frac{2v_d\\zeta }{D}}d\\zeta }{\\left(\\int ^{+\\infty }_{-\\infty }[\\rho _d^{\\prime }(\\zeta )]^2e^{\\frac{v_d\\zeta }{D}}d\\zeta \\right)^2}.$ The higher order derivatives of $\\rho $ appear in the second term due to the integration by parts that is necessary to remove derivatives from the delta function due to $\\langle \\chi (t_1,\\zeta _1)\\chi (t_2,\\zeta _2)\\rangle $ .",
"See Ref.",
"[38] for the original derivation and further details.",
"It is now clear that the qualitative behavior of the two terms in Eq.",
"(REF ) is the same.",
"Indeed, the denominators are identical, and the integrands in the numerators have the same scaling behavior at the front, where divergences could occur.",
"To see this, one can substitute the asymptotic behavior of the population density, $\\rho \\sim e^{-k\\zeta }$ , and confirm that both numerators scale as $e^{-\\zeta (3k-2v_d/D)}$ .",
"Thus, the transition from fully-pushed to semi-pushed waves leads to the divergence of both integrals, and the scaling exponent $\\alpha _{\\mathrm {\\textsc {d}}}$ is the same for both migration and demographic fluctuations.",
"For simplicity, only demographic fluctuations are considered in all other sections of this paper.",
"Results for exactly solvable models In the regime of fully-pushed waves, we can evaluate $D_\\mathrm {f}$ explicitly for the exactly solvable models introduced in section I.",
"The details of these calculations are summarized in the subsection on integral evaluation at the end of section III.",
"For the model specified by Eq.",
"(REF ), we find that $D_{\\mathrm {f}} = \\frac{3}{16\\pi }\\frac{\\gamma _n^0}{N}\\sqrt{\\frac{2D}{g_0}} \\frac{\\left( 1 - 4\\rho ^* \\right) \\left( 3 - 4\\rho ^* \\right)}{\\rho ^* \\left( 1 - 2\\rho ^* \\right)\\left( 1 - \\rho ^* \\right)^2} \\tan {2\\pi \\rho ^*},$ when $\\gamma _m=0$ and $\\gamma _n(n)=\\gamma _n^0=\\mathrm {const}$ .",
"Note that the choice of $\\gamma _n(n)$ is not specified by the deterministic model of population growth and needs to be determined either from the microscopic dynamics or from empirical observations.",
"For models formulated in terms of independent birth and death rates, $\\gamma _n(n)$ is a constant on the order of $1/\\tau $ , where $\\tau $ is the generation time.",
"However, different $\\gamma _n$ are possible.",
"For example, our simulations that are based on the Wright-Fisher model have $\\gamma _f=\\gamma _n^0(1-n/N)$ , and the corresponding theoretical prediction for $D_{\\mathrm {f}}$ reads $D_{\\mathrm {f}} = \\frac{3}{20\\pi }\\frac{\\gamma _n^0}{N}\\sqrt{\\frac{2D}{g_0}} \\frac{\\left( 1 - 4\\rho ^* \\right) \\left( 3 - 4\\rho ^* \\right)}{\\rho ^* \\left( 1 - \\rho ^* \\right) \\left( 1 - 2\\rho ^* \\right)} \\tan {2\\pi \\rho ^*}.$ For the same model of an Allee effect, the contribution of the noise due to migration with $\\gamma _m=\\gamma _m^0=\\mathrm {const}$ is given by $D_{\\mathrm {f}} = \\frac{\\sqrt{2}}{40\\pi }\\frac{\\gamma _m^0}{N}\\sqrt{\\frac{g_0}{D}} \\frac{\\left( 1 + \\rho ^* \\right) \\left( 1 + 7\\rho ^* \\right) \\left( 1 - 4\\rho ^* \\right) \\left( 3 - 4\\rho ^* \\right)}{\\rho ^* \\left( 1 - 2\\rho ^* \\right)\\left( 1 - \\rho ^* \\right)^2} \\tan {2\\pi \\rho ^*}$ assuming $\\gamma _n=0$ .",
"For completeness, we also provide the results for other models and different choices of $\\gamma _n$ and $\\gamma _m$ .",
"$D_{\\mathrm {f}} = \\frac{3}{16\\pi }\\frac{\\gamma _n^0}{N}\\sqrt{\\frac{2D}{r_0 B}} \\frac{B^2\\left( B + 4 \\right) \\left( 3B + 4 \\right)}{\\left( B + 2 \\right) \\left( B + 1 \\right)^2} \\tan {\\frac{2\\pi }{B}}$ for the model of cooperative growth defined in the main text with $\\gamma _n = \\gamma _n^0$ and $\\gamma _m=0$ .",
"$D_{\\mathrm {f}} = \\frac{3}{20\\pi }\\frac{\\gamma _n^0}{N}\\sqrt{\\frac{2D}{r_0B}} \\frac{B \\left( B + 4\\right) \\left( 3B + 4 \\right)}{\\left( B + 2 \\right)\\left( B + 1 \\right)} \\tan {\\frac{2\\pi }{B}}$ for the model of cooperative growth defined in the main text with $\\gamma _n = \\gamma _n^0(1-n/N)$ and $\\gamma _m=0$ .",
"$D_{\\mathrm {f}} = \\frac{\\sqrt{2}}{40\\pi }\\frac{\\gamma _m^0}{N}\\sqrt{\\frac{r_0 B}{D}} \\frac{\\left( B - 1 \\right)\\left( B - 7 \\right)\\left( B + 4 \\right) \\left( 3B + 4 \\right)}{\\left( B + 2 \\right) \\left( B + 1 \\right)^2} \\tan {\\frac{2\\pi }{B}}$ for the model of cooperative growth defined in the main text with $\\gamma _m = \\gamma _m^0=\\mathrm {const}$ and $\\gamma _n = 0$ .",
"VII.",
"Correction to velocity due to demographic fluctuations In this section, we compute the correction to the wave velocity directly from the stochastic formulation in Eq.",
"(REF ) instead of relying on a growth-rate cutoff at low densities.",
"Our main finding is that, for pushed waves, the scaling of $\\Delta v$ with $N$ coincidesThe scaling behavior of $\\Delta v$ is different for pulled waves because $\\Delta v \\sim \\ln ^{-2}N$ ; see Ref.",
"[1], [34], [33], [39].",
"with that of front diffusion constant $D_{\\mathrm {f}}$ and the rate of diversity loss $\\Lambda $ .",
"Note that this result cannot be obtain from the cutoff-based calculation of $\\Delta v$ without knowing the correct dependence of $n_c$ on $v/v_{\\mathrm {\\textsc {f}}}$ .",
"Thus, the calculation of $\\Delta v$ in the stochastic model provides an additional insight in the dynamics of fluctuating fronts.",
"To the best of our knowledge, the results presented in this section are new.",
"Because the first order correction to $v$ is zero (See Eq.",
"(REF )), we proceed to the second order in perturbation theory.",
"In this calculation, it is convenient to distinguish the contributions to $\\zeta $ and $\\Delta \\rho $ that come from the different orders of the perturbative expansion: $\\begin{aligned}&\\zeta = x-v_dt -\\xi _{(1)}(t) -\\xi _{(2)}(t),\\\\& \\rho (t,\\zeta ) = \\rho _d(\\zeta ) + \\Delta \\rho _{(1)}(t,\\zeta ) + \\Delta \\rho _{(2)}(t,\\zeta ),\\end{aligned}$ where the order is indicated by a subscript in brackets.",
"For fully-pushed waves, we expect that the first order corrections $\\xi _{(1)}(t)$ and $\\Delta \\rho _{(1)}(t,\\zeta )$ scale as $1/\\sqrt{N}$ , and the second order corrections $\\xi _{(2)}(t)$ and $\\Delta \\rho _{(2)}(t,\\zeta )$ scale as $1/N$ .",
"Therefore, we expand all terms in Eq.",
"(REF ) up to order $1/N$ .",
"For $\\partial \\rho /\\partial t$ , we obtain $\\frac{\\partial \\rho }{\\partial t} \\approx \\rho ^{\\prime }_d(- v_d - \\xi _{(1)}^{\\prime } - \\xi ^{\\prime }_{(2)}) + \\frac{1}{2}\\rho ^{\\prime \\prime }_d\\left\\langle \\frac{(d\\xi _{(1)})^2}{dt}\\right\\rangle + \\frac{\\partial \\Delta \\rho _{(1)}}{\\partial \\zeta }(-v_d -\\xi ^{\\prime }_{(1)}) + \\frac{\\partial \\Delta \\rho _{(1)}}{\\partial t} + \\frac{\\partial \\Delta \\rho _{(2)}}{\\partial \\zeta }(-v_d ) + \\frac{\\partial \\Delta \\rho _{(2)}}{\\partial t},$ where $\\rho ^{\\prime \\prime }_d/2\\left\\langle (d\\xi _{(1)})^2/dt\\right\\rangle $ arises due to the Itô formula of stochastic calculus, which prescribes how to compute derivatives of nonlinear functions; see Refs.",
"[58], [77], [78], [86].",
"The unusual derivative $\\left\\langle (d\\xi _{(1)})^2/dt\\right\\rangle $ is non-zero because the displacement of a random walk grows as $\\sqrt{dt}$ .",
"Using Eqs.",
"(REF ) and (REF ), we express this derivative in terms of $D_{\\mathrm {f}}$ , which we know to the order $1/N$ from the first order of the perturbation theory: $\\left\\langle (d\\xi _{(1)})^2/dt\\right\\rangle = 2 D_{\\mathrm {f}}.$ The expansion of other terms is more straightforward and does not involve any additional terms due to the special rules of Itô calculus: $\\begin{aligned}& D\\frac{\\partial ^2 \\rho }{\\partial x^2} \\approx D \\rho ^{\\prime \\prime }_d + D\\frac{\\partial ^2 \\Delta \\rho _{(1)}}{\\partial \\zeta ^2} + D\\frac{\\partial ^2 \\Delta \\rho _{(2)}}{\\partial \\zeta ^2},\\\\& r(\\rho )\\rho \\approx [r(\\rho )\\rho ]|_{\\rho =\\rho _d} + [r(\\rho )\\rho ]^{\\prime }|_{\\rho =\\rho _d}\\Delta \\rho _{(1)} + \\frac{1}{2} [r(\\rho )\\rho ]^{\\prime \\prime }|_{\\rho =\\rho _d}(\\Delta \\rho _{(1)})^2 + [r(\\rho )\\rho ]^{\\prime }|_{\\rho =\\rho _d}\\Delta \\rho _{(2)}, \\\\& \\Gamma (\\rho ) \\approx \\Gamma (\\rho _d) +\\Gamma ^{\\prime }(\\rho _d)\\Delta \\rho _{(1)},\\end{aligned}$ where we kept only the terms that scale at most as $1/N$ and used $|_{\\rho =\\rho _d}$ to indicate that the expression to the left is evaluated at $\\rho =\\rho _d$ .",
"Upon choosing $\\Delta \\rho _{(1)}$ and $\\xi _{(1)}$ that satisfy the first order equation, i.e.",
"Eq.",
"(REF ), we obtain the following equation for $\\Delta \\rho _{(2)}$ and $\\xi _{(2)}$ : $\\frac{\\partial \\Delta \\rho _{(2)}}{\\partial t} - \\mathfrak {L}_p \\Delta \\rho _{(2)} = -D_{\\mathrm {f}}\\rho ^{\\prime \\prime }_d +\\frac{\\partial \\Delta \\rho _{(1)}}{\\partial \\zeta }\\xi ^{\\prime }_{(1)} +\\frac{1}{2}[r^{\\prime \\prime }(\\rho _d)\\rho _d+2r^{\\prime }(\\rho _d)](\\Delta \\rho _{(1)})^2 + \\frac{1}{\\sqrt{N}}\\Gamma ^{\\prime }(\\rho _d)\\Delta \\rho _{(1)}\\eta + \\rho ^{\\prime }_d\\xi ^{\\prime }_{(2)}.$ The value of $\\xi ^{\\prime }_{(2)}$ needs to be chosen to satisfy the solvability condition, which we obtain by multiplying both sides of Eq.",
"(REF ) by $L(\\zeta )$ , integrating over $\\zeta $ , and requiring that $\\Delta \\rho _{(2)}$ has zero projection on $L(\\zeta )$ .",
"The result reads $\\xi _{(2)}^{\\prime }(t) = - \\frac{\\int _{-\\infty }^{+\\infty }L(\\zeta ) \\left[ -D_{\\mathrm {f}}\\rho ^{\\prime \\prime }_d + \\frac{\\partial \\Delta \\rho _{(1)}}{\\partial \\zeta }\\xi ^{\\prime }_{(1)} +\\frac{1}{2}[r^{\\prime \\prime }(\\rho _d)\\rho _d+2r^{\\prime }(\\rho _d)](\\Delta \\rho _{(1)})^2 + \\frac{1}{\\sqrt{N}}\\Gamma ^{\\prime }(\\rho _d)\\Delta \\rho _{(1)}\\eta \\right] d\\zeta }{\\int _{-\\infty }^{+\\infty }L(\\zeta )\\rho ^{\\prime }_d(\\zeta )d\\zeta }.$ The correction to the velocity, $\\Delta v$ , can now be obtained by averaging Eq.",
"(REF ) over $\\eta $ and substituting the explicit expression for $L(\\zeta )$ from Eq.",
"(REF ): $\\begin{aligned}&\\Delta v = \\langle \\xi ^{\\prime }_{(2)} \\rangle = D_{\\mathrm {f}}\\frac{\\int _{-\\infty }^{+\\infty } e^{v_d \\zeta /D} \\rho _d^{\\prime }(\\zeta ) \\rho ^{\\prime \\prime }_d(\\zeta ) d\\zeta }{\\int _{-\\infty }^{+\\infty } e^{v_d \\zeta /D} [\\rho _d^{\\prime }(\\zeta )]^2 d\\zeta } - \\frac{\\int _{-\\infty }^{+\\infty } e^{v_d \\zeta /D} \\rho _d^{\\prime }(\\zeta ) \\frac{1}{2}[r^{\\prime \\prime }(\\rho _d)\\rho _d+2r^{\\prime }(\\rho _d)] \\langle [\\Delta \\rho _{(1)}(t,\\zeta )]^2\\rangle d\\zeta }{\\int _{-\\infty }^{+\\infty } e^{v_d \\zeta /D} [\\rho _d^{\\prime }(\\zeta )]^2 d\\zeta }.\\end{aligned}$ Note that $\\langle \\Delta \\rho _{(1)}\\eta \\rangle =0$ and, therefore, $\\langle \\Delta \\rho _{(1)}\\xi ^{\\prime } \\rangle = 0$ because $\\Delta \\rho _{(1)}(t,\\zeta )$ depends only on $\\eta (\\tilde{t},\\zeta )$ with $\\tilde{t}<t$ and $\\langle \\eta (\\tilde{t},\\zeta )\\eta (t,\\zeta )\\rangle =0$ .This simplification is specific to the Itô calculus and does not occur in Stratonovich's formulation.",
"The results of course do not depend on the type of calculus used as long as all calculations are carried using the same calculus and the initial problem statement is correctly formulated.",
"In population dynamics, demographic fluctuations affect only future generations, so Itô's formulation appears naturally.",
"The first term could be further simplified through integration by parts in the numerator, assuming that the integrals converge: $\\begin{aligned}&\\Delta v = -v_d\\frac{D_{\\mathrm {f}}}{2D} - \\frac{\\int _{-\\infty }^{+\\infty } e^{v_d \\zeta /D} \\rho _d^{\\prime }(\\zeta ) \\frac{1}{2}[r^{\\prime \\prime }(\\rho _d)\\rho _d+2r^{\\prime }(\\rho _d)] \\langle [\\Delta \\rho _{(1)}(t,\\zeta )]^2\\rangle d\\zeta }{\\int _{-\\infty }^{+\\infty } e^{v_d \\zeta /D} [\\rho _d^{\\prime }(\\zeta )]^2 d\\zeta }.\\end{aligned}$ To complete the calculation of $\\Delta v$ , we need to obtain $\\Delta \\rho _{(1)}$ by solving Eq.",
"(REF ).",
"Before performing this calculation, let us state the main findings and discuss their implications.",
"For fully-pushed waves, we find that $\\Delta \\rho _{(1)}\\sim 1/\\sqrt{N}$ , and all integrals in Eq.",
"(REF ) converge.",
"Thus, $\\Delta v \\sim 1/N$ in this regime.",
"For semi-pushed waves, one needs to apply a cutoff at large $\\zeta $ to ensure convergence.",
"We show that the divergence of the last term in Eq.",
"(REF ) does not exceed that of $D_{\\mathrm {f}}$ .",
"Thus, the leading behavior is controlled by the first term, and the scaling of $\\Delta v$ coincides with that of $D_{\\mathrm {f}}$ .",
"The scaling behavior of $\\Delta v$ and $D_\\mathrm {f}$ is slightly different for pulled waves: $\\Delta v \\sim \\ln ^{-2}N$ and $D_{\\mathrm {f}}\\sim \\ln ^{-3}N$ ; see Ref.",
"[1], [34], [33], [39].",
"The calculation of $\\Delta \\rho _{(1)}$ can be simplified by transforming Eq.",
"(REF ) into a Hermitian form.",
"This is accomplished by the following change of variables that eliminates the term linear in $\\partial /\\partial \\zeta $ from $\\mathfrak {L}_p$ : $\\Delta \\rho _{(1)}(t,\\zeta ) = e^{-\\frac{v_d\\zeta }{2D}}\\Psi (t,\\zeta ).$ Equation (REF ) then takes the following form $\\frac{\\partial \\Psi }{\\partial t} - \\mathfrak {H}_p \\Psi = e^{\\frac{v_d\\zeta }{2D}}\\left[\\rho ^{\\prime }_d \\xi _{(1)}^{\\prime } + \\frac{1}{\\sqrt{N}}\\Gamma (\\rho _d)\\eta \\right],$ where $\\mathfrak {H}_p$ is a Hermitian operator: $\\mathfrak {H}_p = D\\frac{\\partial ^2}{\\partial \\zeta ^2} - \\frac{v_d^2}{4D} + r(\\rho _d) + r^{\\prime }(\\rho _d)\\rho _d.$ We solve Eq.",
"(REF ) using the method of separation of variables.",
"Let us denote the eigenvalues and normalized eigenvectors of $\\mathfrak {H}_p$ by $\\lambda _l$ and $\\mathfrak {h}_l$ respectively.",
"The index $l$ labels both discrete and continuous parts of the spectrum of $\\mathfrak {H}_p$ such that $\\lambda _l$ are in the decreasing order; $l=0$ corresponds to the zero mode.",
"In the basis of $\\mathfrak {h}_l$ , we express $\\Psi $ as: $\\Psi (t,\\zeta ) = \\sum _l a_l(t)\\mathfrak {h}_l(\\zeta ).$ The unknown coefficients $a_l(t)$ are determined by projecting Eq.",
"(REF ) on $\\mathfrak {h}_l$ : $\\frac{d a_l}{dt} -\\lambda _l a_l = \\int _{-\\infty }^{+\\infty } e^{\\frac{v_d\\zeta }{2D}}\\left[\\rho ^{\\prime }_d \\xi _{(1)}^{\\prime } + \\frac{1}{\\sqrt{N}}\\Gamma (\\rho _d)\\eta \\right] \\mathfrak {h}_l d\\zeta .$ and then solving these linear equations: $a_l(t) = \\int _{-\\infty }^{t} e^{\\lambda _l(t-\\tilde{t})} \\left\\lbrace \\int _{-\\infty }^{+\\infty } e^{\\frac{v_d\\zeta }{2D}}\\left[\\rho ^{\\prime }_d(\\zeta ) \\xi _{(1)}^{\\prime }(\\tilde{t}) + \\frac{1}{\\sqrt{N}}\\Gamma (\\rho _d(\\zeta ))\\eta (\\tilde{t},\\zeta )\\right] \\mathfrak {h}_l(\\zeta ) d\\zeta \\right\\rbrace d\\tilde{t}.$ Here, we assumed that the front has been propagating for a very long time and, therefore, set the lower integration limit of the integral over $\\tilde{t}$ to $-\\infty $ .",
"The next step is to substitute the solution for $\\xi ^{\\prime }_{(1)}$ from Eq.",
"(REF ): $a_l(t) = \\frac{1}{\\sqrt{N}}\\int _{-\\infty }^{t} e^{\\lambda _l(t-\\tilde{t})} \\left\\lbrace \\int _{-\\infty }^{+\\infty } \\left[ e^{\\frac{v_d\\zeta }{2D}}\\Gamma (\\rho _d(\\zeta ))\\eta (\\tilde{t},\\zeta )-\\mathfrak {h}_0(\\zeta )\\int _{-\\infty }^{+\\infty } e^{\\frac{v_d\\tilde{\\zeta }}{2D}}\\Gamma (\\rho _d(\\tilde{\\zeta }))\\eta (\\tilde{t},\\tilde{\\zeta }) \\mathfrak {h}_0(\\tilde{\\zeta })d\\tilde{\\zeta }\\right] \\mathfrak {h}_l(\\zeta ) d\\zeta \\right\\rbrace d\\tilde{t},$ where we used the fact that $\\mathfrak {h}_0(\\zeta ) = \\frac{ e^{\\frac{v_d\\zeta }{2D}}\\rho ^{\\prime }_d(\\zeta )}{\\sqrt{\\int _{-\\infty }^{+\\infty }e^{\\frac{v_d\\zeta }{D}} [\\rho ^{\\prime }_d(\\zeta )]^2d\\zeta }},$ and, therefore, $\\xi ^{\\prime }_{(1)}(t) = -\\frac{1}{\\sqrt{N}} \\frac{\\int _{-\\infty }^{+\\infty } e^{\\frac{v_d\\zeta }{2D}}\\Gamma (\\rho _d(\\zeta ))\\eta (t,\\zeta ) \\mathfrak {h}_0(\\zeta )d\\zeta }{\\sqrt{\\int _{-\\infty }^{+\\infty }e^{\\frac{v_d\\zeta }{D}} [\\rho ^{\\prime }_d(\\zeta )]^2d\\zeta }}.$ Equation (REF ) is further simplified by carrying out the integration over $\\zeta $ in the last term and using the orthogonality of $\\mathfrak {h}_0$ and $\\mathfrak {h}_l$ for $l>0$ : $a_l(t) = \\frac{1-\\delta _{0l}}{\\sqrt{N}}\\int _{-\\infty }^{t} e^{\\lambda _l(t-\\tilde{t})} \\left\\lbrace \\int _{-\\infty }^{+\\infty } e^{\\frac{v_d\\zeta }{2D}}\\Gamma (\\rho _d(\\zeta ))\\eta (\\tilde{t},\\zeta ) \\mathfrak {h}_l(\\zeta ) d\\zeta \\right\\rbrace d\\tilde{t},$ Note that, $a_0=0$ consistent with the solvability condition that $\\Delta \\rho _{(1)}$ has a vanishing projection on the zero mode.",
"With the solution for $\\Delta \\rho _{(1)}$ at hand, we proceed to calculate the average $[\\Delta \\rho _{(1)}]^2$ that enters Eq.",
"(REF ): $\\begin{aligned}& \\langle [\\Delta \\rho _{(1)}(t,\\zeta )]^2\\rangle = \\sum _{l_1>0}\\sum _{l_2>0}\\mathfrak {h}_{l_{1}}(\\zeta )\\mathfrak {h}_{l_{2}}(\\zeta ) \\langle a_{l_{1}}(t)a_{l_{2}}(t)\\rangle \\\\&= \\frac{1}{N} \\sum _{l_1>0}\\sum _{l_2>0} \\mathfrak {h}_{l_{1}}(\\zeta )\\mathfrak {h}_{l_{2}}(\\zeta ) \\frac{-1}{\\lambda _{l_1} + \\lambda _{l_2}} \\int _{-\\infty }^{+\\infty }e^{\\frac{v_d\\zeta }{D}}\\Gamma ^2(\\rho _d(\\zeta )) \\mathfrak {h}_{l_1}(\\zeta ) \\mathfrak {h}_{l_2}(\\zeta ) d\\zeta .\\end{aligned}$ Upon substituting this result into Eq.",
"(REF ), we obtain $\\begin{aligned}&\\Delta v = -v_d\\frac{D_{\\mathrm {f}}}{2D} - \\frac{1}{N} \\frac{1}{\\int _{-\\infty }^{+\\infty } e^{v_d \\zeta /D} [\\rho _d^{\\prime }(\\zeta )]^2 d\\zeta } \\times \\\\ &\\sum _{l_1>0}\\sum _{l_2>0} \\frac{\\left[\\int _{-\\infty }^{+\\infty }e^{\\frac{v_d\\zeta }{D}}\\gamma _n(\\rho _d(\\zeta ))\\rho _d(\\zeta ) \\mathfrak {h}_{l_1}(\\zeta ) \\mathfrak {h}_{l_2}(\\zeta ) d\\zeta \\right]\\left[ \\int _{-\\infty }^{+\\infty } e^{v_d \\zeta /D} \\rho _d^{\\prime }(\\zeta ) \\frac{1}{2}[r^{\\prime \\prime }(\\rho _d)\\rho _d+2r^{\\prime }(\\rho _d)] \\mathfrak {h}_{l_{1}}(\\zeta )\\mathfrak {h}_{l_{2}}(\\zeta )d\\zeta \\right]}{-(\\lambda _{l_1} + \\lambda _{l_2})} .\\end{aligned}$ Since the eigenvectors $\\mathfrak {h}_{l}$ decay at least as fast as $\\mathfrak {h}_{0}\\sim e^{\\frac{v_d\\zeta }{2D}}\\rho ^{\\prime }_d(\\zeta )$ as $\\zeta \\rightarrow +\\infty $ , all the integrands in Eq.",
"(REF ) decay faster than $e^{-(3k-2v_d/D)\\zeta }$ .",
"For fully-pushed waves, all the integrals converge, and the correction to the velocity scales as $1/N$ .",
"For semi-pushed waves, the term with $D_{\\mathrm {f}}$ shows the fastest divergence with the cutoff and, therefore, determines the scaling of $\\Delta v$ with $N$ .",
"VIII.",
"Rate of diversity loss, $\\Lambda $ In this section, we describe how genetic diversity is lost during a range expansion and provide the derivation of Eq.",
"(5) from the main text, which was first derived in Ref. [35].",
"For simplicity, we consider an expansion that started with two neutral genotypes present throughout the population and determine how the probability to sample two different genotypes at the front decreases with time.",
"The calculation of $\\Lambda $ is based on the perturbation theory in $1/N$ and relies on a mean-field assumption that $n(t,x)$ can be approximated by $\\langle n(t,x) \\rangle $ .",
"This analysis is asymptotically exact for fully-pushed waves and could be extended to semi-pushed and pulled waves by applying a cutoff at large $\\zeta $ as we show in section IX.",
"The current section contain no new results except for the calculation of $\\Lambda $ in exactly solvable models of fully-pushed waves.",
"Forward-in-time analysis of the decay of heterozygosity We quantify the genetic diversity in the population by the average heterozygosity: $H(t,\\zeta _1,\\zeta _2) = \\langle f(t,\\zeta _1) [1-f(t,\\zeta _2)] + [1-f(t,\\zeta _1)]f(t,\\zeta _2) \\rangle ,$ which is the probability to sample two different genotypes at positions $\\zeta _1$ and $\\zeta _2$ in the comoving reference frame at time $t$ .",
"Here, $f$ denotes the frequency of one the two genotypes; the frequency of the other genotype is $1-f$ .",
"To obtain a closed equation for the dynamics of $H$ , we assume that $n(t,\\zeta )$ is given by its non-fluctuating stationary limit, $n(\\zeta )$ , from Eq.",
"(REF ).",
"Then, we differentiate Eq.",
"(REF ) with respect to time and use Eq.",
"(REF ) to eliminate the time derivatives of $f$ .",
"The result reads $\\frac{\\partial H}{\\partial t} = \\left( \\mathcal {L}_{\\zeta _1} + \\mathcal {L}_{\\zeta _2} \\right)H - \\delta (\\zeta _1-\\zeta _2)\\frac{\\gamma _f(n)}{n}H,$ where $\\mathcal {L}_{\\zeta } = D\\frac{\\partial ^2}{\\partial \\zeta ^2} + v\\frac{\\partial }{\\partial \\zeta } + 2D\\frac{\\partial \\ln n}{\\partial \\zeta }\\frac{\\partial }{\\partial \\zeta }.$ we note that the first term in Eq.",
"(REF ) follows from the rules of regular calculus, but the last term arises due to the Itô formula of stochastic calculus, which prescribes how to compute derivatives of nonlinear functions; see Refs.",
"[58], [77], [78], [86].",
"This last term encapsulates the effect of genetic drift and ensures that genetic diversity decays to zero due to the fixation of one of the genotypes.",
"Since $H$ obeys a linear equation, it will decay to zero exponentially in time with the decay rate given by the solution of the following eigenvalue problem: $- \\Lambda H= \\left( \\mathcal {L}_{\\zeta _1} + \\mathcal {L}_{\\zeta _2} \\right)H - \\delta (\\zeta _1-\\zeta _2)\\frac{\\gamma _f(n)}{n}H,$ where we seek the smallest $\\Lambda $ or alternatively the largest eigenvalue of the operator on the right hand side.",
"We compute $\\Lambda $ perturbatively by treating $1/N$ as a small parameter.",
"To the zeroth order, we can neglect the last term in Eq.",
"(REF ) because it scales as $1/N$ .",
"Without the sink term, Eq.",
"(REF ) admits a constant stationary solution ($H(t,\\zeta )=\\mathrm {const}$ ), so the smallest decay rate is zero.",
"Thus, the zeroth order solution of Eq.",
"(REF ) reads $\\begin{aligned}\\Lambda &= 0,\\\\R(\\zeta _1,\\zeta _2) & = 1.\\end{aligned}$ Because $\\mathcal {L}_{\\zeta }$ contains terms linear in $\\frac{\\partial }{\\partial \\zeta }$ , the operator in Eq.",
"(REF ) is not Hermitian.",
"Therefore, we also need $L(\\zeta _1, \\zeta _2)$ , the left eigenvector of $ \\mathcal {L}_{\\zeta _1} + \\mathcal {L}_{\\zeta _2}$ , to compute the first order correction.",
"It is not difficult to guess $L(\\zeta _1, \\zeta _2)$ because it corresponds to the right eigenvector of the adjoint operator, and we already obtained the stationary distribution for $\\partial S/ \\partial \\tau = L^{+}_{\\zeta }S$ when we discussed the patterns of ancestry.",
"Since $\\mathcal {L}_{\\zeta _1}$ and $\\mathcal {L}_{\\zeta _2}$ act on different variables, the sought-after eigenfunction is the product of the eigenfunctions of these two operators: $L(\\zeta _1,\\zeta _2) = n^2(\\zeta _1)e^{\\zeta _1 v /D}n^2(\\zeta _2)e^{\\zeta _2 v /D}.$ The first order correction to $\\Lambda $ is given by the standard formula [77], [78]: $\\begin{aligned}\\Lambda = \\frac{ \\int _{-\\infty }^{+\\infty }d\\zeta _1\\int _{-\\infty }^{+\\infty }d\\zeta _2 L(\\zeta _1,\\zeta _2) \\delta (\\zeta _1-\\zeta _2)\\frac{\\gamma _f(n)}{n} R(\\zeta _1,\\zeta _2) }{\\int _{-\\infty }^{+\\infty }d\\zeta _1\\int _{-\\infty }^{+\\infty }d\\zeta _2 L(\\zeta _1,\\zeta _2)R(\\zeta _1,\\zeta _2)}.\\end{aligned}$ We now use the expressions of $L(\\zeta _1,\\zeta _2)$ and $R(\\zeta _1,\\zeta _2)$ from Eqs.",
"(REF ) and (REF ) and obtain the final result: $\\Lambda = \\frac{\\int _{-\\infty }^{+\\infty }\\gamma _{f}(n(\\zeta ))n^{3}(\\zeta )e^{\\frac{2v\\zeta }{D}}d\\zeta }{\\left(\\int ^{+\\infty }_{-\\infty }n^2(\\zeta )e^{\\frac{v\\zeta }{D}}d\\zeta \\right)^2}.$ which becomes identical to Eq.",
"(5) in the main text upon substituting $n = N\\rho $ .",
"This result was first obtained in Ref. [35].",
"For fully-pushed waves, all the integrals in Eq.",
"(REF ) converge and one can obtain the dependence of $\\Lambda $ on model parameters by dimensional analysis.",
"Specifically, each factor of $n$ contributes a factor of $N$ , and each $d\\zeta $ contributes a width of the front (the integrands rapidly tend to zero in the bulk and the leading edge).",
"In total, $\\Lambda $ is inversely proportional to the product of $N$ and front width, i.e.",
"to the number of individuals at the front.",
"This result is quite intuitive because, in well-mixed populations, the rate of diversity loss scales as the total population size, and $\\Lambda ^{-1}$ is often denoted as an effective population size [87].",
"Thus, the neutral evolution in a fully-pushed wave could be approximated by that in a well-mixed population consisting of all the organisms at the front.",
"In contrast, only the very tip of the front drives the evolutionary dynamics in semi-pushed and pulled waves.",
"Equation (REF ) also suggests that the deterministic approximation for $n(t,\\zeta )$ that we made in Eq.",
"(REF ) is asymptotically exact for fully-pushed waves.",
"Indeed, the main contribution to the integrals in Eq.",
"(REF ) comes for the interior regions of the front, where the fluctuations in $n$ are small compared to the mean population density.",
"Finally, we note that one can avoid using the perturbation theory for non-Hermitian operators to derive Eq.",
"(REF ).",
"Specifically, one can recast $(\\mathcal {L}_{\\zeta _1} + \\mathcal {L}_{\\zeta _1} )$ in a Hermitian form by finding a function $\\beta (\\zeta )$ such that $(\\mathcal {L}_{\\zeta _1} + \\mathcal {L}_{\\zeta _1} ) \\beta (\\zeta _1)\\beta (\\zeta _2)\\Psi (t,\\zeta _1,\\zeta _2) = \\beta (\\zeta _1)\\beta (\\zeta _2) \\mathcal {H}\\Psi (t,\\zeta _1,\\zeta _2)$ , where $\\mathcal {H}$ is a Hermitian operator, which contains no terms linear in $\\partial /\\partial \\zeta _1$ or $\\partial /\\partial \\zeta _2$ .",
"Then, the substitution: $H(t,\\zeta _1,\\zeta _2) = \\beta (\\zeta _1)\\beta (\\zeta _2)\\Psi (t,\\zeta _1,\\zeta _2)$ converts Eq.",
"(REF ) into a Hermitian eigenvalue problem.",
"The following equations summarize the main steps in this approach: $\\begin{aligned}& \\beta (\\zeta ) = n^{-1}(\\zeta ) e^{-\\frac{\\zeta v}{2D}},\\\\& \\mathcal {H} = D\\frac{\\partial ^2}{\\partial \\zeta _1^2} - \\left(\\frac{D}{n(\\zeta _1)}\\frac{\\partial ^2 n(\\zeta _1)}{\\partial \\zeta _1^2} + \\frac{v}{n(\\zeta _1)}\\frac{\\partial n(\\zeta _1)}{\\partial \\zeta _1} + \\frac{v^{2}}{4D} \\right) + D\\frac{\\partial ^2}{\\partial \\zeta _2^2} - \\left(\\frac{D}{n(\\zeta _2)}\\frac{\\partial ^2 n(\\zeta _2)}{\\partial \\zeta _2^2} + \\frac{v}{n(\\zeta _2)}\\frac{\\partial n(\\zeta _2)}{\\partial \\zeta _2} + \\frac{v^{2}}{4D} \\right), \\\\& h_0(\\zeta _1,\\zeta _2) = \\frac{1}{\\beta (\\zeta _1)}\\frac{1}{\\beta (\\zeta _2)} = n(\\zeta _1) e^{\\frac{\\zeta _1 v}{2D}}n(\\zeta _2) e^{\\frac{\\zeta _2 v}{2D}},\\end{aligned}$ where $h_0(\\zeta )$ is the eigenvector corresponding to the zero eigenvalue.",
"This eigenvector is easily obtained from the reverse transformation from $H$ to $\\Psi $ and the fact that $H=\\mathrm {const}$ is the right eigenvector of the original operator, $(\\mathcal {L}_{\\zeta _1} + \\mathcal {L}_{\\zeta _1} )$ .",
"Since the eigenvalues of $\\mathcal {H}$ coincide with the eigenvalues of $(\\mathcal {L}_{\\zeta _1} + \\mathcal {L}_{\\zeta _1} )$ , one can compute $\\Lambda $ by the standard formula: $\\Lambda = \\frac{\\int _{-\\infty }^{+\\infty }\\int _{-\\infty }^{+\\infty } h_0(\\zeta _1,\\zeta _2) \\delta (\\zeta _1-\\zeta _2)\\frac{\\gamma _f(n(\\zeta _1))}{n(\\zeta _1)} h_0(\\zeta _1,\\zeta _2) d\\zeta _1d\\zeta _2}{\\int _{-\\infty }^{+\\infty }h^2_0(\\zeta _1,\\zeta _2) d\\zeta _1d\\zeta _2}.$ It is now easy to see that Eqs.",
"(REF ) and (REF ) lead to the same expression for $\\Lambda $ as in Eq.",
"(REF ).",
"Backward-in-time analysis of lineage coalescence To complement the forward-in-time analysis, we show how $\\Lambda $ can be computed by tracing ancestral lineages backward in time.",
"One advantage of this approach is that it provides a more intuitive explanation of Eq.",
"(REF ).",
"The discussion of this approach closely follows Ref. [35].",
"We motivate the backward-in-time approach by considering how $H$ can be estimated from its definition as the probability to sample two different genotypes.",
"To determine whether the genotypes are different, we trace their ancestral lineages backward in time and observe that only two outcomes are possible: Either the lineages never interact with each other until they hit the initial conditions or the lineages coalesce, i.e.",
"converge on the same ancestor, at some point during the range expansion.",
"In the former case, the probability to be different is determined by the initial heterozygosity of the population.",
"In the latter case, the probability to be different is zero because we do not allow mutations.",
"Thus, $H(t,\\zeta _1^d,\\zeta _2^d)$ is intimately related to the probability $S^{(2)}$ that two lineages sampled at time $t$ at positions $\\zeta _1^d$ and $\\zeta _2^d$ have not coalesced up to time $\\tau $ into the past and were present at $\\zeta _1^a$ and $\\zeta _2^a$ at time $t-\\tau $ ; the superscripts distinguish between the positions of the descendants and the ancestors.",
"To simplify the notation, we suppress descendent-related variables, drop the subscripts, and write this probability as $S^{(2)}(\\tau , \\zeta _1, \\zeta _2)$ .",
"We keep the superscript to distinguish $S^{(2)}$ from $S$ , which denotes the position of a single ancestral lineage.",
"The dynamical equation for $S^{(2)}$ can be derived either from the forward-in-time formulation for $H$ or directly from the dynamics of ancestral lineages.",
"The result reads $\\frac{\\partial S^{(2)}}{\\partial \\tau } = \\left( \\mathcal {L}^+_{\\zeta _1} + \\mathcal {L}^+_{\\zeta _2} \\right)S^{(2)} - \\delta (\\zeta _1-\\zeta _2)\\frac{\\gamma _f(n)}{n}S^{(2)},$ where the first term describes the motion of the two ancestral lineages, and the last term accounts for the lineage coalescence.",
"As expected, the rate of coalescence events is inversely proportional to the local effective population size $n/\\gamma _f$ ; see [87], [65], [64].",
"Because the linear operators on the right hand side of Eqs.",
"(REF ) and (REF ) are adjoint to each other, their eigenvalues coincide.",
"Therefore, the temporal decay of $S^{(2)}$ is exponential in $\\tau $ with the decay rate equal to $\\Lambda $ .",
"The expression for $\\Lambda $ that we obtained previously (see Eq.",
"(REF )) is much easier to interpret in the backward-in-time formulation.",
"To show this, let us rewrite Eq.",
"(REF ) as $\\Lambda = \\int _{-\\infty }^{+\\infty } \\frac{\\gamma _f(n(\\zeta ))}{n(\\zeta )}S^{2}(\\zeta )d\\zeta ,$ where we used Eq.",
"(REF ) to express $\\Lambda $ in terms of $S(\\zeta )$ , the stationary distribution of the location of a single ancestral lineage.",
"We can now see that the effective coalescence rate, $\\Lambda $ , is given by the sum of the local coalescence rates, $\\gamma _f/n$ , weighted by the probability that two lineages are present at the same location, $S^2$ .",
"Thus, the first order perturbation theory is equivalent to assuming that the positions of the two ancestral lineages are uncorrelated with each other and distributed according to their stationary distribution $S(\\zeta )$ .",
"The last results also clarifies the difference between pulled, semi-pushed, and fully-pushed waves.",
"For pulled waves, $S(\\zeta )$ is peaked at the leading edge and, since the coalescent rate peaks at the same location, the neutral evolution is driven by the very tip of the front.",
"In semi-pushed waves, $S(\\zeta )$ is peaked in the interior of the front, but the $1/n$ increase in the coalescence rates at the front is sufficiently strong to keep all coalescent events at the front edge.",
"Finally, in fully-pushed waves, the decay of $S^2(\\zeta )$ at the front is stronger than the increase in the coalescence rates, and most coalescence events occur in the interior of the front.",
"Thus, the focus of diversity is located in the interior of the front in fully-pushed waves, but at the front edge in pulled and semi-pushed waves.",
"Explicit results for $\\Lambda $ in exactly solvable models and connection In the regime of fully-pushed waves, we can evaluate $\\Lambda $ explicitly for the exactly solvable models introduced in section I.",
"Specifically, we find that $\\Lambda = \\frac{\\gamma _f^0}{4\\pi N}\\sqrt{\\frac{g_0}{2D}}\\frac{1 - 4\\rho ^*}{\\rho ^*}\\tan {2\\pi \\rho ^*}$ for the model specified by Eq.",
"(REF ) with $\\gamma _f(n)$ that does not depend on $n$ and is equal to $\\gamma _f^0$ .",
"Note that the choice of $\\gamma _{f}(n)$ is not specified by the deterministic model of population growth and needs to be determined from the microscopic dynamics, from phenomenological considerations, or empirically.",
"For most commonly used models, $\\gamma _f(n)$ is a constant.",
"In our simulations, this constant is $1/(a\\tau )$ , where $\\tau $ is the generation time and $a$ is the spatial scale over which genetic drift is correlated.",
"However, different $\\gamma _f$ are possible.",
"For example, $\\gamma _f=\\gamma _f^0(1-n/N)$ could be appropriate for models that allow no births or deaths once the population has reached the carrying capacity.",
"In such models $\\gamma _f(N)=0$ , and genetic drift operates only at the front.",
"For completeness, we also provide the results for other models and different choices of $\\gamma _f(n)$ : $\\Lambda = \\frac{\\gamma _f^0}{6\\pi N}\\sqrt{\\frac{g_0}{2D}}\\frac{(1 - 2\\rho ^*)(1 - 4\\rho ^*)}{\\rho ^*}\\tan {2\\pi \\rho ^*}$ for the model specified by Eq.",
"(REF ) with $\\gamma _f= \\gamma _f^0(1-n/N)$ , $\\Lambda = \\frac{\\gamma _f^0}{4\\pi N}\\sqrt{\\frac{r_0B}{2D}}(B + 4)\\tan {\\frac{2\\pi }{B}}$ for the model of cooperative growth defined in the main text with $\\gamma _f = \\gamma _f^0$ ; $\\Lambda = \\frac{\\gamma _f^0}{6\\pi N}\\sqrt{\\frac{r_0B}{2D}}\\frac{(B + 2)(B + 4)}{B}\\tan {\\frac{2\\pi }{B}}$ for the model of cooperative growth defined in the main text with $\\gamma _f = \\gamma _f^0 (1-n/N)$ .",
"IX.",
"Cutoffs for deterministic and fluctuating fronts The integrands that appear in the expressions for $\\Delta v$ , $D_{\\mathrm {f}}$ , and $\\Lambda $ diverge near the front edge in pulled and semi-pushed waves.",
"Since there are no organisms sufficiently far ahead of the front, these divergences are technical artifacts that do not represent the actual dynamics of the traveling wave.",
"For example, in our calculation of $\\Lambda $ , the divergence appears because we approximate the wave front by the stationary, deterministic profile, $n(\\zeta )$ , from Eq.",
"(REF ).",
"In this section, we show how to remove these divergences by applying a cutoff at large $\\zeta $ .",
"The value of the cutoff, $\\zeta _c$ , scales as $\\ln (N)/q$ for fluctuating fronts, but as $\\ln (N)/k$ for deterministic fronts with $\\gamma _n=0$ ; $k$ and $q$ are given in Eq.",
"(REF ).",
"The derivation of $\\zeta _c\\sim \\ln (N)/q$ is the main new result in this section.",
"Cutoff for deterministic fronts A cutoff for the growth rate was first introduced in the context of pulled waves [1].",
"The primary motivation for the cutoff was to compute the corrections to the wave velocity and to resolve the velocity-selection problem, i.e.",
"to explain why the simulations of discrete entities never exhibit waves with velocities greater than $v_{\\mathrm {\\textsc {f}}}$ even though such solutions are possible in the continuum limit.",
"The naive argument for a cutoff is that there should be no growth in areas where the average number of individuals falls below one per site in lattice models or one per typical dispersal distance in models with continuous space.",
"We denote the relevant spatial scale, i.e.",
"the distance between lattice sites or the dispersal distance, by $a$ , so the cutoff density is $1/a$ .",
"Since, at such low densities, the front shape is well approximated by the asymptotic solution $n\\sim Ne^{-k\\zeta }$ , the value of the cutoff is given by $\\zeta _c = \\frac{1}{k}\\ln (Na).$ While this cutoff regularizes all the integrals and captures the gross effects of the stochastic dynamics, it is not quantitatively accurate for fluctuating fronts.",
"Previous studies showed significant differences between the predictions of Eq.",
"(REF ) and simulations and argued that the factor multiplying $\\ln (N)$ in Eq.",
"(REF ) should be different from $1/k$ [31].",
"The main goal of this section is to derive the correct cutoff for fluctuating fronts.",
"Before proceeding with fluctuating fronts, however, it is important to point out that Eq.",
"(REF ) prescribes the correct cutoff for deterministic models with discrete entities [88].",
"In such models, the main effect of discreteness is simply the absence of growth for $n<1/a$ , and, therefore, Eq.",
"(REF ) does apply.",
"Our simulations show clear differences in the scaling of $\\Delta v$ and $\\Lambda $ with $N$ for deterministic and fluctuating fronts (Fig.",
"REF ).",
"Moreover, these differences are explained entirely by the different cutoffs that one needs to apply for fluctuating and deterministic fronts.",
"Cutoff for fluctuating fronts Analysis of fluctuating fronts is a challenging problem that is typically addressed by matching the nonlinear quasi-deterministic dynamics at the bulk of the front and the linear, but stochastic dynamics at the front edge [22].",
"Recently, a more rigorous approach has been developed in Refs.",
"[37], [40], which relies on modifying the reaction-diffusion model to ensure that the hierarchy of moment equations closes exactly.",
"The details of this approach are sufficiently technical and tangential to the main issues discussed in this paper, so we do not discuss them here.",
"Instead, we refer the readers to Ref.",
"[37] for a self-contained presentation of the new method.",
"For our purpose, the most useful result from Ref.",
"[37] is that the deterministic equation for the steady-state density profile needs to be modified as $Dn^{\\prime \\prime } + vn^{\\prime } + r(n)n - \\frac{\\gamma _n(n)n^2e^{\\frac{v\\zeta }{D}}}{\\int _{-\\infty }^{+\\infty }n^2e^{\\frac{v\\zeta }{D}}d\\zeta }= 0;$ see Eq.",
"(10) in Ref. [37].",
"The only difference between Eq.",
"(REF ) and Eq.",
"(REF ) is an additional term, which, as we show below, effectively imposes a cutoff on the growth rate.",
"To quantify the magnitude of the new term, it is convenient to define a ratio between the terms due to front fluctuations and population growth: $E(\\zeta ) = \\frac{\\frac{\\gamma _n(n)n^2e^{\\frac{v\\zeta }{D}}}{\\int _{-\\infty }^{+\\infty }n^2e^{\\frac{v\\zeta }{D}}d\\zeta }}{r(n)n}= \\frac{\\gamma _n(n)ne^{\\frac{v\\zeta }{D}}}{r(n)\\int _{-\\infty }^{+\\infty }n^2e^{\\frac{v\\zeta }{D}}d\\zeta }.$ Note that the first three terms in Eq.",
"(REF ) are of the same order at the front, so any one of them could be used to define $E$ .",
"Since we are only interested in the behavior of $E$ near the front edge, $E$ can be further simplified as $E(\\zeta ) \\approx \\frac{\\gamma _n(0)ne^{\\frac{v\\zeta }{D}}}{r(0)\\int _{-\\infty }^{+\\infty }n^2e^{\\frac{v\\zeta }{D}}d\\zeta } = \\frac{\\gamma _n(0)v\\rho e^{\\frac{v\\zeta }{D}}}{r(0)DNI},$ where, in the last equality, we made the dependence on all dimensional quantities explicit by using $\\rho =n/N$ and introducing a non-dimensional integral $I = \\int _{-\\infty }^{+\\infty }\\rho ^2(\\zeta )e^{\\frac{v\\zeta }{D}}d\\left(\\frac{v\\zeta }{D}\\right).$ To obtain the scaling behavior of $E$ for large $\\zeta $ , we approximate $\\rho $ as $e^{-k\\zeta }$ and replace $v/D$ by $k+q$ (see Eq.",
"(REF )) $E\\sim e^{-\\zeta (k-v/D)}\\sim e^{q\\zeta }.$ Note that $q$ is defined in Eq.",
"(REF ) as the decay rate for the solution of Eq.",
"(REF ) that is inconsistent with the boundary conditions.",
"Therefore, $q$ corresponds to the unphysical part of the solution for $\\rho $ and does not directly enter the asymptotic behavior of the population density.",
"In the following, $q$ is often used instead of $v$ to make the formulas more compact.",
"We now determine the cutoff $\\zeta _c$ for all subtypes of traveling waves.",
"The main idea is to check whether the solution of Eq.",
"(REF ) is consistent with the addition of a term due to front fluctuations in Eq.",
"(REF ).",
"If the solution is consistent, then no cutoff is necessary.",
"If the solution is not consistent, then it can be valid only up to some critical $\\zeta $ , which acts as an effective cutoff.",
"Cutoff for pushed waves expanding into a metastable state When the invaded state is metastable, the low-density growth rate is negative, and, therefore, $q<0$ ; see Eq.",
"(REF ).",
"In consequence, $E\\rightarrow 0$ as $\\zeta \\rightarrow +\\infty $ , and front fluctuations have a negligible effect on wave dynamics.",
"Thus, no cutoff is necessary, i.e.",
"$\\zeta _c=+\\infty $ .",
"Cutoff for pushed waves expanding into an unstable state When the invaded state is unstable, $q$ is positive, and $E$ diverges as $\\zeta \\rightarrow +\\infty $ .",
"This contradicts Eq.",
"(REF ), where all terms need to cancel out.",
"To satisfy the equation, $\\rho $ must decay faster than $e^{-k\\zeta }$ at the front beyond some critical $\\zeta _c$ , so that $E$ never becomes much greater than one.",
"The value of $\\zeta _c$ is then determined by the solution of $E(\\zeta _c)=1$ with the deterministic approximation for $\\rho $ .",
"Hence, we substitute $\\rho \\sim e^{-k\\zeta }$ in Eq.",
"(REF ) and find that $\\zeta _c = \\frac{1}{q}\\ln \\left(N\\frac{Dr(0)I}{v\\gamma _n(0)}\\right) = \\frac{1}{q}\\ln \\left(\\frac{N}{k+q}\\frac{r(0)I}{\\gamma _n(0)}\\right) \\sim \\frac{1}{q}\\ln N.$ This is the most important result of this section because it determines the novel scaling behavior of $\\Delta v$ , $D_{\\mathrm {f}}$ , and $\\Lambda $ in semi-pushed waves.",
"To the best of our knowledge, Eq.",
"REF is a new finding.",
"Since $q<k$ , fluctuating fronts have a larger $\\zeta _c$ than deterministic fronts and a lower normalized cutoff density $\\rho _c\\sim e^{-k\\zeta _c}\\sim N^{-k/q}<N^{-1}$ .",
"The applicability of the continuum theory for $\\rho <1/N$ seems counter-intuitive, not only because the expected number of organisms at a site falls below one, but also because fluctuations should appreciably modify the profile density at least when $\\rho <1/\\sqrt{N}$ , i.e.",
"well before the naive $1/N$ cutoff.",
"The key problem with this argument is that it assumes a continual front and neglects the possibility that a sufficiently large group of organisms can occasionally expand well ahead of the deterministic front [19], [63].",
"Such front excursions prevent a sharp cutoff in the population density below $1/N$ .",
"More importantly, they significantly amplify both genetic drift and front wandering.",
"In the continuum theory, this increase in fluctuations is captured by greater $\\zeta _c$ , which increases both $\\Lambda $ and $D_{\\mathrm {f}}$ .",
"The probability of front excursions is controlled not only by the intensity of demographic fluctuations, but also by other parameters of the population dynamics.",
"In particular, $\\rho _c$ depends on cooperativity through $v/v_{\\mathrm {\\textsc {f}}}$ .",
"Inclusion of this dependence is necessary to accurately describe the dynamics of semi-pushed waves.",
"Cutoff for pulled waves For pulled waves, $q>0$ , and we obtain the value of the cutoff by solving $E(\\zeta _c)=1$ just as for semi-pushed waves: $\\gamma _n(0) \\rho (\\zeta _c) e^{\\frac{v\\zeta _c}{D}}= r(0)N\\int _{-\\infty }^{\\zeta _c}\\rho ^2(\\zeta )e^{\\frac{v\\zeta }{D}}d\\zeta .$ Note, however, that there are two important differences in the calculation for pulled compared to semi-pushed waves.",
"First, the integral $I$ diverges and needs to be cut off at $\\zeta _c$ .",
"Second, the front shape also acquires corrections due to the cutoff and needs to be determined self-consistently.",
"This sensitivity of the front shape originates from the degeneracy $k=q$ that occurs in pulled waves.",
"In the continuum limit, this degeneracy modifies the scaling of $\\rho $ from $e^{-k_{\\mathrm {\\textsc {f}}}\\zeta }$ to $k_{\\mathrm {\\textsc {f}}}\\zeta e^{-k_{\\mathrm {\\textsc {f}}}\\zeta }$ .",
"For a fluctuating front, however, the wave velocity deviates slightly from $v_{\\mathrm {\\textsc {f}}}$ , and the correction to the front shape is different.",
"The shape of the front can be obtained by setting the growth rate to zero for $\\zeta >\\zeta _c$ and solving the resulting equation for $\\rho (\\zeta )$ ; see Refs.",
"[31], [1], [33].",
"The result reads $\\rho (\\zeta ) \\sim \\frac{k_{\\mathrm {\\textsc {f}}}\\zeta _c}{\\pi }\\sin \\left(\\pi \\frac{\\zeta _c-\\zeta }{\\zeta _c}\\right)e^{-k_{\\mathrm {\\textsc {f}}}\\zeta }.$ Upon substituting this result in Eq.",
"(REF ), we find the following condition on $\\zeta _c$ : $e^{k_{\\mathrm {\\textsc {f}}}\\zeta _c} \\sim \\frac{1}{2\\pi ^2}\\frac{r(0)N}{\\gamma _n(0)k_{\\mathrm {\\textsc {f}}}} (k_{\\mathrm {\\textsc {f}}}\\zeta _c)^3,$ where we shifted $\\zeta _c$ in the argument of the sine by $1/k_{\\mathrm {\\textsc {f}}}$ , which does not change the asymptotic scaling of the exponential term, but avoids setting the left hand side to zero.",
"To solve Eq.",
"(REF ), we treat $(k_{\\mathrm {\\textsc {f}}}\\zeta _c)^3$ as a small perturbation compared to $\\frac{r(0)N}{\\gamma _n(0)k_{\\mathrm {\\textsc {f}}}}$ and obtain that, up to additive numerical factors, the leading behavior of $\\zeta _c$ is given by $\\zeta _c = \\frac{1}{k_{\\mathrm {\\textsc {f}}}}\\ln \\left(\\frac{r(0)N}{\\gamma _n(0)k_{\\mathrm {\\textsc {f}}}}\\right) + \\frac{3}{k_{\\mathrm {\\textsc {f}}}}\\ln \\left[\\ln \\left(\\frac{r(0)N}{\\gamma _n(0)k_{\\mathrm {\\textsc {f}}}}\\right)\\right] \\sim \\frac{1}{k_{\\mathrm {\\textsc {f}}}}\\ln N + \\frac{3}{k_{\\mathrm {\\textsc {f}}}}\\ln \\ln N .$ Note that the second order term cannot be neglected in the calculation of $\\Delta v$ , $D_{\\mathrm {f}}$ , and $\\Lambda $ because these quantities have terms that scale both linearly and exponentially with $\\zeta _c$ .",
"Equation (REF ) was first motivated phenomenologically in Ref.",
"[33] and then derived more rigorously in Ref. [37].",
"X.",
"Scaling of $\\Delta v$ , $D_{\\mathrm {f}}$ , and $\\Lambda $ in pulled, semi-pushed, and fully-pushed waves In this section, we synthesize the results of the perturbation theory for $\\Delta v$ , $D_{\\mathrm {f}}$ , and $\\Lambda $ and supplement them with an appropriate cutoff $\\zeta _c$ when needed.",
"We show that pushed waves consist of two distinct classes.",
"In fully-pushed waves, the fluctuations scale as $1/N$ consistent with the central limit theorem, but, in semi-pushed waves, non-trivial power scaling occurs.",
"The exponent of this power law depends only on $v/v_{\\mathrm {\\textsc {f}}}$ and is the same for $\\Delta v$ , $D_{\\mathrm {f}}$ , and $\\Lambda $ .",
"For completeness, also provide the corresponding results for pulled waves and models with deterministic fronts.",
"While most results of the perturbation theory are not new, the synthesis of these results, the application of an appropriate cutoff, and the discovery of semi-pushed waves are novel contributions of this paper.",
"The main results of the perturbation theory are given by Eq.",
"(REF ) for the correction to wave velocity, by Eq.",
"(REF ) for the effective diffusion constant of the front, and by Eq.",
"(REF ) for the rate of diversity loss.",
"All of these equations, have a similar form and contain a ratio of two integrals.",
"Both integrals converge for $\\zeta \\rightarrow -\\infty $ , but they could diverge for $\\zeta \\rightarrow +\\infty $ .",
"At the front, the integrands in the numerator scale as $e^{-(3k-2v/D)\\zeta }$ , and the integrands in the denominators scale as $e^{-(2k-v/D)\\zeta }$ ; therefore, the integrals in the denominators always converge when the integrals in the numerators converge.",
"Since the ratio of $k$ to $v/D$ depends on the degree to which the growth is cooperative, the integrals could change their behavior as cooperativity is varied.",
"A change from convergence to divergence in either of the integrals corresponds to a transitions between different classes of waves.",
"Below we consider each class separately.",
"$1/N$ scaling in fully-pushed waves The class of fully-pushed waves is defined by the requirement that all integrals converge.",
"In this case, the perturbation theory is well-posed without a cutoff and provides not only the scaling, but also the exact values of $\\Delta v$ , $D_{\\mathrm {f}}$ , and $\\Lambda $ .",
"Straightforward dimensional analysis shows that all three quantities scale as $1/N$ , i.e.",
"the central limit theorem holds.",
"The convergence of integrals requires that $kD/v$ is greater than $2/3$ .",
"For waves expanding into an metastable state, $kD/v>1$ (see Eq.",
"(REF )), so these waves are always fully-pushed.",
"For expansions into an unstable state, it is convenient to express the convergence condition only as a function of $v$ using Eq.",
"(REF ): $v \\ge \\frac{3}{2\\sqrt{2}}v_{\\mathrm {\\textsc {f}}}.$ where $v_{\\mathrm {\\textsc {f}}}=2\\sqrt{Dr(0)}$ is the linear spreading velocity.",
"We emphasize that $v_{\\mathrm {\\textsc {f}}}$ serves only as convenient notation for $2\\sqrt{Dr(0)}$ ; in particular, the wave is not pulled, and the wave velocity is greater than $v_{\\mathrm {\\textsc {f}}}$ .",
"Equation (REF ) immediately implies that not every pushed wave is fully-pushed.",
"Indeed, only $v>v_{\\mathrm {\\textsc {f}}}$ is required for a wave to be pushed, which is a weaker condition than Eq.",
"(REF ).",
"Because $v/v_{\\mathrm {\\textsc {f}}}$ increases with cooperativity, fully-pushed waves occur once cooperativity exceeds a certain threshold.",
"We can also express the condition that $kD/v>2/3$ in terms of $k$ and $q$ using Eq.",
"(REF ).",
"Because $v/D=k+q$ , this convergence condition is equivalent to $k>2q$ .",
"For all pushed waves, $k>q$ , but a stronger inequality is required for fully-pushed waves.",
"Note that $q<0$ for waves propagating into a metastable state, so $k>q$ is satisfied.",
"Finally, we discuss the effects of a cutoff derived in the previous section.",
"For expansions into a metastable state, $\\zeta _c=+\\infty $ , i.e.",
"no cutoff is necessary.",
"For expansions into an unstable state, the theory suggest a finite cutoff: $\\zeta _c\\sim \\ln (N)/q$ .",
"Note, however, that the application of this cutoff in the formulas for $\\Delta v$ , $D_{\\mathrm {f}}$ , and $\\Lambda $ only produces subleading corrections to the $1/N$ scaling because convergent integrals are insensitive to small changes in their upper limit of integration.",
"$N^{\\alpha }$ scaling in semi-pushed waves We now proceed to the second class of pushed waves, for which the integrals in the numerators diverge, i.e.",
"$v<\\sqrt{9/8}v_{\\mathrm {\\textsc {f}}}$ .",
"We term the waves in this class semi-pushed because the fluctuations at the front make a significant contribution to their dynamics.",
"Note that the integrals in the denominators converge for all pushed waves because $kD/v>1/2$ ; see Eq.",
"(REF ).",
"To estimate the scaling of $\\Delta v$ , $D_{\\mathrm {f}}$ , and $\\Lambda $ , we cut off the integrals in the numerators at $\\zeta _c\\sim \\ln (N)/q$ and find that three quantities scale as $N^{\\alpha }$ with $\\alpha $ given by $\\alpha = -2\\frac{\\sqrt{1-v^2_{\\mathrm {\\textsc {f}}}/v^2}}{1-\\sqrt{1-v^2_{\\mathrm {\\textsc {f}}}/v^2}}.$ The details of this calculation for $\\Lambda $ are summarized below $\\begin{aligned}\\Lambda &= \\frac{1}{N} \\frac{\\int _{-\\infty }^{\\zeta _c}\\gamma _{f}(\\rho )\\rho ^{3}(\\zeta )e^{\\frac{2v\\zeta }{D}}d\\zeta }{\\left(\\int ^{+\\infty }_{-\\infty }\\rho ^2(\\zeta )e^{\\frac{v\\zeta }{D}}d\\zeta \\right)^2} \\sim \\frac{1}{N} \\frac{\\gamma _{f}(0)\\int _{-\\infty }^{+\\zeta _c}e^{-(k-2q)\\zeta }d\\zeta }{\\left(I\\frac{D}{v}\\right)^2} \\sim \\frac{(k+q)^2\\gamma _f(0)}{I^2N}\\frac{e^{-\\frac{(k-2q)}{q}\\ln \\left( \\frac{N}{k+q}\\frac{r(0)I}{\\gamma _n(0)}\\right)}}{k-2q}\\\\& \\sim N^{-\\frac{k-q}{q}}I^{-\\frac{k}{q}}\\gamma _f(0)[\\gamma _n(0)]^{\\frac{k-2q}{q}}[r(0)]^{-\\frac{k-2q}{q}}(k+q)^{\\frac{k}{q}}(k-2q)^{-1}\\sim N^{-\\frac{k-q}{q}}\\sim N^{-2\\frac{\\sqrt{v^2 -v^2_{\\mathrm {\\textsc {f}}}}}{v-\\sqrt{v^2-v^2_{\\mathrm {\\textsc {f}}}}}}.\\end{aligned}$ where we used Eqs.",
"(REF ), (REF ), (REF ), and (REF ).",
"The calculations for $D_{\\mathrm {f}}$ and $\\Delta v$ are essentially the same.",
"Logarithmic scaling in pulled waves The remaining possibility is that the integrals diverge in both numerators and denominators.",
"This is the case for pulled waves because $v_{\\mathrm {\\textsc {f}}}=2k_{\\mathrm {\\textsc {f}}}D$ .",
"To compute the asymptotic scaling of $D_{\\mathrm {f}}$ and $\\Lambda $ , we use (REF ) and (REF ) together with the cutoff from Eq.",
"(REF ) and the profile shape from Eq.",
"(REF ).",
"The results read $\\begin{aligned}& D_{\\mathrm {f}} \\sim \\frac{\\gamma _{n}(0)}{k^2_{ \\mathrm {\\textsc {f}} } }\\ln ^{-3}\\left(\\frac{N}{k_{ \\mathrm {\\textsc {f}} }}\\right),\\\\& \\Lambda \\sim \\gamma _{f}(0)\\ln ^{-3}\\left(\\frac{N}{k_{ \\mathrm {\\textsc {f}} }}\\right).\\\\\\end{aligned}$ This completes our discussion of different scaling regimes in fluctuating fronts.",
"Scaling of $\\Lambda $ with $N$ in deterministic fronts Some of our results for fluctuating fronts depend on the specific choice of the cutoff $\\zeta _c=\\ln (N)/q$ .",
"This cutoff is different from the naive expectation that $\\zeta _c=\\ln (N)/k$ because occasional fluctuations establish a small population far ahead of the deterministic front.",
"To understand the effect of such fluctuations, we now examine the properties of deterministic fronts, where $\\gamma _n=0$ , but there is a cutoff on the growth rate below $\\rho _c\\sim 1/N$ .",
"Since deterministic fronts do not fluctuate, their diffusion constant is zero.",
"Genetic drift, however, occurs even without any fluctuations in the total population size, so the rate of diversity loss is well-defined.",
"Therefore, we focus on the scaling of $\\Lambda $ with $N$ in this subsection.",
"Our analysis of fully-pushed waves remains unchanged because all the integrals converge, and a cutoff is not required.",
"Thus, $\\Lambda \\sim N^{-1}$ for fully-pushed waves with or without demographic fluctuations.",
"Moreover, the transition point between fully-pushed and semi-pushed waves remains the same because it depends on the behavior of the integrands in Eq.",
"(REF ) rather than on the value of the cutoff.",
"For semi-pushed waves, $\\zeta _c$ does enter the calculation and changes the value of $\\alpha $ .",
"For deterministic fronts, we find that $\\alpha _{\\mathrm {deterministic}} = -2\\frac{k-q}{k} = - \\frac{4\\sqrt{1 - v_{\\mathrm {\\textsc {F}}}^2/v^2}}{1 + \\sqrt{1 - v_{\\mathrm {\\textsc {F}}}^2/v^2}}.$ Similarly to our results for the fluctuating fronts, $\\alpha _{\\mathrm {deterministic}}$ approaches 0 and $-1$ near the transitions to pulled and fully-pushed waves.",
"Within the class of semi-pushed waves, however, $\\alpha _{\\mathrm {deterministic}}$ is less than $\\alpha $ for fluctuating fronts ($|\\alpha _{\\mathrm {deterministic}}|>|\\alpha _{\\mathrm {fluctuating}}|$ ), that is genetic drift is amplified by front fluctuations.",
"For pulled waves, we find that $\\Lambda \\sim \\gamma _f(0)\\ln ^{-6}(N/k_{\\mathrm {\\textsc {f}}}),$ which further supports the fact that genetic drift is weaker without front fluctuations.",
"The $\\ln ^{-6}N$ scaling was previously suggested for the diffusion constant of pulled waves based on the incorrect application of the naive cutoff [34].",
"Moreover, simulations that limited the extent of demographic fluctuations indeed observed that $D_{\\mathrm {f}}\\sim \\ln ^{-6}N$ [88].",
"Comparison of $\\Delta v$ in deterministic vs. fluctuating fronts We close this section by comparing velocity corrections for deterministic and fluctuating fronts.",
"This comparison highlights the conceptual challenges that we resolved in order to describe the stochastic dynamics of range expansions and provides a useful perspective on the potential pitfalls in approximating a fluctuating front by a deterministic front with a cutoff.",
"Because corrections to velocity have been a subject of intense theoretical study [1], [34], [31], [39], [33], the following discussion also clarifies the connection between our and previous work.",
"The standard approach to computing $\\Delta v$ is to impose a zero growth rate below a certain population density; typically $\\rho _c=1/N$ .",
"The deterministic reaction-diffusion equation is then solved separately for $\\zeta <\\zeta _c$ and $\\zeta >\\zeta _c$ , and the solutions are matched at $\\zeta =\\zeta _c$ .",
"This approach is thought to be largely correct because it yields the right scaling of $\\Delta v\\sim \\ln ^{-2}N$ for pulled waves [39], which have been the primary subject of research.",
"Our calculation of the cutoff, however, shows that the agreement between $\\rho _c$ and $1/N$ for pulled waves is rather accidental because these two quantities are different for all other wave classes.",
"Moreover, further work on pulled waves showed that $1/N$ cutoff is insufficient to describe all of their properties, and the second term on the right hand side of Eq.",
"(REF ) is necessary [33].",
"This result was first obtained from phenomenological considerations [33], but was later derived more rigorously via an approach that also justified the existence of the cutoff [37].",
"The calculation of $\\Delta v$ based on a fixed growth-rate cutoff at $\\rho _c$ was extended to pushed waves by Kessler et al.",
"[31], who found thatRef.",
"[31] computed $\\Delta v$ for an unspecified cutoff at $\\rho =\\rho _c$ ; we substituted $\\rho _c=1/N$ to facilitate the comparison with other results in this paper.",
"$\\Delta v_{\\mathrm {deterministic}} \\sim N^{-1+\\frac{q}{k}}.$ The same result is obtained from the first order perturbation theory (Eq.",
"(REF )) with $\\rho _c=1/N$ .",
"By numerically solving a reaction-diffusion equation with an imposed growth-rate cutoff, Ref.",
"[31] confirmed that Eq.",
"(REF ) provides an accurate prediction for $\\Delta v$ in deterministic fronts, but the applicability of Eq.",
"(REF ) to fluctuating fronts has not been investigated.",
"Our findings show that there are two qualitative differences between the predictions of Eq.",
"(REF ) and the actual behavior of fluctuating fronts (Fig.",
"REF A).",
"First, Eq.",
"(REF ) predicts that the exponent $\\alpha _{\\mathrm {\\textsc {v}}}$ changes gradually from 0 to $-2$ as the strength of the Allee effect increasesAt the boundary with pulled waves, $\\alpha _{\\mathrm {\\textsc {v}}}=0$ , and $\\alpha _{\\mathrm {\\textsc {v}}}=-2$ for the maximal strength of the Allee effect at which the invasion can still proceed ($v>0$ ).",
"Between these two limiting cases, there are no transitions that would indicate the existence of distinct classes of pushed waves.",
"Second, Eq.",
"(REF ) misses the $1/N$ scaling of $\\Delta v$ in the regime of highly cooperative growth, where the central limit theorem applies because the properties of the wave are determined by the dynamics in the interior of the front rather than at the leading edge.",
"The origin of these discrepancies is different for semi-pushed and fully-pushed waves.",
"For semi-pushed waves, the different behavior of deterministic and fluctuating front comes from the dependence of the cutoff on the strength of the Allee effect (Eq.",
"(REF )).",
"Indeed, Eq.",
"(REF ) and the results from Ref.",
"[31] reproduce the correct scaling of $\\Delta v$ with $N$ (Eq.",
"(REF )) once we substitute $\\rho _c=N^{-k/q}$ instead of the $\\rho _c=N^{-1}$ .",
"For fully-pushed waves, this approach still produces unrealistic scaling with $\\alpha _{\\mathrm {\\textsc {v}}}<-1$ because both the first order perturbation theory and the approach in Ref.",
"[31] assume that the main contribution to $\\Delta v$ comes from the stochastic dynamics of the tip of front.",
"The dynamics of fully-pushed waves are, however, controlled by the fluctuations throughout the front, and, therefore, cannot be described by an effective cutoff.",
"This is clearly demonstrated by the second order perturbation theory (Eq.",
"(REF )), which shows how the $1/N$ scaling, expected from the central limit theorem, emerges from the stochastic dynamics at the entire front.",
"Thus, replacing the full stochastic dynamics by a deterministic front with a cutoff can fail to describe population dynamics both because the value of the cutoff has a nontrivial dependence on model parameters and because the dominant contribution of fluctuations may not be restricted to the leading edge of the reaction-diffusion wave.",
"XI.",
"Precise definitions of the foci of growth, ancestry, and diversity In this section, we consolidate the results obtained above on the spatial distribution of growth, ancestry, diversity processes within the wave front.",
"We also provide the precise definitions of the foci of growth, ancestry, and diversity.",
"The spatial distribution of the per capita growth rate is given by $\\mathrm {growth\\;\\;distribution} \\sim r(n(\\zeta )).$ The mode of this distribution is the focus of growth.",
"For monotonically decreasing $r(n)$ , the focus of growth is at the very edge of the front, i.e.",
"at $\\zeta =\\zeta _c$ , but it is in the interior of the front otherwise.",
"The spatial distribution of the most recent common ancestor of the entire population at the front is given by Eq.",
"(REF ): $\\mathrm {ancestry\\;\\;distribution}=S(\\zeta ) \\sim n^{2}(\\zeta )e^{v\\zeta /D}.$ The mode of this distribution is the focus of ancestry, which is the most likely location of the most recent common ancestor.",
"The focus of ancestry is located at $\\zeta =\\zeta _c$ for pulled waves and in the interior of the front for pushed waves.",
"To characterize the contribution of the different regions of the front to genetic diversity, we consider the spatial distribution of the locations where two ancestral lineages coalesce.",
"That is we consider the spatial location of the most recent common ancestor of two randomly sampled individuals.",
"From Eq.",
"(REF ), it follows that this distribution is given by $\\mathrm {diversity\\;\\;distribution}=C(\\zeta ) \\sim \\frac{\\gamma _f(n(\\zeta ))}{n(\\zeta )}S^{2}(\\zeta ) \\sim \\gamma _f(n(\\zeta ))n^{3}(\\zeta )e^{2v\\zeta /D}.$ The mode of this distribution is the focus of ancestry.",
"The focus of diversity is located at $\\zeta =\\zeta _c$ for pulled and semi-pushed waves and in the interior of the front for fully-pushed waves.",
"The definitions above are somewhat arbitrary as one could have used the mean or median of the corresponding distributions rather than the mode in defining the foci of ancestry and diversity.",
"The precise definitions of foci are, however, irrelevant for understanding the differences in wave properties because the spatial distributions fundamentally change at the transitions between different wave classes.",
"For pulled waves, the distribution of ancestor $S(\\zeta )$ becomes independent of $\\zeta $ for large positive $\\zeta $ .",
"Therefore, the distribution is not normalizable, and effectively all the weight of the distribution is concentrated on large positive values of $\\zeta $ .",
"In consequence, both the mean, median, and the mode are at large positive $\\zeta $ .",
"Thus, the transition from pulled to pushed waves is marked by a fundamental change in the distribution and an infinite jump in the focus of ancestry.",
"A similar transition occurs for the focus of diversity as waves transition from semi-pushed to fully-pushed.",
"For fully-pushed waves, $C(\\zeta )$ is normalizable and peaked at a well-defined value of $\\zeta $ .",
"For semi-pushed waves, $C(\\zeta )$ diverges at large $\\zeta $ and is therefore not normalizable.",
"The weight of the distribution shifts to very large $\\zeta $ , so we described this transition as the shift in the focus of diversity from the bulk to the edge of the front.",
"The focus of growth is less informative because waves could still be pulled even when the growth is not maximal at the very edge of the front.",
"Nevertheless, the transition from pulled to pushed waves is marked by a nonzero contribution of growth throughout the front to the wave velocity, so one can loosely speak of a shift in growth from the edge to the bulk of the front.",
"Figure REF graphically summarizes how the locations of different processes change as waves transition from pulled, to semi-pushed, and to fully-pushed waves.",
"XII Prevalence of semi-pushed waves The range of velocities of semi-pushed waves appears to be small from $1.00$ to about $1.06$ times the Fisher velocity.",
"Therefore, one might be tempted to conclude that semi-pushed waves are rare.",
"Below we show that this conclusion is not justified.",
"While the ratio of wave velocity to Fisher velocity is a universal metric of cooperativity, it does not faithfully represent the size of the parameter space.",
"Indeed, the entire region of pulled waves collapses to a single point $v/v_{\\mathrm {\\textsc {f}}}=1$ .",
"Pulled waves of course occur for more than a single parameter: The growth rate could include an arbitrary density-dependence as long as it decreases with population density, and the growth rate could even be mildly cooperative.",
"Because semi-pushed waves are bordering pulled waves, the parameter space also undergoes compression when mapped into the space of $v/v_{\\mathrm {\\textsc {f}}}$ .",
"To illustrate this, we consider three models of the growth rate: the cooperative model from the main text (Eq.",
"(3)), a completely different model with predator satiation, and a model of an experimental system that was recently used to show a transition from pulled to pushed waves [32].",
"For the model in the manuscript, the growth rate is given by $r(n)=r_0(1-n/N)(1+Bn/N)$ .",
"Here, parameter $B$ represents cooperativity in the growth rate and controls the transition from pulled to pushed waves.",
"For this model, pulled waves occurs for $B$ between 0 and 2, semi-pushed waves for $B$ between 2 and 4, and fully-pushed waves for $B$ greater than 4.",
"Thus, the extensively-studied pulled waves and the newly-discovered semi-pushed waves occupy regions in the parameter space of exactly the same size.",
"This model can be parameterized differently, see Eq.",
"REF .",
"For this parameterization, the region of pulled waves occurs for $n^*/N <-0.5$ , semi-pushed waves for $n^*/N$ between $-0.5$ and $-0.25$ , and fully-pushed waves for $n^*/N$ between $-0.25$ and $0.5$ .",
"From this comparison of essentially the same models, it is clear that the size of a region in the parameter spaces depend on the type of parameterization, but, generically, semi-pushed waves occupy about as much parameter space as pulled and fully-pushed waves.",
"To demonstrate, that the above conclusion is not specific to the cooperative model studied in the manuscript, we considered a completely different mechanism behind pushed waves: namely, predator satiation.",
"This type of an Allee effect can be modeled by $r(n)=r_0\\left(1-\\frac{n}{N}\\right) - d \\frac{n^*}{n+n^*}.$ Below the Allee threshold $n^*$ , the population experiences a high per capita death rate $d$ from predation, but, above $n^*$ , the limited number of predators cannot keep up with the prey, and the per capita death rate declines.",
"We found that pulled waves occur for $n^*/N$ greater than $0.35$ , semi-pushed waves for $n^*/N$ between $0.08$ and $0.35$ , and fully-pushed waves for $n^*/N$ less than $0.08$ ; see Fig.",
"REF A.",
"In this model, semi-pushed waves occupy a larger region in the parameter space than fully-pushed waves, supporting the conclusion that all three types of waves are likely to occur in nature.",
"In drawing this conclusion, we assumed that probability distribution of parameters such as $B$ or $n^*/N$ is uniform in the parameter space.",
"While this is certainly a gross approximation, it could be more accurate than the assumption that the values of $v/v_{\\mathrm {\\textsc {f}}}$ are uniformly distributed.",
"Finally, we analyzed the model of cooperative yeast growth in sucrose from Ref. [32].",
"As far as we know, this is the only study that both measured the wave velocity and parameters necessary to determine $v/v_{\\mathrm {\\textsc {f}}}$ and also varied the environmental parameter (sucrose concentration) to change the mode of propagation from pulled to pushed.",
"Because the computational growth model in Ref.",
"[32] showed excellent agreement with the experimental data, we used the model instead of the actual data to compare the regions in the parameter space occupied by the three classes of waves.",
"This model is described in detail in Ref.",
"[32], but is briefly summarized below.",
"The expansions occur in a one-dimensional metapopulation with discrete cycles of migration and growth.",
"The dynamics during the growth cycle is described by the following set of differential equations for the population density $n$ , the glucose concentration $g$ and the sucrose concentration $s$ : $\\begin{aligned}\\frac{dn}{dt} & = \\gamma _{\\mathrm {max}}\\frac{g + g_{\\mathrm {loc}}}{g + g_{\\mathrm {loc}} + k_g}n\\\\\\frac{dg}{dt} & = - Y \\frac{dn}{dt} + nv_s\\frac{s}{s + k_s}\\\\\\frac{ds}{dt} & = - nv_s\\frac{s}{s + k_s},\\\\\\end{aligned}$ where $g_{\\mathrm {loc}}$ is given by $g_{\\mathrm {loc}} = g_{\\mathrm {eff}}v_s\\frac{s}{s + k_s}.$ The behavior of this model is illustrated in Fig.",
"REF B.",
"We found that pulled waves occur for a sucrose concentration between 0 and $0.004\\%$ , semi-pushed waves for a sucrose concentration $0.004$ to $0.4\\%$ , and fully-pushed waves for a sucrose concentration between $0.4\\%$ and $2\\%$ , which was the upper value of the sugar explored in the study; presumably very high concentrations of sucrose become toxic.",
"Thus, semi-pushed waves occur in a substantial part of the parameter space for this experimental population.",
"Overall, we believe all three wave classes could be readily observed in nature, but further empirical work is necessary to test this hypothesis.",
"We also think that this conclusion should hold for physical systems.",
"Indeed, the quadratic $r(n)$ from Eq.",
"(3) corresponds to the quartic potential function $V(n) = -\\frac{d}{dn}(r(n)n)$ , which is a common model for phase transitions.",
"External parameters such as temperature or chemical potential can change $B$ and drive the transition between different wave classes.",
"Since the ranges of $B$ for pulled and semi-pushed waves are the same, so should be the ranges of the external parameter.",
"Therefore, one should be able to observe both types of waves.",
"XIII.",
"Computer simulations In this section, we explain the details of our computer simulations and the subsequent data analysis.",
"Interpretation of the simulations as the Wright-Fisher model with vacancies Deterministic migration between patches followed by the Wright-Fisher sampling provides one of the most efficient ways to simulate population dynamics.",
"In its standard formulation, the Wright-Fisher model cannot simulate population growth because it assumes that the population size is fixed at the carrying capacity.",
"To overcome this difficulty, we generalized the Wright-Fisher model by considering the number of vacancies, i.e.",
"the difference between the carrying capacity $N$ and the total population density $n$ , as the abundance of an additional species.",
"With this modification, the total abundance of the two genotypes can increase at the expense of the number of vacancies.",
"Following Ref.",
"[35], the growth of the population was modeled by introducing a fitness difference between the vacancies and the actual species.",
"Specifically, the fitness of the two genotypes was set to $w_i=1$ and the fitness of the vacancies was set to $w_{\\mathrm {v}}=1-r(n)/(1-n/N)$ .",
"The probability to sample genotype $i$ was then proportional to the ratio of $w_i$ to the mean fitness of the population $\\bar{w}=n/N + w_{\\mathrm {v}}(N-n)/N=1-r(n)$ , which explains why we used $1/(1-r(n)\\tau )$ instead of $1+r(n)\\tau $ in Eq. (11).",
"Simulations of deterministic fronts We also simulated range expansions without demographic fluctuations ($\\gamma _n = 0$ ), but with genetic drift.",
"In these simulations, the total population density was updated deterministically: $n(t+\\tau ,x) = \\left\\lfloor (p_1+p_2)N \\right\\rfloor ,$ where $p_i$ are the same as in Eq.",
"(11), and $\\left\\lfloor y \\right\\rfloor $ denotes the floor function, which is equal to the greast integer less than $y$ .",
"The abundances of the two neutral genotypes were then determined by Binomial sampling with $n(t+\\tau ,x)$ trials and $p_i/(p_1+p_2)$ probability of choosing genotype $i$ .",
"For all simulations $m=0.25$ and $r_0=g_0=0.01$ were used, unless noted otherwise.",
"Boundary and initial conditions The most direct approach to simulating a range expansion is to use a stationary habitat, in which the range expansion proceeds from one end to the other.",
"This approach is however expensive because the computational times grows quadratically with the duration of the simulations.",
"Instead, we took advantage of the fact that all population dynamics are localized to the vicinity of the expansion front and simulated only a region of 300 patches comoving with the expansion.",
"Specifically, every simulation time step, we shifted the front backward if the total population inside the simulation array $n_{\\mathrm {array}}$ exceeded $150N$ , i.e.",
"half of the maximally possible population size.",
"The magnitude of the shift was equal to $\\lfloor (n_{\\mathrm {array}}-150N)/N \\rfloor +1$ .",
"The population density in the patches that were added ahead of the front was set to zero, and the number of the individuals moved outside the box was stored, so that we could compute the total number of individuals $n_{\\mathrm {tot}}$ in the entire population including both inside and outside of the simulation array.",
"Our choice of 300 patches in the simulation array was sufficient to ensure that at least one patch remained always unoccupied ahead of the expansion front and that the patches shifted outside the array were always at the carrying capacity.",
"We initialized all simulations by leaving the right half of the array unoccupied and filling the left half to the carrying capacity.",
"In each occupied patch, we determined the relative abundance of the two neutral genotypes by sampling the binomial distribution with $N$ trials and equal probabilities of choosing each of the genotypes.",
"Duration of simulations and data collection To ensure that we can access the exponential decay of the average heterozygosity, simulations were carried out for $2\\sqrt{N}$ generations for pulled waves and for $N$ generations for pushed waves.",
"Although, for pulled waves, the expected timescale of heterozygosity decay is $\\ln ^3{N}$ , we chose a longer duration of simulations to account for possible deviations from this asymptotic scaling.",
"In all simulations, the minimal simulation time was set to $10^4$ time steps.",
"For each simulation, we saved the total population size $n_{\\mathrm {tot}}$ and the population heterozygosity $h$ at 1000 time points evenly distributed across the simulation time.",
"These were used to compute $\\mathrm {Var}X_{\\mathrm {f}}$ and $H$ by averaging over 1000 independent simulation runs.",
"Computing front velocity The velocity of the front was measured by fitting $n_{\\mathrm {tot}}/N$ to $vt+\\mathrm {const}$ .",
"For this fit, we discarded the first $10\\%$ of the total simulation time (1000 generations for the shortest runs) to account for the transient dynamics.",
"The length of the transient is the largest for pulled waves and is specified by the following result from Ref.",
"[1]: $v(t) = v_{\\mathrm {\\textsc {F}}} \\left[ 1 - \\frac{3}{4r_0 t} + \\mathcal {O}\\left( t^{-3/2} \\right) \\right].$ Thus, discarding time points prior to $t\\sim 1/r_0$ was sufficient to eliminate the transient dynamics in all of our simulations.",
"Computing the diffusion constant of the front To measure $D_{\\mathrm {f}}$ , we discarded early time points as described above and then fitted $\\mathrm {Var}\\lbrace n_{\\mathrm {tot}}/N\\rbrace $ to $2D_{\\mathrm {f}}t+\\mathrm {const}$ .",
"Computing heterozygosity and the rate of its decay For each time point, the heterozygosity $h$ was computed as follows $h(t) = \\frac{1}{300}\\sum _{x}\\frac{2n_1(t,x)n_2(t,x)}{[n_1(t,x)+n_2(t,x)]^2},$ where the sum is over $x$ within the simulation array.",
"The average heterozygosity $H$ was then obtained by averaging over independent simulation runs.",
"To compute $\\Lambda $ we fitted $\\ln H$ to $-\\Lambda t +\\mathrm {const}$ .",
"The transient, non-exponential, decay of $H$ lasted much longer compared to the transient dynamics of $v$ and $\\mathrm {Var}X_{\\mathrm {f}}$ ; in addition, our estimates of $H$ had large uncertainty for large $t$ because only a few simulation runs had non-zero heterozygosity at the final time point.",
"To avoid these sources of error, we restricted the analysis $H(t)$ to $t\\in (t_{\\mathrm {i}},t_{\\mathrm {f}})$ .",
"The value of $t_{\\mathrm {f}}$ was chosen such that at least 50 simulations had non-zero heterozygosity at $t=t_{\\mathrm {f}}$ .",
"The value of $t_{\\mathrm {i}}$ was chosen to maximize the goodness of fit ($R^2$ ) between the fit to $H\\sim e^{-\\Lambda t}$ and the data subject to the constraint that $t_{\\mathrm {f}}-t_{\\mathrm {i}}>1000$ .",
"The latter constraint ensured that we had a sufficient number of uncorrelated data points to carry out the fitting procedure.",
"Computing the scaling exponents for $D_{\\mathrm {f}}$ , $\\Lambda $ , and $v-v_{\\mathrm {d}}$ To quantify the dependence of $D_{\\mathrm {f}}$ , $\\Lambda $ , and $v-v_{\\mathrm {d}}$ on $N$ , we fitted a power-law dependence using linear regression on log-log scale.",
"Because the power-law behavior is only asymptotic and did not match the results for low $N$ , the exponents were calculated using the data only for $N>10^4$ .",
"For the velocity corrections, we also needed to determine the value of $v_{\\mathrm {d}}$ .",
"This was done by maximizing the goodness of fit ($R^2$ ) between the simulation results and theoretical predictions.",
"XIV.",
"Supplemental results and figures In this section, we present additional simulation data that further supports and clarifies the conclusions made in the main text.",
"Of particular interest is the comparison between deterministic and fluctuating fronts and the results for an alternative model of an Allee effect that can describe propagation into a metastable state (strong Allee effect).",
"Figure REF shows that the semi-pushed waves occupy a sizable region in the parameter space for two additional models of an Allee effect: one with predator satiation and one with cooperative breakdown of sucrose by yeast.",
"Figure REF graphically summarizes how the locations of different processes change as waves transition from pulled, to semi-pushed, and to fully-pushed waves.",
"Figure REF shows the data that we used to conclude that fluctuations in semi-pushed waves exhibit different scaling behavior compared to pulled and fully-pushed waves.",
"Figure REF demonstrates that the perturbation theory accurately predicts not only the scaling with $N$ , but also the exact values of $D_{\\mathrm {f}}$ and $\\Lambda $ for fully-pushed waves.",
"The scaling properties of fully-pushed waves that propagate into a metastable state are shown in Fig.",
"REF .",
"This figure also illustrates the transition from pulled to semi-pushed and then to fully-pushed waves in an alternative model of an Allee effect.",
"Figure REF shows that the transition between different wave classes can also be detected from the small corrections to the wave velocity due to demographic fluctuations.",
"Genetic drift in deterministic fronts is examined in Fig.",
"REF , and Fig.",
"REF compares the scaling behavior of $\\Lambda $ with $N$ in deterministic vs. fluctuating fronts.",
"Finally, Fig.",
"REF contrasts the behavior of $\\Delta v$ and $\\Lambda $ in fluctuating vs. deterministic fronts.",
"For $\\Lambda $ , both deterministic and stochastic fronts show a transition between large fluctuations in semi-pushed waves and regular $1/N$ fluctuations in fully-pushed waves.",
"Moreover, both deterministic and stochastic fronts have quite similar values $\\alpha $ for semi-pushed waves.",
"In contrast, the behavior of $\\Delta v$ is qualitatively different.",
"Only stochastic fronts exhibit a transition between large fluctuations and $1/N$ scaling.",
"For deterministic fronts, $\\alpha _{\\mathrm {\\textsc {v}}}$ smoothly decreases with the Allee threshold and does not signal the existence of two types of pushed waves.",
"Thus, neglecting front fluctuations has a fundamentally different effect on $\\Lambda $ and $\\Delta v$ .",
"For $\\Lambda $ , the transition between fully-pushed and semi-pushed waves is indicated by the divergence of the integrals in the perturbation theory.",
"Front fluctuations simply modify the cutoff necessary to regularize these integrals and change $\\alpha $ only quantitatively.",
"For $\\Delta v$ , on the other hand, the cutoff is the sole cause of slower expansion velocity of deterministic fronts.",
"For semi-pushed waves, which are sensitive to the dynamics at the front edge, the cutoff qualitatively captures the nontrivial power law dependence of $\\Delta v$ on N. The cutoff, however, cannot account for velocity corrections in fully-pushed waves because $\\Delta v$ arise due to fluctuations throughout the whole front and the contribution from the front edge is negligible."
]
] | 1709.01601 |
[
[
"A 85 kpc Halpha tail behind 2MASX J11443212+2006238 in A1367"
],
[
"Abstract We report the detection of an Halpha trail of 85 kpc projected length behind galaxy 2MASX J11443212+2006238 in the nearby cluster of galaxies Abell 1367.",
"This galaxy was discovered to possess an extended component in earlier, deeper H$\\alpha$ observations carried out with the Subaru telescope.",
"However, lying at the border of the Subaru field, the extended Halpha tail was cut out, preventing the determination of its full extent.",
"We fully map this extent here, albeit the shallower exposure."
],
[
"Introduction",
"A 1.5 $\\rm deg^2$ region of the nearby cluster of galaxies Abell 1367 ($z\\sim 0.0217$ ) was recently surveyed with deep H$\\alpha $ observations using the Subaru telescope (Yagi et al.",
"2017).",
"These observations, at the limiting surface brightness of $2.5\\times 10^{-18}~\\rm erg ~cm^{-2}~sec^{-1}~arcsec^{-2}$ , revealed the presence of H$\\alpha $ tails behind ten out of the cluster's 26 late-type galaxies (LTG) that were surveyed.",
"This indicates that, when observed with sufficiently deep observations, approximately 40 % of all LTGs in this cluster reveal an associated extended trail of H$\\alpha $ , in agreement with the frequency obtained in the Coma cluster by Yagi et al.",
"(2010).",
"This evidence strengthens previous suggestions that a massive infall of gas-rich, star forming galaxies (100-400 galaxies per Gyr, Adami et al.",
"2005, Boselli et al 2008; Gavazzi et al 2013, 2013b) is occurring at the present epoch onto rich clusters of galaxies such as Coma and Virgo.",
"This estimate derives from the combined evidence that ionized tails arise from the ram pressure stripping (Gunn & Gott, 1972) of galaxies crossing the intracluster medium (ICM) at high speed for the first time, and that the gas ablation produced by such interaction proceeds on timescales as short as 100 Myr (Boselli & Gavazzi, 2006, 2014).",
"Figure: Gray-scale representation of the Hα\\alpha emission from CGCG 97-121 (left) and 2MASX J11443212+2006238 (right),with levels chosen to highlight the bright part of the emission.",
"Smoothed (by 2.8 arcsec) contours of the Hα\\alpha -NET image show the low-brightness,extended tail behind 2MASX J11443212+2006238 whose maximum projected length is 85 kpc.",
"North is at the top and East to the left.Table: Observational parameters of the two target galaxies.The galaxy 2MASX J11443212+2006238 lies very close to the edge of the Subaru field (see Fig.",
"9 of Yagi et al.",
"2017), preventing a robust measurement of the extended H$\\alpha $ flux and of its total length.",
"Guided by these observations, we decided to devote two entire (moon free, clear but not photometric) nights of observations at the 2.1m telescope of the San Pedro Martir observatory to the field containing 2MASX J11443212+2006238 and another cluster member: CGCG 97-121 (Zwicky et al.",
"1961-68).",
"In spite of the three-times-shallower data obtained with our smaller telescope, we could detect the full extent of the H$\\alpha $ emission trailing behind 2MASX J11443212+2006238."
],
[
"Observations",
"On the nights of April 24 and 25, 2017, we used the 2.1m telescope at San Pedro Martir to repeatedly observe a 5 x 5 $\\rm arcmin^2$ region of the nearby cluster of galaxies Abell 1367.",
"We used a narrow (80 Å) band filter centered at 6723 Å to detect the H$\\alpha $ emission (at the mean redshift $z\\sim 0.0217$ of the cluster) and a broad band $r$ (Gunn) filter (effective $\\lambda $ 6231 Å, $\\Delta \\lambda \\sim 1200 ~Å$ ) to recover the continuum emission.",
"The field, which contains both 2MASX J11443212+2006238 and CGCG 97 121, was observed with 33 independent pointings of 900 seconds with the ON-band filter, and with 25 pointings of 180 seconds with the broad band (OFF-band) $r$ filter.",
"After flat-fielding, the aligned observations were combined into a final ON-band frame totalling 8.25 hours exposure, and an OFF-band frame of 1.25 hours exposure (details on the reduction procedures of H$\\alpha $ observations can be found in Gavazzi et al.",
"2012).",
"To calibrate the data, which were observed in clear sky conditions, we used the Sloan Digital Sky Survey (SDSS) nuclear fiber spectrum of 2MASX J11443212+2006238 from which we measure an H$\\alpha $ point-like flux (within the central 3 arcsec) of $2.45 \\times 10^{-14}$ $\\rm erg ~cm^{-2}~sec^{-1} Å^{-1}$ .",
"Assuming this flux within the 3\" central aperture of the galaxy, we derive an effective zero point for our observations of -15.29 $\\rm erg ~cm^{-2}~sec^{-1}$ this should be compared with -15.54 obtained in the previous photometric nights and we estimate a limiting surface brightness at H$\\alpha ~\\approx 7.9\\times 10^{-18}~\\rm erg ~cm^{-2}~sec^{-1}~arcsec^{-2}$ ."
],
[
"Results",
"In Table REF we list the celestial coordinates, redshift, total stellar mass, and total equivalent width (E.W.)",
"and flux separately for 2MASX J11443212+2006238, for the diffuse trailing material, and for CGCG 97 121.",
"The flux of the diffuse gas of 2MASX J11443212+20062 (in parenthesis) is measured in a polygonal aperture that fits the last contour level shown in Fig.",
"REF .",
"Stellar masses were derived from the $i$ -band luminosity and the $g-i$ color according to Zibetti et al (2009), assuming a Chabrier initial mass function (IMF) (Chabrier, 2003).",
"Fig.",
"REF shows the continuum subtracted H$\\alpha $ image of our field smoothed by 2.8 arcsec, from which we measured the extent of the diffuse flux associated to 2MASX J11443212+2006238.",
"Assuming a distance from Abell 1367 of 95 Mpc, the tail extends in the NE direction for $\\approx 85\\rm ~kpc$ .",
"Such a distance can be covered in $\\approx 85$ Myr by a galaxy travelling at approximately 1000 $\\rm km~s^{-1}$ , which is the typical velocity of galaxies in clusters.",
"Given that this length is typical in Abell 1367 (the tail length ranges from 15 to 230 kpc in this cluster, Yagi et al.",
"2017), 85-100 Myr can be assumed as the typical time during which the tails remain visible.",
"This time is longer than the typical electron recombination time, recently estimated as 0.2 Myr for UGC 6697 (Consolandi et al.",
"2017) and even shorter for ESO137-001 (Fossati et al.",
"2016), suggesting the existence of some mechanism capable of keeping the gas ionized for a longer period of time in the wakes of the extended tails.",
"However, 100 Myr is a rather short period of time in cosmic history.",
"This suggests that the ram pressure stripping phenomenon occurs during the first pericenter passage over a short period of time, consistent with the evidence of truncated star formation at the southern side (opposite to the tail) of 2MASX J11443212+20062 (Yagi et al 2017), which could indicate quenching timescales as short as 100 Myr (see Boselli et al.",
"2006, 2016a, 2016b).",
"The very existence of 11 tails of similar typical length allows us to roughly estimate the infall rate of galaxies in A1367 as 130 per Gyr, consistent with other existing estimates for Coma and Virgo.",
"This research has made use of the GOLDmine database (Gavazzi et al.",
"2003, 2014b) and of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration."
]
] | 1709.01735 |
[
[
"Compressive Sensing Techniques for Next-Generation Wireless\n Communications"
],
[
"Abstract A range of efficient wireless processes and enabling techniques are put under a magnifier glass in the quest for exploring different manifestations of correlated processes, where sub-Nyquist sampling may be invoked as an explicit benefit of having a sparse transform-domain representation.",
"For example, wide-band next-generation systems require a high Nyquist-sampling rate, but the channel impulse response (CIR) will be very sparse at the high Nyquist frequency, given the low number of reflected propagation paths.",
"This motivates the employment of compressive sensing based processing techniques for frugally exploiting both the limited radio resources and the network infrastructure as efficiently as possible.",
"A diverse range of sophisticated compressed sampling techniques is surveyed and we conclude with a variety of promising research ideas related to large-scale antenna arrays, non-orthogonal multiple access (NOMA), and ultra-dense network (UDN) solutions, just to name a few."
],
[
"Introduction",
"The explosive growth of traffic demand resulted in gradually approaching the system capacity of the operational cellular networks [1].",
"It is widely recognized that substantial system capacity improvement is required for 5G in the next decade [1].",
"To tackle this challenge, a suite of 5G techniques and proposals have emerged, accompanied by: i) increased spectral efficiency relying on multi-antenna techniques and novel multiple access techniques offering more bits/sec/Hz per node; ii) a larger transmission bandwidth relying on spectrum sharing and extension; iii) improved spectrum reuse relying on network densification having more nodes per unit area.",
"Historically speaking, the transmission bandwidth has increased from 200 kHz in the 2G GSM system to 5 MHz in the 3G, to at most 20 MHz in the 4G.",
"Meanwhile, the number of antennas employed also increases from 1 in the 2G/3G systems to 8 in 4G, along with the increasing density of both the base stations (BSs) deployed and users supported.",
"Despite the gradual quantitative increase of bandwidth, number of antennas, density of BS and users, all previous wireless cellular networks have relied upon the classic Nyquist sampling theorem, stating that any bandwidth-limited signal can be perfectly reconstructed, when the sampling rate is higher than twice the signal's highest frequency.",
"However, the emerging 5G solutions will require at least 100 MHz bandwidth, hundreds of antennas, and ultra-densely deployed BSs to support massive users.",
"These qualitative changes indicate that applying Nyquist's sampling theorem to 5G techniques reminiscent of the previous 2G/3G/4G solutions may result in unprecedented challenges: prohibitively large overheads, unaffordable complexity, and high cost and/or power consumption due to the large number of samples required.",
"On the other hand, compressive sensing (CS) offers a sub-Nyquist sampling approach to the reconstruction of sparse signals of an under-determined linear system in a computationally efficient manner [2].",
"Given the large bandwidth of next-generation systems and the proportionally high Nyquist-frequency, we arrive at an excessive number of resolvable multipath components, even though only a small fraction of them is non-negligible.",
"This phenomenon inspired us to sample the resultant sparse channel impulse response (CIR) as well as other signals under the framework of CS, thus offering us opportunities to tackle the above-mentioned challenges [2].",
"Figure: Three promising technical directions for 5G.To be more specific, in Section , we first introduce the key 5G techniques, while in Section , we present the concept of CS, where three fundamental elements, four models, and the associated recovery algorithms are introduced.",
"Furthermore, in Sections , and , we investigate the opportunities and challenges of applying the CS techniques to those key 5G solutions by exploiting the multifold sparsity inherent, as briefly presented below: We exploit the CIR-sparsity in the context of massive MIMO systems both for reducing the channel-sounding overhead required for reliable channel estimation, as well as the spatial modulation (SM)-based signal sparsity inherent in massive SM-MIMO and the codeword sparsity of non-orthogonal multiple access (NOMA) in order to reduce the signal detection complexity.",
"We exploit the sparse spectrum occupation with the aid of cognitive radio (CR) techniques and the sparsity of the ultra-wide band (UWB) signal for reducing both the hardware cost as well as the power consumption.",
"Similarly, we exploit the CIR sparsity in millimeter-wave (mmWave) communications for improving the transmit precoding performance as well as for reducing the CIR estimation overhead.",
"Finally, we exploit the sparsity of the interfering base stations (BSs) and of the traffic load in ultra-dense networks (UDN), where sparsity can be capitalized on by reducing the overheads required for inter-cell-interference (ICI) mitigation, for coordinated multiple points (CoMP) transmission/reception, for large-scale random access and for traffic prediction.",
"We believe that these typical examples can further inspire the conception of a plethora of potential sparsity exploration and exploitation techniques.",
"Our hope is that you valued colleague might also become inspired to contribute to this community-effort."
],
[
"Key Technical Directions in 5G",
"The celebrated Shannon capacity formula indicates the total network capacity can be approximated as ${C_{{\\rm {network}}}} \\approx \\sum \\limits _i^I {\\sum \\limits _j^J {{W_{i,j}}{{\\log }_2}\\left( {1 + {\\rho _{i,j}}} \\right)} } ,$ where $i$ and $j$ are the indices of cells and channels, respectively, $I$ and $J$ are the numbers of cells and channels, respectively, $W_{i,j}$ and $\\rho _{i,j}$ are the associated bandwidth and signal-to-interference-plus-noise ratio (SINR), respectively.",
"As shown in Fig.",
"REF at a glance, increasing ${C_{{\\rm {network}}}}$ for next-generation systems relies on 1) achieving an increased spectral efficiency with larger number of channels, for example by spatial-multiplexing MIMO; 2) an increased transmission bandwidth including spectrum sharing and extension; and 3) better spectrum reuse relying on more cells per area for improving the area-spectral-efficiency (ASE).",
"To elaborate a little further: 1) Increased spectral efficiency can be achieved for example: first, massive multi-antenna aided spatial-multiplexing techniques can substantially boost the system capacity, albeit both the channel estimation in massive MIMO [3], [4] and the signal detection of massive spatial modulation (SM)-MIMO [5] remain challenging issues; second, NOMA techniques are theoretically capable of supporting more users than conventional orthogonal multiple access (OMA) under the constraint of limited radio resources, but the optimal design of sparse codewords capable of approaching the NOMA capacity remains an open problem at the time of writing [6].",
"2) Larger transmission bandwidth may be invoked relying on both CR [7], [8] and UWB [10], [9] techniques, both of which can coexist with licenced services under the umbrella of spectrum sharing, where the employment of sub-Nyquist sampling is of salient importance.",
"As another promising candidate, mmWave communications is capable of facilitating high data rates with the aid of its wider bandwidth [1], [12], [11].",
"However, due to the limited availability of hardware at a low cost and owing to its high path-loss, both channel estimation and transmit precoding are more challenging in mmWave systems than those in the existing cellular systems.",
"3) Better spectrum reuse can be realized with the aid of small cells [1], which improves the ASE expressed in bits/sec/Hz/km$^2$ .",
"However, how to realize interference mitigation, CoMP transmission/reception and massive random access imposes substantial challenges [13], [14], [15].",
"Table: Typical CS Models"
],
[
"Compressive Sensing Theory",
"Naturally, most continuous signals from the real world exhibit some inherent redundancy or correlation, which implies that the effective amount of information conveyed by them is typically lower than the maximum amount carried by uncorrelated signals in the same bandwidth [2].",
"This is exemplified by the inter-sample correlation of so-called voiced speech segments, by adjacent video pixels, correlated fading channel envelopes, etc.",
"Hence the number of effective degrees of freedom of the corresponding sampled discrete time signals can be much smaller than that potentially allowed by their dimensions.",
"This indicates that these correlated time-domain (TD) signals typically can be represented by much less samples in the frequency-domain (FD) [2], because correlated signals only have a few non-negligible low-frequency FD components.",
"Just to give a simple example, a sinusoidal signal can be represented by a single non-zero frequency-domain tone after the transformation by the Fast Fourier transformation (FFT).",
"Sometimes this is also referred to as the energy-compaction property of the FFT.",
"Against this background, CS theory has been developed and applied in diverse fields, which shows that the sparsity of a signal can indeed be exploited to recover a replica of the original signal from fewer samples than that required by the classic Nyquist sampling theorem.",
"To briefly introduce CS theory, we consider the sparse signal ${\\bf {x}} \\in \\mathbb {C}^{n\\times 1}$ having the sparsity level of $k$ (i.e., $\\bf {x}$ has only $k\\ll n$ non-zero elements), which is characterized by the measurement matrix of ${\\bf {\\Phi }}\\in \\mathbb {C}^{m \\times n}$ associated with $m\\ll n$ , where ${\\bf {y}} ={\\bf {\\Phi }} {\\bf {x}}\\in \\mathbb {C}^{m \\times 1} $ is the measured signal.",
"In CS theory, the key issue is how to recover $\\bf {x}$ by solving the under-determined set of equations ${\\bf {y}} ={\\bf {\\Phi }} {\\bf {x}}$ , given $\\bf {y}$ and $\\bf {\\Phi }$ .",
"Generally, $\\bf {x}$ may not exhibit sparsity itself, but it may exhibit sparsity in some transformed domain, which is formulated as ${\\bf {x}}= {\\bf {\\Psi }} {\\bf {s}}$ , where $ {\\bf {\\Psi }}$ is the transform matrix and $\\bf {s}$ is the sparse signal associated with the sparsity level $k$ .",
"Hence we can formulate the standard CS Model (1) of Table REF .",
"Additionally, we can infer from the standard CS Model (1) of Table I the equally important Models (2), (3), and (4) of Table REF , which can provide more reliable compression and recovery of sparse signals, when some of the specific sparse properties of practical applications are considered.",
"Specifically, Model (2) is capable of separating multiple sparse signals $\\left\\lbrace {{{\\bf {s}}_p}} \\right\\rbrace _{p = 1}^P$ associated with different measurement matrices $\\left\\lbrace {{{\\bf \\Theta } _p}} \\right\\rbrace _{p = 1}^P$ by recovering the aggregate sparse signal ${\\bf {s}} = {\\left[ {{\\bf {s}}_1^{\\rm {T}},{\\bf {s}}_2^{\\rm {T}}, \\cdots ,{\\bf {s}}_P^{\\rm {T}}} \\right]^{\\rm {T}}}$ ; Model (3) has the potential of improving the estimation performance of $\\bf s$ by exploiting the block sparsity of $\\bf s$ , as shown in Table I; Model (4) is capable of enhancing the estimation performance of $P$ sparse signals $\\left\\lbrace {{{\\bf {s}}_p}} \\right\\rbrace _{p = 1}^P$ , when their identical/partially common sparsity pattern is exploited.",
"Considering the standard CS model, we arrive at the three fundamental elements of CS theory as follows.",
"1) Sparse transformation is essential for CS, since finding a suitable transform matrix $\\bf {\\Psi }$ can efficiently transform the original (non-sparse) signal $\\bf {x}$ into the sparse signal $\\bf {s}$ .",
"2) Sparse signal compression refers to the design of $\\bf {\\Phi }$ or ${\\bf {\\Theta }} = {\\bf {\\Phi \\Psi }}$ .",
"$\\bf {\\Phi }$ should reduce the dimension of measurements, while minimizing the information loss imposed, which can be quantified in terms of the coherence or restricted isometry property (RIP) of $\\bf {\\Phi }$ or ${\\bf {\\Theta }}$ [2].",
"3) Sparse signal recovery algorithms are important for the reliable reconstruction of $\\bf {x}$ or $\\bf {s}$ from the measured signal $\\bf {y}$ .",
"Particularly, the CS algorithms widely applied in wireless communications can be mainly divided into three categories as follows.",
"i) Convex relaxation algorithms such as basis pursuit (BP) as well as BP de-noising (BPDN), and so on, can formulate the CS problem as a convex optimization problem and solve them using convex optimization software like CVX [2].",
"For instance, the CS problem for Model (1) of Table I can be formulated as a Lagrangian relaxation of a quadratic program as ${\\bf {\\hat{s}}} = \\arg \\mathop {\\min }\\limits _{\\bf {s}} {\\left\\Vert {\\bf {s}} \\right\\Vert _1} + \\lambda {\\left\\Vert {{\\bf {y}} - {\\bf {\\Theta s}}} \\right\\Vert _2},$ with ${\\left\\Vert \\cdot \\right\\Vert _1}$ and ${\\left\\Vert \\cdot \\right\\Vert _2}$ being ${l_1}$ -norm and ${l_2}$ -norm operators, respectively, and $\\lambda >0$ , and the resultant algorithms belong to the BPDN family.",
"These algorithms usually require a small number of measurements, but they are complex, e.g., the complexity of BP algorithm is on the order of $O\\left( {{m^2}{n^{3/2}}} \\right)$ [2].",
"ii) Greedy iterative algorithms can identify the support set in a greedy iterative manner.",
"They have a low complexity and fast speed of recovery, but suffer from a performance loss, when the signals are not very sparse.",
"The representatives of these algorithms are orthogonal matching pursuit (OMP), CoSaMP, and subspace pursuit (SP), which have the complexity of $O\\left( kmn \\right)$ [2].",
"iii) Bayesian inference algorithms like sparse Bayesian learning and approximate message passing infer the sparse unknown signal from the Bayesian viewpoint by considering the sparse priori.",
"The complexity of these algorithms varies from individual to individual.",
"For example, the complexity of Bayesian compressive sensing via belief propagation is $O\\left( {n{{\\log }^2}n} \\right)$ [2].",
"Note that, the algorithms mentioned above have to be further developed for Models (2)-(4) of Table I.",
"For example, the group-sparse BPDN, the simultaneous OMP (SOMP), and the group-sparse Bayesian CS algorithms tailored for MMV Model (4) are promising future candidates [2].",
"Since the conception of CS theory in 2004, it has been extensively developed, extended and applied to practical systems.",
"Indeed, prototypes for MIMO radar, CR, UWB, and so on based on CS theory have been reported by Eldar's research group [2].",
"Undoubtedly, the emerging CS theory provides us with a revolutionary tool for reconstructing signals, despite using sub-Nyquist sampling rates [2].",
"Therefore, how to exploit CS theory in the emerging 5G wireless networks has become a hot research topic [3], [4], [5], [7], [8], [10], [11], [12], [13], [14], [15], [9].",
"By exploring and exploiting the inherent sparsity in all aspects of wireless networks, we can create more efficient 5G networks.",
"In the following sections, we will explore and exploit the sparsity inherent in future 5G wireless networks in the context of the three specific technical directions discussed in Section ."
],
[
"Higher Spectral Efficiency",
"The first technical direction to support the future 5G vision is to increase the spectral efficiency, where massive MIMO, massive SM-MIMO and NOMA schemes constitute promising candidates.",
"This section will discuss how to explore and exploit the sparsity inherent in these key 5G techniques."
],
[
"Massive MIMO Schemes",
"Massive MIMO employing hundreds of antennas at the BS are capable of simultaneously serving multiple users at an improved spectral- and the energy-efficiency [3], [4].",
"Although massive MIMO indeed exhibit attractive advantages, a challenging issue that hinders the evolution from the current frequency division duplex (FDD) cellular networks to FDD massive MIMO is the indispensible estimation and feedback of the downlink FDD channels to the transmitter.",
"However, for FDD massive MIMO, the users have to estimate the downlink channels associated with hundreds of transmit and receive antenna pairs, which results in a prohibitively high pilot overhead.",
"Moreover, even if the users have succeeded in acquiring accurate downlink channel state information (CSI), its feedback to the BS requires a high feedback rate.",
"Hence the codebook-based CSI-quantization and feedback remains challenging, while the overhead of analog CSI feedback is simply unaffordable [4].",
"By contrast, in time division duplex (TDD) massive MIMO, the downlink CSI can be acquired from the uplink CSI by exploiting the channel's reciprocity, provided that the interference is also similar at both ends of the link.",
"Furthermore, the pilot contamination may significantly degrade the system's performance due to the limited number of orthogonal pilots, which hence have to be reused in adjacent cells [3].",
"Fortunately, recent experiments have shown that due to the limited number of significant scatterers in the propagation environments and owing to the strong spatial correlation inherent in the co-located antennas at the BS, the massive MIMO channels exhibit sparsity either in the delay domain [3] or in the angular domain or in both [4].",
"For massive MIMO channels observed in the delay domain, the number of paths containing the majority of the received energy is usually much smaller than the total number of CIR taps, which implies that the massive MIMO CIRs exhibit sparsity in the delay domain and can be estimated using the standard CS Model (1) of Table I, where $\\bf s$ is the sparse delay-domain CIR, $\\bf \\Theta $ consists of pilot signals, and $\\bf y$ is the received signal [3].",
"Due to the co-located nature of the antenna elements, the CIRs associated with different transmit and receiver antenna pairs further exhibit structured sparsity, which manifests itself in the block-sparsity Model (3) of [3].",
"Moreover, the BS antennas are usually found at elevated location with much few scatterers around, while the users roam at ground-level and experience rich scatterers.",
"Therefore, the massive MIMO CIRs seen from the BS exhibit only limited angular spread, which indicates that the CIRs exhibit sparsity in the angular domain [4].",
"Due to the common scatterers shared by multiple users close to each other, the massive multi-user MIMO channels further have the structured sparsity and can be jointly estimated using the MMV Model (4) of Table I [4].",
"Additionally, this sparsity can also be exploited for mitigating the pilot contamination in TDD massive MIMO, where the CSI of the adjacent cells can be estimated with the aid of the signal separation Model (2) for further interference mitigation or for multi-point cooperation.",
"Remark: Exploiting the sparsity of massive MIMO channels with the aid of CS theory to reduce the overhead required for channel estimation and feedback are expected to solve various open challenges and constitute a hot topic in the field of massive MIMO [3], [4].",
"However, if the pilot signals of CS-based solutions are tailored to a sub-Nyquist sampling rate, ensuring its compatibility with the existing systems based on the classic Nyquist sampling rate requires further research.",
"Figure: The SM signals in massive SM-MIMO systems are sparse."
],
[
"Massive SM-MIMO Schemes",
"In massive MIMO systems, each antenna requires a dedicated radio frequency (RF) chain, which will substantially increase the power consumption of RF circuits, when the number of BS antennas becomes large.",
"To circumvent this issue, as shown in Fig.",
"REF , the BS of massive SM-MIMO employs hundreds of antennas, but a much smaller number of RF chains and antennas is activated for transmission.",
"Explicitly, only a small fraction of the antennas is selected for the transmission of classic modulated signals in each time slot.",
"For massive SM-MIMO, a 3-D constellation diagram including the classic signal constellation and the spatial constellation is exploited.",
"Moreover, massive SM-MIMO can also be used in the uplink [5], where multiple users equipped with a single-RF chain, but multiple antennas can simultaneously transmit their SM signals to the BS.",
"In this way, the uplink throughput can also be improved by using SM, albeit at the cost of having no transmit diversity gain.",
"This problem can be mitigated by activating a limited fraction of the antennas.",
"Due to the potentially higher number of transmit antennas than the number of activated receive antennas, signal detection and channel estimation in massive SM-MIMO can be a large-scale under-determined problem.",
"The family of optimal maximum likelihood or near-optimal sphere decoding algorithms suffers from a potentially excessive complexity.",
"By contrast, the conventional low-complexity linear algorithms, such as the linear minimum mean square error (LMMSE) algorithm, suffer from the obvious performance loss inflicted by under-determined rank-deficient systems.",
"Fortunately, it can be observed that in the downlink of massive SM-MIMO, since only a fraction of the transmit antennas are active in each time slot, the downlink SM signals are sparse in the signal domain.",
"Hence, we can use the standard CS Model (1) of Table I for developing SM signal detection, where $\\bf {s}$ is the sparse SM signal, $\\bf {\\Theta }$ is the MIMO channel matrix, and $\\bf y$ is the received signal.",
"Moreover, observe in Fig.",
"2 that for the uplink of massive SM-MIMO, each user's uplink SM signal also exhibits sparsity, thus the aggregated SM signal incorporating all of the multiple users' uplink SM signals exhibits sparsity.",
"Therefore, it is expected that by exploiting the sparsity of the aggregated SM signals, we can use the signal separation Model (2) of Table I to develop a low-complexity, high-accuracy signal detector for improved uplink signal detection [5].",
"Remark: The sparsity of SM signals can be exploited for reducing the computational complexity of signal detection at the receiver.",
"To elaborate a little further, channel estimation in massive SM-MIMO is more challenging than that in massive MIMO, since only a fraction of the antennas are active in each time slot.",
"Hence, how to further explore the intrinsic sparsity of massive SM-MIMO channels and how to exploit the estimated CSI associated with the active antennas to reconstruct the complete CSI is a challenging problem requiring further investigations.",
"Figure: SCMA is capable of supporting overloaded transmission by sparse code domain multiplexing."
],
[
"Sparse Codewords in NOMA systems",
"Cellular networks of the first four generations obeyed different orthogonal multiple access (OMA) techniques [6].",
"In contrast to conventional OMA techniques, such as frequency division multiple access (FDMA), time division multiple access (TDMA), and orthogonal frequency division multiple access (OFDMA), NOMA systems are potentially capable of supporting more users/devices by using non-orthogonal resources, albeit typically at the cost of increased receiver complexity.",
"As a competitive NOMA candidate, sparse code multiple access (SCMA) supports the users with the aid of their unique user-specific spreading sequence, which are however non-orthogonal to each other - similar to classic m-sequences, as illustrated in Fig.",
"REF [6].",
"Each codeword exhibits sparsity and represents a spread transmission layer.",
"In the uplink, the BS can then uniquely and unambiguously distinguish the different sparse codewords of multiple users relying on non-orthogonal resources.",
"In the downlink, more than one transmission layers can be transmitted to each of the multiple users with the aid of the above-mentioned non-orthogonal codewords.",
"The SCMA signal detection problem can be readily formulated as the signal separation Model (2) of Table I, where the columns of ${{{\\bf \\Theta } _p}} $ consist of the $p$ th user's codewords, and ${\\bf {s}}_p$ is a vector with 0 and 1 binary values and only one non-zero element.",
"Amongst others, the low-complexity message passing algorithm (MPA) can be invoked by the receiver for achieving a near-maximum-likelihood multi-user detection performance.",
"Remark: The optimal codeword design of SCMA and the associated multi-user detector may be designed with the aid of CS theory for improving the performance versus complexity trade-off [6]."
],
[
"Larger Transmission Bandwidth",
"The second technical direction contributing to the 5G vision is based on the larger transmission bandwidth, where the family of promising techniques includes CR, UWB and mmWave communication.",
"How we might explore and exploit sparsity in these key 5G techniques will be addressed in this section."
],
[
"Cognitive Radio",
"It has been revealed in the open literature that large portions of the licensed spectrum remains under-utilized [7], [8], since the licensed users may not be fully deployed across the licensed territory or might not occupy the licensed spectrum all the time, and guard bands may be adopted by primary users (PUs).",
"Due to the sparse spectrum exploitation, CR has been advocated for dynamically sensing the unused spectrum and for allowing the secondary users (SUs) to exploit the spectrum holes, while imposing only negligible interference on the PUs.",
"However, enabling dynamic spectrum sensing and sharing of the entire spectral bandwidth is challenging, due to the high Nyquist sampling rate for SUs to sense a broad spectrum.",
"To exploit the low spectrum occupancy by the licensed activities, as verified by extensive experiments and field tests [7], the compressive spectrum sensing concept, which can be described by the standard CS Model (1) of Table I, has been invoked for sensing the spectrum at sub-Nyquist sampling rates.",
"In CR networks, every SU can sense the spectrum holes, despite using a sub-Nyquist sampling rate.",
"However, this strategy may be susceptible to channel fading, hence collaborative sensing relying on either centralized or distributed processing has also been proposed [7], [8].",
"Due to the collaborative strategy, the sparse spectrum seized by each SU may share common components, which can be readily described by the MMV Model (4) of Table I to achieve spatial diversity [7].",
"Moreover, integrating a geo-location database into compressive CR is capable of further improving the performance attained [8].",
"Remark: CS-based CR can facilitate the employment of low-speed analog-to-digital-converter (ADC) instead of the high-speed ADC required by conventional Nyquist sampling theory.",
"In closing we mention that in addition to sensing the spectrum holes by conventional CR schemes, Xampling is also capable of demodulating the compressed received signals, provided that their transmission parameters, such as their frame structure and modulation modes are known [2]."
],
[
"Ultra-Wide Band Transmission",
"UWB systems are capable of achieving Gbps data rates in short range transmission at a low power consumption [10], [9].",
"Due to the ultra-wide bandwidth utilized at a low power-density, UWB may coexist with licenced services relying on frequency overlay.",
"Meanwhile, the ultra-short duration of time-hopping UWB pulses enables it to enjoy fine time-resolution and multipath immunity, which can be used for wireless location.",
"According to Nyquist's sampling theorem, the GHz bandwidth of UWB signals requires a very high Nyquist sampling rate, which leads to the requirement of high-speed ADC and to the associated strict timing control at the receiver.",
"This increases both the power consumption and the hardware cost.",
"However, the intrinsic time-domain sparsity of the received line-of-sight (LOS) or non-line-of-sight (NLOS) UWB signals inspires the employment of an efficient sampling approach under the framework of CS, where the sparse UWB signals can be recovered by using sub-Nyquist sampling rates.",
"Moreover, the UWB signals received over multipath channels can also be approximately considered as a linear combination of several signal bases, as in the standard CS Model (1) of Table I, where these signal bases are closely related to the UWB waveform, such as the Gaussian pulse or it derivatives [10], [9].",
"Compared to those users, who only exploit the time-domain sparsity of UWB signals, the latter approach can lead to a higher energy-concentration and to the further improvement of the sparse representation of the received UWB signals, hence enhancing the reconstruction performance of the UWB signals received by using fewer measurements.",
"Besides, CS can be further applied to estimate channels in UWB transmission by formulating it as MMV Model (4) of Table 1, where the common sparsity of multiple received pilot signals is exploited [10].",
"Remark: The sparsity of the UWB signals facilitates the reconstruction of the UWB signals from observations sampled by the low-speed and power-saving ADCs relying on sub-Nyquist sampling.",
"The key challenge is how to extract the complete information characterizing the analog UWB signals from the compressed measurements.",
"Naturally, if the receiver only wants to extract the information conveyed by the UWB signals, it may be capable of directly processing the compressed measurements by skipping the reconstruction of the UWB signals [9]."
],
[
"Millimeter-Wave Communications",
"The crowded microwave frequency band and the growing demand for increased data rates motivated researchers to reconsider the under-utilized mmWave spectrum (30$\\sim $ 100 GHz).",
"Compared to existing cellular communications operating at sub-6 GHz frequencies, mmWave communications have three distinctive features: a) the spatial sparsity of channels due to the high path-loss of NLOS paths, b) the low signal-to-noise-ratio (SNR) experienced before beamforming, and c) the much smaller number of RF chains than that of the antennas due to the hardware constraints in mmWave communications [12], [11].",
"Hence, the spatial sparsity of channels can be readily exploited for designing cost-efficient mmWave communications."
],
[
"Hybrid Analog-Digital Precoding",
"The employment of transmit precoding is important for mmWave MIMO systems to achieve a large beamforming gain for the sake of compensating their high pathloss.",
"However, the practical hardware constraint makes the conventional full-digital precoding in mmWave communications unrealistic, since a specific RF chain required by each antenna in full-digital precoding may lead to an unaffordable hardware cost and to excessive power consumption.",
"Meanwhile, conventional analog beamforming is limited to single-stream transmission and hence fails to effectively harness spatial multiplexing.",
"To this end, hybrid analog-digital precoding relying on a much lower number of RF chains than that of the antennas has been proposed, where the phase-shifter network can be used for partial beamforming in the analog RF domain for the sake of an improved spatial multiplexing [11].",
"The optimal array weight vectors of analog precoding can be selected from a set of beamforming vectors prestored according to the estimated channels.",
"Due to the limited number of RF chains and as a benefit of spatial sparsity of the mmWave MIMO channels, the hybrid precoding can be formulated as a sparse signal recovery problem, which was referred to as spatially sparse precoding [Equ.",
"(18) in 11].",
"This problem can be efficiently solved by the modified OMP algorithm.",
"However, the operation of this CS-based hybrid precoding scheme is limited to narrow-band channels, while practical broadband mmWave channels exhibit frequency-selective fading, which leads to a frequency-dependent hybrid precoding across the bandwidth [11].",
"For practical dispersive channels where OFDM is likely to be used, it is attractive to design different digital precoding/combining matrices for the different subchannels, which may then be combined with a common analog precodering/combining matrix with the aid of CS theory."
],
[
"Channel Estimation",
"Hybrid precoding relies on accurate channel estimation, which is practically challenging for mmWave communications relying on sophisticated transceiver algorithms, such as multiuser MIMO techniques.",
"In order to reduce the training overhead required for accurate channel estimation, CS-based estimation schemes have been proposed in [12], [11] by exploiting the sparsity of mmWave channels.",
"Compared to conventional MIMO systems, channel estimation designed for mmWave massive MIMO in conjunction with hybrid precoding can be more challenging due to the much smaller number of RF chains than that of the antennas.",
"The mmWave massive MIMO flat-fading channel estimation can be formulated as the standard CS Model (1) of Table I [Equ.",
"(24) in 11], where $\\bf s$ is the sparse channel vector in the angular domain, the hybrid precoding and combining matrices as well as the angular domain transform matrix compose $\\bf \\Theta $ .",
"While for dispersive mmWave MIMO channels, the sparsity of angle of arrival (AoA), angle of departure (AoD), and multipath delay indicates that the channel has a low-rank property.",
"This property can be leveraged to reconstruct the dispersive mmWave MIMO CIR, despite using a reduced number of observations [12].",
"Remark: By exploiting the sparsity of mmWave channels, CS can be readily exploited both for reducing the complexity of hybrid precoding and for mitigating the training overhead of channel estimation.",
"However, as to how we can extend the existing CS-based solutions from narrow-band systems to broadband mmWave MIMO systems is still under investigation."
],
[
"Better Spectrum Reuse",
"The third technical direction to realize the 5G vision is to improve the frequency reuse, which can be most dramatically improved by reducing the cell-size [1].",
"Ultra-dense small cells including femocells, picocells, visible-light atto-cells are capable of supporting seamless coverage, in a high energy efficiency, and a high user-capacity.",
"Explicitly, they can substantially decrease the power consumption used for radio access, since the shorter distance between the small-cell BSs and the users reduces the path-loss [13], [14], [15].",
"This section will address how to explore and exploit the sparsity in dense networks under the framework of CS theory.",
"Figure: Sparsity in ultra-dense networks: (a) Sparse interfering BSs; (b) Sparsity of active users can be exploited to reduce the overhead for massive random access; (c) Low-rank property of large-scale traffic matrix facilitates its reconstructionwith reduced overhead to dynamically manage the network."
],
[
"BSs Identification",
"The ultra-dense small cells may impose non-negligible ICI, which significantly degrades the received SINR.",
"Thus, efficient interference cancellation is required for such interference-limited systems.",
"In conventional cellular systems, orthogonal time-, frequency-, and code resources can be used for effectively mitigating the ICI.",
"By contrast, in the ultra-dense small cells, mitigating the ICI in the face of limited orthogonal resources remains an open challenge [13].",
"In the ultra-dense small cells of Fig.",
"REF (a), a user will be interfered by multiple interfering BSs.",
"The actual number of interfering BSs for a certain user is usually small, although the number of available BSs can be large.",
"Hence, the identification of the interfering BSs can be formulated based on the CS Model (1) of Table I, where indices of non-zero elements in $\\bf s$ corresponds to the interfering BSs, $\\bf \\Theta $ consists of the training signals, and $\\bf y$ is the received signal at the users.",
"To identify the interfering BSs, each BS transmits non-orthogonal training signals based on their respective cell identity.",
"Then the users detect both the identity and even the CSI of the interfering BSs from the non-orthogonal received signals at a small overhead.",
"Moreover, the ICI may be further mitigated by using the signal separation Model (2) of Table I via the CoMP transmission, where the interfering BSs becomes the coordinated BSs.",
"Additionally, by exploiting the observations from multiple antennas in the spatial domain and/or frames in the temporal domain, the block-sparsity Model (3) and MMV Model (4) of Table I can be further considered for improved performance [13].",
"Remark: The identification of the interfering BSs can be formulated as a CS problem for reducing the associated overhead, by exploiting the fact that the actual number of BSs interfering with a certain user's reception is usually smaller than the total number of BSs.",
"Under the framework of CS, designing optimal non-orthogonal training signals and robust yet low-complexity detection algorithms for identifying these potential BSs with limited resources are still under investigation."
],
[
"Massive Random Access",
"It is a widely maintained consensus that the Internet of Things (IoT) will lead to a plethora of devices connecting to dense networks for cloud services in 5G networks.",
"However, the conventional orthogonal resources used for multiple access impose a hard user-load limit, which may not be able to cope with the massive connectivity ranging from $10^2/\\rm {km}^2$ to $10^7/\\rm {km}^2$ for the IoT [1].",
"It can be observed for each small-cell BS that although the number of potential users in the coverage area can be large, the proportion of active users in each time-slot is likely to remain small due to the random call initiation attempts of the users accessing typical bursty data services, as shown in Fig.",
"REF (b) [14].",
"In this article, this phenomenon is referred to as the sparsity of traffic, which points in the direction of CS-based massive random access.",
"More particularly, in the uplink, the users transmit their unique non-orthogonal training signals to access the cellular networks.",
"As a result, the small-cell BSs have to detect multiple active users based on the limited non-orthogonal resources.",
"This multi-user detection process can be described by the signal separation Model (2) of Table I [Equ.",
"(2) in 14].",
"Moreover, due to the ultra-dense nature of the small cells, the adjacent small-cell BSs can also receive some common signals, which implies that the adjacent small-cell BSs may share some common sparse components.",
"Were considering small-cell BSs constituted by the remote radio head (RRH) of the cloud radio access network (C-RAN) architecture, this sparse active-user detection carried out at the baseband unit (BBU) can be characterized by the MMV Model (4) of Table I.",
"By exploiting this structured sparsity, it is expected that further improved active user detection performance can be achieved.",
"Remark: The sparsity of traffic in UDN can be exploited for mitigating the access overhead with the aid of CS theory.",
"Compared to the identification of BSs in the downlink, supporting large-scale random access in the uplink is more challenging: 1) Since the number of users is much higher than that of the BSs, the design of non-orthogonal training signals under the CS framework may become more difficult; 2) The centralized cooperative processing may be optimal, but the compression of the feedback required for centralized processing may not be trivial; 3) Distributed processing contributes an alternative technique of reducing the feedback overhead, but the design of efficient CS algorithms remains challenging."
],
[
"Traffic Estimation and Prediction for Energy-Efficient Dense Networks",
"It has been demonstrated that the majority of power consumption for the radio access is dissipated by the BSs, but this issue is more challenging in dense networks [1].",
"To dynamically manage the radio access for the sake of improved energy efficiency, the estimation of traffic load is necessary.",
"However, under the classic Nyquist sampling framework, to estimate the large-scale traffic matrix for UDN, the measurements required as well as the associated storage, feedback, and energy consumption may become prohibitively high.",
"Therefore, it is necessary to explore efficient techniques of estimating the traffic load for dense networks.",
"Experiments have shown that the demand for radio access exhibits the obvious periodic variation on a daily basis and it also has a spatial variation due to human activities .",
"The strong spatio-temporal correlation of traffic load indicates that the indicator matrix of traffic load exhibits a low rank, which inspires us to reconstruct the complete indicator matrix with the aid of sub-Nyquist sampling techniques [2].",
"When partial traffic data is missing, a spatio-temporal Kronecker compressive sensing method may be involved for recovering the traffic matrix as the standard CS Model (1) of Table I [Equ.",
"(15) in 15].",
"This may motivate us to exploit the low-rank property for estimating the complete traffic matrix with a reduced number of observations.",
"Furthermore, if the past history of the traffic load has been acquired, traffic prediction may be obtained by exploiting the low-rank property of the indicator matrix, and then the BSs can dynamically manage the network for the improved energy efficiency.",
"This process is illustrated in Fig.",
"REF (c).",
"To achieve global traffic prediction from the different BSs, the estimate of traffic load sampled by different sensors has to be fed back to the fusion center, which may impose a huge overhead.",
"This challenge may be mitigated by using part of the historic data for traffic prediction and by exploiting the low-rank nature of the indicator matrix.",
"Moreover, since the spatial correlation of traffic load is reduced as a function of the distance of different BSs, using distributed CS-based traffic prediction with limited feedback may become a promising alterative approach to be further studied.",
"Remark: The low-rank nature of the traffic-indicator matrix can be exploited for reconstructing the complete indicator matrix with the aid of sub-Nyquist sampling techniques.",
"In this way, the measurements used for traffic prediction or their feedback to the fusion center can be reduced."
],
[
"Conclusions",
"CS has inspired the entire signal processing community and in this treatise we revisited the realms of next-generation wireless communications technologies.",
"On the one hand, the very wide bandwidth, hundreds of antennas, and ultra-densely deployed BSs to support massive users in those 5G techniques will result in the prohibitively large overheads, unaffordable complexity, high cost and/or power consumption due to the large number of samples required by Nyquist sampling theorem.",
"On the other hand, CS theory has provided a sub-Nyquist sampling approach to efficiently tackle the above-mentioned challenges for these key 5G techniques.",
"We have investigated the exploitation of sparsity in key 5G techniques from three technical directions and four typical models.",
"Furthermore, we have discussed a range of open problems and future research directions from the perspective of CS theory.",
"The theoretical research on CS-based next generation communication technologies has made substantial progress, but its applications in practical systems still have to be further investigated.",
"CS algorithms exhibiting reduced complexity and increased reliability, as well as compatibility with the current systems and hardware platforms constitute promising potential future directions.",
"It may be anticipated that CS will play a critical role in the design of future wireless networks.",
"Hence our hope is that you valued colleague might like to join this community-effort."
]
] | 1709.01757 |
[
[
"Close binary evolution. III. Impact of tides, wind magnetic braking, and\n internal angular momentum transport"
],
[
"Abstract Massive stars with solar metallicity lose important amounts of rotational angular momentum through their winds.",
"When a magnetic field is present at the surface of a star, efficient angular momentum losses can still be achieved even when the mass-loss rate is very modest, at lower metallicities, or for lower-initial-mass stars.",
"In a close binary system, the effect of wind magnetic braking also interacts with the influence of tides, resulting in a complex evolution of rotation.",
"We study the interactions between the process of wind magnetic braking and tides in close binary systems.",
"We discuss the evolution of a 10 M$_\\odot$ star in a close binary system with a 7 M$_\\odot$ companion using the Geneva stellar evolution code.",
"The initial orbital period is 1.2 days.",
"The 10 M$_\\odot$ star has a surface magnetic field of 1 kG.",
"Various initial rotations are considered.",
"We use two different approaches for the internal angular momentum transport.",
"In one of them, angular momentum is transported by shear and meridional currents.",
"In the other, a strong internal magnetic field imposes nearly perfect solid-body rotation.",
"The evolution of the primary is computed until the first mass-transfer episode occurs.",
"The cases of different values for the magnetic fields and for various orbital periods and mass ratios are briefly discussed.",
"We show that, independently of the initial rotation rate of the primary and the efficiency of the internal angular momentum transport, the surface rotation of the primary will converge, in a time that is short with respect to the main-sequence lifetime, towards a slowly evolving velocity that is different from the synchronization velocity.",
"(abridged)."
],
[
"Introduction",
"Many massive stars are in binary systems [8], [20] and a significant fraction of these systems [6] are tight enough that interactions between the components, either by tides, mass transfer episodes, or merging events, should be expected during their main sequence (MS) lifetimes.",
"The impact of tidal torques on the evolution of the primary star in close binaries has been studied recently as a path to trigger strong tidally induced shear mixing [4], [23], and homogeneous evolution [4], [24], especially as formation mechanism for coalescing binary stellar black holes [16], [5], [17].",
"According to recent large surveys, Magnetism in Massive Stars [MiMeS] [31], and B fields in OB Stars [BOB] [11], about 7-10% of OB stars show a surface magnetic field above 100 G [30].",
"These magnetic massive stars pose extremely interesting challenges.",
"One of them is to understand the origin of such magnetic fields.",
"Although it is generally agreed that these are quasi-stable remnants of magnetic fields generated earlier in their evolution [1], the originating mechanisms are currently unknown.",
"Another interesting question is whether such strong surface magnetic fields may impact the evolution of their host star.",
"For example, surface magnetic fields may significantly reduce the mass lost through stellar winds [21].",
"As shown by these authors, this may allow \"heavy\" stellar-mass black holes with masses above $\\sim $ 25 M$_\\odot $ to be produced even at solar metallicity where classical mass loss would prevent such massive black holes from being formed.",
"Such an effect may also explain the occurrence of Pair Instability supernovae at solar or higher metallicities [12].",
"A second important consequence of a surface magnetic field is the wind magnetic braking effect [27], [28].",
"By channeling the matter expelled in the wind to large distances from the star, magnetic field lines may exert a torque at the surface of the star.",
"This latter aspect has been explored for massive stars by [18].",
"These authors showed that, depending on the efficiency of the angular momentum transport inside the star, very different surface enrichments in nitrogen are produced.",
"In the case of solid body rotation, the wind magnetic braking effect produces slowly rotating MS stars with no surface nitrogen enrichment.",
"In models for which angular momentum is transported by meridional currents and shear, wind magnetic braking produces slowly rotating and strongly nitrogen enriched MS stars.",
"In the present work we want to address the question of how a massive star with a strong surface magnetic field would evolve in a close binary system.",
"[22] have already studied such systems but without following the evolution of the detailed structure of the star as we have done here[9] have also already studied such effects in the context of cool Algol systems..",
"They obtain that in certain circumstances “the stellar spin tends to reach a quasi-equilibrium state, where the effects of tides and winds are counteracting each other.”.",
"We confirm this interesting effect in the present work.",
"Since we are following the evolution of the structure of the star, we also study how tides and wind magnetic braking affect the evolutionary tracks, the changes of the surface composition, and the evolution of the orbit simultaneously.",
"We also explore the impact of two different ways of treating the transport of angular momentum in the interiors of stars.",
"In one treatment, we apply the theory devised by [34] and [15], where the angular momentum and chemical species are transported by shear turbulence and meridional currents.",
"In a second treatment, by introducing a large diffusive coefficient $D_{\\Omega }$ in the advecto-diffusive equation, we consider a very efficient transport of angular momentum inside the star that would impose solid-body rotation during the MS phase.",
"In this last type of model, chemical mixing through shear mostly disappears but continues to be driven, to a degree, by meridional currents (see the right panel of Fig. 7).",
"In Sect.",
"2, we use an analytic approach to explore the space of initial conditions, allowing tides and magnetic torque to more or less compensate each other on a timescale shorter than the MS.",
"Numerical models computed using the Geneva stellar evolution code, accounting more consistently for all physical aspects of the problem than done in the analytic approach, are presented in Sect. 3.",
"Their results are compared with the analytic solutions obtained in Sect.",
"2, and various consequences of the interactions of the tidal torques and the wind magnetic braking are discussed for both differentially and solid-body rotating models.",
"Section 4 synthesizes the main conclusions and proposes some future perspectives.",
"The change of the spin angular momentum (assuming solid-body rotation) due to tidal interaction is given by [33] $\\left( {dJ \\over dt} \\right)_{\\rm tide}=-3MR^{2}(\\Omega -\\omega _{\\rm orb})\\left( {GM \\over R^{3}} \\right)^{1/2}[q^{2}\\left( {R\\over a} \\right)^{6}]E_{2}s_{22}^{5/3},$ where $J$ is the total angular momentum of the star, $M$ , $R$ and $\\Omega $ , respectively, are its total mass, radius, and spin angular velocity, $\\omega _{\\rm orb}$ is the orbital angular velocity, $a$ is the separation between the two components of the binary system (we assume a circular orbit), and $E_2$ is the tidal coefficient that can be expressed by $E_2\\sim 10^{-1.37}(R_{\\rm conv}/R)^8$ , with $R_{\\rm conv}$ being the radius of the convective core [32] and $s_{22}=2|\\Omega -\\omega _{\\rm orb}| \\left(R^3 \\over GM\\right)^{1/2}$ [33].",
"In differentially rotating models, $J$ is the angular momentum of the region in the star where the tides deposit or remove angular momentum.",
"This region is assumed to be the outer layers of the star comprising a few percent of the total mass [23].",
"The angular momentum evolution due to magnetic braking can be expressed as [27], [28] $\\left( {dJ \\over {\\rm d} t} \\right)_{\\rm mb}\\simeq \\frac{2}{3}\\dot{M}\\Omega R^{2}[0.29+(\\eta _{\\ast }+0.25)^{1/4}]^{2},$ where $\\dot{M}$ is the mass-loss rate, $\\eta _{\\ast }=\\frac{B_{\\rm eq}^2R^2}{\\dot{M}\\upsilon _\\infty }$ , with $B_{\\rm eq}$ being the equatorial magnetic field, which is equal to one-half the polar field in the case of a dipolar magnetic field aligned with the rotational axis, and $\\upsilon _\\infty $ the terminal wind velocity of the wind.",
"The ratio of the tidal to the magnetic timescale is given by: $\\frac{\\tau _{\\rm tides}}{\\tau _{\\rm mb}}={{(J/{dJ \\over dt} )}_{\\rm tide}\\over {(J/{dJ \\over d t} )}_{\\rm mb} } =\\frac{ \\frac{2}{3}\\dot{M} \\Omega R^{2}[0.29+(\\eta _{\\ast }+0.25)^{1/4}]^{2} }{3MR^{2}|\\Omega -\\omega _{\\rm orb}|\\left(\\frac{GM}{R^{3}}\\right)^{1/2}[q^{2}\\left(\\frac{R}{a}\\right)^{6}]E_{2}s_{22}^{5/3} }.$ We have considered here the absolute value of $\\Omega -\\omega _{\\rm orb}$ because timescales are always positive.",
"Figure: Variation of the timescales for the tidal torques (blue continuous lines)and for the wind magnetic braking (black dashed lines) as a function of the orbital periodfor a 10 M ⊙ _\\odot star with Z=0.007.",
"The initial spin angular velocity of the primary is taken to be equal tof ω orb \\omega _{\\rm orb}.The tidal torque has been estimated assuming that thecompanion is a 7 M ⊙ _\\odot star, that means a mass ratio of the secondary to the primary, qq, equal to 0.7.",
"The quantities have been estimated adopting zero-age mainsequence (ZAMS) values for the 10 M ⊙ _\\odot star.",
"The horizontal magenta linecorresponding to 30 Myr, indicates the typical MS lifetime of a 10 M ⊙ _\\odot star.",
"Left panel:The case of spin-up by the tidal torque.",
"Right panel: the case of spin-down."
],
[
"Order-of-magnitude estimates",
"Let us consider the case of a binary system with a mass of the primary $M_1$ equal to 10 M$_\\odot $ , and a mass ratio, $q=M_2/M_1$ equal to 0.7 (thus a mass of secondary, $M_2$ , equal to 7 M$_\\odot $ ).",
"We consider ZAMS values for the mass-loss rate, stellar radius and radius of the convective core of the 10 M$_\\odot $ model.",
"We adopt a mass-loss rate computed from the relation of [29], with the terminal velocity, $\\upsilon _\\infty =2.6\\ \\upsilon _{\\rm esc}$ , given by those same authors (where $\\upsilon _{\\rm esc}$ is the escape velocity).",
"In this work, we have accounted for the possible quenching of the mass-loss rate by the magnetic field as was done by [21].",
"Kepler's law is used for linking $a$ to the orbital period $P_{\\rm orb}$ .",
"In Eq.",
"(1), we shall use $P_{\\rm orb}$ instead of the orbital angular velocity ($P_{\\rm orb}=2\\pi /\\omega _{\\rm orb}$ ).",
"We express the spin angular velocity $\\Omega $ at the stellar surface as a function of $P_{\\rm orb}$ introducing a parameter $f$ such that $\\Omega = f \\omega _{\\rm orb}= f 2\\pi /P_{\\rm orb}$We note that $f$ has an upper limit given by $\\Omega =\\Omega _{\\rm crit}$ , where $\\Omega _{\\rm crit}$ is the angular velocity of the primary such that the centrifugal acceleration at the equator balances the gravity there.",
"The upper limit of $f$ is $3.1944\\ P_{\\rm orb}$ , with $P_{\\rm orb}$ in days..",
"Using these inputs, we obtain $\\frac{\\tau _{\\rm tides}}{\\tau _{\\rm mb}}={4.948\\ \\ 10^{34} {f \\over P_{\\rm orb}} [0.29+(1.537\\ \\ 10^{4} B_{\\rm eq}^2+0.25)^{1/4}]^2\\over 1.0373\\ \\ 10^{40} {|f-1|^{8/3} \\over P_{\\rm orb}^{20/3}}},$ where $P_{\\rm orb}$ is in days and $B_{\\rm eq}$ in kG.",
"The tidal and magnetic wind torque timescales are plotted in Fig.",
"REF .",
"Let us focus on the left panel corresponding to spin-up cases, and follow the evolution of a star beginning with a spin angular velocity equal to 1% the orbital angular velocity ($f=0.01$ ) and having a 10 kG surface equatorial magnetic field.",
"For all the orbital periods considered in this plot, the tidal torque timescale is much shorter than the magnetic torque (see the dashed horizontal lines) as well as the MS timescale (see the magenta continuous horizontal line).",
"Thus tides will rapidly spin up the star, producing an increase of $f$ .",
"The star will therefore move up in this diagram.",
"The orbital period will also change.",
"On one hand, the system is losing mass, making the orbit larger and increasing $P_{\\rm orb}$ .",
"On the other hand, the tidal torque accelerates the star and thus transfers angular momentum from the orbit to the star.",
"This tends to decrease the separation $a$ and therefore $P_{\\rm orb}$ .",
"At a given point however, the timescale for acceleration by the tides becomes of the same order of magnitude as the timescale for braking by the magnetic winds.",
"From this point on, the surface velocity of the star will remain locked around the value where these two torques compensate each other.",
"We shall call this velocity the equilibrium velocity.",
"For an orbital period equal to 1 day, this will correspond to about 85% of the angular orbital velocity.",
"For an orbital period of 2 days, this equilibrium velocity will be around 50% of the angular orbital velocity.",
"Actually this equilibrium velocity will change since the separation of the two stars will change as a result of tidal torques that will continuously transfer angular momentum from the orbit to the star, compensating for the loss by the winds (magnetic or not).",
"However, these changes are slow as will appear in Sect.",
"3 below.",
"In the case of spin-down (see the right panel of Fig.",
"REF ), tides will cause the star to move vertically, as previously, since in both cases of spin-up and spin-down, the tidal timescale increases.",
"Now, in the spin-down case, the magnetic braking will work together with the tides to slow down the star.",
"At the beginning, tides are stronger than the magnetic torque.",
"However the strength of the tides decreases while that of the magnetic braking remains more or less constant (we note that here it remains strictly constant because we assumed ZAMS values for all the ingredients entering the wind magnetic braking timescale).",
"When the tidal torque becomes zero (i.e., when $\\Omega $ = $\\omega _{\\rm orb}$ ), the magnetic braking is still active and will cause $\\Omega $ to overshoot below $\\omega _{\\rm orb}$ , until a point when the tidal torque becomes again larger than the wind magnetic braking.",
"But this time, since $\\Omega $ will be below $\\omega _{\\rm orb}$ , tides will accelerate the star and thus counteract the braking due to the winds.",
"From this stage on we shall have a situation similar to that discussed in the spin-up case.",
"The star will finally settle around an equilibrium velocity that will be similar to the one reached in the spin-up case.",
"The equilibrium velocity depends on many factors (as the masses and separation of the two components, the tidal forces, the mass loss rates, etc.)",
"and in particular on the strength of the surface magnetic field.",
"A strong field causes the rotation of the star to approach a smaller equilibrium velocity than a weak field.",
"For a quite unrealistic value, $B_{\\rm eq}$ =1000 kG, the equilibrium velocities would be between a few percent and 45% of the orbital velocity for $P_{\\rm orb}$ =2 and 1 day, respectively.",
"In the case where $B_{\\rm eq}= $ 1 kG, tides will provide the main torque since the magnetic braking timescale is of the same order of magnitude as the MS timescale.",
"The surface velocity of the star will converge towards values above 90% of the orbital velocity at $P_{\\rm orb}$ = 1 day and around 70% of the orbital velocity at $P_{\\rm orb}$ = 2 days.",
"In Fig.",
"REF , we have plotted lines corresponding to $\\frac{\\tau _{\\rm tides}}{\\tau _{\\rm mb}}=1$ in the plane surface (equatorial) magnetic field versus stellar equilibrium rotation velocities at the equator.",
"We see that a 10 M$_\\odot $ primary star with an equatorial surface magnetic field of 10 kG, in a binary system with a 7 M$_\\odot $ companion and an orbital period equal to 1 day, will reach, after the convergence process, a surface velocity of about 160 km s$^{-1}$ .",
"When the orbital period increases, rotation will converge towards lower values since the tidal torque is smaller.",
"The same is true when the mass ratio decreasesFor a general $q$ value, Eq.",
"(4) has to be multiplied by a factor equal to 0.215 $(10q(1+q))^{1/3}/q^2$ .. We note that there is no difference in Fig.2 whether a spin-up or a spin-down is considered.",
"Indeed, as mentioned above, in case of spin-down, the magnetic braking will, at a given time, overcome the tidal torque until the surface rotation decreases to a value such that tidal torque will again accelerate the star.",
"Thus we shall be back to a situation of spin-up.",
"The numerical models that are discussed in the following section will show however that the two cases of spin-up and spin-down are not completely equivalent.",
"Figure: This plot allows determination of the rotation velocity of the primary (x-axis) when the equilibrium state is reached (τ tides τ mb =1\\frac{\\tau _{\\rm tides}}{\\tau _{\\rm mb}}=1) forvarious values of the surface magnetic field (y-axis).",
"The relations between these two quantities are given by the lines.",
"For example,the rotation of the primary in a system composed of a 10 and a 7 M ⊙ _\\odot star (blue continuous lines) when the orbital period is 1.1 days, assuming the primaryhas a surface magnetic field of 10 kG, would be around 137 km s -1 ^{-1}.The blue continuous lines are for a mass ratio q=M 2 /M 1 q=M_2/M_1=0.7and the red dashed lines are for a mass ratio q=0.2q=0.2."
],
[
"Tides and magnetic braking; a numerical approach",
"The evolution of a 10 M$_\\odot $ stellar model in a close binary system with a 7 M$_\\odot $ companion is computed with the Geneva stellar evolution code.",
"The 10 M$_\\odot $ star has a surface magnetic field of 1 kG.",
"We assume this value to be constant during the MS phase.",
"An initial orbital period equal to 1.2 days was considered For a system of mass $M_{1}+M_{2}=10 M_{\\odot }+7 M_{\\odot }$ , one has a semi-major axis $a=7.52\\ \\ 10^{11} P_{\\rm orb}^{2/3}$ (in units of cm) where $P_{\\rm orb}$ is in days.. Two different initial rotation rates for the 10 M$_\\odot $ model are considered: a value $\\upsilon _{\\rm ini}$ = 310 km s$^{-1}$ for the spin-down cases and $\\upsilon _{\\rm ini}$ = 60 km s$^{-1}$ for the spin-up cases.",
"We assume the secondary star is a point mass with negligible mass loss during the period considered.",
"We used two different prescriptions for the transport of angular momentum inside the star.",
"We considered the case where a moderately efficient transport occurs, allowing a moderate radial differential rotation to develop inside the star.",
"This case corresponds to the shellular rotation model proposed by [34], [15].",
"The same prescriptions for rotation described by [10] are used to compute these models.",
"In the following we label these models using the capital letter D (for `differential rotation').",
"We also consider the case of solid body rotation.",
"This case corresponds to the case of very efficient internal angular momentum transport.",
"For instance, in these models any change of angular momentum at the stellar surface has an instantaneous impact on the angular momentum distribution in the whole star.",
"For computing these models we used the same prescriptions for rotation as described by [24]This very efficient transport is mediated in the present models through the theory devised by [25], [26] that postulates the activity of an efficient dynamo in differentially rotating radiative layers of the stars.",
"Actually for our purpose, the mechanism responsible for this efficient angular momentum transport is not very important provided that it is efficient enough to impose solid body rotation [24]..",
"In the following, we label these models with an S (for `solid-body rotation').",
"In these models, the mixing of the chemical elements is driven by meridional currents."
],
[
"The orbital evolution",
"In our numerical approach, we account for the evolution of the orbit as a function of time in the following way.",
"The orbital angular momentum of the system is $J_{\\rm orb}=M_{1}M_{2}(\\frac{Ga}{M_{1}+M_{2}})^{1/2}$ where $a$ is the orbital separation assuming a circular orbitThis expression implicitly assumes that there is no wind material retained by the system and participating in the orbital evolution.",
"One assumes the winds are sufficiently fast to not feel any significant gravitational attraction when flowing out of the system..",
"The variation of the orbital separation is given by taking the time derivative of $J_{\\rm orb}$ .",
"We deduce that $\\frac{\\dot{a}}{a}=2\\frac{\\dot{J}_{\\rm orb}}{J_{\\rm orb}}-2\\frac{\\dot{M}_{\\rm 1,wind}}{M_{1}}-2\\frac{\\dot{M}_{\\rm 2, wind}}{M_{2}}+\\frac{\\dot{M}_{\\rm 1,wind}+\\dot{M}_{\\rm 2,wind}}{M_{1}+M_{2}},$ where mass-loss rates due to stellar winds for the two components are $\\dot{M}_{\\rm 1,wind}$ and $\\dot{M}_{\\rm 2,wind}$ , respectively.",
"The secondary star is treated as a point mass and its stellar wind has been neglected ($\\dot{M}_{\\rm 2, wind}=0.0$ ).",
"Therefore, the above equation becomes $\\frac{\\dot{a}}{a}=2\\frac{\\dot{J}_{\\rm orb}}{J_{\\rm orb}}-2\\frac{\\dot{M}_{\\rm 1,wind}}{M_{1}}+\\frac{\\dot{M}_{\\rm 1,wind}}{M_{1}+M_{2}}.$ The variation of the orbital angular momentum is given by $\\dot{J}_{orb}=-\\dot{J}_{tide} + \\dot{J}_{wind}$ , where $\\dot{J}_{tide}$ is given by Eq.",
"(1), and $\\dot{J}_{\\rm wind}= \\dot{M}_{\\rm 1,wind} (\\frac{M_{2}}{M_{1}+M_{2}}a)^{2}\\omega _{\\rm orb}$ .",
"This last quantity accounts for the fact that the matter outflowing from the primary star initially has an angular velocity equal to the orbital velocity of the starActually the wind would also have an angular momentum originating from the spin of the star, but we are only interested here in the orbital angular momentum that is lost by the winds.",
"We note that this term is identical to the last term in Eqs.",
"7 and 11 of [9]..",
"The quantity $(\\frac{M_{2}}{M_{1}+M_{2}}a)^{2}\\omega _{\\rm orb}$ is the specific orbital angular momentum of the primary star.",
"At first sight, it might appear strange that the wind magnetic braking does not appear explicitly in this expression.",
"Actually, $a$ depends on the orbital angular momentum content, not on the spin angular momentum of the stars.",
"Any effect that changes the spins of the star (such as the mass loss with or without magnetic braking) is accounted for through $\\dot{J}_{tide}$ (that scales with $\\Omega -\\omega _{\\rm orb}$ ).",
"Expression (6) can be simplified, replacing $\\omega _{\\rm orb}$ by $[\\frac{G(M_{1}+M_{2})}{a^{3}}]^{1/2}$ , and using the above expression for $\\dot{J}_{\\rm wind}$ , $\\frac{\\dot{a}}{a}=-2\\frac{\\dot{J}_{\\rm tide} }{J_{\\rm orb}}-\\frac{\\dot{M}_{\\rm 1,wind}}{M_{1}+M_{2}}.$ For computing $\\dot{M}_{\\rm 1,wind}$ , we have accounted for the fact that, as discussed by [21], the mass-loss rate is significantly reduced when a sufficiently strong surface magnetic field is present.",
"The factor $f_{\\rm B}$ equal to $\\dot{M}_{\\rm 1,wind}/\\dot{M}_{\\rm 1,wind}$ (B=0) is shown in the left panel of Fig.",
"REF .",
"We see that the reduction of the mass-loss rate with respect to the non-magnetic case is significant.",
"On the other hand, as explained further below, this effect should not be very critical for what concerns the results obtained in the present work (i.e., ignoring this reduction factor does not significantly change the results of the present paper.)",
"Another effect that might impact the results is the gravitational influence of the secondary.",
"Indeed, the gravity exerted by the companion changes the orbital angular momentum lost by the wind.",
"For example, some material can be accreted by the secondary, or can be trapped in its gravitational potential well.",
"The secondary can also cause some wind material to have its initial orbital angular momentum decreased or boosted.",
"This issue has been discussed, for example, by [2].",
"They found that the orbital angular momentum that is lost is larger or smaller than that inferred neglecting the presence of the secondary depending on many parameters (such as the mass ratio, the geometry of the surface where matter is injected into the wind, and the injection velocity of wind particles).",
"The average angular momentum of escaped particles may change by factors between 0.5 and 20 (see Table 1 in the above reference).",
"Let us suppose that such an effect would change the loss of orbital angular momentum by a factor 10.",
"In our Eq.",
"7 above, such an effect would multiply the term dependent upon the mass-loss rate by a factor $\\sim $ 12.",
"We note that this factor is valid only for the cases with no magnetic field, since the work by [2] does not consider this effect.",
"Would such a change have strong consequences for the evolution of the separation?",
"The effect would be limited for the following reason: In Eq.",
"7, the term $-2\\frac{\\dot{J}_{\\rm tide} }{J_{\\rm orb}}$ is in general significantly larger than the term $\\frac{\\dot{M}_{\\rm 1,wind}}{M_{1}+M_{2}}$ .",
"Typically, for a difference $|f-1|$ of 0.1 between the spin and orbital angular velocity (a typical value at the equilibrium stage), and for an orbital period of 1 day, the first term is about 260 times larger than the second one.",
"Increasing the mass loss rate term by a factor 12 would keep the first term more than 22 times larger than the second one, and thus tides would remain the dominant term for the evolution of the separation.",
"Now let us discuss the case with a surface magnetic field attached to the primary as studied hereWe note that we assume here that the secondary has no strong surface magnetic field..",
"In case of magnetic wind, the wind material will be less sensitive to the gravitational attraction of the companion than without magnetic field.",
"Indeed if we approximate the evolution of the wind velocity by a $\\beta $ -law [14] then at a distance of about 6 stellar radii, the wind has nearly reached the escape velocity from the primary, namely a value around 1000 km s$^{-1}$ .",
"The ratio of the kinetic energy to the gravitational potential of the secondary at that point is equal to 2.3.",
"The ratio between the magnetic energy and the kinetic energy is 3460, assuming that the magnetic field at the surface of the star is 1 kG and, that for a dipolar magnetic field, the magnetic field decreases with the distance $r$ as $1/r^3$ .",
"Thus magnetism is largely dominant and likely the gravitational field of the secondary can be neglected.",
"There is however another effect that enters into the game when a strong magnetic field is considered: the matter launched into the magnetized wind will keep the orbital angular velocity of the primary star until it reaches the Alvén radius.",
"The matter launched into the equatorial plan may thus carry away an angular momentum that is increased by a factor very roughly equal to the square of the ratio between the Alfvén radius ($\\sim $ 40 R$_\\odot $ ) and the distance to the center of mass $\\frac{M_{2}}{M_{1}+M_{2}}a$ ($\\sim $ 6 R$_\\odot $ ) with respect to the mass-loss rate term considered in Eq.",
"(7).",
"This factor is equal to about $(40/6)^2=44.4$ .",
"This is an upper limit because not all the mass is launched into the equatorial plane.",
"Along the polar axis for instance, the expression used in the present work would apply.",
"Again, the tidal term would still remain the dominant term, being 260/44.4 $\\sim $ 6 times larger than the mass-loss rate term.",
"Therefore this effect would again have rather limited consequence.",
"Figure: Left: Evolution of the ratio, f B f_{\\rm B} between M ˙ 1, wind \\dot{M}_{\\rm 1,wind} and the mass-loss rate without a magnetic field M ˙ 1, wind (B=0)\\dot{M}_{\\rm 1,wind}(B=0) for spin-down and spin-up cases with differential (D) rotation, tidal forces (T) and wind magnetic braking (MB).",
"Right: Evolution of the separation, expressed in units of solar radii, for spin-down and spin-up cases with both solid (S) body and differential (D) rotation as a function of the spin angular velocity of the primary normalized to the orbital angular velocity.",
"Point A denotes the ZAMS.",
"Tidal torques dominate the orbital evolution from point A to point B in both spin-down and spin-up cases (τ tides <τ mb \\tau _{\\rm tides} < \\tau _{\\rm mb}).",
"At point B, one has that τ tides =τ mb \\tau _{\\rm tides}=\\tau _{\\rm mb} The loss of mass due to stellar winds governs the orbital evolution from point B to point C in spin-down cases (τ tides >τ mb \\tau _{\\rm tides} > \\tau _{\\rm mb}).",
"The wind torques (or magnetic torques) are counteracted by tidal torques from point C to D in spin-down cases and from point B to D in spin-up cases.",
"Point D denotes the beginning of Roche lobe overflow.",
"The red and the green curves show the cases where the redistribution of angular momentum inside the star is due to shear instabilities and meridional currents (cases noted D in the box).",
"The blue and the black curves are cases of solid body rotation (cases noted S in the box) R sun _{\\rm sun} is the solar radius.",
"The dashed vertical lines correspond to perfect synchronization (Ω=ω orb \\Omega =\\omega _{\\rm orb} (magenta line) and to 90% and 110% of the orbital velocity (black lines left and right of magenta line, respectively).Fig.",
"REF shows the variation of the orbital separation as a function of the ratio of the spin angular velocity to the orbital angular velocity for the primary star.",
"The evolution along the horizontal axis is determined by many different processes.",
"The spin surface velocity of the primary star depends on its mass-loss rate, the strength of the surface magnetic field, on the change of the stellar radius, and of the angular momentum transport inside the star.",
"The orbital velocity is linked to changes of the orbital angular momentum through mass loss and redistribution of angular momentum between the primary star and the orbit through tides.",
"The evolution along the vertical axis reflects the exchange of angular momentum between the star and the orbit through tides and the impact of the mass lost by the primary (see Eq. 7).",
"We see that in case of spin-up, the orbital separation always decreases, while in the case of spin-down, the orbital separation first increases and then decreases.",
"In the case of spin-up, tides always counteract the wind magnetic braking process, and thus there is continuous transfer of angular momentum from the orbit to the star.",
"This effect dominates the $\\dot{M}_{\\rm 1,wind}$ in Eq.",
"7, and thus the separation reduces.",
"In the case of spin-down, tides and mass loss with or without magnetic braking begin to act together, spinning down the star.",
"The tides transfer angular momentum from the star to the orbit and thus the separation increases.",
"As already noted above, at a given point, the tidal torque becomes very small or - at synchronization - even disappears.",
"On the other hand, the wind (possibly magnetic) braking process remains.",
"Thus $\\Omega $ evolves below $\\omega _{\\rm orb}$ , until tidal acceleration becomes strong enough to keep $\\Omega $ at a given value that will evolve slowly with time (this happens when the net torque resulting from both the wind braking and the tides has a timescale that becomes longer than the evolutionary timescale).",
"From this point on, the separation decreases.",
"Most of the tracks in Fig.",
"3 end with a nearly vertical drop that corresponds to the phase when the spin angular velocity of the primary star has converged towards the near-equilibrium velocity.",
"We see that this equilibrium velocity is around 80% of the orbital velocity for models with magnetic braking.",
"We see that when magnetic braking is accounted for, the variations of the orbital separation are larger than in the non-magnetic case.",
"This is because more angular momentum is transferred from the orbit to the star when the star loses its angular momentum more efficiently."
],
[
"The evolution of the surface velocities",
"The evolutions of the surface velocities for various 10 M$_\\odot $ primary stars are shown in Fig.",
"REF .",
"The two red curves correspond to the evolution of stars with no interaction with a companion (either because the star is single or in a wide binary system).",
"The model allowing an internal radial differential rotation has a lower surface velocity than the solid-body rotating model.",
"This reflects two processes.",
"First, starting from solid-body rotating ZAMS models, strong meridional currents rapidly remove angular momentum from the outer layers and transport it toward the central regions until some equilibrium situation sets in between this inward transport and the outward transport by the shear [7], [19].",
"Secondly, once this large redistribution is terminated, meridional currents change sign and begin to transport angular momentum from the central regions toward the surface.",
"This compensates somewhat for the loss of angular momentum via the stellar wind.",
"When solid body rotation is considered, the internal redistribution of angular momentum described immediately above obviously cannot occur and thus there is no initial strong decrease of the surface velocity.",
"Comparing the left panel (high initial velocity) to the right panel (moderate initial rotation) of Fig.4, we see that at low velocities the initial drop of the surface velocity in differentially rotating models is much more modest than at high velocities.",
"This is because meridional currents approximately scale with $\\Omega $ .",
"Now let us investigate what happens when the rotation of a single star is slowed by the wind magnetic braking mechanism (blue curves in Fig.",
"4).",
"We see that in all cases, the surface velocity rapidly decreases.",
"The decrease is relatively strong for the fast spinning models and more gentle in the cases of the initially slowly rotating models.",
"This illustrates the fact that a moderate equatorial surface magnetic field (1 kG) can prevent a 10 M$_\\odot $ star from being a fast rotator at the end of the MS phase.",
"When a sufficiently nearby companion is present, tidal forces may spin down or spin up the 10 M$_\\odot $ primary star, depending on its initial angular velocity.",
"In Fig.",
"4, the cases with no magnetic braking are shown by the black curves, those with magnetic braking by the magenta curves.",
"A few interesting points may be noted: Whatever the strength of the surface magnetic field between 0 and 1 kG, and whatever choice has been made between the two options for the internal angular momentum transport, after a rapid transition due to tides, the surface velocity evolves slowly.",
"The beginning of the slow phase is characterized by a surface equatorial velocity of about 170 km s$^{-1}$ in the case of spin down, and of about 120 km s$^{-1}$ in the case of spin up.",
"In the case of spin down, the velocity curves are convex, showing a minimum surface velocity.",
"In the case of spin up, we see a continual increase of the surface velocity as a function of time.",
"Let us now try to understand these features.",
"The first point reflects the fact that, as expected from timescale arguments presented in Sect.",
"2, tides are initially the dominant effect.",
"The beginning of the slow phase would correspond to what we call the equilibrium stage in Sect.",
"2, that is, the stage at which the tidal torque becomes comparable to the wind magnetic torque.",
"Looking at Fig.",
"REF , we see that for a value of $B_{\\rm eq}$ = 1 kG and an orbital period of 1.2 days, the equilibrium velocity is 145 km s$^{-1}$ , which is between the velocities quoted above at the beginning of the slow phase, namely 120 and 170 km s$^{-1}$ for the spin down and spin up cases, respectively.",
"Let us stress here that to obtain Fig.",
"REF , we assumed that the star retains the characteristics it had on the ZAMS all along its evolution, which is obviously a rough assumption.",
"Despite this, we obtain a value comparable to that deduced from the more detailed numerical simulations.",
"The difference of velocities between the spin-down and spin-up cases comes from the fact that these two situations are not exactly symmetric.",
"This is also apparent looking at the different shapes of the curves underlined in the second point mentioned above.",
"A first obvious difference between the spin-up and spin-down cases is of course due to the fact that, depending on the initial rotation rate of the primary, its initial angular momentum is different.",
"Stellar winds (magnetic or not) remove angular momentum from the surface.",
"Internal redistribution of angular momentum will then tap from the internal angular momentum of the star to compensate for these losses at the surface (in general angular momentum is transported from the inner regions to the envelope either by the transport due to magnetic fields in the solid-body rotating models, or by meridional currents in the differentially rotating models).",
"Thus, depending on the initial spin angular momentum content of the star, one can expect differences in the outcomes.",
"Another effect is that in the case of spin down, and only in the case of spin down, the synchronization stage is reached before the near-equilibrium state.",
"At synchronization, only the magnetic braking is active.",
"Thus the equilibrium velocity is approached from a situation where the braking (due to winds, magnetic or not) is larger than the tides.",
"When the timescale for the net braking torque, that is, resulting from the difference between the wind and tidal torques, becomes sufficiently large, the spin velocity decreases slowly.",
"This explains that in the case of spin down, the curves show a minimum value.",
"Why do they increase thereafter?",
"Since angular momentum is continuously transferred from the orbit to the star, the separation decreases.",
"This reinforces the tides that eventually accelerate the star and thus the velocity will slowly increase after that minimum.",
"In contrast, when we have a situation of spin up, the surface equatorial velocity is always increasing.",
"This can be explained by the fact that at the beginning, the accelerating tides dominate the magnetic braking.",
"After, tides are evolving as a result of two counteracting effects.",
"On one side, tides decrease due to the fact the $\\Omega $ approaches $\\omega _{\\rm orb}$ .",
"On the other side, the separation decreases, increasing the tides.",
"This last effect always dominates over the first one, hence the surface velocity is always increasing, preventing a perfect synchronization.",
"From the discussion above we conclude that only spin-down cases show a phase during which the stars are deaccelerated, otherwise stars are always accelerated, even if they possess a strong surface magnetic field.",
"This also implies that the possibility to measure the time variation of the spin periods of stars would be an interesting way to distinguish between the spin-down and spin-up cases, provided of course that the system corresponds to the initial period where the difference arises.",
"We see also that the binary systems in Fig.",
"4 evolve very similarly for the differentially and solid body rotating cases.",
"Thus changing between these two transport mechanisms has little influence on the evolution of the surface velocities."
],
[
"The evolution in the HR diagram",
"Figure REF shows the evolutionary tracks for the same models as those shown in Fig.",
"REF .",
"The fast rotating models with no magnetic braking and no tidal forces (red curves) present quite different behaviors depending on the internal rotation profile.",
"The solid-body rotating model follows a homogeneous evolution, while the differentially rotating model follows a more or less normal evolution (to the red part of the HR diagram).",
"The initially slower-rotating models (see the red curves on the right panel) do not present any strong differences between differential and solid-body rotation.",
"This is because in both cases (solid-body and differentially rotating), the initial velocity considered is below the threshold for obtaining a homogeneous evolution.",
"We just see that the solid-body rotating model is slightly more luminous because it undergoes a slightly more efficient mixing.",
"The non-interacting stars with magnetic braking (see the blue curves) present an interesting behavior: in cases of both high and moderate initial rotation, the models reaching the largest luminosities, that is, with the strongest signs of mixing, are those with an internal differential rotation and no longer those with a solid body rotation, in contrast to models with no magnetic braking.",
"This simply illustrates a result already discussed by [18], showing that in a model with internal differential rotation, any braking at the surface reinforces the differential rotation and thus the shear mixing.",
"In a solid-body rotating star, the braking decreases $\\Omega $ and thus the velocity of the meridional currents, which are the main drivers for the mixing of chemical elements.",
"Thus the solid-body rotating tracks for single stars with magnetic braking are less luminous and less strongly mixed.",
"The black and magenta curves show the results when the star belongs to a close binary system.",
"We see that, depending on the internal angular momentum transport, the tracks are different.",
"They are shifted to higher effective temperatures and luminosities when solid body rotation is accounted for.",
"Thus, in close binaries the situation is qualitatively the same as for single stars.",
"This can be understood by the fact that tides cause these stars to evolve as if they have no braking.",
"Actually, they are even slowly accelerated.",
"This slow acceleration is not able to produce strong gradients in the outer layers and thus no additional mixing appears in the differentially rotating models.",
"There are few differences between the models with wind magnetic braking and those without.",
"As already mentioned above, in close binaries tides dominate (at least for the conditions explored here).",
"This discussion indicates that the study of the evolutionary tracks in close binaries might be an interesting way to constrain the transport mechanisms induced by rotation.",
"At a given age before mass transfer occurs, solid-body rotating models predict a larger luminosity-mass relation.",
"Let us note that considering a binary system allows us to somewhat constrain the age of the system in the sense that a single isochrone should pass through the observed HR diagram positions of the two components of the binary system.",
"Of course, the two components should have different initial masses.",
"In case eclipsing binaries are observed, then the study of the orbit yields the individual masses of the components and thus tracks of the appropriate masses should pass through the two observed positions of the stars in the HR diagram at the same age.",
"This makes the system tightly constrained.",
"Cases with and without wind magnetic braking are predicted to give very similar mass-luminosity relations, a characteristic that might be interesting to check as well.",
"Finally, the situations for spin-down and spin-up are similar as well."
],
[
"The evolution of the surface composition",
"The evolution of the surface enrichment in nitrogen is plotted as a function of the surface velocity in Fig.",
"REF for the different models shown in Fig.",
"REFTo simplify our calculations, we assume here that for those models with a surface magnetic field of fossil origin (i.e., not linked to a dynamo operating at the surface), the stronger magnetic field that is expected in the interior does not suppress either the shear mixing, or the meridional currents.",
"Recent results suggest that this assumption may not be altogether accurate (e.g., Briquet et al.",
"2012).. Let us first discuss the spin-down cases (left panel).",
"The single star models with fast rotation and no magnetic braking (red curves) evolve almost vertically in the nitrogen relative abundance versus surface rotation plot.",
"The differentially rotating model shows the rapid initial decrease in velocity already discussed above.",
"It also reaches a smaller nitrogen enrichment at the end of the MS phase.",
"The corresponding single star models with magnetic braking (blue curves) evolve toward slower rotation as expected.",
"We see that before the end of the MS phase, the differentially rotating model reaches higher nitrogen surface enrichment than the corresponding model with no magnetic braking.",
"As already discussed above, this results from the strong shear produced in the outer layers that triggers a stronger mixing compared to the non-magnetic model.",
"Single stars with wind magnetic braking and internal differential rotation might explain MS stars that are slow rotators and are highly nitrogen enriched, for example, the group 2 stars of [13].",
"In the case of solid body rotation, we have the reverse situation.",
"The model with magnetic braking is much less enriched, because the braking process decreases $\\Omega $ , and decreasing $\\Omega $ reduces the meridional currents that, in these models, are the main drivers of the chemical mixing.",
"The close binary models evolve almost vertically, similar to the non-magnetic single star models.",
"The differentially rotating models reach surface enrichments that are higher, at a given age, than the solid body-rotating cases.",
"This comes from the fact that torques applied at the surface of a star trigger stronger shear when the star rotates differentially.",
"However, in the previous section we saw that the solid-body rotating tracks had higher luminosities than the differentially rotating models, indicating a stronger degree of internal mixing.",
"How can we explain this apparent contradiction?",
"This apparent contradiction comes from the fact that evolution in the HRD depends on the distribution of the elements inside the entire star, and not only at the surface.",
"The solid-body rotating models in binaries (black and magenta dashed curves in Fig.",
"REF ) have significantly larger convective cores than in the analogous differentially rotating models (see the black and magenta continuous curves).",
"Thus, globally, solid-body rotating stars are closer to a homogeneous structure than those that are differentially rotating.",
"Why are the convective cores larger and the surface nitrogen enrichments weaker in solid body rotating models compared to differentially rotating ones?",
"To understand this, let us have a look at the right panel of Fig. 7.",
"This plot shows the variation inside our 10 M$_\\odot $ models (S and D models) of two diffusion coefficients, $D_{\\rm shear}$ and $D_{\\rm eff}$ at an age of 7 Myr.",
"The coefficient $D_{\\rm shear}$ describes the diffusion due to the shear turbulence.",
"The coefficient $D_{\\rm eff}$ describes the mixing process accounting for the effects of a strong horizontal turbulence (responsible for the shellular rotation) and of the meridional currents [3].",
"We see that in the solid body rotating model, the mixing of the elements is entirely due to the action of $D_{\\rm eff}$ .",
"In the differentially rotating model, $D_{\\rm eff}$ dominates the mixing immediately above the convective core (between the mass coordinate 0.33-0.43) where the gradients of chemical composition reduce the shear turbulence, while $D_{\\rm shear}$ dominates the transport in the outer parts of the radiative envelope.",
"The fact that $D_{\\rm eff}$ in the solid body rotating model is larger than in the differentially rotating one explains why the convective core is larger in that model.",
"The convective core in solid body rotating models is more efficiently replenished by fresh fuel present in the radiative envelope.",
"On the other hand, the fact that $D_{\\rm shear}$ in the differentially rotating model is larger than $D_{\\rm eff}$ in the solid body rotating case throughout the external radiative envelope explains why the surface enrichments are larger in the differentially rotating model.",
"When lower initial rotation velocities are considered, we obtain the results shown in the right panel of Fig.",
"REF .",
"All the single star models evolve to the left and all the close binary models evolve to the right.",
"The single star models with no magnetic braking show almost no enrichment.",
"Only the single star model with differential rotation and including wind magnetic braking shows a significant surface enrichment (by a factor of 2.5), although the model began with a modest initial rotation, here around 65 km s$^{-1}$ .",
"The corresponding solid-body rotating model shows no enrichment at all.",
"Differentially rotating close binary models present similar levels of enrichment to the single differentially rotating model with magnetic braking.",
"This is related to the fact that a similar torque (although of different sign and of different origin) results from the wind magnetic braking and the tidal torque is produced during the slow equilibrium phase of the evolution of the surface velocity.",
"Similar torques produce similar gradients of $\\Omega $ and thus trigger similar shear mixing.",
"The binary solid-body rotating analogs shows less enrichment than the differentially rotating models, although the enrichment is significantly larger than expected from single star models.",
"The surface enrichment in those models is mainly due to meridional currents.",
"From the evolution of the surface compositions, we conclude the same as for the HR diagram tracks.",
"The evolution of the nitrogen surface abundances of single stars is sensitive to both the internal angular momentum transport and to the wind magnetic braking.",
"In close binaries, the nitrogen surface abundances are much more sensitive to the internal angular momentum transport than to the wind magnetic braking.",
"The variety of outputs obtained makes the interpretation of the positions of the stars in the nitrogen surface-enrichments versus surface-velocity plane rather complicated.",
"The above discussion shows that, for a given initial mass at a given initial metallicity, the position depends at least on the initial rotation, the age, the strength of the wind magnetic braking, the strength of the tidal torques and the efficiency of the internal angular momentum transport.",
"The present numerical experiments show that few parts of that plane cannot be reached by considering some special initial conditions.",
"Population synthesis models accounting in a proper way for the distribution of the initial conditions for all the above quantities are needed to accurately understand whether theory can provide a good fit of reality, at least in a statistical way.",
"We have studied the interactions between tides and wind magnetic braking in both differentially rotating and solid-body rotating stellar models.",
"We show that tides govern the initial phase of the evolution in these systems (at least for reasonable magnetic fields and sufficiently short initial orbital periods).",
"Then a near-equilibrium stage may set in, during which acceleration of the rotation by tides and deceleration by magnetic braking more or less compensate.",
"This quasi-equilibrium may be reached after a short time and in that case it is maintained at least until either the end of the MS phase or until a mass-transfer event occurs.",
"Since during that phase angular momentum is continuously tapped from the orbit, the separation decreases.",
"As a consequence, this increases the accelerating force and thus increases the rotation rate of the star.",
"Close binary evolution may thus produce fast rotating, strongly magnetic main-sequence stars with ages that are well above the timescale for the wind magnetic braking.",
"The evolutionary tracks in the HR diagram of the close binaries do not depend significantly on whether the star undergoes wind magnetic braking or not, but they are sensitive to the efficiency of the internal angular momentum transport.",
"We find that solid-body rotating models show higher luminosity-to-mass ratios than differentially rotating models.",
"Interestingly, we find that solid-body rotating models in close binaries are more globally mixed than analogous differentially rotating ones, but the surface enrichment in nitrogen is larger in differentially rotating models.",
"This difference between the degree of global and surface mixing emerges from the different physics involved in the mixing of the chemical elements in differentially rotating and solid-body rotating models.",
"In differentially rotating models, any change of $\\Omega $ in the outer layers reinforces the shear and thus the mixing in that region.",
"For a similar velocity, solid-body rotation enhances the diffusion coefficient near the convective core, enlarging it.",
"This last effect makes the binary solid-body rotating models more luminous than the differentially rotating models.",
"Of course, a larger domain of the parameter space should be explored.",
"This will be the subject of a forthcoming paper.",
"Observations of the mass-luminosity ratio, together with nitrogen surface enrichment, of close binaries before the first mass transfer would provide interesting clues about the physics of rotation.",
"For those systems with a strong surface magnetic field, these observations would also test whether or not a strong surface magnetic field can suppress the internal mixing of chemical species.",
"As a consequence, confrontation of the predictions of these models using observations of real systems holds significant promise for advancing our understanding of stellar rotation, tidal interaction, and magnetic fields.",
"This work was sponsored by the Swiss National Science Foundation (project number 200020-172505), National Natural Science Foundation of China (Grant No.",
"11463002), the Open Foundation of key Laboratory for the Structure and evolution of Celestial Objects, Chinese Academy of Science (Grant No.",
"OP201405).",
"GAW acknowledges Discovery Grant support from the Natural Sciences and Engineering Research Council (NSERC) of Canada.",
"T.F.",
"acknowledges support from the Ambizione Fellowship of the Swiss National Science Foundation (grant PZ00P2-148123)."
]
] | 1709.01902 |
[
[
"A new perspective on the Fano absorption spectrum in terms of complex\n spectral analysis"
],
[
"Abstract A new aspect of understanding a Fano absorption spectrum is presented in terms of the complex spectral analysis.",
"The absorption spectrum of an impurity embedded in semi-infinite superlattice is investigated.",
"The boundary condition on the continuum causes a large energy dependence of the self-energy, enhances the nonlinearity of the eigenvalue problem of the effective Hamiltonian, yielding several nonanalytic resonance states.",
"The overall spectral features is perfectly reproduced by the direct transitions to these discrete resonance states.",
"Even with a single optical transition path the spectrum exhibits an asymmetric Fano profile, which is enhanced for the transition to the nonanalytic resonance states.",
"Since this is the genuine eigenstates of the total Hamiltonian, there is no ambiguity in the interpretation of the absorption spectrum, avoiding the arbitrary interpretation based on the quantum interference.",
"The spectral change around the exceptional point is well understood when we extract the resonant state component."
],
[
"Introduction",
"The Fano effect is a ubiquitous phenomena in quantum mechanics, recognized as a manifestation of quantum interference [1].",
"In his seminal paper, Fano revealed that the absorption spectrum in the photoionization process of an inner-shell electron shows a characteristic asymmetric spectral profile known as the Fano profile[2], [3], [4], as a result of the interference between the direct photoionization transition and the ionization transition mediated by a resonance state.",
"Since then, a growing number of works have been devoted to study the Fano resonance in various physical systems.",
"However, it has been realized that there are cases that cannot be simply fit to the original interpretation of the interference of multiple transitions to a common continuum.",
"As an example, the absorption spectrum in a quantum well shows a distinct Fano resonance, even though there is no direct transition to the continuum [5].",
"In order to explain the Fano profile for a broader range of physical situations, the phenomenological effective Hamiltonian has been proposed [6], which is represented by a finite non-Hermitian matrix of complex constants.",
"The discrete resonance eigenstates are identified as the eigenstate of the effective Hamiltonian with complex eigenvalues whose imaginary part represents the decay rate of the resonance state.",
"While the idea of the resonance state may go back to the early days of the study of nuclear reactions where the resonance state has been obtained by Feshbach projection method [7], [8], there has recently been much focus on expanding the horizon of quantum mechanics so as to incorporate the irreversible decay process directly into quantum theory[9], [10], [11], [12], [13], [14], [15].",
"In these studies, the starting Hamiltonian itself is taken as a non-Hermitian or a Parity-Time-symmetry (PT-symmetry) Hamiltonian at the beginning[10], [11].",
"Even though the non-Hermitian effective Hamiltonian is useful to reproduce the Fano spectral profile, it is not clear how these matrix elements have been derived from time-reversible microscopic dynamics.",
"Indeed, since the derivation of the effective Hamiltonian usually counts on the Weisskopf-Wigner approximation [16], i.e.",
"Markovian approximation, the validity of the effective Hamiltonian should be reexamined.",
"In addition, the microscopic information of the interaction with the continuum is missing because the effect of the interaction is rewritten into complex constants that are phenomenologically determined.",
"On the other hand, there have been efforts to derive a non-Hermitian effective Hamiltonian from the microscopic total (Hermitian) Hamiltonian with use of the Brillouin-Wigner-Feshbach projection operator method (BWF method)[7], [17], [13], [14], [18], [19], where the effect of the microscopic interaction with the continuum is represented by the energy dependent self-energy function.",
"Prigogine and one of the authors (T.P.)",
"et al.",
"have clarified that the spectrum of the effective Hamiltonian coincides with that of the total Hamiltonian, revealing that the total Hermitian Hamiltonian may have complex eigenvalues due to the resonance singularity if we extend the eigenvector space from the ordinary Hilbert space to the extended Hilbert space, where the Hilbert norm of the eigenvector vanishes[20], [21], [22].",
"It should be emphasized that the complex eigenvalue problem of the effective Hamiltonian becomes nonlinear in this approach in the sense that the effective Hamiltonian depends on its eigenvalue.",
"It has been revealed recently that this nonlinearity of the eigenvalue problem of the effective Hamiltonian causes interesting phenomena, such as the dynamical phase transition[23], bound states in continuum (BIC)[24], [25], as well as non-analytic spectral features [26], [27], [28], and modified time evolution near exceptional points[29].",
"Our aim in this paper is to show that the Fano absorption profile is well explained in terms of the complex spectral analysis, even in the absence of multiple transition paths.",
"As a typical system, we consider the core-level absorption of an impurity atom embedded in a semi-infinite superlattice, where the charge transfer decay following the optical excitation is reflected in the absorption spectrum (see Fig.REF ).",
"The Existence of the boundary with an infinite potential wall at the end of the chain causes striking effects on the resonance states in contrast to the infinite chain case: The self-energy has a strong energy dependence over the entire energy range of the continuum, which enhances the nonlinearity of the eigenvalue problem of the effective Hamiltonian.",
"As a result, even with a single impurity state, there appear several discrete resonance states that are nonanalytic in terms of the coupling constant.",
"We have discovered that the absorption spectrum is essentially determined by a sum of the direct transitions to these discrete resonance states.",
"The spectral profile due to the transition takes an asymmetric Fano shape, even when only a single optical transition channel is present.",
"This is because the transition strength (oscillator strength), which is ordinarily real valued [30], becomes complex as a result of the fact that the resonance state of the total Hamiltonian belongs to the extended Hilbert space.",
"Since our interpretation is based on the genuine eigenstate representation of the total Hamiltonian, there is no ambiguity in the interpretation of the origin of the asymmetry in the absorption spectrum, contrary to the method using quantum interference in terms of the Hilbert space basis.",
"We have also found that because of the nonlinearity in the effective Hamiltonian there appears exceptional points (EP) where two resonance states coalesce in terms of not only energies but also their eigenstates[31].",
"We reveal that the absorption spectrum around the EP shows a broad single peak structure consisting of absorption transitions to the nearby degenerate nonanalytic resonance states.",
"The paper is organized as follows.",
"We introduce our model for a semi-infinite chain including a two-level impurity atom incorporating a single intra-atomic optical transition in Section .",
"The complex eigenvalue problem of the total Hamiltonian is solved in Section and we present the characteristic behaviors of the trajectories of the eigenvalues of the effective Hamiltonian in the complex energy plane.",
"The absorption spectrum is studied in terms of complex spectral analysis in Section , which is followed by some concluding discussions in Section ."
],
[
"Model",
"Our model consists of a semi-infinite one-dimensional superlattice with a two-level impurity atom as shown in Fig.REF , where the two-level atom is located at a distance $n_d \\,a$ from the boundary.",
"In this work we take the lattice constant $a$ as our unit length, i.e.",
"$a=1$ .",
"We first consider a finite chain with the length $N$ , then the Hamiltonian reads $\\hat{H} &= E_{c} | c {\\rangle }{\\langle }c | + E_{d} | d {\\rangle }{\\langle }d | +\\sum _{n=1}^N E_0|n{\\rangle }{\\langle }n| \\\\& - \\frac{B}{2} \\sum _{n=1}^{N-1} \\left( | n+1{\\rangle }{\\langle }n | + {\\rm H.c. }\\right)+ gV \\left( | n_d {\\rangle }{\\langle }d | + {\\rm H.c.} \\right).$ The wavenumber state for the semi-infinite lattice is defined by $| k_j {\\rangle } \\equiv \\sqrt{2\\over N}\\sum _{n=1}^N \\sin ( k_j n) |n {\\rangle }\\;,$ where $k_j$ takes $k_j=\\pi j/ (N+1) \\; (j=1, \\cdots , N)$ under the fixed boundary condition.",
"In the limit $N\\rightarrow \\infty $ , the discrete variable $k_j$ becomes continuous in $0< k <\\pi $ , and the summation becomes an integral over $k$ .",
"Applying the transform for the discrete wave number $k_j$ to the continuous $k$ in the limit $N\\rightarrow \\infty $ , defined by[22] $|k{\\rangle } \\equiv \\left({N\\over \\pi }\\right)^{1/2} |k_j{\\rangle } \\;,$ the continuous unperturbed basis $|k{\\rangle }$ satisfies the orthonormality according to Dirac's delta function ${\\langle }k|k^{\\prime }{\\rangle }=\\delta (k-k^{\\prime })$ and the completeness relation $1=|d{\\rangle }{\\langle }d|+\\int _0^\\pi |k{\\rangle }{\\langle }k| dk \\;.$ With use of this basis, the total Hamiltonian is rewritten as $\\hat{H} = \\hat{H}_0+g \\hat{V} \\;,$ where $\\hat{H}_0&=E_c |c{\\rangle }{\\langle }c| + E_{d} | d {\\rangle }{\\langle } d | + \\int _0^\\pi E_{k} | k {\\rangle }{\\langle } k | dk \\;,\\\\g\\hat{V}&=\\int _0^\\pi gV_{k} \\left( | k \\rangle \\langle d | + |d{\\rangle }{\\langle }k| \\right) dk\\;.$ The energy dispersion of the continuum is given by $E_{k} =E_0-B \\;{\\rm cos} \\;k \\;,$ and the interaction potential $V_k$ in terms of the continuous wave number is given by $V_k\\equiv \\left({2\\over \\pi }\\right)^{1/2}V \\sin (n_d k) \\;.$ Hereafter, we take $E_0=0$ and $B=1$ , as the energy origin and the energy unit, respectively.",
"In this paper, we consider a single intra-atomic optical transition induced by incident light with frequency $\\omega $ under the dipole approximation.",
"The transition operator is given by $\\hat{T} \\equiv \\mu \\bigl ( T_{dc} | d \\rangle \\langle c | + {\\rm H.c.} \\bigr ),$ with a dimensionless coupling constant $\\mu $ , where we have adopted the rotating-wave-approximation (RWA) in the weak coupling case $\\mu \\ll 1$ .",
"Using the first order time-dependent perturbation method for the interaction between light and matter, the absorption spectrum is given by [32], [33] $F(\\omega )&={1\\over \\pi } {\\rm Re}\\int _0^\\infty dt \\, e^{i(\\omega +E_c) t-\\epsilon t} {\\langle }c|\\hat{T} e^{-i\\hat{H} t}\\hat{T}|c{\\rangle } \\\\&=-{\\mu ^2 T_{dc}^2\\over \\pi }\\;{\\rm Im}{\\langle }d| {1\\over \\Omega -\\hat{H} +i\\epsilon } |d{\\rangle } \\;,$ where we denote $\\Omega \\equiv \\omega +E_c$ , and we have used Eq.",
"(REF ).In Eq.",
"(REF ), we have dropped the factor $2\\pi /\\hbar $ for convenience.",
"Even though the Green's function method yields an analytical formula for $F(\\Omega )$ for this system as shown in Appendix , we shall present an alternative way to interpret the absorption profile in terms of resonance states, which are considered to be decaying elementary excitations inherent to a given system."
],
[
"Complex eigenvalue problem ",
"We begin our analysis by solving the complex eigenvalue problem of $\\hat{H}$ [21]: $\\hat{H} | \\phi _\\xi {\\rangle } = z_\\xi | \\phi _\\xi {\\rangle }\\;,\\; {\\langle }\\tilde{\\phi }_\\xi | \\hat{H} = z_\\xi {\\langle } \\tilde{\\phi }_\\xi | \\;,$ where the right- and the left-eigenstates, $|\\phi _\\xi {\\rangle }$ and ${\\langle }\\tilde{\\phi }_\\xi |$ , respectively, share the same eigenvalue $z_\\xi $ ; we use a greek index for the (anti-)resonant states with complex eigenvalues and a roman index for the eigenstates (bound or continuum) with real eigenvalues.",
"These eigenstates satisfy biorthonormality and bicompleteness: $\\delta _{\\xi ,\\xi ^{\\prime }}={\\langle }\\tilde{\\phi }_\\xi |\\phi _{\\xi ^{\\prime }}{\\rangle } \\;,\\; 1=\\sum _\\xi |\\phi _\\xi {\\rangle }{\\langle }\\tilde{\\phi }_\\xi | \\;.$ In the present model, just as in the case that we studied in Ref.",
"[33], the bicomplete basis set of the total Hamiltonian is composed of the discrete resonant states, the continuous state, as well as the stable bound states: $\\sum _{i\\in \\rm {R}^{I}} | \\phi _i {\\rangle }{\\langle } \\phi _i | +\\sum _{\\alpha =1}^{n_0-1} |\\phi _\\alpha {\\rangle }{\\langle }\\tilde{\\phi }_\\alpha |+ \\int _0^\\pi dk | \\phi _k {\\rangle }{\\langle } \\tilde{\\phi }_k | = 1 \\;,$ where the first, the second, and the third terms represent the bound states in the first Riemann sheet, the resonance states, and the continuous states, respectively.",
"Since this decomposition of the identity is represented by the eigenstates of the total Hamiltonian, it is essential to understand the absorption spectra in terms of the irreversible decay process emerging from the time-reversible microscopic dynamics.",
"In order to obtain the discrete resonance states, we remove the infinite number of degrees from the problem by using the Brilluoin-Wigner-Feshbach projection method via the projection operators [7]Since the core level is decoupled from $|d{\\rangle }$ and $|k{\\rangle }$ states in $\\hat{H}$ , we here focus on the other terms than the first one in $\\hat{H}_0$ given by Eq.",
"(a) $\\hat{P}^{(d)} \\equiv | d {\\rangle }{\\langle }d | \\;, \\; \\hat{Q}^{(d)} \\equiv 1 - \\hat{P}^{(d)} =\\int _0^\\pi | k {\\rangle }{\\langle } k | dk \\;,$ where $\\hat{P}^{(d)}$ is the projection for the impurity state and $\\hat{Q}^{(d)}$ is its complement.",
"We apply the projection operators to the right-eigenvalue problem (the first equation of Eq.",
"(REF ) ), which then reads $\\hat{P}^{(d)} \\hat{H}_0\\hat{P}^{(d)} |\\phi _\\alpha {\\rangle }+\\hat{P}^{(d)} g\\hat{V}\\hat{Q}^{(d)} |\\phi _\\alpha {\\rangle }&=z_d\\hat{P}^{(d)}|\\phi _\\alpha {\\rangle } \\;,\\\\\\hat{Q}^{(d)} g\\hat{V}\\hat{P}^{(d)} |\\phi _\\alpha {\\rangle }+\\hat{Q}^{(d)} \\hat{H}\\hat{Q}^{(d)} |\\phi _\\alpha {\\rangle }&=z_d\\hat{Q}^{(d)}|\\phi _\\alpha {\\rangle } \\;.$ The $\\hat{Q}^{(d)}|\\phi _\\alpha {\\rangle }$ component is solved in Eq.",
"(b) as $\\hat{Q}^{(d)}|\\phi _\\alpha {\\rangle }={1\\over z_\\alpha -\\hat{Q}^{(d)} \\hat{H}\\hat{Q}^{(d)}}\\hat{Q}^{(d)}\\hat{H}\\hat{P}^{(d)}|\\phi _\\alpha {\\rangle } \\;,$ which, after substitution into Eq.",
"(a), gives the right-eigenvalue problem of the effective Hamiltonian $\\hat{H}_{\\rm eff}(z_\\xi ) \\hat{P}^{(d)}| \\phi _{\\xi } {\\rangle } = z_{\\xi } \\hat{P}^{(d)} | \\phi _{\\xi } {\\rangle } \\;,$ where the effective Hamiltonian $\\hat{H}_{\\rm eff}(z) $ is defined by $\\hat{H}_{\\rm eff}(z)&=\\hat{P}^{(d)}\\hat{H}_0 P^{(d)} \\\\&\\quad + P^{(d)}\\hat{V}\\hat{Q}^{(d)} {g^2\\over z-\\hat{Q}^{(d)} \\hat{H} \\hat{Q}^{(d)}} \\hat{Q}^{(d)} \\hat{V} P^{(d)} \\\\&=\\left[E_d+g^2 \\Sigma ^+(z)\\right]\\hat{P}^{(d)}\\;,$ and the self-energy $\\Sigma ^+(z)$ is given by $\\Sigma ^+(z)&= \\int _0^\\pi dk {V_k^2 \\over (z-E_k)^+} ={2\\over \\pi } \\int _0^\\pi dk {V^2 \\sin ^2 (n_d k) \\over (z-E_k)^+} \\\\&={V^2\\over \\sqrt{z^2-1}} \\left[ 1-\\left(z-\\sqrt{z^2-1}\\right)^{2n_d} \\right] \\;.$ Note that $\\Sigma ^+(z)$ is defined by the Cauchy integral with the branch cut from -1 to 1 in the energy plane; we define $\\Sigma ^+(z)$ by taking the analytic continuation from the upper half energy plane as denoted by the $+$ superscript[21].",
"This is the same self-energy that brings about the bound-state-in-continuum (BIC) as studied in Ref.[24].",
"(See Eq.",
"(4) in [24].)",
"In general, the self-energy has a strong energy dependence near the lower energy bound of the continuum, which results in non-exponential quantum decay [34], [35].",
"In addition, when the potential $V_k$ changes with the wave number $k$ , the self-energy depends on the energy.",
"In the present system, the latter yields oscillations in $\\Sigma ^+(z)$ with energy as shown in Fig.2 in Ref.",
"[24], in contrast to the infinite chain case [33].",
"In fact, with the change of variables $z=-\\cos \\theta $ , the self-energy is written by $\\Sigma ^+(z)={V^2\\over i \\sin \\theta }\\left(1-e^{i 2n_d \\theta } \\right) \\;,$ leading to the BIC points $E_k^{\\rm BIC}=-\\cos \\left({\\pi k \\over n_d}\\right) \\quad (k=1,\\cdots ,n_d-1) \\;,$ where the wave function is confined in between the boundary and the impurity atom [24].",
"Note that Eq.",
"(REF ) is nonlinear in the sense that the operator itself depends on its eigenvalue due to the energy dependance of the self-energy as pointed out in the Introduction[21], [23], [28].",
"It is only when taking into account this nonlinearlity that the spectrum of the effective Hamiltonian $\\hat{H}_{\\rm eff}$ coincides with that of the total Hamiltonian $\\hat{H}$ , and the effective problem is dynamically justified.",
"Thus, the dispersion equation for $\\hat{H}_{\\rm eff}(z)$ reads $&\\eta ^+(z)\\equiv z-E_d -g^2\\Sigma ^+(z)\\\\&=z-E_d- g^2{V^2 \\over \\sqrt{z^2-1}} \\left\\lbrace 1-\\left( z-\\sqrt{z^2-1}\\right)^{2 n_d} \\right\\rbrace =0 \\;,$ yielding $n_d+1$ discrete solutions, among which there are $n_d-1$ resonance state solutions with a negative imaginary part of the eigenvalues and the two real-valued eigenvalues.",
"Here the boundary condition of the infinite potential wall at the one end introduces strong wave number dependence, which allows several discrete resonance states to appear such that the numbers of the resonance states increases as $n_d$ increases.",
"Figure: (Color online) Resonance state solutions z α z_\\alpha of Eq.",
"() for (a) g=0.16g=0.16, (b) g=0.1728≃g EP g=0.1728\\simeq g_{\\rm EP}, and (c) g=0.2g=0.2 for n d =4n_d=4 in the complex energy plane, where the horizontal and the vertical axes denote the real and the imaginary parts of the eigenvalues.The filled (red) circles and (blue) squares denote the discrete resonance states for E d =-0.5E_d=-0.5 and E d =-0.4≃E EP - E_d=-0.4\\simeq E_{\\rm EP}^-, respectively, while the open ones denote the corresponding bare impurity energies E d E_d.The dotted lines are the trajectory of the resonance state solutions with the change of E d E_d.The arrows indicate the direction of the trajectories as E d E_d increases as shown by the double arrows, and thick arrows indicate the entry at E d =-1E_d=-1.As an illustration, we show the discrete resonance state solutions of Eq.",
"(REF ) for the case $n_d=4$ in the complex energy plane in Fig.REF , where the results are shown for (a) $g=0.16$ , (b) $g=0.1728$ , and (c) $g=0.2$ .",
"There are three BIC points at $E_d=\\pm 1/\\sqrt{2}$ and 0, irrespective of the values of $g$ , as seen from Eq.",
"(REF ).",
"We show the three resonance states for $E_d=-0.5$ and those for $E_d=-0.4$ by the filled circles and squares, respectively, while the corresponding bare impurity energies are also plotted by the open circles and squares on the real axis, respectively.",
"Hereafter we use the index $\\alpha $ to distinguish the discrete resonance states.",
"The position of the three resonance states change with $E_d$ as depicted by the dotted lines, which we shall call the trajectories of the resonance states.",
"In Fig.REF , we show the three trajectories as $E_d$ changes from $-1$ to 1.",
"The arrows indicate the directions of the trajectories with increasing of $E_d$ , where the entries of the three trajectories from $E_d< -1$ are indicated by the thick arrows.",
"It is found that the feature of the trajectories are characteristically different for the three cases, distinguished at the critical value $g_{\\rm EP}=0.1728\\cdots $ , which corresponds to the exceptional point.",
"The exceptional point is a special point in parameter space where not only the complex eigenvalues but also the wave functions coalesce [31].",
"The exceptional points are obtained by looking for parameter values satisfying the double root condition for the dispersion equation[26], [15], which requires in addition to Eq.",
"(REF ) that ${d\\over dz}\\eta ^+(z;E_d,g)=1-g^2{d\\over dz}\\Sigma ^+(z)=0 \\;.$ As shown in Fig.REF (b), we numerically obtained an exceptional point at $E_{\\rm EP}^\\pm =\\pm 0.3981\\cdots $ for $g_{\\rm EP}\\simeq 0.1728\\cdots $ , which is in between the BIC points for $E_d$ : $E_1^{\\rm BIC}=-1/\\sqrt{2}< E_{\\rm EP}^-< E_2^{\\rm BIC}=0$ .",
"There are several other exceptional points satisfying the double root conditions of Eqs.",
"(REF ) and (REF ).",
"But in this work we focus on the spectral change when we change $E_d$ from one of the BIC points to the other: from $E_1^{\\rm BIC}=-1/\\sqrt{2}$ to $E_2^{\\rm BIC}=0$ .",
"For $g=0.16(<g_{\\rm EP})$ in Fig.REF (a), the trajectory numbered by (i) is continuously close to the real axis (except near the band edge) which may be identified as the perturbed solution of Eq.",
"(REF ).",
"On the other hand, the other two trajectories numbered by (ii) and (iii) are separated from the trajectory (i).",
"These two solutions are nonanalytic in terms of $g$ : Indeed, the imaginary part of their complex eigenvalues are the order $\\gtrsim O(g) > O(g^2)$ for $g < 1$ .",
"As $E_d$ increases from $E_d=E_1^{\\rm BIC}=-1/\\sqrt{2}$ the trajectories of (i) and (ii) come close, and around $E_d\\simeq E_{\\rm EP}^-$ they repel each other in parallel to the imaginary axis while moving in opposite directions as indicated by the arrows.",
"A similar repulsion happens between the trajectories (i) and (iii) around $E_d\\simeq E_{\\rm EP}^+$ .",
"In the case of $g=0.2(>g_{\\rm EP})$ shown in Fig.REF (c), the three trajectories instead repel each other in parallel to the real axis while traveling in opposite directions, and there is no continuous trajectory close to the real axis.",
"In this case, as $E_d$ increases from $E_d=E_1^{\\rm BIC}=-1/\\sqrt{2}$ , the trajectory (i) starts from the BIC value and shifts away from the real axis, and becomes strongly nonanalytic, while the trajectory (ii) moves in the opposite direction.",
"The case $g\\simeq g_{\\rm EP}$ shown in Fig.REF (b) is the boundary between these two cases.",
"The trajectories (i) and (ii) coalesce at $E_d=E_{\\rm EP}^-=-0.3981\\cdots $ .",
"As will be shown in the next section, it is found that the characteristic difference of these behaviors in the trajectories are clearly reflected in the absorption spectrum.",
"Once we have solved the eigenvalue problem of the effective Hamiltonian, it is straightforward to obtain the complex eigenstate of the total Hamiltonian, by adding the complement component.",
"The explicit representation of the right-eigenstate is obtained as $|\\phi _\\alpha {\\rangle }={\\langle }d|\\phi _\\alpha {\\rangle }\\left( |d{\\rangle }+g \\int _0^\\pi dk {V_k \\over (z-E_k)^+_{z=z_\\alpha }} |k{\\rangle }\\right) \\;,$ where the $+$ sign in the integrand indicates taking the analytical continuation from the upper energy plane to the resonance pole $z_\\alpha $ as mentioned above.",
"The left-eigenstate is similarly obtained as ${\\langle }\\tilde{\\phi }_\\alpha |={\\langle }\\tilde{\\phi }_\\alpha | d{\\rangle } \\left( {\\langle }d|+g \\int _0^\\pi dk {V_k \\over (z-E_k)^+_{z=z_\\alpha }} {\\langle }k|\\right) \\;.$ Note the analytic continuation should be taken in the same direction as $|\\phi _\\alpha {\\rangle }$ , resulting in $\\left(|\\phi _\\alpha {\\rangle }\\right)^\\dagger \\ne {\\langle }\\tilde{\\phi }_\\alpha | \\;.$ The normalization condition for these eigenstates is given by ${\\langle }\\tilde{\\phi }_\\alpha |\\phi _{\\alpha ^{\\prime }}{\\rangle }=\\delta _{\\alpha ,\\alpha ^{\\prime }} \\;,$ so that ${\\langle }\\tilde{\\phi }_\\alpha |d{\\rangle }{\\langle }d|\\phi _\\alpha {\\rangle }=\\left( 1-g^2 {d\\over dz}\\Sigma ^+(z)\\Big |_{z=z_\\alpha } \\right)^{-1} \\;.$ Taking a derivative of $z_\\alpha (E_d)$ as a function of $E_d$ in Eq.",
"(REF ), we find [23] ${d\\over d E_d}z_\\alpha (E_d)={\\langle }\\tilde{\\phi }_\\alpha |d{\\rangle }{\\langle }d|\\phi _\\alpha {\\rangle } \\;.$ It is clear that the normalization constant can be complex, which is rigorously determined by the normalization condition for the eigenstate of the total Hamiltonian.",
"Note that the normalization constant diverges at the EP[26], [28], [29].",
"As will be seen in the next section, this complex normalization constant determines the absorption spectral shape, while the complex eigenvalues determine the peak position and the spectral width.",
"By using the projection operator $\\hat{P}^{(k)}\\equiv |k{\\rangle }{\\langle }k|$ and its complement, we have similarly obtained the right-eigenstate for the continuous state [21] in terms of the continuous wave number as $|\\phi _k{\\rangle }=|k{\\rangle }+{g V_k \\over \\eta _d^+(E_k)} \\left( |d{\\rangle }+g \\int _0^\\pi dk^{\\prime } {V_{k^{\\prime }}\\over E_k-E_{k^{\\prime }}+i\\epsilon }|k^{\\prime }{\\rangle } \\right) \\;,$ where we have defined the delayed analytic continuation of the Green's function as [21] ${1\\over \\eta ^+_d(E_k)}\\equiv {1\\over \\eta ^+(E_k)}\\prod _\\alpha { E_k-z_\\alpha \\over (E_k-z)^+_{z=z_\\alpha }} \\;,$ with the inverse of the Green's function $\\eta ^+(z)$ given by Eq.",
"(REF ).",
"The left-eigenstate for the continuous state is also given by ${\\langle }\\tilde{\\phi }_k|={\\langle }k|+{g V_k \\over \\eta ^-(E_k)} \\left( {\\langle }d|+g \\int _0^\\pi dk^{\\prime } {V_{k^{\\prime }}\\over E_k-E_{k^{\\prime }}-i\\epsilon }{\\langle }k^{\\prime }| \\right) \\;.$ In Eqs.",
"(REF ) and (REF ), $\\epsilon $ is a positive infinitesimal.",
"Together with the two bound eigenstates with real eigenvalues, we come to the decomposition of the identity as shown in Eq.",
"(REF )."
],
[
"Absorption spectrum",
"Applying the bi-completeness Eq.",
"(REF ) to the absorption spectrum Eq.",
"(REF ), we obtain the representation of the absorption spectrum in terms of the complex eigenstates given in the preceding section as $&F(\\Omega )=\\mu ^2 T_{dc}^2 \\sum _{i\\in \\rm {R}^{I}} \\left| {\\langle }d|\\phi _i{\\rangle }\\right|^2 \\delta ( \\Omega - E_i) \\\\& -{ \\mu ^2 T_{dc}^2 \\over \\pi } {\\rm Im} \\left[ \\sum _\\alpha {{\\langle }d|\\phi _\\alpha {\\rangle }{\\langle }\\tilde{\\phi }_\\alpha |d{\\rangle } \\over \\Omega - z_\\alpha + i \\epsilon } +\\int {{\\langle }d|\\phi _k{\\rangle }{\\langle }\\tilde{\\phi }_k|d{\\rangle } \\over \\Omega - E_k+ i \\epsilon } dk\\right] \\;,$ where the first term is attributed to the bound states, while the second and the third terms are attributed to the resonance states and the continuous eigenstates, respectively, which are directly related to the Fano effect.",
"We have shown in Fig.REF the absorption spectra $F(\\Omega )$ by the (black) solid lines and the resonance state component by the (red) dotted lines, corresponding to the case of Fig.REF : (a) $g=0.16$ , (b) $g=0.1728$ , and (c) $g=0.20$ for $n_d=4$ .",
"In each panel, we show the absorption spectra for $E_d=-0.6, -0.5, -0.4, -0.3$ , and $-0.2$ from top to bottom.",
"Figure: (Color online) Absorption spectra F(Ω)F(\\Omega ) corresponding to the case of Fig.",
": (a) g=0.16g=0.16, (b) g=0.1728g=0.1728, and (c) g=0.20g=0.20 for n d =4n_d=4, where the spectrum intensity is divided by μ 2 T dc 2 \\mu ^2 T_{\\rm dc}^2: F(Ω)/μ 2 T dc 2 F(\\Omega )/\\mu ^2 T_{\\rm dc}^2.In each panel, we show the absorption spectra for E d =-0.6,-0.5,-0.4,-0.3E_d=-0.6, -0.5, -0.4, -0.3, and -0.2-0.2 from the top to bottom.The (black) solid lines represent F(Ω)F(\\Omega ) and the (red) dotted lines represent the resonance state component of Eq.",
"().It is striking that the absorption spectra are almost perfectly reproduced only by the direct transitions to the discrete resonance states given by the second term of Eq.",
"(REF ).",
"We also find that the overall features of the spectral change are similar for the three cases: At $E_d=-0.6$ close to the BIC point $E_1^{\\rm BIC}$ , there is a sharp peak with a high energy tail.",
"As $E_d$ increases, the peak position shifts to the high energy side, and the spectrum is broadened.",
"When $E_d$ comes close to $E_{\\rm EP}$ , the spectrum becomes a nearly-symmetric broad peak.",
"And as $E_d$ further increases toward the second BIC point $E_2^{\\rm BIC}$ , the peak position further shifts to the high energy side and it becomes a sharp peak with a low energy tail.",
"Thus, even with a single intra-atomic optical transition, the absorption profile changes as $E_d$ changes.",
"It will be shown below that even though the overall features of the spectral changes are similar for the three cases, the spectral component is very different reflecting the characteristic difference in the complex eigenvalues in Fig.REF .",
"In order to demonstrate this, we decompose the resonant state component, i.e., the second term of Eq.",
"(REF ), into symmetric and antisymmetric parts according to the real and imaginary parts of the normalization constant ${\\langle }d|\\phi _\\alpha {\\rangle }{\\langle }\\tilde{\\phi }_\\alpha |d{\\rangle }$ given by Eq.",
"(REF ).",
"The resonance state component for $|\\phi _\\alpha {\\rangle }$ $({\\langle }\\tilde{\\phi }_\\alpha |)$ is written by $f_\\alpha (\\Omega )\\equiv -{ \\mu ^2 T_{dc}^2 \\over \\pi } {\\rm Im} {{\\langle }d|\\phi _\\alpha {\\rangle }{\\langle }\\tilde{\\phi }_\\alpha |d{\\rangle } \\over \\Omega - z_\\alpha + i \\epsilon }= f^S_\\alpha (\\Omega )+ f^A_\\alpha (\\Omega ) \\;,$ where $f_\\alpha ^S(\\Omega )&\\equiv { \\mu ^2 T_{dc}^2 \\over \\pi }\\cdot {\\gamma _\\alpha \\over (\\Omega -\\varepsilon _\\alpha )^2 +\\gamma _\\alpha ^2}\\cdot {d\\varepsilon _\\alpha \\over d E_d} \\;, \\\\f_\\alpha ^A(\\Omega )&\\equiv { \\mu ^2 T_{dc}^2 \\over \\pi }\\cdot {(\\Omega -\\varepsilon _\\alpha ) \\over (\\Omega -\\varepsilon _\\alpha )^2 +\\gamma _\\alpha ^2} \\cdot {d\\gamma _\\alpha \\over d E_d} \\;,$ and $z_\\alpha =\\varepsilon _\\alpha -i\\gamma _\\alpha $ $(\\gamma _\\alpha >0)$ .",
"The first factors of Eqs.",
"() are the optical transition strengths.",
"The second factors determine the spectral profiles: symmetric and antisymmetric profiles for Eq.",
"(a) and Eq.",
"(b), respectively, whose maximum values are ${1/ 2\\gamma _\\alpha }$ .",
"The factor $d\\gamma _\\alpha /dE_d$ in Eq.",
"(b) determines the degree of the Fano-type asymmetry.",
"In the weak coupling case under the Markovian approximation, where the energy dependence of the self-energy may be neglected, $\\gamma _\\alpha $ does not depend on $E_d$ , so that the resonance state component comes only from $f_\\alpha ^S(\\Omega )$ and becomes a symmetric Lorentzian.",
"Thus, the degree of the asymmetry (DA) for a particular resonance state $|\\phi _\\alpha {\\rangle }$ is evaluated by the ratio of the third factors ${\\rm DA}_\\alpha \\equiv {d \\gamma _\\alpha \\over d\\varepsilon _\\alpha }\\;,$ i.e.",
"the tangent of the trajectories of Fig.REF .",
"The direction along the trajectory determines the direction of the asymmetry.",
"We gain deeper insight by comparing the resonance state component $f_\\alpha (\\Omega )$ with the ordinary Fano profile $f_{\\rm F}(x)&={(x+q)^2\\over x^2+1}=1+{q^2-1 \\over x^2+1}+ {2q \\, x \\over x^2+1} \\;.$ In Eqs.",
"(), with the definition $x\\equiv {\\Omega -\\varepsilon _\\alpha \\over \\gamma _\\alpha }\\;,$ the symmetric and the antisymmetric parts of the resonance state component are rewritten as $f_\\alpha ^S(\\Omega )&=\\left({ \\mu ^2 T_{dc}^2 \\over \\pi \\gamma _\\alpha }\\right) {1 \\over x^2+1} {d\\varepsilon _\\alpha \\over d E_d} \\;, \\\\f_\\alpha ^A(\\Omega )&=\\left({ \\mu ^2 T_{dc}^2 \\over \\pi \\gamma _\\alpha }\\right) {x \\over x^2+1}{d\\gamma _\\alpha \\over d E_d} \\;.$ By comparing Eqs.",
"(REF ) and (), it follows that we may evaluate the $q$ -factor for a resonance state $|\\phi _\\alpha {\\rangle }$ as $q={1\\over {\\rm DA}_\\alpha }\\left[ 1\\pm \\sqrt{1+{\\rm DA}_\\alpha ^2} \\right] \\;,$ where the sign is chosen according to the sign of $d\\gamma _\\alpha /dE_d$ .",
"From this relation, we find $\\left|q\\right| \\rightarrow 1 &\\text{ as }\\left| DA_\\alpha \\right|\\rightarrow \\infty \\;, \\\\\\left| q\\right|\\rightarrow \\infty & \\text{ as } DA_\\alpha \\rightarrow 0\\;.$ It should be noted that, since the relation Eq.",
"(REF ) is determined by solving the complex eigenvalue problem of the total Hamiltonian, our method enables us to rigorously determine the Fano $q$ -factor based on microscopic dynamics.",
"Figure: (Color online) Resonant state components of the absorption spectrum for g=0.2g=0.2 and E d =-0.5E_d=-0.5 in Fig..In the top panel, the three resonance state components f α (Ω)f_\\alpha (\\Omega ) are shown by the thin (red) lines, and the sum of them are depicted by the (black) thick line.In the three lower panels, we have decomposed f α (Ω)f_\\alpha (\\Omega ) into the symmetric f α S (Ω)f^S_\\alpha (\\Omega ) and antisymmetric f α A (Ω)f^A_\\alpha (\\Omega ) components shown by the (blue) dashed lines and (green) dotted lines, respectively.The values of DA α DA_\\alpha and the Fano qq-factors are also shown in the inset.We show in Fig.REF the three resonance state components of the absorption spectrum for $g=0.2$ and $E_d=-0.5$ in Fig.REF .",
"The corresponding three resonance states are shown by the (red) filled circles in Fig.REF (c).",
"In the top panel, the three resonance state components $f_\\alpha (\\Omega )$ are shown by the thin (red) lines, and the sum of them are depicted by the (black) thick line.",
"In the lower panels of Fig.REF , we have decomposed $f_\\alpha (\\Omega )$ into the symmetric $f^S_\\alpha (\\Omega )$ and antisymmetric $f^A_\\alpha (\\Omega )$ parts shown by the (blue) dashed lines and (green) dotted lines, respectively, where $DA_\\alpha $ and the Fano $q$ -factors evaluated by Eq.",
"(REF ) are also shown in the inset.",
"We see that the main contribution comes from the resonance state component $f_{\\rm (i)}(\\Omega )$ .",
"This is because the maximum values of the spectral profile are given by $1/2\\gamma _\\alpha $ as mentioned above, so that the resonance state with the smallest value of $\\gamma _{\\rm (i)}$ is mostly attributed to the spectrum.",
"The spectrum $f_{\\rm (i)}(\\Omega )$ exhibits a sharp Fano profile with $DA_{\\rm (i)}=0.664$ and $q=3.313$ , while the antisymmetric parts overwhelms the symmetric parts for the nonanalytic resonance states (ii) and (iii) with large DA values.",
"Figure: (Color online) The resonance state components in the case of E d =-0.4E_d=-0.4 for (a) g=0.16g=0.16, (b) g=0.1728g=0.1728, and (c) g=0.2g=0.2, corresponding to the case of E d =-0.4E_d=-0.4 in Fig..The depiction of the spectrum decomposition is the same as in Fig..Next we study the difference of the absorption spectrum reflecting the trajectories of the complex eigenvalues shown in Fig.REF .",
"We show in Fig.REF the resonance state components in the case $E_d=-0.4$ for (a) $g=-0.16$ , (b) $g=-0.1728$ , and (c) $g=-0.2$ , corresponding to the case $E_d=-0.4$ in Fig.REF .",
"The three resonance states for each cases are shown by the filled squares in Fig.REF .",
"The depiction of the spectrum decomposition is the same as in Fig.REF .",
"In the weak coupling case (a) $g=0.16 < g_{\\rm EP}$ , where the repulsion of the trajectories occurs in parallel to the real axis, since $\\gamma _{(i)}$ is much smaller than those for the other nonanalytic resonance states, the spectrum is mostly governed by the resonance state component $f_{\\rm (i)}$ .",
"Since ${\\rm DA}_{\\rm (i),(ii)}\\simeq 0$ as seen from Fig.REF (a), the antisymmetric parts are very small, so that $f_{\\rm (i),(ii)}(\\Omega )$ show almost symmetric Lorentzian profiles.",
"The symmetric component $f_{\\rm (ii)}$ is negative, because $d\\varepsilon _{\\rm (ii)}/dE_d <0$ , as indicated by the arrow in Fig.REF (a).",
"On the other hand in the case of strong coupling (c) $g=0.2 >g_{\\rm EP}$ , since the repulsion of the trajectories occurs in parallel to the imaginary axis as shown in Fig.REF , ${\\rm DA}_{\\rm (i),(ii)}$ becomes large, so that the antisymmetric part $f^A_{\\rm (i),(ii)}$ overwhelms the symmetric part $f^S_{\\rm (i),(ii)}$ as shown in Fig.REF (c).",
"The direction of the asymmetric spectral profiles of $f^A_{\\rm (i)}$ and $f^A_{\\rm (ii)}$ are opposite.",
"Because $d \\gamma _\\alpha /d E_d$ has opposite sign between them as seen in Fig.REF (c), hence the spectral components cancel each other except around $\\Omega \\simeq -0.5$ .",
"As a result, the resonance state component $\\sum _\\alpha f_\\alpha (\\Omega )$ shows a broad single Gaussian-type peak: Even though the absorption spectrum is similarly a single peak to Fig.REF (a), its origin is very different.",
"For the case (b) $g=0.1728 \\simeq g_{\\rm EP}$ , it is striking that both the symmetric and antisymmetric components become very large and yet cancel each other between the resonance states (i) and (ii).",
"(See the vertical scale of Fig.REF (b).)",
"This reflects the fact that the normalization constant diverges at the EP, as mentioned above.",
"As a result of this (partial) cancellation, the sum of the resonance components show a single peak."
],
[
"Discussion",
"In this paper, we have presented a new perspective on the absorption spectrum in terms of the complex spectral analysis.",
"We have studied the specific example of the absorption spectrum of an impurity embedded in semi-infinite superlattice.",
"It is found that due to the boundary condition on the lattice, the self-energy has a strong energy dependence over the entire energy range of the continuum, which enhances the nonlinearity of the eigenvalue problem of the effective Hamiltonian, yielding several nonanalytic resonance states with respect to the coupling constant at $g=0$ .",
"It has been revealed that the overall spectral features are almost perfectly determined by the direct transitions to these discrete resonance states, reflecting the characteristic change in the complex energy spectrum of the total Hamiltonian.",
"Even with only a single optical transition channel present, the absorption spectrum due to the transition to the resonance states, in general, takes an asymmetric Fano profile.",
"The asymmetry of the absorption spectrum is exaggerated for the transition to the nonanalytic resonance state.",
"Since this is a genuine eigenstate of the total Hamiltonian, there is no ambiguity in the interpretation of the origin of the asymmetric profile of the absorption spectrum, avoiding the arbitrary interpretation based on the quantum interference.",
"In order to illustrate the physical impact of the nonlinearity in the present system, it is interesting to compare the present results with the absorption spectrum of the unbounded chain system, studied in Ref.",
"[33], in terms of the resonant state representation.",
"For the unbounded chain, the Hamiltonian is represented by $\\hat{\\cal H} &= E_c |c{\\rangle }{\\langle }c| + E_{d} | d {\\rangle }{\\langle } d | + \\int _{-\\pi }^\\pi E_{k} | k {\\rangle }{\\langle } k | dk \\\\&\\quad + \\int _{-\\pi }^\\pi gV \\left( | k \\rangle \\langle d | + |d{\\rangle }{\\langle }k| \\right) dk\\;,$ where the energy dispersion $E_k$ is the same as Eq.",
"(REF ).",
"The important difference from Eq.",
"() is that in this case the interaction potential does not depend on the wave number.",
"Then the effective Hamiltonian is given by $\\hat{\\cal H}_{\\rm eff}(z)=E_d+g^2 \\sigma ^+(z)$ , where the self-energy is given by $\\sigma ^+(z)= {1\\over 2\\pi } \\int _{-\\pi }^\\pi dk {V^2 \\over (z-E_k)^+} ={V^2\\over \\sqrt{z^2-1}} \\;.$ The dispersion equation $\\eta ^+(z)\\equiv z-E_d-g^2\\sigma ^+(z)=0$ reduces to a fourth order polynomial equation, yielding a resonance state and an anti-resonance state in addition to the two bound states that are called as Persistent Bound States: PBS [25], [33].The definition for PBS is introduced in Ref.[25].",
"We show in Fig.REF the trajectory of the resonance state by the dotted line and the solutions for $E_d=-0.9$ and $E_d=-0.6$ , by filled circle and filled square, respectively.",
"In contrast to the semi-infinite chain system, there is only a single resonance state in the infinite chain system, because the self-energy does not exhibit a strong energy dependence within the energy range of the continuum except for the band edges.",
"Figure: (Color online) Resonance state solutions z α z_\\alpha of the infinite chain system for g=0.2g=0.2 for a fixed value of E d =-0.9E_d=-0.9 (filled circle) and E d =-0.6E_d=-0.6 (filled square) in the complex energy plane, where the horizontal and the vertical axes denote the real and the imaginary parts of the eigenvalues.The open circle and square denote the position of the bare impurity energies E d E_d.The dotted lines are the trajectory of the resonance state solutions with the change of E d E_d.The arrows indicate the direction of the solutions along the trajectories as the bare energy increases as shown by the double arrows.The absorption spectrum is calculated in terms of the complex eigenstate of the total Hamiltonian in a similar manner to the preceding section.",
"We have shown in Fig.REF the absorption spectra for $g=0.2$ , where the bare impurity state energies $E_d$ are taken at $E_d=-0.9, -0.6, -0.3$ , and 0.",
"It is seen that the overall spectral features shown by the (black) solid lines are perfectly reproduced by the resonance state components (shown by the (red) dotted lines), just as in Fig.REF ; Significant deviation appears only for the case $E_d=-0.9$ , where the self-energy changes due to the branch point effect, as explaind just below Eq.",
"(REF ).",
"In contrast to Fig.REF , however, the spectral profiles do not change so much with $E_d$ : The single peak just shifts toward the higher energy side as $E_d$ increases, because the self-energy does not have strong energy dependence within the continuum, as mentioned above.",
"Figure: (Color online) Absorption spectra F(Ω)F(\\Omega ) for the infinite chain for g=0.20g=0.20, where the spectra are divided by μ 2 T dc 2 \\mu ^2 T_{\\rm dc}^2: F(Ω)/μ 2 T dc 2 F(\\Omega )/\\mu ^2 T_{\\rm dc}^2.In each panels, we show the absorption spectra for E d =-0.9,-0.6,-0.3,E_d=-0.9, -0.6, -0.3, and 0 from the top to bottom.The solid lines (black) represent F(Ω)F(\\Omega ) and the dotted lines (red) represent the resonance component of Eq.",
"().In order to confirm that the absorption spectrum of the transition to the resonance states takes an asymmetric Fano shape, even with a single optical transition channel, we decompose the resonance state component for $E_d=-0.9$ and $E_d=-0.6$ into symmetric and antisymmetric parts in Fig.REF .",
"We found that the degree of the asymmetry is always non-zero although it is quite small, therefore the absorption spectrum exhibits a Fano-type asymmetry.",
"However, compared to the semi-infinite lattice case, the degree of the asymmetry is very small so that the absorption spectrum takes an almost symmetric Lorentzian shape.",
"Figure: (Color online) The resonance state components in the cases of E d =-0.9E_d=-0.9 and E d =-0.9E_d=-0.9 for g=0.2g=0.2 corresponding to Fig..The depiction of the spectrum decomposition is the same as in Fig..The present method for interpreting the absorption spectrum in terms of the direct transition to the discrete resonance states is an extension of Bohr's idea for quantum jumps between discrete states of matter under optical transitions.",
"In the usual picture, the spectrum due to the quantum jump just exhibits a symmetric Lorentzian profile, whose peak position and width are determined by the excitation energy and the decay rate, i.e.",
"the real and imaginary parts of the complex eigenvalues, respectively.",
"What we have shown here is that the optical spectrum due to the quantum jump between the resonance states can cause much richer spectral features, representing not only their complex eigenvalues but also the peculiar features of the wave functions belonging to the extended Hilbert space.",
"Therefore, we hope that the present method can be applied to give a new understanding for stationary spectroscopies, such as resonance fluorescence, four-wave mixing, etc., in the frequency domain, but also for time-resolved spectroscopies [36], [37], [38].",
"We are very grateful Dr. Tomio Petrosky for many valuable discussions.",
"We also thank K. Mizoguchi and Y. Kayanuma for fruitful comments.",
"This work was partially supported by JSPS Grant-in-Aid for Scientific Research No.16H04003, 16K05481, and 17K05585."
],
[
"Absorption spectrum in terms of Green's function method",
"In this section, we briefly review the Green's function method to evaluate the absorption spectrum Eq.",
"(REF ).",
"Defining the resolvent operator as $\\hat{G}(z)\\equiv {1\\over z-\\hat{H}}={1\\over z-\\hat{H}_0-g\\hat{V}} \\;,$ where $\\hat{H}_0$ and $g\\hat{V}$ are given by Eq().",
"With use of the Dyson's equation, we have the relations $G_{dd}(z)&\\equiv {\\langle }d|\\hat{G}(z)|d{\\rangle }={1\\over z-E_d}+{1\\over z-E_d} \\int dk g V_k G_{kd} \\;, \\\\G_{kd}(z)&\\equiv {\\langle }k|\\hat{G}(z)|d{\\rangle }={1\\over z-E_k}gV_k G_{dd} \\;,$ where $G_{ij}(z)$ is an element of the resolvent.",
"It immediately follows from Eqs.",
"(REF ) that we obtain $G_{dd}(z)={1\\over z-E_d-g^2 \\Sigma ^+(z)} \\;.$ Therefore, the absorption spectrum is obtained by substituting Eq.",
"(REF ) into Eq.",
"(REF ) as $F(\\Omega )=-{\\mu ^2T_{dc}^2\\over \\pi }{g^2 {\\rm Im}\\Sigma ^+(\\Omega ) \\over \\left(\\Omega -E_d-g^2{\\rm Re}\\Sigma ^+(\\Omega ) \\right)^2+g^4 \\left({\\rm Im}\\Sigma ^+(\\Omega )\\right)^2} \\;,$ where the self-energy is defined by Eq.",
"(REF )."
],
[
"Complex eigenvalue problem with the projection method",
"In this section we briefly summarize the complex eigenvalue problem with use of the BWF projection method.",
"One could refer to the literatures for details[21], [13], [23].",
"First, we consider the right-eigenstate for the discrete resonance state.",
"$\\hat{H}|\\phi _\\alpha {\\rangle }=z_\\alpha |\\phi _\\alpha {\\rangle } \\;.$ The application of the projection operators given in Eq.",
"(REF ) to the above leads to $\\hat{P}^{(d)} \\hat{H}_0\\hat{P}^{(d)} |\\phi _\\alpha {\\rangle }+\\hat{P}^{(d)} g\\hat{V}\\hat{Q}^{(d)} |\\phi _\\alpha {\\rangle }&=z_d\\hat{P}^{(d)}|\\phi _\\alpha {\\rangle } \\;,\\\\\\hat{Q}^{(d)} g\\hat{V}\\hat{P}^{(d)} |\\phi _\\alpha {\\rangle }+\\hat{Q}^{(d)} \\hat{H}\\hat{Q}^{(d)} |\\phi _\\alpha {\\rangle }&=z_d\\hat{Q}^{(d)}|\\phi _\\alpha {\\rangle } \\;.$ The $\\hat{Q}^{(d)}|\\phi _\\alpha {\\rangle }$ is solved in Eq.",
"(b) as $\\hat{Q}^{(d)}|\\phi _\\alpha {\\rangle }={1\\over z_\\alpha -\\hat{Q}^{(d)} \\hat{H}\\hat{Q}^{(d)}}\\hat{Q}^{(d)}\\hat{H}\\hat{P}^{(d)}|\\phi _\\alpha {\\rangle } \\;,$ which is substituted to Eq.",
"(a), we then have the eigenvalue problem of the effective Hamiltonian Eq.",
"(REF ), where the effective Hamiltonian is expressed by $\\hat{H}_{\\rm eff}(z)&=\\hat{P}^{(d)}\\hat{H}_0 P^{(d)} \\\\&\\quad + P^{(d)}\\hat{V}\\hat{Q}^{(d)} {g^2\\over z-\\hat{Q}^{(d)} \\hat{H} \\hat{Q}^{(d)}} \\hat{Q}^{(d)} \\hat{V} P^{(d)} \\\\&=E_d+g^2\\Sigma ^+(z)\\;.$ The discrete resonance state eigenvalues are obtained as the solutions of the dispersion equation Eq.",
"(REF ).",
"The corresponding resonance state is obtained by adding the $\\hat{Q}^{(d)}$ component as $|\\phi _\\alpha {\\rangle }&=\\hat{P}^{(d)}|\\phi _\\alpha {\\rangle }+\\hat{Q}^{(d)}|\\phi _\\alpha {\\rangle } \\\\&={\\langle }d|\\phi _\\alpha {\\rangle }\\left( |d{\\rangle }+g \\int _0^\\pi dk {V_k \\over (z-E_k)^+_{z=z_\\alpha }} |k{\\rangle }\\right)$ where we have used Eq.",
"(REF ).",
"The left-eigenstate problem ${\\langle }\\tilde{\\phi }_\\alpha |\\hat{H}=z_\\alpha {\\langle }\\tilde{\\phi }_\\alpha |$ is similarly solved by applying the projection operators from the right.",
"Next we solve for the continuous eigenstate.",
"For this purpose, we choose the projection operators as $\\hat{P}^{(k)}=|k{\\rangle }{\\langle }k| \\;, \\hat{Q}^{(k)}=1-\\hat{P}^{(k)} \\;.$ The effective Hamiltonian for $k$ -space is given by $\\hat{H}_{\\rm eff}^{(k)}(z)&=\\hat{P}^{(k)}\\hat{H}_0 P^{(k)} \\\\&\\quad + P^{(k)}\\hat{V}\\hat{Q}^{(k)} {g^2\\over z-{\\cal H}^{(k)}} \\hat{Q}^{(k)} \\hat{V} P^{(k)} \\;,$ where we have denoted ${\\cal H}^{(k)}\\equiv \\hat{Q}^{(k)} \\hat{H} \\hat{Q}^{(k)} \\;.$ This is represented by ${\\cal H}^{(k)}&=E_d|d{\\rangle }{\\langle }d|+\\sum _{k^{\\prime }(\\ne k)} E_{k^{\\prime }}|k^{\\prime }{\\rangle }{\\langle }k^{\\prime }| \\\\&+{2 \\over \\sqrt{N}} g \\sum _{k^{\\prime }(\\ne k)}\\sin k^{\\prime } \\left( |d{\\rangle }{\\langle }k^{\\prime }|+|k^{\\prime }{\\rangle }{\\langle }d|\\right) \\\\&\\equiv {\\cal H}_0^{(k)} + {\\cal V}^{(k)} \\;.$ Then the matrix elements in the second term of Eq.",
"(REF ) are represented by ${\\langle }k| \\hat{V} \\hat{Q}^{(k)}{1\\over z_k-{\\cal H}^{(k)} } Q^{(k)} \\hat{V} |k{\\rangle } = {4\\over N}(\\sin k)^2 g^2 G_{dd}^{(k)}(z_k)$ where the Green's function in terms of the $k$ -state is given by $G_{dd}^{(k)}(z_k)= {\\langle }d|{1\\over z_k-{\\cal H}^{(k)}} |d{\\rangle }\\;.$ By using the Dyson equation, we have the relations $G_{dd}^{(k)}(z)&={1\\over z-E_d}+{1\\over z-E_d}{2g \\over \\sqrt{N}}\\sum _{k^{\\prime }(\\ne k)}\\sin k^{\\prime } G_{k^{\\prime } d}^{(k)}(z) \\;, \\\\G_{k^{\\prime }d}^{(k)}(z)&={1\\over z-E_{k^{\\prime }}}{2 \\sin k^{\\prime } \\over \\sqrt{N}} g G_{dd}^{(k)}(z) \\;.$ Substitution of Eq.",
"(b) into (a) yileds $G_{dd}^{(k)}(z)= {1\\over \\eta ^+(z)} \\;,$ where $\\eta ^+(z)$ is given in Eq.",
"(REF ).",
"Using Eqs.",
"(REF ), (REF ), and (REF ), the eigenvalue problem of the effective Hamiltonian $\\hat{H}_{\\rm eff}^{(k)}(z)$ reads $\\hat{H}_{\\rm eff}^{(k)}(z_k)\\hat{P}^{(k)}|\\phi _k{\\rangle }&=\\left[ E_k+ {4(\\sin k)^2 \\over N} {g^2 \\over \\eta ^+(z_k)} \\right]\\hat{P}^{(k)}|\\phi _k{\\rangle } \\\\&=z_k \\hat{P}^{(k)}|\\phi _k{\\rangle } \\;.$ We find that $z_k=E_k$ in the limit $N\\rightarrow \\infty $ .",
"The right-continuous eigenstate for the wave number $k$ is given by adding the $\\hat{Q}^{(k)}$ component $|\\phi _k{\\rangle }&=\\hat{P}^{(k)}|\\phi _k{\\rangle }+\\hat{Q}^{(k)}|\\phi _k{\\rangle } \\\\&=\\left[|k{\\rangle }+\\hat{Q}^{(k)} {1\\over E_k -{\\cal H}^{(k)} } \\hat{Q}^{(k)} \\hat{V} |k{\\rangle } \\right]{\\langle }k|\\phi _k{\\rangle } \\;.$ The second term is written by $&\\hat{Q}^{(k)} {1\\over E_k -{\\cal H}^{(k)} } \\hat{Q}^{(k)} \\hat{V} |k{\\rangle } \\nonumber \\\\&= |d{\\rangle }{\\langle }d|{1\\over E_k - {\\cal H}^{(k)} }|d{\\rangle }{\\langle }d| \\hat{V} |k{\\rangle } \\nonumber \\\\&+\\sum _{k^{\\prime }(\\ne k)}|k^{\\prime }{\\rangle }{\\langle }k^{\\prime }|{1\\over E_k - {\\cal H}^{(k)} }|d{\\rangle }{\\langle }d| \\hat{V} |k{\\rangle } \\\\&={2gV \\sin k \\over \\sqrt{N}}\\left[ |d{\\rangle } G_{dd}^{(k)}(E_k)+\\sum _{k^{\\prime }(\\ne k)} |k^{\\prime }{\\rangle } G_{k^{\\prime }a}^{(k)}(E_k) \\right] \\;.$ Substituting Eq.",
"() into Eq.",
"(REF ), we obtain Eq.",
"(REF ).",
"The left-eigenstate for the continuous state is obtained in the same way."
],
[
"Dispersion equation in a form of a polynomial",
"In this section we reduce the dispersion equation Eq.",
"(REF ) to a $2n_d$ -th order polynomial equation.",
"Using a binomial expansion, it is written as $&z - E_d + \\frac{g^2}{ \\sqrt{z^2 - 1} } \\sum _{m=0}^{n_d {\\!}",
"-1 } \\binom{2 n_d {\\!}",
"}{ 2m+1 } ( -z )^{2n_d {\\!}",
"- (2m + 1) } \\Bigl ( \\sqrt{ z^2 - 1 } \\Bigr )^{2m+1} \\\\&\\qquad =\\frac{g^2}{ \\sqrt{z^2 - 1} } \\biggl [ 1 - \\sum _{m=0}^{n_d {\\!}",
"} \\binom{2 n_d {\\!}",
"}{ 2m } ( -z )^{2n_d {\\!}",
"- 2m } \\Bigl ( \\sqrt{ z^2 - 1 } \\Bigr )^{2m} \\biggr ] \\;.$ Using the identities $&\\sum _{m=0}^{n_d {\\!}",
"-1 } \\binom{2 n_d {\\!}",
"}{ 2m+1 } ( -z )^{2n_d {\\!}",
"- (2m + 1) } \\Bigl ( \\sqrt{ z^2 - 1 } \\Bigr )^{2m+1} = \\frac{1}{2} \\biggl \\lbrace \\Bigl ( -z + \\sqrt{z^2 - 1} \\Bigr )^{2n_d {\\!}",
"} - \\Bigl ( -z - \\sqrt{z^2 - 1} \\Bigr )^{2n_d {\\!}",
"} \\biggr \\rbrace \\;,\\\\&\\sum _{m=0}^{n_d {\\!}",
"} \\binom{2 n_d {\\!}",
"}{ 2m } ( -z )^{2n_d {\\!}",
"- 2m } \\Bigl ( \\sqrt{ z^2 - 1 } \\Bigr )^{2m} = \\frac{1}{2} \\biggl \\lbrace \\Bigl ( -z + \\sqrt{z^2 - 1} \\Bigr )^{2n_d {\\!}",
"} + \\Bigl ( -z - \\sqrt{z^2 - 1} \\Bigr )^{2n_d {\\!}",
"} \\biggr \\rbrace \\;,$ Eq.",
"(REF ) reads $&z - E_d + \\frac{g^2}{ 2\\sqrt{z^2 - 1} } \\biggl [ \\Bigl ( -z + \\sqrt{z^2 - 1} \\Bigr )^{2n_d {\\!}",
"} - \\Bigl ( -z - \\sqrt{z^2 - 1} \\Bigr )^{2n_d {\\!}",
"} \\biggr ] \\\\&\\qquad =\\frac{g^2}{ 2\\sqrt{z^2 - 1} } \\biggl [ 2 - \\Bigl ( -z + \\sqrt{z^2 - 1} \\Bigr )^{2n_d {\\!}",
"} + \\Bigl ( -z - \\sqrt{z^2 - 1} \\Bigr )^{2n_d {\\!}",
"} \\biggr ] \\;.$ Taking the square of Eq.",
"(REF ) and using Eq.",
"(b) again, we obtain the $2n_d$ -th order polynomial equation: $&(z - E_d)^2 + 2 g^2 (z - E_d) \\biggl \\lbrace \\sum _{m=0}^{n_d {\\!}",
"-1 } \\binom{2 n_d {\\!}",
"}{ 2m+1 } ( -z )^{2n_d {\\!}",
"- (2m + 1) } \\Bigl ( z^2 - 1 \\Bigr )^{m} \\biggr \\rbrace \\\\&\\qquad + 2 g^4 \\biggl \\lbrace \\sum _{m=0}^{n_d {\\!}",
"-1} z^{2m} + \\sum _{m=1}^{n_d {\\!}",
"} \\binom{2 n_d {\\!}",
"}{ 2m } ( -z )^{2n_d {\\!}",
"- 2m } \\Bigl ( z^2 - 1 \\Bigr )^{m-1} \\biggr \\rbrace = 0.$"
]
] | 1709.01740 |
[
[
"Semi-Supervised Recurrent Neural Network for Adverse Drug Reaction\n Mention Extraction"
],
[
"Abstract Social media is an useful platform to share health-related information due to its vast reach.",
"This makes it a good candidate for public-health monitoring tasks, specifically for pharmacovigilance.",
"We study the problem of extraction of Adverse-Drug-Reaction (ADR) mentions from social media, particularly from twitter.",
"Medical information extraction from social media is challenging, mainly due to short and highly information nature of text, as compared to more technical and formal medical reports.",
"Current methods in ADR mention extraction relies on supervised learning methods, which suffers from labeled data scarcity problem.",
"The State-of-the-art method uses deep neural networks, specifically a class of Recurrent Neural Network (RNN) which are Long-Short-Term-Memory networks (LSTMs) \\cite{hochreiter1997long}.",
"Deep neural networks, due to their large number of free parameters relies heavily on large annotated corpora for learning the end task.",
"But in real-world, it is hard to get large labeled data, mainly due to heavy cost associated with manual annotation.",
"Towards this end, we propose a novel semi-supervised learning based RNN model, which can leverage unlabeled data also present in abundance on social media.",
"Through experiments we demonstrate the effectiveness of our method, achieving state-of-the-art performance in ADR mention extraction."
],
[
"Introduction",
"Adverse-Drug-Reactions (ADRs) are a leading cause of mortality and morbidity in health care.",
"In a study, it was observed that from a death count in the range of (44,000-98,000) due to medical errors, 7000 deaths occurred due to ADRs.http://bit.ly/2vaWF6e.",
"Postmarket drug surveillance is therefore required to identify such potential adverse reactions.",
"The formal systems for postmarket surveillance can be slow and under-efficient.",
"Studies show that 94% ADRs are under-reported [5].",
"Social media presents a useful platform to conduct such postmarket surveillance, given the large audience and vast reach of such platforms.",
"Such platforms have been used for real-time information retrieval and trends tracking, including digital disease surveillance system [12].",
"Recent study shows that twitter has 3 times more ADRs reported than were reported through FDA.",
"Out of 61,000 tweets collected, 4400 had mention of ADRs as compared to 1400 ADRs reported through FDA during the same time-period [2].",
"This makes Twitter a great source for building a real-time post-marketing drug safety surveillance system.",
"However, information extraction from social media comes with its own set of challenges.",
"Some of them are: 1) Short nature of the text (twitter has a 142 character limit), making the language more ambiguous.",
"2) Sparsity of drug-related tweets 3) Highly colloquial language as compared to more technical and formal medical reports.",
"Consider for example the tweets, 'Cymbalta, you're driving me insane'; '@$<$ USER$>$ Ugh, sorry.",
"This effexor is not making me feel so awesome'.",
"In the first tweet, 'driving me insane' and in the second one, 'not making me feel so awesome' are ADR mentions which indicate some level of discomfort in the user's body.",
"These tweets clearly show how information extraction from social media suffers from above-mentioned problems.",
"Recent work in deep learning has demonstrated its superiority over traditional hand-crafted feature based machine learning models [8], [11].",
"However, due to a large number of free parameters, deep learning models rely heavily on large annotated dataset.",
"In the real-world, it is often the case that labeled data is sparse, making it challenging to train such models.",
"Semi-supervised learning based methods provide a viable solution to this.",
"These methods rely on a small labeled data and a large unlabeled data for training.",
"In this work, we present a novel semi-supervised Recurrent Neural Network (RNN) [4] based method for ADR mention extraction, specifically leveraging a relatively large unlabeled data.",
"We demonstrate the effectiveness of our method through experimentation on ADR mention annotated tweet corpus [1].",
"Our method achieves superior results than the current state-of-the-art in ADR extraction from twitter.",
"Our main contributions are : We propose a novel semi-supervised sequence labeling method based on RNN, specifically Long-Short-Term-Memory Network [6] which are known to capture long-term dependencies better than vanilla RNN.",
"For the unsupervised learning part, we explore a novel problem of drug name prediction given context from tweets.",
"The goal is to predict the drug-name which is masked, given it's context in the tweet.",
"For supervised learning, we explore different word embedding initializing schemes and present results for the same.",
"We demonstrate that by training a semi-supervised model, ADR extraction performance can be improved substantially as compared to current methods.",
"On the twitter dataset with ADR mentions annotated [1], our method achieves an F-score of 0.751 surpassing the current state-of-the-art method by 3.01%."
],
[
"Related Work",
"The problem of ADR mention extraction falls under the category of sequence labeling problem.",
"State-of-the-art method for sequence labeling problem is Conditional Random Fields (CRFs) [10].",
"ADRMine [16], is a CRF-based model for ADR extraction task.",
"It uses a variety of hand-crafted features, including word context, ADR lexicon, POS-tag and word embedding based features as input to CRF.",
"The word embedding based features are trained on a large domain-specific tweet corpus.",
"The problem with the above-mentioned approach is its dependency on hand-crafted features, which is time and effort consuming.",
"A Long-Short-Term-Memory (LSTM) network based model is proposed [1] to get around this problem.",
"Instead of using human-engineered features, word embedding based features are passed to a Bi-directional LSTM model which is trained to generate a sequence of labels, given the input word sequence.",
"State-of-the-art results are achieved, surpassing CRF-based ADRMine results.",
"Some recent work also focuses on the problem of Adverse-Drug-Event (ADE) detection [13], [7].",
"The goal is to identify whether there is an Adverse-Drug-Event mentioned in the tweet based on its textual content."
],
[
"ADR-mention extraction using Semi-supervised Bi-directional LSTM",
"In this section, we present our approach for ADR extraction.",
"Our method is based on a semi-supervised learning method which operates in two phases: 1) Unsupervised learning: In this phase, we train a Bidirectional LSTM model to predict the drug name given its context in the tweet.",
"As training data for this task, we select tweets with exactly one mention of any prescription drug.",
"Since we already know the drug name beforehand, it doesn't need any annotation effort.",
"2) Supervised learning: In this phase, we use the same bidirectional LSTM model from phase 1 and (re)train it to predict the sequence of labels, given the tweet text."
],
[
"Unsupervised learning",
"For this phase, we choose a novel task of drug name prediction from its context in the tweet.",
"For training data, we use a large collection of tweets with exactly one mention of the drug name in them.",
"Since we are predicting the drug name from a tweet which is already present in it, in order to avoid the network to learn a trivial function which maps drug-name in input to drug-name in output without considering the context in account, we mask the drug-name in the tweet with a dummy token.",
"For feature-extraction, we use a Bidirectional LSTM model.",
"The model takes as input, a sequence of continuous word vectors as input and predicts a corresponding sequence of word vectors as output.",
"The equations governing the dynamics of LSTMs are defined as follows: $\\begin{split}& \\vec{g}^u = \\sigma ( W^u * \\vec{h}_{t-1} + I^u * \\vec{{\\mathbf {x}}}_t) \\\\& \\vec{g}^f = \\sigma ( W^f * \\vec{h}_{t-1} + I^f * \\vec{{\\mathbf {x}}}_t) \\\\& \\vec{g}^c = \\tanh ( W^c * \\vec{h}_{t-1} + I^c * \\vec{{\\mathbf {x}}}_t) \\\\& \\vec{m}_t = \\vec{g}^f \\odot + \\vec{g}^u \\odot \\vec{g}^c \\\\& \\vec{g}^o = \\sigma (W^o * \\vec{h}_{t-1} + I^o * \\vec{{\\mathbf {x}}}_t) \\\\& \\vec{h}_t = \\tanh ( \\vec{g}^o \\odot \\vec{m}_{t-1})\\end{split}$ here $\\sigma $ is the logistic sigmoid function, $\\mathbf {W}^u, \\mathbf {W}^f, \\mathbf {W}^o, \\mathbf {W}^c$ are recurrent weight matrices and $\\mathbf {I}^u, \\mathbf {I}^f, \\mathbf {I}^o, \\mathbf {I}^c$ are projection matrices.",
"In a conventional LSTM, the sequence order is from left to right.",
"In Bidirectional LSTM, two sequence directions are considered, one from left to right and the other one opposite to it.",
"The final hidden layers activation is the concatenation of vectors from both directions.",
"Mathematically, $\\mathbf {h}_t = [\\vec{h}_t ; \\overleftarrow{h}_t]$ To generate the final representation of the tweet, average-pooling is applied over all hidden state vectors.",
"$\\mathbf {h} = \\sum _{t=1}^{T} \\mathbf {h}_t$ where T is the maximum time-step.",
"Finally a softmax transformation is applied to generate a probability distribution over all drug-names followed by categorical cross-entropy loss.",
"Table: Performance of various deep neural network methods on ADR extraction task.",
"Results are averaged over 10 trails, and are presented with std.",
"deviation"
],
[
"Supervised Sequence Classification",
"For this phase, we reuse the Bidirectional LSTM trained from the previous phase following the setup similar to state-of-the-art [1].",
"At each time-step of the sequence, a softmax layer is applied which predicts a probability distribution over sequence labels.",
"Formally, $\\mathbf {y}_t = softmax(\\mathbf {W} \\mathbf {h} + \\mathbf {b}) \\\\$ here $\\mathbf {W}$ and $\\mathbf {b}$ are weight matrices for the softmax layer.",
"The final loss for the sequence labeling is sum of categorical cross-entropy loss at each time-step.",
"The hidden state $\\mathbf {h}$ and the parameters $\\mathbf {W}^u$ , $\\mathbf {W}^f$ , $\\mathbf {W}^o$ , $\\mathbf {W}^c$ , $\\mathbf {I}^u$ , $\\mathbf {I}^f$ , $\\mathbf {I}^o$ , $\\mathbf {I}^c$ are shared during training both phases.",
"The intuition around the unsupervised task is that the network can learn the textual context where drug names appear, which can help in identifying Adverse Drug Reactions from drugs."
],
[
"Dataset",
"We use the twitter dataset annotated with ADR mention collected during the period of 2007-2010.",
"Tweets were collected using 81 drug names as keyword search terms.",
"In the original dataset, a total of 960 tweets are annotated with ADR mentions.",
"Due to Twitter's search APIs license, only tweet ids were released.",
"Out of the total of 960, we collected a total of 645 tweets using Python library tweepyhttps://github.com/tweepy/tweepy.",
"According to the given train-test split, 470 tweets are used for training and 170 tweets for testing.",
"For the unlabeled dataset, we used the Twitter's search API https://dev.twitter.com/rest/public/search with the drug names used in the original study as keyword search terms http://diego.asu.edu/Publications/ADRMine.html Some example drug names used as keywords are: humira, dronedarone, lamictal, pradaxa, paxil, zoledronic acid, trazodone, enbrel, cymbalta, quetiapine .",
"We crawled the tweets over a period of two months.",
"For the sake of simplicity, we removed the tweets with more than one drug mentions, , resulting in a total of 0.1 Million tweets."
],
[
"Implementation Details",
"We use Kerashttps://keras.io/ for implementation.",
"For text pre-processing, we applied several pre-processing steps, which are : Normalizing HTML links and user-mentions:We replaced all HTML link mentions with the token \"$<$ LINK$>$ \".",
"Similarly, we replaced all user handle mentions (for ex.",
"@JonDoe) with the token \"$<$ USER$>$ \".",
"Special Character Removal: We removed all punctuations and special symbols like '#' from tweets.",
"Emoticons Removal: We removed all emoticons, in general all non-ascii characters which are special types of emoticons.",
"Stop-word and rare words removal: We removed all stop-words and set the vocabulary size to top-15000 most frequent words in the corpus.",
"We used the word2vec [15] embeddings trained on a large generic twitter corpus [3] as input to the model.",
"Word vector dimension is set to 400.",
"BiLSTM parameters are set to the best reported setting from [1], with hidden unit's dimension equal to 500.",
"For training the supervised model, we use the adam optimizer [9] with batch-size equal to 1 and for training the unsupervised model, we used the batch adam optimizer [9] with batch-size set to 128.",
"The supervised model was trained for a total of 5 epochs, and unsupervised model trained for 30 epochs."
],
[
"Results",
"To convert ADR extraction problem into sequence labeling problem, we need to assign annotated entities with appropriate tag representations.",
"We follow IO encoding scheme where each word belongs to either of the following categories: (1) I-ADR (inside ADR) (2) I-Indication (inside Indication ) (3) O (Outside any mention) (4) $<$ PAD$>$ (if the word is padding token).",
"It should be noted that, similar to the baseline method [1] we report the performance on the ADR label only.",
"This is because the number of Indication annotations are very less in number45 in training, 16 in testing.",
"An example tweet annotated with IO-encoding:@BLENDOSO LamictalO andO trileptalO andO seroquelO ofO courseO theO seroquelO IO takeO inO severeO situationsO becauseO weightI-ADR gainI-ADR isO notO coolO For performance evaluation we use approximate-matching [18], which is used popularly in biomedical entity extraction tasks [1], [16].",
"We report the F1-score, Precision and Recall computed using approximate matching as follows: $\\text{Precision} = \\frac{\\text{\\#ADR approximately matched}}{\\text{\\#ADR spans predicted}}$ $\\text{Recall}= \\frac{\\text{\\#ADR approximately matched}}{\\text{\\#ADRspans in total}}$ Table 1 presents the results of our approach along with comparisons.",
"Since the number of tweets used for training and testing differs from the one used in baseline [1], we re-ran their model using the source-code released by themhttps://github.com/chop-dbhi/twitter-adr-blstm.",
"It should be noted that the original model used RMSProp [17] as an optimizer, so for a fair comparison we also report the baseline results with optimizer as adam instead of RMSProp.",
"Replacing RMSProp with adam, although gives an improvement over the original baseline, still under-performs our method .",
"Our approach gives the state-of-the-art results, giving an improvement of 2.97%F1 over the original baseline and an improvement of 1.88% F1 over the re-implemented baseline."
],
[
"Effect of drug-mask",
"For the unsupervised learning phase, we select the task of drug-name prediction given its context.",
"In order to avoid the network learning a degenerate function which maps input drug-name to output drug-name, we mask all drug-names in input with a single token.",
"In order to verify this, we report the accuracy results without the drug-mask, i.e.",
"with drug-name included in the input.",
"The result is presented in Table 2.",
"It is clear that removing the drug mask from input degrades the end-performance by 0.535% in F-score.",
"This further validates our claim that masking the drug-names is effective."
],
[
"Effect of embeddings and dictionary",
"We experiment with word embeddings trained on different corpus to observe its effect on the end-performance.",
"We experiment with embeddings trained on part of Google News Dataset, which consists of around 100 billion wordshttps://code.google.com/archive/p/word2vec/.",
"It can be observed that using Google News Corpus trained embeddings degrade the performance by 2.038% in F-score.",
"This is due to the fact that these embeddings are trained on a large News Corpus, which is more grammatically sound and formal than the raw social media posts.",
"Conceptually, the shift in the lexical data distribution of the News corpus as compared to tweets containing ADR causes the degradation in performance.",
"We also experiment with word embeddings trained on a large medical-concept terms related tweet corpushttps://zenodo.org/record/27354#.WWYph1ekW4A [14].",
"Intuitively, embeddings trained on similar domain (medical in this case) should perform better, but surprisingly it performs worst amongst all methods.",
"The generic embeddings trained on large tweet corpus captures potentially large variation of semantics and linguistic properties of text and due to the free-style nature of writing on social media, this helps more than domain-knowledge, as captured by medical-domain trained embeddings.",
"We also experimented with a different vocabulary initialization.",
"In our proposed formulation, we construct vocabulary from both unlabeled and labeled corpus, resulting in a larger vocabulary size.",
"When experimented with a restricted vocabulary (only from labeled training data), we observe that the F1-score drops by 0.8%.",
"This suggests the use of a larger vocabulary with more coverage in similar settings."
],
[
"Conclusions",
"We present a novel semi-supervised Bi-directional LSTM based model for ADR mention extraction.",
"We evaluate our method on an annotated twitter corpus.",
"By leveraging potentially large unlabeled corpus, our method outperforms the state-of-the-art method by 3.01% in F1-score.",
"We also demonstrate that word embeddings trained on a large domain-agnostic twitter corpus performs better than more popular Google News Corpus trained word-embeddings and surprisingly even better than medical domain-specific word embeddings trained on tweets, which suggests that language structure and semantics is more important in downstream information extraction tasks, compared to domain knowledge.",
"In future, we will explore drug and side-effect (adverse-effect) mention relation along with ADR extraction and explore if both can be formulated in a multi-task learning setup."
]
] | 1709.01687 |
[
[
"Bose-Einstein correlations of same-sign charged pions in the forward\n region in $pp$ collisions at $\\sqrt{s}$ = 7 TeV"
],
[
"Abstract Bose-Einstein correlations of same-sign charged pions, produced in proton-proton collisions at a 7 TeV centre-of-mass energy, are studied using a data sample collected by the LHCb experiment.",
"The signature for Bose-Einstein correlations is observed in the form of an enhancement of pairs of like-sign charged pions with small four-momentum difference squared.",
"The charged-particle multiplicity dependence of the Bose-Einstein correlation parameters describing the correlation strength and the size of the emitting source is investigated, determining both the correlation radius and the chaoticity parameter.",
"The measured correlation radius is found to increase as a function of increasing charged-particle multiplicity, while the chaoticity parameter is seen to decrease."
],
[
"0.5em"
]
] | 1709.01769 |
[
[
"The active nucleus of the ULIRG IRAS F00183-7111 viewed by NuSTAR"
],
[
"Abstract We present an X-ray study of the ultra-luminous infrared galaxy IRAS F00183-7111 (z=0.327), using data obtained from NuSTAR, Chandra X-ray Observatory, Suzaku and XMM-Newton.",
"The Chandra imaging shows that a point-like X-ray source is located at the nucleus of the galaxy at energies above 2 keV.",
"However, the point source resolves into diffuse emission at lower energies, extending to the east, where the extranuclear [O III] emission, presumably induced by a galactic-scale outflow, is present.",
"The nuclear source is detected by NuSTAR up to the rest-frame 30 keV.",
"The strong, high-ionization Fe K line, first seen by XMM-Newton, and subsequently by Suzaku and Chandra, is not detected in the NuSTAR data.",
"The line flux appears to have been declining continuously between 2003 and 2016, while the continuum emission remained stable to within 30%.",
"The X-ray continuum below 10 keV is characterised by a hard spectrum caused by cold absorption of nH ~1e23 cm-2, compatible to that of the silicate absorption at 9.7 micron, and a broad absorption feature around 8 keV which we attribute to a high-ionization Fe K absorption edge.",
"The latter is best described by a blueshifted, high-ionization (log xi ~3) absorber.",
"No extra hard component, which would arise from a Compton-thick source, is seen in the NuSTAR data.",
"While a pure reflection scenario (with a totally hidden central source) is viable, direct emission from the central source of L(2-10 keV) = 2e44 erg/s, behind layers of cold and hot absorbing gas may be an alternative explanation.",
"In this case, the relative X-ray quietness (Lx/L_AGN ~6e-3), the high-ionization Fe line, strong outflows inferred from various observations, and other similarities to the well-studied ULIRG/QSO Mrk 231 point that the central source in this ULIRG might be accreting close to the Eddington limit."
],
[
"Introduction",
"IRAS F00183–7111 is an ultra-luminous infrared galaxy (ULIRG) with log $L$ (1-1000 $\\mu $ m) $= 9.9\\times 10^{12}L_{\\odot }$ (Spoon et al 2009) at $z= 0.327$ and shows an archetypal absorption-dominated mid-infrared spectrum with no polycyclic aromatic hydrocarbon (PAH) feature detected at 6.2 $\\mu $ m (Spoon et al 2004).",
"The silicate absorption depth at 9.7 $\\mu $ m is however shallower than those in the branch of deeply obscured sources (e.g.",
"NGC 4418, IRAS F08572+3915) in the “fork diagram” of Spoon et al (2007).",
"This apparent shallowness of the silicate depth could be a result of dilution by the continuum which leaks through ruptured absorbing screen, and it led Spoon et al (2007, 2009) to suggest that IRAS F00183–7111 may be in the early phase of disrupting the nuclear obscuration to evolve into the quasar regime.",
"The fast outflow signatures found in the mid-infrared lines (Spoon et al 2009) and the VLBI-scale radio jets (Norris et al 2012) appear to support this picture.",
"The radio luminosity is in the range of powerful radio galaxies and the radio excess with respect to the infrared emission (Roy & Norris 1997; Drake et al 2004) indicates presence of a powerful active galactic nucleus (AGN).",
"A XMM-Newton observation detected a hard-spectrum X-ray source with a 2-10 keV luminosity of $\\sim 10^{44}$ erg s$^{-1}$ .",
"The X-ray spectrum shows a strong Fe K feature, suggesting that a Compton thick AGN with a much larger luminosity is hidden in this ULIRG (Nandra & Iwasawa 2007; Ruiz, Carrera & Panessa 2007).",
"The Fe K line is, however, found at the rest-frame 6.7 keV, indicating Fe xxv, i.e.",
"highly ionized line-emitting medium, which is unusual for an obscured AGN as it normally shows a Fe K feature dominated by a 6.4 keV line from cold Fe (less ionized than Fe xvii).",
"The fast outflow signature of IRAS F00183–7111 was first identified by the optical [Oiii]$\\lambda 5007$ kinematics and the ionized gas extends by $\\sim 10$ arcsec to the east of the nucleus (Heckman, Armus & Miley 1990).",
"Much more enhanced outflow signatures were found in the blueshifted mid-IR lines of [Neii]$\\lambda 12.81 \\mu $ m and [Neiii]$\\lambda 15.56 \\mu $ m with velocity widths of FWZI$\\sim 3000$ km s$^{-1}$ (Spoon et al 2009), which presumably occur in the region close to the nucleus where dust obscuration hides the optical signatures from our view.",
"Weak soft X-ray emission appears to be displaced from the hard X-ray position by $\\sim 5$ arcsec towards the east and is possibly associated with the extended outflow structure.",
"We newly acquired X-ray data from Chandra X-ray Observatory (Chandra) and NuSTAR to study further the X-ray properties of IRAS F00183–7111.",
"The arcsec resolution of Chandra imaging was used to investigate the extended soft X-ray emission hinted by the XMM-Newton observation.",
"The hard X-ray spectrum obtained from NuSTAR was examined for constraining the properties of the obscured active nucleus.",
"The Fe K emission is a key diagnostic for the nuclear obscuration and the physical condition of the nuclear medium, for which data of better quality are desired.",
"The existing XMM-Newton data and the Suzaku data from the public archive are supplemented to study the Fe K band spectrum.",
"The cosmology adopted here is $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _{\\Lambda }=0.72$ , $\\Omega _{\\rm M}=0.28$ (Bennett et al 2013).",
"For the redshift of IRAS F00183–7111 ($z=0.327$ ), the luminosity distance is $D_{\\rm L}= 1723$ Mpc and the angular-scale is 4.7 kpc arcsec$^{-1}$ ."
],
[
"Observations",
"X-ray observations of IRAS F00183–7111 with four X-ray observatories, XMM-Newton, Chandra, Suzaku and NuSTAR are listed in Table 1.",
"Our NuSTAR data of a total exposure time of 105 ks were taken in two occasions separated by four months.",
"The data were calibrated and cleaned, using the NuSTARDAS included in HEASOFT (v6.19).",
"In addition to the default data cleaning, some time intervals near the south atlantic anomaly (SAA) passage with elevated background were discarded by applying the screening with SAAMODE=optimized and TENTACLE=yes, which removed further 4 ks for the first observation.",
"On inspecting the 3-20 keV light curve obtained from the whole field of view, short intervals ($\\le 2$ ks in total) of solar flares were noted during the first observation, which may affect the background subtraction around 10 keV (Lanzuisi et al 2016).",
"However, as no notable effect was found in the spectrum of the first observation, the intervals were not excluded.",
"The NuSTAR spectra obtained from the two detector modules FPMA and FPMB and the two observations agree with each other within error, and the four datasets are combined together for the analysis below.",
"The Chandra and XMM-Newton data were reduced using the standard software CIAO (v4.8) and XMMSAS (v15.0), respectively.",
"The Suzaku data (PI: E. Nardini) were taken from the public archive and reduced using XSELECT of HEASoft.",
"All the spectral data were analysed using the spectral analysis package XSPEC (v12.9)."
],
[
"Chandra imaging",
"The arcsecond resolution of the Chandra image localizes an X-ray source at the position of the IRAS F00183–7111 nucleus (Fig 1a).",
"The source at energies above 2 keV is point-like.",
"However, the point-like source disappears below 1 keV, leaving only faint emission which extends to the east up to $\\sim 15$ arcsec, as suggested by the overlaid contours (Fig.",
"1b).",
"With the exposure time of 23 ks, 9 counts are detected along this eastern extension in the 0.4-1 keV band outside of the 2 arcsec from the nucleus position.",
"When the annulus of 2-18 arcsec from the nuclear position is divided azimuthally to four quadrants, the eastern quadrant contains these 9 counts and the other three contain 2 (northern), 3 (southern) and 1 (western).",
"The background counts are estimated to be $1.7\\pm 0.1$ counts for each quadrant.",
"The eastern region has 5 times excess counts of the background while the other regions have counts comparable to the background.",
"While we can thus only remark that detected counts are clustered on the eastern side of the nucleus, the small detected counts do not warrant any further imaging analysis.",
"The one-sided extension is compatible with the eastern offset seen in the soft X-ray image of XMM-Newton (Nandra & Iwasawa 2007).",
"The mean surface brightness of this extended emission is $\\sim 2\\times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ It has been known that [Oiii]$\\lambda 5007$ emission extends towards the east in this ULIRG (Heckman, Armus & Miley 1990).",
"Our recent VLT VIMOS observation reveals more details of the optical extended nebula (Spoon et al.",
"in prep.",
"), and the [Oiii] contours overlaid onto the continuum image, obtained from the VIMOS data, is shown in Fig.",
"1c.",
"Note that the scale of this image differs from the X-ray images (a factor of $\\sim 2$ smaller).",
"It is unclear whether this optical emission-line nebula is closely related to the extended soft X-ray emission.",
"They both extend towards the east and some soft X-ray photons are detected along the [Oiii] extension.",
"However, the [Oiii] image shows a distinctive, bright knot at 8 arcsec from the nucleus (with PA$\\simeq 270^{\\circ }$ ), which has no X-ray counterpart.",
"In contrast, the soft X-ray emission is diffuse and the general direction of the extension may be slightly tilted towards north (PA$\\sim 250^{\\circ }$ ) from that of [Oiii]."
],
[
"X-ray spectrum",
"We present results on the X-ray spectrum of IRAS F00183–7111 as follows.",
"The iron K emission line was investigated combining the data from XMM-Newton, Chandra and Suzaku, as they have a comparable spectral resolution (Sect.",
"3.2.1).",
"Then the flux variability of the Fe line (Sect.",
"3.2.1) and the underlying 2-8 keV continuum emission (Sect.",
"3.2.2) between the observations was measured including the NuSTAR data.",
"The 2-20 keV continuum spectrum can be described by an absorbed power-law and the cold absorbing column is constrained well, combining the NuSTAR data with the low-energy data from the other instruments.",
"Then the broad-band NuSTAR spectrum was investigated for the hard band spectral properties (Sect.",
"3.2.3)."
],
[
"Fe K line",
"A line feature at 5 keV was detected by the XMM-Newton observation and identified with an iron K line of Fe xxv at the rest-energy of 6.7 keV (Nandra & Iwasawa 2007; Ruiz et al 2007).",
"We inspected the presence of the Fe K line by further adding the data from the Chandra ACIS-S and the Suzaku XIS, which have similar spectral resolutions to that of the XMM-Newton EPIC cameras.",
"The 3.7-7 keV (5-9 keV in the rest-frame) spectrum of a 100-eV resolution, obtained by combining XMM-Newton, Chandra and Suzaku, after correcting for the respective detector responses is shown in Fig.",
"2 (note that this plot is only for displaying purposes).",
"The plotted data were averaged with weighting by the signal to noise ratio.",
"The line is now detected at $4 \\sigma $ significance above the neighbouring continuum in the 3.7-7 keV range with average intensity of $(2.1\\pm 0.5)\\times 10^{-6}$ ph cm$^{-2}$ s$^{-1}$ .",
"Fitting a Gaussian to the line feature in all the spectral data jointly gives the centroid energy of $6.69\\pm 0.04$ keV in the galaxy rest-frame.",
"The line feature is resolved: $\\sigma = 0.16\\pm 0.03$ keV in Gaussian dispersion, which is significantly broader than that expected from a Fe xxv complex ($\\sigma \\sim 0.02$ keV).",
"The calibration errors in energy scale for respective instruments are at the level of $<10$ eV (M. Guainazzi, priv.",
"comm.",
"), which cannot account for the broadening.",
"The line detection and the above results on the line parameters are robust against continuum modelling (see below).",
"The equivalent width of the line feature with respect to the local continuum is $1.3\\pm 0.3$ keV (the mean of the three observations, corrected for the galaxy redshift).",
"No clear 6.4 keV cold Fe K line is seen.",
"The 90% upper limit on a narrow line at rest-frame 6.4 keV is 0.16 keV.",
"Figure: Energy spectra of IRAS F00183–7111, observed with four X-rayobservatories: a) XMM-Newton: EPIC pn (open squares) and EPIC MOS1and MOS2 combined (solid circles); b) Chandra ACIS-S; c) Suzaku XIS0(solid circles); XIS1 and XIS3 combined (open squares); and d)NuSTAR: FPMA and FPMB combined.",
"The dotted-line histogram in eachpanel indicates the best-fit model of an absorbed power-law plus aGaussian for the 2-20 keV data obtained by fitting to all thedatasets jointly (see text).",
"Spectral parameters are identicalbetween the observatories apart from the normalizations of thepower-law and the Gaussian line.",
"Emission below 2 keV originatesfrom an extended extranuclear region (Fig.",
"1).",
"Note that the modelfor Fe K in d) shows a line feature with its intensity observed withXMM-Newton for a comparison with the data.Figure: Light curves of Fe K line (a: Upper panel) and the 2-8 keVcontinuum (b: Bottom panel) of IRAS F00183–7111, observed with fourX-ray observatories over 2003-2016.",
"The continuum flux is in unitsof 10 -14 10^{-14} erg s -1 ^{-1} cm -2 ^{-2}.",
"Note that the joint fit determines thecontinuum shape better than in an individual fit, helping reducingthe uncertainties of the line fluxes.",
"The line flux appears todecrease continuously by factor of ∼4\\sim 4 since the firstobservation with XMM-Newton while the continuum flux remains atcomparable level within 30%.The Fe K line is however not detected in the NuSTAR data.",
"We believe that the lower resolution of the detector (0.4 keV in FWHM) is unlikely to be the reason for the non detection, since it is compatible with the line width measured above and would instead be optimal for a detection.",
"The Fe K band data with, say, 200-eV intervals, which moderately oversample the detector resolution reveal no excess emission at around 5 keV while it would be easily detectable if the line flux remains the same.",
"The 90% upper limit of line intensity is $1.9\\times 10^{-6}$ ph cm$^{-2}$ s$^{-1}$ , if the line energy and width of the XMM, Chandra and Suzaku combined spectrum are assumed.",
"A possible line-like excess (unresolved) is instead seen at 5.4 keV ($7.28\\pm 0.15$ keV in rest energy) with an intensity of $(1.2\\pm 0.5)\\times 10^{-6}$ ph cm$^{-2}$ s$^{-1}$ , but its detection is uncertain ($\\sim 2\\sigma $ ).",
"The apparent lack of the Fe K feature prompted us an investigation of line variability, as the X-ray observations of this ULIRG span over 13 yr (Table 1).",
"We performed a joint fitting of a Gaussian to the Fe K feature again, but the line intensity was left independent between different observations.",
"The continuum was modelled by an absorbed power-law and its normalization was left independent likewise (see Sect.",
"3.2.2) while power-law slope and absorbing column of cold absorption were kept common between the observations.",
"We used the 3-20 keV data from NuSTAR and 2-8 keV data from the other instruments jointly (Fig.",
"3).",
"The photon index and absorbing column density as measured in the galaxy rest-frame are found to be $\\Gamma = 2.3\\pm 0.2$ and $N_{\\rm H}$$=(1.3\\pm 0.3)\\times 10^{23}$ cm$^{-2}$ .",
"The line centroid and the width, common to all the observations, are found identical to those reported above.",
"The line intensity variability is plotted in Fig.",
"4a.",
"The line flux appears to decline monotonically since the first observation with XMM-Newton in 2003.",
"The line flux observed with Suzaku and Chandra in 2012-2013 is about half the flux measured by XMM-Newton and the NuSTAR measurement is further down.",
"With the four data points, the hypothesis of linearly declining line flux, for example, is favoured by F test to a constant line flux at $\\sim 98$ % confidence.",
"However, we have only four measurements, three of which are close together.",
"The putative line variability largely relies on the high flux measured by XMM-Newton.",
"While the consistently strong line fluxes measured with the two different detectors of XMM-Newton, the pn and MOS cameras, support the reliability of the strong XMM-Newton flux, another data point after the NuSTAR observation with good quality is desirable to verify the trend of fading Fe line."
],
[
"2-8 keV continuum",
"In contrast to the Fe K emission, the neighbouring continuum remains at similar flux level (Fig.",
"4b).",
"Fluxes of the four observations derived from the joint fit in which the power-law normalizations were left independent between the observations are used to plot Fig.",
"4b.",
"All the four measurements agree within 30%.",
"Given the statistical uncertainties of individual measurements of $\\sim 15$ -20% and the general cross-calibration error of $\\sim 10$ % (Ishida et al.",
"2011, Kettula, Nevalainen & Miller 2013; Madsen et al 2015), no significant variability in continuum flux can be observed.",
"The most deviated NuSTAR flux is $\\sim 30\\pm 20$ % above the mean value of the other three.",
"Even if this real, its variability is contrary to the declining Fe K line flux."
],
[
"NuSTAR Hard X-ray spectrum",
"Here, we investigate the hard X-ray spectrum of IRAS F00183–7111 obtained from NuSTAR (Fig.",
"5) which has the sensitivity at energies above 8 keV where the other instruments do not cover.",
"With the reflection scenario in mind (Sect.",
"1), the original aim of this NuSTAR observation was to see whether a strongly absorbed component emerges in the hard X-ray band.",
"Apparently, there is no such an extra hard-component above 8 keV (see Sect.",
"4.1 and Fig.",
"6 for details) A question is then whether the observed hard X-ray emission is reflected light of a totally hidden central source or its direct radiation modified by moderate absorption, which we examine here using the NuSTAR data.",
"The 3-20 keV data are binned so that each spectral bin has more than 14 net counts after background correction (Fig.",
"5a), and the $\\chi ^2$ minimisation was used to search for the best fits.",
"The hard spectrum measured with XMM-Newton, Chandra and Suzaku is presumably caused by cold absorption of $N_{\\rm H}$$\\sim 1\\times 10^{23}$ cm$^{-2}$ (Sect.",
"3.2.2).",
"Since this degree of absorption cannot be constrained well by the NuSTAR data alone, because of the limited low-energy coverage (down to 4 keV in the rest frame), we impose cold absorption with $N_{\\rm H}$$= 1.3\\times 10^{23}$ cm$^{-2}$ , as obtained from the joint fitting with the low-energy instruments (Sect.",
"3.2.2), in the following spectral analysis.",
"A simple power-law modified by the fixed cold absorption yields a photon index of $\\Gamma = 2.39\\pm 0.16$ (Table 2).",
"This fit leaves systematic wiggles in the deviation of the data from the model as shown in Fig.",
"5b, indicating possible presence of an Fe K absorption edge, which corresponds to the broad deficit around 8 keV.",
"A sharp deficit at the observed energy of 10 keV may be an artefact due to background (NuSTAR Observatory Guide 2014).",
"Including an absorption edge model, edge, improves the fit and gives an edge threshold energy of $\\sim 7.5$ keV ($9.9\\pm 0.3$ keV in the rest frame) with an optical depth of $\\tau = 0.9\\pm 0.3$ .",
"The edge energy suggests highly ionized Fe.",
"However, a sharp, single edge, as described by the edge model, is inappropriate for high-ionization Fe, as an absorption spectrum would instead consist of a series of absorption features from a range of ionization (Kallman et al 2004), which has to be computed by photoionization models like XSTAR (Kallman & Bautista 2001).",
"Alternatively, such an absorption feature appears also strongly in a reflection spectrum when reflection occurs in an optically thick, ionized medium.",
"Below we test the two hypotheses of ionized reflection and direct emission modified by an ionized absorber.",
"We first compare the data with the ionized reflection spectrum computed with xillver of García et al.",
"(2013).",
"A thick slab assumed for reflecting medium, e.g.",
"an accretion disc, in xillver is an approximation to the relatively thick reflecting medium.",
"In fitting to the XMM-Newton, Chandra and Suzaku spectra, the Fe line feature drives the fit and gives the ionization parameter of the reflecting medium of log $\\xi =3.2\\pm 0.1$ .",
"Fitting it to the NuSTAR data alone finds a higher value of ionization parameter, log $\\xi =3.5^{+0.3}_{-0.2}$ , which has the reduced sharp line and the increased Compton-broadened component, matching better the broad hump around 5 keV and the weak line (IREFL in Table 2, Fig.",
"5).",
"Secondly, the absorption model is tested, introducing an ionized absorber to account for the high-ionization Fe K edge feature in addition to the cold absorber.",
"No Fe emission line is included.",
"We used the analytic XSTAR model warmabs (Kallman 2016) to compute an ionized absorption spectrum to compare with the data.",
"Since no obvious absorption lines of Fe xxv and Fe xxvi are visible in the NuSTAR data, we assumed that these lines are too narrow to be resolved and chose a small turbulent velocity of $v_{\\rm turb} = 200$ km s$^{-1}$ .",
"Fitting the ionization parameter and the column density of the ionized absorber gives log $\\xi = 3.1\\pm 0.2$ and $N_{{\\rm H},i}=1\\times 10^{24}$ cm$^{-2}$ (Note the best-fit column density is just below the maximum value computed for the model and the upper bound of the error is not obtained.).",
"When the ionized absorber is allowed to be blueshifted (IABS2), as expected for a high-velocity outflow, the fit improves with similar absorber parameters and the blueshift of $-0.18^{+0.02}_{-0.01}\\,c$ (Table 2, Fig.",
"5).",
"The residuals of the fits with IREFL and IABS2 are shown in Fig.",
"5c to compare with Fig.",
"5b from the absorbed power-law (APL).",
"Those three spectral models before folding through the detector response are shown in Fig.",
"5d, illustrating that the Fe absorption-edge feature and its shift to the higher energy is a key to match the data.",
"Between the reflection and absorption models, the (blueshifted) absorption model gives a better fit than the reflection model but with more free parameters.",
"Given the fitting quality, we cannot say the absorption model is strongly preferred over the reflection model, e.g.",
"a test by Bayesian information criterion (BIC, Schwarz 1978) favours the absorption model but the difference in BIC ($\\Delta {\\rm BIC}=2.6$ ) indicates its preference is positive but not sufficiently strong (Kass & Raftery 1995).",
"A critical test would be a detection of the Fe absorption lines with higher-resolution data.",
"The CCD resolution, e.g.",
"FWHM$\\sim 150$ eV, would suffice if they exist.",
"We returned to the XMM-Newton, Chandra and Suzaku data to see whether one of the absorption features due to Fe xxv at the observed energy of 6 keV is present.",
"There is marginal ($2\\sigma $ ) evidence of an absorption line at $6.07\\pm 0.05$ keV (the rest-frame $8.05\\pm 0.07$ keV, see Fig.",
"2), when all the datasets are jointly fitted by a Gaussian.",
"The line is unresolved (the 90% upper limit of the dispersion is 0.35 keV) and the intensity is $(-3.2\\pm 1.6)\\times 10^{-7}$ ph cm$^{-2}$ s$^{-1}$ , corresponding to the equivalent width with respect to the neighbouring continuum is EW $=-0.16\\pm 0.08$ keV.",
"The IABS2 model obtained for the NuSTAR data, adjusted to the continuum level and with the Gaussian for the Fe K emission, is shown in Fig.",
"2.",
"The data are compatible in the line energy and its depth predicted by the absorption model.",
"The unresolved line is consistent with the small turbulent velocity chosen for the model.",
"Albeit the detection is inconclusive, the possible absorption line is found where the blueshifted absorption model predicts.",
"Table: NuSTAR spectrum of IRAS F00183–7111"
],
[
"Nuclear obscuration",
"The NuSTAR spectrum of IRAS F00183–7111 reported in this paper shows no spectral hardening at high energies ($>8$ keV, as shown by the spectral fits in Sect.",
"3.2.3), which would be observed if either reflection from cold medium, expected from a Compton thick AGN, or direct emission from a central source modified by a large absorbing column of $N_{\\rm H}$$>10^{24}$ cm$^{-2}$ is present.",
"This is illustrated by Fig.",
"6, where the rest-frame broad-band X-ray spectrum of IRAS F00183–7111 is compared with three obscured AGN with different degrees of absorption.",
"The spectrum of IRAS F00183–7111 is composed of the 0.6-10 keV data, made from the XMM-Newton, Chandra and Suzaku data, and the 8-20 keV NuSTAR data while those of NGC 1068 (Marinucci et al 2016), NGC 4945 (Puccetti et al 2014) and IRAS F05189–2524 (Teng et al 2015) are XMM-Newton and NuSTAR combined data.",
"Observed below 10 keV in NGC 1068 and NGC 4945 are reflected light only and their spectra show upturns at energies above 10 keV due to a reflection hump (Matt et al 1997) and a strongly absorbed continuum ($N_{\\rm H}$$\\simeq 3.5\\times 10^{24}$ cm$^{-2}$ ), respectively.",
"Such a spectral upturn lacks in the hard-band spectrum of IRAS F00183–7111, which rather resembles the moderately absorbed spectrum of IRAS F05189–2524.",
"It leaves two possible interpretations for the origin of the hard X-ray emission from the ULIRG.",
"Firstly, it could be reflected light of a hidden AGN from a highly ionized medium, as originally suggested by Nandra & Iwasawa (2007) and Ruiz et al (2007).",
"Direct emission from the central source has to be totally suppressed by cold gas of an extremely large absorbing column, e.g., $N_{\\rm H}$$\\ge 10^{25}$ cm$^{-2}$ .",
"Reflection off a highly ionized medium would have a continuum spectrum similar to the direct emission (e.g., García et al 2013), keeping the NuSTAR band spectral slope steep, as observed.",
"It naturally explains the strong Fe xxv line observed with XMM-Newton, Chandra and Suzaku.",
"Secondly, the hard X-ray emission could be of a moderately absorbed central source which we see directly, as suggested by resemblance of the overall spectral shape with that of IRAS F05189–2524 with the absorption column density of $N_{\\rm H}$$\\simeq (0.6-0.9)\\times 10^{23}$ cm$^{-2}$ (Severgnini et al 2001; Imanishi & Terashima 2004, Ptak et al 2003, Grimes et al 2005, Iwasawa et al 2011, Teng et al 2015).",
"Spectral fitting based on these two interpretations were made against the NuSTAR spectrum in Sect.",
"3.2.3.",
"Here we discuss their likelihoods, combining with other information.",
"Detection of a strong Fe line is generally considered to be good evidence for a Compton thick AGN, as it leaves a reflection-dominated spectrum below 10 keV.",
"We refer to Nandra & Iwasawa (2007) for detailed discussion of the reflection scenario.",
"Here, we look into a few outstanding problems with this interpretation.",
"One peculiar feature in IRAS F00183–7111 (as a Compton thick AGN) is the high-ionization (Fe xxv) Fe line.",
"Normally, the Fe K feature in Compton thick AGN is dominated by a cold line at 6.4 keV, which originates from a Compton-thick absorber itself.",
"Even when any high-ionization lines are present (e.g.",
"in NGC 1068, Iwasawa et al 1997), they are minor components of the Fe K complex and the 6.4 keV line is always the major feature.",
"On the contrary, a cold Fe line is not detected in IRAS F00183–7111 (see Sect.",
"3.2.1).",
"In the reflection scenario, the central source is assumed to be hidden from our direct view by cold gas of an extreme thickness (thus invisible).",
"Irradiation of the obscuring gas by the central source however produces reprocessed X-ray emission characterised by, e.g.",
"a 6.4 keV Fe line, and part of it enters our view, unless obscuration is very thick and the coverage is complete.",
"This may occur in nuclear sources of some extreme objects like Arp 220 (Scoville et al 2017).",
"However, in IRAS F00183–7111, AGN-heated hot dust emission clearly visible in infrared (Spoon et al 2004), for example, provides evidence against such an extreme form of obscuration (see the SED comparison in Fig.",
"7 for the strong contrast of mid-IR dust emission between IRAS F00183–7111 and Arp 220).",
"Thus the lack of reprocessed X-ray features from cold gas of a putative Compton thick absorber is somewhat puzzling in the reflection scenario.",
"Another critical issue is the line variability (Sect.",
"3.2.1).",
"If the line were indeed variable, the disconnected variability between the line and continuum would give a problem with the reflection scenario, since both originate in the same process and they should vary in unison.",
"A further line flux measurement with, e.g., XMM-Newton, will provide a critical test against it.",
"The fit to the NuSTAR data (Sect.",
"3.2.3) shows that the reflection model is still viable.",
"In that case, the ionized reflecting medium has to be sufficiently optically thick ($N_{\\rm H}$$\\ge 10^{24}$ cm$^{-2}$ ) to produce the edge absorption feature.",
"The observed silicate depth might be diluted by leaked IR continuum emission and the true column density of cold clouds responsible for the silicate absorption could then be much larger than the apparent value $N_{\\rm H}$$\\simeq 1.7\\times 10^{23}$ cm$^{-2}$ (Spoon et al 2004) to provide a Compton-thick absorption towards the central X-ray source."
],
[
"Direct emission scenario",
"In the direct emission scenario, absorption by cold gas is moderate ($N_{\\rm H}$$\\simeq 1.3\\times 10^{23}$ cm$^{-2}$ ), but in addition to that, a blue-shifted, highly ionized absorber with a large column $N_{\\rm H}$$\\sim 10^{24}$ cm$^{-2}$ is required to account for the absorption feature around 8 keV observed in the NuSTAR spectrum (Sect.",
"3.2.3, Table 2).",
"The column density of the cold absorber is compatible with that inferred from the silicate absorption (Spoon et al 2004).",
"The high-ionization absorber has an ionization parameter of log $\\xi \\sim 3$ and is only detectable in X-ray.",
"The inferred blueshift $v\\simeq 0.18\\,c$ lies in the range of high-velocity outflows found in growing number of Seyfert galaxies and quasars (e.g., Tombesi et al 2010).",
"Given the presence of powerful outflow signatures observed in mid-IR (Spoon et al 2009), optical (Heckman et al 1990) and possibly radio (Ruffa et al in prep.)",
"bands, detection of X-ray outflowing gas comes as no surprise.",
"The 2-10 keV luminosity corrected for both cold and high-ionization absorption is found to be $2.1\\times 10^{44}$ erg s$^{-1}$ .",
"Discounting the $\\sim 14$ % starburst contribution (the star formation rate, SFR of $220 M_{\\odot }\\,{\\rm yr}^{-1}$ , Mao et al 2014; $230M_{\\odot }\\,{\\rm yr}^{-1}$ , Ruffa et al in prep.",
"), we estimate the AGN bolometric luminosity of IRAS F00183–7111 to be $3.3\\times 10^{46}f^{-1}$ erg s$^{-1}$ , where $f(<1)$ is a covering factor of obscuring dust shrouds which absorb radiation from the central source and reemit in infrared.",
"This gives $L_{2-10}/L_{\\rm bol,AGN}\\sim 6\\times 10^{-3}f$ (or the X-ray bolometric correction $k_{\\rm bol}\\sim 150f^{-1}$ ).",
"This value is smaller than that of typical AGN, e.g., $\\sim 0.02$ for a normal quasar with a similar bolometric luminosity (Marconi et al 2004), but comparable to those measured in local U/LIRGs, including IRAS F05189–2524 (e.g.",
"Imanishi & Terashima 2004) or in AGN accreting at a high Eddington ratio ($\\lambda = L_{\\rm bol,AGN}/L_{\\rm Edd}$ ).",
"We will return to the latter point later.",
"The absorption model fit applied to the NuSTAR spectrum ignores the Fe emission-line observed with XMM-Newton, Chandra and Suzaku.",
"Unless the line-emitting gas is hidden from our view by some contrived change in the nuclear obscuration, e.g.",
"a passage of a Compton thick cloud (Sanfrutos et al 2016), the Fe line strength needs to be explained by the illumination of the central source.",
"The observed mean line luminosity is $(7\\pm 2)\\times 10^{42}$ erg s$^{-1}$ .",
"XSTAR predicts, with the enhanced fluorescence yield of high-ionization Fe (e.g., Matt, Fabian & Ross 1996), compatible line luminosity can just be produced by photoionization of thick medium of $N_{\\rm H}$$\\sim 1\\times 10^{24}$ cm$^{-2}$ with log $\\xi =3$ by the central source discussed above, under favourable conditions, e.g.",
"a large covering factor, without suppression by the ionized absorber.",
"Unlike the absorbing gas, the line emitter has to be bound in the nucleus as it is observed at the rest energy while their ionization states are similar.",
"The line emitter could be the accretion disc surface or the dense, base part of the outflowing wind.",
"Alternatively, it could also be optically thin gas on a pc-scale.",
"Although the line flux variability is not conclusive (Sect.",
"3.2.1), its decline might be reverberation if a large flare of the central source occured before the 2003 XMM-Newton observation.",
"Besides the high infrared luminosity, strong outflow signatures (Spoon et al 2009; Heckman et al 1990) and the high-ionization gas inferred from the X-ray Fe features hint that the black hole in IRAS F00183–7111 may be accreting close to the Eddington limit ($\\lambda \\sim 1$ ), although there is no reliable way to measure its black hole mass.",
"If the above hypothesis of a moderately absorbed source is correct, the nuclear source is relatively X-ray quiet, which means a large X-ray bolometric correction, $k_{\\rm bol}$ , or a steep optical/UV to X-ray spectral slope.",
"In the correlation diagram of $k_{\\rm bol} - \\lambda $ for the COSMOS Type I AGN (Lusso et al 2010), $k_{\\rm bol}$ of IRAS F00183–7111 corresponds to $\\lambda \\sim 1$ .",
"We noticed that IRAS F00183–7111 shares interesting multiwavelength properties with the well-studied local ULIRG/BAL quasar, Mrk 231.",
"Mrk 231 exhibits strong galactic-scale outflows (Feruglio et al 2010; Fischer et al 2010; Cicone et al 2012; Veilleux et al 2016) as well as X-ray high-velocity winds (Feruglio et al 2015), has an X-ray quiet nuclear source (log $(L_{2-10}/L_{\\rm bol})\\sim -3$ ), shows an Fe xxv emission-line (Reynolds et al 2017; Teng et al 2014), and has a compact pc-scale radio source (Reynolds et al 2013).",
"We note that the $L_{\\rm 2-10}/L_{\\rm bol}$ estimated for IRAS F00183–7111 (Sect 4.1.2) becomes even closer to that of Mrk 231, if the covering factor $f$ is smaller than unity.",
"Mrk 231 was not as radio-loud as IRAS F00183–7111 (Fig.",
"7) but is becoming more radio-loud in recent years with elevated radio activity (Reynolds et al 2017).",
"Curiously, the mid-IR AGN tracer [Ne v]$\\lambda 14.32\\,\\mu $ m is not detected in either objects (Armus et al 2007; Spoon et al 2004, 2009).",
"Contrary to the common wisdom for AGN being variable X-ray sources, both IRAS F00183–7111 and Mrk 231 show stable intrinsic brightness over years.",
"The black hole mass and the AGN bolometric luminosity of Mrk 231 have been estimated in various methods but with large uncertainties.",
"Among the black hole mass measurements ranging from $1.3\\times 10^7M_{\\odot }$ to $6\\times 10^8 M_{\\odot }$ (Tacconi et al 2002; Davies et al 2004; Dasyra et al 2006; Kawakatu et al 2007; Leighly et al 2014) and the AGN bolometric luminosity estimates of (0.4-1.1)$\\times 10^{46}$ erg s$^{-1}$ (Lonsdale et al 2003; Farrah et al 2003; Veilleux et al 2009; Leighly et al 2014), we picked the respective medians ($M_{\\rm BH}=8.7\\times 10^7 M_{\\odot }$ by Kawakatu et al 2007 and $L_{\\rm bol,AGN}=8.4\\times 10^{45}$ erg s$^{-1}$ by Leighly et al 2014) and obtained an Eddington ratio of $\\lambda = 0.76$ for Mrk 231.",
"It indicates that the black hole in Mrk 231 is likely operating close to the critical accretion rate (e.g., Veilleux et al 2016).",
"This may also be supported by radio ejection events observed in the compact radio source (Reynolds et al 2017), if analogy to the stellar mass black holes (e.g., Fender, Belloni & Gallo 2004) applies, as radio ejection events occur only when they are accreting at $\\lambda \\sim 1$ in the activity hysteresis.",
"The production of the radio emission is thus not of the “radio mode” in the low accretion-rate regime but of the critical accretion, as seen in some radio-loud objects like 3C 120 (e.g.",
"Ballantyne et al 2004).",
"The characteristic similarities mentioned above suggest that IRAS F00183–7111 could also be a source of a high accretion rate.",
"The major divider between the two objects is the cold, line-of-sight absorber which imprints the deep silicate absorption in the mid-IR spectrum of IRAS F00183–7111.",
"Instead, the weak silicate absorption and the constraint on the jet angle ($<25^{\\circ }$ , Reynolds et al 2013) suggest that the inner nuclear structure in Mrk 231 is nearly face-on.",
"The AGN bolometric luminosity means a larger black hole mass in IRAS F00183–7111: $M_{\\rm BH}\\sim 3\\times 10^8 f^{-1} M_{\\odot }$ for $\\lambda = 1$ .",
"In fact, Type II AGN at $z\\sim 2.5$ showing high-ionization Fe K feature in the COSMOS field are found to be hosted by Hyperluminous ($L_{\\rm ir}\\sim 10^{13}L_{\\odot }$ ) IR galaxies with their IR SED similar to IRAS F00183–7111 and their nuclei are suspected to accrete close to $\\lambda =1$ (Iwasawa et al 2012).",
"Assuming both objects have a critical accretion disc, we speculate the cause of the strangely stable X-ray luminosity observed in the two sources might be a high optical depth of the thick disc where the central source is located.",
"Photon trapping starts to take effects on an X-ray source around $\\lambda \\sim 0.2$ (e.g., Wyithe & Loeb 2012) and multiple Thomson scatterings would smear out their intrinsic X-ray variability, although long term variability is difficult to wipe out by this effect.",
"Strong outflows observed in both objects are a natural consequence of critical accretion discs (e.g.",
"Ohsuga et al 2002; Begelman 2012).",
"The scientific results reported in this article are based on observations made by Chandra X-ray Observatory, Suzaku, NuSTAR and XMM-Newton, and has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology under contract with NASA.",
"Support for this work was partially provided by NASA through Chandra Award Number GO2-13122X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the NASA under contract NAS8-03060.",
"KI acknowledges support by the Spanish MINECO under grant AYA2016-76012-C3-1-P and MDM-2014-0369 of ICCUB (Unidad de Excelencia 'María de Maeztu').",
"Support from the ASI/INAF grant I/037/12/0 – 011/13 is acknowledged (AC, MB, EP, GL, RG and CV)."
]
] | 1709.01708 |
[
[
"Trace-Based Run-time Analysis of Message-Passing Go Programs"
],
[
"Abstract We consider the task of analyzing message-passing programs by observing their run-time behavior.",
"We introduce a purely library-based instrumentation method to trace communication events during execution.",
"A model of the dependencies among events can be constructed to identify potential bugs.",
"Compared to the vector clock method, our approach is much simpler and has in general a significant lower run-time overhead.",
"A further advantage is that we also trace events that could not commit.",
"Thus, we can infer alternative communications.",
"This provides the user with additional information to identify potential bugs.",
"We have fully implemented our approach in the Go programming language and provide a number of examples to substantiate our claims."
],
[
"Introduction",
"We consider run-time analysis of programs that employ message-passing.",
"Specifically, we consider the Go programming language [4] which integrates message-passing in the style of Communicating Sequential Processes (CSP) [6] into a C style language.",
"We assume the program is instrumented to trace communication events that took place during program execution.",
"Our objective is to analyze program traces to assist the user in identifying potential concurrency bugs.",
"falseblue func reuters(ch chan string) { ch <- \"REUTERS\" } // r!",
"func bloomberg(ch chan string) { ch <- \"BLOOMBERG\" } // b!",
"func newsReader(rCh chan string, bCh chan string) { ch := make(chan string) go func() { ch <- (<-rCh) }() // r?; ch!",
"go func() { ch <- (<-bCh) }() // b?; ch!",
"x := <-ch // ch? }",
"func main() { reutersCh := make(chan string) bloombergCh := make(chan string) go reuters(reutersCh) go bloomberg(bloombergCh) go newsReader(reutersCh, bloombergCh) // N1 newsReader(reutersCh, bloombergCh) // N2 } In Listing we find a Go program implementing a system of newsreaders.",
"The main function creates two synchronous channels, one for each news agency.",
"Go supports (a limited form of) type inference and therefore no type annotations are required.",
"Next, we create one thread per news agency via the keyword go.",
"Each news agency transmits news over its own channel.",
"In Go, we write ch <- \"REUTERS\" to send value \"REUTERS\" via channel ch.",
"We write <-ch to receive a value via channel ch.",
"As we assume synchronous channels, both operations block and only unblock once a sender finds a matching receiver.",
"We find two newsreader instances.",
"Each newsreader creates two helper threads that wait for news to arrive and transfer any news that has arrived to a common channel.",
"The intention is that the newsreader wishes to receive any news whether it be from Reuters or Bloomberg.",
"However, there is a subtle bug (to be explained shortly)."
],
[
"Trace-Based Run-Time Verification",
"We only consider finite program runs and therefore each of the news agencies supplies only a finite number of news (exactly one in our case) and then terminates.",
"During program execution, we trace communication events, e.g.",
"send and receive, that took place.",
"Due to concurrency, a bug may not manifest itself because a certain `bad' schedule is rarely taken in practice.",
"Here is a possible trace resulting from a `good' program run.",
"r!; N1.r?; N1.ch!; N1.ch?; b!; N2.b?; N2.ch!; N2.ch?",
"We write r!",
"to denote that a send event via the Reuters channel took place.",
"As there are two instances of the newsReader function, we write N1.r?",
"to denote that a receive event via the local channel took place in case of the first newsReader call.",
"From the trace we can conclude that the Reuters news was consumed by the first newsreader and the Bloomberg news by the second newsreader.",
"Here is a trace resulting from a bad program run.",
"r!; b!; N1.r?; N1.b?; N1.ch!; N1.ch?",
"; DEADLOCK The helper thread of the first newsreader receives the Reuters and the Bloomberg news.",
"However, only one of these messages will actually be read (consumed).",
"This is the bug!",
"Hence, the second newsreader gets stuck and we encounter a deadlock.",
"The issue is that such a bad program run may rarely show up.",
"So, the question is how can we assist the user based on the trace information resulting from a good program run?",
"How can we infer that alternative schedules and communications may exist?"
],
[
"Event Order via Vector Clock Method",
"A well-established approach is to derive a partial order among events.",
"This is usually achieved via a vector of (logical) clocks.",
"The vector clock method was independently developed by Fidge [1] and Mattern [8].",
"For the above good program run, we obtain the following partial order among events.",
"r! < N1.r? b! < N2.b?",
"N1.r? < N1.ch! N2.b? < N2.ch!",
"(1) N1.ch! < N1.ch? N2.ch! < N2.ch?",
"(2) For example, (1) arises because N2.ch!",
"happens (sequentially) after N2.b?",
"For synchronous send/receive, we assume that receive happens after send.",
"See (2).",
"Based on the partial order, we can conclude that alternative schedules are possible.",
"For example, b!",
"could take place before r!.",
"However, it is not clear how to infer alternative communications.",
"Recall that the issue is that one of the newsreaders may consume both news messages.",
"Our proposed method is able to clearly identify this issue and has the advantage to require a much simpler instrumentation We discuss these points shortly.",
"First, we take a closer look at the details of instrumentation for the vector clock method.",
"Vector clocks are a refinement of Lamport's time stamps [7].",
"Each thread maintains a vector of (logical) clocks of all participating partner threads.",
"For each communication step, we advance and synchronize clocks.",
"In pseudo code, the vector clock instrumentation for event sndR.",
"falseblue vc[reutersThread]++ ch <- (\"REUTERS\", vc, vcCh) vc' := max(vc, <-vcCh) We assume that vc holds the vector clock.",
"The clock of the Reuters thread is incremented.",
"Besides the original value, we transmit the sender's vector clock and a helper channel vcCh.",
"For convenience, we use tuple notation.",
"The sender's vector clock is updated by building the maximum among all entries of its own vector clock and the vector clock of the receiving party.",
"The same vector clock update is carried out on the receiver side."
],
[
"Our Method",
"We propose a much simpler instrumentation and tracing method to obtain a partial order among events.",
"Instead of a vector clock, each thread traces the events that might happen and have happened.",
"We refer to them as pre and post events.",
"In pseudo code, our instrumentation for sndR looks like follows.",
"falseblue pre(hash(ch), \"!\")",
"ch <- (\"REUTERS\", threadId) post(hash(ch), \"!\")",
"The bang symbol (`!')",
"indicates a send operation.",
"Function hash builds a hash index of channel names.",
"The sender transmits its thread id number to the receiver.",
"This is the only intra-thread overhead.",
"No extra communication link is necessary.",
"Here are the traces for individual threads resulting from the above good program run.",
"R: pre(r!); post(r!)",
"N1_helper1: pre(r?); post(R#r?); pre(ch1!); post(ch1!)",
"N1_helper2: pre(b?)",
"N1: pre(ch1?); post(N1_helper1#ch1?)",
"B: pre(b!); post(b!)",
"N2_helper1: pre(r?)",
"N2_helper2: pre(b?); post(B#b?); pre(ch2!); post(ch2!)",
"N2: pre(ch2?); post(N2_helper2#ch2?)",
"We write pre(r!)",
"to indicate that a send via the Reuters channel might happen.",
"We write post(R#r?)",
"to indicate that a receive has happened via thread R. The partial order among events is obtained by a simple post-processing phase where we linearly scan through traces.",
"For example, within a trace there is a strict order and therefore N2_helper2: pre(b?); post(B#b?); pre(ch2!); post(ch2!)",
"implies N2.b?",
"< N2.ch!.",
"Across threads we check for matching pre/post events.",
"Hence, R: pre(r!); post(r!)",
"N1_helper1: pre(r?); post(R#r?); ...",
"implies r!",
"< N1.r?.",
"So, we obtain the same (partial order) information as the vector clock approach but with less overhead.",
"The reduction in terms of tracing overhead compared to the vector clock method is rather drastic assuming a library-based tracing scheme with no access to the Go run-time system.",
"For each communication event we must exchange vector clocks, i.e.",
"$n$ additional (time stamp) values need to be transmitted where $n$ is the number of threads.",
"Besides extra data to be transmitted, we also require an extra communication link because the sender requires the receivers vector clock.",
"In contrast, our method incurs a constant tracing overhead.",
"Each sender transmits in addition its thread id.",
"No extra communication link is necessary.",
"This results in much less run-time overhead as we will see later.",
"The vector clock tracing method can be improved assuming we extend the Go run-time system.",
"For example, by maintaining a per-thread vector clock and having the run-time system carrying out the exchange of vector clocks for each send/receive communication.",
"There is still the $O(n)$ space overhead.",
"Our method does not require any extension of the Go run-time system to be efficient and therefore is also applicable to other languages that offer similar features as found in Go.",
"A further advantage of our method is that we also trace (via pre) events that could not commit (post is missing).",
"Thus, we can easily infer alternative communications.",
"For example, for R: pre(r!",
"); ... there is the alternative match N2_helper1: pre(r?).",
"Hence, instead of r!",
"< N1.r?",
"also r!",
"< N2.r?",
"is possible.",
"This indicates that one newsreader may consume both news message.",
"The vector clock method, only traces events that could commit, post events in our notation.",
"Hence, the above alternative communication could not be derived."
],
[
"Contributions",
"Compared to earlier works based on the vector clock method, we propose a much more light-weight and more informative instrumentation and tracing scheme.",
"Specifically, we make the following contributions: We give a precise account of our run-time tracing method (Section ) for message-passing as found in the Go programming language (Section ) where for space reasons we only formalize the case of synchronous channels and selective communications.",
"A simple analysis of the resulting traces allows us to detect alternative schedules and communications (Section ).",
"For efficiency reasons, we employ a directed dependency graph to represent happens-before relations (Section REF ).",
"We show that vector clocks can be easily recovered based on our tracing method (Section ).",
"We also discuss the pros and cons of both methods for analysis purposes.",
"Our tracing method can be implemented efficiently as a library.",
"We have fully implemented the approach supporting all Go language features dealing with message-passing such as buffered channels, select with default or timeout and closing of channels (Section ).",
"We provide experimental results measuring the often significantly lower overhead of our method compared to the vector clock method assuming based methods are implemented as libraries (Section REF ).",
"The online version of this paper contains an appendix with further details.https://arxiv.org/abs/1709.01588"
],
[
"Syntax",
"For brevity, we consider a much simplified fragment of the Go programming language.",
"We only cover straight-line code, i.e.",
"omitting procedures, if-then-else etc.",
"This is not an onerous restriction as we only consider finite program runs.",
"Hence, any (finite) program run can be represented as a program consisting of straight-line code only.",
"[Program Syntax] $\\begin{array}{lcll}x,y, &\\dots & &\\mbox{Variables, Channel Names}\\\\ i,j, &\\dots && \\mbox{Integers}\\\\b & ::= & x \\mid i \\mid {\\sf hash}(x) \\mid {\\sf head}(b) \\mid {\\sf last}(b) \\mid \\textit {bs} \\mid \\mbox{\\sf {tid}}& \\mbox{Expressions}\\\\ \\textit {bs} & ::= & [] \\mid b : \\textit {bs}\\\\e,f & ::= & x \\leftarrow b \\mid y :=\\leftarrow x & \\mbox{Transmit/Receive}\\\\c & ::= & y :=b \\mid y :=\\mbox{\\sf makeChan}\\mid \\mbox{\\sf go}\\ p \\mid \\mbox{\\sf select}\\ [e_i \\Rightarrow p_i]_{i\\in I} & \\mbox{Commands}\\\\p,q,r & ::= & [] \\mid c : p & \\mbox{Program}\\end{array}$ For our purposes, values are integers or lists (slices in Go terminology).",
"For lists we follow Haskell style notation and write $b : bs$ to refer to a list with head element $b$ and tail $bs$ .",
"We can access the head and last element in a list via primitives ${\\sf head}$ and ${\\sf last}$ .",
"We often write $[b_1,\\dots ,b_n]$ as a shorthand $b_1:\\dots : []$ .",
"Primitive $\\mbox{\\sf {tid}}$ yields the thread id number of the current thread.",
"We assume that the main thread always has thread id number 1 and new thread id numbers are generated in increasing order.",
"Primitive ${\\sf hash}()$ yields a unique hash index for each variable name.",
"Both primitives show up in our instrumentation.",
"A program is a sequence of commands where commands are stored in a list.",
"Primitive $\\mbox{\\sf makeChan}$ creates a new synchronous channel.",
"Primitive $\\mbox{\\sf go}$ creates a new go routine (thread).",
"For send and receive over a channel we follow Go notation.",
"We assume that a receive is always tied to an assignment.",
"For assignment we use symbol $:=$ to avoid confusion with the mathematical equality symbol $=$ .",
"In Go, symbol $:=$ declares a new variable with some initial value.",
"We also use $:=$ to overwrite the value of existing variables.",
"As a message passing command we only support selective communication via select.",
"Thus, we can fix the bug in our newsreader example.",
"falseblue func newsReaderFixed(rCh chan string, bCh chan string) { ch := make(chan string) select { case x := <-rCh: case x := <-bCh: } } The select statement guarantees that at most one news message will be consumed and blocks if no news are available.",
"In our simplified language, we assume that the $x \\leftarrow b$ command is a shorthand for $\\mbox{\\sf select}\\ [x \\leftarrow b \\Rightarrow []]$ .",
"For space reasons, we omit buffered channels, select paired with a default/timeout case and closing of channels.",
"All three features are fully supported by our implementation."
],
[
"Trace-Based Semantics",
"The semantics of programs is defined via a small-step operational semantics.",
"The semantics keeps track of the trace of channel-based communications that took place.",
"This allows us to relate the traces obtained by our instrumentation with the actual run-time traces.",
"We support multi-threading via a reduction relation $(S, [i_1 \\sharp p_1,\\dots ,i_n \\sharp p_n]) 0359{}{T} (S^{\\prime }, [j_1 \\sharp q_1,\\dots ,j_n \\sharp q_n]).$ We write $i \\sharp p$ to denote a program $p$ that runs in its own thread with thread id $i$ .",
"We use lists to store the set of program threads.",
"The state of program variables, before and after execution, is recorded in $S$ and $S^{\\prime }$ .",
"We assume that threads share the same state.",
"Program trace $T$ records the sequence of communications that took place during execution.",
"We write $x !$ to denote a send operation on channel $x$ and $x ?$ to denote a receiver operation on channel $x$ .",
"The semantics of expressions is defined in terms a big-step semantics.",
"We employ a reduction relation $(i, S) \\, \\vdash \\,b \\Downarrow v$ where $S$ is the current state, $b$ the expression and $v$ the result of evaluating $b$ .",
"The formal details follow.",
"[State] $\\begin{array}{lcll}v & ::= & x \\mid i \\mid [] \\mid \\textit {vs} & \\mbox{Values}\\\\\\textit {vs} & ::= & [] \\mid v : \\textit {vs}\\\\ s & ::= & v \\mid \\mathit {Chan}& \\mbox{Storables}\\\\S & ::= & () \\mid (x \\mapsto s) \\mid S \\lhd S & \\mbox{State}\\end{array}$ A state $S$ is either empty, a mapping, or an override of two states.",
"Each state maps variables to storables.",
"A storable is either a plain value or a channel.",
"Variable names may appear as values.",
"In an actual implementation, we would identify the variable name by a unique hash index.",
"We assume that mappings in the right operand of the map override operator $\\lhd $ take precedence.",
"They overwrite any mappings in the left operand.",
"That is, $(x \\mapsto v_1) \\lhd (x \\mapsto v_2) = (x \\mapsto v_2)$ .",
"[Expression Semantics $(i, S) \\, \\vdash \\,b \\Downarrow v$ ] $\\begin{array}{c}{\\begin{array}{c} S(x) = v\\\\ \\hline (i, S) \\, \\vdash \\,x \\Downarrow v \\end{array}}\\ \\ \\ \\ (i, S) \\, \\vdash \\,j \\Downarrow j\\ \\ \\ \\ (i, S) \\, \\vdash \\,[] \\Downarrow []\\ \\ \\ \\ {\\begin{array}{c} (i, S) \\, \\vdash \\,b \\Downarrow v\\ (i, S) \\, \\vdash \\,\\textit {bs} \\Downarrow \\textit {vs}\\\\ \\hline (i, S) \\, \\vdash \\,b:\\textit {bs} \\Downarrow v:\\textit {vs} \\end{array}}\\\\{\\begin{array}{c} (i, S) \\, \\vdash \\,b \\Downarrow v : \\textit {vs}\\\\ \\hline (i, S) \\, \\vdash \\,{\\sf head}(b) \\Downarrow v \\end{array}}\\ \\ \\ \\ {\\begin{array}{c} (i, S) \\, \\vdash \\,b \\Downarrow [v_1,\\dots ,v_n]\\\\ \\hline (i, S) \\, \\vdash \\,{\\sf last}(b) \\Downarrow v_n \\end{array}}\\ \\ \\ \\ (i, S) \\, \\vdash \\,\\mbox{\\sf {tid}} \\Downarrow i\\ \\ \\ \\ (i, S) \\, \\vdash \\,{\\sf hash}(x) \\Downarrow x\\end{array}$ [Program Execution $(S, P) 0359{}{T} (S^{\\prime }, Q)$ ] $\\begin{array}{lcll}i \\sharp p &&& \\mbox{Single program thread}\\\\ P,Q & ::= & [] \\mid i \\sharp p : P & \\text{Program threads}\\\\t & :=& i \\sharp x!",
"\\mid i \\leftarrow j \\sharp x?",
"& \\text{Send and receive event}\\\\T & ::= & [] \\mid t : T & \\mbox{Trace}\\end{array}$ We write $(S, P) 0359{}{} (S^{\\prime }, Q)$ as a shorthand for $(S, P) 0359{}{[]} (S^{\\prime }, Q)$ .",
"[Single Step] $\\begin{array}{c}\\mbox{(Terminate)} \\ (S, i \\sharp [] : P) 0359{}{} (S, P)\\\\\\\\\\mbox{(Assign)} \\ {\\begin{array}{c} (i, S) \\, \\vdash \\,b \\Downarrow v \\ \\ \\ S^{\\prime } = S \\lhd (y \\mapsto v)\\\\ \\hline (S, i \\sharp (y :=b:p):P) 0359{}{} (S^{\\prime }, i \\sharp p:P) \\end{array}}\\\\\\\\\\mbox{(MakeChan)} \\ {\\begin{array}{c} S^{\\prime } = S \\lhd (y \\mapsto \\mathit {Chan}\\\\ \\hline (S, i \\sharp (y :=\\mbox{\\sf makeChan}:p):P) 0359{}{} (S^{\\prime }, i \\sharp p:P) \\end{array}}\\end{array}$ [Multi-Threading and Synchronous Message-Passing] $\\begin{array}{c}\\mbox{(Go)} \\ {\\begin{array}{c} i \\notin \\lbrace i_1,\\dots ,i_n\\rbrace \\\\ \\hline (S, i_1 \\sharp (\\mbox{\\sf go}\\ p:p_1):P) 0359{}{} (S, i \\sharp p:i_1 \\sharp p_1:P) \\end{array}}\\\\\\\\\\mbox{(Sync)} \\ {\\begin{array}{c} \\exists l \\in J, m \\in K. e_l = x \\leftarrow b \\ \\ \\ f_m = y :=\\leftarrow x\\ \\ \\ S(x) = \\mathit {Chan}\\\\ (i_1, S) \\, \\vdash \\,b \\Downarrow v \\ \\ \\ S^{\\prime } = S \\lhd (y \\mapsto v)\\\\ \\hline (S, i_1 \\sharp (\\mbox{\\sf select}\\ [e_j \\Rightarrow q_j]_{j\\in J}:p_1):i_2 \\sharp (\\mbox{\\sf select}\\ [f_k \\Rightarrow r_k]_{k\\in K}:p_2):P)\\\\ 0359{}{[i_1 \\sharp x!, i_2 \\leftarrow i_1 \\sharp x?]}",
"\\\\ (S^{\\prime }, i_1 \\sharp (q_l \\ \\texttt {++}\\ p_1):i_2 \\sharp (r_m \\ \\texttt {++}\\ p_2):P) \\end{array}}\\end{array}$ [Scheduling] $\\begin{array}{c}\\mbox{(Schedule)} \\ {\\begin{array}{c} \\mbox{$\\pi $ permutation on $\\lbrace 1,\\dots ,n\\rbrace $}\\\\ \\hline (S, [i_1 \\sharp p_1, \\dots , i_n \\sharp p_n]) 0359{}{} (S, [\\pi (i_1) \\sharp p_{\\pi (1)}, \\dots , \\pi (i_n) \\sharp p_{\\pi (n)}]) \\end{array}}\\\\\\\\\\mbox{(Closure)} \\ {\\begin{array}{c} (S, P) 0359{}{T} (S^{\\prime }, P^{\\prime }) \\ \\ \\ (S^{\\prime }, P^{\\prime }) 0359{}{T^{\\prime }} (S^{\\prime \\prime }, P^{\\prime \\prime })\\\\ \\hline (S, P) 0359{}{T \\ \\texttt {++}\\ T^{\\prime }} (S^{\\prime \\prime }, P^{\\prime \\prime }) \\end{array}}\\end{array}$"
],
[
"Instrumentation and Run-Time Tracing",
"For each message passing primitive (send/receive) we log two events.",
"In case of send, (1) a pre event to indicate the message is about to be sent, and (2) a post event to indicate the message has been sent.",
"The treatment is analogous for receive.",
"In our instrumentation, we write $x !$ to denote a single send event and $x ?$ to denote a single receive event.",
"These notations are shorthands and can be expressed in terms of the language described so far.",
"We use $\\equiv $ to define short-forms and their encodings.",
"We define $x !",
"\\equiv [{\\sf hash}(x),1]$ and $x ?",
"\\equiv [{\\sf hash}(x),0]$ .",
"That is, send is represented by the number 1 and receive by the number 0.",
"As we support non-deterministic selection, we employ a list of pre events to indicate that one of several events may be chosen For example, $\\mathit {pre}([x !, y ?",
"])$ indicates that there is the choice among sending over channel $x$ and receiving over channel $y$ .",
"This is again a shorthand notation where we assume $\\mathit {pre}([b_1,\\dots ,b_n]) \\equiv [0,b_1,\\dots ,b_n]$ .",
"A post event is always singleton as at most one of the possible communications is chosen.",
"As we also trace communication partners, we assume that the sending party transmits its identity, the thread id, to the receiving party.",
"We write $\\mathit {post}(i \\sharp x ?",
")$ to denote reception via channel $x$ where the sender has thread id $i$ .",
"In case of a post send event, we simply write $\\mathit {post}(x !",
")$ .",
"The above are yet again shorthands where $i \\sharp x ?",
"\\equiv [{\\sf hash}(x),0,i]$ and $\\mathit {post}(b) \\equiv [1,b]$ .",
"Pre and post events are written in a fresh thread local variable, denoted by $x_{tid}$ where $tid$ refers to the thread's id number.",
"At the start of the thread the variable is initialized by $x_{tid} :=[]$ .",
"Instrumentation ensures that pre and post events are appropriately logged.",
"As we keep track of communication partners, we must also inject and project messages with additional information (the sender's thread id).",
"We consider instrumentation of $\\mbox{\\sf select}\\ [x \\leftarrow 1 \\Rightarrow [], y :=\\leftarrow x \\Rightarrow [ z \\leftarrow y]].$ We assume the above program text is part of a thread with id number 1.",
"We non-deterministically choose between a send an receive operation.",
"In case of receive, the received value is further transmitted.",
"Instrumentation yields the following.",
"$\\begin{array}{l}[x_1 :=x_1 \\ \\texttt {++}\\ \\mathit {pre}([x !, x ?",
"]),\\\\\\ \\mbox{\\sf select}\\ [x \\leftarrow [\\mbox{\\sf {tid}}, 1] \\Rightarrow [x_1 :=x_1 \\ \\texttt {++}\\ \\mathit {post}(x !",
")],\\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ y^{\\prime } :=\\leftarrow x \\Rightarrow [ x_1 :=x_1 \\ \\texttt {++}\\ \\mathit {post}({\\sf head}(y^{\\prime }) \\sharp x ?",
"), y :={\\sf last}(y^{\\prime }),\\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ z \\leftarrow [\\mbox{\\sf {tid}}, y]]]\\end{array}$ We first store the pre events, either a read or send via channel $x$ .",
"The send is instrumented by additionally transmitting the senders thread id.",
"The post event for this case simply logs that a send took place.",
"Instrumentation of receive is slightly more involved.",
"As senders supply their thread id, we introduce a fresh variable $y^{\\prime }$ .",
"Via ${\\sf head}(y^{\\prime })$ we extract the senders thread id to properly record the communication partner in the post event.",
"The actual value transmitted is accessed via ${\\sf last}(y^{\\prime })$ .",
"[Instrumentation of Programs] We write $\\mathit {instr}(p) = q$ to denote the instrumentation of program $p$ where $q$ is the result of instrumentation.",
"Function $\\mathit {instr}(\\cdot )$ is defined by structural induction on a program.",
"We assume a similar instrumentation function for commands.",
"$\\begin{array}{lcl}\\mathit {instr}([]) & = & []\\\\\\mathit {instr}(c:p) & = & \\mathit {instr}(c) : \\mathit {instr}(p)\\\\\\\\\\mathit {instr}(y :=b) & = & [y :=b]\\\\\\mathit {instr}(y :=\\mbox{\\sf makeChan}) & = & [y :=\\mbox{\\sf makeChan}]\\\\\\mathit {instr}(\\mbox{\\sf go}\\ p) & = & [\\mbox{\\sf go}\\ ([x_{tid} :=[] \\ \\texttt {++}\\ \\mathit {instr}(p)])]\\\\\\mathit {instr}(\\mbox{\\sf select}\\ [e_i \\Rightarrow p_i]_{i\\in \\lbrace 1,\\dots ,n\\rbrace })& = & [x_{tid} :=x_{tid} \\ \\texttt {++}\\ [\\mathit {pre}([\\mathit {retr}(e_1),\\dots ,\\mathit {retr}(e_n)])],\\\\ & &\\mbox{\\sf select}\\ [\\mathit {instr}(e_i \\Rightarrow p_i)]_{i \\in \\lbrace 1,\\dots ,n\\rbrace }]\\\\\\mathit {instr}(x \\leftarrow b \\Rightarrow p)& = & x \\leftarrow [\\mbox{\\sf {tid}},b] \\Rightarrow (x_{tid} :=x_{tid} \\ \\texttt {++}\\ [\\mathit {post}(x !)])",
"\\ \\texttt {++}\\ \\mathit {instr}(p)\\\\\\mathit {instr}(y :=\\leftarrow x \\Rightarrow p)& = & y^{\\prime } :=\\leftarrow x \\Rightarrow [x_{tid} :=x_{tid} \\ \\texttt {++}\\ [\\mathit {post}({\\sf head}(y^{\\prime }) \\sharp x ?",
")],\\\\ && \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ y :={\\sf last}(y^{\\prime })] \\ \\texttt {++}\\ \\mathit {instr}(p)\\end{array}$ $\\begin{array}{c}\\mathit {retr}(x \\leftarrow b) = x !\\ \\ \\ \\ \\mathit {retr}(y = \\leftarrow x) = x ?\\end{array}$ Run-time tracing proceeds as follows.",
"We simply run the instrumented program and extract the local traces connected to variables $x_{tid}$ .",
"We assume that thread id numbers are created during program execution and can be enumerated by $1\\dots n$ for some $n>0$ where thread id number 1 belongs to the main thread.",
"[Run-Time Tracing] Let $p$ and $q$ be programs such that $\\mathit {instr}(p) = q$ .",
"We consider a specific instrumented program run where $((), [1 \\sharp [x_1 :=[]] \\ \\texttt {++}\\ q]) 0359{}{T} (S, 1 \\sharp []:P)$ for some $S$ , $T$ and $P$ .",
"Then, we refer to $T$ as $p$ 's actual run-time trace.",
"We refer to the list $[1 \\sharp S(x_1),\\dots ,n \\sharp S(x_n)]$ as the local traces obtained via the instrumentation of $p$ .",
"Command $x_1 :=[]$ is added to the instrumented program to initialize the trace of the main thread.",
"Recall that main has thread id number 1.",
"This extra step is necessary because our instrumentation only initializes local traces of threads generated via go.",
"The final configuration $(S,1 \\sharp []:P)$ indicates that the main thread has run to full completion.",
"This is a realistic assumption as we assume that programs exhibit no obvious bug during execution.",
"There might still be some pending threads, in case $P$ differs from the empty list."
],
[
"Trace Analysis",
"We assume that the program has been instrumented and after some program run we obtain a list of local traces.",
"We show that the actual run-time trace can be recovered and we are able to point out alternative behaviors that could have taken place.",
"Alternative behaviors are either due alternative schedules or different choices among communication partners.",
"We consider the list of local traces $[1 \\sharp S(x_1), \\dots , n \\sharp S(x_n)]$ .",
"Their shape can be characterized as follows.",
"[Local Traces] $\\begin{array}{lcl}U,V & ::= & [] \\mid i \\sharp L : U\\\\L & ::= & [] \\mid \\mathit {pre}(as) : M\\\\as & ::= & [] \\mid x !",
": as \\mid x ?",
": as\\\\M & ::= & [] \\mid \\mathit {post}(x !)",
": L\\mid \\mathit {post}(i \\sharp x ?)",
": L\\end{array}$ We refer to $U=[1 \\sharp L_1,\\dots ,n \\sharp L_n]$ as a residual list of local traces if for each $L_i$ either $L_i = []$ or $L_i=[\\mathit {pre}(\\dots )]$ .",
"To recover the communications that took place we check for matching pre and post events recorded in the list of local traces.",
"For this purpose, we introduce a relation $U 0359{}{T} V$ to denote that `replaying' of $U$ leads to $V$ where communications $T$ took place.",
"Valid replays are defined via the following rules.",
"[Replay $U 0359{}{T} V$ ] $\\begin{array}{c}\\end{array}\\mbox{(Sync)} \\ {\\begin{array}{c} L_1 = \\mathit {pre}([\\dots ,x !,\\dots ]) : \\mathit {post}(x !)",
": L_1^{\\prime }\\\\ L_2 = \\mathit {pre}([\\dots ,x ?, \\dots ]) : \\mathit {post}(i_1 \\sharp x ?)",
": L_2^{\\prime }\\\\ \\hline i_1 \\sharp L_1 :i_2 \\sharp L_2 : U 0359{}{[i_1 \\sharp x!, i_2 \\leftarrow i_1 \\sharp x?]}",
"i_1 \\sharp L_1^{\\prime } : i_2 \\sharp L_2^{\\prime } : U \\end{array}}\\\\\\\\\\mbox{(Schedule)} \\ {\\begin{array}{c} \\mbox{$\\pi $ permutation on $\\lbrace 1,\\dots ,n\\rbrace $}\\\\ \\hline [i_1 \\sharp L_1,\\dots ,i_n \\sharp L_n] 0359{}{[]} [i_{\\pi (1)} \\sharp L_{\\pi (1)},\\dots ,i_{\\pi (n)} \\sharp L_{\\pi (n)}] \\end{array}}\\\\\\\\\\mbox{(Closure)} \\ {\\begin{array}{c} U 0359{}{T} U^{\\prime } \\ \\ U^{\\prime } 0359{}{T^{\\prime }} U^{\\prime \\prime }\\\\ \\hline U 0359{}{T \\ \\texttt {++}\\ T^{\\prime }} U^{\\prime \\prime } \\end{array}}$ $$ Rule (Sync) checks for matching communication partners.",
"In each trace, we must find complementary pre events and the post events must match as well.",
"Recall that in the instrumentation the sender transmits its thread id to the receiver.",
"Rule (Schedule) shuffles the local traces as rule (Sync) only considers the two leading local traces.",
"Via rule (Closure) we perform repeated replay steps.",
"We can state that the actual run-time trace can be obtained via the replay relation $U 0359{}{T} V$ but further run-time traces are possible.",
"This is due to alternative schedules.",
"[Replay Yields Run-Time Traces] Let $p$ be a program and $q$ its instrumentation where for a specific program run we observe the actual behavior $T$ and the list $[1 \\sharp L_1,\\dots ,n \\sharp L_n]$ of local traces.",
"Let ${\\cal T} = \\lbrace T^{\\prime } \\mid [1 \\sharp L_1,\\dots ,n \\sharp L_n] 0359{}{T^{\\prime }} 1 \\sharp []:U\\ \\mbox{for some residual $U$} \\rbrace $ .",
"Then, we find that $T \\in {\\cal T}$ and for each $T^{\\prime } \\in {\\cal T}$ we have that $((), p) 0359{}{T^{\\prime }} (S, 1 \\sharp [] : P)$ for some $S$ and $P$ .",
"[Alternative Schedules] We say $[1 \\sharp L_1,\\dots ,n \\sharp L_n]$ contains alternative schedules iff the cardinality of the set $\\lbrace T^{\\prime } \\mid [1 \\sharp L_1,\\dots ,n \\sharp L_n] 0359{}{T^{\\prime }} 1 \\sharp []:U\\ \\mbox{for some residual $U$} \\rbrace $ is greater than one.",
"We can also check if even further run-time traces might have been possible by testing for alternative communications.",
"[Alternative Communications] We say $[1 \\sharp L_1,\\dots ,n \\sharp L_n]$ contains alternative matches iff for some $i,j, x, L, L^{\\prime }$ we have that (1) $L_i = \\mathit {pre}([\\dots ,x !,\\dots ]) : L$ , (2) $L_j = \\mathit {pre}([\\dots ,x ?, \\dots ]) : L^{\\prime }$ , and (3) if $L = \\mathit {post}(x !",
"):L^{\\prime \\prime }$ for some $L^{\\prime \\prime }$ then $L^{\\prime } \\ne \\mathit {post}(j \\sharp x ?)",
": L^{\\prime \\prime \\prime }$ for any $L^{\\prime \\prime \\prime }$ .",
"We say $U=[1 \\sharp L_1,\\dots ,n \\sharp L_n]$ contains alternative communications iff $U$ contains alternative matches or there exists $T$ and $V$ such that $U 0359{}{T} V$ and $V$ contains alternative matches.",
"The alternative match condition states that a sender could synchronize with a receiver (see (1) and (2)) but this synchronization did not take place (see (3)).",
"For an alternative match to result in an alternative communication, the match must be along a possible run-time trace."
],
[
"Dependency Graph for Efficient Trace Analysis",
"Instead of replaying traces to check for alternative schedules and communications, we build a dependency graph where the graph captures the partial order among events.",
"It is much more efficient to carry out the analysis on the graph than replaying traces.",
"Figure REF shows a simple example.",
"We find a program that makes use of two channels and four threads.",
"For reference, send/receive events are annotated (as subscript) with unique numbers.",
"We omit the details of instrumentation and assume that for a specific program run we find the list of given traces on the left.",
"Pre events consist of singleton lists as there is no select.",
"Hence, we write $\\mathit {pre}((y ?",
")_{6})$ as a shorthand for $\\mathit {pre}([(y ?",
")_{6}])$ .",
"Replay of the trace shows that the following locations synchronize with each other: $(4,6)$ , $(3,1)$ and $(5,2)$ .",
"This information as well as the order among events can be captured by a dependency graph.",
"Nodes are obtained by a linear scan through the list of traces.",
"To derive edges, we require another scan for each element in a trace as we need to find pre/post pairs belonging to matching synchronizations.",
"This results overall in $O(m*m)$ for the construction of the graph where $m$ is the number of elements found in each trace.",
"To avoid special treatment of dangling pre events (with not subsequent post event), we assume that some dummy post events are added to the trace.",
"[Construction of Dependency Graph] Each node corresponds to a send or a receive operation in the program text.",
"Edges are constructed by observing events recorded in the list of traces.",
"We draw a (directed) edge among nodes if either the pre and post events of one node precede the pre and post events of another node in the trace, or the pre and post events belonging to both nodes can be synchronized.",
"See rule (Sync) in Definition .",
"We assume that the edge starts from the node with the send operation.",
"Applied to our example, this results in the graph on the right.",
"See Figure REF .",
"For example, $x !|3$ denotes a send communication over channel $x$ at program location 3.",
"As send precedes receive we find an edge from $x !|3$ to $x ?|1$ .",
"In general, there may be several initial nodes.",
"By construction, each node has at most one outgoing edge but may have multiple incoming edges.",
"The trace analysis can be carried out directly on the dependency graph.",
"To check if one event happens-before another event we seek for a path from one event to the other.",
"This can be done via a depth-first search and takes time $O(v+e)$ where $v$ is the number of nodes and $e$ the number of edges.",
"Two events are concurrent if neither happens-before the other.",
"To check for alternative communications, we check for matching nodes that are concurrent to each other.",
"By matching we mean that one of the nodes is a send and the other is a receive over the same channel.",
"For our example, we find that $x !|5$ and $x ?|1$ represents an alternative communication as both nodes are matching and concurrent to each other.",
"To derive (all) alternative schedules, we perform a backward traversal of the graph.",
"Backward in the sense that we traverse the graph by moving from children to parent node.",
"We start with some final node (no outgoing edge).",
"Each node visited is marked.",
"We proceed to the parent if all children are marked.",
"Thus, we guarantee that the happens-before relation is respected.",
"For our example, suppose we visit first $y ?",
"{6}$ .",
"We cannot visit its parent $y !",
"{4}$ until we have visited $x ?",
"{2}$ and $x !",
"{5}$ .",
"Via a (backward) breadth-first search we can `accumulate' all schedules."
],
[
"Comparison to Vector Clock Method",
"Via a simple adaptation of the Replay Definition we can attach vector clocks to each send and receive event.",
"Hence, our tracing method strictly subsumes the vector clock method as we are also able to trace events that could not commit.",
"[Vector Clock] $\\begin{array}{lcll}cs & ::= & [] \\mid n : cs\\end{array}$ For convenience, we represent a vector clock as a list of clocks where the first position belongs to thread 1 etc.",
"We write $cs[i]$ to retrieve the $i$ -th component in $cs$ .",
"We write ${\\sf inc}(i,cs)$ to denote the vector clock obtained from $cs$ where all elements are the same but at index $i$ the element is incremented by one.",
"We write ${\\sf max}(cs_1,cs_2)$ to denote the vector clock where we per-index take the greater element.",
"We write $i^{cs}$ to denote thread $i$ with vector clock $cs$ .",
"We write $i \\sharp x!^{cs}$ to denote a send over channel $x$ in thread $i$ with vector clock $cs$ .",
"We write $i \\leftarrow j \\sharp x?^{cs}$ to denote a receive over channel $x$ in thread $i$ from thread $j$ with vector clock $cs$ .",
"[From Trace Replay to Vector Clocks] $\\begin{array}{c}\\end{array}\\mbox{(Sync)} \\ {\\begin{array}{c} L_1 = \\mathit {pre}([\\dots ,x !,\\dots ]) : \\mathit {post}(x !)",
": L_1^{\\prime }\\\\ L_2 = \\mathit {pre}([\\dots ,x ?, \\dots ]) : \\mathit {post}(i_1 \\sharp x ?)",
": L_2^{\\prime }\\\\ cs = {\\sf max}({\\sf inc}(i_1,cs_1),{\\sf inc}(i_2,cs_2))\\\\ \\hline i_1^{cs_1} \\sharp L_1 :i_2^{cs_2} \\sharp L_2 : U 0359{}{[i_1 \\sharp x!^{cs}, i_2 \\leftarrow i_1 \\sharp x?^{cs}]} i_1^{cs} \\sharp L_1^{\\prime } : i_2^{cs} \\sharp L_2^{\\prime } : U \\end{array}}$ $$ Like the construction of the dependency graph, the (re)construction of vector clocks takes time $O(m*m)$ where $m$ is the number of elements found in each trace.",
"To check for an alternative communication, the vector clock method seeks for matching events.",
"This incurs the same (quadratic in the size of the trace) cost as for our method.",
"However, the check that these two events are concurrent to each other can be performed more efficiently via vector clocks.",
"Comparison of vector clocks takes time $O(n)$ where $n$ is the number of threads.",
"Recall that our graph-based method requires time $O(v+e)$ where $v$ is the number of nodes and $e$ the number of edges.",
"The number $n$ is smaller than $v+e$ .",
"However, our dependency graph representation is more efficient in case of exploring alternative schedules.",
"In case of the vector clock method, we need to continuously compare vector clocks whereas we only require a (backward) traversal of the graph.",
"We believe that the dependency graph has further advantages in case of user interaction and visualization as it is more intuitive to navigate through the graph.",
"This is something we intend to investigate in future work."
],
[
"Implementation",
"We have fully integrated the approach laid out in the earlier sections into the Go programming language and have built a prototype tool.",
"We give an overview of our implementation which can be found here [5].",
"A detailed treatment of all of Go's message-passing features can be found in the extended version of this paper."
],
[
"Library-Based Instrumentation and Tracing",
"We use a pre-processor to carry out the instrumentation as described in Section .",
"In our implementation, each thread maintains an entry in a lock-free hashmap where each entry represents a thread (trace).",
"The hashmap is written to file either at the end of the program or when a deadlock occurs.",
"We currently do not deal with the case that the program crashes as we focus on the detection of potential bugs in programs that do not show any abnormal behavior."
],
[
"Measurement of Run-Time Overhead Library-Based Tracing",
"We measure the run-time overhead of our method against the vector clock method.",
"Both methods are implemented as libraries assuming no access to the Go run-time system.",
"For experimentation we use three programs where each program exercises some of the factors that have an impact on tracing.",
"For example, dynamic versus static number of threads and channels.",
"Low versus high amount of communication among threads.",
"The Add-Pipe (AP) example uses $n$ threads where the first $n-1$ threads receive on an input channel, add one to the received value and then send the new value on their output channel to the next thread.",
"The first thread sends the initial value and receives the result from the last thread.",
"In the Primesieve (PS) example, the communication among threads is similar to the Add-Pipe example.",
"The difference is that threads and channels are dynamically generated to calculate the first $n$ prime numbers.",
"For each found prime number a `filter' thread is created.",
"Each thread has an input channel to receive new possible prime numbers $v$ and an output channel to report each number for which $v \\mathit {}\\mod {\\mathit {p}rime} \\ne 0$ where ${\\mathit {p}rime}$ is the prime number associated with this filter thread.",
"The filter threads are run in a chain where the first thread stores the prime number 2.",
"The Collector (C) example creates $n$ threads that produce a number which is then sent to the main thread for collection.",
"This example has much fewer communications compared to the other examples but uses a high number of threads.",
"Figure REF summarizes our results.",
"Results are carried out on some commodity hardware (Intel i7-6600U with 12 GB RAM, a SSD and Go 1.8.3 running on Windows 10 was used for the tests).",
"Our results show that a library-based implementation of the vector clock method does not scale well for examples with a dynamic number of threads and/or a high amount communication among threads.",
"See examples Primesieve and Add-Pipe.",
"None of the vector clock optimizations [3] apply here because of the dynamic number of threads and channels.",
"Our method performs much better.",
"This is no surprise as we require less (tracing) data and no extra communication links.",
"We believe that the overhead can still be further reduced as access to the thread id in Go is currently rather cumbersome and expensive."
],
[
"Conclusion",
"One of the challenges of run-time verification in the concurrent setting is to establish a partial order among recorded events.",
"Thus, we can identify potential bugs due to bad schedules that are possible but did not take place in some specific program run.",
"Vector clocks are the predominant method to achieve this task.",
"For example, see work by Vo [11] in the MPI setting and work by Tasharofi [10] in the actor setting.",
"There are several works that employ vector clocks in the shared memory setting For example, see Pozniansky's and Schuster's work [9] on data race detection.",
"Some follow-up work by Flanagan and Freund [2] employs some optimizations to reduce the tracing overhead by recording only a single clock instead of the entire vector.",
"We leave to future work to investigate whether such optimizations are applicable in the message-passing setting and how they compare to existing optimizations such as [3].",
"We have introduced a novel tracing method that has much less overhead compared to the vector clock method.",
"Our method can deal with all of Go's message-passing language features and can be implemented efficiently as a library.",
"We have built a prototype that can automatically identify alternative schedules and communications.",
"In future work we plan to conduct some case studies and integrate heuristics for specific scenarios, e.g.",
"reporting a send operation on a closed channel etc."
],
[
"Acknowledgments",
"We thank some HVC'17 reviewers for their constructive feedback on an earlier version of this paper."
],
[
"Overview",
"Besides selective synchronous message-passing, Go supports some further message passing features that can be easily dealt with by our approach and are fully supported by our implementation.",
"Figure REF shows such examples where we put the program text in two columns."
],
[
"Buffered Channels",
"Go also supports buffered channels where send is asynchronous assuming sufficient buffer space exists.",
"See function buffered in Figure REF .",
"Depending on the program run, our analysis reports that either A1 or A2 are alternative matches for the receive operation.",
"In terms of the instrumentation and tracing, we treat each asynchronous send as if the send is executed in its own thread.",
"This may lead to some slight inaccuracies.",
"Consider the following variant.",
"falseblue func buffered2() { x := make(chan int,1) x <- 1 // B1 go A(x) // B2 <-x // B3 } Our analysis reports that B2 and B3 form an alternative match.",
"However, in the Go semantics, buffered messages are queued.",
"Hence, for every program run the only possibility is that B1 synchronizes with B3.",
"B3 never takes place!",
"As our main objective is bug finding, we argue that this loss of accuracy is justifiable.",
"How to eliminate such false positives is subject of future work."
],
[
"Select with default/timeout",
"Another feature in Go is to include a default/timeout case to select.",
"See selDefault in Figure REF .",
"The purpose is to avoid (indefinite) blocking if none of the other cases are available.",
"For the user it is useful to find out if other alternatives are available in case the default case is selected.",
"The default case applies for most program runs.",
"Our analysis reports that A1 and A3 are an alternative match.",
"To deal with default/timeout we introduce a new post event $\\mathit {post}(\\mathit {select})$ .",
"To carry out the analysis in terms of the dependency graph, each subtrace $\\dots ,\\mathit {pre}([\\dots ,\\mathit {select},\\dots ]),\\mathit {post}(\\mathit {select}),\\dots $ creates a new node.",
"Construction of edges remains unchanged."
],
[
"Closing of Channels",
"Another feature in Go is the ability to close a channel.",
"See closedChan in Figure REF .",
"Once a channel is closed, each send on a closed channel leads to failure (the program crashes).",
"On the other hand, each receive on a closed channel is always successful, as we receive a dummy value.",
"A run of is successful if the close operation of the main thread happens after the send in thread A.",
"As the close and send operations happen concurrently, our analysis reports that the send A1 may take place after close.",
"For instrumentation/tracing, we introduce event $\\mathit {close}(x)$ .",
"It is easy to identify a receive on a closed channel, as we receive a dummy thread id.",
"So, for each subtrace $[\\dots ,\\mathit {pre}([\\dots ,x ?,\\dots ]),\\mathit {post}(i \\sharp x ?",
"),\\dots ]$ where $i$ is a dummy value we draw an edge from $\\mathit {close}(x)$ to $x ?$ .",
"Here are the details of how to include buffered channels, select and closing of channels."
],
[
"Buffered Channels",
"Consider the following Go program.",
"falseblue x := make(chan, 2) x <- 1 // E1 x <- 1 // E2 <- x // E3 <- x // E4 We create a buffer of size 2.",
"The two send operations will then be carried out asynchronously and the subsequent receive operations will pick up the buffered values.",
"We need to take special care of buffered send operations.",
"If we would treat them like synchronous send operations, their respective pre and post events would be recorded in the same trace as the pre and post events of the receive operations.",
"This would have the consequence that our trace analysis does not find out that events E1 and E2 happen before E3 and E4.",
"Our solution to this issue is to treat each send operation on a buffered channel as if the send operation is carried out in its own thread.",
"Thus, our trace analysis is able to detect that E1 and E2 take place before E3 and E4.",
"This is achieved by marking each send on a buffered channel in the instrumentation.",
"After tracing, pre and post events will then be moved to their own trace.",
"From the viewpoint of our trace analysis, a buffered channel then appears as having infinite buffer space.",
"Of course, when running the program a send operation may still block if all buffer space is occupied.",
"Here are the details of the necessary adjustments to our method.",
"During instrumentation/tracing, we simply record if a buffered send operation took place.",
"The only affected case in the instrumentation of commands (Definition ) is $x \\leftarrow b \\Rightarrow p$ .",
"We assume a predicate $\\sf {isBuffered}(\\cdot )$ to check if a channel is buffered or not.",
"In terms of the actual implementation this is straightforward to implement.",
"We write $\\mathit {postB}(x,n)$ to indicate a buffered send operation via $x$ where $n$ is a fresh thread id.",
"We create fresh thread id numbers via $\\mbox{\\sf {tidB}}$ .",
"[Instrumentation of Buffered Channels] Let $x$ be a buffered channel.",
"$\\begin{array}{lcl}\\mathit {instr}(x \\leftarrow b \\Rightarrow p)&&\\\\\\ \\ \\mid \\sf {isBuffered}(x) & = & x \\leftarrow [n,b] \\Rightarrow (x_{tid} :=x_{tid} \\ \\texttt {++}\\ [\\mathit {postB}(x !",
"{n},])) \\ \\texttt {++}\\ \\mathit {instr}(p)\\\\ & & \\mbox{where} \\ \\ n = \\mbox{\\sf {tidB}}\\\\\\ \\ \\mid \\mbox{otherwise} & = & x \\leftarrow [\\mbox{\\sf {tid}},b] \\Rightarrow (x_{tid} :=x_{tid} \\ \\texttt {++}\\ [\\mathit {post}(x !)])",
"\\ \\texttt {++}\\ \\mathit {instr}(p)\\end{array}$ The treatment of buffered channels has no overhead on the instrumentation and tracing.",
"However, we require a post-processing phase where marked events will be then moved to their own trace.",
"This can be achieved via a linear scan through each trace.",
"Hence, requires time complexity $O(k)$ where $k$ is the overall size of all (initially recorded) traces.",
"For the sake of completeness, we give below a declarative description of post-processing in terms of relation $U \\Rightarrow V$ .",
"[Post-Processing for Buffered Channels $U \\Rightarrow V$ ] $\\begin{array}{c}\\mbox{(MovePostB)} \\ {\\begin{array}{c} L = \\mathit {pre}(as) : \\mathit {postB}(x !,n) : L^{\\prime }\\\\ \\hline i \\sharp L : U \\Rightarrow i \\sharp L^{\\prime } : n \\sharp [\\mathit {pre}(as), \\mathit {postB}(x !,n)] : U \\end{array}}\\\\\\\\\\mbox{(Shift)} \\ {\\begin{array}{c} L = \\mathit {pre}(as) : \\mathit {post}(a) : L^{\\prime }\\\\ (a = x !",
"\\ \\vee \\ a = j \\sharp x ?",
")\\\\ i \\sharp L^{\\prime }:U \\Rightarrow i \\sharp L^{\\prime \\prime }:U^{\\prime } \\\\ \\hline i \\sharp L : U \\Rightarrow i \\sharp \\mathit {pre}(as) : \\mathit {post}(a) : L^{\\prime \\prime } : U^{\\prime } \\end{array}}\\\\\\\\\\mbox{(Schedule)} \\ {\\begin{array}{c} \\mbox{$\\pi $ permutation on $\\lbrace 1,\\dots ,n\\rbrace $}\\\\ \\hline [i_1 \\sharp L_1,\\dots ,i_n \\sharp L_n] \\Rightarrow [i_{\\pi (1)} \\sharp L_{\\pi (1)},\\dots ,i_{\\pi (n)} \\sharp L_{\\pi (n)}] \\end{array}}\\\\\\\\\\mbox{(Closure)} \\ {\\begin{array}{c} U \\Rightarrow U^{\\prime } \\ \\ U^{\\prime } \\Rightarrow U^{\\prime \\prime }\\\\ \\hline U \\Rightarrow U^{\\prime \\prime } \\end{array}}\\end{array}$ Subsequent analysis steps will be carried out on the list of traces obtained via post-processing.",
"There is some space for improvement.",
"Consider the following program text.",
"falseblue func A(x chan int) { x <- 1 // A1 } func buffered2() { x := make(chan int,1) x <- 1 // B1 go A(x) // B2 <-x // B3 } Our analysis (for some program run) reports that B2 and B3 is an alternative match.",
"However, in the Go semantics, buffered messages are queued.",
"Hence, for every program run the only possibility is that B1 synchronizes with B3.",
"B3 never takes place.",
"As our main objective is bug finding, we can live with this inaccuracy.",
"We will investigate in future work how to eliminate this false positive."
],
[
"Select with default/timeout",
"In terms of the instrumentation/tracing, we introduce a new special post event $\\mathit {post}(\\mathit {select})$ .",
"For the trace analysis (Definition ), we require a new rule.",
"$\\begin{array}{c}\\mbox{(Default/Timeout)} \\ i \\sharp \\mathit {pre}([\\dots ]): \\mathit {post}(\\mathit {select}) : L : U 0359{}{[i \\sharp \\mathit {select}]} i \\sharp L : U\\end{array}$ This guarantees that in case default or timeout is chosen, select acts as if asynchronous.",
"The dependency graph construction easily takes care of this new feature.",
"For each default/timeout case we introduce a node.",
"Construction of edges remains unchanged."
],
[
"Closing of Channels",
"For instrumentation/tracing of the $\\mathit {close}(x)$ operation on channel $x$ , we introduce a special pre and post event.",
"Our trace analysis keeps track of closed channels.",
"As a receive on a closed channel yields some dummy values, it is easy to distinguish this case from the regular (Sync).",
"Here are the necessary adjustments to our replay relation from Definition .",
"$\\begin{array}{lcl}C & ::= & [] \\mid i \\sharp \\mathit {close}(x) : C\\end{array}$ $\\begin{array}{c}\\mbox{(Close)} \\ (i \\sharp \\mathit {pre}(\\mathit {close}(x)) : \\mathit {post}(\\mathit {close}(x)) : L : U \\mid C) 0359{}{[]} (i \\sharp L : U \\mid i \\sharp \\mathit {close}(x) : C)\\\\\\\\\\mbox{(RcvClosed)} \\ {\\begin{array}{c} Q = j \\sharp \\mathit {close}(x) : Q^{\\prime }\\\\ \\hline (i \\sharp \\mathit {pre}([\\dots ,x ?,\\dots ]) : \\mathit {post}(j^{\\prime } \\sharp x ?)",
": L : U \\mid Q) 0359{}{[j \\sharp \\mathit {close}(x), i \\leftarrow j \\sharp x?]}",
"(i \\sharp L : U \\mid Q) \\end{array}}\\end{array}$ For the construction of the dependency graph, we create a node for each close statement.",
"For each receive on a closed channel $x$ at program location $l$ , we draw an edge from $\\mathit {close}(x)$ to $x ?|l$ ."
],
[
"Add-Pipe",
"falseblue func add1(in chan int) chan int { out := make(chan int) go func() { for { n := <-in out <- n + 1 } }() return out } func main() { in := make(chan int) c1 := add1(in) for i := 0; i < 19; i++ { c1 = add1(c1) } for n := 1; n < 1000; n++ { in <- n <-c1 } } falseblue func generate(ch chan int) { for i := 2; ; i++ { ch <- i } } func filter(in chan int, out chan int, prime int) { for { tmp := <-in if tmpout <- tmp } } } func main() { ch := make(chan int) go generate(ch) for i := 0; i < 100; i++ { prime := <-ch ch1 := make(chan int) go filter(ch, ch1, prime) ch = ch1 } } falseblue func collect(x chan int, v int) { x <- v } func main() { x := make(chan int) for i := 0; i < 1000; i++ { go collect(x, i) } for i := 0; i < 1000; i++ { <-x } }"
]
] | 1709.01588 |
[
[
"Constraints on Mirror Models of Dark Matter from Observable\n Neutron-Mirror Neutron Oscillation"
],
[
"Abstract The process of neutron-mirror neutron oscillation, motivated by symmetric mirror dark matter models, is governed by two parameters: $n-n'$ mixing parameter $\\delta$ and $n-n'$ mass splitting $\\Delta$.",
"For neutron mirror neutron oscillation to be observable, the splitting between their masses $\\Delta$ must be small and current experiments lead to $\\delta \\leq 2\\times 10^{-27}$ GeV and$\\Delta \\leq 10^{-24}$ GeV.",
"We show that in mirror universe models where this process is observable, this small mass splitting constrains the way that one must implement asymmetric inflation to satisfy the limits of Big Bang Nucleosynthesis on the number of effective light degrees of freedom.",
"In particular we find that if asymmetric inflation is implemented by inflaton decay to color or electroweak charged particles, the oscillation is unobservable.",
"Also if one uses SM singlet fields for this purpose, they must be weakly coupled to the SM fields."
],
[
"1. Introduction",
"The possibility that there may be a mirror sector of the standard model (called MSM) with identical particle content and gauge symmetry as the standard model (called SM) [1] has received a great deal of attention for the reason that the dark matter of the universe may be the lightest baryon (or atom) of the mirror sector.",
"These models assume the existence of $Z_2$ symmetry between the two sectors which keeps the same number of parameters even though the number of particles is doubled.",
"Depending on the status of the $Z_2$ symmetry there arise two possibilities: (i) The mirror symmetry is unbroken so that particle masses in the MSM are the same as in the SM [2]and (ii) The symmetry is spontaneously broken [3] and mirror sector particles to have similar masses compared to the SM.",
"An important problem for both classes of models is the large number of light particle degrees of freedom at the epoch of big bang nucleosynthesis $(\\nu _a, \\gamma ; e, \\nu ^\\prime _a, \\gamma ^{\\prime })$ and possibly $e^\\prime $ making $N_{eff}=21.5$ in clear contradiction to the current CMB bound from Planck and WMAP of $N_{eff} \\le 11.5$ .",
"A solution to this problem of mirror models was suggested in [7] where it was proposed that inflation in mirror models is different from that in conventional cosmology.",
"The inflaton is assumed to be a mirror odd scalar field that couples to particles of both sectors in such a way that after inflation is completed, the reheat temperature of the mirror sector becomes less than that of the SM sector i.e.",
"$T^\\prime _{reheat} \\simeq \\frac{1}{2} T_{reheat} $ , so that the contribution of the light mirror particles to energy density (which goes like $\\sim T^4$ ) at the BBN epoch is highly suppressed.",
"An important implication of this proposal, called asymmetric inflation, is that at some high scale, there must be spontaneous breaking of the $Z_2$ mirror symmetry.",
"This breaking should have some effect on low energy physics of the model.",
"This is what we investigate in this paper in the context of the symmetric mirror model.",
"In the symmetric mirror models, the neutron ($n$ ) and the mirror neutron ($n^\\prime $ ) have same mass in the tree approximation.",
"It has been proposed that if there is a small mixing, $\\delta $ , between $n$ and $n^\\prime $ arising from some higher order induced dimension six operators of the form $uddu^{\\prime }d^{\\prime }d^{\\prime }$ , there can be oscillations between neutron and the mirror neutron [8].",
"Observation of this process would have dramatic implications for particle physics and cosmology.",
"This process has been looked for in experiments [9] and currently there is a lower bound on the transition time for $n\\leftrightarrow n^{\\prime }$ of 448 sec.",
"which translates to an upper limit on the mixing parameter $\\delta \\le 2\\times 10^{-27}$ GeV.",
"Further experiments are being planned [10] to search for this process.",
"For $n\\leftrightarrow n^{\\prime }$ oscillation to be observable, a high degree of degeneracy between $n$ and $n^{\\prime }$ is essential.",
"This is similar to the case of neutron-anti-neutron oscillation [11] where this degeneracy is guaranteed at the particle physics level by CPT invariance.",
"On the other hand, in the case of $n\\leftrightarrow n^{\\prime }$ oscillation, the only symmetry that guarantees their mass equality is a $Z_2$ symmetry which is necessarily broken by the requirement of asymmetric inflation.",
"In viable mirror models where asymmetric inflation is enforced by choice of fields, one would expect some degree of mass splitting between the $n$ and $n^{\\prime }$ masses.",
"In this paper, we show that unless one is careful in choosing the fields required to implement asymmetric inflation, the $n-n^{\\prime }$ mass splitting can exceed the value required for having observable $n-n^{\\prime }$ oscillation.",
"It is known that both symmetric and asymmetric mirror models can have viable dark matter candidates [4], [5], [6].",
"We comment on the implications of observable $n-n^{\\prime }$ oscillation on the properties of dark matter in the former case.",
"This paper is organized as follows: in sec.",
"2, we discuss the upper limit on the splitting of $n,n^{\\prime }$ for the oscillation to be observable; in sec.",
"3, we review the field theoretic model that implements asymmetric inflation in the mirror model; in sec.",
"4, we discuss the implications of the asymmetric inflation model for the magnitude of $n-n^{\\prime }$ mass splitting.",
"In sec.",
"5, we comment on some aspects of the dark matter physics in this model."
],
[
"2. Required level of $n-n^{\\prime }$ mass degeneracy for observable {{formula:8cfa97f6-2871-44b1-9fc2-24f9f0192ecf}} oscillation",
"To find the maximal value of the splitting between $m_n$ and $m_{n^{\\prime }}$ which allows observable $n-n^{\\prime }$ oscillation, we ignore spin and write the evolution equation of the $n$ and $n^{\\prime }$ system: $\\frac{d}{dt}\\left(\\begin{array}{c} n \\\\ {n^{\\prime }}\\end{array}\\right)~=~\\left(\\begin{array}{cc} m & \\delta \\\\ \\delta & m^{\\prime }\\end{array}\\right)\\left(\\begin{array}{c} n \\\\ {n^{\\prime }}\\end{array}\\right).$ Starting initially with neutrons, the probability that an mirror neutron will appear after a time of $t$ is given by: $P_{n-{n^{\\prime }}}~=~\\frac{4\\delta ^2}{\\Delta ^2+4\\delta ^2}{\\rm sin}^2 \\frac{\\sqrt{\\Delta ^2+4\\delta ^2} ~t/2}{\\hbar }$ where $\\Delta =m^{\\prime }-m$ .",
"This difference could arise from higher order quantum corrections or a difference between magnetic fields ($B$ and $B^{\\prime }$ ).",
"Here we focus on the first source.",
"Note that assuming a typical neutron transit time $\\sim 1$ sec., we would expect that for $n-n^{\\prime }$ oscillation to be observable, we must have $\\Delta \\le 10^{-24}$ GeV.",
"If the transit time is shorter, requiring a more intense neutron beam, the bound would be proportionately weaker.",
"The neutrons mass is given by $m_n =M_0 +2m_d+m_u$ with $M_0$ being a QCD generated mass and $m_i= y_i.v$ are the bare quark masses generated by coupling to the Higgs VeV $v$ .",
"Since the quark masses contribute only 1% to the $m_n$ requiring $\\Delta < 10 ^{-24}$ GeV implies that $\\delta y/y\\equiv (y-y^{\\prime })/y \\le 10^{-22}$ GeV.",
"We now discuss the impact of $Z_2$ mirror parity breaking for asymmetric inflation and its impact on $\\Delta $ .",
"We envision a general set-up [7], where we have a $Z_2$ odd gauge singlet field $\\eta $ which acts as the inflaton and $X, X^{\\prime }$ be the two mirror partner complex scalar fields.",
"We assume the following inflaton field potential: $V(\\eta , X, X^{\\prime })~=~m^2_\\eta \\eta ^2+\\lambda _\\eta \\eta ^4+\\mu \\eta (X^\\dagger X-{X^{\\prime }}^{\\dagger } X^{\\prime })+\\lambda \\eta ^2(X^\\dagger X+{X^{\\prime }}^{\\dagger } X^{\\prime })$ Different inflation pictures arise by modifying the $m^2_\\eta \\eta ^2+\\lambda _\\eta \\eta ^4$ part, the details of which are not crucial to our conclusion.",
"The part which leads to asymmetric inflation comes from the last two terms in Eq.",
"(3).",
"To recap the argument of [7], we note that after $\\eta $ field acquires a vev, the inflaton field decays asymmetrically to the $X$ and $X^{\\prime }$ fields with the ratio of the decay widths given by $\\frac{\\Gamma _{\\eta \\rightarrow XX}}{\\Gamma _{\\eta \\rightarrow X^{\\prime }X^{\\prime }}}=\\left(\\frac{\\mu +\\lambda <\\eta >}{\\mu -\\lambda <\\eta >}\\right)^2$ Noticing that reheat temperature is $T_R\\simeq \\sqrt{\\Gamma _\\eta M_{Pl}}$ , we find that $T^{\\prime }_R < T_R$ if all the parameters are positive.",
"By appropriately choosing the parameters, we can make $T^{\\prime }_R/T_R\\simeq 1/2$ as required in order to satisfy BBN constraints.",
"We also note that the required symmetry breaking potential in Eq.",
"(3) leads to different masses for $X,X^{\\prime }$ and to the relation (assuming the bare masses for $X,X^{\\prime }$ to be small): $M_{X}/M_{X^{\\prime }}\\propto T_R/{T^{\\prime }}_R$ For a detailed discussion of reheating in such scenarios, see [12].",
"The $X$ and $X^{\\prime }$ must couple to the known SM fields ( and to their mirrors , respectively) in order to generate the latter with the required different temperatures , starting the usual Hubble expansion in each sector.",
"The mass difference of the $X,X^{\\prime }$ fields causes the mirror symmetry breaking manifesting in the different $M_X$ and $M_{X^{\\prime }}$ to trickle down to lower energies and , in particular to generate different Yukawa couplings $y_i$ for the quarks."
],
[
"4. Implications for $n-n^{\\prime }$ mass splitting",
"In this section, we give several examples for the fields $X,X^{\\prime }$ and note the implications for $\\delta y/y\\equiv (y-y^{\\prime })/y$ for each case.",
"The correction to $\\delta y$ arise from two loop self energy corrections to the Yukawa couplings of quarks of the type shown in Fig.1: Figure: Typical two loop diagram that contributes mass difference between nn and n ' n^{\\prime }.",
"This graph corresponds to the case where XX is a colored particle.",
"If $X$ and $X^{\\prime }$ are color non-singlet fields, then they will have color gauge couplings which will induce different wave function renormalization for the quarks in the Yukawa couplings which through RGE running lead to an estimate $\\frac{\\delta y}{y}\\simeq \\frac{\\alpha ^2_c}{16\\pi ^2}\\ell n\\frac{M^2_X}{{M}^2_{X^{\\prime }}}=\\frac{\\alpha ^2_c}{16\\pi ^2}\\ell n\\frac{T^2_R}{{T^{\\prime }}^2_{R}}$ (where $\\alpha _c$ is the QCD fine structure constant).",
"This splitting is of order $\\sim 10^{-4}$ , which is clearly much larger than what is allowed in order to make $n-n^{\\prime }$ oscillation observable.",
"Therefore if $n-n^{\\prime }$ oscillation is to be observable, one cannot allow $X,X^{\\prime }$ to have color.",
"Instead if $X,X^{\\prime }$ were color singlet but $SU(2)_L$ and $SU(2)^{\\prime }_L$ non-singlets, we replace the $\\alpha _c$ above by $\\alpha _W$ yielding $\\frac{\\delta y}{y}\\sim 10^{-6}$ , which is also much larger than $\\Delta $ estimated above.",
"This difficulty can be ameliorated if $X,X^{\\prime }$ are SM and MSM singlets e.g.",
"a singlet scalar,$S$ and $S^{\\prime }$ .",
"In this case the scalar must couple to SM fields to generate the familiar Hubble expansion with SM matter fields.",
"This implies that $S$ couples to the Higgs fields (of SM and MSM) via the coupling $\\mu _S (SH^\\dagger H+S^{\\prime }{H^{\\prime }}^\\dagger H^{\\prime })$ .",
"In this case, we have the $\\frac{\\delta y}{y} \\sim \\frac{y^2_u\\mu ^2_S}{(16\\pi ^2)^2 M^2_S}\\ell n\\frac{T^2_R}{{T^{\\prime }}^2_{R}}$ and we can bring down this value to the acceptable level for observability of $n-n^{\\prime }$ oscillation by tuning $\\mu _S$ somewhat.",
"Taking $y_u\\sim 10^{-5}$ and $\\mu _S/M_S\\sim 10^{-4}$ , we get the desired small splitting $\\Delta $ .",
"The first factor $y_u$ corresponds to the first generation quark Yukawa couplings since $n$ and $n^{\\prime }$ involve only the first generation quarks (Fig 2).",
"The singlet fields could also be singlet fermions in which case the the inflation coupling to them would contain nonrenormalizable terms i.e.",
"${\\cal L}_\\eta \\sim \\eta (NN-N^{\\prime }N^{\\prime })+\\frac{\\eta ^2}{\\Lambda }(NN+N^{\\prime }N^{\\prime })$ and all the above considerations go through.",
"In discussing all these cases we have assumed that the bare mass contribution to $X.X^{\\prime }$ masses is small compared to the $<\\eta >$ contribution.",
"However, if we assume that the bare masses are much larger than $<\\eta >$ , the log in the splitting corrections simplifies to $\\frac{M^2_X-M^2_{X^{\\prime }}}{M^2_X}$ and for $\\delta y/y$ to be less than $10^{-22}$ , we must have $\\frac{M^2_X-M^2_{X^{\\prime }}}{M^2_X}\\le 10^{-18}$ (in the colored $X,X^{\\prime }$ and slightly weaker bound in the SM non-singlet case.).",
"This requires that $<\\eta >/M_X \\le 10^{-18}$ which is a very high degree of fine tuning.",
"Figure: Typical two loop diagram that contributes mass difference between nn and n ' n^{\\prime }.",
"This graph corresponds to the case where XX is a SM singlet scalar field.From the above discussion, we see that the structure of the inflation sector of the mirror model must be very restrictive i.e.",
"the inflaton field must couple to only to SM and MSM singlets.",
"Furthermore the coupling of the singlet fields to SM and MSM fields must be very weak."
],
[
"5. Comments on dark matter properties in symmetric vs asymmetric mirror models",
"Observable $n-n^{\\prime }$ oscillation also has implications for the nature of dark matter as we see now.",
"This is due to the fact that dark matter has self interactions and there are limits on the DM-DM cross section from bullet cluster observations [14].",
"In mirror scenarios with appreciable mirror symmetry breaking (e.g.",
"the asymmetric mirror model), the fermion spectrum in the SM and MSM sector need not be same and therefore the lightest mirror nucleon can be $n^{\\prime }$ and serve as the DM particle.The cross-section for low energy $n^{\\prime }-n^{\\prime }$ scattering is $\\sigma (n^{\\prime }n^{\\prime }) \\sim \\sigma ( nn) = 4\\pi a^2 \\sim {\\rm Barn}= 10 ^{-24} {\\rm cm}^2.$ This is an optimal value close to what may be required to solve the CDM cusp problem [13] and yet consistent with the Bullet cluster upper bound.",
"This is not the case in models with the very precise mirror symmetry required for the observability of n-n' oscillations.",
"To have a kinematically allowed $p^{\\prime }\\rightarrow n^{\\prime } +\\nu ^{\\prime } +{e^{\\prime }}^{+}$ decay so as to ensure that $n^{\\prime }$ is the dark matter, the masses of some mirror particles should be shifted relative to the masses of their SM counter-parts by several MeV- exceeding by twenty orders of magnitude the allowed $n^{\\prime }-n$ splitting $\\Delta $ .",
"The dark matter should then consist of dark atoms i.e.",
"$H^{\\prime }$ and $He^{\\prime }$ - mirror Hydrogen and mirror Helium and it was pointed out that the latter may be more abundant relative to the former than in our ordinary baryonic matter [5].",
"However in either case the low energy DM-DM elastic scattering is fixed by the atomic sizes to be $\\sim {\\rm Angstrom}^2 = 10 ^{-16}$ cm$^2$ which will exceed the upper bound by 7-8 orders of magnitude.",
"Due to the large impact parameter in the atomic collisions we do not have - as in the case of $n-n $ (or $n^{\\prime }-n^{\\prime }$ ) scattering -only S wave contribution and the differential elastic cross-section is non-isotropic and forward peaked.",
"This reduces the effective transport cross-section which as noted in [6] is the one relevant here.",
"At the relatively high O( keV) collision energies which are $\\sim 100$ times the Ionization threshold), there can also be many atomic excitations and ionization processes and theoretical estimates of the total stopping power of the gas are difficult to make.",
"Actual data on atomic beam scattering [16] suggest about 1/100 reduction of the effective cross-sections from the naive Angstrom $^2$ estimate leaving us with values which are still considerably higher than the upper bound from bullet cluster observation.",
"This would suggest that to have a viable model we need that - unlike for ordinary galaxies and galaxies clusters most of the mirror matter should reside in collisonless stars and not in gas- a rather non-trivial constraint when added to other requirements which DM has to satisfy.",
"In summary, we have pointed out that for $n-n^{\\prime }$ to be observable, one must have a specific structure for the inflation reheat sector of the Lagrangian in mirror models- the inflation reheating must proceed via fields that are standard model and mirror standard model gauge singlets.We also make some general comments on the dark matter properties in symmetric vs asymmetric mirror picture, which has bearing on the issue of $n-n^{\\prime }$ oscillation to be observable."
],
[
"Acknowledgement",
"R.N.M.",
"was supported by the US National Science Foundation under Grant No.",
"PHY1620074."
]
] | 1709.01637 |
[
[
"Set-theoretical entropies of generalized shifts"
],
[
"Abstract In the following text for arbitrary $X$ with at least two elements, nonempty set $\\Gamma$ and self-map $\\varphi:\\Gamma\\to\\Gamma$ we prove the set-theoretical entropy of generalized shift $\\sigma_\\varphi:X^\\Gamma\\to X^\\Gamma$ ($\\sigma_\\varphi((x_\\alpha)_{\\alpha\\in\\Gamma})=(x_{\\varphi(\\alpha)})_{\\alpha\\in\\Gamma}$ (for $(x_\\alpha)_{\\alpha\\in\\Gamma}\\in X^\\Gamma$)) is either zero or infinity, moreover it is zero if and only if $\\varphi$ is quasi-periodic.",
"We continue our study on contravariant set-theoretical entropy of generalized shift and motivate the text using counterexamples dealing with algebraic, topological, set-theoretical and contravariant set-theoretical positive entropies of generalized shifts."
],
[
"Introduction",
"Amongst the most powerful tools in ergodic theory and dynamical systems we may mention one-sided shift $\\mathop {\\lbrace 1,\\ldots ,k\\rbrace ^{\\mathbb {N}}\\rightarrow \\lbrace 1,\\ldots ,k\\rbrace ^{\\mathbb {N}}}\\limits _{\\:\\:\\:\\:(x_n)_{n\\ge 1}\\mapsto (x_{n+1})_{n\\ge 1}}$ and two-sided shift $\\mathop {\\lbrace 1,\\ldots ,k\\rbrace ^{\\mathbb {Z}}\\rightarrow \\lbrace 1,\\ldots ,k\\rbrace ^{\\mathbb {Z}}}\\limits _{\\:\\:\\:\\:(x_n)_{n\\in {\\mathbb {Z}}}\\mapsto (x_{n+1})_{n\\in {\\mathbb {Z}}}}$ [8].",
"Now suppose $X$ is an arbitrary set with at least two elements, $\\Gamma $ is a nonempty set, and $\\varphi :\\Gamma \\rightarrow \\Gamma $ is arbitrary, then $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ with $\\sigma _\\varphi ((x_\\alpha )_{\\alpha \\in \\Gamma })=(x_{\\varphi (\\alpha )})_{\\alpha \\in \\Gamma }$ (for $(x_\\alpha )_{\\alpha \\in \\Gamma }\\in X^\\Gamma $ ) is a generalized shift.",
"Generalized shifts have been introduced for the first time in [3].",
"It's evident that for self-map $\\varphi :\\Gamma \\rightarrow \\Gamma $ and generalized shift $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ if $X$ has a group (resp.",
"vector space, topological) structure, then $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ is a group homomorphism (resp.",
"linear map, continuous (in which $X^\\Gamma $ considered under product topology)), so many dynamical [4] and non-dynamical [7] properties of generalized shifts have been studied in several texts.",
"In this text our main aim is to study set-theoretical and contravariant set-theoretical entropy of generalized shifts.",
"We complete our investigations with a comparative study regarding set-theoretical, contravariant set-theoretical, topological and algebraic entropies of generalized shifts.",
"For self-map $g:A\\rightarrow A$ and $x,y\\in A$ let $x\\le _gy$ if and only if there exists $n\\ge 0$ with $g^n(x)=y$ , then $(A,\\le _g)$ is a preordered (reflexive and transitive) set.",
"Note that for set $A$ by $|A|$ we mean the cardinality of $A$ if it is finite and $\\infty $ otherwise.",
"Although one may obtain the following lemma using [7], we establish it here directly.",
"Note 1.1 (Bounded self-map, Quasi-periodic self-map) For self-map $g:A\\rightarrow A$ the following statements are equivalent (consider preordered set $(A,\\le _g)$ ): 1. there exists $N\\ge 1$ such that for all totally preordered subset $I$ of $A$ (reflexive, transitive and for all $x,y\\in A$ we have $x\\le _gy$ or $y\\le _g x$ ) we have $|I|\\le N$ (i.e., $g:A\\rightarrow A$ is bounded [7]), 2.",
"$\\sup \\left\\lbrace |\\lbrace g^n(x):n\\ge 0\\rbrace |:x\\in A\\right\\rbrace <\\infty $ , 3. there exists $n>m\\ge 1$ with $g^n=g^m$ ($g$ is quasi-periodic).",
"“(1) $\\Rightarrow $ (2)” Suppose there exists $N\\ge 1$ such that for all totally preordered subset $I$ of $A$ we have $|I|\\le N$ .",
"Choose $x\\in A$ , then $\\lbrace g^n(x):n\\ge 0\\rbrace $ is a totally preordered subset of $A$ , thus $|\\lbrace g^n(x):n\\ge 0\\rbrace |\\le N$ , hence $\\sup \\left\\lbrace |\\lbrace g^n(y):n\\ge 0\\rbrace |:y\\in A\\right\\rbrace $ $\\le N<\\infty $ .",
"“(2) $\\Rightarrow $ (3)” Suppose $\\sup \\left\\lbrace |\\lbrace g^n(x):n\\ge 0\\rbrace |:x\\in A\\right\\rbrace =N<\\infty $ , then for all $x\\in A$ we have $\\lbrace g^n(x):n\\ge 0\\rbrace =\\lbrace x,g(x),\\ldots ,g^{N-1}(x)\\rbrace $ and there exists $n_x\\in \\lbrace 0,\\ldots ,N-1\\rbrace $ with $g^N(x)=g^{n_x}(x)$ , thus $g^{N-n_x}(g^N(x))=g^N(x)$ and $\\forall z\\in g^N(A)\\:\\:\\exists i\\in \\lbrace 1,\\ldots ,N\\rbrace \\:(g^i(z)=z)$ so for all $z\\in g^N(A)$ we have $g^{N!",
"}(z)=z$ , thus for all $x\\in A$ we have $g^{N!+N}(x)=g^N(x)$ .",
"“(3) $\\Rightarrow $ (1)” Suppose there exist $n>m\\ge 1$ with $g^n=g^m$ and $I$ is a totally preordered subset of $A$ , choose distinct $x_1,\\ldots ,x_k\\in I$ and suppose $x_1\\le _gx_2\\le _g\\cdots \\le _gx_k$ .",
"For all $i\\in \\lbrace 1,\\ldots ,k\\rbrace $ there exists $p_i\\ge 0$ with $x_i=g^{p_i}(x_1)$ , so $\\lbrace x_1,\\ldots ,x_k\\rbrace \\subseteq \\lbrace g^i(x_1):i\\ge 0\\rbrace =\\lbrace g^i(x_1):i\\in \\lbrace 0,\\ldots ,n\\rbrace \\rbrace $ and $k\\le n+1$ .",
"Hence $|I|\\le n+1$ which completes the proof.",
"Convention.",
"In the following text suppose $X$ is an arbitrary set with at least two elements, $\\Gamma $ is a nonempty set, and $\\varphi :\\Gamma \\rightarrow \\Gamma $ is arbitrary."
],
[
"Background on set-theoretical entropy",
"For self-map $g:A\\rightarrow A$ and $a\\in A$ , the set $\\lbrace g^n(a):n\\ge 0\\rbrace $ is the orbit of $a$ , we say $a\\in A$ is a wandering point (or non-quasi periodic point) of $g$ , if $\\lbrace g^n(a):n\\ge 0\\rbrace $ is infinite, or equivalently $\\lbrace g^n(a)\\rbrace _{n\\ge 1}$ is a one-to-one sequence.",
"We denote the collection of all wandering points of $g:A\\rightarrow A$ with $W(g)$ .",
"For $g:A\\rightarrow A$ denote the infinite orbit number of $g$ by $\\mathfrak {o}(g)$ and define it with $\\sup (\\lbrace 0\\rbrace \\cup \\lbrace k\\ge 1:\\exists a_1,\\ldots ,a_k\\in W(g)\\:(\\lbrace g^n(a_1)\\rbrace _{n\\ge 1},\\ldots ,\\lbrace g^k(a)\\rbrace _{n\\ge 1}$ are pairwise disjoint sequences$)\\rbrace )$ , i.e.",
"$\\mathfrak {o}(g)=\\sup (\\lbrace 0\\rbrace \\cup \\lbrace k\\ge 1:$ there exists $k$ pairwise disjoint infinite orbits$\\rbrace )$ .",
"So $W(g)\\ne \\varnothing $ if and only if $\\mathfrak {o}(g)\\ge 1$ .",
"On the other hand for finite subset $D$ of $A$ the following limit exists [2]: ${\\rm ent}_{\\rm set}(g,D)={\\displaystyle \\lim _{n\\rightarrow \\infty }\\dfrac{|D\\cup g(D)\\cup \\cdots \\cup g^{n-1}(D)|}{n}}\\:.$ Now we call $\\sup \\lbrace {\\rm ent}_{\\rm set}(g,D):D$ is a finite subset of $A\\rbrace $ the set-theoretical entropy of $g$ and denote it with ${\\rm ent}_{\\rm set}(g)$ .",
"Moreover ${\\rm ent}_{\\rm set}(g)=\\mathfrak {o}(g)$ [2]."
],
[
"Background on contravariant set-theoretical entropy",
"Suppose self-map $g:A\\rightarrow A$ is onto and finite fibre (i.e., for all $a\\in A$ , $g^{-1}(a)$ is finite), then for finite subset $D$ of $A$ the following limit exists [5]: ${\\rm ent}_{\\rm cset}(g,D)={\\displaystyle \\lim _{n\\rightarrow \\infty }\\dfrac{|D\\cup g^{-1}(D)\\cup \\cdots \\cup g^{-(n-1)}(D)|}{n}}$ Now let ${\\rm ent}_{\\rm cset}(g):=\\sup \\lbrace {\\rm ent}_{\\rm cset}(g,D):D$ is a finite subset of $A\\rbrace $ .",
"If $k:A\\rightarrow A$ is an arbitrary finite fibre map, then for surjective cover of $k$ , i.e.",
"${\\rm sc}(k):=\\bigcap \\lbrace k^n(A):n\\ge 1\\rbrace $ , the map $k\\upharpoonright _{{\\rm sc}(k)}:{\\rm sc}(k)\\rightarrow {\\rm sc}(k)$ is an onto finite fibre map and ${\\rm ent}_{\\rm cset}(k):={\\rm ent}_{\\rm cset}(k\\upharpoonright _{{\\rm sc}(k)})$ is the contravariant set-theoretical entropy of $k$ .",
"Moreover we say $\\lbrace x_n\\rbrace _{n\\ge 1}$ is a $k-$anti-orbit sequence (or simply anti-orbit sequence) if for all $n\\ge 1$ we have $k(x_{n+1})=x_n$ and define infinite anti-orbit number of $k$ as $\\mathfrak {a}(k)=\\sup (\\lbrace 0\\rbrace \\cup \\lbrace j\\ge 1:$ there exists $j$ pairwise disjoint infinite anti-orbits$\\rbrace )$ .",
"Moreover ${\\rm ent}_{\\rm cset}(k)=\\mathfrak {a}(k)$ [5]."
],
[
"Set-theoretical entropy of $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $",
"In this section we prove that for generalized shift $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ , ${\\rm ent}_{\\rm set}( \\sigma _\\varphi )\\in \\lbrace 0,\\infty \\rbrace $ and ${\\rm ent}_{\\rm set}( \\sigma _\\varphi )=0$ if and only if $\\varphi $ is quasi-periodic.",
"Lemma 2.1 If $W(\\varphi )\\ne \\varnothing $ , then $W(\\sigma _\\varphi )\\ne \\varnothing $ .",
"Consider distinct points $p,q\\in X$ and $\\theta \\in W(\\varphi )$ , thus $(\\varphi ^n(\\theta ))_{n\\ge 0}$ is a one-to-one sequence.",
"Let: $x_\\alpha :=\\left\\lbrace \\begin{array}{lc} p & \\alpha \\in \\lbrace \\varphi ^{2^n}(\\theta ):n\\ge 1\\rbrace \\:, \\\\q & {\\rm otherwise\\:,}\\end{array}\\right.$ then $(x_\\alpha )_{\\alpha \\in \\Gamma }\\in W(\\sigma _\\varphi )$ , otherwise there exists $s>t\\ge 1$ such that $\\sigma _{\\varphi }^s((x_\\alpha )_{\\alpha \\in \\Gamma })=\\sigma _{\\varphi }^t((x_\\alpha )_{\\alpha \\in \\Gamma })$ , thus $x_{\\varphi ^s(\\alpha )}=x_{\\varphi ^t(\\alpha )}$ for all $\\alpha \\in \\Gamma $ .",
"In particular, $x_{\\varphi ^{s+i}(\\theta )}=x_{\\varphi ^{t+i}(\\theta )}$ for all $i\\ge 0$ .",
"Choose $j\\ge 1$ with $j+s\\in \\lbrace 2^n:n\\ge 1\\rbrace $ .",
"We have the following cases: Case 1: $j+t\\notin \\lbrace 2^n:n\\ge 1\\rbrace $ .",
"In this case we have $p=x_{\\varphi ^{s+j}(\\theta )}=x_{\\varphi ^{t+j}(\\theta )}=q$ , which is a contradiction.",
"Case 2: $j+t\\in \\lbrace 2^n:n\\ge 1\\rbrace $ .",
"In this case using $j+t>j+s\\in \\lbrace 2^n:n\\ge 1\\rbrace $ we have $j+t\\ge 3$ .",
"There exist $k\\ge 1$ and $l\\ge 2$ with $j+s=2^k$ and $j+t=2^l$ .",
"Let $i=2j+s$ , then $i+s=2(j+s)=2^{k+1}\\in \\lbrace 2^n:n\\ge 1\\rbrace $ and $i+t=2^l+2^k=2^l(1+2^{k-l})\\notin \\lbrace 2^n:n\\ge 1\\rbrace $ (note that $k>l$ and $1+2^{k-l}$ is odd).",
"So $p=x_{\\varphi ^{s+i}(\\theta )}=x_{\\varphi ^{t+i}(\\theta )}=q$ , which is a contradiction.",
"Using the above two cases, we have $(x_\\alpha )_{\\alpha \\in \\Gamma }\\in W(\\sigma _\\varphi )$ .",
"Lemma 2.2 If $W(\\varphi )\\ne \\varnothing $ , then $\\mathfrak {o}(\\sigma _\\varphi )=\\infty $ .",
"Consider $\\theta \\in \\Gamma $ with infinite $\\lbrace \\varphi ^n(\\theta ):n\\ge 0\\rbrace $ and choose distinct $p,q\\in X$ , thus $(\\varphi ^n(\\theta ))_{n\\ge 0}$ is a one-to-one sequence.",
"For $s\\ge 1$ let: $x_\\alpha ^s:=\\left\\lbrace \\begin{array}{lc} p & \\alpha \\in \\lbrace \\varphi ^n(\\theta ):\\exists k\\ge 0\\:(ks+\\frac{k(k+1)}{2}<i\\le ks+\\frac{k(k+1)}{2}+s)\\rbrace \\:, \\\\q & {\\rm otherwise}\\:,\\end{array}\\right.$ so: $(x_{\\varphi (\\theta )}^s,x_{\\varphi ^2(\\theta )}^s,x_{\\varphi ^3(\\theta )}^s,\\cdots )$ $=(\\:\\underbrace{p,\\cdots ,p}_{s\\:{\\rm times}},q,\\underbrace{p,\\cdots ,p}_{s\\:{\\rm times}},q,q,\\underbrace{p,\\cdots ,p}_{s\\:{\\rm times}},q,q,q,\\underbrace{p,\\cdots ,p}_{s\\:{\\rm times}},q,q,q,q,\\cdots ).$ Let $x^s:=(x_\\alpha ^s)_{\\alpha \\in \\Gamma }$ .",
"Now we have the following steps: Step 1.",
"For $s\\ge 1$ , the sequence $(\\sigma _\\varphi ^n(x^s))_{n\\ge 1}$ is one-to-one: Consider $j>i\\ge 0$ , then $i<js+\\frac{j(j+1)}{2}+s$ , so there exists $t\\ge 1$ with $i+t=js+\\frac{j(j+1)}{2}+s$ moreover $js+\\frac{j(j+1)}{2}+s<\\underbrace{js+\\frac{j(j+1)}{2}+s+(j-i)}_{j+t}<(j+1)s+\\frac{(j+1)(j+2)}{2}\\:,$ which show $x^s_{\\varphi ^{i+t}(\\theta )}=p$ and $x^s_{\\varphi ^{j+t}(\\theta )}=q$ and: $x^s_{\\varphi ^{i+t}(\\theta )}\\ne x^s_{\\varphi ^{j+t}(\\theta )}\\:.\\qquad \\mathrm {(*)}$ Using (*) we have $\\sigma _\\varphi ^i((x_\\alpha ^s)_{\\alpha \\in \\Gamma })\\ne \\sigma _\\varphi ^j((x_\\alpha ^s)_{\\alpha \\in \\Gamma })$ , thus $(\\sigma _\\varphi ^n(x^s))_{n\\ge 1}$ is a one-to-one sequence.",
"Step 2.",
"$(\\sigma _\\varphi ^n(x^1))_{n\\ge 1}$ , $(\\sigma _\\varphi ^n(x^2))_{n\\ge 1}$ , $(\\sigma _\\varphi ^n(x^3))_{n\\ge 1}$ , ... are pairwise disjoint sequences: consider $s\\ge r\\ge 1$ and $i,j\\ge 0$ with $\\sigma _\\varphi ^i(x^s)=\\sigma _\\varphi ^j(x^r)$ .",
"Choose $m\\ge 0$ with $i+m=is+\\frac{i(i+1)}{2}+1$ , now we have: $\\sigma _\\varphi ^i(x^s)=\\sigma _\\varphi ^j(x^r) & \\Rightarrow &(\\forall \\alpha \\in \\Gamma \\:(x^s_{\\varphi ^i(\\alpha )}=x^r_{\\varphi ^j(\\alpha )})) \\\\& \\Rightarrow & (\\forall n\\ge 0\\: (x^s_{\\varphi ^{i+n}(\\theta )}=x^r_{\\varphi ^{j+n}(\\theta )})) \\\\& \\Rightarrow & (\\forall k\\ge 0\\: (x^s_{\\varphi ^{i+m+k}(\\theta )}=x^r_{\\varphi ^{j+m+k}(\\theta )})) \\\\& \\Rightarrow & (\\forall k\\in \\lbrace 0,\\ldots ,s-1\\rbrace \\:(p=x^s_{\\varphi ^{i+m+k}(\\theta )}=x^r_{\\varphi ^{j+m+k}(\\theta )}))$ using $x^r_{\\varphi ^{j+m}(\\theta )}=x^r_{\\varphi ^{j+m+1}(\\theta )}=\\cdots =x^r_{\\varphi ^{j+m+{s-1}}(\\theta )}=p$ and the way of definition of $x^r$ we have $s\\le r$ , thus $s=r$ , and $\\sigma _\\varphi ^i(x^s)=\\sigma _\\varphi ^j(x^s)$ which leads to $i=j$ by Step 1.",
"Using the above two steps $(\\sigma _\\varphi ^n(x^1))_{n\\ge 1}$ , $(\\sigma _\\varphi ^n(x^2))_{n\\ge 1}$ , $(\\sigma _\\varphi ^n(x^3))_{n\\ge 1}$ , ... are pairwise disjoint infinite sequences which leads to $\\mathfrak {o}(\\sigma _\\varphi )=\\infty $ .",
"Lemma 2.3 Let $W(\\varphi )=\\varnothing $ , and $\\varphi $ is not quasi-periodic, then $\\mathfrak {o}(\\sigma _\\varphi )=\\infty $ .",
"Since $W(\\varphi )=\\varnothing $ , for all $\\alpha \\in \\Gamma $ , $\\lbrace \\varphi ^n(\\alpha ):n\\ge 0\\rbrace $ is finite.",
"Since $\\varphi $ is not quasi-periodic we have $\\sup \\left\\lbrace |\\lbrace \\varphi ^n(\\alpha ):n\\ge 0\\rbrace |:\\alpha \\in \\Gamma \\right\\rbrace =\\infty $ .",
"Thus there exist $\\theta _1,\\theta _2,\\ldots \\in \\Gamma $ such that for all $i\\ge 1$ the set $\\lbrace \\theta _i,\\varphi (\\theta _i),\\ldots ,\\varphi ^i(\\theta _i)\\rbrace $ has $i+1$ elements, moreover for all $j\\ne i$ we have $\\lbrace \\theta _i,\\varphi (\\theta _i),\\ldots ,\\varphi ^i(\\theta _i)\\rbrace \\cap \\lbrace \\theta _j,\\varphi (\\theta _j),\\ldots ,\\varphi ^j(\\theta _j)\\rbrace =\\varnothing $ .",
"For $n\\ge 1$ suppose $u_n$ is the $n$ th prime number and choose distinct $p,q\\in X$ , now for $m\\ge 1$ let: $x^m_\\alpha =\\left\\lbrace \\begin{array}{lc} p & \\alpha \\in \\lbrace \\varphi ^n(\\theta _{u_m^t}):t\\ge 1,1\\le n< u_m^t\\rbrace \\:,\\\\q & {\\rm otherwise}\\:,\\end{array}\\right.$ so: $\\begin{array}{rcl}(p,q) & = & (x^1_{\\varphi (\\theta _2)},x^1_{\\varphi ^2(\\theta _2)}) \\\\(p,p,p,q) & = & (x^1_{\\varphi (\\theta _4)},x^1_{\\varphi ^2(\\theta _4)},x^1_{\\varphi ^3(\\theta _4)},x^1_{\\varphi ^4(\\theta _4)}) \\\\(p,p,p,p,p,p,p,q) & = & (x^1_{\\varphi (\\theta _8)},x^1_{\\varphi ^2(\\theta _8)},x^1_{\\varphi ^3(\\theta _8)},x^1_{\\varphi ^4(\\theta _8)},x^1_{\\varphi ^5(\\theta _8)},x^1_{\\varphi ^6(\\theta _8)},x^1_{\\varphi ^7(\\theta _8)},x^1_{\\varphi ^8(\\theta _8)}) \\\\& \\vdots & \\\\(p,p,q) & = & (x^2_{\\varphi (\\theta _3)},x^2_{\\varphi ^2(\\theta _3)},x^2_{\\varphi ^3(\\theta _3)}) \\\\(p,p,p,p,p,p,p,p,q) & = & (x^2_{\\varphi (\\theta _9)},x^2_{\\varphi ^2(\\theta _9)},x^2_{\\varphi ^3(\\theta _9)},x^2_{\\varphi ^4(\\theta _9)},x^2_{\\varphi ^5(\\theta _9)},x^2_{\\varphi ^6(\\theta _9)},x^2_{\\varphi ^7(\\theta _9)},x^2_{\\varphi ^8(\\theta _9)},x^2_{\\varphi ^9(\\theta _9)}) \\\\& \\vdots & \\\\(p,p,p,p,q) & = & (x^3_{\\varphi (\\theta _5)},x^3_{\\varphi ^2(\\theta _5)},x^3_{\\varphi ^3(\\theta _5)},x^3_{\\varphi ^4(\\theta _5)},x^3_{\\varphi ^5(\\theta _5)}) \\\\& \\vdots & \\\\(\\underbrace{p,\\cdots ,p}_{u_m-1{\\rm \\: times}},q) & = & (x^m_{\\varphi (\\theta _{u_m})},x^m_{\\varphi ^2(\\theta _{u_m})},\\cdots ,x^m_{\\varphi ^{u_m}(\\theta _{u_m})}) \\\\(\\underbrace{p,\\cdots ,p}_{u_m^2-1{\\rm \\: times}},q) & = &(x^m_{\\varphi (\\theta _{u^2_m})},x^m_{\\varphi ^2(\\theta _{u^2_m})},\\cdots ,x^m_{\\varphi ^{u^2_m}(\\theta _{u^2_m})}) \\\\(\\underbrace{p,\\cdots ,p}_{u_m^3-1{\\rm \\: times}},q) & = & (x^m_{\\varphi (\\theta _{u^3_m})},x^m_{\\varphi ^2(\\theta _{u^3_m})}, \\cdots ,x^m_{\\varphi ^{u^3_m}(\\theta _{u^3_m})}) \\\\& \\vdots &\\end{array}$ For $m\\ge 1$ , let $x^m:=(x_\\alpha ^m)_{\\alpha \\in \\Gamma }$ .",
"Now we have the following steps: Step 1.",
"For $m\\ge 1$ , the sequence $(\\sigma _\\varphi ^n(x^m))_{n\\ge 1}$ is one-to-one: Consider $j\\ge i\\ge 1$ with $\\sigma _\\varphi ^i(x^m)=\\sigma _\\varphi ^j(x^m)$ , choose $t,l\\ge 1$ such that $j+l=u_m^t$ , so: $\\sigma _\\varphi ^i(x^m)=\\sigma _\\varphi ^j(x^m) & \\Rightarrow &(\\forall \\alpha \\in \\Gamma \\:(x^m_{\\varphi ^i(\\alpha )}=x^m_{\\varphi ^j(\\alpha )})) \\\\& \\Rightarrow & (\\forall k\\ge 0\\:\\forall s\\ge 1\\:(x^m_{\\varphi ^{i+k}(\\theta _{u_m^s})}=x^m_{\\varphi ^{j+k}(\\theta _{u_m^s})})) \\\\& \\Rightarrow & (x^m_{\\varphi ^{i+l}(\\theta _{u_m^t})}=x^m_{\\varphi ^{j+l}(\\theta _{u_m^t})}=x^m_{\\varphi ^{u_m^t}(\\theta _{u_m^t})}=q)$ using $x^m_{\\varphi ^{i+l}(\\theta _{u_m^t})}=q$ and $1\\le i+l\\le j+l=u_m^t$ considering the way of definition of $x^m$ we have $i+l=u_m^t=j+l$ , thus $i=j$ and the sequence $(\\sigma _\\varphi ^n(x^m))_{n\\ge 1}$ is one-to-one.",
"Step 2.",
"$(\\sigma _\\varphi ^n(x^1))_{n\\ge 1}$ , $(\\sigma _\\varphi ^n(x^2))_{n\\ge 1}$ , $(\\sigma _\\varphi ^n(x^3))_{n\\ge 1}$ , ... are pairwise disjoint sequences: Consider $r,m\\ge 1$ and $i\\ge j\\ge 1$ with $\\sigma _\\varphi ^i(x^m)=\\sigma _\\varphi ^j(x^r)$ .",
"Choose $l,t\\ge 1$ with $i+l=u_m^t-1$ , so: $\\sigma _\\varphi ^i(x^m)=\\sigma _\\varphi ^j(x^r) & \\Rightarrow &(\\forall \\alpha \\in \\Gamma \\:(x^m_{\\varphi ^i(\\alpha )}=x^r_{\\varphi ^j(\\alpha )})) \\\\& \\Rightarrow & (\\forall k\\ge 0\\:\\forall s\\ge 1\\:(x^m_{\\varphi ^{i+k}(\\theta _{u_m^s})}=x^m_{\\varphi ^{j+k}(\\theta _{u_r^s})})) \\\\& \\Rightarrow & p=x^m_{\\varphi ^{i+l}(\\theta _{u_m^t})}= x^r_{\\varphi ^{j+l}(\\theta _{u_m^t})} \\\\& \\Rightarrow & \\varphi ^{j+l}(\\theta _{u_m^t})\\in \\lbrace \\varphi ^v(u_r^w):w\\ge 1,1\\le v<u_r^w\\rbrace \\\\& \\Rightarrow & (\\exists w\\ge 1\\:(\\varphi ^{j+l}(\\theta _{u_m^t})\\in \\lbrace \\varphi ^v(u_r^w):1\\le v< u_r^w\\rbrace )) \\\\& \\Rightarrow & (\\exists w\\ge 1\\:\\lbrace \\varphi ^v(\\theta _{u_m^t}):1\\le v< u_m^t\\rbrace \\cap \\lbrace \\varphi ^v(u_r^w):1\\le v< u_r^w\\rbrace ) \\\\& & \\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:({\\rm since \\:} 1\\le j+l\\le i+l<u_m^t) \\\\& \\Rightarrow & (\\exists w\\ge 1\\:u_m^t=u_r^w)\\:({\\rm use \\: the \\: way \\: of \\: choosing \\:}\\theta _v{\\rm s}) \\\\& \\Rightarrow & u_m=u_r \\: ({\\rm since \\:}u_m{\\rm \\: and \\:}u_r{\\rm \\: are \\: prime \\: numbers}) \\\\& \\Rightarrow & m=r$ thus $m=r$ and $\\sigma _\\varphi ^i(x^m)=\\sigma _\\varphi ^j(x^m)$ which leads to $i=j$ by Step 1.",
"By the above two steps $(\\sigma _\\varphi ^n(x^1))_{n\\ge 1}$ , $(\\sigma _\\varphi ^n(x^2))_{n\\ge 1}$ , $(\\sigma _\\varphi ^n(x^3))_{n\\ge 1}$ , ... are pairwise disjoint infinite sequences which leads to $\\mathfrak {o}(\\sigma _\\varphi )=\\infty $ .",
"Theorem 2.4 The following statements are equivalent: 1.",
"$W(\\sigma _\\varphi )=\\varnothing $ (i.e., $\\mathfrak {o}(\\sigma _\\varphi )={\\rm ent}_{\\rm set}( \\sigma _\\varphi )=0$ ), 2.",
"$\\varphi $ is quasi-periodic, 3.",
"${\\rm ent}_{\\rm set}( \\sigma _\\varphi )<\\infty $ (i.e., $\\mathfrak {o}(\\sigma _\\varphi )<\\infty $ ).",
"“(1) $\\Rightarrow $ (2)”: Suppose $W(\\sigma _\\varphi )=\\varnothing $ , thus by Lemma REF we have $W(\\varphi )=\\varnothing $ .",
"Since $W(\\sigma _\\varphi )=\\varnothing $ , we have $\\mathfrak {o}(\\sigma _\\varphi )=0$ .",
"Using $\\mathfrak {o}(\\sigma _\\varphi )=0$ , $W(\\varphi )=\\varnothing $ and Lemma REF , $\\varphi $ is quasi-periodic.",
"“(2) $\\Rightarrow $ (1)”: If there exist $n>m\\ge 1$ with $\\varphi ^n=\\varphi ^m$ , then for all $(x_\\alpha )_{\\alpha \\in \\Gamma }\\in X^\\Gamma $ we have $\\sigma _\\varphi ^n((x_\\alpha )_{\\alpha \\in \\Gamma })=(x_{\\varphi ^n(\\alpha )})_{\\alpha \\in \\Gamma }=(x_{\\varphi ^m(\\alpha )})_{\\alpha \\in \\Gamma }=\\sigma _\\varphi ^m((x_\\alpha )_{\\alpha \\in \\Gamma })$ which shows $(x_\\alpha )_{\\alpha \\in \\Gamma }\\notin W(\\sigma _\\varphi )$ .",
"“(3) $\\Rightarrow $ (2)”: By Lemmas REF and REF , if $\\varphi $ is not quasi-periodic, then $\\mathfrak {o}(\\sigma _\\varphi )=\\infty $ .",
"Corollary 2.5 By Theorem REF we have: ${\\rm ent}_{\\rm set}( \\sigma _\\varphi )=\\left\\lbrace \\begin{array}{lc} 0 & \\varphi {\\rm \\: is \\: quasi-periodic}\\:,\\\\\\infty & {\\rm otherwise}\\:.\\end{array}\\right.$"
],
[
"Contravariant set-theoretical entropy of $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $",
"For $\\alpha ,\\beta \\in \\Gamma $ let $\\alpha \\Re \\beta $ if and only if there exists $n\\ge 1$ with $\\varphi ^n(\\alpha )=\\varphi ^n(\\beta )$ .",
"Then $\\Re $ is an equivalence relation on $\\Gamma $ , moreover it's evident that $\\alpha \\Re \\beta $ if and only if $\\varphi (\\alpha )\\Re \\varphi (\\beta )$ .",
"In this section we prove that for all $x\\in X^\\Gamma $ , $\\sigma _\\varphi ^{-1}(x)$ is finite, if and only if either $\\Gamma =\\varphi (\\Gamma )$ or “$X$ and $\\Gamma \\setminus \\varphi (\\Gamma )$ are finite”.",
"Moreover if for all $x\\in X^\\Gamma $ , $\\sigma _\\varphi ^{-1}(x)$ is finite, then ${\\rm ent}_{\\rm cset}( \\sigma _\\varphi )\\in \\lbrace 0,\\infty \\rbrace $ with ${\\rm ent}_{\\rm cset}( \\sigma _\\varphi )=0$ if and only if there exists $n\\ge 1$ such that $\\varphi ^n(\\alpha )\\Re \\alpha $ for all $\\alpha \\in \\Gamma $ .",
"Remark 3.1 The generalized shift $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ is on-to-one (resp.",
"onto) if and only if $\\varphi :\\Gamma \\rightarrow \\Gamma $ is onto (resp.",
"one-to-one) [3], [2].",
"Note 3.2 Consider $\\widetilde{\\varphi }:\\mathop {\\frac{\\Gamma }{\\Re }\\rightarrow \\frac{\\Gamma }{\\Re }}\\limits _{[\\alpha ]_\\Re \\mapsto [\\varphi (\\alpha )]_\\Re }$ and note that (for $\\alpha \\in \\Gamma $ let $[\\alpha ]_\\Re =\\lbrace y\\in \\Gamma :x\\Re y\\rbrace $ and $\\frac{\\Gamma }{\\Re }=\\lbrace [\\lambda ]_\\Re :\\lambda \\in \\Gamma \\rbrace $ ): $\\mathop {\\mathfrak {f}:{\\rm sc}(\\sigma _\\varphi )\\rightarrow X^{\\frac{\\Gamma }{\\Re }}}\\limits _{\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:(x_\\alpha )_{\\alpha \\in \\Gamma }\\mapsto (x_\\alpha )_{[\\alpha ]_\\Re \\in \\frac{\\Gamma }{\\Re }}}$ is well-defined, since for $(x_\\alpha )_{\\alpha \\in \\Gamma }\\in {\\rm sc}(\\sigma _\\varphi )$ and $\\theta ,\\beta \\in \\Gamma $ with $\\theta \\Re \\beta $ , there exists $n\\ge 1$ and $(y_\\alpha )_{\\alpha \\in \\Gamma }\\in X^\\Gamma $ with $\\varphi ^n(\\theta )=\\varphi ^n(\\beta )$ and $(y_{\\varphi ^n(\\alpha )})_{\\alpha \\in \\Gamma }=\\sigma _\\varphi ^n((y_\\alpha )_{\\alpha \\in \\Gamma })=(x_\\alpha )_{\\alpha \\in \\Gamma }$ , thus $x_\\theta =y_{\\varphi ^n(\\theta )}=y_{\\varphi ^n(\\beta )}=x_\\beta $ .",
"Now we have: 1.",
"$\\mathfrak {f}:{\\rm sc}(\\sigma _\\varphi )\\rightarrow X^{\\frac{\\Gamma }{\\Re }}$ is one-to-one.",
"2.",
"The following diagram commutes: ${{\\rm sc}(\\sigma _\\varphi ) [rr]^{\\sigma _\\varphi \\upharpoonright _{{\\rm sc}(\\sigma _\\varphi )}}[d]_{\\mathfrak {f}} && {\\rm sc}(\\sigma _\\varphi ) [d]^{\\mathfrak {f}} \\\\X^{\\frac{\\Gamma }{\\Re }} [rr]^{\\sigma _{\\widetilde{\\varphi }}} && X^{\\frac{\\Gamma }{\\Re }}}$ 3.",
"Using Remark REF , since $\\widetilde{\\varphi }:\\frac{\\Gamma }{\\Re }\\rightarrow \\frac{\\Gamma }{\\Re }$ is one-to-one, $\\sigma _{\\widetilde{\\varphi }}:X^{\\frac{\\Gamma }{\\Re }}\\rightarrow X^{\\frac{\\Gamma }{\\Re }}$ is onto.",
"Lemma 3.3 The generalized shift $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ is finite fibre, if and only if at least one of the following conditions hold: $X$ and $\\Gamma \\setminus \\varphi (\\Gamma )$ are finite, $\\Gamma =\\varphi (\\Gamma )$ .",
"First suppose for all $x\\in X^\\Gamma $ , $\\sigma _\\varphi ^{-1}(x)$ is finite and $\\Gamma \\ne \\varphi (\\Gamma )$ .",
"Choose $p\\in X$ , for all $q=(q_\\alpha )_{\\alpha \\in \\Gamma \\setminus \\varphi (\\Gamma )}\\in X^{\\Gamma \\setminus \\varphi (\\Gamma )}$ let: $x_\\alpha ^q:=\\left\\lbrace \\begin{array}{lc} q_\\alpha & \\alpha \\in \\Gamma \\setminus \\varphi (\\Gamma )\\:, \\\\p & {\\rm otherwise}\\:,\\end{array}\\right.$ then $\\sigma _\\varphi ((x_\\alpha ^q)_{\\alpha \\in \\Gamma })=(p)_{\\alpha \\in \\Gamma }$ .",
"So $\\mathop {X^{\\Gamma \\setminus \\varphi (\\Gamma )}\\rightarrow \\sigma _\\varphi ^{-1}((p)_{\\alpha \\in \\Gamma })}\\limits _{q\\mapsto (x_\\alpha ^q)_{\\alpha \\in \\Gamma }}$ is one-to-one, using finiteness of $\\sigma _\\varphi ^{-1}((p)_{\\alpha \\in \\Gamma })$ , $X^{\\Gamma \\setminus \\varphi (\\Gamma )}$ is finite too.",
"Both sets $\\Gamma \\setminus \\varphi (\\Gamma ),X$ are finite since $X^{\\Gamma \\setminus \\varphi (\\Gamma )}$ is finite, $X$ has at least two elements and $\\Gamma \\setminus \\varphi (\\Gamma )\\ne \\varnothing $ .",
"Conversely, if $\\Gamma =\\varphi (\\Gamma )$ , then by Remark REF , $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ is one-to-one, so for all $x\\in X^\\Gamma $ the set $\\sigma _\\varphi ^{-1}(x)$ has at most one element and is finite.",
"Now suppose $X$ and $\\Gamma \\setminus \\varphi (\\Gamma )$ are finite.",
"For all $(x_\\alpha )_{\\alpha \\in \\Gamma },(y_\\alpha )_{\\alpha \\in \\Gamma }\\in X^\\Gamma $ we have: $(y_\\alpha )_{\\alpha \\in \\Gamma }\\in \\sigma _\\varphi ^{-1}(\\sigma _\\varphi ((x_\\alpha )_{\\alpha \\in \\Gamma })) & \\Rightarrow & \\sigma _\\varphi ((y_\\alpha )_{\\alpha \\in \\Gamma })=\\sigma _\\varphi ((x_\\alpha )_{\\alpha \\in \\Gamma }) \\\\& \\Rightarrow & (y_{\\varphi (\\alpha )})_{\\alpha \\in \\Gamma }=(x_{\\varphi (\\alpha )})_{\\alpha \\in \\Gamma } \\\\& \\Rightarrow & \\forall \\beta \\in \\varphi (\\Gamma )\\: y_\\beta =x_\\beta \\\\& \\Rightarrow & (y_\\alpha )_{\\alpha \\in \\Gamma }\\in \\lbrace (z_\\alpha )_{\\alpha \\in \\Gamma }\\in X^\\Gamma :\\forall \\alpha \\in \\varphi (\\Gamma )\\:z_\\alpha =x_\\alpha \\rbrace $ Hence $|\\sigma _\\varphi ^{-1}(\\sigma _\\varphi ((x_\\alpha )_{\\alpha \\in \\Gamma }))|\\le |\\lbrace (z_\\alpha )_{\\alpha \\in \\Gamma }\\in X^\\Gamma :\\forall \\alpha \\in \\varphi (\\Gamma )\\:z_\\alpha =x_\\alpha \\rbrace |=|X^{\\Gamma \\setminus \\varphi (\\Gamma )}|<\\infty \\:.$ Thus for all $w\\in X^\\Gamma $ , $\\sigma _\\varphi ^{-1}(w)$ is finite.",
"Lemma 3.4 If $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ is finite fibre, then $\\sigma _{\\widetilde{\\varphi }}:X^{\\frac{\\Gamma }{\\Re }}\\rightarrow X^{\\frac{\\Gamma }{\\Re }}$ is finite fibre.",
"Suppose for all $w\\in X^\\Gamma $ , $\\sigma _\\varphi ^{-1}(w)$ is finite, then by Lemma REF we have the following cases: Case 1.",
"$\\Gamma =\\varphi (\\Gamma )$ : In this case we have $\\widetilde{\\varphi }(\\frac{\\Gamma }{\\Re })=\\lbrace [\\varphi (\\alpha )]_\\Re :\\alpha \\in \\Gamma \\rbrace =\\lbrace [\\alpha ]_\\Re :\\alpha \\in \\varphi (\\Gamma )\\rbrace =\\lbrace [\\alpha ]_\\Re :\\alpha \\in \\Gamma \\rbrace =\\frac{\\Gamma }{\\Re }$ .",
"Case 2.",
"$X$ and $\\Gamma \\setminus \\varphi (\\Gamma )$ are finite: For all $A\\in \\frac{\\Gamma }{\\Re }\\setminus \\widetilde{\\varphi }(\\frac{\\Gamma }{\\Re })$ we have $A\\subseteq \\Gamma \\setminus \\varphi (\\Gamma )$ which leads to $|\\frac{\\Gamma }{\\Re }\\setminus \\widetilde{\\varphi }(\\frac{\\Gamma }{\\Re })|\\le |\\bigcup (\\frac{\\Gamma }{\\Re }\\setminus \\widetilde{\\varphi }(\\frac{\\Gamma }{\\Re })|\\le |\\Gamma \\setminus \\varphi (\\Gamma )|$ and $\\frac{\\Gamma }{\\Re }\\setminus \\widetilde{\\varphi }(\\frac{\\Gamma }{\\Re })$ is finite in this case.",
"Using the above two cases and Lemma REF , $\\sigma _{\\widetilde{\\varphi }}$ is finite fibre.",
"Lemma 3.5 We have $\\mathfrak {a}( \\sigma _\\varphi )=\\mathfrak {a}( \\sigma _{\\widetilde{\\varphi }})$ .",
"In particular by Lemma REF , if $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ is finite fibre, then ${\\rm ent}_{\\rm cset}( \\sigma _\\varphi )={\\rm ent}_{\\rm cset}( \\sigma _{\\widetilde{\\varphi }})$ .",
"For $s\\ge 1$ suppose $(y_n^1)_{n\\ge 1},\\ldots ,(y_n^s)_{n\\ge 1}$ are pairwise disjoint infinite $\\sigma _\\varphi -$ anti-orbit sequences, then for all $i\\in \\lbrace 1,\\ldots ,s\\rbrace $ and $n\\ge 1$ we have $y^i_n\\in {\\rm sc}(\\sigma _\\varphi )$ .",
"Using Note REF , $\\mathfrak {f}$ is one-to-one, thus $(\\mathfrak {f}(y_n^1))_{n\\ge 1},\\ldots ,(\\mathfrak {f}(y_n^s))_{n\\ge 1}$ are infinite pairwise disjoint sequences in $X^{\\frac{\\Gamma }{\\Re }}$ , moreover for all $i\\in \\lbrace 1,\\ldots ,s\\rbrace $ and $n\\ge 1$ we have $\\sigma _{\\widetilde{\\varphi }}(\\mathfrak {f}(y_{n+1}^i))=\\mathfrak {f}(\\sigma _\\varphi (y_{n+1}^i))=\\mathfrak {f}(y_n^i)$ .",
"Thus $(\\mathfrak {f}(y_n^1))_{n\\ge 1},\\ldots ,(\\mathfrak {f}(y_n^s))_{n\\ge 1}$ are infinite pairwise disjoint $\\sigma _{\\widetilde{\\varphi }}-$ anti-orbit sequences.",
"Therefore $\\mathfrak {a}( \\sigma _{\\widetilde{\\varphi }})\\ge \\mathfrak {a}( \\sigma _\\varphi )$ .",
"Now for $x=(x_\\beta )_{\\beta \\in \\frac{\\Gamma }{\\Re }}\\in X^{\\frac{\\Gamma }{\\Re }}$ let $w^x=(x_{[\\alpha ]_\\Re })_{\\alpha \\in \\Gamma }\\in X^\\Gamma $ .",
"So for all $x,y\\in X^{\\frac{\\Gamma }{\\Re }}$ if $w^x=w^y$ , then $x=y$ .",
"Moreover for $x=(x_\\beta )_{\\beta \\in \\frac{\\Gamma }{\\Re }}\\in X^{\\frac{\\Gamma }{\\Re }}$ we have: $w^{\\sigma _{\\widetilde{\\varphi }}(x)} & = &w^{(x_{\\widetilde{\\varphi }(\\beta )})_{\\beta \\in \\frac{\\Gamma }{\\Re }}}=(x_{\\widetilde{\\varphi } ([\\alpha ]_\\Re )})_{\\alpha \\in \\Gamma } \\\\& = & (x_{[\\varphi (\\alpha )]_\\Re })_{\\alpha \\in \\Gamma }=\\sigma _\\varphi ((x_{[\\alpha ]_\\Re })_{\\alpha \\in \\Gamma })=\\sigma _\\varphi (w^x)\\:.$ For $t\\ge 1$ suppose $(y_n^1)_{n\\ge 1},\\ldots ,(y_n^t)_{n\\ge 1}$ are pairwise disjoint infinite $\\sigma _{\\widetilde{\\varphi }}-$ anti-orbit sequences, then $(w^{y_n^1})_{n\\ge 1},\\ldots ,(w^{y_n^t})_{n\\ge 1}$ are pairwise disjoint infinite $\\sigma _\\varphi -$ anti-orbit sequences.",
"Thus $\\mathfrak {a}( \\sigma _\\varphi )\\ge \\mathfrak {a}( \\sigma _{\\widetilde{\\varphi }})$ .",
"Lemma 3.6 Suppose $\\psi :\\Gamma \\rightarrow \\Gamma $ is one-to-one and has at least one non-periodic point, then $\\mathfrak {a}(\\sigma _\\psi )=\\infty $ , thus if $\\sigma _\\psi :X^\\Gamma \\rightarrow X^\\Gamma $ is finite fibre too, then ${\\rm ent}_{\\rm cset}( \\sigma _\\psi )=\\infty $ .",
"Suppose $\\varphi :\\Gamma \\rightarrow \\Gamma $ is one-to-one, and $\\theta \\in \\Gamma $ is a non-periodic point of $\\varphi $ .",
"Choose distinct $p,q\\in X$ and for $m,n\\ge 1$ let: $(x_n^m(0),x_n^m(1),x_n^m(2),\\cdots ):=(\\:\\underbrace{p,\\cdots ,p}_{m {\\rm \\: times}},\\underbrace{q,\\cdots ,q}_{n {\\rm \\: times}}, p,p,p,\\cdots )\\:,$ now let: $z^{(n,m)}_\\alpha :=\\left\\lbrace \\begin{array}{lc} x_n^m(k) & k\\ge 0,\\alpha =\\varphi ^k(\\theta ) \\: , \\\\p & {\\rm otherwise}\\:.",
"\\end{array}\\right.",
"$ Then for $z^{(n,m)}:=(z^{(n,m)}_\\alpha )_{\\alpha \\in \\Gamma }$ , considering the sequences $(z^{(1,m)})_{m\\ge 1},(z^{(2,m)})_{m\\ge 1},(z^{(3,m)})_{m\\ge 1},\\ldots $ we have: For $k,n,i,j\\ge 1$ if $z^{(n,i)}=z^{(k,j)}$ , then: $\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:z^{(n,i)}=z^{(k,j)} & \\Rightarrow &(\\forall \\alpha \\in \\Gamma \\: z_\\alpha ^{(n,i)}=z_\\alpha ^{(k,j)}) \\\\& \\Rightarrow & (z_\\theta ^{(n,i)},z_{\\varphi (\\theta )}^{(n,i)},z_{\\varphi ^2(\\theta )}^{(n,i)},\\cdots )(z_\\theta ^{(k,j)},z_{\\varphi (\\theta )}^{(k,j)},z_{\\varphi ^2(\\theta )}^{(k,j)},\\cdots ) \\\\& \\Rightarrow & (x_n^i(0),x_n^i(1),\\cdots )=(x_k^j(0),x_k^j(1),\\cdots ) \\\\& \\Rightarrow & (\\:\\underbrace{p,\\cdots ,p}_{i {\\rm \\: times}},\\underbrace{q,\\cdots ,q}_{n {\\rm \\: times}}, p,\\cdots )=(\\:\\underbrace{p,\\cdots ,p}_{j {\\rm \\: times}},\\underbrace{q,\\cdots ,q}_{k {\\rm \\: times}}, p,\\cdots ) \\\\& \\Rightarrow & (i=j\\wedge n=k)$ Thus $(z^{(1,m)})_{m\\ge 1},(z^{(2,m)})_{m\\ge 1},(z^{(3,m)})_{m\\ge 1},\\ldots $ are paiwise disjoint infinite sequences.",
"For all $n,m\\ge 1$ and $\\alpha \\in \\Gamma $ we have: $z^{(n,m+1)}_{\\varphi (\\alpha )}=q & \\Leftrightarrow & \\varphi (\\alpha )\\in \\lbrace \\varphi ^i(\\theta ):m+1\\le i<m+1+n\\rbrace \\\\& \\Leftrightarrow &\\alpha \\in \\lbrace \\varphi ^i(\\theta ):m\\le i<m+n\\rbrace \\\\&\\Leftrightarrow & z^{(n,m)}_\\alpha =q$ thus $\\sigma _\\varphi (z^{(n,m+1)})=z^{(n,m)}$ and $(z^{(n,k)})_{k\\ge 1}$ is an anti-orbit Hence $(z^{(1,m)})_{m\\ge 1},(z^{(2,m)})_{m\\ge 1},(z^{(3,m)})_{m\\ge 1},\\ldots $ are pairwise disjoint infinite $\\sigma _\\varphi -$ anti-orbit sequences and $\\mathfrak {a}(\\sigma _\\varphi )=\\infty $ .",
"Note 3.7 Suppose all points of $\\Gamma $ are periodic points of $\\varphi :\\Gamma \\rightarrow \\Gamma $ , then $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ is bijective (note that $\\varphi :\\Gamma \\rightarrow \\Gamma $ is bijective and apply Remark REF ) and using Corollary REF we have: ${\\rm ent}_{\\rm cset}( \\sigma _\\varphi )={\\rm ent}_{\\rm set}( \\sigma _\\varphi ^{-1})={\\rm ent}_{\\rm set}( \\sigma _{\\varphi ^{-1}})=\\left\\lbrace \\begin{array}{lc} 0 & \\exists n\\ge 1\\:\\varphi ^n={\\rm id}_\\Gamma \\: , \\\\ \\infty & {\\rm otherwise}\\:.\\end{array}\\right.$ where for arbitrary $A$ we have $\\mathop {{\\rm id}_A:A\\rightarrow A}\\limits _{\\: \\: \\: \\: \\:\\: \\: \\: \\: \\:\\:\\:x\\mapsto x}$ .",
"Corollary 3.8 Suppose $\\varphi :\\Gamma \\rightarrow \\Gamma $ is one-to-one and $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ is finite fibre, then ${\\rm ent}_{\\rm cset}( \\sigma _\\varphi )={\\rm ent}_{\\rm set}( \\sigma _\\varphi )=\\left\\lbrace \\begin{array}{lc} 0 & \\exists n\\ge 1\\:\\varphi ^n={\\rm id}_\\Gamma \\: , \\\\ \\infty & {\\rm otherwise}\\:.\\end{array}\\right.$ Use Corollary REF , Lemma REF and Note REF .",
"Corollary 3.9 If $\\sigma _\\varphi :X^\\Gamma \\rightarrow X^\\Gamma $ is finite fibre, then: ${\\rm ent}_{\\rm cset}( \\sigma _\\varphi )={\\rm ent}_{\\rm cset}( \\sigma _{\\widetilde{\\varphi }})=\\left\\lbrace \\begin{array}{lc} 0 & \\exists n\\ge 1\\:(\\widetilde{\\varphi })^n={\\rm id}_\\frac{\\Gamma }{\\Re }\\: , \\\\ \\infty & {\\rm otherwise}\\:.\\end{array}\\right.$ First we recall that $\\widetilde{\\varphi }:\\frac{\\Gamma }{\\Re }\\rightarrow \\frac{\\Gamma }{\\Re }$ is one-to-one by Note REF .",
"Use Corollary REF and Lemma REF to complete the proof."
],
[
"Other entropies: counterexamples",
"The main aim of this section is to compare positive topological, algebraic, set-theoretical and contravariant set-theoretical entropies in generalized shifts.",
"Remark 4.1 If $G$ is an abelian group, $\\theta :G\\rightarrow G$ is a group homomorphism and $H$ is a finite subset of $G$ , then ${\\rm ent}_{\\rm alg}(\\theta ,H)={\\displaystyle \\lim _{n\\rightarrow \\infty }\\dfrac{\\log (|H\\cup \\theta (H)\\cup \\cdots \\cup \\theta ^{n-1}(H)|)}{n}}$ exists [5], [6] and we call ${\\rm ent}_{\\rm alg}(\\theta ):=\\sup \\lbrace {\\rm ent}_{\\rm alg}(\\theta ,H):H$ is a finite subgroup of $G\\rbrace $ the algebraic entropy of $\\theta $ .",
"Moreover if $\\varphi :\\Gamma \\rightarrow \\Gamma $ is finite fibre and $X$ is a finite nontrivial group with identity $e$ , then ${\\rm ent}_{\\rm alg}(\\sigma _\\varphi \\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Gamma }X})={\\rm ent}_{\\rm cset}(\\varphi )\\log |X|$ (as it has been mentioned in [1] ${\\rm ent}_{\\rm alg}(\\sigma _\\varphi \\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Gamma }X})$ is equal to the product of string number of $\\varphi $ and $\\log |X|$ this result has been evaluated in [5] in the above form), where $\\mathop {\\oplus }\\limits _{\\Gamma }X=\\lbrace (x_\\alpha )_{\\alpha \\in \\Gamma }\\in X^\\Gamma :\\exists \\alpha _1,\\ldots ,\\alpha _n\\in \\Gamma \\:\\forall \\alpha \\in \\Gamma \\setminus \\lbrace \\alpha _1,\\ldots ,\\alpha _n\\rbrace \\:(x_\\alpha =e)\\rbrace $ .",
"Also by [7], ${\\rm ent}_{\\rm alg}(\\sigma _\\varphi )\\in \\lbrace 0,\\infty \\rbrace $ with ${\\rm ent}_{\\rm alg}(\\sigma _\\varphi )=0$ if and only if there exists $n>m\\ge 1$ with $\\varphi ^n=\\varphi ^m$ (thus ${\\rm ent}_{\\rm alg}(\\sigma _\\varphi )={\\rm ent}_{\\rm set}(\\sigma _\\varphi )$ by Corollary REF ).",
"Remark 4.2 Suppose $Y$ is a compact topological space and $\\mathcal {U},\\mathcal {V}$ are open covers of $Y$ , let $\\mathcal {U}\\vee \\mathcal {V}:=\\lbrace U\\cap V:U\\in \\mathcal {U},V\\in \\mathcal {V}\\rbrace $ and $N(\\mathcal {U}):=\\min \\lbrace |\\mathcal {W}|:\\mathcal {W}$ is a finite subcover of $\\mathcal {U}\\rbrace $ .",
"Now suppose $T:Y\\rightarrow Y$ is continuous, then ${\\rm ent}_{\\rm top}(T,\\mathcal {U}):={\\displaystyle \\lim _{n\\rightarrow \\infty }\\dfrac{\\log N(\\mathcal {U}\\vee T^{-1}(\\mathcal {U})\\vee \\cdots \\vee T^{-(n-1)}(\\mathcal {U}))}{n}}$ exists [8] and we call ${\\rm ent}_{\\rm top}(T):=\\sup \\lbrace {\\rm ent}_{\\rm top}(T,\\mathcal {W}):\\mathcal {W}$ is a finite open cover of $Y\\rbrace $ the topological entropy of $T$ .",
"If $X$ is a finite discrete topological space with at least two elements and $X^\\Gamma $ considered with product (pointwise convergence) topology, then ${\\rm ent}_{\\rm top}(\\sigma _\\varphi )={\\rm ent}_{\\rm set}(\\varphi )\\log |X|$ [2].",
"In the rest let: $\\mathcal {C}$ is the collection of all generalized shifts $\\sigma _\\psi :Y^\\Gamma \\rightarrow Y^\\Gamma $ such that $Y$ is a nontrivial finite discrete topological group (so $2\\le |Y|<\\infty $ ), $\\Gamma $ is a nonempty set and both maps $\\psi :\\Gamma \\rightarrow \\Gamma $ , $\\sigma _\\psi :Y^\\Gamma \\rightarrow Y^\\Gamma $ are finite fibre, $\\mathcal {C}_{\\rm top}$ is the collection of all elements of $\\mathcal {C}$ like $\\sigma _\\psi :Y^\\Gamma \\rightarrow Y^\\Gamma $ such that ${\\rm ent}_{\\rm top}(\\sigma _\\psi )>0$ , $\\mathcal {C}_{\\rm dalg}$ is the collection of all elements of $\\mathcal {C}$ like $\\sigma _\\psi :Y^\\Gamma \\rightarrow Y^\\Gamma $ such that ${\\rm ent}_{\\rm alg}(\\sigma _\\psi \\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Gamma }Y})>0$ , $\\mathcal {C}_{\\rm cset}$ is the collection of all elements of $\\mathcal {C}$ like $\\sigma _\\psi :Y^\\Gamma \\rightarrow Y^\\Gamma $ such that ${\\rm ent}_{\\rm cset}(\\sigma _\\psi )>0$ , $\\mathcal {C}_{\\rm set}$ is the collection of all elements of $\\mathcal {C}$ like $\\sigma _\\psi :Y^\\Gamma \\rightarrow Y^\\Gamma $ such that ${\\rm ent}_{\\rm set}(\\sigma _\\psi )>0$ (i.e., ${\\rm ent}_{\\rm alg}(\\sigma _\\psi )>0$ by Remark REF ).",
"Lemma 4.3 We have $\\mathcal {C}_{\\rm top}\\subseteq \\mathcal {C}_{\\rm cset}\\subseteq \\mathcal {C}_{\\rm set}$ and $\\mathcal {C}_{\\rm dalg}\\subseteq \\mathcal {C}_{\\rm set}$ .",
"As a matter of fact for an element of $\\mathcal {C}$ like $\\sigma _\\varphi :Y^\\Gamma \\rightarrow Y^\\Gamma $ we have: ${\\rm ent}_{\\rm top}(\\sigma _\\varphi )\\le {\\rm ent}_{\\rm cset}(\\sigma _\\varphi )\\le {\\rm ent}_{\\rm set}(\\sigma _\\varphi )$ and ${\\rm ent}_{\\rm alg}(\\sigma _\\varphi \\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Gamma }Y})\\le {\\rm ent}_{\\rm set}(\\sigma _\\varphi )$ .",
"$\\:$ “${\\rm ent}_{\\rm top}(\\sigma _\\varphi )\\le {\\rm ent}_{\\rm cset}(\\sigma _\\varphi )$ ” Suppose ${\\rm ent}_{\\rm top}(\\sigma _\\varphi )>0$ , then $\\mathfrak {o}(\\varphi )\\log |Y|={\\rm ent}_{\\rm set}(\\varphi )\\log |Y|={\\rm ent}_{\\rm top}(\\sigma _\\varphi )>0$ , thus $\\mathfrak {o}(\\varphi )>0$ and $W(\\varphi )\\ne \\varnothing $ .",
"Choose $\\alpha \\in W(\\varphi )$ , then $\\lbrace \\varphi ^n(\\alpha )\\rbrace _{n\\ge 0}$ is a one-to-one sequence thus for all $n>m\\ge 0$ , $[\\varphi ^n(\\alpha )]_\\Re \\ne [\\varphi ^m(\\alpha )]_\\Re $ , so for all $n\\ge 1$ we have $\\widetilde{\\varphi }^n([\\alpha ]_\\Re )\\ne [\\alpha ]_\\Re $ , hence by Corollary REF , ${\\rm ent}_{\\rm cset}(\\sigma _\\varphi )>0$ and ${\\rm ent}_{\\rm cset}(\\sigma _\\varphi )=\\infty (\\ge {\\rm ent}_{\\rm top}(\\sigma _\\varphi )$ ).",
"“${\\rm ent}_{\\rm cset}(\\sigma _\\varphi )\\le {\\rm ent}_{\\rm set}(\\sigma _\\varphi )$ ” Suppose ${\\rm ent}_{\\rm set}(\\sigma _\\varphi )\\ne \\infty $ , then ${\\rm ent}_{\\rm set}(\\sigma _\\varphi )=0$ and there exists $n>m\\ge 1$ with $\\varphi ^n=\\varphi ^m$ , thus $\\widetilde{\\varphi }^n=\\widetilde{\\varphi }^m$ , and using the fact that $\\widetilde{\\varphi }$ is one-to-one we lave $\\widetilde{\\varphi }^{n-m}={\\rm id}_{\\frac{\\Gamma }{\\Re }}$ , thus ${\\rm ent}_{\\rm cset}(\\sigma _\\varphi )=0$ by Corollary REF .",
"“${\\rm ent}_{\\rm alg}(\\sigma _\\varphi \\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Gamma }Y})\\le {\\rm ent}_{\\rm set}(\\sigma _\\varphi )$ ” Suppose ${\\rm ent}_{\\rm alg}(\\sigma _\\varphi \\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Gamma }Y})>0$ , then $\\mathfrak {a}(\\varphi )\\log |Y|={\\rm ent}_{\\rm cset}(\\varphi )\\log |Y|={\\rm ent}_{\\rm alg}(\\sigma _\\varphi \\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Gamma }Y})>0$ , thus $\\mathfrak {a}(\\varphi )>0$ and there exists a one-to-one anti-orbit sequence $\\lbrace \\alpha _n\\rbrace _{n\\ge 1}$ in $\\Gamma $ .",
"For all $n>m\\ge 1$ we have $\\varphi ^n(\\alpha _{n+m})=\\alpha _m\\ne \\alpha _n=\\varphi ^m(\\alpha _{n+m})$ and $\\varphi ^n\\ne \\varphi ^m$ , thus ${\\rm ent}_{\\rm set}(\\sigma _\\varphi )=\\infty (\\ge {\\rm ent}_{\\rm alg}(\\sigma _\\varphi \\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Gamma }Y})$ ) by Corollary REF .",
"Table 4.4 We have the following table, in which the mark “$\\surd $ ” means $p\\le q$ for the corresponding case for all $\\sigma _\\psi :Y^\\Gamma \\rightarrow Y^\\Gamma $ in $\\mathcal {C}$ , also the mark “$\\times $ ” indicates that there exists $\\sigma _\\psi :Y^\\Gamma \\rightarrow Y^\\Gamma $ in $\\mathcal {C}$ with $p>q$ in the corresponding case.",
"Table: NO_CAPTIONFor all “$\\surd $ ” marks use Lemma REF .",
"In order to establish “$\\times $ ” marks use the following counterexamples.",
"Define $\\lambda _1,\\lambda _2,\\lambda _3:{\\mathbb {Z}}\\rightarrow {\\mathbb {Z}}$ with the following diagrams: Table: NO_CAPTION So: $\\lambda _1(n)=\\left\\lbrace \\begin{array}{lc} n-1 & n\\ge 1\\:, \\\\ 0 & n=0\\:, \\\\ n+1 & n\\le -1\\:, \\end{array}\\right.\\: \\: \\: \\: \\:\\lambda _2(n)=\\left\\lbrace \\begin{array}{lc} n+1 & n\\ge 1\\:, \\\\ 0 & n=0\\:, \\\\ n-1 & n\\le -1\\:, \\end{array}\\right.\\: \\: \\: \\: \\:\\lambda _3(n)=\\left\\lbrace \\begin{array}{lc} n+1 & n\\le -1\\:, \\\\ 0 & n=0,1\\:, \\\\ 3 & n=2\\:, \\\\3 & n=3 \\:, \\\\ 5 & n=4 \\:, \\\\ 6 & n=5 \\: \\\\ 6 & n=6\\: , \\\\ \\vdots & \\end{array}\\right.$ Then for discrete finite abelian group $G$ with $|G|\\ge 2$ and $\\sigma _{\\lambda _i}:G^{\\mathbb {Z}}\\rightarrow G^{\\mathbb {Z}}$ we have: $\\bullet $ $\\mathfrak {o}(\\lambda _1)=\\mathfrak {o}(\\lambda _3)=0$ , $\\mathfrak {o}(\\lambda _2)=2$ , $\\mathfrak {a}(\\lambda _1)=2$ , $\\mathfrak {a}(\\lambda _2)=0$ , $\\mathfrak {a}(\\lambda _3)=1$ , $\\bullet $ ${\\rm ent}_{\\rm top}(\\sigma _{\\lambda _1})={\\rm ent}_{\\rm top}(\\sigma _{\\lambda _3})=0$ , ${\\rm ent}_{\\rm top}(\\sigma _{\\lambda _2})=2\\log |G|$ , $\\bullet $ ${\\rm ent}_{\\rm alg}(\\sigma _{\\lambda _1}\\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Gamma }G})=2\\log |G|$ , ${\\rm ent}_{\\rm alg}(\\sigma _{\\lambda _2}\\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Gamma }G})=0$ , ${\\rm ent}_{\\rm alg}(\\sigma _{\\lambda _3}\\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Gamma }G})=\\log |G|$ , $\\bullet $ ${\\rm ent}_{\\rm cset}(\\sigma _{\\lambda _1})={\\rm ent}_{\\rm cset}(\\sigma _{\\lambda _3})=0$ , ${\\rm ent}_{\\rm cset}(\\sigma _{\\lambda _2})=\\infty $ , $\\bullet $ ${\\rm ent}_{\\rm set}(\\sigma _{\\lambda _1})={\\rm ent}_{\\rm set}(\\sigma _{\\lambda _2})={\\rm ent}_{\\rm set}(\\sigma _{\\lambda _3})=\\infty $ , which complete the proof.",
"Diagram 4.5 We have the following diagram: $\\mathcal {C}$$\\mathcal {C}_{\\rm set}$$\\mathcal {C}_{\\rm cset}$$\\mathcal {C}_{\\rm top}$E1E5E4E3E6E7E2$\\mathcal {C}_{\\rm dalg}$where by “Ei” we mean counterexample $\\sigma _{\\mu _i}:G^{\\Lambda _i}\\rightarrow G^{\\Lambda _i}$ for finite discrete abelian group $G$ with $|G|\\ge 2$ .",
"for $\\Lambda _1:={\\mathbb {Z}}$ and $\\mu _1:=\\lambda _2$ as in Table REF , we have ${\\rm ent}_{\\rm top}(\\sigma _{\\mu _1})=2\\log |G|>0$ and ${\\rm ent}_{\\rm alg}(\\sigma _{\\mu _1}\\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Lambda _1}G})=0$ , for $\\Lambda _2:={\\mathbb {Z}}$ and $\\mu _2:=\\lambda _1$ as in Table REF , we have ${\\rm ent}_{\\rm alg}(\\sigma _{\\mu _2}\\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Lambda _2}G})=2\\log |G|>0$ and ${\\rm ent}_{\\rm cset}(\\sigma _{\\mu _2})=0$ , for $\\Lambda _3:={\\mathbb {Z}}\\times \\lbrace 0,1\\rbrace $ and $\\mu _3(n,i)=\\left\\lbrace \\begin{array}{lc} \\mu _1(n) & i=0\\:, \\\\ \\mu _2(n) & i=1\\:, \\end{array}\\right.$ we have ${\\rm ent}_{\\rm top}(\\sigma _{\\mu _3})= {\\rm ent}_{\\rm alg}(\\sigma _{\\mu _3}\\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Lambda _3}G})=2\\log |G|>0$ , for $\\Lambda _4:={\\mathbb {N}}$ and $\\mu _4=\\lambda _3\\upharpoonright _{\\mathbb {N}}$ we have $ {\\rm ent}_{\\rm alg}(\\sigma _{\\mu _4}\\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Lambda _4}G})={\\rm ent}_{\\rm cset}(\\sigma _{\\mu _4})=0$ and ${\\rm ent}_{\\rm set}(\\sigma _{\\mu _4})=\\infty $ , for $\\Lambda _5:={\\mathbb {Z}}$ and $\\mu _5(n)=-n$ ($n\\in \\mathbb {Z}$ ) we have ${\\rm ent}_{\\rm set}(\\sigma _{\\mu _5})=0$ , for $\\Lambda _6:={\\mathbb {N}}$ and $\\mu _6=(1,2)(3,4,5)(6,7,8,9)(10,11,12,13,14)\\cdots $ we have $ {\\rm ent}_{\\rm alg}(\\sigma _{\\mu _6}\\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Lambda _6}G})={\\rm ent}_{\\rm top}(\\sigma _{\\mu _6})=0$ and ${\\rm ent}_{\\rm cset}(\\sigma _{\\mu _6})=\\infty $ , for $\\Lambda _7=(\\mathbb {N}\\times \\lbrace 0\\rbrace )\\cup (\\mathbb {Z}\\times \\lbrace 1\\rbrace )$ and $\\mu _7(n,i)=\\left\\lbrace \\begin{array}{lc} \\mu _6(n) & i=0\\:, \\\\ \\mu _2(n) & i=1\\:, \\end{array}\\right.$ we have ${\\rm ent}_{\\rm alg}(\\sigma _{\\mu _7}\\upharpoonright _{\\mathop {\\oplus }\\limits _{\\Lambda _7}G})=2\\log |G|>0$ , ${\\rm ent}_{\\rm cset}(\\sigma _{\\mu _7})=\\infty $ , ${\\rm ent}_{\\rm set}(\\sigma _{\\mu _7})=0$ .",
"Zahra Nili Ahmadabadi, Islamic Azad University, Science and Research Branch, Tehran, Iran (e-mail: [email protected]) Fatemah Ayatollah Zadeh Shirazi, Faculty of Mathematics, Statistics and Computer Science, College of Science, University of Tehran , Enghelab Ave., Tehran, Iran (e-mail: [email protected])"
]
] | 1709.01579 |
[
[
"Vaisman solvmanifolds and relations with other geometric structures"
],
[
"Abstract We characterize unimodular solvable Lie algebras with Vaisman structures in terms of K\\\"ahler flat Lie algebras equipped with a suitable derivation.",
"Using this characterization we obtain algebraic restrictions for the existence of Vaisman structures and we establish some relations with other geometric notions, such as Sasakian, coK\\\"ahler and left-symmetric algebra structures.",
"Applying these results we construct families of Lie algebras and Lie groups admitting a Vaisman structure and we show the existence of lattices in some of these families, obtaining in this way many examples of new solvmanifolds equipped with invariant Vaisman structures."
],
[
"Introduction",
"Let $(M,J,g)$ be a $2n$ -dimensional Hermitian manifold, where $J$ is a complex structure and $g$ is a Hermitian metric, and let $\\omega $ denote its fundamental 2-form, that is, $\\omega (X,Y)=g(JX,Y)$ for any $X,Y$ vector fields on $M$ .",
"The manifold $(M,J,g)$ is called locally conformally Kähler (LCK) if $g$ can be rescaled locally, in a neighborhood of any point in $M$ , so as to be Kähler, i.e., there exists an open covering $\\lbrace U_i\\rbrace _{i\\in I}$ of $M$ and a family $\\lbrace f_i\\rbrace _{i\\in I}$ of $C^{\\infty }$ functions, $f_i:U_i \\rightarrow \\mathbb {R}$ , such that each local metric $g_i=\\exp (-f_i)\\,g|_{U_i}$ is Kähler.",
"These manifolds are a natural generalization of the class of Kähler manifolds, and they have been much studied by many authors since the work of I. Vaisman in the '70s (see for instance [16], [20], [36], [47]).",
"An equivalent characterization of an LCK manifold can be given in terms of the fundamental form $\\omega $ , which is defined by $\\omega (X,Y)=g(JX,Y)$ , for all $X,Y \\in \\mathfrak {X}(M)$ .",
"Indeed, a Hermitian manifold $(M,J,g)$ is LCK if and only if there exists a closed 1-form $\\theta $ globally defined on $M$ such that $d\\omega =\\theta \\wedge \\omega .$ This closed 1-form $\\theta $ is called the Lee form (see [26]).",
"Furthermore, the Lee form $\\theta $ is uniquely determined by the following formula: $\\theta =-\\frac{1}{n-1}(\\delta \\omega )\\circ J,$ where $\\omega $ is the fundamental 2-form, $\\delta $ is the codifferential operator and $2n$ is the dimension of $M$ .",
"A Hermitian $(M,J,g)$ is called globally conformally Kähler (GCK) if there exists a $C^{\\infty }$ function, $f:M\\rightarrow \\mathbb {R}$ , such that the metric $\\exp (-f)g$ is Kähler, or equivalently, the Lee form is exact.",
"Therefore a simply connected LCK manifold is GCK.",
"It is well known that LCK manifolds belong to the class $\\mathcal {W}_4$ of the Gray-Hervella classification of almost Hermitian manifolds [21].",
"Also, an LCK manifold $(M,J,g)$ is Kähler if and only if $\\theta =0$ .",
"Indeed, $\\theta \\wedge \\omega =0$ and $\\omega $ non degenerate imply $\\theta =0$ .",
"It is known that if $(M,J,g)$ is a Hermitian manifold with $\\dim M\\ge 6$ such that (REF ) holds for some 1-form $\\theta $ , then $\\theta $ is automatically closed, and therefore $M$ is LCK.",
"The Hopf manifolds are examples of LCK manifolds, and they are obtained as a quotient of $\\mathbb {C}^n-\\lbrace 0\\rbrace $ with the Boothby metric by a discrete subgroup of automorphisms.",
"These manifolds are diffeomorphic to $S^1\\times S^{2n-1}$ and have first Betti number equal to 1, so that they do not admit any Kähler metric.",
"The LCK structures on these Hopf manifolds have a special property, as shown by Vaisman in [45].",
"Indeed, the Lee form is parallel with respect to the Levi-Civita connection of the Hermitian metric.",
"The LCK manifolds sharing this property form a distinguished class, which has been much studied since Vaisman's seminal work [20], [24], [37], [38], [45], [46].",
"Definition 1.1 $(M,J,g)$ is a Vaisman manifold if it is LCK and the Lee form $\\theta $ is parallel with respect to the Levi-Civita connection.",
"A Vaisman manifold satisfies stronger topological properties than general LCK manifolds.",
"For instance, a compact Vaisman non-Kähler manifold $(M,J,g)$ has $b_1(M)$ odd ([24], [46]), whereas in [34] an example is given of a compact LCK manifold with even $b_1(M)$ .",
"This also implies that a compact Vaisman manifold cannot admit Kähler metrics, since the odd Betti numbers of a compact Kähler manifold are even.",
"Moreover, it was proved in [37] and [38] that any compact Vaisman manifold admits a Riemannian submersion to a circle such that all fibers are isometric and admit a natural Sasakian structure.",
"It was shown in [48] that any compact complex submanifold of a Vaisman manifold is Vaisman, as well.",
"In [8] the classification of compact complex surfaces admitting a Vaisman structure is given.",
"It is known that a homogeneous LCK manifold is Vaisman in any of the following cases: (i) when the manifold is compact ([20]), and (ii) when the manifold is a quotient of a reductive Lie group such that the normalizer of the isotropy group is compact ([1]).",
"In this article we are interested in invariant Vaisman structures on solvmanifolds, that is, compact quotients $\\Gamma \\backslash G$ where $G$ is a simply connected solvable Lie group and $\\Gamma $ is a lattice in $G$ .",
"We begin by studying left invariant Vaisman structures on Lie groups, or equivalently, Vaisman structures on a Lie algebra.",
"Let $G$ be a Lie group with a left invariant complex structure $J$ and a left invariant metric $g$ .",
"If $(G,J,g)$ satisfies the LCK condition (REF ), then $(J,g)$ is called a left invariant LCK structure on the Lie group $G$ .",
"In this case, it follows from (REF ) that the corresponding Lee form $\\theta $ on $G$ is also left invariant.",
"This fact allows us to define LCK structures on Lie algebras.",
"Recall that a complex structure J on a Lie algebra $\\mathfrak {g}$ is an endomorphism $J: \\mathfrak {g}\\rightarrow \\mathfrak {g}$ satisfying $J^2=-\\operatorname{Id}$ and $ N_J=0, \\quad \\text{where} \\quad N_J(x,y)=[Jx,Jy]-[x,y]-J([Jx,y]+[x,Jy]),$ for any $x,y \\in \\mathfrak {g}$ .",
"Let $\\mathfrak {g}$ be a Lie algebra, $J$ a complex structure and $\\langle \\cdot ,\\cdot \\rangle $ a Hermitian inner product on $\\mathfrak {g}$ , with $\\omega \\in $$$ 2g*$ the fundamental $ 2$-form.",
"We say that $ (g,J,,)$is \\textit {locally conformally Kähler} (LCK) if there exists $ g*$, with $ d=0$, such that\\begin{equation} d\\omega =\\theta \\wedge \\omega .\\end{equation}Here $ d$ denotes the coboundary operator of the Chevalley-Eilenberg complex of $ g$ corresponding to the trivial representation.$ If the Lie group $G$ is simply connected then any left invariant Vaisman structure on $G$ turns out to be globally conformal to a Kähler structure.",
"Therefore we will study compact quotients of such a Lie group by discrete subgroups (if they exist); these quotients will be non simply connected and will inherit a Vaisman structure.",
"Recall that a discrete subgroup $\\Gamma $ of a simply connected Lie group $G$ is called a lattice if the quotient $\\Gamma \\backslash G$ is compact.",
"According to [32], if such a lattice exists then the Lie group must be unimodular.",
"The quotient $\\Gamma \\backslash G$ is known as a solvmanifold if $G$ is solvable and as a nilmanifold if $G$ is nilpotent, and in these cases we have that $\\pi _1(\\Gamma \\backslash G)\\cong \\Gamma $ .",
"Moreover, the diffeomorphism class of solvmanifolds is determined by the isomorphism class of the corresponding lattices, as the following results show: Theorem 1.2 [39] Let $G_1$ and $G_2$ be simply connected solvable Lie groups and $\\Gamma _i$ , $i=1,2$ , a lattice in $G_i$ .",
"If $f:\\Gamma _1\\rightarrow \\Gamma _2$ is an isomorphism, then there exists a diffeomorphism $F:G_1\\rightarrow G_2$ such that $F|_{\\Gamma _1}=f$ , $F(g\\gamma )=F(g)f(\\gamma )$ , for any $\\gamma \\in \\Gamma _1$ and $g\\in G_1$ .",
"Corollary 1.3 [33] Two solvmanifolds with isomorphic fundamental groups are diffeomorphic.",
"LCK and Vaisman structures on Lie groups and Lie algebras and also in their compact quotients by discrete subgroups have been studied by several authors lately (see for instance [1], [5], [6], [15], [25], [40], [41], [42], [43]).",
"For instance, it was shown in [40] that if an LCK Lie algebra is nilpotent then it is isomorphic to $\\mathfrak {h}_{2n+1}\\times \\mathbb {R}$ and the LCK structure is Vaisman.",
"In [1] the authors prove that if a reductive Lie algebra admits an LCK structure then it is isomorphic to either $\\mathfrak {u}(2)$ or $\\mathfrak {gl}(2,\\mathbb {R})$ .",
"In [25] it is proved the non-existence of Vaisman metrics on some solvmanifolds with left invariant complex structures.",
"In [6] it is proved that if a nilmanifold $\\Gamma \\backslash G$ admits a Vaisman structure (not necessarily invariant), then $G$ is isomorphic to the cartesian product of a Heisenberg group $H_{2n+1}$ with $\\mathbb {R}$ .",
"In [43] it is shown that if a completely solvable solvmanifold equipped with an invariant complex structure admits a Vaisman metric, then the solvmanifold is again a quotient of $H_{2n+1}\\times \\mathbb {R}$ .",
"In [29] the authors obtain a Vaisman structure on the total space of certain $S^1$ -bundles over compact coKähler manifolds, and all the examples they exhibit are diffeomorphic to compact solvmanifolds.",
"Despite these results, there is not yet a thorough description of the Lie algebras admitting Vaisman structures, and consequently there are not many known examples.",
"In this article we deal with this problem, and we obtain a characterization of the unimodular solvable Lie algebras admitting Vaisman structures in terms of Kähler flat Lie algebras equipped with suitable derivations (see Theorems REF and REF ).",
"Indeed, we show that any unimodular solvable Vaisman Lie algebra is a double extension of a Kähler flat Lie algebra.",
"In order to do this, we use the fact that Vaisman structures are closely related to Sasakian structures.",
"Moreover, we establish also a relation with other geometric structures, namely with coKähler Lie algebras and left-symmetric algebras.",
"More precisely, we show that any unimodular solvable Vaisman Lie algebra is a central extension of a coKähler flat Lie algebra, and using this we prove the existence of a complete left-symmetric algebra structure on the Vaisman Lie algebra.",
"This gives rise to a complete flat torsion-free connection on any associated solvmanifold.",
"The article is organized as follows.",
"In §2 we prove a general result about unimodular LCK Lie algebras and we recall some basic definitions.",
"In §3 we review some properties about Vaisman Lie algebras and we give the proof of the main theorems (Theorems REF and REF ).",
"As a consequence of these theorems, we need to study derivations of a Kähler Lie algebra, and we do this in §4.",
"In §5, we obtain a strong restriction for the existence of Vaisman structures, namely, if a unimodular solvable Lie algebra admits such a structure then the spectrum of $\\operatorname{ad}_X$ is contained in $i\\mathbb {R}$ for any $X$ in the Lie algebra (see Theorem REF ).",
"In §6 we prove the relation mentioned above with coKähler Lie algebras (Theorem REF ) and left-symmetric algebras (Corollary REF ).",
"Finally, in §7, using the characterization obtained previously we provide families of new examples of Vaisman Lie algebras in any even dimension and determine the existence of lattices in many of the corresponding Lie groups."
],
[
"Preliminaries",
"Let $(\\mathfrak {g},J,\\langle \\cdot ,\\cdot \\rangle )$ be a Lie algebra with an LCK structure.",
"We have the following orthogonal decomposition for $\\mathfrak {g}$ , $\\mathfrak {g}=\\mathbb {R}A \\oplus \\ker \\theta $ where $\\theta $ is the Lee form and $\\theta (A)=1$ .",
"Since $d\\theta =0$ , we have that $\\mathfrak {g}^{\\prime }=[\\mathfrak {g},\\mathfrak {g}]\\subset \\ker \\theta $ .",
"It is clear that $JA\\in \\ker \\theta $ , but when $\\mathfrak {g}$ is unimodular we may state a stronger result.",
"Recall that a Lie algebra is unimodular if $\\operatorname{tr}(\\operatorname{ad}_x)=0$ for all $x$ in the Lie algebra.",
"Proposition 2.1 If $\\mathfrak {g}$ is unimodular and $(J,\\langle \\cdot ,\\cdot \\rangle )$ is an LCK structure on $\\mathfrak {g}$ , then $JA\\in \\mathfrak {g}^{\\prime }$ .",
"Let $\\lbrace e_1,\\dots , e_{2n}\\rbrace $ be an orthonormal basis of $\\mathfrak {g}$ .",
"Recall from [9] the following formula $\\delta \\eta =-\\sum _{j=1}^{2n} \\iota _{e_j} (\\nabla _{e_j} \\eta ),$ where $\\eta $ is a $p$ -form and $\\delta $ is the codifferential operator.",
"Note that the Koszul formula for the Levi-Civita connection in this setting is given simply by $ \\langle \\nabla _xy,z\\rangle = \\frac{1}{2} \\left(\\langle [x,y],z\\rangle -\\langle [y,z],x\\rangle +\\langle [z,x],y\\rangle \\right), \\qquad x,y,z\\in \\mathfrak {g}.$ Using this formula we compute $\\delta \\omega $ , where $\\omega $ is the fundamental 2-form.",
"For $x\\in \\mathfrak {g}$ , $\\delta \\omega (x) = & -\\sum _i (\\nabla _{e_i} \\omega )(e_i,x) \\\\= & \\sum _i \\omega (\\nabla _{e_i}e_i,x)+\\omega (e_i,\\nabla _{e_i}x) \\\\= & \\sum _i -\\langle \\nabla _{e_i}e_i,Jx\\rangle +\\langle Je_i,\\nabla _{e_i}x\\rangle \\\\= & \\frac{1}{2}\\left\\lbrace \\sum _i \\langle [e_i,Jx],e_i\\rangle -\\langle [Jx,e_i],e_i\\rangle +\\langle [e_i,x],Je_i\\rangle -\\langle [x,Je_i],e_i\\rangle +\\langle [Je_i,e_i],x\\rangle \\right\\rbrace \\\\= & \\frac{1}{2} \\left\\lbrace -2\\operatorname{tr}(\\operatorname{ad}_{Jx})+\\operatorname{tr}(J\\circ \\operatorname{ad}_x)-\\operatorname{tr}(\\operatorname{ad}_x\\circ J)+ \\sum _i\\langle [Je_i,e_i],x\\rangle \\right\\rbrace \\\\= & \\frac{1}{2}\\sum _i\\langle [Je_i,e_i],x\\rangle $ It follows from (REF ) that $\\theta (x)=\\frac{1}{2(n-1)}\\sum \\langle J[Je_i,e_i],x\\rangle $ .",
"On the other hand, the Lee form can be written in terms of the inner product as $\\theta (x)=\\frac{\\langle A,x\\rangle }{|A|^2}$ .",
"If we compare both expressions we obtain that $A=\\frac{|A|^2}{2(n-1)}\\sum _i J[Je_i,e_i].$ Therefore $JA\\in \\mathfrak {g}^{\\prime }$ .",
"We will see in forthcoming sections that Vaisman structures on Lie algebras are closely related to certain almost contact metric structures on lower-dimensional Lie algebras.",
"Moreover, when the Vaisman Lie algebra is unimodular and solvable we will show that it is a double extension of a Kähler flat Lie algebra.",
"Let us recall the relevant definitions."
],
[
"Almost contact metric Lie algebras",
"An almost contact metric structure on a Lie algebra $\\mathfrak {h}$ is a quadruple $(\\langle \\cdot ,\\cdot \\rangle , \\phi , \\xi , \\eta )$ , where $\\langle \\cdot ,\\cdot \\rangle $ is an inner product on $\\mathfrak {g}$ , $\\phi $ is an endomorphism $\\phi :\\mathfrak {h}\\rightarrow \\mathfrak {h}$ , and $\\xi \\in \\mathfrak {h}$ , $\\eta \\in \\mathfrak {h}^*$ satisfy the following conditions: $\\eta (\\xi )=1$ , $\\phi ^2=-\\operatorname{Id}+\\eta \\otimes \\xi $ , $\\langle \\phi x,\\phi y\\rangle =\\langle x,y\\rangle -\\eta (x)\\eta (y)$ , for all $x,y\\in \\mathfrak {h}$ .",
"It follows that $|\\xi |=1$ , $\\phi (\\xi )=0$ , $\\eta \\circ \\phi =0$ , and $\\phi $ is skew-symmetric.",
"The fundamental 2-form $\\Phi $ associated to $(\\langle \\cdot ,\\cdot \\rangle , \\phi , \\xi , \\eta )$ is defined by $\\Phi (x,y)=\\langle \\phi x, y\\rangle $ , for $x,y\\in \\mathfrak {h}$ .",
"The almost contact metric structure is called: normal if $N_\\phi =-d\\eta \\otimes \\xi $ ; Sasakian if it is normal and $d\\eta =2\\Phi $ ; almost coKähler if $d\\eta =d\\Phi =0$ ; coKähler if it is almost coKähler and normal (hence $N_\\phi =0$ ).",
"Equivalently, $\\phi $ is parallel (see [11]).",
"Here $N_\\phi $ denotes the Nijenhuis tensor associated to $\\phi $ , which is defined, for $x,y\\in \\mathfrak {h}$ , by $N_\\phi (x,y) = [\\phi x,\\phi y] +\\phi ^2[x,y] -\\phi ([\\phi x,y] +[x,\\phi y]).$ CoKähler structures are also known as “cosymplectic”, following the terminology introduced by Blair in [10] and used in many articles since then, but their striking analogies with Kähler manifolds have led Li (see [27]) and other authors to use the term “coKähler” for these structures, and this is becoming common practice.",
"In the present article we follow this terminology.",
"Remark 2.2 Let $\\mathfrak {h}$ be a Lie algebra equipped with a Sasakian structure $(\\langle \\cdot ,\\cdot \\rangle , \\phi , \\eta , \\xi )$ .",
"it follows from $d\\eta =2\\Phi $ that $\\eta $ is a contact form on $\\mathfrak {h}$ , and consequently $\\xi $ is called the Reeb vector.",
"It is easy to verify that the center of $\\mathfrak {h}$ has dimension at most 1.",
"Moreover, if $\\dim \\mathfrak {z}(\\mathfrak {h})=1$ then the center is generated by the Reeb vector (see [4]).",
"We recall next a result about Sasakian Lie algebras, which will be necessary to prove the main result.",
"Proposition 2.3 ([4]) Let $(\\phi ,\\eta ,\\xi ,\\langle \\cdot ,\\cdot \\rangle )$ be a Sasakian structure on a Lie algebra $\\mathfrak {h}$ with non trivial center $\\mathfrak {z}(\\mathfrak {h})$ generated by $\\xi $ .",
"If $\\mathfrak {k}:=\\ker \\eta $ , then the quadruple $(\\mathfrak {k},[\\cdot ,\\cdot ]_{\\mathfrak {k}},\\phi |_{\\mathfrak {k}},\\langle \\cdot ,\\cdot \\rangle |_{\\mathfrak {k}\\times \\mathfrak {k}})$ is a Kähler Lie algebra, where $[\\cdot ,\\cdot ]_{\\mathfrak {k}}$ is the component of the Lie bracket of $\\mathfrak {h}$ on $\\mathfrak {k}$ ."
],
[
"Double extension of Lie algebras",
"Let $\\mathfrak {h}$ be a real Lie algebra and $\\beta \\in $$$ 2 h*$ a closed $ 2$-form.",
"If we consider $ R$ as the $ 1$-dimensional abelian Lie algebra, generated by an element $ R$, wemay define on the vector space $ Rh$ the following bracket:$$ [x,y]_\\beta =\\beta (x,y)\\xi + [x,y]_\\mathfrak {h}, \\qquad [\\xi ,x]_\\beta =0, \\qquad x,y\\in \\mathfrak {h}.",
"$$It is readily verified that this bracket satisfies the Jacobi identity, and $ Rh$ will be called the central extension of $ h$ by the closed $ 2$-form $$.",
"It willbe denoted $ h()$.$ Given a derivation $D$ of $\\mathfrak {h}_\\beta (\\xi )$ , the double extension of $\\mathfrak {h}$ by the pair $(D,\\beta )$ is defined as the semidirect product $\\mathfrak {h}(D,\\beta ):=\\mathbb {R}\\ltimes _D \\mathfrak {h}_\\beta (\\xi )$ (see [2] for more details).",
"Note that $\\xi \\in \\mathfrak {z}(\\mathfrak {h}(D,\\beta ))$ , the center of the double extension, and the derived ideal satisfies $[\\mathfrak {h}(D,\\beta ),\\mathfrak {h}(D,\\beta )]\\subset \\mathfrak {h}_\\beta (\\xi )$ .",
"Lemma 2.4 Let $\\mathfrak {h}(D,\\beta )$ be the double extension of $\\mathfrak {h}$ by the pair $(D,\\beta )$ .",
"Then $\\mathfrak {h}(D,\\beta )$ is unimodular if and only if $\\mathfrak {h}$ is unimodular and $\\operatorname{tr}D=0$ .",
"Let us denote $\\mathfrak {g}=\\mathfrak {h}(D,\\beta )$ .",
"Let $A$ be a generator of $\\mathbb {R}$ , so that $[A,x]=Dx$ for all $x\\in \\mathfrak {g}$ .",
"Fix any inner product $\\langle \\cdot ,\\cdot \\rangle $ on $\\mathfrak {g}$ such that $\\text{span}\\lbrace A,\\xi \\rbrace $ is orthogonal to $\\mathfrak {h}$ , $\\langle A,\\xi \\rangle =0$ and $|A|=|\\xi |=1$ .",
"Given an orthonormal basis $\\lbrace e_1,\\dots ,e_{n}\\rbrace $ of $\\mathfrak {h}$ , we have that $ \\lbrace A, \\xi \\rbrace \\cup \\lbrace e_1,\\cdots ,e_n\\rbrace $ is an orthonormal basis of $\\mathfrak {g}$ .",
"For any $x\\in \\mathfrak {h}_\\beta (\\xi )$ , we compute $\\operatorname{tr}(\\operatorname{ad}_x^\\mathfrak {g}) =& \\langle [x,A],A\\rangle +\\langle [x,\\xi ]_\\beta ,\\xi \\rangle + \\sum _{i=1}^n \\langle [x,e_i]_\\beta ,e_i\\rangle \\\\=& \\sum _{i=1}^n \\left( \\langle [x,e_i]_\\mathfrak {h},e_i\\rangle + \\langle \\beta (x,e_i)\\xi ,e_i\\rangle \\right) \\\\=& \\operatorname{tr}(\\operatorname{ad}_x^\\mathfrak {h}).$ From this and the fact that $\\operatorname{tr}(\\operatorname{ad}^\\mathfrak {g}_A)=\\operatorname{tr}D$ , the result follows."
],
[
"Vaisman structures on Lie algebras",
"In this section we show the main results of this article, namely, a characterization of unimodular solvable Lie algebras admitting a Vaisman structure (Theorems and REF ).",
"In order to prove them, we establish first basic properties of Vaisman Lie algebras and later we exploit the close relation between Vaisman and Sasakian structures.",
"A Vaisman structure on a Lie algebra $\\mathfrak {g}$ is an LCK structure $(J,\\langle \\cdot ,\\cdot \\rangle )$ such that the associated Lee form $\\theta $ satisfies $\\nabla \\theta =0$ .",
"If $\\mathfrak {g}=\\mathbb {R}A\\oplus \\ker \\theta $ with $A\\in (\\ker \\theta )^\\perp $ such that $\\theta (A)=1$ , then the Vaisman condition is equivalent to $\\nabla A=0$ , and since $d\\theta =0$ , this is in turn equivalent to $A$ being a Killing vector field (considered as a left invariant vector field on the associated Lie group with left invariant metric).",
"Recalling that a left invariant vector field is Killing if and only if the corresponding adjoint operator on the Lie algebra is skew-symmetric, we have: Proposition 3.1 ([3]) If $(J,\\langle \\cdot ,\\cdot \\rangle )$ is an LCK structure on $\\mathfrak {g}$ , then it is Vaisman if and only if the endomorphism $\\operatorname{ad}_A$ is skew-symmetric.",
"Remark 3.2 In [42] it was proved that an LCK structure on a unimodular solvable Lie algebra is Vaisman if and only if $\\langle [A,JA],JA\\rangle =0$ , where $A$ is the metric dual of $\\theta $ .",
"However, the characterization given in Proposition REF will be more useful for our purposes.",
"Example 3.3 Let $\\mathfrak {g}=\\mathbb {R}\\times \\mathfrak {h}_{2n+1}$ , where $\\mathfrak {h}_{2n+1}$ is the $(2n+1)$ -dimensional Heisenberg Lie algebra.",
"There is a basis $\\lbrace x_1,\\dots ,x_n,y_1,\\dots ,y_n,z,w\\rbrace $ of $\\mathfrak {g}$ with Lie brackets given by $[x_i,y_i]=z$ for $i=1,\\dots ,n$ and $w$ in the center.",
"We define an inner product $\\langle \\cdot ,\\cdot \\rangle $ on $\\mathfrak {g}$ such that the basis above is orthonormal.",
"Let $J$ be the almost complex structure on $\\mathfrak {g}$ given by: $Jx_i=y_i, \\quad Jz=-w \\; \\; \\; \\text{for $i=1,\\dots ,n$}.$ It is easily seen that $J$ is a complex structure on $\\mathfrak {g}$ compatible with $\\langle \\cdot ,\\cdot \\rangle $ .",
"If $\\lbrace x^i,y^i,z^*,w^*\\rbrace $ denote the 1-forms dual to $\\lbrace x_i,y_i,z,w\\rbrace $ respectively, then the fundamental 2-form is: $\\omega =\\sum _{i=1}^n(x^i\\wedge y^i) - z^*\\wedge w^*.$ Thus, $ d\\omega =w^*\\wedge \\omega ,$ and therefore $(\\mathfrak {g},J,\\langle \\cdot ,\\cdot \\rangle )$ is LCK.",
"It follows from Proposition REF with $A=w$ that this structure is Vaisman.",
"This example appeared in [15] (see also [40], [3]).",
"It is known that $\\mathfrak {g}$ is the Lie algebra of the Lie group $\\mathbb {R}\\times H_{2n+1}$ , where $H_{2n+1}$ is the $(2n+1)$ -dimensional Heisenberg group.",
"The Lie group $H_{2n+1}$ admits a lattice $\\Gamma $ and therefore the nilmanifold $N= S^1 \\times \\Gamma \\backslash H_{2n+1}$ admits an LCK structure which is Vaisman.",
"The nilmanifold $N$ is a primary Kodaira surface, and it cannot admit any Kähler metric.",
"The following important properties of Vaisman Lie algebras follow from Proposition REF and the integrability of the complex structure (see also [45]): Proposition 3.4 Let $(\\mathfrak {g},J,\\langle \\cdot ,\\cdot \\rangle )$ be a Vaisman Lie algebra, then $[A,JA]=0$ , $J\\circ \\operatorname{ad}_A=\\operatorname{ad}_A\\circ J$ , $J\\circ \\operatorname{ad}_{JA}=\\operatorname{ad}_{JA}\\circ J$ , $\\operatorname{ad}_{JA}$ is skew-symmetric.",
"Without loss of generality, we will assume from now on that $|A|=1$ (rescaling the metric if necessary).",
"Let us denote $W=(\\text{span}\\lbrace A,JA\\rbrace )^\\perp $ , so that $\\ker \\theta =\\mathbb {R}JA\\oplus ^\\perp W$ .",
"The following proposition, which shows the close relation between Vaisman and Sasakian structures, follows from general results proved by I. Vaisman, but we include a proof at the Lie algebra level for the sake of completeness.",
"We will use the following convention for the action of a complex structure on a 1-form: if $\\alpha $ is a 1-form, then $J\\alpha :=-\\alpha \\circ J$ .",
"Proposition 3.5 Set $\\xi :=JA$ , $\\eta :=J\\theta |_{\\ker \\theta }$ , and define an endomorphism $\\phi \\in \\operatorname{End}(\\ker \\theta )$ by $\\phi (a\\xi +x)=Jx$ for $a\\in \\mathbb {R}$ and $x\\in W$ .",
"Then the following relations hold: $\\phi ^2=-\\operatorname{Id}+\\eta \\otimes \\xi $ , $\\langle \\phi x,\\phi y\\rangle =\\langle x,y\\rangle -\\eta (x)\\eta (y)$ , for all $x,y\\in \\ker \\theta $ , $N_\\phi =-d\\eta \\otimes \\xi $ , $d\\eta (x,y)=-\\langle \\phi x, y \\rangle $ , for all $x,y\\in \\ker \\theta $ , where $N_\\phi $ is defined as in (REF ).",
"Note that $W=\\ker \\eta $ and $\\eta (\\xi )=1$ .",
"$(1)$ and $(2)$ follow from the fact that $(J,\\langle \\cdot ,\\cdot \\rangle )$ is a Hermitian structure on $\\mathfrak {g}$ and $|A|=1$ .",
"Since $\\mathfrak {g}$ is Vaisman, using Proposition REF and Proposition REF , it can be seen that $N_\\phi (x,y)=N_J(x,y)- d\\eta (x,y)\\xi ,$ for all $x,y\\in \\ker \\theta $ .",
"Since $J$ is integrable, we have that $N_J=0$ and then we obtain $(3)$ .",
"In order to prove $(4)$ we compute, for all $X,Y\\in \\ker \\theta $ , $d\\eta (x,y) = \\theta (J[x,y])=\\langle A, J[x,y] \\rangle =\\omega ([x,y],A).$ On the other hand, $\\langle Jx,y\\rangle &= \\omega (x,y)\\\\&= \\theta \\wedge \\omega (x,y,A)\\\\&= d\\omega (x,y,A)\\\\&= -\\omega ([x,y],A)-\\omega ([y,A],x)-\\omega ([A,x],y)\\\\&= -\\omega ([x,y],A),$ where we have used Proposition REF in the last step.",
"It is easy to verify that $\\langle Jx,y\\rangle =\\langle \\phi x, y\\rangle $ for any $x,y\\in \\ker \\theta $ , thus the proof is complete.",
"The quadruple $(\\langle \\cdot ,\\cdot \\rangle |_{\\ker \\theta },\\phi ,\\eta ,\\xi )$ on $\\ker \\theta $ from Proposition REF does not satisfy exactly the equations of a Sasakian structure given in §, but it is easy to show that if we modify it as follows: $\\langle \\cdot ,\\cdot \\rangle ^{\\prime }=\\frac{1}{4}\\langle \\cdot ,\\cdot \\rangle , \\quad \\phi ^{\\prime }=\\phi , \\quad \\eta ^{\\prime }=-\\frac{1}{2} \\eta , \\quad \\xi ^{\\prime }=-2\\xi ,$ then $(\\langle \\cdot ,\\cdot \\rangle ^{\\prime },\\phi ^{\\prime },\\eta ^{\\prime },\\xi ^{\\prime })$ is a Sasakian structure on $\\ker \\theta $ .",
"However, in this article, for simplicity, we shall call $(\\langle \\cdot ,\\cdot \\rangle |_{\\ker \\theta },\\phi ,\\eta ,\\xi )$ a Sasakian structure on $\\ker \\theta $ .",
"More generally, when we refer to a Sasakian structure on a Lie algebra we will be assuming that it satisfies the equations on Proposition REF .",
"Therefore, we may rewrite Proposition REF as Corollary 3.6 If $(\\mathfrak {g},J,\\langle \\cdot ,\\cdot \\rangle )$ is a Vaisman Lie algebra with Lee form $\\theta $ , then $\\ker \\theta $ has a Sasakian structure.",
"Conversely, let $\\mathfrak {h}$ be a Lie algebra equipped with a Sasakian structure $(\\langle \\cdot ,\\cdot \\rangle , \\phi , \\eta , \\xi )$ .",
"Taking into account Propositions REF and REF we define the Lie algebra $\\mathfrak {g}=\\mathbb {R}A\\ltimes _D\\mathfrak {h}$ where $D$ is a skew-symmetric derivation of $\\mathfrak {h}$ such that $D(\\xi )=0$ and $D^{\\prime }:=D|_{\\ker \\eta }$ satisfies $D\\phi =\\phi D$ .",
"We consider on $\\mathfrak {g}$ the almost complex structure $J$ given by $J|_{\\ker \\eta }:=\\phi |_{\\ker \\eta }$ , $JA=\\xi $ , and we extend $\\langle \\cdot ,\\cdot \\rangle $ to an inner product on $\\mathfrak {g}$ such that $A$ is orthogonal to $\\mathfrak {h}$ and $|A|=1$ .",
"Note that $(J,\\langle \\cdot ,\\cdot \\rangle )$ is an almost hermitian structure on $\\mathfrak {g}$ .",
"It is easy to prove that $(J,\\langle \\cdot ,\\cdot \\rangle )$ is in fact a Vaisman structure on $\\mathfrak {g}$ .",
"From now on, we assume that $\\mathfrak {g}$ is solvable and unimodular (this is a necessary condition for the associated simply connected Lie group to admit lattices, according to [32]).",
"The next step in order to characterize the Lie algebras admitting Vaisman structures is to prove that $JA$ is a central element of $\\mathfrak {g}$ .",
"Moreover, the dimension of $\\mathfrak {z}(\\mathfrak {g})$ , the center of $\\mathfrak {g}$ , is at most 2.",
"Theorem 3.7 Let $\\mathfrak {g}$ be a unimodular solvable Lie algebra equipped with a Vaisman structure $(J,\\langle \\cdot ,\\cdot \\rangle )$ .",
"Then $JA\\in \\mathfrak {z}(\\mathfrak {g})$ .",
"Moreover, $\\mathfrak {z}(\\mathfrak {g})\\subset \\text{span}\\lbrace A,JA\\rbrace $ .",
"It follows from Proposition REF that $JA\\in \\mathfrak {g}^{\\prime }$ .",
"As $\\mathfrak {g}$ is solvable, it follows that $\\mathfrak {g}^{\\prime }$ is nilpotent and hence $\\operatorname{ad}_{JA}:\\mathfrak {g}\\rightarrow \\mathfrak {g}$ is a nilpotent endomorphism.",
"On the other hand, we know that $\\operatorname{ad}_{JA}$ is skew-symmetric, according from Proposition REF , and therefore $\\operatorname{ad}_{JA}=0$ , that is, $JA\\in \\mathfrak {z}(\\mathfrak {g})$ .",
"Now we will see that $\\mathfrak {z}(\\mathfrak {g})\\subset \\lbrace A,JA\\rbrace $ .",
"For $z\\in \\mathfrak {z}(\\mathfrak {g})$ , we may assume that $z=aA+z^{\\prime }$ with $a\\in \\mathbb {R}$ and $z^{\\prime }\\in W$ , since $JA\\in \\mathfrak {z}(\\mathfrak {g})$ .",
"We have that $0=\\operatorname{ad}_z=a\\operatorname{ad}_A+\\operatorname{ad}_{z^{\\prime }}$ , and it follows from Proposition REF that $\\operatorname{ad}_{z^{\\prime }}$ is a skew-symmetric endomorphism of $\\mathfrak {g}$ .",
"If $[z^{\\prime },Jz^{\\prime }]=cJA + u$ for some $c\\in \\mathbb {R}, u\\in W$ , then $c=\\langle [z^{\\prime },Jz^{\\prime }], JA\\rangle =-\\langle Jz^{\\prime }, [z^{\\prime },JA]\\rangle =0$ .",
"On the other hand, taking into account Proposition REF , we have that $c &=\\langle [z^{\\prime },Jz^{\\prime }], JA\\rangle =\\eta ([z^{\\prime },Jz^{\\prime }])\\\\&= -d\\eta (z^{\\prime },Jz^{\\prime }) = \\langle Jz^{\\prime },Jz^{\\prime }\\rangle \\\\&=|z^{\\prime }|^2,$ Therefore $z^{\\prime }=0$ , and then $z=aA$ .",
"Corollary 3.8 Any unimodular solvable Lie algebra admitting a Sasakian structure has non trivial center.",
"Let $\\mathfrak {h}$ be a unimodular solvable equipped with a Sasakian structure.",
"Therefore, the semidirect product $\\mathfrak {g}=\\mathbb {R}A\\ltimes _D \\mathfrak {h}$ , where $D$ is a suitable skew-symmetric derivation of $\\mathfrak {h}$ , admits a Vaisman structure.",
"It is clear that $\\mathfrak {g}$ is unimodular and solvable.",
"It follows from Theorem REF that $\\mathfrak {h}=\\ker \\theta $ has non trivial center.",
"According to Corollary REF the Sasakian Lie algebra $\\ker \\theta $ has non trivial center, which is therefore generated by the Reeb vector $JA$ , and $\\ker \\theta $ can be decomposed orthogonally as $ \\ker \\theta =\\mathbb {R}JA \\oplus W. $ For $x,y\\in W $ , it is easy to show that $[x,y]=\\omega (x,y)JA + [x,y]_W,$ where $[x,y]_W \\in W$ denotes the component of $[x,y]$ in $W$ .",
"It follows from Proposition REF that $(W,[\\cdot ,\\cdot ]_W,J|_W,\\langle \\cdot ,\\cdot \\rangle |_{W\\times W})$ is a Kähler Lie algebra.",
"We will denote by $\\mathfrak {k}$ the Lie algebra $(W,[\\cdot ,\\cdot ]_W)$ .",
"We point out that $\\mathfrak {k}$ is not a Lie subalgebra from either $\\ker \\theta $ or $\\mathfrak {g}$ , however, it is clear from (REF ) that $\\ker \\theta $ is the central extension $\\ker \\theta =\\mathfrak {k}_{\\omega ^{\\prime }}(JA)$ , where $\\omega ^{\\prime }=\\omega |_{\\mathfrak {k}\\times \\mathfrak {k}}$ is the fundamental 2-form on $\\mathfrak {k}$ , which is closed since $\\mathfrak {k}$ is Kähler.",
"Moreover, if we denote $D:=\\operatorname{ad}_A|_{\\ker \\theta }$ , then $D$ is a skew-symmetric derivation of $\\ker \\theta $ , according to Proposition REF .",
"Therefore, $\\mathfrak {g}$ can be decomposed orthogonally as $\\mathfrak {g}=\\mathbb {R}A\\ltimes _D(\\mathbb {R}JA \\oplus _{\\omega ^{\\prime }}\\mathfrak {k}),$ thus $\\mathfrak {g}$ can be regarded as a double extension $\\mathfrak {g}=\\mathfrak {k}(D,\\omega ^{\\prime })$ .",
"Note that, according to Proposition REF , $D(JA)=0$ .",
"According to Lemma REF , $\\mathfrak {k}$ is unimodular, therefore $\\mathfrak {k}$ is a unimodular Lie algebra admitting a Kähler structure $(J|_\\mathfrak {k},\\langle \\cdot ,\\cdot \\rangle |_\\mathfrak {k})$ .",
"Due to a classical result from Hano [22], the metric $\\langle \\cdot ,\\cdot \\rangle |_\\mathfrak {k}$ is flat.",
"Thus, $\\mathfrak {g}$ is a double extension of a Kähler flat Lie algebra.",
"Let us denote $D^{\\prime }:=D|_{\\mathfrak {k}}$ .",
"Then it follows from Proposition REF that $D^{\\prime }$ commutes with $J|_{\\mathfrak {k}}$ and therefore $D^{\\prime }\\in \\mathfrak {u}(\\mathfrak {k},J|_{\\mathfrak {k}}, \\langle \\cdot ,\\cdot \\rangle |_\\mathfrak {k})$ .",
"Furthermore, $D^{\\prime }$ is a derivation of $\\mathfrak {k}$ .",
"In fact, given $x,y\\in \\mathfrak {k}$ we have that $D^{\\prime }[x,y]_\\mathfrak {k}& = D([x,y]-\\omega (x,y)JA)\\\\& =D[x,y]\\\\& =[Dx,y]+[x,Dy]\\\\& =[D^{\\prime }x,y]+[x,D^{\\prime }y]\\\\& =\\omega (D^{\\prime }x,y)JA+[D^{\\prime }x,y]_\\mathfrak {k}+\\omega (x,D^{\\prime }y)JA+[x,D^{\\prime }y]_\\mathfrak {k}\\\\& =[D^{\\prime }x,y]_\\mathfrak {k}+[x,D^{\\prime }y]_\\mathfrak {k},$ since $\\omega (D^{\\prime }x,y)=-\\omega (x,D^{\\prime }y)$ .",
"Therefore we have associated to any unimodular solvable Vaisman Lie algebra a Kähler flat Lie algebra equipped with a skew-symmetric derivation which commutes with the complex structure.",
"This is summarized in the following theorem.",
"Theorem 3.9 Let $\\mathfrak {g}$ be a unimodular solvable Lie algebra equipped with a Vaisman structure $(J,\\langle \\cdot ,\\cdot \\rangle )$ , with fundamental 2-form $\\omega $ and Lee form $\\theta $ .",
"Then there exists a Kähler flat Lie algebra $\\mathfrak {k}$ such that $\\ker \\theta =\\mathfrak {k}_{\\omega ^{\\prime }}(JA)$ and $\\mathfrak {g}=\\mathfrak {k}(D,\\omega ^{\\prime })$ , where $D:=\\operatorname{ad}_A|_{\\ker \\theta }$ and $\\omega ^{\\prime }:=\\omega |_{\\mathfrak {k}\\times \\mathfrak {k}}$ is the fundamental 2-form of $\\mathfrak {k}$ .",
"Moreover, $D^{\\prime }:=D|_\\mathfrak {k}$ is a skew-symmetric derivation of $\\mathfrak {k}$ that commutes with its complex structure.",
"Next, we will prove the converse of Theorem REF , namely, we show that beginning with a Kähler flat Lie algebra and a suitable derivation we are able to produce a Vaisman structure on a double extension of this Lie algebra.",
"Theorem 3.10 Let $(\\mathfrak {k},J^{\\prime },\\langle \\cdot ,\\cdot \\rangle ^{\\prime })$ be a Kähler flat Lie algebra with $\\omega ^{\\prime }$ its fundamental 2-form and let $D^{\\prime }$ be a skew-symmetric derivation of $\\mathfrak {k}$ such that $J^{\\prime }D^{\\prime }=D^{\\prime }J^{\\prime }$ .",
"Let $D$ be the skew-symmetric derivation of the central extension $\\mathfrak {k}_{\\omega ^{\\prime }}(B)$ defined by: $D(B)=0$ , $D|_{\\mathfrak {k}}=D^{\\prime }$ .",
"Then the double extension $\\mathfrak {g}:=\\mathfrak {k}(D,\\omega ^{\\prime })=\\mathbb {R}A\\ltimes _D \\mathfrak {k}_{\\omega ^{\\prime }}(B)$ admits a Vaisman structure $(J,\\langle \\cdot ,\\cdot \\rangle )$ , where $JA=B$ , $J|_\\mathfrak {k}=J^{\\prime }$ , and $\\langle \\cdot ,\\cdot \\rangle $ extends $\\langle \\cdot ,\\cdot \\rangle ^{\\prime }$ in the following way: $|A|=|B|=1$ , $\\langle A,B\\rangle =0, \\, \\langle A,\\mathfrak {k}\\rangle =\\langle B,\\mathfrak {k}\\rangle =0$ .",
"The Lie algebra $\\mathfrak {g}$ is unimodular and solvable, and the Lee form $\\theta $ is the metric dual of $A$ .",
"Recall that the Lie bracket of $\\mathfrak {g}$ is given by: $\\operatorname{ad}_A|_{\\mathfrak {k}}=D^{\\prime }$ , $B\\in \\mathfrak {z}(\\mathfrak {g})$ and for any $x,y\\in \\mathfrak {k}$ , $[x,y]=\\omega ^{\\prime }(x,y)B + [x,y]_\\mathfrak {k},$ where $\\omega ^{\\prime }(x,y)=\\langle J^{\\prime }x,y\\rangle ^{\\prime }$ and $[\\cdot ,\\cdot ]_\\mathfrak {k}$ denotes the Lie bracket on $\\mathfrak {k}$ .",
"It is easy to see that $(J,\\langle \\cdot ,\\cdot \\rangle )$ in the statement is an almost Hermitian structure on $\\mathfrak {g}$ , and we will call $\\omega $ its Kähler 2-form.",
"Since $(\\mathfrak {k},\\langle \\cdot ,\\cdot \\rangle ^{\\prime })$ is flat then $\\mathfrak {k}$ is unimodular and solvable (see Proposition REF below), and it follows from Lemma REF that $\\mathfrak {g}$ is unimodular and solvable as well.",
"Now we show that $J$ is a complex structure on $\\mathfrak {g}$ .",
"It is enough to show that $N_J(x,y)=0$ and $N_J(A,y)=0$ for all $x, y\\in \\mathfrak {k}$ .",
"Firstly, $N_J(x,y) & = [Jx,Jy] -[x,y] -J([Jx,y] +[x,Jy])\\\\& = N_{J^{\\prime }}^{\\mathfrak {k}}(x,y)+ \\omega ^{\\prime }(Jx,Jy)B-\\omega ^{\\prime }(x,y)B - J(\\omega ^{\\prime }(x,Jy)B+\\omega ^{\\prime }(Jx,y)B)\\\\& = N_{J^{\\prime }}^{\\mathfrak {k}}(x,y) + (-\\langle x,J^{\\prime }y\\rangle ^{\\prime }-\\langle J^{\\prime }x,y\\rangle ^{\\prime })B + (\\langle J^{\\prime }x,J^{\\prime }y\\rangle ^{\\prime }-\\langle x,y\\rangle ^{\\prime })A\\\\& = 0,$ for all $x, y\\in \\mathfrak {k}$ since $J^{\\prime }$ is a complex structure on $\\mathfrak {k}$ , i.e.",
"$N_{J^{\\prime }}^{\\mathfrak {k}}=0$ .",
"Secondly, $N_J(A,y) & = [JA,Jy] -[A,y] -J([JA,y] + [A,Jy])\\\\& = -Dy-JDJy\\\\& = -D^{\\prime }y-J^{\\prime }D^{\\prime }J^{\\prime }y\\\\& = 0,$ for any $y\\in \\mathfrak {k}$ since $JA=B$ is central and $D^{\\prime }$ commutes with $J^{\\prime }$ .",
"Thus $J$ is integrable on $\\mathfrak {g}$ .",
"The next step is to show that $(J,\\langle \\cdot ,\\cdot \\rangle )$ is LCK, to do this we have to verify that $d\\omega =\\theta \\wedge \\omega $ where $\\omega $ is the fundamental form on $\\mathfrak {g}$ and $\\theta $ is the dual 1-form of the vector $A$ .",
"If $x,y,z\\in \\mathfrak {k}$ , then $d\\omega (x,y,z) & = -\\omega ([x,y],z)-\\omega ([y,z],x)-\\omega ([z,x],y)\\\\& = \\langle [x,y]_\\mathfrak {k} +\\omega ^{\\prime }(x,y)B,Jz\\rangle + \\langle [y,z]_\\mathfrak {k} +\\omega ^{\\prime }(y,z)B,Jx\\rangle + \\langle [z,x]_\\mathfrak {k} +\\omega ^{\\prime }(z,x)B,Jy\\rangle \\\\& = \\langle [x,y]_\\mathfrak {k} ,J^{\\prime }z\\rangle ^{\\prime }+ \\langle [y,z]_\\mathfrak {k} ,J^{\\prime }x\\rangle ^{\\prime }+ \\langle [z,x]_\\mathfrak {k} +,J^{\\prime }y\\rangle ^{\\prime }\\\\& = d^\\mathfrak {k}\\omega ^{\\prime }(x,y,z)\\\\& = 0.$ On the other hand, $\\theta \\wedge \\omega (x,y,z)=0$ since $\\mathfrak {k}\\subset \\ker \\theta $ .",
"If $y,z\\in \\mathfrak {k}$ , then $d\\omega (B,y,z) & = -\\omega ([B,y],z)-\\omega ([y,z],B)-\\omega ([z,B],y)\\\\& = \\langle [y,z]_\\mathfrak {k} +\\omega ^{\\prime }(y,z)B,JB\\rangle \\\\& = 0.$ On the other hand, $\\theta \\wedge \\omega (\\xi ,y,z)=0$ since $\\mathbb {R}B\\oplus \\mathfrak {k} = \\ker \\theta .$ If $y,z\\in \\mathfrak {k}$ , then $d\\omega (A,y,z) & = -\\omega ([A,y],z)-\\omega ([y,z],A)-\\omega ([z,A],y)\\\\& = \\langle Dy,Jz\\rangle + \\langle \\omega ^{\\prime }(y,z)B+[y,z]_\\mathfrak {k},JA\\rangle + \\langle -Dz,Jy\\rangle \\\\& = \\langle D^{\\prime }y,J^{\\prime }z\\rangle ^{\\prime } + \\omega ^{\\prime }(y,z) -\\langle D^{\\prime }z,J^{\\prime }y\\rangle ^{\\prime }\\\\& = \\omega ^{\\prime }(y,z),$ since $D^{\\prime }$ is skew-symmetric and commutes with $J^{\\prime }$ .",
"On the other hand, $\\theta \\wedge \\omega (A,y,z)=\\omega (y,z)=\\omega ^{\\prime }(y,z)$ .",
"If $z\\in \\mathfrak {k}$ , then $d\\omega (A,B,z) & = -\\omega ([A,B],z)-\\omega ([B,z],A)-\\omega ([z,A],B)\\\\& = -\\langle D^{\\prime }z,A\\rangle ^{\\prime } \\\\& = 0,$ since $B$ is central and $D^{\\prime }z\\in \\mathfrak {k}$ .",
"On the other hand, $\\theta \\wedge \\omega (A,B,z)=\\omega (B,z)=0$ .",
"Thus we have that $(J,\\langle \\cdot ,\\cdot \\rangle )$ is an LCK structure on $\\mathfrak {g}$ .",
"Moreover, $(J,\\langle \\cdot ,\\cdot \\rangle )$ is Vaisman since $\\operatorname{ad}_A=D$ is a skew-symmetric endomorphism of $\\mathfrak {g}$ (Proposition REF ).",
"As a by-product of this analysis, we obtain the following stronger version of Corollary REF : Corollary 3.11 Any unimodular solvable Lie algebra admitting a Sasakian structure is a central extension of a Kähler flat Lie algebra.",
"It follows from Theorems REF and REF that there is a one-to-one correspondence between unimodular solvable Lie algebras equipped with a Vaisman structure and pairs $(\\mathfrak {k}, D^{\\prime })$ where $\\mathfrak {k}$ is a Kähler flat Lie algebra and $D^{\\prime }$ is a skew-symmetric derivation of $\\mathfrak {k}$ which commutes with the complex structure.",
"Moreover, Theorem REF provides a way to construct all unimodular solvable Lie algebras carrying Vaisman metrics.",
"In order to do this, we need a better understanding of the derivations of Kähler flat Lie algebras, and we pursue this in the following section."
],
[
"Derivations of Kähler flat Lie algebras",
"Let us recall the following result which describes the structure of any Lie algebra equipped with a flat metric.",
"The original version was proved by Milnor in [32], and it was later refined in [7].",
"Proposition 4.1 ([32], [7]) Let $(\\mathfrak {k},\\langle \\cdot ,\\cdot \\rangle )$ be a flat Lie algebra.",
"Then $\\mathfrak {k}$ decomposes orthogonally as $\\mathfrak {k}=\\mathfrak {z}\\oplus \\mathfrak {h}\\oplus \\mathfrak {k}^{\\prime }$ (direct sum of vector spaces) where $\\mathfrak {z}$ is the center of $\\mathfrak {k}$ and the following properties are satisfied: $\\mathfrak {k}^{\\prime }=[\\mathfrak {k},\\mathfrak {k}]$ and $\\mathfrak {h}$ are abelian.",
"$\\operatorname{ad}:\\mathfrak {h}\\rightarrow \\mathfrak {so(k^{\\prime })}$ is injective and $\\mathfrak {k}^{\\prime }$ is even dimensional.",
"In particular, $\\dim \\mathfrak {h}\\le \\frac{\\dim \\mathfrak {k}^{\\prime }}{2}$ .",
"$\\operatorname{ad}_x=\\nabla _x$ for any $x\\in \\mathfrak {z}\\oplus \\mathfrak {h}$ .",
"$\\nabla _x=0$ if and only if $x\\in \\mathfrak {z}\\oplus \\mathfrak {k}^{\\prime }$ .",
"Remark 4.2 It follows easily from Proposition REF that $\\mathfrak {k}$ is a unimodular solvable Lie algebra, whose nilradical is given by $\\mathfrak {z}\\oplus \\mathfrak {k}^{\\prime }$ .",
"We will use this proposition in order to better describe Kähler flat Lie algebras.",
"First we give necessary and sufficient conditions for an almost complex structure on a flat Lie algebra to be Kähler (cf.",
"[28]).",
"Proposition 4.3 Let $(\\mathfrak {k},\\langle \\cdot ,\\cdot \\rangle )$ be a flat Lie algebra and let $J$ be an almost complex structure on $\\mathfrak {k}$ , compatible with $\\langle \\cdot ,\\cdot \\rangle $ .",
"Then $J$ is Kähler if and only if the following two properties are satisfied: $\\mathfrak {z}\\oplus \\mathfrak {h}$ and $\\mathfrak {k}^{\\prime }$ are $J$ -invariant.",
"$\\operatorname{ad}_H\\circ J=J\\circ \\operatorname{ad}_H$ , for any $H\\in \\mathfrak {h}$ .",
"Assume first that $J$ is Kähler, that is, $\\nabla J=0$ , or equivalently $\\nabla _x J=J \\nabla _x$ for any $x\\in \\mathfrak {k}$ .",
"For $x\\in \\mathfrak {k^{\\prime }}$ , it follows from Proposition REF that $x=\\displaystyle \\sum _i [H_i,y_i]$ for some $H_i\\in \\mathfrak {h}$ and $y_i\\in \\mathfrak {k^{\\prime }}$ .",
"Then $Jx=\\displaystyle \\sum _i J[H_i,y_i]=\\displaystyle \\sum _i J\\nabla _{H_i}Y_i=\\displaystyle \\sum _i \\nabla _{H_i}Jy_i= \\displaystyle \\sum _i [H_i,Jy_i]\\in \\mathfrak {k^{\\prime }},$ where we have used Proposition REF (c).",
"Therefore $\\mathfrak {k}^{\\prime }$ is $J$ -invariant.",
"Thus $J$ also preserves the orthogonal complement of $\\mathfrak {k^{\\prime }}$ , that is, $\\mathfrak {z}\\oplus \\mathfrak {h}$ is $J$ -invariant, and this proves ${\\rm (i)}$ .",
"Finally ${\\rm (ii)}$ follows from the fact that $\\nabla _H J=J \\nabla _H$ and $\\nabla _H=\\operatorname{ad}_H$ for any $H\\in \\mathfrak {h}$ .",
"If we assume now that ${\\rm (i)}$ and ${\\rm (ii)}$ hold then it follows easily from Proposition REF that $\\nabla _x J=J \\nabla _x$ for any $x\\in \\mathfrak {k}$ , i.e.",
"$J$ is Kähler.",
"Remark 4.4 Given any even-dimensional flat Lie algebra, it is easy to define an almost complex structure which satisfies the conditions of Proposition REF , thus it becomes a Kähler flat Lie algebra (see [28], [7]).",
"Our next aim is to study unitary derivations of a Kähler Lie algebra, i.e.",
"skew-symmetric derivations which commute with the complex structure.",
"In order to do so, we prove the following lemma.",
"Lemma 4.5 Let $(\\mathfrak {k},\\langle \\cdot ,\\cdot \\rangle )$ be a flat Lie algebra with decomposition $\\mathfrak {k}=\\mathfrak {z}\\oplus \\mathfrak {h}\\oplus \\mathfrak {k}^{\\prime }$ as in Proposition REF .",
"If $D$ is a skew-symmetric derivation of $\\mathfrak {k}$ , then $D(\\mathfrak {h})=0$ .",
"Since $D$ is a derivation of $\\mathfrak {k}$ , then $D(\\mathfrak {k}^{\\prime })\\subset \\mathfrak {k}^{\\prime }$ and $D(\\mathfrak {z})\\subset \\mathfrak {z}$ .",
"Moreover, since $D$ is skew-symmetric and the sum is orthogonal we have that $D(\\mathfrak {h})\\subset \\mathfrak {h}$ .",
"It follows from Proposition REF (b) that $\\operatorname{ad}_H\\in \\mathfrak {so(\\mathfrak {k}^{\\prime })}$ for any $H\\in \\mathfrak {h}$ .",
"Then, since $\\mathfrak {h}$ is abelian, we get that $\\mathfrak {F}=\\lbrace \\operatorname{ad}_H:\\mathfrak {k}^{\\prime }\\rightarrow \\mathfrak {k}^{\\prime }: H\\in \\mathfrak {h}\\rbrace $ is a commutative family of skew-symmetric endomorphisms of $\\mathfrak {k}^{\\prime }$ .",
"Therefore, $\\mathfrak {F}$ is contained in a maximal abelian subalgebra $\\mathfrak {a}$ of $\\mathfrak {so}(\\mathfrak {k}^{\\prime })$ .",
"The subalgebra $\\mathfrak {a}$ is conjugated by an element of $SO(\\mathfrak {k}^{\\prime })$ to the following maximal abelian subalgebra of $\\mathfrak {so}(\\mathfrak {k}^{\\prime })$ : $\\left\\lbrace \\left(\\begin{array}{ccccc}0 &-a_1 & & & \\\\a_1 & 0 & & & \\\\& & \\ddots & & \\\\& & & 0 & -a_n \\\\& & & a_n & 0 \\\\\\end{array}\\right): a_i\\in \\mathbb {R}\\right\\rbrace ,$ for some orthonormal basis $\\lbrace e_1,f_1,\\dots ,e_n,f_n\\rbrace $ of $\\mathfrak {k}^{\\prime }$ .",
"In this basis the elements of the family $\\mathfrak {F}$ can be represented by matrices $\\operatorname{ad}_H=\\left(\\begin{array}{ccccc}0 & -\\lambda _1(H) & & & \\\\\\lambda _1(H) & 0 & & & \\\\& & \\ddots & & \\\\& & & 0 & -\\lambda _n(H)\\\\& & & \\lambda _n(H) & 0 \\\\\\end{array}\\right), $ for some $\\lambda _i\\in \\mathfrak {h}^*$ , $i=1,\\dots ,n$ .",
"We compute $D[H,e_i]=[DH,e_i]+[H,De_i]=\\lambda _i(DH)f_i+[H,De_i],$ while, on the other hand, $D[H,e_i]=D(\\lambda _i(H)f_i)=\\lambda _i(H)Df_i.$ Comparing the components in the direction of $f_i$ in both expressions we obtain that $0=\\langle \\lambda _i(H)Df_i,f_i\\rangle = \\langle \\lambda _i(DH)f_i+[H,De_i],f_i\\rangle = \\lambda _i(DH),$ since $D$ and $\\operatorname{ad}_H$ are skew-symmetric.",
"Therefore, $\\lambda _i(DH)=0$ for all $i$ , that is, $\\operatorname{ad}_{DH}=0$ .",
"It follows from Proposition REF (b) that $DH=0$ .",
"As a consequence we have the following result, which will be an important tool to produce examples of unimodular solvable Lie algebras with Vaisman structures in §.",
"Theorem 4.6 If $(\\mathfrak {k}, J,\\langle \\cdot ,\\cdot \\rangle )$ is a Kähler flat Lie algebra and $D$ is a unitary derivation of $\\mathfrak {k}$ , then $D(\\mathfrak {h}+J\\mathfrak {h})=0$ and there exists an orthonormal basis $\\lbrace e_1,f_1,\\dots ,e_n,f_n\\rbrace $ of $\\mathfrak {k}^{\\prime }$ such that $J|_{\\mathfrak {k}^{\\prime }}=\\left(\\begin{array}{ccccc}0 &-1 & & & \\\\1 & 0 & & & \\\\& & \\ddots & & \\\\& & & 0 & -1 \\\\& & & 1 & 0 \\\\\\end{array}\\right), \\quad D|_{\\mathfrak {k}^{\\prime }}=\\left(\\begin{array}{ccccc}\\end{array}0 &-a_1 & & & \\\\a_1 & 0 & & & \\\\& & \\ddots & & \\\\& & & 0 & -a_n \\\\& & & a_n & 0 \\\\\\right.,$ $\\operatorname{ad}_H|_{\\mathfrak {k}^{\\prime }}=\\left(\\begin{array}{ccccc}\\end{array}0 & -\\lambda _1(H) & & & \\\\\\lambda _1(H) & 0 & & & \\\\& & \\ddots & & \\\\& & & 0 & -\\lambda _n(H)\\\\& & & \\lambda _n(H) & 0 \\\\\\right., $ for some $a_i\\in \\mathbb {R}$ and $\\lambda _i\\in \\mathfrak {h}^*$ for all $i=1,\\dots ,n$ .",
"These linear functionals satisfy $\\lambda _i\\ne 0$ for all $i=1,\\ldots ,n$ , and $\\bigcap _{i=1}^n\\ker \\lambda _i=\\lbrace 0\\rbrace $ .",
"It follows immediately from Lemma REF and $DJ=JD$ that $D(\\mathfrak {h}+J\\mathfrak {h})=0$ .",
"From Lemma REF and the fact that $D$ is a derivation of $\\mathfrak {k}$ we obtain that $D$ commutes with $\\operatorname{ad}_H$ for all $H\\in \\mathfrak {h}$ .",
"Also, Proposition REF implies that $J|_{\\mathfrak {k}^{\\prime }}$ commutes with $\\operatorname{ad}_H|_{\\mathfrak {k}^{\\prime }}$ for all $H\\in \\mathfrak {h}$ .",
"Therefore the family $\\mathfrak {F}^{\\prime }=\\lbrace \\operatorname{ad}_H:\\mathfrak {k}^{\\prime }\\rightarrow \\mathfrak {k}^{\\prime }: H\\in \\mathfrak {h}\\rbrace \\cup \\lbrace J|_{\\mathfrak {k}^{\\prime }},D|_{\\mathfrak {k}^{\\prime }}\\rbrace $ is a commutative family of skew-symmetric endomorphisms of $\\mathfrak {k}^{\\prime }$ .",
"The existence of the basis in the statement follows as in the proof of Lemma REF .",
"We analyze next the linear functionals $\\lambda _i\\in \\mathfrak {h}^*$ , $i=1,\\ldots ,n$ .",
"If $\\lambda _i=0$ for some $i$ then $[H,e_i]=0=[H,f_i]$ for all $H\\in \\mathfrak {h}$ , and this implies that $e_i,f_i\\in \\mathfrak {z}$ , which is a contradiction since $\\mathfrak {z}\\cap \\mathfrak {k}^{\\prime }=\\lbrace 0\\rbrace $ .",
"Now, if $H\\in \\bigcap _{i=1}^n\\ker \\lambda _i=\\lbrace 0\\rbrace $ , we have that $\\operatorname{ad}_H|_{\\mathfrak {k}^{\\prime }}=0$ and therefore $H=0$ according to Proposition REF (b)."
],
[
"Further properties of Vaisman Lie algebras",
"In this section we continue our study of unimodular solvable Vaisman Lie algebras, applying the results obtained in § to the Kähler flat Lie algebra given in Theorem REF .",
"In this way we obtain algebraic restrictions for the existence of Vaisman structures.",
"In Corollary REF we determined the center of a unimodular solvable Lie algebra $\\mathfrak {g}$ admitting a Vaisman structure.",
"In what follows we will derive other algebraic properties of these Lie algebras, in particular we will analyze its commutator $\\mathfrak {g}^{\\prime }$ and its nilradical $\\mathfrak {n}$ .",
"Recall that, since $\\mathfrak {g}$ is solvable, its nilradical is given by $\\mathfrak {n}=\\lbrace x\\in \\mathfrak {g}\\, : \\, \\operatorname{ad}_x:\\mathfrak {g}\\rightarrow \\mathfrak {g}\\text{ is nilpotent}\\rbrace $ and $\\mathfrak {g}^{\\prime }\\subseteq \\mathfrak {n}$ .",
"Let us set some notation.",
"We denote by $\\mathfrak {u}$ the largest $J$ -invariant subspace of the center of $\\mathfrak {k}$ , that is, $\\mathfrak {u}=\\mathfrak {z}\\cap J\\mathfrak {z}$ , and we define $2r=\\dim \\mathfrak {u}+\\dim \\mathfrak {k^{\\prime }}$ and $s=\\dim \\mathfrak {z}- \\dim \\mathfrak {u}$ .",
"Note that $\\mathfrak {z}\\oplus \\mathfrak {h}$ decomposes orthogonally as $\\mathfrak {z}\\oplus \\mathfrak {h}=\\mathfrak {u}\\oplus (\\mathfrak {h}+J\\mathfrak {h})$ .",
"Proposition 5.1 Let $\\mathfrak {g}$ be a unimodular solvable Lie algebra admitting a Vaisman structure and consider the decomposition $\\mathfrak {g}=\\mathbb {R}A\\ltimes _D(\\mathbb {R}JA\\oplus _{\\omega ^{\\prime }}\\mathfrak {k})$ with $\\mathfrak {k}=\\mathfrak {z}\\oplus \\mathfrak {h}\\oplus \\mathfrak {k}^{\\prime }$ as in Proposition REF .",
"Then the commutator ideal $\\mathfrak {g}^{\\prime }$ of $\\mathfrak {g}$ is given by $\\mathfrak {g}^{\\prime }=\\mathbb {R}JA\\oplus \\operatorname{Im}(D|_\\mathfrak {u})\\oplus \\mathfrak {k^{\\prime }}$ , while the nilradical $\\mathfrak {n}$ of $\\mathfrak {g}$ is given by: If $D|_\\mathfrak {u}\\ne 0$ or $D|_{\\mathfrak {k}^{\\prime }}\\notin \\operatorname{ad}(\\mathfrak {h})\\subset \\mathfrak {so}(\\mathfrak {k}^{\\prime }) $ , then $\\mathfrak {n}=\\mathbb {R}JA\\oplus \\mathfrak {z}\\oplus \\mathfrak {k^{\\prime }}\\simeq \\mathbb {R}^{s}\\times \\mathfrak {h}_{2r+1}$ .",
"If $D|_\\mathfrak {u}=0$ and $D|_{\\mathfrak {k}^{\\prime }}=0$ , i.e., $A\\in \\mathfrak {z}(\\mathfrak {g})$ , then $\\mathfrak {n}=\\mathbb {R}A\\oplus \\mathbb {R}JA\\oplus \\mathfrak {z}\\oplus \\mathfrak {k^{\\prime }}\\simeq \\mathbb {R}^{s+1}\\times \\mathfrak {h}_{2r+1}.$ If $D|_\\mathfrak {u}=0$ and $0\\ne D|_{\\mathfrak {k}^{\\prime }}\\in \\operatorname{ad}(\\mathfrak {h})$ , i.e., $D|_{\\mathfrak {k}^{\\prime }}=-\\operatorname{ad}_H|_{\\mathfrak {k}^{\\prime }}$ for a unique $0\\ne H\\in \\mathfrak {h}$ , then $\\mathfrak {n}=\\mathbb {R}(A+H)\\oplus \\mathbb {R}JA\\oplus \\mathfrak {z}\\oplus \\mathfrak {k}^{\\prime }$ .",
"Furthermore, if $JH\\in \\mathfrak {h}$ , then $\\mathfrak {n}\\simeq \\mathbb {R}^{s+1}\\times \\mathfrak {h}_{2r+1}$ ; and if $JH\\notin \\mathfrak {h}$ , then $\\mathfrak {n}\\simeq \\mathbb {R}^{s-1}\\times \\mathfrak {h}_{2(r+1)+1}$ .",
"Since $JA\\in \\mathfrak {g}^{\\prime }$ (see Proposition REF ), it follows from (REF ) that $\\mathfrak {k}^{\\prime }=[\\mathfrak {k},\\mathfrak {k}]_\\mathfrak {k}\\subset \\mathfrak {g}^{\\prime }$ .",
"Clearly, $D(\\mathfrak {h}+J\\mathfrak {h})=(D|_{\\mathfrak {k}})(\\mathfrak {h}+J\\mathfrak {h})$ and, as $D|_{\\mathfrak {k}}$ is a unitary derivation of $\\mathfrak {k}$ (Theorem REF ), we have $D(\\mathfrak {h}+J\\mathfrak {h})=0$ , due to Theorem REF .",
"As a consequence, $\\mathfrak {g}^{\\prime }=\\mathbb {R}JA\\oplus \\operatorname{Im}(D|_\\mathfrak {u})\\oplus \\mathfrak {k^{\\prime }}$ .",
"In order to determine the nilradical note first that, for any $z\\in \\mathfrak {z}$ , $\\operatorname{Im}(\\operatorname{ad}_z)\\subset \\mathbb {R}JA\\oplus \\mathfrak {z}$ and $\\operatorname{Im}(\\operatorname{ad}_z^2)\\subset \\mathbb {R}JA$ .",
"Since $JA\\in \\mathfrak {z}(\\mathfrak {g})$ , we obtain that $\\operatorname{ad}_z$ is a nilpotent operator on $\\mathfrak {g}$ , which implies that $\\mathfrak {z}\\subset \\mathfrak {n}$ .",
"Therefore $\\mathbb {R}JA\\oplus \\mathfrak {z}\\oplus \\mathfrak {k^{\\prime }}\\subset \\mathfrak {n}$ .",
"Moreover, it follows from Proposition REF (b) that $\\mathfrak {h}\\cap \\mathfrak {n}= \\lbrace 0\\rbrace $ .",
"Suppose that $\\mathbb {R}JA\\oplus \\mathfrak {z}\\oplus \\mathfrak {k^{\\prime }}\\subsetneq \\mathfrak {n}$ , then there exists $0\\ne A+H\\in \\mathfrak {n}$ for some $H\\in \\mathfrak {h}$ .",
"Therefore the operator $\\operatorname{ad}_{A+H}$ is nilpotent.",
"This operator can be written, for some orthonormal basis of $\\mathbb {R}JA\\oplus \\mathfrak {u}\\oplus (\\mathfrak {h}+J\\mathfrak {h})\\oplus \\mathfrak {k}^{\\prime }$ , as $\\operatorname{ad}_{A+H}=\\left(\\begin{array}{c|ccc|ccc|ccc}\\; & & & & & v^t & & & & \\\\\\hline & & & & & & & & & \\\\& & D|_\\mathfrak {u}& & & & & & & \\\\& & & & & & & & & \\\\\\hline & & & & & & & & & \\\\& & & & & & & & & \\\\& & & & & & & & & \\\\\\hline & & & & & & & & & \\\\& & & & & & & & D|_{\\mathfrak {k}^{\\prime }} +\\operatorname{ad}_H|_{\\mathfrak {k}^{\\prime }} & \\\\& & & & & & & & & \\\\\\end{array}\\right),$ with both $D|_\\mathfrak {u}$ and $D|_{\\mathfrak {k}^{\\prime }} +\\operatorname{ad}_H|_{\\mathfrak {k}^{\\prime }}$ skew-symmetric.",
"It follows that $\\operatorname{ad}_{A+H}$ is nilpotent if and only if $D|_\\mathfrak {u}=0 \\quad \\text{ and } \\quad D|_{\\mathfrak {k}^{\\prime }}=- \\operatorname{ad}_H|_{\\mathfrak {k}^{\\prime }}.$ It follows that $H\\in \\mathfrak {h}$ satisfying (REF ) is unique, since $\\operatorname{ad}:\\mathfrak {h}\\rightarrow \\mathfrak {\\mathfrak {k}^{\\prime }}$ is injective (Proposition REF (b)).",
"We have two possibilities: if $H=0$ , then $A\\in \\mathfrak {z}(\\mathfrak {g})$ and $\\mathfrak {n}=\\mathbb {R}A\\oplus \\mathbb {R}JA\\oplus \\mathfrak {z}\\oplus \\mathfrak {k^{\\prime }}$ .",
"On the other hand, if $H\\ne 0$ , then $\\mathfrak {n}=\\mathbb {R}(A+H)\\oplus \\mathbb {R}JA\\oplus \\mathfrak {z}\\oplus \\mathfrak {k}^{\\prime }$ .",
"Assume now that $\\mathfrak {n}=\\mathbb {R}JA\\oplus \\mathfrak {z}\\oplus \\mathfrak {k^{\\prime }}$ ; according to (REF ) this happens if and only if $D|_\\mathfrak {u}\\ne 0$ or $D|_{\\mathfrak {k}^{\\prime }}\\notin \\operatorname{ad}(\\mathfrak {h})$ .",
"The isomorphism classes of $\\mathfrak {n}$ in the different cases above follow easily from the description of the Lie bracket of $\\mathfrak {g}$ .",
"Remark 5.2 Note that the nilradical $\\mathfrak {n}$ in Proposition REF is isomorphic to $\\mathbb {R}^{p-1}\\times \\mathfrak {h}_{2q+1},$ for some $p,q\\in \\mathbb {N}$ and, as a consequence, it is never abelian.",
"On the other hand, it follows from Proposition REF and Lemma REF that $\\mathfrak {n}^\\perp $ is an abelian subalgebra of $\\mathfrak {g}$ if and only if $\\mathfrak {h}\\cap J\\mathfrak {h}=\\lbrace 0\\rbrace $ .",
"We can give next a strong algebraic obstruction to the existence of Vaisman structures on a unimodular solvable Lie algebra.",
"Theorem 5.3 If the unimodular solvable Lie algebra $\\mathfrak {g}$ admits a Vaisman structure, then the eigenvalues of the operators $\\operatorname{ad}_x$ with $x\\in \\mathfrak {g}$ are all imaginary (some of them are 0).",
"Recall that $\\mathfrak {g}$ is a double extension $\\mathfrak {g}=\\mathbb {R}A\\ltimes _D(\\mathbb {R}JA\\oplus _{\\omega ^{\\prime }}\\mathfrak {k})$ with $\\mathfrak {k}=\\mathfrak {z}\\oplus \\mathfrak {h}\\oplus \\mathfrak {k}^{\\prime }$ and $\\mathfrak {z}\\oplus \\mathfrak {h}=\\mathfrak {z}\\cap J\\mathfrak {z}\\oplus (\\mathfrak {h}+J\\mathfrak {h})$ .",
"Let $\\lbrace A,JA,u_1,v_1,\\dots ,u_p,v_p,x_1,y_1,\\dots ,x_q,y_q,e_1,f_1,\\dots ,e_m,f_m,\\rbrace $ be an orthonormal basis of $\\mathfrak {g}$ adapted to this decomposition.",
"Moreover, $Ju_i=v_i$ , $Jx_i=y_i$ and $Je_i=f_i$ .",
"Given $x=aA+bJA+z+h+y\\in \\mathfrak {g}$ with $a,b\\in \\mathbb {R}$ , $z\\in \\mathfrak {z}\\cap J\\mathfrak {z}$ , $h\\in \\mathfrak {h}+J\\mathfrak {h}$ , $y\\in \\mathfrak {k}^{\\prime }$ , we compute the matrix of the operator $\\operatorname{ad}_x$ in such basis: $\\operatorname{ad}_x=\\left(\\begin{array}{c|c|ccc|ccc|ccc}& & & & & & & & & & \\\\\\hline & & & \\alpha ^t & & & \\beta ^t & & & & \\\\\\hline & & & & & & & & & & \\\\& & & aD|_\\mathfrak {u}& & & & & & & \\\\& & & & & & & & & & \\\\\\hline & & & & & & & & & & \\\\\\gamma & & & & & & & & & \\\\& & & & & & & & & & \\\\\\hline & & & & & & & & & & \\\\\\delta & & & & & & C& & & aD|_{\\mathfrak {k}^{\\prime }} + B & \\\\& & & & & & & & & & \\\\\\end{array}\\right),$ for some $\\alpha \\in \\mathbb {R}^{2p}$ , $\\beta ,\\gamma \\in \\mathbb {R}^{2q}$ , $\\delta \\in \\mathbb {R}^{2m}$ , $C\\in \\operatorname{Mat}(m\\times q,\\mathbb {R})$ and $B\\in \\mathfrak {u}(m)$ .",
"Since $aD|_\\mathfrak {u}$ and $aD|_{\\mathfrak {k}^{\\prime }} + B$ are skew-symmetric, it is easy to see that the eigenvalues of $\\operatorname{ad}_x$ for $x\\in \\mathfrak {g}$ are all imaginary.",
"Remark 5.4 A Lie algebra $\\mathfrak {g}$ satisfying the condition that the eigenvalues of the operators $\\operatorname{ad}_x$ are all imaginary for all $x\\in \\mathfrak {g}$ is called a Lie algebra of type I (see [35]).",
"A consequence of Theorem REF is the following result, proved recently by H. Sawai in [43].",
"We recall that a Lie group is called completely solvable if it is solvable and all the eigenvalues of the adjoint operators $\\operatorname{ad}_x$ are real, for all $x$ in its Lie algebra.",
"Corollary 5.5 Let $G$ be a simply connected completely solvable Lie group and $\\Gamma \\subset G$ a lattice.",
"If the solvmanifold $\\Gamma \\backslash G$ admits a Vaisman structure $(J,g)$ such that the complex structure $J$ is induced by a left invariant complex structure on $G$ , then $G=H_{2n+1}\\times \\mathbb {R}$ , where $H_{2n+1}$ denotes the $(2n+1)$ -dimensional Heisenberg Lie group.",
"Using the symmetrization process of Belgun ([8]) together with results in [42], one can produce a left invariant Riemannian metric $\\tilde{g}$ on $G$ such that $(J,\\tilde{g})$ is again a Vaisman structure.",
"This gives rise to a Vaisman structure on $\\operatorname{Lie}(G)$ and, taking into account Theorem REF , we have that $G$ is a nilpotent Lie group.",
"It follows from [40] that $G=H_{2n+1}\\times \\mathbb {R}$ ."
],
[
"CoKähler Lie algebras",
"In previous sections we have seen that any unimodular solvable Lie algebra admitting a Vaisman structure is the semidirect product of $\\mathbb {R}$ with a Sasakian Lie algebra and this Sasakian Lie algebra, in turn, is a central extension of a Kähler flat Lie algebra.",
"In what follows we will establish a relation with another type of almost contact metric Lie algebras, namely, coKähler ones.",
"In fact, we will show that a unimodular solvable Lie algebra admitting a Vaisman structure is a central extension of a coKähler Lie algebra which is, in turn, a semidirect product of $\\mathbb {R}$ with a Kähler flat Lie algebra.",
"Let $\\mathfrak {g}$ be a unimodular solvable Lie algebra that admits a Vaisman structure $(J,\\langle \\cdot ,\\cdot \\rangle )$ , with $\\omega $ its fundamental 2-form, $\\theta $ the corresponding Lee form and $A\\in \\mathfrak {g}$ its metric dual, as before.",
"We know from Theorem REF that $\\mathfrak {g}$ is a double extension $\\mathfrak {g}=\\mathfrak {k}(D,\\omega ^{\\prime })$ of the Kähler flat Lie algebra $\\mathfrak {k}$ by certain derivation $D$ of $\\mathfrak {k}_{\\omega ^{\\prime }}(\\xi )$ such that $D^{\\prime }:=D|_\\mathfrak {k}$ is a unitary derivation of $\\mathfrak {k}$ .",
"Theorem 6.1 With notation as above, the Lie algebra $\\mathfrak {d}=\\mathbb {R}A\\ltimes _{D^{\\prime }}\\mathfrak {k}$ admits a coKähler structure $(\\langle \\cdot ,\\cdot \\rangle |_{\\mathfrak {d}\\times \\mathfrak {d}},\\phi , \\xi , \\eta )$ , where $\\phi \\in \\operatorname{End}({\\mathfrak {d}})$ is defined by $\\phi (aA+x)=Jx$ for $a\\in \\mathbb {R},\\,x\\in \\mathfrak {k}$ , and $\\eta :=\\theta |_{\\mathfrak {d}}$ , $\\xi :=A$ .",
"Moreover, if $\\Phi $ denotes the (closed) fundamental 2-form on $\\mathfrak {d}$ associated to this coKähler structure, then $\\mathfrak {g}$ is isomorphic to the central extension $\\mathfrak {d}_{\\Phi }(JA)$ .",
"It is readily verified that $(\\langle \\cdot ,\\cdot \\rangle |_{\\mathfrak {d}\\times \\mathfrak {d}},\\phi , \\xi , \\eta )$ is an almost contact metric structure on $\\mathfrak {d}$ .",
"Let us prove now that it is almost coKähler, i.e., $d\\eta =0$ and $d\\Phi =0$ , where $\\Phi $ is the fundamental 2-form defined by $\\Phi (x,y)=\\langle \\phi x, y\\rangle $ , $x,y\\in \\mathfrak {d}$ .",
"Since $[\\mathfrak {d},\\mathfrak {d}]\\subset \\mathfrak {k}=\\ker \\eta $ , we have that $d\\eta =0$ .",
"Now, for $a,b,c\\in \\mathbb {R}$ and $x,y,z\\in \\mathfrak {k}$ , we compute easily $d\\Phi (aA+x,bA+y,cA+z) & =d^\\mathfrak {k}\\omega ^{\\prime }(x,y,z)+a(\\langle D^{\\prime }y,Jz\\rangle -\\langle D^{\\prime }z,Jy\\rangle )\\\\ \\nonumber & \\qquad +b(\\langle D^{\\prime }z,Jx\\rangle -\\langle D^{\\prime }x,Jz\\rangle )+c(\\langle D^{\\prime }x,Jy\\rangle -\\langle D^{\\prime }y,Jx\\rangle ),$ where $\\omega ^{\\prime }=\\omega |_{\\mathfrak {k}\\times \\mathfrak {k}}$ .",
"Since $\\omega ^{\\prime }$ is the Kähler form on $\\mathfrak {k}$ , we have that $d^\\mathfrak {k}\\omega ^{\\prime }=0$ .",
"Since both $D$ and $J$ are skew-symmetric and $D^{\\prime }J|_\\mathfrak {k}=J|_\\mathfrak {k}D^{\\prime }$ , we obtain that the last terms in (REF ) vanish, and therefore $d\\Phi =0$ .",
"To verify the normality of this structure, since $d\\eta =0$ we only have to check that $N_\\phi =0$ .",
"For $a,b\\in \\mathbb {R}$ and $x,y\\in \\mathfrak {k}$ , we compute $ N_\\phi (aA+x,bA+y)=N^\\mathfrak {k}_{J|_\\mathfrak {k}}(x,y)-a(D^{\\prime }y+JD^{\\prime }Jy)+b(D^{\\prime }x+JD^{\\prime }Jx).$ Since $N^\\mathfrak {k}_{J|_\\mathfrak {k}}=0$ and $D^{\\prime }J|_\\mathfrak {k}=J|_\\mathfrak {k}D^{\\prime }$ , it follows that $N_\\phi =0$ .",
"To prove the last statement we compute the Lie bracket $[\\cdot ,\\cdot ]^{\\prime }$ on the central extension $\\mathfrak {d}_{\\Phi }(JA)$ .",
"We have that $JA$ is central and for $a,b\\in \\mathbb {R}$ , $x,y\\in \\mathfrak {k}$ we compute $[aA+x,bA+y]^{\\prime } & = \\Phi (aA+x,bA+y)JA+[aA+x,bA+y]_{\\mathfrak {d}} \\\\& = \\langle \\phi (aA+x),bA+y\\rangle JA + aD^{\\prime }y-bD^{\\prime }x+[x,y]_\\mathfrak {k}\\\\& = \\langle Jx,y\\rangle JA + aDy-bDx+[x,y]_\\mathfrak {k}\\\\& =\\omega (x,y)JA+aDy-bDx+[x,y]_\\mathfrak {k},$ which coincides with the Lie bracket on $\\mathfrak {g}$ , according to Theorem REF and (REF ).",
"This completes the proof.",
"Remark 6.2 The first part of Theorem REF follows also from [18].",
"Moreover, according to [18], $(\\mathfrak {d},\\langle \\cdot ,\\cdot \\rangle )$ is a flat Lie algebra, since $\\mathfrak {d}$ is unimodular.",
"If, as above, $\\mathfrak {k}=\\mathfrak {z}\\oplus \\mathfrak {h}\\oplus \\mathfrak {k}^{\\prime }$ is the decomposition of $\\mathfrak {k}$ given by Proposition REF , then the corresponding decomposition of $\\mathfrak {d}$ is $\\mathfrak {d}=\\widetilde{\\mathfrak {z}}\\oplus \\widetilde{\\mathfrak {h}}\\oplus \\mathfrak {d}^{\\prime }$ , where $\\widetilde{\\mathfrak {z}}=\\ker (D^{\\prime }|_{\\mathfrak {z}\\cap J\\mathfrak {z}})$ , $\\widetilde{\\mathfrak {h}}=\\mathbb {R}A\\oplus \\mathfrak {h}$ and $\\mathfrak {d}^{\\prime }=\\mathfrak {k}^{\\prime }\\oplus \\operatorname{Im}(D^{\\prime }|_{\\mathfrak {z}\\cap J\\mathfrak {z}})$ , whenever $D^{\\prime }\\ne 0$ .",
"If $D^{\\prime }=0$ , we have that $\\widetilde{\\mathfrak {z}}=\\mathbb {R}A\\oplus \\mathfrak {z}$ , $\\widetilde{\\mathfrak {h}}=\\mathfrak {h}$ and $\\mathfrak {d}^{\\prime }=\\mathfrak {k}^{\\prime }$ ."
],
[
"Left-symmetric algebra structures",
"In this section we show that any unimodular solvable Lie algebra equipped with either a Sasakian or a Vaisman structure admits also another kind of algebraic structure with a geometrical interpretation.",
"A left-symmetric algebra (LSA) structure on a Lie algebra $\\mathfrak {a}$ is a bilinear product $\\mathfrak {a}\\times \\mathfrak {a} \\longrightarrow \\mathfrak {a},\\,(x,y)\\mapsto x\\cdot y$ , which satisfies $[x,y]=x\\cdot y-y\\cdot x$ and $x\\cdot (y\\cdot z)-(x\\cdot y)\\cdot z=y\\cdot (x\\cdot z)-(y\\cdot x)\\cdot z,$ for any $x,y,z\\in \\mathfrak {a}$ .",
"See [13] for a very interesting review on this subject.",
"LSA structures have the following well known geometrical interpretation.",
"If $G$ is a Lie group and $\\mathfrak {g}$ is its Lie algebra, then LSA structures on $\\mathfrak {g}$ are in one-to-one correspondence with left invariant flat torsion-free connections $\\nabla $ on $G$ .",
"Indeed, this correspondence is given as follows: $\\nabla _xy=x\\cdot y$ for $x,y\\in \\mathfrak {g}$ .",
"Since this connection is left invariant, any quotient $\\Gamma \\backslash G$ of $G$ by a discrete subgroup $\\Gamma $ also inherits a flat torsion-free connection.",
"It is well known that one can study the completeness of the connection $\\nabla $ on $G$ in terms only of the corresponding LSA structure on $\\mathfrak {g}$ .",
"Indeed, $\\nabla $ is geodesically complete if and only if the endomorphisms $\\operatorname{Id}+\\rho (x)$ of $\\mathfrak {g}$ are bijective for all $x\\in \\mathfrak {g}$ , where $\\rho (x)$ denotes right-multiplication by $x$ , i.e., $\\rho (x)y=y\\cdot x$ (see for instance [44]).",
"In this case, it is said that the LSA structure is complete.",
"Note that the Levi-Civita connection of a flat metric on a Lie algebra is an example of a complete LSA structure, since the corresponding left invariant metric on any associated Lie group is homogeneous and therefore complete.",
"Theorem 6.3 Let $(\\mathfrak {h},\\langle \\cdot ,\\cdot \\rangle )$ be a flat Lie algebra and let $\\beta $ denote a 2-form which is parallel with respect to the Levi-Civita connection $\\nabla $ of $\\langle \\cdot ,\\cdot \\rangle $ (hence $\\beta $ is closed).",
"Then the central extension $\\mathfrak {g}=\\mathfrak {h}_\\beta (\\xi )$ admits an LSA structure defined by $ (a\\xi +x)\\cdot (b\\xi +y)=\\frac{1}{2} \\beta (x,y)\\xi +\\nabla _xy, \\quad a,b\\in \\mathbb {R},\\, x,y\\in \\mathfrak {h}.$ Furthermore, this LSA structure is complete.",
"Taking into account that $\\nabla $ is a torsion-free connection on $\\mathfrak {h}$ and the fact that $\\xi $ is central in $\\mathfrak {g}$ , it is easily verified that $ (a\\xi +x)\\cdot (b\\xi +y)-(b\\xi +y)\\cdot (a\\xi +x)=[a\\xi +x,b\\xi +y]_\\beta .",
"$ Therefore, (REF ) holds for this product.",
"In order to prove (REF ), let us compute $(a\\xi +x)\\cdot ((b\\xi +y)\\cdot (c\\xi +z)) & -(b\\xi +y)\\cdot ((a\\xi +x)\\cdot (c\\xi +z)) = \\\\ & = \\frac{1}{2}\\beta (x,\\nabla _yz)+\\nabla _x\\nabla _yz - \\frac{1}{2}\\beta (y,\\nabla _xz) -\\nabla _y\\nabla _xz\\\\& = -\\frac{1}{2}\\beta (\\nabla _yx,z) + \\frac{1}{2}\\beta (\\nabla _xy,z)+\\nabla _x\\nabla _yz -\\nabla _y\\nabla _xz\\\\& = \\frac{1}{2}\\beta ([x,y],z)+\\nabla _{[x,y]}z \\\\& =[a\\xi +x,b\\xi +y]_\\beta \\cdot (c\\xi +z),$ where we have used $\\nabla \\beta =0$ in the third line and the fact that $\\nabla $ is torsion-free and flat in the fourth line.",
"This is equivalent to (REF ), thus this product is an LSA structure on $\\mathfrak {g}$ .",
"Let us prove the completeness.",
"Fix $b\\xi +y\\in \\mathfrak {g}$ and assume that $(\\operatorname{Id}+\\rho (b\\xi +y))(a\\xi +x)=0$ .",
"Then $0 & = a\\xi +x + \\frac{1}{2} \\beta (x,y)\\xi +\\nabla _xy \\\\& = \\left(a+\\frac{1}{2} \\beta (x,y)\\right)\\xi + \\left(x+\\nabla _xy\\right),$ hence $a+\\frac{1}{2} \\beta (x,y)=0$ and $x+\\nabla _xy=0$ .",
"But, since $\\nabla $ itself is a complete LSA structure on $\\mathfrak {h}$ , it follows that $x=0$ .",
"This implies that $a=0$ , and the completeness follows.",
"Corollary 6.4 Let $\\mathfrak {g}$ be a unimodular solvable Lie algebra.",
"If $\\mathfrak {g}$ carries a Vaisman structure, then $\\mathfrak {g}$ admits a complete LSA structure.",
"If $\\mathfrak {g}$ carries a Sasakian structure, then $\\mathfrak {g}$ admits a complete LSA structure.",
"If $\\mathfrak {g}$ admits a Vaisman structure then, according to Theorem REF , $\\mathfrak {g}$ is a central extension of a coKähler Lie algebra $\\mathfrak {d}$ by the fundamental 2-form $\\Phi $ , which is parallel.",
"As mentioned in Remark REF , $\\mathfrak {d}$ is flat, and therefore (1) follows from Theorem REF .",
"If $\\mathfrak {g}$ admits a Sasakian structure, then it follows from Corollary REF that $\\mathfrak {g}$ is a central extension of a Kähler flat Lie algebra.",
"Hence, (2) follows from Theorem REF again.",
"Corollary 6.5 Any solvmanifold admitting either an invariant Vaisman structure or an invariant Sasakian structure carries a geodesically complete flat torsion-free connection."
],
[
"Examples",
"Using the results in the previous sections we will construct many examples of unimodular solvable Lie algebras equipped with Vaisman structures.",
"In particular, we will provide two infinite families of such examples, and also the classification of such Lie algebras in dimensions 4 and 6.",
"We will also analyze the existence of lattices in these examples.",
"Firstly, we would like to prove a general result concerning lattices in semidirect products, which will be frequently used later in this section.",
"Lemma 7.1 Let $H$ be a simply connected Lie group equipped with a Lie group homomorphism $\\phi : \\mathbb {R}\\rightarrow \\operatorname{Aut}(H)$ , and let $G=\\mathbb {R}\\ltimes _\\phi H$ be the corresponding semidirect produt.",
"Let $\\Gamma $ be a lattice on $H$ .",
"If there exists $a\\in \\mathbb {R}$ , $a\\ne 0$ , such that $\\phi (a)(\\Gamma )\\subset \\Gamma $ , then $a\\mathbb {Z}\\ltimes _\\phi \\Gamma $ is a lattice on $G$ .",
"If $a\\in \\mathbb {R}$ , $a\\ne 0$ , satisfies $\\phi (a)(\\Gamma )\\subset \\Gamma $ , then $\\phi (ak)(\\Gamma )\\subset \\Gamma $ for all $k\\in \\mathbb {Z}$ and therefore the semidirect product $\\tilde{\\Gamma }:=a\\mathbb {Z}\\ltimes _\\phi \\Gamma $ is well defined.",
"Clearly, $\\tilde{\\Gamma }$ is a discrete subgroup of $G$ .",
"We only have to show that it is co-compact.",
"Let us recall that two elements $(t_1,g_1)$ and $(t_2,g_2)$ in $G$ belong to the same left-coset with respect to $\\tilde{\\Gamma }$ if and only if there exist $k\\in \\mathbb {Z}$ and $\\gamma \\in \\Gamma $ such that $(t_2,g_2) =& (ak,\\gamma )(t_1,g_1) \\\\=& (ak + t_1, \\gamma \\phi (a)^kg_1).",
"\\nonumber $ Let $\\lbrace a_n\\rbrace $ be a sequence in $\\tilde{\\Gamma }\\backslash G$ , with $a_n=[(t_n,g_n)]$ .",
"It follows easily from (REF ) that we can choose $t_n\\in [0,a]$ for all $n\\in \\mathbb {N}$ ; moreover, as $\\Gamma \\backslash H$ is compact, we may assume that $[g_n]$ converges to $[g]$ in $\\Gamma \\backslash H$ , for some $g\\in H$ .",
"The canonical projection $\\pi : H\\rightarrow \\Gamma \\backslash H$ is a local diffeomorphism, therefore we can choose a representative $g^{\\prime }_n$ of $[g_n]$ such that $g^{\\prime }_n$ converges to $g$ in $H$ .",
"Taking into account that also $[0,a]$ is compact, we may assume that $a_n=[(t_n,g^{\\prime }_n)]$ satisfies $t_n\\rightarrow t$ in $[0,a]$ and $g^{\\prime }_n\\rightarrow g$ in $H$ .",
"It is easy to see that $[(t_n,g^{\\prime }_n)]$ converges to $[(t,g)]$ in $\\tilde{\\Gamma }\\backslash G$ .",
"Since $\\lbrace a_n\\rbrace $ is arbitrary, we thus obtain that $\\tilde{\\Gamma }\\backslash G$ is compact."
],
[
"Example 1: Oscillator solvmanifolds",
"We start with the abelian Lie algebra $\\mathfrak {k}=\\mathbb {R}^{2n}$ with its canonical Kähler flat structure $(J,\\langle \\cdot ,\\cdot \\rangle )$ .",
"Let $\\lbrace e_1,f_1,\\dots ,e_n,f_n\\rbrace $ be an orthonormal basis of $\\mathfrak {k}$ where $Je_i=f_i$ and let $\\omega =\\sum _{i=1}^n e^i\\wedge f^i$ be the fundamental form.",
"Then we consider the central extension $ \\mathfrak {k}_\\omega (B)=\\mathbb {R}B\\oplus _\\omega \\mathfrak {k},$ which is easily seen to be isomorphic to $\\mathfrak {h}_{2n+1}$ , the $(2n+1)$ -dimensional Heisenberg Lie algebra.",
"Next, we consider the double extension $\\mathfrak {g}=\\mathfrak {k}(D,\\omega )=\\mathbb {R}A\\ltimes _D\\mathfrak {h}_{2n+1},$ where the action is given by $ D=\\left(\\begin{array}{cccccc}0 & & & & & \\\\& 0 & -a_1 & & & \\\\& a_1 & 0 & & & \\\\& & & \\ddots & & \\\\& & & & 0 & -a_n\\\\& & & & a_n & 0 \\\\\\end{array}\\right), $ in the basis $\\lbrace B,e_1,f_1,\\dots ,e_n,f_n\\rbrace $ where $[e_i,f_i]=B$ , for some $a_i\\in \\mathbb {R}$ .",
"Since $D|_{\\mathfrak {k}}$ satisfies the conditions of Theorem REF , $\\mathfrak {g}$ admits a Vaisman structure.",
"If $a_i=0$ for all $i=1,\\ldots ,n$ , then we recover the well known Vaisman structure on $\\mathbb {R}\\times \\mathfrak {h}_{2n+1}$ (see Example REF ).",
"Let us consider now the case when not all the $a_i$ 's vanish.",
"We may reorder the elements of the basis of $\\mathfrak {g}$ , if necessary, and we may assume that the constants $a_i$ satisfy $0\\le a_1\\le a_2\\le \\cdots \\le a_n$ .",
"We will denote this Lie algebra by $\\mathfrak {g}_{(a_1,\\ldots ,a_n)}$ .",
"Note that $\\mathfrak {g}_{(a_1,\\ldots ,a_n)}$ is an almost nilpotent Lie algebra, i.e, it contains a codimension one nilpotent ideal.",
"The associated simply connected Lie group will be denoted $G_{(a_1,\\ldots ,a_n)}$ , and it is known as an oscillator group.",
"Oscillator groups have many geometric properties, for instance, it was proved in [31] that they are the only simply connected and non simple Lie groups which admit an indecomposable bi-invariant Lorentz metric.",
"Other geometric features of oscillator groups in dimension 4 have been studied in [14].",
"In this example we have shown: Theorem 7.2 Any solvmanifold of an oscillator group admits invariant Vaisman structures.",
"Lattices in oscillator groups have been characterized in [19].",
"Here we will provide an explicit construction of some families of lattices in $G_{(a_1,\\ldots ,a_n)}$ , and later we will determine the first homology group and first Betti number of the associated solvmanifolds.",
"We begin by recalling the following result, which deals with the isomorphism classes of these Lie algebras.",
"Proposition 7.3 ([19]) Let $\\mathfrak {g}_{(a_1,\\ldots ,a_n)}$ and $\\mathfrak {g}_{(b_1,\\ldots ,b_n)}$ be two Lie algebras as defined above.",
"Then $\\mathfrak {g}_{(a_1,\\ldots ,a_n)} \\cong \\mathfrak {g}_{(b_1,\\ldots ,b_n)}$ if and only if there exists $c\\in \\mathbb {R}-\\lbrace 0\\rbrace $ such that $a_j=cb_j$ for all $j=1,\\ldots ,n$ .",
"The Lie group $G_{(a_1,\\ldots ,a_n)}$ can be described as follows.",
"We consider the Lie group homomorphism $\\varphi : \\mathbb {R}\\longrightarrow \\operatorname{Aut}(H_{2n+1})$ given by $ \\varphi (t)=e^{tD}=\\left(\\begin{array}{cccccc}1 & & & & & \\\\& \\cos (a_1t) &-\\sin (a_1t) & & & \\\\& \\sin (a_1t) & \\cos (a_1t) & & & \\\\& & & \\ddots & & \\\\& & & & \\cos (a_nt) &-\\sin (a_nt)\\\\& & & & \\sin (a_nt) & \\cos (a_nt)\\\\\\end{array}\\right) $ where $H_{2n+1}$ is the $(2n+1)$ -dimensional Heisenberg Lie group, i.e.",
"the Euclidean manifold $\\mathbb {R}^{2n+1}$ equipped with the following product: $(z,x_1,y_1,\\dots ,x_n,y_n)\\cdot (z^{\\prime },x^{\\prime }_1,y^{\\prime }_1,\\dots ,x^{\\prime }_n,y^{\\prime }_n)=(z+z^{\\prime }+\\frac{1}{2} \\sum _{j=1}^n(x_jy^{\\prime }_j-x^{\\prime }_jy_j),x_1+x^{\\prime }_1,\\dots ,y_n+y^{\\prime }_n).$ Then $G_{(a_1,\\ldots ,a_n)}$ is the semidirect product $G_{(a_1,\\ldots ,a_n)}=\\mathbb {R}\\ltimes _\\varphi H_{2n+1}$ .",
"We show next the existence of lattices for some choice of the parameters $a_i$ (compare [30]): Proposition 7.4 If $a_i\\in \\mathbb {Q}$ for $i=1,\\ldots ,n$ , then $G_{(a_1,\\ldots ,a_n)}$ admits lattices.",
"If $a_i\\in \\mathbb {Q}$ , then $a_i=\\frac{p_i}{q_i}$ for some $p_i\\in \\mathbb {Z},\\,q_i\\in \\mathbb {N}$ with $(p_i,q_i)=1$ .",
"Setting $t_0:=2\\pi \\prod q_i$ , we obtain that $\\varphi (t_0)$ is an integer matrix.",
"Moreover, the structure constants of $\\mathfrak {g}_{(a_1,\\ldots ,a_n)}$ corresponding to the basis $\\lbrace B,e_1,f_1,\\dots ,e_n,f_n\\rbrace $ are all rational.",
"As $G_{(a_1,\\ldots ,a_n)}$ is an almost nilpotent Lie group, it follows from [12] (see also [17]) that $G_{(a_1,\\ldots ,a_n)}$ admits lattices.",
"Therefore we will consider $a_i\\in \\mathbb {Q}$ for all $i=1,\\ldots , n$ ; moreover, it follows from Proposition REF that we may assume $a_i\\in \\mathbb {Z}$ for all $i=1,\\ldots , n$ , with $\\operatorname{gcd}(a_1,\\ldots ,a_n)=1$ .",
"Beginning with a lattice in $H_{2n+1}$ we may extend it to a lattice in $G_{(a_1,\\ldots ,a_n)}$ .",
"Consider the following lattices in $H_{2n+1}$ : for each $k \\in \\mathbb {N}$ there exists a lattice $\\Gamma _k$ in $H_{2n+1}$ given by $\\Gamma _{k}=\\frac{1}{2k}\\mathbb {Z}\\times \\mathbb {Z}\\times \\cdots \\times \\mathbb {Z}$ .",
"It can be shown that $\\Gamma _k/[\\Gamma _k, \\Gamma _k]$ is isomorphic to $\\mathbb {Z}^{2n}\\oplus \\mathbb {Z}_{2k}$ .",
"Hence, $\\Gamma _r$ and $\\Gamma _s$ are non-isomorphic for $r\\ne s$ .",
"Any lattice $\\Gamma _k$ in $H_{2n+1}$ is invariant under the subgroups generated by $\\varphi (\\pi /2)$ , $\\varphi (\\pi )$ and $\\varphi (2\\pi )$ .",
"According to Lemma REF we have three families of lattices in $G_{(a_1,\\ldots ,a_n)}$ : $\\Lambda _{k,\\frac{\\pi }{2}} & = \\frac{\\pi }{2}\\mathbb {Z}\\ltimes _\\varphi \\Gamma _k,\\\\\\Lambda _{k,\\pi } & = \\pi \\mathbb {Z}\\ltimes _\\varphi \\Gamma _k,\\\\\\Lambda _{k,2\\pi } & = 2\\pi \\mathbb {Z}\\ltimes _\\varphi \\Gamma _k.$ We analyze next some topological properties of the oscillator solvmanifolds $\\Lambda _{k,j}\\backslash G_{(a_1,\\ldots ,a_n)}$ for $j=2\\pi ,\\pi ,\\pi /2$ : $\\bullet $ Note that $\\varphi (2\\pi )=\\operatorname{Id}$ , therefore $\\Lambda _{k,2\\pi }=2\\pi \\mathbb {Z}\\times \\Gamma _k$ , which is isomorphic to a lattice in $\\mathbb {R}\\times H_{2n+1}$ .",
"According to Corollary REF , we have that the solvmanifold $\\Lambda _{k,2\\pi }\\backslash G_{(a_1,\\ldots ,a_n)}$ is isomorphic to the nilmanifold $S^1\\times \\Gamma _k\\backslash H_{2n+1}$ , for any choice of $(a_1,\\ldots ,a_n)$ .",
"It is easy to see that the first homology group of this nilmanifold is $\\mathbb {Z}^{2n+1}\\oplus \\mathbb {Z}_{2k}$ and hence its first Betti number is $b_1=2n+1$ .",
"$\\bullet $ For the family $\\Lambda _{k,\\pi }$ , note that the isomorphism class of this lattice depends only on the parity of the integers $a_j$ , therefore according to Corollary REF , we may assume that $a_j\\in \\lbrace 0,1\\rbrace $ for all $j$ , not all of them equal to 0.",
"After reordering, we have that there exists $p\\in \\lbrace 0,1,\\ldots ,n-1\\rbrace $ such that $a_j=0$ if $j\\le p$ and $a_j=1$ if $j>p$ .",
"Then it can be seen that $[\\Lambda _{k,\\pi },\\Lambda _{k,\\pi }] & = \\lbrace (0,r,s_1,t_1,\\ldots ,s_n,t_n)\\in \\Lambda _{k,\\pi }\\,: r\\in \\mathbb {Z};\\; \\\\& \\qquad \\qquad s_j=t_j=0\\, (j=1,\\ldots ,p); \\, s_j,t_j\\in 2\\mathbb {Z}\\, (j=p+1,\\ldots ,n)\\rbrace ,$ thus the first homology group of the solvmanifold $\\Lambda _{k,\\pi }\\backslash G_{(a_1,\\ldots ,a_n)}$ is $ \\Lambda _{k,\\pi }/[\\Lambda _{k,\\pi },\\Lambda _{k,\\pi }] \\cong \\mathbb {Z}\\oplus \\mathbb {Z}_{2k}\\oplus (\\mathbb {Z}\\oplus \\mathbb {Z})^p \\oplus (\\mathbb {Z}_2\\oplus \\mathbb {Z}_2)^{n-p}.$ The first Betti number is $b_1=2p+1$ .",
"$\\bullet $ For the family $\\Lambda _{k,\\pi /2}$ , note that the isomorphism class of this lattice depends only on the congruence class of the integers $a_j$ modulo 4, therefore according to Corollary REF , we may assume that $a_j\\in \\lbrace 0,1,2,3\\rbrace $ for all $j$ , not all of them even.",
"After reordering, we have that there exist $c,d\\in \\lbrace 0,1,\\ldots ,n-1\\rbrace $ with $0\\le c+d\\le n-1$ such that $a_1=\\cdots =a_c=0$ , $a_{c+1}=\\cdots =a_{c+d}=2$ and $a_j\\in \\lbrace 1,3\\rbrace $ for $j>c+d$ .",
"It can be seen that $[\\Lambda _{k,\\frac{\\pi }{2}},\\Lambda _{k,\\frac{\\pi }{2}}] & = \\lbrace (0,r,s_1,t_1,\\ldots ,s_n,t_n)\\in \\Lambda _{k,\\frac{\\pi }{2}}\\,: r\\in \\mathbb {Z},\\; s_j=t_j=0\\, (j=1,\\ldots ,c); \\\\& \\qquad \\qquad \\quad s_j,t_j\\in 2\\mathbb {Z}\\, (j=c+1,\\ldots ,c+d);\\; s_j+t_j\\in 2\\mathbb {Z}\\,(j>c+d) \\rbrace ,$ thus the first homology group of the solvmanifold $\\Lambda _{k,\\frac{\\pi }{2}}\\backslash G_{(a_1,\\ldots ,a_n)}$ is $ \\Lambda _{k,\\frac{\\pi }{2}}/[\\Lambda _{k,\\frac{\\pi }{2}},\\Lambda _{k,\\frac{\\pi }{2}}] \\cong \\mathbb {Z}\\oplus \\mathbb {Z}_{2k}\\oplus (\\mathbb {Z}\\oplus \\mathbb {Z})^c \\oplus (\\mathbb {Z}_2\\oplus \\mathbb {Z}_2)^d \\oplus (\\mathbb {Z}_2)^{n-(c+d)}.$ The first Betti number is $b_1=2c+1$ .",
"Note that for any $n\\in \\mathbb {N}$ and $r\\in \\lbrace 0,1,\\ldots ,n-1\\rbrace $ , we can find a $(2n+2)$ -dimensional Vaisman solvmanifold $\\Lambda _{k,j}\\backslash G_{(a_1,\\ldots ,a_n)}$ with first Betti number $b_1=2r+1$ .",
"Remark 7.5 In [29] the authors provide examples of compact Vaisman manifolds which are obtained as the total spaces of a principal $S^1$ -bundle over coKähler manifolds, and they show that they are diffeomorphic to solvmanifolds.",
"The Lie algebras associated to these solvmanifolds are isomorphic to some $\\mathfrak {g}_{(a_1,\\ldots ,a_n)}$ , but the lattices that they consider are different from ours."
],
[
"Example 2",
"We start with a Kähler flat Lie algebra $(\\mathfrak {k}, J, \\langle \\cdot ,\\cdot \\rangle )$ such that $\\dim \\mathfrak {h}=1$ , where $\\mathfrak {k}=\\mathfrak {z}\\oplus \\mathfrak {h}\\oplus \\mathfrak {k}^{\\prime }$ is the orthogonal decomposition of $\\mathfrak {k}$ given by Proposition REF .",
"Let $\\mathfrak {h}=\\mathbb {R}H$ with $|H|=1$ , and let us set $2m:=\\dim \\mathfrak {k}^{\\prime }$ and $2l+1:=\\dim \\mathfrak {z}$ .",
"According to Proposition REF , if $Z:=JH$ then $Z\\in \\mathfrak {z}$ and there exists a $J$ -invariant subspace $\\mathfrak {u}$ of $\\mathfrak {z}$ such that $\\mathfrak {z}=\\mathbb {R}Z\\oplus ^\\perp \\mathfrak {u}$ .",
"If $\\omega $ denotes the fundamental 2-form of $(J,\\langle \\cdot ,\\cdot \\rangle )$ , then it is easy to verify that the Sasakian central extension of $\\mathfrak {k}$ by $\\omega $ can be decomposed as: $\\mathfrak {k}_\\omega (B)=\\mathbb {R}H\\ltimes _M(\\mathbb {R}Z\\times \\mathfrak {h}_{2(m+l)+1}),$ where $M$ is the operator given by the following matrix $ M=\\left(\\begin{array}{cccccccc}0 & 0 & & & & & & \\\\1& 0 & & & & & & \\\\& & 0_{l\\times l} & & & & & \\\\& & & 0 & -a_1 & & & \\\\& & &a_1 & 0 & & & \\\\& & & & & \\ddots & & \\\\& & & & & & 0 & -a_m\\\\& & & & & & a_m & 0 \\\\\\end{array}\\right),$ in an orthonormal basis $\\lbrace Z,B,e_1,f_1,\\dots ,e_l,f_l,u_1,v_1,\\dots ,u_m,v_m\\rbrace $ such that: $\\lbrace e_1,f_1,\\cdots ,e_l,f_l\\rbrace $ is a basis of $\\mathfrak {u}$ , $\\lbrace u_1,v_1,\\cdots ,u_m,v_m\\rbrace $ is a basis of $\\mathfrak {k}^{\\prime }$ and $Je_i=f_i$ , $Ju_i=v_i$ .",
"Moreover, $[e_j,f_j]=B$ and $[u_j,v_j]=B$ for all $j$ .",
"It follows from Proposition REF that $a_j\\ne 0$ for all $j=1,\\ldots ,m$ .",
"Let $\\mathfrak {g}$ be the double extension $\\mathfrak {g}=\\mathfrak {k}(D,\\omega )=\\mathbb {R}A \\ltimes _D(\\mathbb {R}H\\ltimes _M(\\mathbb {R}Z\\times \\mathfrak {h}_{2(m+l)+1})),$ where $D$ is the derivation of $\\mathfrak {k}_\\omega (B)$ given by $ D= \\begin{pmatrix}0 & & & & & & & \\\\& 0 & & & & & & \\\\& & 0 & & & & & \\\\& & & 0 & -\\alpha _1 & & & \\\\& & &\\alpha _1 & 0 & & & \\\\& & & & &\\ddots & & \\\\& & & & & & 0 & -\\alpha _{m+l}\\\\& & & & & & \\alpha _{m+l} & 0 \\\\\\end{pmatrix}, $ for some $\\alpha _j\\in \\mathbb {R}$ , in the basis $\\lbrace H,Z, B,e_1,f_1,\\dots ,e_l,f_l,u_1,v_1,\\cdots ,u_m,v_m\\rbrace $ .",
"Since $D|_{\\mathfrak {k}}$ satisfies the conditions of Theorem REF we have that $\\mathfrak {g}$ admits a Vaisman structure.",
"Now we study lattices on the simply connected Lie group $G$ associated to $\\mathfrak {g}$ when $a_i, \\alpha _i\\in \\mathbb {Z}$ for any $i$ .",
"Set $n:=m+l$ and consider the Lie group homomorphism $\\psi : \\mathbb {R}\\longrightarrow \\operatorname{Aut}(\\mathbb {R}\\times H_{2n+1})$ given by $ \\psi (t)=e^{tM}=\\begin{pmatrix}1 & 0 & & & & & & \\\\t & 1 & & & & & & \\\\& & \\operatorname{Id}_{l\\times l} & & & & & \\\\& & & \\cos (a_1t) & -\\sin (a_1t) & & & \\\\& & & \\sin (a_1t) & \\cos (a_1t) & & & \\\\& & & & & \\ddots & & \\\\& & & & & & \\cos (a_mt) & -\\sin (a_mt)\\\\& & & & & & \\sin (a_mt) & \\cos (a_mt)\\\\\\end{pmatrix}.$ On $\\mathbb {R}^{2n+3}$ consider the algebraic structure given by the semidirect product of $\\mathbb {R}$ and $\\mathbb {R}\\times H_{2n+1}$ via $\\psi $ and we obtain the simply connected Lie group $S=\\mathbb {R}\\ltimes _\\psi (\\mathbb {R}\\times H_{2n+1}).$ If $\\Gamma _k$ is the lattice in $H_{2n+1}$ considered in the examples in §REF , then the group $L_k=a\\mathbb {Z}\\times \\Gamma _k$ is a lattice in $\\mathbb {R}\\times H_{2n+1}$ for any $a\\in \\mathbb {R}$ , $a\\ne 0$ .",
"In particular, according to Lemma REF , for each $j= 2\\pi , \\pi , \\frac{\\pi }{2}$ we obtain a lattice $\\Gamma _{k,j}$ in $S$ defined by $\\Gamma _{k,j}=j\\mathbb {Z}\\ltimes (j^{-1}\\mathbb {Z}\\times \\Gamma _k).",
"$ Now we consider the Lie group homomorphism $\\varphi : \\mathbb {R}\\longrightarrow \\operatorname{Aut}(S)$ given by $ \\varphi (t)=e^{tD}=\\left(\\begin{array}{cccccccc}1 & & & & & & & \\\\& 1 & & & & & & \\\\& & 1& & & & & \\\\& & & \\cos (\\alpha _1t) &-\\sin (\\alpha _1t) & & & \\\\& & & \\sin (\\alpha _1t) & \\cos (\\alpha _1t) & & & \\\\& & & & & \\ddots & & \\\\& & & & & & \\cos (\\alpha _nt) &-\\sin (\\alpha _nt)\\\\& & & & & & \\sin (\\alpha _nt) & \\cos (\\alpha _nt)\\\\\\end{array}\\right) $ On $\\mathbb {R}^{2n+4}$ consider the algebraic structure given by the semidirect product of $\\mathbb {R}$ and $S$ via $\\varphi $ and we obtain the simply connected Lie group $G=\\mathbb {R}\\ltimes _\\varphi S.$ Since $\\Gamma _{k,j}$ is invariant under the subgroups generated by $\\varphi (2\\pi ), \\varphi (\\pi ), \\varphi (\\frac{\\pi }{2})$ , then for each $\\Gamma _{k,j}$ we have three new lattices in $G$ , given by $\\Lambda _{k,j,i}=i\\mathbb {Z}\\ltimes \\Gamma _{k,j},$ for $i= 2\\pi , \\pi , \\frac{\\pi }{2}$ .",
"Therefore we get new examples of solvmanifolds $M_{k,j,i}=\\Lambda _{k,j,i}\\backslash \\mathbb {R}\\ltimes _\\varphi (\\mathbb {R}\\ltimes _\\psi (\\mathbb {R}\\times H_{2n+1}))$ equipped with a Vaisman structure arising from a Vaisman structure on the Lie algebra."
],
[
"Classification of Vaisman Lie algebras in low dimensions",
"In this section we determine all unimodular solvable Lie algebras of dimension 4 and 6 that admit a Vaisman structure.",
"(i) According to Theorem REF , any 4-dimensional unimodular solvable with a Vaisman structure is a double extension $\\mathfrak {g}=\\mathfrak {k}(D,\\omega )$ , where $\\mathfrak {k}=\\mathbb {R}^2$ .",
"It is easy to see that $(\\mathfrak {g},J,\\langle \\cdot ,\\cdot \\rangle )$ is equivalent, via a holomorphic and isometric isomorphism, to a Lie algebra which has an orthonormal basis $\\lbrace A,B,e,f\\rbrace $ that satisfies $ [A,e]=cf,\\quad [A,f]=-cf, \\quad [e,f]=B, \\quad JA=B, \\; Je=f \\qquad (c=0 \\text{ or } c=1).$ Both Lie algebras belong to the family of examples constructed in §REF , and the associated simply connected Lie groups admit lattices.",
"If $c=0$ , then $\\mathfrak {g}$ is isomorphic to the nilpotent Lie algebra $\\mathbb {R}\\times \\mathfrak {h}_3$ , equipped with its canonical Vaisman structure (see Example REF ), and any nilmanifold obtained as a quotient of $\\mathbb {R}\\times H_3$ is a primary Kodaira surface.",
"If $c=1$ , then $\\mathfrak {g}$ is a solvable non-nilpotent Lie algebra $\\mathbb {R}\\ltimes \\mathfrak {h}_3$ , and any solvmanifold obtained as a quotient of $\\mathbb {R}\\ltimes H_3$ is a secondary Kodaira surface (see [23]).",
"(ii) According to Theorem REF , any 6-dimensional unimodular solvable Lie algebra with a Vaisman structure is a double extension $\\mathfrak {g}=\\mathfrak {k}(D,\\omega )$ , where $\\mathfrak {k}$ is a 4-dimensional Kähler flat Lie algebra which decomposes orthogonally as $\\mathfrak {k}=\\mathfrak {z}\\oplus \\mathfrak {h}\\oplus \\mathfrak {k}^{\\prime }$ , according to Proposition REF .",
"In particular, the dimension of $\\mathfrak {k^{\\prime }}$ must be 0 or 2 and, as a consequence, $\\dim \\mathfrak {h}$ is equal to 0 or 1.",
"Therefore there are only two cases for $\\mathfrak {k}$ , namely, $\\mathfrak {k}_1=\\mathfrak {z}=\\mathbb {R}^4$ or $\\mathfrak {k}_2=\\mathbb {R}Z\\oplus \\mathbb {R}H\\oplus \\mathbb {R}^2$ for some $Z\\in \\mathfrak {z},\\, H\\in \\mathfrak {h}$ .",
"In the latter case, the action of $H$ on $\\mathfrak {k}^{\\prime }$ is given by a matrix $\\begin{pmatrix}0 & -a \\\\ a & 0\\end{pmatrix}$ , with $a\\ne 0$ , and it is easy to see that these Lie algebras are all isomorphic to the one with $a=1$ .",
"The central extension of $\\mathfrak {k}_1$ is isomorphic to $\\mathfrak {h}_5$ , while the central extension of $\\mathfrak {k}_2$ is the Lie algebra $\\mathfrak {s}_5$ given by $ [e_1,e_2]=e_3, \\; [e_1,e_3]=-e_2,\\; [e_1,e_4]=[e_2,e_3]=B,$ in a basis $\\lbrace B, e_1,e_2,e_3,e_4\\rbrace $ .",
"It is a consequence of Corollary REF that $\\mathfrak {h}_5$ and $\\mathfrak {s}_5$ are the only unimodular solvable Sasakian Lie algebras of dimension 5 (this follows also from [4]).",
"The corresponding unimodular solvable Vaisman Lie algebras, obtained as double extensions of $\\mathfrak {k}_1$ and $\\mathfrak {k}_2$ , are given in the next result.",
"Note that the double extensions $\\mathfrak {k}_1(D,\\omega )$ belong to the family in §REF , while the double extensions $\\mathfrak {k}_2(D,\\omega )$ belong to the family in §REF .",
"Proposition 7.6 Let $\\mathfrak {g}$ be a 6-dimensional unimodular solvable Lie algebra.",
"If $\\mathfrak {g}$ admits a Vaisman structure, then $\\mathfrak {g}$ is isomorphic to one of the following Lie algebras: $\\mathbb {R}\\times \\mathfrak {h}_5$ , $\\mathbb {R}\\ltimes _{D_r}\\mathfrak {h}_5$ with $r\\in [0,1]$ , $\\mathbb {R}\\times \\mathfrak {s}_5$ , $\\mathbb {R}\\ltimes _{D_0}\\mathfrak {s}_5$ , where $ D_r=\\left(\\begin{array}{ccccc}0 & & & & \\\\& 0 & -r & & \\\\& r & 0 & & \\\\& & & 0 & -1 \\\\& & & 1 & 0 \\\\\\end{array}\\right)$ in the basis $\\lbrace B, e_1,e_2,e_3,e_4\\rbrace $ as above.",
"Moreover, all these Lie algebras are pairwise non-isomorphic and the corresponding simply connected Lie groups admit lattices (with $r\\in \\mathbb {Q}$ for $\\mathbb {R}\\ltimes _{D_r}\\mathfrak {h}_5$ ).",
"It is easy to see, taking into account Proposition REF , that $\\mathfrak {k}_1(D,\\omega )$ is isomorphic to either $\\mathbb {R}\\times \\mathfrak {h}_5$ or $\\mathbb {R}\\ltimes _{D_r}\\mathfrak {h}_5$ with $r\\in [0,1]$ .",
"On the other hand, for $\\mathfrak {k}_2(D,\\omega )$ , it can be shown that this Lie algebra is isomorphic to either $\\mathbb {R}\\times \\mathfrak {s}_5$ or $\\mathbb {R}\\ltimes _{D_0}\\mathfrak {s}_5$ .",
"Comparing the nilradical of these Lie algebras and Proposition REF again, it follows that these Lie algebras are pairwise non-isomorphic.",
"Note that $\\mathbb {R}\\ltimes _{D_r}\\mathfrak {h}_5$ corresponds to the Lie algebra $\\mathfrak {g}_{(r,1)}$ with $r\\in [0,1]$ studied in §REF .",
"Therefore the existence of lattices for the oscillator group $G_{(r,1)}$ with $r\\in \\mathbb {Q}$ was established in Proposition REF .",
"In the case of $\\mathbb {R}\\times \\mathfrak {s}_5$ and $\\mathbb {R}\\ltimes _{D_0}\\mathfrak {s}_5$ the existence of lattices in the associated simply connected Lie group follows from §REF .",
"It follows from Proposition REF that the Lie algebra $\\mathfrak {g}_{(r,1)}=\\mathbb {R}\\ltimes _{D_r}\\mathfrak {h}_5,\\, r\\in \\mathbb {Q},$ from Proposition REF is isomorphic to $\\mathfrak {g}_{(a,b)}$ for some $a,b\\in \\mathbb {Z}$ , $(a,b)=1$ .",
"Next we determine the first homology group and the first Betti number of the 6-dimensional oscillator solvmanifolds $\\Lambda _{k,j}\\backslash G_{(a,b)}$ for $j=2\\pi ,\\pi ,\\pi /2$ , following §REF .",
"$\\bullet $ For the familiy $\\Lambda _{k,2\\pi }=2\\pi \\mathbb {Z}\\times \\Gamma _k$ , the solvmanifold $\\Lambda _{k,2\\pi }\\backslash G_{(a,b)}$ is diffeomorphic to a nilmanifold $S^1\\times \\Gamma _k\\backslash H_5$ , for all $(a,b)$ .",
"The first homology group of this nilmanifold is $\\mathbb {Z}^5\\oplus \\mathbb {Z}_{2k}$ and hence its first Betti number is $b_1=5$ .",
"$\\bullet $ For the family $\\Lambda _{k,\\pi }$ , the isomorphism class of this lattice depends only on the parity of the integers $a, b$ .",
"Then it is straightforward to verify that the first homology group of the solvmanifold $M:=\\Lambda _{k,\\pi }\\backslash G_{(a,b)}$ and its first Betti number are given by: Table: NO_CAPTION$\\bullet $ For the family $\\Lambda _{k,\\pi /2}$ , the isomorphism class of this lattice depends only on the congruence class of the integers $a, b$ modulo 4, and it can be seen that the first homology group of the solvmanifold $M:=\\Lambda _{k,\\frac{\\pi }{2}}\\backslash G_{(a,b)}$ and its first Betti number are given by: Table: NO_CAPTION"
]
] | 1709.01567 |
[
[
"Foundation for a series of efficient simulation algorithms"
],
[
"Abstract Compute the coarsest simulation preorder included in an initial preorder is used to reduce the resources needed to analyze a given transition system.",
"This technique is applied on many models like Kripke structures, labeled graphs, labeled transition systems or even word and tree automata.",
"Let (Q, $\\rightarrow$) be a given transition system and Rinit be an initial preorder over Q.",
"Until now, algorithms to compute Rsim , the coarsest simulation included in Rinit , are either memory efficient or time efficient but not both.",
"In this paper we propose the foundation for a series of efficient simulation algorithms with the introduction of the notion of maximal transitions and the notion of stability of a preorder with respect to a coarser one.",
"As an illustration we solve an open problem by providing the first algorithm with the best published time complexity, O(|Psim |.|$\\rightarrow$|), and a bit space complexity in O(|Psim |^2.",
"log(|Psim |) + |Q|.",
"log(|Q|)), with Psim the partition induced by Rsim."
],
[
"Introduction",
"The simulation relation has been introduced by Milner [11] as a behavioural relation between process.",
"This relation can also be used to speed up the test of inclusion of languages [2] or as a sufficient condition when this test of inclusion is undecidable in general [5].",
"Another very helpful use of a simulation relation is to exhibit an equivalence relation over the states of a system.",
"This allows to reduce the state space of the given system to be analyzed while preserving an important part of its properties, expressed in temporal logics for examples [7].",
"Note that the simulation equivalence yields a better reduction of the state space than the better known bisimulation equivalence."
],
[
"State of the Art",
"The paper that has most influenced the literature is that of Henzinger, Henzinger and Kopke [10].",
"Their algorithm, designed over Kripke structures, and here named HHK, to compute ${R}_{\\mathrm {sim}}$ , the coarsest simulation, runs in $O(|Q|.|{\\rightarrow }|)$ -time, with $\\rightarrow $ the transition relation over the state space $Q$ , and uses $O(|Q|^2.\\log (|Q|))$ bits.",
"But it happens that ${R}_{\\mathrm {sim}}$ is a preorder.",
"And as such, it can be more efficiently represented by a partition-relation pair $(P,R)$ with $P$ a partition, of the state space $Q$ , whose blocks are classes of the simulation equivalence relation and with $R\\subseteq P\\times P$ a preorder over the blocks of $P$ .",
"Bustan and Grumberg [3] used this to propose an algorithm, here named BG, with an optimal bit-space complexity in $O(|P_{\\mathrm {sim}}|^2 +|Q|.\\log (|P_{\\mathrm {sim}}|))$ with $|P_{\\mathrm {sim}}|$ (in general significantly smaller than $|Q|$ ) the number of blocks of the partition $P_{\\mathrm {sim}}$ associated with ${R}_{\\mathrm {sim}}$ .",
"Unfortunately, BG suffers from a very bad time complexity.",
"Then, Gentilini, Piazza and Policriti [8] proposed an algorithm, here named GPP, with a better time complexity, in $O(|P_{\\mathrm {sim}}|^2.|{\\rightarrow }|)$ , and a claimed bit space complexity like the one of BG.",
"This algorithm had a mistake and was corrected in [16].",
"It is very surprising that none of the authors citing [8], including these of [16], [14], [15], [6] and [9], realized that the announced bit space complexity was also not correct.",
"Indeed, as shown in [4] and [13] the real bit space complexity of GPP is $O(|P_{\\mathrm {sim}}|^2.\\log (|P_{\\mathrm {sim}}|)+|Q|.\\log (|Q|))$ .",
"In a similar way, [15] and [6] did a minor mistake by considering that a bit space in $O(|Q|.\\log (|P_{\\mathrm {sim}}|))$ was sufficient to represent the partition in their algorithms while a space in $O(|Q|.\\log (|Q|))$ is needed.",
"Ranzato and Tapparo [14], [15] made a major breakthrough with their algorithm, here named RT, which runs in $O(|P_{\\mathrm {sim}}|.|{\\rightarrow }|)$ -time but uses $O(|P_{\\mathrm {sim}}|.|Q|.\\log (|Q|))$ bits, which is more than GPP.",
"The difficulty of the proofs and the abstract interpretation framework put aside, RT is a reformulation of HHK but with a partition-relation pair instead of a mere relation between states.",
"Over unlabelled transition systems, this is the best algorithm regarding the time complexity.",
"Since [14] a question has emerged: is there an algorithm with the time complexity of RT while preserving the space complexity of GPP ?",
"Crafa, Ranzato and Tapparo [6], modified RT to enhance its space complexity.",
"They proposed an algorithm with a time complexity in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |+|P_{\\mathrm {sim}}|^2.|\\rightarrow _{P_{\\mathrm {sp}},P_{\\mathrm {sim}}}|)$ and a bit space complexity in $O(|P_{\\mathrm {sp}}|.|P_{\\mathrm {sim}}|.\\log (|P_{\\mathrm {sp}}|)+ |Q|.\\log (|Q|))$ with $|P_{\\mathrm {sp}}|$ between $|P_{\\mathrm {sim}}|$ and $|P_{\\mathrm {bis}}|$ , the number of bisimulation classes, and $\\rightarrow _{P_{\\mathrm {sp}},P_{\\mathrm {sim}}}$ a smaller abstraction of $\\rightarrow $ .",
"Unfortunately (although this algorithm provided new insights), for 22 examples, out of the 24 they provided, there is no difference between $|P_{\\mathrm {bis}}|$ , $|P_{\\mathrm {sp}}|$ and $|P_{\\mathrm {sim}}|$ .",
"For the two remaining examples the difference is marginal.",
"With a little provocation, we can then consider that $|P_{\\mathrm {sp}}|\\approx |P_{\\mathrm {bis}}|$ and compute the bisimulation equivalence (what should be done every time as it produces a considerable speedup) then compute the simulation equivalence with GPP on the obtained system is a better solution than the algorithm in [6] even if an efficient computation of the bisimulation equivalence requires, see [12], a bit space in $O(|\\rightarrow |.\\log (|Q|))$ .",
"Ranzato [13] almost achieved the challenge by announcing an algorithm with the space complexity of GPP but with the time complexity of RT multiplied by a $\\log (|Q|)$ factor.",
"He concluded that the suppression of this $\\log (|Q|)$ factor seemed to him quite hard to achieve.",
"Gentilini, Piazza and Policriti [9] outperformed the challenge by providing an algorithm with the space complexity of BG and the time complexity of RT, but only in the special case of acyclic transition systems."
],
[
"Our Contributions",
"In this paper, we respond positively to the question and propose the first simulation algorithm with the time complexity of RT and the space complexity of GPP.",
"Our main sources of inspiration are [12] for its implicit notion of stability against a coarser partition, that we generalize in the present paper for preorders, and for the counters it uses, [10] for the extension of these counters for simulation algorithms, [3] for its use of little brothers to which we prefer the use of what we define as maximal transitions, [13] for its implicit use of maximal transitions to split blocks and for keeping as preorders the intermediate relations of its algorithm and [4] for its equivalent definition of a simulation in terms of compositions of relations.",
"Note that almost all simulation algorithms are defined for Kripke structures.",
"However, in each of them, after an initial step which consists in the construction of an initial preorder ${R}_{\\mathrm {init}}$ , the algorithm is equivalent to calculating the coarsest simulation inside ${R}_{\\mathrm {init}}$ over a classical transition system.",
"We therefore directly start from a transition system $(Q,\\rightarrow )$ and an initial preorder ${R}_{\\mathrm {init}}$ inside which we compute the coarsest simulation."
],
[
"Preliminaries",
"Let $Q$ be a set of elements, or states.",
"The number of elements of $Q$ is denoted $|Q|$ .",
"A relation over $Q$ is a subset of $Q\\times Q$ .",
"Let ${R}$ be a relation over $Q$ .",
"For all $q,q^{\\prime }\\in Q$ we may write $q\\,{R}\\,q^{\\prime }$ , or $q \\mathbin {[baseline] [dashed,->] (0pt,.5ex)-- node[font=\\footnotesize ,fill=white,inner sep=2pt] {{R}}(6ex,.5ex);} q^{\\prime }$ in the figures, when $(q,q^{\\prime })\\in {R}$ .",
"We define ${R}(q)\\triangleq \\lbrace q^{\\prime }\\in Q\\,\\big |\\,q\\,{R}\\,q^{\\prime }\\rbrace $ for $q\\in Q$ , and ${R}(X)\\triangleq \\cup _{q\\in X}{R}(q)$ for $X\\subseteq Q$ .",
"We write $X\\mathrel {R}Y$ , or $X \\mathbin {[baseline] [dashed,->](0pt,.7ex) -- node[font=\\footnotesize ,fill=white,inner sep=2pt]{{R}} (6ex,.7ex);} Y$ in the figures, when $X\\times Y\\cap \\,{R}\\ne \\emptyset $ .",
"For $q\\in Q$ and $X\\subseteq Q$ , we also write $X\\mathrel {R} q$ (resp.",
"$q\\mathrel {R} X$ ) for $X\\mathrel {R} \\lbrace q\\rbrace $ (resp.",
"$\\lbrace q\\rbrace \\mathrel {R} X$ ).",
"A relation ${R}$ is said coarser than another relation ${R}^{\\prime }$ when ${R}^{\\prime }\\subseteq {R}$ .",
"The inverse of ${R}$ is ${{R}}^{-1}\\triangleq \\lbrace (y,x)\\in Q\\times Q\\,\\big |\\,(x,y)\\in {R}\\rbrace $ .",
"The relation ${R}$ is said symmetric if ${{R}}^{-1}\\subseteq {R}$ and antisymmetric if $q\\,{R}\\,q^{\\prime }$ and $q^{\\prime }\\,{R}\\,q$ implies $q=q^{\\prime }$ .",
"Let ${S}$ be a second relation over $Q$ , the composition of ${R}$ by ${S}$ is ${S}\\mathrel {\\circ }{R}\\triangleq \\lbrace (x,y)\\in Q\\times Q\\,\\big |\\,y\\in {S}({R}(x))\\rbrace $ .",
"The relation ${R}$ is said reflexive if for all $q\\in Q$ we have $q\\,{R}\\,q$ , and transitive if ${R}\\mathrel {\\circ }{R}\\subseteq {R}$ .",
"A preorder is a reflexive and transitive relation.",
"A partition $P$ of $Q$ is a set of non empty subsets of $Q$ , called blocks, that are pairwise disjoint and whose union gives $Q$ .",
"A partition-relation pair is a pair $(P,R)$ with $P$ a partition and $R$ a relation over $P$ .",
"To a partition-relation pair $(P,R)$ we associate a relation ${R}_{(P,R)}\\triangleq \\bigcup _{(C,D)\\in R}C\\times D$ .",
"Let ${R}$ be a preorder on $Q$ and $q\\in Q$ , we define $[q]_{R} \\triangleq \\lbrace q^{\\prime }\\in Q \\,\\big |\\,q \\,{R}\\, q^{\\prime } \\wedge q^{\\prime }\\,{R}\\, q\\rbrace $ and $P_{{R}} \\triangleq \\lbrace [q]_{R}\\subseteq Q\\,\\big |\\,q\\in Q\\rbrace $ .",
"It is easy to show that $P_{{R}}$ is a partition of $Q$ .",
"Therefore, given any preorder ${R}$ and a state $q\\in Q$ , we also call block, the block of $q$ , the set $[q]_{R}$ .",
"A symmetric preorder ${P}$ is totally represented by the partition $P_{{P}}$ since ${P}=\\cup _{E\\in P_{P}} E\\times E$ .",
"Let us recall that a symmetric preorder is traditionally named an equivalence relation.",
"Conversely, given a partition $P$ , there is an associated equivalence relation ${P}_P\\triangleq \\cup _{E\\in P}E\\times E$ .",
"In the general case, a preorder ${R}$ is efficiently represented by the partition-relation pair $(P_{{R}}, R_{{R}})$ with $R_{{R}}\\triangleq \\lbrace ([q]_{R},[q^{\\prime }]_{R})\\in P_{{R}}\\times P_{{R}}\\,\\big |\\,q\\,{R}\\,q^{\\prime }\\rbrace $ a reflexive, transitive and antisymmetric relation over $P_{{R}}$ .",
"Furthermore, for a preorder ${R}$ , we note $[\\cdot ]_{{R}}$ the relation over $Q$ which associates to a state the elements of its block.",
"Said otherwise: $[\\cdot ]_{{R}}\\triangleq \\cup _{q\\in Q}\\lbrace q\\rbrace \\times [q]_{{R}}$ .",
"Finally, for a set $X$ of sets we note $\\cup X$ for $\\cup _{E\\in X}E$ .",
"Proposition 1 Let $X$ and $Y$ be two blocks of a preorder ${R}$ .",
"Then $(X^{\\prime }\\subseteq X\\wedge Y^{\\prime }\\subseteq Y\\wedge X^{\\prime }\\mathrel {R}Y^{\\prime })\\Rightarrow X\\times Y\\subseteq {R}.$ Said otherwise, when two subsets of two blocks of ${R}$ are related by ${R}$ then all the elements of the first block are related by ${R}$ with all the elements of the second block.",
"Thanks to the transitivity of ${R}$ .",
"A finite transition systems (TS) is a pair $(Q,\\rightarrow )$ with $Q$ a finite set of states, and $\\rightarrow $ a relation over $Q$ called the transition relation.",
"A relation ${S}$ is a simulation over $(Q,\\rightarrow )$ if: ${S}\\mathrel \\circ \\rightarrow ^{-1}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {S}$ For a simulation ${S}$ , when we have $q\\mathrel {S}q^{\\prime }$ , we say that $q$ is simulated by $q^{\\prime }$ (or $q^{\\prime }$ simulates $q$ ).",
"A relation ${B}$ is a bisimulation if ${B}$ and ${B}^{-1}$ are both simulations.",
"The interesting bisimulations, such as the coarsest one included in a preorder, are equivalence relations.",
"It is easy to show that an equivalence relation ${B}$ is a bisimulation iff : ${B}\\mathrel \\circ \\rightarrow ^{-1}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {B}$ Remark The classical definition is to say that a relation ${S}$ is a simulation if: $q_1\\mathrel {S} q_2\\wedge q_1\\rightarrow q^{\\prime }_1\\Rightarrow \\exists q^{\\prime }_2\\;.\\;q_2\\rightarrow q^{\\prime }_2\\wedge {q^{\\prime }_1\\mathrel {S} q^{\\prime }_2} $ .",
"However, we prefer the formula (REF ), which is equivalent, because it is more global and to design efficient simulation algorithms we must abstract from individual states.",
"In the remainder of the paper, all relations are over the same finite set $Q$ and the underlying transition system is $(Q,\\rightarrow )$ ."
],
[
"Key Ideas",
"Let us start from equation (REF ).",
"If a relation ${R}$ is not a simulation, we have ${R}\\mathrel \\circ {\\rightarrow ^{-1}}\\nsubseteq \\rightarrow ^{-1}\\mathrel {\\circ }{R}$ .",
"This implies the existence of a relation $Remove$ such that: ${R}\\mathrel \\circ \\,\\rightarrow ^{-1}\\subseteq {(\\rightarrow ^{-1}\\mathrel {\\circ }{R)}\\cup Remove}$ .",
"It can be shown that most of the simulation algorithms cited in the introduction, like HHK, GPP and RT, are based on this last equation.",
"In this paper, like in [4], we make the following different choice.",
"When ${R}$ is not a simulation, we reduce the problem of finding the coarsest simulation inside ${R}$ to the case where there is a relation $\\mathit {NotRel}$ such that: ${R}\\mathrel \\circ \\rightarrow ^{-1}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ ({R}\\cup \\mathit {NotRel})$ .",
"Let us note ${U}\\triangleq {R}\\cup \\mathit {NotRel}$ .",
"We will say that ${R}$ is ${U}$ -stable since we have: ${R}\\mathrel \\circ \\rightarrow ^{-1}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {U}$ Our definition of stability is new.",
"However, it is implicit in the bisimulation algorithm of [12] where, with the notations from [12], a partition $Q$ is said stable with every block of a coarser partition $X$ .",
"Within our formalism we can say the same thing with the formula ${P}_{Q}\\mathrel \\circ \\rightarrow ^{-1}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {P}_{X}$ .",
"Figure: R{R} is U{U}-stable and V{V}, obtainedafter a split of blocks of R{R} and arefinement of R{R}, is R{R}-stable.Consider the transition $c\\rightarrow b$ in Figure REF .",
"The preorder ${R}$ is assumed to be ${U}$ -stable and we want to find the coarsest simulation included in ${R}$ .",
"Since ${R}$ is a preorder, the set ${R}(c)$ is a union of blocks of ${R}$ .",
"A state $d$ in ${R}(c)$ which doesn't have an outgoing transition to ${R}(b)$ belongs to $\\rightarrow ^{-1}\\mathrel \\circ {U}(b)$ , thanks to (REF ), but cannot simulate $c$ .",
"Thus, we can safely remove it from ${R}(c)$ .",
"But to do this effectively, we want to manage blocks of states and not the individual states.",
"Hence, we first do a split step by splitting the blocks of ${R}$ such that a resulting block, included in both ${R}(c)$ and $\\rightarrow ^{-1}\\mathrel \\circ {U}(b)$ , is either completely included in $\\rightarrow ^{-1}\\mathrel \\circ {R}(b)$ , which means that its elements still have a chance to simulate $c$ , or totally outside of it, which means that its elements cannot simulate $c$ .",
"Let us call ${P}$ the equivalence relation associated to the resulting partition.",
"We will say that ${P}$ is ${R}$ -block-stable.",
"Then, to test whether a block, $E$ of ${P}$ , which has an outgoing transition in $\\rightarrow ^{-1}\\mathrel \\circ ({U}\\setminus {R})(b)$ , is included in $\\rightarrow ^{-1}\\mathrel \\circ {R}(b)$ , it is sufficient to do the test for only one of its elements, arbitrarily choosen, we call the representative of $E$ : $E.$ .",
"To do this test in constant time we manage a counter which, at first, count the number of transitions from $E.$ to ${U}(b)={U}([b]_{{P}})$.",
"By scanning the transitions whose destination belongs to $({U}\\setminus {R})(b)$ this counter is updated to count the transitions from $E.$ to ${R}(b)={R}([b]_{{P}})$ .",
"Therefore we get the equivalences: there is no transition from $E$ to ${R}(b)$ iff there is no transition from $E.$ to ${R}(b)$ iff this counter is null.",
"Remark that the total bit size of all the counters is in $O(|P_{\\mathrm {sim}}|^2.\\log (|Q|))$ since there is at most $|P_{\\mathrm {sim}}|$ blocks like $E$ , $|P_{\\mathrm {sim}}|$ blocks like $[b]_{{P}}$ and $|Q|$ transitions from a state like $E.$ .",
"The difference is not so significative in practice but we will reduce this size to $O(|P_{\\mathrm {sim}}|^2.\\log (|P_{\\mathrm {sim}}|))$ , at a cost of $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ elementary steps, which is hopefully within our time budget.",
"Removing from ${R}(c)$ the blocks of $\\rightarrow ^{-1}\\mathrel \\circ ({U}\\setminus {R})(b)$ , like $[d]_{{P}}$ , which do not have an outgoing transition to ${R}(b)$ is called the refine step.",
"After this refine step, ${R}(c)$ has been reduced to ${V}(c)$ .",
"Doing these split and refine steps for all transitions $c\\rightarrow b$ results in the relation ${V}$ that we will prove to be a ${R}$ -stable preorder.",
"In summary, from an initial preorder we will build a strictly decreasing series of preorders $({R}_i)_{i\\ge 0}$ such that ${R}_{i+1}$ is ${R}_i$ -stable and contains, by construction, all simulations included in ${R}_i$ .",
"Since all the relations are finite, this series has a limit, reached in a finite number of steps.",
"Let us call ${R}_{\\mathrm {sim}}$ this limit.",
"We have: ${R}_{\\mathrm {sim}}$ is ${R}_{\\mathrm {sim}}$ -stable.",
"Therefore, with (REF ) and (REF ), ${R}_{\\mathrm {sim}}$ is a simulation and by construction contains all simulations included in the initial preorder: this is the coarsest one.",
"Remark The counters which are used in the previous paragraphs play a similar role as the counters used in [12].",
"Without them, the time complexity of the algorithm of the present paper would have been multiplied by a $|P_{\\mathrm {sim}}|$ factor and would have been this of GPP: $O(|P_{\\mathrm {sim}}|^2.|{\\rightarrow }|)$ ."
],
[
"Underlying Theory",
"In this section we give the necessary theory to define what should be the ideal split step and we justify the correctness of our refine step which allows to treat blocks as if they were single states.",
"We begin by introducing the notion of maximal transition.",
"This is the equivalent concept for transitions from that of little brothers, introduced in [3], for states.",
"The main difference is that little brothers have been defined relatively to the final coarsest simulation in a Kripke structure.",
"Here we define maximal transitions relatively to a current preorder ${R}$ .",
"Definition 2 Let ${R}$ be a preorder.",
"The transition $q\\rightarrow q^{\\prime }$ is said maximal for ${R}$ , or ${R}$ -maximal, which is noted $q\\rightarrow _{{R}}q^{\\prime }$ , when: $\\forall q^{\\prime \\prime }\\in Q \\;.\\;(q\\rightarrow q^{\\prime \\prime } \\wedge q^{\\prime }\\mathrel {R}q^{\\prime \\prime })\\Rightarrow q^{\\prime \\prime }\\in [q^{\\prime }]_{{R}}$ The set of ${R}$ -maximal transitions and the induced relation are both noted $\\rightarrow _{{R}}$ .",
"Figure: Illustration of the left property of Lemma .Lemma 3 (Figure REF ) For a preorder ${R}$ , the two following properties are verified: $\\rightarrow ^{-1}\\,\\subseteq \\,\\rightarrow _{{R}}^{-1}\\mathrel {\\circ }{R} \\text{ and }\\rightarrow ^{-1}\\mathrel {\\circ }{R} \\,=\\, \\rightarrow ^{-1}_{{R}}\\mathrel {\\circ }{R}$ Let $(q,q^{\\prime })\\in \\rightarrow $ and $X=\\lbrace q^{\\prime \\prime }\\in Q\\,\\big |\\,q\\rightarrow q^{\\prime \\prime }\\wedge q^{\\prime }{R}q^{\\prime \\prime }\\rbrace $ .",
"Since ${R}$ is reflexive, this set is not empty because it contains $q^{\\prime }$ .",
"Let $Y$ be the set of blocks of ${R}$ which contain an element from $X$ .",
"Since this set is finite (there is a finite number of blocks) there is at least a block $G$ maximal in $Y$ .",
"Said otherwise, there is no $G^{\\prime }\\in Y$ , different from $G$ , such that $G\\,{R}\\, G^{\\prime }$ .",
"Let $q^{\\prime \\prime }\\in G$ such that $q\\rightarrow q^{\\prime \\prime }$ .",
"From what precedes, the transition $(q,q^{\\prime \\prime })$ is maximal and $q^{\\prime }{R}q^{\\prime \\prime }$ .",
"Hence: $(q^{\\prime },q)\\in \\rightarrow _{{R}}^{-1}\\mathrel {\\circ }{R}$ .",
"So we have: $\\rightarrow ^{-1} \\,\\subseteq \\, \\rightarrow ^{-1}_{{R}}\\mathrel {\\circ }{R}$ Now, from (REF ) we get $\\rightarrow ^{-1}\\mathrel {\\circ }{R} \\subseteq \\rightarrow ^{-1}_{{R}}\\mathrel {\\circ }{R}\\mathrel {\\circ }{R}$ and thus $\\rightarrow ^{-1}\\mathrel {\\circ }{R} \\subseteq \\rightarrow ^{-1}_{{R}}\\mathrel {\\circ }{R}$ since ${R}$ is a preorder.",
"The relation $\\rightarrow _{{R}}$ is a subset of $\\rightarrow $ .",
"Therefore we also have $\\rightarrow ^{-1}_{{R}}\\mathrel {\\circ }{R}\\subseteq \\rightarrow ^{-1}\\mathrel {\\circ }{R}$ which concludes the proof.",
"In the last section, we introduced the notions of stability and of block-stability.",
"Let us define them formaly.",
"Definition 4 Let ${R}$ a preorder.",
"${R}$ is said ${U}$ -stable, with ${U}$ a coarser preorder than ${R}$ , if: ${R}\\mathrel \\circ \\rightarrow ^{-1}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {U}$ An equivalence relation ${P}$ included in ${R}$ , is said ${R}$ -block-stable if: $\\hfill \\forall b,d,d^{\\prime }\\in Q\\,.\\, d\\,{P}\\,d^{\\prime }\\Rightarrow (d\\in \\rightarrow ^{-1}\\mathrel {\\circ }{R}(b) \\Leftrightarrow d^{\\prime }\\in \\rightarrow ^{-1}\\mathrel {\\circ }{R}(b)) \\hfill $ Remark Say that ${P}$ is included in ${R}$ means that each block of ${P}$ is included in a block of ${R}$ .",
"As seen in the following lemma we have a nice equivalence: an equivalence relation ${P}$ is ${R}$ -block-stable iff it is ${R}$ -stable.",
"Lemma 5 Let ${P}$ be an equivalence relation included in a preorder ${R}$ .",
"Then (REF ) is equivalent with: ${P}\\mathrel \\circ \\rightarrow ^{-1}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {R}$ Figure: Illustration of .To show the equivalence of (REF ) and (REF ) we use an intermediate property: ${P}\\mathrel \\circ \\rightarrow ^{-1}\\mathrel \\circ {R}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {R}$ With the help of Figure REF it is straightforward to see the equivalence of (REF ) and (REF ).",
"It remains therefore to show the equivalence of (REF ) and (REF ).",
"(REF ) $\\Rightarrow $ (REF ).",
"From (REF ) we get ${P}\\mathrel \\circ \\rightarrow ^{-1}\\mathrel \\circ {R}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {R}\\mathrel \\circ {R}$ and thus (REF ) since, as a preorder ${R}$ is transitive.",
"(REF ) $\\Rightarrow $ (REF ).",
"Let ${I}=\\lbrace (q,q)\\,\\big |\\,q\\in Q\\rbrace $ be the identity relation.",
"We have ${P}\\mathrel \\circ \\rightarrow ^{-1}={P}\\mathrel \\circ \\rightarrow ^{-1}\\mathrel \\circ {I}$ and thus ${P}\\mathrel \\circ \\rightarrow ^{-1}\\mathrel \\circ {I}\\,\\subseteq \\,{P}\\mathrel \\circ \\rightarrow ^{-1}\\mathrel \\circ {R}$ since ${R}$ is a preorder and as such contains ${I}$ .",
"With (REF ) we thus get (REF ).",
"With (REF ) and (REF ) the reader should now be convinced by the interest of (REF ) to define a simulation.",
"Following the keys ideas given in Section there is an interest, for the time complexity, of having a coarse ${R}$ -block-stable equivalence relation ${P}$ .",
"Hopefully there is a coarsest one.",
"Proposition 6 Given a preorder ${R}$ , there is a coarsest ${R}$ -stable equivalence relation.",
"With Lemma REF and by an easy induction based on the two following properties: the identity relation, ${I}=\\lbrace (q,q)\\,\\big |\\,q\\in Q\\rbrace $ , is a ${R}$ -stable equivalence relation.",
"the reflexive and transitive closure $({P}_1\\cup {P}_2)^*$ of the union of two ${R}$ -stable equivalence relations, ${P}_1$ and ${P}_2$ , is also a ${R}$ -stable equivalence relation, coarser than them.",
"We are now ready to introduce the main result of this section.",
"It is a formalization, and a justification, of the refine step given in Section .",
"In the following theorem, the link with the decreasing sequence of relations $({R}_i)_{i\\ge 0}$ mentioned at the end of Section is: if ${R}_i$ is the current value of ${R}$ then ${R}_{i-1}$ is ${U}$ and ${R}_{i+1}$ will be ${V}$ .",
"The reader can also ease its comprehension of the theorem by considering Figure REF .",
"Theorem 7 Let ${U}$ be a preorder, ${R}$ be a ${U}$ -stable preorder and ${P}$ be the coarsest ${R}$ -stable equivalence relation.",
"Let $\\mathit {NotRel}={U}\\mathrel \\setminus {R}$ and ${V}={R}\\mathrel \\setminus \\mathit {NotRel}^{\\prime }$ with $\\mathit {NotRel}^{\\prime } =\\bigcup _{\\begin{array}{c}b,c,d\\,\\in \\, Q,\\; c\\,\\rightarrow \\, b,\\;c\\,{R}\\, d,\\\\d\\,\\in \\, \\rightarrow ^{-1}\\circ \\, \\mathit {NotRel}(b),\\;\\\\d\\,\\notin \\, \\rightarrow ^{-1}\\,\\circ \\,{R}(b)\\end{array}}[c]_{{P}}{\\times } [d]_{{P}}$ Then: $\\mathit {NotRel}^{\\prime } = X$ with $X =\\bigcup _{\\begin{array}{c}b,c,d\\,\\in \\, Q,\\; c\\,\\rightarrow _{\\mbox{\\tiny ${R}$}}\\, b,\\;c\\,{R}\\, d,\\\\d\\,\\in \\, {\\rightarrow }_{ \\mbox{\\tiny ${R}$}}^{-1}\\circ \\, \\mathit {NotRel}(b),\\;\\\\d\\,\\notin \\, \\rightarrow _{\\mbox{\\tiny ${R}$}}^{-1}\\,\\circ \\,{R}(b)\\end{array}}\\lbrace (c,d)\\rbrace $ Any simulation ${S}$ included in ${R}$ is also included in ${V}$ .",
"${V}\\mathrel \\circ \\rightarrow ^{-1}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {R}$ ${V}$ is a preorder.",
"${V}$ is ${R}$ -stable.",
"Blocks of ${V}$ are blocks of ${P}$ (i.e.",
"$P_{{V}}= P_{{P}}$ ).",
"Since $(c,d)$ belongs to $[c]_{{P}}{\\times } [d]_{{P}}$ , $\\rightarrow _{{R}} \\;\\subseteq \\; \\rightarrow $ and $\\rightarrow ^{-1}\\mathrel {\\circ }{R} \\,=\\,\\rightarrow ^{-1}_{{R}}\\mathrel {\\circ }{R}$ , from Lemma REF , we get $X\\subseteq \\mathit {NotRel}^{\\prime }$ .",
"For the converse, let $(c^{\\prime },d^{\\prime })\\in \\mathit {NotRel}^{\\prime }$ .",
"By definition, there are $b,c,d\\in Q$ such that $c\\rightarrow b$ , $c\\,{R}\\, d$ , $d\\in \\rightarrow ^{-1}\\circ \\,\\mathit {NotRel}(b)$ , $d\\notin \\rightarrow ^{-1}\\mathrel \\circ {R}(b)$ , $c^{\\prime }{\\in }[c]_{{P}}$ and $d^{\\prime }{\\in } [d]_{{P}}$ .",
"From $d\\notin \\rightarrow ^{-1}\\mathrel \\circ {R}(b)$ and Lemma REF we have $d\\notin \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(b)$ .",
"From $c\\rightarrow b$ , Lemma REF , Lemma REF , and the hypothesis that ${P}$ is ${R}$ -stable, we have $c^{\\prime }\\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(b)$ .",
"Therefore, there is a state $b^{\\prime }$ such that $c^{\\prime }\\rightarrow _{{R}}b^{\\prime }$ and $b\\,{R}\\, b^{\\prime }$ .",
"Les us suppose $d^{\\prime }\\in \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(b^{\\prime })$ .",
"Since $b\\,{R}\\, b^{\\prime }$ , we would have had $d\\in \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(b)$ .",
"Thus $d^{\\prime }\\notin \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(b^{\\prime })$ .",
"We have $c^{\\prime }\\,{P}\\, c$ , $c\\,{R}\\, d$ , $d\\,{P}\\, d^{\\prime }$ , and thus $c^{\\prime }\\,{R}\\, d^{\\prime }$ since ${R}$ is a preorder and ${P}\\subseteq {R}$ .",
"With $c^{\\prime }\\rightarrow _{{R}}b^{\\prime }$ and the hypothesis that ${R}$ is ${U}$ -stable, we get $d^{\\prime }\\in \\rightarrow ^{-1}\\mathrel \\circ {U}(b^{\\prime })$ and thus, with Lemma REF , $d^{\\prime }\\in \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}\\mathrel \\circ {U}(b^{\\prime })$ and thus $d^{\\prime }\\in \\rightarrow _{{R}}^{-1}\\mathrel \\circ {U}(b^{\\prime })$ since ${U}$ is a preorder and ${R}\\subseteq {U}$ .",
"As seen above, $d^{\\prime }\\notin \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(b^{\\prime })$ .",
"So we have $d^{\\prime }\\in \\rightarrow _{{R}}^{-1}\\mathrel \\circ ({U}\\setminus {R})(b^{\\prime })$ .",
"In summary: $c^{\\prime }\\rightarrow _{{R}}b^{\\prime }$ , $c^{\\prime }\\,{R}\\, d^{\\prime }$ , $d^{\\prime }\\in \\rightarrow _{{R}}^{-1}\\mathrel \\circ \\mathit {NotRel}(b^{\\prime })$ and $d^{\\prime }\\notin \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(b^{\\prime })$ .",
"All of this implies that $(c^{\\prime },d^{\\prime })\\in X$ .",
"So we have $\\mathit {NotRel}^{\\prime }\\subseteq X$ and thus $\\mathit {NotRel}^{\\prime }=X$ .",
"By contradiction.",
"Let $(c,d)\\in {S}$ such that $(c,d)\\notin {V}$ .",
"This means that $(c,d)\\in \\mathit {NotRel}^{\\prime }$ .",
"From REF ) and the hypothesis ${S}\\subseteq {R}$ there is $b\\in Q$ such that $c\\rightarrow _{{R}} b$ , $d\\notin \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(b)$ and thus $d\\notin \\rightarrow ^{-1}\\mathrel \\circ {R}(b)$ , from Lemma REF .",
"From $c\\rightarrow _{{R}} b$ , thus $c\\rightarrow b$ , and the assumption that ${S}$ is a simulation there is $d^{\\prime }\\in Q$ with $d\\rightarrow d^{\\prime }$ and $b\\mathrel {S}d^{\\prime }$ thus $b\\mathrel {R}d^{\\prime }$ .",
"This contradicts $d\\notin \\rightarrow ^{-1}\\mathrel \\circ {R}(b)$ .",
"Therefore ${S}\\subseteq {V}$ .",
"If this is not the case, there are $b,c,d\\in Q$ such that $c\\mathrel {V}d$ , $c\\rightarrow b$ and $d\\notin \\rightarrow ^{-1}\\mathrel \\circ {R}(b)$ .",
"Since ${V}\\subseteq {R}$ and ${R}$ is a ${U}$ -stable relation there is $d^{\\prime }\\in Q$ such that $d\\rightarrow d^{\\prime }$ and $b\\mathrel {U}d^{\\prime }$ .",
"The case $b\\mathrel {R}d^{\\prime }$ would contradict $d\\notin \\rightarrow ^{-1}\\mathrel \\circ {R}(b)$ .",
"Therefore $d\\in \\rightarrow ^{-1}\\mathrel \\circ \\mathit {NotRel}(b)$ and all the conditions are met for $(c,d)$ belonging in $\\mathit {NotRel}^{\\prime }$ which contradicts $c\\mathrel {V}d$ .",
"Let us show that ${V}$ is both reflexive and transitive.",
"If it is not reflexive, since ${R}$ is reflexive, from REF ) there is $(c,d)$ in $X$ and a state $b$ such that $c\\rightarrow _{{R}} b$ and $d\\notin \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(b)$ and $c=d$ .",
"But this is impossible since ${R}$ is reflexive.",
"Hence, ${V}$ is reflexive.",
"We also prove by contradiction that ${V}$ is transitive.",
"If it is not the case, there are $c,e,d\\in Q$ such that $c\\mathrel {V}e$ , $e\\mathrel {V}d$ but $\\mathrel \\lnot c\\mathrel {V}d$ .",
"Since ${V}\\subseteq {R}$ and ${R}$ is transitive then $c\\mathrel {R}d$ .",
"With $\\mathrel \\lnot c\\mathrel {V}d$ and REF ), there is $b$ such that $c\\rightarrow _{{R}} b$ and $d\\notin \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(b)$ .",
"But from REF ), there is $b^{\\prime }\\in Q$ such that $b\\mathrel {R}b^{\\prime }$ and $e\\rightarrow b^{\\prime }$ .",
"With $e\\mathrel {V}d$ and the same reason, there is $b^{\\prime \\prime }$ such that $b^{\\prime }\\mathrel {R}b^{\\prime \\prime }$ and $d\\rightarrow b^{\\prime \\prime }$ .",
"By transitivity of ${R}$ we get $b\\mathrel {R}b^{\\prime \\prime }$ and thus $d\\in \\rightarrow ^{-1}\\mathrel \\circ {R}(b)$ .",
"With Lemma REF this contradicts $d\\notin \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(b)$ .",
"Hence, ${V}$ is transitive.",
"This is a direct consequence of the two preceding items and the fact that by construction ${V}\\subseteq {R}$ .",
"By hypothesis, ${P}\\subseteq {R}$ .",
"This means that blocks of ${R}$ are made of blocks of ${P}$ .",
"By definition, ${V}$ is obtained by deleting from ${R}$ relations between blocks of ${P}$ .",
"This implies that blocks of ${V}$ are made of blocks of ${P}$ .",
"To proove that a block of ${V}$ is made of a single block of ${P}$ , let us assume, by contradiction, that there are two different blocks, $B_1$ and $B_2$ , of ${P}$ in a block of ${V}$ .",
"We show that ${P}$ is not the coarsest ${R}$ -block-stable equivalence relation.",
"Let ${P}^{\\prime }={P}\\cup B_1\\times B_2\\cup B_2\\times B_1$ .",
"Then ${P}^{\\prime }$ is an equivalence relation strictly coarser than ${P}^{\\prime }$ .",
"Furthermore, since $B_1$ and $B_2$ are blocks of ${V}$ , we get ${P}^{\\prime }\\subseteq {V}$ .",
"With REF ) we get that ${P}^{\\prime }$ is ${R}$ -stable and thus ${R}$ -block-stable with Lemma REF .",
"This contradicts the hypothesis that ${P}$ was the coarsest one.",
"Therefore, blocks of ${V}$ are blocks of ${P}$ .",
"Remark REF ) means that blocks of ${P}$ are sufficiently small to do the refinement step efficiently, as if they were states.",
"REF ) means that these blocks cannot be bigger.",
"REF ) means that we are ready for next split and refinement steps.",
"In what precedes, we have assumed that for the preorder ${R}$ , inside which we want to compute the coarsest simulation, there is another preorder ${U}$ such that condition (REF ) holds.",
"The fifth item of Theorem REF says that if this true at a given iteration (made of a split step and a refinement step) of the algorithm then this is true at the next iteration.",
"For the end of this section we show that we can safely modify the initial preorder such that this is also initially true.",
"This is indeed a simple consequence of the fact that a state with an outgoing transition cannot be simulated by a state with no outgoing transition.",
"Definition 8 Let ${R}_{\\mathrm {init}}$ be a preorder.",
"We define $\\operatorname{InitRefine}({R}_{\\mathrm {init}})$ such that: $\\hfill \\operatorname{InitRefine}({R}_{\\mathrm {init}})\\triangleq {R}_{\\mathrm {init}}\\cap \\lbrace (c,d)\\in Q\\times Q\\,\\big |\\, \\exists c^{\\prime }\\in Q\\,.\\,c\\rightarrow c^{\\prime } \\;\\Rightarrow \\;\\exists d^{\\prime }\\in Q\\,.\\,d\\rightarrow d^{\\prime } \\rbrace \\hfill $ Proposition 9 Let ${R} =\\operatorname{InitRefine}({R}_{\\mathrm {init}})$ with ${R}_{\\mathrm {init}}$ a preorder.",
"Then: ${R}$ is $(Q\\times Q)$ -stable, a simulation ${S}$ included in ${R}_{\\mathrm {init}}$ is also included in ${R}$ .",
"$Q\\times Q$ is trivially a preorder.",
"It remains to show that ${R}$ is also a preorder and that (REF ) is true with ${U}=Q\\times Q$ .",
"Since ${R}_{\\mathrm {init}}$ is a preorder and thus reflexive, ${R}$ is also trivially reflexive.",
"Now, by contradiction, let us suppose that ${R}$ is not transitive.",
"There are three states $c,e,d\\in Q$ such that: $c\\,{R}\\,e\\wedge e\\,{R}\\,d \\wedge \\lnot \\;c\\,{R}\\,d$ .",
"From the fact that ${R}\\subseteq {R}_{\\mathrm {init}}$ and ${R}_{\\mathrm {init}}$ is a preorder, we get $c\\,{R}_{\\mathrm {init}}\\,d$ .",
"With $\\lnot \\; c\\,{R}\\,d$ and the definition of ${R}$ this means that $c$ has a successor while $d$ has not.",
"But the hypothesis that $c$ has a successor and $c\\,{R}\\,e$ implies that $e$ has a successor.",
"With $e\\,{R}\\,d$ we also get that $d$ has also a successor, which contradicts what is written above.",
"Hence, ${R}$ is transitive and thus a preorder.",
"The formula ${R}\\mathrel {\\circ }\\rightarrow ^{-1}\\subseteq \\rightarrow ^{-1}\\circ (Q\\times Q)$ just means that the two hypotheses, $c\\,{R}\\,d$ and $c$ has a successor, imply that $d$ has also a successor.",
"This is exactly the meaning of the second part of the intersection in the definition of ${R}$ .",
"By contradiction, if this is not true there is $(c,d)$ a pair of states which belongs to ${S}$ and ${R}_{\\mathrm {init}}$ but does not belong to ${R}$ .",
"By definition of ${R}$ this means that $c$ has a successor while $d$ has not.",
"But the hypotheses $c\\,{S}\\,d$ , $c$ has a successor and ${S}$ is a simulation imply that $d$ has also a successor which contradicts what is written above.",
"Hence, ${S}$ is also included in ${R}$ .",
"The total relation $Q\\times Q$ will thus play the role of the initial ${U}$ in the algorithm.",
"Remark In [12] there is also a similar preprocessing of the initial partition where states with no output transition are put aside."
],
[
"The Algorithm",
"The approach of the previous section can be applied to several algorithms, with different balances between time complexity and space complexity.",
"It can also be extended to labelled transition systems.",
"In this section the emphasis is on the, theoretically, most efficient in memory of the fastest simulation algorithms of the moment."
],
[
"Counters and Splitter Transitions",
"Let us remember that a partition $P$ and its associated equivalence relation ${P}_P$ (or a equivalence relation ${P}$ and its associated partition $P_{{P}}$ ) denote essentially the same thing.",
"The difference is that for a partition we focus on the set of blocks whereas for an equivalence relation we focus on the relation which relates the elements of a same block.",
"For a first reading, the reader may consider that a partition is an equivalence relation, and vice versa.",
"From a preorder ${R}$ that satisfies (REF ) we will need to split its blocks in order to find its coarsest ${R}$ -block-stable equivalence relation.",
"Then, Theorem REF will be used for a refine step.",
"For all this, we first need the traditional function.",
"Definition 10 Given a partition $P$ and a set of states $Marked$ , the function $(P,Marked)$ returns a partition similar to $P$ , but with the difference that each block $E$ of $P$ such that $E\\cap Marked \\ne \\emptyset $ and $E\\lnot \\subseteq Marked$ is replaced by two blocks: $E_1=E\\cap Marked$ and $E_2=E\\mathrel \\setminus E_1$ .",
"To efficiently perform the split and refine steps we need a set of counters which associates to each representative state of a block, of an equivalence relation, the number of blocks, of that same equivalence relation, it reaches in ${R}(B)$ , for $B$ a block of ${R}$ .",
"Definition 11 Let ${P}$ be an equivalence relation included in a preorder ${R}$ .",
"We assume that for each block $E$ of ${P}$ , a representative state $E.$ has been chosen.",
"Let $E$ be a block of ${P}$ , $B$ be a block of ${R}$ and $B^{\\prime }\\subseteq B$ .",
"We define: $\\hfill _{({P},{R})}(E,B^{\\prime })\\triangleq |\\lbrace E^{\\prime }\\in P_{{P}}\\,\\big |\\,E.\\rightarrow E^{\\prime }\\wedge B^{\\prime }\\mathrel {R} E^{\\prime }\\rbrace | \\hfill $ Proposition 12 Let ${P}$ be an equivalence relation included in a preorder ${R}$ , $E$ be a block of ${P}$ , $B$ be a block of ${R}$ and $B^{\\prime }$ be a non empty subset of $B$ .",
"Then: $\\begin{split}_{({P},{R})}(E,B)=_{({P},{R})}(E,B^{\\prime })\\end{split}$ Thanks to the transitivity of ${R}$ .",
"Following Section , the purpose of these counters is to check in constant time whether a block $E$ of an equivalence relation ${P}$ is included in $\\rightarrow ^{-1}\\mathrel {\\circ }{R}(b)$ for a given state $b$ .",
"But this is correct only if ${P}$ is already ${R}$ -block-stable.",
"If this is not the case, its underlying partition should be split accordingly.",
"We thus introduce the first condition which necessitates a split of the current equivalence relation ${P}$ to approach the coarsest ${R}$ -block-stable equivalence relation.",
"For this, we take advantage of the existence of $_{({P},{R})}$ .",
"Definition 13 Let ${P}$ be an equivalence relation included in a preorder ${R}$ , $E$ be a block of ${P}$ and $B$ be a block of ${R}$ such that $E\\rightarrow B$ and $_{({P},{R})}(E,B)=0$ .",
"The transition $E\\rightarrow B$ is called a $({P},{R})$ -splitter transition of type 1.",
"The intuition is as follows.",
"With a block $E$ of ${P}$ and a block $B$ of ${R}$ , if ${P}$ was ${R}$ -block-stable, with $E\\rightarrow B$ and Lemma REF we would have had $E\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ .",
"But $_{({P},{R})}(E,B)=0$ denies this.",
"So we have to split $P_{P}$ .",
"Lemma 14 Let $E\\rightarrow B$ be a $({P},{R})$ -splitter transition of type 1.",
"Let $P^{\\prime }=(P_{{P}},\\rightarrow ^{-1}\\mathrel \\circ {R}(B))$ .",
"Then ${P}_{P^{\\prime }}$ is strictly included in ${P}$ and contains all ${R}$ -block-stable equivalence relations included in ${P}$ .",
"To prove the strict inclusion, let us show that $E$ is split.",
"From $E\\rightarrow B$ there is $e\\in E$ such that $e\\rightarrow B$ and thus $e\\in \\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ since ${R}$ is reflexive.",
"Furthermore, by definition, $_{({P},{R})}(E,B)=0$ means that $E.\\notin \\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ .",
"This implies that $e$ and $E.$ do not belong to the same block of $P^{\\prime }$ .",
"Thus, at least $E$ has been split.",
"Now, let ${P}^{\\prime \\prime }$ be a ${R}$ -block-stable equivalence relation included in ${P}$ .",
"If ${P}^{\\prime \\prime }$ is not included in ${P}^{\\prime }$ there are two states, $h_1$ and $h_2$ , from a block $H$ of ${P}^{\\prime \\prime }$ such that $h_1\\in \\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ and $h_2\\notin \\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ .",
"But, by definition, if ${P}^{\\prime \\prime }$ is ${R}$ -block-stable, $h_1\\in \\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ implies $h_2\\in \\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ which leads to a contradiction.",
"Therefore ${P}^{\\prime \\prime }$ is included in ${P}^{\\prime }$ .",
"Saying that ${P}_{P^{\\prime }}$ is strictly included in ${P}$ means that at least one block of ${P}$ , here $E$ , has been split to obtain $P^{\\prime }$ .",
"Lemma 15 Let ${P}$ be an equivalence relation included in a preorder ${R}$ such that there is no $({P},{R})$ -splitter transition of type 1.",
"Let $E$ be a block of ${P}$ and $B$ be a block of ${R}$ such that $E\\rightarrow B$ .",
"Then, $E.\\in \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(B)$ .",
"Let $E\\rightarrow B$ be a transition with $E$ and $B$ satisfying the hypotheses of the lemma.",
"Since there is no $({P},{R})$ -splitter transition of type 1, $_{({P},{R})}(E,B)\\ne 0$ .",
"Therefore, $E.\\in \\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ and thus $E.\\in \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}(B)$ by Lemma REF .",
"Now, for $E$ to be really a representative, we need the following implication: $E.\\rightarrow B \\Rightarrow E\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ .",
"But to check this property effectively, taking advantage of the counters, we need a stronger property equivalent with the one, (REF ), defining block-stability.",
"Lemma 16 Let ${P}$ be an equivalence relation included in a preorder ${R}$ .",
"Then (REF ) is equivalent with: ${P}\\mathrel \\circ \\rightarrow _{{R}}^{-1}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ [\\cdot ]_{{R}}$ With Lemma REF it suffices to show the equivalence between (REF ) and (REF ): (REF ) $\\Rightarrow $ (REF ).",
"Let $(c^{\\prime },d)\\in {P}\\mathrel \\circ \\rightarrow _{{R}}^{-1}$ .",
"There is a state $c$ such that $(c,d)\\in {P}$ and $c\\rightarrow _{{R}}c^{\\prime }$ .",
"With (REF ) there is a state $d^{\\prime }$ such that $c^{\\prime }\\,{R}\\,d^{\\prime }$ and $d\\rightarrow d^{\\prime }$ .",
"With $(d,c)\\in {P}$ , since ${P}$ is symmetric, and (REF ) again there is a state $c^{\\prime \\prime }$ such that $d^{\\prime }\\,{R}\\,c^{\\prime \\prime }$ and $c\\rightarrow c^{\\prime \\prime }$ .",
"But $c\\rightarrow c^{\\prime }$ being a ${R}$ -maximal transition implies that $c^{\\prime \\prime }\\in [c^{\\prime }]_{{R}}$ .",
"So we have $c^{\\prime \\prime }\\,{R}\\,c^{\\prime }$ , $c^{\\prime }\\,{R}\\,d^{\\prime }$ , $d^{\\prime }\\,{R}\\,c^{\\prime \\prime }$ and thus $d^{\\prime }\\in [c^{\\prime }]_{{R}}$ .",
"This means that $d\\in \\rightarrow ^{-1}\\mathrel \\circ [\\cdot ]_{{R}}(c^{\\prime })$ and thus $(c^{\\prime },d)\\in \\rightarrow ^{-1}\\mathrel \\circ [\\cdot ]_{{R}}$ .",
"Therefore ${P}\\mathrel \\circ \\rightarrow _{{R}}^{-1}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ [\\cdot ]_{{R}}$ .",
"(REF ) $\\Rightarrow $ (REF ).",
"With Lemma REF we have ${P}\\mathrel \\circ \\rightarrow ^{-1}\\,\\subseteq \\,{P}\\mathrel \\circ \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}$ .",
"With (REF ) we have ${P}\\mathrel \\circ \\rightarrow _{{R}}^{-1}\\mathrel \\circ {R}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ [\\cdot ]_{{R}}\\mathrel \\circ {R}$ .",
"But ${R}$ being a preorder we have $[\\cdot ]_{{R}}\\mathrel \\circ {R}\\,\\subseteq \\,{R}$ .",
"With all of this: ${P}\\mathrel \\circ \\rightarrow _{{R}}^{-1}\\,\\subseteq \\,\\rightarrow ^{-1}\\mathrel \\circ {R}$ .",
"Definition 17 Let ${P}$ be an equivalence relation included in a preorder ${R}$ .",
"Let $E$ be a block of ${P}$ and $B$ be a block of ${R}$ such that $E.\\rightarrow B$ , $_{({P},{R})}(E,B)= |\\lbrace [b]_{{P}}\\subseteq B\\,\\big |\\,E.\\rightarrow b\\rbrace |$ and $E\\nsubseteq \\rightarrow ^{-1}(B)$ .",
"The transition $E\\rightarrow B$ is called a $({P},{R})$ -splitter transition of type 2.",
"Remark The conditions in Definition REF are inspired from those used in [13] for its split step.",
"The intuition is as follows.",
"If $E.\\rightarrow B$ and the condition on the counter is true (all transitions from $E.$ that reach states greater, relatively to ${R}$ , than $B$ actually have their destination states in $B$ ), this means that the transition $E.\\rightarrow B$ is maximal.",
"With Lemma REF , if ${P}$ is ${R}$ -block-stable this should imply $E\\subseteq \\rightarrow ^{-1}(B)$ .",
"Since this is not the case, ${P}$ must be split.",
"Lemma 18 Let ${P}$ be an equivalence relation included in a preorder ${R}$ such that there is no $({P},{R})$ -splitter transition of type 1 and let $E\\rightarrow B$ be a splitter transition of type 2.",
"Let $P^{\\prime }=(P_{{P}},E\\cap \\rightarrow ^{-1}(B))$ .",
"Then ${P}_{P^{\\prime }}$ is strictly included in ${P}$ and contains all ${R}$ -block-stable equivalence relations included in ${P}$ .",
"Figure: Proof of second statement of Lemma .To prove the strict inclusion, let us show that $E$ is split.",
"From $E\\rightarrow B$ being a $({P},{R})$ -splitter transition of type 2 we have $|\\lbrace [b]_{{P}}\\subseteq B\\,\\big |\\,E.\\rightarrow b\\rbrace |\\ne 0$ and $E\\nsubseteq \\rightarrow ^{-1}(B)$ .",
"The first property implies that $ E.\\in {\\rightarrow ^{-1}(B)}$ and the second one implies the existence of a state $e\\in E$ which does not belong to $\\rightarrow ^{-1}(B)$ .",
"Therefore $E$ has been split.",
"The second statement is proved by contradiction.",
"Let ${P}^{\\prime \\prime }$ be a ${R}$ -block-stable equivalence relation included in ${P}$ .",
"Consider Figure REF .",
"If ${P}^{\\prime \\prime }$ is not included in ${P}^{\\prime }$ there are two states, $h_1$ and $h_2$ , from a block $H$ of ${P}^{\\prime \\prime }$ such that $h_1\\in \\rightarrow ^{-1}(B)$ and $h_2\\notin \\rightarrow ^{-1}(B)$ .",
"With Lemma REF there is a block $G_1$ of ${R}$ such that: $B\\mathrel {R}G_1$ and $h_1\\rightarrow _{{R}}G_1$ .",
"Since ${P}^{\\prime \\prime }$ is ${R}$ -block-stable this implies $h_2\\rightarrow G_1$ .",
"With Lemma REF there is a block $G_2$ of ${R}$ such that $E.\\rightarrow _{{R}} G_2$ and $G_1\\mathrel {R} G_2$ .",
"But the condition $_{({P},{R})}(E,B)=|\\lbrace [b]_{{P}}\\subseteq B\\,\\big |\\,E.\\rightarrow b\\rbrace |$ implies that the transition from $E.$ to $B$ is maximal which implies that $G_2=B=G_1$ .",
"So we have $h_2\\in \\rightarrow ^{-1}(B)$ which contradicts an above assumption.",
"Therefore ${P}^{\\prime \\prime }$ is included in ${P}^{\\prime }$ .",
"Theorem 19 Let ${P}$ be an equivalence relation included in a preorder ${R}$ such that there is no $({P},{R})$ -splitter transition of type 1 or $({P},{R})$ -splitter transition of type 2.",
"Then ${P}$ is ${R}$ -block-stable.",
"Figure: Proof of Theorem .Consider Figure REF .",
"Let us consider a transition $E\\rightarrow _{{R}} B$ with $E$ a block of ${P}$ and $B$ a block of ${R}$ .",
"By definition there is a state $e\\in E$ such that $e\\rightarrow _{{R}} B$ and by Lemma REF there is a block $G$ such that $B\\mathrel {R} G$ and $E.\\rightarrow _{{R}} G$ .",
"From ${E.\\rightarrow _{{R}}G}$ it is easy to show that $_{({P},{R})}(E,G)\\ne 0$ and $_{({P},{R})}(E,G)=|\\lbrace [g]_{{P}}\\subseteq G\\,\\big |\\,E.\\rightarrow g\\rbrace |$ since the transition $E.\\rightarrow G$ is maximal.",
"But since there is neither $({P},{R})$ -splitter transition of type 1 nor $({P},{R})$ -splitter transition of type 2 then $E\\subseteq \\rightarrow ^{-1}(G)$ and thus $e\\rightarrow G$ .",
"With $e\\rightarrow _{{R}} B$ and $B\\mathrel {R} G$ we necessarily get $B=G$ and thus $E\\subseteq \\rightarrow ^{-1}(B)$ .",
"So we have $E\\rightarrow _{{R}} B$ implies $E\\subseteq \\rightarrow ^{-1}(B)$ .",
"This is equivalent of saying that (REF ) is true.",
"With Lemma REF this implies that ${P}$ is ${R}$ -block-stable.",
"Therefore, in the algorithm, before a refine step on ${R}$ using Theorem REF , we will start from the partition $P_{{R}}$ and split it in conformity with Lemma REF and Lemma REF .",
"By doing so, we will obtain the coarsest ${R}$ -block-stable equivalence relation.",
"The next proposition shows where to search splitter transitions: those who ends in blocks $B$ of ${R}$ such that $\\mathit {NotRel}(B)$ is not empty.",
"Proposition 20 Let ${U}$ be a preorder, ${R}$ be a ${U}$ -stable preorder, ${P}$ be an equivalence relation included in ${R}$ and let $\\mathit {NotRel}={U}\\mathrel \\setminus {R}$ .",
"Then If $E\\rightarrow B$ is a $({P},{R})$ -splitter transition of type 1 then $\\mathit {NotRel}(B)\\ne \\emptyset $ .",
"Under the absence of $({P},{R})$ -splitter transition of type 1, if $E\\rightarrow B$ is a $({P},{R})$ -splitter transition of type 2 then $\\mathit {NotRel}(B)\\ne \\emptyset $ .",
"Figure: Proof of second item of Proposition .",
"Let $E\\rightarrow B$ be a $({P},{R})$ -splitter transition of type 1.",
"By definition there is $e\\in E$ such that $e\\rightarrow B$ .",
"Since $E$ is a block of ${P}$ we have $e\\mathrel {P} E.$ and thus $e\\mathrel {R} E.$ .",
"With the hypothesis that ${R}$ is ${U}$ -stable we get $E.\\in \\rightarrow ^{-1}\\mathrel \\circ {U}(B)$ .",
"But the hypothesis that $E\\rightarrow B$ is a $({P},{R})$ -splitter transition of type 1 implies $E.\\notin \\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ .",
"From these two last constraints on $E.$ we get $E.\\in \\rightarrow ^{-1}\\mathrel \\circ ({U}\\mathrel \\setminus {R})(B)$ and thus $\\mathit {NotRel}(B)\\ne \\emptyset $ .",
"Consider Figure REF .",
"Let $E\\rightarrow B$ be a $({P},{R})$ -splitter transition of type 2.",
"By definition we get $E.\\rightarrow _{{R}}B$ and $E_2=E\\mathrel \\setminus (\\rightarrow ^{-1}(B))$ is not empty.",
"Let $e\\in E_2$ , with a similar argument than the first item, we get $e\\in \\rightarrow ^{-1}\\mathrel \\circ {U}(B)$ .",
"By contradiction, let us assume $e\\in \\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ .",
"There is $G_1$ a block of ${R}$ such that $B\\mathrel {R} G_1$ and $e\\rightarrow G_1$ .",
"From Lemma REF there is $G_2$ a block of ${R}$ such that $E.\\rightarrow G_2$ and $G_1\\mathrel {R} G_2$ , and thus $B\\mathrel {R} G_2$ .",
"Since the transition from $E.$ to $B$ is maximal this implies that $B=G_1=G_2$ which contradicts $e\\notin \\rightarrow ^{-1}(B)$ .",
"Therefore $e\\notin \\rightarrow ^{-1}\\mathrel \\circ {R}(B)$ .",
"With $e\\in \\rightarrow ^{-1}\\mathrel \\circ {U}(B)$ we get $e\\in \\rightarrow ^{-1}\\mathrel \\circ ({U}\\mathrel \\setminus {R})(B)$ and thus $\\mathit {NotRel}(B)\\ne \\emptyset $ .",
"We have now everything to propose an efficient algorithm."
],
[
"Data Structures and Space Complexity",
"Remark Since the final partition $P_{\\mathrm {sim}}$ is obtained after several splits of the initial partition $P_{\\mathrm {init}}$ we have $|P|\\le |P_{\\mathrm {sim}}|$ with $P$ the current partition at a given step of the algorithm.",
"The current relation ${R}$ is represented in the algorithm by a partition-relation pair $(P,Rel)$ .",
"The data structure used to represent a partition is traditionally a (possibly) doubly linked list of blocks, themselves represented by a doubly linked list of states.",
"But in practice, each node of a list contains a reference, to the next node of the list.",
"The size (typically 64 bits nowadays) of these references is static and does not depend on the size of the list.",
"We therefore prefer the use of arrays because we can control the size of the slots and manipulate arrays is faster than manipulate lists.",
"The idea, see [17] and [4] for more details, is to identify a state with its index in $Q$ and to distribute these indexes in an array, let us name it $T$ , such that states belonging to the same block are in consecutive slots in $T$ .",
"A block could thus be represented by the indexes of its first and last elements in $T$ , the two indexes defining a contiguous subarray.",
"If a block is split, the two subblocks form two contiguous subarrays of that subarray.",
"By playing with these arrays (other arrays are needed, like the one giving the position in $T$ of a state) we obtain a representation of a partition which allows splitting (some elements of a block are removed from their original block and form a new block) and scanning of a block in linear time.",
"However, as seen in the previous sections, we need two generations of blocks at the same time: the first one corresponds to blocks of ${R}$ and the second one corresponds to blocks of the next generation, ${V}$ , of this preorder.",
"Hence, we need an intermediate between blocks and their corresponding states: nodes.",
"A node corresponds to a block or to an ancestor of a block in the family tree of the different generations of blocks issued from the split steps of the algorithm.",
"To simplify the writing, we associate in the present paper, a node to the set of its corresponding states.",
"As an example, consider a block $B$ which consists of the following three states $\\lbrace q_1,q_2,q_3\\rbrace $ .",
"In reality, we will associate $B$ to a node $N=\\lbrace q_1,q_2,q_3\\rbrace $ .",
"In this way, if $B$ is split in $B_1$ and $B_2$ , corresponding respectively to $\\lbrace q_1,q_2\\rbrace $ and $\\lbrace q_3\\rbrace $ , we create two new nodes $N_1=\\lbrace q_1,q_2\\rbrace $ and $N_2=\\lbrace q_3\\rbrace $ and we associate $B_1$ to $N_1$ and $B_2$ to $N_2$ .",
"By doing so, $N$ remains bounds to the set $\\lbrace q_1,q_2,q_3\\rbrace $ .",
"To represent a node we just need to keep in memory the index of its first element and the index of its last element in the array $T$ (see the previous paragraph).",
"When a block which corresponds to a node is split, the corresponding states change their places in $T$ but keep in the same subarray.",
"Let us note that when a block is split, it is necessarily in two parts.",
"Therefore, the number of nodes is at most twice the number of blocks and the bit space needed to represent the partition and the nodes is in $O(|Q|.\\log (|Q|))$ since there is less blocks than states.",
"To be more precise about the relations between states, blocks and nodes: at any time of the algorithm, the index of a state $q$ is associated to the index of its block $q.$ , the index of a block $E$ is associated to the index of its node $E.$ and the index of a node is associated to the states it contains (via two indexes of the array $T$ ).",
"A node which is not linked by a block is an ancestor of, at least two, blocks.",
"By the data structure chosen to represent the partition it is easy to see that given a node $N$ we can scan in linear time the states it contains (this corresponds to the scan of a contiguous subarray) and the blocks it contains (by a similar process).",
"The function $(N)$ which arbitrarily choose one block whose set of elements is included in those of $N$ is executed in constant time (we choose $e.$ , with $e$ the first element of $N$ ).",
"Similarly, the function ${E}$ which returns a state of a block $E$ , used to defined $E.$ a representative of $E$ , is also executed in constant time (we choose the first element in $E.$ ).",
"The relation $Rel$ is distributed on the blocks.",
"To each block $C$ we associate an array of booleans, $C.$ , such that $(C,D)\\in Rel$ , what we note $D\\in C.$ , iff the boolean at the index of the block $D$ in $C.$ is true.",
"These arrays are resizable arrays whose capacities are doubled as needed.",
"Therefore the classical operations on these arrays, like get and set, take constant amortized time.",
"We use this type of array wherever necessary.",
"The bit size needed to represent $Rel$ is therefore in $O(|P_{\\mathrm {sim}}|^2)$ .",
"The relation ${U}$ that appears in the previous sections is not directly represented.",
"We use instead the equality ${U} = {R}\\cup \\mathit {NotRel}$ and represent $\\mathit {NotRel}$ .",
"Since ${U}$ is a coarser preorder than ${R}$ , for a given node $B$ which represents a block of ${R}$ , the set $\\mathit {NotRel}(B)$ is represented in the algorithm by $B.$ a set of nodes (encoded by a resizable array of the indexes of the corresponding nodes) which represent blocks of ${R}$ .",
"As explained earlier, we have to use nodes instead of blocks because nodes never change whereas blocks can be split afterwards.",
"The bit space representation of $\\mathit {NotRel}$ is thus in $O(|P_{\\mathrm {sim}}|^2.\\log (|P_{\\mathrm {sim}}|))$ .",
"Remember, the number of nodes is linear in the maximal number of blocks: $|P_{\\mathrm {sim}}|$ .",
"In Section REF we introduced a counter, $_{({P},{R})}(E,B)$ , for each pair made of a block $E$ of the current equivalence relation ${P}$ represented in the algorithm by the current partition $P$ , and a block $B$ of the current relation ${R}$ represented in the algorithm by $(P,Rel)$ .",
"As seen in Proposition REF , for any subblock $B^{\\prime }\\in P_{{P}}$ of $B$ we have $_{({P},{R})}(E,B) =_{({P},{R})}(E,B^{\\prime })$ .",
"Therefore, we can limit these counters to any pair of blocks of the current partition $P$ in the algorithm.",
"Such a counter counts a number of blocks.",
"This means that the total bit size of these counters is in $O(|P_{\\mathrm {sim}}|^2.\\log (|P_{\\mathrm {sim}}|))$ .",
"In practice, we associate to each block $B^{\\prime }$ a resizable array, $B^{\\prime }.$ , of $|P|$ elements such that $B^{\\prime }.",
"(E)=_{({P},{R})}(E,B^{\\prime })$ .",
"At several places in the algorithm we use a data structure to manage a set of indexed elements.",
"This is the case for $Touched$ , $Touched^{\\prime }$ , $Marked$ , $RefinerNodes$ , $RefinerNodes^{\\prime }$ , $PreB^{\\prime }$ , $Remove$ , $PreE$ and $PreE^{\\prime }$ .",
"Such a set is implemented by a resizable boolean array, to know in constant time whether a given element belongs to the set, and another resizable array to store the indexes of the elements which belongs to the set.",
"This last array serves to scan in linear time the elements of the set or to emptied the set in linear time of the number of the elements in the set.",
"So the operations, add an element in the set and test whether an element belongs to the set are done in constant time or amortized constant time.",
"Also, scanning the elements of the set and emptying the set are executed in linear time of the size of the set.",
"We use a finite number of these sets for states, blocks or nodes.",
"The overall bit space used for them is therefore in $O(|Q|.\\log (|Q|))$ .",
"The other variables used in the algorithm, some booleans and a counter, $E.$ , associated to each block $E$ in Function REF are manipulated by constant time operations and need a bit space in $O(|Q|.\\log (|Q|))$ since $|P|$ and the number of nodes are both in $O(|Q|)$ .",
"From all of this, we derive the following theorem.",
"Theorem 21 The overall bit space used by the presented simulation algorithm is in $O(|P_{\\mathrm {sim}}|^2.\\log (|P_{\\mathrm {sim}}|) + |Q|.\\log (|Q|))$ .",
"Remark We assume one can iterate through the transition relation $\\rightarrow $ in linear time of its size.",
"From each state $q$ we also assume one can iterate through the set $\\rightarrow ^{-1}(q)$ in linear time of its size.",
"It is a tradition in most articles dealing with simulation to not count the space used to represent the transition relation since it is considered as an input data.",
"If it was to be counted it would cost $O(|\\rightarrow |.\\log (|Q|))$ bits."
],
[
"Procedures and Time Complexity",
"In this section we analyze the different functions of the algorithm and give their overall time complexities.",
"The reader should remember that, in the algorithm, a block is just an index and the set of states corresponding to a block $E\\in P$ is $E.$ ."
],
[
"Function ",
"[!t] Sim($Q,\\rightarrow ,P_{\\mathrm {init}}, R_{\\mathrm {init}}$ ) $RefinerNodes$ : the set of nodes $B$ corresponding to blocks of ${R}$ such that $B.\\ne \\emptyset $ $RefinerNodes:=\\emptyset $ $P:=$REF ($Q,\\rightarrow ,P_{\\mathrm {init}},R_{\\mathrm {init}}, RefinerNodes$ ) $RefinerNodes\\ne \\emptyset $ Sim:3 REF ($P, RefinerNodes$ ) Sim:4 REF ($P, RefinerNodes$ ) REF ($P, RefinerNodes$ ) REF ($RefinerNodes$ ) $P_{\\mathrm {sim}} := P$ ; $R_{\\mathrm {sim}}:= \\lbrace (C,D)\\in P\\times P\\,\\big |\\,D\\in C.\\rbrace $ $(P_{\\mathrm {sim}}$ , $R_{\\mathrm {sim}})$ This is the main function of the algorithm.",
"It takes as input a transition system $(Q,\\rightarrow )$ and an initial preorder, ${R}_{\\mathrm {init}}$ , represented by the partition-relation pair $(P_{\\mathrm {init}},R_{\\mathrm {init}})$ .",
"Let us define the two following relations: ${R}&\\triangleq \\bigcup _{\\lbrace (E,E^{\\prime })\\in P^2\\,\\big |\\,E^{\\prime }\\in E.\\rbrace }E.\\times E^{\\prime }.\\\\\\mathit {NotRel}&\\triangleq \\bigcup _{B\\;\\in \\; RefinerNodes} B\\times (\\cup B.",
")$ Let ${R}_0=Q\\times Q$ and ${R}_i$ (resp.",
"$\\mathit {NotRel}_i$ ) be the value of ${R}$ (resp.",
"$\\mathit {NotRel}$ ) at the $i^{\\text{th}}$ iteration of the while loop at line of Function REF .",
"We will show (in the analysis of Function REF for the base case and procedures REF and REF for the inductive step) that, at this line , we maintain the five following properties at the $i^{\\text{th}}$ iteration of the while loop: ${R}_{i}\\text{ is } {R}_{i-1}\\text{-stable}$ $\\text{A simulation included in }{R}_{i-1}\\text{ is included in }{R}_{i}$ $ \\mathit {NotRel}_{i}={R}_{i-1}\\mathrel \\setminus {R}_{i} \\text{ and thus }{R}_{i-1} = {R}_{i} \\cup \\mathit {NotRel}_{i}$ $\\hfill \\text{$RefinerNodes$ is the set of nodes $B$ corresponding to}\\hfill \\\\\\hfill \\text{blocks of ${R}_i$ such that $B.\\ne \\emptyset $}\\hfill $ $\\hfill \\forall E,B^{\\prime }\\in P\\;.\\; B^{\\prime }.",
"{E} =|\\lbrace E^{\\prime }\\in P\\,\\big |\\,E.\\rightarrow E^{\\prime }.\\wedge B^{\\prime }.\\times E^{\\prime }.\\subseteq \\mathrel {{R}_{i-1}} \\rbrace |\\hfill $ From (REF ), and thus ${R}_{i}\\subseteq {R}_{i-1}$ , (REF ), (REF ) and the condition of the while loop we get that $({R}_i)_{i\\ge 0}$ is a strictly decreasing sequence of relations.",
"Since the underlying set of states is finite, this sequence reaches a limit in a finite number of iterations.",
"Furthermore, if this limit is reached at the $k^{\\text{th}}$ iteration, then, from (REF ) and the condition of the while loop, we have $\\mathit {NotRel}_{k+1}=\\emptyset $ and from (REF ) and (REF ) we obtain that ${R}_{k}={R}_{k+1}$ and ${R}_{k+1}$ is ${R}_{k}$ -stable.",
"Which means that ${R}_{k}$ is a simulation.",
"From (REF ) and the fact, to be shown in the analysis of function Init, that all simulation included in ${R}_{\\mathrm {init}}$ is also included in ${R}_{1}$ , we deduce that ${R}_{k}$ contains all simulation included in ${R}_{\\mathrm {init}}$ .",
"Therefore, Function REF returns a partition-relation pair that corresponds to ${R}_{\\mathrm {sim}}$ , the coarsest simulation included in ${R}_{\\mathrm {init}}$ .",
"The fact that $({R}_i)_{i\\ge 0}$ is a strictly decreasing sequence of relations and (REF ) imply the following lemma that will be used as a key argument to analyze the time complexity of the algorithm.",
"Lemma 22 Two states, and thus two blocks or two nodes, are related by $\\mathit {NotRel}$ in at most one iteration of the while loop in function REF ."
],
[
"Procedure ",
"[!t] SimUpdateData($P, RefinerNodes$ ) $PreE^{\\prime } := \\emptyset $ $B\\in RefinerNodes$ $B^{\\prime }={B}$ At this stage, $B^{\\prime }$ is the only block of $B$ $E^{\\prime }\\in P \\,\\big |\\,E^{\\prime }.\\subseteq \\cup B.$ $e\\in \\rightarrow ^{-1}(E^{\\prime }.",
")$ SimUpdateData:5 $E := e.$ $e=E.$ $PreE^{\\prime } := PreE^{\\prime } \\cup \\lbrace E\\rbrace $ $E \\in PreE^{\\prime }$ $B^{\\prime }.",
"(E)--$ $PreE^{\\prime } := \\emptyset $ Assuming (REF ) is true, the role of this procedure is to render the following formula true after line of Function REF during the $i^{\\text{th}}$ iteration of the while loop.",
"In this way, the counters are made consistent with (REF ): $\\hfill \\forall E,B^{\\prime }\\in P\\;.\\; B^{\\prime }.",
"{E} =|\\lbrace E^{\\prime }\\in P\\,\\big |\\,E.\\rightarrow E^{\\prime }.\\wedge B^{\\prime }.\\times E^{\\prime }.\\subseteq \\mathrel {{R}_i} \\rbrace | \\hfill $ From (), (REF ), (REF ) and (REF ) we just have, for each node $B$ in $RefinerNodes$ , to scan the blocks in $\\cup B.$ in order to identify their predecessor blocks.",
"The corresponding counters are then decreased.",
"The lines which are the most executed are those in the loop starting at line .",
"They are executed once for each pair of $(e\\rightarrow e^{\\prime },B)$ with $e^{\\prime }\\in \\cup B.$ .",
"But from Lemma REF such a pair can be considered only once during the life time of the algorithm and thus the overall time complexity of this procedure is in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ ."
],
[
"Procedure ",
"[!t] Split($P, Marked$ ) $Touched := \\emptyset $ $r\\in Marked$ $Touched := Touched \\cup \\lbrace r.\\rbrace $ $C\\in Touched \\,\\big |\\,C.\\nsubseteq Marked$ $D:=$ ; $P := P \\cup \\lbrace D\\rbrace $ $D.",
":= C.\\cap Marked$ $q\\in D.$ $q.",
":= D$ The subblock which is disjoint from $Marked$ keeps the identity of $C$ .",
"$C.",
":= C.\\mathrel \\setminus D.$ REF ($P,C,D$ ) $Touched := \\emptyset $ This procedure corresponds to Definition REF .",
"The differences are that Procedure REF transforms the current partition (it is not a function) and each time a block is split, Procedure REF is called to update the data structures (mainly and ).",
"Apart from the call of REF , which we discuss right after, it is known that, with a correct implementation found in most articles from the bibliography of the present paper, a call of REF is done in $O(|Marked|)$ -time.",
"Therefore, we only give a high level presentation of the procedure."
],
[
"Procedure ",
"[!t] SplitUpdateData($P,C,D$ ) $C$ keeps the identity of the parent block ; $D$ is the new block $D.",
":= 0$ Update of the 's $D.",
":= {C.}$ SplitUpdateData:2 $E\\in P\\,\\big |\\,C\\in E.$ SplitUpdateData:3 $E.",
":= E.\\cup \\lbrace D\\rbrace $ Update of the 's $D.",
":= C.$ SplitUpdateData:4 $D.",
":= {C.}$ SplitUpdateData:5 $C..= C$ SplitUpdateData:6 $X := D$ $B^{\\prime }\\in P$$B^{\\prime }.",
"{D} := B^{\\prime }.",
"{C}$ $X := C$ Update of the 's from predecessors of both $C$ and $D$ $Touched := \\emptyset $ ; $Touched^{\\prime } := \\emptyset $ SplitUpdateData:12 $e\\in \\rightarrow ^{-1}(C.)\\,\\big |\\,e = e..$ $Touched := Touched \\cup \\lbrace e.\\rbrace $ ; $e\\in \\rightarrow ^{-1}(D.)\\,\\big |\\,{e = e..}\\wedge {e.\\in Touched} \\wedge e.\\notin Touched^{\\prime }$ SplitUpdateData:15 $Touched^{\\prime } := Touched^{\\prime } \\cup \\lbrace e.\\rbrace $ SplitUpdateData:16; $E\\in Touched^{\\prime }$ SplitUpdateData:17 $B^{\\prime }\\in P\\,\\big |\\,C\\in B^{\\prime }.$ SplitUpdateData:18$B^{\\prime }.",
"{E}++ $ SplitUpdateData:19 $Touched := \\emptyset $ ; $Touched^{\\prime } :=\\emptyset $ SplitUpdateData:20 Compute the 's from $X$ ($C$ or $D$ ) which did not inherited the old $C.$ .",
"$X.",
":= {X}$ SplitUpdateData:21 $B^{\\prime }\\in P$ $B^{\\prime }.",
"{X} := 0$ $ q \\rightarrow q^{\\prime } \\,\\big |\\,q = X.$ $Touched := Touched \\cup \\lbrace q^{\\prime }.\\rbrace $ $E^{\\prime }\\in Touched$ SplitUpdateData:25 $B^{\\prime }\\in P\\,\\big |\\,E^{\\prime }\\in B^{\\prime }.$$B^{\\prime }.",
"{X}++$ ;SplitUpdateData:27 $Touched := \\emptyset $ SplitUpdateData:28 $C..:=\\emptyset $ ; $C..^{\\prime }:=\\emptyset $ $D..:=\\emptyset $ ; $D..^{\\prime }:=\\emptyset $ The purpose of this procedure is to maintain the data structures coherent after a split of a block $C$ in two subblocks, the new $C$ and a new $D$ .",
"Since this procedure only modifies the $$ 's and $$ 's, we will only look at (REF ) and (REF ).",
"The 's are updated at lines and .",
"Let ${R}^{\\prime }$ be the value of ${R}$ before the split, let ${R}^{\\prime \\prime }$ be its value after the split and let $N$ be the node associated with the block $C$ before it was split.",
"Before the split, we have $N=C.$ and after the split we have $N=C.\\cup D.$ .",
"With line , we have, for each block $E\\in P$ , the equivalence $N\\times E.\\subseteq {R}^{\\prime }\\Leftrightarrow (C.\\cup D.)\\times E.\\subseteq {R}^{\\prime \\prime }$ and with line we have the equivalence $E.\\times N\\subseteq {R}^{\\prime }\\Leftrightarrow E.\\times (C.\\cup D.)\\subseteq {R}^{\\prime \\prime }$ .",
"And thus ${R}$ has not changed since the other products $E.\\times E^{\\prime }.$ in its definition have not changed.",
"For one call of REF these two lines are executed in $O(|P|)$ -time.",
"Since REF is called only when a block is split and since there is, at the end, at most $|P_{\\mathrm {sim}}|$ blocks.",
"The overall time complexity of these two lines is in $O(|P_{\\mathrm {sim}}|^2)$ .",
"The 's are updated in the other lines of the procedure.",
"In (REF ), the split can involve three blocks: $B^{\\prime }$ , $E$ or $E^{\\prime }$ .",
"Let us remember that ${R}$ is not changed during this procedure.",
"Line treats the case where the split block is a $B^{\\prime }$ .",
"Since ${R}$ is a preorder, after lines and , we have $C\\in D.$ and $D\\in C.$ .",
"It is therefore normal that for any $E\\in P$ we have $D.",
"{E} = C.{E}$ since ${R}$ is a preorder.",
"The overall time complexity of line is thus in $O(|P_{\\mathrm {sim}}|^2)$ .",
"If in (REF ) the split block is $E$ , the test at line determines the block $X$ among the new $C$ and $D$ that did not inherit $E.$ (which is $C.$ at this line since $C$ keeps the identity of the parent block, the old $C$ and thus $E$ ).",
"This means that we will have to initialise $B^{\\prime }.",
"{X}$ for all $B^{\\prime }\\in P$ .",
"This is done after at lines to .",
"For now, if $D$ has inherited $E.$ we do $B^{\\prime }.",
"{D} :=B^{\\prime }.",
"{E}$ for all $B^{\\prime }\\in P$ .",
"Remember, at this stage, we have $B^{\\prime }.",
"{E} =B^{\\prime }.",
"{C}$ .",
"Lines to treat the case where the split block is $E^{\\prime }$ .",
"We thus have $E^{\\prime }.=C.\\cup D.$ .",
"There is three alternatives for a given block $E$ : either $E.\\rightarrow C.$ , or $E.\\rightarrow D.$ , or both.",
"For the two first alternatives $B^{\\prime }.",
"{E}$ does not change.",
"But for the third one, we have to increment this count by one.",
"Apart for lines to the overall time complexity of these lines is in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ .",
"Remember, REF is called only when a block is split and this occurs at most $|P_{\\mathrm {sim}}|$ times.",
"To correctly analyze the time complexity of lines to , for a state $e$ let us first define $\\rightarrow (e)_P\\triangleq \\lbrace E^{\\prime }\\in P\\,\\big |\\,e\\rightarrow E^{\\prime }\\rbrace $ .",
"This set $\\rightarrow (e)_P$ is a partition of $\\rightarrow (e)$ , the set of the successors of $e$ .",
"Then, each time a state $e$ is involved at line this means that a block $E^{\\prime }$ in $\\rightarrow (e)_P$ has been split in $C$ and in $D$ .",
"For a given state $e$ there can be at most $|\\rightarrow (e)|$ splits of $\\rightarrow (e)_P$ .",
"Hence, the sum of the sizes of $Touched^{\\prime }$ for all executions of REF is in $O(|\\rightarrow |)$ .",
"Furthermore, for one execution of this procedure, the time complexity of lines – is in $O(|P_{\\mathrm {sim}}|)$ .",
"Therefore, the overall time complexity of lines to is in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ .",
"Lines to treat the case where the split block is $E$ .",
"The block $E$ has been split in $C$ and $D$ .",
"One of them contains $E.rep$ .",
"It is therefore not necessary to recompute the counters associated with it (although these counters have been possibly updated at lines to ).",
"The variable $X$ represents the block, $C$ or $D$ , which has not inherited $E.$ .",
"Therefore, we have to initialize the counters associated with it.",
"Note that lines to are executed at most once for each block.",
"Therefore, apart from the nested loops at lines –, the overall time complexity of them is in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ .",
"For the nested loops, we have to observe that the size of $Touched$ is less than the number of outgoing transitions from $X.$ and those transitions are considered only once during the execution of the algorithm.",
"Therefore the overall time complexity of the nested loops is also in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ .",
"All of this implies that the overall time complexity of Procedure REF is in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ ."
],
[
"Function ",
"[!t] Init($Q,\\rightarrow ,P_{\\mathrm {init}},R_{\\mathrm {init}}, RefinerNodes$ ) $P := (P_{\\mathrm {init}})$ Init:1 $E\\in P$ $E.",
":= 0$ ; $E.",
":= {E}$ $E.",
":= \\lbrace q\\in Q\\,\\big |\\,q.= E\\rbrace $ $E.",
":= \\lbrace E^{\\prime } \\in P \\,\\big |\\,(E,E^{\\prime }) \\in R_{\\mathrm {init}}\\rbrace $ Init:5 Initialization to take into account Proposition REF .",
"$Marked := \\emptyset $ ; $Touched := \\emptyset $ Init:6 $q\\rightarrow q^{\\prime }$ $Marked := Marked \\cup \\lbrace q\\rbrace $ REF $(P,Marked)$Init:8 $E\\in P\\,\\big |\\,E.\\in Marked$ $Touched := Touched \\cup \\lbrace E\\rbrace $ ; $C\\in Touched$ $D\\notin Touched$ $C.",
":= C.\\mathrel \\setminus \\lbrace D\\rbrace $ $Marked := \\emptyset $ ; $Touched := \\emptyset $ Init:13 Initialization of $RefinerNodes$ and the $$ 's.",
"$C\\in P$ Init:14 $C..^{\\prime } :=\\emptyset $ $C..:= \\lbrace D.node \\,\\big |\\,D \\notin C.\\rbrace $ $C..\\ne \\emptyset $ $RefinerNodes:=RefinerNodes\\cup \\lbrace C.\\rbrace $ Initialization of the 's.",
"$E,B^{\\prime }\\in P$ $B^{\\prime }.",
"{E} := 0$ Init:19 $PreE^{\\prime } := \\emptyset $ $E^{\\prime }\\in P$ $e\\in \\rightarrow ^{-1}(E^{\\prime }.",
")\\,\\big |\\,e=e..$ Init:22 $PreE^{\\prime } := PreE^{\\prime }\\cup \\lbrace e.\\rbrace $ ; $E\\in PreE^{\\prime }, B^{\\prime }\\in P$ Init:24 $B^{\\prime }.",
"{E}++$ ; $PreE^{\\prime } := \\emptyset $ Init:26 $(P)$ This function initializes the data structures and transforms the initial preorder such that we start from a preorder stable with the total relation ${R}_0=Q\\times Q$ .",
"The first lines require no special comment except that the $$ array for each block $E$ is initialized according to (REF ).",
"The time complexity of these lines – is in $O(|P_{\\mathrm {sim}}|^2)$ .",
"Lines – transform the initial preorder according to Proposition REF .",
"Note that the call of Function REF at line has the side effect to transform the counters before their initialization.",
"This is not a problem since this does not change the overall time complexity and the real initialization of the counters is done after, at lines –.",
"Just after line , with Proposition REF , the invariants (REF ) is true for $i=1$ and each simulation included in the preorder represented by the partition-relation pair $(P_{\\mathrm {init}},R_{\\mathrm {init}})$ is included in ${R}_1$ .",
"Apart from the inner call of Procedure REF , whose we know the overall time complexity, $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ , the time complexity of these lines is in $O(|\\rightarrow | + |P_{\\mathrm {sim}}|^2)$ .",
"The loop starting at line initializes the 's according to () such that (REF ) is also true for $i=1$ .",
"The set $RefinerNodes$ is also initialized according to (REF ) for $i=1$ .",
"The time complexity of this loop is in $O(|P_{\\mathrm {sim}}|^2)$ .",
"Lines – initialize the counters such that for all $E,B^{\\prime }\\in P$ we have: $B^{\\prime }.",
"{E} = |\\lbrace E^{\\prime }\\in P\\,\\big |\\,E.\\rightarrow E^{\\prime }.\\rbrace |$ The invariant (REF ) is thus true for $i=1$ since $B^{\\prime }.\\times E^{\\prime }.\\subseteq {R}_0$ is always true, with ${R}_0=Q\\times Q$ .",
"The time complexity of the loop at line is in $O(|P_{\\mathrm {sim}}|^2)$ .",
"The time complexity of the loop at line is in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ .",
"An iteration of the loop at line corresponds to a meta-transition $E.\\rightarrow E^{\\prime }.$ and a block $B^{\\prime }\\in P$ .",
"Its time complexity is therefore in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ ."
],
[
"Procedure ",
"[!t] Split1($P, RefinerNodes$ ) $Marked := \\emptyset $ ; atLeastOneSplit := $B\\in RefinerNodes $ $B^{\\prime }=(B)$ Split1:3 $e\\in \\rightarrow ^{-1}(B)$ Split1:4 $B^{\\prime }.(e.",
")=0$ $atLeastOneSplit := $ $atLeastOneSplit = $ $q\\rightarrow q^{\\prime } \\,\\big |\\,q^{\\prime }.\\in B^{\\prime }.$ Split1:8 $Marked := Marked\\cup \\lbrace q\\rbrace $ REF ($P,Marked$ ) $Marked := \\emptyset $ ; atLeastOneSplit := Split1:11 The purpose of this procedure is to apply Lemma REF to split the current partition $P$ until there is no $({P},{R})$ -splitter transition of type 1, with ${P}={P}_P$ and ${R}$ defined by (REF ).",
"This is done such that the coarsest ${R}$ -block-stable equivalence relation included in ${P}_P$ before the execution of REF is still included in ${P}_P$ after its execution.",
"Proposition REF guarantees that all splitter transitions of type 1 have been treated since we assume (REF ).",
"Note that at line , $B$ is a node, a set of states, which corresponds to a block of the relation ${R}$ .",
"As explained in section REF , the counters are defined between two blocks of the current partition $P$ .",
"We thus need to choose one of this block included in $B$ to represent $B$ since we have Proposition REF .",
"A transition $e\\rightarrow e^{\\prime }$ with $e^{\\prime }\\in B$ at line is considered only if there is a block in $\\cup B.$ .",
"From Lemma REF this can happen only $|P_{\\mathrm {sim}}|$ times.",
"Therefore, the overall time complexity of the loop at line is in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ .",
"From Lemma REF , lines – are executed only when at least one block is split.",
"Therefore, their overall time complexity is in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ ."
],
[
"Procedure ",
"[!t] Split2($P, RefinerNodes$ ) $PreB^{\\prime } := \\emptyset $ ; $Touched := \\emptyset $ ; $Touched^{\\prime } := \\emptyset $ ; $Marked := \\emptyset $ $B\\in RefinerNodes $ $B^{\\prime } \\in P \\,\\big |\\,B^{\\prime }.\\subseteq B$ Split2:3 $e\\in \\rightarrow ^{-1}(B^{\\prime }.",
")$ Split2:4 $E := e.$ $e=E.$ $PreB^{\\prime } := PreB^{\\prime } \\cup \\lbrace E\\rbrace $ $E \\in PreB^{\\prime }$ Split2:8 $E.$ ++ $Touched := Touched \\cup \\lbrace E\\rbrace $ $PreB^{\\prime } := \\emptyset $ $B^{\\prime }=(B)$ $E\\in Touched$ Split2:13 $B^{\\prime }.",
"(E) = E.$ The transition $E.\\rightarrow B$ is maximal $Touched^{\\prime } := Touched^{\\prime } \\cup \\lbrace E\\rbrace $ $E.\\mathrel {:=} 0$ $e\\in \\rightarrow ^{-1}(B) \\,\\big |\\,e.\\in Touched^{\\prime }$ Split2:17 $Marked := Marked\\cup \\lbrace e\\rbrace $ REF ($P,Marked$ ) Split2:19 $Touched := \\emptyset $ ; $Touched^{\\prime } := \\emptyset $ ; $Marked := \\emptyset $ This procedure is applied after REF .",
"We can therefore assume that there is no more $({P},{R})$ -splitter transition of type 1, with ${P}$ and ${R}$ defined in the analysis of Split1.",
"The aim of this procedure is to implement Lemma REF .",
"It works as follows.",
"For a given node $B$ which corresponds to a block of ${R}$ such that $B.\\ne \\emptyset $ we scan the blocks $B^{\\prime }$ of $P$ which are included in $B$ .",
"For each of those $B^{\\prime }$ we scan, loop at line , the incoming transitions from representatives states of blocks $E$ .",
"For each of these blocks we increment $E.$ , loop at line .",
"Therefore, at the end of the loop at line , for each block $E\\in P$ such that $E\\rightarrow B$ , we have $E.= |\\lbrace [b]_{{P}}\\subseteq B\\,\\big |\\,E.\\rightarrow b\\rbrace |$ .",
"This allows us to identify, loop at line , the blocks that may be split according to Lemma REF .",
"The split is done thanks to lines–.",
"For reasons similar to those for Procedure REF , we have: the overall time complexity of Procedure REF is in $O(|P_{\\mathrm {sim}}|.",
"{|\\rightarrow |})$ ."
],
[
"Procedure ",
"[!t] Refine($RefinerNodes$ ) $Remove := \\emptyset $ ; $RefinerNodes^{\\prime }:=\\emptyset $ $B\\in RefinerNodes $ Refine:2 $B^{\\prime }=(B)$ $d\\in \\rightarrow ^{-1}( \\cup B.",
")$ Refine:4 $D := d.$ $B^{\\prime }.",
"{D} = 0$ Refine:6 $Remove := Remove \\cup \\lbrace D\\rbrace $ $c\\in \\rightarrow ^{-1}(B)$ Refine:8 $C := c.$ $D\\in Remove \\,\\big |\\,D\\in C.$ Refine:10 $C.",
":= C.\\mathrel \\setminus \\lbrace D\\rbrace $ $C..^{\\prime } := C..^{\\prime } \\cup \\lbrace D.\\rbrace $ $RefinerNodes^{\\prime } := RefinerNodes^{\\prime } \\cup \\lbrace C.\\rbrace $ $Remove := \\emptyset $ ; $B.",
":= \\emptyset $ $RefinerNodes:=\\emptyset $ Refine:15 $B\\in RefinerNodes^{\\prime }$ $(B., B.^{\\prime })$ $(RefinerNodes, RefinerNodes^{\\prime })$ Refine:18 Procedures REF and REF possibly change the current partition $P$ , but not ${R}$ , and preserve (), (REF ), (REF ), (REF ), (REF ) and (REF ).",
"Thanks to Lemma REF and Lemma REF all ${R}$ -block-stable equivalence relation presents in ${P}_P$ before the execution of REF and REF are still presents after.",
"Furthermore, with Theorem REF , we know that ${P}_P$ , after the execution of REF , is ${R}$ -block-stable.",
"From all of this, before the execution of REF , ${P}_P$ is the coarsest ${R}$ -block-stable equivalence relation.",
"The conditions are thus met to apply Theorem REF to do a refine step of the algorithm.",
"Note that, thanks to (REF ) and Proposition REF , we have the equivalence: $d\\notin \\rightarrow ^{-1}\\mathrel \\circ {R}(B)\\Leftrightarrow B^{\\prime }.",
"{D} = 0$ at line .",
"At the end of the while loop at line , the relation ${R}$ has been refined by $\\mathit {NotRel}^{\\prime }$ .",
"In lines – $\\mathit {NotRel}$ is set to $\\mathit {NotRel}^{\\prime }$ and $RefinerNodes$ is set to $RefinerNodes^{\\prime }$ to prepare the next iteration of the while loop in Procedure REF .",
"By construction, see the loop starting at line , (REF ) and (REF ) are set for the next iteration of the while loop in REF .",
"In a similar way, the 's are not modified by REF , (REF ) is thus still true and (REF ) will be true for the next iteration of the while loop in REF .",
"Thanks to Theorem REF , (REF ) and (REF ) are also preserved.",
"From Lemma REF , a node $B$ and a transition $d\\rightarrow d^{\\prime }$ with $d^{\\prime }\\in \\cup B.$ are considered at most once during the execution of the algorithm.",
"Therefore, the overall time complexity of the loop at line is in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ .",
"Let us now consider a block $D$ in $Remove$ and a transition $c\\rightarrow b$ with $b\\in B$ .",
"By contradiction, let us suppose this pair $(D,c\\rightarrow b)$ can happen twice, at iteration $i$ and at iteration $j$ of the while loop of Function REF , with $i<j$ .",
"From (REF ) and line we have for $k=i$ and $k=j$ : $ D\\rightarrow {R}_{k-1}(b)$ and $D\\lnot \\rightarrow {R}_{k}(b)$ .",
"But this is not possible since $({R}_i)_{i\\ge 0}$ is a strictly decreasing sequence of relations.",
"This means that the overall time complexity of the loop at line is also in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ .",
"The other lines have a lower overall time complexity.",
"From all of this, the overall time complexity of this procedure is in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ ."
],
[
"Time Complexity of the Algorithm",
"From the analysis of the functions and procedures of the algorithm, we derive the following theorem.",
"Theorem 23 The time complexity of the presented simulation algorithm is in $O(|P_{\\mathrm {sim}}|.|\\rightarrow |)$ ."
],
[
"Improvements and Future Work",
"With the new notions of maximal transition, stable preorder, block-stable equivalence relation and representative state, we have introduced new foundations that we will use to design some efficient simulation algorithms.",
"This formalism has been illustrated with the presentation of the most efficient in memory of the fastest simulation algorithms of the moment.",
"It is possible to increase in practice the time efficiency of procedures REF , REF and REF if we allow the use of both $\\rightarrow $ and $\\rightarrow ^{-1}$ (or if we calculate one from the other, which requires an additional bit space in $O({|\\rightarrow |.",
"}\\log (|Q|))$ to store the result).",
"Procedures REF and REF can be further improved in practice, but this changes $O(|P_{\\mathrm {sim}}|^2.\\log (|P_{\\mathrm {sim}}|))$ to $O(|P_{\\mathrm {sim}}|^2.\\log (|Q|))$ , which is not really noticeable in practice, in the bit space complexity, if we count states instead of blocks in (REF ), (REF ) and (REF ).",
"Simulation algorithms are generally extended for labeled transition systems (LTS) by embedding them in normal transition systems.",
"This is what is proposed in [15], [8] and [9] for example.",
"By doing this, the size of the alphabet is introduced in both time and space complexities.",
"Even in [1], where a more specific algorithm is proposed for LTS, the size of the alphabet still matter.",
"In [4] we proposed three extensions of [15] for LTS with significant reduction of the incidence of the size of the alphabet.",
"We will therefore propose the same extensions but from the foundations given in the present paper.",
"Note that the bit space complexity of the algorithm presented here is in fact in $\\Theta (|P_{\\mathrm {sim}}|^2.\\log (|P_{\\mathrm {sim}}|) + |Q|.\\log (|Q|))$ .",
"It is therefore interesting to propose other algorithms with better compromises between time and space complexities.",
"We will therefore compare in practice different propositions.",
"Then, we will have all the prerequisites to address a more challenging problem that we open here: the existence of a simulation algorithm with a time complexity in $O(|\\rightarrow |.\\log (|Q|) +|P_{\\mathrm {sim}}|.|{\\rightarrow _{\\mathrm {sim}}}|)$ , with $\\rightarrow _{\\mathrm {sim}}$ the relation over $P_{\\mathrm {sim}}$ induced by $\\rightarrow $ , and a bit space complexity in $O(|P_{\\mathrm {sim}}|^2.\\log (|P_{\\mathrm {sim}}|)+{|\\rightarrow |.",
"}\\log (|Q|))$ .",
"What is surprising is that the biggest challenge is not the part in $O(|P_{\\mathrm {sim}}|.|{\\rightarrow _{\\mathrm {sim}}}|)$ but the part in $O(|\\rightarrow |.\\log (|Q|))$ in the time complexity.",
"Such an algorithm will lead to an even greater improvement from the algorithm of the present paper than that of the passage from HHK to RT since there are, in general, many more transitions than states in a transition system."
],
[
"Acknowledgements",
"I thank the anonymous reviewers of the paper to LICS'17.",
"Their questions and suggestions have helped to improve the presentation of the paper.",
"I am also grateful to Philippe Canalda for his helpful advice."
]
] | 1709.01826 |
[
[
"Spectral Calibration of the Fluorescence Telescopes of the Pierre Auger\n Observatory"
],
[
"Abstract We present a novel method to measure precisely the relative spectral response of the fluorescence telescopes of the Pierre Auger Observatory.",
"We used a portable light source based on a xenon flasher and a monochromator to measure the relative spectral efficiencies of eight telescopes in steps of 5 nm from 280 nm to 440 nm.",
"Each point in a scan had approximately 2 nm FWHM out of the monochromator.",
"Different sets of telescopes in the observatory have different optical components, and the eight telescopes measured represent two each of the four combinations of components represented in the observatory.",
"We made an end-to-end measurement of the response from different combinations of optical components, and the monochromator setup allowed for more precise and complete measurements than our previous multi-wavelength calibrations.",
"We find an overall uncertainty in the calibration of the spectral response of most of the telescopes of 1.5% for all wavelengths; the six oldest telescopes have larger overall uncertainties of about 2.2%.",
"We also report changes in physics measureables due to the change in calibration, which are generally small."
],
[
"Introduction",
"The Pierre Auger Observatory [1] has been designed to study the origin and the nature of ultra high-energy cosmic rays, which have energies above $10^{18}$ eV.",
"The construction of the complete observatory following the original design finished in 2008.",
"The observatory is located in Malargüe, Argentina, and consists of two complementary detector systems, which provide independent information on the cosmic ray events.",
"Extensive Air Showers (EAS) initiated by cosmic rays in the Earth's atmosphere are measured by the Surface Detector (SD) and the Fluorescence Detector (FD).",
"The SD is composed of 1660 water Cherenkov detectors located mostly on a triangular array of 1.5 km spacing covering an area of roughly 3000 km$^{2}$ .",
"The SD measures the EAS secondary particles reaching ground level [2].",
"The FD is designed to measure the nitrogen fluorescence light produced in the atmosphere by the EAS secondary particles.",
"The FD is composed of 27 telescopes overlooking the SD array from four sites, Los Leones (LL), Los Morados (LM), Loma Amarilla (LA), and Coihueco (CO) [3].",
"The SD takes data continuously, but the FD operates only on clear nights, and care is taken to avoid exposure to too much moonlight.",
"The energy of the primary cosmic ray is a key measurable for the science of the observatory, and the FD measurement of the energy, with lower independent systematic uncertainties, is used to calibrate the SD energy scale using events observed by both detectors.",
"The work described here explains how the FD calibration at wavelengths across the nitrogen fluorescence spectrum has recently been improved, resulting in smaller related systematic uncertainties.",
"The buildings at the four FD sites each have six independent telescopes, and each telescope has a 30°$\\times $ 30° field of view, leading to a 180° coverage in azimuth and from 2° to 32° in elevation at each building.",
"Additionally, three specialized telescopes called HEAT [4] are located near Coihueco to overlook a portion of the SD array at higher elevations, from 32° to 62°, to register EAS of lower energies.",
"All these telescopes are housed in climate-controlled buildings, isolated from dust and day light.",
"The layout of the observatory is shown schematically in Figure REF .",
"Figure: A schematic of the Pierre Auger Observatory where each black dot is a water Cherenkov detector.",
"Locations of the fluorescence telescopes are shown along the perimeter of the surface detector array, where the blue lines indicate their individual field of view.",
"The field of view of the HEAT telescopes are indicated with red lines.Each FD telescope is composed of several optical components as shown in Figure REF : a 2.2 m aperture diaphragm, a UV filter to reduce the background light, a Schmidt corrector annulus, a 3.5 m $\\times $ 3.5 m tessellated spherical mirror, and a camera formed by an array of 440 hexagonal photomultipliers (PMT) each with a field of view of 1.5° full angle.",
"Each PMT has a light concentrator approximating a hexagonal Winston cone to reduce dead spaces between PMTs [3].",
"Figure: The optical components of an individual fluorescence telescope.The energy calibration of the data [5], [6] for the Pierre Auger Observatory, including events observed by the SD only, relies on the calibration of the FD.",
"Events observed by both FD and SD provide the link from the FD, which is absolutely calibrated, to the SD data.",
"To calibrate the FD three different procedures are performed: the absolute [7], the relative [8], and the spectral (or multi-wavelength) calibrations [9].",
"We focus here on the spectral calibration, which is a relative measurement that relates the absolute calibration performed at 365 nm to wavelengths across the nitrogen fluorescence spectrum, which is shown in Figure REF .",
"Figure: The nitrogen fluorescence spectrum as measured by the AIRFLY collaboration showing the 21 major transitions.To perform this measurement the drum-shaped portable light source used for the absolute calibration [7] was adapted to emit UV light across the wavelength range of interest.",
"The drum light source is designed to uniformly illuminate all 440 PMTs in a single camera simultaneously when it is placed at the aperture of the FD telescope, enabling the end-to-end calibration.",
"The FD response as a function of wavelength was initially calculated as a convolution of separate reflection or transmission measurements of each optical component used in the first Los Leones telescopes [11].",
"The first end-to-end spectral calibration of the FD was performed using the drum light source with a xenon flasher and filter wheel to provide five points across the FD wavelength response [9].",
"This measurement represented an improvement for the energy estimation of all events observed by the Pierre Auger Observatory as it has been shown to increase the reconstructed energy of events by nearly 4% for all energies [12].",
"However, that result has two limitations: first, the differences in FD optical components were not measured since only one telescope was calibrated; and second, determining the FD spectral response curve using only five points involved a complicated fitting procedure, and was particularly difficult considering the large width of the filters, which resulted in relatively large systematic uncertainties.",
"The aim of the work described in this paper was to measure the FD efficiency at many points across the nitrogen fluorescence spectrum with a reduced wavelength bite at each point, and to do it at enough telescopes to cover the different combinations of optical components making up all the telescopes within the Auger Observatory.",
"The spectral calibration described here proceeds in three steps.",
"First, the relative drum emission spectrum is measured in the dark hall lab in Malargüe with a specific calibration PMT, called the “Lab-PMT”, observing the drum at a large distance, in a similar fashion to the absolute calibration of the drum; see [7] and explanatory drawings therein.",
"Knowing the intensity of the drum at each wavelength, we next measure the response of the FD telescopes to the output of the multi-wavelength drum over the course of several nights, while recording data from a monitoring photodiode (PD) exposed to the narrow-band light at each point to ensure knowledge of the relative drum spectrum.",
"Finally, the FD telescope response is normalized by the measured relative drum emission spectrum at every wavelength, and we evaluate the associated systematic uncertainties in the final calculation of the efficiency.",
"This following sections describe the measurements and analysis of data taken during March 2014: FD optical components in section 2; the new drum light source in section 3; measurements of the drum light source spectrum in section 4; calibrations performed at the FD telescopes in section 5; FD efficiency as a function of wavelength in section 6; and final calibration results in section 7.",
"Effects on physics measurables due to changing calibrations are discussed in section 8."
],
[
"Optical Components of the Fluorescence Telescopes",
"There are two types of mirrors used in the telescopes, and the glass used for the corrector rings was produced using two different glass-making procedures.",
"The 12 mirrors at Los Leones and Los Morados are aluminum with a 2 mm AlMgSiO$_5$ layer glued on as the reflective surface, and the 12 mirrors at Coihueco and Loma Amarilla are composed of a borosilicate glass with a 90 nm Al layer and then a 110 nm SiO$_2$ layer (see [3] for more details).",
"Two different procedures were used to grow the borosilicate glass used in the corrector rings, both by Schott Glass ManufacturesSchott Glass, http://www.us.schott.com/english/index.html.",
"One type is called Borofloat 33Borofloat, http://www.us.schott.com/borofloat/english/attribute/optical, and the other is a crown glass labeled P-BK7P-BK7, http://www.schott.com/advanced_optics/us/abbe_datasheets/schott_datasheet_all_us.pdf, and the transmission of UV light differs for these two products.",
"Given the different wavelength dependencies of the above components, our aim was to measure the four combinations of mirrors and corrector rings present in the FDs.",
"This meant calibrating at three of the four FD buildings.",
"Table REF shows the eight telescopes we calibrated at the three FD sites along with which components make up each telescope.",
"Calibration of these eight telescopes gives a complete coverage of the different components and a duplicate measure of each combination.",
"Table: List of the FD telescopes we calibrated and their respective optical components.",
"Calibration at these eight FD telescopes gives a complete coverage of the different components and a duplicate measure of each combination.",
"The last column indicates all other (unmeasured) telescopes with the same optical components.As seen in Table REF , the telescopes CO 4/5 are the only ones that have same nominal components as those located at other FD buildings, which have different construction dates.",
"It is usualy the case that optical components degrade their properties when exposed to light and ambient conditions (ageing), whose effect depends on exposure time.",
"Even if FD telescopes are kept in climate-controlled buildings, an analysis of ageing follows.",
"Regarding the spectral calibration, what has to be evaluated is the change in the spectral response of a given FD telescope, i.e.",
"the shape of the response curve vs wavelength.",
"This kind of differential degradation is not obviously seen at the FD telescopes.",
"One way to evaluate whether there is any change in the spectral response is to track the absolute calibration done periodically at 375 nm [1], [2].",
"The absolute calibration is scaled at any given date by using the nightly relative calibration, which is done at 470 nm [1], [2].",
"Because these two calibrations are done at different wavelengths, any change in the spectral response would translate in a drift of the absolute calibration with time.",
"In Table REF we show the variations of the ratio (R) of absolute calibrations performed in 2010 and 2013, where $R = (Abs_\\mathrm {2013}-Abs_\\mathrm {2010})/Abs_\\mathrm {2013}$ , along with the date of finished construction for telescopes at a given building.",
"As seen in the table, the variations do not respond to any ageing pattern, e.g.",
"for the oldest telescopes there is a positive variation for LL and a negative variation for CO.",
"Moreover, the overall effect that telescopes could have in data analysis do not change the final reconstructed energy significantly (see Fig.",
"49 in [1]).",
"For these reasons, we consider that different time of telescope construction do not play a role in the spectral calibration described in this paper and, consequently, CO 4/5 can be taken as representative of LA and HEAT.",
"Table: List of FD buildings and dates when construction was finished and operation started.",
"Δ\\Delta t is the elapsed time until measurements done for this work (March 2014).",
"R is the ratio of absolute calibrations performed in 2010 and 2013 (see text)."
],
[
"Monochromator Drum Setup",
"The work described in [9] was the first in-situ end-to-end measurement of the FD efficiency as a function of wavelength.",
"It limited the measurement to only five points across the $\\sim $ 150 nm wide acceptance of the FDs, and the filters had a fairly wide spectral width, about $\\sim $ 15 nm FWHM, as shown in the bottom of Figure REF .",
"The large spectral width led to a complicated procedure of accounting for the width effects along the rising and falling edges of the efficiency curve [9].",
"In addition, since there were only five measured points, the resulting calibration curve had to be interpolated between the points, and the original piece-wise efficiency curve [11] was used as the starting point.",
"In the five-point measurement [9] the efficiency was assumed to go to zero below 290 nm and above 425 nm since the filters did not extend to these wavelengths, thus the values resulting from the piece-wise convolution of the component efficiencies [11] were the only data for wavelengths below 290 nm and above 425 nm.",
"Figure: A comparison showing the spectral width of the output of the monochromator sampled every 5 nm (top, this work) and the notch filter spectral transmission (bottom, ).",
"The y-axes are the intensity in arbitrary units for the monochromator and the normalized transmission for the notch filters.These reasons are the motivation for using a monochromator to select the wavelengths out of a UV spectrum.",
"A monochromator allows for a high resolution probe across the FD acceptance, and a far more detailed measurement can be performed.",
"The top of Figure REF shows the output of the monochromator in 5 nm steps from 275 nm to 450 nm with a xenon flasher as the input, each step with a 2 nm FWHM.",
"The xenon flasher is an Excelitas PAX-10 modelPAX-10 10-Watt Precision-Aligned Pulsed Xenon Light Source - http://www.excelitas.com/downloads/dts_pax10.pdf with improved EM noise reduction and variable flash intensity.",
"The monochromator output width was chosen to provide a reasonable compromise between wavelength resolution and the drum intensity required for use at the FDs.",
"For the work described here, an enclosure housing the monochromator and xenon flasher was mounted onto the rear of the drum.",
"The enclosure was insulated and contained a heater and associated controlling circuitry to maintain a stable 20$\\pm $ 2 ℃ temperature for monochromator reliability.",
"A custom 25.4 mm diameter aluminum tube was fabricated and attached to the output of the monochromator; it protrudes into the interior of the drum.",
"At the end of the tube a 0.23 mm thick Teflon diffuser ensured that the illumination of the front face of the drum was uniform as measured with long-exposure CCD images, similar to what had been measured previously [3], [13].",
"A photodiode (PD) was mounted near the output of the monochromator, but upstream of the tube that protruded into the drum, allowing for pulse-by-pulse monitoring of the emission spectrum from the monochromator.",
"The monochromator and xenon flasher were controlled with the same common gateway interface (CGI) web page and calibration electronics that have been used in the absolute calibration [7].",
"Scanning of the monochromator, triggering of the flasher, and data acquisition from monitoring devices and the FD were all fully automated using CGI code and cURLcURL Documentation - http://curl.haxx.se/ scripts over the wireless LAN used for drum calibrations."
],
[
"Lab Measurements and the Drum Spectrum",
"To characterize the drum emission as a function of wavelength, several measurements were needed in the laboratory.",
"For the one-week calibration campaign described here, four measurements were performed in the lab, two prior to any field work at the FD telescopes, one two days later and the last one at the end of the week."
],
[
"Drum Emission",
"With the automated scanning of the monochromator and data acquisition we took measurements of the relative drum emission spectrum as viewed by the calibration Lab-PMT.",
"The monitoring PD detector measured the monochromator output as described above.",
"The setup for these measurements had the drum at the far end of the dark hall and the Lab-PMT inside the darkbox in the calibration room, about 16 m away from the Teflon face of the drum.",
"See [7] for a detailed description of the dark hall calibration setup.",
"The average response of the Lab-PMT to 100 pulses of the drum was recorded as a function of wavelength from 250 nm to 450 nm, in steps of 1 nm.",
"The uncertainty in the average for a given wavelength was calculated as the standard deviation of the mean, $\\frac{\\sigma _\\mathrm {Drum}}{\\sqrt{100}}$ .",
"The solid grey line in Figure REF shows an example of one of these spectra.",
"We took averages of the four spectra at each wavelength as the final measurement of the drum spectrum, which is shown in the same figure as blue dots.",
"This final drum spectrum used measurements in steps of 5 nm corresponding to the step size used when calibrating the FD telescopes.",
"For wavelengths between 320 nm and 390 nm, the four measurements were generally statistically consistent.",
"But for wavelengths at the low and high ends of the spectrum there was disagreement; section REF explains how we introduce a systematic uncertainty to account for this disagreement.",
"Figure: Drum emission spectra.",
"Solid grey line: one of the measured spectra taken with the Lab-PMT; the line shows the average responses to 100 pulses of the drum as a function of wavelength, in steps of 1 nm.Blue points: the averaged drum spectrum as measured by the Lab-PMT throughout the calibration campaign; the spectrum is taken in steps of 5 nm as this is what is used to measure the FD responses; error bars are the statistical uncertainties, which are generally smaller than the plotted points.Black line and points: the averaged drum spectrum as measured by the monitoring photodiode (PD) throughout the calibration campaign, in steps of 5 nm.For each of these four spectra measured with the Lab-PMT there are data from the monitoring PD.",
"The monitoring PD data were handled in the same way; we made an average of the four spectra recorded by the PD and an associated error based on the spread of the four measurements.",
"These data are shown in Figure REF as black line and points."
],
[
"Lab-PMT Quantum Efficiency",
"A measurement of the quantum efficiency (QE) of the Lab-PMT, which is used to measure the relative drum emission spectrum, has to be performed to measure the relative response of a given FD telescope at different wavelengths.",
"The method used here is similar to what was done previously [9] except, instead of a DC deuterium lamp, we used the xenon flasher as the UV light source into the monochromator.",
"For the work reported here we only needed a relative measurement of the QE, and so several uncertainties associated with an absolute QE measurement are not included in this work.",
"Figure: Shown in black squares is the measured relative Lab-PMT QE.The error bars are the statistical uncertainty associated with the distribution of the response of the PMT at each wavelength.",
"The blue circles are a fourth order polynomial fit to the data that serves to smooth out the measurement.Several measurements of the Lab-PMT QE were performed prior to and after the FD spectral calibration campaign, and these measurements typically yielded curves consistent with the data shown as black squares in Figure REF .",
"The error bars are the statistical uncertainty associated with the spread in the response of the PMT to 100 pulses at each wavelength.",
"The variations in the QE from point to point are typical when this kind of measurement is performed (e.g.",
"see [9]), although they are not expected.",
"In an attempt to smooth out these variations we fit the PMT QE curve with a fourth order polynomial shown as blue circles in the figure.",
"The error bars in the fit are the relative statistical uncertainty for a given wavelength applied to the interpolated values in the fit.",
"Deviations of the fit from measured points are largest at both the lower and upper ends of the wavelength range.",
"However, the FD response is significant only in the range 310-410 nm (see Figure REF ) where the deviations are less than 2% with RMS of approximately 1%.",
"We take this 1% as a conservative estimate of the systematic uncertainty in the measurement of the Lab-PMT QE: $\\delta ^\\mathrm {Drum}_\\mathrm {QESyst}(\\lambda )\\approx 1\\%$ .",
"Changing the nature of the fitted curve or using simply the measured black points from Figure REF has little effect on measurements of EAS events.",
"For example a change of order 0.1% on the reconstructed energy would result from using the measured QE points instead of the smoothed curve.",
"The small effect on energy occurs because in the region at high and low wavelengths where the fit deviates most from the measured points the FD efficiency is very low and the nitrogen fluorescence spectrum has no large features."
],
[
"Uncertainties in Lab Measurements",
"The estimate of the statistical uncertainties for the various response distributions to the xenon flasher are taken as the standard deviation of the mean.",
"Figure REF shows the response distribution of the Lab-PMT to 100 flashes of the drum at 375 nm where $\\delta ^\\mathrm {Drum}_\\mathrm {PMTStat}(\\lambda =375 \\text{ nm})=\\frac{\\sigma (\\lambda =375 \\text{ nm})}{\\sqrt{N}}\\approx 1\\%$ of the average response $\\overline{S^\\mathrm {Drum}}(\\lambda =375 \\text{nm})$ .",
"The intensity at the monochromator output is known to be stable (with associated statistical uncertainties) through the monitoring PD spectra taken at the same time as the Lab-PMT data.",
"A similar distribution was produced for each wavelength in the Lab-PMT QE measurement and gives $\\delta ^\\mathrm {Drum}_\\mathrm {QEStat}(\\lambda )\\approx 1\\%$ .",
"Figure: Distribution of the response of the Lab-PMT to 100 flashes of the drum at 375 nm.Estimating the systematic uncertainties associated with the relative drum emission spectrum is done by comparing the different drum emission spectra measured using the Lab-PMT over the course of the one-week campaign.",
"Prior to the comparison, the Lab-PMT data are normalized by the simultaneous PD data at each wavelength to account for changes in the monochromator emission spectrum.",
"Shown in the top panel of Figure REF are the four drum spectra measured with the Lab-PMT that are used to calculate an average spectrum of the drum, and the middle plot shows the residuals from the average in percent as a function of wavelength.",
"Over most of the wavelength region where the FD efficiency is nonzero, 300 nm to 420 nm, the residuals plotted in Figure REF are close to agreement with each other within the statistical uncertainties.",
"To estimate the systematic uncertainty of the drum emission at each wavelength we introduce an additive parameter, $\\varepsilon ^\\mathrm {Drum}_\\mathrm {Syst}(\\lambda )$ , such that calculating a $\\chi ^2$ per degree of freedom comparison via equation (REF ) gives $\\chi ^2_\\mathrm {ndf}\\lesssim 1$ , and then this parameter is taken as the systematic uncertainty: $\\chi ^2_\\mathrm {ndf}(\\lambda )=\\frac{1}{3}\\sum _{n=1}^4\\frac{\\big (S^\\mathrm {Drum}(\\lambda )_n-\\overline{S^\\mathrm {Drum}}(\\lambda )\\big )^2}{\\big (\\delta ^\\mathrm {Drum}_\\mathrm {PMTStat}(\\lambda )_n\\big )^2+\\big (\\varepsilon ^\\mathrm {Drum}_\\mathrm {Syst}(\\lambda )\\big )^2}\\lesssim 1 ~.$ In equation (REF ) the Lab-PMT response (or drum emission) at a given wavelength is $S^\\mathrm {Drum}(\\lambda )$ , the associated statistical uncertainty is $\\delta ^\\mathrm {Drum}_\\mathrm {PMTStat}(\\lambda )$ , and the average spectrum is $\\overline{S^\\mathrm {Drum}}(\\lambda )$ .",
"Figure: Four drum emission spectra as measured by the Lab-PMT (top), residuals from the average in percent (middle), the resulting systematic uncertainty ε Syst Drum (λ)\\varepsilon ^\\mathrm {Drum}_\\mathrm {Syst}(\\lambda ) shown as a percent of the PMT response at a given wavelength (bottom).For a few wavelengths $ \\chi ^2_\\mathrm {ndf}(\\lambda ) <1 $ without adding the systematic term in the denominator, and the corresponding systematic uncertainty is set to zero.",
"But most wavelengths result in $ \\chi ^2_\\mathrm {ndf}(\\lambda )>1$ without the added term, so we calculate the systematic uncertainty for those wavelengths.",
"The result of this procedure is that the non-zero Lab-PMT systematic uncertainties vary from less than 1% to approximately 3%, and in the important region from 300 nm to 400 nm the average systematic uncertainty is, conservatively, about 1%, see the bottom panel of Figure REF .",
"As a check, the PD spectra were treated with a similar evaluation of a systematic uncertainty at each wavelength as in equation (REF ).",
"The corresponding systematic uncertainty estimates for the PD would all be approximately 1% or smaller.",
"But there is no need to assess a systematic uncertainty on the drum intensity due to the PD since the PD data are only used to normalize the PMT data to reduce the spread in PMT measurements, and we use the spread in (normalized) PMT data for the systematic uncertainty.",
"We estimate the overall systematic uncertainty on the intensity of the drum at each wavelength based on the QE measurement of the Lab-PMT ($\\delta ^\\mathrm {Drum}_\\mathrm {QESyst}(\\lambda )~\\approx 1\\%$ ) and the four measurements of the drum spectrum ($ \\varepsilon ^\\mathrm {Drum}_\\mathrm {Syst}(\\lambda ) ~\\approx 1\\%$ ).",
"Each of these uncertainties is conservatively about 1% in the main region of the FD efficiency and nitrogen fluorescence spectrum, so a reasonable estimate of the overall systematic uncertainty of the drum intensity is found by adding them in quadrature: 1.4%."
],
[
"FD Measurements",
"During the March 2014 calibration campaign we measured the response of the eight telescopes, as specified in Table REF , in steps of 5 nm, over the course of five days.",
"Data from the monitoring PD were also acquired during the FD measurements to be able to control for changes in the drum spectral emission.",
"The procedure for measuring the telescope response to the multi-wavelength drum was to first scan from 255 nm to 445 nm in steps of 10 nm, and then scan from 250 nm to 450 nm in steps of 10 nm.",
"At each wavelength a series of 100 pulses from the drum was recorded by the FD data acquisition at a rate of 1 Hz.",
"A full telescope response is then an interleaving of these scans.",
"Later in analysis, wavelengths that result in essentially zero FD efficiency - at low and high wavelengths in the scan corresponding to the edges of the nitrogen spectrum - were dropped and set to zero.",
"In the previous sections we evaluated the systematic uncertainty in the drum light source intensity as a function of wavelength.",
"The contributions to this uncertainty are the spread in the four measurements of the drum intensity over the week of the calibration campaign and the systematic uncertainty in the quantum efficiency of the PMT used to measure the drum output.",
"In this section we evaluate the uncertainty in the responses of the telescopes to the drum by comparing the responses of telescopes with the same optical components - see Table REF .",
"We do this comparison because we have not measured every telescope in the observatory, so we have to develop a single calibration constant for each wavelength for each of the four sets of optical components in the table.",
"Then we use these calibration constants for all telescopes with like components (again, see Table REF ), including those not measured.",
"Combining this uncertainty on the FD response, described below, with the drum emission systematic uncertainty will give the overall systematic uncertainties on the spectral calibration of the telescopes.",
"As we will see below, of the four combinations of optical components in Table REF three will result in systematics on FD response well below the systematics from the drum emission, but one pair of telescopes will have significantly different responses to the drum resulting in a systematic uncertainty larger than that from the drum intensity."
],
[
"FD Systematic Uncertainties Evaluated by Comparing Similar Telescopes",
"We assume that the FDs built with like components - the same mirror and corrector ring types - should give similar responses, and to test that assumption we make a comparison between them to derive a meaningful systematic uncertainty.",
"To that end we perform a $\\chi ^2$ test and introduce parameters to minimize the $\\chi ^2$ such that $\\chi ^2_\\mathrm {ndf}\\lesssim 1$ for $ndf=34$ , where there are 35 wavelengths used in the comparison.",
"The parameters introduced are an overall scale factor $\\beta $ that is applied to one of the FD responses, and then $\\varepsilon _\\mathrm {FD}$ which is an estimate of a systematic uncertainty that would be needed to account for the difference between the two telescopes.",
"Thus the raw response of one of the FDs as a function of wavelength is then $\\beta *\\big (FD_\\mathrm {Resp}(\\lambda )\\pm \\delta ^{\\mathrm {FD}}_\\mathrm {Stat}(\\lambda )\\pm \\varepsilon _\\mathrm {FD}\\big )$ in the comparison, where $\\delta ^\\mathrm {FD}_\\mathrm {Stat}(\\lambda )$ is the statistical uncertainty (small) as mentioned at the end of this section.",
"The scale factor $\\beta $ does not represent a systematic uncertainty, it just accounts for any overall difference in response between the two telescopes.",
"This is similar to performing a relative calibration analysis as in [14] between the two telescopes.",
"The minimization is done in two steps according to equation (REF ) where the sum is over the $N_{\\lambda }$ measured wavelength points: $\\chi ^2_\\mathrm {ndf}=\\frac{1}{34}\\sum _\\mathrm {n=1}^\\mathrm {N_{\\lambda }}\\frac{\\Big (FD_1(\\lambda )_\\mathrm {n}-\\beta *FD_2(\\lambda )_\\mathrm {n}\\Big )^2}{\\Big (\\delta ^\\mathrm {FD_1}_\\mathrm {Stat}(\\lambda )_\\mathrm {n}\\Big )^2+\\Big (\\beta *\\delta ^\\mathrm {FD_2}_\\mathrm {Stat}(\\lambda )_\\mathrm {n}\\Big )^2+\\Big (\\beta *\\varepsilon _\\mathrm {FD}\\Big )^2}{} ~.$ First a minimum in $\\chi ^2$ is found by setting $\\varepsilon _\\mathrm {FD}=0$ and allowing the scale factor $\\beta $ to vary.",
"Once $\\beta $ has been determined, $\\varepsilon _\\mathrm {FD}$ is allowed to vary until $\\chi ^2_\\mathrm {ndf}\\lesssim 1$ .",
"Prior to the minimization the $FD_2(\\lambda )$ response data are normalized by the ratio of the monitoring PD response as measured at $FD_1$ and $FD_2$ for a given wavelength.",
"This serves to divide out any change in intensity of the light source as measured by the PD just downstream of the monochromator, and this normalization does, as expected, improve the agreement in response for some telescope pairs.",
"Table: ε FD \\varepsilon _\\mathrm {FD} and β\\beta values obtained via equation () for the similarly constructed telescopes.",
"The ε FD \\varepsilon _\\mathrm {FD} for a given pair of telescopes is given in percentage relative to the averaged response of the pair of telescopes at 375 nm.The values for $\\varepsilon _\\mathrm {FD}$ and $\\beta $ are listed in Table REF for each pair of telescopes that are constructed with nominally identical components, and the systematic uncertainties, $\\varepsilon _\\mathrm {FD}$ , are reported as a percentage of the average response of the two telescopes at 375 nm.",
"Aside from Los Leones telescopes 3 and 4, the $\\varepsilon _\\mathrm {FD}$ values derived through this minimization technique are all less than 0.5 % , and the $\\beta $ scale factors are all within about 3 % of unity.",
"The values obtained for Los Leones, although larger than the others, are still small.",
"By trying to find a reason for this difference we note that telescope 4 was part of the Engineering Array (EA, [15]) together with telescope 5.",
"However, both telescopes were rebuilt after the EA operation, particularly the mirrors were all replaced by new ones after re-setting the design parameters.",
"So, the discrepancy between LL 3 and 4 is highly probably not caused by any difference in used materials and, in any case, is included in the uncertainties.",
"We use the $\\varepsilon _\\mathrm {FD}$ calculated for a given pair of FD telescopes as a systematic uncertainty across all wavelengths for all telescopes of the corresponding construction; see Table REF .",
"These systematic uncertainties are then normalized by the telescope response at 375 nm and are added in quadrature with the uncertainties associated with the spread in Lab-PMT measurements of the drum (about 1% in important wavelength range, a function of wavelength), and the Lab-PMT QE (1%, not wavelength dependent) to calculate the overall systematic uncertainty on telescopes of each combination of optical components.",
"An example result is plotted as the red brackets in Figure REF for the Los Morados telescopes 4 and 5; for this pair (and like telescopes) the overall systematic uncertainty on the FD response is approximately $\\sqrt{1^2+1^2+0.14^2}=1.4 \\%$ , and it is dominated by the uncertainty in the drum intensity.",
"For the Coihueco instruments the overall systematic uncertainty is about 1.5%.",
"For the telescopes at Los Leones the uncertainty from the difference in response between the two telescopes is larger than the drum-related systematic uncertainties, and the overall systematic uncertainty on all of the Los Leones telescopes is about 2.2%.",
"The statistics of the data taken with the drum light source at the FD telescopes also contribute to the uncertainties on the calibration constants.",
"The typical spread in the average response of the 440 PMTs to the 100 drum pulses at a given wavelength is 0.4% RMS, which is much smaller than the systematic uncertainties.",
"Adding the statistical uncertainty in quadrature with the systematic uncertainties yields the overall uncertainties on the calibration constants listed in Table REF , which are the main result of this work.",
"Table: Overall uncertainties on spectral calibration constants for the pairs of telescopes measured and all other (unmeasured) telescopes with the same optical components."
],
[
"Photodiode Monitor Data",
"We performed a comparison between the average dark hall PD spectrum and each of the spectra measured for the data-taking nights at the FDs to ensure that the light source was stable and was consistent with what had been measured in the lab.",
"An overall correction of $1.00\\pm 0.01$ night to night was found as the average ratio of the PD response at the FD to that at the lab to accommodate any overall variations in intensity or response due to temperature effects, and then we performed a $\\chi ^2$ comparison for all the measured wavelengths.",
"For all measuring nights at the FDs the PD spectra agree very well, the comparison gives a $\\chi ^{2}_\\mathrm {ndf}\\sim 1$ where $ndf=34$ for each, implying that the spectrum as observed by the PD was the same at all locations."
],
[
"Calculation of the FD Efficiency",
"We calculate the relative FD efficiency for each telescope by dividing the measured telescope response to the drum by the measured drum emission spectrum.",
"The relative drum emission spectrum is measured as described in section and takes into account the Lab-PMT quantum efficiency over the range from 250 nm to 450 nm.",
"The relative efficiency for a given telescope at a given wavelength, $FD_\\mathrm {eff}^\\mathrm {Rel}(\\lambda )$ , is calculated for each wavelength from 280 nm to 440 nm in steps of 5 nm: $FD_\\mathrm {eff}^\\mathrm {Rel}(\\lambda )=\\frac{FD_\\mathrm {Resp}(\\lambda )*{QE^\\mathrm {Lab}_\\mathrm {PMT}}(\\lambda )}{\\overline{S^\\mathrm {Drum}}(\\lambda )}*\\frac{1}{FD_\\mathrm {eff}(\\lambda =375~\\text{nm})}{} ~.$ The curves are taken relative to the efficiency of the telescope at 375 nm since this is what is used in the Pierre Auger Observatory reconstruction software [16] for all FD calculations.",
"The range in wavelength from 280 nm to 440 nm used for evaluating the FD efficiency is smaller than the range measured in the lab because below 280 nm and above 440 nm the light level is near zero intensity for the nitrogen emission spectrum and the FD response is also very near zero.",
"As an example, Figure REF shows the relative efficiency for the Los Morados telescopes 4 and 5 based on this work compared with the previous measurement [9].",
"Figure: Relative efficiencies for the average of Los Morados telescopes 4 and 5.",
"Top: the five filter curve shown as a dashed blue line and the monochromator result shown as the solid line.",
"Error bars are statistical uncertainties, and the red brackets are the systematic uncertainties calculated as described in section .",
"Bottom: difference between the five filter result and this work.",
"The error bars and brackets are the same as in the top plot, shown here for clarity.The uncertainties in the FD efficiencies have statistical and systematic components associated with the measurement of the relative emission spectrum of the drum, the Lab-PMT QE, and the FD response to the multi-wavelength drum.",
"The statistical uncertainties associated with the lab work and the FD responses are propagated through the calculation of the FD efficiency via equation (REF ) as a function of wavelength.",
"All systematic uncertainties described above associated with the lab work, the Lab-PMT and its QE, the FD response, and $\\varepsilon _\\mathrm {FD}$ for a given FD telescope, are added together in quadrature as a function of wavelength.",
"For much of the wavelength range the new results agree with the older five-point scan.",
"The disagreement at the shortest and longest wavelengths is perhaps not surprising since the previous lowest and highest measurements were at 320 nm and 405 nm, and the efficiency was extrapolated to zero from those points following the piecewise curve [11].",
"The efficiency was assumed to go to zero below 295 nm and above 425 nm."
],
[
"Comparison of telescopes with differing optical components",
"After estimating the systematic uncertainties for each measured FD telescope, $\\varepsilon _\\mathrm {FD}$ , we made a $\\chi ^2$ comparison between the six combinations of unlike FD optical components listed in Table REF to determine whether the unlike components result in any different telescope responses.",
"In calculating the $\\chi ^2_\\mathrm {ndf}$ for the differently constructed FD telescopes we use the ratio of the PD data taken at the corresponding FDs to normalize the average response of one of the FD types.",
"The PD data from the two FD data-taking nights are averaged as a function of wavelength and the ratio of the PD averages from the two types of FDs are applied to the combined FD response along with the statistical and systematic uncertainties.",
"Using this normalization serves to divide out any differences in the drum spectrum between the two measurements of the FD types.",
"An example for calculating the $\\chi ^2$ between the average of Coihueco 2 and Coihueco 3 ($\\overline{S_\\mathrm {CO23}(\\lambda )_\\mathrm {n}}$ ) and the average of Coihueco 4 and 5 follows.",
"The uncertainties in equation REF have obvious labels; for example $\\varepsilon _\\mathrm {FD}^\\mathrm {CO23}$ is the systematic uncertainty for the Coihueco telescopes 2 and 3 from Table REF .",
"$\\begin{split}&\\chi ^{2}_\\mathrm {ndf}=\\frac{1}{34}\\sum _\\mathrm {n=1}^\\mathrm {N_\\lambda }\\frac{\\Big (\\overline{S_\\mathrm {CO23}(\\lambda )_\\mathrm {n}}-PD_\\mathrm {Ratio}(\\lambda )_\\mathrm {n}*\\overline{S_\\mathrm {CO45}(\\lambda )_\\mathrm {n}}\\Big )^2}{\\big (\\delta ^\\mathrm {CO23}_\\mathrm {Stat}(\\lambda )_\\mathrm {n}\\big )^2+\\big (PD_\\mathrm {Ratio}(\\lambda )_\\mathrm {n}*\\delta ^\\mathrm {CO45}_\\mathrm {Stat}(\\lambda )_\\mathrm {n}\\big )^2 + \\big (\\varepsilon _\\mathrm {FD}^\\mathrm {CO23}\\big )^2 + \\big (\\varepsilon _\\mathrm {FD}^\\mathrm {CO45}\\big )^2} ~, \\\\& \\\\&PD_\\mathrm {Ratio}(\\lambda )\\equiv \\frac{\\overline{S^\\mathrm {PD}_\\mathrm {CO23}}(\\lambda )}{\\overline{S^\\mathrm {PD}_\\mathrm {CO45}}(\\lambda )} ~.\\end{split}$ The results of the comparisons are shown in Table REF .",
"The telescopes with different components are all significantly different from each other except when comparing the average of Los Leones telescopes 3 and 4 to the average of Los Morados telescopes 4 and 5.",
"In principle this low $\\chi ^2$ could indicate that all telescopes constructed with components like those at Los Leones 3 and 4 and those constructed like Los Morados telescopes 4 and 5 have the same response, and therefore could share the relative calibration constants that are the goal of this work.",
"However, the two detectors at Los Leones have a much greater difference in response between them than do the two telescopes from Los Morados: the systematic uncertainty in Table REF for Los Leones telescopes is more than a factor of 10 larger than that for Los Morados ones.",
"The large systematic uncertainty for the telescopes at Los Leones could be masking a real difference with those at Los Morados.",
"For this reason we feel it is reasonable not to combine the Los Leones and Los Morados telescopes to calculate the final spectral calibration constants.",
"We conclude that all four sets of FD telescopes listed in Table REF need different spectral calibrations, and four sets of calibration constants have been computed.",
"Examining the results in Table REF and Table REF we note that the largest $\\chi ^2_\\mathrm {ndf}$ values in Table REF are associated with changing mirrors not changing corrector rings.",
"For example, comparing Coihueco 4/5 with Los Morados 4/5 changes only the mirror and gives a $\\chi ^2_\\mathrm {ndf}$ of 55, but comparing Coihueco 4/5 with Coihueco 2/3, which changes only the corrector ring, yields a $\\chi ^2_\\mathrm {ndf}$ of 5.6.",
"Changing both components by comparing Coihueco 2/3 with Los Morados 4/5 gives a $\\chi ^2_\\mathrm {ndf}$ of 161, but we note that the telescopes at Los Morados have a very small systematic uncertainty in Table REF .",
"These examples have so far left out the Los Leones telescopes.",
"The large systematic uncertainty derived by comparing the two Los Leones telescopes reduces the $\\chi ^2_\\mathrm {ndf}$ values when comparing to other telescopes, but the idea that the mirrors are the main effect is still present when comparing the Los Leones telescopes to the others.",
"Table: Comparison of spectral response for FD telescopes with different components.",
"χ ndf 2 \\chi ^2_\\mathrm {ndf} values obtained for the sets in Table , where ndf=34ndf=34."
],
[
"Effect on Physics Measurables",
"To evaluate the effect a new calibration has on physics measurables, we reconstructed a set of events using the new calibration and compare to results from that same set of events using the prior calibration.",
"When we did this exercise upon changing from initial piecewise to the five-point calibration, the reconstructed energies increased about 4% at $10^{18}$ eV, and the increase lessened slightly to 3.6% at $10^{19}$ eV [12].",
"The lessening of the energy increase due the five-point calibration is understood because much of the change in calibration was at low wavelengths, and the five-point calibration makes the FDs less efficient at short wavelengths making the reconstructed energy higher.",
"The higher energy events make more light, and they can be detected at greater distances than lower energy events.",
"But at greater distances more of the short wavelength light will be Rayleigh scattered away, so the lower wavelengths - and the change in calibration there - do not affect the higher energy events as much when we change to the new calibration.",
"When we change from the five-point to the calibration described here, the reconstructed energies increase on average over all FD telescopes by about 1%, and that increase is relatively flat in energy.",
"However, this increase is not the same at all the telescopes.",
"The increase in reconstructed energy is greatest at Los Leones, about 2.8% at $10^{18}$ eV falling to 2.5% at $10^{19}$ eV.",
"For Los Morados the reconstructed event energies increase by about 1.8% without much energy dependence.",
"For all other telescopes the energy increases, but those increases are less than 0.35% for all energies.",
"All these changes in the reconstructed energy are important to know to fully characterize the telescopes.",
"Regarding the associated uncertainties, they are all significantly smaller than the uncertainties involved in the energy scale for the FD telescopes (see Table 3 in [1]), particularly the 3.6% from the Fluorescence yield and the 9.9% from the FD calibration."
],
[
"Conclusions",
"Determining the spectral response of the Pierre Auger Observatory fluorescence telescopes is essential to the success of the experiment.",
"A method using a monochromator-based portable light source has been used for eight FD telescopes with measurements performed every five nanometers from 280 nm to 440 nm.",
"With the calibration of these eight telescopes, the four possible combinations of different optical components in the FD were covered, thus assuring the spectral calibration of all FD telescopes at the observatory.",
"The uncertainty associated with the emission spectrum of the drum light source used for the calibration was found to be 1.4%, which is an improvement on our previous 3.5% [9].",
"For the present work we compared telescopes with nominally the same optical components, and we find that such pairs have the same spectral response within a fraction of a percent - as expected - for three out of the four pairs of like telescopes.",
"But one pair with like components, the oldest telescopes in the observatory, shows a significant difference in spectral response.",
"The overall uncertainty in the FD spectral response is 1.5% for 21 of the 27 telescopes.",
"The overall systematic uncertainty for the remaining six telescopes is 2.2%, and is somewhat larger on account of the larger difference between the two telescopes measured.",
"We also compared the differently constructed telescopes.",
"These comparisons show significantly different efficiencies as a function of wavelength, with differences mainly in the rising edge of the efficiency curve between 300 nm and 340 nm.",
"The differences seem to come mostly from the two different mirror types, and they are reflected in different calibration constants for each of the four combinations of optical components.",
"The new calibration constants affect the reconstruction of EAS events, and we looked at two important quantities.",
"The primary cosmic ray energy increases by 1.8% to 2.8% for half of the telescopes in the observatory, and for the other half the change in energy is negligible.",
"The position of the maximum in shower development in the atmosphere, $X_\\mathrm {max}$ , is not changed significantly by the change in calibration."
],
[
"Acknowledgments",
"The successful installation, commissioning, and operation of the Pierre Auger Observatory would not have been possible without the strong commitment and effort from the technical and administrative staff in Malargüe.",
"We are very grateful to the following agencies and organizations for financial support: Argentina – Comisión Nacional de Energía Atómica; Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT); Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET); Gobierno de la Provincia de Mendoza; Municipalidad de Malargüe; NDM Holdings and Valle Las Leñas; in gratitude for their continuing cooperation over land access; Australia – the Australian Research Council; Brazil – Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); Financiadora de Estudos e Projetos (FINEP); Fundação de Amparo à Pesquisa do Estado de Rio de Janeiro (FAPERJ); São Paulo Research Foundation (FAPESP) Grants No.",
"2010/07359-6 and No.",
"1999/05404-3; Ministério de Ciência e Tecnologia (MCT); Czech Republic – Grant No.",
"MSMT CR LG15014, LO1305, LM2015038 and CZ.02.1.01/0.0/0.0/16_013/0001402; France – Centre de Calcul IN2P3/CNRS; Centre National de la Recherche Scientifique (CNRS); Conseil Régional Ile-de-France; Département Physique Nucléaire et Corpusculaire (PNC-IN2P3/CNRS); Département Sciences de l'Univers (SDU-INSU/CNRS); Institut Lagrange de Paris (ILP) Grant No.",
"LABEX ANR-10-LABX-63 within the Investissements d'Avenir Programme Grant No.",
"ANR-11-IDEX-0004-02; Germany – Bundesministerium für Bildung und Forschung (BMBF); Deutsche Forschungsgemeinschaft (DFG); Finanzministerium Baden-Württemberg; Helmholtz Alliance for Astroparticle Physics (HAP); Helmholtz-Gemeinschaft Deutscher Forschungszentren (HGF); Ministerium für Innovation, Wissenschaft und Forschung des Landes Nordrhein-Westfalen; Ministerium für Wissenschaft, Forschung und Kunst des Landes Baden-Württemberg; Italy – Istituto Nazionale di Fisica Nucleare (INFN); Istituto Nazionale di Astrofisica (INAF); Ministero dell'Istruzione, dell'Universitá e della Ricerca (MIUR); CETEMPS Center of Excellence; Ministero degli Affari Esteri (MAE); Mexico – Consejo Nacional de Ciencia y Tecnología (CONACYT) No.",
"167733; Universidad Nacional Autónoma de México (UNAM); PAPIIT DGAPA-UNAM; The Netherlands – Ministerie van Onderwijs, Cultuur en Wetenschap; Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO); Stichting voor Fundamenteel Onderzoek der Materie (FOM); Poland – National Centre for Research and Development, Grants No.",
"ERA-NET-ASPERA/01/11 and No.",
"ERA-NET-ASPERA/02/11; National Science Centre, Grants No.",
"2013/08/M/ST9/00322, No.",
"2013/08/M/ST9/00728 and No.",
"HARMONIA 5–2013/10/M/ST9/00062, UMO-2016/22/M/ST9/00198; Portugal – Portuguese national funds and FEDER funds within Programa Operacional Factores de Competitividade through Fundação para a Ciência e a Tecnologia (COMPETE); Romania – Romanian Authority for Scientific Research ANCS; CNDI-UEFISCDI partnership projects Grants No.",
"20/2012 and No.194/2012 and PN 16 42 01 02; Slovenia – Slovenian Research Agency; Spain – Comunidad de Madrid; Fondo Europeo de Desarrollo Regional (FEDER) funds; Ministerio de Economía y Competitividad; Xunta de Galicia; European Community 7th Framework Program Grant No.",
"FP7-PEOPLE-2012-IEF-328826; USA – Department of Energy, Contracts No.",
"DE-AC02-07CH11359, No.",
"DE-FR02-04ER41300, No.",
"DE-FG02-99ER41107 and No.",
"DE-SC0011689; National Science Foundation, Grant No.",
"0450696; The Grainger Foundation; Marie Curie-IRSES/EPLANET; European Particle Physics Latin American Network; European Union 7th Framework Program, Grant No.",
"PIRSES-2009-GA-246806; European Union's Horizon 2020 research and innovation programme (Grant No.",
"646623); and UNESCO."
]
] | 1709.01537 |
[
[
"Ultra-massive black hole feedback in compact galaxies"
],
[
"Abstract Recent observations confirm the existence of ultra-massive black holes (UMBH) in the nuclei of compact galaxies, with physical properties similar to NGC 1277.",
"The nature of these objects poses a new puzzle to the `black hole-host galaxy co-evolution' scenario.",
"We discuss the potential link between UMBH and galaxy compactness, possibly connected via extreme active galactic nucleus (AGN) feedback at early times ($z > 2$).",
"In our picture, AGN feedback is driven by radiation pressure on dust.",
"We suggest that early UMBH feedback blows away all the gas beyond a $\\sim$kpc or so, while triggering star formation at inner radii, eventually leaving a compact galaxy remnant.",
"Such extreme UMBH feedback can also affect the surrounding environment on larger scales, e.g.",
"the outflowing stars may form a diffuse stellar halo around the compact galaxy, or even escape into the intergalactic or intracluster medium.",
"On the other hand, less massive black holes will drive less powerful feedback, such that the stars formed within the AGN feedback-driven outflow remain bound to the host galaxy, and contribute to its size growth over cosmic time."
],
[
"Introduction",
"Scaling relations between supermassive black holes and their host galaxies, such as the black hole mass-stellar velocity dispersion ($M_{BH} - \\sigma $ ) or black hole mass-bulge mass ($M_{BH} - M_b$ ) relations, have been extensively studied over the past decades [15].",
"These observational correlations point towards some form of `co-evolution', in which the central black hole is connected to its host galaxy, through the active galactic nucleus (AGN) feedback [26], [3], [14].",
"The recent discovery of ultra-massive black holes (UMBH with $M_{BH} \\gtrsim 10^{10} M_{\\odot }$ ) in compact galaxies (with sizes $R \\lesssim 2$ kpc), poses a new puzzle to the co-evolutionary scenario.",
"The first example is NGC 1277 in the Perseus cluster, which is reported to have a central black hole of mass $M_{BH} \\sim 1.7 \\times 10^{10} M_{\\odot }$ located at the centre of a compact galaxy with a half-light radius of $R_e \\sim 1.6$ kpc [31].",
"The corresponding black hole-to-total stellar mass ratio is of the order of $\\sim 14\\%$ , much larger than the usually expected values.",
"We note that there are some uncertainties concerning the black hole mass estimate in NGC 1277 [7], [23]; nonetheless, the system still harbours one of the most massive black holes known, remaining a major outlier in the black hole-host galaxy scaling relations [32].",
"Recent observations confirm the existence of further sources in the local Universe, with physical properties very similar to NGC 1277 [6].",
"Interestingly, several UMBH candidates are found to reside in nearby galaxy clusters, implying very large mean black hole mass densities [4].",
"What is unique about these systems is that exceptionally massive black holes are hosted in compact galaxies, suggesting an intriguing connection between the presence of a UMBH at the centre and the host galaxy's compactness.",
"Here we consider the possibility of a direct link between UMBH and compact galaxy formation, through extreme AGN feedback at early times.",
"In fact, powerful AGN feedback may be expected at high redshifts ($z \\gtrsim 2$ ), when the accreting UMBH were likely shining as bright quasars.",
"It is now becoming clear that the effects of AGN feedback on its host galaxy can be quite complex, with both positive and negative feedback playing a role [10], [25], [37].",
"We have previously discussed how AGN feedback, driven by radiation pressure on dust, may trigger star formation in the host galaxy [10]; and how such star formation induced in the feedback-driven outflow may account for the size evolution of massive galaxies observed over cosmic time [13].",
"The possibility of star formation occurring within galactic outflows has now been observationally confirmed, with the first direct detection obtained for a nearby source [17].",
"In the following, we discuss the case of UMBH feedback and its potential effects on the host galaxy, both in terms of star formation triggering and gas removal.",
"We assume AGN feedback driven by radiation pressure on dust.",
"Radiative feedback sweeps up the surrounding material into an outflowing shell.",
"The equation of motion of the shell is given by: $\\frac{d}{dt} [M_{g}(r) v] = \\frac{L}{c} (1 + \\tau _{IR} - e^{-\\tau _{UV}}) - \\frac{G M(r) M_{g}(r)}{r^2} \\, ,$ where $L$ is the central luminosity, $M(r)$ is the total mass distribution, and $M_{g}(r)$ is the gas mass, assuming a thin shell approximation [28], [12].",
"The infrared (IR) and ultraviolet (UV) optical depths are given by: $\\tau _{IR,UV}(r) = \\frac{\\kappa _\\mathrm {IR,UV} M_{g}(r)}{4 \\pi r^2}$ , where $\\kappa _{IR}$ and $\\kappa _{UV}$ are the IR and UV opacities, respectively.",
"The gravitational potential is modelled by a Navarro-Frenk-White (NFW) profile $\\rho (r) = \\frac{\\rho _\\mathrm {c} \\delta _\\mathrm {c}}{(r/r_\\mathrm {s})(1+r/r_\\mathrm {s})^2} \\, ,$ where $\\rho _\\mathrm {c} = \\frac{3 H^2}{8 \\pi G}$ is the critical density, $\\delta _\\mathrm {c}$ is a characteristic density, and $r_\\mathrm {s}$ is a characteristic scale radius.",
"The corresponding mass profile is given by $M(r) = 4 \\pi \\delta _\\mathrm {c} \\rho _\\mathrm {c} r_\\mathrm {s}^3 \\, \\left[ \\ln \\left(1+\\frac{r}{r_\\mathrm {s}}\\right) - \\frac{r}{r+r_\\mathrm {s}} \\right] \\, .$ The ambient gas density distribution can be parametrized as a power law of radius $r$ , with slope $\\alpha $ : $n(r) = n_0 \\left( \\frac{r}{R_0} \\right)^{-\\alpha } \\, ,$ where $n_0$ is the density of the external medium, and $R_0$ is the initial radius.",
"The case $\\alpha = 2$ corresponds to the isothermal distribution, $n(r) \\propto 1/r^2$ .",
"This may be a good approximation in the inner regions, but should break down on larger scales, where the gas density is likely to fall off more steeply with radius.",
"We thus consider a slightly modified form: $n(r) \\propto \\frac{1}{r^{2} (r+r_a)^{\\gamma }} \\, ,$ where $r_a$ is a scale radius and $\\gamma > 0$ .",
"Below, we assume the case of $\\gamma = 1$ and $r_a \\sim $ 1 kpc.",
"The corresponding gas mass is given by: $M_{g}(r) = 4 \\pi m_p \\int n(r) r^2 dr= K \\ln \\frac{r+r_a}{r_a} \\, ,$ where $K = 4 \\pi m_p n_0 R_0^2 (R_0+r_a)$ .",
"From Eq.",
"REF , a critical luminosity can be defined by equating the outward force due to radiation pressure and the inward force due to gravity: $L_\\mathrm {E}^{^{\\prime }} = \\frac{Gc}{r^2} M(r) M_\\mathrm {{g}}(r) (1 + \\tau _\\mathrm {{IR}} - e^{-\\tau _\\mathrm {{UV}}})^{-1}$ .",
"This can be considered as a generalised form of the Eddington luminosity.",
"The corresponding effective Eddington ratio ($\\Gamma = L/L_E^{\\prime }$ ) is given by: $\\Gamma = \\frac{L r^2}{c G M(r) M_{g}(r)} (1 + \\tau _{IR} - e^{-\\tau _{UV}}) \\, ,$ which basically corresponds to the ratio of the radiative force to the gravitational force.",
"As the ambient medium is swept up by the passage of the outflowing shell, the resulting compression of the gas may induce local density enhancements, which in turn may trigger star formation [10].",
"As previously discussed, we adopt a simple parametrisation for the star formation rate (SFR) in the outflow: $\\dot{M}_{\\star } = \\epsilon _{\\star } \\frac{M_g(r)}{t_{flow}(r)} \\, ,$ where $\\epsilon _{\\star }$ is the star formation efficiency, and $t_{flow} = r/v(r)$ is the local flow time."
],
[
"UMBH feedback-induced formation of compact galaxies",
"Here we focus on the case of UMBH ($M_{BH} \\sim 10^{10} M_{\\odot }$ , with standard bolometric Eddington luminosities of the order of $L_E \\sim 10^{48}$ erg/s), actively accreting at high redshifts.",
"Figure REF shows the radial velocity profile of the outflowing shell, derived by integrating the equation of motion (Eq.",
"REF ).",
"The shell velocity can be compared with the local escape velocity, which is defined as $v_\\mathrm {esc} = \\sqrt{2 \\vert \\Phi (r) \\vert }$ , where the gravitational potential is obtained by integrating the Poisson equation $\\nabla ^2 \\Phi = 4 \\pi G \\rho $ .",
"We assume typical opacities of $\\kappa _{IR} = 3 \\mathrm {cm^2/g}$ and $\\kappa _{UV} = 10^3 \\mathrm {cm^2/g}$ , with the initial radius set to $R_0 = 200$ pc.",
"We see that for a UMBH with a high central luminosity ($L = 3 \\times 10^{47}$ erg/s), the shell velocity exceeds the local escape velocity beyond $r \\gtrsim 1$ kpc (green solid curve); while for a less massive black hole, with a correspondingly lower central luminosity ($L = 8 \\times 10^{46}$ erg/s), the shell velocity is below the escape speed, and the outflowing shell remains bound to the galaxy (blue dashed curve).",
"In Figure REF , the escape velocity (cyan dotted line) is computed for NFW dark halo parameters appropriate for NGC 1277 [35], while the escape speed may be somewhat different for the less massive black hole.",
"But even adopting the conservative case of the isothermal potential, for which the escape velocity is given by $v_{esc} = 2 \\sigma $ , the corresponding speed is of the order of $v_{esc} \\sim 500$ km/s, which should not change the qualitative result.",
"As the radiation pressure-driven shell expands outwards, new stars can form within the outflowing shell.",
"Figure REF shows the associated star formation rates in the outflow, for a star formation efficiency of $\\epsilon _{\\star } = 0.1$ .",
"The newly formed stars are assumed to initially share the velocity of the outflowing shell in which they are formed (although, once decoupled from the shell, they will be slowed down by gravity).",
"From Figure REF , we observe that stars formed at small radii have low velocities, below the local escape speed, and thus remain bound to the galaxy; while stars formed at larger radii are born with higher velocities (since the shell is accelerated outwards).",
"In the case of UMBH, the latter stars may exceed the escape speed on $\\gtrsim $ kpc scales, and hence be unbound.",
"As a result, a compact galaxy remnant may be left behind.",
"In contrast, for less massive black holes, with lower central luminosities, the newly formed stars never reach the escape speed, and thus remain bound to the galaxy.",
"We note that the prescription for the star formation rate (Eq.",
"REF ) is simply a parametrisation of the overall rate of conversion of gas into stars within the outflowing shell, induced by AGN feedback; and Figure REF shows the corresponding radial profile (SFR(r) at radius $r$ ).",
"The total stellar mass added in the process can be obtained by integrating the star formation rate over the AGN feedback timescale, and assuming some value for the star formation efficiency.",
"There are several observational indications that the star formation efficiency increases at high redshifts, and is higher in starburst-like systems [27], [24].",
"Assuming a characteristic timescale ($\\sim 10^7$ yr, shorter than the Salpeter time), we obtain an order-of-magnitude estimate for the stellar mass, from a few times $\\sim 10^9 M_{\\odot }$ possibly up to $\\sim 10^{10} M_{\\odot }$ .",
"This is to be compared with the typical host stellar mass ($M_{\\star } \\sim 10^{11} M_{\\odot }$ ) of compact galaxies [6].",
"We see that the increase in stellar mass due to AGN feedback-triggered star formation is not dramatic, but may still form a non-negligible fraction of the total stellar mass.",
"In cases where the gas reservoir is not completely removed by AGN feedback, several re-accretion events can occur at later times (with associated feedback-driven star formation episodes), further contributing to the growth of the host galaxy [13].",
"The formation of a compact galaxy requires a very massive black hole, with a very high luminosity, capable of accelerating the shell to escape speed.",
"In physical terms, an increase in the luminosity implies a higher effective Eddington ratio (Eq.",
"REF ), which leads to efficient acceleration and resulting high velocity.",
"For a given central luminosity, a decrease in the ambient density will naturally facilitate the shell escape.",
"Provided that the black hole is massive enough and that the surrounding gas density is not too high, UMBH feedback may blow away all the gas beyond a $\\sim $ kpc-scale, eventually leaving a compact galaxy remnant.",
"On the other hand, if the black hole is less massive and the ambient gas density is high, the stars formed in the outflowing shell remain bound to the galaxy, contributing to the development of its spheroidal component and size evolution over cosmic time [11].",
"Furthermore, if the outflowing shell remains trapped in the galactic halo, gas may later fall back, providing fuel for further accretion and star formation.",
"Therefore, extreme UMBH feedback coupled with moderate ambient densities, may favour the formation of compact galaxies; while less massive black holes in higher density environments may drive weaker feedback, resulting in more `normal' extended galaxies.",
"The central luminosity may play a major role in determining the radial velocity profile of the outflowing shell, and hence the galaxy compactness; while, the ambient density may be the dominant factor in setting the triggered star formation rate (see Appendix ).",
"Figure: Velocity as a function of radius: L=3×10 47 L = 3 \\times 10^{47}erg/s, n 0 =5×10 2 cm -3 n_0 = 5 \\times 10^2 cm^{-3} (green solid); L=8×10 46 L = 8 \\times 10^{46}erg/s, n 0 =2×10 3 cm -3 n_0 = 2 \\times 10^3 cm^{-3} (blue dashed); local escape velocity (cyan dotted).",
"Fiducial parameters: κ IR =3cm 2 /g\\kappa _{IR} = 3 cm^2/g, κ UV =10 3 cm 2 /g\\kappa _{UV} = 10^3 cm^2/g, r a =1r_a = 1kpc, R 0 =200R_0 = 200pc.Figure: Star formation rate as a function of radius, with ϵ ☆ =0.1\\epsilon _{\\star } = 0.1.",
"Same physical parameters as in Figure ."
],
[
"Discussion",
"Observations have uncovered a number of compact UMBH host galaxies in the local Universe, which share similarities with NGC 1277 [31], [5], [6].",
"The UMBH host galaxies are characterised by uniformly old stellar populations (with ages $> 10$ Gyr), high metallicities, and high $[\\alpha /Fe]$ abundance ratios [29], [6].",
"This indicates that the bulk of the stars formed in a short burst event at high redshift ($z > 2$ ), with correspondingly high star formation rates.",
"The observed kinematics of compact galaxies is also peculiar, with centrally peaked velocity dispersions and high radial velocities ($v > 200$ km/s) [18], [6].",
"The set of particular features characterising the compact galaxies suggest that these are `relic galaxies', which remained basically unaltered since their formation at high redshift, i.e.",
"without undergoing merger episodes at later times [29], [5].",
"In our picture, powerful UMBH feedback can blow away the surrounding material, eventually leaving a compact galaxy remnant.",
"Such extreme feedback may be expected at high redshifts ($z > 2$ ), close to the peak epoch of both AGN and star formation activities, when UMBH were rapidly accreting (e.g.",
"on timescales comparable to the Salpeter time).",
"Since the gas reservoir is removed by early UMBH feedback, the formation of new stars is inhibited at later times.",
"However, star formation may be triggered within the AGN feedback-driven outflow itself.",
"As the high-speed outflow sweeps through the galaxy on a short timescale, a strong radial stellar age gradient is not expected.",
"This may be compatible with the uniformly old stellar populations observed in compact galaxies [29], [6].",
"Stars formed in the outflowing shell are also expected to have high radial velocities; radial velocities of several hundred km/s are observed in compact galaxies [18], [6].",
"Some of the outflowing stars, with the highest radial velocities, will be ejected from the galaxy, and even escape into the intergalactic or intracluster medium.",
"Such UMBH feedback-ejected stars may form a diffuse stellar halo around the compact galaxy, also contributing to the so-called intracluster light (ICL).",
"In fact, observations indicate that a significant fraction of stars in galaxy clusters is not bound to individual galaxies, but rather reside in a common diffuse component [1], [19], [36].",
"We recall that several compact galaxies are found in galaxy clusters, like NGC 1277, which is located in the core of the Perseus cluster [4].",
"Recent HST observations seem to suggest that a significant population of intracluster stars is also present in the Perseus cluster [8].",
"In addition to the different physical mechanisms proposed for the formation of intergalactic stars [30], AGN feedback-driven stellar outflows may also play a role.",
"The UMBH host galaxies could then be surrounded by extended distributions of stars.",
"An interesting observational implication would be the direct detection of diffuse stellar haloes around compact galaxies, which may be achievable with future instruments.",
"In this framework, the observational appearance of the source (e.g.",
"the galaxy compactness) would be mainly determined by internal processes associated with the central UMBH, rather than environmental effects such as tidal stripping.",
"Outflowing stars may also be produced in wind-shock outflows, and this scenario has been investigated in numerical simulations [21], [37].",
"In this picture, rapid cooling of the shocked gas leads to shell fragmentation, eventually triggering the formation of high-velocity stars on radial trajectories.",
"In contrast, in the radiative feedback scenario, we do not expect the development of strong shocks, except possibly an outer shock in the swept-up material (which subsequently cools down).",
"In fact, radiative cooling of dense gas is very efficient and radiation pressure acts on the bulk of the mass (the dust being mainly located in the innermost regions).",
"Furthermore, recent observations indicate that large amounts of dust can be formed in core-collapse supernovae, with a significant fraction of the dust grains surviving in the medium, due to their large sizes [22], [34].",
"In our picture, fresh dust can be produced, when massive stars formed in the outflowing shell explode as supernovae.",
"The release of such additional dust, spread into the surrounding environment, may contribute to sustain the overall feedback process.",
"An important requirement in our model is the presence of dust at early times, since the whole AGN feedback process relies on radiation pressure on dust.",
"Indeed, the large dust masses observed in submillimetre galaxies and quasars at $z > 4$ [20] require prompt dust production, possibly by early supernovae.",
"Rapid dust formation is also needed in order to account for the detection of dust at the highest redshifts ($z \\sim 7$ ), indicating early enrichment [33], [16].",
"Following their fast formation event at high redshifts, the compact galaxies may evolve differently depending on the subsequent merger history.",
"According to the popular `two-phase' galaxy formation scenario [9], [2], the first phase of in-situ star formation is followed by a sequence of merger episodes, which account for the increase in size observed from $z \\sim 2$ to the present.",
"In this scenario, the relic galaxies are galaxies that did not experience the second phase (due to the stochastic nature of mergers) and survived unchanged to the present day [29], [5], [6].",
"While the second growth phase is usually attributed to minor mergers, the observed size evolution of massive galaxies may also be explained by several episodes of star formation triggered by AGN feedback [13].",
"In this context, we have previously discussed how such AGN feedback-driven star formation mainly contributes to the development of the spheroidal component of galaxies (with the most massive black holes associated with elliptical galaxies, while the smaller ones are linked to the central bulges of disc galaxies).",
"In particular, we have shown that the relation between characteristic radius and mass (set by the action of radiation pressure on dust), of the form $R \\propto \\sqrt{M}$ , may naturally account for the `mass-radius' scaling relation observed in early-type galaxies [11].",
"Finally, we recall that compact galaxies hosting UMBH are found to be major outliers in the local black hole mass-bulge mass ($M_{BH}-M_b$ ) relation [5], [6].",
"In the case of UMBH feedback, the gas reservoir is removed at early times, such that further star formation is inhibited, and only a compact stellar bulge is left behind.",
"At high redshifts, most UMBH host galaxies would then be similar to NGC 1277, with a high $M_{BH}/M_b$ ratio.",
"There are observational indications that the $M_{BH}/M_b$ ratios are larger at higher redshifts compared to local values, suggesting a higher normalisation in the $M_{BH}-M_b$ relation at earlier cosmic epochs [15].",
"Intriguingly, the stellar mass density profile of NGC 1277 is found to be in remarkable agreement with that of massive compact galaxies at high redshifts [29], further supporting the `relic' nature of compact galaxies.",
"As discussed in [4], the subsequent evolution may depend on the location within the galaxy cluster environment: if the compact galaxy sits at the centre of the potential well, it can re-accrete gas and stars, and may develop into a brightest cluster galaxy (BCG); whereas, if the galaxy is rapidly orbiting in the core of the cluster, it is unable to accrete further material and may survive as a compact remnant, like NGC 1277."
],
[
"Acknowledgements ",
"We acknowledge Roberto Maiolino for helpful comments on the manuscript.",
"We thank the anonymous referee for a constructive report."
],
[
"Luminosity and density dependence",
"As discussed in Sect.",
"REF , the shell radial profiles are mainly determined by two parameters: the central luminosity and the ambient density.",
"For a given density distribution, an increase in the luminosity leads to a smaller crossing radius (where the shell velocity exceeds the local escape velocity).",
"In fact, a higher luminosity implies a higher effective Eddington ratio (Eq.",
"REF ), leading to efficient shell acceleration and resulting high velocity.",
"Likewise, for a given central luminosity, a decrease in the external density leads to a smaller crossing radius, since the effective Eddington ratio increases with decreasing $n_0$ in the single scattering regime ($\\Gamma _{SS} \\propto 1/n_0$ , cf Eq.",
"REF ).",
"Concerning triggered star formation, we note that an increase in the ambient density leads to higher star formation rates (directly scaling as $\\dot{M}_ {\\star } \\propto n_0$ ); while a higher luminosity also implies higher star formation rates (as $\\dot{M}_{\\star } \\propto v$ ).",
"In Figures REF and REF , we quantify the dependence on the luminosity and ambient density, by varying both $L$ and $n_0$ by a factor of two.",
"From Figure REF , we see that a doubling in $L$ (green dash) has a larger effect on the velocity profile than a change by a factor of 2 in $n_0$ (blue dash-dot).",
"Thus the central luminosity is more important in determining the crossing radius, and hence the galaxy compactness.",
"Conversely, from Figure REF , we observe that a doubling in $n_0$ (blue dash-dot) has a larger effect on the star formation rate than a doubling in $L$ (green dash).",
"Thus the external density is the primary parameter governing the star formation rate.",
"In fact, it is the combination of the two main parameters, luminosity and density, that ultimately determine the galaxy compactness and the importance of AGN feedback-triggered star formation.",
"Figure: Velocity as a function of radius.",
"Variations in luminosity and density: L=5×10 47 L = 5 \\times 10^{47}erg/s, n 0 =10 3 cm -3 n_0 = 10^3 cm^{-3} (black solid, fiducial); L=10 48 L = 10^{48}erg/s, n 0 =10 3 cm -3 n_0 = 10^3 cm^{-3} (green dashed); L=5×10 47 L = 5 \\times 10^{47}erg/s, n 0 =5×10 2 cm -3 n_0 = 5 \\times 10^2 cm^{-3} (blue dash-dot); local escape velocity (cyan dotted).Figure: Star formation rate as a function of radius, with ϵ ☆ =0.1\\epsilon _{\\star } = 0.1.",
"Same parameters as in Figure ."
]
] | 1709.01551 |
[
[
"HST Detection of Extended Neutral Hydrogen in a Massive Elliptical at z\n = 0.4"
],
[
"Abstract We report the first detection of extended neutral hydrogen (HI) gas in the interstellar medium (ISM) of a massive elliptical galaxy beyond z~0.",
"The observations utilize the doubly lensed images of QSO HE 0047-1756 at z_QSO = 1.676 as absorption-line probes of the ISM in the massive (M_star ~ 10^11 M_sun) elliptical lens at z = 0.408, detecting gas at projected distances of d = 3.3 and 4.6 kpc on opposite sides of the lens.",
"Using the Space Telescope Imaging Spectrograph (STIS), we obtain UV absorption spectra of the lensed QSO and identify a prominent flux discontinuity and associated absorption features matching the Lyman series transitions at z = 0.408 in both sightlines.",
"The HI column density is log N(HI) = 19.6-19.7 at both locations across the lens, comparable to what is seen in 21 cm images of nearby ellipticals.",
"The HI gas kinematics are well-matched with the kinematics of the FeII absorption complex revealed in ground-based echelle data, displaying a large velocity shear of 360 km/s across the galaxy.",
"We estimate an ISM Fe abundance of 0.3-0.4 solar at both locations.",
"Including likely dust depletions increases the estimated Fe abundances to solar or supersolar, similar to those of the hot ISM and stars of nearby ellipticals.",
"Assuming 100% covering fraction of this Fe-enriched gas,we infer a total Fe mass of M_cool(Fe)~(5-8)x10^4 M_sun in the cool ISM of the massive elliptical lens, which is no more than 5% of the total Fe mass observed in the hot ISM."
],
[
" Reference sheet for natbib usage (Describing version 7.1 from 2003/06/06) For a more detailed description of the natbib package, the source file natbib.dtx.",
"The natbib package is a reimplementation of the \\cite command, to work with both author–year and numerical citations.",
"It is compatible with the standard bibliographic style files, such as plain.bst, as well as with those for harvard, apalike, chicago, astron, authordate, and of course natbib.",
"Load with \\usepackage[options]{natbib}.",
"See list of options at the end.",
"I provide three new .bst files to replace the standard numerical ones: plainnat.bst abbrvnat.bst unsrtnat.bst The natbib package has two basic citation commands, \\citet and \\citep for textual and parenthetical citations, respectively.",
"There also exist the starred versions \\citet* and \\citep* that print the full author list, and not just the abbreviated one.",
"All of these may take one or two optional arguments to add some text before and after the citation.",
"Table: NO_CAPTION Multiple citations may be made by including more than one citation key in the \\cite command argument.",
"Table: NO_CAPTION These examples are for author–year citation mode.",
"In numerical mode, the results are different.",
"Table: NO_CAPTION As an alternative form of citation, \\citealt is the same as \\citet but without parentheses.",
"Similarly, \\citealp is \\citep without parentheses.",
"Multiple references, notes, and the starred variants also exist.",
"Table: NO_CAPTION The \\citetext command allows arbitrary text to be placed in the current citation parentheses.",
"This may be used in combination with \\citealp.",
"In author–year schemes, it is sometimes desirable to be able to refer to the authors without the year, or vice versa.",
"This is provided with the extra commands Table: NO_CAPTION If the first author's name contains a von part, such as “della Robbia”, then \\cite{dRob98} produces “della Robbia (1998)”, even at the beginning of a sentence.",
"One can force the first letter to be in upper case with the command \\Citet instead.",
"Other upper case commands also exist.",
"Table: NO_CAPTION These commands also exist in starred versions for full author names.",
"Sometimes one wants to refer to a reference with a special designation, rather than by the authors, i.e.",
"as Paper I, Paper II.",
"Such aliases can be defined and used, textual and/or parenthetical with: Table: NO_CAPTION These citation commands function much like \\citet and \\citep: they may take multiple keys in the argument, may contain notes, and are marked as hyperlinks.",
"Use the command \\bibpunct with one optional and 6 mandatory arguments: the opening bracket symbol, default = ( the closing bracket symbol, default = ) the punctuation between multiple citations, default = ; the letter `n' for numerical style, or `s' for numerical superscript style, any other letter for author–year, default = author–year; the punctuation that comes between the author names and the year the punctuation that comes between years or numbers when common author lists are suppressed (default = ,); the opening bracket symbol, default = ( the closing bracket symbol, default = ) the punctuation between multiple citations, default = ; the letter `n' for numerical style, or `s' for numerical superscript style, any other letter for author–year, default = author–year; the punctuation that comes between the author names and the year the punctuation that comes between years or numbers when common author lists are suppressed (default = ,); The optional argument is the character preceding a post-note, default is a comma plus space.",
"In redefining this character, one must include a space if one is wanted.",
"Example 1, \\bibpunct{[}{]}{,}{a}{}{;} changes the output of \\cite{jon90,jon91,jam92} into [Jones et al.",
"1990; 1991, James et al.",
"1992].",
"Example 2, \\bibpunct[; ]{(}{)}{,}{a}{}{;} changes the output of \\cite{jon90} into (Jones et al.",
"1990; and references therein).",
"Redefine \\bibsection to the desired sectioning command for introducing the list of references.",
"This is normally \\section* or \\chapter*.",
"Define \\bibpreamble to be any text that is to be printed after the heading but before the actual list of references.",
"Define \\bibfont to be a font declaration, e.g.",
"\\small to apply to the list of references.",
"Define \\citenumfont to be a font declaration or command like \\itshape or \\textit.",
"Redefine \\bibnumfmt as a command with an argument to format the numbers in the list of references.",
"The default definition is [#1].",
"The indentation after the first line of each reference is given by \\bibhang; change this with the \\setlength command.",
"The vertical spacing between references is set by \\bibsep; change this with the \\setlength command.",
"If one wishes to have the citations entered in the .idx indexing file, it is only necessary to issue \\citeindextrue at any point in the document.",
"All following \\cite commands, of all variations, then insert the corresponding entry to that file.",
"With \\citeindexfalse, these entries will no longer be made.",
"The natbib package is compatible with the chapterbib package which makes it possible to have several bibliographies in one document.",
"The package makes use of the \\include command, and each \\included file has its own bibliography.",
"The order in which the chapterbib and natbib packages are loaded is unimportant.",
"The chapterbib package provides an option sectionbib that puts the bibliography in a \\section* instead of \\chapter*, something that makes sense if there is a bibliography in each chapter.",
"This option will not work when natbib is also loaded; instead, add the option to natbib.",
"Every \\included file must contain its own \\bibliography command where the bibliography is to appear.",
"The database files listed as arguments to this command can be different in each file, of course.",
"However, what is not so obvious, is that each file must also contain a \\bibliographystyle command, preferably with the same style argument.",
"Do not use the cite package with natbib; rather use one of the options sort or sort&compress.",
"These also work with author–year citations, making multiple citations appear in their order in the reference list.",
"Use option longnamesfirst to have first citation automatically give the full list of authors.",
"Suppress this for certain citations with \\shortcites{key-list}, given before the first citation.",
"Any local recoding or definitions can be put in natbib.cfg which is read in after the main package file.",
"round (default) for round parentheses; square for square brackets; curly for curly braces; angle for angle brackets; colon (default) to separate multiple citations with colons; comma to use commas as separaters; authoryear (default) for author–year citations; numbers for numerical citations; super for superscripted numerical citations, as in Nature; sort orders multiple citations into the sequence in which they appear in the list of references; sort&compress as sort but in addition multiple numerical citations are compressed if possible (as 3–6, 15); longnamesfirst makes the first citation of any reference the equivalent of the starred variant (full author list) and subsequent citations normal (abbreviated list); sectionbib redefines \\thebibliography to issue \\section* instead of \\chapter*; valid only for classes with a \\chapter command; to be used with the chapterbib package; nonamebreak keeps all the authors' names in a citation on one line; causes overfull hboxes but helps with some hyperref problems.",
"(default) for round parentheses; for square brackets; for curly braces; for angle brackets; (default) to separate multiple citations with colons; to use commas as separaters; (default) for author–year citations; for numerical citations; for superscripted numerical citations, as in Nature; orders multiple citations into the sequence in which they appear in the list of references; as sort but in addition multiple numerical citations are compressed if possible (as 3–6, 15); makes the first citation of any reference the equivalent of the starred variant (full author list) and subsequent citations normal (abbreviated list); redefines \\thebibliography to issue \\section* instead of \\chapter*; valid only for classes with a \\chapter command; to be used with the chapterbib package; keeps all the authors' names in a citation on one line; causes overfull hboxes but helps with some hyperref problems."
]
] | 1709.01563 |
[
[
"A Semi-Supervised Approach to Detecting Stance in Tweets"
],
[
"Abstract Stance classification aims to identify, for a particular issue under discussion, whether the speaker or author of a conversational turn has Pro (Favor) or Con (Against) stance on the issue.",
"Detecting stance in tweets is a new task proposed for SemEval-2016 Task6, involving predicting stance for a dataset of tweets on the topics of abortion, atheism, climate change, feminism and Hillary Clinton.",
"Given the small size of the dataset, our team created our own topic-specific training corpus by developing a set of high precision hashtags for each topic that were used to query the twitter API, with the aim of developing a large training corpus without additional human labeling of tweets for stance.",
"The hashtags selected for each topic were predicted to be stance-bearing on their own.",
"Experimental results demonstrate good performance for our features for opinion-target pairs based on generalizing dependency features using sentiment lexicons."
],
[
"=1 pdfinfo= Title=NLDS-UCSC at SemEval-2016 Task 6: A Semi-Supervised Approach to Detecting Stance in Tweets, Author=Amita Misra, Brian Ecker, Theodore Handleman, Nicolas Hahn and Marilyn Walker, Subject=, Keywords=stance classification, tweets ,social media [pages=1-last]stance.pdf"
]
] | 1709.01895 |
[
[
"Regular characters of classical groups over complete discrete valuation\n rings"
],
[
"Abstract Let $\\mathfrak{o}$ be a complete discrete valuation ring with finide residue field $\\mathsf{k}$ of odd characteristic, and let $\\mathbf{G}$ be a symplectic or special orthogonal group scheme over $\\mathfrak{o}$.",
"For any $\\ell\\in\\mathbb{N}$ let $G^\\ell$ denote the $\\ell$-th principal congruence subgroup of $\\mathbf{G}(\\mathfrak{o})$.",
"An irreducible character of the group $\\mathbf{G}(\\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{\\ell+1}$ for some $\\ell$, and if its restriction to $G^\\ell/G^{\\ell+1}\\simeq \\mathrm{Lie}(\\mathbf{G})(\\mathsf{k})$ consists of characters of minimal $\\mathbf{G}(\\mathsf{k}^{\\rm alg})$ stabilizer dimension.",
"In the present paper we consider the regular characters of such classical groups over $\\mathfrak{o}$, and construct and enumerate all regular characters of $\\mathbf{G}(\\mathfrak{o})$, when the characteristic of $\\mathsf{k}$ is greater than two.",
"As a result, we compute the regular part of their representation zeta function."
],
[
"Introduction",
"Let $K$ be a non-archimedean local field, and let $\\mathfrak {o}$ be its valuation ring, with maximal ideal $\\mathfrak {p}$ and finite residue field $\\mathsf {k}$ of odd characteristic.",
"Let $q$ and $p$ denote the cardinality and characteristic of $\\mathsf {k}$ , respectively.",
"Fix $\\pi $ to be a uniformizer of $\\mathfrak {o}$ .",
"Let $\\mathbf {G}\\subseteq \\mathrm {SL}_N$ be a symplectic or a special orthogonal group scheme over $\\mathfrak {o}$ , i.e.",
"the group of automorphisms of determinant 1, preserving a fixed non-degenerate anti-symmetric or symmetric $\\mathfrak {o}$ -defined bilinear form.",
"In this article we study the set of irreducible regular characters of the group of $\\mathfrak {o}$ -points $G=\\mathbf {G}(\\mathfrak {o})$ , the definition of which we now present.",
"Let $\\mathrm {Irr}(G)$ denote the set of irreducible complex characters of $G$ which afford a continuous representation with respect to the profinite topology.",
"The level of a character $\\chi \\in \\mathrm {Irr}(G)$ is the minimal number $\\ell \\in \\mathbb {N}_0=\\mathbb {N}\\cup \\left\\lbrace {0}\\right\\rbrace $ such that the restriction of any representation associated to $\\chi $ to the principal congruence subgroup $G^{\\ell +1}=\\mathrm {Ker}\\left(G\\rightarrow \\mathbf {G}(\\mathfrak {o}/\\mathfrak {p}^{\\ell +1})\\right)$ is trivial.",
"For example, the set of characters of level 0 is naturally identified with the set of irreducible complex characters of $\\mathbf {G}(\\mathsf {k})$ .",
"Let ${\\mathbf {g}}=\\mathrm {Lie}(\\mathbf {G})\\subseteq \\mathfrak {gl}_N$ denote the Lie algebra scheme of $\\mathbf {G}$ .",
"The smoothness of $\\mathbf {G}$ implies the equality $G/G^{\\ell +1}=\\mathbf {G}(\\mathfrak {o}/\\mathfrak {p}^{\\ell +1})$ , and moreover, the existence of an isomorphism of abelian groups between ${\\mathbf {g}}(\\mathsf {k})$ and the quotient group $G^\\ell /G^{\\ell +1}$ , for any $\\ell \\ge 1$ (see [11]).",
"In the notation of [11], this isomorphism is denoted $x\\mapsto e^{\\pi ^{\\ell }x}$ .",
"The action of $G$ by conjugation on the quotient $G^\\ell /G^{\\ell +1}$ factors through its quotient $\\mathbf {G}(\\mathsf {k})$ , making the isomorphism above $\\mathbf {G}(\\mathsf {k})$ -equivariant, with respect to the action given by $\\mathrm {Ad}\\circ \\alpha _\\ell $ , where $\\mathrm {Ad}$ denotes the adjoint action of $\\mathbf {G}(\\mathsf {k})$ on ${\\mathbf {g}}(\\mathsf {k})$ , and $\\alpha _\\ell :\\mathbf {G}(\\mathsf {k})\\rightarrow \\mathbf {G}(\\mathsf {k})$ is a bijective endomorphism of $\\mathbf {G}(\\mathsf {k})$ , determined by a field automorphism of $\\mathsf {k}$ (see, e.g.",
"[25] and the reference therein).",
"Additionally, by the assumption ${\\mathrm {char}}(\\mathsf {k})\\ne 2$ and [32], the underlying additive group of ${\\mathbf {g}}(\\mathsf {k})$ can be naturally identified with its Pontryagin dual in a $\\mathbf {G}(\\mathsf {k})$ -equivariant manner.",
"Consequently, there exists an isomorphism of $\\mathbf {G}(\\mathsf {k})$ -spaces ${\\mathbf {g}}(\\mathsf {k})\\xrightarrow{}\\mathrm {Irr}\\left(G^\\ell /G^{\\ell +1}\\right).$ Let $\\chi \\in \\mathrm {Irr}(G)$ have level $\\ell >0$ .",
"Consider the restriction $\\chi _{G^\\ell }$ of $\\chi $ to $G^\\ell $ .",
"By Clifford's Theorem and the definition of level, the restricted character $\\chi _{G^\\ell }$ is equal to a multiple of the sum over a single $\\mathbf {G}(\\mathsf {k})$ -orbit of characters of $G^{\\ell }/G^{\\ell +1}$ .",
"Using (REF ), this orbit corresponds to a single $\\mathbf {G}(\\mathsf {k})$ -orbit in ${\\mathbf {g}}(\\mathsf {k})$ , which we call the residual orbit of $\\chi $, and denote $\\Omega _1(\\chi )\\in ~\\mathrm {Ad}\\circ \\alpha _\\ell (\\mathbf {G}(\\mathsf {k}))\\backslash {\\mathbf {g}}(\\mathsf {k})=\\mathrm {Ad}\\left(\\mathbf {G}(\\mathsf {k})\\right)\\backslash {\\mathbf {g}}(\\mathsf {k})$ ."
],
[
"Regular characters",
"Let ${\\mathsf {k}^{\\mathrm {alg}}}$ be a fixed algebraic closure of $\\mathsf {k}$ .",
"An element of ${\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ is said to be regular if its centralizer in $\\mathbf {G}({\\mathsf {k}^{\\mathrm {alg}}})$ has minimal dimension among such centralizers (cf.",
"[36]).",
"By extension, an element of ${\\mathbf {g}}(\\mathsf {k})$ is said to be regular if its image under the natural inclusion of ${\\mathbf {g}}(\\mathsf {k})$ into ${\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ is regular.",
"Definition (Regular Characters) A character $\\chi \\in \\mathrm {Irr}(G)$ of positive level is said to be regular if its residual orbit $\\Omega _1(\\chi )$ consists of regular elements of ${\\mathbf {g}}(\\mathsf {k})$ .",
"For a general overview of regular elements in reductive algebraic groups over algebraically closed fields, we refer to [32].",
"The definition of regular characters goes back to Shintani [31] and Hill [18].",
"An overview of the history of regular characters of $\\mathrm {GL}_N(\\mathfrak {o})$ can be found in [34].",
"Also- see [22], [35] and [37] for the analysis of regular characters of isotropic groups of type $\\mathsf {A}_n$ , as well as [30], for a partial treatment of anisotropic groups of type $\\mathsf {A}_n$ .",
"Following [18], we begin our investigation of regular characters with the study of regular elements in the finite Lie rings ${\\mathbf {g}}(\\mathfrak {o}_r)$ , where $\\mathfrak {o}_r=\\mathfrak {o}/\\mathfrak {p}^r$ (see defi:regular-elements).",
"A central feature of the analysis undertaken in [18] is the introduction and application of geometric methods to the study of regular characters.",
"Given $x\\in \\mathrm {M}_N(\\mathfrak {o})$ and $r\\in \\mathbb {N}$ , let $x_r$ denote the image of $x$ in $\\mathrm {M}_N(\\mathfrak {o}_r)$ under the reduction map.",
"The condition of commuting with $x_r$ defines a closed $\\mathfrak {o}_r$ -group subscheme of the fiber productThe notation $\\mathbf {G}\\times \\mathfrak {o}_r$ is shorthand for the fiber product $\\mathbf {G}\\times _{\\mathrm {Spec\\:}{\\mathfrak {o}}}\\mathrm {Spec\\:}{\\mathfrak {o}_r}$ .",
"Similar notation is used whenever the base change being performed is between spectra of rings, and the base ring of the given schemes is understood from context.",
"$\\mathrm {GL}_N\\times \\mathfrak {o}_r$ , which, upon application of the Greenberg functor, defines a $\\mathsf {k}$ -group scheme [16].",
"The element $x_r$ is said to be regular if the group scheme thus obtained is of minimal dimension among such group schemes (see [18]).",
"In [18], Hill proved that $x_r\\in \\mathrm {M}_n(\\mathfrak {o}_r)$ is regular if and only if its image $x_1\\in \\mathrm {M}_n(\\mathsf {k})$ is regular.",
"Additionally, regularity of $x_r$ was shown to be equivalent to the cyclicity of the module $\\mathfrak {o}_r^N$ over the ring $\\mathfrak {o}_r[x_r]\\subseteq \\mathrm {M}_N(\\mathfrak {o}_r)$ .",
"We note that Hill's definition of regularity is equivalent to Shintani's definition of quasi-regularity [31].",
"The equivalence of regularity over the ring $\\mathfrak {o}_r$ and over $\\mathsf {k}$ was recently extended to general semisimple groups of type $\\mathsf {A}_n$ in [22].",
"In subsection:reg-characters we further extend this equivalence of to the generality of classical groups of type $\\mathsf {B}_n,\\:\\mathsf {C}_n$ and $\\mathsf {D}_n$ in odd characteristic.",
"However, the equivalence of regularity of an element $x_r\\in {\\mathbf {g}}(\\mathfrak {o}_r)$ with the cyclicity of the module $\\mathfrak {o}_r^N$ over $\\mathfrak {o}_r[x_r]$ , while true in $\\mathrm {GL}_N$ and generically true in $\\mathbf {G}$ (see corol:reg-g1-reg-glN-nonsing), is not a general phenomenon and in fact fails in certain cases (see lem:nilpotent-reg-glN-not-reg-soN).",
"Nevertheless, in the present setting, it is possible to prove a supplementary result (theo:inverse-lim), which specializes to the above equivalence in the case of $\\mathbf {G}=\\mathrm {GL}_N$ , and provides us with the information needed in order to describe the inertia subgroup of such a character and enumerate the characters of $G$ lying above a given regular orbit.",
"Consequently, we deduce the first main result of this article.",
"Theorem I Let $\\mathfrak {o}$ be a discrete valuation ring with finite residue field of odd characteristic, and let $\\mathbf {G}$ be a symplectic or a special orthogonal group over $\\mathfrak {o}$ with generic fiber of absolute rank $n$ .",
"Let $\\Omega \\subseteq {\\mathbf {g}}(\\mathsf {k})$ be an $\\mathrm {Ad}(\\mathbf {G}(\\mathsf {k}))$ -orbit consisting of regular elements and let $\\ell \\in \\mathbb {N}$ .",
"The number of regular characters $\\chi \\in \\mathrm {Irr}(G)$ of level $\\ell $ whose residual orbit is equal to $\\Omega $ is $\\frac{\\left|\\mathbf {G}(\\mathsf {k})\\right|}{\\left|\\Omega \\right|}\\cdot q^{(\\ell -1)n}$ .",
"Any such character has degree $\\left|\\Omega \\right|\\cdot q^{(\\ell -1)\\alpha },$ where $\\alpha =\\frac{\\dim \\mathbf {G}-n}{2}$ ."
],
[
"Regular representation zeta functions",
"Taking the perspective of representation growth, given a group $H$ , one is often interested in understanding the asymptotic behaviour of the sequence $\\left\\lbrace {r_m(H)}\\right\\rbrace _{m=1}^\\infty $ , where $r_m(H)\\in \\mathbb {N}\\cup \\left\\lbrace {0,\\infty }\\right\\rbrace $ denotes the number of elements of $\\mathrm {Irr}(H)$ of degree $m$ .",
"In the case where the sequence $r_m(H)$ is bounded above by a polynomial in $m$ , the representation zeta function of $H$ is defined to be the Dirichlet generating function $\\zeta _H(s)=\\sum _{m=1}^\\infty r_m(H)m^{-s},\\quad (s\\in \\mathbb {C}).$ In the specific case $H=G=\\mathbf {G}(\\mathfrak {o})$ , one may initially restrict to a description of the regular representation zeta function, i.e.",
"the Dirichlet function counting only regular characters of $G$ .",
"In this respect, mainthm:enumeration+dimension implies that the rate of growth of regular characters of $G$ is polynomial of degree $\\frac{2n}{\\dim \\mathbf {G}-n}$ .",
"Furthermore, we obtain the following.",
"Corollary Let $X\\subseteq \\mathrm {Ad}(\\mathbf {G}(\\mathsf {k}))\\backslash {\\mathbf {g}}(\\mathsf {k})$ denote the set of orbits consisting of regular elements, and let $\\mathfrak {D}_{{\\mathbf {g}}(\\mathfrak {o})}(s)=\\sum _{\\Omega \\in X}{\\left|\\mathbf {G}(\\mathsf {k})\\right|}\\cdot \\left|\\Omega \\right|^{-(s+1)},\\quad (s\\in \\mathbb {C}).$ The regular zeta function of $G=\\mathbf {G}(\\mathfrak {o})$ is of the form $\\zeta ^\\mathrm {reg}_G(s)=\\frac{\\mathfrak {D}_{{\\mathbf {g}}(\\mathfrak {o})}(s)}{1-q^{n-\\alpha s}}$ where $n$ and $\\alpha $ are as in mainthm:enumeration+dimension."
],
[
"Classification of regular orbits in ${\\mathbf {g}}(\\mathsf {k})$",
"The second goal of this article is to compute the regular representation zeta function of the symplectic and special orthogonal groups over $\\mathfrak {o}$ .",
"In view of corol:reg-zeta-function, to do so, one must classify and enumerate the regular orbits in ${\\mathbf {g}}(\\mathsf {k})$ , under the adjoint action of $\\mathbf {G}(\\mathsf {k})$ .",
"This classification is undertaken in section:classical-groups, and its consequences are summarized in theo:orbits-sp2nso2n+1 and theo:orbits-so2n.",
"The classification of regular adjoint classes in the ${\\mathbf {g}}(\\mathsf {k})$ is closely related to the question of classifying conjugacy classes in classical groups over a finite field, a question which was solved in complete generality by Wall in [39].",
"Taking a enumerative prespective, the regular semisimple conjugacy classes in finite classical groups were enumerated in [14], using generating functions.",
"Another closely related question is that of enumerating cyclic conjugacy classes in finite classical groups.",
"This question is addressed in [15], [28]; see subsection:summary-classical for further discussion.",
"The enumeration carried out in this paper yields uniform formulae for the function $\\mathfrak {D}_{{\\mathbf {g}}(\\mathfrak {o})}$ (and, consequently, for the regular representation zeta function) of each of the classical groups in question, which are independent of the cardinality of $\\mathsf {k}$ .",
"Given $n\\in \\mathbb {N}$ let $\\mathcal {X}_n$ denote the set of triplets $\\tau =(r,S,T)$ , in which $r\\in \\mathbb {N}_0$ and $S=~(S_{d,e})$ and $T=~(T_{d,e})$ are $n\\times n$ matrices with non-negative integer entires, satisfying the condition $r+\\sum _{d,e=1}^n de\\left(S_{d,e}+T_{d,e}\\right)=n.$ Given $\\tau =(r,S,T)\\in \\mathcal {X}_n$ , define the following polynomial in $\\mathbb {Z}[t]$ $c^{\\tau }(t)&=t^n\\prod _{d,e}(1+t^{-d})^{S_{d,e}}(1-t^{-d})^{T_{d,e}}$ and let $u_1(q)=\\left|\\mathrm {Sp}_{2n}(\\mathsf {k})\\right|=\\left|\\mathrm {SO}_{2n+1}(\\mathsf {k})\\right|$ .",
"Note that the value $u_1(q)$ is given by evaluation at $t=q$ of a polynomial $u_1(t)\\in \\mathbb {Z}[t]$ , which is independent of $q$ (see, e.g., [42]).",
"Additionally, for any $\\tau \\in \\mathcal {X}_n$ , let $M_{\\tau }(q)$ denote the number of polynomials of type $\\tau $ over a field of $q$ elements; see defi:poly-type.",
"An explicit formula for $M_{\\tau }(q)$ is computed in subsubsection:enumerative.",
"We remark that the value of $M_{\\tau }(q)$ is given by evaluation at $t=q$ of a uniform polynomial formula which is independent of $q$ as well; see (REF ).",
"Theorem II Let $\\mathfrak {o}$ be a complete discrete valuation ring of odd residual characteristic.",
"Let $n\\in \\mathbb {N}$ and $\\mathbf {G}$ be one of the $\\mathfrak {o}$ -defined algebraic group schemes $\\mathrm {Sp}_{2n}$ or $\\mathrm {SO}_{2n+1}$ , with ${\\mathbf {g}}=\\mathrm {Lie}(\\mathbf {G})$ .",
"Given $\\tau =(r,S,T)\\in \\mathcal {X}_n$ let $\\nu (\\tau )=\\nu _{\\mathbf {G}}(\\tau )={\\left\\lbrace \\begin{array}{ll}1&\\text{if }\\mathbf {G}=\\mathrm {Sp}_{2n}\\text{ and }r>0,\\\\0&\\text{otherwise}.\\end{array}\\right.",
"}$ The Dirichlet polynomial $\\mathfrak {D}_{{\\mathbf {g}}(\\mathfrak {o})}(s)$ (see (REF )) is given by $\\mathfrak {D}_{{\\mathbf {g}}(\\mathfrak {o})}(s)=\\sum _{\\tau \\in \\mathcal {X}_n} 4^{\\nu (\\tau )} M_{\\tau }(q)\\cdot c^{\\tau }(q)\\cdot \\left(\\frac{ u_1(q)}{2^{\\nu (\\tau )}c^{\\tau }(q)}\\right)^{-s}.$ Recall that a symmetric bilinear form over a finite field of odd characteristic is determined by the Witt index of the form, i.e.",
"the dimension of a maximal totally isotropic subspace with respect to the form.",
"Following standard notation, we write $\\mathrm {SO}_{2n}^+$ and $\\mathrm {SO}_{2n}^-$ to the group schemes whose group of $\\mathsf {k}$ -points are associated with a symmetric bilinear form of Witt index $n$ and $n-1$ respectively.",
"Also, for convenience, we often use the notation $\\mathrm {SO}_{2n}^{\\pm 1}$ for $\\mathrm {SO}_{2n}^\\pm $ .",
"Given $\\epsilon \\in \\left\\lbrace {\\pm 1}\\right\\rbrace $ , let $u_2^\\epsilon (q)=\\left|\\mathrm {SO}_{2n}^\\epsilon (\\mathsf {k})\\right|$ .",
"As in the previous case, note that the value $u_2^{\\epsilon }(q)$ is given by evaluation at $t=q$ of a polynomial $u_2^\\epsilon (t)\\in \\mathbb {Z}[t]$ , which is independent of $q$ (see [42]) Theorem III Let $\\mathfrak {o}$ be a complete discrete valuation ring of odd residual characteristic and whose residue field has more than 3 elements.",
"Let $n\\in \\mathbb {N}$ and $\\epsilon \\in \\left\\lbrace {\\pm 1}\\right\\rbrace $ .",
"Let $\\mathbf {G}^\\epsilon =\\mathrm {SO}_{2n}^\\epsilon $ be the $\\mathfrak {o}$ -defined special orthogonal group scheme, as described above, and let ${\\mathbf {g}}^\\epsilon =\\mathrm {Lie}(\\mathbf {G}^\\epsilon )$ .",
"Let $\\mathcal {X}_n^{0}$ denote the set of triplets $\\tau =(r,S,T)\\in \\mathcal {X}_n$ with $r=0$ , and let $\\mathcal {X}_n^{0,+1}$ denote the subset of $\\mathcal {X}_n^0$ consisting of elements $(0,S,T)$ such that $\\sum _{d,e}eS_{d,e}$ is even and $\\mathcal {X}_n^{0,-1}=\\mathcal {X}_n^0\\setminus \\mathcal {X}_n^{0,+1}$ .",
"The Dirichlet polynomial $\\mathfrak {D}_{{\\mathbf {g}}(\\mathfrak {o})}$ (see (REF )) is given by $\\mathfrak {D}_{{\\mathbf {g}}^\\epsilon (\\mathfrak {o})}(s)=\\sum _{\\tau \\in \\mathcal {X}_n^{0,\\epsilon }} M_{\\tau }(q) \\cdot c^{\\tau }(q)\\cdot \\left(\\frac{u_2^\\epsilon (q)}{c^\\tau (q)}\\right)^{-s}+\\sum _{\\tau \\in \\mathcal {X}_n\\setminus \\mathcal {X}_n^{0}} 4\\cdot M_{\\tau }(q) \\cdot c^{\\tau }(q)\\cdot \\left(\\frac{u_2^\\epsilon (q)}{2\\cdot c^\\tau (q)}\\right)^{-s}.$"
],
[
"Organization",
"section:notation gathers necessary preliminary results and sets up notation.",
"section:reg-elements-and-chars contains basic structural results regarding the regular orbits of ${\\mathbf {g}}(\\mathfrak {o})$ and regular characters of $\\mathbf {G}(\\mathfrak {o})$ , and the proof of mainthm:enumeration+dimension.",
"Finally, in section:classical-groups we classify the regular adjoint orbits of ${\\mathbf {g}}(\\mathsf {k})$ and compute the regular representation zeta function of $\\mathbf {G}(\\mathfrak {o})$ ."
],
[
"The symplectic and orthogonal groups",
"Fix $N\\in \\mathbb {N}$ and a matrix ${\\mathbf {J}}\\in \\mathrm {GL}_N(\\mathfrak {o})$ such that ${\\mathbf {J}}^t=\\epsilon {\\mathbf {J}}$ , with $\\epsilon =-1$ in the symplectic case and $\\epsilon =1$ in the special orthogonal case.",
"The group scheme $\\mathbf {G}$ is defined by $\\mathbf {G}(R)=\\left\\lbrace {\\mathbf {x}\\in \\mathrm {M}_N(R)\\mid \\mathbf {x}^t{\\mathbf {J}}\\mathbf {x}={\\mathbf {J}}\\:\\text{and }\\det (\\mathbf {x})=1}\\right\\rbrace ,$ where $R$ is a commutative $\\mathfrak {o}$ -algebra and the notation $\\mathbf {x}^t$ stands for the transpose matrix of $\\mathbf {x}$ .",
"A standard computation (see, e.g.",
"[40]) shows that the Lie-algebra scheme ${\\mathbf {g}}=\\mathrm {Lie}(\\mathbf {G})$ is given by ${\\mathbf {g}}(R)=\\left\\lbrace {\\mathbf {x}\\in \\mathrm {M}_N(R)\\mid \\mathbf {x}^t{\\mathbf {J}}+{\\mathbf {J}}\\mathbf {x}=0}\\right\\rbrace .$ Let $n$ and $d$ denote the dimension and the absolute rank of the generic fiber of $\\mathbf {G}$ .",
"Note that the absolute rank and dimension of the generic fiber of $\\mathbf {G}$ are equal to those of its special fiber, by flatness of $\\mathbf {G}$ and of its maximal tori (see [1]).",
"Let $R^N$ denote the $N$ -th cartesian power of $R$ , identified with the space $\\mathrm {M}_{N\\times 1}(R)$ of column vectors, and define a non-degenerate bilinear form on $R^N$ by $B_R(u,v)=u^t{\\mathbf {J}}v$ .",
"One defines an $R$ -anti-involution on $\\mathrm {M}_N(R)=\\mathrm {End}_R(R^N)$ by $A^\\star ={\\mathbf {J}}^{-1}A^t{\\mathbf {J}}\\quad (A\\in \\mathrm {M}_N(R)),$ or equivalently, by letting $A^\\star $ be the unique matrix satisfying $B_R(A^\\star u,v)=B_R(u,Av)$ , for all $u,v\\in R^N$ .",
"In this notation, we have that $A\\in \\mathbf {G}(R)$ if and only if $\\det (A)=1$ and $A^\\star A= 1$ , and that $A\\in {\\mathbf {g}}(R)$ if and only if $A^\\star +A=0$ ."
],
[
"Maximal tori and centralizers over algebraically closed fields",
"Let $\\mathbf {T}$ be a maximal torus of $\\mathbf {G}$ and let $\\mathbf {t}\\subseteq {\\mathbf {g}}$ be its Lie-algebra.",
"Given an algebraically closed field $L$ , which is an $\\mathfrak {o}$ -algebra, we may assume that $\\mathbf {T}(L)$ is the group of $N\\times N$ diagonal matrices.",
"Moreover, upto possibly replacing ${\\mathbf {J}}$ with a congruent matrix, which amounts to conjugation of the given embedding $\\mathbf {G}\\subseteq \\mathrm {GL}_N$ by a fixed matrix over $\\mathfrak {o}$ , we may assume that $\\mathbf {T}(L)$ is mapped onto the subgroup of diagonal matrices $\\mathrm {diag}(\\nu _1,\\ldots ,\\nu _{N})$ , satisfying $\\nu _{2i}=\\nu _{2i-1}^{-1}$ for all $i=1,\\ldots ,\\lfloor N/2\\rfloor $ , and with $\\nu _N=1$ if $N$ is odd.",
"In particular, the absolute rank of the generic fiber of $\\mathbf {G}$ is $n=\\dim ( \\mathbf {T}\\times _{\\mathrm {Spec\\:}\\mathfrak {o}} \\mathrm {Spec\\:}L)=\\lfloor N/2\\rfloor $ , for $L=K^{\\mathrm {alg}}$ the algebraic closure of $K$ .",
"Under this embedding, the Lie-algebra $\\mathbf {t}(L)$ consists of diagonal matrices of the form $\\mathrm {diag}(\\nu _1,\\ldots ,\\nu _N)$ , with $\\nu _{2i}=-\\nu _{2i-1}$ for all $i=1,\\ldots ,n$ and $\\nu _N=0$ if $N$ is odd.",
"We require the following well-known result.",
"Proposition Let $s\\in {\\mathbf {g}}(L)$ be a semisimple element.",
"The centralizer of $s$ under the adjoint action of $\\mathbf {G}(L)$ is of the form $\\mathbf {C}_{\\mathbf {G}(L)}(s)\\simeq \\prod _{j=1}^t\\mathrm {GL}_{m_j}\\left(L\\right)\\times \\Delta (L),$ where $\\Delta $ is the $L$ -algebraic group of isometries of the restriction of $B_L$ to (a non-degenerate bilinear form on) $\\mathrm {Ker}(s)$ , the eigenspace associated with the eigenvalue 0, and the values $m_1,\\ldots ,m_t$ are the algebraic multiplicities of all non-zero eigenvalues of $s$ such that for any such eigenvalue $\\lambda $ , there exists a unique $j=1,\\ldots ,t$ such that $m_j$ is the algebraic multiplicity of $\\lambda $ and $-\\lambda $ .",
"Let $V=L^{N}$ the be the fixed $L$ -vector space on which $\\mathbf {G}(L)\\subseteq \\mathrm {GL}_N(L)$ acts.",
"The element $s$ is thus considered as an endomorphism of $V$ .",
"The decomposition of $V$ into eigenspaces of $s$ gives rise to a direct decomposition into isotypic $\\mathbf {C}_{\\mathrm {GL}_N(L)}(s)$ -modules, $V=\\oplus _{\\lambda \\in L} W_\\lambda $ , where $W_\\lambda =\\mathrm {Ker}(s-\\lambda \\mathbf {1})$ .",
"For any non-zero $\\lambda \\in L$ , put $W_{[\\lambda ]}=W_\\lambda \\oplus W_{-\\lambda }$ .",
"A simple computation reveals that the spaces $W_0$ and $W_{[\\lambda ]}$ are non-degenerate with respect to the ambient symmetric or anti-symmetric bilinear form.",
"Since $\\mathbf {C}_{\\mathbf {G}(L)}(s)=\\mathbf {C}_{\\mathrm {GL}_N(L)}(s)\\cap \\mathbf {G}(L)$ , it holds that $x\\in \\mathbf {C}_{\\mathbf {G}(L)}(s)$ if and only if $x\\in \\mathbf {C}_{\\mathrm {GL}_N(L)}(s)$ and $x$ acts as an isometry with respect to the restriction of $B$ to the spaces $W_0$ and $W_{[\\lambda ]}$ (for $\\lambda \\ne 0$ ).",
"Arguing as in [4], one verifies that for any $\\lambda \\ne 0$ the decomposition $W_{[\\lambda ]}=W_{\\lambda }\\oplus W_{-\\lambda }$ is into maximal isotropic subspaces, and in particular $\\dim _L W_{\\lambda }=\\dim _{L} W_{-\\lambda }=m_j$ , for some $j=1,\\ldots ,t$ .",
"Invoking Witt's Theorem [39], and the $\\mathbf {C}_{\\mathrm {GL}_N(L)}(s)$ -isotipicity of the decomposition, we obtain that any automorphism of $W_{\\lambda }$ extends uniquely to an isometric automorphism of $W_{[\\lambda ]}$ which commutes with the action of $s$ , and that the action of $\\mathbf {C}_{\\mathbf {G}(L)}(s)$ on $W_{[\\lambda ]}$ is determined in this manner.",
"Furthermore, it holds that any automorphism of $W_0=\\mathrm {Ker}(s)$ which preserves the restriction of $B$ to $W_0$ necessarily commutes with $s$ , and that the action of $\\mathbf {C}_{\\mathbf {G}(L)}(s)$ on this subspace is by such automorphisms.",
"The proposition follows.",
"Let $K^{\\mathrm {alg}}$ be a fixed algebraic closure of $K$ and let $K^\\mathrm {unr}$ be the maximal unramified extension of $K$ in $K^{\\mathrm {alg}}$ .",
"Let $\\mathfrak {O}$ be the valuation ring of $K^\\mathrm {unr}$ , and $\\mathfrak {P}=\\pi \\mathfrak {O}$ its maximal ideal.",
"The residue field of $\\mathfrak {O}$ is identified with the algebraic closure ${\\mathsf {k}^{\\mathrm {alg}}}$ of $\\mathsf {k}$ .",
"Given $r\\in \\mathbb {N}$ we put $\\mathfrak {o}_r:=\\mathfrak {o}/\\mathfrak {p}^r$ and $\\mathfrak {O}_r:=\\mathfrak {O}/\\mathfrak {P}^r$ and write $\\eta _r:\\mathfrak {O}\\rightarrow \\mathfrak {O}_r$ and $\\eta _{r,m}:\\mathfrak {O}_r\\rightarrow \\mathfrak {O}_m$ for the reduction maps, for any $1\\le m\\le r$ .",
"The notation $\\eta _r$ and $\\eta _{r,m}$ is also used to denote the coordinatewise reduction map on $\\mathrm {M}_N(\\mathfrak {O})$ and $\\mathrm {M}_N(\\mathfrak {O}_r)$ , respectively.",
"The map $\\eta _1$ admits a canonical splitting map $s:{\\mathsf {k}^{\\mathrm {alg}}}\\rightarrow \\mathfrak {O}$ , which restricts to a homomorphic embedding of $({\\mathsf {k}^{\\mathrm {alg}}})^\\times $ into $\\mathfrak {O}^\\times $ , and satisfies $s(0)=0$ ; see [29].",
"Let $\\sigma :K^{\\mathrm {unr}}\\rightarrow K^{\\mathrm {unr}}$ be the local Frobenius map whose fixed field is $K$ .",
"Then $\\sigma $ restricts to a ring automorphism of $\\mathfrak {O}$ , with fixed subring $\\mathfrak {O}^\\sigma =\\mathfrak {o}$ , and induces a map $\\mathfrak {O}_r\\rightarrow \\mathfrak {O}_r$ for any $r\\ge 1$ whose fixed subring is $\\mathfrak {o}_r$ .",
"In the special case $r=1$ , the map $\\sigma :{\\mathsf {k}^{\\mathrm {alg}}}\\rightarrow {\\mathsf {k}^{\\mathrm {alg}}}$ is given by the $q$ -power map $x\\mapsto x^q$ , where $q=\\left|\\mathsf {k}\\right|$ ."
],
[
"The Greenberg functor",
"The Greenberg functor was introduced in [16] and [17], as a generalization of Shimura's reduction mod $\\mathfrak {p}$ functor to higher powers of $\\mathfrak {p}$ .",
"Given an artinian local principal ideal ring $R$ (or more generally, an artinian local ring) with a perfect residue field $\\mathfrak {k}$ , the Greenberg functor $\\mathcal {F}_R$ associates to any $R$ -scheme $\\mathbf {Y}$ locally of finite type a scheme $\\mathcal {F}_R(\\mathbf {Y})$ locally of finite type over the residue field $\\mathfrak {k}$ .",
"Given another such ring $R^{\\prime }$ with residue field $\\mathfrak {k}$ and a ring homomorphism $R\\rightarrow R^{\\prime }$ , the functors $\\mathcal {F}_R$ and $\\mathcal {F}_{R^{\\prime }}$ are related via connecting morphisms, on which we expand further below.",
"A defining property of the functor is the existence of a canonical bijection $\\mathcal {F}_{R}(\\mathbf {Y})(\\mathfrak {k})=\\mathbf {Y}(R).$ More generally, if $A$ is a perfect commutative unital $\\mathfrak {k}$ -algebra, then either $\\mathcal {F}_R(\\mathbf {Y})(A)=\\mathbf {Y}(R\\otimes _{\\mathfrak {k}} A)$ , in the case where $R$ is a $\\mathfrak {k}$ -algebra, or otherwise $\\mathcal {F}_{R}(\\mathbf {Y})(A)=\\mathbf {Y}(R\\otimes _{W(\\mathfrak {k})}W(A))$ where $W(\\cdot )$ denotes the ring of $p$ -typical Witt vectors [29].",
"For further introduction we refer to [6].",
"Our application of the Greenberg functor is focused on the artininan principal ideal rings $\\mathfrak {O}_r$ .",
"For any $r$ , we let $\\mathbf {G}_{\\mathfrak {O}_r}=\\mathbf {G}\\times \\mathfrak {O}_r$ and ${\\mathbf {g}}_{\\mathfrak {O}_r}={\\mathbf {g}}\\times \\mathfrak {O}_r$ denote the base change of the group and Lie-algebra schemes $\\mathbf {G}$ and ${\\mathbf {g}}$ .",
"Put $\\mathbf {\\Gamma }_r=\\mathcal {F}_{\\mathfrak {O}_r}(\\mathbf {G}_{\\mathfrak {O}_r})$ and $_r=\\mathcal {F}_{\\mathfrak {O}_r}({\\mathbf {g}}_{\\mathfrak {O}_r})$ .",
"Given $m\\le r$ , we write $\\eta _{r,m}^*$ to denote the connecting maps $\\mathbf {\\Gamma }_r\\rightarrow \\mathbf {\\Gamma }_m$ and $_r\\rightarrow _m$ , and put $\\mathbf {\\Gamma }^m_r=(\\eta _{r,m}^*)^{-1}(1)=\\mathrm {Spec\\:}(\\kappa (1))\\times _{\\mathbf {\\Gamma }_m}\\mathbf {\\Gamma }_r$ (the scheme-theoretic group kernel) and $_{r}^m=(\\eta _{r,m}^*)^{-1}(0)=\\mathrm {Spec\\:}(\\kappa (0))\\times _{_m}_r$ (the scheme-theoretic Lie-algebra kernel).",
"Here, the notation $\\kappa (\\cdot )$ stands for the residue field at a rational point of a scheme.",
"Note that, a priori, the connecting morphism between a scheme and its base change is dependent on the scheme in question as well.",
"The apparent abuse of notation in writing $\\eta _{r,m}^*$ for the connecting morphisms of different schemes is permissible by [16], applied for $g$ the inclusion morphism (see also Assertion 2 of the Main Theorem of loc.",
"cit.).",
"The main properties which we require are summarized in the following lemma.",
"Lemma For $r\\in \\mathbb {N}$ fixed, we have The rings $\\mathfrak {O}_r$ are the ${\\mathsf {k}^{\\mathrm {alg}}}$ -points of an $r$ -dimensional algebraic ring scheme $\\mathbf {O}_r$ over ${\\mathsf {k}^{\\mathrm {alg}}}$ .",
"The canonical map $s:{\\mathsf {k}^{\\mathrm {alg}}}\\rightarrow \\mathfrak {O}_r$ defines a closed embedding $s^*:\\mathbb {A}_{\\mathsf {k}^{\\mathrm {alg}}}^1\\rightarrow \\mathbf {O}_r$ of the affine line over ${\\mathsf {k}^{\\mathrm {alg}}}$ into this ring variety.",
"The restriction of $s^*$ to the multiplicative group $\\mathbb {G}_m\\subseteq \\mathbb {A}_{{\\mathsf {k}^{\\mathrm {alg}}}}^1$ is a monomorphism of ${\\mathsf {k}^{\\mathrm {alg}}}$ -linear algebraic groups, satisfying $\\eta _{r,1}^*\\circ s^*=\\mathbf {1}_{\\mathbb {A}_{{\\mathsf {k}^{\\mathrm {alg}}}}^1}$ .",
"The group $\\mathbf {\\Gamma }_r$ is a $d\\cdot r$ -dimensional linear algebraic group over ${\\mathsf {k}^{\\mathrm {alg}}}$ .",
"The Greenberg functor maps smooth closed sub-$\\mathfrak {O}_r$ -group schemes of $\\mathbf {G}_{\\mathfrak {O}_r}$ to closed algebraic ${\\mathsf {k}^{\\mathrm {alg}}}$ -subgroups of $\\mathbf {\\Gamma }_r$ .",
"The scheme $_r$ is a $d\\cdot r$ -dimensional affine space over ${\\mathsf {k}^{\\mathrm {alg}}}$ , which is naturally endowed with the structure of a Lie-algebra scheme over the ring scheme $\\mathbf {O}_r$ .",
"The connecting morphisms $\\eta _{r,m}^*$ , for $m\\le r$ , give rise to surjective ${\\mathsf {k}^{\\mathrm {alg}}}$ -group scheme $\\mathbf {\\Gamma }_r\\rightarrow ~\\mathbf {\\Gamma }_m$ morphisms.",
"Similarly, for $_r\\rightarrow _m$ , these are surjective Lie-ring morphisms.",
"The adjoint action of $\\mathbf {G}_{\\mathfrak {O}_r}$ on the Lie-ring scheme ${\\mathbf {g}}_{\\mathfrak {O}_m}$ with $m\\le ~r$ , induces an action of the algebraic group $\\mathbf {\\Gamma }_r$ on $_m$ .",
"The application of $\\mathcal {F}_{\\mathfrak {O}_r}$ preserves centralizers of $\\mathfrak {O}_m$ -rational points of ${\\mathbf {g}}_{\\mathfrak {O}_m}$ .",
"1.",
"See [16].",
"2.",
"The group $\\mathbf {\\Gamma }_r$ is a smooth affine group scheme of finite type over ${\\mathsf {k}^{\\mathrm {alg}}}$ (see [16] and [17]).",
"Thus, by [40], $\\mathbf {\\Gamma }_r$ is a linear algebraic ${\\mathsf {k}^{\\mathrm {alg}}}$ -group (see also [33]).",
"The dimension of $\\mathbf {\\Gamma }_r$ may be computed by induction on $r$ , using Greenberg's Structure Theorem [17] (see remark on p. 266 of loc.",
"cit.",
"; also, see [3] for an explicit argument in the case where $r$ is divisible by the absolute ramification index of $\\mathfrak {o}$ ).",
"3.",
"Let $\\Delta \\subseteq \\mathbf {G}_{\\mathfrak {O}_r}$ be a closed smooth sub-$\\mathfrak {O}_r$ -scheme.",
"The argument of the previous assertion shows that $\\mathcal {F}_{\\mathfrak {O}_r}(\\Delta )$ is a linear algebraic group over ${\\mathsf {k}^{\\mathrm {alg}}}$ .",
"That $\\mathcal {F}_{\\mathfrak {O}_r}(\\Delta )$ is a closed subgroup of $\\mathbf {\\Gamma }_r$ follows from Assertions (2) and (5) of the Main Theorem of [16].",
"4.",
"The Lie-algebra scheme ${\\mathbf {g}}_{\\mathfrak {O}_r}$ is isomorphic to the affine $d$ -dimensional space $\\mathbb {A}^d_{\\mathfrak {O}_r}$ over $\\mathfrak {O}_r$ , and is endowed with $\\mathfrak {O}_r$ -regular maps, defining an $\\mathfrak {O}_r$ -module structure and an $\\mathfrak {O}_r$ -bilinear Lie-bracket on ${\\mathbf {g}}_{\\mathfrak {O}_r}$ .",
"It follows from the Main Theorem [16], that $_r=\\mathcal {F}_{\\mathfrak {O}_r}({\\mathbf {g}}_{\\mathfrak {O}_r})$ is isomorphisc to $\\mathbb {A}_{{\\mathsf {k}^{\\mathrm {alg}}}}^{dr}$ , the affine space of dimension $d\\cdot r$ over ${\\mathsf {k}^{\\mathrm {alg}}}$ .",
"Multiplication by scalars from $\\mathbf {O}_r$ , the Lie-bracket and addition on ${\\mathbf {g}}_{\\mathfrak {O}_r}$ are transported by $\\mathcal {F}_{\\mathfrak {O}_r}$ to schematic morphisms $\\mathbf {O}_r\\times _r\\rightarrow _r$ and ${_r\\times _r}\\rightarrow _r$ by [16].",
"The Lie-axioms on $\\mathcal {F}_{\\mathfrak {O}_r}(_r)$ may be verified using compatibility of the Greenberg functor with preimages [16].",
"For example, the Jacobi identity can be reformulated using the equality ${\\mathbf {g}}_{\\mathfrak {O}_r}\\times {\\mathbf {g}}_{\\mathfrak {O}_r}\\times {\\mathbf {g}}_{\\mathfrak {O}_r}=J^{-1}(0)$ , where $J:{\\mathbf {g}}_{\\mathfrak {O}_r}\\times {\\mathbf {g}}_{\\mathfrak {O}_r}\\times {\\mathbf {g}}_{\\mathfrak {O}_r}\\rightarrow {\\mathbf {g}}_{\\mathfrak {O}_r}$ is the morphism satisfying $J(R)(x,y,z)=[[x,y],z]+[[y,z],x]+[[z,x],y]$ for any $\\mathfrak {O}_r$ -algebra $R$ and $x,y,z\\in {\\mathbf {g}}_{\\mathfrak {O}_r}(R)$ .",
"5.",
"The connecting map is shown to be a group homomorphism in [16], and the preservation of the Lie-bracket follows similarly from [16].",
"Its surjectivity follows from the smoothness of $\\mathbf {G}_{\\mathfrak {O}_r}$ (resp.",
"$_{\\mathfrak {O}_r}$ ), and [17].",
"6.",
"The action of $\\mathbf {\\Gamma }_r$ on $_m$ is given by $\\mathcal {F}_{\\mathfrak {O}_m}(\\alpha _{m})\\circ (\\eta _{r,m}^*\\times \\mathbf {1}_{_m}):\\mathbf {\\Gamma }_r\\times \\gamma _m\\rightarrow _m$ , where $\\alpha _{m}:\\mathbf {G}_{\\mathfrak {O}_m}\\times {\\mathbf {g}}_{\\mathfrak {O}_m}\\rightarrow {\\mathbf {g}}_{\\mathfrak {O}_m}$ is the adjunction map; see [33].",
"One notes easily that, since the group $\\mathbf {\\Gamma }^m_r$ acts trivially on $_m$ , this action commutes pointwise with the bijection (REF ).",
"The preservation of centralizers follows from [33], by taking $\\mathbf {Y}$ and $\\mathbf {Z}$ to be the sub-schemes defined by the spectrum of the reside field of ${\\mathbf {g}}_{\\mathfrak {O}_m}$ at the given rational point.",
"Remark In the case where $\\mathfrak {O}_r$ is a ${\\mathsf {k}^{\\mathrm {alg}}}$ -algebra, lem:greenberg-props.",
"(3) may be somewhat strengthened, as in this case $\\gamma _r$ can be shown to coincide with the Lie-algebra of $\\mathbf {\\Gamma }_r$ .",
"In the case of unequal characteristic, the equality $_r=\\mathrm {Lie}(\\mathbf {\\Gamma }_r)$ is generally false.",
"For example, in the case of $\\mathbf {G}=\\mathbb {G}_a$ , the additive group scheme, we have that $_2({\\mathsf {k}^{\\mathrm {alg}}})=\\mathrm {Lie}(\\mathbb {G}_a)(W_2({\\mathsf {k}^{\\mathrm {alg}}}))=W_2({\\mathsf {k}^{\\mathrm {alg}}})$ is a ring of characteristic $p^2$ , while $\\mathrm {Lie}(\\mathbf {\\Gamma }_2)({\\mathsf {k}^{\\mathrm {alg}}})$ is a two-dimensional ${\\mathsf {k}^{\\mathrm {alg}}}$ -Lie-algebra and, in particular, has $p$ -torsion.",
"We also require the following lemma.",
"Lemma For any $m,r\\in \\mathbb {N}$ with $m\\le r$ , there exists an injective homomorphism of the underlying additive group schemes $v_{r,m}^*:_{r-m}\\rightarrow _r$ , such that for any $y\\in _{r-m}({\\mathsf {k}^{\\mathrm {alg}}})={\\mathbf {g}}(\\mathfrak {O}_{r-m})$ , it holds that $v_{r,m}^*({\\mathsf {k}^{\\mathrm {alg}}})(y)=\\pi ^{m}\\tilde{y}$ , where $\\tilde{y}\\in {\\mathbf {g}}(\\mathfrak {O}_r)$ is such that $\\eta _{r,r-m}(\\tilde{y})=y$ ; the sequence $0\\rightarrow _{r-m}\\xrightarrow{}_{r}\\xrightarrow{}_{m}\\rightarrow 0$ is exact; for any $y\\in _{r-m}({\\mathsf {k}^{\\mathrm {alg}}})$ the square (REF ) commutes $\\begin{tikzpicture}(m) [matrix of math nodes,row sep=2em,column sep=4.5em,minimum width=2em,nodes={minimum height=2em,text height=1.5ex,text depth=.25ex,anchor=center}]{_r&_r\\\\_{r-m}&_{r-m},\\\\};[->]{(m-1-1) edge[thin] node[auto] {\\mathrm {ad}(v_{r,m}^*(y))} (m-1-2)(m-2-1) edge[thin] node[auto] {\\mathrm {ad}(y)} (m-2-2)(m-1-1) edge[thin] node[left] {\\eta _{r,r-m}^*} (m-2-1)(m-2-2) edge[thin] node[right] {v_{r,m}^*} (m-1-2)};\\end{tikzpicture}$ where $\\mathrm {ad}(z):_j\\times _j\\rightarrow _j$ (for $j\\in \\mathbb {N}$ and $z\\in _j({\\mathsf {k}^{\\mathrm {alg}}})$ ) is the map defined by $\\mathrm {ad}(z)(A)(x)=[z,x]$ for any commutative unital ${\\mathsf {k}^{\\mathrm {alg}}}$ -algebra $A$ and $x\\in _j(A)$ ; The equality $\\eta _{r,m+1}^*\\circ v_{r,m}^*=v_{m+1,1}^*\\circ \\eta _{r-m,1}^*$ holds.",
"The map $x\\mapsto \\pi ^{m}x\\colon {\\mathbf {g}}(\\mathfrak {o}_r)\\rightarrow {\\mathbf {g}}(\\mathfrak {o}_r)$ gives rise to an injective $\\mathfrak {o}_r$ -module map $v_{r,m}:{\\mathbf {g}}(\\mathfrak {o}_{r-m})\\rightarrow {\\mathbf {g}}(\\mathfrak {o}_r)$ , which in turn extends to a map of $\\mathfrak {O}_r$ -modules, giving rise to the exact sequence $0\\rightarrow {\\mathbf {g}}_{\\mathfrak {O}_{r-m}}(\\mathfrak {O}_r)\\xrightarrow{}{\\mathbf {g}}_{\\mathfrak {O}_r}(\\mathfrak {O}_r)\\xrightarrow{}{\\mathbf {g}}_{\\mathfrak {O}_{m}}(\\mathfrak {O}_r)\\rightarrow 0.$ Applying [16] to both maps of the above sequence, these define $\\mathsf {k}$ -regular maps between associated module variety structures over ${\\mathsf {k}^{\\mathrm {alg}}}$ of the modules above, which, in turn, define an exact sequence of ${\\mathsf {k}^{\\mathrm {alg}}}$ -schemes $0\\rightarrow _{r-m}\\xrightarrow{}_r\\xrightarrow{}_{m}\\rightarrow 0,$ where the right-most map coincides with $\\eta _{r,m}^*$ by same proposition and by [16].",
"The first and second assertions of the lemma follow.",
"As for the third assertion, in order to prove that the morphisms ${v_{r,m}^*\\circ \\mathrm {ad}(y)\\circ \\eta _{r,r-m}^*}$ and $\\mathrm {ad}(v_{r,m}(y))$ coincide, it is enough to show that, upon passing to their associated comorphisms, they induce the same endomorphism of the coordinate ring of $_r$ .",
"Invoking the isomorphism $_r\\simeq \\mathbb {A}_{{\\mathsf {k}^{\\mathrm {alg}}}}^{dm}$ of lem:greenberg-props.",
"(3), since an endomorphism of a polynomial algebra in $dm$ variables is determined by specifying the images of $t_1,\\ldots ,t_{dm}$ in ${\\mathsf {k}^{\\mathrm {alg}}}[t_1,\\ldots ,t_{dm}]$ , by Nullstellensatz, it is enough to show that the two endomorphisms above coincide pointwise on $_r({\\mathsf {k}^{\\mathrm {alg}}})$ .",
"This is immediate by the first two assertions and the linearity of $\\mathrm {ad}(\\cdot )$ over $\\mathfrak {O}_r$ .",
"The fourth assertion may be proved in a similar vein as Assertion (3).",
"Remark In the case where $\\mathfrak {O}$ is either a ${\\mathsf {k}^{\\mathrm {alg}}}$ -algebra, or is absolutely unramified (i.e.",
"$\\mathfrak {P}=p\\mathfrak {O}$ ), and thus isomorphic to the ring $W({\\mathsf {k}^{\\mathrm {alg}}})$ , the map $v_{r,m}^*$ of the lemma may be described explicitly, by fixing a suitable coordinate system for $_r$ over ${\\mathsf {k}^{\\mathrm {alg}}}$ and taking $v^*_{r,m}$ to be either a coordinate shift in the former case, or given by successive applications of the verschiebung and Frobenius maps coordinatewise (see [29]) in the latter."
],
[
"The Cayley map",
"Let $D$ be the affine $\\mathfrak {o}$ -scheme $\\mathrm {Spec\\:}(\\mathfrak {o}[t_{1,1},\\ldots ,t_{N,N},(\\det (\\mathbf {t}+1))^{-1}])$ , where $t_{1,1},\\ldots ,t_{N,N}$ are indeterminates and $\\mathbf {t}+1$ is the $N\\times N$ matrix whose $(i,j)$ -th entry is $t_{i,j}+\\delta _{i,j}$ , with $\\delta _{i,j}$ the Kronecker delta function.",
"Note that for any commutative unital $\\mathfrak {o}$ -algebra $R$ , the set of $R$ -points of $D$ is naturally identified with the set $\\left\\lbrace {\\mathbf {x}\\in \\mathrm {M}_N(R)\\mid \\det \\left(1+\\mathbf {x}\\right)\\in R^\\times }\\right\\rbrace .$ Let $\\mathrm {cay}:D\\rightarrow D$ be the $\\mathfrak {o}$ -scheme morphism with associated comorphism $\\mathrm {cay}^\\sharp $ given on generators of $\\mathfrak {o}[t_{1,1},\\ldots ,t_{N,N},(\\det (1+\\mathbf {t}))^{-1}]$ by mapping $t_{i,j}$ to the $(i,j)$ -th entry of the matrix $(1-\\mathbf {t})(1+\\mathbf {t})^{-1}$ .",
"Note that $\\mathrm {cay}^\\sharp (\\det (1+\\mathbf {t})^{-1})=2^{-N}\\det (1+\\mathbf {t})$ .",
"A direct computation shows that, as 2 is invertible in $\\mathfrak {o}$ , the map $\\mathrm {cay}^\\sharp $ is its own inverse and thus $\\mathrm {cay}$ is an isomorphism of $D$ onto itself.",
"Under the identification (REF ) for $R$ an $\\mathfrak {o}$ -algebra as above, the action of $\\mathrm {cay}$ on the set of $R$ -points of $D$ is given explicitly by $\\mathrm {cay}(R)(\\mathbf {x})=(1-\\mathbf {x})(1+\\mathbf {x})^{-1}.$ In the specific case $R={\\mathsf {k}^{\\mathrm {alg}}}$ , the sets $(D\\cap {\\mathbf {g}})({\\mathsf {k}^{\\mathrm {alg}}})$ and $(D\\cap \\mathbf {G})({\\mathsf {k}^{\\mathrm {alg}}})$ are principal open subsets of $({\\mathsf {k}^{\\mathrm {alg}}})$ and $\\mathbf {\\Gamma }({\\mathsf {k}^{\\mathrm {alg}}})$ respectivelyHere $\\cap $ denotes the scheme-theoretic intersection, $D\\cap {\\mathbf {g}}=D\\times _{\\mathrm {Spec\\:}(\\mathfrak {o}[t_{1,1},\\ldots ,t_{N,N}])}$ .",
"Note that, in the present setting, as ${\\mathbf {g}}\\subseteq \\mathrm {M}_N\\times \\mathfrak {o}=\\mathrm {Spec\\:}(\\mathfrak {o}[t_{1,1},\\ldots ,t_{N,N}])$ , for any $\\mathfrak {o}$ -algebra $R$ , $(D\\cap {\\mathbf {g}})(R)$ is simply the set of matrices $\\mathbf {x}\\in {\\mathbf {g}}(R)$ such that the matrix $\\mathbf {1}+\\mathbf {x}$ is invertible.",
"Likewise for $D\\cap \\mathbf {G}$ , using the inclusions $\\mathbf {G}\\subseteq \\mathrm {SL}_N\\times \\mathfrak {o}\\subseteq \\mathrm {M}_N\\times \\mathfrak {o}$ ..",
"Using the description of ${\\mathbf {g}}$ and $\\mathbf {G}$ given in subsubsection:adjoint-operators, one verifies that the restriction of $\\mathrm {cay}({{\\mathsf {k}^{\\mathrm {alg}}}})$ to $(D\\cap {\\mathbf {g}})({\\mathsf {k}^{\\mathrm {alg}}})$ defines an algebraic isomorphism of affine varieties onto $(D\\cap \\mathbf {G})({\\mathsf {k}^{\\mathrm {alg}}})$ , and hence a birational map $\\mathrm {cay}({{\\mathsf {k}^{\\mathrm {alg}}}}):{\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})\\dashrightarrow \\mathbf {G}({\\mathsf {k}^{\\mathrm {alg}}})$ .",
"Additionally, being given by a rational function in $\\mathbf {x}$ on $(D\\cap {\\mathbf {g}})({\\mathsf {k}^{\\mathrm {alg}}})$ , the map $\\mathrm {cay}({\\mathsf {k}^{\\mathrm {alg}}})$ is equivariant with respect to the conjugation action of $\\mathbf {G}({\\mathsf {k}^{\\mathrm {alg}}})$ .",
"The properties listed in this paragraph carry over to the associated ${\\mathsf {k}^{\\mathrm {alg}}}$ -group schemes described in the previous section, as noted in lem:props-of-cayley-map below.",
"The Cayley map was introduced in [10].",
"Its generalization to groups arising as the set of unitary transformations with respect an anti-involution of an associative algebra is attributed to A. Weil [41].",
"See also [24] for a more generalized treatment of the Cayley map."
],
[
"Properties of the Cayley map",
"Given $r\\in \\mathbb {N}$ , put $D_r=D\\times \\mathfrak {O}_r$ and let $\\mathrm {cay}_r=\\mathrm {cay}\\times \\mathbf {1}_{\\mathfrak {O}_r}$ be the base change of $\\mathrm {cay}$ .",
"Let $\\Delta _r=\\mathcal {F}_{\\mathfrak {O}_r}(D_r)$ and $\\widehat{\\mathrm {cay}}_r=\\mathcal {F}_{\\mathfrak {O}_r}(\\mathrm {cay}_r)$ .",
"Note that, by construction and by the Main Theorem of [16], $\\Delta _{r}$ is an open affine subscheme of $\\mathbb {A}_{{\\mathsf {k}^{\\mathrm {alg}}}}^{N^2m}$ .",
"Lemma Let $1\\le m\\le r$ .",
"The map $\\widehat{\\mathrm {cay}}_r$ has the following properties.",
"The map $\\widehat{\\mathrm {cay}}_r$ is a birational equivalence $_r\\dashrightarrow \\mathbf {\\Gamma }_r$ .",
"Furthermore, its restriction to the kernel $_r^m$ is an isomorphism of ${\\mathsf {k}^{\\mathrm {alg}}}$ -varieties onto $\\mathbf {\\Gamma }_r^m$ , and is an isomorphism of abelian groups if $2m\\ge r$ .",
"The map $\\widehat{\\mathrm {cay}}_r$ is $\\mathbf {\\Gamma }_r$ -equivariant with respect to the action given in lem:greenberg-props.",
"(6) on $_r$ and with respect to group conjugation on $\\mathbf {\\Gamma }_r$ .",
"The diagram in (REF ) commutes.",
"$\\begin{tikzpicture}(m) [matrix of math nodes,row sep=3em,column sep=4.5em,minimum width=2em,nodes={minimum height=2em,text height=1.5ex,text depth=.25ex,anchor=center}]{_r&\\mathbf {\\Gamma }_r\\\\_m&\\mathbf {\\Gamma }_m.\\\\};[->]{(m-1-1) edge[dashed] node[auto] {\\widehat{\\mathrm {cay}}_r} (m-1-2)(m-2-1) edge[dashed] node[auto] {\\widehat{\\mathrm {cay}}_m} (m-2-2)(m-1-1) edge[thin] node[left] {\\eta _{r,m}^*} (m-2-1)(m-1-2) edge[thin] node[auto] {\\eta _{r,m}^*} (m-2-2)};\\end{tikzpicture}$ 1.",
"The inclusion map $D_r\\cap {\\mathbf {g}}_{\\mathfrak {O}_r}\\subseteq {\\mathbf {g}}_{\\mathfrak {O}_r}$ is an open immersion, and thus by Assertion (2) and (3) of the Main Theorem of [16], the ${\\mathsf {k}^{\\mathrm {alg}}}$ -scheme $\\Delta _r\\cap _r$ is immersed as an open subscheme of $_r$ .",
"Similarly for $\\Delta _r \\cap \\mathbf {\\Gamma }_r$ .",
"By functoriality, the morphism $\\widehat{\\mathrm {cay}}_r$ is an isomorphism of $\\Delta _r\\cap _r$ onto $\\Delta _r\\cap \\mathbf {\\Gamma }_r$ , and hence $_r$ and $\\mathbf {\\Gamma }_r$ are birationally equivalent.",
"To prove that $\\widehat{\\mathrm {cay}}_r$ restricts to an isomorphism of $^m_r$ onto $\\mathbf {\\Gamma }^m_r$ , it would be enough that both are embedded as sub-schemes of $\\Delta _r$ under the given inclusions into $\\mathbb {A}_{{\\mathsf {k}^{\\mathrm {alg}}}}^{N^2m}$ .",
"Note that by applying Greenberg's Structure Theorem [17] inductively, both $^m_r$ and $\\mathbf {\\Gamma }_r^m$ are reduced, and thus are ${\\mathsf {k}^{\\mathrm {alg}}}$ -varieties.",
"Thus, by Nullstellensatz, they are determined by their ${\\mathsf {k}^{\\mathrm {alg}}}$ -points and it suffices to show they are included in the reduced subscheme $(\\Delta _r)_{\\mathrm {red}}\\subseteq \\Delta _r$ .",
"This follows from the bijection (REF ), as $^m_r({\\mathsf {k}^{\\mathrm {alg}}})={\\mathbf {g}}_{\\mathfrak {O}_r}(\\mathfrak {O}_r)\\cap \\eta _{r,m}^{-1}(0)$ is included in the nilradical of the matrix algebra $\\mathrm {M}_N(\\mathfrak {O}_r)$ , and hence included in $D_r(\\mathfrak {O}_r)$ , and since $\\mathbf {\\Gamma }_r^m({\\mathsf {k}^{\\mathrm {alg}}})=\\mathbf {G}_{\\mathfrak {O}_r}(\\mathfrak {O}_r)\\cap \\eta _{r,m}^{-1}(1)\\subseteq 1+\\pi \\mathrm {M}_N(\\mathfrak {O}_r)\\subseteq \\mathrm {GL}_N(\\mathfrak {O}_r)$ , and thus (since ${\\mathrm {char}}( {\\mathsf {k}^{\\mathrm {alg}}})\\ne 2$ ) included in $D_r(\\mathfrak {O}_r)$ .",
"Lastly, to prove that $\\widehat{\\mathrm {cay}}_r$ is a group homomorphism whenever $2m\\ge r$ , it is equivalent to show that it preserves comultiplication in the Hopf-algebra structure of the coordinate ring of $_r^m$ in this case.",
"Arguing as in the proof lem:verschiebung, it sufficient to verify this on the ${\\mathsf {k}^{\\mathrm {alg}}}$ -points of the variety.",
"This follows from the definition of $\\mathrm {cay}$ (REF ), as in this case $_r^m({\\mathsf {k}^{\\mathrm {alg}}})\\subseteq {\\mathbf {g}}_{\\mathfrak {O}_r}(\\mathfrak {O}_r)$ is included in an ideal of vanishing square in $\\mathrm {M}_N(\\mathfrak {O}_r)$ and the map $\\widehat{\\mathrm {cay}}_r$ coincides with the map $\\mathbf {x}\\mapsto 1-2\\mathbf {x}$ .",
"2.",
"Property (REF ) holds since $\\mathcal {F}_{\\mathfrak {O}_r}$ maps the cartesian square (REF ), which states the $\\mathbf {G}_{\\mathfrak {O}_r}$ -equivariance of $\\mathrm {cay}_r$ , to a corresponding cartesian square, stating the $\\mathbf {\\Gamma }_r$ -equivariance of $\\widehat{\\mathrm {cay}}_r$ .",
"$\\begin{tikzpicture}(m) [matrix of math nodes,row sep=3em,column sep=7em,minimum width=2em,nodes={minimum height=2em,text height=1.5ex,text depth=.25ex,anchor=center}]{\\mathbf {G}_{\\mathfrak {O}_r}\\times {\\mathbf {g}}_{\\mathfrak {O}_r}&{\\mathbf {g}}_{\\mathfrak {O}_r}\\\\\\mathbf {G}_{\\mathfrak {O}_r}\\times \\mathbf {G}_{\\mathfrak {O}_r}&\\mathbf {G}_{\\mathfrak {O}_r}\\\\};\\node (1,1) [right] {\\square };[->]{(m-1-1) edge[thin] node[auto] {\\alpha _{\\mathbf {G}_{\\mathfrak {O}_r},{\\mathbf {g}}_{\\mathfrak {O}_r}}} (m-1-2)(m-2-1) edge[thin] node[auto] {\\alpha _{\\mathbf {G}_{\\mathfrak {O}_r},\\mathbf {G}_{\\mathfrak {O}_r}}} (m-2-2)(m-1-1) edge[dashed] node[left] {\\mathbf {1}_{\\mathbf {G}_{\\mathfrak {O}_r}}\\times \\mathrm {cay}_r} (m-2-1)(m-1-2) edge[dashed] node[auto] {\\mathrm {cay}_r} (m-2-2)};\\end{tikzpicture}$ Here $\\alpha _{\\mathbf {G}_{\\mathfrak {O}_r},\\mathbf {X}}$ denotes the action map of $\\mathbf {G}_{\\mathfrak {O}_r}$ on $\\mathbf {X}\\in \\left\\lbrace {\\mathbf {G}_{\\mathfrak {O}_r},{\\mathbf {g}}_{\\mathfrak {O}_r}}\\right\\rbrace $ by either conjugation or by the adjoint action.",
"3.",
"Finally, property (REF ) is simply an application of [16], to the case $R=~\\mathfrak {o}_r$ , $R^{\\prime }=\\mathfrak {o}_{m},\\:\\varphi =\\eta _{r,m}$ , $X_1=_r\\cap D_r$ , $X_2=\\mathbf {\\Gamma }_r\\cap D_r$ and $g=\\mathrm {cay}_r$ ."
],
[
"Groups, Lie algebras and characters",
"In general, given finite groups $\\Delta \\subseteq \\Gamma $ and characters $\\sigma \\in \\mathrm {Irr}(\\Delta )$ and $\\chi \\in \\mathrm {Irr}(\\Gamma )$ , we denote by $\\chi _\\Delta $ the restriction of $\\chi $ to $\\Delta $ , and by $\\sigma ^\\Gamma $ the character induced from $\\sigma $ in $\\Gamma $ .",
"Group commutators are denoted by $(x,y)=xyx^{-1}y^{-1}$ .",
"Lie-algebra commutators are denoted by $[x,y]=xy-yx$ .",
"The center of a group $\\Gamma $ is denoted by $\\mathbf {Z}(\\Gamma )$ .",
"The Pontryagin dual of a finite abelian group $\\Delta $ is denoted by $\\widehat{\\Delta }=\\mathrm {Hom}(\\Delta ,\\mathbb {C}^\\times )$ .",
"If $\\Delta $ is endowed with an additional structure (e.g.",
"a ring or a Lie-algebra), then $\\widehat{\\Delta }$ refers to the Pontryagin dual of the abelian group underlying $\\Delta $ ."
],
[
"Regular elements",
"We begin our analysis of regular characters by inspecting the group $\\mathbf {G}(\\mathfrak {O})$ .",
"To do so, we first consider the regular orbits for the action of $\\mathbf {G}(\\mathfrak {O}_r)$ on ${\\mathbf {g}}(\\mathfrak {O}_r)$ , or, equivalently (see [33]), of $\\mathbf {\\Gamma }_r({\\mathsf {k}^{\\mathrm {alg}}})$ on $_r({\\mathsf {k}^{\\mathrm {alg}}})$ , via the action described in lem:greenberg-props.(6).",
"The methods which we apply are influenced by [18].",
"Recall that an element of a reductive algebraic group over an algebraically closed field is said to be regular if its centralizer is an algebraic group of minimal dimension among such centralizers [36].",
"Following [18], this definition is extended to elements of $_r$ .",
"Definition Let $r\\ge 1$ .",
"An element $x\\in {\\mathbf {g}}(\\mathfrak {O}_r)$ is said to be regular if the group scheme $\\mathcal {F}_{\\mathfrak {O}_r}\\left(\\mathbf {C}_{\\mathbf {G}_{\\mathfrak {O}_r}}(x)\\right)=\\mathbf {C}_{\\mathbf {\\Gamma }_r}(x)$ , obtained by applying the Greenberg functor to the centralizer group scheme of $x$ in $\\mathbf {G}_{\\mathfrak {O}_r}$ , is of minimal dimension among such group schemes.",
"The following theorem lists the main properties of regular elements of $_r$ , which are proved in this section.",
"Theorem 1.1 Let $\\mathbf {G}$ be a symplectic or a special orthogonal group scheme over a complete discrete valuation ring $\\mathfrak {o}$ of odd residue field characteristic, with Lie-algebra ${\\mathbf {g}}=\\mathrm {Lie}(\\mathbf {G})$ .",
"Fix $r\\in \\mathbb {N}$ and let $x\\in {\\mathbf {g}}(\\mathfrak {O}_r)$ .",
"If $x_r$ is a regular element of ${\\mathbf {g}}_{\\mathfrak {O}_r}(\\mathfrak {O}_r)$ , then $\\mathbf {C}_{\\mathbf {\\Gamma }_{r}}(x_r)$ is a ${\\mathsf {k}^{\\mathrm {alg}}}$ -group scheme of dimension $r\\cdot n$ , where $n=~\\mathrm {rk}(\\mathbf {G}\\times ~ K^{{\\mathrm {alg}}})$ .",
"The element $x_r$ is regular if and only if $x_1=\\eta _{r,1}(x_r)$ is a regular element of ${\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ .",
"Suppose $x_r\\in {\\mathbf {g}}(\\mathfrak {O}_r)$ is regular.",
"The restriction of the reduction map $\\eta _{r,1}$ to $\\mathbf {C}_{\\mathbf {G}(\\mathfrak {O}_r)}(x_r)$ is surjective onto $\\mathbf {C}_{\\mathbf {G}({\\mathsf {k}^{\\mathrm {alg}}})}(x_1)$ .",
"Remark Assertions (1) and (3) of theo:properties, as well as Assertion (1) of theo:inverse-lim below, are formal consequences of the stronger statement that the centralizer group scheme $\\mathbf {C}_{\\mathbf {G}_{\\mathfrak {O}_r}}(x)$ is smooth over $\\mathfrak {O}_r$ , whenever $x\\in {\\mathbf {g}}(\\mathfrak {O}_r)$ is regular.",
"This statement, while plausible, is not proved in this article.",
"The proofs of Assertions (1),(2) and (3) of theo:properties are given, respectively, in sections REF , REF and REF below.",
"Once the proof of theo:properties is complete, we return to analyze the case of regular elements of $\\mathfrak {g}_r=_r(\\mathfrak {O}_r)^\\sigma $ .",
"Proposition Let $\\mathbf {G}$ be a symplectic or a special orthogonal group over $\\mathfrak {o}$ with ${\\mathbf {g}}=\\mathrm {Lie}(\\mathbf {G})$ and let $x\\in \\mathfrak {g}={\\mathbf {g}}(\\mathfrak {o})$ .",
"Assume $x_r=\\eta _r(x)$ is regular for some $r\\in \\mathbb {N}$ .",
"Then $\\mathbf {C}_{G}(x)=\\varprojlim _r\\mathbf {C}_{G_r}(x_r),$ where $G=\\mathbf {G}(\\mathfrak {o})$ and $G_r=\\mathbf {G}(\\mathfrak {o}_r)$ Furthermore, $x$ is a regular element of ${\\mathbf {g}}(K^{\\mathrm {alg}})$ .",
"theo:inverse-lim has the following corollary.",
"Corollary In the notation of theo:inverse-lim, let $x\\in \\mathfrak {g}$ such that $x_r=\\eta _r(x)$ is a regular element of $\\mathfrak {g}_{r}$ , for some $r\\in \\mathbb {N}$ .",
"Then $\\mathbf {C}_{G_r}(x_r)$ is abelian.",
"We begin by examining some basic properties of the group $\\mathbf {\\Gamma }_r$ ($r\\in \\mathbb {N}$ ) and of centralizers of elements of $_r$ , when considered as algebraic group schemes over ${\\mathsf {k}^{\\mathrm {alg}}}$ .",
"The following lemma summarizes the necessary components for the proof of theo:properties.",
"(1), and is mostly included in [33].",
"Lemma The group scheme $\\mathbf {\\Gamma }_r$ is a connected linear algebraic group over ${\\mathsf {k}^{\\mathrm {alg}}}$ .",
"The unipotent radical of $\\mathbf {\\Gamma }_r$ is $\\mathbf {\\Gamma }_r^1$ .",
"Let $\\mathbf {T}$ be a maximal torus of $\\mathbf {G}$ , defined over $\\mathfrak {O}$ , and let $\\mathbf {T}_1=\\mathbf {T}\\times {\\mathsf {k}^{\\mathrm {alg}}}\\subseteq \\mathbf {\\Gamma }_1$ .",
"The restriction of the map $s^*:~\\mathbb {A}_{{\\mathsf {k}^{\\mathrm {alg}}}}^1\\rightarrow ~\\mathbf {O}_r$ of lem:greenberg-props.",
"(1) to $\\mathbb {G}_m$ extends to an embedding of $\\mathbf {T}_1$ as a maximal torus in $\\mathbf {\\Gamma }_r$ .",
"The centralizer of $s^*(\\mathbf {T}_1)$ in $\\mathbf {\\Gamma }_r$ is the Cartan subgroup $\\mathcal {F}_{\\mathfrak {O}_r}(\\mathbf {T}\\times \\mathfrak {O}_r)$ .",
"Moreover, $\\mathcal {F}_r(\\mathbf {T}\\times ~\\mathfrak {O}_r)$ is a linear algebraic ${\\mathsf {k}^{\\mathrm {alg}}}$ -group of dimension $n\\cdot r$ .",
"1.",
"Connectedness is proved in [33].",
"The fact that $\\mathbf {\\Gamma }_r$ is linear algebraic over ${\\mathsf {k}^{\\mathrm {alg}}}$ is shown in lem:greenberg-props.(1).",
"2.",
"See [33].",
"3.",
"May be proved by following the argument of [18], practically verbatim, making use of the fact that $\\mathbf {\\Gamma }_r$ and $\\mathbf {\\Gamma }_1=\\eta _{r,1}^*(\\mathbf {\\Gamma }_r)$ are of the same rank by the previous assertion, and that $s^*(\\mathbf {T}_1)$ is a connected abelian subgroup of $\\mathbf {\\Gamma }_r$ of dimension $n=\\mathrm {rk}(\\mathbf {\\Gamma }_1)$ .",
"4.",
"The inclusion $\\mathbf {T}(\\mathfrak {O}_r)\\subseteq \\mathbf {C}_{\\mathbf {\\Gamma }_r({\\mathsf {k}^{\\mathrm {alg}}})}(s^*(\\mathbf {T}_1)({\\mathsf {k}^{\\mathrm {alg}}}))$ is clear, since $\\mathbf {T}(\\mathfrak {O}_r)$ is abelian and contains $s^*(\\mathbf {T}_1)({\\mathsf {k}^{\\mathrm {alg}}})$ , by lem:greenberg-props.(1).",
"The inclusion $\\mathbf {T}\\times \\mathfrak {O}_r\\subseteq \\mathbf {C}_{\\mathbf {\\Gamma }_r}(s^*(\\mathbf {T}_1))$ follows (see [33]).",
"By [33], $\\mathcal {F}_{\\mathfrak {O}_r}(\\mathbf {T}\\times \\mathfrak {O}_r)$ is a Cartan subgroup of $\\mathbf {\\Gamma }_r$ and hence is equal to the centralizer of $s^*(\\mathbf {T}_1)$ .",
"Finally, the statement regarding the dimension of $\\mathcal {F}_r(\\mathbf {T}\\times \\mathfrak {O}_r)$ follows from lem:greenberg-props.(1).",
"[Proof of theo:properties.",
"(1)] The alternative proof of [36] shows that the minimal centralizer dimension of an element of ${\\mathbf {g}}(\\mathfrak {O}_r)$ is equal to that of a Cartan subgroup of $\\mathbf {\\Gamma }_r$ , provided that the Cartan subgroups of $\\mathbf {\\Gamma }_r$ are abelian and that their union forms a dense subset of $\\mathbf {\\Gamma }_r$ .",
"The former of these conditions holds by [33], and the latter by [5]."
],
[
"Regularity and the reduction maps",
"The first step towards the proof of the second assertion of theo:properties is an analogous result to [18] in the Lie-algebra setting.",
"Following this, we use the properties of the Cayley map in order to transfer the result to the group setting and to deduce the equivalence of regularity of an element of $_r$ and of its image in $_1$ .",
"Lemma Let $x\\in {\\mathbf {g}}(\\mathfrak {O})$ be fixed, and for any $r\\in \\mathbb {N}$ put $x_r=\\eta _{r}(x)\\in {\\mathbf {g}}(\\mathfrak {O}_r)$ .",
"Let $\\mathbf {C}_{_r}(x_r)$ denote the Lie-algebra centralizer of $x_r$ , i.e.",
"$\\mathbf {C}_{_r}(x_r)(A)=\\left\\lbrace {y\\in _r(A)\\mid \\mathrm {ad}(x_r)(A)(y)=0}\\right\\rbrace ,$ for any commutative unital ${\\mathsf {k}^{\\mathrm {alg}}}$ -algebra $A$ .",
"The image of $\\mathbf {C}_{_r}(x_r)$ under the connecting morphism $\\eta _{r,1}^*$ is a ${\\mathsf {k}^{\\mathrm {alg}}}$ -group scheme of dimension greater or equal to $n$ .",
"Assume towards a contradiction that the statement of the lemma is false, and let $r$ be minimal such that $\\dim \\eta _{r,1}^*\\left(\\mathbf {C}_{_r}(x_r)\\right)<n$ .",
"Note that, since $\\eta _{r,1}^*\\circ \\eta _{m,r}^*=\\eta _{m,1}^*$ for all $m>r$ (by [16]) we also have that $\\dim \\eta _{m,1}^*\\left(\\mathbf {C}_{_m}(x_m)\\right)<n$ for all $m\\ge r$ .",
"Fix $m\\ge r$ , and consider the sequence of immersions $\\mathbf {C}_{_m}(x_m)\\supseteq \\mathbf {C}_{_m^1}(x_m)\\supseteq \\ldots \\supseteq \\mathbf {C}_{_{m}^{m-1}}(x_m)\\supseteq 0,$ where $\\mathbf {C}_{_m^i}(x_m)=\\mathbf {C}_{_m}(x_m)\\cap ^i_m$ .",
"Then $\\dim \\mathbf {C}_{_m}(x_m)=\\sum _{i=0}^{m-1}\\left(\\dim \\mathbf {C}_{^{i}_m}(x_m) -\\dim \\mathbf {C}_{^{i+1}_m}(x_m)\\right),$ where $_m^0=_m$ and $_m^m=\\mathrm {Spec\\:}(\\kappa (0))$ .",
"For any $0\\le i\\le m-1$ , the map $v_{i,m}^*:_{m-i}\\rightarrow \\gamma _{m}$ of lem:verschiebung restricts, by Assertion (3) of the lemma, to an isomorphism of abelian ${\\mathsf {k}^{\\mathrm {alg}}}$ -group schemes $\\mathbf {C}_{_{m-i}}(x_{m-i})\\simeq \\mathbf {C}_{_{m}^i}(x_m)$ , which restricts further, by Assertion (4) of the lemma, to an isomorphism $\\mathbf {C}_{^{i+1}_m}(x_m)\\simeq ~\\mathbf {C}_{_{m-i}^1}(x_{m-i})$ .",
"Using these isomorphisms and the exact sequence $0\\rightarrow \\mathbf {C}_{_{m-i}^1}(x_{m-i})\\rightarrow \\mathbf {C}_{_{m-i}}(x_{m-i})\\xrightarrow{}\\eta _{m-i,1}^*\\left(\\mathbf {C}_{_{m-i}}(x_{m-i})\\right)\\rightarrow 0,$ we deduce $\\dim \\mathbf {C}_{_m}(x_m)&=\\sum _{i=0}^{m-1}\\left(\\dim \\mathbf {C}_{_{m-i}}(x_{m-i})-\\dim \\mathbf {C}_{_{m-i}^1}(x_{m-i})\\right)=\\sum _{i=0}^{m-1}\\dim \\eta _{m-i,1}^*\\left(\\mathbf {C}_{_{m-i}}(x_{m-i})\\right)\\\\&=\\sum _{i=1}^{r-1}\\dim \\eta _{i,1}^*\\left(\\mathbf {C}_{_i}(x_i)\\right)+\\sum _{i=r}^{m}\\dim \\eta _{i,1}^*\\left(\\mathbf {C}_{_{i}}(x_i)\\right)\\\\&\\le d\\cdot (r-1)+(n-\\alpha )\\cdot (m-r),$ for some integer $\\alpha \\ge 1$ , where $d=\\dim _1=\\dim \\mathbf {\\Gamma }_1$ .",
"For any $m\\in \\mathbb {N}$ , by Property (REF ) of the Cayley map and the preservation of open immersions of the Greenberg functor, the Cayley map restricts to a birational equivalence of the Lie-centralizer $\\mathbf {C}_{_m}(x_m)$ and the group-centralizer $\\mathbf {C}_{\\mathbf {\\Gamma }_m}(x_m)$ of $x_m$ .",
"In particular, by theo:properties.",
"(1), we have that $\\dim \\mathbf {C}_{_m}(x_m)=\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_m}(x_m)\\ge m\\cdot n$ .",
"Manipulating the inequality (REF ), we get that $\\alpha \\cdot m\\le d\\cdot (r-1)-r\\cdot (n-\\alpha )$ for all $m>r$ .",
"A contradiction, since $m$ can be chosen to be arbitrarily large while the right-hand side of (REF ) remains constant.",
"Using lem:props-of-cayley-map, we now pass to the group setting.",
"Proposition Let $x\\in $ and $x_r=\\eta _{r}(x)$ for all $r\\in \\mathbb {N}$ .",
"The group scheme $\\eta _{r,1}^*\\left(\\mathbf {C}_{\\mathbf {\\Gamma }_r}(x_r)\\right)$ is a linear algebraic ${\\mathsf {k}^{\\mathrm {alg}}}$ -group of dimension greater or equal to $n$ .",
"Properties (REF ) and (REF ) of the Cayley map imply the commutativity of the square (REF ) $\\begin{tikzpicture}(m) [matrix of math nodes,row sep=2em,column sep=4.5em,minimum width=2em,nodes={minimum height=2em,text height=1.5ex,text depth=.25ex,anchor=center}]{\\mathbf {C}_{_r}(x_r)&\\mathbf {C}_{\\mathbf {\\Gamma }_r}(x_r)\\\\\\eta _{r,1}^*\\left(\\mathbf {C}_{_r}(x_r)\\right)&\\eta _{r,1}^*\\left(\\mathbf {C}_{\\mathbf {\\Gamma }_r}(x_r)\\right).\\\\};[->]{(m-1-1) edge[dashed] node[auto] {\\widehat{\\mathrm {cay}}_r} (m-1-2)(m-2-1) edge[dashed] node[auto] {\\mathrm {cay}_{\\mathsf {k}^{\\mathrm {alg}}}} (m-2-2)(m-1-1) edge[thin] node[left] {\\eta _{r,1}^*} (m-2-1)(m-1-2) edge[thin] node[auto] {\\eta _{r,1}^*} (m-2-2)};\\end{tikzpicture}$ A short computation, using Property (REF ), shows that this square is cartesian.",
"Thus, by (REF ), and the properties of the fiber product, it follows that the two terms of the bottom row are of the same dimension.",
"[Proof of theo:properties.",
"(2)]The assertion is proved by induction on $r$ , similarly to [18], the case $r=1$ being trivially true.",
"Consider the following exact sequence ${1[r]& \\mathbf {C}_{\\mathbf {\\Gamma }^1_r}(x_r)[r]&\\mathbf {C}_{\\mathbf {\\Gamma }_r}(x_r)[r]^{\\eta _{r,1}^*}&\\mathbf {C}_{\\mathbf {\\Gamma }_1}(x_1).",
"}$ Properties (REF ) and (REF ) imply that the map $\\widehat{\\mathrm {cay}}_r$ is defined on $\\mathbf {C}_{\\mathbf {\\Gamma }^1_r}(x_r)$ and is mapped onto $\\mathbf {C}_{^1_{r}}(x_{r})$ .",
"Combined with lem:verschiebung, we get that $\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_r^1}(x_r)=\\dim \\mathbf {C}_{_{r-1}}(x_{r-1})$ .",
"Moreover, since $\\Delta _{r-1}\\cap \\mathbf {C}_{_{r-1}}(x_{r-1})$ is a non-trivial open subscheme of $\\mathbf {C}_{_{r-1}}(x_{r-1})$ , and is mapped by $\\widehat{\\mathrm {cay}}_r$ to an open subscheme of $ \\mathbf {C}_{\\mathbf {\\Gamma }_{r-1}}(x_{r-1})$ , we deduce the equality $\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_{r}^1}(x_r)=\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_{r-1}}(x_{r-1}).$ If $x_1$ is regular then by induction we have that $\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_{r-1}}(x_{r-1})=n(r-1)$ and hence, by (REF ) and (REF ), $\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_r}(x_r)\\le \\dim \\mathbf {C}_{\\mathbf {\\Gamma }_{r-1}}(x_{r-1})+\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_1}(x_1)=r\\cdot n.$ Conversely, if $x_1$ is not regular, then by induction $x_{r-1}$ is not regular, and the dimension of $\\mathbf {C}_{\\mathbf {\\Gamma }_{r-1}}(x_{r-1})$ is strictly greater than $n(r-1)$ .",
"By propo:reg-element-reduction-groups and (REF ), have $\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_r}(x_r)=\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_{r-1}}(x_{r-1})+\\dim \\eta _1\\left(\\mathbf {C}_{\\mathbf {\\Gamma }_r}(x_r)\\right)>n(r-1)+n=n\\cdot r,$ and $x_r$ is not regular.",
"Before discussing the final assertion of theo:properties, let us observe a simple corollary of lem:reg-element-reduction, which is the Lie-algebra version of the assertion.",
"Corollary Let $r\\in \\mathbb {N}$ and $x_r\\in _r({\\mathsf {k}^{\\mathrm {alg}}})$ be regular.",
"The restriction of $\\eta _{r,1}$ to $\\mathbf {C}_{{\\mathbf {g}}(\\mathfrak {O}_r)}(x_r)$ is onto $\\mathbf {C}_{{\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})}(x_1)$ , where $x_1=\\eta _{r,1}(x_r)\\in {\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ .",
"theo:properties.",
"(2) implies that $x_1$ is regular and hence $\\mathbf {C}_{_1}(x_1)({\\mathsf {k}^{\\mathrm {alg}}})=\\mathbf {C}_{{\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})}(x_1)$ is a ${\\mathsf {k}^{\\mathrm {alg}}}$ -vector space of dimension $n=\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_1}(x_1)$ .",
"By lem:reg-element-reduction, the ${\\mathsf {k}^{\\mathrm {alg}}}$ -points of the image of $\\mathbf {C}_{_r}(x_r)$ under $\\eta _{r,1}^*$ comprise a subspace of $\\mathbf {C}_{{\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})}(x_1)$ of the same dimension."
],
[
"The image of $\\eta _{r,1}$ on {{formula:cb73d789-1bd0-4e4b-878e-d183293ac85f}}",
"To complete the proof of the third assertion of theo:properties we require the following proposition, which is stated here in a slightly more general setting than necessary at the moment, and will also be applied later on in the proof of corol:fin-index-abelian.",
"Proposition Let $L$ be either ${\\mathsf {k}^{\\mathrm {alg}}}$ or $K^{\\mathrm {alg}}$ , and let $\\mathbf {H}=\\mathbf {G}\\times \\mathrm {Spec\\:}(L)$ and $\\mathbf {h}=\\mathrm {Lie}(\\mathbf {H})$ its Lie-algebra.",
"Put $H=\\mathbf {H}(L)$ and $\\mathfrak {h}=\\mathbf {h}(L)$ .",
"Let $x\\in \\mathbf {h}(L)$ be regular.",
"Then $\\mathbf {C}_{H}(x)=\\mathbf {C}_{\\mathbf {H}}(x)^\\circ (L)\\cdot \\mathbf {Z}(H),$ where $\\mathbf {C}_{\\mathbf {H}}(x)^\\circ $ is the connected component of 1.",
"In particular, $\\left|\\mathbf {C}_{H}(x):\\mathbf {C}_{\\mathbf {H}}(x)^\\circ (L)\\right|\\le 2$ and $\\mathbf {C}_{H}(x)$ is abelian.",
"Let $x=s+h$ be the Jordan decomposition of $x$ , with $s,h\\in \\mathfrak {h}$ , $s$ semisimple, $h$ nilpotent and $[s,h]=0$ .",
"Note that, as an element of $H$ commutes with $x$ if and only if it commutes with both $s$ and $h$ , we have that $\\mathbf {C}_{H}(x)=\\mathbf {C}_{\\mathbf {C}_H(s)}(h)$ .",
"From propo:centralizer-semisimple-element, it follows that $\\mathbf {C}_{H}(x)=\\mathbf {C}_{\\mathbf {C}_H(s)}(h)=\\prod _{j=1}^t\\mathbf {C}_{\\mathrm {GL}_{m_j}\\left(L\\right)}\\left(h\\mid _{W_{\\lambda _j}}\\right)\\times \\mathbf {C}_{\\Delta (L)}\\left(h\\mid _{\\mathrm {Ker}(s)}\\right),$ where $\\Delta $ is a classical linear algebraic group over $L$ of automorphisms preserving a non-degenerate bilinear form on a subspace of $L^N$ , and $\\pm \\lambda _1,\\ldots ,\\pm \\lambda _t$ are the non-zero eigenvalues of $s$ , as described in propo:centralizer-semisimple-element, with respective multiplicities $m_1,\\ldots ,m_t$ , and $W_{\\lambda _j}=\\mathrm {Ker}(s-\\lambda _j\\mathbf {1})$ .",
"Additionally, by [36], the restricted operators $h\\mid _{W(\\lambda _j)}$ and $h\\mid _{\\mathrm {Ker}(s)}$ are regular as elements of the Lie-algebras of $\\mathrm {GL}_{m_j}$ and of $\\Delta $ over $L$ , respectively.",
"By [32] it is known that all factors in (REF ), apart from $\\mathbf {C}_{\\Delta }(h\\mid _{\\mathrm {Ker}(s)})$ , are connected.",
"Furthermore, by [32] and the assumption ${\\mathrm {char}}(L)\\ne 2$ , we have $\\mathbf {C}_{\\Delta }\\left(h\\mid _{\\mathrm {Ker}(s)}\\right)=\\mathbf {C}_{\\Delta }\\left(h\\mid _{\\mathrm {Ker}(s)}\\right)^\\circ \\cdot \\mathbf {Z}(\\Delta ),$ (see [32]).",
"Taking into account the fact that, as ${\\mathrm {char}}(L)\\ne 2$ , $\\mathbf {Z}(\\Delta (L))$ is the finite group $\\left\\lbrace {\\pm 1}\\right\\rbrace $ , one easily deduces from this the equality $\\mathbf {C}_{H}(x)=\\mathbf {C}_{\\mathbf {H}}(x)^\\circ (L)\\cdot \\mathbf {Z}(H).$ Lastly, $\\mathbf {C}_{H}(x)^\\circ $ is abelian by [32], and $\\left|\\mathbf {C}_H(x):\\mathbf {C}_{H}(x)^\\circ \\right|\\le \\left|\\mathbf {Z}(H)\\right|=2$ .",
"[Proof of theo:properties.",
"(3)] By propo:reg-element-reduction-groups and Chevalley's Theorem [12], the image of $\\mathbf {C}_{\\mathbf {\\Gamma }_r}(x_r)$ under $\\eta _{r,1}^*$ contains the connected component $\\mathbf {C}_{\\mathbf {\\Gamma }_1}(x_1)^\\circ $ of the identity in $\\mathbf {C}_{\\mathbf {\\Gamma }_1}(x)$ .",
"Additionally, the center $\\mathbf {Z}(\\mathbf {\\Gamma }_r)$ of $\\mathbf {\\Gamma }_r$ is clearly contained in $\\mathbf {C}_{\\mathbf {\\Gamma }_r}(x_r)$ and is mapped by $\\eta _{r,1}^*$ onto $\\mathbf {Z}(\\mathbf {\\Gamma }_1)$ .",
"This implies the inclusion $\\mathbf {C}_{\\mathbf {\\Gamma }_1}(x_1)\\supseteq \\eta _{r,1}^*\\left(\\mathbf {C}_{\\Gamma _r}(x_r)\\right)\\supseteq \\left(\\mathbf {C}_{\\Gamma _1}(x_1)\\right)^\\circ \\cdot \\mathbf {Z}(\\mathbf {\\Gamma }_1).$ Evaluating the above inclusions at ${\\mathsf {k}^{\\mathrm {alg}}}$ -points, by propo:centralizer-of-reg-over-bkk, we deduce the equality."
],
[
"Returning to the $\\mathfrak {o}$ -rational setting",
"In this section we prove theo:inverse-lim.",
"An initial step towards this goal is to show that the third assertion of theo:properties remains true when replacing the groups $\\mathbf {G}(\\mathfrak {O}_r)$ and Lie-rings ${\\mathbf {g}}(\\mathfrak {O}_r)$ with the group and Lie-rings of $\\mathfrak {o}_r$ -rational points, i.e.",
"$G_r=\\mathbf {G}(\\mathfrak {o}_r)$ and $\\mathfrak {g}_r={\\mathbf {g}}(\\mathfrak {o}_r)$ .",
"Given $1\\le m\\le r$ , we write $G^m_r$ and $\\mathfrak {g}^m_r$ to denote the congruence subgroup $\\mathrm {Ker}(G_r\\xrightarrow{}G_m)=G_r\\cap \\eta _{r,m}^{-1}(1)$ and congruence subring $\\mathrm {Ker}(\\mathfrak {g}_r\\xrightarrow{}\\mathfrak {g}_m)=\\mathfrak {g}_r\\cap \\eta _{r,m}^{-1}(0)$ , respectively.",
"Recall that $\\sigma :\\mathfrak {O}\\rightarrow \\mathfrak {O}$ was defined in subsubsection:artinian-rings to be the local Frobenius automorphism of $\\mathfrak {O}$ over $\\mathfrak {o}$ , given on its quotient ${\\mathsf {k}^{\\mathrm {alg}}}$ by $\\sigma (\\xi )=\\xi ^{\\left|\\mathsf {k}\\right|}$ .",
"This automorphism gives rise to an automorphism of $\\mathbf {G}(\\mathfrak {O})$ , and of its quotients $\\mathbf {G}(\\mathfrak {O}_r)$ and their Lie-algebras.",
"By definition, an element $x\\in \\mathfrak {g}_r$ is regular if and only if it is a regular $\\sigma $ -fixed element of ${\\mathbf {g}}(\\mathfrak {O}_r)=_r({\\mathsf {k}^{\\mathrm {alg}}})$ .",
"We require the following variant of Lang's Theorem.",
"Lemma Let $r\\in \\mathbb {N}$ and let $x_r\\in \\mathfrak {g}_r$ be a regular element and $x_1=\\eta _{r,1}(x_r)$ .",
"Given $g\\in \\mathbf {C}_{G_1}(x_1)=\\mathbf {C}_{\\mathbf {G}({\\mathsf {k}^{\\mathrm {alg}}})}(x_1)\\cap G_1$ , let $F_g=\\eta _{r,1}^{-1}(g)\\cap \\mathbf {C}_{\\mathbf {G}(\\mathfrak {O}_r)}(x_r)$ , and let $\\mathcal {L}_g$ be the map defined by $h\\mapsto h\\cdot \\sigma (h)^{-1}.$ Then $\\mathcal {L}_g:F_g\\rightarrow F_1$ is a well-defined surjective map.",
"The sets $F_{g^{\\prime }}$ ($g^{\\prime }\\in \\mathbf {C}_{G_1}(x_1)$ ) are simply cosets of the subgroup $F_1=\\mathbf {C}_{\\mathbf {\\Gamma }^1_r({\\mathsf {k}^{\\mathrm {alg}}})}(x_r)$ .",
"In particular, by (REF ) and (REF ), the $F_{g^{\\prime }}$ 's are the ${\\mathsf {k}^{\\mathrm {alg}}}$ -points of algebraic varieties, isomorphic to $\\mathbf {C}_{^1_r({\\mathsf {k}^{\\mathrm {alg}}})}(x_r)$ and hence affine ${(r-1)n}$ -dimensional spaces over ${\\mathsf {k}^{\\mathrm {alg}}}$ .",
"Since the reduction map $\\eta _{r,1}$ commutes with the Frobenius maps, and since $g$ is assumed fixed by $\\sigma $ , we have that $\\mathcal {L}_g$ is well-defined.",
"The surjectivity of $\\mathcal {L}_g$ now follows as in the proof of the classical Lang Theorem [23], using the fact that $F_1$ is a connected linear algebraic group over ${\\mathsf {k}^{\\mathrm {alg}}}$ (see also [32] and [17]).",
"Corollary Let $x_r\\in \\mathfrak {g}_r$ be regular and $x_1=\\eta _{r,1}(x_r)$ .",
"The restriction of $\\eta _{r,1}$ to $\\mathbf {C}_{G_r}(x_r)$ is onto $\\mathbf {C}_{G_1}(x_1)$ .",
"lem:lang and theo:properties.",
"(3) imply that for any $g\\in \\mathbf {C}_{G_1}(x_1)$ , there exists an element $h\\in \\mathbf {C}_{\\mathbf {\\Gamma }_r}(x_r)$ such that $\\eta _{r,1}(h)=g$ and such that $\\mathcal {L}_g(h)=h\\sigma (h)^{-1}=1$ .",
"In particular, $h$ is fixed under $\\sigma $ and hence $h\\in \\mathbf {C}_{G_r}(x_r)\\cap \\eta _{r,1}^{-1}(g)$ .",
"Another necessary ingredient in the proof of theo:inverse-lim is the connection between the groups $\\mathbf {C}_{G_r}(x_r)$ and $\\mathbf {C}_{G_m}(x_m)$ , where $m\\le r$ and $x\\in \\mathfrak {g}$ is such that $x_r$ is regular.",
"Lemma Let $r\\in \\mathbb {N}$ and $x_r\\in \\mathfrak {g}_r$ be regular.",
"For any $1\\le m\\le r$ write $x_m=\\eta _{r,m}(x_r)$ .",
"The map $\\eta _{r,m}:\\mathbf {C}_{\\mathfrak {g}_r}(x_r)\\rightarrow \\mathbf {C}_{\\mathfrak {g}_m}(x_m)$ is surjective.",
"The map $\\eta _{r,m}:\\mathbf {C}_{G_r}(x_r)\\rightarrow \\mathbf {C}_{G_m}(x_m)$ is surjective.",
"We prove both assertions by induction on $m$ .",
"1.",
"The case $m=1$ follows in corol:reduction-surjective-lie-alg and Lang's Theorem, as $\\mathbf {C}_{_1}(x_1)$ and $\\eta ^*_{r,1}(\\mathbf {C}_{_r}(x_r))$ are both affine $n$ -spaces over ${\\mathsf {k}^{\\mathrm {alg}}}$ .",
"Consider the commutative diagram in (REF ), in which both rows are exact by induction hypothesis.",
"$\\begin{tikzpicture}(m) [matrix of math nodes,row sep=1.5em,column sep=3em,minimum width=2em,nodes={minimum height=2em,text height=1.5ex,text depth=.25ex,anchor=center}]{\\mathbf {C}_{\\mathfrak {g}^{m-1}_{r}}(x_r)&\\mathbf {C}_{\\mathfrak {g}_r}(x_r)&\\mathbf {C}_{\\mathfrak {g}_{m-1}}(x_{m-1})&0\\\\\\mathbf {C}_{\\mathfrak {g}^{m-1}_{m}}(x_m)&\\mathbf {C}_{\\mathfrak {g}_m}(x_m)&\\mathbf {C}_{\\mathfrak {g}_{m-1}}(x_{m-1})&0\\\\};[->,font=\\scriptsize ](m-1-1) edge[thin] node[auto] {} (m-1-2)(m-1-2) edge[thin] node[auto] {\\eta _{r,m-1}} (m-1-3)(m-1-3) edge[thin] node[auto] {} (m-1-4)(m-2-1) edge[thin] node[auto] {} (m-2-2)(m-2-2) edge[thin] node[below] {\\eta _{m,m-1}} (m-2-3)(m-2-3) edge[thin] node[auto] {} (m-2-4)(m-1-2) edge[thin] node[auto] {\\eta _{r,m}} (m-2-2)(m-1-1) edge[thin] node[auto] {} (m-2-1);[-](m-1-3) edge[double equal sign distance, thin] node {} (m-2-3)(m-1-4) edge[double equal sign distance, thin] node[auto] {} (m-2-4);\\end{tikzpicture}$ By the Four Lemma (on epimorphisms), in order to prove the surjectivity of the map $\\eta _{r,m}:\\mathbf {C}_{\\mathfrak {g}_r}(x_r)\\rightarrow \\mathbf {C}_{\\mathfrak {g}_m}(x_m)$ , it suffices to show that the restricted map $\\eta _{r,m}:\\mathbf {C}_{\\mathfrak {g}^{m-1}_r}(x_r)\\rightarrow \\mathbf {C}_{\\mathfrak {g}_m^{m-1}}(x_m)$ is surjective.",
"This follows from the commutativity of the square in (REF ), in which the maps on the top and bottom rows are given the $\\mathfrak {o}$ -module isomorphism $y\\mapsto \\pi ^{m-1}y$ (cf.",
"lem:verschiebung), and the map on the left column is surjective by the base of induction.",
"$\\begin{tikzpicture}(m) [matrix of math nodes, row sep=1.5em, column sep=3em, minimum width=2em,nodes={minimum height=2em,text height=1.5ex,text depth=.25ex,anchor=center}]{\\mathbf {C}_{\\mathfrak {g}_{r-m+1}}(x_{r-m+1})&\\mathbf {C}_{\\mathfrak {g}^{m-1}_r}(x_r)\\\\\\mathbf {C}_{\\mathfrak {g}_1}(x_1)&\\mathbf {C}_{\\mathfrak {g}_m^{m-1}}(x_m)\\\\};[->,font=\\scriptsize ](m-1-1) edge[thin] node[auto] {\\sim } (m-1-2)(m-2-1) edge[thin] node[auto] {\\sim } (m-2-2)(m-1-2) edge[thin] node[right] {\\eta _{r,m}} (m-2-2);\\end{tikzpicture}[->>,font=\\scriptsize ](m-1-1) edge[thin] node[left] {\\eta _{r-m+1,1}} (m-2-1);$ 2.",
"In the current setting, one invokes lem:lang in order to prove the induction base $m=1$ .",
"The case $m>1$ is handled in a manner completely analogous to the first case, applying the Four Lemma for a suitable diagram of groups.",
"The main difference from the previous case is that in proving the surjectivity of the map $\\eta _{r,m}:\\mathbf {C}_{G_r^{m-1}}(x_r)\\rightarrow \\mathbf {C}_{G_m^{m-1}}(x_m)$ , one considers the commutative square in (REF ) in which the leftmost vertical arrow is shown to be surjective in the previous case, and the horizontal arrows are given by the suitable Cayley maps.",
"Note that the fact that the top horizontal arrow in (REF ) is not necessarily a group homomorphism does not affect the proof of the assertion.",
"$\\begin{tikzpicture}(m) [matrix of math nodes, row sep=1.5em, column sep=3em, minimum width=2em,nodes={minimum height=2em,text height=1.5ex,text depth=.25ex,anchor=center}]{\\mathbf {C}_{\\mathfrak {g}^{m-1}_r}(x_{r})&\\mathbf {C}_{G^{m-1}_r}(x_r)\\\\\\mathbf {C}_{\\mathfrak {g}^{m-1}_m}(x_m)&\\mathbf {C}_{G_m^{m-1}}(x_m)\\\\};[->,font=\\scriptsize ](m-1-1) edge[thin] node[auto] {\\mathrm {cay}_{r}} (m-1-2)(m-2-1) edge[thin] node[auto] {\\mathrm {cay}_{m}} (m-2-2)(m-1-2) edge[thin] node[right] {\\eta _{r,m}} (m-2-2);\\end{tikzpicture}[->>,font=\\scriptsize ](m-1-1) edge[thin] node[left] {\\eta _{r,m}} (m-2-1);$ [Proof of theo:inverse-lim] 1.",
"Given $g_r\\in \\mathbf {C}_{G_r}(x_r)$ one inductively invokes lem:surjective-r-to-m to construct a converging sequence $(g_m)_{m\\ge r}$ such that $g_m\\in \\mathbf {C}_{G_m}(\\eta _m(x))$ and such that $\\eta _{m^{\\prime },m}(g_{m^{\\prime }})=g_m$ for all $m^{\\prime }\\ge m\\ge r$ .",
"The limit $g=\\lim _m g_m$ is easily verified to be an element of $\\mathbf {C}_{G}(x)$ , which is mapped by $\\eta _r$ to $g_r$ .",
"2.",
"By theo:properties, it suffices to consider the case where $x_1=\\eta _{1}(x)\\in {\\mathbf {g}}(\\mathsf {k})$ is regular.",
"By [26], under this assumption, we have that $\\dim \\mathbf {C}_{\\mathbf {G}\\times K^{{\\mathrm {alg}}}}(x)=\\dim \\left(\\mathbf {C}_{\\mathbf {G}\\times \\mathfrak {O}}(x)\\times K^{\\mathrm {unr}}\\right)\\le \\dim \\left(\\mathbf {C}_{\\mathbf {G}\\times \\mathfrak {O}}(x)\\times {\\mathsf {k}^{\\mathrm {alg}}}\\right)=\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_1}(x_1)=n,$ as $\\mathbf {C}_{\\mathbf {G}\\times \\mathfrak {O}}(x)\\times K^{\\rm unr}$ and $\\mathbf {C}_{\\mathbf {G}\\times \\mathfrak {O}}(x)\\times {\\mathsf {k}^{\\mathrm {alg}}}$ are, respectively, the generic and special fiber of $\\mathbf {C}_{\\mathbf {G}\\times \\mathfrak {O}}(x)$ .",
"On the other hand, by [36], the minimum value of centralizer dimension of an element of $\\mathfrak {g}$ is $n=\\mathrm {rk}(\\mathbf {G})$ .",
"Hence, $x$ is regular.",
"Finally, we deduce corol:fin-index-abelian.",
"[Proof of corol:fin-index-abelian] The regularity of $x$ in $\\mathfrak {g}$ , and propo:centralizer-of-reg-over-bkk (applied for $L=K^{{\\mathrm {alg}}}$ ), imply that the centralizer of $x$ in $\\mathbf {G}(K^{\\mathrm {alg}})$ is an abelian group.",
"In particular, it follows from this that the group $\\mathbf {C}_{G}(x)$ is abelian as well, and consequently, by theo:inverse-lim.",
"(1), so are its quotient groups $\\mathbf {C}_{G_r}(x_r)$ for all $r\\in \\mathbb {N}$ ."
],
[
"Regular characters",
"At this point, our description of the regular elements of the Lie-algebras $\\mathfrak {g}_r$ is sufficient in order to initiate the description of regular characters of $G$ and to prove mainthm:enumeration+dimension and corol:reg-zeta-function.",
"To do so, we prove the following variant of [22].",
"Theorem 2.1 Let $\\Omega \\subseteq \\mathfrak {g}_1$ be a regular orbit and let $r\\in \\mathbb {N}$ and $m=\\lfloor \\frac{r}{2}\\rfloor $ .",
"The set $\\mathrm {Irr}(G^m_r\\mid \\Omega )$ of characters of $G^m_r=\\mathrm {Ker}(G_r\\rightarrow G_m)$ which lie above the regular orbit $\\Omega $ consists of exactly $q^{n(r-m-1)}$ orbits for the coadjoint action of $G_r$ .",
"Given a character $\\sigma \\in \\mathrm {Irr}(G^m_r\\mid \\Omega )$ , the set of irreducible characters of $G_r$ whose restriction to $G^m_r$ has $\\sigma $ as a constituent is in bijection with the Pontryagin dual of $\\mathbf {C}_{G_m}(x_m)$ , for $x_m\\in \\mathfrak {g}_m$ any element such that $\\eta _{m,1}(x_m)\\in \\Omega $ .",
"Any such character $\\sigma \\in \\mathrm {Irr}(G^m_r\\mid \\Omega )$ extends to its inertia group $I_{G_r}(\\sigma )$ .",
"In particular, each such extension induces to a regular character of $G_r$ .",
"Note that the first assertion of mainthm:enumeration+dimension follows from Assertions (1) and (2) of theo:charcaters-of-Grm and corol:surjective-Gm.",
"The second assertion of mainthm:enumeration+dimension follows from the Assertion (3) of theo:charcaters-of-Grm and (REF ) below.",
"The proof of theo:charcaters-of-Grm follows the same path as [22].",
"For the sake of brevity, rather then rehashing the proof appearing in great detail in loc.",
"cit., our focus for the remainder of this section would be on setting up the necessary preliminaries and state the necessary modification required in order to adapt the construction of [22] to the current setting.",
"Recall that the group $G=\\mathbf {G}(\\mathfrak {o})$ and $\\mathfrak {g}={\\mathbf {g}}(\\mathfrak {o})$ are naturally embedded in the matrix algebra $\\mathrm {M}_N(\\mathfrak {o})$ (see subsection:group-notation).",
"Similarly, the congruence quotients $G_r$ and $\\mathfrak {g}_r$ are embedded in $\\mathrm {M}_{N}(\\mathfrak {o}_r)$ .",
"From here on, all computation are to be understood in the framework of the embedding of the given groups and Lie-rings in their respective matrix algebras."
],
[
"Duality for Lie-rings",
"The Lie-algebra $\\mathfrak {g}={\\mathbf {g}}(\\mathfrak {o})\\subseteq \\mathrm {M}_N(\\mathfrak {o})$ is endowed with a symmetric bilinear $G(\\mathfrak {o})$ -invariant form $\\kappa :\\mathfrak {g}\\times \\mathfrak {g}\\rightarrow \\mathfrak {o},\\quad (x,y)\\mapsto \\mathrm {Tr}(xy).$ Note that, by the assumption $p={\\mathrm {char}}(\\mathsf {k})$ is odd, the form $\\kappa $ reduces to a non-degenerate form on $\\mathfrak {g}_1$ (see [32]), and hence $\\left\\lbrace {x\\in \\mathfrak {g}\\mid \\kappa (x,y)\\in \\mathfrak {p}\\text{ for all }y\\in \\mathfrak {g}}\\right\\rbrace =\\pi \\mathfrak {g}$ .",
"Fixing a non-trivial character $\\psi :K\\rightarrow \\mathbb {C}^\\times $ with conductor $\\mathfrak {o}$ (see e.g.",
"[2]), for any $r\\in \\mathbb {N}$ , we have a well-defined map $\\mathfrak {g}_r\\rightarrow \\widehat{\\mathfrak {g}_r},\\quad y\\mapsto \\varphi _y\\text{ where }\\varphi _y(x)=\\psi (\\pi ^{-r}\\kappa (x,y)).$ Furthermore, by the assumption $\\pi ^{-1}\\mathfrak {o}\\lnot \\subseteq \\mathrm {Ker}(\\psi )$ , the map above induces a $G_r$ -equivariant bijection of $\\mathfrak {g}_r$ with its Pontryagin dual $\\widehat{\\mathfrak {g}_r}$ ."
],
[
"Exponential and logarithm",
"Let $m,r\\in \\mathbb {N}$ with $\\frac{r}{3}\\le m\\le r$ .",
"The truncated exponential map, defined by $\\exp (x)=1+x+\\frac{1}{2}x^2\\quad (x\\in \\mathfrak {g}^m_r),$ is a well-defined bijection of $\\mathfrak {g}^m_r$ onto the group $G^m_r$ , and is equivariant with respect to the adjoint action of $G_r$ , with an inverse map given by $\\log (1+x)=x-\\frac{1}{2}x^2\\quad (1+x\\in G^m_r).$ In the case where $\\frac{r}{2}\\le m$ , the exponential map is simply given by $\\exp (x)=1+x$ and defines an isomorphism of abelian groups $\\mathfrak {g}^m_r\\xrightarrow{}G^m_r$ .In the more general setting we have the following.",
"Lemma Let $m,r\\in \\mathbb {N}$ be such that $\\frac{r}{3}\\le m\\le r$ .",
"For any $x,y\\in \\mathfrak {g}^m_r$ , $\\log \\left(\\left(\\exp (x),\\exp (y)\\right)\\right)=\\left[x,y\\right],$ where $(\\exp (x),\\exp (y))$ denotes the group commutator of $\\exp (x)$ and $\\exp (y)$ in $G^m_r$ .",
"Furthermore, the following truncated version of the Baker-Campbell-Hausdorff formula holds $\\log \\left(\\exp (x)\\cdot \\exp (y)\\right)=x+y+\\frac{1}{2}\\left[x,y\\right].$ The formulae in lem:ad-relations may be verified by direct computation; their proof is omitted."
],
[
"Characters of $G^{\\lfloor {r/2}\\rfloor }_r$",
"Fix $r\\in \\mathbb {N}$ and put ${m^{\\prime }}=\\lfloor \\frac{r}{2}\\rfloor $ and $m=\\lceil \\frac{r}{2}\\rceil =r-{m^{\\prime }}$ .",
"As mentioned above, the exponential map on $\\mathfrak {g}^{m}_r$ is given by $x\\mapsto 1+x:\\mathfrak {g}^{m}_r\\rightarrow G^{m}_r$ and defines a $G_r$ -equivariant isomorphism of abelian groups.",
"Taking into account the module isomorphism $x\\mapsto \\pi ^{m} x:\\mathfrak {g}_{{m^{\\prime }}}\\rightarrow \\mathfrak {g}_{r}^{m}$ and (REF ) we obtain a $G_r$ -equivariant bijection $\\Phi \\quad :\\quad \\mathfrak {g}_{m^{\\prime }}\\rightarrow \\widehat{\\mathfrak {g}_{m^{\\prime }}}\\rightarrow \\widehat{\\mathfrak {g}_r^{m}}\\rightarrow \\mathrm {Irr}(G^{m}_r),$ given explicitly by $\\Phi (y)(1+x)=\\varphi _y(\\pi ^{-m}x)$ , for $y\\in \\mathfrak {g}_{m^{\\prime }}$ and $x\\in \\mathfrak {g}^{m}_r$ .",
"In the case where $r=2m^{\\prime }$ deduce the following.",
"Lemma Assume $r=2m$ is even.",
"The map $\\Phi $ defined in (REF ) is a $G_r$ -equivariant bijection of $\\mathrm {Irr}(G^{m^{\\prime }}_r)$ and $\\mathfrak {g}_{m^{\\prime }}$ .",
"In the case where $r=2m^{\\prime }+1$ , the irreducible characters of $G^{m^{\\prime }}_r$ are classified in terms of their restriction to $G^{m}_r$ , using the method of Heisenberg lifts, which we briefly recall here.",
"For a more elaborate survey we refer to [22] and [9].",
"Let $\\vartheta \\in \\mathrm {Irr}(G^{m}_r)$ be given, and let $y\\in \\mathfrak {g}_{m^{\\prime }}$ be such that $\\vartheta =\\Phi (y)$ .",
"Note that, as the group $G^{m}_r$ is central in $G^{m^{\\prime }}_r$ and $(G^{m^{\\prime }}_r,G^{m^{\\prime }}_r)\\subseteq G^{m}_r$ , the following map is a well defined alternating $\\mathbb {C}^\\times $ -valued bilinear form $B_{\\vartheta }:G^{m^{\\prime }}_{r}/G^{m}_r\\times G^{m^{\\prime }}_r/G^{m}_r\\rightarrow \\mathbb {C}^\\times ,\\quad B_{\\vartheta }(x_1 G^{m}_r,x_2G^{m}_r)=\\vartheta \\left(\\left(x_1 ,x_2\\right)\\right).$ Using the definition of $\\Phi (y)=\\vartheta $ and the explicit isomorphism $x\\mapsto \\exp (\\pi ^rx):\\mathfrak {g}_1\\rightarrow G^{m^{\\prime }}_{m}=G^{m^{\\prime }}_r/G^{m}_r$ , we obtain an alternating bilinear form $\\beta _y:\\mathfrak {g}_1\\times \\mathfrak {g}_1\\rightarrow \\mathsf {k}$ given by $\\beta _y(x_1,x_2)=\\mathrm {Tr}(\\eta _{{m^{\\prime }},1}(y)\\cdot [x_1,x_2])$ , such that the diagram in (REF ) commutes.",
"$\\begin{tikzpicture}(m) [matrix of math nodes, row sep=1.5em, column sep=.25em, minimum width=2em,nodes={minimum height=2em,text height=1.5ex,text depth=.25ex,anchor=center}]{G^{m^{\\prime }}_{m}&\\times &G^{m^{\\prime }}_{m}&~&~&\\mathbb {C}^\\times \\\\\\mathfrak {g}_1&\\times &\\mathfrak {g}_1&~&~&\\mathsf {k}\\\\};[->,font=\\scriptsize ](m-2-1) edge[thin] node[right] {\\rotatebox {90}{\\sim }} (m-1-1)(m-2-3) edge[thin] node[right] {\\rotatebox {90}{\\sim }} (m-1-3)(m-1-3) edge[thin] node[above] {B_\\vartheta } (m-1-6)(m-2-3) edge[thin] node[above] {\\beta _y} (m-2-6)(m-2-6) edge[thin] node[right] {\\psi (\\pi ^{-1}(\\cdot ))} (m-1-6);\\end{tikzpicture}$ A short computation, using the non-degeneracy of the trace and the definition of $\\beta _y$ , shows that the radical of this form coincides with the centralizer sub-algebra $\\mathbf {C}_{\\mathfrak {g}_1}(\\eta _{{m^{\\prime }},1}(y))$ of $\\mathfrak {g}_1$ (see [22]).",
"Let $\\mathfrak {R}_y$ and $R_y$ denote the preimages of $\\mathfrak {m}_y$ in $\\mathfrak {g}_r^{m^{\\prime }}$ and in $G^{m^{\\prime }}_r$ under the associated quotient maps.",
"Let $\\mathfrak {m}_y\\subseteq \\mathfrak {j}\\subseteq \\mathfrak {g}_1$ be a maximal subspace such that $\\beta _y(\\mathfrak {j},\\mathfrak {j})=\\left\\lbrace {0}\\right\\rbrace $ (i.e.",
"such that $\\mathfrak {j}/\\mathfrak {m}_y$ is a maximal isotropic subspace of $\\mathfrak {g}_1/\\mathfrak {m}_y$ ), and let $\\mathfrak {J}\\subseteq \\mathfrak {g}^{m^{\\prime }}_r$ and $J\\subseteq G^{m^{\\prime }}_r$ be the corresponding preimages of $\\mathfrak {j}$ ; see (REF ).",
"Figure: NO_CAPTIONLet $\\theta =\\vartheta \\circ \\exp $ be the pull-back of $\\vartheta $ to $\\mathfrak {g}^{m}_r$ .",
"By virtue of the commutativity of $\\mathfrak {R}_y$ , the character $\\theta ^{\\prime }$ extends to a character of $\\mathfrak {R}_y$ in $\\left|\\mathfrak {R}_y:G^{m}_r\\right|=\\left|\\mathfrak {m}_y\\right|$ many ways.",
"By lem:ad-relations, given such an extension $\\theta ^{\\prime }\\in \\widehat{\\mathfrak {R}_y}$ , the map $\\vartheta ^{\\prime }:R_y\\rightarrow \\mathbb {C}^\\times $ is a character of $R_y$ .",
"Thus, the character $\\vartheta $ admits $\\left|\\mathfrak {m}_y\\right|$ many extensions to $R_y$ .",
"Lemma Any extension $\\vartheta ^{\\prime }\\in \\mathrm {Irr}(R_y)$ of $\\vartheta $ extends further to a character $\\vartheta ^{\\prime \\prime }\\in \\mathrm {Irr}(J)$ .",
"The induced character $\\sigma =(\\vartheta ^{\\prime \\prime })^{G^{m^{\\prime }}_r}$ is irreducible and is independent of the choice of extension $\\vartheta ^{\\prime \\prime }$ and of $\\mathfrak {j}$ .",
"The character $\\sigma $ is the unique character of $G^{m^{\\prime }}_r$ whose restriction to $R_y$ contains $\\vartheta ^{\\prime }$ .",
"Furthermore, all irreducible characters of $G^{m^{\\prime }}_r$ which lie above $\\vartheta $ are obtained in this manner.",
"The triple $(G^{m^{\\prime }}_r,R_y,\\vartheta ^{\\prime })$ satisfies the hypothesis of [9], and the alternating bilinear form $\\beta _y$ (which corresponds to $h_\\chi $ in the notation of loc.",
"cit.)",
"reduces to a non-degenerate form on the elementary abelian group $G^{m^{\\prime }}_r/R_y\\simeq \\mathfrak {g}_1/\\mathfrak {r}_y$ .",
"The subgroup $J\\subseteq G^{m^{\\prime }}_r$ may be identified with the group denoted in [9] by $G_1$ , and the extension of $\\vartheta ^{\\prime }$ to an irreducible character of $J$ exists by virtue of $J/\\mathrm {Ker}(\\vartheta ^{\\prime })$ being finite and abelian.",
"The irreducibility and independence of the choice of $J$ are shown within the proof of [9], as well as uniqueness of $\\sigma $ as the only irreducible character of $G^{m^{\\prime }}_r$ whose restriction to $R_y$ contains $\\vartheta ^{\\prime }$ .",
"The final assertion, that all characters of $G^{m^{\\prime }}_r$ lying above $\\vartheta $ are obtained in this manner is obvious, as the restriction of any such character of $G^{m^{\\prime }}_r$ to $R_y$ necessarily contains an extension $\\vartheta ^{\\prime }$ of $\\vartheta $ ."
],
[
"Inertia subgroups in $\\mathbf {G}(\\mathfrak {o}_r)$ of regular characters",
"The final ingredient required in order to implement the construction of [22] to the current setting is a structural description of the inertia subgroup of a character of $G^{\\lceil r/2\\rceil }_r$ lying below a regular character of level $\\ell =r+1$ .",
"As in the previous section, put ${m^{\\prime }}=\\lfloor \\frac{r}{2}\\rfloor $ and $m=\\lceil \\frac{r}{2}\\rceil $ , and let $\\vartheta \\in \\mathrm {Irr}(G^{m}_r)$ .",
"Recall that the inertia subgroup of $\\vartheta $ in $G_r$ is defined by $I_{G_r}(\\vartheta )=\\left\\lbrace {g \\in G_r\\mid \\vartheta (g^{-1}xg)=\\vartheta (x)\\text{ for all }x\\in G^{m}_r}\\right\\rbrace .$ By subsubsection:character-Grm, there exists a unique $y\\in \\mathfrak {g}_{{m^{\\prime }}}$ such that $\\vartheta =\\Phi (y)$ .",
"Moreover, letting $\\hat{y}_r\\in \\mathfrak {g}_r$ be an arbitrary lift of $y_{m^{\\prime }}$ to $\\mathfrak {g}_r$ , we have that $I_{G_r}(\\vartheta )=G^{m}_r\\cdot \\mathbf {C}_{G_r}(\\hat{y}_r).$ Indeed, the only non-trivial step to proving (REF ) is the inclusion $\\subseteq $ , which from follows lem:surjective-r-to-m, as both hands of the equation are mapped by $\\eta _{r,m}$ onto the group $\\mathbf {C}_{G_{m}}(\\eta _{r,m}(\\hat{y}_r))$ .",
"[Proof of theo:charcaters-of-Grm] A short computation, proves that the set $\\tilde{\\Omega }=\\eta _{{m^{\\prime }},1}^{-1}(\\Omega )$ consists of $q^{n({m^{\\prime }}-1)}$ distinct adjoint orbits for the action of $G_{m^{\\prime }}$ , and hence for the action of $G_r$ as well.",
"Indeed, $\\tilde{\\Omega }$ is a $G_{m^{\\prime }}$ -stable set of order $\\left|\\Omega \\right|\\cdot \\left|G^1_{m^{\\prime }}\\right|=\\left|\\Omega \\right|q^{d(m^{\\prime }-1)}$ , invoking the bijection $\\mathfrak {g}^1_{m^{\\prime }}\\rightarrow G^1_{m^{\\prime }}$ induced by the Cayley map, and each of the orbits $G_{m^{\\prime }}$ -orbits in $\\tilde{\\Omega }$ has cardinality $\\left|G_{m^{\\prime }}:\\mathbf {C}_{G_{m^{\\prime }}}(x)\\right|=\\left|G_1:\\mathbf {C}_{G_1}\\left(\\eta _{m^{\\prime },1}\\left(x\\right)\\right)\\right|\\cdot \\left|G^1_{m^{\\prime }}:\\mathbf {C}_{G^1_{m^{\\prime }}}(x)\\right|=\\left|\\Omega \\right|\\cdot q^{(d-n)(m^{\\prime }-1)},$ by corol:surjective-Gm, for any $x\\in \\tilde{\\Omega }$ .",
"By the $G_r$ -equivariance of the map $\\Phi $ , defined in subsubsection:character-Grm, it follows that the set $\\mathrm {Irr}(G^{m}_r\\mid \\Omega )$ consists of $q^{n({m^{\\prime }}-1)}$ coadjoint orbits of $G_r$ .",
"In the case where $r$ is even, the first assertion of theo:charcaters-of-Grm follows from lem:half-level-characters-even-case, since $m=r-m$ .",
"In the case of $r$ odd, by lem:heisenberg, and by regularity of the elements of $\\Omega $ , any character in $\\mathrm {Irr}(G_r^{m}\\mid \\Omega )$ extends to $G^{m^{\\prime }}_r$ in exactly $q^n$ -many ways.",
"Thus, the number of coadjoint $G_r$ -orbits in $\\mathrm {Irr}(G^{m^{\\prime }}_r)$ is $q^{n({m^{\\prime }}-1)+n}=q^{n(r-{m^{\\prime }}-1)}$ , whence the first assertion.",
"The second assertion of theo:charcaters-of-Grm follows from the third assertion, (REF ) and [21].",
"Lastly, for the proof of the third assertion of theo:charcaters-of-Grm, we refer to [22] for the explicit construction, in the analogous case of $\\mathrm {GL}_n(\\mathfrak {o})$ and $\\mathrm {U}_n(\\mathfrak {o})$ , of an extension of a character $\\sigma \\in \\mathrm {Irr}(G^{m^{\\prime }}_r)$ to its inertia subgroup $I_{G_r}(\\sigma )$ .",
"Note that the construction of loc.",
"cit.",
"can be applied verbatim to the present setting, invoking the fact the $I_{G_r}(\\sigma )$ is generated by two abelian subgroups, one of which is normal in $G_r$ ((REF ) and corol:fin-index-abelian) in the generality of classical groups."
],
[
"Summary of section",
"In this section we compute the regular representation zeta function of classical groups of types $\\mathsf {B}_n,\\mathsf {C}_n$ and $\\mathsf {D}_n$ .",
"Following corol:reg-zeta-function, to do so, we classify the regular orbits in the space of orbits $\\mathrm {Ad}(G_1)\\backslash \\mathfrak {g}_1$ and compute their cardinalities, in order to obtain a formula for the Dirichlet polynomial $\\mathfrak {D}_{\\mathfrak {g}}(s)=\\sum _{\\Omega \\in X}{\\left|G_1\\right|}\\cdot \\left|\\Omega \\right|^{-(s+1)}.$ As it turns out, the cases where $\\mathbf {G}$ is of type $\\mathsf {B}_n$ or $\\mathsf {C}_n$ , i.e.",
"$\\mathbf {G}=\\mathrm {SO}_{2n+1}$ or $\\mathbf {G}=\\mathrm {Sp}_{2n}$ , can be handled simultaneously, and are analyzed in subsection:sp2n-so2n+1.",
"The case of the groups of the form $\\mathsf {D}_n$ , i.e.",
"even-dimensional orthogonal groups, is slightly more intricate.",
"The analysis of this case is carried out in subsection:so2n.",
"The main difference between the two cases lies in the fact that regularity of an element of the Lie-algebras $\\mathfrak {sp}_{2n}(\\mathsf {k})$ and $\\mathfrak {so}_{2n+1}(\\mathsf {k})$ is equivalent to it being give by a regular matrix in $\\mathrm {M}_N(\\mathsf {k})$ ; see propo:reg-equiavlent (also, cf.",
"[38]).",
"This equivalence fails to hold for even-orthogonal groups; see lem:nilpotent-reg-glN-not-reg-soN below.",
"Nevertheless, in both cases, we obtain a classification of the regular orbits in the Lie-algebra $\\mathfrak {g}_1$ in terms of the minimal polynomial of the elements within the orbit.",
"Recall that two matrices $x,y\\in \\mathrm {M}_N(\\mathsf {k})$ are said to be similar if there exists a matrix $g\\in \\mathrm {GL}_N(\\mathsf {k})$ such that $y=gxg^{-1}$ .",
"Our description of regular orbits of $\\mathfrak {g}_1$ follows the following steps.",
"Classification of all similarity classes in $\\mathfrak {gl}_N(\\mathsf {k})$ which intersect the set of regular elements in $\\mathfrak {g}_1$ non-trivially; Description of the intersection of such a similarity class with $\\mathfrak {g}_1$ as a union of $\\mathrm {Ad}(G_1)$ -orbits; Computation of the cardinality of the $\\mathrm {Ad}(G_1)$ -orbit of each regular element.",
"A rich theory of centralizers and conjugacy classes in classical groups over finite fields already exists, most notably Wall's extensive analysis in [39].",
"The enumeration of elements of a finite classical group $G_1$ whose representing matrix is cyclic (i.e.",
"regular when considered as an element of $\\mathrm {GL}_N({\\mathsf {k}^{\\mathrm {alg}}})$ ) was addressed in [28] and [15] where the proportion of such elements in $G_1$ , for all classical groups, was estimated and its limit as $\\mathrm {rk}(\\mathbf {G})$ tends to infinit was computed using generating functions.",
"The precise number of regular semisimple conjugacy classes was computed, again using generating functions, in [14], where the discrepancy between regularity of semisimple elements of the even dimensional orthogonal groups and of regularity of their representing matrices in $\\mathrm {GL}_N({\\mathsf {k}^{\\mathrm {alg}}})$ is determined (see [14]).",
"In the case of the symplectic group, the equivalence of regularity of an element of $\\mathrm {Sp}_{2n}(\\mathsf {k})$ and of its representing matrix in $\\mathrm {GL}_{2n}({\\mathsf {k}^{\\mathrm {alg}}})$ was noted in [15].",
"Examples of regular elements of $\\mathrm {SO}_{2n}(\\mathsf {k})$ which do not satisfy this equivalence appear in [27].",
"The setting considered in the present manuscript, while akin to, is rather simpler than the one dealt with in [39].",
"Namely, the relatively simpler theory of centralizers for the adjoint action of $\\mathbf {G}({\\mathsf {k}^{\\mathrm {alg}}})$ on ${\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ , in comparison with that of $\\mathbf {G}({\\mathsf {k}^{\\mathrm {alg}}})$ on itself by conjugation (compare, for example, propo:centralizer-semisimple-element and [20]), allows one to retrace much of Wall's analysis in the Lie-algebra setting, without having to invoke the notion of multipliers (see [39]).",
"Furthermore, the focus on regular adjoint classes results in a fairly “well-behaved” elementary divisor decomposition of the elements in the orbits under inspection.",
"We also remark that steps (1),(2) and (3) above are in direct parallel with items (i), (ii) and (iv), respectively, of [39], and may be derived from loc.",
"cit.",
"by the following procedure.",
"Given $x\\in \\mathrm {M}_N(\\mathsf {k})$ let $\\lambda \\in \\mathsf {k}$ be such that $x-\\lambda \\mathbf {1}$ is a non-singular matrix, and consider the dilated Cayley transform $g_x=(x-\\lambda \\mathbf {1})^{-1}(x+\\lambda \\mathbf {1})$ .",
"Then $x$ is similar to an element of $\\mathfrak {g}_1$ if and only if $g_x$ is similar to an element of $G_1$ , and the map $x^{\\prime }\\mapsto (x^{\\prime }-\\lambda \\mathbf {1})(x^{\\prime }+\\lambda \\mathbf {1})$ is a bijection between the adjoint orbit of $x$ under $\\mathrm {GL}_N(\\mathsf {k})$ (resp.",
"under $G_1$ ), and the similarity (resp.",
"adjoint) class of $g_x$ .",
"However, applying such an argument necessitates imposing additional restrictions on the characteristic of $\\mathsf {k}$ , and is somewhat less suitable for the purpose of enumeration of regular classes.",
"Given these complications, and the relative simplicity of the adjoint classes in question, we have opted to present a self-contained and independent analysis of the regular adjoint classes in $\\mathfrak {g}_1$ , which is presented in Sections REF -REF below.",
"Definition (Type of a polynomial) Let $f(t)\\in \\mathsf {k}[t]$ be a polynomial of degree $N$ and $n=\\lfloor \\frac{N}{2}\\rfloor $ .",
"For any $1\\le d,e\\le n$ , let $S_{d,e}(f)$ denote the number of distinct monic irreducible even polynomials of degree $2d$ which occur in $f$ with multiplicity $e$ , and let $T_{d,e}(f)$ denote the number of pairs $\\left\\lbrace {\\tau (t),\\tau (-t)}\\right\\rbrace $ , with $\\tau (t)$ monic, irreducible and coprime to $\\tau (-t)$ , such that $\\tau $ is of degree $d$ and occurs in $f$ with multiplicity $e$ .",
"Let $r(f)$ be the maximal integer such that $t^{2r(f)}$ divides $f$ .",
"The type of $f$ is defined to be the triplet $\\tau (f)=(r(f),S(f),T(f))$ , where $S(f)$ and $T(f)$ are the matrices $(S_{d,e}(f))_{d,e}$ and $(T_{d,e}(f))_{d,e}$ respectively.",
"Recall that $\\mathcal {X}_n$ denotes the set of triplets $\\tau =(r,S,T)\\in \\mathbb {N}_0\\times \\mathrm {M}_n(\\mathbb {N}_0)\\times \\mathrm {M}_{n}(\\mathbb {N}_0)$ , with $S=(S_{d,e})$ and $T=(T_{d,e})$ which satisfy $r+\\sum _{d,e=1}^n de\\cdot (S_{d,e}+T_{d,e})=n.$ Note that, for $n=\\lfloor \\frac{N}{2}\\rfloor $ , it holds that $\\tau (f)\\in \\mathcal {X}_n$ whenever $f$ is monic and satisfies $f(-t)=(-1)^{N}f(t)$ .",
"The number of monic irreducible polynomials of degree $d$ over $\\mathsf {k}$ is given by evaluation at $t=q$ of the function $w_d(t)=\\frac{1}{d}\\sum _{r\\mid d}\\mu \\left(\\frac{d}{r}\\right)t^d$ , where $\\mu (\\cdot )$ is the Möbius function (see, e.g., [13]).",
"A polynomial $f\\in \\mathsf {k}[t]$ is said to be even (resp.",
"odd) if it satisfies the condition $f(-t)=f(t)$ (resp.",
"$f(-t)=-f(t)$ ).",
"Note that, by assumption the $\\mathsf {k}$ is of odd characteristic, the only monic irreducible odd polynomial over $\\mathsf {k}$ is $f(t)=t$ .",
"The number of monic irreducible even polynomials of degree $d$ over $\\mathsf {k}$ is given by evaluation at $t=q$ of the function $E_d(t)={\\left\\lbrace \\begin{array}{ll}\\frac{1}{d}\\displaystyle \\sum _{\\begin{array}{c}m\\mid d\\:,\\:m\\text{ odd}\\end{array}}\\mu \\left(m\\right)(t^{d/2m}-1)&\\text{if $d$ is even}\\\\0&\\text{otherwise};\\end{array}\\right.",
"}$ cf.",
"[8], noting that the set of monic irreducible even polynomials of degree $d$ is in bijection with the set $N^{*1}(d,q)\\subseteq \\mathsf {k}[t]$ , defined in loc.",
"cit., via the map $f(t)\\mapsto \\frac{(1+t)^{\\deg f}}{f(-1)}f(\\frac{1-t}{1+t})$ .",
"Put $P_d(t)={\\left\\lbrace \\begin{array}{ll}w_d(t)-E_d(t)&\\text{if }d>1\\\\t-1&\\text{if }d=1.\\end{array}\\right.",
"}$ Note that, for $q$ odd, $P_d(q)$ is the number of irreducible polynomials of degree $d$ which are neither odd nor even over a field of cardinality $q$ .",
"Given $N\\in \\mathbb {N}$ , $n=\\lfloor \\frac{N}{2}\\rfloor $ , and $\\tau \\in \\mathcal {X}_n$ , the number of polynomials $f\\in \\mathsf {k}[t]$ of type $\\tau (f)$ such that $f(-t)=(-1)^Nf(t)$ is given by evaluation at $t=q$ of the polynomial $M_{\\tau }(t)=\\left(\\frac{1}{2}\\right)^{\\sum _{d,e} T_{d,e}}\\prod _{d=1}^n{\\sum _e S_{d,e}\\atopwithdelims ()S_{d,1},S_{d,2},\\ldots ,S_{d,n}}\\cdot {E_{2d}(t)\\atopwithdelims ()\\sum _e S_{d,e}}\\cdot {\\sum _e T_{d,e}\\atopwithdelims ()T_{d,1},T_{d,2},\\ldots ,T_{d,n}}\\cdot {P_{d}(t)\\atopwithdelims ()\\sum _e T_{d,e}}.$ The combinatorial data described above is utilized in mainthm:dirichlet-polnomials-sp-odd-orth and mainthm:dirichlet-polnomials-even-orth, where it allows to enumerate the similarity classes in $\\mathrm {M}_N(\\mathsf {k})$ which meet the Lie-algebra $\\mathfrak {g}_1$ non-trivially in terms of the minimal polynomial of the class elements.",
"The classification of such similarity classes and their decomposition into $\\mathrm {Ad}(G_1)$ is described theo:orbits-sp2nso2n+1 and theo:orbits-so2n below.",
"Once Theorems REF and REF are proved, the proof of mainthm:dirichlet-polnomials-sp-odd-orth and of mainthm:dirichlet-polnomials-even-orth may be completed by direct computation."
],
[
"Statement of results- symplectic and odd-dimensional special orthogonal groups",
"Theorem 1.1 Assume ${\\mathrm {char}}(\\mathsf {k})\\ne 2$ .",
"Let $V=\\mathsf {k}^N$ and let $B$ be a non-degenerate bilinear form which is anti-symmetric if $N=2n$ is even, and symmetric if $N=2n+1$ .",
"Let $\\mathbf {G}\\in \\left\\lbrace {\\mathrm {Sp}_{2n},\\mathrm {SO}_{2n+1}}\\right\\rbrace $ be the algebraic group of isometries of $V$ with respect to $B$ and put $G_1=\\mathbf {G}(\\mathsf {k})$ and $\\mathfrak {g}_1={\\mathbf {g}}(\\mathsf {k})$ where ${\\mathbf {g}}=\\mathrm {Lie}(\\mathbf {G})$ .",
"Let $x\\in \\mathrm {M}_N(\\mathsf {k})$ have minimal polynomial $m_x\\in \\mathsf {k}[t]$ .",
"The element $x$ is similar to a regular element of $\\mathfrak {g}_1$ if and only if $m_x$ has degree $N$ and satisfies $m_x(-t)=(-1)^Nm_x(t)$ .",
"Furthermore, assume $x\\in \\mathfrak {g}_1$ is a regular element and let $\\Omega =\\mathrm {Ad}(G_1)x$ denote its orbit under $G_1$ .",
"If $N$ is even and $m_x(0)=0$ , then the intersection $\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1$ is the union of two distinct $\\mathrm {Ad}(G_1)$ -orbits.",
"Otherwise, $\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1=\\Omega $ .",
"Let $\\tau =\\tau (m_x)=\\left(r(m_x),S(m_x),T(m_x)\\right)$ as in defi:poly-type.",
"Then $\\left|\\Omega \\right|=q^{2n^2}\\cdot \\left(\\frac{1}{2}\\right)^\\nu \\frac{\\prod _{i=1}^n(1-q^{-2i})}{\\prod _{1\\le d,e\\le n}(1+q^{-d})^{S_{d,e}(m_x)}\\cdot (1-q^{-d})^{T_{d,e}(m_x)}},$ where $\\nu =1$ if $N=2n$ is even and $m_x(0)=0$ , and $\\nu =0$ otherwise.",
"The proofs of Assertions (1), (2) and (3) of the theorem are carried out in sections REF ,REF and REF respectively."
],
[
"Statement of results- even-dimensional special orthogonal groups",
"Theorem 1.2 Assume $\\left|\\mathsf {k}\\right|>3$ and ${\\mathrm {char}}(\\mathsf {k})\\ne 2$ .",
"Let $N=2n$ with $n\\ge 2$ .",
"Let $V=\\mathsf {k}^N$ and let $B^+$ and $B^-$ be non-degenerate symmetric forms on $V$ of Witt index $n$ and $n-1$ , respectively.",
"Given $\\epsilon \\in \\left\\lbrace {\\pm 1}\\right\\rbrace $ , let $\\mathbf {G}^\\epsilon =\\mathrm {SO}_{2n}^\\epsilon $ be the $\\mathsf {k}$ -algebraic group of isometries of $V$ with respect to $B^\\epsilon $ and put $G^\\epsilon _1=\\mathbf {G}_1^\\epsilon (\\mathsf {k})$ and let $\\mathfrak {g}^\\epsilon _1={\\mathbf {g}}^\\epsilon (\\mathsf {k})$ , where ${\\mathbf {g}}^\\epsilon =\\mathrm {Lie}(\\mathbf {G})$ .",
"Let $x\\in \\mathrm {M}_N(\\mathsf {k})$ have minimal polynomial $m_x(t)$ .",
"If $m_x(0)=0$ (i.e.",
"$x$ is a singular matrix) then the following are equivalent.",
"The polynomial $m_x$ has degree $N-1$ and satisfies $m_x(-t)=-m_x(t)$ .",
"The element $x$ is similar to a regular element of $\\mathfrak {g}_1^+$ .",
"The element $x$ is similar to a regular element of $\\mathfrak {g}_1^-$ .",
"Otherwise, if $m_x(0)\\ne 0$ , let $\\epsilon =\\epsilon (x)=(-1)^{\\sum _{e}eS_{d,e}(m_x)}$ where $S=(S_{d,e}(m_x))$ is as in defi:poly-type.",
"Then $x$ is similar to a regular element of $\\mathfrak {g}_1^\\epsilon $ if and only if $m_x$ has degree $N$ and satisfies $m_x(-t)=m_x(t)$ .",
"Moreover, in this case $x$ is not similar to an element of $\\mathfrak {g}_1^{-\\epsilon }$ .",
"Furthermore, assume $x\\in \\mathfrak {g}^\\epsilon _1$ is a regular element and let $\\Omega ^\\epsilon =\\mathrm {Ad}(G_1^\\epsilon )x$ denote its orbit under $G_1^\\epsilon $ , for $\\epsilon \\in \\left\\lbrace {\\pm 1}\\right\\rbrace $ fixed.",
"In the case where $m_x(0)=0$ , the intersection $\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1^\\epsilon $ is the disjoint union of two distinct $\\mathrm {Ad}(G_1^\\epsilon )$ -orbits.",
"Otherwise, $\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1^\\epsilon =\\Omega ^\\epsilon $ .",
"Assume $m_x(0)=0$ and let $\\tau =\\tau (t\\cdot m_x)$ .",
"Then $\\left|\\Omega ^\\epsilon \\right|=q^{2n^2}\\cdot \\frac{1}{2}\\cdot \\frac{(1+\\epsilon q^{-n})\\prod _{i=1}^{n-1}(1-q^{-2i})}{\\prod _{1\\le d,e\\le n}(1+q^{-d})^{S_{d,e}(m_x)}\\cdot (1-q^{-d})^{T_{d,e}(m_x)}}.$ Otherwise, let $\\tau =\\tau (m_x)$ .",
"Then $\\left|\\Omega ^\\epsilon \\right|=q^{2n^2}\\cdot \\frac{(1+\\epsilon q^{-n})\\prod _{i=1}^{n-1}(1-q^{-2i})}{ \\prod _{1\\le d,e\\le n}(1+q^{-d})^{S_{d,e}(m_x)}\\cdot (1-q^{-d})^{T_{d,e}(m_x)}}.$ The proofs of Assertions (1),(2) and (3) of the theorem appear in sections REF ,REF and REF .",
"The exclusion of the specific case of $\\mathsf {k}=\\mathbb {F}_3$ is done for technical reasons, and may possibly be undone by replacement of the argument in lem:difference-of-squares below."
],
[
"Regularity for non-singular elements",
"Lemma Let $x\\in {\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})\\subseteq \\mathfrak {gl}_N({\\mathsf {k}^{\\mathrm {alg}}})$ be non-singular.",
"Then $x$ is regular in ${\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ if and only if $x$ is a regular element of $\\mathfrak {gl}_N({\\mathsf {k}^{\\mathrm {alg}}})$ .",
"Let $W=({\\mathsf {k}^{\\mathrm {alg}}})^N$ , so that ${\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ is given as the Lie-algebra of anti-symmetric operators with respect to a non-degenerate bilinear form $B=B_{{\\mathsf {k}^{\\mathrm {alg}}}}$ on $W$ (see subsection:group-notation).",
"Note that the existence of non-singular elements in ${\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ implies that $N=2n$ is even.",
"Indeed, $x\\in {\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ if and only if $x^\\star =-x$ (notation of subsubsection:adjoint-operators), and $\\det (x)=\\det (x^\\star )=(-1)^N\\det (x)$ is possible if and only if $N$ is even, since ${\\mathrm {char}}({\\mathsf {k}^{\\mathrm {alg}}})\\ne 2$ .",
"Let $x=s+h$ be the Jordan decomposition of $x$ , with $s,h\\in {\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ , $s$ semisimple, $h$ nilpotent and $[s,h]=0$ .",
"Let $\\lambda _1,\\ldots ,\\lambda _t\\in {\\mathsf {k}^{\\mathrm {alg}}}$ be non-zero and such that $\\left\\lbrace {\\pm \\lambda _1,\\ldots ,\\pm \\lambda _t}\\right\\rbrace $ is the set of all eigenvalues of $s$ with $\\lambda _i\\ne \\pm \\lambda _j$ whenever $i\\ne j$ .",
"As in propo:centralizer-semisimple-element, the space $W$ decomposes as a direct sum $W=\\bigoplus _{i=1}^t (W_{\\lambda _i}\\oplus W_{-\\lambda _i})$ , where, for any $i=1,\\ldots ,t$ , the subspace $W_{[\\lambda _i]}=W_{\\lambda _i}\\oplus W_{-\\lambda _i}$ is non-degenerate, and its subspaces $W_{\\lambda _i}$ and $W_{-\\lambda _i}$ are maximal isotropic.",
"Comparing centralizer dimension, and invoking [36], we have that $x$ is regular if and only if the restriction of $x$ to each of the subspaces $W_{[\\lambda _i]}$ ($i=1,\\ldots ,t$ ) is regular in $\\mathfrak {gl}_N(W_{[\\lambda _i]})$ .",
"Likewise, $x$ is regular in ${\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ if and only if the restriction of $x$ to each subspace $W_{[\\lambda _i]}$ is regular within the Lie-algebra of anti-symmetric operators with respect to the restriction of $B_{\\mathsf {k}^{\\mathrm {alg}}}$ to $W_{[\\lambda _i]}$ .",
"Thus, it is sufficient to prove the lemma in the case where $s$ has precisely two eigenvalues $\\lambda ,-\\lambda $ .",
"Representing $s$ in a suitable eigenbasis, it be identified with the block-diagonal matrix $\\mathrm {diag}(\\lambda \\mathbf {1}_n,-\\lambda \\mathbf {1}_n)$ .",
"Under this identification, the centralizer of $s$ in $\\mathfrak {gl}_N({\\mathsf {k}^{\\mathrm {alg}}})$ is identified with the subgroup of block diagonal matrices consisting of two $n\\times n$ blocks.",
"Moreover, the involution $\\star $ maps an element $\\mathrm {diag}(y_1,y_2)\\in \\mathbf {C}_{\\mathfrak {gl}_N({\\mathsf {k}^{\\mathrm {alg}}})}(s)$ , with $y_1,y_2\\in \\mathfrak {gl}_n({\\mathsf {k}^{\\mathrm {alg}}})$ to the matrix $\\mathrm {diag}(y_2^t,y_1^t)$ .",
"In particular, it follows that $h\\in \\mathbf {C}_{\\mathfrak {gl}_N}({\\mathsf {k}^{\\mathrm {alg}}})\\cap {\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ is of the form $h=\\mathrm {diag}(h_1,-h_1^t)$ , where $h_1\\in \\mathfrak {gl}_n({\\mathsf {k}^{\\mathrm {alg}}})$ is nilpotent.",
"Arguing as in [32], we have that $\\mathbf {C}_{\\mathrm {GL}_N}(x)=\\mathbf {C}_{\\mathbf {C}_{\\mathrm {GL}_N}(s)}(h)\\simeq \\mathbf {C}_{\\mathrm {GL}_n}(h_1)\\times \\mathbf {C}_{\\mathrm {GL}_n}(-h_1^t)\\simeq \\mathbf {C}_{\\mathrm {GL}_n}(h_1)\\times \\mathbf {C}_{\\mathrm {GL}_n}(h_1)$ where the final isomorphism utilizes the isomorphism $y\\mapsto (y^t)^{-1}:\\mathbf {C}_{\\mathrm {GL}_n}(-h_1^t)\\rightarrow \\mathbf {C}_{\\mathrm {GL}_n}(h_1)$ .",
"Finally, since the group $\\mathbf {G}$ is embedded in $\\mathrm {GL}_N$ as the group of unitary elements with respect to $\\star $ , we have $\\mathrm {diag}(y_1,y_2)\\in \\mathbf {C}_{\\mathrm {GL}_N({\\mathsf {k}^{\\mathrm {alg}}})}(s)\\cap \\mathbf {G}({\\mathsf {k}^{\\mathrm {alg}}})$ if and only if $y_2=(y_1^{t})^{-1}$ , and hence the map $y\\mapsto \\mathrm {diag}(y,(y^t)^{-1})$ is an isomorphism of $\\mathbf {C}_{\\mathrm {GL}_n}(h_1)$ onto $\\mathbf {C}_{\\mathbf {C}_\\mathbf {G}(s)}(h)$ and hence $\\mathbf {C}_\\mathbf {G}(x)=\\mathbf {C}_{\\mathbf {C}_\\mathbf {G}(s)}(h)\\simeq \\mathbf {C}_{\\mathrm {GL}_n}(h_1).$ Thus $\\dim \\mathbf {C}_{\\mathrm {GL}_N}(x)=2\\dim \\mathbf {C}_{\\mathrm {GL}_n}(h_1)=2\\dim \\mathbf {C}_{\\mathbf {G}}(x),$ and the lemma follows.",
"Remark The assumption that $x$ is non-singular in corol:reg-g1-reg-glN-nonsing is crucial, as the proof relies heavily on the fact that the centralizer of a non-singular semisimple element of $_1({\\mathsf {k}^{\\mathrm {alg}}})$ in $\\mathbf {\\Gamma }_1({\\mathsf {k}^{\\mathrm {alg}}})$ is a direct product of groups of the form $\\mathrm {GL}_{m_j}({\\mathsf {k}^{\\mathrm {alg}}})$ (see propo:centralizer-semisimple-element).",
"The same argumentation would not apply in the case where $x$ is singular, and in fact fails in certain cases; see lem:nilpotent-reg-glN-not-reg-soN below."
],
[
"From similarity classes to adjoint orbits",
"In this section develop some the tools required in order to analyze the decomposition of the set $\\Pi _x=\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1$ , for $x\\in \\mathfrak {g}_1$ regular, in to $\\mathrm {Ad}(G_1)$ -orbits.",
"The results appearing below can also be derived from [39].",
"However, as the case of regular elements of the Lie-algebra $\\mathfrak {g}_1$ allows for a much more transparent argument, we present it here for completeness.",
"Let $\\mathrm {Sym}(\\star ;x)$ be the set of elements $Q\\in \\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)$ such that $Q^\\star =Q$ and define an equivalence relation on $\\mathrm {Sym}(\\star ;x)$ by $Q_1\\sim Q_2\\quad \\text{if there exists }a\\in \\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)\\text{ such that }Q_1=a^\\star Q_2a.$ Let $\\Theta _x$ to be the set of equivalence classes of $\\sim $ in $\\mathrm {Sym}(\\star ;x)$ .",
"In the case where $\\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)$ is abelian (e.g., when $x$ is a regular element of $\\mathfrak {gl}_N(\\mathsf {k})$ ), the set $\\mathrm {Sym}(\\star ;x)$ is a subgroup and the set $\\Theta _x$ is simply its quotient by the image of restriction of $w\\mapsto w^\\star w$ to $\\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)$ .",
"Proposition Let $x\\in \\mathfrak {g}_1$ and let $\\Pi _x$ denote the intersection $\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1$ .",
"There exists a map $\\Lambda :\\Pi _x\\rightarrow \\Theta _x$ such that $y_1,y_2\\in \\Pi _x$ are $\\mathrm {Ad}(G_1)$ -conjugate if and only if $\\Lambda (y_1)=\\Lambda (y_2)$ .",
"1.",
"Construction of $\\Lambda $ .",
"Let $y\\in \\Pi _x$ and let $w\\in \\mathrm {GL}_N(\\mathsf {k})$ be such that $y=wxw^{-1}$ .",
"Put $Q=w^\\star w$ .",
"Note that, as $x,y\\in \\mathfrak {g}_1$ , by applying the anti-involution $\\star $ to the equation $y=wxw^{-1}$ , we deduce that $(w^\\star )^{-1}xw^\\star =y$ as well and consequently, that $Q=w^\\star w$ commutes with $x$ .",
"Since $Q^\\star =Q$ , we get that $Q\\in \\mathrm {Sym}(\\star ;x)$ .",
"Define $\\Lambda (y)$ to be the equivalence class of $Q$ in $\\Theta _x$ .",
"To show that $\\Lambda $ is well-defined, let $w^{\\prime }\\in \\mathrm {GL}_N(\\mathsf {k})$ be another element such that $y=w^{\\prime }xw^{\\prime -1}$ and $Q^{\\prime }=w^{\\prime \\star }w^{\\prime }$ .",
"Put $a=w^{-1}w^{\\prime }$ .",
"Then $a$ commutes with $x$ , and $a^\\star Q a=w^{\\prime \\star }(w^\\star )^{-1}Qw^{-1}w^{\\prime }=w^{\\prime \\star }w^{\\prime }=Q^{\\prime },$ hence $Q\\sim Q^{\\prime }$ .",
"2.",
"Proof that $y_1,y_2\\in \\Pi _x$ are $\\mathrm {Ad}(G_1)$ -conjugate if $\\Lambda (y_1)=\\Lambda (y_2)$ .",
"Let $w_1,w_2\\in \\mathrm {GL}_N(\\mathsf {k})$ be such that $y_i=w_ixw_i^{-1}$ , and let $Q_i=w_i^\\star w_i$ ($i=1,2$ ).",
"Then, by assumption, there exists $a\\in \\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)$ such that $Q_2=a^\\star Q_1a$ .",
"Put $z=w_1aw_2^{-1}$ .",
"Note that $zy_2z^{-1}=y_1$ .",
"We claim that $z\\in G_1$ .",
"This holds since for any $u,v\\in V$ $B(zu,zv)&=B(w_1aw_2^{-1}u,w_1aw_2^{-1}v)=B(a^\\star (w_1^\\star w_1)aw_2^{-1}u,w_2^{-1}v)\\\\&=B(a^\\star Q_1 aw_2^{-1}u,w_2^{-1}v)=B(Q_2 w_2^{-1} u,w_2^{-1}v)&\\text{(since $Q_2=a^\\star Q_1 a$)}\\\\&=B(w_2^\\star u,w_2^{-1}v)=B(u,v).$ 3.",
"Proof that $y_1,y_2\\in \\Pi _x$ are $\\mathrm {Ad}(G_1)$ -conjugate only if $\\Lambda (y_1)=\\Lambda (y_2)$ .",
"Assume now that $z\\in G_1$ is such that $y_1=zy_2z^{-1}$ , and let $w_1,w_2\\in \\mathrm {GL}_N(\\mathsf {k})$ be such that $y_i=w_ixw_i^{-1}$ ($i=1,2$ ).",
"Then $w_1$ and $zw_2$ both conjugate $x$ to $y_1$ , and hence, by the unambiguity of the definition of $\\Lambda $ and fact that $z\\in G_1$ , we have that $\\Lambda (y_1)=[w_1^\\star w_1]=[w_2^\\star ( z^\\star z) w_2]=[w_2^\\star w_2]=\\Lambda (y_2).$ A crucial property of the set $\\Theta _x$ in the case $x$ is regular, which makes the analysis of adjoint orbits feasible, is that it may be realized within the quotient of an étale algebra over $\\mathsf {k}$ by the image of the algebra under an involution.",
"As a consequence, the set $\\Pi _x$ decomposes into $\\left|\\mathrm {Im}\\Lambda \\right|$ many $\\mathrm {Ad}(G_1)$ -orbits, a quantity which does not exceed the value four in the regular case.",
"Let us state another general lemma, which will be required in the description of $\\Theta _x$ .",
"Lemma Let $\\mathcal {C}\\subseteq \\mathrm {M}_N(\\mathsf {k})$ be the ring of matrices commuting with a matrix $x$ , with $x^\\star =-x$ (or $x^\\star =x$ ), and let $\\mathcal {N}\\triangleleft \\mathcal {C}$ be a nilpotent ideal, invariant under $\\star $ .",
"The following are equivalent, for any $Q_1,Q_2\\in \\mathrm {Sym}(\\star ;x)$ .",
"There exists $a\\in \\mathcal {C}$ such that $a^\\star Q_1 a=Q_2$ ; There exists $a\\in \\mathcal {C}$ such that $a^\\star Q_1 a\\equiv Q_2\\mathcal {}\\pmod {\\mathcal {N}}$ .",
"The argument of [39] applies to the case where $\\mathcal {N}$ is any nilpotent ideal which is invariant under $\\star $ , provided that the required trace condition holds.",
"In the present case the condition holds since ${\\mathrm {char}}(\\mathsf {k})\\ne 2$ ."
],
[
"Similarity classes via bilinear forms",
"We recall a basic lemma which would allow us to determine when an element of $\\mathfrak {gl}_N(\\mathsf {k})$ is similar to an element of $\\mathfrak {g}_1$ .",
"Here and in the sequel, given a non-degenerate bilinear form $C$ on a finite dimensional vector space $V$ over $\\mathsf {k}$ , we call an operator $x\\in \\mathrm {End}(V)$ $C$ -anti-symmetric, if $C(xu,v)+C(u,xv)=0$ holds for all $u,v\\in V$ .",
"Lemma Let $C_1,C_2$ be two non-degenerate bilinear forms on a vector space $V=\\mathsf {k}^N$ , and assume there exists $g\\in \\mathrm {End}(V)$ and $\\delta \\in \\mathsf {k}$ such that $C_1(gu,gv)=\\delta C_2(u,v)$ for all $u,v\\in V$ .",
"Let $x\\in \\mathfrak {gl}_N(\\mathsf {k})$ be $C_2$ -anti-symmetric.",
"Then $gxg^{-1}$ is $C_1$ -anti-symmetric.",
"The proof of lem:conj-by-forms is by direct computation, and is omitted.",
"Throughout subsection:sp2n-so2n+1, we assume $\\mathbf {G}=\\mathrm {Sp}_{2n}$ or $\\mathbf {G}=\\mathrm {SO}_{2n+1}$ .",
"The following well-known fact is very useful in the classification of regular adjoint classes in the Lie-algebra $\\mathfrak {g}_1$ .",
"Lemma Let $\\epsilon =-1$ and $N=2n$ in the symplectic case, or $\\epsilon =1$ and $N=2n+1$ in the special orthogonal case.",
"Let $C_1,C_2$ be two non-degenerate forms on $V=\\mathsf {k}^N$ such that $C_i(u,v)=\\epsilon C_i(v,u)$ for all $u,v\\in V$ and $i=1,2$ .",
"There exists $\\delta \\in \\mathsf {k}$ and $g\\in \\mathrm {End}(V)$ such that $C_1(gu,gv)=\\delta C_2(u,v)$ for all $u,v\\in V$ .",
"Additionally, if $\\epsilon =-1$ then $\\delta $ can be taken to be 1.",
"See, e.g., [42] in the symplectic case and [42] in the special orthogonal case."
],
[
"Similarity classes of regular elements",
"The following lemma gives a criterion for a regular matrix to be similar to an element of $\\mathfrak {g}_1$ .",
"Lemma Let $x\\in \\mathfrak {gl}_N(\\mathsf {k})$ with minimal polynomial $m_x(t)\\in \\mathsf {k}[t]$ .",
"If $x$ is similar to an element of $\\mathfrak {g}_1$ then $m_x(t)$ satisfies $m_x(-t)=(-1)^{\\deg m_x}m_x(t)$ .",
"If $x$ is a regular element of $\\mathfrak {gl}_N(\\mathsf {k})$ (and hence $\\deg m_x=N$ ) such that $m_x(t)=(-1)^Nm_x(t)$ , then $x$ is similar to an element of $\\mathfrak {g}_1$ .",
"For the first assertion, we may assume $x\\in \\mathfrak {g}_1$ .",
"Note that for any $r\\in \\mathbb {N}$ we have that $B(x^ru,v)=B(u,(-1)^r x^rv)$ for all $u,v\\in V=\\mathsf {k}^N$ .",
"Invoking the non-degeneracy of $B$ , we deduce that $(-1)^{\\deg m_x}m_x(-t)$ is a monic polynomial of degree $\\deg m_x$ which vanishes at $x$ , and hence equal to $m_x(t)$ .",
"By lem:conj-by-forms, to prove the second assertion it would suffice to construct a non-degenerate bilinear form $C$ on $V$ such that $B$ and $C$ satisfy the hypothesis of lem:conj-by-forms.",
"In view of lem:eseential-uniqueness, in the present case it suffices to construct some non-degenerate bilinear form $C$ on $V$ such that $C(u,v)=\\epsilon C(v,u)$ , where $\\epsilon =(-1)^N$ , and such that $x$ is $C$ -anti-symmetric.",
"By [36], the assumption that $x$ is a regular matrix is equivalent to $V$ being a cyclic module over the ring $\\mathsf {k}[x]$ (which, in turn, is equivalent to $\\deg m_x=N$ ).",
"In particular, there exists $v_0\\in V$ such that $(v_0,xv_0,\\ldots ,x^{N-1}v_0)$ is a $\\mathsf {k}$ -basis for $V$ .",
"Let $\\mathrm {Prj}_{N-1}:V\\rightarrow \\mathsf {k}$ denote the projection onto $\\mathsf {k}\\cdot x^{N-1}v_0$ .",
"Given $u_1,u_2\\in V$ let $p_1,p_2\\in \\mathsf {k}[t]$ be polynomials such that $u_i=p_i(x)v_0$ and define $ C(u_1,u_2)=\\mathrm {Prj}_{N-1}\\left(p_1(x)p_2(-x)v_0\\right).$ The fact that $C$ is well-defined, bilinear and satisfies $C(u,v)=\\epsilon C(c,u)$ follows by direct computation.",
"Let us verify that $C$ is non-degenerate.",
"Let $u\\in V$ be non-zero, and let $p(t)$ be such that $p(x)v_0=u$ .",
"By unambiguity of the definition of $C$ , we may assume that $\\deg p(t)<N$ .",
"Let $v=x^{N-1-\\deg p} v_0\\in V$ .",
"Then $C(u,v)=\\mathrm {Prj}_{N-1}((-1)^{N-1-\\deg p}x^{N-1-\\deg p}p(x)v_0)$ is non-zero, since $t^{N-1-\\deg p} p(t)$ is a polynomial of degree $N-1$ .",
"Finally, for $u_i=p_i(x)v_0$ as above, we have that $C(xu,v)+X(u,xv)=\\mathrm {Prj}_{N-1}(xp_1(x)p_2(x)v_0)+\\mathrm {Prj}_{N-1}(p_1(x)\\cdot (-xp_2(-x))v_0)=\\mathrm {Prj}_{N-1}(0)=0,$ and hence $x$ is $C$ -anti-symmetric.",
"Note that lem:conj-to-g1 gives a criterion for a regular element of $\\mathfrak {gl}_N(\\mathsf {k})$ to be similar to an element of $\\mathfrak {g}_1$ , but a-priori, not necessarily to a regular element of $\\mathfrak {g}_1$ .",
"We will shortly see that it is indeed the case that the similarity class of such $x$ meets $\\mathfrak {g}_1$ at a regular orbit.",
"Before proving this, let us consider an important example.",
"Example (Regular nilpotent elements) Let $x\\in \\mathfrak {gl}_N(\\mathsf {k})$ be a regular nilpotent element, i.e.",
"$m_x(t)=t^N$ .",
"Picking a generator $v_0$ for $V$ over $\\mathsf {k}[x]$ and putting $\\mathcal {E}=(v_0,xv_0,\\ldots ,x^{N-1}v_0)$ , the element $x$ is represented in the basis $\\mathcal {E}$ by the matrix $\\Upsilon $ , given by an $N\\times N$ nilpotent Jordan block.",
"The bilinear form $C$ of lem:conj-to-g1 is represented in this basis by the matrix $\\mathbf {c}=\\left(\\begin{matrix}&&&1\\\\&&-1\\\\&{\\ddots }\\\\(-1)^{N-1}\\end{matrix}\\right).$ To show that $\\Upsilon $ is similar to a regular element of $\\mathfrak {g}_1$ , by [36] and [20], it suffices to show that the centralizer of $\\Upsilon $ within the Lie-algebra $\\mathfrak {h}\\subseteq \\mathrm {M}_N({\\mathsf {k}^{\\mathrm {alg}}})$ , of matrices $y$ satisfying the condition $y^t\\mathbf {c}+\\mathbf {c} y$ (i.e.",
"the Lie-algebra of the linear algebraic ${\\mathsf {k}^{\\mathrm {alg}}}$ -group of isometries of $C(\\cdot ,\\cdot )$ ), is of dimension $n$ over ${\\mathsf {k}^{\\mathrm {alg}}}$ .",
"By direct computation, one shows that $\\mathbf {C}_{\\mathfrak {h}}(\\Upsilon )=\\left\\lbrace {\\left(\\begin{matrix}a_1&a_2&\\ldots &a_{N}\\\\&\\ddots &\\ddots &\\vdots \\\\&&a_1&a_2\\\\&&&a_1\\end{matrix}\\right)\\in \\mathrm {M}_N({\\mathsf {k}^{\\mathrm {alg}}})\\mid 2a_{2i-1}=0\\text{ for all }i=1,\\ldots ,\\lceil {N}/{2}\\rceil }\\right\\rbrace .$ Recalling that ${\\mathrm {char}}({\\mathsf {k}^{\\mathrm {alg}}})\\ne 2$ , it follows that $a_{2i+1}=0$ for all $i=0,\\ldots ,\\lceil {N/2}\\rceil $ and hence $\\dim _{{\\mathsf {k}^{\\mathrm {alg}}}}\\mathbf {C}_{\\mathfrak {h}}(\\Upsilon )=\\lfloor N/2\\rfloor = n.$ Proposition Let $x\\in \\mathfrak {g}_1$ .",
"Then $x$ is a regular element of $\\mathfrak {g}_1$ if and only if $x$ is regular in $\\mathfrak {gl}_N(\\mathsf {k})$ .",
"By [36], we need to show $\\dim \\mathbf {C}_{\\mathbf {\\Gamma }_1}(x)=n$ if and only if $\\dim \\mathbf {C}_{\\mathrm {GL}_N\\times {\\mathsf {k}^{\\mathrm {alg}}}}(x) =N$ .",
"Let $x=s+h$ be the Jordan decomposition of $x$ over ${\\mathsf {k}^{\\mathrm {alg}}}$ , with $s,h\\in {\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ , $s$ semisimple, $h$ nilpotent, and $[s,h]=0$ .",
"As seen in the proof of propo:centralizer-semisimple-element, the space $W=({\\mathsf {k}^{\\mathrm {alg}}})^N$ decomposes as an orthogonal direct sum $W_1\\oplus W_0$ with respect to the ambient bilinear form $B_{{\\mathsf {k}^{\\mathrm {alg}}}}$ , where $W_0=\\mathrm {Ker}(s)$ and $s\\mid _{W_1}$ is non-singular.",
"Let $\\Sigma \\subseteq \\mathbf {G}({\\mathsf {k}^{\\mathrm {alg}}})$ be the subgroup of elements acting trivially on $W_0$ and preserving $W_1$ , and let $\\Delta $ be as in propo:centralizer-semisimple-element.",
"Then $\\mathbf {C}_{\\mathbf {\\Gamma }_1}(x)&=\\mathbf {C}_{\\Sigma }(x)\\times \\mathbf {C}_{\\left\\lbrace {1_{W_1}}\\right\\rbrace \\times \\Delta }(x)\\multicolumn{2}{l}{\\text{and}}\\\\ \\mathbf {C}_{\\mathrm {GL}_N({\\mathsf {k}^{\\mathrm {alg}}})}(x)&=\\mathbf {C}_{\\mathrm {GL}(W_1)\\times \\left\\lbrace {1_{W_0}}\\right\\rbrace }(x)\\times \\mathbf {C}_{\\left\\lbrace {1_{W_1}}\\right\\rbrace \\times \\mathrm {GL}(W_0)}(x)$ and therefore the proof reduces to the cases where $x$ is non-singular and where $x$ is a nilpotent element acting on $W_0$ .",
"The first case follows from corol:reg-g1-reg-glN-nonsing, whereas the second case follows from exam:reg-nil-elt-so2n+1-sp2n and from the uniqueness of a regular nilpotent orbit over algebraically closed fields [36].",
"[Proof of theo:orbits-sp2nso2n+1.",
"(1)] propo:reg-equiavlent implies that any element $x\\in \\mathrm {M}_N(\\mathsf {k})$ which is similar to a regular element of $\\mathfrak {g}_1$ is regular as an element of $\\mathfrak {gl}_N(\\mathsf {k})$ .",
"It follows easily that $\\deg m_x=N$ and $m_x(-t)=(-1)^N m_x(t)$ (see lem:conj-to-g1.(1)).",
"The converse implication is given by lem:conj-to-g1.",
"(2)."
],
[
"From similarity classes to adjoint orbits",
"In this section is to we analyze decomposition of the set $\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1$ , for $x\\in \\mathfrak {g}_1$ regular, into $\\mathrm {Ad}(G_1)$ -orbits, and prove theo:orbits-sp2nso2n+1.(2).",
"Notation Given a polynomial $f(t)\\in \\mathsf {k}[t]$ we write $\\mathsf {k}\\langle f\\rangle $ for the quotient ring $\\mathsf {k}[t]/(f)$ .",
"For example, if $f$ is an irreducible polynomial over $\\mathsf {k}$ then $\\mathsf {k}\\langle f\\rangle $ stands for the splitting field of $f$ .",
"We write $\\mathrm {GL}_1(\\mathsf {k}\\langle f\\rangle )$ for the group of units of $\\mathsf {k}\\langle f\\rangle $ .",
"Assuming further that $f(t)=\\pm f(-t)$ , let $\\sigma _f$ denote the $\\mathsf {k}$ -involution of $\\mathsf {k}\\langle f\\rangle $ , induced from the $\\mathsf {k}[t]$ -involution $t\\mapsto -t$ , and let $\\mathrm {U}_1(\\mathsf {k}\\langle f\\rangle )$ be the group of elements $\\xi \\in \\mathsf {k}\\langle f\\rangle $ such that $\\sigma _f(\\xi )\\cdot \\xi =~1$ .",
"Proposition Let $x\\in \\mathfrak {g}_1$ be a regular element, and put $\\Pi _x=\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1$ .",
"If $x$ is singular and $N$ is even, then the intersection $\\Pi _x$ is the disjoint union of two distinct $\\mathrm {Ad}(G_1)$ -orbits.",
"Otherwise, $\\Pi _x=\\mathrm {Ad}(G_1)x$ .",
"The notation of propo:orbit-decomposition is used freely throughout the proof.",
"We proceed in the following steps.",
"Computation of the cardinality of $\\Theta _x$ .",
"Namely, we show that $\\left|\\Theta _x\\right|=2$ if $x$ is singular and equals 1 otherwise.",
"Description of the image of the map $\\Lambda $ in $\\Theta _x$ .",
"By Lemma REF , the minimal polynomial $m_x$ of $x$ is of degree $N$ and satisfies $m_x(-t)=(-1)^{N} m_x(t)$ .",
"Thus, it can be expressed uniquely as the product of pairwise coprime factors $m_x(t)=t^{d_1}\\cdot \\prod _{i=1}^{d_2}\\varphi _i(t)^{l_i}\\cdot \\prod _{i=1}^{d_3}\\theta _i(t)^{r_i},$ where the polynomials $\\varphi _1,\\ldots ,\\varphi _{d_2}$ are irreducible, monic and even, and $\\theta _1,\\ldots ,\\theta _{d_3}$ are of the form $\\theta _i(t)=\\tau _i(t)\\cdot \\tau _i(-t)$ with $\\tau _i(t)$ monic, irreducible and coprime to $\\tau (-t)$ .",
"The centralizer $\\mathcal {C}=\\mathbf {C}_{\\mathrm {M}_N(\\mathsf {k})}(x)$ is isomorphic to the ring $\\mathsf {k}\\langle m_x\\rangle $ and the restriction of the involution $\\star $ to $\\mathcal {C}$ is transferred via this isomorphism to the map $\\sigma _{m_x}$ , defined in nota:even-polys-splitting-rings.",
"By the Chinese remainder theorem, we get $\\mathcal {C}\\simeq \\mathsf {k}\\langle t^{d_1}\\rangle \\times \\prod _{i=1}^{d_2}\\mathsf {k}\\langle \\varphi _i(t)^{m_1}\\rangle \\times \\prod _{i=1}^{d_3}\\mathsf {k}\\langle \\theta _i(t)^{r_i}\\rangle .$ Furthermore, the restriction of the involution $\\sigma _{m_x}$ to each of the factors $\\mathsf {k}\\langle f\\rangle $ , for $f\\in \\left\\lbrace { t^{d_1},\\varphi _i^{l_i},\\theta _j^{r_j}}\\right\\rbrace $ coincides with the respective involution $\\sigma _f$ , induced from $t\\mapsto -t$ .",
"A short computation shows that the nilpotent radical of $\\mathcal {C}$ is isomorphic to the direct product of the nilpotent radicals of all factors on the right hand side of (REF ), and that the quotient $\\mathcal {C}/\\mathcal {N}$ is isomorphic to the étale algebra $\\mathcal {K}=\\mathsf {k}^r\\times \\prod _{i=1}^{d_2}\\mathsf {k}\\langle \\varphi _i\\rangle \\times \\prod _{i=1}^{d_3}\\mathsf {k}\\langle \\theta _i\\rangle ,$ where $r=1$ if $d_1>0$ (i.e.",
"if $x$ is singular) and equals 0 otherwiseHere it is understood that the ring $\\mathsf {k}^0$ is the trivial algebra $\\left\\lbrace {0}\\right\\rbrace $ .. Let $\\dagger $ denote the involution induced on the $\\mathsf {k}$ -algebra $\\mathcal {K}$ in (REF ) from the restriction of $\\star $ to $\\mathcal {C}$ .",
"From the observation regarding the action of $\\star $ on $\\mathcal {C}$ above, we deduce the following properties of the involution $\\dagger $ on $\\mathcal {K}$ .",
"The involution $\\dagger $ preserves the factor $\\mathsf {k}^r$ and acts trivially on it.",
"The involution $\\dagger $ preserves the factors $\\mathsf {k}\\langle \\varphi _i\\rangle $ and coincides with the non-trivial field involution $\\sigma _{\\varphi _i}$ .",
"The involution $\\dagger $ preserves the factors $\\mathsf {k}\\langle \\theta _i\\rangle \\simeq \\mathsf {k}\\langle \\tau _i(t)\\rangle \\times \\mathsf {k}\\langle \\tau _i(-t)\\rangle $ and maps a pair $(\\xi ,\\nu )\\in \\mathsf {k}\\langle \\tau _i(t)\\rangle \\times \\mathsf {k}\\langle \\tau _i(-t)\\rangle $ to the pair $(\\iota ^{-1}(\\nu ),\\iota (\\xi ))$ , where $\\iota :\\mathsf {k}\\langle \\tau _i(t)\\rangle \\rightarrow \\mathsf {k}\\langle \\tau _i(-t)\\rangle $ is the isomorphism induced from $t\\mapsto -t$ .",
"Let $\\mathrm {Sym}(\\dagger )$ be subgroup of $\\mathcal {K}^\\times $ of elements fixed by $\\dagger $ .",
"Note that, as $\\mathcal {K}\\simeq \\mathcal {C}/\\mathcal {N}$ is a commutative ring, by lem:CmodN-suff-cond, the set $\\Theta _x$ can be identified with the quotient of $\\mathrm {Sym}(\\dagger )$ by the image of the map $z\\mapsto z^\\dagger z:\\mathcal {K}\\rightarrow \\mathrm {Sym}(\\dagger )$ .",
"By (REF ) and the theory of finite fields, the restriction of the map $z\\mapsto z^\\dagger z$ to the factors $\\mathsf {k}\\langle \\varphi _i\\rangle $ coincides with the field norm onto the subfield of element fixed by $\\dagger $ , and is surjective onto this subfield.",
"Furthermore, by (REF ), it is evident that an element $(\\xi ,\\nu )\\in \\mathsf {k}\\langle \\tau _i(t)\\rangle \\times \\mathsf {k}\\langle \\tau _i(-t)\\rangle $ is fixed by $\\dagger $ if an only if $\\nu =\\iota (\\xi )$ , in which case $(\\xi ,\\nu )=(\\xi ,1)^\\dagger \\cdot (\\xi ,1)$ .",
"Lastly, by (REF ) it holds that the image of the restriction of $z\\mapsto z^\\dagger z$ to the multiplicative group of $\\mathsf {k}^r$ is either trivial, if $r=0$ , or the group of squares in $\\mathsf {k}^\\times $ , otherwise.",
"It follows from this that the set $\\Theta _x$ is either in bijection with the quotient $\\left(\\mathsf {k}^\\times /(\\mathsf {k}^\\times )^2\\right)$ , and hence of cardinality 2, if $x$ singular, or otherwise trivial.",
"This completes the first step of the proof.",
"For the second step, we divide the analysis according to the parity of $N$ , in order to describe the image of $\\Lambda $ .",
"$N$ even.",
"In this case we show that $\\Lambda $ is surjective.",
"To do so, let $Q\\in \\mathrm {Sym}(\\star ; x)$ .",
"Note that, by assumption, $Q^\\star = Q$ and $Q\\in \\mathrm {GL}_N(\\mathsf {k})$ , and hence the form $(u,v)\\mapsto B(u,Qv)$ is alternating and non-degenerate.",
"By lem:eseential-uniqueness, there exists $w\\in \\mathrm {GL}_N(\\mathsf {k})$ such that $Q=w^\\star w$ .",
"To show that $Q\\in \\mathrm {Im}\\Lambda $ we only need to verify that $y=wxw^{-1}\\in \\mathfrak {g}_1$ .",
"This holds, as $y^\\star =(w^\\star )^{-1 }x^\\star w^\\star =-(w^{\\star })^{-1}(Q x Q^{-1})w^\\star =-wxw^{-1}=-y,$ since $Q$ is assumed to commute with $x$ .",
"$N$ odd.",
"Note that in this case, all elements of $\\mathfrak {g}_1$ are non-singular and hence $\\left|\\Theta _x\\right|=2$ for all $x\\in \\mathfrak {g}_1$ .",
"In this case we prove that the map $\\Lambda $ is not surjective.",
"Note that by definition of the equivalence class $\\sim $ , if $Q_1,Q_2\\in \\mathrm {Sym}(\\star ;x)$ are such that $Q_1\\sim Q_2$ , then $\\det (Q_1)^{-1}\\det (Q_2)$ is a square in $\\mathsf {k}^\\times $ .",
"This holds since $\\det (a^\\star )=\\det (a)$ for all $a\\in \\mathrm {M}_N(\\mathsf {k})$ .",
"By the same token, it follows that the $\\det (w^\\star w)$ is a square in $\\mathsf {k}^\\times $ for all $w\\in \\mathrm {GL}_N(\\mathsf {k})$ .",
"Therefore, to show that $\\Lambda $ is not surjective, it suffices to show that $\\mathrm {Sym}(\\star ;x)$ contains elements whose determinant is not a square in $\\mathsf {k}$ .",
"One may take, for example, the element $Q=\\delta \\cdot 1_N$ , for $\\delta \\in \\mathsf {k}^\\times $ non-square."
],
[
"Centralizers of regular elements",
"Finally, we compute the order of the centralizer of a regular element of $\\mathfrak {g}_1$ .",
"The analysis we propose is analogous to [22].",
"Lemma Let $x\\in \\mathfrak {g}_1$ be regular with minimal polynomial $m_x(t)=t^{d_1}\\prod _{i=1}^{d_2}\\varphi _i(t)^{l_i}\\prod _{i=1}^{d_3}\\theta _i(t)^{r_i},$ where the product on the right hand side is as in (REF ), with $\\theta _i(t)=\\tau _i(t)\\tau _i(-t)$ .",
"The determinant map induces a short exact sequence $1\\rightarrow \\mathbf {C}_{G_1}(x)\\rightarrow \\mathrm {U}_1(\\mathsf {k}\\langle t^{d_1}\\rangle )\\times \\prod _{i=1}^{d_2}\\mathrm {U}_1(\\mathsf {k}\\langle \\varphi _i^{l_i}\\rangle )\\times \\prod _{i=1}^{d_3}\\mathrm {GL}_1(\\mathsf {k}\\langle \\tau _i^{r_i}\\rangle )\\xrightarrow{} Z\\rightarrow 1$ where $Z\\subseteq \\mathsf {k}^\\times $ is the subgroup of order 2 if $N$ is odd and trivial otherwise.",
"As shown in the proof on propo:decomposition-of-sim-class1, the centralizer of $x$ in $\\mathrm {GL}_N(\\mathsf {k})$ is isomorphic to the group of units of the ring $\\mathcal {C}$ , i.e.",
"the direct product $\\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)\\simeq \\mathrm {GL}_1(\\mathsf {k}\\langle t^{d_1}\\rangle )\\times \\prod _{i=1}^{d_2}\\mathrm {GL}_1(\\mathsf {k}\\langle \\varphi _i^{l_i}\\rangle )\\times \\prod _{i=1}^{d_3} \\mathrm {GL}_1(\\mathsf {k}\\langle \\theta _i^{r_i}\\rangle ).$ Furthermore, the involution $\\star $ of $\\mathrm {GL}_N(\\mathsf {k})$ restricts to an involution of $\\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)$ which is transferred via this isomorphism to the involution $\\sigma _{m_x}$ , induced by $t\\mapsto -t$ , and restricts to the involution $\\sigma _f$ on each of the factors $\\mathrm {GL}_1(\\mathsf {k}\\langle f\\rangle )$ for $f\\in \\left\\lbrace {t^{d_1},\\varphi _i^{l_i},\\theta _i^{r_i}}\\right\\rbrace $ .",
"The additional condition $z^\\star z=1$ , and the fact that $\\star $ preserves all factors in the decomposition (REF ), imply that the centralizer of $x$ in $G_1$ is embedded in the group $ \\mathrm {U}_1(\\mathsf {k}\\langle t^{d_1}\\rangle )\\times \\prod _{i=1}^{d_2}\\mathrm {U}_1(\\mathsf {k}\\langle \\varphi _i(t)^{l_i}\\rangle \\times \\prod _{i=1}^{d_3}\\mathrm {U}_1(\\mathsf {k}\\langle \\theta _i(t)^{r_i}\\rangle ).$ Similarly to propo:decomposition-of-sim-class1, the map $\\sigma _{\\theta _i^{r_i}}$ acts on the factors $ \\mathrm {GL}_1(\\mathsf {k}\\langle \\theta _i(t)^{r_i}\\rangle )\\simeq \\mathrm {GL}_1(\\mathsf {k}\\langle \\tau _i(t)^{r_i}\\rangle )\\times \\mathrm {GL}_1(\\mathsf {k}\\langle \\tau _i(-t)^{r_i}\\rangle )$ as $(\\xi ,\\nu )\\mapsto (\\iota ^{-1}(\\nu ),\\iota (\\xi ))$ , where $\\iota :\\mathsf {k}\\langle \\tau _i(t)^{r_i}\\rangle \\rightarrow \\mathsf {k}\\langle \\tau _i(-t)^{r_i}\\rangle $ is the isomorphism induced from $t\\mapsto -t$ .",
"It follows from this that $(\\xi ,\\nu )\\in \\mathrm {U}_1(\\mathsf {k}\\langle \\theta _i^{r_i}\\rangle )$ if and only if $\\iota (\\xi )=\\nu ^{-1}$ , and hence that $\\mathrm {U}_1(\\mathsf {k}\\langle \\theta _i^{r_i}\\rangle )\\simeq \\mathrm {GL}_1(\\mathsf {k}\\langle \\tau _i^{r_i}\\rangle )$ .",
"Lastly, we compute order of the group $Z$ .",
"Since for any $w\\in \\mathrm {GL}_N(\\mathsf {k})$ we have that $\\det (w^\\star )=\\det (w)$ , it follows that the condition $w^\\star w=1$ implies that $\\det (w)\\in \\left\\lbrace {\\pm 1}\\right\\rbrace $ .",
"Thus, to complete the lemma, we need to show that both values occur in the case of $N$ odd, and that only 1 is possible for $N$ even.",
"Both statements are well-known.",
"The former can be proved simply by considering the elements $\\pm 1\\in \\mathrm {GL}_N(\\mathsf {k})$ , while the latter can be deduced by considering the Pfaffian of the matrix $w^t {\\mathbf {J}}w={\\mathbf {J}}$ .",
"Lemma Let $f\\in \\mathsf {k}[t]$ be a monic irreducible polynomial with $f(-t)=\\pm f(t)$ and let $r\\in \\mathbb {N}$ .",
"Let $E_{f^r}$ denote the image of the map $z\\mapsto \\sigma _{f^r}(z)\\cdot z:\\mathrm {GL}_1(\\mathsf {k}\\langle f^r\\rangle )\\rightarrow \\mathrm {GL}_1(\\mathsf {k}\\langle f^r\\rangle )$ .",
"Given $y\\in \\mathrm {GL}_1(\\mathsf {k}\\langle f^r\\rangle )$ it holds that $y\\in E_{f^r}$ if and only if $\\sigma _{f^r}(y)=y$ , and there exists $z\\in \\mathrm {GL}_1(\\mathsf {k}\\langle f(t)^r\\rangle )$ such that $y\\equiv z\\sigma _{f^r}(z)\\pmod {f}$ .",
"In particular, we have $\\left|E_{f^r}\\right|={\\left\\lbrace \\begin{array}{ll}q^{\\frac{1}{2}r\\deg f}(1-q^{-\\frac{1}{2}\\deg f})&\\text{if }f(t)\\ne t\\\\\\frac{q-1}{2}q^{\\lceil \\frac{r}{2}\\rceil -1}&\\text{if }f(t)=t.\\end{array}\\right.",
"}$ Let $W$ denote the vector space underlying the ring $\\mathsf {k}\\langle f^r\\rangle $ and let $C$ be the bilinear form defined on $W$ as in lem:conj-to-g1.",
"Let $x$ be the linear operator defined on $W$ by multiplication by $t$ .",
"The map $t\\mapsto x$ sets up a ring isomorphism of $\\mathsf {k}\\langle f^r\\rangle $ with the ring $\\mathcal {C}\\subseteq \\mathrm {M}_{r\\cdot \\deg f}(\\mathsf {k})$ of matrices commuting with $x$ , and the involution $\\star $ on $\\mathcal {C}$ is identified with the ring involution $\\sigma _{f^r}$ .",
"Note that, in the current setting, if $y\\in \\mathsf {k}\\langle f^r\\rangle $ is the image modulo $(f^r)$ of a polynomial $\\tilde{y}(t)$ , then the assumption $\\sigma _{f^r}(y)=y$ is equivalent to $\\tilde{y}(x)\\in \\mathcal {C}$ satisfying $\\tilde{y}(x)^\\star =\\tilde{y}(x)$ or, in the notation of subsubsection:wall, to $\\tilde{y}(x)\\in \\mathrm {Sym}(\\star ;x)$ .",
"Also, the nilpotent radical of $\\mathcal {C}$ is given as the image of the ideal $(f)\\subseteq \\mathsf {k}\\langle f^r\\rangle $ .",
"The equivalence stated in the lemma now follows from lem:CmodN-suff-cond, by taking $Q_1=1$ and $Q_2=\\tilde{y}(x)\\in \\mathrm {Sym}(\\star ;x)$ .",
"We now compute the cardinality of $E_{f^r}$ .",
"In the case $f(t)=t$ , the equivalence proved above implies that $E_{f^r}$ can be identified with the subgroup of the ring $\\mathsf {k}[t]/(t^{r})$ of truncated polynomials of degree no greater than $r-1$ , which consists of even polynomials whose constant term is an invertible square of $\\mathsf {k}$ .",
"Hence $\\left|E_{f^r}\\right|=\\frac{q-1}{2}q^{\\lceil \\frac{r}{2}\\rceil -1}$ .",
"In the complementary case, by irreducibility, necessarily $f(t)=f(-t)$ and has even degree.",
"In this case, by the Jordan-Chevalley Decomposition Theorem, there exist polynomials $S,H\\in ~ \\mathsf {k}[t]$ such that the endomorphism $S(x)$ (resp.",
"$H(x)$ ) acts semisimply (resp.",
"nilpotently) on the vector space $W=\\mathsf {k}\\langle f^r\\rangle $ , on which $x$ acts by multiplication by $t$ , and such that $H(t)+S(t)\\equiv t\\pmod {f(t)^r}$ (see [19]; note that $S,H\\in \\mathsf {k}[t]$ is possible since $\\mathsf {k}$ is perfect).",
"It follows that $\\mathsf {k}\\langle f^r\\rangle \\simeq \\mathsf {k}[x]=\\mathsf {k}[S(x)][H(x)]$ .",
"A quick computation shows that the minimal polynomials of $S(x)$ and $H(x)$ are $f(t)$ and $t^r$ respectively, and thus $\\mathsf {k}\\langle f\\rangle \\simeq \\mathsf {k}[S(x)][H(x)]\\simeq \\mathsf {k}\\langle f\\rangle \\otimes _\\mathsf {k}(\\mathsf {k}[h]/(h^r))$ .",
"Moreover, by the properties of the Jordan-Chevalley decomposition, both $S(t)$ and $H(t)$ satisfy $S(-x)=-S(x)$ and $H(-x)=-H(x)$ [7].",
"Thus, under this identification, the involution $\\sigma _{f^r}$ is transferred to an involution of $ \\mathsf {k}\\langle f\\rangle \\otimes _\\mathsf {k}(\\mathsf {k}[h]/(h^r))$ , mapping $h$ to $-h$ and acting as $\\sigma _f$ on the field $\\mathsf {k}\\langle f\\rangle $ .",
"By the equivalence in the lemma, and the theory of finite fields, the group $E_{f^r}$ is identified with the subgroup of $(\\mathsf {k}\\langle f^r\\rangle )^\\times $ of elements fixed by $\\sigma _{f^r}$ .",
"Using the identification above, this subgroup consists of elements of the form $\\sum _{i=0}^{r-1}a_i\\otimes h^i$ , with $a_0,\\ldots ,a_{r-1}\\in \\mathsf {k}\\langle f\\rangle $ , $a_0\\ne 0$ , and $\\sigma _f(a_i)={\\left\\lbrace \\begin{array}{ll}a_i&\\text{if $i$ is even}\\\\-a_i&\\text{if $i$ is even}.\\end{array}\\right.",
"}$ The equality $\\left|E_{f^r}\\right|=q^{\\frac{1}{2}r \\deg f}(1-q^{-\\frac{1}{2}\\deg f})$ now follows by direct computation.",
"Proposition Let $x\\in \\mathfrak {g}_1$ be a regular element with minimal polynomial $m_x\\in \\mathsf {k}[t]$ .",
"Let $\\tau (m_x)=(r(m_x),S(m_x),T(m_x))\\in \\mathcal {X}_n$ be the type of $m_x$ (see defi:poly-type).",
"Then $\\left|\\mathbf {C}_{G_1}(x)\\right|=2^{\\nu }q^n\\prod _{d,e}\\left(1+q^{-d}\\right)^{S_{d,e}(m_x)}\\cdot \\left(1-q^{-d}\\right)^{T_{d,e}(m_x)},$ where $\\nu =1$ in the case where $N=2n$ is even and $r(m_x)>0$ , and $\\nu =0$ otherwise.",
"Let $m_x=t^{d_1}\\prod _{i=1}^{d_2}\\varphi _i^{l_i}\\prod _{i=1}^{d_3}\\theta _i^{r_i}$ be a decomposition of $m_x$ as in (REF ), with $\\varphi _i$ even and irreducible, and $\\theta _i(t)=\\tau _i(t)\\tau _i(-t)$ with $\\tau _i(t),\\tau _i(-t)$ irreducible and coprime.",
"Note that by definition of $\\tau (m_x)$ we have that $r(m_x)=\\lfloor \\frac{d_1}{2}\\rfloor $ .",
"In view of lem:short-exact-sequence it suffices to show the following three assertions.",
"$\\left|\\mathrm {U}_1(\\mathsf {k}\\langle t^{d_1}\\rangle )\\right|=2q^{r(m_x)}$ ; $\\left|\\mathrm {U}_1(\\mathsf {k}\\langle \\varphi _i^{l_i}\\rangle )\\right|=q^{\\frac{1}{2}l_i\\cdot \\deg \\varphi _i}(1+q^{-\\frac{1}{2}\\deg \\varphi _i})$ ; $\\left|\\mathrm {GL}_1(\\mathsf {k}\\langle \\tau _i^{r_i}\\rangle )\\right|=q^{r_i\\cdot \\deg \\tau _i}(1-q^{-\\deg \\tau _i})$ .",
"Note that for any irreducible polynomial $f(t)\\in \\mathsf {k}[t]$ and $r\\in \\mathbb {N}$ , invoking the Jordan-Chevalley Decomposition as in lem:unitary-of-kphi, the group $\\mathrm {GL}_1(\\mathsf {k}\\langle f^r\\rangle )$ is isomorphic to the group of units of the ring $\\mathsf {k}\\langle f\\rangle [u]/(u^r)$ , and hence $\\left|\\mathrm {GL}_1(\\mathsf {k}\\langle f^r\\rangle )\\right|=q^{r\\cdot \\deg f}\\left(1-q^{-\\deg f}\\right)$ .",
"Assertion (3) follows by taking $f(t)=\\tau _i(t)$ and $r=r_i$ .",
"Assertions (1) and (2) follow from the exactness of the sequence $1\\rightarrow \\mathrm {U}_1(\\mathsf {k}\\langle f^r\\rangle )\\rightarrow \\mathrm {GL}_1(\\mathsf {k}\\langle f^r\\rangle )\\xrightarrow{} E_{f^r}\\rightarrow 1,$ which holds for any irreducible $f\\in \\mathsf {k}[t]$ with $f(-t)=\\pm f(t)$ and $r\\in \\mathbb {N}$ , and from the computation of $\\left|E_{f^r}\\right|$ in lem:unitary-of-kphi and $\\left|\\mathrm {GL}_1(\\mathsf {k}\\langle f^r\\rangle )\\right|$ for the case where $f(t)=t$ and $r=d_1$ , and the cases $f(t)=\\varphi _i(t)$ and $r=l_i$ .",
"The final assertion of theo:orbits-sp2nso2n+1 follows directly from corol:centralizer-size-sp2nso2n+1."
],
[
"Even dimensional special orthogonal groups",
"The following lemma demonstrates the failure of the first assertion of theo:orbits-sp2nso2n+1 in the even orthogonal case.",
"Lemma Let $N=2n$ be even and let $x\\in \\mathfrak {gl}_N({\\mathsf {k}^{\\mathrm {alg}}})$ be a regular nilpotent element.",
"Then $x$ is not anti-symmetric with respect to any non-degenerate symmetric bilinear form on $V=({\\mathsf {k}^{\\mathrm {alg}}})^N$ .",
"Note that, as $x$ is conjugate to an $N\\times N$ nilpotent Jordan block, the kernel of $x$ is one dimensional.",
"Assume towards a contradiction that $C$ is a symmetric non-degenerate bilinear form on $V$ such that $x$ is $C$ -anti-symmetric.",
"Consider the form $F(u,v)=C(u,xv)$ on $V$ .",
"By assumption the $C(xu,v)+C(u,xv)=0$ , we have that $F$ is anti-symmetric.",
"Additionally, the radical of $F$ coincides with the kernel of $x$ , by non-degeneracy of $C$ .",
"By properties of antisymmetric forms, it follows that the kernel of $x$ is even-dimensional.",
"A contradiction.",
"Nonetheless, regular nilpotent elements in the case of even-dimensional special orthogonal groups are well-known to exist [32].",
"In lem:nilpotent-reg-element-so2n below we shall construct such an element and compute its centralizer.",
"Recall that non-degenerate symmetric bilinear forms on $V=\\mathsf {k}^N$ are classified by the dimension of a maximal totally isotropic subspace of $V$ with respect to the given form (i.e.",
"its Witt index), and that over a finite field of odd characteristic there are exactly two such forms, upto isometry.",
"We fix $B^+$ and $B^-$ to be bilinear forms on $V$ of Witt index $n$ and $n-1$ , respectively.",
"In suitable bases, the forms $B^+$ and $B^-$ are represented by the matrices ${\\mathbf {J}}^+=\\left(\\begin{matrix}0&1\\\\1&0\\\\&&\\ddots \\\\&&&0&1\\\\&&&1&0\\\\&&&&&0&1\\\\&&&&&1&0\\end{matrix}\\right)\\quad \\text{or}\\quad {\\mathbf {J}}^-=\\left(\\begin{matrix}0&1\\\\1&0\\\\&&\\ddots \\\\&&&0&1\\\\&&&1&0\\\\&&&&&1&0\\\\&&&&&0&\\delta \\end{matrix}\\right),$ where $\\delta \\in \\mathfrak {o}^\\times $ is a fixed non-square.",
"Given $\\epsilon \\in \\left\\lbrace {\\pm 1}\\right\\rbrace $ , let $G_1^\\epsilon =\\mathrm {SO}_N^\\epsilon (\\mathsf {k})$ and $\\mathfrak {g}_1^\\epsilon =\\mathfrak {so}_{N}^\\epsilon (\\mathsf {k})$ be the group of isometries of determinant 1 and the Lie-algebra of anti-symmetric operators with respect to the form $B^\\epsilon $ .",
"We will also occasionally use the colloquial notation $G_1^{\\pm }=G_1^{+} \\cup G_1^-$ and $\\mathfrak {g}_1^{\\pm }=\\mathfrak {g}_1^+\\cup \\mathfrak {g}_1^-$ .",
"For example, the phrase $x$ is a regular element of $\\mathfrak {g}_1^\\pm $ indicates that $x$ is either a regular element of $\\mathfrak {g}_1^+$ or of $\\mathfrak {g}_1^-$ ."
],
[
"Similarity classes of regular elements",
"In this section we prove the first assertion of theo:orbits-so2n, which classifies the similarity classes of $\\mathfrak {gl}_N(\\mathfrak {o})$ whose intersection with $\\mathfrak {g}_1^\\pm $ consists of regular elements.",
"Following this, we differentiate whether such a similarity class intersects $\\mathfrak {g}_1^+$ or $\\mathfrak {g}_1^-$ .",
"Note that if $x\\in \\mathfrak {gl}_N(\\mathsf {k})$ is a non-singular element whose minimal polynomial $m_x$ is even and has degree $N$ then, by applying the argument of lem:conj-to-g1.",
"(2) verbatim, we have that $x$ is anti-symmetric with respect to a non-degenerate symmetric bilinear form and hence similar to an element of $\\mathfrak {g}_1^\\pm $ .",
"By corol:reg-g1-reg-glN-nonsing, all non-singular regular elements of $\\mathfrak {g}_1$ are obtained in this manner.",
"Thus, for $x\\in \\mathfrak {gl}_N(\\mathsf {k})$ non-singular, it holds that $x$ is similar to a regular element of $\\mathfrak {g}^{\\pm }$ if and only if the minimal polynomial of $x$ is even and of degree $N$ .",
"As explained below (see propo:orbit-decomposition=so2n), the case of singular regular elements of $\\mathfrak {g}_1^\\pm $ is essentially reduced to the study of nilpotent regular elements.",
"These elements are considered in the following lemma.",
"Lemma Let $x\\in \\mathfrak {gl}_N(\\mathsf {k})$ have minimal polynomial $m_x(t)=t^{N-1}$ .",
"Then $x$ is similar to a regular nilpotent element of $\\mathfrak {g}_1^+$ , as well as to a regular nilpotent element of $\\mathfrak {g}_1^-$ .",
"By considering the Jordan normal form of such an element $x$ , there exist elements $v_0,u_0\\in V$ with $u_0\\in \\mathrm {Ker}(x)$ and such that $\\mathcal {E}=\\left\\lbrace {v_0,xv_0,\\ldots ,x^{N-2}v_0,u_0}\\right\\rbrace $ is a $\\mathsf {k}$ -basis for $V$ .",
"Let $\\mathcal {E}^{\\prime }=\\left\\lbrace {v_0,\\ldots x^{N-2}v_0}\\right\\rbrace $ and $V^{\\prime }=\\textstyle \\mathop {Span}_\\mathsf {k}\\mathcal {E}^{\\prime }$ .",
"Since the element $x\\mid _{V^{\\prime }}\\in \\mathfrak {gl}(V^{\\prime })$ has minimal polynomial $t^{N-1}=t^{\\dim V^{\\prime }}$ , it is regular in $\\mathfrak {gl}(V^{\\prime })$ .",
"By the proof of lem:conj-to-g1, there exists a non-degenerate symmetric bilinear form $C^{\\prime }$ on $V^{\\prime }$ , with respect to which $x\\mid _{V^{\\prime }}$ is anti-symmetric.",
"We wish to extend $C^{\\prime }$ to a non-degenerate symmetric bilinear form on $V$ , with respect to which $x$ is anti-symmetric.",
"This is equivalent to finding an invertible matrix $\\mathbf {d}\\in \\mathrm {M}_N(\\mathsf {k})$ , whose top-left $(N-1)\\times (N-1)$ submatrix coincides with the matrix $\\mathbf {c}$ of exam:reg-nil-elt-so2n+1-sp2n (see (REF )), and such that $\\mathbf {d}^t\\Upsilon +\\Upsilon \\mathbf {d}=0\\quad \\text{where}\\quad \\Upsilon =[x]_\\mathcal {E}=\\left(\\begin{matrix}0&1\\\\&\\ddots &\\ddots \\\\&&0&1\\\\&&&0&0\\\\&&&&0\\end{matrix}\\right).$ A short computation shows that the matrix $\\mathbf {d}=\\mathbf {d}_\\eta =\\left(\\begin{matrix}&&&1\\\\&&-1\\\\&{\\ddots }\\\\1\\\\&&&&\\eta \\end{matrix}\\right),$ where $\\eta \\in \\mathsf {k}^\\times $ satisfies the required equality.",
"Furthermore, by applying a signed permutation to $\\mathcal {E}$ , one may verify easily that $\\mathbf {d}_\\eta $ is congruent to the matrix ${\\mathbf {J}}^+$ of (REF ) if $\\eta $ is a square, and to ${\\mathbf {J}}^-$ otherwise.",
"Thus, $x$ is similar in this case to elements of both $\\mathfrak {g}_1^+$ and of $\\mathfrak {g}_1^-$ .",
"Lastly, we need to verify that $x$ is similar to a regular element of $\\mathfrak {g}_1^\\pm $ .",
"To do so, we pass to the algebraic closure ${\\mathsf {k}^{\\mathrm {alg}}}$ of $\\mathsf {k}$ and compute the centralizer in $\\mathbf {G}({\\mathsf {k}^{\\mathrm {alg}}})$ of an element $zxz^{-1}\\in \\mathfrak {g}_1$ .",
"Working in the basis $\\mathcal {E}$ , by direct computation, one sees that the centralizer of $x$ in $\\mathrm {M}_N({\\mathsf {k}^{\\mathrm {alg}}})$ can be identified with the set of matrices $\\mathbf {y}=\\left({\\begin{matrix}\\mathbf {A}&\\mathbf {v}\\\\\\mathbf {u}^t&r\\end{matrix}}\\right)$ , where $\\mathbf {A}\\in \\mathrm {M}_{N-1}({\\mathsf {k}^{\\mathrm {alg}}})$ and commutes with the restriction of $\\Upsilon $ to $V^{\\prime }=\\textstyle \\mathop {Span}_{{\\mathsf {k}^{\\mathrm {alg}}}}{\\mathcal {E}^{\\prime }}$ ; $\\mathbf {u},\\mathbf {v}\\in ({\\mathsf {k}^{\\mathrm {alg}}})^{N-1}$ are elements of the kernel of $\\Upsilon $ and $\\Upsilon ^t$ , respectively, and hence of the form $\\mathbf {v}=\\left(\\begin{matrix}v_1&0&\\ldots &0\\end{matrix}\\right)^t$ and $\\mathbf {u}=\\left(\\begin{matrix}0&\\ldots &0&u_{N-1}\\end{matrix}\\right)^t$ ; and $r\\in {\\mathsf {k}^{\\mathrm {alg}}}$ is arbitrary.",
"As in exam:reg-nil-elt-so2n+1-sp2n, the centralizer of $z x z^{-1}\\in \\mathfrak {g}_1$ is conjugated in $\\mathrm {GL}_N({\\mathsf {k}^{\\mathrm {alg}}})$ to the group $\\left\\lbrace {\\mathbf {y}\\in \\mathbf {C}_{\\mathrm {GL}_N({\\mathsf {k}^{\\mathrm {alg}}})}(\\Upsilon )\\mid \\mathbf {y}^t\\mathbf {d}\\mathbf {y}=\\mathbf {d}}\\right\\rbrace .$ Computing its Lie-algebra, which consists of matrices $\\mathbf {y}\\in \\mathbf {C}_{\\mathrm {M}_N({\\mathsf {k}^{\\mathrm {alg}}})}(\\Upsilon )$ satisfying $\\mathbf {y}^t\\mathbf {d}+\\mathbf {d}\\mathbf {y}=0$ , we get the additional three conditions $\\mathbf {A}^t\\mathbf {c}+\\mathbf {c}\\mathbf {A}=0$ , where $\\mathbf {c}$ is as in exam:reg-nil-elt-so2n+1-sp2n; $\\eta \\mathbf {u}+\\mathbf {c}\\mathbf {v}=0$ , i.e.",
"$v_1=-\\eta u_{N-1}$ ; and $2\\eta r=0$ , and hence $r=0$ .",
"It follows that $\\mathbf {C}_{\\mathbf {\\Gamma }_1}(zx z^{-1})$ is at most $n$ -dimensional, and hence $x$ is regular.",
"To streamline the analysis of nilpotent regular orbits, let us fix some notation.",
"Notation Given a matrix $\\mathbf {A}\\in \\mathrm {M}_{N-1}(\\mathsf {k})$ , column vectors $\\mathbf {v},\\mathbf {u}\\in \\mathsf {k}^{N-1}$ and $r\\in \\mathsf {k}$ , let $\\Xi (\\mathbf {A},\\mathbf {v},\\mathbf {u},r)$ denote the $N\\times N$ matrix $\\Xi (\\mathbf {A},\\mathbf {v},\\mathbf {u},r)=\\left(\\begin{matrix}\\mathbf {A}&\\mathbf {v}\\\\\\mathbf {u}^t&r\\end{matrix}\\right).$ We also write $\\mathbf {A}^{\\flat }$ for the matrix $\\mathbf {c}\\mathbf {A}^t\\mathbf {c}$ , where $\\mathbf {c}$ is as in exam:reg-nil-elt-so2n+1-sp2n.",
"Note that, in the case where $\\mathbf {d}=\\mathbf {d}_\\eta $ is the representing matrix for the symmetric bilinear form given on $V$ , we have that $\\Xi (\\mathbf {A},\\mathbf {v},\\mathbf {u},r)^\\star =\\left(\\begin{matrix}\\mathbf {A}^{\\flat }&\\eta \\mathbf {cu}\\\\\\eta ^{-1}\\mathbf {v}^t\\mathbf {c}&r\\end{matrix}\\right)=\\Xi (\\mathbf {A}^{\\flat },\\eta \\mathbf {cu},\\eta ^{-1}\\mathbf {cv},r).$ The next step of the computation is to differentiate whether a given element $x\\in \\mathfrak {gl}_N(\\mathsf {k})$ , which is similar to a regular element of $\\mathfrak {g}_1^\\pm $ , is similar to either $\\mathfrak {g}_1^+$ or $\\mathfrak {g}_1^-$ .",
"We first consider two specific cases, depending on the minimal polynomial of $x$ .",
"Lemma (cf.",
"[39]) Let $x\\in \\mathfrak {gl}_N(\\mathsf {k})$ have minimal polynomial $m_x$ .",
"Assume $x$ is similar to a regular element of $\\mathfrak {g}_1^\\pm $ .",
"If $m_x(t)=f(t)f(-t)$ for some polynomial $f\\in \\mathsf {k}[t]$ with $f(0)\\ne 0$ , then $x$ is similar to an element of $\\mathfrak {g}_1^+$ , and not to an element of $\\mathfrak {g}_1^-$ .",
"If $m_x=\\varphi ^r$ for $\\varphi \\in \\mathsf {k}[t]$ an even irreducible polynomial and $r\\in \\mathbb {N}$ odd, then $x$ is similar to a regular element of $\\mathfrak {g}_1^-$ and not to an element of $\\mathfrak {g}_1^+$ .",
"Let $C$ be a non-degenerate symmetric bilinear, with respect to which $x$ is $C$ -anti-symmetric.",
"We will show that $C$ necessarily has Witt index $n$ in the first case and $n-1$ in the second case.",
"1.",
"By the assumption $m_x(0)\\ne 0$ and corol:reg-g1-reg-glN-nonsing, it follows that $x$ is also a regular element of $\\mathfrak {gl}_N(\\mathsf {k})$ , and hence the space $V$ is cyclic as a $\\mathsf {k}[x]$ module.",
"Put $W=f(x) V$ .",
"Then $W$ is isomorphic, as a $\\mathsf {k}[x]$ -module, to $V/f(-x)V$ , and hence is of dimension $n=\\frac{N}{2}$ over $\\mathsf {k}$ .",
"Additionally, for any $u,v\\in V$ we have $C(f(x)u,f(x)v)=C(f(x)f(-x)u,v)=0$ , and hence $W$ is totally isotropic.",
"2.",
"Let us first consider the case where $r=1$ , and hence $V$ is isomorphic to the field extension $\\mathsf {k}\\langle \\varphi \\rangle $ of $\\mathsf {k}$ .",
"Furthermore, the map $\\sigma _{\\varphi }\\in \\mathrm {Aut}_\\mathsf {k}(\\mathsf {k}\\langle \\varphi \\rangle )$ , induced from $t\\mapsto -t$ is a field involution of $\\mathsf {k}\\langle \\varphi \\rangle $ over $\\mathsf {k}$ , with fixed field $\\mathsf {K}$ , such that $\\left|\\mathsf {k}\\langle \\varphi \\rangle :\\mathsf {K}\\right|=2$ .",
"Note that in this setting, without loss of generality, we may assume that $C(u,v)= \\mathrm {Tr}_{\\mathsf {k}\\langle \\varphi \\rangle /\\mathsf {k}}(\\sigma _\\varphi (u)v)$ for all $u,v\\in V$ .",
"Indeed, invoking the separability of the extension $\\mathsf {k}\\langle \\varphi \\rangle /\\mathsf {k}$ , there exists an element $c\\in \\mathsf {k}\\langle \\varphi \\rangle $ such that $C(u,1)=\\mathrm {Tr}_{\\mathsf {k}\\langle \\varphi \\rangle /\\mathsf {k}}(c\\cdot u)$ for all $u\\in \\mathsf {k}\\langle \\varphi \\rangle $ .",
"From the symmetry of $C$ and the invariance of $\\mathrm {Tr}_{\\mathsf {k}\\langle \\varphi \\rangle /\\mathsf {k}}$ under $\\sigma _\\varphi $ , it can be deduced that in fact $c\\in \\mathsf {K}$ .",
"By the theory of finite fields, there exists an element $d\\in \\mathsf {k}\\langle \\varphi \\rangle $ such that $c=\\sigma _{\\varphi }(d)d$ .",
"It follows that multiplication by $d$ is an isometry of $C$ with the trace pairing $(u,v)\\mapsto \\mathrm {Tr}_{\\mathsf {k}\\langle \\varphi \\rangle /\\mathsf {k}}(\\sigma _\\varphi (u)v)$ .",
"Note that an element $u\\in \\mathsf {k}\\langle \\varphi \\rangle $ is isotropic if and only if $\\sigma _\\varphi (u) u$ is a traceless element of $\\mathsf {K}$ .",
"Since the number of non-zero traceless elements in the extension $\\mathsf {K}/\\mathsf {k}$ is $q^{n-1}-1$ , and by the surjectivity of the norm map $\\mathrm {Nr}_{\\mathsf {k}\\langle \\varphi \\rangle /\\mathsf {K}}$ , it follows that the number of non-zero isotropic element of $\\mathsf {k}\\langle \\varphi \\rangle $ is $(q^n+1)(q^{n-1}-1)$ .",
"The fact that $C$ is of Witt index $n-1$ now follows as in [42].",
"For the case $r>1$ , put $l=\\lfloor \\frac{r}{2}\\rfloor $ and $U=\\varphi (x)^{l+1}V$ .",
"Then, similarly to (1), $U$ is an isotropic subspace of $V$ , with perpendicular space $U^\\perp =\\varphi (x)^{l}V$ .",
"Moreover, the form $C$ reduces to a non-degenerate symmetric bilinear form on the quotient space $U^\\perp /U$ , on which $x$ acts as an anti-symmetric operator with minimal polynomial $\\varphi $ .",
"By the case $r=1$ , we find a two-dimensional anisotropic subspace $\\bar{L}\\subseteq U^\\perp /U$ , whose pull-back to $U^\\perp $ is contains a two-dimensional anisotropic subspace of $V$ .",
"It follows that the Witt index of $C$ is necessarily $n-1$ .",
"Having lem:basic-cases at hand, we need one more basic tool in order to complete the classification of similarity classes containing regular elements of $\\mathfrak {g}_1^\\pm $ .",
"Notation Given a finite, even-dimensional vector space $U$ over $\\mathsf {k}$ with a non-degenerate symmetric bilinear form $C$ , put $\\delta _U=1$ if $U$ is of Witt index $\\frac{1}{2}\\dim _\\mathsf {k}U$ and $\\delta _U=-1$ otherwise.",
"Lemma Let $U,W$ be finite, even dimensional vector spaces over $\\mathsf {k}$ with non-degenerate symmetric bilinear forms $C_U$ and $C_V$ respectively.",
"Endow the space $U\\oplus W$ with the non-degenerate symmetric bilinear form $C_{U\\oplus W}(u+w,u^{\\prime }+w^{\\prime })=C_U(u,u^{\\prime })+C_W(w,w^{\\prime })$ where $u,u^{\\prime }\\in U$ and $w,w^{\\prime }\\in W$ .",
"Then $\\delta _{U\\oplus W}=\\delta _U\\cdot \\delta _W.$ The lemma follows, e.g., from [42], noting that the direct product of the groups of isometries of $C_U$ and $C_W$ is embedded in the group of isometries of $C_{U\\oplus W}$ .",
"We are now ready to complete the proof of the first and second assertions of theo:orbits-so2n.",
"Proposition Let $x\\in \\mathfrak {gl}_N$ have minimal polynomial $m_x$ .",
"Assume $m_x(-t)=(-1)^{\\deg m_x}m_x(t)$ and let $m_x(t)=t^{d_1}\\prod _{i=1}^{d_2}\\varphi _i^{l_i}\\prod _{i=1}^{d_3}\\theta _i^{r_i}$ a decomposition as in (REF ), with $\\varphi _i(t)$ even and irreducible, and $\\theta _i(t)=\\tau _i(t)\\tau _i(-t)$ with $\\tau _i(t)$ monic, irreducible and coprime to $\\tau _i(-t)$ .",
"If $d_1>0$ then $x$ is similar to a regular element of $\\mathfrak {g}_1^\\pm $ if and only if $\\deg m_x=N-1$ .",
"Moreover, in this case $x$ is similar to an element of $\\mathfrak {g}_1^+$ as well as to an element of $\\mathfrak {g}_1^-$ .",
"Otherwise, if $d_1=0$ then $x$ is similar to a regular element of $\\mathfrak {g}_1^\\pm $ if and only if $\\deg m_x=N$ .",
"In this case, put $\\omega (m_x)=\\sum _{i=1}^d l_i$ .",
"If $\\omega (m_x)$ is even, then $x$ is similar to an element of $\\mathfrak {g}_1^+$ and not to an element of $\\mathfrak {g}_1^-$ .",
"Otherwise, if $\\omega (m_x)$ is odd, then $x$ is similar to an element of $\\mathfrak {g}_1^-$ and not to an element of $\\mathfrak {g}_1^-$ .",
"Considering the primary canonical form of $x$ , the space $V$ decomposes as a $\\mathsf {k}[x]$ -invariant direct sum $V=W_{t^{d_1}}\\oplus \\bigoplus _{i=1}^{d_2} W_{\\varphi _i^{l_i}}\\oplus \\bigoplus _{i=1}^{d_3} W_{\\theta _i^{r_i}}$ , where the restriction of $x$ to the spaces $W_{f}$ has minimal polynomial $f(t)$ , with $f\\in \\left\\lbrace {t^{d_1},\\varphi _i^{l_i},\\theta _i^{r_i}}\\right\\rbrace $ .",
"For any $f(t)\\ne t^{d_1}$ , the restriction of $x$ to $W_{f}$ is a regular element of $\\mathfrak {gl}(W_f)$ .",
"By corol:reg-g1-reg-glN-nonsing, the space $W_f$ is endowed with a non-degenerate symmetric bilinear form on which $x\\mid _{W_f}$ acts as an anti-symmetric operator.",
"Furthermore, by lem:basic-cases, in the case where $f=\\theta _i^{r_i}$ for $i=1,\\ldots ,d_3$ or $f=\\varphi _i^{l_i}$ with $l_i$ even, then $\\delta _{W_f}=+1$ .",
"Otherwise, if $f=\\varphi _i^{l_i}$ with $l_i$ odd, $\\delta _{W_f}=-1$ .",
"Assertion (2), in which $d_1=0$ is assumed, now follows from lem:direct-sum-of-forms.",
"In the case where $d_1>0$ , the assumption $\\deg m_x=N-1$ implies that $t\\cdot m_x(t)=c_x$ , where $c_x(t)$ is the characteristic polynomial of $x$ .",
"It follows that the restriction of $x$ to $W_{t^{d_1}}$ has minimal polynomial $t^{d-1}$ , and hence, by lem:nilpotent-reg-element-so2n, is antisymmetric with respect to non-degenerate symmetric forms of Witt index $\\frac{d_1}{2}$ as well as $\\frac{d_1}{2}-1$ .",
"Thus $\\delta _{W_{t^{d_1}}}$ can be taken to be either $+1$ or $-1$ .",
"By the case where $x$ is non-singular, and by lem:direct-sum-of-forms, $x$ is similar to an element of $\\mathfrak {g}_1^+$ as well as to an element of $\\mathfrak {g}_1^-$ ."
],
[
"From Similarity classes to adjoint orbits",
"Our next goal, once the similarity classes containing regular elements of $\\mathfrak {g}_1^{\\pm }$ have been classified, is to describe the set $\\Pi _x=\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1^\\epsilon $ into $\\mathrm {Ad}(G_1^\\epsilon )$ -orbits, for $\\epsilon \\in \\left\\lbrace {\\pm 1}\\right\\rbrace $ fixed.",
"In order to complete the description, we require the following lemma, whose proof is appears after propo:orbit-decomposition=so2n.",
"Lemma Assume $\\left|\\mathsf {k}\\right|>3$ and ${\\mathrm {char}}(\\mathsf {k})\\ne 2$ .",
"For any element $\\gamma \\in \\mathsf {k}^\\times $ there exist $\\nu ,\\delta \\in \\mathsf {k}^\\times $ such that $\\nu \\in (\\mathsf {k}^\\times )^2,\\:\\delta \\in \\mathsf {k}^\\times \\setminus (\\mathsf {k}^\\times )^2$ and such that $\\gamma =\\nu -\\delta $ .",
"Proposition Assume $\\left|\\mathsf {k}\\right|>3$ .",
"Fix $\\epsilon \\in \\left\\lbrace {\\pm 1}\\right\\rbrace $ and let $x\\in \\mathfrak {g}^\\epsilon _1$ be regular.",
"If $x$ is singular, then the intersection $\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1^\\epsilon $ is the disjoint union of two distinct $\\mathrm {Ad}(G_1^\\epsilon )$ -orbits.",
"Otherwise, $\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1^\\epsilon =\\mathrm {Ad}(G_1^\\epsilon )x$ .",
"In the notation of propo:orbit-decomposition, let $\\Pi _x=\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k})) x\\cap \\mathfrak {g}_1$ and $\\Theta _x$ the set of equivalence classes in $\\mathrm {Sym}(\\star ;x)=\\left\\lbrace {Q\\in \\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)\\mid Q^\\star = Q}\\right\\rbrace $ under the equivalence relation $\\sim $ , defined in (REF ).",
"Let $\\Lambda :\\Pi _x\\rightarrow \\Theta _x$ be the map $wxw^{-1}\\mapsto [w^\\star w]\\in \\Theta _x$ , for $y=wxw^{-1}\\in \\Pi _x$ .",
"In the case where $x$ is non-singular, by applying the argument of propo:decomposition-of-sim-class1 for non-singular elements verbatim, we have that $\\Theta _x$ consists of a single element and therefore that $\\Pi _x=\\mathrm {Ad}(G_1^\\epsilon ) x$ .",
"Furthermore, in the case where $x$ is singular, by considering the decomposition of $x$ into primary rational canonical forms, one may restrict $x$ to a maximal subspace of $\\mathsf {k}^N$ on which $x$ acts as a regular nilpotent element.",
"This subspace is even-dimensional and admits an orthogonal complement, on which $x$ acts as a non-singular regular element.",
"Additionally, any operator commuting with $x$ must preserve this subspace as well as its orthogonal complement.",
"It follows that to prove the proposition in the case where $x$ is singular it is sufficient to consider the case where $x$ is a nilpotent regular element of $\\mathfrak {g}_1^\\epsilon $ .",
"In this case, by the uniqueness of a nilpotent regular element in ${\\mathbf {g}}({\\mathsf {k}^{\\mathrm {alg}}})$ [32], we may invoke lem:nilpotent-reg-element-so2n and fix a basis $\\mathcal {E}$ , with respect to which $x$ is represented by the matrix $\\Upsilon $ , defined in (REF ), and that the ambient non-degenerate symmetric bilinear form is represented in $\\mathcal {E}$ by the matrix $\\mathbf {d}=\\mathbf {d}_\\eta $ of (REF ), where $\\eta \\in \\mathsf {k}^\\times $ is a square if $\\epsilon =1$ and non-square otherwise.",
"The centralizer $\\mathcal {C}$ of $\\Upsilon $ in $\\mathrm {M}_N(\\mathsf {k})$ is isomorphic to the ring of $\\mathsf {k}[x]$ -endomorphisms of $\\mathsf {k}[x]\\times \\mathsf {k}$ , and can be realized as the set of matrices $\\Xi (\\mathbf {A},\\mathbf {v},\\mathbf {u},r)$ (see nota:regular-nilpotent-centralizer) with $\\mathbf {v}$ and $\\mathbf {u}$ elements of the kernel of $\\Upsilon $ and $\\Upsilon ^t$ respectively, $\\mathbf {A}\\in \\mathrm {M}_{N-1}(\\mathsf {k})$ is an upper triangular Töplitz matrix, and $r\\in \\mathsf {k}$ .",
"Note that the ideal generated by elements of the form $\\Xi (0_{N-1},\\mathbf {v},\\mathbf {u},0)\\in \\mathcal {C}$ is nilpotent and in particular is contained in the nilpotent radical $\\mathcal {N}$ of $\\mathcal {C}$ .",
"It follows that the quotient ring $\\mathcal {C}/\\mathcal {N}$ is isomorphic to the étale algebra $\\mathsf {k}\\times \\mathsf {k}$ .",
"Additionally, by lem:CmodN-suff-cond, we have that $\\Xi (\\mathbf {A},\\mathbf {v},\\mathbf {u},r)\\sim \\Xi (\\mathbf {A}^{\\prime },\\mathbf {v}^{\\prime },\\mathbf {u}^{\\prime },r^{\\prime })$ if and only if there exists a block matrix $\\Xi (\\mathbf {q},0,0,s)$ such that $\\left(\\begin{matrix}\\mathbf {q}&\\\\&s\\end{matrix}\\right)^\\star \\left(\\begin{matrix}\\mathbf {A}&\\mathbf {v}\\\\\\mathbf {u}^t&r\\end{matrix}\\right)\\left(\\begin{matrix}\\mathbf {q}&\\\\&s\\end{matrix}\\right)\\equiv \\left(\\begin{matrix}\\mathbf {A}^{\\prime }&\\mathbf {v}^{\\prime }\\\\\\mathbf {u}^{\\prime t}&r^{\\prime }\\end{matrix}\\right)\\mathcal {}\\pmod {\\mathcal {N}}.$ Applying a similar argument as in the nilpotent case of propo:decomposition-of-sim-class1, we have that the involution $\\star $ restricts to the identity map on $\\mathcal {C}/\\mathcal {N}$ and hence that the quotient $\\Theta _x$ of $\\mathrm {Sym}(\\star ;x)$ by the relation $\\sim $ , defined in subsubsection:wall, is isomorphic to the quotient group $\\mathsf {k}^\\times /(\\mathsf {k}^\\times )^2\\times \\mathsf {k}^\\times /(\\mathsf {k}^\\times )^2$ and is of order 4.",
"The final step of the proof is to compute the image of the map $\\Lambda $ .",
"Recall that $\\Lambda $ maps an element $wxw^{-1}\\in \\Pi _x=\\mathrm {Ad}(\\mathrm {GL}_N(\\mathsf {k}))x\\cap \\mathfrak {g}_1^\\epsilon $ to the equivalence class of $w^\\star w$ in $\\Theta _x$ .",
"As in the odd orthogonal case, two elements which are equivalent with respect to $\\sim $ must have determinant in the same coset of $\\mathsf {k}^{\\times }/(\\mathsf {k}^\\times )^2$ .",
"In particular, as $w^\\star w$ has square determinant, the image of $\\Lambda $ in $\\Theta _x$ is contained in the subset of equivalence classes in $\\Theta _x$ , containing block matrices $\\Xi (\\mathbf {A},0,0,r)$ with $\\det {\\mathbf {A}}\\equiv r\\mathsf {}\\pmod {(\\mathsf {k}^\\times )^2}$ .",
"To complete the proof that $\\left|\\mathrm {Im}(\\Lambda )\\right|=2$ it suffices to find an element $w\\in \\mathrm {GL}_N(\\mathsf {k})$ such that $wxw^{-1}\\in \\mathfrak {g}_1$ and such that $w^\\star w$ is a block matrix of the form $\\Xi (\\mathbf {A},0,0,r)$ with $\\det \\mathbf {A},r~\\notin ~(\\mathsf {k}^\\times )^2$ .",
"Let $\\eta \\in \\mathsf {k}^\\times $ be as above put $\\alpha =(-1)^{(N-2)/2}$ .",
"Let $\\nu \\in (\\mathsf {k}^\\times )^2$ and $\\delta \\in \\mathsf {k}^\\times \\setminus (\\mathsf {k}^\\times )^2$ be such that $\\alpha \\eta =\\nu -\\delta $ ; see lem:difference-of-squares.",
"Let $\\nu _1\\in \\mathsf {k}^\\times $ be such that $\\nu _1^2=\\nu $ , and put $z=\\eta \\cdot \\nu _1^{-1}$ .",
"Let $w\\in \\mathrm {GL}_N(\\mathsf {k})$ be represented in $\\mathcal {E}$ by the matrix $\\mathbf {w}$ of (REF ), in which the upper-left scalar block with $\\delta $ on the diagonal is $\\left(\\frac{N-2}{2}\\right)\\times \\left(\\frac{N-2}{2}\\right)$ .",
"Recalling that $w^\\star $ is represented by the matrix $\\mathbf {d}^{-1}\\mathbf {w}^t\\mathbf {d}$ , one verifies by direct computation that $w^\\star w$ is given by the diagonal matrix $\\Xi (\\delta 1_{N-1},0,0,\\nu ^{-1}\\delta )$ , and consequently, that $w^\\star w\\in \\mathrm {Sym}(\\star ;x)$ and $wxw^{-1}\\in \\mathfrak {g}^\\epsilon _1$ , and that $w^\\star w$ is not equivalent to $1_N$ under the relation $\\sim $ .",
"$\\mathbf {w}=\\left(\\begin{matrix}\\delta \\\\&\\ddots \\\\&&\\delta \\\\&&&\\nu _1&&&&\\alpha z\\\\&&&&1\\\\&&&&&\\ddots \\\\&&&&&&1\\\\&&&-\\eta ^{-1}\\nu _1z&&&&1\\end{matrix}\\right).$ [Proof of lem:difference-of-squares] Let $\\xi \\in \\mathsf {k}^\\times $ be a non-square, and let $\\mathsf {K}=\\mathsf {k}\\langle t^2-\\xi \\rangle $ be the splitting field of $t^2-\\xi $ , with $\\xi _1\\in \\mathsf {K}^\\times $ a square root of $\\xi $ .",
"The norm map $\\mathrm {Nr}_{\\mathsf {K}\\mid \\mathsf {k}}:\\mathsf {K}^\\times \\rightarrow \\mathsf {k}^\\times $ is surjective and has fibers of order $q+1$ .",
"In particular, there exist $\\nu _1,\\delta _1\\in \\mathsf {k}$ such that $\\mathrm {Nr}_{\\mathsf {K}\\mid \\mathsf {k}}(\\nu _1+\\xi _1\\delta _1)=\\nu _1^2-\\xi \\delta _1^2=\\gamma .$ We claim that $\\nu _1$ and $\\delta _1$ can be taken to be both non-zero.",
"Case 1, $\\gamma \\in \\mathsf {k}^\\times \\setminus (\\mathsf {k}^\\times )^2$.",
"Note that in this case we must have that $\\delta _1\\ne 0$ , as otherwise $\\gamma =\\nu _1^2\\in (\\mathsf {k}^\\times )^2$ .",
"Furthermore, if $\\nu _1=0$ for any pair $(\\nu _1,\\delta _1)$ such that $\\nu _1^2-\\xi \\delta _1^2=\\gamma $ then $\\mathrm {Nr}_{\\mathsf {K}\\mid \\mathsf {k}}^{-1}(\\gamma )\\subseteq \\xi _1\\mathsf {k}^\\times $ , and in particular has order smaller than $q$ .",
"A contradiction.",
"Case 2, $\\gamma \\in (\\mathsf {k}^\\times )^2$.",
"Consider the set $\\mathrm {Nr}_{\\mathsf {K}\\mid \\mathsf {k}}^{-1}(\\gamma )\\setminus \\mathsf {k}^\\times $ .",
"Note that, as $\\left|\\mathrm {Nr}_{\\mathsf {K}\\mid \\mathsf {k}}^{-1}(\\gamma )\\cap \\mathsf {k}^\\times \\right|=2$ (namely, it consists of the two roots of $\\gamma $ in $\\mathsf {k}$ ), the order of $\\mathrm {Nr}_{\\mathsf {K}\\mid \\mathsf {k}}^{-1}(\\gamma )\\setminus \\mathsf {k}^\\times $ is exactly $q-1$ .",
"Assume towards a contradiction that there is no solution $(\\nu _1,\\delta _1)\\in \\mathsf {k}^\\times \\times \\mathsf {k}^\\times $ for the equation $\\nu _1^2-\\xi \\delta _1^2=\\mathrm {Nr}_{\\mathsf {K}\\mid \\mathsf {k}}(\\nu _1-\\xi _1\\delta _1)=\\gamma .$ This implies that any solution not in $\\mathsf {k}^\\times \\times \\left\\lbrace {0}\\right\\rbrace $ is an element of $\\left\\lbrace {0}\\right\\rbrace \\times \\mathsf {k}^\\times $ , or in other words, that $\\mathrm {Nr}_{\\mathsf {K}\\mid \\mathsf {k}}^{-1}(\\gamma )\\setminus \\mathsf {k}^\\times \\subseteq \\xi _1\\mathsf {k}^\\times $ .",
"By considering the cardinality of the two sets, we deduce that this inclusion is in fact an equality.",
"In particular, this implies that for any $\\delta _1\\in \\mathsf {k}^\\times $ , $\\mathrm {Nr}_{\\mathsf {K}\\mid \\mathsf {k}}(\\xi _1\\delta _1)=-\\xi \\delta _1^2=\\gamma .$ Thus, the set of squares in $\\mathsf {k}^\\times $ equals the singleton set $\\left\\lbrace {-\\xi ^{-1}\\gamma }\\right\\rbrace $ .",
"This contradicts the assumption $\\left|\\mathsf {k}\\right|>3$ .",
"The lemma follows by taking $\\nu =\\nu _1^2$ and $\\delta =\\xi \\delta _1^2$ ."
],
[
"Centralizers of regular elements",
"Lemma Let $\\epsilon \\in \\left\\lbrace {\\pm 1}\\right\\rbrace $ .",
"Let $x\\in \\mathfrak {g}_1^\\epsilon $ be regular, with minimal polynomial $m_x(t)=t^{d_1}\\prod _{i=1}^{d_2}\\varphi _i^{l_i}\\prod _{i=1}^{d_3}\\theta _i^{r_i},$ a decomposition as in (REF ), with $\\theta _i=\\tau _i(t)\\tau _i(-t)$ and $\\tau _i(t)$ irreducible and coprime to $\\tau _i(-t)$ .",
"If $d_1>0$ , then there exists a short exact sequence $1\\rightarrow \\mathbf {C}_{G_1^\\epsilon }(x)\\rightarrow \\mathcal {A}^\\epsilon \\times \\prod _{i=1}^{d_2}\\mathrm {U}_1(\\mathsf {k}\\langle \\varphi _i^{l_i}\\rangle )\\times \\prod _{i=1}^{d_3}\\mathrm {GL}_1(\\mathsf {k}\\langle \\tau _i^{r_i}\\rangle )\\xrightarrow{} \\left\\lbrace {\\pm 1}\\right\\rbrace \\rightarrow 1.",
"$ where $\\mathcal {A}^\\epsilon =\\left\\lbrace {\\mathbf {w}\\in \\mathbf {C}_{\\mathrm {GL}_{d_1+1}(\\mathsf {k})}{(\\Upsilon )}\\mid \\mathbf {w}^t\\mathbf {d}_\\eta \\mathbf {w}=\\mathbf {d}_\\eta }\\right\\rbrace ,$ with $\\Upsilon $ and $\\mathbf {d}_\\eta $ the $(d_1+1)\\times (d_1+1)$ matrices defined as in (REF ) and (REF ).",
"Otherwise, the group $\\mathbf {C}_{G_1^\\epsilon }(x)$ is isomorphic to $\\prod _{i=1}^{d_2}\\mathrm {U}_1(\\mathsf {k}\\langle \\varphi _i^{l_i}\\rangle )\\times \\prod _{i=1}^{d_3}\\mathrm {GL}_1(\\mathsf {k}\\langle \\tau _i^{r_i}\\rangle )$ .",
"Similarly to lem:short-exact-sequence, in order to prove the lemma, it is sufficient to compute the possible determinants of the middle term of (REF ).",
"For the first assertion it is sufficient to verify that both $+1$ and $-1$ are obtained as determinant of elements from $\\mathcal {A}^\\epsilon $ , for which it is enough to consider block diagonal matrices of the form $\\left({\\begin{matrix}1_{d_1}&0\\\\0&\\pm 1\\end{matrix}}\\right)\\in \\mathcal {A}^\\epsilon $ .",
"For the second assertion, we need to verify that any element $w\\in \\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)$ such that $w^\\star w=1$ has determinant 1.",
"Since any element of $\\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)$ preserves the invariant factors of the decomposition of $V$ as a $\\mathsf {k}[x]$ -module, it is sufficient to consider the following cases of the minimal polynomial of $x$ .",
"Case 1.",
"Assume $m_x(t)=\\varphi _i(t)^m$ , with $\\varphi _i\\in \\mathsf {k}[t]$ irreducible and even and $m\\in \\mathbb {N}$ .",
"Let $x=s+h$ be the Jordan decomposition of $x$ , with $s,h\\in \\mathfrak {g}_1^\\epsilon $ , $s$ semisimple, $h$ nilpotent and $[s,h]=0$ .",
"As $m_x(0)\\ne 0$ , by propo:orbit-decomposition=so2n.",
"(2), the space $V$ is cyclic as a $\\mathsf {k}[x]$ -module and hence $\\mathbf {C}_{\\mathrm {M}_N(\\mathsf {k})}(x)\\simeq \\mathsf {k}[x]=\\mathsf {k}[s][h]\\simeq \\mathsf {k}\\langle \\varphi _i\\rangle [u]/({u}^{m})$ (see lem:unitary-of-kphi).",
"Let $\\rho : \\mathsf {k}\\langle \\varphi _i\\rangle [u]/(u^m)\\rightarrow \\mathbf {C}_{\\mathrm {M}_N(\\mathsf {k})}(x)$ be a $\\mathsf {k}$ -linear isomorphism.",
"The $\\mathsf {k}$ -linearity of $\\rho $ and the nilpotency of $u$ imply that $\\det (\\rho (\\alpha _0+\\alpha _1 u+\\ldots +\\alpha _{m-1} u^{m-1}))=\\mathrm {Nr}_{\\mathsf {k}\\langle \\varphi _i\\rangle /\\mathsf {k}}(\\alpha _0)^m.$ Furthermore, the restriction of the involution $\\star $ to the image of $\\rho $ induces a $\\mathsf {k}$ -automorphism $\\sigma _{\\varphi _i^m}$ of $\\mathsf {k}\\langle \\varphi _i^m\\rangle $ which acts on $\\mathsf {k}\\langle \\varphi _i\\rangle $ as the involution $\\sigma _{\\varphi _i}$ , and maps $u$ to $-u$ .",
"Consequently, if $z\\in \\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)$ is given by $z=\\rho (\\alpha _0+\\alpha _1 u+\\ldots +\\alpha _{m-1}u^{m-1})$ and satisfies $z^\\star z=1$ then necessarily $\\mathrm {Nr}_{\\mathsf {k}\\langle \\varphi _i\\rangle /\\mathsf {K}}(\\alpha _0)=\\sigma _{\\varphi _i}(\\alpha _0)\\alpha _0=\\rho ^{-1}(z^\\star z)\\mid _{u=0}=1$ and $\\det (z)&=\\det (\\rho (\\alpha _0+\\alpha _1 u+\\ldots + \\alpha _{m-1}u^{m-1}))\\\\&=\\mathrm {Nr}_{\\mathsf {k}\\langle \\varphi _i\\rangle /\\mathsf {k}}(\\alpha _0)^m=\\left(\\mathrm {Nr}_{\\mathsf {K}/\\mathsf {k}}\\circ \\mathrm {Nr}_{\\mathsf {k}\\langle \\varphi _i\\rangle /\\mathsf {K}}(\\alpha _0)\\right)^m=1.$ Case 2.",
"Assume $m_x(t)=\\left(\\tau _i(t)\\cdot \\tau _i(-t)\\right)^{r}$ , for $\\tau _i(t)$ irreducible and coprime to $\\tau (-t)$ .",
"In this case, by the cyclicity of the $\\mathsf {k}[x]$ module $V$ , we have that $\\mathbf {C}_{\\mathrm {GL}_N(\\mathsf {k})}(x)\\simeq \\mathrm {GL}_1(\\mathsf {k}\\langle \\tau (t)^r\\rangle )\\times \\mathrm {GL}_1(\\mathsf {k}\\langle \\tau (-t)^r\\rangle )$ .",
"Moreover, the map $\\star $ restricts to the map $(\\xi ,\\nu )\\mapsto (\\iota ^{-1}(\\xi ),\\iota (\\nu ))$ , where $\\iota :\\mathsf {k}\\langle \\tau (t)^r\\rangle \\rightarrow \\mathsf {k}\\langle \\tau (-t)^r\\rangle $ is the isomorphism induced from $t\\mapsto -t$ .",
"Furthermore, since $\\iota $ is a ring-isomorphism which preserves $\\mathsf {k}$ , we have that $\\det (\\iota (\\xi ))=\\det (\\xi )$ for all $\\xi \\in \\mathsf {k}\\langle \\tau (t)^r\\rangle $ .",
"In particular, if $(\\xi ,\\nu )^\\star (\\xi ,\\nu )=1$ then $\\nu =\\iota (\\xi )^{-1}$ and hence, $\\det ((\\xi ,\\nu ))=\\det (\\xi )\\cdot \\det (\\xi )^{-1}=1$ .",
"Proposition Let $x\\in \\mathfrak {g}_1^\\pm $ be regular with minimal polynomial $m_x(t)$ .",
"Let $c_x$ denote the characteristic polynomial of $x$ , i.e.",
"$c_x=m_x$ if $x$ is non-singular, and $c_x(t)=t\\cdot m_x(t)$ otherwise.",
"Let $\\tau (c_x)=(r(c_x),S(c_x),T(c_x))\\in \\mathcal {X}_n$ be the type of $c_x$ (see defi:poly-type).",
"Then $\\left|\\mathbf {C}_{G_1^\\epsilon }(x)\\right|=2^{\\nu }q^n\\prod _{d,e}\\left(1+q^{-d}\\right)^{S_{d,e}(m_x)}\\cdot \\left(1-q^{-d}\\right)^{T_{d,e}(m_x)},$ where $\\epsilon \\in \\left\\lbrace {\\pm }\\right\\rbrace $ and $\\nu =1$ if $r(m_x)>0$ and 0 otherwise.",
"In the case where $x$ is non-singular the assertion follows verbatim as in corol:centralizer-size-sp2nso2n+1.",
"Otherwise, if $x$ is singular, by decomposing $x$ into its primary rational canonical forms, it is sufficient to consider the case where $x$ is a regular nilpotent element, with minimal polynomial $m_x(t)=t^{2n-1}$ , and show that $\\left|\\mathbf {C}_{G_1}(x)\\right|=2q^n$ .",
"Without loss of generality, we fix the basis $\\mathcal {E}$ of lem:nilpotent-reg-element-so2n, with respect to which the ambient symmetric form $B^\\epsilon $ is represented by the matrix $\\mathbf {d}=\\mathbf {d}_\\eta $ , for some $\\eta \\in \\mathsf {k}^\\times $ , and $x$ is represented by the matrix $\\Upsilon $ .",
"Let $\\mathcal {A}^\\epsilon =\\left\\lbrace {z\\in \\mathbf {C}_{\\mathrm {GL}_{N}(\\mathsf {k})}(\\Upsilon )\\mid z^t\\mathbf {d}z=\\mathbf {d}}\\right\\rbrace $ , as in lem:short-exact-sequence-so2n.",
"Let $\\mathcal {N}\\subseteq \\mathcal {A}^\\epsilon $ be the subgroup consisting of elements of the form $\\mathfrak {X}(\\xi )=\\left(\\begin{matrix}1&&&2\\eta \\xi ^2&2\\xi \\\\&1\\\\&&\\ddots \\\\&&&1\\\\&&&2\\eta \\xi &1\\end{matrix}\\right)\\quad (\\xi \\in \\mathsf {k}).$ Note that $\\mathfrak {X}$ defines a one-parameter subgroup of $\\mathcal {A}^\\epsilon $ of order $\\left|\\mathsf {k}\\right|=q$ .",
"Additionally, $\\mathcal {N}=\\mathrm {Im}(\\mathfrak {X})$ is the image under the Cayley map of the Lie-ideal generated by elements of the form $\\Xi (0_{N-1},\\mathbf {u},\\mathbf {v},0)\\in \\mathfrak {g}_1$ , and hence is normal in $\\mathcal {A}^\\epsilon $ .",
"Let $\\mathcal {H}\\subseteq \\mathcal {A}^\\epsilon $ be the subgroup of block diagonal matrices $\\Xi {(\\mathbf {A},0,0,r)}$ .",
"Note that, by (REF ) and the assumption $\\Xi (\\mathbf {A},0,0,r)^\\star \\Xi (\\mathbf {A},0,0,r)=1_N$ , we have that $\\mathbf {A}^{\\flat }\\mathbf {A}=1_{N-1}$ and $r^2=1$ .",
"Additionally, since $\\mathbf {A}$ commutes with the restriction of $\\Upsilon $ to the subspace spanned by the first $N-1$ elements of $\\mathcal {E}$ , we have that $\\left|\\mathcal {H}\\right|=\\left|\\mathrm {U}_1(\\mathsf {k}\\langle t^{2n-1}\\rangle )\\times \\left\\lbrace {\\pm 1}\\right\\rbrace \\right|=4q^{n-1}$ (by the first assertion in the proof of corol:centralizer-size-sp2nso2n+1).",
"Given an arbitrary element $\\Xi (\\mathbf {A},\\mathbf {v},\\mathbf {u},r)\\in \\mathcal {A}^\\epsilon $ , it holds that $\\mathbf {A}$ must be invertible, and that $\\mathbf {v}=\\gamma \\mathbf {d}\\mathbf {u}$ for some $\\gamma \\in \\mathsf {k}$ .",
"In particular, $\\mathbf {v}=0$ if and only if $\\mathbf {u}=0$ .",
"It follows from this, and by direct computation, that $\\mathfrak {X}\\left(-\\frac{v_1}{a_{1,1}\\eta }\\right)\\left(\\begin{matrix}\\mathbf {A}&\\mathbf {v}\\\\\\mathbf {u}^t&r\\end{matrix}\\right)\\in \\mathcal {H},$ where $v_1$ is the first entry of $\\mathbf {v}$ , and $a_{1,1}$ is the $(1,1)$ -th entry of $\\mathbf {A}$ .",
"Therefore, we have that $\\mathcal {A}^\\epsilon =\\mathcal {H}\\cdot \\mathcal {N}$ and hence, as $\\mathcal {H}\\cap \\mathcal {N}=\\left\\lbrace {1}\\right\\rbrace $ , that $\\left|\\mathcal {A}^\\epsilon /\\mathcal {N}\\right|=\\left|\\mathcal {H}\\right|=4q^{n-1}.$ To conclude, we have that $\\left|\\mathcal {A}^\\epsilon \\right|=4q^{n}$ , and the result follows from lem:short-exact-sequence-so2n.",
"The final assertion of theo:orbits-so2n follows from corol:size-of-centralizer-so2n."
],
[
"Acknowledgements",
"This paper is part of the author's doctoral thesis.",
"I wish to thank Uri Onn for guiding and advising this research.",
"I also wish to thank Alexander Stasinski for carefully reading through a preliminary version of this article and offering some essential remarks.",
"Finally, I wish to acknowledge the valuable input offered by the anonymous referee.",
"The present research was supported by the Israel Science Foundation (ISF) grant 1862."
]
] | 1709.01685 |
[
[
"Using Posters to Recommend Anime and Mangas in a Cold-Start Scenario"
],
[
"Abstract Item cold-start is a classical issue in recommender systems that affects anime and manga recommendations as well.",
"This problem can be framed as follows: how to predict whether a user will like a manga that received few ratings from the community?",
"Content-based techniques can alleviate this issue but require extra information, that is usually expensive to gather.",
"In this paper, we use a deep learning technique, Illustration2Vec, to easily extract tag information from the manga and anime posters (e.g., sword, or ponytail).",
"We propose BALSE (Blended Alternate Least Squares with Explanation), a new model for collaborative filtering, that benefits from this extra information to recommend mangas.",
"We show, using real data from an online manga recommender system called Mangaki, that our model improves substantially the quality of recommendations, especially for less-known manga, and is able to provide an interpretation of the taste of the users."
],
[
"Introduction",
"Recommender systems are useful to help users decide what to enjoy next.",
"In the case of anime and mangas, users, easily overwhelmed by the ever-growing amount of works, end up barely scratching the surface of what Japanese animation has to offer.",
"Collaborative filtering is a popular technique that relies on existing rating data from users on items in order to predict unseen ratings aggarwal2016recommender.",
"However, it is still hard to recommend items for which little information is available, e.g., items for which few or no ratings have been provided by the community.",
"This problem has been referred to as the item cold-start problem.",
"In order to alleviate this problem, it is possible to rely on extra information about the items, such as metadata (e.g., for movies: directors, composers, release date).",
"However, such information is not always available: new anime projects may only have a poster or a trailer, and a title.",
"Such a poster is usually the first contact that a user has with an item and plays a large role in the user's decision to watch it or not.",
"Especially in the manga and anime industry, posters contain a lot of information about the characters, in order to maximize the visual appeal for the consumers.",
"Hence, it is natural to consider posters as a source for additional metadata.",
"In recent years, convolutional neural networks (CNNs) have established themselves as the de-facto method for extracting semantic information from image content in a wide variety of tasks.",
"We propose using a CNN for extracting meaningful tags directly from the item's poster.",
"Such extracted tags can help draw links between items, which can be useful when few ratings are available.",
"In this paper, we present BALSEhttp://knowyourmeme.com/memes/events/balse (Blended Alternate Least Squares with Explanation), a new method leveraging tag information extracted from the posters for tackling the item cold-start problem and improving the recommendation performance for little-known items.",
"We show using real data that our method provides better rating predictions than existing techniques, and gives interpretable insight about the user's taste.",
"To the best of our knowledge, this is the first research work that uses tag prediction on item posters in order to improve the accuracy of a recommender system and explain to users why they are recommended little-known items.",
"This paper is organized as follows.",
"We first present existing work related to this research.",
"Then, we expose the context of collaborative filtering and item cold-start, together with a few common assumptions.",
"We then describe our model, BALSE, and present some experimental results on a real dataset.",
"We finish by discussing the results and future work."
],
[
"Related Work",
"Using side information in order to improve recommendations has been the core of existing research kula2015metadata and several approaches have been developed to take into account extra data about users or items, whether coming from text alexandridis2017parvecmffang2011matrix, social networks delporte2013socially, images or other types of data nedelec2017specializingkim2014scalablexu2013speedup.",
"More recently, deep learning techniques have been used for this purpose.",
"YouTube is extracting features from the videos browsed within a user history in order to improve their recommendations Covington2016.",
"Researchers have also analyzed music content as extra information Van2013.",
"They managed to recover explainable latent features, corresponding to certain types of music, without any human annotation.",
"Such side information is particularly useful in order to mitigate the cold-start problem wei2017collaborativekula2015metadatabiswas2017combatingbobadilla2012collaborative.",
"In the exact context of movies, Zhao2016 extract latent features from the posters using CNNs and improve the recommendations using those latent features.",
"However, those extracted features do not have semantic meaning, therefore they cannot be used to explain to the user why extra works are recommended to them.",
"Several approaches have tried to bridge the gap between content-based approaches and collaborative filtering burke2002hybrid.",
"The main idea behind those so-called hybrid methods is to combine different recommendation models in order to overcome their limitations and build more robust models with higher predictive power.",
"Existing techniques can take on several names: blending or stacking roda2011optimaljahrer2010combining, or the general ensemble methods for machine learning estimators.",
"These techniques use the output of different models as features for a higher-level model.",
"This higher-level model is usually a linear model sill2009feature.",
"Such blended methods have played an important role in achieving top performance in challenges such as the Netflix Prize sill2009featurekoren2009bellkor.",
"The approach described in this paper builds upon these ideas, as we are presenting a blended model, but the combination we present is nonlinear.",
"We complement a classical collaborative filtering recommender system with a fallback model that will compensate the prediction error on hard-to-predict data points, i.e.",
"items with few ratings."
],
[
"Context",
"We assume the typical setting for collaborative filtering: we have access to a $n \\times m$ rating matrix $R$ containing the ratings of $n$ users on $m$ items that can be either manga or anime: $r_{ij}$ represents the rating that user $i$ gave to item $j$ .",
"In practice, since users only rate the few items that they have interacted with, the rating matrix $R$ tend to be very sparse: in the dataset considered in this paper, less than 1% of the entries are known; other popular datasets in the field aggarwal2016recommenderwang2016learning report similar levels of sparsity.",
"Therefore, it is challenging to infer the missing entries of $R$ .",
"Another assumption is that the whole rating matrix can be explained by few latent profiles, i.e.",
"each user rating vector can be decomposed as a combination of few latent vectors.",
"Therefore, matrix completion is usually performed using matrix factorization: we try to devise a factorization $R \\approx UV^T$ where a $n \\times r$ matrix $U$ represents the user feature vectors and a $m \\times r$ matrix $V$ represents the item feature vectors.",
"Once this model is trained, that is, when the available entries of $R$ match their counterparts in $UV^T$ , computing a missing entry $(i, j)$ of the rating matrix $R$ is simply performed by looking at the corresponding $(i, j)$ entry of $UV^T$ , or, equivalently, computing the dot product $U_i^T V_j$ where $U_i$ in the $i$ -th row of $U$ and $V_j$ is the $j$ -th row of $V$ .",
"Finally, we also assume that we have access to the posters of some of the items.",
"This is all the content we have."
],
[
"Our Model: BALSE",
"We now describe BALSE (Blended Alternate Least Squares with Explanation), our model for recommending anime and mangas.",
"The main idea is to rely on the rating matrix when possible, and on the posters when rating information barely exists.",
"We expect a nonlinear blending of two models considering these sources of information to achieve higher performance than any of the models.",
"BALSE is composed of several blocks: an Illustration2Vec block, which is a convolutional neural network that takes a poster as input and outputs tag predictions; an ALSALS stands for Alternate Least Squares.",
"block, that performs a matrix factorization of the rating matrix for collaborative filtering using alternate least squares with $\\lambda $ -weighted regularization; a LASSOLASSO stands for Least Absolute Shrinkage and Selection Operator.",
"block, that performs a regularized linear regression of each row of the rating matrix, using tag predictions, in order to infer explainable user preferences; a Steins gate, that performs a blending of the outputs of ALS and LASSO models, in order to overcome their limitations and provide a final rating value.",
"The main architecture of our model is presented in Figure REF .",
"Both posters and ratings are used for the predictions."
],
[
"Illustration2Vec",
"This block extracts tag information from the posters, such as “1girl” or “weapon”.",
"Such tags are associated with confidence weights that represent how likely a certain tag appears in a certain poster.",
"Formally, from the poster database, we want to extract a $m \\times t$ matrix $T$ where $m$ is the number of items and $t$ is the number of possible tags, such that $t_{jk} \\in [0, 1]$ represents how likely a tag $k$ appears in the poster of item $j$ .",
"$T$ is computed using Illustration2Vec Saito2015, a VGG-16 neural network simonyan2014very that predicts a variety of tags based on illustrations, pre-trained on ImageNet and trained on manga illustrations labeled with tags from the community website Danbooru.",
"We use the implementation provided by the authors, which is freely available.",
"The output of the network is for each poster $j$ , a vector $T_j = (t_{j1}, \\ldots , t_{jt})$ where for tag $k = 1, \\ldots , t$ , component $t_{jk} \\in [0, 1]$ represents how likely tag $k$ describes the poster of item $j$ .",
"In other words, the output of Illustration2Vec is a row of matrix $T$ .",
"We will call such a vector a tag prediction.",
"See Fig.",
"REF for an example of an output of the Illustration2Vec model."
],
[
"LASSO",
"The LASSO block approximates the rating matrix $R$ with a regularized linear regression model called LASSO tibshirani1996regression, using the tag predictions as features for the items.",
"We train a LASSO model for every user in the train set.",
"$R \\approx P T^T$ where: $P$ contains the parameters to learn, a $n \\times t$ matrix of user preferences, of which the $i$ -th row is denoted as $P_i$ (likewise, $R_i$ denotes the $i$ -th row of $R$ ); $T$ is the given $m \\times t$ matrix of tag predictions for each item.",
"LASSO comes with an $\\alpha $ parameter which induces a L1 regularization term to prevent overfitting, and to provide explanation of user preferences as we will show later.",
"Therefore, for every user $i$ of the train set, we estimate the parameters $P_i$ that minimize: $\\frac{1}{2\\mathcal {N}_i} {R_i - P_i T^T}_2^2 + \\alpha {P_i}_1$ where $\\mathcal {N}_i$ is the number of items rated by user $i$ .",
"The output of the LASSO block is a rating prediction for each pair $(i, j)$ : $\\hat{r}_{ij}^{LASSO} = \\tau (P_i^T T_j).$ where $\\tau : x \\mapsto \\max (\\min (x, 2), -2)$ is a function that shrinks its input to values between -2 and 2.",
"Such a function prevents the regressor from providing invalid predictions that are outside the range of rating values."
],
[
"ALS",
"The ALS block performs matrix factorization of the $n \\times m$ sparse rating matrix $R$ , in order to provide an estimate $\\hat{r}_{ij}^{ALS}$ for the missing entries $(i, j)$ .",
"Thus, we learn the parameters of the following factorization: $R \\approx U V^T$ where: $U$ is the $n \\times r$ matrix of user latent vectors; $V$ is the $m \\times r$ matrix of item latent vectors.",
"In order to avoid overfitting, we regularize the parameters to estimate.",
"Therefore, as we want to minimize the squared error, the loss function to minimize has the following form: $\\sum _{i, j | r_{ij} \\ne 0} (r_{ij} - U_i^T V_j)^2 + \\lambda \\left( {U_i}_2^2 + {V_j}^2_2 \\right)$ where $U_i$ for every $i = 1, \\ldots , n$ are the rows of $U$ and $V_j$ for every $j = 1, \\ldots , m$ are the rows of $V$ , and $\\lambda $ is a regularization parameter.",
"This estimation is made by using alternate least squares with weighted $\\lambda $ -regularization (ALS-WR) Zhou2008.",
"Once the parameters have been learned, the prediction for rating of user $i$ on item $j$ is: $\\hat{r}_{ij}^{ALS} = U_i^T V_j.$"
],
[
"Steins Gate",
"At this step, we have predictions from two different blocks: ALS trained on the ratings and LASSO trained on the tag predictions of the posters.",
"We want to improve the predictive power of the overall model, thus we learn a rule that would automatically choose the best model according to the number of ratings of the item considered.",
"Formally, we want to learn parameters $\\beta $ and $\\gamma $ such that: $\\hat{r}_{ij}^{BALSE} & = \\sigma (\\beta (\\mathcal {R}_j - \\gamma )) \\hat{r}_{ij}^{ALS}\\\\& \\quad + \\left(1 - \\sigma (\\beta (\\mathcal {R}_j - \\gamma ))\\right) \\hat{r}_{ij}^{LASSO}$ where: $\\mathcal {R}_j$ is the number of ratings of the item $j$ ; $\\hat{r}_{ij}^{ALS}$ is the rating prediction of ALS model for user $i$ on item $j$ ; $\\hat{r}_{ij}^{LASSO}$ is the rating prediction of LASSO model for user $i$ on item $j$ ; $\\sigma : x \\mapsto 1/(1 + e^{-x})$ is the sigmoid function.",
"The intuition behind this formula is the following: we want to find a threshold $\\gamma $ such that when the number of ratings of item $j$ verifies $R_j \\gg \\gamma $ , BALSE mimics ALS, e.g., $\\hat{r}_{ij}^{BALSE} \\approx \\hat{r}_{ij}^{ALS}$ , while when $R_j \\ll \\gamma $ , i.e.",
"in a cold-start setting, BALSE mimics LASSO, e.g., $\\hat{r}_{ij}^{BALSE} \\approx \\hat{r}_{ij}^{LASSO}$ .",
"$\\beta $ is just a scaling parameter that indicates how sharp the passage from LASSO to ALS will be.",
"Formally, we want to estimate the parameters $\\beta $ and $\\gamma $ that minimize: $\\sum _{i, j | r_{ij} \\ne 0} \\left(\\hat{r}_{ij}^{BALSE} - r_{ij}\\right)^2.$ This formula is differentiable with respect to $\\gamma $ , thus it makes its optimization easier.",
"It can be seen as a soft switch between the two possible predictions (ALS and LASSO), according to the number of ratings of the item.",
"The parameters $\\beta $ and $\\gamma $ are learned using gradient descent."
],
[
"Mangaki dataset",
"Mangakihttps://mangaki.fr Vie2015 is a website where people can rate items that represent either manga or anime, and receive recommendations based on their ratings.",
"Mangaki can be seen as an open source version of Movielens movielens2015dataset for manga and anime.",
"The Mangaki dataset is a $2079 \\times 9979$ anonymized matrix of 334390 ratings from 2079 users on 9979 items.",
"80% of the items have a poster.",
"Users can either rate an item with {favorite, like, neutral, dislike} if they watched it, or {willsee, wontsee} if they did not watch it, i.e.",
"testify whether they want to watch it or not, based on the content presented: poster, possibly synopsis, or some statistics."
],
[
"Models",
"The models considered in this benchmark are: ALS: alternate least squares with weighted $\\lambda $ -regularization from Zhou2008, that ignores posters; LASSO: regularized linear regression using ratings and the tag predictions from Illustration2Vec, that is content-based; BALSE: the proposed method.",
"In practice, we use $\\lambda = 0.1$ and rank $r = 20$ for every ALS model trained and $\\alpha = 0.01$ for every LASSO model trained.",
"Ratings are mapped into custom values: (favorite, like, neutral, dislike) = (4, 2, 0.1, -2) and (willsee, wontsee) = (0.5, -0.5).",
"The Steins gate is optimized using gradient descent with exponential decay implemented in TensorFlow.",
"The learning rate starts at $0.9$ and decays every 20 steps with a base of $0.997$ .",
"All the code is available on our GitHub repositoryhttps://github.com/mangaki/balse."
],
[
"5-fold cross validation",
"We perform a 5-fold cross validation over the triplets $(i, j, r_{ij})$ of the database, keeping 30% of the train set as a validation set.",
"Therefore, our data is split into a train set (56%), a validation set (24%) and a test set (20%).",
"The vanilla models ALS and LASSO are trained on both the train set and the validation set.",
"For BALSE, the ALS and LASSO blocks are first trained using the train set only, and the Steins gate parameters $\\beta $ and $\\gamma $ are trained using the validation setPlease also note that in Steins gate, the number of ratings $R_j$ of item $j$ is computed over the train set., in order to prevent overfitting.",
"For the final predictions of BALSE, blending is performed using the learned $\\beta $ and $\\gamma $ parameters, and the vanilla ALS and LASSO models.",
"Finally, the root mean squared error (RMSE) is computed over the test set.",
"We distinguish the performance of all three models on three sets: the whole test set, a set of little-known items that received less than 3 ratings in the train and validation set (that represents 1000 ratings, therefore 3% of the test set), and cold-start items, i.e.",
"items that were never seen in the train and validation sets."
],
[
"Results",
"BALSE achieves a comparable performance than ALS overall, but substantially improves the recommendations on little-known items, see Table REF .",
"Table: Results of RMSE on various subsets of the test set.Figure: This is Steins gate's choice: γ=0.79040\\gamma = 0.79040.",
"For items having at least one rating, it is better to rely more on the ratings predicted by ALS than by LASSO.The learned parameter $\\gamma $ of the Steins gate was less than 1, see Figure REF , which means that items having at least 1 rating can start to rely on ALS (their ratings) more than LASSO (their poster) for the predictions.",
"However, BALSE provides better predictions than ALS for cold-start items, because ALS was not trained on them in the train set, therefore it outputs constant predictions.",
"ALS converges after 10 iterations.",
"Steins gate takes 15k iterations to converge.",
"LASSO is the bottleneck of the proposed approach because one LASSO model should be trained per user that appears in the train set."
],
[
"Explanation of user taste",
"Using the tags, it is possible to provide an explanation of the taste of any user $i$ using the preference matrix $P$ learned by LASSO, because the columns of $P_i$ are labeled with tags.",
"LASSO has been appreciated for its explainability tibshirani1996regression: the row preferences of each user are sparse, allowing to capture the tags that explain best the ratings of every user.",
"As an example, for a certain user among the authors, LASSO or BALSE report that his six most preferred tags are: kneehighs, cat, serafukuSerafuku means “Japanese school uniform”., twin braids, japanese clothes and angry whereas his six most disliked tags are: pleated skirt, standing, silver hair, window, torn clothes and skirt.",
"Using this information, LASSO or BALSE can explain a recommendation: “We recommend to you the anime Chivalry of a Failed Knight, because there is a girl with twin braids, serafuku and japanese clothes” or a warning: “You might not like the anime The Asterisk War: The Academy City on the Water because there is a girl with a pleated skirt, even though there are kneehighs and serafuku.”"
],
[
"Conclusion and Future Work",
"We proposed BALSE, a model for recommending anime and manga that makes use of information that is automatically extracted from posters.",
"We showed that our model performs better than the baseline models, especially in the item cold-start scenario.",
"This paper is a proof a concept and the obtained results are very encouraging.",
"Indeed, the blending Steins gate is such that any improvement made on any block would improve the overall performance of the approach.",
"As future work, we plan to replace blocks in our architecture with more complex models: Illustration2Vec could be replaced with residual networks he2016deep, ALS could be replaced with factorization machines rendle2010factorization or co-factorization fang2011matrix, LASSO could be replaced with Localized Lasso yamada2016localized, a variant that works well for few samples, many features.",
"We also to integrate more side information, for instance the drawing style of the image, or tags coming from open databases such as AniDBhttp://anidb.net or AniListhttps://anilist.co, in order to improve the explanation of the users' preferences.",
"For the sake of simplicity, we mapped the categorical ratings like, dislike, etc.",
"to ad-hoc values, but we could instead use ordinal regression methods pedregosa2017consistency.",
"However, they require more computation to be trained properly.",
"Ensemble methods that blend more than two models could be considered sill2009feature, or that rely not also on the number of ratings provided for a certain item, but on the number of ratings provided by a certain user, or the number of works that contain a certain tag.",
"Here, we mitigated the problem of item cold-start recommendation through the use of extra information on the item side.",
"Obviously, similar results could be obtained for the user cold-start problem, provided enough data is available to describe the users.",
"Using BALSE, recommender systems can automatically replenish their database, where new items go through the tag prediction track and the explainable model in order to justify the recommendations for their first users, and automatically go to the main track when sufficient ratings have been collected."
],
[
"Acknowledgments",
"tocsectionAcknowledgments This work was carried out while Florian Yger was a visiting researcher at RIKEN Center for AIP, and Kévin Cocchi and Thomas Chalumeau were interns at Mangaki.",
"We would like to thank Nicolas Hug and Étienne Simon for their helpful comments and Solène Pichereau for kindly providing the example illustration of Figure REF ."
]
] | 1709.01584 |
[
[
"MAGIC observations on Pulsar Wind Nebulae around high spin-down power\n Fermi-LAT pulsars"
],
[
"Abstract Pulsar Wind Nebulae (PWNe) represent the most numerous population of TeV sources in our Galaxy.",
"These sources, some of which emit very-high-energy (VHE) gamma-rays, are believed to be related to the young and energetic pulsars that power highly magnetized nebulae (a few $\\mu$G to a few hundred $\\mu$G).",
"In this scenario, particles are accelerated to VHE along their expansion into the pulsar surroundings, or at the shocks produced in collisions of the winds with the surrounding medium.",
"Those energetic pulsars can be traced using observations with the Fermi-LAT detector.",
"The MAGIC Collaboration has carried out deep observations of PWNe around high spin-down power Fermi pulsars.",
"We study the PWN features in the context of the already known TeV PWNe.",
"We present here the analysis accomplished with three selected PWNe: PSR J0631+1036, PSR J1954+2838 and PSR J1958+2845."
],
[
"Introduction",
"Pulsars, highly magnetized rotating neutron stars, are constantly releasing their rotational energy in the form of relativistic Poynting and particles flux, the so-called pulsar wind.",
"This wind interacts with the interstellar medium (ISM), giving rise to a termination shock in which particles are accelerated.",
"When flowing out, the relativistic particles can, in turn, interact with the surrounding medium generating a magnetized bubble known as Pulsar Wind Nebula (PWN).",
"For the first thousand years, emission from this nebula is mainly synchrotron dominated and detected from radio to X-rays.",
"In the gamma-ray regime, emission is produced through inverse Compton (IC) up-scattering of low-energy photons, composed by CMB, far infrared (FIR) or near infrared (NIR) and optical photons [1].",
"Based on observational criteria, TeV PWNe are expected when hosting high-spin down power pulsars ($\\gtrsim 10^{34}$ erg s$^{-1}$ ) (see e.g.",
"[2]).",
"PWNe correspond to the most numerous galactic very-high-energy (VHE; $E>100$ GeV) population in the Milky Way.",
"The MAGIC telescopes deeply studied the most luminous one, the Crab Nebula [3], and discovered the least luminous PWN, 3C 58 [4].",
"The southern hemisphere Imaging Atmospheric Cherenkov Telescopes (IACTs) H.E.S.S.",
"extensively studied this type of source as well, and were able to firmly identify 14 PWNe during their Galactic Plane Survey (HGPS; [2]) towards the inner part of our Galaxy.",
"In this work, we aim to prove particle acceleration at the outer side of the Galaxy, for which we selected promising PWN candidates based on the high spin-down power (between $\\sim 10^{35}$ –$10^{37}$ erg s$^{-1}$ ) and characteristic age (few tens of kyr) of the hosted Fermi-LAT pulsars: PSR J0631+1036, PSR J1954+2838 and PSR J1958+2845.",
"Basic information from these pulsars can be found in Table REF , taken from the ATNF pulsar cataloguehttp://www.atnf.csiro.au/research/pulsar/psrcat/ [5].",
"If no distance information is available, this parameter is taken from the literature.",
"The pseudo-distance is also provided, which is estimated making use of the spin-down energy loss rate and the gamma-ray luminosity [6].",
"Table: Characteristics of the selected PWN candidates for the study.",
"From left to right: Spin-down power, characteristic age, distance and pseudo-distance.",
"Information taken from the ATNF catalog if not specified otherwise.",
"E ˙/Distance\\dot{E}/Distance is computed using the so-called Distance.",
"a ^{a} , b ^{b} ."
],
[
"Observations and data analysis",
"MAGIC is a stereoscopic system consisting of two 17 m diameter IACTs situated in El Roque de los Muchachos observatory in the Canary island of La Palma, Spain ($28.8^{\\circ }$ N, $17.8^{\\circ }$ W, 2225 m a.s.l.).",
"After a major upgrade that involved digital trigger, readout system and MAGIC I camera, the integral sensitivity in stereoscopic mode achieved at low zenith angles is $0.66\\pm 0.03$ % of the Crab Nebula flux in 50 hr above 220 GeV [8].",
"Data analysis was performed making use of the MAGIC standard software (MARS, [9]).",
"The significance is calculated using Equation 17 from [10], while upper limits (ULs) are obtained following the Rolke method [11] at a 95% confidence level (CL), assuming a Gaussian background and 30% systematics on the effective area.",
"All observations presented in this proceeding were carried out under different moonlight conditions, divided according to the $DC$ level during observations.",
"Thus, data was classified as dark (if $DC<2.0~\\mu $ A), moderate moon ($2.0~\\mu $ A$<DC<4.0~\\mu $ A) and decent moon ($4.0~\\mu $ A$<DC<8.0~\\mu $ A).",
"The cleaning levels applied in each case correspond to those of $1\\times NSB_{\\textrm {dark}}$ , 2–3$\\times NSB_{\\textrm {dark}}$ and 5–8$\\times NSB_{\\textrm {dark}}$ (with nominal HV), respectively, presented in [12]."
],
[
"PSR J0631+1036",
"PSR J0631+1036 belongs to the population of radio-loud gamma-ray pulsars, first detected in a radio search for counterparts of X-ray sources found in Einstein IPC images.",
"VERITAS observed this source in their first year of operation in 2007 for 13 hours, finding no significant excess.",
"Following the Fermi's Bright Source List, eight years of data from the ground based water Cherenkov observatory Milagro were re-analyzed giving rise to a 3.7$\\sigma $ hotspot at the position of this source [13].",
"Nevertheless, in the recently published second HAWC catalog the hint of possible TeV emission from the direction of PSR J0631+1036 could not be confirmed [14].",
"MAGIC conducted deep observations of this source for a total of 37 hours during the winter seasons of 2014/15 and 2015/16, within a zenith range of 15 to 50$^{\\circ }$ and under different moon conditions."
],
[
"PSR J1954+2838 and PSR J1958+2845",
"These two pulsars, reported in the First Fermi-LAT Source Catalog (1FGL) [15], are located in a very dense and crowded region, in which the association of different structures within the field of view is still under debate.",
"PSR J1954+2838 is positionally coincident with SNR G65.1+06, which is a very faint supernova remnant (SNR) at a distance of 9.2 kpc with an estimated age between 40–140 kyr [7].",
"This SNR seems to be associated with another pulsar in the FoV, PSR J1957+2831.",
"At the south of the remnant, an IR source is detected, IRAS 19520+2759, which was found to be related to CO, H$_{2}$ O and OH emission lines at a distance similar to SNR G65.1+06, suggesting interaction with molecular clouds.",
"The re-analysis of 8 years of Milagro data revealed hot spots at the level of 4.3$\\sigma $ and $4.0\\sigma $ in the direction of PSR J1954+2838 and PSR J1958+2845, respectively [13].",
"This emission may originate from the corresponding PWN or interaction of the SNR and molecular cloud, in the case of PSR J1954+2838.",
"In 2010, MAGIC observed these two pulsars in the stand-alone mode with MAGIC I for $\\sim 25$ hours, resulting in a non-detection [16].",
"Nevertheless, the major upgrade between 2011–2012 that both telescopes underwent allowed to improve MAGIC sensitivity with respect to former observations.",
"In the new campaign, MAGIC observed PSR J1954+2838 for a total of $\\sim 16$ hours between April and November 2015, in a zenith range between 5$^{\\circ }$ to 50$^{\\circ }$ .",
"In the case of PSR J1958+2845, only moon data were available, amounting $\\sim 4$ hours of good quality data, in a zenith range of 10$^{\\circ }$ to 40$^{\\circ }$ ."
],
[
"PSR J0631+1036",
"We did not find any significant excess in the direction of PSR J061+1036 after $\\sim 37$ hours.",
"Following Table A.2.",
"from [2], this source is expected to be extended for MAGIC, taken into account its characteristic age and relative proximity to our solar system.",
"Nevertheless, since its extension was not confirmed observationally so far, we computed ULs for both point-like and disk profile with 0.3$^{\\circ }$ radius (maximum possible extension given our observational settings).",
"The corresponding integral ULs above 300 GeV are $6.0 \\times 10^{-13}$ cm$^{-2}$ s$^{-1}$ and $2.8 \\times 10^{-12}$ cm$^{-2}$ s$^{-1}$ , respectively.",
"Both ULs are not in agreement with the Milagro measurement, assuming a power-law spectrum with photon index $\\Gamma =2.2$ , which corroborates the non-detection reported by HAWC [14].",
"For the following discussion we will adopt the more conservative limit, but note that a larger extension than 0.3$^{\\circ }$ could affect our background estimation and render the ULs too optimistic."
],
[
"PSR J1954+2838 and PSR J1958+2845",
"No gamma-ray excess was found in the direction of either PSR J1954+2838 or PSR J1958+2845.",
"The measured signal is compatible with background at energies greater than 300 GeV and 1 TeV (the latter motivated by Milagro hotspots).",
"A hotspot situated in the FoV of PSR J1954+2838 appeared at an offset of $\\sim 0.23^{\\circ }$ from the nominal source at the level of $\\sim 3.5\\sigma $ , although its position is not coincident with any known system (see Figure REF ).",
"Figure: MAGIC significance skymap for the observations of PSR J1954+2838 (white diamond).",
"The pulsar PSR J1957+2831 associated to the SNR G65.1+06 is marked in blue, while the IR source, IRAS 19520+2759, located at the south of the remnant, is shown in green.The corresponding integral ULs for energies above 300 GeV assuming a power-law distribution with photon index $\\Gamma =2.6$ are $1.1 \\times 10^{-12}$ ph cm$^{-2}$ s$^{-1}$ and $2.5 \\times 10^{-12}$ ph cm$^{-2}$ s$^{-1}$ ($\\sim 2.6$ % CU) for PSR J1954+2838 and PSR J1958+2845, respectively.",
"Differential ULs are listed in Table REF and depicted in the spectral energy distribution (SED) shown in Figure REF .",
"Table: MAGIC 95% CL differential flux ULs for PSR J1954+2838 and PSR J1958+2845 assuming a power-law spectrum with spectral index of Γ=2.6\\Gamma =2.6.Figure: SED for the observation on PSR J1954+2838 and PSR J1958+2845.",
"Results for the Fermi-LAT pulsar as well as the Milagro re-analysis are shown, taken from and , respectively."
],
[
"Discussion and conclusions",
"Despite observing promising candidates for emitting VHE gamma rays based on observational criterion, no detection was achieved for any PWN, even though all of them, except for PSR J1958+2845, were observed for a large amount of hours.",
"Given the location of these PWNe, at the outer parts of the Galaxy, a non-detection could be explained if different behaviors are found in the MAGIC candidates with respect to those shown by detected PWNe located mostly in the inner regions.",
"Our results, along with all detected PWNe (inside and outside of the HGPS), HGPS candidates and the ULs obtained for the undetected HGPS PWNe (see [2]) are shown in Figure REF , where the PWN luminosity between 1–10 TeV is plotted versus the characteristic age and spin-down power of the hosted pulsars.",
"The luminosity of the MAGIC PWNe are also quoted in Table REF .",
"MAGIC results are in agreement with the fit obtained by [2] using detected TeV PWNe and ULs.",
"Therefore, it confirms the initial assumption that gamma-ray emission is expected from these candidates given their features.",
"Figure: TeV luminosity (1–10 TeV) with respect to the characteristic age (top) and the spin-down power of the pulsar (bottom).",
"The candidates included in this project are marked with squares, while external PWNe are shown with circles.",
"In the latter, detected PWN from inside and outside of the HGPS, candidates and non-detected nebulae from it are included.",
"The fit obtained in the study of PWN by is depicted as a blue band.Table: From left to right: Integral UL above 300 GeV, UL on the TeV luminosity (1–10 TeV), efficiency converting rotational energy into TeV gamma rays (L γ,1-10TeV /E ˙L_{\\gamma ,1-10 TeV}/\\dot{E}).",
"To compute the luminosity, the values from the Distance column in Table were used.Possible reasons for the non-detection can be either a larger extension than expected from these sources or lower density photon fields.",
"Both scenarios are currently being investigated and their results will be set in context with other PWNe studied by MAGIC."
],
[
"Acknowledgement",
"We would like to thank the IAC for the excellent working conditions at the ORM in La Palma.",
"We acknowledge the financial support of the German BMBF, DFG and MPG, the Italian INFN and INAF, the Swiss National Fund SNF, the European ERDF, the Spanish MINECO, the Japanese JSPS and MEXT, the Croatian CSF, and the Polish MNiSzW."
]
] | 1709.01754 |
[
[
"A M\\\"obius scalar curvature rigidity on compact conformally flat\n hypersurfaces in $\\mathbb{S}^{n+1}$"
],
[
"Abstract In this paper, we study conformally flat hypersurfaces of dimension $n(\\geq 4)$ in $\\mathbb{S}^{n+1}$ using the framework of M\\\"obius geometry.",
"First, we classify and explicitly express the conformally flat hypersurfaces of dimension $n(\\geq 4)$ with constant M\\\"obius scalar curvature under the M\\\"obius transformation group of $\\mathbb{S}^{n+1}$.",
"Second, we prove that if the conformally flat hypersurface with constant M\\\"obius scalar curvature $R$ is compact, then $$R=(n-1)(n-2)r^2, ~~0<r<1,$$ and the compact conformally flat hypersurface is M\\\"obius equivalent to the torus $$\\mathbb{ S}^1(\\sqrt{1-r^2})\\times \\mathbb{S}^{n-1}(r)\\hookrightarrow \\mathbb{S}^{n+1}.$$"
],
[
"Introduction",
"A Riemannian manifold $(M^n, g)$ is conformally flat, if every point has a neighborhood which is conformal to an open set in the Euclidean space $\\mathbb {R}^n$ .",
"A hypersurface of the sphere $\\mathbb {S}^{n+1}$ is said to be conformally flat if so it is with respect to the induced metric.",
"Due to conformal invariant objects, the theory of conformally flat hypersurfaces is essentially the same whether it is considered in the space forms $\\mathbb {R}^{n+1}$ , $\\mathbb {S}^{n+1}$ or $\\mathbb {H}^{n+1}$ .",
"In fact, there exists conformal diffeomorphism between the space forms.",
"The $(n+1)$ -dimensional hyperbolic space $\\mathbb {H}^{n+1}$ defined by $\\mathbb {H}^{n+1}=\\lbrace (y_0,y_1,\\cdots ,y_{n+1})|-y_0^2+y_1^2+\\cdots +y_{n+1}^2=-1,y_0>0\\rbrace .$ The conformal diffeomorphisms $\\sigma ,\\tau $ are defined by $\\begin{split}&\\sigma :\\mathbb {R}^{n+1}\\rightarrow \\mathbb {S}^{n+1}\\backslash \\lbrace (-1,\\vec{0})\\rbrace ,~~~~\\sigma (u)=(\\frac{1-|u|^2}{1+|u|^2},\\frac{2u}{1+|u|^2}),\\\\&\\tau :\\mathbb {H}^{n+1}\\rightarrow \\mathbb {S}^{n+1}_+\\subset \\mathbb {S}^{n+1},~~~\\tau (y)=(\\frac{1}{y_0},\\frac{\\vec{y}}{y_0}), ~~y=(y_0,\\vec{y})\\in \\mathbb {H}^{n+1},\\end{split}$ where $\\mathbb {S}^{n+1}_+$ is the hemisphere in $\\mathbb {S}^{n+1}$ whose the first coordinate is positive.",
"By conformal diffeomorphisms $\\sigma ,\\tau $ , the conformally flat hypersurfaces in space forms are equivalent to each other.",
"The dimension of the hypersurface seems to play an important role in the study of conformally flat hypersurfaces.",
"For $n\\ge 4$ , the immersed hypersurface $f: M^n \\rightarrow \\mathbb {S}^{n+1}$ is conformally flat if and only if at least $n-1$ of the principal curvatures coincide at each point by the result of Cartan-Schouten ([1],[10]).",
"Cartan-Schouten's result is no longer true in dimension 3.",
"Lancaster ([6]) gave some examples of conformally flat hypersurfaces in $\\mathbb {R}^4$ having three different principal curvatures.",
"For $n=2$ , the existence of isothermal coordinates means that any Riemannian surface is conformally flat.",
"Do Carmo, Dajczer and Mercuri in [2] have studied Diffeomorphism types of the compact conformally flat hypersurfaces in $\\mathbb {R}^{n+1}$ .",
"Pinkall in [9] was studied the intrinsic conformal geometry of compact conformally flat hypersurfaces.",
"Suyama in [11] explicitly constructs compact conformally flat hypersurfaces in space forms using codimension one foliation by $(n-1)$ -spheres.",
"Standard examples of the conformally flat hypersurfaces come from cones, cylinders, or rotational hypersurfaces over a curve in Euclidean 2-space $\\mathbb {R}^2$ , 2-sphere $\\mathbb {S}^2$ , or hyperbolic 2-space $\\mathbb {R}^2_+$ , respectively (see section 3).",
"In [4], Lin and Guo showed that if the conformally flat hypersurface has closed Möbius form, then it is Möbius equivalent to one of the standard examples.",
"It is known that the conformal transformations group of a sphere is isomorphic to its Möbius transformation group.",
"As conformal invariant objects, conformally flat hypersurfaces are investigated in this paper using the framework of Möbius geometry.",
"If the conformally flat hypersurface is no umbilical point everywhere, then there exists a global Möbius metric (see section 2), which is invariant under the Möbius transformation group of $\\mathbb {S}^{n+1}$ .",
"The scalar curvature with respect to the Möbius metric is called Möbius scalar curvature.",
"First, we classify locally the conformally flat hypersurfaces of dimension $n(\\ge 4)$ with constant Möbius scalar curvature under the Möbius transformation group of $\\mathbb {S}^{n+1}$ .",
"Theorem 1.1 Let $f:M^n\\rightarrow \\mathbb {S}^{n+1}$ , $n\\ge 4$ , be a conformally flat hypersurface without umbilical points.",
"If the Möbius scalar curvature is constant, then the Möbius form is closed and $f$ is Möbius equivalent to one of the following hypersurfaces in $\\mathbb {S}^{n+1}$ , $(i)$ the image of $\\sigma $ of a cylinder over a curvature-spiral in $\\mathbb {R}^2\\subset \\mathbb {R}^{n+1}$ ; $(ii)$ the image of $\\sigma $ of a cone over a curvature-spiral in $\\mathbb {S}^2\\subset \\mathbb {R}^3\\subset \\mathbb {R}^{n+1}$ ; $(iii)$ the image of $\\sigma $ of a rotational hypersurface over a curvature-spiral in $\\mathbb {R}^2_+\\subset \\mathbb {R}^{n+1}$ .",
"Here the so-called curvature-spiral in a 2-dimensional space form $N^2(\\epsilon )=\\mathbb {S}^2,\\mathbb {R}^2,\\mathbb {R}^2_+$ (of Gaussian curvature $\\epsilon =1,0,-1$ respectively) is determined by the intrinsic equation $-\\frac{\\kappa _{ss}}{\\kappa ^3}+\\frac{(n+2)\\kappa _s^2}{2\\kappa ^4}+\\epsilon \\frac{n-2}{2\\kappa ^2}=R,~~~~\\kappa _s=\\frac{d}{ds}\\kappa .$ Here $s$ is the arc-length parameter, $\\kappa $ denotes the geodesic curvature of the curve $\\gamma $ , and $R$ is a real constant.",
"In [3], authors classified locally the hypersurfaces with constant Möbius sectional curvature, which is some special conformally flat hypersurfaces with Möbius scalar curvature by the equation (REF ).",
"For compact conformally flat hypersurfaces, we obtain the following Möbius scalar curvature rigidity theorem, which means that the closed curve in $\\mathbb {R}^2_+$ satisfying the intrinsic equation (REF ) with geodesic curvature $\\kappa >0$ is circle $\\mathbb {S}^1$ .",
"Theorem 1.2 Let $f:M^n\\rightarrow \\mathbb {S}^{n+1}$ , $n\\ge 4$ , be a compact conformally flat hypersurface without umbilical points everywhere.",
"If the Möbius scalar curvature $R$ is constant, then $R=(n-1)(n-2)r^2, ~~0<r<1,$ and the compact conformally flat hypersurface is Möbius equivalent to the torus $f:\\mathbb { S}^1(\\sqrt{1-r^2})\\times \\mathbb {S}^{n-1}(r)\\rightarrow \\mathbb {S}^{n+1}.$ Remark 1.1 Theorem REF and Theorem REF is true for $n=3$ provided that the 3-dimensional conformally flat hypersurface has only two distinct principal curvatures.",
"The paper is organized as follows.",
"In section 2, we review the elementary facts about Möbius geometry of hypersurfaces in $\\mathbb {S}^{n+1}$ .",
"In section 3, we prove the theorem REF .",
"In section 4, we prove the theorem REF ."
],
[
"Möbius invariants of hypersurfaces in $\\mathbb {S}^{n+1}$",
"In this section, we recall some facts about the Möbius invariants of hypersurfaces in $\\mathbb {S}^{n+1}$ .",
"For details we refer to [12].",
"Let $f:M^{n}\\rightarrow \\mathbb {S}^{n+1}$ be a hypersurface without umbilical points.",
"In this section we use the range of indices: $1\\le i,j,k,l\\le n.$ We assume that $\\lbrace e_i\\rbrace $ is an orthonormal basis with respect to the induced metric with $\\lbrace \\theta _i\\rbrace $ the dual basis.",
"Let $II=\\sum _{ij}h_{ij}\\theta _i\\theta _j$ and $H=\\sum _i\\frac{h_{ii}}{n}$ be the second fundamental form and the mean curvature of $x$ , respectively.",
"We define the Möbius metric $g$ , the Möbius second fundamental form $B$ , the Blaschke tensor $A$ and the Möbius form $C$ as follows, respectively, $\\begin{split}g=&\\rho ^2dx\\cdot dx,~~~~~~~\\rho ^2=\\frac{n}{n-1}(|h|^2-nH^2),\\\\B=&\\rho \\sum _{ij}(h_{ij}-H\\delta _{ij})\\theta _i\\otimes \\theta _j,\\\\C=&-\\rho ^{-1}\\sum _i[e_i(H)+\\sum _j(h_{ij}-H\\delta _{ij})e_j]\\theta _i,\\\\A=&\\sum _{ij}\\Big \\lbrace e_i(\\log \\rho )e_j(\\log \\rho )-\\nabla _{e_i}\\nabla _{e_j}\\log \\rho +Hh_{ij}+\\\\&\\frac{1}{2}[1-H^2-|\\bigtriangledown \\log \\rho |^2]\\delta _{ij}\\Big \\rbrace \\theta _i\\otimes \\theta _j.\\end{split}$ Note that the conformal compactification space $\\mathbb {S}^{n+1}$ unifies the space forms $\\mathbb {S}^{n+1},$ $\\mathbb {R}^{n+1},\\mathbb {H}^{n+1}$ and the formula above defining the Möbius metric $g$ and the Möbius second fundamental form $B$ are the same for any of them.",
"Theorem 2.1 $\\cite {w}$ Two hypersurfaces $f: M^n\\rightarrow \\mathbb {S}^{n+1}$ and $\\bar{f}:M^n\\rightarrow \\mathbb {S}^{n+1} (n\\ge 3)$ are Möbius equivalent if and only if there exists a diffeomorphism $\\varphi : M^n\\rightarrow M^n$ which preserves the Möbius metric and the Möbius second fundamental form.",
"Let $E_i=\\rho ^{-1}e_i, \\omega _i=\\rho \\theta _i$ , then $\\lbrace E_1,\\cdots ,E_n\\rbrace $ is an orthonormal basis with respect to the Möbius metric $g$ with the dual basis $\\lbrace \\omega _1,\\cdots ,\\omega _n\\rbrace $ .",
"Let $\\lbrace \\omega _{ij}\\rbrace $ be the connection 1-form of the Möbius metric under the orthonormal basis $\\lbrace \\omega _i\\rbrace $ , and $A=\\sum _{ij}A_{ij}\\omega _i\\otimes \\omega _j,~~B=\\sum _{ij}B_{ij}\\omega _i\\otimes \\omega _j,~~C=\\sum _iC_i\\omega _i.$ The covariant derivative of $C_i, A_{ij}, B_{ij}$ are defined by $&&\\sum _jC_{i,j}\\omega _j=dC_i+\\sum _jC_j\\omega _{ji},\\\\&&\\sum _kA_{ij,k}\\omega _k=dA_{ij}+\\sum _kA_{ik}\\omega _{kj}+\\sum _kA_{kj}\\omega _{ki},\\\\&&\\sum _kB_{ij,k}\\omega _k=dB_{ij}+\\sum _kB_{ik}\\omega _{kj}+\\sum _kB_{kj}\\omega _{ki}.$ The integrability conditions of the Möbius invariants are given by $&&A_{ij,k}-A_{ik,j}=B_{ik}C_j-B_{ij}C_k,\\\\&&C_{i,j}-C_{j,i}=\\sum _k(B_{ik}A_{kj}-B_{jk}A_{ki}),\\\\&&B_{ij,k}-B_{ik,j}=\\delta _{ij}C_k-\\delta _{ik}C_j,\\\\&&R_{ijkl}=B_{ik}B_{jl}-B_{il}B_{jk}+\\delta _{ik}A_{jl}+\\delta _{jl}A_{ik}-\\delta _{il}A_{jk}-\\delta _{jk}A_{il},.$ where $R_{ijkl}$ denote the curvature tensor of $g$ .",
"Moreover, $\\begin{split}&\\sum _iB_{ii}=0, ~~\\sum _{ij}(B_{ij})^2=\\frac{n-1}{n},\\\\&\\sum _iA_{ii}=\\frac{1}{2n}+\\frac{R}{2(n-1)},~~\\sum _jB_{ij,j}=-(n-1)C_i,\\end{split}$ where $R=\\sum _{i>j}R_{ijij}$ is the Möbius scalar curvature.",
"By equation (), we have $dC=0\\Leftrightarrow \\sum _k(B_{ik}A_{kj}-B_{jk}A_{ki})=0,$ which implies that the matrix $(B_{ij})$ and $(A_{ij})$ can be diagonalizable simultaneously."
],
[
"Local geometry of conformally flat hypersurfaces",
"In this section, we will give the Möbius invariants of the standard examples of conformally flat hypersurfaces in $\\mathbb {R}^{n+1}$ .",
"Then we prove that the conformally flat hypersurfaces with constant Möbius scalar curvature come from these examples.",
"A key observation is that the Möbius metric of those standard examples are of the form $g=\\kappa ^2(s)\\left(ds^2+I_{-\\epsilon }^{n-1}\\right)~,$ where $I_{-\\epsilon }^{n-1}$ is the metric of $n-1$ dimensional space form of constant curvature $-\\epsilon $ .",
"For such metric forms we have Lemma 3.1 The metric $g=\\kappa ^2(s)(ds^2+I_{-\\epsilon }^{n-1})$ given above is of constant scalar curvature $R$ if and only if the function $\\kappa (s)$ satisfies $ -\\frac{\\kappa _{ss}}{\\kappa ^3}+\\frac{(n+2)\\kappa _s^2}{2\\kappa ^4}+\\epsilon \\frac{n-2}{2\\kappa ^2}=R,~~~~\\kappa _s=\\frac{d}{ds}\\kappa .$ This lemma is easy to prove using exterior differential forms and we omit the proof at here.",
"Below we give the explicit construction of the standard examples of conformally flat hypersurfaces as well as their Möbius metric.",
"Example 3.1 Let $\\gamma :I\\rightarrow \\mathbb {R}^2$ be a regular curve, and $s$ denote the arclength of $\\gamma (s)$ .",
"we define cylinder in $\\mathbb {R}^{n+1}$ over $\\gamma $ , $f(s,y)=(\\gamma (s),y):I\\times \\mathbb {R}^{n-1}\\longrightarrow \\mathbb {R}^{n+1},$ where $y:\\mathbb {R}^{n-1}\\longrightarrow \\mathbb {R}^{n-1}$ is identical maping.",
"The first fundamental form $I$ and the second fundamental form $II$ of the cylinder $f$ are, respectively, $ I=ds^2+I_{\\mathbb {R}^{n-1}}, \\;\\; II=\\kappa ds^2~,$ where $\\kappa (s)$ is the geodesic curvature of $\\gamma $ , $I_{\\mathbb {R}^{n-1}}$ is the standard Euclidean metric of $\\mathbb {R}^{n-1}$ .",
"So we have $(h_{ij})=\\operatorname{diag}(\\kappa ,0,\\cdots ,0)~,~H=\\frac{\\kappa }{n}~,~\\rho =\\kappa ~.$ Thus the Möbius metric $g$ of the cylinder $f$ is $g=\\rho ^2 I=\\kappa ^2(ds^2+I_{\\mathbb {R}^{n-1}}).$ where $I_{\\mathbb {R}^{n-1}}$ is the standard hyperbolic metric of $\\mathbb {R}^{n-1}(-1)$ .",
"Because at least $n-1$ of the principal curvatures coincide at each point, the cylinder $f$ is a conformally flat hypersurface.",
"When $\\gamma =\\mathbb {S}^1$ , the cylinder $f$ is the isoparametric hypersurface $\\mathbb {S}^1\\times \\mathbb {R}^{n-1}\\rightarrow \\mathbb {R}^{n+1}$ .",
"Example 3.2 Let $\\gamma :I\\rightarrow \\mathbb {S}^2(1)\\subset \\mathbb {R}^3$ be a regular curve, and $s$ denote the arclength of $\\gamma (s)$ .",
"we define cone in $R^{n+1}$ over $\\gamma $ , $f(s,t,y)=(t\\gamma (s),y):I\\times \\mathbb {R}^{+}\\times \\mathbb {R}^{n-2}\\longrightarrow \\mathbb {R}^{n+1},$ where $y:\\mathbb {R}^{n-2}\\longrightarrow \\mathbb {R}^{n-2}$ is identical mapping and $\\mathbb {R}^{+}=\\lbrace t|t>0\\rbrace $ .",
"The first and second fundamental forms of the cone $f$ are, respectively, $I=t^2ds^2+I_{R^{n-1}}~, \\;\\; II=t\\kappa ds^2~.$ So we have $(h_{ij})=\\operatorname{diag}\\left(\\frac{\\kappa }{t},0,\\cdots ,0\\right)~,~H=\\frac{\\kappa }{nt}~,~\\rho =\\frac{\\kappa }{t}~.$ Thus the Möbius metric $g$ of the cone $f$ is $g=\\rho ^2I=\\frac{\\kappa ^2}{t^2}\\left(t^2ds^2+I_{\\mathbb {R}^{n-1}}\\right)=\\kappa ^2(ds^2+I_{\\mathbb {H}^{n-1}})~,$ where $I_{\\mathbb {H}^{n-1}}$ is the standard hyperbolic metric of $\\mathbb {H}^{n-1}(-1)$ .",
"Clearly the cone $f$ is a conformally flat hypersurface.",
"When $\\gamma =\\mathbb {S}^1$ , the cone $f$ is the image of $\\tau ^{-1}\\circ \\sigma $ of the isoparametric hypersurface $\\mathbb {S}^1(r)\\times \\mathbb {H}^{n-1}(\\sqrt{1+r^2})\\rightarrow \\mathbb {H}^{n+1}$ .",
"Example 3.3 Let $\\mathbb {R}^2_+=\\lbrace (x,y)\\in \\mathbb {R}^2|y>0\\rbrace $ be the upper half-space endowed with the standard hyperbolic metric $ds^2=\\frac{1}{y^2}[dx^2+dy^2]~.$ Let $\\gamma =(x,y):I\\longrightarrow \\mathbb {R}^2_+$ be a regular curve, and $s$ denote the arclength of $\\gamma (s)$ .",
"we define rotational hypersurface in $\\mathbb {R}^{n+1}$ over $\\gamma $ , $f:I\\times \\mathbb {S}^{n-1}\\longrightarrow \\mathbb {R}^{n+1},~~~~~f(x,y,\\theta )=(x,y\\theta ),$ where $\\theta :\\mathbb {S}^{n-1}\\longrightarrow \\mathbb {R}^{n}$ is a standard immersion of a round sphere.",
"In the Poincare half plane $\\mathbb {R}^2_+$ we denote the covariant differential of the hyperbolic metric as $D$ .",
"Choose orthonormal frames $e_1=y\\frac{\\partial }{\\partial x},e_2=y\\frac{\\partial }{\\partial y}$ .",
"It is easy to find $D_{e_1}e_1=e_2~,~D_{e_1}e_2=-e_1~,~D_{e_2}e_1=D_{e_2}e_2=0.$ For $\\gamma (s)=((x(s),y(s))\\subset \\mathbb {R}^2_+$ let $x^{\\prime }$ denote derivative $\\partial x/\\partial s$ and so on.",
"Choose the unit tangent vector $\\alpha =\\frac{1}{y}(x^{\\prime }(s)e_1+y^{\\prime }(s)e_2)$ and the unit normal vector $\\beta =\\frac{1}{y}(-y^{\\prime }(s)e_1+x^{\\prime }(s)e_2)$ .",
"The geodesic curvature is computed via $\\kappa =\\langle D_\\alpha \\alpha ,\\beta \\rangle =\\frac{x^{\\prime }y^{\\prime \\prime }-x^{\\prime \\prime }y^{\\prime }}{y^2}+\\frac{x^{\\prime }}{y}~.$ After these preparation, we see that the rotational hypersurface $f(x,y,\\theta )=(x,y\\theta )$ has differential $df=(x^{\\prime }ds,y^{\\prime }\\theta ds+y d\\theta )$ and unit normal vector $\\eta =\\frac{1}{y}(-y^{\\prime },x^{\\prime }\\theta ).$ Thus the first and second fundamental forms of hypersurface $f$ are, respectively, $I=df\\cdot df=y^2(ds^2+I_{\\mathbb {S}^{n-1}})~,~ II=-df\\cdot d\\eta =(y\\kappa -x^{\\prime })ds^2-x^{\\prime }I_{\\mathbb {S}^{n-1}}~,$ where $I_{\\mathbb {S}^{n-1}}$ is the standard metric of $\\mathbb {S}^{n-1}(1)$ .",
"Thus principal curvatures are $\\frac{\\kappa y-x^{\\prime }}{y^2},\\frac{-x^{\\prime }}{y^2},\\cdots ,\\frac{-x^{\\prime }}{y^2}.$ So $\\rho =\\frac{\\kappa }{y}$ , and the Möbius metric of $f$ is $g=\\rho ^2I=\\kappa ^2(ds^2+I_{\\mathbb {S}^{n-1}}).$ Clearly the hypersurface $f$ is a conformally flat hypersurface.",
"When $\\gamma =\\mathbb {S}^1$ , the cone $f$ is the image of $\\sigma $ of the isoparametric hypersurface $\\mathbb {S}^1(\\sqrt{1-r^2})\\times \\mathbb {S}^{n-1}(r)\\rightarrow \\mathbb {S}^{n+1}$ .",
"Next, we compute the Möbius invariant of the conformally flat hypersurfaces.",
"From (REF ), We can choose a local orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ with respect to the Möbius metric $g$ such that $(B_{ij})=diag(\\frac{n-1}{n},\\frac{-1}{n},\\cdots ,\\frac{-1}{n}).$ In the following section we make use of the following convention on the ranges of indices: $1\\le i,j,k \\le n; ~~2\\le \\alpha ,\\beta ,\\gamma \\le n.$ Since $B_{\\alpha \\beta }=\\frac{1}{n}\\delta _{\\alpha \\beta }$ , we can rechoose a local orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ with respect to the Möbius metric $g$ such that $(B_{ij})=diag(\\frac{n-1}{n},\\frac{-1}{n},\\cdots ,\\frac{-1}{n}),~~(A_{ij})=\\left(\\begin{array}{ccccc}A_{11} & A_{12} & A_{13} & \\cdots & A_{1n} \\\\A_{21} & a_2 & 0 & \\cdots & 0 \\\\A_{31}& 0 & a_3& \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\A_{n1} & 0 & 0 & \\cdots &a_n \\\\\\end{array}\\right)$ Let $\\lbrace \\omega _1,\\cdots ,\\omega _n\\rbrace $ be the dual basis, and $\\lbrace \\omega _{ij}\\rbrace $ the connection forms.",
"Lemma 3.2 Let $f: M^n\\rightarrow \\mathbb {S}^{n+1}$ $(n\\ge 4)$ be a conformally flat hypersurface without umbilical points.",
"If the Möbius scalar curvature is constant, then we can choose a local orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ with respect to the Möbius metric $g$ such that $(B_{ij})=diag\\lbrace \\frac{n-1}{n},\\frac{-1}{n},\\cdots ,\\frac{-1}{n}\\rbrace ,~~(A_{ij})=diag\\lbrace a_1,a_2,\\cdots ,a_2\\rbrace .$ Moreover, the distribution $\\mathbb {D}=span\\lbrace E_2,\\cdots ,E_n\\rbrace $ is integrable.",
"Using $dB_{ij}+\\sum _kB_{kj}\\omega _{ki}+\\sum _kB_{ik}\\omega _{kj}=\\sum _kB_{ij,k}\\omega _k$ , the equation (), we get $\\begin{split}&B_{1\\alpha ,\\alpha }=-C_1, ~~otherwise, ~~~B_{ij,k}=0;\\\\&\\omega _{1\\alpha }=-C_1\\omega _{\\alpha },~~~~C_{\\alpha }=0.\\end{split}$ Thus $d\\omega _1=0$ and the distribution $\\mathbb {D}=span\\lbrace E_2,\\cdots ,E_n\\rbrace $ is integrable.",
"Using $dC_i+\\sum _kC_k\\omega _{ki}=\\sum _kC_{i,k}\\omega _k$ and (REF ), we can obtain $C_{\\alpha ,\\alpha }=-C_1^2, ~~~C_{\\alpha ,k}=0,\\alpha \\ne k.$ From (REF ), $\\begin{split}&d\\omega _{1\\alpha }=-dC_1\\wedge \\omega _{\\alpha }-C_1d\\omega _{\\alpha }\\\\&=-dC_1\\wedge \\omega _{\\alpha }+C_1^2\\omega _1\\wedge \\omega _{\\alpha }-C_1\\sum _{\\gamma }\\omega _{\\gamma }\\wedge \\omega _{\\gamma \\alpha },\\end{split}$ and $d\\omega _{1\\alpha }-\\sum _j\\omega _{1j}\\wedge \\omega _{j\\alpha }=-\\frac{1}{2}\\sum _{kl}R_{1\\alpha kl}\\omega _k\\wedge \\omega _l$ , we get that $R_{1\\alpha 1\\alpha }=C_{1,1}-C_1^2,~~~~R_{1\\alpha \\beta \\alpha }-C_{1,\\beta }=0.$ Since $R_{1\\alpha 1\\alpha }=-\\frac{n-1}{n^2}+a_1+a_{\\alpha }=C_{1,1}-C_1^2$ and $R_{1\\alpha \\beta \\alpha }=A_{1\\beta },\\alpha \\ne \\beta $ , thus we have $a_2=a_3=\\cdots =a_n,~~~~ A_{1\\beta }=C_{1,\\beta }.$ Thus $A|_{\\mathbb {D}}=aI,~~a=a_2$ .",
"Since $E_1$ is principal vector field, then vector $E=A_{12}E_2+\\cdots +A_{1n}E_n$ is well defined.",
"If $E=0$ , then $A_{12}=\\cdots =A_{1n}=0$ .",
"If $E\\ne 0$ , we can rechoose a local orthonormal basis $\\lbrace \\tilde{E}_2=\\frac{E}{|E|},\\tilde{E}_3,\\cdots ,\\tilde{E}_n\\rbrace $ of $\\mathbb {D}$ with respect to the Möbius metric $g$ such that $(B_{ij})=diag(\\frac{n-1}{n},\\frac{-1}{n},\\cdots ,\\frac{-1}{n}),~~(A_{ij})=\\left(\\begin{array}{ccccc}A_{11} & A_{12} & 0 & \\cdots & 0 \\\\A_{21} & a_2 & 0 & \\cdots & 0 \\\\0& 0 & a_2& \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots &a_2 \\\\\\end{array}\\right)$ To finish the proof of the Lemma, we need to prove that $A_{12}=0$ .",
"Using $dA_{ij}+\\sum _kA_{kj}\\omega _{ki}+\\sum _kA_{ik}\\omega _{kj}=\\sum _kA_{ij,k}\\omega _k$ , the equation (REF ) and (REF ), we get $\\begin{split}&\\sum _mA_{12,m}\\omega _m=dA_{12}+(A_{11}-A_{22})\\omega _{12},\\\\&\\sum _mA_{1\\alpha ,m}\\omega _m=(A_{11}-a_2)\\omega _{1\\alpha }+A_{12}\\omega _{2\\alpha },~~\\sum _mA_{2\\alpha ,m}\\omega _m=A_{12}\\omega _{1\\alpha },~~\\alpha \\ge 3,\\\\&\\sum _kA_{11,k}\\omega _k=dA_{11}+2A_{12}\\omega _{21}, ~~\\sum _kA_{22,k}\\omega _k=dA_{22}+2A_{12}\\omega _{12},\\\\&\\sum _kA_{\\alpha \\alpha ,k}\\omega _k=dA_{\\alpha \\alpha },~~A_{\\alpha \\beta ,k}=0, ~~\\alpha \\ne \\beta , ~~\\alpha ,\\beta \\ge 3.\\end{split}$ From the fourth and seventh equation in (REF ), we get $E_{\\alpha }(a_2)=A_{\\beta \\beta ,\\alpha }=A_{\\beta \\alpha ,\\beta }=0, ~~~\\alpha \\ge 3.$ Since the Möbius scalar curvature is constant, $tr(A)=A_{11}+(n-1)a_2$ is constant.",
"Thus $A_{1\\alpha ,1}=A_{11,\\alpha }=E_{\\alpha }(A_{11})=0,~~~\\alpha \\ge 3.$ From the first, second and third equation in (REF ), we get $A_{12,2}=E_2(A_{12})-(A_{11}-a_2)C_1,~~~A_{1\\beta ,\\beta }=-(A_{11}-a_2)C_1+A_{12}\\omega _{2\\beta }(E_{\\beta }).$ On the other hand, From (REF ), we have $\\begin{split}&E_1(A_{22})=A_{22,1}=A_{12,2}+\\frac{1}{n}C_1=E_2(A_{12})-(A_{11}-a_2)C_1+\\frac{1}{n}C_1,\\\\&E_1(A_{\\alpha \\alpha })=A_{\\alpha \\alpha ,1}=A_{1\\alpha ,\\alpha }+\\frac{1}{n}C_1=-(A_{11}-a_2)C_1+A_{12}\\omega _{2\\beta }(E_{\\beta })+\\frac{1}{n}C_1,\\\\\\end{split}$ which implies that $E_2(A_{12})=A_{12}\\omega _{2\\beta }(E_{\\beta }).$ Since $A_{1\\alpha ,\\beta }=A_{\\alpha \\beta ,1}=A_{1\\beta ,1}=A_{12,\\alpha }=0,~\\alpha \\ne \\beta $ , from the second equation in (REF ) we can obtain $A_{12}\\omega _{2\\beta }(E_k)=0, ~~\\beta \\ge 3,~~k\\ne \\beta .$ From the first and third equation in (REF ), we get $\\begin{split}E_2(a_2)&=E_2(A_{\\beta \\beta })=A_{\\beta \\beta ,2}=A_{2\\beta ,\\beta }=-A_{12}C_1,\\\\E_1(A_{12})&=A_{12,1}=A_{11,2}=E_2(A_{11})+2A_{12}C_1\\\\&=E_2(-(n-1)a_2)+2A_{12}C_1=(n+1)A_{12}C_1.\\end{split}$ Now we assume that $A_{12}\\ne 0,$ From (REF ) and (REF ), we have $\\omega _{2\\alpha }=\\frac{E_2(A_{12})}{A_{12}}\\omega _{\\alpha }:=\\mu \\omega _{\\alpha },~~\\alpha \\ge 3.$ Thus $\\begin{split}&d\\omega _{2\\alpha }=d\\mu \\wedge \\omega _{\\alpha }+\\mu d\\omega _{\\alpha }\\\\&=d\\mu \\wedge \\omega _{\\alpha }-\\mu C_1^2\\omega _1\\wedge \\omega _{\\alpha }+\\mu ^2\\omega _2\\wedge \\omega _{\\alpha }+\\mu \\sum _{\\gamma \\ge 3}\\omega _{\\gamma }\\wedge \\omega _{\\gamma \\alpha }.\\end{split}$ Using $d\\omega _{2\\alpha }-\\sum _j\\omega _{2j}\\wedge \\omega _{j\\alpha }=-\\frac{1}{2}\\sum _{kl}R_{2\\alpha kl}\\omega _k\\wedge \\omega _l$ , we get that $E_(\\mu )-\\mu C_1=-A_{12}.$ On the other hand, using (REF ) and (REF ), we have $\\begin{split}E_1(\\mu )&=E_1\\big [\\frac{E_2(A_{12})}{A_{12}}\\big ]=\\frac{E_1E_2(A_{12})}{A_{12}}-\\frac{E_2(A_{12})E_1(A_{12})}{A_{12}^2}\\\\&=\\frac{E_1E_2(A_{12})}{A_{12}}-(n+1)\\frac{E_2(A_{12})C_1}{A_{12}}\\\\&=\\frac{(E_2E_1+C_1E_2)(A_{12})}{A_{12}}-(n+1)\\frac{E_2(A_{12})C_1}{A_{12}}\\\\&=\\frac{E_2[(n+1)A_{12}C_1]}{A_{12}}-n\\frac{E_2(A_{12})C_1}{A_{12}}\\\\&=(n+1)C_{1,2}+\\frac{E_2(A_{12})C_1}{A_{12}},\\end{split}$ which implies that $(n+1)C_{1,2}=-A_{12}.$ This is a contradiction by $A_{12}=C_{1,2}$ .",
"Therefore $A_{12}=0$ and we finish the proof.",
"By Lemma REF and equation (REF ), we can derive that $dC=0$ .",
"Combining the results in [4] and Lemma REF we finish the proof of Theorem REF ."
],
[
"Global rigidity of Möbius scalar curvature",
"A hypersurface in $\\mathbb {S}^{n+1}$ is called a Möbius isoparametric hypersurface if its Möbius form vanishes and all the eigenvalues of the Möbius second fundamental form $B$ with respect to $g$ are constants.",
"In [5], authors gave the following classification theorem.",
"Theorem 4.1 [5] Let $f: M^n\\rightarrow \\mathbb {S}^{n+1}$ be a Möbius isoparametric hypersurface with two distinct principal curvatures.",
"Then $f$ is Möbius equivalent to an open part of one of the following Möbius isoparametric hypersurfaces in $\\mathbb {S}^{n+1}$ : (i) the standard torus $\\mathbb {S}^k(r)\\times \\mathbb {S}^{n-k}(\\sqrt{1-r^2})$ ; (ii) the image of $\\sigma $ of the standard cylinder $\\mathbb {S}^k(1)\\times \\mathbb {R}^{n-k}\\subset \\mathbb {R}^{n+1}$ ; (iii) the image of $\\tau $ of the standard $\\mathbb {S}^k(r)\\times \\mathbb {H}^{n-k}(\\sqrt{1+r^2})$ in $\\mathbb {H}^{n+1}$ .",
"To prove Theorem REF , we only need to prove $C=0$ .",
"The way of the proof is to use divergence theorem.",
"First, we need some local computation.",
"Lemma 4.1 Let $f: M^n\\rightarrow \\mathbb {S}^{n+1}$ $(n\\ge 4)$ be a conformally flat hypersurface without umbilical points everywhere.",
"If the Möbius scalar curvature $R$ is constant.",
"then under the local orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ in Lemma REF , we have $\\begin{split}&a_1=\\frac{2n-1}{2n^2}-\\frac{R}{2(n-1)(n-2)}+\\frac{n-1}{n-2}(C_{1,1}-C_1^2),\\\\&a_2=\\frac{R}{2(n-1)(n-2)}-\\frac{1}{2n^2}-\\frac{1}{n-2}(C_{1,1}-C_1^2),\\\\&A_{\\alpha \\alpha ,1}=\\frac{R}{(n-1)(n-2)}C_1-\\frac{n}{n-2}(C_1C_{1,1}-C_1^3).\\end{split}$ Except these coefficients $A_{11,1}, A_{1\\alpha ,\\alpha }$ and $A_{\\alpha \\alpha ,1}$ the coefficients of $\\nabla A$ are equal to zero.",
"The first and second equation in (REF ) can derive directly from the equation $tr(A)=a_1+(n-1)a_2=\\frac{1}{2n}+\\frac{R}{2(n-1)}$ and $R_{1\\alpha 1\\alpha }=-\\frac{n-1}{n^2}+a_1+a_2=C_{1,1}-C_1^2$ in (REF ).",
"From (REF ), we can get $A_{1\\alpha ,\\alpha }=(a_2-a_1)C_1=[\\frac{R}{(n-1)(n-2)}-\\frac{1}{n}]C_1-\\frac{n}{n-2}(C_1C_{1,1}-C_1^3).$ By (REF ), we have $A_{\\alpha \\alpha ,1}=A_{1\\alpha ,\\alpha }+\\frac{1}{n}C_1$ .",
"Combining above equation we get the third equation in (REF ).",
"Since $tr(A)=a_1+(n-1)a_2$ is constant, we have $A_{11,1}=-(n-1)A_{\\alpha \\alpha ,1}.$ Thus, by lemma REF , we know that except these coefficients $A_{11,1}, A_{1\\alpha ,\\alpha }$ and $A_{\\alpha \\alpha ,1}$ the coefficients of $\\nabla A$ are equal to zero.",
"Lemma 4.2 Let $f: M^n\\rightarrow \\mathbb {S}^{n+1}$ $(n\\ge 4)$ be a conformally flat hypersurface without umbilical points everywhere.",
"If the Möbius scalar curvature $R$ is constant.",
"then under the local orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ in Lemma REF , we have $\\begin{split}&C_{1,11}=E_1(C_{1,1})=(n+2)C_1C_{1,1}-nC_1^3-\\frac{R}{n-1}C_1,\\\\&C_{\\alpha ,\\alpha 1}=E_1(C_{\\alpha ,\\alpha })=-2C_1C_{1,1},~~C_{1,\\alpha \\alpha }=C_{\\alpha ,1\\alpha }=-(C_{1,1}+C_1^2)C_1.\\end{split}$ Except these coefficients $C_{1,11}, C_{1,\\alpha \\alpha }$ and $C_{\\alpha ,\\alpha 1}$ the coefficients of $\\nabla ^2C$ are equal to zero.",
"Since $(A_{ij})=diag\\lbrace a_1,a_2,\\cdots ,a_2\\rbrace $ under the local orthonormal basis, we have $A_{\\alpha \\alpha ,1}=E_1(A_{\\alpha \\alpha })=E_1(a_2)=-\\frac{1}{n-2}(C_{1,11}-2C_1C_{1,1})$ by Lemma REF , combining the first equation in (REF ), we get the first equation in (REF ).",
"By the equation (REF ) and the equation (REF ), $(C_{i,j})=diag\\big (C_{1,1}, -C_1^2,\\cdots ,-C_1^2\\big )$ under the local orthonormal basis, thus we have $C_{\\alpha ,\\alpha 1}=E_1(C_{\\alpha ,\\alpha })=-2C_1C_{1,1},~~C_{1,\\alpha \\alpha }=C_{\\alpha ,1\\alpha }=-(C_{1,1}+C_1^2)C_1.$ And the rest coefficients of $\\nabla ^2C$ are zero.",
"Since the hypersurface is conformally flat, the Schouten tensor $S=\\sum _{ij}S_{ij}\\omega _i\\otimes \\omega _j$ is a Codazzi tensor (i.e., $S_{ij,k}=S_{ik,j}$ ), which defined by $S_{ij}=R_{ij}-\\frac{R}{2(n-1)}\\delta _{ij}.$ Noting that the scaler curvature $R$ is constant, $tr(A)$ and $tr(S)$ are constant by the equation (REF ).",
"Furthermore, we have $\\sum _jA_{ij,j}=-\\sum _jB_{ij}C_j,~~~\\sum _jS_{ij,j}=0.$ Under the local orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ in Lemma REF , we have $(S_{ij})=diag(S_1,S_2,\\cdots ,S_2),$ $S_1=-\\frac{(2n-1)(n-2)}{2n^2}+(n-2)a_1,~~~~S_2=\\frac{n-2}{2n}+(n-2)a_2.$ Thus we have $\\begin{split}&S_{\\alpha \\alpha ,1}=S_{1\\alpha ,\\alpha }=(n-2)A_{\\alpha \\alpha ,1}=\\frac{R}{(n-1)}C_1-n(C_1C_{1,1}-C_1^3),\\\\&S_{11,1}=-(n-1)(n-2)A_{\\alpha \\alpha ,1}=-RC_1+n(n-1)(C_1C_{1,1}-C_1^3).\\end{split}$ Lemma 4.3 Let $f: M^n\\rightarrow \\mathbb {S}^{n+1}$ $(n\\ge 4)$ be a conformally flat hypersurface without umbilical points everywhere.",
"If the Möbius scalar curvature $R$ is constant.",
"then under the local orthonormal basis $\\lbrace E_1,\\cdots ,E_n\\rbrace $ in Lemma REF , the coefficients of $\\nabla ^2S$ satisfy $\\begin{split}&S_{11,11}=-RC_{1,1}+n(n-1)C_{1,1}^2+n(n-1)^2C_1^2C_{1,1}-n^2(n-1)C_1^4-nRC_1^2,\\\\&S_{11,\\alpha \\alpha }=\\frac{(n+1)R}{n-1}C_1^2-n(n+1)[C_1^2C_{1,1}-C_1^4],~~S_{\\alpha \\alpha ,11}=-(n-1)S_{11,11},\\\\&S_{\\alpha \\alpha ,\\alpha \\alpha }=3S_{\\alpha \\alpha ,\\beta \\beta }=3\\lbrace \\frac{-R}{n-1}C_1^2+n[C_1^2C_{1,1}-C_1^4]\\rbrace ,~~\\alpha \\ne \\beta .\\end{split}$ Since $(S_{ij})=diag(S_1,S_2,\\cdots ,S_2),$ we know that except these coefficients $S_{11,1},$ $ S_{1\\alpha ,\\alpha }$ and $S_{\\alpha \\alpha ,1}$ the coefficients of $\\nabla S$ are equal to zero.",
"Using the definition of the second covariant derivative of $S$ , we can compute these equations in (REF ).",
"Since $E_1$ is principal vector corresponding the eigenvalue $\\frac{n-1}{n}$ of the Möbius second fundamental form $B$ , the $C_1=C(E_1)$ , $C_{1,1}=\\nabla C(E_1,E_1)$ are well-defined functions on $M^n$ up to a sign.",
"Lemma 4.4 Let $f: M^n\\rightarrow \\mathbb {S}^{n+1}$ $(n\\ge 4)$ be a compact conformally flat hypersurface without umbilical points everywhere.",
"If the Möbius scalar curvature $R$ is constant.",
"then $\\begin{split}&\\int _{M^n}C_1^2C_{1,1}dV_g=\\frac{n-1}{3}\\int _{M^n}|C|^4dV_g,\\\\&\\int _{M^n}C_{1,1}^2dV_g=\\int _{M^n}|C|^4dV_g+\\frac{R}{n-1}\\int _{M^n}|C|^2dV_g.\\end{split}$ Using the coefficients of the tensor $C$ and $S$ , we define two smooth vector fields $X_S=\\sum _{ij}C_iS_{ij}E_j, ~~X_C=\\sum _{ij}C_iE_i.$ From Lemma REF and the equation (REF ), we can get the divergence of $X_S, X_C$ , $\\begin{split}&div X_C=\\sum _iC_{i,i}=C_{1,1}-(n-1)C_1^2,\\\\&div X_S=(n-1)(C_{1,1}^2-C_1^4-\\frac{R}{n-1}C_1^2)+\\frac{R}{2(n-1)}div X_C.\\end{split}$ Since the hypersurface is compact, we have $\\begin{split}&\\int _{M^n}C_{1,1}dV_g=(n-1)\\int _{M^n}|C|^2dV_g,\\\\&\\int _{M^n}C_{1,1}^2dV_g=\\int _{M^n}|C|^4dV_g+\\frac{R}{n-1}\\int _{M^n}|C|^2dV_g,\\end{split}$ On the other hand, we compute $\\triangle |C|^2$ , $\\begin{split}&\\triangle |C|^2=\\sum _i(E_iE_i-\\nabla _{E_i}E_i) |C|^2=\\sum _i(E_iE_i-\\nabla _{E_i}E_i)C_1^2\\\\&=E_1E_1(C_1^2)-\\sum _i\\nabla _{E_i}E_i(C_1^2)=2C_{1,1}^2+2C_1C_{1,11}-2(n-1)C_1^2C_{1,1}\\\\&=2C_{1,1}^2+6C_1^2C_{1,1}-2nC_1^4-\\frac{2R}{n-1}C_1^2.\\end{split}$ Since the hypersurface is compact, we have $\\int _{M^n}C_{1,1}^2dV_g+3\\int _{M^n}C_1^2C_{1,1}dV_g-n\\int _{M^n}C_1^4dV_g-\\frac{R}{n-1}\\int _{M^n}|C|^2dV_g=0.$ Combining the equation (REF ), we can derive the first equation in (REF ).",
"Lemma 4.5 Let $f: M^n\\rightarrow \\mathbb {S}^{n+1}$ $(n\\ge 4)$ be a compact conformally flat hypersurface without umbilical points everywhere.",
"If the Möbius scalar curvature $R$ is constant.",
"then $\\int _{M^n}\\big \\lbrace C_{1,1}^3+(n+5)C_1^2C_{1,1}^2-(2n+5)C_1^4C_{1,1}+(n-1)C_1^6-\\frac{2R}{3}C_1^4\\big \\rbrace dV_g=0.$ Using the coefficients of the tensor $C$ and $S$ , we define a smooth vector field $Y_S=\\sum _{ijk}C_{i,j}S_{ij,k}E_k.$ Using Lemma REF and the equation (REF ), we compute the divergence of $Y_S$ , $\\begin{split}&\\frac{1}{n(n-1)}div Y_S=\\sum _{ijk}C_{i,jk}S_{ij,k}+\\sum _{ijk}C_{i,j}S_{ij,kk},\\\\&=C_{1,1}^3+(n+5)C_1^2C_{1,1}^2-(2n+5)C_1^4C_{1,1}+(n-1)C_1^6+\\frac{(2n-1)R}{n(n-1)}C_1^4\\\\&-\\frac{R}{n(n-1)}C_{1,1}^2-\\frac{2(n+3)R}{n(n-1)}C_1^2C_{1,1}+\\frac{R^2}{n(n-1)^2}C_1^2.\\end{split}$ Integrating this equation and using (REF ), we can derive the second equation in (REF ).",
"Lemma 4.6 Let $f: M^n\\rightarrow \\mathbb {S}^{n+1}$ $(n\\ge 4)$ be a compact conformally flat hypersurface without umbilical points everywhere.",
"If the Möbius scalar curvature $R$ is constant.",
"then $\\begin{split}&\\int _{M^n}\\big \\lbrace C_1^2C_{1,1}^2+C_1^4C_{1,1}-\\frac{n}{3}C_1^6-\\frac{R}{3(n-1)}C_1^4\\big \\rbrace dV_g=0,\\\\&\\int _{M^n}\\big \\lbrace C_{1,1}^3+(n-1)C_1^2C_{1,1}^2-(2n+1)C_1^4C_{1,1}+(n+1)C_1^6+\\frac{2(n+1)R}{3(n-2)}C_1^4\\big \\rbrace dV_g=0.\\end{split}$ Using (REF ), $\\begin{split}\\frac{1}{4}\\triangle |C|^4&=\\frac{1}{4}\\sum _i(E_iE_i-\\nabla E_i{E_i})|C|^4=3C_1^2C_{1,1}^2+C_1^3C_{1,11}-(n-1)C_1^4C_{1,1}\\\\&=3C_1^2C_{1,1}^2+3C_1^4C_{1,1}-nC_1^6-\\frac{R}{n-1}C_1^4.\\end{split}$ Since the hypersurface is compact, then $\\int _{M^n}\\lbrace 3C_1^2C_{1,1}^2+3C_1^4C_{1,1}-nC_1^6-\\frac{R}{n-1}C_1^4\\rbrace dV_g=0.$ Since $S_{ij}$ is a Codazzi tensor and $tr(S)$ is constant, we can compute $\\triangle |S|^2$ by (REF ), $\\begin{split}&\\frac{1}{n^2(n-1)}\\triangle |S|^2=\\frac{1}{2n^2(n-1)}\\lbrace \\sum _{ijk}|S_{ij,k}|^2+\\frac{1}{2}\\sum _{ij}(S_i-S_j)^2R_{ijij}\\rbrace \\\\&=C_{1,1}^3-(2n+1)C_1^4C_{1,1}+(n-1)C_1^2C_{1,1}^2-\\frac{2R}{n-1}C_1^2C_{1,1}-\\frac{2R}{n(n-1)}C_{1,1}^2\\\\&+(n+1)C_1^6+\\frac{2(n+1)R}{n(n-1)}C_1^4+\\frac{(n+1)R^2}{n^2(n-1)^2}C_1^2+\\frac{R^2}{n^2(n-1)^2}C_{1,1}.\\end{split}$ Integrating this equation and using (REF ), we can derive the second equation in (REF ).",
"Lemma 4.7 Let $f: M^n\\rightarrow \\mathbb {S}^{n+1}$ $(n\\ge 4)$ be a compact conformally flat hypersurface without umbilical points everywhere.",
"If the Möbius scalar curvature $R$ is constant.",
"then $\\begin{split}&\\int _{M^n}\\big \\lbrace C_{1,1}^3+\\frac{5n+4}{2}C_1^2C_{1,1}^2-3(n+1)C_1^4C_{1,1}+\\frac{n}{2}C_1^6-\\frac{(7n-10)R}{3(n-2)}C_1^4\\big \\rbrace dV_g=0£¬\\\\&\\int _{M^n}\\big \\lbrace C_{1,1}^3+\\frac{5n+16}{2}C_1^2C_{1,1}^2-3(n-1)C_1^4C_{1,1}-\\frac{3n}{2}C_1^6-\\frac{(7n+2)R}{6(n-1)}C_1^4\\big \\rbrace dV_g=0.\\end{split}$ Using the coefficients of the tensor $C$ and $A$ , we have two following smooth functions, $|C|^2_A=\\sum _{ij}C_iA_{ij}C_j=C_1^2a_1, ~~|C|^2_C=\\sum _{ij}C_iC_{i,j}C_j=C_1^2C_{1,1}.$ Next we compute $\\triangle (|C|^2_A)$ and $\\triangle (|C|^2_C)$ .",
"$\\begin{split}&\\frac{n-2}{2(n-1)}\\triangle (|C|^2_A)=\\frac{n-2}{2(n-1)}(E_1E_1(C_1^2a_1)-(n-1)C_1E_1(C_1^2a_1))\\\\&=C_{1,1}^3-3(n+1)C_1^4C_{1,1}+\\frac{(5n+4)}{2}C_1^2C_{1,1}^2+\\frac{n}{2}C_1^6\\\\&+\\frac{n-2}{2(n-1)}[\\frac{2n-1}{n^2}-\\frac{R}{(n-1)(n-2)}]C_{1,1}^2-\\frac{n-2}{2(n-1)}[\\frac{2n-1}{n}-\\frac{(2n-1)R}{(n-1)(n-2)}]C_1^4\\\\&\\frac{n-2}{2(n-1)}[\\frac{3(2n-1)}{n^2}-\\frac{(7n-4)R}{(n-1)(n-2)}]C_1^2C_{1,1}\\\\&-\\frac{n-2}{2(n-1)}[\\frac{2n-1}{n^2}-\\frac{R}{(n-1)(n-2)}]\\frac{R}{n-1}C_1^2.\\end{split}$ Integrating this equation and using (REF ), we can derive the first equation in (REF ).",
"$\\begin{split}&\\frac{1}{2}\\triangle (|C|^2_C)=\\frac{1}{2}(E_1E_1(C_1^2C_{1,1})-(n-1)C_1E_1(C_1^2C_{1,1}))\\\\&=C_{1,1}^3-3(n-1)C_1^4C_{1,1}+\\frac{(5n+16)}{2}C_1^2C_{1,1}^2-\\frac{3n}{2}C_1^6\\\\&-\\frac{7R}{2(n-1)}C_1^2C_{1,1}-\\frac{3R}{2(n-1)}C_1^4.\\end{split}$ Integrating this equation and using (REF ), we can derive the second equation in (REF ).",
"Now we combine these equation system in (REF ), (REF ) and (REF ), we can derive that $\\int _{M^n}C_1^4dV_g=0,$ which implies that $C_1=0$ and the Möbius form vanishes.",
"Thus we finish the proof of Theorem REF ."
]
] | 1709.01658 |
[
[
"Entropy Control Architectures for Next-Generation Supercomputers"
],
[
"Abstract Position paper for US strategic computing initiative."
],
[
"Progress in high-performance computing (HPC) fundamentally requires effective thermal dissipation.",
"At present this challenge is viewed in terms of two distinct architectural components: a computational substrate and a heat-exchange substrate.",
"Typically these two components are viewed as performing conceptually different functions that are independent up to optimization constraints applied when they are jointly implemented.",
"Next-generation computing architectures, by contrast, must be designed from the perspective of computing as a thermodynamic process that can be managed in such a way as to actively control when and where heat is generated.",
"The fundamental connection between computation and thermodynamics is not new.",
"It dates back to Landaur's principle from 1961 [1], which says that a computational process is thermodynamically neutral until information is destroyed.",
"More specifically, what is thought of as an erasure of a bit is actually a process by which information is irreversibly transformed to heat.",
"According to Landaur's principle a fully reversible computational process will not produce heat because no information is destroyed (if information were destroyed then by definition the process could not be reversed).",
"Reversible logic gates have been developed and studied [2], [3], especially recently in the context of quantum computing, and are designed to physically store information that would otherwise be lost.",
"Said another way, they store the ancillary information needed for a conventional logic operation to be undone/reversed.",
"The limitation of reversible computing is, of course, that the amount of stored information will tend to increase without bound during execution of any nontrivial algorithm.",
"What is important to notice is that this ancillary information can be destroyed – i.e., converted to heat – or not at the discretion of the system.",
"Beyond simple erase or don't-erase discretion, the ancillary information retained to allow reversibility can, at the discretion of the system, be transported like any other information from one physical memory location to another.",
"In principle, therefore, information can be scheduled for erasure and then be transported freely to physical locations where it can be converted to heat (erased) upon arrival with minimal impact on the ongoing computational process.",
"To summarize, conventional computing hardware generates heat as a continuous by-product of the operation of logic gates during execution of a program.",
"The precise distribution of heat is therefore a function of the particular algorithm and is not generally knowable a priori, thus the heat-dissipation substrate must be designed under an assumption that pernicious heat accumulation (spikes) can occur anywhere across the physical area in which logic operations may be performed.",
"This need to uniformly maintain separate computation and heatsink functionality within the same physical space is why heat dissipation continues to limit the density and/or frequency of logic operations.",
"An entropy-controlled computing architecture (ECCA) would permit the time and location at which heat is generated to be actively controlled during the computational process rather than letting it be indiscriminately generated at gate locations with an expectation that a separate physical system will soak it up.",
"In the ECCA model the ancillary bits that are conventionally thought of as being stored as part of the operation of each reversible logic gate are now treated as bits of unneeded information that must be taken away in a manner somewhat analogous to garbage collection.",
"More specifically, an unneeded bit could be sent without erasure to a nearby unused memory location or it could be sent to a location at which there is available capacity to accommodate the heat resulting from its erasure.",
"Continuing with the analogy to garbage collection for memory management, the ECCA processing of a given program could progress without heat generation until storage for waste bits has been exhausted, at which point the computational process could be halted until the bits are erased in batch and the resulting heat is dissipated.",
"Alternatively, waste bits could be incrementally transported for erasure at locations selected so that heat is generated uniformly across a physical area and can dissipate without localized accumulation.",
"Batch-mode ECCA is potentially attractive for space-based applications in which heat distribution is easier to monitor than to control and the objective is to maximize the average rate of computation.",
"For example, a satellite-based computing system could potentially afford to accumulate waste bits during periods of sun exposure and free them later during periods of shade.",
"In applications for which a fixed rate of computation is required, e.g., for continuous real-time dynamical control, incremental-mode ECCA is needed.",
"More generally, however, both modes can be applied jointly: batch-mode to exploit a known model for the availability (or absence) of heatsink resources and incremental-mode to actively optimize the management of waste bits with respect to dynamically monitored parameters of the system (including its environment).",
"The ECCA model is most applicable to supercomputing because the rate of computation is fundamentally limited by the speed of light, so any physical separation of logic gates to accommodate heat dissipation increases the time required for information to propagate between them.",
"Increasing the density of gates greatly increases the distribution volatility of heat generation, i.e., the magnitude and frequency of localized heat-spikes.",
"Current technologies for non-specific bulk heat dissipation using static heat sinks or dynamic fluid transport cannot be scaled to the decreasing length-scales needed to maintain the current rate of improvement in supercomputing power.",
"Eventually a coherent solution to the compute-vs-heat problem must be applied.",
"Developing a fully-functional ECCA solution requires physical gates capable of creating ancillary bits like standard reversible logic gates but with means for those bits to be accessed and transported for heat management.",
"Algorithms to exploit this available means for effectively managing the incremental and/or batch erasure of waste bits are clearly also needed.",
"Fortunately, neither the hardware nor software aspects of the proposed solution involve any potentially fundamental limiting theoretical or practical obstacles, so the case for moving quickly toward this next generation of supercomputing architectures is strong."
]
] | 1709.01570 |
[
[
"Ages and Age Spreads in Young Stellar Clusters"
],
[
"Abstract I review progress towards understanding the time-scales of star and cluster formation and of the absolute ages of young stars.",
"I focus in particular on the areas in which Francesco Palla made highly significant contributions - interpretation of the Hertzsprung-Russell diagrams of young clusters and the role of photospheric lithium as an age diagnostic."
],
[
"Introduction",
"Estimating the absolute ages of young stars and ascertaining the extent of age spreads in young clusters is crucial in understanding the mechanisms and timescales upon which stars form, upon which circumstellar disks disperse and planetary systems assemble, and for understanding the role of varying stellar birth environments on these issues.",
"Francesco Palla produced highly influential work in these areas; my review focuses on two key aspects: (i) the interpretation of the Hertzsprung-Russell diagrams (HRDs) and colour-magnitude diagrams (CMDs) of young clusters and star forming regions [27], [28], [29], and (ii) the use of photospheric lithium abundance measurements as an orthogonal method to estimate and calibrate young stellar ages [25], [26], [32]."
],
[
"Ages from H-R diagrams",
"Low-mass stars ($\\le 2M_{\\odot }$ ) take significant time ($\\sim 10$ –200 Myr) to evolve from newly revealed T-Tauri stars to the zero-age-main-sequence (ZAMS).",
"This pre-main-sequence (PMS) evolution occurs on mass-dependent timescales (faster for higher mass stars); stars initially descend fully convective Hayashi tracks followed by, for higher mass objects ($\\ge 0.4M_{\\odot }$ ), the development of radiative cores and a blueward traverse along the Henyey track before settling onto the ZAMS [14].",
"In principle, the construction of grids of mass-dependent evolutionary tracks and corresponding isochrones in the HRD can be used with estimates of luminosity and effective temperature ($T_{\\rm eff}$ ) or equivalently (given appropriate bolometric corrections), absolute magnitude and colour, to yield ages and masses for PMS stars.",
"An advantage to using low-mass stars when studying young clusters, rather than their higher mass siblings, is they are much more populous, allowing statistical analyses, and their movement in the HRD can be much larger for a given age change.",
"In a series of papers, Francesco (and Steven Stahler) noted that, when plotted on the HRD, stars are dispersed around the single isochrones predicted by PMS models.",
"This indicated a substantial age spread of at least a few Myr, and in some cases $>10$ Myr.",
"The pattern was repeated in several young clusters and when ages inferred from HRD position were turned into a star formation history, suggested an accelerating star formation rate as a function of (linear) time.",
"This highly-cited result has launched a thousand telescope proposals and is still hotly debated.",
"Does an extended star formation history indicate inefficient star formation moderated by turbulence and magnetic fields, or can the spreads be explained by observational uncertainties and problems with PMS models so that actually, star formation is rapid and efficient, taking place on dynamical timescales?",
"Opponents of the idea of large age spreads have pointed to the role of astrophysical effects and observational uncertainties in scattering stars in the HRD, giving the impression of a large age dispersion.",
"[13] noted that the apparent age distribution was lognormal, with $\\sigma \\sim 0.4$ dex, perhaps reflecting the logarithmic nature of uncertainties in luminosity estimates and that age $\\propto L^{-3/2}$ on Hayashi tracks.",
"There are uncertainties in distance, extinction, and also due to intrinsic variability, accretion and the presence of binaries that must certainly be accounted for in estimating a true age dispersion.",
"Detailed simulations by [31] and [30] concluded that whilst these effects were important, they probably do not explain the entire extent of observed dispersions.",
"It seems certain that the very old ages assigned to at least some PMS stars in young clusters are due to mis-estimated luminosities and temperatures associated with an incorrect or at least incomplete treatment of extinction and accretion [21].",
"On the other hand, support for genuine dispersions in luminosity (or radius at a given $T_{\\rm eff}$ ) has been found by considering the distribution of projected radii (rotation period multiplied by projected rotation velocity) in the Orion Nebula cluster (ONC) and IC 348 [16], [7] and from the IN-SYNC APOGEE survey that finds a significant correlation between increasing age and spectroscopic gravity in the same clusters [7], [8].",
"There seems little doubt that a fraction of the observed age dispersion must be due to sources of astrophysical and observational uncertainty, but also strong evidence that at least some of the luminosity and radius spread is real.",
"Whether this implies genuine age spreads requires evidence from other observations and independent astrophysics."
],
[
"Lithium as an age indicator",
"Lithium is ephemeral in low-mass stellar photospheres.",
"As PMS stars contract, their cores reach Li-burning temperatures before reaching the ZAMS.",
"If the convection zone base is also above the Li-burning temperature (which it would be in fully convective stars) then photospheric Li is also depleted on timescales less than a few Myr.",
"The age at which core Li burning begins is mass-dependent (later for lower mass stars), but the development of a radiative core can arrest photospheric Li depletion in more massive objects.",
"These phenomena lead to a complex, but age-dependent, behaviour for Li abundance as a function of luminosity, $T_{\\rm eff}$ or colour.",
"Palla et al.",
"(2005, 2007) were among the first to suggest Li depletion could serve as an independent test of ages in very young low-mass stars.",
"Li depletion is expected to begin in stars of $\\sim 0.5M_{\\odot }$ at an age of about 5 Myr and subsequently develops at higher and lower masses.",
"Since the physics of Li depletion is comparatively simple, it has been argued that this currently provides the least model-dependent means of estimating young stellar ages [34], however masses cannot be measured directly so one relies on colours, $T_{\\rm eff}$ or (better) luminosities as proxies.",
"Palla et al.",
"(2005) and Sacco et al.",
"(2007) found examples of Li-depleted low-mass stars that appeared older than 10 Myr in the ONC and the $\\sigma $ Ori cluster, and much older than the bulk of their siblings, perhaps supporting the notion of large age spreads $>10$ Myr.",
"Subsequent work by [33] on the ONC and NGC 2264 confirmed the presence of a dispersion in Li abundance, but noted the difficulty in assessing Li abundances for PMS stars that are often accreting.",
"Any veiling continuum weakens the Li i 6708Å line that is exclusively used; this combined with the saturated nature of this strong resonance line can lead to the mistaken inference of significant Li depletion.",
"[19] took the expedient option of excluding stars with signs of accretion from their analysis (which one might presume were younger stars), still finding evidence for some age dispersion in NGC 2264, but with an absolute value $\\le 4$ Myr and smaller than the spread implied by the HRD.",
"Taken at face value, the combined information from Li depletion, the HRD and spectroscopic indicators of radii suggests that some dispersion in age is present, but probably no more than a few Myr and not as much as suggested by the HRD alone.",
"However, there are problems that have emerged even with this simple interpretation that may betray interesting facets of PMS evolution that have yet to be correctly incorporated."
],
[
"Problems with evolutionary models",
"(i) Why is Li depletion correlated with rotation?",
"That rapidly rotating low-mass stars appear to preserve their Li longer, has been established in older clusters and becomes clearer with better data [4].",
"This trend is now becoming apparent at even younger ages and may be responsible for some of the Li depletion dispersion previously claimed to be associated with an age spread [6].",
"Since PMS stars are expected to spin-up as they contract, then older stars ought to be faster rotating and more Li depleted if the age dispersion were genuine.",
"(ii) Why do Li-depletion ages disagree with isochronal ages from the HRD?",
"[17] have pointed out that Li depletion ages and HRD/CMD ages are not completely independent; Li depletion takes places when the core temperature, and hence mass to radius ratio, reaches a certain threshold, whilst HRD/CMD ages also depend on radius at a given $T_{\\rm eff}$ , though not as sensitively.",
"The same evolutionary models give significantly younger ages for low-mass PMS stars in the $\\gamma ^2$ Velorum cluster than implied by the strong Li depletion seen in its M-dwarfs.",
"The Li depletion also takes place at much redder colours and lower inferred $T_{\\rm eff}$ than expected.",
"The CMD and Li-depletion pattern cannot be explained simultaneously by any commonly used evolutionary codes at any age.",
"(iii) Why are more massive stars in young clusters judged to be older than the low-mass stars?",
"The ages of clusters with PMS stars can also be estimated by looking at how far from the ZAMS towards the terminal-age main-sequence their high mass ($>5M_{\\odot }$ ) stars have progressed.",
"When done with a self-consistent and accurate treatment of reddening [24] suggested that the high-mass stars were significantly older than their low-mass siblings by a factor of two.",
"This was followed-up with a larger sample by [5], who demonstrated that the low-mass ages could be brought into agreement with the high-mass ages (and ages from Li depletion) with systematic changes in the bolometric corrections adopted by the models.",
"(iv) Why do current models fail to correctly predict the location of PMS eclipsing binary components in the HRD?",
"New examples found in star forming regions provide challenges to evolutionary models.",
"Their masses and radii are not well predicted from their estimated luminosities and $T_{\\rm eff}$ [18], [9].",
"The PMS binary components appear colder than predicted by the models and more luminous than predicted at the age of higher mass stars in the same clusters.",
"These problems have lead to consideration of whether PMS evolutionary models are yielding the correct absolute masses, ages and hence age spreads at all.",
"An idea that has gained some traction is that episodic accretion during the first million years of a star's life can significantly influence both the HRD position and Li depletion [1], [3].",
"Variations in accretion rate and the exact timing of accretion could lead to apparent age dispersions and to the occasional star appearing much older in the HRD and/or exhibiting significant Li depletion.",
"An alternative that is also gaining support is that magnetic activity may “inflate” low-mass stars (or at least slow their contraction), either through magnetic inhibition of convection [23], [11] or the blocking of radiative flux by cool starspots [15], [35].",
"These ideas have the attraction that we know young low-mass stars are magnetically active and that they have extensive starspot coverage [12].",
"Let us suppose then that active low-mass PMS stars are inflated by $\\sim 10$ % compared to the predictions of “standard” evolutionary models at a given mass and age.",
"This is roughly the level suggested by recent modelling work that attempts to incorporate the effects of suppressed convection or starspots.",
"Jeffries et al.",
"(2017) [10], [22] have shown that such stars become cooler and only slightly less luminous.",
"The net result is that stars move almost horizontally in the HRD resulting in severely underestimated ages and masses when using “standard” models (see Fig. 1).",
"At the same time their core temperatures are reduced, delaying the onset of Li depletion and decreasing the $T_{\\rm eff}$ of stars in which Li depletion is first seen.",
"If magnetic models such as those of Jackson & Jeffries (2014), [36] or Feiden (2016) are adopted, then HRD/CMD ages are brought into much closer agreement with the Li depletion ages, but at the expense of doubling the ages inferred from the HRD (see Fig. 1).",
"This also brings ages from low-mass and high-mass stars into broad agreement, potentially solves the problems with eclipsing binary parameters [20] and could introduce a dispersion into the HRD and Li-depletion patterns of young stars that is correlated with rotation and/or magnetic activity [36].",
"If correct, such a large shift has considerable implications for the timescales of PMS evolution, the dispersal of circumstellar matter and hence the time available to form planetary systems, all of which are keyed-in to the absolute timescales set by age estimates for young, low-mass stars."
],
[
"Summary",
"The investigation of ages and age spreads in young clusters using the HRD and Li-depletion, begun by Francesco Palla and colleagues, remains a vibrant and controversial topic.",
"Current evidence suggests that age spreads are a lot smaller than 10 Myr (within a single cluster), but that not all the dispersion in cluster HRD/CMDs and Li depletion can be explained by observational and astrophysical uncertainties.",
"Some of the observed spread does appear to be due to a genuine distribution of radius among stars with similar $T_{\\rm eff}$ and mass, which might be attributable to a modest age spread of a few Myr.",
"We are now moving into an era of more sophisticated stellar modelling that questions the veracity of both the absolute ages of PMS stars and the inferred age spreads in young star forming regions."
]
] | 1709.01736 |
[
[
"Systematic corrections to the Thomas-Fermi approximation without a\n gradient expansion"
],
[
"Abstract We improve on the Thomas-Fermi approximation for the single-particle density of fermions by introducing inhomogeneity corrections.",
"Rather than invoking a gradient expansion, we relate the density to the unitary evolution operator for the given effective potential energy and approximate this operator by a Suzuki-Trotter factorization.",
"This yields a hierarchy of approximations, one for each approximate factorization.",
"For the purpose of a first benchmarking, we examine the approximate densities for a few cases with known exact densities and observe a very satisfactory, and encouraging, performance.",
"As a bonus, we also obtain a simple fourth-order leapfrog algorithm for the symplectic integration of classical equations of motion."
],
[
"Introduction",
"All practical applications of density-functional theory (DFT) to systems of interacting particles require trustworthy approximations to the functionals for the kinetic energy and the interaction energy or—more relevant for the set of equations that one needs to solve self-consistently—to their functional derivatives.",
"While the Kohn–Sham (KS) scheme avoids approximations for the kinetic-energy functional (KEF), this comes at the high price of a CPU-costly solution of the eigenvalue-eigenstate problem for the effective single-particle Hamiltonian.",
"The popular alternative to KS-DFT proceeds from the KEF in Thomas–Fermi (TF) approximation , and improves on that by the inclusion of inhomogeneity corrections in the form of gradient terms, with the von Weizsäcker term as the leading correction.",
"This gradient expansion is notorious for its lack of convergence (see, e.g., ), the wrong sign of the von Weizsäcker term for one-dimensional systems , and the vanishing of all corrections for two-dimensional systems , , —or so it seems .",
"While cures have been suggested, such as the use of a Padé approximant rather than the power series (see, e.g., ), or the partial re-summation of the series with the aid of Airy-averaging techniques , , , , , the situation is hardly satisfactory.",
"Recently, however, Ribeiro et al.",
"demonstrated that one can improve very substantially on the TF approximation without any gradient expansion at all.",
"True, the method employed in and related papers , is designed for one-dimensional problems and has so far resisted all attempts of extending it to two and three-dimensional situations.",
"But this work strongly encourages the search for other approximation schemes that do not rely on gradient expansions and are not subject to the said limitations.",
"We are here reporting one scheme of this kind.",
"Our approach is based on the reformulation of DFT in which the effective single-particle potential energy $V({r})$ is a variational variable on equal footing with the single-particle density $n({r})$ , .",
"All functionals of $V({r})$ can be stated in terms of the effective single-particle Hamiltonian, and we relate them to the corresponding unitary evolution operator.",
"That, in turn, is then systematically approximated by products of simpler unitary operators of the Suzuki–Trotter (ST) kind—known as split-operator approximations; see, e.g., .",
"There is, in particular, a relatively simple five-factor approximation that is correct to fourth order.",
"It promises a vast improvement over the TF approximation without the high costs of the KS method and without the dimensional limitation of the Ribeiro et al.",
"method.",
"The unitary ST approximations directly provide symplectic approximants for classical time evolution.",
"The here developed higher-order ST approximations therefore have the potential to significantly improve upon standard second-order leapfrog algorithms , or even fourth-order Runge–Kutta methods."
],
[
"Single-particle density and evolution operator",
"We consider a system of $N$ unpolarized spin-$\\frac{1}{2}$ fermions of mass $m$ , subject to external forces that derive from the potential energy $V_{\\rm {ext}}({r})$ .",
"The energy functional $\\rule {3em}{0pt}E[V,n,\\mu ]=E_1[V-\\mu ]-\\int ({r})\\,[V({r})-V_{\\mathrm {ext}}({r})]\\,n({r})+E_{\\mathrm {int}}[n]+\\mu N\\,,$ which has $n({r})$ and $V({r})$ as well as the chemical potential $\\mu $ as variables, is stationary at the actual values.",
"Therefore, we have the three variational equations $\\makebox{[}5em][s]{{\\delta V:} n({r})}&=&\\frac{\\delta }{\\delta V({r})}E_1[V-\\mu ]\\,,\\\\\\makebox{[}5em][s]{{\\delta n:} V({r})}&=&V_{\\mathrm {ext}}({r})+\\frac{\\delta }{\\delta n({r})}E_{\\mathrm {int}}[n]\\,,\\\\\\makebox{[}5em][s]{{\\delta \\mu :} N}&=&-\\frac{\\partial }{\\partial \\mu }E_1[V-\\mu ]\\,,$ which we need to solve jointly.",
"Here, $E_{\\mathrm {int}}[n]$ is the standard interaction-energy functional (IEF) of the Hohenberg–Kohn (HK) theorem , and the single-particle energy functional $E_1[V-\\mu ]$ is related to the HK-KEF by a Legendre transformation, $E_1[V-\\mu ]=E_{\\mathrm {kin}}[n]+ \\int ({r})\\,[V({r})-\\mu ]\\,n({r})\\,.$ Equation (REF ) yields $n({r})$ for given $V({r})$ and $\\mu $ , whereas we obtain $V({r})$ for given $n({r})$ from equation (); and the correct normalization of $n({r})$ to the particle number $N$ is ensured by equation () because it implies $\\int ({r})\\,n({r})=N$ when combined with equation (REF ).",
"In the KS formalism, we have $n({r})=2{\\left\\langle {{r}}\\right|}\\eta {\\bigl (\\mu -H{\\left({P},{R}\\right)}\\bigr )}{\\left|{{r}}\\right\\rangle }$ with the eigenbras ${\\left\\langle {{r}}\\right|}$ and eigenkets ${\\left|{{r}}\\right\\rangle }$ of the position operator ${R}$ and the single-particle Hamiltonian $H({P},{R})=\\frac{1}{2m}{P}^2+V({R})\\,.$ Here and throughout the paper, the upper-case letters ${P},{R}$ denote the quantum mechanical momentum and position operators, whereas the lower-case letters ${p},{r}$ stand for their classical counterparts.",
"We can either accept equation (REF ) as determining the right-hand side in equation (REF ), which is exact if we include the difference between the KEFs for interacting and non-interacting particles in the IEF, or we regard equation (REF ) as stating an approximation for the right-hand side in equation (REF )—an approximation that has a very good track record.",
"Whichever point of view we adopt, we shall have to deal with equation (REF ).",
"The task, then, is to evaluate, in a good approximation, the diagonal matrix element of the step function of $\\mu -H$ without computing the eigenvalues of $H$ and the corresponding single-particle wave functions (“orbitals”).",
"In the tradition of the TF approximation and its refinements by gradient terms, and also in the spirit of the Ribeiro et al.",
"work, we target an orbital-free (OF) formalism.",
"In a first step toward this goal, we relate $n({r})$ to the unitary evolution operator $\\exp {\\left(-\\frac{It}{\\hbar } H{\\left({P},{R}\\right)}\\right)}$ , $n({r})=2\\int \\limits _{\\begin{picture}(16,3)(-8,-3)\\put (0,0){\\curve (-3,0,-8,0)\\curve (3,0,8,0)}\\put (8,0){\\curve (0,0,-1.5,1.5)\\curve (0,0,-1.5,-1.5)}\\put (0,0){\\arc (-3,0){180}}\\put (0,0){\\makebox{(}0,0){\\cdot }}\\end{picture}}\\hspace*{-5.0pt}\\frac{t̥}{2\\pi It}\\,\\exp {\\left(\\frac{It}{\\hbar }\\mu \\right)}\\,{\\left\\langle {{r}}\\right|}\\exp {\\left(-\\frac{It}{\\hbar }H({P},{R})\\right)}{\\left|{{r}}\\right\\rangle }\\,,$ where the integration path from $t=-\\infty $ to $t=\\infty $ crosses the imaginary $t$ axis in the lower half-plane , , .",
"Therefore, an approximation for $\\exp {\\left(-\\frac{It}{\\hbar } H{\\left({P},{R}\\right)}\\right)}$ provides a corresponding approximation for $n({r})$ ."
]
] | 1709.01719 |
[
[
"Traveling Waves in the Euler-Heisenberg Electrodynamics"
],
[
"Abstract We examine the possibility of travelling wave solutions within the nonlinear Euler-Heisenberg electrodynamics.",
"Since this theory resembles in its form the electrodynamics in matter, it is a priori not clear if there exist travelling wave solutions with a new dispersion relation for $\\omega(k)$ or if the Euler-Heisenberg theory stringently imposes $\\omega=k$ for any arbitrary ansatz $\\mathbf{E}(\\xi)$ and $\\mathbf{B}(\\xi)$ with $\\xi \\equiv \\mathbf{k}\\cdot\\mathbf{r} -\\omega t$.",
"We show that the latter scheme applies for the Euler-Heisenberg theory, but point out the possibility of new solutions with $\\omega \\neq k$ if we go beyond the Euler-Heisenberg theory, allowing strong fields.",
"In case of the Euler-Heisenberg theory the quantum mechanical effect of the travelling wave solutions remains in $\\hbar$ corrections to the energy density and the Poynting vector."
],
[
"Introduction",
"In the presence of intense electromagnetic fields, Quantum Electrodynamics predicts that the vacuum behaves like a material medium.",
"This happens since starting from the one-loop level, light-light interaction becomes possible for even number of photons.",
"Due to this quantum effect, the linear Maxwell theory receives non-linear corrections.",
"If the electromagnetic field does not change too fast and the fields are below the so-called critical field $B{}_{c}=\\frac{m_{e}^{2}}{e}$ , then the lowest order quantum corrections to classical Electrodynamics are encoded in the Euler-Heisenberg Lagrangian [1], [2], [3], [4], [5] $ \\mathcal {L}_{EH}=a\\left(\\left(\\mathbf {E}^{2}-\\mathbf {B}^{2}\\right)^{2}+7\\left(\\mathbf {E\\cdot B}\\right)^{2}\\right),$ where $a=\\frac{e^{4}}{360\\pi ^{2}m_{e}^{4}}.$ The breakdown of linearity is predicted to give rise to plenty of new effects which do not exist in classical Electrodynamics in vacuum.",
"At the optical level the polarization dependent refractive index of the vacuum in the presence of a magnetic or electric field is calculated in [6].",
"Calculations related to the change of the polarization of a wave due to the birefringence of the vacuum can be found in [6], [7], [8], [9].",
"Other effects include vacuum dichroism [10], second harmonic generation [11], [12], [13], [14], parametric amplification [7], [15], quantum vacuum reflection[16], [17], slow light [18], photon acceleration in vacuum [19], pulse collapse [20], [21] and more (see [22], [23] for comprehensive reviews).",
"Examples of waves that are solutions to the Euler-Heisenberg equations but not to the classical Maxwell's equations are solitons [24], [25] and shockwaves [26], [27].",
"Both these solutions are not travelling waves.",
"Worth mentioning are new developments concerning the equation of motion for a test body with either a charged massive particle giving rise to corrections in the Lorentz force [28], or massless photons who now \"feel\" the presence of an electromagnetic field and mimic, in a certain sense, the motion of a massless particle in general relativity [29], [30], [31], [32], [33].",
"Such a self-interaction of the electromagnetic quanta or the interaction of the photon with the field raises the question “what is the role of a plane wave within such a theory” or, more generally, what the role of travelling waves is.",
"Comparing the non-linear Electrodynamics with general relativity, where plane waves as solutions exist only in the linearized version of the theory, it is a priori not clear as to what kind of travelling waves exist in the Euler-Heisenberg theory and what happens to the dispersion relation.",
"It is evident that solutions for which the two gauge invariants $\\mathbf {E}^2 -\\mathbf {B}^2$ and $\\mathbf {E}\\cdot \\mathbf {B}$ are zero, are also solutions of the Maxwell theory with $\\omega =k$ .",
"More generally, keeping $\\omega =k$ , the Maxwell solution itself allows for non-zero values of the gauge invariants.",
"The first question that we can put forward in such a context is whether these Maxwellian solutions are also solutions in the Euler-Heisenberg theory.",
"We will show that the answer is affirmative if we impose a restriction.",
"The second question of interest is if travelling wave solutions exist in the Euler-Heisenberg theory which have no connection to the Maxwellian case, i.e., waves with a new dispersion relation, $\\omega (k) \\ne k$ .",
"We present a lengthy proof demonstrating that the only travelling wave solutions in the Euler-Heisenberg theory are waves with $\\omega (k)=k$ , i.e., they are of Maxwellian type but with a restriction on the integration constants.",
"Interestingly, this result is not due to some physical principle which would exclude all other solutions.",
"From a purely mathematical point of view travelling waves exist with a new dispersion relation, but we have to reject them on physical grounds as in these solutions the strength of the fields exceeds the critical value allowed in the weak field approximation.",
"We touch upon the possibility that such a restriction can, in principle, be avoided by going beyond the Euler-Heisenberg theory.",
"As far as the Euler-Heisenberg theory is concerned, the physical effect of travelling wave solutions is a quantum mechanical contribution to the energy density of the waves of the Poynting vector.",
"The paper is organized as follows.",
"In section 2 we review in full generality the Maxwellian case allowing for non-zero integration constants.",
"In section 3 we recall the salient features of the Euler-Heisenberg theory.",
"In section 4 we present the algebraic equations of the Euler-Heisenberg theory with the traveling waves as an ansatz.",
"Section 5 probes into the existence of travelling wave solutions with $\\omega =k$ .",
"In the appendix we prove that this is the only viable case.",
"In section 6 we discuss a mathematically viable but physically not acceptable solution with $\\omega \\ne k$ .",
"We present the case in order to argue in section 7 that a more general Lagrangian allowing strong fields would make a similar and analog solution possible."
],
[
"Maxwell's travelling waves",
"The method of obtaining solutions in vacuum for the four Maxwell's equations of classical electrodynamics is well known.",
"It starts by taking the Maxwell's equations, four linear first order differential equations that involve the electric and magnetic fields, and combining them to form two waves equations, which are second order differential equations and then solving the wave equations.",
"The answer is given by fields of the form $\\mathbf {E} & = & \\mathbf {E}(\\xi ),\\\\\\mathbf {B} & = & \\mathbf {B}(\\xi ),$ with $\\xi \\equiv \\mathbf {k\\cdot r}-\\omega t.$ Waves with such a dependency on the space and time coordinates are called travelling waves.",
"In this paper we are interested in the travelling wave solutions in the Euler-Heisenberg electrodynamics.",
"In the Euler-Heisenberg case solving the wave equation is not the most useful approach to the problem.",
"As a preparation for the next section and for the sake of comparison, we present a different way to solve the Maxwell's equation in vacuum which does not make use of the wave equation.",
"The same approach will be used later on to deal with the Euler-Heisenberg equations.",
"The magnetic Gauss's, Faraday's, electric Gauss's and Ampere-Maxwell's laws for classical electrodynamics are $\\nabla \\cdot \\mathbf {B} & = & 0,\\\\\\nabla \\times \\mathbf {E} & = & -\\frac{\\partial \\mathbf {B}}{\\partial t},\\\\\\nabla \\cdot \\mathbf {E} & = & 0,\\\\\\nabla \\times \\mathbf {B} & = & \\frac{\\partial \\mathbf {E}}{\\partial t}.$ Using a travelling wave condition as an ansatz, we can write the Maxwell's equation as $\\mathbf {k}\\cdot \\frac{d\\mathbf {B}}{d\\xi } & = & 0,\\\\\\mathbf {k}\\times \\frac{d\\mathbf {E}}{d\\xi } & = & \\omega \\frac{d\\mathbf {B}}{d\\xi },\\\\\\mathbf {k}\\cdot \\frac{d\\mathbf {E}}{d\\xi } & = & 0,\\\\\\mathbf {k}\\times \\frac{d\\mathbf {B}}{d\\xi } & = & \\omega \\frac{d\\mathbf {E}}{d\\xi }.$ These equations can be directly integrated to give the following algebraic relations for the fields $\\mathbf {k}\\cdot \\mathbf {B} & = & C_{B},\\\\\\mathbf {B} & = & \\frac{\\mathbf {k}\\times \\mathbf {E}}{\\omega }+\\mathbf {d}_{B},\\\\\\mathbf {k}\\cdot \\mathbf {E} & = & C_{E},\\\\\\mathbf {E} & = & -\\frac{\\mathbf {k}\\times \\mathbf {B}}{\\omega }+\\mathbf {d}_{E}.$ where $C_{B}$ , $C_{E}$ , $\\mathbf {d}_{B}$ and $\\mathbf {d}_{E}$ are integration constants.",
"Multiplying equations (REF ) and () by $\\mathbf {k}\\cdot $ , we see these constants are not independent, but instead obey the relations $C_{B} & = & \\mathbf {k}\\cdot \\mathbf {d}_{B},\\\\C_{E} & = & \\mathbf {k}\\cdot \\mathbf {d}_{E}.$ To find further relations among the quantities involved, we now replace equation () into (REF ) $\\mathbf {B}=\\frac{\\mathbf {k}\\times }{\\omega }\\left(-\\frac{\\mathbf {k}\\times \\mathbf {B}}{\\omega }+\\mathbf {d}_{E}\\right)+\\mathbf {d}_{B},$ and after some rearranging of the terms we obtain $\\mathbf {B}(1-\\frac{k^{2}}{\\omega ^{2}})=-\\frac{C_{B}}{\\omega ^{2}}\\mathbf {k}+\\mathbf {d}_{B}+\\frac{\\mathbf {k}\\times \\mathbf {d_{E}}}{\\omega }.$ Similarly, we can replace equation (REF ) into equation () to obtain for the electric field $\\mathbf {E}(1-\\frac{k^{2}}{\\omega ^{2}})=-\\frac{C_{E}}{\\omega ^{2}}\\mathbf {k}+\\mathbf {d}_{B}-\\frac{\\mathbf {k}\\times \\mathbf {d_{B}}}{\\omega }.$ A similar algebraic equation will emerge in the Euler-Heisenberg theory when we make the travelling wave ansatz.",
"The right hand side of equations (REF ) and (REF ) are constants.",
"Therefore the only way these equations do not lead to trivial constant solutions is to have the well known dispersion relation for the classical travelling wave $k=\\omega $ .",
"In this way the equations (REF ) and (REF ) become algebraic equations that relate the constants which appear in the problem, namely $\\mathbf {d}_{B} & = & \\frac{C_{B}}{\\omega ^{2}}\\mathbf {k}-\\frac{\\mathbf {k}\\times \\mathbf {d_{E}}}{\\omega },\\\\\\mathbf {d}_{E} & = & \\frac{C_{E}}{\\omega ^{2}}\\mathbf {k}+\\frac{\\mathbf {k}\\times \\mathbf {d_{B}}}{\\omega }.$ Note that if $\\mathbf {d}_{B}=\\mathbf {d}_{E}=0$ , the equations (REF ) and () reduce to $\\mathbf {B} & = & \\mathbf {k}\\times \\mathbf {E},\\\\\\mathbf {E} & = & -\\mathbf {k}\\times \\mathbf {B},$ which is the well known result that $\\mathbf {k}$ and the undulatory parts of $\\mathbf {E}$ and $\\mathbf {B}$ form a right handed triplet of orthogonal vectors.",
"This fact together with the dispersion relations are the main results for the classical waves.",
"Finally, we want to find expressions for the quantities $\\mathbf {E\\cdot B}$ and $B^{2}-E^{2}$ , which are of great importance for the generalizations of classical electrodynamics.",
"The first one can be obtained by direct computation.",
"Multiplying (13) by $\\mathbf {E}$ or (15) by $\\mathbf {B}$ we get $\\mathbf {E\\cdot B}=\\mathbf {E\\cdot }\\mathbf {d_{B}}=\\mathbf {\\mathbf {d_{E}}\\cdot B}.$ For $B^{2}-E^{2}$ we can start by squaring equation (REF ) $B^{2} & = & \\left(\\frac{\\mathbf {k}\\times \\mathbf {E}}{\\omega }+\\mathbf {d}_{B}\\right)^{2}\\nonumber \\\\& = & E^{2}-\\frac{C_{E}}{\\omega ^{2}}+d_{B}^{2}-2\\mathbf {E}\\cdot \\left(\\widehat{\\mathbf {k}}\\times \\mathbf {d_{B}}\\right)\\nonumber \\\\& = & E^{2}+\\frac{C_{E}}{\\omega ^{2}}+d_{B}^{2}-\\mathbf {E\\cdot d_{E}},$ or we can square equation () to have $E^{2} & = & B^{2}-\\frac{C_{B}^{2}}{\\omega ^{2}}+d_{E}+2\\mathbf {B}\\cdot \\left(\\widehat{\\mathbf {k}}\\times \\mathbf {d_{E}}\\right)\\nonumber \\\\& = & B^{2}+\\frac{C_{B}^{2}}{\\omega ^{2}}+d_{E}^{2}-\\mathbf {B\\cdot d_{B}}.$ With this at hand we can write $B^{2}-E^{2}$ in a few different ways $B^{2}-E^{2} & = & \\frac{C_{E}}{\\omega ^{2}}-d_{B}^{2}+2\\mathbf {E}\\cdot \\left(\\widehat{\\mathbf {k}}\\times \\mathbf {d_{B}}\\right)\\nonumber \\\\& = & -\\frac{C_{E}}{\\omega ^{2}}-d_{B}^{2}+\\mathbf {E\\cdot d_{E}}\\nonumber \\\\& = & -\\frac{C_{B}^{2}}{\\omega ^{2}}+d_{E}+2\\mathbf {B}\\cdot \\left(\\widehat{\\mathbf {k}}\\times \\mathbf {d_{E}}\\right)\\nonumber \\\\& = & \\frac{C_{B}^{2}}{\\omega ^{2}}+d_{E}^{2}-\\mathbf {B\\cdot d_{B}}.$ As we will encounter a similar situation in the Euler-Heisenberg case, a comment on the integration constants $\\mathbf {d_E}$ and $\\mathbf {d_B}$ is in order.",
"First, we mention that due to the superposition principle in the linear Maxwell equations we can interpret these constants as part of constant fields which then enter the full solutions.",
"The fact that, e.g., $\\mathbf {d_E}$ is part of a constant field can be seen by writing $\\mathbf {B}=\\mathbf {B}_0(\\xi ) + \\mathbf {d_B}^{\\prime }$ and $\\mathbf {E}=\\mathbf {E}_0(\\xi ) + \\mathbf {d_E}^{\\prime }$ .",
"Using Faraday's law we obtain $\\mathbf {B}=\\mathbf {k} \\times \\mathbf {E}_0 + \\mathbf {d_B} + \\mathbf {k} \\times \\mathbf {d_E}^{\\prime }$ where $\\mathbf {d_B} + \\mathbf {k} \\times \\mathbf {d_E}^{\\prime }$ is the constant magnetic field (a similar consideration can be done for the electric field).",
"Therefore, even if $\\mathbf {k} \\times \\mathbf {d_E}^{\\prime }$ is zero, we are left with a constant magnetic contribution.",
"Thus we can interpret the integration constants as parts of constant fields in which the electromagnetic wave propagates.",
"Secondly, we recall that the photon represented by $\\mathbf {A}=\\mathbf {\\epsilon }e^{ikx}$ with $\\mathbf {k} \\cdot \\mathbf {\\epsilon }=0$ has two degrees of freedom with respect to $\\mathbf {k}$ (two independent polarization vectors $\\mathbf {\\epsilon }$ ).",
"Classically this is in correspondence with the number of parameters required to specify a plane wave in classical electrodynamics.",
"Keeping the constant fields increases the number of parameters required to specify the classical field since every constant arbitrary vector has three free directions.",
"This, however, does not imply that the degrees of freedom for the photon have changed as a photon which moves in a classical electromagnetic field (and every constant electromagnetic field can be considered as classical, see page 15 of [34]) still has only two polarization modes [7].",
"There might exist yet another interpretation regarding the integration constants which introduce additional degrees of freedom if we drop our previous interpretation of a wave in constant fields.",
"One such degree of freedom could be accounted for by the breaking of the conformal symmetry at quantum level [35].",
"A detailed examination of this possibility will be attempted elsewhere."
],
[
"Euler-Heisenberg Electrodynamics ",
"As in the classical electrodynamics, the Euler-Heisenberg theory consists of four equations that determine the evolution of the electric and the magnetic fields.",
"The magnetic Gauss's and Faraday's laws remain the same as in the classical case, namely $\\nabla \\cdot \\mathbf {B} & = & 0,\\\\\\nabla \\times \\mathbf {E} & = & -\\frac{\\partial \\mathbf {B}}{\\partial t},$ These equations serve to define the electromagnetic potentials and are independent of any Lagrangian.",
"The second set of equations, ones that replace the classical electric Gauss's and the Ampere-Maxwell's laws, are derived after a variation of the Lagrangian [34].",
"They can be written, in the absence of electric charges and currents, as $\\nabla \\cdot \\mathbf {D} & = & 0,\\\\\\nabla \\times \\mathbf {H} & = & \\frac{\\partial \\mathbf {D}}{\\partial t}, $ where the auxiliary fields $\\mathbf {D}$ and $\\mathbf {H}$ are given by $\\mathbf {D} & = & \\mathbf {E}+4\\pi \\frac{\\partial \\mathcal {L}_{EH}}{\\partial \\mathbf {E}}\\nonumber \\\\& = & \\mathbf {E}+\\eta \\left[2\\mathbf {E}(E^{2}-B^{2})+7\\mathbf {B}(\\mathbf {E\\cdot B}),\\right]\\\\\\mathbf {H} & = & \\mathbf {B}-4\\pi \\frac{\\partial \\mathcal {L}_{EH}}{\\partial \\mathbf {B}}\\nonumber \\\\& = & \\mathbf {B}+\\eta \\left[2\\mathbf {B}(E^{2}-B^{2})-7\\mathbf {E}(\\mathbf {E\\cdot B}),\\right]$ with $\\eta =\\frac{e^{4}}{45\\pi m_{e}^{4}}.$ As is customary in classical electrodynamics, the four first order differential equations can be combined to create two second order wave equations [25].",
"In this work we will not use the wave equations, we will focus in the first order equations (31)-(34).",
"The symmetric gauge invariant energy-momentum tensor of this theory [36], [37] is $T_{\\mu \\nu } & = & H^{\\mu \\nu }F_{\\nu }^{\\alpha }-\\mathcal {L}g_{\\mu \\nu },$ where the dielectric tensor $H^{\\mu \\nu }$ is given by $H^{\\mu \\nu }=\\frac{\\partial \\mathcal {L}}{\\partial F^{\\mu \\nu }},$ and can be obtained in a simple way from $F^{\\mu \\nu }$ by the replacement $E_{i}\\rightarrow D_{i}$ and $B_{i}\\rightarrow H_{i}$ .",
"We follow [38] and write the energy and momentum components of the energy-momentum tensor as $T^{00} & = & A\\left(\\frac{E^{2}+B^{2}}{8\\pi }\\right)+\\frac{\\tau }{4},\\\\T^{0i} & = & A\\frac{\\left(\\mathbf {E}\\times \\mathbf {B}\\right)_{i}}{4\\pi },$ where, for the weak field Euler-Heisenberg Lagrangian, the dielectric function $A$ and the trace $\\tau $ are $A & \\equiv & 1+2\\eta \\left(E^{2}-B^{2}\\right),\\\\\\tau & \\equiv & a\\left(\\left(E^{2}-B^{2}\\right)^{2}+7\\left(\\mathbf {E\\cdot B}\\right)^{2}\\right).$"
],
[
"Travelling waves in Euler-Heisenberg theory",
"Our procedure is again a straightforward one, i.e., trying the ansatz $\\mathbf {E}=\\mathbf {E}(\\xi )$ and $\\mathbf {B}=\\mathbf {B}(\\xi )$ into the differential Euler-Heisenberg equations.",
"Since the classical dispersion relation is not a priori guaranteed to be obeyed, we look for what conditions $\\mathbf {k}$ and $\\omega $ must satisfy.",
"We can integrate the Euler-Heisenberg equations in the same way as we did for the Maxwell's equations in section 1.",
"We obtain $\\mathbf {k\\cdot B} & = & C_{B},\\\\\\mathbf {B} & = & \\frac{\\mathbf {k}\\times \\mathbf {E}}{\\omega }+\\mathbf {d_{B}},\\\\\\mathbf {k\\cdot }\\mathbf {D} & = & C_{D},\\\\\\mathbf {D} & = & -\\frac{\\mathbf {k}\\times \\mathbf {H}}{\\omega }+\\mathbf {d_{D}},$ where $C_{B}$ ,$C_{D}$ , $\\mathbf {d}_{\\mathbf {D}}$ and $\\mathbf {d}_{\\mathbf {B}}$ are constants related by taking the scalar product of () and () with $\\mathbf {k}$ : $C_{B} & = & \\mathbf {k}\\cdot \\mathbf {d}_{B},\\\\C_{D} & = & \\mathbf {k}\\cdot \\mathbf {d}_{D}.$ We look for the Euler-Heisenberg equivalent of equation (REF ).",
"Let us start by noticing that the auxiliary fields can be written as $\\mathbf {D} & = & A\\mathbf {E}+7\\eta (\\mathbf {E}\\cdot \\mathbf {d_{B}})\\mathbf {B},\\\\\\mathbf {H} & = & A\\mathbf {B}-7\\eta (\\mathbf {E}\\cdot \\mathbf {d_{B}})\\mathbf {E},$ where $A$ is the dielectric function defined in (REF ).",
"With (REF ) and () the equation () can be written as $A\\mathbf {E}+7\\eta (\\mathbf {E}\\cdot \\mathbf {d_{B}})\\mathbf {d_{B}}=-A\\frac{\\mathbf {k}}{\\omega }\\times \\mathbf {B}+\\mathbf {d_{D}},$ where we have used () to transform the terms $7\\eta (\\mathbf {E}\\cdot \\mathbf {d_{B}})\\mathbf {B}$ and $7\\eta (\\mathbf {E}\\cdot \\mathbf {d_{B}})\\frac{\\mathbf {k}}{\\omega }\\times \\mathbf {E}$ into $7\\eta (\\mathbf {E}\\cdot \\mathbf {d_{B}})\\mathbf {d_{B}}$ .",
"Replacing $\\mathbf {B}$ using () we arrive at an algebraic equation in which only the electric field appears $A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E} & =\\mathbf {d_{D}}-A\\lbrace \\frac{\\left(\\mathbf {k\\cdot E}\\right)}{\\omega ^{2}}\\mathbf {k}+\\frac{\\mathbf {k\\times d_{B}}}{\\omega }\\rbrace -7\\eta \\mathbf {d_{B}}\\left(\\mathbf {E\\cdot d_{B}}\\right).$ The dielectric function can also be put solely in terms of $\\mathbf {E}$ as $A=1+2\\eta \\left(E^{2}\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}+\\frac{2\\mathbf {E}\\cdot \\left(\\mathbf {k}\\times \\mathbf {d_{B}}\\right)}{\\omega }-d_{B}^{2}\\right).$ Let us note that equation (REF ) reduces to (REF ) in the limit $\\eta \\rightarrow 0$ , as it should be."
],
[
"Maxwellian case ($k=\\omega $ ) in Euler-Heisenberg Theory",
"It is well known that some solution of the Maxwell's equations are also solutions of the Euler-Heisenberg equations [6].",
"The simplest examples are waves with $E^{2}-B^{2}=\\mathbf {E\\cdot B}=0$ , where the Euler-Heisenberg equations trivially reduce to the classical Maxwell's ones (physically this corresponds to the fact that in QED a single free photon can propagate undisturbed [41]).",
"We shall now see that this fact can be obtained directly from (REF ).",
"Looking for Maxwellian solutions we put $k=\\omega $ into equation (REF ) to obtain $0=\\mathbf {d_{D}}-A\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)\\mathbf {\\widehat{k}}-A\\mathbf {\\widehat{k}\\times d_{B}}-7\\eta \\mathbf {d_{B}}\\left(\\mathbf {E\\cdot d_{B}}\\right).$ Let us first assume that $\\mathbf {d_{B}}$ is not parallel to $\\mathbf {\\widehat{k}}$ , then we can take the scalar product of (REF ) with $\\mathbf {\\widehat{k}}$ , $\\mathbf {d_{B}}$ and $\\mathbf {\\widehat{k}\\times d_{B}}$ (which we take as basis) to obtain the following three equations $0 & = & \\mathbf {d_{D}}\\cdot \\mathbf {\\widehat{k}}-A\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)-7\\eta \\left(\\mathbf {\\widehat{k}\\cdot \\mathbf {d_{B}}}\\right)\\left(\\mathbf {E\\cdot d_{B}}\\right),\\\\0 & = & \\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}-A\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)\\left(\\mathbf {\\widehat{k}\\cdot \\mathbf {d_{B}}}\\right)-7\\eta d_{B}^{2}\\left(\\mathbf {E\\cdot d_{B}}\\right).\\\\0 & = & \\mathbf {d_{D}}\\cdot \\mathbf {\\widehat{k}\\times d_{B}}-A\\left(d_{B}^{2}-\\left(\\mathbf {\\widehat{k}\\cdot \\mathbf {d_{B}}}\\right)^{2}\\right).\\cdot $ From () it follows that $A=constant$ .",
"Meanwhile, equations (REF ) and () have $\\mathbf {\\widehat{k}\\cdot E}$ and $\\mathbf {E\\cdot d_{B}}$ as unknowns.",
"Since (REF ) and () are algebraically independent (due to our choice $\\mathbf {\\widehat{k}\\times d_{B}}\\ne 0$ ), we can solve $\\mathbf {\\widehat{k}\\cdot E}$ and $\\mathbf {E\\cdot d_{B}}$ in terms of constants.",
"Finally, from (REF ) $\\mathbf {E}\\cdot \\left(\\mathbf {k}\\times \\mathbf {d_{B}}\\right)$ is also a constant.",
"We have a case where there is no undulatory solution at all.",
"If, on the other hand, $\\mathbf {k}$ and $\\mathbf {d_{B}}$ are parallel then equation (REF ) reduces to $0=\\mathbf {d_{D}}-\\left(A-7\\eta d_{B}^{2}\\right)\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)\\mathbf {\\widehat{k}}.$ Equation (REF ) tells us that $\\mathbf {d_{D}}$ has to be parallel to $\\mathbf {\\widehat{k}}$ .",
"Furthermore, using (REF ) we can write for $A$ $A=1+2\\eta \\left(\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)^{2}-d_{B}^{2}\\right).$ Then equation (REF ) together with equation (REF ) implies that $\\mathbf {\\widehat{k}\\cdot E}$ and $A$ are constants.",
"This still leave us with enough freedom for the components of $\\mathbf {E}$ orthogonal to $\\mathbf {\\widehat{k}}$ .",
"Since $\\mathbf {\\widehat{k}\\cdot E}$ and $A$ are constants, it can be checked that the Euler-Heisenberg equations reduces to the Maxwell's equations.",
"For example, the following set $\\mathbf {E} & = & \\mathbf {E}_{0}(\\xi )+d_{E}\\mathbf {\\widehat{k}},\\\\\\mathbf {B} & = & \\mathbf {B}_{0}(\\xi )+d_{B}\\mathbf {\\widehat{k}},$ with $\\mathbf {\\widehat{k}}\\cdot \\mathbf {E}_{0}=\\mathbf {\\widehat{k}}\\cdot \\mathbf {B}_{0}=0$ and $\\mathbf {B}_{0}=\\mathbf {\\widehat{k}}\\times \\mathbf {E}_{0}$ , is a solution of both the Maxwell's and Euler-Heisenberg equations.",
"Notice, however, a subtle difference.",
"Whereas $\\mathbf {d_{B}}$ was an arbitrary constant, in the Euler-Heisenberg theory its direction is fixed by $\\mathbf {d_{B}} \\propto \\hat{\\mathbf {k}}$ .",
"At the end of section II we have commented on the interpretation of integration constants in the Maxwell case.",
"In the Euler-Heisenberg theory constant fields are also solutions of the corresponding equations.",
"What we do not have here is a general superposition principle due to the non-linearities of the equations.",
"Interpreting the constants in (REF ) and () as constant fields, we could say that these equations represent a restricted superposition principle where a travelling wave and constant field can be added together to form a new solution if and only if the direction of the constant field is parallel to $\\mathbf {k}$ .",
"An analog situation exists for two or more waves, in the sense that they can be added together to form a new solution to the Euler-Heisenberg equations only if they travel in the same direction [41].",
"The physical interpretation given to this last effect is that the photons which travel in the same direction do not scatter from each other.",
"We can then interpret (REF ) and () as a photon propagating undisturbed through a constant electromagnetic field if and only if the photon's motion is parallel to the direction of the background field.",
"Although waves (REF ) and () are also present in the classical theory, their energy and momentum content are different in the Euler-Heisenberg theory.",
"For example, using () we can write their momentum components as $T^{0i}=\\left(1+2\\eta (d_{E}^{2}-d_{B}^{2})\\right)\\frac{\\left(\\mathbf {E}\\times \\mathbf {B}\\right)_{i}}{4\\pi }.$ We can see from (REF ) that the photon-photon interaction codified in the Euler-Heisenberg Lagrangian implies that the wave's momentum density is slightly bigger when compared to the classical Poynting vector $T_{Maxwell}^{00}=\\frac{\\left(\\mathbf {E}\\times \\mathbf {B}\\right)_{i}}{4\\pi }$ , if $d_{E}^{2}$ is bigger than $d_{B}^{2}$ and vice versa.",
"The energy density is also changed from the classical $T_{Maxwell}^{0i}=\\frac{E^{2}+B^{2}}{8\\pi }$ to $T^{00}=\\left(1+2\\eta (d_{E}^{2}-d_{B}^{2})\\right)\\left(\\frac{E^{2}+B^{2}}{8\\pi }\\right)+\\frac{a}{4}\\left((d_{E}^{2}-d_{B}^{2})^{2}+7\\left(d_{E}d_{B}\\right)^{2}\\right).$ The new terms in the energy density and the Poynting vector proportional to $\\eta $ and $a$ are quantum mechanical in origin.",
"They are small unless the fields become very strong, but that takes us outside the weak field limit of the Euler-Heisenberg Lagrangian.",
"In the appendix we examine all cases with $\\omega \\ne k$ and $ A\\ne 0$ and show that they lead to trivial constant field solutions.",
"The proof makes use of the fact that we can use the integration constant vectors and $\\mathbf {k}$ (or some other combinations involving cross products) as basis and decompose the electric and magnetic fields in terms of projections in this basis."
],
[
"Off the light cone waves ($A=0$ )",
"There is a formal way to invalidate the proof presented in the appendix (this proof demonstrates that no travelling wave solutions with $\\omega \\ne k$ exist in the Euler-Heisenberg theory).",
"Indeed it suffices to put the dielectric function $A$ to zero.",
"However, it is important to bring to attention that $A=0$ is physically not viable.",
"Indeed, such an equation would result in strong fields violating the restriction on the theory.",
"On the other hand, if the weak field restriction is the only obstacle to obtain physically valid solutions, it makes sense to generalize the $A=0$ condition to more general Lagrangians where the weak field restriction is not implemented.",
"This seems, in principle, possible as the Euler-Heisenberg Lagrangian (1) is a weak field version of a more general one.",
"As shown below, $A=0$ , goes hand in hand with $\\omega \\ne k$ , i.e., we have travelling wave solutions off the light cone.",
"For these reasons it is illustrative to consider here the $A=0$ case as in the more general Lagrangian the steps would be similar.",
"Taking $A=0$ in the algebraic equation (REF ) gives us the conditions $1+2\\eta (E^{2}-B^{2}) & = & 0,\\\\\\mathbf {E\\cdot B}=\\mathbf {E\\cdot d_{B}} & = & \\beta =constant.$ We will call “off light cone waves” the waves that obey conditions (REF ) and ().",
"It is easy to check that conditions (REF ) and () give us a solution to the full set of Euler-Heisenberg equations.",
"Using (REF ) and () the auxiliary fields become $\\mathbf {D} & = & 7\\eta \\beta \\mathbf {B},\\\\\\mathbf {H} & = & -7\\eta \\beta \\mathbf {E},$ and we have the strange case where the vector $\\mathbf {D}$ is associated with the magnetic field while the vector $\\mathbf {H}$ is associated with the electric field, the opposite of what one would usually expect in electrodynamics (see, however, [39] ).",
"With the vectors (REF ) and (), the modified Electric Gauss's law (REF ) and the Ampere-Maxwell's law () become the classical magnetic Gauss's and Faraday's laws $7\\eta \\beta \\nabla \\cdot \\mathbf {B} & = & 0,\\\\7\\eta \\beta \\nabla \\times \\mathbf {E} & = & -7\\eta \\beta \\frac{\\partial \\mathbf {B}}{\\partial t}.$ Notice that choosing $\\beta =0$ we end up with $\\mathbf {D}=\\mathbf {H}=0$ .",
"Provided $A=0$ , this configuration is mathematically a solution of the Euler-Heisenberg equations.",
"Finally, the condition (REF ) gives us an intensity dependent dispersion relation.",
"Indeed, using (REF ) we can write $0=1+2\\eta \\left(E^{2}\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}+\\frac{2\\mathbf {E}\\cdot \\left(\\mathbf {k}\\times \\mathbf {d_{B}}\\right)}{\\omega }-d_{B}^{2}\\right).$ As an example, consider the fields $\\mathbf {E} & = & E_{0}\\left(\\cos \\left(\\xi \\right)\\widehat{\\mathbf {x}}\\text{+}\\sin \\left(\\xi \\right)\\widehat{\\mathbf {y}}\\right),\\\\\\mathbf {B} & = & \\frac{kE_{0}}{\\omega }\\left(-\\sin \\left(\\xi \\right)\\widehat{\\mathbf {x}}+\\cos \\left(\\xi \\right)\\right)\\widehat{\\mathbf {y}}.$ with $\\mathbf {k}=\\widehat{\\mathbf {z}}$ .",
"The fields form an off light cone wave solution to the Euler-Heisenberg equations as long as (REF ) is true.",
"Since for this example $d_{B}^{2}=\\mathbf {k\\cdot E}=0,$ we can calculate a dispersion relation of the form $\\frac{k^{2}}{\\omega ^{2}}=1+\\frac{1}{2\\eta E_{0}^{2}}.$ Though unusual, the relevant energy-momentum components would simply read $T^{00} & = & \\frac{\\tau }{4},\\\\T^{0i} & = & 0.$ However, as previously stated, the off the light-cone waves are not well-defined physical solutions.",
"The vanishing of the dielectric function (REF ) implies fields stronger than allowed by the weak field approximation of the Euler-Heisenberg Lagrangian, i. e., $\\frac{B^{2}}{\\eta }>1,$ whereas physically acceptable fields should range below the critical limit $B{}_{c}=\\frac{m_{e}^{2}}{e}$ .",
"However, a more general Lagrangian, like the full Euler-Heisenberg case, can lift this restriction."
],
[
"More General Lagrangian",
"The Euler-Heisenberg Lagrangian (REF ) is not the only proposed modification to the laws of classical electrodynamics.",
"Indeed, we could consider the full version of the nonlinear electrodynamics arising from quantum corrections.",
"To avoid the problem of pair production in such a case we could hypothetically consider an electric field below the pair production threshold and a strong magnetic field.",
"Let the correction to the Maxwell's Lagrangian be given by the non-linear Lagrangian $\\mathcal {L}_{NL}=\\mathcal {L}_{NL}(\\mathcal {F},\\mathcal {G}^{2}),$ where the electromagnetic invariants are given by $\\mathcal {F} & = & \\frac{B^{2}-E^{2}}{2},\\\\\\mathcal {G} & = & \\mathbf {E\\cdot B}.$ The pseudoscalar $\\mathcal {G}$ always appears squared in the Lagrangian to preserve the parity invariance of the theory.",
"In a generic form, the auxiliary fields are $\\mathbf {D} & = & \\mathbf {E}+4\\pi \\frac{\\partial \\mathcal {L}_{NL}}{\\partial \\mathbf {E}}\\nonumber \\\\& = & \\mathbf {E}+4\\pi \\frac{\\partial \\mathcal {L}_{NL}}{\\partial \\mathcal {F}}\\frac{\\partial \\mathcal {F}}{\\partial \\mathbf {E}}+4\\pi \\frac{\\partial \\mathcal {L}_{NL}}{\\partial \\mathcal {G}^{2}}\\frac{\\partial \\mathcal {G}^{2}}{\\partial \\mathbf {E}}\\nonumber \\\\& = & \\mathbf {E}\\left(1-4\\pi \\frac{\\partial \\mathcal {L}_{NL}}{\\partial \\mathcal {F}}\\right)+8\\pi \\frac{\\partial \\mathcal {L}_{NL}}{\\partial \\mathcal {G}^{2}}\\mathbf {B}\\left(\\mathbf {E\\cdot B}\\right),\\\\\\mathbf {H} & = & \\mathbf {B}\\left(1-4\\pi \\frac{\\partial \\mathcal {L}_{NL}}{\\partial \\mathcal {F}}\\right)-8\\pi \\frac{\\partial \\mathcal {L}_{NL}}{\\partial \\mathcal {G}^{2}}\\mathbf {E}\\left(\\mathbf {E\\cdot B}\\right).$ We can again make the travelling wave ansatz and look for solutions of the modified Maxwell equations (31) - (34).",
"Let us define $A \\equiv 1-4\\pi \\frac{\\partial \\mathcal {L}_{NL}}{\\partial \\mathcal {F}}$ .",
"Remembering that for travelling waves $\\mathcal {G}=\\mathbf {E\\cdot B}=\\mathbf {E\\cdot d_{B}}$ , we can see that the conditions $A=0$ and $\\mathbf {E\\cdot d_{B}}=0$ guarantee vanishing auxiliary fields $\\mathbf {D}=\\mathbf {H}=0,$ and this is an immediate solution to the modified Maxwell equations.",
"This generalizes the situation discussed in the last section without violating the weak field restriction.",
"Since the full Lagrangian is given in terms of an integral, it is difficult to derive analytical expressions.",
"Moreover, we speculate that as in section VI, this solution would lead to physically realizable waves with a new dispersion relation.",
"We leave the details to a future investigation.",
"We mention here that in [38] the dielectric function has been calculated to all orders for strong fields analytically up to an integral for $\\mathbf {E}=0$ , $\\mathbf {B} \\ne 0$ and vice versa for $\\mathbf {E} \\ne 0$ and $\\mathbf {B}=0$ .",
"However, if in the Maxwell Lagrangian we also set e.g.",
"$\\mathbf {E}=0$ we would not obtain travelling wave solutions and end up with static cases.",
"A generalization of the results in [38] would be required.",
"$$ $$ $\\mathbf {APPENDIX}$ In this appendix we investigate all cases of different choices of the integration constants and $\\mathbf {k}$ assuming always $\\omega \\ne k$ .",
"We rely on the following equations derived in the main text.",
"$A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E} & =\\mathbf {d_{D}}-A\\lbrace \\frac{\\left(\\mathbf {k\\cdot E}\\right)}{\\omega ^{2}}\\mathbf {k}+\\frac{\\mathbf {k\\times d_{B}}}{\\omega }\\rbrace -7\\eta \\mathbf {d_{B}}\\left(\\mathbf {E\\cdot d_{B}}\\right),\\\\A= & 1+\\eta \\left(E^{2}\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}+\\frac{2\\mathbf {E}\\cdot \\left(\\mathbf {k}\\times \\mathbf {d_{B}}\\right)}{\\omega }-d_{B}^{2}\\right).$ Case 1: If $\\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}=\\mathbf {k}\\cdot \\mathbf {d_{B}}=\\mathbf {k}\\cdot \\mathbf {d_{D}}=0$ We first analyze the case where $\\mathbf {k}$ , $\\mathbf {d_{B}}$ and $\\mathbf {d_{D}}$ form an orthogonal basis.",
"Multiplying (REF ) by $\\mathbf {k}$ , $\\mathbf {d_{B}}$ and $\\mathbf {d_{D}}$ we respectively get $A\\left(\\mathbf {k\\cdot E}\\right) & = & 0,\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right) & = & -7\\eta d_{B}^{2},\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\left(\\mathbf {E}\\cdot \\mathbf {d_{D}}\\right) & = & d_{D}^{2}-A\\mathbf {d_{D}}\\cdot \\left(\\frac{\\mathbf {k\\times d_{B}}}{\\omega }\\right).$ We see from () that $A$ is given by a constant, hence we infer from (REF ) that $\\mathbf {k\\cdot E}=0$ and from () we get that $\\mathbf {E}\\cdot \\mathbf {d_{D}}$ is given in terms of constants.",
"As $\\mathbf {k}$ , $\\mathbf {d_{B}}$ and $\\mathbf {d_{D}}$ form an orthogonal basis, $E^{2}$ can be written as $E^{2}=\\left(\\mathbf {E}\\cdot \\mathbf {\\widehat{d_{B}}}\\right)^{2}+\\left(\\mathbf {E}\\cdot \\mathbf {\\widehat{d_{D}}}\\right)^{2}$ Since $\\mathbf {E}\\cdot \\mathbf {d_{D}}$ and $A$ are constants, when we insert (REF ) into () we find that $\\mathbf {E}\\cdot \\mathbf {d_{B}}$ is a constant.",
"This case allows only trivial constants solutions.",
"Case 2: $\\mathbf {k}\\cdot \\mathbf {d_{B}}=\\mathbf {k}\\cdot \\mathbf {d_{D}}=0$ , $\\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}\\ne 0$ Taking the scalar product of (REF ) with $\\mathbf {k}$ , $\\mathbf {d_{B}}$ , $\\mathbf {d_{D}}$ , $\\mathbf {E}$ and $\\mathbf {k\\times d_{B}}$ we obtain respectively $A\\left(\\mathbf {k\\cdot E}\\right) & = & 0,\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E}\\cdot \\mathbf {d_{B}} & = & \\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}-7\\eta d_{B}^{2},\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\left(\\mathbf {E}\\cdot \\mathbf {d_{D}}\\right) & = & d_{D}^{2}-A\\mathbf {d_{D}}\\cdot \\left(\\frac{\\mathbf {k\\times d_{B}}}{\\omega }\\right)\\nonumber \\\\& & -7\\eta \\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}\\left(\\mathbf {E\\cdot d_{B}}\\right),\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E}\\cdot (\\mathbf {k\\times d_{B}}) & = & \\mathbf {d_{D}}\\cdot (\\mathbf {k\\times d_{B}})-A(\\mathbf {k\\times d_{B}})^{2}$ Now we take a look at the projection.",
"First, if $A\\ne 0$ then from (REF ) $\\mathbf {k\\cdot E}=0.$ Since $\\mathbf {d_{D}}$ is orthogonal to $\\mathbf {k}$ , we can write $\\mathbf {d_{D}}=a\\mathbf {d_{B}}+b\\left(\\mathbf {k\\times d_{B}}\\right),$ for some constant numbers $a$ and $b$ .",
"Then, $\\mathbf {E}\\cdot \\mathbf {d_{D}}=a\\mathbf {E}\\cdot \\mathbf {d_{B}}+b\\mathbf {E}\\cdot \\left(\\mathbf {k\\times d_{B}}\\right).$ We can insert (REF ) into () to obtain $A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)a\\mathbf {E}\\cdot \\mathbf {d_{B}}+bA\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E}\\cdot \\left(\\mathbf {k\\times d_{B}}\\right) & = & d_{D}^{2}-A\\mathbf {d_{D}}\\cdot \\left(\\frac{\\mathbf {k\\times d_{B}}}{\\omega }\\right)\\nonumber \\\\&-& 7\\eta \\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}\\left(\\mathbf {E\\cdot d_{B}}\\right).$ We can use now () and () in (REF ) to transform its left hand side and obtain $& a\\left(\\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}-7\\eta d_{B}^{2}\\right)+b\\left(\\mathbf {d_{D}}\\cdot (\\mathbf {k\\times d_{B}})-A(\\mathbf {k\\times d_{B}})^{2}\\right)\\nonumber \\\\= & d_{D}^{2}-A\\mathbf {d_{D}}\\cdot \\left(\\frac{\\mathbf {k\\times d_{B}}}{\\omega }\\right)-7\\eta \\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}\\left(\\mathbf {E\\cdot d_{B}}\\right).$ Our next step consists in using () to write () only in terms of $\\mathbf {d_{B}\\cdot E}$ .",
"The final equations read $& \\left(\\mathbf {E\\cdot d_{B}}\\right)\\left(\\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}-7\\eta d_{B}^{2}\\right)\\nonumber \\\\& +b\\left(\\left(\\mathbf {E\\cdot d_{B}}\\right)\\mathbf {d_{D}}\\cdot (\\mathbf {k\\times d_{B}})-\\frac{\\left(\\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}-7\\eta d_{B}^{2}\\right)}{1-\\frac{k^{2}}{\\omega ^{2}}}(\\mathbf {k\\times d_{B}})^{2}\\right)\\nonumber \\\\& =\\left(\\mathbf {E\\cdot d_{B}}\\right)d_{D}^{2}-\\frac{\\left(\\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}-7\\eta d_{B}^{2}\\right)}{1-\\frac{k^{2}}{\\omega ^{2}}}\\mathbf {d_{D}}\\cdot \\left(\\frac{\\mathbf {k\\times d_{B}}}{\\omega }\\right)-7\\eta \\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}\\left(\\mathbf {E\\cdot d_{B}}\\right)^{2}.$ Equation (REF ) is a polynomial equation with constant coefficients.",
"Its solution gives $\\mathbf {E\\cdot d_{B}}$ in terms of constants.",
"The only way to avoid this conclusion is to have all the coefficients of each power in $\\mathbf {E\\cdot d_{B}}$ to be zero individually.",
"But it is impossible for the coefficient of the $\\left(\\mathbf {E\\cdot d_{B}}\\right)^{2}$ to be zero by the very same assumption we used at the beginning of this case.",
"$$ Case 3: If $\\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}=\\mathbf {k}\\cdot \\mathbf {d_{B}}=0,$ and $\\mathbf {k}\\cdot \\mathbf {d_{D}}\\ne 0$ Multiplying (REF ) by $\\mathbf {k}$ , $\\mathbf {d_{B}}$ and $\\mathbf {d_{D}}$ we respectively get $A(\\mathbf {E}\\cdot \\mathbf {k}) & = & \\mathbf {d_{D}}\\cdot \\mathbf {k}\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right) & = & -7\\eta d_{B}^{2}\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\left(\\mathbf {E}\\cdot \\mathbf {d_{D}}\\right) & = & d_{D}^{2}-A\\left\\lbrace \\frac{(\\mathbf {E}\\cdot \\mathbf {k})}{\\omega ^{2}}\\mathbf {k\\cdot d_{D}}-\\frac{\\mathbf {d_{D}}\\cdot \\left(\\mathbf {k\\times d_{B}}\\right)}{\\omega }\\right\\rbrace $ We immediately obtain from () that $A$ is a constant and we can use this fact in (REF ) to find that $(\\mathbf {E}\\cdot \\mathbf {k})$ is a constant.",
"These two results together with () tell us that $\\mathbf {E}\\cdot \\mathbf {d_{D}}$ is a constant.",
"As $\\mathbf {d_{B}}$ is orthogonal to $\\mathbf {k}$ and $\\mathbf {d_{D}}$ we can write $E^{2}=\\left(\\mathbf {E}\\cdot \\mathbf {d_{B}}\\right)^{2}+F\\left((\\mathbf {E}\\cdot \\mathbf {k}),\\left(\\mathbf {E}\\cdot \\mathbf {d_{D}}\\right)\\right)$ where $F\\left((\\mathbf {E}\\cdot \\mathbf {k}),\\left(\\mathbf {E}\\cdot \\mathbf {d_{D}}\\right)\\right)$ is just a constant.",
"We now replace (REF ) into () to arrive at an expression for $A$ $& A=1\\nonumber \\\\& +\\eta \\left(\\left(\\left(\\mathbf {E}\\cdot \\mathbf {d_{B}}\\right)^{2}+F\\right)\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}+\\frac{2\\mathbf {E}\\cdot \\left(\\mathbf {k}\\times \\mathbf {d_{B}}\\right)}{\\omega }-d_{B}^{2}\\right)$ The expression $\\mathbf {E}\\cdot \\left(\\mathbf {k}\\times \\mathbf {d_{B}}\\right)^{2}$ is a constant since it can be written in terms of $(\\mathbf {E}\\cdot \\mathbf {k}),$ and $\\mathbf {E}\\cdot \\mathbf {d_{D}}$ .",
"Therefore using (REF ) we reach the conclusion that $\\mathbf {E}\\cdot \\mathbf {d_{B}}$ is also a constant.",
"Case 4: If $\\mathbf {d_{D}}\\cdot \\mathbf {d_{B}}=\\mathbf {k}\\cdot \\mathbf {d_{D}}=0,$ and $\\mathbf {k}\\cdot \\mathbf {d_{B}}\\ne 0$ First note that $\\mathbf {k\\times d_{B}}$ is proportional to $\\mathbf {d_{D}}$ .",
"Hence we will write $\\mathbf {k\\times d_{B}}=a\\mathbf {d_{D}}$ .",
"The scalar product of (REF ) with $\\mathbf {k}$ , $\\mathbf {d_{B}}$ and $\\mathbf {d_{D}}$ gives respectively $A(\\mathbf {E}\\cdot \\mathbf {k}) & = & -7\\eta \\left(\\mathbf {k}\\cdot \\mathbf {d_{B}}\\right)\\left(\\mathbf {E\\cdot d_{B}}\\right),\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\left(\\mathbf {E}\\cdot \\mathbf {d_{B}}\\right) & = & -A\\frac{\\left(\\mathbf {k\\cdot E}\\right)}{\\omega ^{2}}\\left(\\mathbf {k}\\cdot \\mathbf {d_{B}}\\right)-7\\eta d_{B}^{2}\\left(\\mathbf {E\\cdot d_{B}}\\right),\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\left(\\mathbf {E}\\cdot \\mathbf {d_{D}}\\right) & = & d_{D}^{2}-\\frac{A}{\\omega }ad_{D}^{2},\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)E^{2} & = & \\mathbf {E}\\cdot \\mathbf {d_{D}}-A\\lbrace \\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}\\mathbf {k}+\\frac{a}{\\omega }\\mathbf {E}\\cdot \\mathbf {d_{D}}\\rbrace \\nonumber \\\\& & -7\\eta \\left(\\mathbf {E\\cdot d_{B}}\\right)^{2}$ Replacing equation (REF ) into () leads to $A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)=\\frac{7\\eta }{\\omega ^{2}}\\left(\\mathbf {k}\\cdot \\mathbf {d_{B}}\\right)^{2}-7\\eta d_{B}^{2},$ and it follows that $A$ is a constant.",
"By virtue of () this implies that $\\mathbf {E}\\cdot \\mathbf {d_{D}}$ is a constant.",
"By using equation () to write $\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)E^{2}=\\frac{A-1}{\\eta }-\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}-2\\frac{a}{\\omega }\\mathbf {E}\\cdot \\mathbf {d_{D}}+d_{B}^{2}.$ and replacing (REF ) into () $A\\left(\\frac{A-1}{\\eta }+d_{B}^{2}-\\frac{a}{\\omega }\\mathbf {E}\\cdot \\mathbf {d_{D}}\\right)=\\mathbf {E}\\cdot \\mathbf {d_{D}}-7\\eta \\left(\\mathbf {E\\cdot d_{B}}\\right)^{2},$ we conclude that $\\mathbf {E\\cdot d_{B}}$ is a constant.",
"Case 5: if $d_{D}=0$ but $d_{B}\\ne 0$ The scalar product of (REF ) with $\\mathbf {k}$ , $\\mathbf {d_{B}}$ $\\cdot \\left(\\mathbf {k\\times d_{B}}\\right)$ results into the following equations $0 & = & A(\\mathbf {k\\cdot E})+7\\eta \\left(\\mathbf {k}\\cdot \\mathbf {d_{B}}\\right)\\left(\\mathbf {E\\cdot d_{B}}\\right),\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\left(\\mathbf {E}\\cdot \\mathbf {d_{B}}\\right) & = & A\\frac{(\\mathbf {E}\\cdot \\mathbf {k})}{\\omega ^{2}}\\mathbf {k\\cdot d_{B}}-7\\eta d_{B}^{2}\\left(\\mathbf {E\\cdot d_{B}}\\right),\\\\\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E}\\cdot \\left(\\mathbf {k\\times d_{B}}\\right) & = & \\frac{1}{\\omega }\\left(\\mathbf {k\\times d_{B}}\\right)^{2}\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)E^{2} & = & -A\\left\\lbrace \\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}-\\frac{\\mathbf {E}\\cdot \\left(\\mathbf {k\\times d_{B}}\\right)}{\\omega }\\right\\rbrace \\nonumber \\\\& & -7\\eta \\left(\\mathbf {E\\cdot d_{B}}\\right)^{2}$ We can solve for $A(\\mathbf {k\\cdot E})$ in (REF ) and insert it in () to obtain $A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)=\\frac{7\\eta }{\\omega ^{2}}\\left(\\mathbf {k}\\cdot \\mathbf {d_{B}}\\right)\\mathbf {k\\cdot d_{B}}-7\\eta d_{B}^{2}.$ Again we arrive at the conclusion that A has to be a constant.",
"Moreover, we can read directly from () that $\\mathbf {E}\\cdot \\left(\\mathbf {k\\times d_{B}}\\right)$ is a constant.",
"From () we can write $\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)E^{2}=\\frac{A-1}{\\eta }-\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}-2\\frac{1}{\\omega }\\mathbf {E}\\cdot \\left(\\mathbf {k\\times d_{B}}\\right)+d_{B}^{2},$ and replacing (REF ) into () $-7\\eta \\left(\\mathbf {E\\cdot d_{B}}\\right)^{2}=A\\left[(\\frac{A-1}{\\eta })-\\frac{1}{\\omega }\\mathbf {E}\\cdot \\left(\\mathbf {k\\times d_{B}}\\right)+d_{B}^{2}\\right].$ Independent of the numerical value of the right hand side, we easily see that $\\mathbf {E\\cdot d_{B}}$ is a constant.",
"Case 6: If $\\mathbf {d_{B}}=d_{B}\\mathbf {k}$ and $\\mathbf {d_{D}}\\ne 0.$ In this case the equation (REF ) reduces to $A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E}=\\mathbf {d_{D}}-\\lbrace \\frac{A}{\\omega ^{2}}-7\\eta d_{B}^{2}\\rbrace \\left(\\mathbf {k\\cdot E}\\right)\\mathbf {k}.$ We can choose $\\mathbf {k}$ , $\\mathbf {k}\\times \\mathbf {d_{D}}$ and $\\mathbf {k}\\times \\mathbf {\\left(\\mathbf {k}\\times \\mathbf {d_{D}}\\right)}$ as a basis.",
"To make the notation more concise, let us define $\\mathbf {k}_{\\perp }=\\mathbf {k}\\times \\mathbf {\\left(\\mathbf {k}\\times \\mathbf {d_{D}}\\right)}$ .",
"It is clear from (REF ) that $\\mathbf {E}$ does not have components in the $\\mathbf {k}\\times \\mathbf {d_{D}}$ direction, and hence $\\mathbf {E}$ can be written in the following form $\\mathbf {E}=\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)\\widehat{\\mathbf {k}}+\\left(\\mathbf {\\widehat{k}_{\\perp }\\cdot E}\\right)\\widehat{\\mathbf {k}}_{\\perp },$ By the same token we have $\\mathbf {d_{D}}=a\\widehat{\\mathbf {k}}+b\\widehat{\\mathbf {k}}_{\\perp }$ for some numbers $a$ and $b$ .",
"Equation (REF ) allows us to write $E^{2}=\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)^{2}+\\left(\\mathbf {\\widehat{k}_{\\perp }\\cdot E}\\right)^{2},$ and therefore $A=1+\\eta \\left(\\left(\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)^{2}+\\left(\\mathbf {\\widehat{k}_{\\perp }\\cdot E}\\right)^{2}\\right)\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}-d_{B}^{2}\\right).$ The scalar product of (REF ) with $\\mathbf {\\widehat{k}}$ , and $\\mathbf {\\widehat{k}}_{\\perp }$ leads to the following set of equations $& \\left[1+\\eta \\left(\\left(\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)^{2}+\\left(\\mathbf {\\widehat{k}_{\\perp }\\cdot E}\\right)^{2}\\right)\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}-d_{B}^{2}\\right)\\right]\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)\\nonumber \\\\& =a-7\\eta d_{B}^{2}\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)k^{2}\\\\& \\left[1+\\eta \\left(\\left(\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)^{2}+\\left(\\mathbf {\\widehat{k}_{\\perp }\\cdot E}\\right)^{2}\\right)\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}-d_{B}^{2}\\right)\\right]\\nonumber \\\\& \\times \\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\left(\\mathbf {\\widehat{k}_{\\perp }\\cdot E}\\right)\\nonumber \\\\& =b & .$ Equations (REF ) and () are algebraic independent polynomials for any (non zero) value of the constants.",
"This means that we cannot choose any relation among $k,\\, d_{B},\\, a$ and $b$ to make (REF ) proportional to ().",
"By Bézout's theorem [40] the systems (REF ) and () have a finite number of solutions.",
"These solutions will be functions of the coefficients of the polynomials, i.e., of constants.",
"Therefore we have trivial constant solutions at hand.",
"On the other hand, if $\\mathbf {d_{B}}$ is parallel to $\\mathbf {k}$ , then equation (REF ) reduces further to $A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E}=-\\lbrace \\frac{A}{\\omega ^{2}}\\left(\\mathbf {k\\cdot E}\\right)-7\\eta d_{B}^{2}\\left(\\mathbf {k\\cdot E}\\right)-d_{D}\\rbrace \\mathbf {k}.$ There are two ways to solve equation (REF ).",
"The first is letting $k=\\omega $ that leads to the condition $\\mathbf {k\\cdot E}=constant$ which is identical to the classical Gauss law and also leads to a classical solution to the Maxwell's equations.",
"The other solution is to set $A=0$ which also leads to $\\mathbf {k\\cdot E}=constant$ , but we know from section 6 that this kind of waves are not viable solutions.",
"$$ Case 7: $\\mathbf {d_{D}}=d_{D}\\mathbf {k}$ and $\\mathbf {d_{B}}\\ne 0.$ For this case, equation (REF ) reduces to $A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E}=d_{D}\\mathbf {k}-A\\lbrace \\frac{\\left(\\mathbf {k\\cdot E}\\right)}{\\omega ^{2}}\\mathbf {k}+\\frac{\\mathbf {k\\times d_{B}}}{\\omega }\\rbrace -7\\eta \\mathbf {d_{B}}\\left(\\mathbf {E\\cdot d_{B}}\\right).$ By taking the dot product with $\\mathbf {k\\times d_{B}}$ we get $\\mathbf {\\mathbf {E}\\cdot \\left(k\\times d_{B}\\right)}=C=constant.$ Similar to the previous case, if $\\mathbf {d_{B}}$ is not parallel to $\\mathbf {k}$ , then we can choose as a basis the vectors $\\mathbf {k}$ , $\\mathbf {k}\\times \\mathbf {d_{B}}$ and, $\\mathbf {\\widehat{k}_{\\perp }}$ where $\\mathbf {\\widehat{k}_{\\perp }}=\\mathbf {k}\\times \\mathbf {\\left(\\mathbf {k}\\times \\mathbf {d_{B}}\\right)}$ .",
"In this way we can write $\\mathbf {E}=\\left(\\mathbf {E\\cdot \\mathbf {\\widehat{k}}}\\right)\\mathbf {\\widehat{k}}+\\left(\\mathbf {E}\\cdot \\mathbf {\\widehat{k}_{\\perp }}\\right)\\mathbf {\\widehat{k}_{\\perp }}$ , and therefore $E^{2} & = & \\left(\\mathbf {\\widehat{k}\\cdot E}\\right)^{2}+\\left(\\mathbf {\\widehat{k}_{\\perp }\\cdot E}\\right)^{2}+C,\\\\A & = & 1\\nonumber \\\\& & +\\eta [\\left(\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)^{2}+\\left(\\mathbf {\\widehat{k}_{\\perp }\\cdot E}\\right)^{2}+C\\right)\\\\& & \\times (\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}-d_{B}^{2})],\\\\\\mathbf {E\\cdot d_{B}} & = & \\left(\\mathbf {E\\cdot \\mathbf {\\widehat{k}}}\\right)\\mathbf {\\widehat{k}\\cdot d_{B}}+\\left(\\mathbf {E}\\cdot \\mathbf {\\widehat{k}_{\\perp }}\\right)\\mathbf {\\widehat{k}_{\\perp }\\cdot d_{B}}.$ We can then write the equations for the projections in $\\mathbf {\\widehat{k}}$ , and $\\mathbf {\\widehat{k}}_{\\perp }$ to get $& \\left[1+\\eta \\left(\\left(\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)^{2}+\\left(\\mathbf {\\widehat{k}_{\\perp }\\cdot E}\\right)^{2}+C^{2}\\right)\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}-d_{B}^{2}\\right)\\right]\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)\\nonumber \\\\& =d_{D}k-7\\eta \\left(\\mathbf {\\mathbf {\\widehat{k}\\cdot d_{B}}}\\right)\\left(\\left(\\mathbf {E\\cdot \\mathbf {\\widehat{k}}}\\right)\\mathbf {\\widehat{k}\\cdot d_{B}}+\\left(\\mathbf {E}\\cdot \\mathbf {\\widehat{k}_{\\perp }}\\right)\\mathbf {\\widehat{k}_{\\perp }\\cdot d_{B}}\\right)\\\\& \\left[1+\\eta \\left(\\left(\\left(\\mathbf {\\widehat{k}\\cdot E}\\right)^{2}+\\left(\\mathbf {\\widehat{k}_{\\perp }\\cdot E}\\right)^{2}+C^{2}\\right)\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}-d_{B}^{2}\\right)\\right]\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\left(\\mathbf {\\widehat{k}_{\\perp }\\cdot E}\\right)\\nonumber \\\\& =7\\eta \\left(\\mathbf {\\mathbf {\\widehat{k}_{\\perp }\\cdot d_{B}}}\\right)\\left(\\left(\\mathbf {E\\cdot \\mathbf {\\widehat{k}}}\\right)\\mathbf {\\widehat{k}\\cdot d_{B}}+\\left(\\mathbf {E}\\cdot \\mathbf {\\widehat{k}_{\\perp }}\\right)\\mathbf {\\widehat{k}_{\\perp }\\cdot d_{B}}\\right) & .$ As in the previous case, equations (REF ) and () are algebraically independent, and therefore only admit a finite number of constant solutions.",
"For $\\mathbf {d_{B}}$ parallel to $\\mathbf {k}$ we can write (REF ) as $A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E}=d_{D}\\mathbf {k}-A\\frac{\\left(\\mathbf {k\\cdot E}\\right)}{\\omega ^{2}}\\mathbf {k}-7\\eta d_{B}^{2}\\left(\\mathbf {E\\cdot k}\\right)\\mathbf {k},$ but $\\mathbf {E}=\\frac{\\left(\\mathbf {k\\cdot E}\\right)}{k^{2}}\\mathbf {k}$ and $A=1+\\eta \\left(E^{2}\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}-d_{B}^{2}\\right)=1+\\eta \\left(\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{k^{2}}-d_{B}^{2}\\right)$ and therefore we can write $\\left(1+\\eta \\left(\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{k^{2}}-d_{B}^{2}\\right)\\right)\\frac{\\left(\\mathbf {k\\cdot E}\\right)}{k}=d_{D}k-7\\eta d_{B}^{2}k\\left(\\mathbf {E\\cdot k}\\right),$ which is an algebraic equation for $\\left(\\mathbf {k\\cdot E}\\right)$ in terms of constant coefficients and therefore we again haave a trivial constant solution for the fields.",
"$$ Case 8: $\\mathbf {k}$ , $\\mathbf {d_{B}}$ , $\\mathbf {d_{D}}$ are parallel.",
"This case is trivial.",
"When $\\mathbf {k}$ , $\\mathbf {d_{B}}$ , $\\mathbf {d_{D}}$ are parallel and neither $A$ nor $1-\\frac{k^{2}}{\\omega ^{2}}$ vanish, then we can write (REF ) as $A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E}=d_{D}\\mathbf {k}-A\\frac{\\left(\\mathbf {k\\cdot E}\\right)}{\\omega ^{2}}\\mathbf {k}-7\\eta d_{B}^{2}\\left(\\mathbf {E\\cdot k}\\right)\\mathbf {k}.$ But $\\mathbf {E}=\\frac{\\left(\\mathbf {k\\cdot E}\\right)}{k^{2}}\\mathbf {k}$ and $A=1+\\eta \\left(E^{2}\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)+\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{\\omega ^{2}}-d_{B}^{2}\\right)=1+\\eta \\left(\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{k^{2}}-d_{B}^{2}\\right)$ and therefore we can write $\\left(1+\\eta \\left(\\frac{\\left(\\mathbf {k\\cdot E}\\right)^{2}}{k^{2}}-d_{B}^{2}\\right)\\right)\\frac{\\left(\\mathbf {k\\cdot E}\\right)}{k}=d_{D}k-7\\eta d_{B}^{2}k\\left(\\mathbf {E\\cdot k}\\right),$ which is an algebraic equation for $\\left(\\mathbf {k\\cdot E}\\right)$ in terms of constant coefficients and therefore we again have a trivial constant solution for the fields.",
"$$ Case 9: None of $\\mathbf {k}$ , $\\mathbf {d_{B}}$ and $\\mathbf {d_{D}}$ are parallel or orthogonal to any of the others.",
"Taking the scalar product of (REF ) with $\\mathbf {k}$ , $\\mathbf {d_{B}}$ , $\\mathbf {k\\times d_{B}}$ and $\\mathbf {E}$ we respectively get $A(\\mathbf {E}\\cdot \\mathbf {k}) & = & \\mathbf {d_{D}\\cdot k}-7\\eta \\left(\\mathbf {k}\\cdot \\mathbf {d_{B}}\\right)\\left(\\mathbf {E\\cdot d_{B}}\\right),\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\left(\\mathbf {E}\\cdot \\mathbf {d_{B}}\\right) & = & \\mathbf {d_{D}\\cdot \\mathbf {d_{B}}}+A\\frac{(\\mathbf {E}\\cdot \\mathbf {k})}{\\omega ^{2}}\\mathbf {k\\cdot d_{B}}\\nonumber \\\\& & -7\\eta d_{B}^{2}\\left(\\mathbf {E\\cdot d_{B}}\\right),\\\\A\\left(1-\\frac{k^{2}}{\\omega ^{2}}\\right)\\mathbf {E}\\cdot \\left(\\mathbf {k\\times d_{B}}\\right) & = & \\mathbf {d_{D}\\cdot k\\times d_{B}}+\\frac{A}{\\omega }\\left(\\mathbf {k\\times d_{B}}\\right)^{2}.$ As $\\mathbf {k}$ , $\\mathbf {d_{B}}$ and $\\mathbf {k\\times d_{B}}$ are not parallel they form a basis and we can write any other vector, like $\\mathbf {E}$ and $\\mathbf {d_{D}}$ , as a linear combination of them.",
"This means that $E^{2}$ (and therefore $A$ ) can be written in terms of $\\mathbf {E}\\cdot \\mathbf {k}$ , $\\mathbf {E}\\cdot \\mathbf {d_{B}}$ and $\\mathbf {E}\\cdot \\left(\\mathbf {k\\times d_{B}}\\right)$ .",
"Moreover, $E^{2}$ (and therefore $A$ ) will contain a term $\\left(\\mathbf {E}\\cdot \\widehat{\\left(\\mathbf {k\\times d_{B}}\\right)}\\right)^{2}$ , and therefore equation () will have a term $\\left(\\mathbf {E}\\cdot \\widehat{\\left(\\mathbf {k\\times d_{B}}\\right)}\\right)^{3}$ .",
"This cubic term cannot be eliminated by any choice of the constants, and therefore equation () cannot be reduced to equation (REF ) or ().",
"Using the same argument, equations () will have a cubic term of the form $\\left(\\mathbf {E}\\cdot \\mathbf {d_{B}}\\right)^{3}$ that cannot be eliminated and therefore equation () cannot be reduced to equation (REF ).",
"We have then a system of three algebraically independent equations for the three unknowns.",
"We can use Bézout's theorem to say that the system allows only for a finite number of solutions that will be given in terms of constants.",
"Therefore, this case also leads to a trivial constant solution.",
"This completes our proof that all $\\omega \\ne k$ cases lead to trivial constant solutions assuming $A \\ne 0$ .",
"We thank the Faculty of Science at the Universidad de los Andes and the administrative department of science, technology and innovation of Colombia (Colciencias) for financial support."
]
] | 1709.01617 |
[
[
"Photoionization in the time and frequency domain"
],
[
"Abstract Ultrafast processes in matter, such as the electron emission following light absorption, can now be studied using ultrashort light pulses of attosecond duration ($10^{-18}$s) in the extreme ultraviolet spectral range.",
"The lack of spectral resolution due to the use of short light pulses may raise serious issues in the interpretation of the experimental results and the comparison with detailed theoretical calculations.",
"Here, we determine photoionization time delays in neon atoms over a 40 eV energy range with an interferometric technique combining high temporal and spectral resolution.",
"We spectrally disentangle direct ionization from ionization with shake up, where a second electron is left in an excited state, thus obtaining excellent agreement with theoretical calculations and thereby solving a puzzle raised by seven-year-old measurements.",
"Our experimental approach does not have conceptual limits, allowing us to foresee, with the help of upcoming laser technology, ultra-high resolution time-frequency studies from the visible to the x-ray range."
],
[
"=1 patterns arrows.meta fadings spy >=Latex[width=2mm,length=2mm] compat=newest plot coordinates/math parser=false Photoionization in the time and frequency domain M. IsingerDepartment of Physics, Lund University, P.O.",
"Box 118, SE-22 100 Lund, Sweden R.J. SquibbDepartment of Physics, Gothenburg University, Origovägen 6B, SE-41 296 Göteborg, Sweden D. BustoDepartment of Physics, Lund University, P.O.",
"Box 118, SE-22 100 Lund, Sweden S. ZhongDepartment of Physics, Lund University, P.O.",
"Box 118, SE-22 100 Lund, Sweden A. HarthDepartment of Physics, Lund University, P.O.",
"Box 118, SE-22 100 Lund, Sweden D. KroonDepartment of Physics, Lund University, P.O.",
"Box 118, SE-22 100 Lund, Sweden S. NandiDepartment of Physics, Lund University, P.O.",
"Box 118, SE-22 100 Lund, Sweden C. L. ArnoldDepartment of Physics, Lund University, P.O.",
"Box 118, SE-22 100 Lund, Sweden M. MirandaDepartment of Physics, Lund University, P.O.",
"Box 118, SE-22 100 Lund, Sweden J.M.",
"DahlströmDepartment of Physics, Lund University, P.O.",
"Box 118, SE-22 100 Lund, Sweden Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden E. LindrothDepartment of Physics, Stockholm University, SE-106 91 Stockholm, Sweden R. FeifelDepartment of Physics, Gothenburg University, Origovägen 6B, SE-41 296 Göteborg, Sweden M. GisselbrechtDepartment of Physics, Lund University, P.O.",
"Box 118, SE-22 100 Lund, Sweden A. L'HuillierDepartment of Physics, Lund University, P.O.",
"Box 118, SE-22 100 Lund, Sweden Ultrafast processes in matter, such as the electron emission following light absorption, can now be studied using ultrashort light pulses of attosecond duration ($10^{-18}$ s) in the extreme ultraviolet spectral range.",
"The lack of spectral resolution due to the use of short light pulses may raise serious issues in the interpretation of the experimental results and the comparison with detailed theoretical calculations.",
"Here, we determine photoionization time delays in neon atoms over a 40 eV energy range with an interferometric technique combining high temporal and spectral resolution.",
"We spectrally disentangle direct ionization from ionization with shake up, where a second electron is left in an excited state, thus obtaining excellent agreement with theoretical calculations and thereby solving a puzzle raised by seven-year-old measurements.",
"Our experimental approach does not have conceptual limits, allowing us to foresee, with the help of upcoming laser technology, ultra-high resolution time-frequency studies from the visible to the x-ray range.",
"While femtosecond lasers allow the study and control of the motion of nuclei in molecules, attosecond light pulses give access to even faster dynamics, such as electron motion induced by light-matter interactions [1].",
"During the last ten years, seminal experiments with sub-femtosecond temporal resolution have allowed the observation of the electron valence motion [2], monitoring of the birth of an autoionizing resonance [3], [4] and tracking the motion of a two-electron wavepacket [5], to cite only a few examples.",
"Fast electron motion occurs even when electrons are directly emitted from materials upon absorption of sufficiently energetic radiation (the photoelectric effect).",
"The time for the photoelectron emission [6], which, in free atoms, represents the time for the electron to escape the potential, called photoionization time delay [7], [9], is typically of the order of tens of attoseconds, depending on the excitation energy and on the underlying core structure.",
"Photoemission has traditionally been studied in the frequency domain, using high-resolution photoelectron spectroscopy with x-ray synchrotron radiation sources, and such methods have provided a detailed understanding of the electronic structure of matter [10], [11].",
"Absorption of light in the 60-100 eV range by Ne atoms, for example, leads to direct ionization in the $2\\mathrm {s}$ or $2\\mathrm {p}$ shells and to processes mediated by electron-electron interaction, leaving the residual ion in an excited state (often called shake-up) or doubly ionized [12], [13], [14].",
"It may be argued that the high temporal resolution achieved in attosecond experiments prevents any spectral accuracy and thus may affect the interpretation of experimental results.",
"This is especially true when different processes can be induced simultaneously and lead to photoelectrons with kinetic energies within the bandwidth of the excitation pulse.",
"In fact, the natural trade-off between temporal and spectral resolution may be overcome, as beautifully shown in the visible spectrum using high-resolution frequency combs based upon phase-stable femtosecond pulse trains [15].",
"Figure: NO_CAPTIONIn this work, we bridge the gap between high-resolution photoelectron spectroscopy and attosecond dynamics, making use of the high-order harmonic spectrum obtained by phase-stable interferences between attosecond pulses in a train.",
"We present a study of photoionization time delays of the $2\\mathrm {s}$ and $2\\mathrm {p}$ shells in neon over a broad energy range from 65 to 100 eV, using an interferometric technique combining high temporal (20 as) and spectral (200 meV) accuracy, originally introduced for characterizing attosecond pulses in a train [16], [17] and called RABITT (Reconstruction of Attosecond Beating by Interference of Two-photon Transitions).",
"Remarkably, our temporal and spectral resolution depends only partly on the properties of the extreme ultraviolet (XUV) pulses.",
"In the limit of long infrared (IR) pulses leading to trains with reproducible attosecond pulses, the temporal resolution is only limited by the stability of our interferometer and the resolving power of the electron spectrometer.",
"In the present work, our spectral resolution, limited both by the harmonic bandwidths and by the spectrometer resolution, estimated to be $\\simeq 200$ meV, allows us to disentangle direct $2\\mathrm {s}$ ionization from shake up processes, where a $2\\mathrm {p}$ electron is ionized while a second is excited to a $3\\mathrm {p}$ state.",
"As shown in Fig.",
"REF (a), our experimental results for the difference between $2\\mathrm {s}$ and $2\\mathrm {p}$ time delays, as indicated by the red and blue dots, agree very well with theoretical calculations performed within the framework of many-body perturbation theory (the solid black line).",
"Our experimental observation of a shake up process due to electron correlation also provides a possible explanation for the discrepancy between the pioneering result of Schultze et al.",
"[7] (green dot) and theoretical calculations [18], [19], [20].",
"Photoionization time delays.",
"In general, experimentally measured delays can be considered as the sum of two contributions, $\\tau _{\\textsc {XUV}} + \\tau _\\mathrm {A}$ , where the first term is the group delay of the broadband excitation $\\textsc {XUV}$ field [17] and the second term reflects the influence of the atomic system.",
"To eliminate the influence of the excitation pulse, two measurements can be performed simultaneously, for example, on different ionization processes [7], [9], [21] or in different target species [22], [23].",
"This enables the determination of relative photoionization time delays.",
"Absolute photoionization delays can be deduced if we assume that one of the delays can by sufficiently accurately calculated to serve as an absolute reference [24].",
"In nonresonant conditions, the atomic delay $\\tau _\\mathrm {A}$ can in turn be approximated as the sum of two contributing delays, $\\tau _\\mathrm {W} + \\tau _{\\mathrm {cc}}$ .",
"The first term is the group delay of the electronic wavepacket created by absorption of XUV radiation, also called photoionization time delay or, shortly, Wigner delay.",
"Already in 1955, E. Wigner interpreted the derivative of the scattering phase as the group delay of the outgoing electronic wavepacket in a collision process [25].",
"This interpretation also applies to photoionization with a dominant outgoing channel, with a factor one half to account for the fact that photoionization is a half collision.",
"The second term, $\\tau _{\\mathrm {cc}}$ , is a correction to the photoionization time delay due to the interaction of the IR field with the Coulomb potential, which is required for the measurement.",
"At high kinetic energies, larger than $\\approx 10$ eV, $\\tau _{\\mathrm {cc}}$ can be accurately calculated using either the asymptotic form of the wave function [26] or by classical trajectories [27].",
"The index “cc” refers to the fact that the involved IR transitions are between two continuum states.",
"In other works [27], [24], it is denoted $\\tau _{\\mathrm {CLC}}$ (where the index is an abbreviation of “Coulomb-laser coupling”).",
"In the case of multiple angular channels with comparable amplitude [23], as is the case close to resonances [28] or with angle-resolved detection away from the XUV light polarization axis [29], the separation of the two contributions $\\tau _\\mathrm {W}$ and $\\tau _{\\mathrm {cc}}$ may become ambiguous.",
"Figure: Principle of the interferometric technique.",
"(a) Kinetic energy diagram for ionization from the 2s2\\mathrm {s} and 2p2\\mathrm {p} subshells using XUV (blue arrows) and IR (red arrows) radiation; (b) Time-averaged photoelectron spectrum obtained with Al-Zr-filtered harmonics.",
"For both the 2s2\\mathrm {s} and 2p2\\mathrm {p} shell ionization results in three peaks due to absorption of harmonics (H41, H43 and H45) and two sidebands peaks (S42 and S44) reachable via two-color two-photon transitions.",
"(c) Photoelectron spectrum as function of delay between the XUV pulse train and the IR field.",
"The sideband amplitudes strongly oscillate as a function of delay.",
"The electron yield from 2s2\\mathrm {s} ionization has been multiplied by a factor of 5 for visibility.Figure: Energy-resolved interferometric technique and identification of shake up process.",
"(a) Kinetic energy diagram for 2s2\\mathrm {s} ionization and 2p2\\mathrm {p}-ionization accompanied by 2p→3p2\\mathrm {p} \\rightarrow 3\\mathrm {p} excitation (shake up).",
"The difference in threshold energy for these two processes is approximately 7.4 eV .",
"(b) Photoelectron spectra for XUV only (blue) and XUV + IR (red).",
"The electron peak due to shake-up induced by absorption of H61 partly overlaps with S56 from 2s2\\mathrm {s}-ionization.",
"The shoulder on the S56 (red spectrum) can be attributed to one-photon induced shake up.",
"(c) Energy-resolved amplitude and phase of the oscillation from the RABBITT-spectrogram.",
"The harmonics oscillate out of phase with the sidebands, causing a sudden drop in the energy-resolved phase.",
"The sideband originating from the shake-up state can be distinguished on the right side, allowing for a separate analysis of its time delay.Experimental method.",
"The experiments were carried out using a Ti:Sapphire femtosecond laser system, delivering 20-fs pulses at 1 kHz repetition rate, 800 nm central wavelength with a pulse energy up to 5 mJ.",
"The pulses are fed to an actively stabilized Mach-Zehnder interferometer [30].",
"In one arm, high harmonics of the fundamental laser frequency were generated from a pulsed gas cell filled with neon.",
"The other arm contained a piezoelectric delay stage as well as a half-wave-plate and a broadband polarizer used for adjusting the probe pulse energy.",
"Metallic filters placed in the XUV beam path limited the bandwidth of the XUV-pulses and eliminated the residual IR field present in the pump arm.",
"Two sets of filters were used: a combination of zirconium and aluminum foils of 200 nm thickness each, yielding a narrow band-pass filter over the 60-75 eV range [red spectrum in Fig.",
"REF (b)] and a set of two Zr foils, resulting in a sharp edged high-pass filter above 70 eV [blue spectrum in Fig.",
"REF (b)].",
"The recombined pump and probe pulses were focused by a toroidal mirror into a magnetic bottle electron spectrometer similar to that described previously in [31], with a 2 m long time-of-flight tube and a $4 \\pi $ sr collection angle, and incorporating a set of retarding lenses.",
"This spectrometer design combines a high collection efficiency with good spectral resolution ($\\le $ 100 meV) for low photon energies.",
"Interferometric technique.",
"Fig.",
"REF illustrates the principle of our interferometric measurement when using the Al-Zr filter combination.",
"Two-photon ionization leads to sidebands which can be reached by two pathways: absorption of one harmonic and an IR photon, and by absorption of the next harmonic together with emission of one IR photon [Fig.",
"REF (a)].",
"Ionization of one sub-shell by the high-order harmonics and the IR field results in five electron peaks: three peaks due to single-photon ionization by harmonics 41, 43 and 45 and two sidebands 42 and 44.",
"Since for this filter set the XUV spectrum spans less than 15 eV and the difference in the ionization energies of the Ne $2\\mathrm {s}$ and $2\\mathrm {p}$ subshells is 26.8 eV [12], [13], the spectra generated from the two subshells are energetically well separated [Fig.",
"REF (b)].",
"Fig.",
"REF (c) shows the variation of the spectrum as a function of the delay $\\tau $ between the XUV and IR fields.",
"The intensity of the sidebands oscillates according to [9] $S(\\tau )=\\alpha + \\beta \\cos [2\\omega (\\tau -\\tau _\\textsc {XUV}-\\tau _\\mathrm {A})],$ where $\\alpha $ and $\\beta $ are delay-independent and $\\omega $ denotes the IR frequency ($\\pi /\\omega =1.3$ fs in our experiment).",
"Our analysis consists in determining the phase and amplitude of the signal oscillating at $2\\omega $ by fitting Eq.",
"REF to the experimental data.",
"The delay $\\tau _\\textsc {XUV}$ depends only on the excitation pulse, which is the same for the $2\\mathrm {s}$ and $2\\mathrm {p}$ -ionization paths.",
"The difference in the photoionization time delays can therefore be obtained by comparing the oscillations of the sidebands corresponding to the same absorbed energy (e.g.",
"S42), involving the same harmonics (H41 and H43).",
"This analysis is performed over the bandwidth of the excitation pulse, from 60 to 75 eV in the experiment with the Al-Zr filters [red spectrum in Fig.",
"REF (b)] and from 80 to 100 eV using the Zr-filters [blue spectrum].",
"Shake up.",
"If the different energy components of the sideband are in phase, the analysis can be performed on the energy-integrated signal.",
"In the present work, following the method described in [3], we analyze the sideband oscillations across its spectrum, in steps of 50 meV.",
"Fig.",
"REF illustrates how this method allows us to identify shake up processes and eliminate their influence on the $2\\mathrm {s}$ -time delay measurement.",
"In Fig.",
"shakeup(a), we indicate two competing ionization pathways leading to overlapping electron spectra: $2\\mathrm {s}$ -ionization by absorption of H57 and emission of one IR photon (S56); $2\\mathrm {p}$ -ionization and excitation $2\\mathrm {p} \\rightarrow 3\\mathrm {p}$ by absorption of H61; Similarly, $2\\mathrm {s}$ -ionization by absorption of H57 and two-photon shake up (H61+IR) overlap.",
"Although a number of shake up processes come into play at photon energies above 50 eV, shake up to the $2\\mathrm {p}^4(^1\\mathrm {D})3\\mathrm {p}(^2\\mathrm {P}^0)$ state, with binding energy equal to 55.8 eV, is the most intense [12], [32], reaching one sixth of the amplitude of $2\\mathrm {s}$ -ionization, and is thus comparable to a $2\\mathrm {s}$ -sideband.",
"A comparison between the photoelectron spectra with and without IR shown in Fig.",
"shakeup(b) shows the effect of shake up on the right side of the $2\\mathrm {s}$ -sideband.",
"In Fig.",
"REF (c), the amplitude and phase of the $2\\omega $ oscillation is shown as a function of energy.",
"The phase is strongly modified in the region of overlap between $\\mathrm {H}61_{\\mathrm {su}}$ and $\\mathrm {S}56_{2\\mathrm {s}}$ .",
"In general, harmonic and sideband oscillate out of phase, so that, with poor spectral resolution, even a weak shake up harmonic signal strongly influences the phase of a partially overlapping $2\\mathrm {s}$ -ionization sideband signal.",
"The spectrally-resolved phase of the $2\\mathrm {p}$ -sidebands (not shown) is completely flat, owing to the fact that this region is void of resonances [28] or shake-up states [32].",
"The time delays indicated in Fig.",
"REF (b) have been obtained by selecting a flat spectral region for the $2\\mathrm {s}$ -phase determination, avoiding shake-up processes.",
"We could also estimate the difference in time delay between shake-up and $2\\mathrm {p}$ -ionization to $-70\\pm 25$ as, by analyzing the shake-up sidebands amplitude and phase [see $\\mathrm {S}62_{\\mathrm {su}}$ on the right side in Fig.",
"REF (c)].",
"Comparison of theory and experiment.",
"The key results obtained in the present work are summarized in Fig.",
"REF (a).",
"For the experimental results [red and blue dots, corresponding to the spectra shown in (b)], the indicated error bars correspond to the standard deviation from ten spectrograms, weighted with the quality of the fitted sideband oscillations.",
"The difference in time delay is negative, which indicates that $2\\mathrm {p}$ -ionization is slightly delayed compared to $2\\mathrm {s}$ -ionization, and decreases as the excitation energy increases.",
"Unfortunately, we could not determine delays at energies higher than 100 eV due to overlap between electrons created by $2\\mathrm {s}$ -ionization with 100 eV photon energy and those by $2\\mathrm {p}$ -ionization with 70 eV.",
"The difference in ionization energy between the two subshells corresponds almost exactly to 17$\\omega $ , which hinders any spectral analysis.",
"Fig.",
"REF (a) also presents calculations using a many-body perturbation theory approach for the treatment of electron correlation effects [33], [20].",
"Here, we calculate $\\tau _\\mathrm {A}$ by using lowest-order perturbation theory for the radiation fields.",
"The interaction with the XUV photon is assumed to initiate the photoionization process with many-body effects included to the level of the random phase approximation with exchange.",
"The laser photon is then assumed to act perturbatively on the photoelectron to drive a transition to an uncorrelated final state.",
"The final state is computed by solving an approximate Schrödinger equation with a static spherical potential of the final ion.",
"Special care is taken that the laser dipole interaction of these two continuum waves is computed to radial infinity.",
"Using this method with ab-initio Hartree-Fock energies, it has been predicted [33] that the atomic delay from the $2\\mathrm {p}$ state in neon is rather insensitive to interorbital correlation, while the coupling of the $2\\mathrm {s}$ orbital is advanced by a few attoseconds due to coupling to the $2\\mathrm {p}$ orbital.",
"Here the calculations are improved further by using the experimental binding energies of $2\\mathrm {p}$ and $2\\mathrm {s}$ .",
"The two-photon ionization amplitude is averaged over all emission angles ($\\theta $ ) to mimic the experimental conditions.",
"We emphasize that the excellent agreement obtained between theory and experiment for the difference in time delays between $2\\mathrm {s}$ and $2\\mathrm {p}$ ionization requires the careful energy-resolved analysis presented above and the disentanglement between $2\\mathrm {s}$ -ionization and shake up.",
"Figure: NO_CAPTIONAbsolute photoionization time delays.",
"Fig.",
"REF presents more details about the calculations and illustrates the contributions to the measured time delay differences.",
"In Fig.",
"REF (a), the black curve represents the Wigner delay $\\tau _\\mathrm {W}$ for $2\\mathrm {p}$ -ionization, calculated for an emission angle in the direction of polarization, while the dashed green curve is the angle-averaged time delay, defined as $\\tau _1=\\tau _\\mathrm {A}-\\tau _{\\mathrm {cc}}$ (for the calculation of $\\tau _{\\mathrm {cc}}$ , see [26]).",
"The difference between the two curves is at most two attoseconds, which indicates a very small angle-dependence of the $2\\mathrm {p}$ time delay [28].",
"Indeed, in this energy region, ionization towards the $\\mathrm {s}$ continuum is much lower than towards the $\\mathrm {d}$ continuum, which justifies our interpretation of $\\tau _1$ in terms of Wigner delay for the $\\mathrm {d}$ -channel.",
"Fig.",
"REF (b) shows the same quantities for $2\\mathrm {s}$ -ionization.",
"Here, the difference between $\\tau _1$ and $\\tau _\\mathrm {W}$ is not visible, which also justifies the interpretation of $\\tau _1$ as Wigner delay.",
"The red, blue and green dots have been obtained by subtracting from the experimental data [see Fig.",
"REF (a)] the calculated $\\tau _\\mathrm {A}(2\\mathrm {p})=\\tau _1(2\\mathrm {p})+\\tau _{\\mathrm {cc}}(2\\mathrm {p})$ and the continuum-continuum contribution $\\tau _{\\mathrm {cc}}(2\\mathrm {s})$ , thus extracting absolute Wigner delays for $2\\mathrm {s}$ -ionization.",
"The $2\\mathrm {s}$ and $2\\mathrm {p}$ ionization time delays at 100 eV are approximately -5 and +3 attoseconds, leading to a difference of -8 as.",
"The energy-increasing, larger delays observed in Fig.",
"REF (a) reflect essentially the energy dependence of $\\tau _{\\mathrm {cc}}(2\\mathrm {s})-\\tau _{\\mathrm {cc}}(2\\mathrm {p})$ , which itself is dominated by the variation of $\\tau _{\\mathrm {cc}}(2\\mathrm {s})$ .",
"Other calculations of the Wigner delays [34] agree to within a few as with our theoretical results and therefore with the experimental data.",
"We have also compared our results with theoretical calculations in the conditions of a streaking experiment, i.e.",
"with a stronger IR field and a single attosecond pulse [32].",
"The calculated Wigner delay agrees very well with the data presented here.",
"In summary, we have presented experimental data and numerical calculations of the photoionization time delays from the $2\\mathrm {s}$ and $2\\mathrm {p}$ shells in neon for photon energies ranging from 65 eV up to 100 eV and retrieved the Wigner delay of the electronic $2\\mathrm {s}$ wave-packet.",
"The good agreement obtained gives us confidence in this type of measurement, and point out the necessity for keeping high frequency resolution in addition to high temporal resolution.",
"We also carried out an energy-integrated instead of energy-resolved analysis of the sideband oscillations and obtained time delay differences which were often below those indicated in Fig.",
"REF (a), actually close to that retrieved by Schultze et al. [7].",
"This leads us to suggest that the discrepancy of the latter result with theory [18], [19], [20] might be due to the influence of shake up processes, not spectrally resolved in the experiment and not included in the theory (see, however, a detailed theoretical analysis including shake up processes in [32]).",
"Our method can be significantly improved by using attosecond pulse trains generated with long laser pulses and/or in the mid infrared region.",
"The long pulse duration allows the generation of stable attosecond pulse trains with many pulses (and thus narrow harmonic bandwidth) while the long wavelength leads to broad XUV spectra [36], [37] and better energy sampling.",
"The door is open to the study and control of photo-induced processes both in the time and frequency domain from the visible to the x-ray range.",
"This research was supported by the European Research Council (Advanced grant PALP), the Swedish Research Council and the Knut and Alice Wallenberg Foundation.",
"J.M.D.",
"was funded by the Swedish Research Council, Grant No.",
"2014-3724."
]
] | 1709.01780 |
[
[
"Detection of a repeated transit signature in the light curve of the\n enigma star KIC 8462852: a 928-day period?"
],
[
"Abstract As revealed by its peculiar Kepler light curve, the enigmatic star KIC 8462852 undergoes short and deep flux dimmings at a priori unrelated epochs.",
"It presents nonetheless all other characteristics of a quiet 1 Gyr old F3V star.",
"These dimmings resemble the absorption features expected for the transit of dust cometary tails.",
"The exocomet scenario is therefore most commonly advocated.",
"We reanalyzed the Kepler data and extracted a new high-quality light curve to allow for the search of shallow signature of single or a few exocomets.",
"We discovered that among the 22 flux dimming events that we identified, two events present a striking similarity.",
"These events occurred 928.25 days apart, lasted for 4.4 days with a drop of the star brightness by 1000 ppm.",
"We show that the light curve of these events is well explained by the occultation of the star by a giant ring system, or the transit of a string of half a dozen of exocomets with a typical dust production rate of 10$^5$-10$^6$ kg/s.",
"Assuming that these two similar events are related to the transit of the same object, we derive a period of 928.25 days.",
"The following transit was expected in March 2017 but bad weather prohibited us to detect it from ground-based spectroscopy.",
"We predict that the next event will occur from the 3rd to the 8th of October 2019."
],
[
"Introduction",
"KIC 8462852 is a peculiar and intriguing source that caught a lot of attention from the astronomic community in the recent years [8], [31], [26], [41], [5], [13], [35], [27], [36], [38], [1], [28], [14], [33].",
"KIC 8462852 is an F3V star located at about 454$\\pm $ 35 pc away from us [14].",
"The Kepler Spacecraft photometric data revealed an enigmatic lightcurve for this star, with erratic, up to $\\sim $ 20% deep, stellar flux dimmings [8].",
"Being otherwise considered a standard F-star, stellar unstabilities could be excluded to explain its strange behavior.",
"More recently, it was found that KIC 8462852's flux dropped by $\\sim $ 2.5% over 200 days [31], while it was suspected from a thorough analysis of photographic plates taken over the last century to continuously decrease by about 0.3% yr$^{-1}$ [35].",
"The most popular scenario advocated to explain the frequent but aperiodic dips is that of many uncorrelated circumstellar objects transiting at different epochs; either comets [8], [33] or planetesimal fragments [5].",
"This is reminiscent of the case of $\\beta $ Pictoris, on the spectra of which many variable narrow absorptions were observed at high-resolution in Ca II doublet, best explained by extrasolar comets, or exocomets [11], [4], [18].",
"It should be noted however that contrary to $\\beta $ Pictoris, KIC 8462852 is not young (1 Gyr), and any circumstellar gas or dust remain unobserved at infrared wavelengths.",
"About 20 years ago, [23] published innovative simulations of photometric signatures produced by the transit of the dusty tail of exocomets.",
"The shape of the theoretical absorption signatures obtained has some unique specificities: a peaky core for the transit of the head of the coma, and a long trailing slope.",
"Nonetheless, the only direct evidence for comets around stars other than the Sun came from high-resolution spectroscopy, observing the atomic gas counterpart of the cometary tail, with e.g.",
"$\\beta $ Pic [11], HD172555 [19], HR10 [21] or 49 Ceti [32], [30].",
"Before KIC 8462852, photometry never revealed any direct observations of exocomets around any star.",
"The level of precision needed to detect the transit of a single $\\beta $ Pic like exocomet is about several 100 ppm [23].",
"Detecting such object is a difficult task, since a single solar-like exocomet cannot be expected to transit several times during the lifetime of Kepler (in the Solar System, the comets have period typically larger than 3 years).",
"Nevertheless, the opportunity of detecting repeated transits should not be completely excluded.",
"We selected KIC 8462852 for thorough analysis of its Kepler lightcurve with the goal of finding single object 100 ppm-deep transit signatures.",
"We report in the present paper the detection of a 1000 ppm deep signature repeating twice at 928 days interval in KIC 8462852 lightcurve.",
"We successfully modeled this signature by a string of exocomets crossing the line of sight one after the other at 0.3 AU from the central star.",
"Alternatively, we found that at least one other scenario could provide a good fit of the lightcurve: the transit of a wide ring system surrounding a planet orbiting at 2.1 au from the star.",
"Hill-sphere could indeed become much wider than the star itself at distances larger than 1 au, and contain transiting materials such as rings [17], [25], [2].",
"Moreover, while plausible and simplistic, such scenario was recently proposed by [3] to explain the smooth and solitary D800 dip of [8], with a transiting ring planet on a 12 years orbit around the star.",
"Thus, instead of being a comets host, KIC 8462852 might just be a planetary system with at least two ring planets.",
"Kepler data reduction is presented in Section 2.",
"The two identitical 1000 ppm deep events are presented in Section 3.",
"In Section 4 we show that these events are real and not instrumental or due to background objects.",
"The modelisation of these events are presented in Section 5.",
"Finally, we present in Sect.",
"6, an attempt to observe that event in March 2017 which failed due to bad weather, and a prediction on its future realisations in October 2019 and later."
],
[
"The Kepler photometric data reduction",
"The Kepler spatial observatory [6] followed KIC 8462852 in long cadence mode (30-min sampling) during about 4 years from the 2nd of May, 2009 to the 11th of May, 2013, separated into 17 quarters of continuous integration.",
"The Kepler pipeline produced raw (simple aperture photometry, SAP) and reduced (pre-search data conditioning, PDC-SAP) lightcurves of the full 4 years time range [37].",
"The SAP data essentially consist in calibrated flux but uncorrected of cosmic ray absorption, systematic behaviours, jumps etc.",
"The PDCSAP lightcurve are systematically corrected for every trend of non-astrophysical origin by the reduction pipeline.",
"While the PDCSAP data are certainly good enough for detecting short 0.1-1% deep transits, they do not reach the level of precision needed to detect 0.1% deep, possibly day-long, absorption signatures that could be typically produced by transiting exocomets [23], [18].",
"We thus wrote our own MATLAB-routine to carefully reduce the SAP lightcurve, which principles are explained below.",
"The Kepler pipeline determines for each quarter, in each CCD-channel, an ensemble of 16 Cotrending Basis Vector (CBV) calculated from the lightcurve of the brightest stars in the channel using Principal Component Analysis (PCA) [20], [37].",
"These CBVs represent the main systematic behaviours common in the lightcurves collected on the same channel during the same quarter.",
"For each quarter, we fitted to KIC 8462852's SAP lightcurve a linear combination of the 13 first CBVs.",
"Following the recommended process [20], we increased iteratively the number of fitted CBVs.",
"We stopped when the resulting lightcurve baseline was the flattest in the quiet periods, and that adding more CBVs led to no significant improvement.",
"In order to avoid fitting out the physical dips, we iteratively excluded from the fit any data points below the continuum minus 2$\\sigma $ , with $\\sigma $ defined as $\\sigma = 1.48\\times \\text{MAD(data-continuum)}$ The median absolute deviation (MAD) times 1.48 is an estimation of the standard deviation, not biased by outliers.",
"Jumps and spikes (cosmics etc.)",
"were carefully filtered out before applying the fit.",
"This is done by studying the first-derivative of the light curve and identifying spikes and jumps signatures.",
"They appear as single or P-cygni shaped 3-5 cadences-long peaks, with an amplitude at least 4-times larger than the local typical cadence-to-cadence variations.",
"None-to-ten measurements are found in such discontinuities per quarter.",
"If a spike is encountered, the bad cadences are first removed and then the light curve is linearly interpolated through the resultant gap.",
"If a jump is encountered, the bad cadences are removed and the lightcurve separated in two pieces around the gap; in this case, the CBVs are fitted out to each piece separatively.",
"More generally, anytime there is missing data (typically more than 25 adjacent cadences) we separated the lightcurve in two pieces around the gap and fitted out CBVs independently for these two pieces.",
"Most of these large discontinuities are due to monthly Earth downlink and usually followed by thermal relaxation [20].",
"Even though most of the time the CBVs capture such variation, simple fitting ignoring cadences within the gap was not accurate enough.",
"Separating the lightcurve around these discontinuities led to better results.",
"Several examples of the detrending results are displayed on Figs.",
"REF .",
"We compare them to the pipeline automatic PDC reduction, which in general presents quarter-long low amplitude variations along the curve, and especially around strong dips.",
"With our reduction the continuum is flat, allowing the shallower dips to emerge more evidently than in the PDCSAP data.",
"This shows the positive effect of excluding the dips measurements when fitting out the systematics.",
"The full detrended lightcurve is plotted on Fig.",
"REF ."
],
[
"Two identical photometric shallow events",
"Using the light curve obtained in the previous section, we can identify the photometric events that happened during the four years of observation.",
"The observed photometric variations shows two different patterns: there are (1) periodic-like variations and (2) short-time decreases of the star brightness.",
"With a period close to 1 day, the periodic modulation corresponds to the 0.88-day signal due to stellar rotation and already noticed by [8].",
"Beyond these variations, the stars shows significant short-time and sporadic variations, all of them are dips of the star brightness below the mean brightness observed during the quiet period.",
"We screen the entire light curve and identified a total of twenty-two significant dips.",
"Apart from the strong dips already listed by [8], we found several shallower dips, some of them also identified by [27].",
"Table REF summarizes these detections.",
"Table: List of detected photometric dips in KIC 8462852 lightcurve.",
"Events 2 and 13 are renamed respectively A and B in the rest of the paper.Among the detected features, two events show a remarkable similarity in shape, duration and depth : the events #2 and #13 in Table REF .",
"Hereafter, we label these events as “event A” and “event B”.",
"The light curves of these two events are plotted in Fig REF .",
"In this figure, we superimposed the raw SAPs, fitted CBVs and corrected SAPs, showing that the two photometric dips are real and not produced by the data analysis procedure.",
"As indicated in Table REF , events A and B were already noticed by [27] but they were suspected to be due to either PSF centroid modulation (event A) or instrumental jitter (event B).",
"In section 4, we show that these events are of astrophysical origin and not related to instrumental systematics.",
"Fitting events A and B together, we derived a time separation between them of $\\Delta t$ =928.25$\\pm $ 0.25 days.",
"The errorbars on the flux were scaled to obtain a reduced $\\chi ^2$ of 1.",
"Shifting the second event light curve by -$\\Delta t$ , i.e.",
"on top of the first event light curve, we obtained the strikingly almost perfect superimposition of the two events as displayed in Fig.",
"REF .",
"To characterize the similarity of these two photometric events and compare them among the 22 detected events, we plotted the depth and the duration of each of them (Fig.",
"REF ).",
"The depths are measured between the continuum level fixed to 1 and the bottom of the light curve defined as the 5$^\\text{th}$ lowest pixel.",
"The durations are measured by calculating the second moments of the variations, which are then multiplied by $2\\sqrt{2\\ln 2}$ to roughly correspond to the full width at half maximum.",
"While other photometric events show a wide diversity in duration and depth, events A and B are remarkably identical.",
"To emphasize this result, we superimposed the light curves of the photometric events #6 and #9, which are, after the events A and B, the closest in the depth-duration diagram (Fig.",
"REF ).",
"It is clear that these photometric events do not show similar shapes of the light curves, as the events A and B do.",
"The shape of the light curves of the events A and B can be obtained by fitting a simple 4-vertices polygon to each curve.",
"We measure the quantities such as ingress, egress and centroid timings, slopes of the left and right wings, and transit depth (Table REF ).",
"These simple fits quantitatively confirm that the two events are strikingly similar.",
"The average duration of the two events from ingress to egress is measured to be 4.44$\\pm $ 0.11 days.",
"The bottom of the light curves are flat with a duration of about 1 day.",
"The two slopes on each side of the flat bottom are straight, with comparable duration between 1.5 and 2 days.",
"The right wings are steeper than the left wings, with respective slopes of about 700 ppm/day and -500 ppm/day.",
"The transit depths are similar in both events at about 1010$\\pm $ 40 ppm.",
"Table: The 4-vertices polygon parameters of the fit to the events A and B light curves.We used a 4-degree polynomial to fit out the baseline, and assumed that the bottom of the lightcurve is flat.Beginning-of-ingress, centroid and end-of-egress timings are given in days past Kepler initial epoch at MJD 2454833.If real, these similar events could be the repeated observation of a same, identical, and periodic phenomenon.",
"After checking for potential reduction artefacts and other systematics in the next section, we will discuss interpretations of this repeating event in Sect. .",
"Figure: Light curves at the time of the photometric events A (upper panel) and B (lower panel).The curves show the detrended (red) and undetrended (blue) data sets.",
"The fitted CBVs continuum is plotted with a green line.Figure: The 2 events superimposed with 3-pixels binning.",
"The bottom x-axis shows the time at the first photometric event A (red line).",
"The top x-axis shows the time at the second photometric event B (blue line), with a shift of 928.25 days relative to the bottom axis.Figure: Plot of the depth and duration of the 22 events cataloged in Table .The events A and B are shown by red and blue symbols, respectively.The events #6 and #9 are shown by green and orange symbols, respectively.Figure: The events #6 and #9 superimposed with 3-pixels binning.",
"The bottom x-axis shows the time at the first photometric event #6 (green line).",
"The top x-axis shows the time at the second photometric event #9 (orange line), with a shift of 316.3 days relative to the bottom axis."
],
[
"Possible bias",
"[27] argued that some of the small amplitude features could be of instrumental or background origin.",
"We thus carefully inspected the pixel tables collected by Kepler around the star's point spread function (PSF) and used to produce the raw SAP lightcurve about the epochs of the two events, in order to exclude any instrumental origin for these events, or the possibility of close background stars contamination."
],
[
"Background stars",
"The closest known stars in the field are Gaia 2081900738645631744 (at 5.4\" with $m_G$ =18.1), referred to as Gaia-208 in the following, and the infrared sources 2MASS J20061551+4427330 (at 8.9\" with $m_J$ =16.1, $m_G$ =18.9) and 2MASS J20061594+4427365 (at 11.83\" with $m_J$ =16.4, $m_G$ =18.7).",
"Their high visual magnitude measured by Gaia [39] implies a $\\Delta V$$>$$6.4$ with KIC 8462852 ($m_G$ =11.7), i.e.",
"a flux ratio $<$ 0.3%.",
"As can be seen on Fig.",
"REF , the pixels corresponding to the theoretical location of the 2 IR sources on the CCD channel are out of the aperture used to calculate the raw lightcurve.",
"We see that KIC 8462852's flux is smeared on an area about 10$\\times $ 10 arcsec$^2$ wide.",
"Since the PSF of the two red stars is likely of similar extension, a bit less than half the flux of 2MASS J20061551+4427330 enters the PSF, while there is almost none for 2MASS J20061594+4427365.",
"Consequently, any flux variation of these stars of order 100% will contaminate the flux to a level lower than 0.1%.",
"However, since the PSF of Gaia-208 almost fully overlap with the PSF of KIC 8462852, the photometric variations of this polluting star might induce variations in the lightcurve, but in any case not higher than 0.3%.",
"At this stage, while we are able to exclude contamination from the two IR background stars, contamination from Gaia 208 cannot be ruled out, although likely negligible.",
"In section REF , we show that no significant and correlated PSF motion of KIC 8462852 is observed during the two events; this advocates for rejecting contamination from any background stars."
],
[
"Lightcurve of closest neighbour KIC 8462934",
"KIC 8462934 is the closest bright star (about 89\" with V$\\sim $ 11.5) to KIC 8462852 (V=12) with a recorded lightcurve in the Kepler Database.",
"Applying our previously introduced detrending method, we recovered a detrended lightcurve using the 13 first CBVs of each quarter/channel, as was applied to KIC 8462852's lightcurve (see Section ).",
"We found no peculiar behaviour, neither strong nor shallow absorptions similar to what observed on KIC 8462852.",
"No features were detected in the lightcurve beyond a few 10$^{-4}$ in normalized flux.",
"This indicates that the CBVs derived by the Kepler pipeline did not miss any small local variations in e.g.",
"pixel sensitivity in the CCD-channel.",
"Therefore, events A and B are indeed features from the local pixels in the photometric aperture shown in Fig.",
"REF ."
],
[
"KIC 8462852's PSF motion on CCD pixels",
"[27] studied the correlation of the PSF centroid motion and the flux dimming in KIC 8462852.",
"They show that a few features could be artifact of background objects occultation, or instrumental jitter.",
"Repeating a similar analyzis on the pixel images collected by Kepler at each cadence, such as presented in Fig.",
"REF , we derived about the epochs of each identified event the centroid motion of KIC 8462852's PSF.",
"Apart from an expected slow shift (0.004 pixel day$^{-1}$ ) of the star's location on the CCD channel during the quarter, we observed that the centroid oscillates with a period of about 3 days and an amplitude of at most a few 0.001 pixel (Fig.",
"REF ).",
"This oscillation is likely related to a vibration mode of the instrument.",
"Consistently, it was found that a few pixels show shallow flux modulations of a few 0.1% in correlation or anti-correlation with the PSF centroid oscillations.",
"Impressively, these instrumental modulations exactly cancels out.",
"We found no counterpart for these modulations in the raw lightcurve, demonstrating the excellent quality of the flat field determination made by Kepler on the aperture.",
"Since the amplitude of the 3 days-modulation is similar to the amplitude of the two events discussed in this paper, instrumental PSF variations cannot be at their origin.",
"Indeed, in such case, we would have observed 0.1% deep 3 days-modulation rather than only single dips.",
"We have seen in Section REF that the closest background star, Gaia-208, is located at 5.4\" from KIC 8462852.",
"This is a bit more than 1 pixel apart, on the aperture (Fig.",
"REF ).",
"The luminosity variation due to the occultation of a third of Gaia-208 stellar disk would be close to 0.1%, leading to a PSF centroid variation of about 10$^{-3}$ pixels.",
"We can exclude from Fig.",
"REF a PSF centroid variation of this amplitude during event A, between day-213.3 and day-217.8.",
"Repeating this analysis for event B led to the same conclusion.",
"We thus exclude for both events any significant PSF motion correlated with the lightcurve, eliminating background star pollution, and instrumental variations as possible origin.",
"Figure: Pixel table of flux around KIC 8462852, collected during quarter # 2 on CCD-channel 32.",
"The colors are logarithmically scaled with the flux.",
"The aperture is displayed in solid red line.",
"A right ascension and declination grid is superimposed in dotted black lines.",
"The theoretical position of KIC 8462852 on the CCD is depicted as a red star; the position of the two faint IR sources 2MASS J20061551+4427330 & 2MASS J20061594+4427365 are marked by white stars; and Gaia 2081900738645631744 appears as a black star.Figure: Upper panel: position of the PSF centroid around event A, from epochs 208 to 225 (in blue), and highlighted in red, from ingress (213) to egress (220).",
"Time arrow goes from top right to bottom left.",
"Lower panel: PSF centroid motion around the 4th degree polynomial fit of the main trend in the upper panel.",
"The modulation period is about 3 days."
],
[
"Local pixel variations",
"As a final check, we verified the collective variations of the local flux in each pixel around the PSF of KIC 8462852 at different times between the beginning and the end of both events.",
"It clearly appeared that the whole image of the star was fainting during the dimming events, thus confirming that its origin is neither related to background objects occultation, nor associated to any instrumental PSF motion."
],
[
"Models",
"We can try to explain the repeated photometric event, observed 928 days apart.",
"The observed variations correspond to a dip in the star brightness by about 0.1%, which lasted for about 4.4 days.",
"A decrease in the star brightness (moreover in a context of a star showing multiple photometric variations, always in the form of brightness decrease) suggests an explanation by the transit of a partially occulting body.",
"With that in mind, the duration of the event ($\\sim $ 5 days) is puzzling.",
"With a possible period of 928 days, and assuming a mass of 1.4 solar mass for the F3V central star, the corresponding semi-major axis is 2.1 au and the orbital velocity on a circular orbit is 24.4 km s$^{-1}$ .",
"At this transiting velocity, the maximum transit time in front of a $R_*=1.3R_\\odot $ star is about 10.3 hours.",
"Even on an highly eccentric orbit and observed at apoastron, the transit of a body on a 928-days period orbit cannot last longer than 14.6 hours.",
"Therefore, the photometric events of 4.4-days can be explained by the transit of an occulting body only if this body is significantly larger in size that the star ; in this case, the duration of the transit is related to the size of the object itself.",
"The main scenario for explaining the other deeper dips in the KIC 8462852 light curve invokes the transit of trains of extrasolar comets [8], [5] or planet fragments [29].",
"In fact, the photometric variations observed in KIC 8462852 light curve look like the spectroscopic variations observed in $\\beta $ Pictoris, which can last several days and are interpreted by the transit of exocomets [11], [22], [40], [18].",
"We will explore this scenario in Sect.",
"REF .",
"Nevertheless, keeping the idea of a transiting body, we can imagine another possible scenario to explain the repeated photometric events A and B.",
"The straight ingress and egress slopes, and the flat bottom of the light curve point toward the possibility that the transiting body can be a single body with a simple shape.",
"Acknowledging that the Hill-spheres of a massive planet can extend to several stellar radius in size, the transit of a ring system surrounding a giant planet could explain the observed photometric event A and B, as done for the light curve of 1SWASP J140747.93-394542.6 (also named J1407), an old star in the Sco-Cen OB association [17]; see also [25] and [2] for other case studies of exo-planetary ring systems.",
"This scenario will be discussed in Sect.",
"REF ."
],
[
"The comets string model",
"In the exocomets scenario, the duration of the transit event in the light curve implies that several comets passed in front of the star, within an extended string long of several millions of kilometers.",
"While we do not aim at exploring the whole range of possibilities to fit the events A and B light curves, we could use some of the exocomet tail transit signatures given e.g.",
"in the library of [24] to show that a generic transit model of a few trailing exocomets can easily provide a satisfactory fit to the data.",
"As a reference light curve, we decided to use the light curve labeled '20_F_50_03_p4_00' in [24], and plotted on Fig.",
"REF .",
"It is obtained through the simulation of cometary tails orbiting an F star with a periastron of 0.3 au, a longitude of periastron of 90, and a production rate of 10$^5$ kg/s at 1 au [23].",
"It assumes a grain size distribution given by $dn(s) = (1-s_0/s)^m s^{-n} \\, ds$ , with $s_0$ =0.05 $\\mu $ m, $n$ =4.2, $m$ =$n(s_p-s_0)/s_0$ , and peaking at $s_p$ =0.2 $\\mu $ m. This distribution is derived from observations in solar system comets at less than 0.5 au from the Sun.",
"The physical model used to calculate the photometric transit signatures of exocomet tails is discussed in depth in [23].",
"The choice of the characteristics (the orbit and the dust production rate) of the specific exocomet for the reference light curve is not critical, because all the transit light curves show a similar triangular shape.",
"At a fixed distance to the star, the transit depth of an individual light curve is constrained by the dust production rate, and the duration is mainly related to the longitude of the periastron.",
"The depth of the global light curve resulting from the transit of a string of several exocomets therefore depends on the production rate of each exocomet.",
"However, the duration of the global light curve is not constrained by the duration of each individual transit, but by the spread of the transit time of each exocomets.",
"To simplify the fit to the data, we approximated the reference light curve of a single comet by a piecewise linear function.",
"Each individual exocomet lightcurve is defined by two parameters: the time of mid-transit $T_0$ , and the maximum occultation depth, $\\Delta F/F$ .",
"Exploring the library of [24], we find that in the range 10$^4$ -10$^6$ kg/s the maximum occultation depth is related to the dust production rate, $\\dot{M}$ , by $\\log _{10} \\dot{M}/(1\\,{\\rm kg\\,s^{-1}}) =5+1.25\\times \\log _{10}(\\Delta F/F /10^{-4})$ .",
"We fitted the average light curve of events A and B with a combination of several individual light curves defined by $T_{0k}$ and $\\dot{M}_k$ for each comet $k$ of the string.",
"With N comets in the string, the total number of parameters reaches $2N+1$ with 2 parameters per comet and one for the baseline level (slightly larger than 1).",
"Given the possibly large number of parameters, we used a Markov-chain Monte Carlo (MCMC) algorithm as a fitting procedure.",
"The best fit is obtained for 7 comets, including the feature at the top of the signature left wing.",
"It is plotted in Fig.",
"REF with the parameters given in Table REF and plotted in Fig.",
"REF .",
"The dust production rates obtained for the comets are typical of Hale-Bopp type comets in the Solar System, i.e.",
"between 10$^5$ and 10$^6$ kg s$^{-1}$ [15].",
"If we consider we are actually overfitting stellar variations, we could accept a poorer fit with residuals in the order of the mean amplitude of the stellar variations.",
"In this case, 5 comets are enough to fit satisfyingly the average light curve.",
"An example of such a fit is shown in Fig.",
"REF with the parameters given in Table REF and plotted in Fig.",
"REF .",
"Here we used a different longitude of periastron of 112.5 and a different grain size distribution labeled '50' in [24], peaking at 0.5 $\\mu $ m. This shows that the observations cannot constrain the comets properties and that the comets model can easily explain the data without any fine tuning of parameters.",
"Therefore, the values given in Table REF should not be considered as measurements on existing bodies, but as possible values for a generic model of a string of exocomets.",
"Interestingly, both resulting models are reminiscent of the case of the Solar System comet Shoemaker-Levy 9 (SL9).",
"Figure REF , bottom-panel, shows the distribution of diameters, $D$ (in log-space), with respect to timing of impact with Jupiter of all 21 fragments of SL9 [12], [9], [10].",
"Since the dust production rate is proportional to the surface of the nucleus (all other things equal), $\\log D$ of SL9 fragments could be compared to $\\log \\dot{M}$ of events A and B comets (Fig.",
"REF , top-panel).",
"We see that in both cases, the distribution of size (evaporation rate) is mainly flat with decreasing size of the comet nuclei at the head and tail of the fragments string.",
"This tentatively suggests events A and B could be the break-up remnants of a bigger body along its orbit.",
"If the periodicity of this transit is later confirmed, non-gravitational effects should be properly modelized to take into account a slow relative drift of the fragments.",
"Figure: Fit to the light curve using a string of 7 exocomets.The light curve of each exocomet is given by the thin black lines.The data of the events A and B are plotted with the red and blue thin lines,and the co-addition of the two light curves is given by the thick black line.The best fit is plotted with the thick green line.Figure: Same as Fig.",
"using a string of 5 exocomets.Figure: Top:The dust production rates and the transit times of the transiting bodies for the 7 comets model(black dots; Figure )and the 5 comets model (red squares; Figure ).Bottom: Shoemaker-Levy 9 fragments diameter versus epochs of impact with Jupiter's atmosphere.Table: Best fit parameters for the models with the transit of 7 or 5 exocomets.",
"The error bars correspond to 3-σ\\sigma and have been estimated using MCMC.",
"The central transit time, T 0 T_0, are given for the event A, a constant of 928.25 days must be added for the event B."
],
[
"The planetary ring model",
"Here we discuss another possible scenario consisting in the transit of a giant ring system surrounding a planet with a 928-days orbital period (2.1 au semi-major axis).",
"Indeed, a ring system can be stable within half a Hill-sphere radius of a planet.",
"Around a massive planet the Hill-sphere can extend up to several stellar radii in size ; therefore rings [17], [25], [2] or dust envelope like e.g.",
"Fomalhaut b [16] can be large enough that the transit duration can reach up to a few days.",
"For instance, the Hill-sphere of Jupiter extends up to 0.34 au (73 $R_\\odot $ ).",
"To model this scenario, we take the reference frame linked to the planet, and consider that the star transits behind the rings.",
"To simplify the problem, we assume that i) the planet moves on a circular orbit at 2.1 au (v$_\\text{transit}$ =24.4 km s$^{-1}$ ); and ii) that the rings are seen face-on.",
"We consider two simple models of rings, and fitted them to the data using Levenberg-Marquardt minimization of the $\\chi ^2$ : 1.",
"The first model consists in a large circular homogenous, constant opacity, ring with a non-zero impact parameter of the star's trajectory behind the ring during the transit (Fig.",
"REF , left panel).",
"In this case, the signature of the transit is round-shaped.",
"The data are best fitted with a ring exterior diameter of 8.8 $R_\\star $ , an impact parameter of 8.5 $R_\\star $ and an extinction $\\tau $ =0.0014.",
"Nonetheless, this model do not provide a good fit to the data, which show straight wings and a flat bottom.",
"2.",
"In the second model, the ring is made of an inner core of constant opacity for $r$$<$$R_{\\rm const}$ and an external ring with an extinction decreasing with the distance to the star following $\\propto $ $r^{-\\alpha }$ for $r$$>$$R_{\\rm const}$ (Fig.",
"REF , right panel).",
"As can be seen in the figure, this model provides a much better fit to the data.",
"Using a zero impact parameter, the best fit is found with an outer radius of 4.86$\\pm $ 0.15 $R_\\star $ , an interior core of radius $R_{\\rm const}$ =1.91$\\pm $ 0.03 $R_\\star $ with constant extinction $\\tau $ =(9.9$\\pm $ 0.1)$\\times $ 10$^{-4}$ , and an extinction parameter $\\alpha $ =1.70$\\pm $ 0.06.",
"We tried more sophisticated models by introducing elliptical rings, non-zero impact parameter and a non-zero position angle of the ellipse major-axis with respect to the transit direction (model #3 in Table REF ).",
"The improvement of the fit is significant but only indicates that the rings as seen for Earth are likely elliptic ($e$ >$0.8$ ) and not aligned with the transit direction.",
"This is in accordance with the observed asymmetry on the slopes of the left and right wings, as explained in Section .",
"Since the projection of an inclined circle is an ellipse, the eccentric solution corresponds to a circular ring system inclined with respect to the plane-of-the-sky at angle $\\theta $ ($=$$\\arcsin e$ )$>$$53^\\circ $ .",
"Interestingly, [3] recently proposed that the two main dips (D800 and D1500) of KIC 8462852 could be related to a ring planet on a 12 years orbit, with trailing trojans at the L5 point.",
"If true, KIC 8462852 might be the first exoplanetary system with two ring planets detected.",
"Table: Table of χ 2 \\chi ^2 and BIC of the different ring models proposed in the text.",
"τ\\tau is the extinction.Figure: Fit to the average light curve of events A and B (in red)by the two models of KIC 8462852's occultation by a planetary ring,as presented in Section (black curves).Upper panel: Homogeneous circular ring with a constant opacity and a non-zero impact parameter.Lower panel: Circular ring with a constant opacity in the center (red area) and a decreasing opacity with distance following a r -α r^{-\\alpha } law in the external ring(blue area).",
"The impact parameter is fixed to 0.The black circle figures the edge of stellar disk."
],
[
"Observing the future events",
"With the last event on BJD 2455977.15, and assuming periodicity with $P$ =$t_{B}-t_A$ =928.25$\\pm $ 0.25 days, the phenomenon is expected to repeat itself every $t_{B} + N \\times P$ .",
"The occurence timing closest to the present date is for $N$ =2 (event D) with TD = 2457833.65 0.80 or between the 20th of March 2017 at 07:55 UT and the 21st of March 2017 at 23:17 UT.",
"The beginnning-of-ingress and end-of-egress timings were also estimated.",
"Table REF summarizes these informations.",
"Table: Timing and ephemeris of transit events with PP=928.25928.25 days starting from event B at t B t_B=1144 days past Kepler initial epoch at MJD 2454833.We planned observing KIC 8462852 between the 19th of March and the 23rd of March 2017 in photometry and/or spectroscopy.",
"Unfortunately, HST and Spitzer were both unable to point at KIC 8462852 on these dates.",
"State-of-the-art ground-based photometry is not sensitive and stable enough to confirm a 0.1% deep transit signature lasting several days.",
"Ground-based spectroscopy has been tried, since in case of exocomet transit, variable Na I or Ca II features could be expected in the KIC 8462852 spectrum [18], [19], [4], [11].",
"We therefore planned observations of the star with the SOPHIE spectrograph installed on the 1.93m telescope of Observatoire de Hautes-Provence [34], [7] between the 15th and the 26th of March 2017.",
"Unfortunately, bad weather conditions prevented us to observe KIC 8462852 after the 19th of March 2017.",
"We could collect good quality spectra of KIC 8462852 on the 15th, 16th, 17th and 19th of March between 03:30 and 03:45 UT.",
"The median Na I spectrum of KIC 8462852 observed with SOPHIE between these 4 dates is plotted on Fig.",
"REF .",
"At the right hand side of the stellar Na I doublet lines, we detect an emission feature, which is also observed in the simultaneous sky-background spectrum obtained through the second aperture of the spectrograph (Fiber B).",
"It is identified to geocoronal emission from Earth atmosphere.",
"We subtracted this feature from all Na I spectra by fitting out the sky spectrum.",
"As can be seen in Fig REF , the resulting Na I spectrum is totally quiet through the 4 days.",
"It only presents a stable double peak absorption line, which is most likely of interstellar origin, since no counterpart is observed in the Ca II spectrum at the star radial velocity (Fig.",
"REF ).",
"Similarly the Ca II doublet spectrum of KIC 8462852 does not present any variable features between the 15th and the 19th of March.",
"Nevertheless, the predicted time of ingress is just after the observation dates.",
"The 19th of March at 03:45 (UT) is at the top of the signature left wing, before the predicted timing of ingress (19th of March 04:48 UT).",
"Therefore, the absence of observed features cannot exclude that significant absorption occurred in KIC 8462852 spectrum during the transit.",
"The observed spectra could neither infirm or confirm the periodicity of these transit events.",
"Assuming periodicity, the next event is predicted to occur between the 3rd and the 8th of October 2019 with ingress, centroid and egress timings given in Table REF .",
"New observations of KIC 8462852 between the 3rd and the 8th of October 2019 in both photometry and spectroscopy, with Spitzer, Cheops, HST, JWST and ground-based spectroscopes, are strongly encouraged.",
"They should allow confirmation or infirmation of the periodicity in the observed photometric event.",
"Figure: Na I spectra of KIC 8462852.",
"In blue, the average of the spectra collected through Fiber A of the SOPHIE spectrograph, on the 15, 16, 17 and 19th of March 2017.",
"In grey, the average sky-background spectrum taken simultaneously with each star's spectrum on Fiber B.",
"The emission line seen on Fiber A and B is clearly identified as geocoronal sodium emission.",
"The double peak feature on the left of the telluric emission is most probably of interstellar absorption origin, since no counterpart is observed in the Ca II spectrum at the star radial velocity (see Fig.",
").Figure: Comparison of the Na I spectra of KIC 8462852 ordinated by increasing dates from top to bottom.",
"The sky spectrum obtained simultaneously in fiber B has been fitted out of the original spectra (see Fig.",
").Figure: Comparison of the Ca II spectra of KIC 8462852 ordinated by increasing dates from top to bottom, with 3-pixels binning.",
"As can be seen, there are no spectral signatures of transient phenomenon in these spectra."
],
[
"Conclusions",
"After a careful detrending of the Kepler lightcurve of the peculiar star KIC 8462852, we identified among 22 signatures, two strickingly similar shallow absorptions with a separation of 928.25 days (event A & B).",
"These two events presented 0.1% deep stellar flux variations with duration of 4.4 days, consistent with the transit of a single or a few objects with a 928-days orbital period.",
"We thoroughly verified the different possible sources of systematics that could have produced the transit-like signatures of event A and B.",
"We conclude that these two events are certainly of astrophysical origin, and occurred in the system of KIC 8462852.",
"We found that two scenarios could well reproduce the transit lightcurve of events A and B.",
"They consist in the occultation of the star by two kind of objects: 1.",
"Either a string of half-a-dozen of exocomets orbiting at a distance $\\gtrsim $ 0.3 au, with evaporation rates similar to comet Hale-Bopp, and scattered along their common orbit much like the 1994 Shoemaker-Levy 9 fragments.",
"2.",
"Either an extended ring system surrounding a planet orbiting at 2.1 au from the star, and composed of a constant opacity interior ring and an exterior ring with decreasing opacity towards larger radius.",
"It should be mentioned that the main argument against the exocomet scenario for KIC 8462852 dimming events is the absence of any detectable IR excess.",
"This is an important problem that will always lead to risky comparison with other emblematic exocomet hosts such as $\\beta $ Pic.",
"These stars are all young ($<$ 100 Myr) with strong Vega-like excess, and thus massive debris disk.",
"The age of KIC 8462852 (1 Gyr) would well explain the lack of IR excess, still it must be explained how the vaporization of the remaining small bodies would fit below the detection level.",
"In fact, [8] showed that dust clouds of the mass of a fully vaporized Hale-Bopp comets, as needed to explain the strongest dips of KIC 8462852 lightcurve, are not expected to produce visible IR emission as long as the distance of the clouds is greater than 0.2 au.",
"This happens to be the case in the exocomets string model proposed here.",
"This is the first strong evidence for a periodic signal coming from KIC 8462852.",
"All the other dimmings present irregular behavior with apparently uncorrelated timings.",
"If periodic, our discovery opens a gate for in-depth characterization of a collection of objects present around this star.",
"Assuming periodicity, we predict that the next event to happen will occur between the 3$^\\text{rd}$ and the 8$^\\text{th}$ of October 2019.",
"The observation of KIC 8462852 at these dates will confirm or infirm the 928.25 day period, and hopefully will allow us to discriminate between the two scenarios proposed in this paper.",
"This work has been supported by the Centre National des Etudes Spatiales (CNES).",
"We acknowledge the support of the French Agence Nationale de la Recherche (ANR), under program ANR-12-BS05-0012 “Exo-Atmos”.",
"V.B.",
"acknowledges the financial support of the Swiss National Science Foundation.",
"This paper includes data collected by the Kepler NASA mission, and with the SOPHIE spectrograph on the 1.93-m telescope at Observatoire de Haute-Provence (CNRS).",
"Funding for the Kepler mission is provided by the NASA Science Mission directorate.",
"We thank the anonymous referee for his careful reading of our manuscript and his insightful comments and suggestions."
]
] | 1709.01732 |
[
[
"Atomically flat two-dimensional silicon crystals with versatile\n electronic properties"
],
[
"Abstract Silicon (Si) is one of the most extensively studied materials owing to its significance to semiconductor science and technology.",
"While efforts to find a new three-dimensional (3D) Si crystal with unusual properties have made some progress, its two-dimensional (2D) phases have not yet been explored as much.",
"Here, based on a newly developed systematic $ab$ $initio$ materials searching strategy, we report a series of novel 2D Si crystals with unprecedented structural and electronic properties.",
"The new structures exhibit perfectly planar outermost surface layers of a distorted hexagonal network with their thicknesses varying with the atomic arrangement inside.",
"Dramatic changes in electronic properties ranging from semimetal to semiconducting with indirect energy gaps and even to one with direct energy gaps are realized by varying thickness as well as by surface oxidation.",
"Our predicted 2D Si crystals with flat surfaces and tunable electronic properties will shed light on the development of silicon-based 2D electronics technology."
],
[
"Introduction",
"Recently, various 2D materials with weak van der Waals (vdW) interlayer interaction have been extensively studied due to their unusual properties [1], [2], [3].",
"Examples of these include graphene, hexagonal boron nitride, black phosphorous, and transition metal dichalcogenides.",
"Not only that they have shown superior physical and chemical properties, some of the models in theoretical physics such as massless Dirac fermions have also been realized in experiments, which otherwise have not been observed in conventional materials [1].",
"Nevertheless, many practical issues about large-scale synthesis, processing for defects, and contaminant control still need to be resolved [1], [4], which are critical for them to be realized as next generation electronic devices and energy applications.",
"Silicon, on the other hand, has served as a mainstay of semiconductor technologies, and a vast amount of advanced processing techniques have been accumulated for decades.",
"It is mainly due to its abundance on the Earth surface as well as the existence of a single oxide form (SiO$_2$ ) which is advantageous to the mass production of a single-element device that is free from phase separations.",
"These make undoubtedly Si be unique in current semiconductor technologies.",
"Therefore, despite very active researches on the aforementioned 2D materials as a next generation platform for various applications, the best candidate may still be Si itself.",
"This leads us to believe that discovering a novel 2D phase of Si materials with desirable physical properties would be important.",
"Compared with the number of efforts for new bulk phase of Si [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], however, searching for a new 2D Si crystalline phase has not been remarkably succeeded yet, and only a few theoretical predictions [20], [21], [22], [23], [24], [25], [26], [27] and experimental reports [28], [29], [30], [31], [32], [33] exist in the literature.",
"Silicene [34], [35], [36], [37], [38], [39], a monolayer form of the 2D Si crystals which is analogous to graphene, cannot form a stable layered structure by itself since surface of the silicene is chemically reactive, so that the adjacent silicene layers form strong covalent bonds [1], [37].",
"This is due to the strong preference of Si for the sp$^3$ hybridization over the sp$^2$ in contrast to carbon with the same number of valence electrons.",
"Thus, silicene may not be a good candidate for a scalable 2D phase of Si [1].",
"In addition, due to the strong covalent bonding character of Si, as-cleaved surfaces inevitably have unpaired electrons localized at dangling bonds on the surface, which makes an energetically unfavorable situation.",
"This is evidenced by prevalent severe surface reconstruction to reduce the number of unpaired electrons as can be seen in most of the previously reported 2D Si crystals [21], [22], [23], [24], [25], [26], [27], [33].",
"In all the cases, however, some of the surface atoms still remain under-coordinated even after the reconstruction, implying that those atoms prone to form strong covalent bonding with one another as pointed out in the case of silicene [1], [37].",
"In this work, we theoretically predict a series of novel 2D allotrope of Si crystals constructed by a concrete ab initio materials search strategy.",
"The predicted 2D crystals show characteristic structural features as follows.",
"The crystal is composed of two parts: (1) the atomically flat surface layers and (2) the inner layer connecting them through sp$^3$ -like covalent bonds as seen in FIG.",
"REF .",
"The surface layer features perfectly planar stable hexagonal framework unlike other 2D Si crystals that have hitherto been studied [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], in which buckled surfaces are predominant.",
"Moreover, the crystal is completely free from coordination number (CN) defects.",
"The structures composed of two parts are revealed stable against serious perturbations as will be discussed later."
],
[
"Computational Details",
"We concisely describe a novel 2D crystal structure prediction method: namely, Search by Ab initio Novel Design via Wyckoff position Iterations in the Conformational Hypersurface (abbreviated as SANDWICH), which explores the conformational hyperspace to find various local minima systematically.",
"The method is especially suited for predicting 2D phases of covalent materials by designing the 2D crystals free of CN defects.",
"This can be achieved by building surface and inner parts with different symmetries from each other, and by joining the two parts in such a way that under-coordinated atoms at the interface are compensated by one another.",
"By doing so, the crystal becomes stabilized by eliminating dangling bonds.",
"Specifically, we chose surface layers to have a space group of P6/mmm (No.",
"191), while the bulk maintains a sp$^3$ bonding character.",
"Among special positions in the given space group, we find that Wyckoff sites of $e$ (0, 0, $\\pm $ z) with a point group of 6mm and $i$ [(1/2, 0, $\\pm $ z), (0, 1/2, $\\pm $ z) and (1/2, 1/2, $\\pm $ z)] with the group of 2mm are suitable for building the CN defect-free crystals.",
"Also, we considered two relative positions of the surface layers in this study, represented by displacement vectors of $\\vec{d}$ =$\\mathbf {0}$ and $\\vec{d}$ = 0.5 $\\vec{a}_{1}$ + 0.5 $\\vec{a}_{2}$ , where $\\vec{a}_1$ and $\\vec{a}_2$ are lattice vectors.",
"With all the settings described so far, we generated structures by varying the number of atoms in the inner layer ($n$ ) consecutively from 0 up to 9.",
"Our method exhausts all the possible combinations for atomic positions of 2D crystals with given constraints: surface symmetry, choice of Wyckoff positions and thickness as schematically shown in FIG. S1.",
"We note that the method is undoubtedly advantageous since it explores almost all the structures from a highly probable subset of the entire search space for given conditions.",
"More detailed descriptions of our SANDWICH method can be found in the supplementary material.",
"We performed a series of first-principles calculations to obtain optimized structures by using Vienna ab initio simulation package (VASP) code [40], [41].",
"Conjugate gradient method was used to find the equilibrium structures with a force criterion of 1 meV/Å.",
"For Kohn-Sham orbital, core part was treated by using projector augmented wave method [42], while the valence part was approximated by linear expansion of a plane wave basis set with the kinetic energy cutoff of 450 eV.",
"Self-consistent field of DFT was iterated until the differences of the total energy and eigenvalues are less than 10$^{-7}$ eV.",
"Numerical integration in the first Brillouin zone (BZ) was done on the $\\Gamma $ -centered 12$\\times $ 12$\\times $ 1 grid meshes generated by a Monkhorst-Pack scheme.",
"The exchange-correlation functional of Perdew-Burke-Ernzerhof was used to build the Hamiltonian of an electron-ion system [43].",
"For a better description of electronic structures with a band gap, a hybrid functional of Heyd-Scuseria-Ernzerhof [44] as implemented in the VASP code was used.",
"Dynamical stability was also checked on each of the relaxed structures.",
"Phonon dispersion spectra were generated by using a direct method [45] as implemented in phonopy package [46].",
"To obtain force constants, we used 4$\\times $ 4$\\times $ 1 and 5$\\times $ 5$\\times $ 1 supercells to generate displaced configurations.",
"In this case, the k-point sampling in the BZ was done on the 24$\\times $ 24$\\times $ 1 and 25$\\times $ 25$\\times $ 1 Monkhorst-Pack grids."
],
[
"Results and Discussion",
"The 2D Si crystals constructed by the SANDWICH method demonstrate unique structural features.",
"As we intentionally put together two symmetrically distinct parts (surface and inner layers) so that CN defects on both components are fully compensated by each other, all the atoms in the crystal have a CN of 4 without any dangling bonds as shown in FIG.",
"REF (a).",
"This condition is particularly favored for Si atoms which show strong preference for sp$^3$ bonding.",
"We find that the four-coordinated networks are maintained in the fully relaxed equilibrium structures.",
"Moreover, the crystals exhibit atomically flat surface structures without buckling or reconstruction, which is uncommon for 2D Si crystals except for silicene bilayers [21], [27].",
"Note that those bilayers with planar surfaces are nothing but the cases in our model with ($\\vec{d}$ =$\\mathbf {0}$ , $n$ =0) [21] and ($\\vec{d}$ = 0.5 $\\vec{a}_{1}$ + 0.5 $\\vec{a}_{2}$ , $n$ =0) [27].",
"Figure: Classification of the 2D Si crystals.",
"(a) Type of distortions in a tetrahedron building block.",
"On the undistorted regular tetrahedron, skew lines along in-plane directions are marked as thick dashed lines.",
"Bond stretch, bending, and twist are shown aside.",
"The undistorted configurations are drawn in dotted lines for comparison, and restoration forces due to the distortion are shown in blue arrows.",
"(b) Classification of the crystals due to various lattice distortions.",
"Local and global stresses are shown in blue and black arrows, respectively.",
"Group 1: local distortions in the surface layers mainly due to bond bending lead to the global stresses of positive (negative) normal stress along x (y) axis.",
"Group 2: the unitcell is deformed by twist-like distortions in the inner layer, resulting in negative (positive) normal stress along x (y) axis.",
"Group 3: dihedral angle distortions from different inner layers yield nonzero shear stress, so that the deformation of the unitcell becomes asymmetric (|a → 1 |≠|a → 2 ||\\vec{a}_{1}|\\ne |\\vec{a}_{2}|).Looking into the structures in detail, we find that the surface and inner layers have different bonding characteristics as expected from the fact that they initially had different symmetries.",
"The surface layers show distorted hexagonal lattices with additional bonds toward the inner layer, while the atoms in the inner layer (hereafter called as bridge atoms as denoted in FIG.",
"REF (b)) form distorted tetrahedral bonding with its two opposite edges parallel to the plane of the surface (FIG.",
"REF ).",
"The atomic arrangement of the inner layer is similar to that of the {100} surfaces of the cubic diamond phase (dSi) distorted by an in-plane shear strain.",
"We note that the key role of the outermost bridge atom located on the bond center of the surface atoms for stabilizing the characteristic planar surface structure of the crystals, which would not have been realized otherwise.",
"Due to the discrepancy of the preferred local environments for the two parts in the common unitcell, some of the local structures must be distorted for compatibility.",
"For instance, the angle between those bridge atoms and surface atoms is largely deviated from the ideal value of $\\sim $ 109.5 degrees ($$ ) to $\\sim $ 60$$ .",
"Also, the angles between the two skew lines (marked as thick dashed lines in FIG.",
"REF (a)) are deviated from the right angle (90$$ ) for some of the tetrahedrons made by bridge atoms, which creates torsional restoration forces as explained in FIG.",
"REF (a).",
"Among the unique structural features shared by the crystals in this study (namely, flat surfaces without CN defects), we find that a set of 2D crystals constructed with the same $\\vec{d}$ and $n$ display a variety of microstructures depending on the atomic arrangement in the inner layer.",
"This comes from the various relative orientations of the distorted sp$^3$ bonds as explained above.",
"The global stress of the crystals is also affected critically by the $\\vec{d}$ , $n$ and the atomic arrangement in the inner layer, making the unitcells distorted in various ways (Table S1).",
"Thus, for the sake of discussion, we classify the crystals into three distinct groups.",
"(1) The crystals in the first group (group 1) are characterized by the same magnitudes of the lattice vectors ($|\\vec{a}_1|$ =$|\\vec{a}_2|$ ) and reduced cell angle ($\\gamma <60$ ).",
"In this case, the unitcell is distorted by the normal components ($\\sigma _x$ and $\\sigma _y$ ) only, as the shear component ($\\tau _{xy}$ ) is vanished due to symmetry as illustrated in FIG.",
"REF (b).",
"All the crystals in this group have $\\vec{d}$ = 0.5 $\\vec{a}_1$ + 0.5 $\\vec{a}_2$ .",
"Those crystals fall into the C222 space group, except for the case of $n$ =2 which is in the Cmme group as in the Table S1.",
"The magnitude of the global stress seems to be decreasing for thicker crystals (or with the increased $n$ ), indicated by the $\\gamma $ approaching to 60$$ .",
"(2) The crystals in the second group (group 2) feature the symmetric lattice vectors ($|\\vec{a}_1|=|\\vec{a}_2|$ ) with $\\gamma >$ 60$$ .",
"As in the case of group 1, only the $\\sigma _x$ and $\\sigma _y$ distort the unitcell without shear strain because of the symmetry; the crystals also fall into the space group of C222.",
"Crystals in group 1 and 2 differ by the signs of the normal stresses as indicated in FIG.",
"REF (b).",
"The crystals in group 2 have two subgroups of $\\vec{d}$ = $\\mathbf {0}$ and $\\vec{d}$ = 0.5 $\\vec{a}_1$ + 0.5 $\\vec{a}_2$ .",
"They behave differently in terms of the $\\gamma $ with respect to the $n$ as can be seen in the Table S1.",
"In the former case, symmetry of the surface layers overtakes that of the inner layer, and the $\\gamma $ approaches to 60$$ as the $n$ increases.",
"On the other hand, the crystals with the $\\vec{d}$ = 0.5 $\\vec{a}_1$ + 0.5 $\\vec{a}_2$ subgroup are dominated by the symmetry of the inner layer, making the $\\gamma $ approach to 90$$ as the $n$ increases.",
"(3) Lastly, group 3 crystals are constructed with nonzero $\\tau _{xy}$ , resulting in the asymmetric lattice vectors ($|\\vec{a}_1|\\ne |\\vec{a}_2|$ ) as well as the space group of P2 with lower symmetry.",
"In this case, the principal stresses ($\\sigma _1$ and $\\sigma _2$ ) do not agree with the $\\sigma _x$ and $\\sigma _y$ due to the $\\tau _{xy}$ .",
"Figure: Stability of the crystals.",
"(a) Cohesive energy (E coh _{\\mathrm {coh}}) of the 2D crystals with respect to the number density.",
"(b) Harmonic phonon dispersion spectra for a representative structure in each group.Stability of the crystals is confirmed as shown in FIG.",
"REF .",
"Cohesive energies (E$_{\\mathrm {coh}}$ ) for each crystal are plotted with respect to the number density, defined as the number of Si atoms in the unitcell divided by the unitcell area ($|\\vec{a}_1\\times \\vec{a}_2|$ ).",
"In all the proposed cases, well-defined energy minima are shown as in FIG.",
"REF (a).",
"When compared with other Si structures, the crystals in this study show slightly higher E$_{\\mathrm {coh}}$ .",
"For example, with a number density of $\\sim $ 0.4 Si/Å$^2$ , the crystal with a thickness of 0.5 nm shows a higher E$_{\\mathrm {coh}}$ by 0.02 eV/atom than that of 2$\\times $ 1-dSi (100) with a thickness of 0.6 nm.",
"We ascribe this to the significant distortion in bond angle especially near the surface as mentioned above.",
"We note, however, that the 2D crystals in this study may stay more stable than other crystals in a chemical environment because the dangling bonds on the surface were eliminated.",
"This chemical stability undoubtedly becomes a critical factor for device realization [1], [4].",
"Moreover, a recently synthesized allotrope of the 3D Si crystals, namely Si$_{24}$ , also shows distorted bond angles ranging from 93.73$$ to 123.17$$ , and extended bond distances with a higher total energy than that of the ground state dSi by 0.09 eV/Si [11].",
"Thus, for realization of stable Si crystals, it could be more critical for the individual Si atoms to satisfy the ground state CN than to maintain the exact bond angles and distances of the ideal sp$^3$ bonding found in dSi.",
"We also provide harmonic phonon dispersion spectra in FIG.",
"REF (b) for the 2D crystals in the three groups.",
"In all the cases, we confirm that the crystals are dynamically stable.",
"Furthermore, we find that the new 2D Si crystals are quite stable against strong perturbations beyond a usual harmonic interatomic force regime.",
"To check such a structural stability, we generated computationally “shaken” structures that have been proved useful to check the stabilities of other crystal structures [47] (see supplementary material).",
"Starting from the fully relaxed crystal in group 1 with $n$ =2, we displaced every Si atom in a random direction with a fixed amount of 0.5 Å in the 2$\\times $ 2 supercell.",
"Note that we used the significantly large displacement when compared with that of 0.01 Å used to obtain harmonic interatomic force constants (FIG.",
"REF (b)).",
"Then, each of the perturbed structures was relaxed to the corresponding energy minimum configurations by using the conjugate gradient method.",
"By comparing $\\sim $ 5,500 randomly generated configurations, we found that our proposed structure was retained in the $\\sim $ 3,500 samples after relaxation, proving that the structure is robust against severe thermal fluctuations as shown in the FIG. S2.",
"The fundamental electronic structures of the crystals are closely related to their thickness ($n$ ) and the crystal classification defined above, displaying a wide variety of electronic properties ranging from metallic to semiconducting.",
"For instance, all the crystals categorized as group 1 are semiconductors, showing a finite bandgap ($\\Delta >0$ ) with their sizes decreasing with an increasing $n$ as shown in FIG.",
"REF (a).",
"The maximum size of the energy gap is $\\sim $ 0.5 eV for the thinnest crystal ($n$ =1).",
"On the other hand, most of the crystals in group 2 and group 3 are metallic.",
"Note that the $\\Delta $ in this case is a measure of the overlap between the conduction and valence energy bands.",
"For group 2, the $\\Delta $ behaves differently with respect to the $n$ depending on the subgroup ($\\vec{d}$ ) as discussed above.",
"Figure: Electronic structures of the 2D crystals.",
"(a) Bandgap (Δ\\Delta ) as a function of thickness (nn) is shown for the 2D crystals in each group.",
"In group 2 and 3, empty and filled symbols indicate the displacement vectors d →\\vec{d} = 0\\mathbf {0} and d →\\vec{d} = 0.5 a → 1 \\vec{a}_1 + 0.5 a → 2 \\vec{a}_2.",
"Electronic band dispersion of (b) group 1 with nn=2, (c) group 2 with nn=3 and (d) group 3 with nn=4.",
"A schematic diagram of the 1st BZ with high symmetry points is shown above (d).Figure: Oxidation of the 2D Si crystal.",
"(a) Potential energy surface of the 2D crystal by an O probe atom with a height of 2 Å.",
"Atoms are superimposed on the map: large blue and small white balls indicate surface silicon atoms and underneath bridge silicon atoms, respectively.",
"The minimum energy for adsorption site at b 1 _1 is set to be zero in the color bar shown below.",
"(b) Electronic band dispersion and (c) atomic structure of the fully oxidized crystal in ground state.",
"Adsorbed oxygen atoms are denoted by small red balls.To reveal the origin of various electronic phases in the crystal families, we provide electronic band dispersions for each group in FIG.",
"REF (b)-(d).",
"Edges of the valence and the conduction bands are located at different momenta ($\\vec{k}$ ), indicating that the crystals are either indirect semiconductors or semimetals.",
"For the crystals with a space group of C222 (group 1 and 2), we note that the relative energetic positions of the bands at $\\vec{k}$ = 0.5 $\\vec{b}_1$ ($\\epsilon _{M_1}$ ) and $\\vec{k}$ = 0.5 $\\vec{b}_1$ + 0.5 $\\vec{b}_2$ ($\\epsilon _{M_2}$ ) are directly responsible for the electronic phase transition.",
"Here, $\\vec{b}_1$ and $\\vec{b}_2$ indicate the reciprocal lattice vectors in the BZ as shown in FIG.",
"REF .",
"That is to say, when the conduction band minimum (CBM) is located at M$_1$ , the crystals represent finite energy gaps ($\\Delta >0$ ), while semimetallic phases with $\\Delta <0$ are realized for crystals with the CBM located at M$_2$ as seen in FIG.",
"REF (b) and (c).",
"Note that the crystals in group 1 and 2 belong to the former and the latter cases, respectively.",
"Because crystals are classified by the lattice parameters of the unitcell (FIG.",
"REF ), we ascribe the electronic phase transition to the changes in the local atomic structures due to the lattice parameters, especially to the $\\gamma $ .",
"We firstly confirm that the electronic wave functions for the CBM are highly localized near the surface layers (FIG.",
"S3), indicating that the surface geometry is mainly responsible for the electronic phase transition.",
"We further verify this by pure shear deformation of the unitcell only to change the $\\gamma $ without changing $|\\vec{a}_1|$ and $|\\vec{a}_2|$ , and see that the crossover between $\\epsilon _{M_1}$ and $\\epsilon _{M_2}$ indeed occurs as shown in FIG. S4.",
"Based on those facts, we can construct a concise but essential model for the whole crystal explaining the characteristic variation of electronic structures from semiconductor to semimetal as functions of thickness and group classification (see FIG. S5).",
"In addition, these crystals provide good transport properties when compared to the dSi.",
"We note that the transverse effective masses for electrons near the CBM are reduced by half when compared with the dSi, while similar values of other components are shown as in the Table S2.",
"The structures with lower crystal symmetry (group 3) always show semimetallic electronic structures (FIG.",
"REF (d)).",
"Interestingly, we find that the surface oxidation can extremely widen the applicability of the new crystals.",
"With the oxygen (O) adsorption on the surface, the crystal in group 1 with $n$ =2 is significantly stabilized by 1.98 eV/O$_2$ , and with the subsequent dissociation of the adsorbed O$_2$ molecule on the surface, the system becomes even more stabilized by 5.89 eV/O$_2$ (See supplementary material for details).",
"We confirm that adsorption and dissociation of O$_2$ molecules do not affect much on the characteristic planar structure of the surface in a wide range of O coverage from an isolated limit (FIG.",
"S8) to the full coverage (FIG.",
"REF (c)).",
"We find that the ground state occurs when the adsorbed O atom is located on the middle of the surface Si-Si bond just on top of the underneath bridge Si atom marked as b$_1$ in FIG.",
"REF (a).",
"Moreover, electronic properties vary notably via surface oxidation from the indirect energy gap of $\\sim $ 0.2 eV (FIG.",
"4(a)) to the direct bandgap of $\\sim $ 1.2 eV (FIG.",
"REF (b)), which significantly widens the versatility of the 2D Si crystals in this study.",
"Furthermore, we confirm that the oxidized 2D crystals can form stable layered structure by itself, suggesting this Si material as a feasible candidate for a component in vdW heterojunction [1]."
],
[
"Conclusions",
"In this work, using a newly developed ab initio computational method, we propose a series of two-dimensional silicon crystals with versatile electronic properties.",
"The surface layer of the new 2D Si crystals exhibits atomically flat distorted hexagonal structure without buckling, and the inner layer silicon atoms fill up the space between the flat surface layers.",
"We classified 2D Si structures into three groups and each of the groups possesses distinct electronic properties originated from structural variations such as semiconductor as well as semimetals.",
"Moreover, their oxidized forms are shown to be a direct bandgap semiconductor.",
"Therefore, we believe that our new 2D Si crystals satisfy highly desirable characteristics of next generation electronic technology platforms only with a single atomic element and their oxides, very similar with the current 3D Si electronic devices.",
"The authors thank Dr. In-Ho Lee for discussions.",
"D. Y. K. acknowledges financial support from the NSAF (U1530402).",
"Y.-W. S. was supported by the NRF of Korea funded by the MSIP (QMMRC, No.",
"R11-2008-053-01002-0 and vdWMRC, No.",
"2017R1A5A1014862).",
"The computing resources were supported by the Center for Advanced Computation of KIAS.",
"Supplementary Material for: Atomically flat two-dimensional silicon crystals with versatile electronic properties Kisung Chae,$^1$ Duck Young Kim,$^2$ and Young-Woo Son$^1$ $^1$ Korea Institute for Advanced Study, Seoul 02455, South Korea $^2$ Center for High Pressure Science and Technology Advanced Research, Shanghai 201203, P. R. China (Dated: 2023/01/06 00:25:53)"
],
[
"A new structure search method for 2D crystal prediction",
"Here, we describe in detail the structure searching method used to predict the 2D Si crystals in this study, named SANDWICH (Search by Ab initio Novel Design via Wyckoff positions Iteration in Conformational Hypersurface).",
"The main idea of the method is to put together two symmetrically distinctive parts to compensate unpaired electrons at the interface.",
"Thus, the method is particularly suitable for predicting 2D crystals which might favor electronic compensation over local distortion.",
"By doing so, we believe that a new series of 2D crystals can be efficiently searched in a highly confined conformational space near the local minimum structures.",
"The number of distinct sets of crystals that can be constructed by this method can be as many as the possible combinations of surfaces and inner parts, which varies depending on the molecular geometries of the ground state structure (i.e., tetrahedral building block for Si).",
"General procedures of the SANDWICH method are summarized as a flowchart in FIG.",
"REF , and structural parameters of 2D crystals in this study are listed in the Table REF .",
"Figure: A flowchart of the SANDWICH method.To be more specific, the first step in the SANDWICH method begins with the choice of geometry of surface layers, which determines the space group of the 2D crystals.",
"Note that the two surface layers can be displaced relative to each other by a fractional lattice translation vector $\\vec{d}$ , and only specific $\\vec{d}$ vectors are allowed only when the set of Wyckoff positions in the given space group remain invariant after a translation operation by $\\vec{d}$ .",
"For instance, the two $\\vec{d}$ vectors of $\\mathbf {0}$ and 0.5 $\\vec{a}_{1}$ + 0.5 $\\vec{a}_{2}$ were considered in this study because a set of Wyckoff positions of $e$ (0, 0, $\\pm $ z) and $i$ [(1/2, 0, $\\pm $ z), (0, 1/2, $\\pm $ z) and (1/2, 1/2, $\\pm $ z), respectively] under the space group of P6/mmm (No.",
"191) are invariant with respect to both of the $\\vec{d}$ vectors.",
"For the same surface layers, if Wyckoff positions of $e$ and $h$ [(1/3, 1/3, $\\pm $ z) and (2/3, 2/3, $\\pm $ z)] are chosen to construct the initial structures, there must be three $\\vec{d}$ vectors of $\\mathbf {0}$ and $\\pm $ 1/3 $\\vec{a}_{1}$ + $\\pm $ 1/3 $\\vec{a}_{2}$ .",
"The next step is to select some of the Wyckoff positions for inner layer construction from the list given for the space group.",
"Here, a rule of thumb is that all the atoms in the constructed crystals have to be free of CN defects.",
"At this point, for a given number of atomic layers in the inner layer n (or thickness), a set of crystals can be constructed.",
"Then, starting with those initial guess structures, the corresponding ground states are sought, followed by dynamic stability tests.",
"Table: Structural parameters of 2D crystals in various groups.We note that some of the 2D Si crystals already reported elsewhere can also be found by the method proposed here.",
"For instance, the crystals, so called a bilayer silicene [21], [27], in AA (ontop) and orthorhombic (displaced along the zigzag chain direction) stacking of the two silicene monolayers are nothing but the ones with ($\\vec{d}$ =0, n=0) and ($\\vec{d}$ =0.5 $\\vec{a}_{1}$ + 0.5 $\\vec{a}_{2}$ , n=0), respectively.",
"Similarly, it is theoretically shown that other group IV elements such as germanium (Ge) and tin (Sn) with the same valence electron configuration as Si can form similar bilayer structures as well [48], [49]."
],
[
"Structural stability test",
"In addition to calculating harmonic phonon dispersion to check the dynamic stability, we also tested the stability beyond a harmonic regime.",
"We started from one of our proposed 2D silicon crystals (group 1, n=2), and then randomly moved each atomic position from the equilibrium within a pre-determined spherical space with a fixed radius.",
"This method was used to predict a low energy models of SiNF [50], and it is one important operation in ab initio random structure searching [47] strategy, which have been successfully applied to many compounds under pressure.",
"We moved every Si atom in a (2$\\times $ 2) supercell by 0.5 Å, and then allowed a full relaxation.",
"Note that the magnitude of the displacement is significantly larger than that used in harmonic limit of 0.01 Å.",
"This indicates that the crystals were tested on a very harsh condition or very high temperature.",
"We examined $\\sim $ 5,500 structures and the results are shown in FIG.",
"REF .",
"We find that majority of relaxed structures after random distortion are recovered back to the original structure, which reaches $\\sim $ 68 % out of the total samples, and possess the lowest total energy.",
"Rest of the distorted structures were relaxed to local minimum structures with buckled surfaces with higher total energy as shown in FIG.",
"REF ."
],
[
"Origin of a variety of electronic properties",
"We demonstrate that the versatile electronic properties (FIG.",
"4) are primarily attributed to the surface layer structure.",
"Simple shear deformation, together with the fact that electronic wave functions near the Fermi energy are localized on the surface layers (FIG.",
"REF ), shows that the variation of electronic properties is solely due to the surface geometry, and rules out the role of inner layer (FIG.",
"REF ).",
"Specifically, shift of the band at M$_2$ ($\\vec{k}_{M_2}$ = 0.5 $\\vec{b}_{1}$ + 0.5 $\\vec{b}_{2}$ ) with respect to the shear strain (or the cell angle $\\gamma $ ) is clearly shown, altering the electronic structures ranging from semimetallic to indirect semiconducting.",
"Figure: Charge densities for a 2D crystal in group 2 with n=9 at valence band maximum (VBM) and conduction band minimum (CBM) rendered in red and yellow, respectively.",
"An isovalue of 0.005 electron per Å 3 ^3 was used.",
"Unitcell is drawn as black dashed line.",
"Charge density projected along the normal direction to the crystal is plotted for CBM and VBM.To reveal the mechanism of the band shift at M$_2$ , we consider a minimal surface model.",
"The model is composed of essential parts: a hexagonal framework of Si with a bridge Si atom with hydrogen passivation as shown in FIG.",
"REF (a).",
"Based on the fact that the band shift is primarily due to the surface layer structure, we varied the $\\gamma $ from 50$$ to 70$$ .",
"The length of the surface bond with the bridge atom (denoted as b in FIG.",
"REF (a)) does not vary monotonously with respect to the $\\gamma $ , but becomes diminished as the unitcell is distorted as seen in FIG.",
"REF (b).",
"Instead, the bond angle at the surface ($\\theta $ ) decreases monotonously with an increasing $\\gamma $ , so does the relative band shift between M$_1$ and M$_2$ ($\\epsilon _{M_2}-\\epsilon _{M_1}$ ) as seen in FIG.",
"REF (c).",
"So, the $\\theta $ appears mainly responsible for the band shift.",
"Figure: Electronic band dispersion of a 2D Si crystal in group 1 with n=2 as a function of γ\\gamma : (a) 54.45, (b) 56.72 and (c) 58.99.For more rigorous discussion, we show the electronic band structures projected on atomic orbitals of the minimal model in FIG.",
"REF (d).",
"The band near the valence band maximum seems to be composed mainly of the s, p$_x$ and p$_z$ states of the surface atoms.",
"On the other hand, orbital characters of the band at M$_1$ and M$_2$ vary significantly.",
"The band at M$_1$ contains a significant portion of atomic orbital perpendicular to the surface layer such as p$_z$ , d$_{xz}$ and d$_{yz}$ .",
"On the other hand, the band at M$_2$ is mainly composed of orbital lying on the surface layer such as s, p$_x$ and d$_{xy}$ .",
"We note that d$_{xy}$ orbital contributes 22 % to the band at M$_2$ .",
"Thus, we find out that the sensitive band shift at M$_2$ is partly due to the enhanced (diminished) contribution of d$_{xy}$ orbital with the decreased (increased) $\\gamma $ as shown in FIG.",
"REF (c).",
"The bands at M$_1$ and M$_3$ do not vary as much as M$_2$ because the bridge bond varies less sensitively to the $\\gamma $ , compared to the surface bonds.",
"Therefore, the $\\Delta $ remains almost constant even with the $\\gamma $ smaller than 60$$ .",
"Figure: (a) Atomic configuration of the model.",
"(b) Bond length and bond angle marked in (a) with respect to γ\\gamma .",
"(c) Δ\\Delta and energy level difference at M 1 _1 and M 2 _2 due to γ\\gamma are shown.",
"The d xy _{xy} orbital overlaid on the surface layer with a corresponding angle is shown below.",
"(d) Dispersion of the electronic band structure with atomic orbital projection.",
"Orbitals for surface and bridge atoms are denoted as s and b, respectively, on the superscript.",
"Size of the symbols represents the amount of contribution."
],
[
"Effective masses",
"Effective masses for electron and hole are calculated by interpolating energy band dispersion to a parabolic band around the band extrema for selected 2D Si crystals.",
"In general, the 2D crystals in this study have elliptical Fermi surfaces, indicating anisotropic effective masses as seen in FIG.",
"REF .",
"The calculated effective masses are summarized in the Table REF .",
"Compared to cubic diamond phase (dSi) with longitudinal ($m^{*}_{e,L}$ ) and transverse ($m^{*}_{e,T}$ ) electronic effective masses of 0.92 $m_0$ and 0.19 $m_0$ ($m_0$ : mass of a free electron), respectively [51], our predicted crystals in group 1 and 2 show 50 % lighter $m^{*}_{e,L}$ .",
"For hole effective masses, our results are comparable to the dSi: 0.49 $m_0$ and 0.16 $m_0$ for heavy and light holes, respectively.",
"We note that the crystal in group 1 with n=2 can be a good n-type semiconductor with high mobility due to reduced effective mass.",
"Figure: Eigenvalue maps in the reciprocal space.",
"Top and bottom panes indicate conduction and valence bands, respectively, for (a, d) group 1, n=2, (b, e) group 2, n=3, and (c, f) group 3, n=4.",
"Band extrema are marked as x, and the first BZ is drawn as black lines.",
"Effective masses around the band extrema were calculated along longitudinal and transverse direction as marked by dissecting yellow lines.Table: Electron (e) and hole (h) effective masses (m * ^*) of the 2D Si crystals with varying group and n. Longitudinal (L) and transverse (T) directions for each case are shown in FIG.",
"."
],
[
"Oxygen adsorption",
"In general, Si surface is vulnerable to an oxygen (O) ambient environment due to strong interaction between Si and O, forming a stable oxide film.",
"In addition to their mechanical and dynamical stability as confirmed in FIG.",
"3, our crystals are expected to show a better chemical stability when compared with other Si crystals, because all the surface Si atoms are fully compensated.",
"To validate this, we attached an isolated O$_2$ molecule on a (3$\\times $ 2) supercell of an orthogonal unitcell as seen in FIG.",
"REF .",
"As a diatomic molecule, three orientations for O$_2$ alignment along x, y and z were considered for adsorption on the following sites: ontop (o), bridge 1 (b$_1$ ) and bridge 2 (b$_2$ ) (FIG.",
"REF ).",
"Note that b$_1$ and b$_2$ are distinguished whether the bridge bond possesses a bridge Si atom on the opposite side or not.",
"We observe negative adsorption energies ($E_{\\mathrm {ads}}$ ) as $E_{\\mathrm {ads}}=E(\\mathrm {Si}+\\mathrm {O_2})-E(\\mathrm {Si})-E(\\mathrm {O_2}),$ indicating that the O$_2$ adsorption is a thermodynamically spontaneous process.",
"Moreover, we observe that internal energy of the system is further reduced in a great deal by dissociation of the adsorbed O$_2$ molecule.",
"Interestingly, the characteristic flat surface of the crystals is retained in most cases throughout adsorption and subsequent dissociation processes (FIG.",
"REF ).",
"This indicates that an oxidation process would make the crystals even more stable without disturbing the original framework, and prevent the structure from being degraded by further oxidation.",
"Figure: A xy-projection of (3×\\times 2) supercell (red) of the conventional Bravais unit cell shown in magenta lines.",
"Only top surface layer (blue balls) and the subsurface bridge atoms (white) are represented.",
"Three possible adsorption sites of ontop (o), bridge 1 (b 1 _1) and bridge 2 (b 2 _2) are marked on the corresponding sites.Figure: Initial and relaxed structures of the three low-energy configurations of O 2 _2 (red balls) adsorption.In some cases, we observe a significant deformation of the crystal when an O$_2$ molecule is adsorbed on the o site as shown in FIG.",
"REF (b).",
"We found that the one of the dissociated O atoms is bound to the Si atom, of which the bond orientation is perpendicular to the surface.",
"This changes the local bonding character of the Si to sp$^3$ similar to the dSi (111) surface, so that the flat surface is not preserved anymore.",
"Also, the bridge Si atom, which was originally bound to that Si atom is significantly relaxed, making a new bond with another bridge Si atom.",
"Consequently, the surface Si atom on the other side becomes protruded as seen in FIG.",
"REF (b).",
"This process seems irreversible because of the strong Si-O bonding, and the protruded Si atom on the other side is likely to be chemically reactive since one of the valence electrons is not fully compensated anymore.",
"However, this process is energetically less favorable by 1.35 eV per a O$_2$ molecule (Table REF ), and O atoms will sit on the most stable b$_1$ site without disturbing the framework in a quasi-static oxidation condition.",
"Table: Adsorption energy (E ads E_{\\mathrm {ads}}) of an isolated O 2 _2 molecule on various adsorption sites."
]
] | 1709.01675 |
[
[
"Inverse Obstacle Scattering for Elastic Waves in Three Dimensions"
],
[
"Abstract Consider an exterior problem of the three-dimensional elastic wave equation, which models the scattering of a time-harmonic plane wave by a rigid obstacle.",
"The scattering problem is reformulated into a boundary value problem by introducing a transparent boundary condition.",
"Given the incident field, the direct problem is to determine the displacement of the wave field from the known obstacle; the inverse problem is to determine the obstacle's surface from the measurement of the displacement on an artificial boundary enclosing the obstacle.",
"In this paper, we consider both the direct and inverse problems.",
"The direct problem is shown to have a unique weak solution by examining its variational formulation.",
"The domain derivative is studied and a frequency continuation method is developed for the inverse problem.",
"Numerical experiments are presented to demonstrate the effectiveness of the proposed method."
],
[
"Introduction",
"The obstacle scattering problem, which concerns the scattering of a time-harmonic incident wave by an impenetrable medium, is a fundamental problem in scattering theory [7].",
"It has played an important role in many scientific areas such as geophysical exploration, nondestructive testing, radar and sonar, and medical imaging.",
"Given the incident field, the direct obstacle scattering problem is to determine the wave field from the known obstacle; the inverse obstacle scattering problem is to determine the shape of the obstacle from the measurement of the wave field.",
"Due to the wide applications and rich mathematics, the direct and inverse obstacle scattering problems have been extensively studied for acoustic and electromagnetic waves by numerous researchers in both the engineering and mathematical communities [8], [30], [31].",
"Recently, the scattering problems for elastic waves have received ever-increasing attention because of the significant applications in geophysics and seismology [2], [5], [21].",
"The propagation of elastic waves is governed by the Navier equation, which is complex due to the coupling of the compressional and shear waves with different wavenumbers.",
"The inverse elastic obstacle scattering problem is investigated mathematically in [6], [9], [11] for the uniqueness and numerically in [14], [19] for the shape reconstruction.",
"We refer to for some more related direct and inverse scattering problems for elastic waves [1], [3], [13], [15], [17], [18], [22], [24], [25], [26], [27], [28], [29], [33].",
"In this paper, we consider the direct and inverse obstacle scattering problems for elastic waves in three dimensions.",
"The goal is fourfold: (1) develop a transparent boundary condition to reduce the scattering problem into a boundary value problem; (2) establish the well-posedness of the solution for the direct problem by studying its variational formulation; (3) characterize the domain derivative of the wave field with respect to the variation of the obstacle's surface; (4) propose a frequency continuation method to reconstruct the obstacle's surface.",
"This paper significantly extends the two-dimensional work [23].",
"We need to consider more complicated Maxwell's equation and associated spherical harmonics when studying the transparent boundary condition.",
"Computationally, it is also more intensive.",
"The rigid obstacle is assumed to be embedded in an open space filled with a homogeneous and isotropic elastic medium.",
"The scattering problem is reduced into a boundary value problem by introducing a transparent boundary condition on a sphere.",
"We show that the direct problem has a unique weak solution by examining its variational formulation.",
"The proofs are based on asymptotic analysis of the boundary operators, the Helmholtz decomposition, and the Fredholm alternative theorem.",
"The calculation of domain derivatives, which characterize the variation of the wave field with respect to the perturbation of the boundary of an medium, is an essential step for inverse scattering problems.",
"The domain derivatives have been discussed by many authors for the inverse acoustic and electromagnetic obstacle scattering problems [10], [16], [32].",
"Recently, the domain derivative is studied in [20] for the elastic wave by using boundary integral equations.",
"Here we present a variational approach to show that it is the unique weak solution of some boundary value problem.",
"We propose a frequency continuation method to solve the inverse problem.",
"The method requires multi-frequency data and proceed with respect to the frequency.",
"At each frequency, we apply the descent method with the starting point given by the output from the previous step, and create an approximation to the surface filtered at a higher frequency.",
"Numerical experiments are presented to demonstrate the effectiveness of the proposed method.",
"A topic review can be found in [4] for solving inverse scattering problems with multi-frequencies to increase the resolution and stability of reconstructions.",
"The paper is organized as follows.",
"Section 2 introduces the formulation of the obstacle scattering problem for elastic waves.",
"The direct problem is discussed in section 3 where well-posedness of the solution is established.",
"Section 4 is devoted to the inverse problem.",
"The domain derivative is studied and a frequency continuation method is introduced for the inverse problem.",
"Numerical experiments are presented in section 5.",
"The paper is concluded in section 6.",
"To avoid distraction from the main results, we collect in the appendices some necessary notation and useful results on the spherical harmonics, functional spaces, and transparent boundary conditions."
],
[
"Problem formulation",
"Consider a bounded and rigid obstacle $D\\subset \\mathbb {R}^3$ with a Lipschitz boundary $\\partial D$ .",
"The exterior domain $\\mathbb {R}^3\\setminus \\bar{D}$ is assumed to be filled with a homogeneous and isotropic elastic medium, which has a unit mass density and constant Lamé parameters $\\lambda , \\mu $ satisfying $\\mu >0, \\lambda +\\mu >0$ .",
"Let $B_R=\\lbrace x\\in \\mathbb {R}^3:\\,|x|< R\\rbrace $ , where the radius $R$ is large enough such that $\\bar{D}\\subset B_R$ .",
"Define $\\Gamma _R=\\lbrace x\\in \\mathbb {R}^3:\\,|x|=R\\rbrace $ and $\\Omega =B_R\\setminus \\bar{D}$ .",
"Let the obstacle be illuminated by a time-harmonic plane wave $u^{\\rm inc}=d e^{{\\rm i}\\kappa _{\\rm p}x\\cdot d}\\quad \\text{or}\\quad u^{\\rm inc}=d^\\perp e^{{\\rm i}\\kappa _{\\rm s}x\\cdot d},$ where $d$ and $d^\\perp $ are orthonormal vectors, $\\kappa _{\\rm p}=\\omega /\\sqrt{\\lambda +2\\mu }$ and $\\kappa _{\\rm s}=\\omega /\\sqrt{\\mu }$ are the compressional wavenmumber and the shear wavenumber.",
"Here $\\omega >0$ is the angular frequency.",
"It is easy to verify that the plane incident wave (REF ) satisfies $\\mu \\Delta u^{\\rm inc}+(\\lambda +\\mu )\\nabla \\nabla \\cdot u^{\\rm inc}+\\omega ^2u^{\\rm inc}=0\\quad \\text{in}~ \\mathbb {R}^3\\setminus \\bar{D}.$ Let $u$ be the displacement of the total wave field which also satisfies $\\mu \\Delta u+(\\lambda +\\mu )\\nabla \\nabla \\cdot u+\\omega ^2u=0\\quad \\text{in}~ \\mathbb {R}^3\\setminus \\bar{D}.$ Since the obstacle is elastically rigid, we have $u=0\\quad \\text{on}~ \\partial D.$ The total field $u$ consists of the incident field $u^{\\rm inc}$ and the scattered field $v$ : $u=u^{\\rm inc}+v.$ Subtracting (REF ) from (REF ) yields that $v$ satisfies $\\mu \\Delta v+(\\lambda +\\mu )\\nabla \\nabla \\cdot v+\\omega ^2v=0\\quad \\text{in}~ \\mathbb {R}^3\\setminus \\bar{D}.$ For any solution $v$ of (REF ), we introduce the Helmholtz decomposition by using a scalar function $\\phi $ and a divergence free vector function $\\psi $ : $v=\\nabla \\phi +\\nabla \\times \\psi , \\quad \\nabla \\cdot \\psi =0.$ Substituting (REF ) into (REF ), we may verify that $\\phi $ and $\\psi $ satisfy $\\Delta \\phi +\\kappa ^2_{\\rm p}\\phi =0,\\quad \\Delta \\psi +\\kappa ^2_{\\rm s}\\psi =0.$ In addition, we require that $\\phi $ and $\\psi $ satisfy the Sommerfeld radiation condition: $\\lim _{r\\rightarrow \\infty }r\\left(\\partial _r\\phi -{\\rm i}\\kappa _{\\rm p}\\phi \\right)=0, \\quad \\lim _{r\\rightarrow \\infty }r\\left(\\partial _r\\psi -{\\rm i}\\kappa _{\\rm s}\\psi \\right)=0, \\quad r=|x|.$ Using the identity $\\nabla \\times (\\nabla \\times \\psi )=-\\Delta \\psi +\\nabla (\\nabla \\cdot \\psi ),$ we have from (REF ) that $\\psi $ satisfies the Maxwell equation: $\\nabla \\times (\\nabla \\times \\psi )-\\kappa ^2_{\\rm s}\\psi =0.$ It can be shown (cf.",
"[8]) that the Sommerfeld radiation for $\\psi $ in (REF ) is equivalent to the Silver–Müller radiation condition: $\\lim _{r\\rightarrow \\infty }\\left( (\\nabla \\times \\psi )\\times x-{\\rm i}\\kappa _{\\rm s}r\\psi \\right)=0,\\quad r=|x|.$ Given $u^{\\rm inc}$ , the direct problem is to determine $u$ for the known obstacle $D$ ; the inverse problem is to determine the obstacle's surface $\\partial D$ from the boundary measurement of $u$ on $\\Gamma _R$ .",
"Hereafter, we take the notation of $a\\lesssim b$ or $a\\gtrsim b$ to stand for $a\\le Cb$ or $a\\ge Cb$ , where $C$ is a positive constant whose specific value is not required but should be clear from the context."
],
[
"Direct scattering problem",
"In this section, we study the variational formulation for the direct problem and show that it admits a unique weak solution."
],
[
"Transparent boundary condition",
"We derive a transparent boundary condition on $\\Gamma _R$ .",
"Given $v\\in L^2(\\Gamma _R)$ , it has the Fourier expansion: $v(R, \\theta ,\\varphi )=\\sum _{n=0}^{\\infty }\\sum _{m=-n}^{n}v_{1n}^mT_{n}^{m}(\\theta , \\varphi )+ v_{2n}^mV_{n}^{m}(\\theta ,\\varphi )+v_{3n}^mW_{n}^{m}(\\theta , \\varphi ),$ where $\\lbrace (T_n^m, V_n^m, W_n^m): n=0,1,\\dots , m=-n,\\dots , n\\rbrace $ is an orthonormal system in $L^2(\\Gamma _R)$ and $v_{jn}^m$ are the Fourier coefficients of $v$ on $\\Gamma _R$ .",
"Define a boundary operator ${B}v=\\mu \\partial _rv+(\\lambda +\\mu )(\\nabla \\cdot v)e_r\\quad \\text{on}~\\Gamma _R,$ which is assumed to have the Fourier expansion: $({B}v)(R, \\theta ,\\varphi )=\\sum _{n=0}^{\\infty }\\sum _{m=-n}^{n}w_{1n}^mT_{n}^{m}(\\theta ,\\varphi )+w_{2n}^mV_{n}^{m}(\\theta ,\\varphi )+w_{3n}^mW_{n}^{m}(\\theta , \\varphi ).$ Taking $\\partial _{r}$ of $v$ in (REF ), evaluating it at $r=R$ , and using the spherical Bessel differential equations [34], we get $\\partial _{r}&v(R,\\theta , \\varphi )=\\sum _{n=0}^{\\infty }\\sum _{m=-n}^{n}\\Bigg [\\frac{\\sqrt{n(n+1)}\\phi _{n}^{m}}{R^2} (z_{n}(\\kappa _{\\rm p}R)-1)-\\frac{\\psi _{2n}^m}{R^2}\\Big (1+z_{n}(\\kappa _{\\rm s}R)\\\\&+(R\\kappa _{\\rm s})^2-n(n+1)\\Big )\\Bigg ]T_{n}^{m}+\\Bigg [\\frac{\\kappa _{\\rm s}^2\\psi _{3n}^m}{\\sqrt{n(n+1)}}z_{n}(\\kappa _{\\rm s}R)\\Bigg ]V_{n}^{m}+\\Bigg [\\frac{\\phi _{n}^{m}}{R^2}\\big (n(n+1)\\\\&-(R\\kappa _{\\rm p})^2-2z_{n}(\\kappa _{\\rm p} R)\\big )+\\frac{\\sqrt{n(n+1)} \\psi _{2n}^m}{R^2}(z_{n}(\\kappa _{\\rm s}R)-1)\\Bigg ] W_{n}^m,$ where $z_n(t)=t h_n^{(1)^{\\prime }}(t)/h_n^{(1)}(t), h_n^{(1)}$ is the spherical Hankel function of the first kind with order $n$ , $\\phi _n^m$ and $\\psi _{jn}^m$ are the Fourier coefficients for $\\phi $ and $\\psi $ on $\\Gamma _R$ , respectively.",
"Noting (REF ) and using $\\nabla \\cdot v=\\Delta \\phi =\\frac{2}{r}\\partial _{r}\\phi +\\partial _ {r}^2\\phi +\\frac{1}{r}\\Delta _{\\Gamma _R}\\phi ,$ we have $\\nabla \\cdot v(r,\\theta , \\varphi )=\\sum _{n=0}^{\\infty }\\sum _{m=-n}^{n}\\frac{\\phi _{n}^{m}}{h_{n}^{(1)}(\\kappa _{\\rm p}R)}\\Bigg [\\frac{2}{r}\\frac{\\rm d}{{\\rm d}r}h_{n}^{(1)}(\\kappa _{\\rm p} r) +\\frac{{\\rm d}^2}{{\\rm d}r^2}h_{n}^{(1)}(\\kappa _{\\rm p}r)\\\\-\\frac{n(n+1)}{r^2}h_{n}^{(1)}(\\kappa _{\\rm p}r)\\Bigg ] X_{n}^{m},$ where $\\Delta _{\\Gamma _R}$ is the Laplace–Beltrami operator on $\\Gamma _R$ .",
"Combining (REF ) and (REF )–(REF ), we obtain ${B}&v=\\sum _{n=0}^{\\infty }\\sum _{m=-n}^{n}\\frac{\\mu }{R^2}\\Big [\\sqrt{n(n+1)} (z_n(\\kappa _{\\rm p}R)-1) \\phi _n^m-\\big (1+z_n(\\kappa _{\\rm s}R)+(R\\kappa _{\\rm s})^2\\\\&-n(n+1)\\big )\\psi _{2n}^m\\Big ]T_n^m+\\frac{\\mu \\kappa _{\\rm s}^2}{\\sqrt{n(n+1)}}z_n (\\kappa _{\\rm s}R)\\psi _{3n}^mV_n^m+ \\frac{1}{R^2}\\Big [ \\mu \\big (n(n+1)-(R\\kappa _{\\rm p})^2\\\\&-2z_n(\\kappa _{\\rm p} R)\\big ) \\phi _n^m+\\mu \\sqrt{n(n+1)} (z_n(\\kappa _{\\rm s}R)-1)\\psi _{2n}^m-(\\lambda +\\mu )(\\kappa _{\\rm p}R)^2\\phi _n^m\\Big ]W_n^m.$ Comparing (REF ) with (REF ), we have $(w_{1n}^m, w_{2n}^m, w_{3n}^m)^\\top =\\frac{1}{R^2}G_n (\\phi ^{m}_n, \\psi _{2n}^m, \\psi _{3n}^m)^\\top ,$ where the matrix $G_n=\\begin{bmatrix}0 & 0 & G_{13}^{(n)}\\\\G_{21}^{(n)}& G_{22}^{(n)}&0\\\\G_{31}^{(n)}& G_{32}^{(n)} & 0\\end{bmatrix}.$ Here $G_{13}^{(n)}&=\\frac{\\mu (\\kappa _{\\rm s}R)^{2}z_{n}(\\kappa _{\\rm s}R)}{\\sqrt{n(n+1)}},\\quad G_{21}^{(n)}=\\mu \\sqrt{n(n+1)} (z_{n}(\\kappa _{\\rm p}R)-1),\\\\G_{22}^{(n)}&=\\mu \\left(n(n+1)-(\\kappa _{\\rm s}R)^2-1-z_{n}(\\kappa _{\\rm s}R)\\right),\\\\G_{31}^{(n)}&=\\mu \\left( n(n+1)-(\\kappa _{\\rm p}R)^2-2z_{n}(\\kappa _{\\rm p}R)\\right)-(\\lambda +\\mu )(\\kappa _{\\rm p} R)^2,\\\\G_{32}^{(n)}&=\\mu \\sqrt{n(n+1)}(z_{n}(\\kappa _{\\rm s}R)-1).$ Let $v_n^m=(v_{1n}^m, v_{2n}^m, v_{3n}^m)^\\top , \\quad M_nv_n^m=b_n^m=(b_{1n}^m, b_{2n}^m, b_{3n}^m)^\\top $ , where the matrix $M_n=\\begin{bmatrix}M^{(n)}_{11} & 0 & 0\\\\0 & M^{(n)}_{22} & M^{(n)}_{23}\\\\0& M^{(n)}_{32} & M^{(n)}_{33}\\end{bmatrix}.$ Here $M^{(n)}_{11}&=\\left(\\frac{\\mu }{R}\\right) z_{n}(\\kappa _{\\rm s}R),\\quad M^{(n)}_{22}=-\\left(\\frac{\\mu }{R}\\right)\\left(1+\\frac{(\\kappa _{\\rm s}R)^2 z_{n}(\\kappa _{\\rm p}R)}{\\Lambda _n}\\right),\\\\[5pt]M^{(n)}_{23}&=\\sqrt{n(n+1)}\\left(\\frac{\\mu }{R}\\right)\\left(1+\\frac{(\\kappa _{\\rm s}R)^2}{\\Lambda _n}\\right),\\\\[5pt]M^{(n)}_{32}&=\\sqrt{n(n+1)}\\left(\\frac{\\mu }{R}+\\frac{(\\lambda +2\\mu )}{R}\\frac{(\\kappa _ { \\rm p}R)^2}{\\Lambda _n}\\right),\\\\[5pt]M^{(n)}_{33}&=-\\frac{(\\lambda +2\\mu )}{R}\\frac{(\\kappa _{\\rm p}R)^2}{\\Lambda _n} (1+z_n(\\kappa _{\\rm s}R))-2\\left(\\frac{\\mu }{R}\\right),$ where $\\Lambda _n =z_n(\\kappa _{\\rm p}R)(1+z_n(\\kappa _{\\rm s} R))-n(n+1)$ .",
"Using the above notation and combining (REF ) and (REF ), we derive the transparent boundary condition: ${B}v={T}v:=\\sum _{n=0}^{\\infty }\\sum _{m=-n}^{n} b_{1n}^mT_n^m +b_{2n}^mV_n^m+b_{3n}^m W_n^m\\quad \\text{on} ~\\Gamma _R.$ The matrix $\\hat{M}_n=-\\frac{1}{2}(M_n+M_n^*)$ is positive definite for sufficiently large $n$ .",
"Using the asymptotic expansions of the spherical Bessel functions [34], we may verify that $z_n(t)&=-(n+1)+\\frac{1}{16n}t^4+\\frac{1}{2n}t^2+O\\left(\\frac{1}{n^2}\\right),\\\\\\Lambda _n(t)&=-\\frac{1}{16}(\\kappa _{\\rm p}t)^4-\\frac{1}{16}(\\kappa _{\\rm s}t)^4-\\frac{1}{2}(\\kappa _{\\rm p}t)^2-\\frac{1}{2}(\\kappa _{\\rm s}t)^2+O\\left(\\frac{1}{n}\\right).$ It follows from straightforward calculations that $\\hat{M}_n=\\begin{bmatrix}\\hat{M}^{(n)}_{11} & 0& 0\\\\0 & \\hat{M}^{(n)}_{22} & \\hat{M}^{(n)}_{23}\\\\0 & \\hat{M}^{(n)}_{32} & \\hat{M}^{(n)}_{33}\\end{bmatrix},$ where $\\hat{M}^{(n)}_{11} &=\\left(\\frac{\\mu }{R}\\right)(n+1)+O\\left(\\frac{1}{n}\\right),\\quad \\hat{M}^{(n)}_{22} = -\\left(\\frac{\\omega ^2R}{\\Lambda _n}\\right)(n+1)+O\\left(1\\right),\\\\\\hat{M}^{(n)}_{23} &=-\\left(\\frac{\\mu }{R}+\\frac{\\omega ^2R}{\\Lambda _n}\\right) \\sqrt{n(n+1)}+O(1),\\\\\\hat{M}^{(n)}_{32}&=-\\left(\\frac{\\mu }{R}+\\frac{\\omega ^2R}{\\Lambda _n}\\right) \\sqrt{n(n+1)}+O(1),\\\\\\hat{M}^{(n)}_{33}&=\\frac{2\\mu }{R}+\\frac{\\omega ^2R}{\\Lambda _n}(1+z_n(\\kappa _{\\rm s} R))=-\\left(\\frac{\\omega ^2 R}{\\Lambda _n}\\right)n+O(1).$ For sufficiently large $n$ , we have $\\hat{M}^{(n)}_{11}>0\\quad \\text{and}\\quad \\hat{M}^{(n)}_{22}>0,$ which gives ${\\rm det}[(\\hat{M}_n)_{(1:2,1:2)}]=\\hat{M}^{(n)}_{11}\\hat{M}^{(n)}_{22}>0.$ Since $\\Lambda _n<0$ for sufficiently large $n$ , we have $\\hat{M}^{(n)}_{22}\\hat{M}^{(n)}_{33}-\\left(\\hat{M}^{(n)}_{23}\\right)^2=n(n+1)\\left[\\left(\\frac{\\omega ^2 R}{\\Lambda _n}\\right)^2-\\left(\\frac{\\mu }{R}+\\frac{\\omega ^2R}{\\Lambda _n}\\right)^2\\right]+O(n)>0.$ A simple calculation yields ${\\rm det}[\\hat{M}_{n}]=\\hat{M}^{(n)}_{11}\\left(\\hat{M}^{(n)}_{22}\\hat{M}^{(n)}_{33}-\\left(\\hat{M}^{(n)}_{23}\\right)^2\\right)>0,$ which completes the proof by applying Sylvester's criterion.",
"The boundary operator ${T}: H^{1/2}(\\Gamma _R)\\rightarrow H^{-1/2}(\\Gamma _R)$ is continuous, i.e., $\\Vert {T}u\\Vert _{H^{-1/2}(\\Gamma _R)} \\lesssim \\Vert u\\Vert _{H^{1/2}(\\Gamma _R)},\\quad \\forall \\,u\\in H^{1/2}(\\Gamma _R).$ For any given $u\\in H^{1/2}(\\Gamma _R)$ , it has the Fourier expansion $u(R, \\theta , \\varphi )=\\sum _{n=0}^\\infty \\sum _{m=-n}^n u_{1n}^mT_n^m(\\theta , \\varphi ) +u_{2n}^mV_n^m(\\theta ,\\varphi )+u_{3n}^m W_n^m(\\theta , \\varphi ).$ Let $u_n^m=(u_{1n}^m, u_{2n}^m, u_{3n}^m)^\\top $ .",
"It follows from (REF ) and the asymptotic expansions of $M_{ij}^{(n)}$ that $\\Vert {T} u \\Vert ^2_{H^{1/2}(\\Gamma _R)} &=\\sum _{n=0}^{\\infty }\\sum _{m=-n}^{n}\\left( 1 + n(n+1) \\right)^{-1/2} |M_n u^{m}_{n}|^2 \\\\&\\lesssim \\sum _{n=0}^{\\infty }\\sum _{m=-n}^{n}\\left(1 +n(n+1)\\right)^{1/2}|u^{m}_{n}|^2 = \\Vert u\\Vert ^2_{H^{1/2}(\\Gamma _R)},$ which completes the proof."
],
[
"Uniqueness",
"It follows from the Dirichlet boundary condition (REF ) and the Helmholtz decomposition (REF ) that $v=\\nabla \\phi + \\nabla \\times \\psi =-u^{\\rm inc}\\quad \\text{on} ~ \\partial D.$ Taking the dot product and the cross product of (REF ) with the unit normal vector $\\nu $ on $\\partial D$ , respectively, we get $\\partial _{\\nu }\\phi +(\\nabla \\times \\psi )\\cdot \\nu =-u_1,\\quad (\\nabla \\times \\psi )\\times \\nu +\\nabla \\phi \\times \\nu =-u_2,$ where $u_1=u^{\\rm inc}\\cdot \\nu ,\\quad u_2=u^{\\rm inc}\\times \\nu .$ We obtain a coupled boundary value problem for the potential functions $\\phi $ and $\\psi $ : ${\\left\\lbrace \\begin{array}{ll}\\Delta \\phi + \\kappa _{\\rm p}^2\\phi = 0 ,\\quad \\nabla \\times (\\nabla \\times \\psi ) -\\kappa _{\\rm s}^2\\psi = 0 ,&\\quad {\\rm in}~\\Omega ,\\\\\\partial _{\\nu }\\phi +(\\nabla \\times \\psi )\\cdot \\nu =-u_ 1, \\quad (\\nabla \\times \\psi )\\times \\nu +\\nabla \\phi \\times \\nu =-u_2 &\\quad {\\rm on}~\\partial D,\\\\\\partial _r \\phi - {T}_1\\phi =0 ,\\quad (\\nabla \\times \\psi )\\times e_r-{\\rm i}\\kappa _{\\rm s}{T}_2\\psi _{\\Gamma _R} =0 &\\quad {\\rm on} ~ \\Gamma _R.\\end{array}\\right.",
"}$ where $T_1$ and $T_2$ are the transparent boundary operators given in (REF ) and (REF ), respectively.",
"Multiplying test functions $(p, q)\\in H^1(\\Omega )\\times H({\\rm curl}, \\Omega )$ , we arrive at the weak formulation of (REF ): To find $(\\phi , \\psi )\\in H^1(\\Omega )\\times H({\\rm curl}, \\Omega )$ such that $a(\\phi , \\psi ; p, q)=\\langle u_1,p\\rangle _{\\partial D}+\\langle u_2,q\\rangle _{\\partial D},\\quad \\forall \\, (p, q)\\in H^1(\\Omega )\\times H({\\rm curl}, \\Omega ),$ where the sesquilinear form $a(\\phi , \\psi ; p, q)=(\\nabla \\phi ,\\nabla p)+ (\\nabla \\times \\psi ,\\nabla \\times q)-\\kappa ^2_{\\rm p}(\\phi ,p)-\\kappa ^2_{\\rm s}(\\psi ,q)-\\langle (\\nabla \\times \\psi )\\cdot \\nu ,p\\rangle _{\\partial D}\\\\-\\langle \\nabla \\phi \\times \\nu ,q\\rangle _{\\partial D}-\\langle {T}_1\\phi ,p\\rangle _{\\Gamma _R}-{\\rm i}\\kappa _{\\rm s}\\langle {T}_2\\psi _{\\Gamma _R},q_{\\Gamma _R}\\rangle _{\\Gamma _R}.$ The variational problem (REF ) has at most one solution.",
"It suffices to show that $\\phi =0, \\psi =0$ in $\\Omega $ if $u_1=0, u_2=0$ on $\\partial D$ .",
"If $(\\phi , \\psi )$ satisfy the homogeneous variational problem (REF ), then we have $(\\nabla \\phi , \\nabla \\phi )+(\\nabla \\times \\psi ,\\nabla \\times \\psi )-\\kappa ^2_{\\rm p}(\\phi ,\\phi )-\\kappa ^2_{\\rm s}(\\psi , \\psi )-\\langle (\\nabla \\times \\psi )\\cdot \\nu ,\\phi \\rangle _{\\partial D}\\\\-\\langle \\nabla \\phi \\times \\nu ,\\psi \\rangle _{\\partial D}-\\langle {T}_1\\phi ,\\phi \\rangle _{\\Gamma _R}-{\\rm i}\\kappa _{\\rm s}\\langle {T}_2\\psi _{\\Gamma _R},\\psi _{\\Gamma _R}\\rangle _{\\Gamma _R}=0.$ Using the integration by parts, we may verify that $\\langle (\\nabla \\times \\psi )\\cdot \\nu , \\phi \\rangle _{\\partial D}=-\\langle \\psi , \\nu \\times \\nabla \\phi \\rangle _{\\partial D}=\\langle \\psi , \\nabla \\phi \\times \\nu \\rangle _{\\partial D},$ which gives $\\langle (\\nabla \\times \\psi )\\cdot \\nu , \\phi \\rangle _{\\partial D}+\\langle \\nabla \\phi \\times \\nu , \\psi \\rangle _{\\partial D}=2{\\rm Re}\\langle \\nabla \\phi \\times \\nu ,\\psi \\rangle _{\\partial D}.$ Taking the imaginary part of (REF ) and using (REF ), we obtain ${\\rm Im}\\langle {T}_1\\phi , \\phi \\rangle _{\\Gamma _R}+\\kappa _{\\rm s}{\\rm Re}\\langle {T}_2\\psi _{\\Gamma _R},\\psi _{\\Gamma _R}\\rangle _{\\Gamma _R}=0,$ which gives $\\phi =0, \\psi =0$ on $\\Gamma _R$ , due to Lemma and Lemma .",
"Using (REF ) and (REF ), we have $\\partial _r\\phi =0, (\\nabla \\times \\psi )\\times e_r=0$ on $\\Gamma _R$ .",
"By the Holmgren uniqueness theorem, we have $\\phi =0, \\psi =0$ in $\\mathbb {R}^3\\setminus \\bar{B}$ .",
"A unique continuation result concludes that $\\phi =0, \\psi =0$ in $\\Omega $ ."
],
[
"Well-posedness",
"Using the transparent boundary condition (REF ), we obtain a boundary value problem for $u$ : ${\\left\\lbrace \\begin{array}{ll}\\mu \\Delta u+(\\lambda +\\mu )\\nabla \\nabla \\cdot u+\\omega ^2u=0 &\\quad \\text{in} ~ \\Omega ,\\\\u=0 &\\quad \\text{on} ~ \\partial D,\\\\{B}u={T}u+g&\\quad \\text{on} ~\\Gamma _R,\\end{array}\\right.",
"}$ where $g=({B}-{T})u^{\\rm inc}$ .",
"The variational problem of (REF ) is to find $u\\in H^1_{\\partial D}(\\Omega )$ such that $b(u, v)=\\langle g,v\\rangle _{\\Gamma _R},\\quad \\forall \\,v\\in H^1_{\\partial D}(\\Omega ),$ where the sesquilinear form $b: H^1_{\\partial D}(\\Omega )\\times H^1_{\\partial D}(\\Omega )\\rightarrow \\mathbb {C}$ is defined by $b(u, v)=\\mu \\int _\\Omega \\nabla u:\\nabla \\bar{v}\\,{\\rm d}x+(\\lambda +\\mu )\\int _\\Omega (\\nabla \\cdot u)(\\nabla \\cdot \\bar{v})\\,{\\rm d}x\\\\-\\omega ^2\\int _\\Omega u\\cdot \\bar{v}\\,{\\rm d}x-\\langle {T}u,v\\rangle _{\\Gamma _R}.$ Here $A:B={\\rm tr}(A B^\\top )$ is the Frobenius inner product of square matrices $A$ and $B$ .",
"The following result follows from the standard trace theorem of the Sobolev spaces.",
"The proof is omitted for brevity.",
"It holds the estimate $\\Vert u\\Vert _{H^{1/2}(\\Gamma _R)}\\lesssim \\Vert u\\Vert _{H^1(\\Omega )}, \\quad \\forall \\,u \\in H_{\\partial D}^1(\\Omega ).$ For any $\\varepsilon >0$ , there exists a positive constant $C(\\varepsilon )$ such that $\\Vert u\\Vert _{L^2(\\Gamma _R)} \\le \\varepsilon \\Vert u\\Vert _{H^1(\\Omega )}+C(\\varepsilon )\\Vert u\\Vert _{L^2(\\Omega )}, \\quad \\forall \\, u \\in H_{\\partial D}^1(\\Omega ).$ Let $B^{\\prime }$ be the ball with radius $R^{\\prime }>0$ such that $\\bar{B}^{\\prime }\\subset D$ .",
"Denote $\\tilde{\\Omega }=B \\setminus \\bar{B}^{\\prime }$ .",
"Given $u\\in H^1_{\\partial D}(\\Omega )$ , let $\\tilde{u}$ be the zero extension of $u$ from $\\Omega $ to $\\tilde{\\Omega }$ , i.e., $\\tilde{u} (x) ={\\left\\lbrace \\begin{array}{ll}u (x), & x\\in \\Omega ,\\\\[5pt]0, & x\\in \\tilde{\\Omega } \\setminus \\bar{\\Omega }.\\end{array}\\right.",
"}$ The extension of $\\tilde{u}$ has the Fourier expansion $\\tilde{u}(r,\\theta ,\\varphi )=\\sum _{n=0}^{\\infty }\\sum _{m=-n}^{n}\\tilde{u}_{1n}^m(r)T_{n}^{m}(\\theta ,\\varphi )+\\tilde{u}_{2n}^m(r)V_{n}^{m}(\\theta ,\\varphi )+\\tilde{u}_{3n}^m(r)W_{n}^{m}(\\theta ,\\varphi ).$ A simple calculation yields $\\Vert \\tilde{u}\\Vert ^2_{L^2(\\Gamma _R)}=\\sum _{n=0}^{\\infty }\\sum _{m=-n}^{n} |\\tilde{u}_{1n}^{m}(R)|^2+ |\\tilde{u}_{2n}^{m}(R)|^2+|\\tilde{u}_{3n}^{m}(R)|^2.$ Since $\\tilde{u}(R^{\\prime }, \\theta , \\varphi )=0$ , we have $\\tilde{u}_{jn}^m(R^{\\prime })=0$ .",
"For any given $\\varepsilon >0$ , it follows from Young's inequality that $|\\tilde{u}_{jn}^{m}(R)|^2&=\\int _{R^{\\prime }}^{R}\\frac{{\\rm d}}{{\\rm d}r}|\\tilde{u}_{jn}^m(r)|^2{\\rm d}r\\le \\int _{R^{\\prime }}^{R}2|\\tilde{u}_{jn}^m(r)|\\left|\\frac{{\\rm d}}{{\\rm d}r}\\tilde{u}_{jn}^m(r)\\right|{\\rm d}r\\\\&\\le \\left(R^{\\prime }\\varepsilon \\right)^{-2}\\int _{R^{\\prime }}^{R}|\\tilde{u}_{jn}^m(r)|^2{\\rm d}r+\\left(R^{\\prime }\\varepsilon \\right)^2\\int _{R^{\\prime }}^{R}\\left|\\frac{{\\rm d}}{{\\rm d}r}\\tilde{u}_{jn}^m(r)\\right|^2{\\rm d}r,$ which gives $|\\tilde{u}_{jn}^{m}(R)|^2\\le C(\\varepsilon )\\int _{R^{\\prime }}^{R}|\\tilde{u}_{jn}^m(r)|^2r^2{\\rm d}r+\\varepsilon ^2\\int _{R^{\\prime }}^{R}\\left|\\frac{{\\rm d}}{{\\rm dr}}\\tilde{u}_{jn}^m(r)\\right|^2 r^2{\\rm d}r.$ The proof is completed by noting that $\\Vert \\tilde{u}\\Vert _{L^2(\\Gamma _R)}=\\Vert u\\Vert _{L^2(\\Gamma _R)},\\quad \\Vert \\tilde{u}\\Vert _{L^2(\\tilde{\\Omega })}=\\Vert u\\Vert _{L^2(\\Omega )}, \\quad \\Vert \\tilde{u}\\Vert _{H^1(\\tilde{\\Omega })}=\\Vert u\\Vert _{H^1(\\Omega )}.$ It holds the estimate $\\Vert u\\Vert _{H^1(\\Omega )} \\lesssim \\Vert \\nabla u\\Vert _{L^2(\\Omega )},\\quad \\forall \\,u\\in H_{\\partial D}^1(\\Omega ).$ As is defined in the proof of Lemma REF , let $\\tilde{u}$ be the zero extension of $u$ from $\\Omega $ to $\\tilde{\\Omega }$ .",
"It follows from the Cauchy–Schwarz inequality that $|\\tilde{u}(r, \\theta , \\varphi )|^2 = \\left| \\int _{R^{\\prime }}^r \\partial _r\\tilde{u}(r, \\theta , \\varphi ){\\rm d}r \\right|^2 \\lesssim \\int _{R^{\\prime }}^R \\left|\\partial _r \\tilde{u}(r, \\theta , \\varphi ) \\right|^2{\\rm d}r.$ Hence we have $\\Vert \\tilde{u}\\Vert ^2_{L^2(\\tilde{\\Omega })} &=\\int _{R^{\\prime }}^{R}\\int _{0}^{2\\pi }\\int _{0}^{\\pi }|\\tilde{u}(r, \\theta , \\varphi )|^2r^2{\\rm d}r{\\rm d}\\theta {\\rm d}\\varphi \\lesssim \\int _{R^{\\prime }}^{R}\\int _{0}^{2\\pi }\\int _{0}^{\\pi }\\int _{R^{\\prime }}^{R}|\\partial _r \\tilde{u}(r, \\theta ,\\varphi )|^2{\\rm d}r{\\rm d}\\theta {\\rm d}\\varphi {\\rm d}r\\\\&\\lesssim \\int _{R^{\\prime }}^{R}\\int _{0}^{2\\pi }\\int _{0}^{\\pi }|\\partial _r \\tilde{u}(r, \\theta , \\varphi )|^2{\\rm d}r{\\rm d}\\theta {\\rm d}\\varphi \\lesssim \\Vert \\nabla \\tilde{u}\\Vert ^2_{L^2(\\tilde{\\Omega })}.$ The proof is completed by noting that $\\Vert u \\Vert _{L^2(\\Omega )}=\\Vert \\tilde{u}\\Vert _{L^2(\\tilde{\\Omega })},\\quad \\Vert \\nabla u\\Vert _{L^2(\\Omega )}=\\Vert \\nabla \\tilde{u}\\Vert _{L^2(\\tilde{\\Omega })},\\quad \\Vert u\\Vert _{H^1(\\Omega )}^2 =\\Vert u\\Vert _{L^2(\\Omega )}^2 +\\Vert \\nabla u\\Vert _{L^2(\\Omega )}^2.$ The variational problem (REF ) admits a unique weak solution $u\\in H^1_{\\partial D}(\\Omega )$ .",
"Using the Cauchy–Schwarz inequality, Lemma REF , and Lemma REF , we have $|b(u, v)| \\le &\\mu \\Vert \\nabla u\\Vert _{L^2(\\Omega )}\\Vert \\nabla v\\Vert _{L^2(\\Omega )}+(\\lambda +\\mu )\\Vert \\nabla \\cdot u\\Vert _{0,\\Omega }\\Vert \\nabla \\cdot v\\Vert _{L^2(\\Omega )} + \\omega ^2\\Vert u\\Vert _{L^2(\\Omega )}\\Vert v\\Vert _{L^2(\\Omega )} \\\\&+\\Vert {T}u\\Vert _{H^{-1/2}(\\Gamma _R)}\\Vert v\\Vert _{H^{1/2}(\\Gamma _R) }\\\\\\lesssim &\\Vert u\\Vert _{H^1(\\Omega )}\\Vert v\\Vert _{H^1(\\Omega )},$ which shows that the sesquilinear form $b(\\cdot , \\cdot )$ is bounded.",
"It follows from Lemma REF that there exists an $N_0 \\in \\mathbb {N}$ such that $\\hat{M}_n$ is positive definite for $n>N_0$ .",
"The sesquilinear form $b$ can be written as $b(u, v) =& \\mu \\int _\\Omega (\\nabla u:\\nabla \\bar{v})\\,{\\rm d}x +(\\lambda +\\mu )\\int _\\Omega (\\nabla \\cdot u) (\\nabla \\cdot \\bar{v}) \\, {\\rm d}x - \\omega ^2 \\int _\\Omega u\\cdot \\bar{v}\\,{\\rm d}x\\\\&\\qquad -\\sum _{|n| > N_0}\\sum _{m=-n}^{n} \\left\\langle M_nu_n^m, v_n^m \\right\\rangle - \\sum _{|n| \\le N_0}\\sum _{m=-n}^{n} \\left\\langle M_nu_n^m, v_n^m \\right\\rangle .$ Taking the real part of $b$ , and using Lemma REF , Lemma REF , Lemma REF , we obtain ${\\rm Re}\\, b(u, u) &= \\mu \\Vert \\nabla u \\Vert _{L^2(\\Omega )}^2 + (\\lambda +\\mu ) \\Vert \\nabla \\cdot u\\Vert _{L^2(\\Omega )}^2 + \\sum _{|n| > N_0}\\sum _{m=-n}^{n}\\langle \\hat{M}_n u_n^m, u_n^m\\rangle \\\\&\\qquad -\\omega ^2 \\Vert u\\Vert _{L^2(\\Omega )} + \\sum _{|n|\\le N_0} \\sum _{m=-n}^{n}\\langle \\hat{M}_nu_n^m, u_n^m\\rangle \\\\&\\ge C_1\\Vert u\\Vert _{H^1(\\Omega )}-\\omega ^2\\Vert u\\Vert _{L^2(\\Omega )}-C_2\\Vert u\\Vert _{L^2(\\Gamma _R)}\\\\&\\ge C_1\\Vert u\\Vert _{H^1(\\Omega )}-\\omega ^2\\Vert u\\Vert _{L^2(\\Omega )} -C_2\\varepsilon \\Vert u\\Vert _{H^1(\\Omega )}-C(\\varepsilon )\\Vert u\\Vert _{L^2(\\Omega )}\\\\&=(C_1-C_2\\varepsilon )\\Vert u\\Vert _{H^1(\\Omega )}-C_3\\Vert u\\Vert _{L^2(\\Omega )}.$ Letting $\\varepsilon >0$ to be sufficiently small, we have $C_1-C_2\\varepsilon >0$ and thus Gårding's inequality.",
"Since the injection of $H^1_{\\partial D}(\\Omega )$ into $L^2(\\Omega )$ is compact, the proof is completed by using the Fredholm alternative (cf.",
"[31]) and the uniqueness result in Theorem REF ."
],
[
"Inverse scattering",
"In this section, we study a domain derivative of the scattering problem and present a continuation method to reconstruct the surface."
],
[
"Domain derivative",
"We assume that the obstacle has a $C^2$ boundary, i.e., $\\partial D\\in C^2$ .",
"Given a sufficiently small number $h>0$ , define a perturbed domain $\\Omega _h$ which is surrounded by $\\partial D_h$ and $\\Gamma _R$ , where $\\partial D_h=\\lbrace x+hp(x):x\\in \\partial D\\rbrace .$ Here the function $p\\in C^2(\\partial D)$ .",
"Consider the variational formulation for the direct problem in the perturbed domain $\\Omega _h$ : To find $u_h\\in H^{1}_{\\partial D_h}(\\Omega _h)$ such that $b^h(u_h,v_h)=\\langle g,v_h\\rangle _{\\Gamma _R},\\quad \\forall \\, v_h\\in H^1_{\\partial D_h}(\\Omega _h),$ where the sesquilinear form $b^h: H^1_{\\partial D_h}(\\Omega _h)\\times H^1_{\\partial D_h}(\\Omega _h)\\rightarrow \\mathbb {C}$ is defined by $b^h(u_h, v_h)=\\mu \\int _{\\Omega _h}\\nabla u_h: \\nabla \\bar{v}_h\\,{\\rm d}y+(\\lambda +\\mu )\\int _{\\Omega _h}(\\nabla \\cdot u_h)(\\nabla \\cdot \\bar{v}_h)\\,{\\rm d}y\\\\-\\omega ^2\\int _{\\Omega _h} u_h\\cdot \\bar{v}_h\\,{\\rm d}y-\\langle {T}u_h,v_h\\rangle {\\Gamma _R}.$ Similarly, we may follow the proof of Theorem REF to show that the variational problem (REF ) has a unique weak solution $u_h\\in H^1_{\\partial D_h}(\\Omega _h)$ for any $h>0$ .",
"Since the variational problem (REF ) is well-posed, we introduce a nonlinear scattering operator: ${S}:\\partial D_h\\rightarrow u_h|_{\\Gamma _R},$ which maps the obstacle's surface to the displacement of the wave field on $\\Gamma _R$ .",
"Let $u_h$ and $u$ be the solution of the direct problem in the domain $\\Omega _h$ and $\\Omega $ , respectively.",
"Define the domain derivative of the scattering operator ${S}$ on $\\partial D$ along the direction $p$ as ${S}^{\\prime }(\\partial D;p):=\\lim _{h\\rightarrow 0}\\frac{{S}(\\partial D_h)-{S}(\\partial D)}{h}=\\lim _{h\\rightarrow =0}\\frac{u_h|_{\\Gamma _R}-u|_{\\Gamma _R}}{h} .$ For a given $p\\in C^2(\\partial D)$ , we extend its domain to $\\bar{\\Omega }$ by requiring that $p\\in C^2(\\Omega )\\cap C(\\bar{\\Omega }), p=0$ on $\\Gamma _R$ , and $y=\\xi ^h(x)=x+hp(x)$ maps $\\Omega $ to $\\Omega _h$ .",
"It is clear to note that $\\xi ^h$ is a diffeomorphism from $\\Omega $ to $\\Omega _h$ for sufficiently small $h$ .",
"Denote by $\\eta ^h(y):\\Omega _h\\rightarrow \\Omega $ the inverse map of $\\xi ^h$ .",
"Define $\\breve{u}(x)=(\\breve{u}_1,\\breve{u}_2, \\breve{u}_{3}):=(u_h\\circ \\xi ^h)(x)$ .",
"Using the change of variable $y=\\xi ^h(x)$ , we have from straightforward calculations that $\\int _{\\Omega _h}(\\nabla u_h: \\nabla \\overline{v}_h)\\,{\\rm d}y&=\\sum _{j=1}^3\\int _\\Omega \\nabla \\breve{u}_jJ_{\\eta ^h} J_{\\eta ^h}^\\top \\nabla \\bar{\\breve{v}}_j\\,{\\rm det}(J_{\\xi ^h})\\,{\\rm d}x,\\\\\\int _{\\Omega _h}(\\nabla \\cdot u_h)(\\nabla \\cdot \\bar{v}_h)\\,{ \\rm d}y&=\\int _\\Omega (\\nabla \\breve{u}:J_{\\eta ^h}^\\top ) (\\nabla \\bar{\\breve{v}}:J_{\\eta ^h}^\\top )\\,{\\rm det}(J_{\\xi ^h})\\,{\\rm d}x,\\\\\\int _{\\Omega _h} u_h\\cdot \\bar{v}_h\\,{ \\rm d}y&=\\int _\\Omega \\breve{u}\\cdot \\bar{\\breve{v}}\\,{\\rm det}(J_{\\xi ^h})\\,{\\rm d}x,$ where $\\breve{v}(x)=(\\breve{v}_1,\\breve{v}_2,\\breve{v}_3):=(v_h\\circ \\xi ^h)(x)$ , $J_{\\eta ^h}$ and $J_{\\xi ^h}$ are the Jacobian matrices of the transforms $\\eta ^h$ and $\\xi ^h$ , respectively.",
"For a test function $v_h$ in the domain $\\Omega _h$ , it follows from the transform that $\\breve{v}$ is a test function in the domain $\\Omega $ .",
"Therefore, the sesquilinear form $b^h$ in (REF ) becomes $b^h(\\breve{u}, v)=\\sum _{j=1}^3 \\mu \\int _\\Omega \\nabla \\breve{u}_j J_{\\eta ^h} J_{\\eta ^h}^\\top \\nabla \\bar{v}_j\\,{\\rm det}(J_{\\xi ^h})\\,{\\rm d}x+(\\lambda +\\mu )\\int _\\Omega (\\nabla \\breve{u}: J_{\\eta ^h}^\\top )(\\nabla \\bar{v}: J_{\\eta ^h}^\\top )\\\\\\times {\\rm det}(J_{\\xi ^h})\\,{\\rm d}x-\\omega ^2 \\int _\\Omega \\breve{u}\\cdot \\bar{v}\\,{\\rm det}(J_{\\xi ^h})\\,{\\rm d}x-\\langle {T}\\breve{u},v\\rangle _{\\Gamma _R},$ which gives an equivalent variational formulation of (REF ): $b^h(\\breve{u}, v)=\\langle g,v\\rangle _{\\Gamma _R},\\quad \\forall \\, v\\in H^1_{\\partial D}(\\Omega ).$ A simple calculation yields $b(\\breve{u}-u, v)=b(\\breve{u}, v)-\\langle g,v\\rangle _{\\Gamma _R}=b(\\breve{u},v)-b^h(\\breve{u}, v)=b_1 + b_2 + b_3,$ where $b_1&=\\sum _{j=1}^3 \\mu \\int _\\Omega \\nabla \\breve{u}_j \\left(I-J_{\\eta ^h} J_{\\eta ^h}^\\top \\,{\\rm det}(J_{\\xi ^h})\\right) \\nabla \\bar{v}_j\\,{\\rm d}x,\\\\b_2&=(\\lambda +\\mu )\\int _\\Omega (\\nabla \\cdot \\breve{u})(\\nabla \\cdot \\bar{v})-(\\nabla \\breve{u}:J_{\\eta ^h}^\\top ) (\\nabla \\bar{v}:J_{\\eta ^h}^\\top )\\,{\\rm det}(J_{\\xi ^h})\\,{\\rm d}x,\\\\b_3&=\\omega ^2\\int _\\Omega \\breve{u}\\cdot \\bar{v}\\,\\left({ \\rm det}(J_{\\xi ^h})-1\\right)\\,{\\rm d}x.$ Here $I$ is the identity matrix.",
"Following the definitions of the Jacobian matrices, we may easily verify that ${\\rm det}(J_{\\xi ^h})&=1+h\\nabla \\cdot p+O(h^2),\\\\J_{\\eta ^h}&=J^{-1}_{\\xi ^h}\\circ \\eta ^h=I-hJ_{p}+O(h^2),\\\\J_{\\eta ^h} J^\\top _{\\eta ^h} {\\rm det}(J_{\\xi ^h})&=I-h(J_{p}+J^\\top _{p})+h(\\nabla \\cdot p)I+O(h^2),$ where the matrix $J_{p}=\\nabla p$ .",
"Substituting the above estimates into (REF )–(), we obtain $b_1&=\\sum _{j=1}^3 \\mu \\int _\\Omega \\nabla \\breve{u}_j \\left( h(J_{p}+J^\\top _{p})-h(\\nabla \\cdot p)I+O(h^2)\\right)\\nabla \\bar{v}_j\\,{\\rm d}x,\\\\b_2&=(\\lambda +\\mu )\\int _\\Omega h (\\nabla \\cdot \\breve{u})(\\nabla \\bar{v}: J^\\top _{p})+h(\\nabla \\cdot \\bar{v})(\\nabla \\breve{u}:J^\\top _{p})\\\\&\\hspace{85.35826pt}-h(\\nabla \\cdot p)(\\nabla \\cdot \\breve{u})(\\nabla \\cdot \\bar{v})+O(h^2)\\,{\\rm d}x,\\\\b_3&=\\omega ^2\\int _\\Omega \\breve{u}\\cdot \\bar{v}\\,\\left(h\\nabla \\cdot p+O(h^2)\\right)\\,{\\rm d}x.$ Hence we have $b\\left(\\frac{\\breve{u}-u}{h}, v\\right)=g_1(p)(\\breve{u},v)+g_2(p)(\\breve{u},v)+g_3(p)(\\breve{u},v)+O(h),$ where $g_1&=\\sum _{j=1}^3 \\mu \\int _\\Omega \\nabla \\breve{u}_j \\left( (J_{p}+J^\\top _{p} )-(\\nabla \\cdot p)I\\right)\\nabla \\bar{v}_j\\,{\\rm d}x,\\\\g_2&=(\\lambda +\\mu )\\int _\\Omega (\\nabla \\cdot \\breve{u})(\\nabla \\bar{v}: J^\\top _{p})+(\\nabla \\cdot \\bar{v})(\\nabla \\breve{u}:J^\\top _{p})-(\\nabla \\cdot p)(\\nabla \\cdot \\breve{u})(\\nabla \\cdot \\bar{v})\\,{\\rm d}x,\\\\g_3&=\\omega ^2\\int _\\Omega (\\nabla \\cdot p)\\breve{u}\\cdot \\bar{v}\\,{\\rm d}x.$ Given $p\\in C^2(\\partial D)$ , the domain derivative of the scattering operator ${S}$ is ${S}^{\\prime }(\\partial D;p)=u^{\\prime }|_{\\Gamma _R}$ , where $u^{\\prime }$ is the unique weak solution of the boundary value problem: ${\\left\\lbrace \\begin{array}{ll}\\mu \\Delta u^{\\prime }+(\\lambda +\\mu )\\nabla \\nabla \\cdot u^{\\prime }+\\omega ^2u^{\\prime }=0 &\\quad \\text{in} ~ \\Omega ,\\\\u^{\\prime }=-({p}\\cdot \\nu )\\partial _{\\nu }u &\\quad \\text{on} ~ \\partial D,\\\\{B}u^{\\prime }={T}u^{\\prime }&\\quad \\text{on} ~\\Gamma _R,\\end{array}\\right.",
"}$ and $u$ is the solution of the variational problem (REF ) corresponding to the domain $\\Omega $ .",
"Given $p\\in C^2(\\partial D)$ , we extend its definition to the domain $\\bar{\\Omega }$ as before.",
"It follows from the well-posedness of the variational problem (REF ) that $\\breve{u}\\rightarrow u$ in $H^1_{\\partial D}(\\Omega )$ as $h\\rightarrow 0$ .",
"Taking the limit $h\\rightarrow 0$ in (REF ) gives $b\\left(\\lim _{h\\rightarrow 0}\\frac{\\breve{u}-u}{h},v\\right)=g_1(p)(u,v)+g_2(p)(u,v)+g_3(p)(u,v),$ which shows that $(\\breve{u}-u)/h$ is convergent in $H^1_{\\partial D}(\\Omega )$ as $h\\rightarrow 0$ .",
"Denote the limit by $\\dot{u}$ and rewrite (REF ) as $b(\\dot{u}, v)=g_1(p)(u,v)+g_2(p)(u,v)+g_3(p)(u,v).$ First we compute $g_1(p)(u, v)$ .",
"Noting $p=0$ on $\\partial B$ and using the identity $\\nabla u \\left( (J_{p}+J^\\top _{p})-(\\nabla \\cdot p)I \\right)\\nabla \\bar{v}=&\\nabla \\cdot \\left[(p\\cdot \\nabla u)\\nabla \\bar{v}+(p\\cdot \\nabla \\bar{v})\\nabla u -(\\nabla u\\cdot \\nabla \\bar{v})p\\right]\\\\&-(p\\cdot \\nabla u)\\Delta \\bar{v}-(p\\cdot \\nabla \\bar{v})\\Delta u,$ we obtain from the divergence theorem that $g_1(p)(u, v)&=-\\sum _{j=1}^3\\mu \\int _\\Omega (p\\cdot \\nabla u_j)\\Delta \\bar{v}_j+(p\\cdot \\nabla \\bar{v}_j)\\Delta u_j\\,{\\rm d}x\\\\&\\qquad -\\sum _{j=1}^3 \\mu \\int _{\\partial D}(p\\cdot \\nabla u_j)(\\nu \\cdot \\nabla \\bar{v}_j)+(p\\cdot \\nabla \\bar{v}_j)(\\nu \\cdot \\nabla u_j) - (p\\cdot \\nu )(\\nabla u_j\\cdot \\nabla \\bar{v}_j)\\, {\\rm d}\\gamma \\\\&=-\\mu \\int _\\Omega (p\\cdot \\nabla u)\\cdot \\Delta \\bar{v}+(p\\cdot \\nabla \\bar{v})\\cdot \\Delta u\\,{\\rm d}x\\\\&\\qquad -\\mu \\int _{\\partial D}(p\\cdot \\nabla u)\\cdot (\\nu \\cdot \\nabla \\bar{v})+(p\\cdot \\nabla \\bar{v})\\cdot (\\nu \\cdot \\nabla u) -(p\\cdot \\nu )(\\nabla u:\\nabla \\bar{v})\\, {\\rm d}\\gamma .$ Noting $\\mu \\Delta u+(\\lambda +\\mu )\\nabla \\nabla \\cdot u+\\omega ^2u=0\\quad \\text{in} ~\\Omega ,$ we have from the integration by parts that $& \\mu \\int _\\Omega (p\\cdot \\nabla \\bar{v})\\cdot \\Delta u\\,{\\rm d}x=-(\\lambda +\\mu )\\int _\\Omega (p\\cdot \\nabla \\bar{v})\\cdot (\\nabla \\nabla \\cdot u)\\,{\\rm d}x-\\omega ^2\\int _\\Omega (p\\cdot \\nabla \\bar{v})\\cdot u\\,{\\rm d}x\\\\&=(\\lambda +\\mu )\\int _\\Omega (\\nabla \\cdot u)\\nabla \\cdot (p\\cdot \\nabla \\bar{v})\\,{\\rm d}x+(\\lambda +\\mu )\\int _{\\partial D}(\\nabla \\cdot u)(\\nu \\cdot (p\\cdot \\nabla \\bar{v}) )\\,{\\rm d}\\gamma \\\\&\\hspace{113.81102pt}-\\omega ^2\\int _\\Omega (p\\cdot \\nabla \\bar{v})\\cdot u\\,{\\rm d}x.$ Using the integration by parts again yields $\\mu \\int _\\Omega (p\\cdot \\nabla u)\\cdot \\Delta \\bar{v}\\,{\\rm d}x=-\\mu \\int _\\Omega \\nabla (p\\cdot \\nabla u): \\nabla \\bar{v}\\,{\\rm d}x+\\mu \\int _{\\partial D} (p\\cdot \\nabla u)\\cdot (\\nu \\cdot \\nabla \\bar{v})\\,{\\rm d}\\gamma .$ Let $\\tau _1(x), \\tau _2(x)$ be any two linearly independent unit tangent vectors on $\\partial D$ .",
"Since $u=v=0$ on $\\partial D$ , we have $\\partial _{\\tau _1}u_j=\\partial _{\\tau _2}u_j=\\partial _{\\tau _1}v_j=\\partial _{\\tau _2}v_j=0.$ Using the identities $\\nabla u_j&=\\tau _1\\partial _{\\tau _1}u_j+\\tau _2\\partial _{\\tau _2} u_j+\\nu \\partial _{\\nu }u_j=\\nu \\partial _{\\nu } u_j,\\\\\\nabla v_j&=\\tau _1\\partial _{\\tau _1}v_j+\\tau _2\\partial _{\\tau _2} v_j+\\nu \\partial _{\\nu }v_j=\\nu \\partial _{\\nu } v_j,$ we have $(p\\cdot \\nabla \\bar{v}_j)(\\nu \\cdot \\nabla u_j)=(p\\cdot \\nu \\partial _{\\nu }\\bar{v}_j)(\\nu \\cdot \\nu \\partial _{\\nu } u_j)=(p\\cdot \\nu )(\\partial _{\\nu }\\bar{v}_j\\partial _{\\nu }u_j),$ which gives $\\int _{\\partial D}(p\\cdot \\nabla \\bar{v})\\cdot (\\nu \\cdot \\nabla u)-(p\\cdot \\nu )(\\nabla u:\\nabla \\bar{v})\\, {\\rm d}\\gamma =0.$ Noting $v=0$ on $\\partial D$ and $(\\nabla \\cdot p)(u\\cdot \\bar{v})+(p\\cdot \\nabla \\bar{v})\\cdot u=\\nabla \\cdot ((u\\cdot \\bar{v})p)-(p\\cdot \\nabla u)\\cdot \\bar{v},$ we obtain by the divergence theorem that $\\int _\\Omega (\\nabla \\cdot p)(u\\cdot \\bar{v})+(p\\cdot \\nabla \\bar{v})\\cdot u\\,{\\rm d}x=-\\int _\\Omega (p\\cdot \\nabla u)\\cdot \\bar{v}\\,{\\rm d}x.$ Combining the above identities, we conclude that $g_1(p)(u, v)+g_3(p)(u, v)=\\mu \\int _\\Omega \\nabla (p\\cdot \\nabla u): \\nabla \\bar{v}\\,{\\rm d}x-(\\lambda +\\mu )\\int _\\Omega (\\nabla \\cdot u)\\nabla \\cdot (p\\cdot \\nabla \\bar{v})\\,{\\rm d}x\\\\-\\omega ^2\\int _\\Omega (p\\cdot \\nabla u)\\cdot \\bar{v}\\,{\\rm d}x+(\\lambda +\\mu )\\int _{\\partial D}(\\nabla \\cdot u)(\\nu \\cdot (p\\cdot \\nabla \\bar{v}))\\,{ \\rm d}\\gamma .$ Next we compute $g_2(p)(u, v)$ .",
"It is easy to verify that $\\int _\\Omega &(\\nabla \\cdot u) (\\nabla \\bar{v}:J_{p}^\\top )+(\\nabla \\cdot \\bar{v})(\\nabla u: J_{p}^\\top )\\,{\\rm d}x=\\int _\\Omega (\\nabla \\cdot u)\\nabla \\cdot (p\\cdot \\nabla \\bar{v})\\,{\\rm d}x\\\\&\\quad -\\int _\\Omega (\\nabla \\cdot u)(p\\cdot (\\nabla \\cdot (\\nabla \\bar{v})^\\top ) )\\,{\\rm d}x+\\int _\\Omega (\\nabla \\cdot \\bar{v})\\nabla \\cdot (p\\cdot \\nabla u)\\,{\\rm d}x\\\\&\\hspace{113.81102pt}-\\int _\\Omega (\\nabla \\cdot \\bar{v})(p \\cdot (\\nabla \\cdot (\\nabla u)^\\top ))\\,{\\rm d}x.$ Using the integration by parts, we obtain $&\\int _\\Omega (\\nabla \\cdot p)(\\nabla \\cdot u)(\\nabla \\cdot \\bar{v})\\,{\\rm d}x=-\\int _\\Omega p\\cdot \\nabla ((\\nabla \\cdot u)(\\nabla \\cdot \\bar{v}))\\,{\\rm d}x\\\\&\\hspace{170.71652pt}-\\int _{\\partial D}(\\nabla \\cdot u)(\\nabla \\cdot \\bar{v})(\\nu \\cdot p)\\,{\\rm d}\\gamma \\\\&=-\\int _\\Omega (\\nabla \\cdot \\bar{v})(p\\cdot (\\nabla \\cdot (\\nabla u)^\\top ))\\,{\\rm d}x-\\int _\\Omega (\\nabla \\cdot u)(p\\cdot (\\nabla \\cdot (\\nabla v)^\\top ))\\,{\\rm d}x\\\\&\\hspace{170.71652pt}-\\int _{\\partial D}(\\nabla \\cdot u)(\\nabla \\cdot \\bar{v})(\\nu \\cdot p)\\,{\\rm d}\\gamma .$ Let $\\tau _1=(-\\nu _3,0,\\nu _1)^\\top ,\\tau _2=(0,-\\nu _3,\\nu _2)^\\top ,\\tau _3=(-\\nu _2,\\nu _1,0)^\\top .$ It follows from $\\tau _j\\cdot \\nu =0$ that $\\tau _j$ are tangent vectors on $\\partial D$ .",
"Since $v=0$ on $\\partial D$ , we have $\\partial _{\\tau _j}v=0$ , which yields that $& \\nu _1\\partial _{x_3}v_1=\\nu _3\\partial _{x_1} v_1,\\quad \\nu _1\\partial _{x_3}v_2=\\nu _3\\partial _{x_1} v_2, \\quad \\nu _1\\partial _{x_2}v_1=\\nu _2\\partial _{x_1} v_1,\\\\& \\nu _1\\partial _{x_3}v_3=\\nu _3\\partial _{x_1} v_3,\\quad \\nu _1\\partial _{x_2}v_2=\\nu _2\\partial _{x_1} v_2,\\quad \\nu _1\\partial _{x_2}v_3=\\nu _2\\partial _{x_1} v_3,\\\\& \\nu _2\\partial _{x_3}v_1=\\nu _3\\partial _{x_2} v_1,\\quad \\nu _2\\partial _{x_3}v_2=\\nu _3\\partial _{x_2} v_2,\\quad \\nu _2\\partial _{x_3}v_3=\\nu _3\\partial _{x_2} v_3.$ Hence we get $\\int _{\\partial D}(\\nabla \\cdot u)(\\nabla \\cdot \\bar{v})(\\nu \\cdot p)\\,{\\rm d}\\gamma =\\int _{\\partial D}(\\nabla \\cdot u)(\\nu \\cdot (p\\cdot \\nabla \\bar{v}))\\,{ \\rm d}\\gamma .$ Combining the above identities gives $g_2(p)(u, v)=(\\lambda +\\mu )&\\int _\\Omega (\\nabla \\cdot u)\\nabla \\cdot (p\\cdot \\nabla \\bar{v})\\,{\\rm d}x + (\\lambda +\\mu )\\int _\\Omega \\nabla \\cdot (p\\cdot \\nabla u)(\\nabla \\cdot \\bar{v})\\,{\\rm d}x\\\\&-(\\lambda +\\mu )\\int _{\\partial D}(\\nabla \\cdot u)(\\nu \\cdot (p\\cdot \\nabla \\bar{v}))\\,{ \\rm d}\\gamma .$ Noting (REF ), adding (REF ) and (REF ), we obtain $b(\\dot{u}, v)=\\mu \\int _\\Omega \\nabla (p\\cdot \\nabla u): \\nabla \\bar{v}\\,{\\rm d}x + (\\lambda +\\mu )\\int _\\Omega \\nabla \\cdot (p\\cdot \\nabla u)(\\nabla \\cdot \\bar{v})\\,{\\rm d}x-\\omega ^2\\int _\\Omega (p\\cdot \\nabla u)\\cdot \\bar{v}\\,{\\rm d}x.$ Define $u^{\\prime }=\\dot{u}-p\\cdot \\nabla u$ .",
"It is clear to note that $p\\cdot \\nabla u=0$ on $\\Gamma _R$ since $p=0$ on $\\Gamma _R$ .",
"Hence, we have $b({u}^{\\prime }, v)=0,\\quad \\forall \\,v\\in H^1_{\\partial D}(\\Omega ),$ which shows that $u^{\\prime }$ is the weak solution of the boundary value problem (REF ).",
"To verify the boundary condition of $u^{\\prime }$ on $\\partial D$ , we recall the definition of $u^{\\prime }$ and have from $\\breve{u}=u=0$ on $\\partial D$ that $u^{\\prime }=\\lim _{h\\rightarrow 0}\\frac{\\breve{u}-u}{h}-p\\cdot \\nabla u=-p\\cdot \\nabla u\\quad \\text{on} ~ \\partial D.$ Noting $u=0$ on $\\partial D$ , we have ${p}\\cdot \\nabla u=({p}\\cdot \\nu )\\partial _{\\nu }u,$ which completes the proof by combining (REF ) and (REF )."
],
[
"Reconstruction method",
"Assume that the surface has a parametric equation: $\\partial D=\\lbrace r(\\theta ,\\varphi )=(r_1(\\theta , \\varphi ),r_2(\\theta ,\\varphi ), r_3(\\theta ,\\varphi ))^\\top ,~\\theta \\in (0, \\pi ),\\,\\varphi \\in (0, 2\\pi )\\rbrace ,$ where $r_j$ are biperiodic functions of $(\\theta , \\varphi )$ and have the Fourier series expansions: $r_j(\\theta ,\\phi )=\\sum _{n=0}^{\\infty }\\sum _{m=-n}^{n} a_{jn}^m {\\rm Re}Y_n^m(\\theta ,\\varphi )+ b_{jn}^m {\\rm Im}Y_n^m(\\theta ,\\varphi ),$ where $Y_n^m$ are the spherical harmonics of order $n$ .",
"It suffices to determine $a_{jn}^m, b_{jn}^m$ in order to reconstruct the surface.",
"In practice, a cut-off approximation is needed: $r_{j,N}(\\theta , \\varphi )=\\sum _{n=0}^{N}\\sum _{m=-n}^{n} a_{jn}^m {\\rm Re}Y_n^m(\\theta ,\\varphi )+ b_{jn}^m {\\rm Im}Y_n^m(\\theta ,\\varphi ).$ Denote by $D_N$ the approximated obstacle with boundary $\\partial D_N$ , which has the parametric equation $\\partial D_N=\\lbrace r_N(\\theta ,\\varphi )=(r_{1,N}(\\theta ,\\varphi ),r_{2,N}(\\theta , \\varphi ),r_{3,N}(\\theta ,\\varphi ))^\\top ,~\\theta \\in (0,\\pi ),\\,\\phi \\in (0,2\\pi )\\rbrace .$ Let $\\Omega _N=B_R\\setminus \\bar{D}_N$ and $a_j=(a_{j0}^0, \\cdots , a_{jn}^m, \\cdots , a_{jN}^N),\\quad b_j=(b_{j0}^0, \\cdots , b_{jn}^m, \\cdots , b_{jN}^N),$ where $n=0, 1, \\dots , N, ~ m=-n, \\dots , n.$ Denote the vector of Fourier coefficients $C=(a_1, b_1, a_2,b_2, a_3, b_3)^\\top =(c_1, c_2, \\dots ,c_{6(N+1)^2})^\\top \\in \\mathbb {R}^{6(N+1)^2}$ and a vector of scattering data $U=(u(x_1),\\dots ,u(x_K))^\\top \\in \\mathbb {C}^{3K},$ where $x_k\\in \\Gamma _R, k=1,\\dots ,K$ .",
"Then the inverse problem can be formulated to solve an approximate nonlinear equation: ${F}(C)=U,$ where the operator ${F}$ maps a vector in $\\mathbb {R}^{6(N+1)^2}$ into a vector in $\\mathbb {C}^{3K}$ .",
"Let $u_N$ be the solution of the variational problem (REF ) corresponding to the obstacle $D_N$ .",
"The operator ${F}$ is differentiable and its derivatives are given by $\\frac{\\partial {F}_k(C)}{\\partial c_i}=u^{\\prime }_i(x_k),\\quad i=1, \\dots , 6(N+1)^2, ~k=1, \\dots , K,$ where $u^{\\prime }_i$ is the unique weak solution of the boundary value problem ${\\left\\lbrace \\begin{array}{ll}\\mu \\Delta u^{\\prime }_i +(\\lambda +\\mu )\\nabla \\nabla \\cdot u^{\\prime }_i+\\omega ^2u^{\\prime }_i=0 &\\quad \\text{in}~\\Omega _N,\\\\u^{\\prime }_i=-q_i \\partial _{\\nu _N} u_N&\\quad \\text{on} ~\\partial D_N.\\\\{B}u^{\\prime }_i={T}u^{\\prime }_i&\\quad \\text{on} ~\\Gamma _R.\\end{array}\\right.",
"}$ Here $\\nu _N=(\\nu _{N 1}, \\nu _{N 2}, \\nu _{N 3})^\\top $ is the unit normal vector on $\\partial D_N$ and $q_i(\\theta , \\varphi )={\\left\\lbrace \\begin{array}{ll}\\nu _{N 1} {\\rm Re}Y_{n}^{m}(\\theta , \\varphi ), & i=n^2+n+m+1,\\\\\\nu _{N 1} {\\rm Im}Y_{n}^{m}(\\theta , \\varphi ), & i=(N+1)^2+n^2+n+m+1,\\\\\\nu _{N 2} {\\rm Re}Y_{n}^{m}(\\theta , \\varphi ), & i=2(N+1)^2+n^2+n+m+1,\\\\\\nu _{N 2} {\\rm Im}Y_{n}^{m}(\\theta , \\varphi ), & i=3(N+1)^2+n^2+n+m+1,\\\\\\nu _{N 3} {\\rm Re}Y_{n}^{m}(\\theta , \\varphi ), & i=4(N+1)^2+n^2+n+m+1,\\\\\\nu _{N 3} {\\rm Im}Y_{n}^{m}(\\theta , \\varphi ), & i=5(N+1)^2+n^2+n+m+1,\\end{array}\\right.",
"}$ where $n=0, 1, \\dots , N, m=-n, \\dots , n$ .",
"Fix $i\\in \\lbrace 1, \\dots , 6(N+1)^2\\rbrace $ and $k\\in \\lbrace 1, \\dots , K\\rbrace $ , and let $\\lbrace e_1, \\dots , e_{6(N+1)^2}\\rbrace $ be the set of natural basis vectors in $\\mathbb {R}^{6(N+1)^2}$ .",
"By definition, we have $\\frac{\\partial {F}_k(C)}{\\partial c_i}=\\lim _{h\\rightarrow 0}\\frac{{F}_k(C+he_i)-{F}_k(C)}{h}.$ A direct application of Theorem REF shows that the above limit exists and the limit is the unique weak solution of the boundary value problem (REF ).",
"Consider the objective function $f({C})=\\frac{1}{2}\\Vert {F}(C)-U\\Vert ^2=\\frac{1}{2}\\sum _{k=1}^K|{F}_k(C)-u(x_k)|^2.$ The inverse problem can be formulated as the minimization problem: $\\min _{C}f(C),\\quad {C}\\in \\mathbb {R}^{6(N+1)^2}.$ In order to apply the descend method, we have to compute the gradient of the objective function: $\\nabla f(C)=\\left(\\frac{\\partial f(C)}{\\partial c_1}, \\dots , \\frac{f(C)}{\\partial c_{6(N+1)^2}}\\right)^\\top .$ We have from Theorem REF that $\\frac{\\partial f(C)}{\\partial c_i}={ \\rm Re}\\sum _{k=1}^Ku^{\\prime }_i(x_k)\\cdot (\\bar{{F}}_k(C)-\\bar{u}(x_k)).$ We assume that the scattering data ${U}$ is available over a range of frequencies $\\omega \\in [\\omega _{\\rm min},\\,\\omega _{\\rm max}]$ , which may be divided into $\\omega _{\\rm min}=\\omega _0<\\omega _1<\\cdots <\\omega _J=\\omega _{\\rm max}$ .",
"We now propose an algorithm to reconstruct the Fourier coefficients $c_i, i=1,\\dots , 6(N+1)^2$ .",
"Algorithm: Frequency continuation algorithm for surface reconstruction.",
"Initialization: take an initial guess $c_{2}=-c_{4}=1.44472 R_0$ and $c_{3(N+1)^2+2}=c_{3(N+1)^2+4}=1.44472 R_0$ , $c_{4(N+1)^2+3}=2.0467 R_0$ and $c_i=0$ otherwise.",
"The initial guess is a ball with radius $R_0$ under the spherical harmonic functions; First approximation: begin with $\\omega _0$ , let $k_0=[\\omega _0]$ , seek an approximation to the functions $r_{j, N}$ : $r_{j,k_0}=\\sum _{n=0}^{k_0}\\sum _{m=-n}^{n} a_{jn}^m {\\rm Re}Y_n^m(\\theta ,\\phi )+ b_{jn}^m {\\rm Im}Y_n^m(\\theta ,\\phi ).$ Denote $C^{(1)}_{k_0}=(c_1, c_2, \\dots ,c_{6(k_0+1)^2})^\\top $ and consider the iteration: ${\\bf C}_{k_0}^{(l+1)}={\\bf C}_{k_0}^{(l)}-\\tau \\nabla f({\\bf C}_{k_0}^{(l)}),\\quad l=1, \\dots , L,$ where $\\tau >0$ and $L>0$ are the step size and the number of iterations for every fixed frequency, respectively.",
"Continuation: increase to $\\omega _1$ , let $k_1=[\\omega _1]$ , repeat Step 2 with the previous approximation to $r_{j, N}$ as the starting point.",
"More precisely, approximate $r_{j, N}$ by $r_{j, k_1}=\\sum _{n=0}^{k_1}\\sum _{m=-n}^{n} a_{jn}^m {\\rm Re}Y_n^m(\\theta ,\\phi )+b_{jn}^m {\\rm Im} Y_n^m(\\theta ,\\phi ),$ and determine the coefficients $\\tilde{c}_i, i=1, \\dots , 6(k_1+1)^2$ by using the descent method starting from the previous result.",
"Iteration: repeat Step 3 until a prescribed highest frequency $\\omega _J$ is reached."
],
[
"Numerical experiments",
"In this section, we present two examples to show the effectiveness of the proposed method.",
"The scattering data is obtained from solving the direct problem by using the finite element method with the perfectly matched layer technique, which is implemented via FreeFem++ [12].",
"The finite element solution is interpolated uniformly on $\\Gamma _R$ .",
"To test the stability, we add noise to the data: $u^\\delta (x_k)=u(x_k)(1+\\delta \\,{\\rm rand}),\\quad k=1,\\dots , K,$ where rand are uniformly distributed random numbers in $[-1,\\,1]$ and $\\delta $ is the relative noise level, $x_k$ are data points.",
"In our experiments, we pick 100 uniformly distributed points $x_k$ on $\\Gamma _R$ , i.e., $K=100$ .",
"In the following two examples, we take $\\lambda =2,\\mu =1$ , $R=1$ .",
"The radius of the initial $R_0=0.5$ .",
"The noise level $\\delta =5\\%$ .",
"The step size in (REF ) is $\\tau =0.005/k_i$ where $k_i=[\\omega _i]$ .",
"The incident field is taken as a plane compressional wave.",
"Example 1.",
"Consider a bean-shaped obstacle: $r(\\theta , \\varphi )=(r_1(\\theta , \\varphi ), r_2(\\theta , \\varphi ),r_3(\\theta , \\varphi ))^\\top ,~\\theta \\in [0, \\pi ],\\,\\varphi \\in [0, 2\\pi ],$ where $r_1(\\theta ,\\varphi )&=0.75\\left((1-0.05\\cos (\\pi \\cos \\theta ))\\sin \\theta \\cos \\varphi \\right)^{1/2},\\\\r_2(\\theta , \\varphi )&=0.75\\left((1-0.005\\cos (\\pi \\cos \\theta ))\\sin \\theta \\sin \\varphi +0.35\\cos (\\pi \\cos \\theta )\\right)^{1/2},\\\\r_3(\\theta , \\varphi ) &= 0.75\\cos \\theta .$ The exact surface is plotted in Figure REF (a).",
"This obstacle is non-convex and is usually difficult to reconstruct the concave part of the obstacle.",
"The obstacle is illuminated by the compressional wave sent from a single direction $d=(0,1,0)^\\top $ ; the frequency ranges from $\\omega _{\\rm min}=1$ to $\\omega _{\\rm max}=5$ with increment 1 at each continuation step, i.e., $\\omega _i=i+1, i=0, \\dots , 4$ ; for any fixed frequency, repeat $L=100$ times with previous result as starting points.",
"The step size for the decent method is $0.005/\\omega _i$ .",
"The number of recovered coefficients is $6(\\omega _i+2)^2$ for corresponding frequency.",
"Figure REF (b) shows the initial guess which is the ball with radius $R_0=0.5$ ; Figure REF (c) shows the final reconstructed surface; Figures REF (d)–(f) show the cross section of the exact surface along the plane $x_1=0, x_2=0, x_3=0$ , respectively; Figures REF (g)–(i) show the corresponding cross section for the reconstructed surface along the plane $x_1=0, x_2=0, x_3=0$ , respectively.",
"As is seen, the algorithm effectively reconstructs the bean-shaped obstacle.",
"Figure: Example 1: A bean-shaped obstacle.",
"(a) the exact surface; (b)the initial guess; (c) the reconstructed surface;(d)–(f) the corresponding cross section of the exact surface along planex 1 =0,x 2 =0,x 3 =0x_1=0, x_2=0, x_3=0, respectively; (g)–(i) the corresponding crosssection of the reconstructed surface along plane x 1 =0,x 2 =0,x 3 =0x_1=0, x_2=0, x_3=0,respectively.Example 2.",
"Consider a cushion-shaped obstacle: $r(\\theta , \\varphi )=r(\\theta , \\varphi )(\\sin (\\theta )\\cos (\\varphi ),\\sin (\\theta )\\sin (\\varphi ), \\cos (\\theta ))^\\top ,~\\theta \\in [0,\\pi ],\\,\\varphi \\in [0, 2\\pi ],$ where $r(\\theta ,\\varphi )=\\left(0.75+0.45(\\cos (2\\varphi )-1)(\\cos (4\\theta )-1)\\right)^{1/2}.$ Figure REF (a) shows the exact surface.",
"This example is much more complex than the bean-shaped obstacle due to its multiple concave parts.",
"Multiple incident directions are needed in order to obtain a good result.",
"In this example, the obstacle is illuminated by the compressional wave from 6 directions, which are the unit vectors pointing to the origin from the face centers of the cube.",
"The multiple frequencies are the same as the first example, i.e., the frequency ranges from $\\omega _{\\rm min}=1$ to $\\omega _{\\rm max}=5$ with $\\omega _i=i+1, i=0, \\dots , 4$ .",
"For each fixed frequency and incident direction, repeat $L=50$ times with previous result as starting points.",
"The step size for the decent method is $0.005/\\omega _i$ and number of recovered coefficients is $6(\\omega _i+2)^2$ for corresponding frequency.",
"Figure REF (b) shows the initial guess ball with radius $R_0=0.5$ ; Figure REF (c) shows the final reconstructed surface; Figure REF (d)–(f) show the cross section of the exact surface along the plane $x_1=0, x_2=0, x_3=0$ , respectively; while Figure REF (g)–(i) show the corresponding cross section for the reconstructed surface along the plane $x_1=0, x_2=0, x_3=0$ , respectively.",
"It is clear to note that the algorithm can also reconstruct effectively the more complex cushion-shaped obstacle.",
"Figure: Example 2: A cushion-shaped obstacle.",
"(a) the exact surface; (b)the initial guess; (c) the reconstructed surface; (d)–(f) the correspondingcross section of the exact surface along the plane x 1 =0,x 2 =0,x 3 =0x_1=0, x_2=0, x_3=0,respectively; (d)–(f) the corresponding cross section of the reconstructedsurface along the plane x 1 =0,x 2 =0,x 3 =0x_1=0, x_2=0, x_3=0, respectively."
],
[
"Conclusion",
"In this paper, we have studied the direct and inverse obstacle scattering problems for elastic waves in three dimensions.",
"We develop an exact transparent boundary condition and show that the direct problem has a unique weak solution.",
"We examine the domain derivative of the total displacement with respect to the surface of the obstacle.",
"We propose a frequency continuation method for solving the inverse scattering problem.",
"Numerical examples are presented to demonstrate the effectiveness of the proposed method.",
"The results show that the method is stable and accurate to reconstruct surfaces with noise.",
"Future work includes the surfaces of different boundary conditions and multiple obstacles where each obstacle's surface has a parametric equation.",
"We hope to be able to address these issues and report the progress elsewhere in the future."
],
[
"Spherical harmonics and functional spaces",
"The spherical coordinates $(r, \\theta , \\varphi )$ are related to the Cartesian coordinates $x=(x_1, x_2, x_3)$ by $x_1=r\\sin \\theta \\cos \\varphi ,x_2=r\\sin \\theta \\sin \\varphi , x_3=r\\cos \\theta $ .",
"The local orthonormal basis is $e_r &=(\\sin \\theta \\cos \\varphi , \\sin \\theta \\sin \\varphi ,\\cos \\theta )^\\top , \\\\e_\\theta &=(\\cos \\theta \\cos \\varphi ,\\cos \\theta \\sin \\varphi ,-\\sin \\theta )^\\top , \\\\e_\\varphi &=(-\\sin \\varphi , \\cos \\varphi , 0)^\\top .$ Let $\\lbrace Y_n^m(\\theta , \\varphi ): n=0, 1, 2, \\dots , m=-n, \\dots , n\\rbrace $ be the orthonormal sequence of spherical harmonics of order $n$ on the unit sphere.",
"Define rescaled spherical harmonics $X_n^m(\\theta , \\varphi )=\\frac{1}{R}Y_n^m(\\theta , \\varphi ).$ It can be shown that $\\lbrace X_n^m(\\theta , \\varphi ): n=0, 1, \\dots , m=-n,\\dots , n\\rbrace $ form a complete orthonormal system in $L^2(\\Gamma _R)$ .",
"For a smooth scalar function $u(R, \\theta , \\varphi )$ defined on $\\Gamma _R$ , let $\\nabla _{\\Gamma _R}u=\\partial _\\theta u\\,e_\\theta +(\\sin \\theta )^{-1}\\partial _\\varphi u\\,e_\\varphi $ be the tangential gradient on $\\Gamma _R$ .",
"Define a sequence of vector spherical harmonics: $T_n^m(\\theta , \\varphi )&=\\frac{1}{\\sqrt{n(n+1)}}\\nabla _{\\Gamma _R}X_n^m(\\theta , \\varphi ),\\\\V_n^m(\\theta , \\varphi )&=T_n^m(\\theta ,\\varphi )\\times e_r,\\quad W_n^m(\\theta , \\varphi )=X_n^m(\\theta , \\varphi )e_r,$ where $n=0, 1, \\dots , m=-n, \\dots , n$ .",
"Using the orthogonality of the vector spherical harmonics, we can also show that $\\lbrace (T_n^m, V_n^m,W_n^m): n=0, 1, 2, \\dots , m=-n, \\dots , n\\rbrace $ form a complete orthonormal system in $L^2(\\Gamma _R)=L^2(\\Gamma _R)^3$ .",
"Let $L^2(\\Omega )=L^2(\\Omega )^3$ be equipped with the inner product and norm: $(u, v)=\\int _\\Omega u\\cdot \\bar{v}\\,{\\rm d}x,\\quad \\Vert u\\Vert _{L^2(\\Omega )}=(u,u)^{1/2}.$ Denote by $H^1(\\Omega )$ the standard Sobolev space with the norm given by $\\Vert u\\Vert _{H^1(\\Omega )}=\\left(\\int _\\Omega |u(x)|^2+|\\nabla u(x)|^2\\,{\\rm d}x\\right)^{1/2}.$ Let $H^1_{\\partial D}(\\Omega )=H^1_{\\partial D}(\\Omega )^3$ , where $H^1_{\\partial D}(\\Omega ):=\\lbrace u\\in H^1(\\Omega ): u=0~\\text{on}~\\partial D\\rbrace $ .",
"Introduce the Sobolev space $H({\\rm curl}, \\Omega )=\\lbrace u\\in L^2(\\Omega ),\\nabla \\times u\\in L^2(\\Omega )\\rbrace ,$ which is equipped with the norm $\\Vert u\\Vert _{H({\\rm curl},\\Omega )}=\\left(\\Vert u\\Vert ^2_{L^2(\\Omega )}+\\Vert \\nabla \\times u\\Vert ^2_{L^2(\\Omega )}\\right)^{1/2}.$ Denote by $H^s(\\Gamma _R)$ the trace functional space which is equipped with the norm $\\Vert u\\Vert _{H^s(\\Gamma _R)}=\\left(\\sum _{n=0}^\\infty \\sum _{m=-n}^n (1+n(n+1))^s|u_n^m|^2\\right)^{1/2},$ where $u(R, \\theta , \\varphi )=\\sum _{n=0}^\\infty \\sum _{m=-n}^n u_n^m X_n^m(\\theta ,\\varphi ).$ Let $H^s(\\Gamma _R)=H^s(\\Gamma _R)^3$ which is equipped with the normal $\\Vert u\\Vert _{H^s(\\Gamma _R)}=\\left(\\sum _{n=0}^\\infty \\sum _{m=-n}^n (1+n(n+1))^s|u_n^m|^2\\right)^{1/2},$ where $u_n^m=(u_{1n}^m, u_{2n}^m, u_{3n}^m)^\\top $ and $u(R, \\theta , \\varphi )=\\sum _{n=0}^\\infty \\sum _{m=-n}^nu_{1n}^mT_n^m(\\theta , \\varphi )+u_{2n}^mV_n^m(\\theta ,\\varphi )+u_{3n}^mW_n^m(\\theta , \\varphi ).$ It can be verified that $H^{-s}(\\Gamma _R)$ is the dual space of $H^s(\\Gamma _R)$ with respect to the inner product $\\langle u, v\\rangle _{\\Gamma _R}=\\int _{\\Gamma _R}u\\cdot \\bar{v}\\,{\\rm d}\\gamma =\\sum _{n=0}^\\infty \\sum _{m=-n}^n u_{1n}^m\\bar{v}_{1n}^m+u_{2n}^m\\bar{v}_{2n}^m+u_{3n}^m \\bar{v}_{3n}^m,$ where $v(R, \\theta , \\varphi )=\\sum _{n=0}^\\infty \\sum _{m=-n}^nv_{1n}^mT_n^m(\\theta , \\varphi )+v_{2n}^mV_n^m(\\theta ,\\varphi )+v_{3n}^mW_n^m(\\theta , \\varphi ).$ Introduce three tangential trace spaces: $H_{\\rm t}^s(\\Gamma _R)&=\\lbrace u\\in H^s(\\Gamma _R), ~u\\cdot e_r=0\\rbrace ,\\\\H^{-1/2}({\\rm curl}, \\Gamma _R)&=\\lbrace u\\in H^{-1/2}_{\\rm t}(\\Gamma _R), ~ {\\rm curl}_{\\Gamma _R}u\\in H^{-1/2}(\\Gamma _R)\\rbrace ,\\\\H^{-1/2}({\\rm div}, \\Gamma _R)&=\\lbrace u\\in H^{-1/2}_{\\rm t}(\\Gamma _R), ~ {\\rm div}_{\\Gamma _R}u\\in H^{-1/2}(\\Gamma _R)\\rbrace .$ For any tangential field $u\\in H^s_{\\rm t}(\\Gamma _R)$ , it can be represented in the series expansion $u(R, \\theta , \\varphi )=\\sum _{n=0}^\\infty \\sum _{m=-n}^n u_{1n}^mT_n^m(\\theta , \\varphi )+u_{2n}^m V_n^m(\\theta , \\varphi ).$ Using the series coefficients, the norm of the space $H^s_{\\rm t}(\\Gamma _R)$ can be characterized by $\\Vert u\\Vert ^2_{H^s_{\\rm t}(\\Gamma _R)}=\\sum _{n=0}^\\infty \\sum _{m=-n}^n (1+n(n+1))^s\\left(|u_{1n}^m|^2+|u_{2n}^m|^2 \\right);$ the norm of the space $H^{-1/2}({\\rm curl}, \\Gamma _R)$ can be characterized by $\\Vert u\\Vert ^2_{H^{-1/2}({\\rm curl},\\Gamma _R)}=\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\frac{1}{\\sqrt{1+n(n+1)}}|u_{1n}^m|^2+\\sqrt{1+n(n+1)}|u_{2n}^m|^2;$ the norm of the space $H^{-1/2}({\\rm div}, \\Gamma _R)$ can be characterized by $\\Vert u\\Vert ^2_{H^{-1/2}({\\rm div},\\Gamma _R)}=\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\sqrt{1+n(n+1)}|u_{1n}^m|^2+\\frac{1}{\\sqrt{1+n(n+1)}}|u_{2n}^m|^2.$ Given a vector field $u$ on $\\Gamma _R$ , denote by $ u_{\\Gamma _R}=-e_r\\times (e_r\\times u) $ the tangential component of $u$ on $\\Gamma _R$ .",
"Define the inner product in $\\mathbb {C}^3$ : $\\langle u, v\\rangle =v^*u,\\forall \\, u, v\\in \\mathbb {C}^3, $ where $v^*$ is the conjugate transpose of $v$ ."
],
[
"Transparent boundary conditions",
"Recall the Helmholtz decomposition (REF ): $v=\\nabla \\phi +\\nabla \\times \\psi , \\quad \\nabla \\cdot \\psi =0,$ where the scalar potential function $\\phi $ satisfies (REF ) and (REF ): ${\\left\\lbrace \\begin{array}{ll}\\Delta \\phi +\\kappa ^2_{\\rm p}\\phi =0&\\quad \\text{in}~ \\mathbb {R}^3\\setminus \\bar{D},\\\\\\partial _r\\phi -{\\rm i}\\kappa _{\\rm p}\\phi =o(r^{-1})&\\quad \\text{as}~r\\rightarrow \\infty ,\\end{array}\\right.",
"}$ the vector potential function $\\psi $ satisfies (REF ) and (REF ): ${\\left\\lbrace \\begin{array}{ll}\\nabla \\times (\\nabla \\times \\psi )-\\kappa ^2_{\\rm s}\\psi =0&\\quad \\text{in}~ \\mathbb {R}^3\\setminus \\bar{D},\\\\(\\nabla \\times \\psi )\\times \\hat{x}-{\\rm i}\\kappa _{\\rm s}\\psi =o(r^{-1})&\\quad \\text{as} ~ r\\rightarrow \\infty ,\\end{array}\\right.",
"}$ where $r=|x|$ and $\\hat{x}=x/r$ .",
"In the exterior domain $\\mathbb {R}^3\\setminus \\bar{B}_R$ , the solution $\\phi $ of (REF ) satisfies $\\phi (r, \\theta , \\varphi )=\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\frac{h^{(1)}_n(\\kappa _{\\rm p}r)}{h^{(1)}_n(\\kappa _{\\rm p}R)}\\phi _n^mX_n^m(\\theta , \\varphi ),$ where $h^{(1)}_n$ is the spherical Hankel function of the first kind with order $n$ and $\\phi _n^m=\\int _{\\Gamma _R} \\phi (R, \\theta , \\varphi )\\bar{X}_n^m(\\theta ,\\varphi ){\\rm d}\\gamma .$ We define the boundary operator $T_1$ such that $({T}_1\\phi )(R, \\theta ,\\varphi )=\\frac{1}{R}\\sum _{n=0}^\\infty \\sum _{m=-n}^n z_n(\\kappa _{\\rm p}R)\\phi _n^mX_n^m(\\theta , \\varphi ),$ where $ z_n(t)=th_n^{(1)^{\\prime }}(t)/h_n^{(1)}(t)$ satisfies (cf.",
"[31]) $-(n+1) \\le {\\rm Re}z_n(t)\\le -1,\\quad 0<{\\rm Im}z_n(t)\\le t.$ Evaluating the derivative of (REF ) with respect to $r$ at $r=R$ and using (REF ), we get the transparent boundary condition for the scalar potential function $\\phi $ : $\\partial _r\\phi ={T}_1\\phi \\quad \\text{on}~ \\Gamma _R.$ The following result can be easily shown from (REF )–(REF ).",
"The operator $T_1$ is bounded from $H^{1/2}(\\Gamma _R)$ to $H^{-1/2}(\\Gamma _R)$ .",
"Moreover, it satisfies ${\\rm Re}\\langle T_1 u, u\\rangle _{\\Gamma _R}\\le 0,\\quad {\\rm Im}\\langle T_1 u, u\\rangle _{\\Gamma _R}\\ge 0,\\quad \\forall u\\in H^{1/2}(\\Gamma _R).$ If ${\\rm Re}\\langle T_1 u, u\\rangle _{\\Gamma _R}=0$ or ${\\rm Im}\\langle T_1 u, u\\rangle _{\\Gamma _R}=0$ , then $u=0$ on $\\Gamma _R$ .",
"Define an auxiliary function $\\varphi =({\\rm i}\\kappa _{\\rm s})^{-1}\\nabla \\times \\psi $ .",
"We have from (REF ) that $\\nabla \\times \\psi -{\\rm i}\\kappa _{\\rm s}\\varphi =0, \\quad \\nabla \\times \\varphi +{\\rm i}\\kappa _{\\rm s}\\psi =0,$ which are Maxwell's equations.",
"Hence $\\phi $ and $\\psi $ plays the role of the electric field and the magnetic field, respectively.",
"Introduce the vector wave functions ${\\left\\lbrace \\begin{array}{ll}M_n^m(r, \\theta , \\varphi )=\\nabla \\times (xh_n^{(1)}(\\kappa _{\\rm s}r)X_n^m(\\theta , \\varphi )),\\\\N_n^m(r, \\theta , \\varphi )=({\\rm i}\\kappa _{\\rm s})^{-1}\\nabla \\times M_n^m(r, \\theta , \\varphi ),\\end{array}\\right.",
"}$ which are the radiation solutions of (REF ) in $\\mathbb {R}^3\\setminus \\lbrace 0\\rbrace $ (cf.",
"[30]): $\\nabla \\times M_n^m(r, \\theta , \\varphi )-{\\rm i}\\kappa _{\\rm s}N_n^m(r, \\theta , \\varphi )=0,\\quad \\nabla \\times N_n^m(r, \\theta , \\varphi )+{\\rm i}\\kappa _{\\rm s}M_n^m(r, \\theta ,\\varphi )=0.$ Moreover, it can be verified from (REF ) that they satisfy $M_n^m=h_n^{{(1)}}(\\kappa _{\\rm s}r)\\nabla _{\\Gamma _R}X_n^m\\times e_r$ and $N_n^m=\\frac{\\sqrt{n(n+1)}}{{\\rm i}\\kappa _{\\rm s}r}(h_n^{(1)}(\\kappa _{\\rm s}r)+\\kappa _{\\rm s}r h_n^{(1)^{\\prime }}(\\kappa _{\\rm s}r))T_n^m+\\frac{n(n+1)}{{\\rm i}\\kappa _{\\rm s}r}h_n^{(1)}(\\kappa _{\\rm s}r)W_n^m.$ In the domain $\\mathbb {R}^3\\setminus \\bar{B}_R$ , the solution of $\\psi $ in (REF ) can be written in the series $\\psi =\\sum _{n=0}^\\infty \\sum _{m=-n}^n \\alpha _n^mN_n^m+\\beta _n^mM_n^m,$ which is uniformly convergent on any compact subsets in $\\mathbb {R}^3\\setminus \\bar{B}_R$ .",
"Correspondingly, the solution of $\\varphi $ in (REF ) is given by $\\varphi =({\\rm i}\\kappa _{\\rm s})^{-1}\\nabla \\times \\psi =\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\beta _n^mN_n^m-\\alpha _n^mM_n^m.$ It follows from (REF )–(REF ) that $-e_r\\times (e_r\\times M_n^m)&=-\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}r)V_n^m,\\\\-e_r\\times (e_r\\times N_n^m)&=\\frac{\\sqrt{n(n+1)}}{{\\rm i}\\kappa _{\\rm s}r}(h_n^{(1)}(\\kappa _{\\rm s}r)+\\kappa _{\\rm s}r h_n^{(1)^{\\prime }}(\\kappa _{\\rm s}r))T_n^m$ and $e_r\\times M_n^m&=\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}r)T_n^m,\\\\e_r\\times N_n^m&=\\frac{\\sqrt{n(n+1)}}{{\\rm i}\\kappa _{\\rm s}r}(h_n^{(1)}(\\kappa _{\\rm s}r)+\\kappa _{\\rm s}rh_n^{(1)^{\\prime }}(\\kappa _{\\rm s}r))V_n^m.$ Therefore, by (REF ), the tangential component of $\\psi $ on $\\Gamma _R$ is $\\psi _{\\Gamma _R}=\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\frac{\\sqrt{n(n+1)}}{{\\rm i}\\kappa _{\\rm s}R}(h_n^{(1)}(\\kappa _{\\rm s}R)+\\kappa _{\\rm s}R h_n^{(1)^{\\prime }}(\\kappa _{\\rm s}R))\\alpha _n^m T_n^m +\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}R)\\beta _n^m V_n^m.$ Similarly, by (REF ), the tangential trace of $\\varphi $ on $\\Gamma _R$ is $\\varphi \\times e_r &=\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}R)\\alpha _n^m T_n^m\\\\&\\qquad -\\frac{\\sqrt{n(n+1)}}{{\\rm i}\\kappa _{\\rm s}R}(h_n^{(1)}(\\kappa _{\\rm s}R)+\\kappa _{\\rm s}R h_n^{(1)^{\\prime }}(\\kappa _{\\rm s}R))\\beta _n^m V_n^m.$ Given any tangential component of the electric field on $\\Gamma _R$ with the expression $u=\\sum _{n=0}^\\infty \\sum _{m=-n}^n u_{1n}^m T_n^m+u_{2n}^mV_n^m,$ we define ${T}_2u=\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\frac{{\\rm i}\\kappa _{\\rm s}R}{1+z_n(\\kappa _{\\rm s}R)}u_{1n}^m T_n^m+\\frac{1+z_n(\\kappa _{\\rm s}R)}{{\\rm i}\\kappa _{\\rm s}R} u_{2n}^m V_n^m.$ Using (REF ), we obtain the transparent boundary condition for $\\psi $ : $(\\nabla \\times \\psi )\\times e_r={\\rm i}\\kappa _{\\rm s}{T}_2\\psi _{\\Gamma _R}\\quad \\text{on}~ \\Gamma _R.$ The following result can also be easily shown from (REF ) and (REF ) The operator $T_2$ is bounded from $H^{1/2}({\\rm curl},\\Gamma _R)$ to $H^{-1/2}({\\rm div}, \\Gamma _R)$ .",
"Moreover, it satisfies ${\\rm Re}\\langle T_2 u, u\\rangle _{\\Gamma _R}\\ge 0,\\quad \\forall u\\in H^{1/2}({\\rm curl}, \\Gamma _R).$ If ${\\rm Re}\\langle T_2 u, u\\rangle _{\\Gamma _R}=0$ , then $u=0$ on $\\Gamma _R$ ."
],
[
"Fourier coefficients",
"We derive the mutual representations of the Fourier coefficients between $v$ and $(\\phi , \\psi )$ .",
"First we have from (REF ) that $\\phi (r, \\theta , \\varphi )=\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\frac{h^{(1)}_n(\\kappa _{\\rm p}r)}{h^{(1)}_n(\\kappa _{\\rm p}R)}\\phi _n^mX_n^m(\\theta , \\varphi ).$ Substituting (REF )–(REF ) into (REF ) yields $\\psi (r, \\theta ,\\varphi )&=\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\frac{\\sqrt{n(n+1)}}{{\\rm i}\\kappa _{\\rm s}r}(h_n^{(1)}(\\kappa _{\\rm s}r)+\\kappa _{\\rm s}r h_n^{(1)^{\\prime }}(\\kappa _{\\rm s}r))\\alpha _n^m T_n^m\\\\&\\qquad +\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}r)\\beta _n^m V_n^m+\\frac{n(n+1)}{{\\rm i}\\kappa _{\\rm s}r}h_n^{(1)}(\\kappa _{\\rm s}r)\\alpha _n^m W_n^m.$ Given $\\psi $ on $\\Gamma _R$ , it has the Fourier expansion: $\\psi (R, \\theta , \\varphi )=\\sum _{n=0}^\\infty \\sum _{m=-n}^n \\psi _{1n}^mT_n^m(\\theta , \\varphi )+\\psi _{2n}^mV_n^m(\\theta ,\\varphi )+\\psi _{3n}^mW_n^m(\\theta , \\varphi ).$ Evaluating (REF ) at $r=R$ and then comparing it with (REF ), we get $\\alpha _n^m=\\frac{{\\rm i}\\kappa _{\\rm s}R}{n(n+1)h_n^{(1)}(\\kappa _{\\rm s}R)}\\psi _{3n}^m,\\quad \\beta _n^m=\\frac{1}{\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}R)}\\psi _{2n}^m.$ Plugging (REF ) back into (REF ) gives $\\psi (r, \\theta , \\varphi )&=\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\left(\\frac{R}{r}\\right)\\left(\\frac{h_n^{(1)}(\\kappa _{\\rm s}r)+\\kappa _{\\rm s}rh_n^{(1)^{\\prime }}(\\kappa _{\\rm s}r)}{\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}R)}\\right)\\psi _{3n}^mT_n^m\\\\&\\qquad +\\left(\\frac{h_n^{(1)}(\\kappa _{\\rm s}r)}{h_n^{(1)}(\\kappa _{\\rm s}R)}\\right)\\psi _{2n}^mV_n^m+\\left(\\frac{R}{r}\\right)\\left(\\frac{h_n^ {(1)}(\\kappa _{\\rm s}r)}{h_n^{(1)}(\\kappa _{\\rm s} R)}\\right)\\psi _{3n}^mW_n^m.$ Noting $\\nabla \\phi =\\partial _r\\phi \\, e_r+\\frac{1}{r}\\nabla _{\\Gamma _R}\\phi $ , we have from (REF ) and (REF ) that $\\nabla \\phi &=\\sum _{n=0}^\\infty \\sum _{m=-n}^n \\left(\\frac{\\kappa _{\\rm p}h^{(1)^{\\prime }}_n(\\kappa _{\\rm p}r)}{h^{(1)}_n(\\kappa _{\\rm p}R)}\\right)\\phi _n^m X_n^me_r +\\left(\\frac{h^{(1)}_n(\\kappa _{\\rm p}r)}{r h^{(1)}_n(\\kappa _{\\rm p}R)}\\right)\\phi _n^m \\nabla _{\\Gamma _R}X_n^m\\\\&=\\sum _{n=0}^\\infty \\sum _{m=-n}^n \\left(\\frac{\\kappa _{\\rm p}h^{(1)^{\\prime }}_n(\\kappa _{\\rm p}r)}{h^{(1)}_n(\\kappa _{\\rm p}R)}\\right)\\phi _n^mW_n^m+\\left(\\frac{\\sqrt{n(n+1)} h^{(1)}_n(\\kappa _{\\rm p}r)}{r h^{(1)}_n(\\kappa _{\\rm p}R)}\\right)\\phi _n^mT_n^m.$ and $\\nabla \\times \\psi =\\sum _{n=0}^\\infty \\sum _{m=-n}^n I_{1n}^m+ I_{2n}^m + I_{3n}^m,$ where $I_{1n}^m&=\\nabla \\times \\left[\\left(\\frac{R}{r}\\right)\\left(\\frac{h_n^{(1)}(\\kappa _{\\rm s}r)+\\kappa _{\\rm s}rh_n^{(1)^{\\prime }}(\\kappa _{\\rm s}r)}{\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}R)}\\right)\\psi _{3n}^mT_n^m\\right]\\\\&=\\frac{R h_n^{(1)}(\\kappa _{\\rm s}r)}{\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}R)}\\left(\\kappa ^2_{\\rm s}-\\frac{n(n+1)}{r^2}\\right)\\psi _{3n}^mV_n^m,\\\\I_{2n}^m &=\\nabla \\times \\left[\\left(\\frac{h_n^{(1)}(\\kappa _{\\rm s}r)}{h_n^{(1)}(\\kappa _{\\rm s}R)}\\right)\\psi _{2n}^mV_n^m\\right]\\\\&=\\left(\\frac{h_n^{(1)}(\\kappa _{\\rm s}r)+\\kappa _{\\rm s}rh_n^{(1)^{\\prime }}(\\kappa _{\\rm s}r)}{r h_n^{(1)}(\\kappa _{\\rm s}R)}\\right)\\psi _{2n}^mT_n^m +\\frac{\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}r)}{rh_n^{(1)}(\\kappa _{\\rm s}R)}\\psi _{2n}^m W_n^m,\\\\I_{3n}^m&=\\nabla \\times \\left[\\left(\\frac{R}{r}\\right)\\left(\\frac{h_n^{(1)}(\\kappa _{\\rm s}r)}{h_n^{(1)}(\\kappa _{\\rm s} R)}\\right)\\psi _{3n}^mW_n^m\\right]=\\frac{R\\sqrt{n(n+1)} h_n^{(1)}(\\kappa _{\\rm s}r)}{r^2 h_n^{(1)}(\\kappa _{\\rm s}R)}\\psi _{3n}^mV_n^m.$ Combining the above equations and noting $v=\\nabla \\phi +\\nabla \\times \\psi $ , we obtain $& v(r, \\theta , \\varphi )=\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\left(\\frac{\\sqrt{n(n+1)} h^{(1)}_n(\\kappa _{\\rm p}r)}{r h^{(1)}_n(\\kappa _{\\rm p}R)}\\phi _n^m+\\frac{(h_n^{(1)}(\\kappa _{\\rm s}r)+\\kappa _{\\rm s}r h_n^{(1)^{\\prime }}(\\kappa _{\\rm s}r))}{rh_n^{(1)}(\\kappa _{\\rm s}R)}\\psi _{2n}^m\\right) T_n^m\\\\&\\qquad +\\frac{\\kappa ^2_{\\rm s}R h_n^{(1)}(\\kappa _{\\rm s}r) }{\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}R)}\\psi _{3n}^mV_n^m +\\left(\\frac{\\kappa _{\\rm p}h^{(1)^{\\prime }}_n(\\kappa _{\\rm p}r)}{h^{(1)}_n(\\kappa _{\\rm p}R)}\\phi _n^m+\\frac{\\sqrt{n(n+1)}h_n^{(1)}(\\kappa _{\\rm s}r)}{rh_n^{(1)}(\\kappa _{\\rm s}R)}\\psi _{2n}^m\\right) W_n^m,$ which gives $v(R, \\theta ,\\varphi )=&\\sum _{n=0}^\\infty \\sum _{m=-n}^n\\frac{1}{R}\\left(\\sqrt{n(n+1)}\\phi _n^m+(1+z_n(\\kappa _{\\rm s}R))\\psi _{2n}^m\\right)T_n^m\\\\&+\\frac{\\kappa ^2_{\\rm s} R }{\\sqrt{n(n+1)}}\\psi _{3n}^mV_n^m+\\frac{1}{R}\\left(z_n(\\kappa _{\\rm p}R)\\phi _n^m+\\sqrt{n(n+1)}\\psi _{2n}^m\\right) W_n^m.$ On the other hand, $v$ has the Fourier expansion: $v(R, \\theta , \\varphi )=\\sum _{n=0}^\\infty \\sum _{m=-n}^n v_{1n}^mT_n^m+v_{2n}^mV_n^m+v_{3n}^mW_n^m.$ Comparing (REF ) with (REF ), we obtain ${\\left\\lbrace \\begin{array}{ll}v_{1n}^m =\\dfrac{\\sqrt{n(n+1)}}{R}\\phi _n^m+\\dfrac{(1+z_n(\\kappa _{\\rm s}R))}{R}\\psi _{2n}^m,\\\\[5pt]v_{2n}^m =\\dfrac{ \\kappa ^2_{\\rm s}R}{\\sqrt{n(n+1)}}\\psi _{3n}^m,\\\\[5pt]v_{3n}^m =\\dfrac{z_n(\\kappa _{\\rm p}R)}{R}\\phi _n^m+\\dfrac{\\sqrt{n(n+1)}}{R}\\psi _{2n}^m,\\end{array}\\right.",
"}$ and ${\\left\\lbrace \\begin{array}{ll}\\phi _n^m =\\dfrac{R(1+z_n(\\kappa _{\\rm s}R))}{\\Lambda _n}v_{3n}^m-\\dfrac{R\\sqrt{n(n+1)}}{\\Lambda _n}v_{1n}^m,\\\\[5pt]\\psi _{2n}^m =\\dfrac{R z_n(\\kappa _{\\rm p}R)}{\\Lambda _n}v_{1n}^m-\\dfrac{R\\sqrt{n(n+1)}}{\\Lambda _n}v_{3n}^m,\\\\[5pt]\\psi _{3n}^m =\\dfrac{\\sqrt{n(n+1)}}{\\kappa _{\\rm s}^2 R}v_{2n}^m,\\end{array}\\right.",
"}$ where $\\Lambda _n=z_n(\\kappa _{\\rm p}R)(1+z_n(\\kappa _{\\rm s}R))-n(n+1).$ Noting (REF ), we have from a simple calculation that ${\\rm Im}\\Lambda _n={\\rm Re}z_n(\\kappa _{\\rm p}R){\\rm Im}z_n(\\kappa _{\\rm s}R)+(1+{\\rm Re}z_n(\\kappa _{\\rm s}R)){\\rm Im}z_n(\\kappa _{\\rm p}R)<0,$ which implies that $\\Lambda _n\\ne 0$ for $n=0, 1, \\dots .$"
]
] | 1709.01845 |
[
[
"Shaping and Trimming Branch-and-bound Trees"
],
[
"Abstract We present a new branch-and-bound type search method for mixed integer linear optimization problems based on the concept of offshoots (introduced in this paper).",
"While similar to a classic branch-and-bound method, it allows for changing the order of the variables in a dive (shaping) and removing unnecessary branching variables from a dive (trimming).",
"The regular branch-and-bound algorithm can be seen as a special case of our new method.",
"We also discuss extensions to our new method such as choosing to branch from the top or the bottom of an offshoot.",
"We present several numerical experiments to give a first impression of the potential of our new method."
],
[
"Introduction",
"In this paper we are discussing mixed integer linear optimization problems (MILP), i.e., optimization problems with a linear objective function, linear constraints, and integrality restrictions on some or all of the variables.",
"Most of the techniques described in the paper generalize naturally to problems with non-linear constraints as well (MINLP).",
"The typical approach to solve any optimization problem with an integrality restriction on the feasible domain involves variants of the branch-and-bound algorithm first described for general integer optimization by Land and Doig in [9].",
"For details about the origins of branch-and-bound, see also [4].",
"The branch-and-bound method is at the core of every software to solve mixed integer optimization problems and is successfully used to solve a variety of practical problems.",
"But it is also known that in the worst case the branch-and-bound method will enumerate all possible solutions, leading to a disastrous performance.",
"In Example REF we show such a case.",
"Example 1 Consider the following integer optimization problem with three binary variables: $\\min \\ x_1 - 2 x_2 - 6 x_3 \\\\-3x_1 - 4 x_2 - 2 x_3 &\\ge -8 \\\\ 3x_1 - 4 x_2 - 2 x_3 &\\ge -5 \\\\-3x_1 + 4 x_2 - 2 x_3 &\\ge -4 \\\\3x_1 + 4 x_2 - 2 x_3 &\\ge -1 \\\\x \\in \\lbrace 0, 1\\rbrace ^3$ We solve this example with a depth-first branch-and-bound algorithm where in the left nodes variables are fixed to one.",
"In this example, it does not matter which branching variable selection method is used since there is only one fractional variable in each node and a traditional branch-and-bound method chooses the branching variable only from the fractional variables.",
"In Figure REF we show the branch-and-bound tree resulting from this example.",
"We show the objective value of the LP relaxation below each node and the solution values for the three variables to the right of each node.",
"The number inside the nodes shows the order in which they are processed.",
"Figure: The branch-and-bound tree for the optimization problem .Note that the optimal objective value is $-2$ but it is obtained only after 12 nodes have been processed and proving its optimality requires 15 nodes, the maximal number of nodes possible, which corresponds to enumerating all possible solutions, i.e., the leaf nodes of the tree in Figure REF .",
"Note that the leaf nodes are alternating between infeasible and integral nodes.",
"This is the reason why we have such a bad tree in this example; the enumeration ended up with different types of leaf nodes next to each other.",
"Hence the branch-and-bound method has no chance to prune several leaf nodes together early on at a common ancestor.",
"Research on branch-and-bound algorithms has put a huge emphasis on making the decisions in the algorithm in such a way as to avoid enumeration of large parts of the solution space.",
"Especially selecting the branching variables has been studied extensively (see, for example, [1]), because which branching variable is chosen determines the shape of the tree of a branch-and-bound method and thus how many nodes need to be processed.",
"In this paper we will present an implementation for a branch-and-bound method that follows a different approach in which the shape of the tree can be changed.",
"This means that we want to have a branch-and-bound algorithm where the decisions on the branching variables can be deferred to a time when potentially more information is available to make these decisions.",
"Example REF demonstrates that the same problem from Example REF can be solved more efficiently if the branch-and-bound tree has a different shape.",
"Example 2 In Figure REF we show the branch-and-bound tree for the same mixed integer optimization problem as in Example REF with the only difference that we reversed the order in which we branched on the variables, i.e., we branched on $x_3$ first although it is not even fractional.",
"In this case we only need to process 9 nodes; we show the remaining nodes (without a number) just to highlight the structure of the tree.",
"Figure: Problem () solved with a different branch-and-bound tree.Note that now the infeasible leaf nodes are all on the left side of the tree and all the integral nodes are on the right side of the tree.",
"The integral leaf nodes do not all have to be processed because the common ancestor, node 9, covers them all.",
"Also, if we can detect the integer infeasibility of node 2 (for example with probing or some other node presolver technique), then we can prune all the nodes below it and solve the problem even faster.",
"The lesson is that it is better to have a tree in which nodes with similar properties are the leaf nodes of subtrees so that they can be dealt with at a common ancestor.",
"The question here is twofold.",
"First, we need to reshape the tree if we believe that the current tree structure is inefficient.",
"The second question is equally important: we need to do this in an efficient way, so that we can preserve most of the advanced techniques that make branch-and-bound implementations perform well in practice.",
"While it is possible to simply throw away most of the tree and roll back to the last known good decision point (or variants thereof called restarting as, for example, discussed in [1]), we want to explicitly look into other possibilities here.",
"There has been some research in branch-and-bound methods with an adjustable (sometimes called dynamic) tree.",
"The earliest we are aware of is by Glover and Tangedahl [6].",
"Chvátal in [3] and then Hanafi and Glover in [7] revisited the topic.",
"These papers give valuable insights into alternative methods for solving mixed integer optimization problems but unfortunately do not discuss the implementational challenges.",
"Furthermore, resolution search from [3], for example, is not similar enough to a classic branch-and-bound method such that many of the methods modern solvers successfully use to solve problems today are not directly applicable."
],
[
"Diving, Shaping, and Trimming",
"In this section we discuss three very important concepts for the remainder of this paper: diving, shaping, and trimming.",
"We do so using a depth-first branch-and-bound method because these concepts are easier to explain in this method and are also a natural extension to it.",
"Depth first branch-and-bound (sometimes also called last-in-first-out, i.e.",
"LIFO, branch-and-bound) is a variant of branch-and-bound where the next node processed is always the most recent node added to a stack of open nodes.",
"In practice it is possible to store the open nodes with a stack of bound changes.",
"The depth-first branch-and-bound method also minimizes the number of open nodes.",
"The result is a very memory-efficient branch-and-bound method.",
"In the depth-first branch-and-bound method we repeatedly go down the tree only changing one variable at a time.",
"We call this process of going down a tree diving.",
"Another advantage of the depth-first branch-and-bound method is that the LP relaxations during diving can be solved very efficiently using a dual simplex algorithm where most data structures (most importantly the factorization of the basis matrix) can be kept up to date.",
"We call this hotstarting the dual simplex to express that it is even better than warmstarting, which typically implies that a known dual feasible basis is used to initialize the dual simplex algorithm.",
"When backtracking in the depth-first branch-and-bound we cannot use hotstarting, but since the difference between nodes is typically small we can warmstart from the last basis instead of resolving from scratch.",
"Since diving is much more efficient, current implementations of non-depth-first branch-and-bound methods also use it to process nodes quickly and only do a full node selection if the current dive does not seem promising anymore.",
"The disadvantage of the depth-first branch-and-bound method is that the problem described in the introduction is aggravated: a bad decision early on can result in a very bad enumeration tree and thus long running time or a failure to solve the problem within some resource limitation.",
"But it is also much easier to revise earlier decisions and change the order of bound changes in a dive.",
"Notice that for the status of the final node in a dive the order in which the variables were fixed does not matter.",
"The order of the bound changes in a dive in some sense defines the shape of the branch-and-bound tree.",
"Hence we call changing the order of the bound changes in a dive shaping.",
"Since in a depth-first branch-and-bound we store the bound changes in a stack we can decide to undo them in a different order than we did them during the dive.",
"The only thing we have to keep in mind is that we can only change the order of the bound changes up to the last node where we have already explored the other side of the bound change.",
"In a depth-first branch-and-bound algorithm a dive has to end in a pruned node.",
"A node is pruned either because the LP relaxation is infeasible or because the objective value exceeds the cutoff The case of an integral solution can be seen as first establishing a new cutoff and then pruning the node..",
"It is possible that a situation occurs where a dive contains more bound changes than are strictly necessary to prune a node.",
"In this case it is possible to remove the unneeded bound changes from the dive before backtracking.",
"Since this trims the dive down to a smaller set of bound changes we call this trimming.",
"There are a number of ways to trim a dive.",
"For problems with general integer variables it is possible to remove multiple bound changes on the same bound of the same variable and keep only the tightest one.",
"It is also possible to use reduced cost or Farkas certificate values to trim dives.",
"In fact, this problem is identical to the one we are facing when trying to identify an irreducible infeasible system (IIS), so all the reduction techniques in that domain apply readily to our setup; see [2] for details.",
"In the following sections we will sample a few methods.",
"Shaping and trimming clearly can improve a depth-first branch-and-bound implementation a lot, and the implementational complexity is very low.",
"For shaping, the obvious difficulty is to come up with good rules on which bound change should be undone first.",
"But our experience has been that even simple rules already lead to an improvement.",
"For trimming, the trade-off is between time spent trimming the tree and simply processing nodes.",
"But here as well simple strategies already yielded benefits so that it should be possible to improve any depth-first branch-and-bound implementation not making use of trimming significantly.",
"The only downside is that if node presolving techniques are used in a depth-first branch-and-bound method it is necessary to keep track of implied bound changes separately from the actual branching decisions.",
"As a result, during backtracking some tightenings from node presolve have to be redone.",
"The concepts of shaping and trimming the tree already appear in principle in [6], but that paper does not include any implementational considerations."
],
[
"A New Branch-and-bound Method",
"In this section we present the basic idea of a new branch-and-bound method that allows for shaping and trimming but is not a depth-first method.",
"The fundamental idea is to perform a branch-and-bound method on objects we call offshoots instead of performing it on individual nodes.",
"An offshoot (see Figure REF ) is an object that represents a collection of nodes in a tree.",
"It consists of a top node $s$ , represented by a set $F$ of initial bound changes, with an attached set $D$ of bound changes representing a dive in the branch-and-bound tree.",
"Applying both the initial bound changes in $F$ and the bound changes in the dive $D$ has to result in a node $t$ that can be pruned, either because it is infeasible or because its objective value exceeds the current cutoff.The cutoff is derived from the currently best known primal feasible solution.",
"Note that the order of the bound changes in $D$ is not determined,In a practical implementation we can remember the original order of bound changes so that we can use the intermediate objective values to prune undisturbed nodes inside an offshoot.",
"only the set of all the bound changes needed to reach a terminal node.",
"Figure: The structure of an offshoot.Instead of storing a set of open nodes that still need to be processed we store a set of open offshoots.",
"An offshoot is considered open if it has bound changes in its dive that have not been processed.",
"Once the list of open offshoots is empty the problem is solved.",
"This new method begins with creating a first offshoot for which the set $F_0$ of initial bound changes is empty.",
"Then it performs a dive until it reaches a node that can be pruned and stores the bound changes of this first dive in the set $D_0$ of the first offshoot.",
"Then the first offshoot is added to the list of open offshoots.",
"From now on, in each iteration, the method selects an offshoot from the list of open offshoots as parent offshoot $p$ for a new offshoot $k$ to create.",
"The method also needs to select a bound change to process associated with an offshoot variable $i$ from the list $D_p$ of unprocessed dive bound changes of its parent.",
"The initial set of bound changes for the new offshoot $k$ is $F_k = F_p \\cup (D_k \\setminus \\left\\lbrace x_i \\le b\\right\\rbrace ) \\cup \\left\\lbrace x_i \\ge b+1\\right\\rbrace $ if the bound change for the selected variable was branching down or $F_k = F_p \\cup ( D_k \\setminus \\left\\lbrace x_i \\ge b\\right\\rbrace ) \\cup \\left\\lbrace x_i \\le b-1\\right\\rbrace $ if it was branching up.",
"The new offshoot starts with a node that corresponds to a right node of the dive but since we can freely choose from all bound changes in the dive it might be a right node that does not correspond to any of the dive nodes that were processed when the offshoot was created.",
"This choosing of the variable from the dive corresponds to shaping the tree.",
"After creating the initial node of the new offshoot we solve the LP relaxation of the top node in the new offshoot.",
"If the top node can be pruned we proceed by selecting a new parent offshoot right away.",
"Otherwise we store the objective value as the top bound $z_k^*$ of the new offshoot.",
"Then we perform a dive until we reach a node that can be pruned either because it is infeasible or because its objective value exceeds the current cutoff.",
"If we encounter a new primal feasible solution we update the cutoff.",
"When updating the cutoff we can also remove all open offshoots for which the top bound exceeds the cutoff.In addition, we can also remove those nodes inside offshoots that have not been disturbed yet if their objective value exceeds the cutoff.",
"In this setup we can also easily perform trimming.",
"As mentioned before, this can be done, for example, by removing multiple bound changes on the same bound of a variable (only in the case of general integer variables) or by inspecting the dual information of the pruned node.",
"To specify in more detail: the dual information vector $r$ is either the reduced cost vector for cutoff nodes or the Farkas certificate for infeasible nodes.",
"An upper bound change on variable $i$ can be removed if $r_i \\ge 0$ , and a lower bound change on variable $i$ can be removed if $r_i \\le 0$ .",
"After trimming the dive we can store the new offshoot in the list of open offshoots and remove the parent offshoot if all the bound changes in its dive have been processed.",
"This continues until the list of open offshoots is empty.",
"Figure REF shows an example where the new method is applied to Example REF .",
"Figure: A step-by-step example of the new method."
],
[
"Improvements and Extensions",
"As with many similar methods it is necessary to improve and extend our new method to get the best possible performance.",
"In this section we list some more or less obvious ways to overcome some of the weaknesses of the new method."
],
[
"Branching From the Top",
"In the description of the method in the previous section we only added new offshoots below their parent.",
"This can be seen as branching from the bottom of an offshoot.",
"It is also possible to branch from the top of an offshoot.",
"Then the new offshoot inherits only the bound changes its parent had at the top and additionally exactly one bound change from the dive flipped to the other side.",
"The parent is then adjusted as well and one of the bound changes is moved from the dive to the top.",
"To be precise, the initial set of bound changes for the new offshoot $k$ is $F_k = F_p \\cup \\left\\lbrace x_i \\ge b+1\\right\\rbrace $ if the bound change for the selected variable was branching down or $F_k = F_p \\cup \\left\\lbrace x_i \\le b-1\\right\\rbrace $ if it was branching up.",
"Figure REF illustrates both types of branching next to each other.",
"Figure: The two ways to branch illustrated.The big advantage of this additional level of flexibility is that we can decide which type of branching to use based on how sure we are that an offshoot variable is a good choice.",
"If we are not sure whether a bound change will have large impact and hence should be at the top of the tree we can choose to branch from the bottom to minimize the effect if we made a bad choice.",
"If, on the other hand, we have a strong indication that a bound change will have a huge impact and should be at the top of the tree, then we can branch from the top."
],
[
"Creating Offshoots and Advanced Trimming",
"For the correctness of the method it is not necessary to create offshoots by diving.",
"Any method that creates a set of bound changes that results in a pruned node can be used for the diving set in an offshoot.",
"One slight modification to the method is to apply several bound changes at once before solving an LP relaxation.",
"We call this plunging.",
"This can go as far as fixing all integer variables since the result is guaranteed to be pruned and trimming can then be used to reduce the set of bound changes.",
"Another possibility is to use conflict analysis as described in [1] to obtain a clause.",
"For offshoots that end in an infeasible node the dive set of bound changes is precisely a clause.",
"Hence it is also possible to apply the method described by Karzan et al.",
"in [8] to obtain a minimal clause using a MIPing approach."
],
[
"Improved Pruning",
"One of the disadvantages of the new method is that pruning by bound after a new primal feasible solution has been found is complicated.",
"Obviously, we can prune whole offshoots as soon as their top bound exceeds the new bound.",
"But it can happen that for some offshoots the top bound is not large enough although applying some of the bound changes from the dive would result in an LP bound that would lead to a pruning.",
"This issue can be overcome partially by storing the objective values obtained during a dive.",
"As long as no new offshoot is created from the dive (or new offshoots are created only from the bottom and in the order of the original dive) we can use the objective values to trim the dives after a new bound has been found.",
"Since trimming also invalidates the bounds from the dive it is advisable to delay trimming until we first want to create an offshoot.",
"This requires slightly more memory since more bound changes and dual information might have to be stored, but it could result in significantly better performance."
],
[
"Bounding Offshoots",
"When branching from the top, the top bound of an offshoot remains a valid bound on all the nodes below this offshoot.",
"But since we add a bound change to the top of the offshoot the bound is obviously not as strong as it could be.",
"Hence it might be worthwhile updating the top bound after branching from the top.",
"We propose three methods of increasing computational effort to strengthen the bound.",
"The first method is to derive a bound on the top node of the parent offshoot by using the reduced cost of the just-solved top node of the new offshoot.",
"The second method is a bit more general but also requires more computational effort.",
"It involves simply evaluating the dual solution of the new offshoot's top node for the bounds of the parent offshoot.",
"The third method is to solve the LP for the new top node of the parent offshoot.",
"Since we have a warmstart basis from the top node of the new offshoot this can be done using a very good warmstart basis.",
"Obviously, the first two methods provide only a lower bound on the new optimal objective value."
],
[
"Shortening Dives",
"It is possible to implement the new method in a way that traditional branch-and-bound is just a special case.",
"To this end we only need to ensure that in addition to storing open offshoots we can also store open nodes.",
"This can be achieved, for example, by treating offshoots without a dive as normal nodes, which means when we select them we do not select an offshoot variable.",
"Instead we treat it as the top node of a new offshoot directly.",
"With this in place we can also have a limit on the number of bound changes in a dive.",
"When the limit is hit, we store the last node as an open node in addition to storing the offshoot.",
"The offshoot in this case does not end in a pruned node, but the method works regardless.",
"If we set the limit of bound changes in a dive to zero, the method reverts to a traditional branch-and-bound method."
],
[
"Splitting offshoots",
"Branching from the top on a bound change that was not the initial bound change of an offshoot invalidates all the internal objective values of the original nodes of an offshoot.",
"This prevents us from pruning them and hurts the performance for very deep dives.",
"Therefore it seems advantageous to split very long dives.",
"However, this creates a non-terminal offshoot, so it extends our depth-first framework a little.",
"For maximum efficiency, we need to resolve the linear relaxation of the bottom offshoot, so that we can have a valid objective value useful for pruning by bound.",
"Which bound changes should be in the top of bottom half of the split is an interesting research question."
],
[
"Computational Evaluation",
"The new method was prototyped using the MILP solver in SAS/OR.",
"The prototype was meant as a way to evaluate the correctness and practicability of the method described and as such does not contain all the features and tricks of a full MILP solver.",
"Nevertheless we present results using this prototype to give an impression of the capabilities of the new method.",
"The prototype plugs into the MILP solver after its root node when the actual branch-and-bound phase begins.",
"It features a standard reliability branching strategy with a dynamic strong branching limit and a reliability limit of 5.",
"For selecting the next offshoot we choose the best top bound first without explicit tie breaking.",
"The prototype also features basic node presolve and reduced cost fixing techniques (only at the top of an offshoot), and also using the root reduced cost to fix columns globally as new incumbent solutions are found.",
"What it notably lacks are more advanced node presolver techniques, local or global cuts in the nodes of the branch-and-bound tree, and primal heuristics.",
"We conduct our experiments on 96 machines running 2 jobs each on 16-core/2-socket Intel® Xeon® E5-2630 v3 @ 2.40GHz CPUs.",
"All experiments are done with default settings, a memory limit of 62 GB, and a time limit of 2 hours.",
"We use 798 instances that are the internal test set used to develop the SAS MILP solver.",
"To evaluate our results we use performance profiles as described in [5]."
],
[
"Offshoot variable selection",
"The first experiment evaluates several methods we implemented for choosing the offshoot variable, i.e., it is meant to judge the importance of shaping in the new method.",
"We implemented four different methods: bottom: always branch from the bottom of an offshoot without changing the order of the variables; top: always branch from the top of an offshoot without changing the order of the variables; pseudo: choose the offshoot variable with the best reliable pseudocost score and branch from the top.",
"If there are no offshoot variables with reliable pseudocost available, then choose the variable with the worst pseudocost score and branch from the bottom.",
"pseudodual: like pseudo, except if there are no variables with reliable pseudocost then use the variable with the worst dual information score and branch from the bottom.",
"The dual information score is the reduced cost or the Farkas certificate of the pruned node when the offshoot was first processed.",
"The first two strategies do not shape the tree so they can be seen as a baseline for the performance of the method.",
"The default method is pseudodual.",
"Figure: Performance profile comparing offshoot variable selection strategies.Figure REF shows the performance profile comparing the different offshoot variable selection strategies.",
"In addition we would like to mention that the pseudodual strategy is about 13% faster in the geometric mean of the solve times than the bottom strategy and solves 9 instances more within the time limit.",
"We argue that this shows that shaping, at least in the context of this new method, has a clear impact on the performance.",
"More advanced selection strategies can probably be developed that will demonstrate this even more profoundly."
],
[
"Trimming and pruning",
"Our second experiment is designed to show the combined importance of trimming and pruning.",
"In our prototype implementation we delay trimming an offshoot until we need to choose an offshoot variable for the first time.",
"Since we can either prune the bottom of the offshoot using the current cutoff or apply trimming using the dual information, we analyze how many reductions we get from either and choose the method that yields the most.",
"In this experiment we compare the default version of our prototype that does this delayed pruning or trimming with a version where this feature has been disabled.",
"The performance profile can be seen in Figure REF .",
"The version with trimming and pruning is about 5% faster in the geometric mean of solve times and solves 1 instance fewer within the time limit.",
"Since this effect seems to be rather small we think that it would be necessary to look into better ways to trim dives.",
"Some ideas are described in Section REF .",
"Figure: Performance profile comparing a version of the prototype that does pruning and trimming with a version that does not."
],
[
"Comparison against branch-and-bound",
"In our final experiment we compare our default method that does not limit the depths of the dives to a version where the limit is 0.",
"This means that the method with the limit is essentially a traditional branch-and-bound method.",
"The comparison is not completely fair since a pure branch-and-bound method could be implemented more efficiently, especially regarding memory requirements.",
"But it gives a first impression of how much could be gained by using our new method instead of a traditional branch-and-bound method.",
"Figure REF shows the performance profile.",
"Our new method is 38% faster in the geometric mean of the solve times and solves 47 instances more within the time limit.",
"We consider this an encouraging result.",
"Figure: Performance profile comparing the default version with version with a maximum depth of 0, i.e., that resembles a traditional branch-and-bound."
],
[
"Conclusions",
"It will obviously take more research and a more elaborate implementation to see if our new method is superior to a traditional branch-and-bound method.",
"From a theoretical perspective and from our preliminary experiments it seems likely that shaping and trimming the tree will result in improved run times.",
"Even if the performance gains end up being very small there is also hope that our new method will result in a more stable performance.",
"So far we have not investigated other areas of application for our new method such as mixed integer non-linear optimization problems or branch-and-price algorithms.",
"Since in these areas more flexibility in the tree might be even more advantageous we hope that it will find application there as well."
]
] | 1709.01583 |
[
[
"$Z_N$ Berry Phases in Symmetry Protected Topological Phases"
],
[
"Abstract We show that the $Z_N$ Berry phase (Berry phase quantized into $2\\pi/N$) provides a useful tool to characterize symmetry protected topological phases with correlation that can be directly computed through numerics of a relatively small system size.",
"The $Z_N$ Berry phase is defined in a $N-1$ dimensional parameter space of local gauge twists, which we call \"synthetic Brillouin zone\", and an appropriate choice of an integration path consistent with the symmetry of the system ensures exact quantization of the Berry phase.",
"We demonstrate the usefulness of the $Z_N$ Berry phase by studying two 1D models of bosons, SU(3) and SU(4) AKLT models, where topological phase transitions are captured by $Z_3$ and $Z_4$ Berry phases, respectively.",
"we find that the exact quantization of the $Z_N$ Berry phase at the topological transitions arises from a gapless band structure (e.g., Dirac cones or nodal lines) in the synthetic Brillouin zone."
],
[
"$Z_N$ Berry Phases in Symmetry Protected Topological Phases Toshikaze [email protected] International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science, Tsukuba, 305-0044, Japan Takahiro Morimoto Department of Physics, University of California, Berkeley Yasuhiro Hatsugai Division of Physics, University of Tsukuba We show that the $Z_N$ Berry phase (Berry phase quantized into $2\\pi /N$ ) provides a useful tool to characterize symmetry protected topological phases with correlation that can be directly computed through numerics of a relatively small system size.",
"The $Z_N$ Berry phase is defined in a $N-1$ dimensional parameter space of local gauge twists, which we call “synthetic Brillouin zone”, and an appropriate choice of an integration path consistent with the symmetry of the system ensures exact quantization of the Berry phase.",
"We demonstrate the usefulness of the $Z_N$ Berry phase by studying two 1D models of bosons, SU(3) and SU(4) AKLT models, where topological phase transitions are captured by $Z_3$ and $Z_4$ Berry phases, respectively.",
"We find that the exact quantization of the $Z_N$ Berry phase at the topological transitions arises from a gapless band structure (e.g., Dirac cones or nodal lines) in the synthetic Brillouin zone.",
"In the past decades, topology has come to the fore of the condensed matter research and it has been realized that it serves as a guiding principle to explore novel phases of matter without relying on the symmetry breaking [1].",
"Meanwhile, symmetry still plays an important role in an interplay with topology.",
"For example, topological phases of noninteracting fermions have been classified according to the generic internal symmetries, i.e., time-reversal, particle-hole, and chiral symmetries [1], [2], [3], [4], [5].",
"The topological classification of noninteracting fermions has been further extended by incorporating crystal symmetries [6], [7], [8], [9], [4], [10], [11].",
"On the other hand, characterization of topological phases becomes a more difficult problem for systems of strongly interacting particles [12].",
"There have been active studies on classification and characterization of symmetry protected topological (SPT) phases that are supported with strong correlation effects [1], [13], [14], [15], [16], [17], [18], [19], [20], [21].",
"However, the characterization of SPT phases for a given Hamiltonian remains a highly nontrivial problem.",
"In particular, a concise way to characterize them through fairly cheep numerics has been desired.",
"In characterizing SPT phases, the notion of adiabatic continuation plays an essential role [13], [14], [15], [22].",
"By adiabatically continuing a given system into a simple reference system, the topological character in the original system is easily diagnosed by studying the simple reference system.",
"For example, a system that can be adiabatically decomposed into a set of the elementary units in the system (an atomic insulator in the case of free fermions) is identified as a trivial phase.",
"In contrast, the requirement for keeping a finite gap and the symmetry of the system sometimes excludes possibility of “atomic insulators”, and leaves a set of finite-size entangled clusters [13], [16], [23], which indicates that the state is in an SPT phase.",
"A representative example is Haldane phase in a spin-1 Heisenberg chain [24], [25], [14], [26], [17], where the entangled clusters are intersite singlets of emergent spin-1/2 degrees of freedom.",
"In the search of adiabatic continuation into the embedded entangled clusters, it is useful to study Berry phase defined through the local gauge twist [as schematically illustrated in Fig.",
"REF (a)] [13], [15], [27], [22], [28].",
"Since the Berry phase can be quantized by symmetry in some cases, it provides a conserved quantity in the process of the adiabatic continuation that encodes the topological nature of the system.",
"The Berry phase for the entangled cluster is easily obtained in the simple reference system, and gives a characterization for the original system.",
"For instance, the spin-1/2 singlet in Haldane phase in spin chain is characterized by Berry phase $\\pi $ [13].",
"While the analysis based on Berry phase is useful in characterizing SPT phases, studies of correlated systems so far have mainly focused on those phases characterized by Berry phase $\\pi $ .",
"Figure: (Color online) (a) Schematic picture of the biquadratic model and the local gauge twist.",
"The bonds J i J_i represent the biquadratic interactions.",
"(b)“Synthetic” Brillouin zone and the integration path CC leading tothe Berry phase quantization.",
"(c-e) The energy spectra for the groundstate and the first excited state on the Brillouin zone.",
"The gap atΓ\\Gamma point is always nonzero due to the finite size effect.",
"At thephase transition, the gap closes at K and K' points forming Diraccones.In this paper, we generalize the characterization of SPT phases with correlation based on Berry phase by using fractionally quantized Berry phase $2\\pi /N$ ($Z_N$ Berry phase), and propose that such $Z_N$ Berry phase provides a useful tool to diagnose general topological phases of interacting particles [23].",
"We demonstrate that the $Z_N$ Berry phase is useful in characterizing one-dimensional SPT phases classified by general $Z_N$ topological number.",
"Specifically, we extend the Berry phase analysis so that it can detect entangled clusters other than the conventional spin-1/2 singlets.",
"We demonstrate that spin-1 singlets can be detected with the appropriate redefinition of the Berry phase.",
"This can be applied to a bond alternating spin-1 chain with biquadratic interaction (hereafter, called the biquadratic model), which supports a $Z_3$ SPT phase.",
"In this case, the phase transition is captured by the $Z_3$ Berry phase (0 or $2\\pi /3$ ), instead of the conventional one $\\pi (=2\\pi /2)$ .",
"We also show that an SU(4) symmetric spin chain supports a topological phase with an SU(4) fully antisymmetrized state being the entangled cluster, which can be diagnosed by $Z_4$ Berry phase.",
"These generalizations of the Berry phase into fractional ones involve “synthetic” Brillouin zone (BZ) [see Fig.",
"REF (b)] that parameterizes local gauge twists for a particular bonds.",
"When there exist $N$ kinds of local gauge twists [$N=3$ for the SU(3) chain and $N=4$ for the SU(4) chain], such synthetic BZ is given by a $N-1$ dimensional space.",
"(Note that the system itself is one-dimensional.)",
"We find that the phase transition is governed by a gapless structure appears in the effective band structure in the synthetic Brillouin zone such as Dirac cones shown in Fig.",
"REF (d).",
"Thus the $Z_N$ Berry phase analysis allows us to understand the topological phase transition in the many-body system by using an analogy to that in free-fermion system.",
"Let us begin with the formulation of the Berry phase.",
"For simplicity, we focus on a one-dimensional periodic system with Hamiltonian $H=\\sum _{ij}H_{ij}$ .",
"For finite size systems (either open or periodic) that are studied by numerical calculations in practice, the Berry phase is defined in the following way.",
"First, we pick up a term on a certain bond, $H_{nm}$ , out of the terms in the Hamiltonian.",
"Then, it is replaced by $U_m(\\phi )H_{nm}U^\\dagger _m(\\phi )$ , where $U_m(\\phi )=e^{i\\hat{A}\\phi }$ (the local gauge twist) acts on the $m$ th site.",
"While it looks like a unitary transformation, it actually is not, since the operation is selectively acting on the chosen bond.",
"Therefore, the eigenvalues and eigenvectors change as $|G_0\\rangle \\rightarrow |G_\\phi \\rangle $ .",
"Using the set of these wave functions, the Berry phase $\\gamma $ is defined as $i\\gamma = \\int _0^{2\\pi }d\\phi \\langle G_\\phi |\\partial _\\phi G_\\phi \\rangle .$ The choice of the gauge twist $\\hat{A}$ is the most important part of this scheme.",
"It should make $U_m(\\phi )$ periodic in $\\phi $ , and should properly capture the underlying entangled cluster.",
"Table: Extra phase factors associated with each term of thebiquadratic model induced by the dipolar and quadrupolar twist.In the previous studies of spin systems, $\\hat{A}=S-\\hat{S}_z$ has been the standard choice [13], [15], [27], which is suitable for detecting a spin-1/2 singlet.",
"In this case, some symmetries constrain the Berry phase $\\gamma $ to quantize into 0 or $\\pi $ , where $\\gamma =\\pi $ signals the existence of a spin-1/2 singlet at the chosen bond.",
"This is indeed the case for the Haldane phase in the spin-1 Heisenberg chain, which is a representative SPT phase.",
"The topological nature of the Haldane phase is captured by the valence bond solid picture, where pairs of spin-1/2 obtained from fractionalization of the original spin-1 form intersite spin-1/2 singlets [29].",
"The Berry phase quantizes into $\\gamma =\\pi $ in the Haldane phase, while it quantized into $\\gamma =0$ in the topologically trivial large-D phase where fractional spin-1/2's form intrasite spin-1/2 singlets.",
"Such quantization of the Berry phase (into 0 or $\\pi $ ) allows us to observe the sharp transition between the Haldane and the large-D phases even for a chain of a relatively small number of sites.",
"This observation can be generalized to the case of $Z_N$ Berry phase.",
"Namely, the quantization of the general $Z_N$ Berry phase indicates a sharp phase transition even for a small size system (without extrapolation to the thermodynamic limit) that can be studied in practical numerical calculations.",
"Figure: (Color online) (a) The Berry phase with e iA ^ 3 φ e^{i\\hat{A}_3\\phi } for severalsystem size (LL denotes the number of the spins).",
"The right panel shows theenergy spectra as functions of φ\\phi atΔ=0\\Delta =0.",
"(b) The Berryphase withe i∑ i A ^ i φ i e^{i\\sum _i\\hat{A}_i\\phi _i} using CC in Fig.",
"(b)as an integration path.",
"The right panel shows the energy spectrum along thehigh symmetry lines in the synthetic Brillouin zone.",
"LL is set to12.",
"The energies are in the unit of JJ.Next, we study a case where the entangled cluster is not a conventional spin-1/2 singlet.",
"To this end, we consider a spin-1 chain with bond-alternating biquadratic interaction [30], which is described by the Hamiltonian, $\\hat{H}=-J_1\\sum _i(\\hat{\\mathbf {S}}_{2i}\\cdot \\hat{\\mathbf {S}}_{2i+1})^2-J_2\\sum _i(\\hat{\\mathbf {S}}_{2i+1}\\cdot \\hat{\\mathbf {S}}_{2i+2})^2.",
"$ It is known that this model supports the SU(3) AKLT state [31].",
"In the language of the SU(3) AKLT state, the elementary object is a quark (and antiquark) and the entangled cluster characterizing SPT phase is a meson (specifically, $\\eta $ -meson).",
"In the language of the spin-1 biquadratic model, the entangled cluster is mapped to a spin-1 “singlet” (two-spin state with zero total angular momentum).",
"By writing $J_{1,2}$ as $J_1=J+\\Delta $ and $J_2=J-\\Delta $ , the parameter $\\Delta $ controls how the entangled cluster are formed.",
"Therefore, once we fix the position of the gauge-twisted bond, the transition has to be observed by changing $\\Delta $ .",
"However, the standard choice of $\\hat{A}=S-\\hat{S}_z$ is inadequate for detecting the spin-1 singlet.",
"Instead, we use the twist operator $\\hat{A}=\\hat{A}_3\\equiv 1-\\hat{S}_z^2$ .",
"If the bond with the biquadratic interaction is twisted by $e^{i\\hat{A}_3\\phi }$ , each term acquires the phase as summarized in Table REF .",
"For comparison, we list the phase factors acquired in the conventional twist with $e^{i(S-\\hat{S}_z)\\phi }$ .",
"Since $\\hat{S}_z$ ($\\hat{S}_z^2$ ) is the part of the dipole (quadrupole) moment, we call $e^{i(S-\\hat{S}_z)\\phi }$ ($e^{i\\hat{A}_3\\phi }$ ) dipolar (quadrupolar) twist.",
"An important feature that we can see from the phase factors in Table REF is their symmetry.",
"If they are symmetric with respect to the combined operation of the complex conjugation and $n\\leftrightarrow m$ , the Berry phase should be quantized into 0 or $\\pi $ [13].",
"Indeed the dipolar twist obeys this symmetry (e.g., the operation on the term 2 results in the term 3, leaving the term unchanged in total).",
"On the other hand, the quadrupolar twist breaks this symmetry, and hence, it does not show $Z_2$ quantization, but may quantize into other fractions of $2\\pi $ .",
"The numerically obtained Berry phase as a function of $\\Delta $ is summarized in Fig.",
"REF .",
"We identify two phases, which are characterized by $\\gamma =0$ for $\\Delta >0$ and $\\gamma =2\\pi /3$ for $\\Delta <0$ .",
"The embedded spin-1 singlet exists on the twisted bond if $\\gamma =2\\pi /3$ , since an isolated singlet with a $\\hat{A}_3$ -twist is described by the wave function, $|\\psi _\\phi \\rangle =(|{+1},{-1}\\rangle -e^{i\\phi }|0,0\\rangle +|{-1},{+1}\\rangle )/\\sqrt{3}$ (by using a representation for the state of two spins, $|s^{z}_1,s^{z}_2\\rangle =|s^{z}_1\\rangle \\otimes |s^{z}_2\\rangle $ with $\\hat{S}_{zi}|s^{z}_i\\rangle =s^{z}_i|s^z_{i}\\rangle $ ), and the second term $e^{i\\phi }|0,0\\rangle $ contributes to the Berry phase by $2\\pi /3$ .",
"Note that with the dipolar twist, the Berry phase is zero regardless of the sign of $\\Delta $ , which means that the phase transition in Fig.",
"REF is captured only with our new method.",
"The system size dependence in Fig.",
"REF (a) suggests that the transition gets sharper as we approach the thermodynamic limit.",
"However, the quantization of the Berry phase is not perfect.",
"Thus it does not ensure the advantage of using the Berry phase, i.e., quantization even for a relatively small size system.",
"Fortunately, we have a remedy to this deviation from perfect quantization.",
"The reason why it does not show quantization is that the symmetry of the system is not fully appreciated.",
"The key symmetry of Eq.",
"(REF ) is the spin rotational symmetry, in particular, the symmetry under the interchange of $x$ , $y$ , and $z$ -directions in the spin space.",
"(This corresponds to the interchange of three flavors of quarks which forms a $Z_3$ subgroup of the SU(3) symmetry [31].)",
"Accordingly, our formulation of Berry phase can be symmetrized by considering the other twist operators $\\hat{A}_1=1-\\hat{S}_x^2$ and $\\hat{A}_2=1-\\hat{S}_y^2$ in addition to $\\hat{A}_3$ , and we define the generalized local gauge twist as $\\exp [i\\sum _i\\hat{A}_i\\phi _i]$ with three parameters $\\phi _{1,2,3}$ .",
"Because of $\\hat{A}_1+\\hat{A}_2+\\hat{A}_3=\\hat{1}$ , only two of three parameters are independent, namely, a twist by $e^{i\\hat{A}_3\\phi }$ has the same effect as a twist by $e^{-i(\\hat{A}_1+\\hat{A}_2)\\phi }$ since $e^{i\\hat{1}\\phi }$ is trivial.",
"This means that the generalized local gauge twist is defined on the two-dimensional periodic parameter space, which we call “synthetic Brillouin zone”, with the hexagonal symmetry as shown in Fig.",
"REF (b).",
"In terms of the synthetic BZ, we can see that the Berry phase defined for a straight line along the $\\phi _3$ axis in the synthetic BZ leads to deviation from the quantization [Fig.",
"REF (a)].",
"Instead, we now consider the path $C$ (K$_1$ -$\\Gamma $ -K$_2$ ) in Fig.",
"REF (b) which is more symmetric in the synthetic BZ.",
"Figure REF (b) shows the Berry phase obtained with the path $C$ .",
"In this case, the Berry phase shows an exact quantization into 0 and $2\\pi /3$ , leading to the sharp transition.",
"The origin of the quantization is understood by considering the Berry phases defined with three different paths, $\\gamma _1$ with K$_1$ -$\\Gamma $ -K$_2$ , $\\gamma _2$ with K$_2$ -$\\Gamma $ -K$_3$ , and $\\gamma _3$ with K$_3$ -$\\Gamma $ -K$_1$ .",
"By the three-fold rotational symmetry in the synthetic BZ, we obtain $\\gamma _1=\\gamma _2=\\gamma _3$ .",
"At the same time, if the three paths are combined, they result in a trivial path, giving us $\\sum _i\\gamma _i=0$ (mod $2\\pi $ ).",
"The consequence of this symmetry consideration is that the Berry phase $\\gamma _i$ should quantize into $2\\pi /3$ [23].",
"The introduction of the synthetic BZ reveals another notable aspect of the transition, i.e., an emergent gapless structure in the effective band structure.",
"Generally speaking, quantization of the Berry phase indicates a jump in the value of $\\gamma $ at the phase transition, and such a jump requires a singularity in the wave function which is associated with gap closing.",
"In this case, the energy gap above the ground state should close somewhere on the integration path.",
"Conversely, no sharp transition is expected when the gap remains finite over the entire integration path.",
"The right panel of Fig.",
"REF (a) plots the energy spectrum as a function of $\\phi $ for the $e^{i\\hat{A}_3\\phi }$ twist at $\\Delta =0$ , which shows the absence of any gap closure.",
"This accounts for the smooth change of $\\gamma $ at $\\Delta $ in Fig.",
"REF (a).",
"In contrast, we indeed have a gap closing point on the path $C$ at $\\Delta =0$ .",
"More specifically, the gapless points are found at K and K' points in the synthetic Brillouin zone.",
"[See Figs.",
"REF (c-e) and the right panel of Fig.",
"REF (b).]",
"Interestingly, the energy spectrum at $\\Delta =0$ shows Dirac cones, in a similar way to the band structure of graphene.",
"This reminds us the fact that the topological transition in free fermion systems is often associated with a gapless band structure such as Dirac cones.",
"In an analogy, the topological phase transition in our model, although it is a correlated one-dimensional model, is associated with the Dirac cones that appear in the “synthetic” Brillouin zone.",
"Note that the gap at $\\Gamma $ point, representing the state without any twist, is always finite including the case with $\\Delta =0$ .",
"In passing, we note that it is known that the ground state is doubly degenerate in the thermodynamic limit for $\\Delta =0$ [30].",
"This means that the “band structures” in Figs.",
"REF (c-e) collapse in the infinite size limit, and the jump in $\\gamma $ gets sharper with $L\\rightarrow \\infty $ in any case.",
"However, as we have stressed earlier, the advantage of the quantized Berry phase lies in the usefulness in the finite size calculation of a relatively small system size.",
"Figure: (Color online) (a) Schematic picture of the SU(4) model.",
"qq and q ¯\\bar{q}denote the fundamental representation and the conjugate representation,respectively.",
"(b) The Berry phase obtained with the pathW 1 _1-Γ\\Gamma -W 2 _2.",
"(c) The Berry phase obtained using the straightintegration path along φ 1 \\phi _1 axis.",
"(d,e) Synthetic Brillouin zoneand the integration path.Next we show the usefulness of the $Z_N$ Berry phase by applying it to another example of 1D SPT phases.",
"We consider an SU(4) symmetric Hamiltonian [32], $H=-\\sum _{i}\\sum _{a=1}^{15}\\bigl [J_1\\Lambda _a(2i)\\bar{\\Lambda }_a(2i+1)+J_2\\bar{\\Lambda }_a(2i+1)\\Lambda _a(2i+2)\\bigr ].$ Here, the fundamental representations of SU(4) and its conjugate representations are assigned on the $(2i)$ th sites and $(2i+1)$ th sites, respectively [see Fig.",
"REF (a)].",
"The explicit form of the $\\Lambda _a$ is found in Ref. su4.",
"For convenience, we parameterize $J_{1,2}$ as $J_1=J_0+\\delta J$ and $J_2=J_0-\\delta J$ .",
"With the appropriate parameter choice, the ground state of this Hamiltonian becomes to share the majority of properties with the SU(4) AKLT state [31].",
"In this case, the entangled cluster is the completely antisymmetrized state formed by a pair of the fundamental and its conjugate representations (which is analogous to the $\\eta $ -meson in the SU(3) case).",
"In a similar manner to the case of $\\Delta $ for the biquadratic model, $\\delta J$ controls how the entangled clusters are formed, and the phase transition takes place by changing $\\delta J$ .",
"For the detection of the pattern of entangled clusters, we adopt $U(\\mathbf {\\phi })=\\exp [i\\sum _{n=1}^4\\check{A}_n\\phi _n]$ as a gauge twist, where $(\\check{A}_n)_{ij}=\\delta _{ij}\\delta _{in}$ .",
"By using $\\sum _n\\check{A}_n=\\hat{1}$ , we notice that a twist $e^{i\\check{A}_4\\phi }$ is essentially equivalent to a twist $e^{-i(\\check{A}_1+\\check{A}_2+\\check{A}_3)\\phi }$ , and consequently, the local gauge twist is defined on the three-dimensional synthetic BZ with the symmetry of the fcc BZ.",
"Figure: (Color online) Energy spectrum (in the unit of J 0 J_0) on thesymmetric lines in the syntheticBrillouin zone for δJ=0\\delta {J}=0.",
"The upper right inset shows the bandstructure of thesingle orbital tight-binding model on the diamond lattice.",
"The leftinset shows the location of the nodal lines.The numerically obtained Berry phase is plotted in Fig.",
"REF .",
"Again, the exact quantization of the Berry phase is achieved for a symmetric integration path W$_1$ -$\\Gamma $ -W$_2$ in the synthetic BZ as shown in Figs.",
"REF (d) and REF (e).",
"With this setup, the phase transition is captured by a jump from $\\gamma =0$ to $\\gamma =\\pi /2=(2\\pi /4)$ [23], [34].",
"Similarly to the SU(3) case, the symmetry protecting the quantization of $Z_4$ Berry phase is the invariance under the interchange of the four components of the fundamental representation of SU(4).",
"When the straight integration path along one of the $\\phi _i$ axis is used naively, the Berry phase is no longer quantized and it does not show a sharp transition at $\\delta {J}=0$ [Fig.",
"REF (c)].",
"The jump in the $Z_4$ Berry phase is again associated with the gapless point on the integration path.",
"In this case, the gap closes on the X-W line, i.e., nodal lines appear in the three-dimensional Brillouin zone for $\\delta {J}=0$ [see Fig.",
"REF ].",
"Interestingly, the energy spectrum resembles the band structure for the single orbital tight-binding model on the diamond lattice.",
"To summarize, we have demonstrated the usefulness of the $Z_N$ Berry phase as a topological invariant for SU(N) symmetric SPT phases.",
"The key ingredient is a suitable choice of $\\hat{A}$ for the local gauge twist, and the introduction of the synthetic Brillouin zone reflecting the symmetry of the system.",
"The topological transitions are captured by jumps in the Berry phase, and the associated singularities (Dirac cones/nodal lines) in the synthetic Brillouin zone.",
"It would be an interesting future problem to explore the relationship between Dirac cones/nodal lines found here and those in free fermion systems at the topological transition, for example, in terms of the criticality.",
"Another promising direction would be an extension of the $Z_N$ Berry phase to topological phases in higher spatial dimensions.",
"The major task in doing so will be finding proper ways of applying the local gauge twist to ensure exact quantization.",
"Once they are found, it will provide a tractable way to characterize general SPT phases based on the Hamiltonians explicitly.",
"TK thanks the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo for the use of the facilities.",
"The work is partially supported by a Grant-in-Aid for Scientific Research No.",
"17K14358 (TK), No.",
"17H06138 (TK, YH) and No.",
"16K13845 (YH).",
"TM was supported by the Gordon and Betty Moore Foundation's EPiQS Initiative Theory Center Grant."
]
] | 1709.01546 |
[
[
"Energy-aware Mode Selection for Throughput Maximization in RF-Powered\n D2D Communications"
],
[
"Abstract Doubly-near-far problem in RF-powered networks can be mitigated by choosing appropriate device-to-device (D2D) communication mode and implementing energy-efficient information transfer (IT).",
"In this work, we present a novel RF energy harvesting architecture where each transmitting-receiving user pair is allocated a disjoint channel for its communication which is fully powered by downlink energy transfer (ET) from hybrid access point (HAP).",
"Considering that each user pair can select either D2D or cellular mode of communication, we propose an optimized transmission protocol controlled by the HAP that involves harvested energy-aware jointly optimal mode selection (MS) and time allocation (TA) for ET and IT to maximize the sum-throughput.",
"Jointly global optimal solutions are derived by efficiently resolving the combinatorial issue with the help of optimal MS strategy for a given TA for ET.",
"Closed-form expressions for the optimal TA in D2D and cellular modes are also derived to gain further analytical insights.",
"Numerical results show that the joint optimal MS and TA, which significantly outperforms the benchmark schemes in terms of achievable RF-powered sum-throughput, is closely followed by the optimal TA scheme for D2D users.",
"In fact, about $2/3$ fraction of the total user pairs prefer to follow the D2D mode for efficient RF-powered IT."
],
[
"Introduction",
"With the advent of 5G radio access technologies [1], the ubiquitous deployment of low power wireless devices has led to the emergence of device-to-device (D2D) communications as a promising technology for performance enhancement by exploiting the proximity gains.",
"Despite these merits, the underlying challenge is to provide sustainable network operation by overcoming the finite battery life bottleneck of these devices.",
"Recently, the efficacy of energy harvesting (EH) from dedicated radio frequency (RF) energy transfer (ET) has been investigated to enable controlled energy replenishment of battery-constrained wireless devices [2].",
"However, due to fundamental bottlenecks [2], such as low energy reception sensitivity and poor end-to-end ET efficiency, there is a need for novel RF-EH based D2D communication protocols.",
"In the pioneering work on wireless powered communication network (WPCN) [3], the optimal time allocation (TA) for downlink ET and uplink information transfer (IT) from multiple EH users was investigated to maximize the system throughput.",
"It was shown that WPCN suffers from the doubly-near-far-problem which limits its practical deployment.",
"Although optimal cooperative resource allocation strategies [2], [4] have been proposed, in this work we focus on the efficacy of D2D communications in resolving this issue by exploiting proximity.",
"Despite the pioneering research on WPCN [5], [3], [4], the investigation on EH-assisted D2D communications, where both the information source and the destination are energy constrained, is still in its infancy [6], [7], [8], [9].",
"In [6], performance of the D2D transmission powered solely by ambient interference in cellular networks was investigated using stochastic geometry.",
"The joint resource block and power allocation for maximizing the sum-rate of D2D links in EH-assisted D2D communication underlying downlink cellular networks was studied in [7].",
"Recently, D2D communication powered by ambient interference in cellular networks for relaying machine-type communication traffic was investigated in [8].",
"An integrated information relaying and energy supply assisted RF-EH communication model was presented in [9] to maximize the RF-powered throughput between two energy constrained devices.",
"However none of the above mentioned works [6], [7], [8], [9] investigated the fundamental problem of mode selection (MS) and resource sharing between cellular and D2D users, as done in conventional D2D communications [10].",
"In the recent work [11], without considering EH, joint optimization of MS, uplink/downlink TA, and power allocation was performed to minimize the energy consumption in meeting the traffic demands of a D2D communication network.",
"To the best of our knowledge, the harvested energy-aware joint MS and TA has not been investigated for RF-powered D2D communications.",
"Here, different from conventional D2D systems [10], [11], the optimal decision-making is strongly influenced by the TA for EH during downlink ET from the hybrid access point (HAP).",
"Figure: Modes of operation in RF-powered D2D communications.In comparison to RF-powered cellular networks, RF-EH assisted D2D communication also provides pairing gain, where, as shown in Fig.",
"1, each user pair can select either D2D or cellular mode of communication based on the harvested energy and radio propagation environment.",
"D2D mode enables short-range and low-power links saving the power consumption at the transmitter, and thus assisting in overcoming the doubly-near-far problem [3].",
"On the other hand, communicating via an energy-rich HAP in cellular mode can help in increasing the end-to-end IT range.",
"Apart from MS, the total transmission time needs to be allocated optimally for IT and ET in D2D and cellular modes to maximize RF-powered throughput.",
"So, we need to tackle two trade-offs: (i) choosing between D2D and cellular mode; and (ii) time sharing between ET and IT.",
"The key contributions of this work are four fold.",
"(1) Novel RF-EH communication model and transmission protocol are presented to enable efficient sustainable D2D communications.",
"(2) A joint optimization framework is proposed to maximize sum-throughput of the system by optimally selecting the transmission mode (D2D or cellular) along with TA for ET and IT.",
"Although this is a combinatorial problem which is NP-hard in general [11], we demonstrate that it can be effectively decoupled into equivalent convex problems to obtain both individual and joint global optimal MS and TA solutions.",
"(3) Closed-form expressions are derived for: (i) optimal TA for D2D and cellular modes of communication between a single EH pair; and (ii) tight approximation for optimal TA with fixed D2D mode for all nodes.",
"We also provide analytical insights on the harvested energy-aware optimal MS strategy for a given TA for ET.",
"(4) Throughput gains achieved with help of proposed optimization framework over the benchmark schemes involving fixed TA and MS are quantified by numerical simulations.",
"These results incorporating the impact of practical RF-EH system constraints also validate the accuracy of the analysis and provide useful insights on the jointly optimal solutions."
],
[
"System Model",
"We consider a heterogeneous small cell orthogonal frequency division multiple access (OFDMA) network with a single HAP and multiple RF-EH users that are fully-powered by dedicated RF energy broadcast from the HAP.",
"Without loss of generality, we focus on the RF-powered IT among $N$ RF-EH users, denoted by $\\left\\lbrace \\mathcal {U}_1,\\mathcal {U}_2,\\ldots ,\\mathcal {U}_N\\right\\rbrace $ , that are interested in communicating within this small cell.",
"We assume that $\\frac{N}{2}$ non-overlapping frequency channels are available to enable simultaneous communication between these possible $\\frac{N}{2}$ user pairs.",
"The transmitter and receiver in each user pair are pre-decided and allocated a disjoint frequency channel [10], [11].",
"Each RF-EH user is composed of a single omnidirectional antenna.",
"To enable simultaneous reception and transmission of information at different frequencies in the uplink and downlink, the HAP is equipped with two omnidirectional antennas, one being dedicated for information reception and the other antenna is dedicated for energy or information transmission.",
"Each $\\frac{N}{2}$ user pair chooses either (i) D2D mode where nodes directly communicate with each other, or (ii) cellular mode where they communicate via HAP in a two-hop decode-and-forward (DF) fashion.",
"Both D2D and cellular modes of communication are solely powered by RF-ET from the HAP.",
"As the cell size is very small due to low RF-ET range of the HAP [2], we have not considered the reuse mode [10] to avoid strong co-channel interference.",
"All the links are assumed to follow independent quasi-static block fading; for simplicity, we consider this block duration as $T=1$ sec.",
"The instantaneous channel gains between the HAP and user $\\mathcal {U}_i$ for downlink ET and uplink IT are denoted by $H_{\\mathcal {U}_i}$ and $\\rho _{_{\\mathcal {U}_i}} H_{\\mathcal {U}_i}$ , respectively, where $\\rho _{_{\\mathcal {U}_i}}$ is a positive scalar with $\\rho _{_{\\mathcal {U}_i}}=1$ representing the channel reciprocity case.",
"Similarly, the channel gain between $\\mathcal {U}_j$ and $\\mathcal {U}_k$ is denoted by $G_{\\mathcal {U}_j,\\mathcal {U}_k}$ .",
"These channel gains $\\lbrace H_{\\mathcal {U}_i},G_{\\mathcal {U}_j,\\mathcal {U}_k}\\rbrace $ can be defined as $&\\hspace{-8.53581pt}H_{\\mathcal {U}_i}=\\frac{p\\,h_{_{0,\\mathcal {U}_i}}}{d_{_{0,\\mathcal {U}_i}}^n},\\quad G_{\\mathcal {U}_j,\\mathcal {U}_k}=\\frac{p\\,g_{_{\\mathcal {U}_j,\\mathcal {U}_k}}}{d_{_{\\mathcal {U}_j,\\mathcal {U}_k}}^n},\\;\\forall i,j,k\\le N, j\\ne k,$ where $p$ is path-loss coefficient; $h_{_{0,\\mathcal {U}_i}}$ and $g_{_{\\mathcal {U}_j,\\mathcal {U}_k}}$ respectively denote the channel fading components for HAP-to-$\\mathcal {U}_i$ and $\\mathcal {U}_j$ -to-$\\mathcal {U}_k$ links; $n$ is path-loss exponent; $d_{_{0,\\mathcal {U}_i}}$ and $d_{_{\\mathcal {U}_j,\\mathcal {U}_k}}$ respectively denote HAP-to-$\\mathcal {U}_i$ and $\\mathcal {U}_j$ -to-$\\mathcal {U}_k$ euclidean distances.",
"We assume that the HAP has the full acquisition of instantaneous channel state information (CSI) of links to all users, including the CSI between all possible $\\frac{N\\left(N-1\\right)}{2}$ user pairs.",
"This CSI is estimated by the respective receivers and then fed back to the HAP via the control channel [10], [11].",
"Although the proposed HAP-controlled optimization algorithm is based on instantaneous CSI which incurs a lot of signalling overhead, it can serve as a benchmark for distributed algorithms [11].",
"Figure: Proposed transmission protocols for cellular and D2D modes.As the desired frequency characteristics for efficient ET and IT are different [2] and the maximum end-to-end ET efficiency is achieved over narrow-band transmission [5], we consider a pessimistic scenario by ignoring EH during downlink IT from the HAP.",
"Also, since the received uplink signal strength from energy constrained users is relatively very weak in comparison to the transmit signal strength in the downlink IT from the energy-rich HAP, we do not consider full-duplex operation at the HAP.",
"Instead, we propose an enhanced half-duplex information relaying where orthogonal frequencies can be received and transmitted simultaneously using two antennas."
],
[
"Proposed Transmission Protocol",
"As shown in Fig.",
"REF , the proposed transmission protocols for both cellular and D2D modes of communication are divided into RF-ET of duration $t_e$ sec and IT phase of duration $1-t_e$ sec.",
"This downlink RF-ET based IT forms a WPCN [3].",
"Next, we discuss the sub-operations in these two phases in detail."
],
[
"Phase 1): RF-ET from the HAP to all the EH users",
"During RF-ET phase, HAP broadcasts a single-tone RF energy signal $x_e$ [5], having zero mean and variance $P_0$ , to all EH users.",
"The energy signal $y_{e,\\mathcal {U}_i}$ thus received at user $\\mathcal {U}_i$ is given by $&y_{e,\\mathcal {U}_i}=\\sqrt{H_{\\mathcal {U}_i}} x_e + z_{_{\\mathcal {U}_i}}, \\quad \\forall i\\in \\left\\lbrace 1,2,\\ldots , N\\right\\rbrace ,$ where $z_{_{\\mathcal {U}_i}}$ is the received Additive White Gaussian Noise (AWGN) at $\\mathcal {U}_i$ .",
"Ignoring EH from noise signal $z_{_{\\mathcal {U}_i}}$ due to very low energy reception sensitivity [2], the energy $E_{_{H,\\mathcal {U}_i}}$ harvested at $\\mathcal {U} _i$ over the RF-ET duration of $t_e$ sec is given by $E_{_{H,\\mathcal {U}_i}}=\\eta _{_{\\mathcal {U}_i}} P_0 H_{\\mathcal {U}_i} t_e, \\quad \\forall i\\in \\left\\lbrace 1,2,\\ldots , N\\right\\rbrace ,$ where $\\eta _{_{\\mathcal {U}_i}}$ is RF-to-DC rectification efficiency of RF-EH unit at $\\mathcal {U}_i$ which is in general a nonlinear function of the received RF power [12].",
"If $\\mathcal {U}_i$ is a receiver, then harvested energy $E_{_{H,\\mathcal {U}_i}}$ is stored for carrying out internal node operations.",
"Otherwise, $E_{_{H,\\mathcal {U}_i}}$ is solely used for carrying out IT from $\\mathcal {U}_i$ to HAP, if it follows cellular mode, or to its receiving partner in D2D mode.",
"The operations in this phase depend on the transmission mode, i.e., cellular or D2D."
],
[
"Cellular Mode",
"The IT phase is divided into uplink and downlink subphases.",
"By exploiting the availability of disjoint channels and two antennas at the HAP, we consider different uplink and downlink IT times, as denoted by $1-t_e-t_{d,T_i}$ and $t_{d,T_i}$ , for efficient cellular mode communication between transmitter $T_i$ and receiver $R_i$ of $i$ th user pair with $T_i,R_i\\in \\lbrace \\mathcal {U}_1,\\ldots ,\\mathcal {U}_N\\rbrace $ and $1\\le i\\le N/2$ .",
"The transmit power $P_{T_i}^C$ of $T_i$ for IT to HAP using its harvested energy $E_{_{H,T_i}}$ is $&P_{T_i}^C=\\frac{\\theta E_{_{H,T_i}}}{1-t_e-t_{d,T_i}}=\\frac{\\theta \\eta _{_{T_i}} P_0 H_{T_i} t_e}{1-t_e-t_{d,T_i}},$ where $\\theta $ is the fraction of $E_{_{H,T_i}}$ available for IT after excluding the internal energy losses.",
"Although we only consider the energy harvested during the current block for carrying out IT, the proposed optimization can be easily extended to the scenario where transmit power is given as $P_{T_i}^C=\\frac{\\theta \\big (E_{_{S,T_i}}+E_{_{H,T_i}}\\big )}{1-t_e-t_{d,T_i}}$ with $E_{_{S,T_i}}$ denoting the stored energy available at $T_i$ for IT at the beginning of the current block.",
"The information signal received at the HAP due to this uplink IT from $T_i$ is given by $& y_{u,T_i}= \\sqrt{P_{T_i}^C \\rho _{T_i} H_{T_i}} x_{u,T_i}+z_0,$ where $x_{u,T_i}$ is the normalized zero mean information symbol transmitted by $T_i$ having unit variance and $z_0$ is the received AWGN at the HAP having zero mean and variance $\\sigma ^2$ .",
"After decoding the message signal from $y_{u,T_i}$ , the HAP forwards the decoded message signal $\\widehat{x}_{u,T_i}$ to the receiving cell user $R_i$ in the second subphase of phase 2.",
"Cellular communication between $T_i$ and $R_i$ completes upon downlink IT from the HAP to $R_i$ .",
"The information signal $y_{d,R_i}$ as received at $R_i$ is $& y_{d,R_i}=\\sqrt{\\phi _{_{R_i}} H_{R_i} P_0} \\widehat{x}_{u,T_i} + z_{_{R_i}},$ where $P_0$ is transmit power of HAP and $z_{_{R_i}}$ denotes zero mean AWGN having variance $\\sigma ^2$ .",
"$\\phi _{_{R_i}}$ incorporates the difference in HAP-to-$R_i$ channel characteristics for downlink ET and IT [5].",
"Since the HAP acts like a DF relay [10] with $1-t_e-t_{d,T_i}$ and $t_{d,T_i}$ being time allocations for uplink and downlink IT, the RF-powered throughput for cellular mode communication between $T_i$ and $R_i$ , as obtained using (REF ) and (REF ), is given by $&\\tau _{T_i,R_i}^C=\\textstyle \\min \\left\\lbrace \\left(1-t_e-t_{d,T_i}\\right)\\,\\log _2\\left(1+\\frac{\\theta \\eta _{_{T_i}} P_0 \\rho _{_{T_i}} H_{T_i}^2\\,t_e}{\\sigma ^2\\left( 1-t_e-t_{d,T_i}\\right)}\\right)\\right.,\\nonumber \\\\&\\qquad \\qquad \\qquad \\;\\;\\textstyle \\left.t_{d,T_i}\\,\\log _2\\left(1+{P_0\\,\\phi _{_{R_i}} H_{R_i}}{\\sigma ^{-2}}\\right)\\right\\rbrace .$ With $\\tau _{T_i,R_i}^{\\text{UL}}=\\left(1-t_e-t_{d,T_i}\\right)\\,\\log _2\\left(1+\\frac{\\theta \\eta _{_{T_i}} P_0 \\rho _{_{T_i}} H_{T_i}^2\\,t_e}{\\sigma ^2\\left( 1-t_e-t_{d,T_i}\\right)}\\right)$ and $\\tau _{T_i,R_i}^{\\rm {DL}}=t_{d,T_i}\\,\\log _2\\left(1+{P_0\\,\\phi _{_{R_i}} H_{R_i}}{\\sigma ^{-2}}\\right)$ respectively representing uplink and downlink rates, throughput $\\tau _{T_i,R_i}^C$ defined in (REF ) can be rewritten as: $\\tau _{T_i,R_i}^C=\\min \\left\\lbrace \\tau _{T_i,R_i}^{\\text{UL}},\\tau _{T_i,R_i}^{\\rm {DL}}\\right\\rbrace $ ."
],
[
"D2D Mode",
"Under this mode, $T_i$ directly communicates with $R_i$ by forming a D2D link.",
"With IT phase of $1-t_e$ duration, the transmit power $P_{T_i}^D$ of D2D transmitter $T_i$ is $& P_{T_i}^D=\\frac{\\theta E_{_{H,T_i}}}{1-t_e}=\\frac{\\theta \\eta _{_{T_i}} P_0 H_{T_i} t_e}{1-t_e}.$ The corresponding information signal $y_{d2,R_i}$ received at $R_i$ is $&y_{d2,R_i}= \\sqrt{P_{T_i}^D G_{T_i,R_i}} x_{d2,T_i}+z_{_{R_i}},$ where $x_{d2,T_i}$ is the zero mean and unit variance information symbol transmitted by $T_i$ of $i$ th user pair choosing D2D mode.",
"So using (REF ) and (REF ), the RF-powered throughput for the D2D mode communication between $i$ th user pair is given by $\\tau _{T_i,R_i}^D&=\\textstyle \\left(1-t_e\\right)\\,\\log _2\\left(1+\\frac{\\theta \\eta _{_{T_i}} P_0 H_{T_i} G_{T_i,R_i}\\,t_e}{\\sigma ^2\\left(1-t_e\\right)}\\right).$ With the above two throughput definitions, we investigate the joint optimal MS (between D2D and cellular) and TA (for different phases) policy for the sum-throughput $\\tau _{\\rm {S}}$ maximization."
],
[
"Optimal Time Allocation",
"In this section we first investigate the concavity of the RF-powered throughput in TA for both cellular and D2D modes.",
"Then, we derive the expressions for globally optimal TA."
],
[
"Concavity of Throughput in TA",
"We first show that the throughput $\\tau _{T_i,R_i}^C$ in cellular mode communication between users $T_i$ and $R_i$ is jointly concave in TAs $t_e$ and $t_{d,T_i}$ by proving the concavity of $\\tau _{T_i,R_i}^{\\text{UL}}$ and $\\tau _{T_i,R_i}^{\\text{DL}}$ .",
"The Hessian matrix of $\\tau _{T_i,R_i}^{\\text{UL}}$ is given by $\\textstyle \\mathbb {H}\\left(\\tau _{T_i,R_i}^{\\text{UL}}\\right)\\hspace{-2.84526pt}=\\hspace{-2.84526pt}\\left[\\hspace{-4.2679pt}\\begin{array}{ccc}\\frac{\\partial ^2 \\tau _{T_i,R_i}^{\\text{UL}}}{\\partial t_e^2} & \\frac{\\partial ^2 \\tau _{T_i,R_i}^{\\text{UL}}}{\\partial {t_e}\\partial {t_{d,T_i}}} \\\\\\frac{\\partial ^2 \\tau _{T_i,R_i}^{\\text{UL}}}{\\partial {t_{d,T_i}}\\partial {t_e}} & \\frac{\\partial ^2 \\tau _{T_i,R_i}^{\\text{UL}}}{\\partial {t_{d,T_i}}^2} \\end{array}\\hspace{-4.2679pt}\\right]\\hspace{-2.84526pt}=\\hspace{-1.42262pt}-\\mathcal {Z}_i\\left[\\hspace{-4.2679pt} \\begin{array}{ccc}1-t_{d,T_i}& t_e\\\\t_e & \\frac{t_e^2}{1-t_{d,T_i}} \\end{array}\\hspace{-4.2679pt} \\right]$ where $\\mathcal {Z}_i\\triangleq \\frac{\\left({1-t_{d,T_i}}\\right)\\left(\\eta _{_{T_i}}\\theta \\rho _{_{T_i}}P_0 H_{T_i}^2 \\right)^2}{\\ln (2) \\left(1-t_e-t_{d,T_i}\\right) \\left(\\eta _{_{T_i}}\\theta P_0 \\rho _{_{T_i}} H_{T_i}^2\\, t_e+\\sigma ^2 \\left(1-t_e-t_{d,T_i}\\right)\\right)^2}$ .",
"As $\\mathcal {Z}_i,t_e,$ and $ t_{d,T_i}$ are positive with $t_e+t_{d,T_i}<1$ , we notice that $\\frac{\\partial ^2 \\tau _{T_i,R_i}^{\\text{UL}}}{\\partial t_e^2},\\frac{\\partial ^2 \\tau _{T_i,R_i}^{\\text{UL}}}{\\partial t_{d,T_i}^2}<0$ , and the determinant of $\\mathbb {H}\\left(\\tau _{T_i,R_i}^{\\text{UL}}\\right)$ is zero.",
"This proves that $\\mathbb {H}\\left(\\tau _{T_i,R_i}^{\\text{UL}}\\right)$ is negative semi-definite; hence, $\\tau _{T_i,R_i}^{\\text{UL}}$ is concave in $t_e$ and $t_{d,T_i}$ .",
"Also, the downlink rate $\\tau _{T_i,R_i}^{\\rm {DL}}$ is linear in $t_{d,T_i}$ and independent of $t_e$ .",
"Finally, with the minimum of two concave functions being concave [13], the joint concavity of $\\tau _{T_i,R_i}^C$ in $t_e$ and $t_{d,T_i}$ is proved.",
"The strict-concavity of throughput for D2D mode in $t_e$ is shown by $\\frac{\\partial ^2 \\tau _{T_i,R_i}^D}{\\partial t_e^2}=\\frac{-1}{{(1-t_e)\\,t_e^2 \\ln (2) \\left(1+\\frac{\\sigma ^2 (1-t_e)}{\\eta \\theta H_{T_i} G_{T_i,R_i} P_0\\, t_e}\\right)^2}}<0$ ."
],
[
"Global Optimal TA Solution",
"Using (REF ), the throughput maximization problem for cellular mode communication between $T_i$ and $R_i$ can be defined as $\\begin{aligned}(\\mathcal {OP}):\\;&\\underset{x_i,t_e,t_{d,T_i}}{\\text{maximize}} \\quad \\tau _{T_i,R_i}^C=x_i,\\quad &&\\text{subject to}:\\\\& (\\mathrm {C1}): x_i\\le \\tau _{T_i,R_i}^{\\text{UL}},&& (\\mathrm {C2}): x_i\\le \\tau _{T_i,R_i}^{\\rm {DL}},\\\\& (\\mathrm {C3}): t_e+t_{d,T_i} \\le 1,&&(\\mathrm {C4}): t_e,t_{d,T_i} \\ge 0.\\end{aligned}$ Keeping the positivity constraint $(\\mathrm {C4})$ implicit and associating Lagrange multipliers $\\lambda _1,\\lambda _2,\\lambda _3$ with remaining constraints, the Lagrangian function for $\\mathcal {OP}$ is given by $\\mathcal {L}\\triangleq x_i-\\lambda _1\\left[t+t_{d,T_i}-1\\right] -\\lambda _2\\left[x_i-\\tau _{T_i,R_i}^{\\text{UL}}\\right] -\\lambda _3\\left[x_i- \\tau _{T_i,R_i}^{\\rm {DL}}\\right]$ .",
"The corresponding optimality (KKT) conditions [13] are given by constraints $(\\mathrm {C1})$ –$(\\mathrm {C4})$ , dual feasibility conditions $\\lambda _1,\\lambda _2,\\lambda _3\\ge 0$ , subgradient conditions $\\frac{\\partial \\mathcal {L}}{\\partial x_i},\\frac{\\partial \\mathcal {L}}{\\partial t_e},\\frac{\\partial \\mathcal {L}}{\\partial t_{d,T_i}}=0$ , and Complementary Slackness Conditions (CSC) [13] $\\lambda _1\\left[t+t_{d,T_i}-1\\right]=0,\\lambda _2\\left[x_i-\\tau _{T_i,R_i}^{\\text{UL}}\\right]=0,\\lambda _3\\left[x_i- \\tau _{T_i,R_i}^{\\rm {DL}}\\right]=0$ .",
"To allocate non-zero time for uplink IT, $t_e+t_{d,T_i}<1$ , which results in $\\lambda _1^*=0$ .",
"Using this in $\\frac{\\partial \\mathcal {L}}{\\partial x_i},\\frac{\\partial \\mathcal {L}}{\\partial t_e},\\frac{\\partial \\mathcal {L}}{\\partial t_{d,T_i}}=0$ , implies that for $\\tau _{T_i,R_i}^C>0$ , $\\lambda _1^*=0, \\lambda _2^*>0,$ and $\\lambda _3^*>0$ .",
"Using $\\lambda _2^*>0$ and $\\lambda _3^*>0$ in CSC deduces to $x_i=\\tau _{T_i,R_i}^{\\text{UL}}=\\tau _{T_i,R_i}^{\\rm {DL}}$ .",
"This implies that the optimal TA $t_{d,T_i}$ for downlink IT from the HAP to receiver $R_i$ is such that the uplink and downlink rates become equal, i.e., $\\tau _{T_i,R_i}^C=\\tau _{T_i,R_i}^{\\text{UL}}=\\tau _{T_i,R_i}^{\\rm {DL}}$ .",
"On solving this, the global optimal TA $t_{d,T_i}^*$ for downlink IT is given by $&\\hspace{-14.22636pt}t_{d,T_i}^*\\triangleq \\textstyle \\left(1-t_e\\right) \\Bigg (1+\\frac{\\ln \\left(\\mathcal {Y}_{2,i}\\right)}{\\mathrm {W}_{-1}\\left(-\\frac{\\mathcal {Y}_{1,i} \\ln \\left(\\mathcal {Y}_{2,i}\\right)}{\\mathcal {Y}_{2,i}^{\\mathcal {Y}_{1,i}+1}}\\right)+\\mathcal {Y}_{1,i} \\ln \\left(\\mathcal {Y}_{2,i}\\right)}\\Bigg )\\!,$ where $\\mathcal {Y}_{1,i}=\\frac{\\sigma ^2 (1-t_e)}{\\eta _{_{T_i}} \\theta P_0\\,\\rho _{_{T_i}} H_{T_i}^2 \\, t_e}$ , $\\mathcal {Y}_{2,i}=1+\\frac{P_0\\,\\phi _{_{R_i}} H_{R_i}}{\\sigma ^2}$ , and $\\mathrm {W}_{-1}\\left(\\cdot \\right)$ is the Lambert function [14].",
"Using optimal $t_{d,T_i}^*$ along with $\\tau _{T_i,R_i}^{\\text{UL}}=\\tau _{T_i,R_i}^{\\rm {DL}}$ , the optimal cellular rate $\\tau _{T_i,R_i}^{C^*}\\triangleq t_{d,T_i}^*\\,\\log _2\\left(1+{P_0\\,\\phi _{_{R_i}} H_{R_i}}{\\sigma ^{-2}}\\right)$ is a unimodal function of $t_e$ .",
"Since throughput $\\tau _{T_i,R_i}^D$ in D2D communication between $T_i$ and $R_i$ is strictly-concave in $t_e$ , the global optimal TA $t_e$ for maximizing it, as obtained by solving $\\frac{\\partial \\tau _{T_i,R_i}^D}{\\partial t_e}\\!=\\!0$ , is given by $&\\hspace{-17.07164pt}t_{e,i}^{\\rm {D}^*}\\!\\triangleq \\!\\left[1\\!-\\!\\frac{\\mathcal {Y}_{3,i}\\mathrm {W}_0\\left(\\frac{\\mathcal {Y}_{3,i}-1}{\\exp (1)}\\right)}{\\mathrm {W}_0\\left(\\frac{\\mathcal {Y}_{3,i}-1}{\\exp (1)}\\right)-\\mathcal {Y}_{3,i}+1}\\right]^{\\!-1}\\hspace{-8.53581pt}\\text{ with }\\mathcal {Y}_{3,i}=\\frac{ P_0 H_{T_i} G_{T_i,R_i}}{\\left[ \\theta \\eta _{_{T_i}}\\right]^{-1}\\sigma ^2}$ Here $\\mathrm {W}_{0}\\left(\\cdot \\right)$ is the Lambert function of principal branch [14]."
],
[
"Joint Mode Selection and Time Allocation",
"Using the definition for $\\tau _{T_i,R_i}^{C^*}$ , the mathematical formulation of the joint optimization of MS and TA for maximizing the sum-throughput $\\tau _{\\rm {S}}$ of the considered system is given by $\\begin{aligned}(\\mathcal {JP})\\!\\!",
":\\,&\\underset{t_e,\\lbrace m_i\\rbrace _{i=1}^{N/2}}{\\text{maximize}} &&\\hspace{-8.53581pt}\\tau _{\\rm {S}}\\triangleq \\textstyle \\sum \\limits _{i=1}^{N/2}\\!",
"\\left(1-m_i\\right)t_{d,T_i}^* \\log _2\\!\\left(\\!1+\\!\\frac{P_0\\,\\phi _{_{R_i}} H_{R_i}}{\\sigma ^2}\\!\\right)\\\\& &&\\hspace{-14.22636pt}\\textstyle +m_i\\left(1-t_e\\right) \\log _2\\left(1+\\frac{\\theta \\eta _{_{T_i}} P_0 H_{T_i} G_{T_i,R_i}\\,t_e}{\\sigma ^2\\left(1-t_e\\right)}\\right)\\\\&\\text{subject to:}&&(\\mathrm {C5}): m_i\\in \\left\\lbrace 0,1\\right\\rbrace ,\\quad (\\mathrm {C6}): 0\\le t_e\\le 1,\\end{aligned}$ where constraint $(\\mathrm {C5})$ is defined $\\forall \\,i=1,2,\\ldots ,N/2$ and $m_i$ is the MS based binary decision variable which is defined as $m_i={\\left\\lbrace \\begin{array}{ll}0, & \\tau _{_{T_i,R_i}}^{C^*} > \\tau _{_{T_i,R_i}}^D \\text{ (i.e., Cellular mode)}\\\\1, & \\tau _{_{T_i,R_i}}^{C^*}\\le \\tau _{_{T_i,R_i}}^D \\text{ (i.e., D2D mode)}.\\end{array}\\right.",
"}$ In general $\\mathcal {JP}$ is a combinatorial problem as it involves binary variable $\\lbrace m_i\\rbrace $ .",
"However, we next present a novel harvested energy-aware optimal MS strategy that resolves this issue and helps in obtaining the jointly global optimal solution of $\\mathcal {JP}.$"
],
[
"Optimal Mode Selection Strategy",
"For a given TA $t_e$ , the analytical condition for throughput of RF-powered communication between $T_i$ and $R_i$ of $i$ th user pair in D2D mode to be higher than that in cellular mode is: $\\tau _{T_i,R_i}^D>\\tau _{T_i,R_i}^{C^*}$ .",
"Using $t_{d,T_i}^*$ defined in (REF ), this reduces to: $&\\hspace{-11.38109pt}1+\\frac{\\theta \\eta _{_{T_i}} P_0 H_{T_i} G_{T_i,R_i}\\,t_e}{\\sigma ^2\\left(1-t_e\\right)}>\\left(1+\\frac{\\theta \\eta _{_{T_i}} P_0 \\rho _{_{T_i}} H_{T_i}^2\\,t_e}{\\sigma ^2\\left( 1-t_e\\right)\\left(1-\\mathcal {F}_i\\right)}\\right)^{1-\\mathcal {F}_i},$ where $\\mathcal {F}_i\\triangleq \\frac{t_{d,T_i}^*}{\\left(1-t_e\\right)}\\le 1$ .",
"We note that $\\frac{t_e}{1-t_e}$ and $\\mathcal {F}_i$ are strictly increasing functions of $t_e,$ where the latter holds because $\\frac{\\partial \\mathcal {F}_i}{\\partial t_e}=\\textstyle \\frac{-\\left(1-t_e-t_{d,T_i}^*\\right)\\sigma ^2\\left[\\mathrm {W}_{-1}\\left(-\\mathcal {Y}_{2,i}^{-\\mathcal {Y}_{1,i}-1}\\ln \\left(\\mathcal {Y}_{2,i}^{\\mathcal {Y}_{1,i}}\\right)\\right)+1\\right]^{-1}}{\\eta _{_{T_i}} \\theta P_0 \\rho _{_{T_i}} H_{T_i}^2\\,t_e^2\\left(1-t_e\\right)\\mathcal {Y}_{1,i}}>0$ .",
"This result is obtained by knowing ${\\mathrm {W}_{-1}}\\left(x\\right)+1<0, \\forall x\\in \\left[\\frac{-1}{\\exp (1)},0\\right]$ and $\\mathcal {Y}_{2,i}>1$ along with $\\frac{-1}{\\exp (1)}\\!\\le \\!\\frac{-\\ln \\left(\\mathcal {Y}_{2,i}^{\\mathcal {Y}_{1,i}}\\right)}{\\mathcal {Y}_{2,i}^{\\mathcal {Y}_{1,i}}}\\!<\\!0$ .",
"Now if $G_{T_i,R_i}\\ge \\rho _{_{T_i}}H_{T_i}$ , implying that direct link quality is better than the uplink quality, then from (REF ) we note that $\\tau _{T_i,R_i}^D\\ge \\tau _{T_i,R_i}^{C^*}$ or $m_i=1$ for any feasible $t_e$ because D2D mode is more energy-efficient and provides higher throughput for the same amount of harvested energy.",
"Whereas if $G_{T_i,R_i}<\\rho _{_{T_i}}H_{T_i}$ and $\\exists \\,t_{e,i}^{\\text{th}}\\triangleq \\left\\lbrace t_e\\mathrel {}|\\mathrel {}\\tau _{T_i,R_i}^D=\\tau _{T_i,R_i}^{C^*}\\right\\rbrace $ , then it can be shown that for $t_e>t_{e,i}^{\\text{th}}$ , D2D mode is preferred by the $i$ th user pair over the cellular mode, and vice-versa.",
"This holds because if $\\exists \\, t_{e,i}^{\\text{th}}\\in \\left(0,1\\right)$ , then $\\forall t_e\\in \\left(t_{e,i}^{\\text{th}},1\\right)$ , $\\tau _{T_i,R_i}^D>\\tau _{T_i,R_i}^{C^*}$ due to strictly decreasing nature of $\\tau _{T_i,R_i}^{C^*}$ in $\\mathcal {F}_i$ , the rate of increase of $\\tau _{T_i,R_i}^{C^*}$ is lower than that of $\\tau _{T_i,R_i}^D,\\forall t_e>t_{e,i}^{\\text{th}}.$ The value of this threshold $t_{e,i}^{\\text{th}}$ implies that even when uplink quality is better than direct link between $T_i$ and $R_i$ , D2D is better mode than cellular for $t_e> t_{e,i}^{\\text{th}}$ because the resulting harvested energy makes it more spectrally-efficient as compared to the cellular mode involving redundant transmission during downlink IT."
],
[
"Jointly Global Optimization Algorithm",
"Using thresholds $\\lbrace t_{e,i}^{\\text{th}}\\rbrace _{_{i=1}}^{^{N/2}}$ , the optimal MS for each EH user pair can be decided based on TA $t_e$ for ET.",
"So, we divide the feasible range for $t_e$ into $\\frac{N}{2}+1$ subranges based on $\\frac{N}{2}$ thresholds $\\lbrace t_{e,i}^{\\text{th}}\\rbrace $ .",
"For each of these subranges, the optimal MS policy $\\lbrace m_i^*\\rbrace _{_{i=1}}^{^{N/2}}$ can be obtained using the discussion given in Section REF .",
"Further, using the concavity of $\\lbrace \\tau _{T_i,R_i}^D\\rbrace _{_{i=1}}^{^{N/2}}$ and $\\lbrace \\tau _{T_i,R_i}^{C^*}\\rbrace _{_{i=1}}^{^{N/2}}$ in TA, as proved in Section REF , we note that the sum-throughput $\\tau _{\\rm {S}}$ for optimal MS $\\lbrace m_i^*\\rbrace $ is concave in TA $t_e$ because it is a sum of $\\frac{N}{2}$ concave functions comprising of either $\\lbrace \\tau _{T_i,R_i}^D\\rbrace $ or $\\lbrace \\tau _{T_i,R_i}^{C^*}\\rbrace $ .",
"Hence, the global optimal MS and TA can be obtained by selecting the subrange that gives the maximum sum throughput.",
"We have summarized these steps of the proposed joint global optimization strategy in Algorithm REF .",
"[!h] Joint global optimal MS and TA to maximize $\\tau _{\\rm {S}}$ [1] Set $\\mathbb {S}=\\left\\lbrace \\left(T_i,\\,R_i\\right)\\mathrel {}|\\mathrel {} T_i, R_i\\in \\lbrace \\mathcal {U}_1,\\ldots \\mathcal {U}_{N}\\rbrace \\wedge 1\\le i\\le \\frac{N}{2}\\right\\rbrace $ ; channel and system parameters $H_{T_i}, H_{R_i}, G_{T_i,R_i},\\eta _{_{T_i}},\\rho _{_{T_i}},\\phi _{_{R_i}}$ for each user pair in $\\mathbb {S}$ , along with $\\theta ,P_0,\\sigma ^2$ ; and tolerances $\\epsilon ,\\xi $ Optimal MS $\\lbrace m_i^*\\rbrace $ and TA $t_e^*,\\lbrace t_{d,T_i}^*\\rbrace $ along with $\\tau _{\\rm {S}}^*$ Initialize $\\mathbb {C}=\\emptyset $ and define a very small positive quantity $\\epsilon \\approx 0$ $i \\in \\lbrace 1,2,\\dots ,\\frac{N}{2}\\rbrace $ $\\tau _{T_i,R_i}^D>\\tau _{T_i,R_i}^{C^*}$ for both $t_e=\\epsilon $ and $t_e=1-\\epsilon $ Set $t_{e,i}^{\\text{th}}=0$ Represents D2D mode $\\tau _{T_i,R_i}^D<\\tau _{T_i,R_i}^{C^*}$ for both $t_e=\\epsilon $ and $t_e=1-\\epsilon $ Set $t_{e,i}^{\\text{th}}=1,\\;\\mathbb {C}=\\mathbb {C}\\cup i$ Represents Cellular mode Set $t_{e,i}^{\\text{th}}=\\left\\lbrace t_e\\mathrel {}|\\mathrel {} \\tau _{T_i,R_i}^D=\\tau _{T_i,R_i}^{C^*}\\right\\rbrace ,\\;\\mathbb {C}=\\mathbb {C}\\cup i$ Sort $t_{e,i}^{\\text{th}}$ in descending to store values in $\\mathcal {V}$ and indexes in $\\mathcal {I}$ Set $k_1\\!=\\!\\!\\underset{1\\le i\\le N/2}{\\rm {argmin}}\\!\\left[\\!\\eta _{_{T_i}} H_{T_i} G_{T_i,R_i}\\!\\right]$ , $k_2\\!=\\!\\!\\underset{1\\le i\\le N/2}{\\rm {argmax}}\\!\\left[\\!\\eta _{_{T_i}} H_{T_i} G_{T_i,R_i}\\!\\right]$ Set $\\tau _1\\left(t_e\\right)=\\textstyle \\sum _{j=1}^{N/2} \\left(1-t_e\\right)\\,\\log _2\\Big (1+\\frac{\\theta \\eta _{_{T_j}} P_0 H_{T_j} G_{T_j,R_j}\\,t_e}{\\sigma ^2\\left(1-t_e\\right)}\\Big )$ $\\mid \\mathbb {C}\\mid =\\emptyset $ Represents all Nodes in D2D mode Set $\\text{lb}_1=t_{e,k_2}^{\\rm {D}^*}$ and $\\text{ub}_1=t_{e,k_1}^{\\rm {D}^*}$ by using (REF ) $\\mid \\mathbb {C}\\mid =\\frac{N}{2}$ Represents all Nodes in cellular mode Set $\\text{lb}_1=0$ and $\\text{ub}_1=t_{e,k_1}^{\\rm {D}^*}$ Set $\\tau _1\\left(t_e\\right)=\\textstyle \\sum _{j=1}^{N/2}t_{d,T_j}^*\\,\\log _2\\big (1+{P_0\\,\\phi _{_{R_j}} H_{R_j}}{\\sigma ^{-2}}\\big )$ Set $\\text{lb}_1=\\mathcal {V}_1$ and $\\text{ub}_1=t_{e,k_1}^{\\rm {D}^*}$ $i \\in \\lbrace 1,2,\\dots ,\\mid \\mathbb {C}\\mid \\rbrace $ Set ${\\tau _{i+1}} \\left(t_e\\right)={\\tau _i} \\left(t_e\\right)+\\tau _{T_{\\mathcal {I}_i},R_{\\mathcal {I}_i}}^{C^*}-\\tau _{T_{\\mathcal {I}_i},R_{\\mathcal {I}_i}}^D$ $i<\\mid \\mathbb {C}\\mid $ Set $\\text{lb}_{i+1}=\\mathcal {V}_{i+1}$ and $\\text{ub}_{i+1}=\\mathcal {V}_{i}$ Set $\\text{lb}_{i+1}=0$ and $\\text{lb}_{i+1}=\\mathcal {V}_{i}$ $i \\in \\left\\lbrace 1,2,\\dots , \\mid \\mathbb {C}\\mid +1 \\right\\rbrace $ Set $t_l=\\text{lb}_i$ and $t_u=\\text{ub}_i$ Set $t_p=t_u-0.618\\left(t_u-t_l\\right),t_q=t_l+0.618\\left(t_u-t_l\\right)$ $t_u-t_i>\\xi $ ${\\tau _{i}}\\left(t_p\\right)\\ge {\\tau _{i}}\\left(t_q\\right)$ Set $t_u=t_q$ , $t_q=t_p,t_p=t_u-0.618\\left(t_u-t_l\\right)$ Set $t_l=t_p$ , $t_p=t_q,t_q=t_l+0.618\\left(t_u-t_l\\right)$ Set $t_i^*=\\frac{t_u+t_l}{2}$ and $\\tau _{{\\rm {S}},i}^*=\\tau _i\\left(t_i^*\\right)$ $\\left(\\mid \\mathbb {C}\\mid =\\emptyset \\right)\\vee \\left(\\mid \\mathbb {C}\\mid =\\frac{N}{2}\\right)$ break Set $\\text{opt}=\\underset{1\\le i\\le \\mid \\mathbb {C}\\mid +1}{\\operatornamewithlimits{argmax}}\\;\\tau _{{\\rm {S}},i}^*$ , $\\tau _{\\rm {S}}^*=\\tau _{{\\rm {S}},\\text{opt}}^*$ , and $t_e^*=t_{\\text{opt}}^*$ $\\left(\\mid \\mathbb {C}\\mid =\\emptyset \\right)$ Set $m_i^*=1,\\;t_{d,T_i}^*=0,\\;\\forall i=1,2,\\ldots ,\\frac{N}{2}$ $\\left(\\mid \\mathbb {C}\\mid =\\frac{N}{2}\\right)$ $\\forall i, m_i^*\\hspace{-1.42262pt}=\\hspace{-1.42262pt}0$ and $t_{d,T_i}^*$ is obtained using $t_e\\hspace{-2.13394pt}=\\hspace{-2.13394pt}t_e^*$ in (REF ) $\\forall i\\!=\\!1,2,\\ldots ,\\frac{N}{2},$ set $m_i^*\\!=\\!",
"{\\left\\lbrace \\begin{array}{ll}1, & \\text{$i\\!=\\!\\mathcal {I}_{\\text{opt}+1},\\mathcal {I}_{\\text{opt}+2},\\ldots ,\\mathcal {I}_{\\mid \\mathbb {C}\\mid }$}\\\\0, & \\text{otherwise},\\end{array}\\right.",
"}$ $t_{d,T_i}^*={\\left\\lbrace \\begin{array}{ll}\\text{obtained by substituting $t_e^*$ in (\\ref {eq:opt-td})}, & \\text{$m_i^*=0$}\\\\0, \\qquad \\qquad \\text{ (i.e., D2D mode,) }& \\text{otherwise},\\end{array}\\right.",
"}$ In Algorithm REF after dividing the feasible $t_e$ range, optimal TA $t_e^*$ for each of the possible $\\frac{N}{2}+1$ MS scenarios is obtained using the Golden Section (GS) based one dimensional (1D) search with acceptable tolerance $\\xi \\ll 1$ (implemented in steps REF to REF ).",
"The upper and lower bounds for each search are either based on the thresholds $\\lbrace t_{e,i}^{\\text{th}}\\rbrace $ or on $\\lbrace t_{e,i}^{\\rm {D}^*}\\rbrace $ , as defined in (REF ) for pair $i$ having worst and best average link qualities, respectively.",
"So, we conclude that Algorithm REF returns the global optimal MS and TA along with maximum $\\tau _{\\rm {S}}$ after running GS-based 1D-search for at most $\\frac{N}{2}+1$ times."
],
[
"Numerical Results and Discussion",
"For generating numerical results we consider that $N$ nodes are deployed randomly following Poisson Point Process (PPP) over a square field of area $L\\times L$ m$^2$ with the HAP positioned at the center.",
"This field size $L$ ensures that the average received power for a given path-loss exponent $n$ and average fading parameter $H_{T_i}$ is higher than the minimum received power sensitivity of $-20$ dBm for practical RF-EH circuits [9].",
"So, as $n$ is varied from 2 to 5, maximum field size $L$ decreases from $23.4$ m to $4.4$ m. For user pairing we have considered that node $i$ pairs with node $N-i+1,\\forall i=\\lbrace 1,2,\\ldots , N/2\\rbrace $ .",
"The graphs in this section are obtained by plotting the average results for multiple random channel realizations and multiple random node deployments with unit average channel fading components $\\lbrace H_{T_i},H_{R_i},G_{T_i,R_i}\\rbrace $ and path-loss coefficient $p$ .",
"We have assumed $P_0\\!=\\!4$ W, $\\sigma ^2\\!=\\!-100$ dBm, $\\theta \\!=\\!0.8$ , $\\rho _{_{T_i}}\\!=\\!\\phi _{_{R_i}}\\!=\\!1,\\eta _{_{T_i}}\\!=\\!0.5,\\,\\forall \\, i,$ with tolerances $\\xi \\!=\\!10^{-3}$ and $\\epsilon \\!=\\!10^{-6}$ .",
"Figure: Throughput performance comparison of different schemes.Figure: Variation of optimal MS (D2D versus cellular) with L,nL,n.Figure: Variation of optimal TA t e * t_e^* for ET in different modes.Firstly, we investigate the throughput gain achieved with the help of proposed joint MS and TA optimization over the benchmark schemes that include fixed communication mode (all nodes are either in cellular or D2D mode) and uniform TA for each phase where $t_e=t_{d,T_i}=\\frac{1}{3}$ for cellular mode and $t_e=\\frac{1}{2}$ for D2D mode.",
"Results in Fig.",
"REF show that the joint optimization scheme provides significant gains over the fixed TA and MS schemes.",
"These gains which scale with increased system size $N$ , get enhanced with diminishing field size $L$ due to increased path-loss exponent $n$ for both 20 and 40 user systems.",
"Furthermore, the throughput performance of D2D mode for both optimal TA and fixed TA is much better than that for the corresponding cellular mode communication.",
"In fact, the performance of optimal TA with all nodes selecting D2D mode is very close to that of joint optimal strategy.",
"The reason for this can be observed from the results plotted in Figs.",
"REF and REF .",
"From Fig.",
"REF we observe that irrespective of field size $L$ , about $2/3$ fraction of the total users prefer D2D mode.",
"As the D2D mode involves direct IT, it is more spectrally-efficient and can allocate higher $t_e^*$ for ET as shown in Fig.",
"REF .",
"Since joint MS and TA involves both D2D and cellular modes, its $t_e^*$ lies between that of all-D2D and all-cellular scenarios.",
"We numerically quantify the average optimal uplink and downlink IT times in cellular mode.",
"Fig.",
"REF (a) shows that optimal uplink IT time is much higher than optimal downlink IT time due to significantly low link quality for uplink IT from energy constrained users.",
"On the contrary, downlink involves IT from energy-rich HAP.",
"Further, the IT times (of both uplink and downlink) for joint MS and TA are relatively lower than that for scenario where all nodes follow cellular mode because the former selects only the pairs that have better uplink and downlink qualities for cellular mode communication.",
"Figure: Approximation validationFrom Figs.",
"REF and REF , it was noted that optimal TA with all nodes selecting D2D performs very close to jointly optimal MS and TA.",
"So, in Fig.",
"REF (b) we have compared the variation of $\\tau _{\\rm {S}}$ achieved by optimal TA scheme with fixed D2D mode for all nodes against that achieved by considering tight analytical approximation $\\widehat{t_e^{\\rm {D}^*}}$ for $t_e^*$ as obtained by substituting the average channel gain for the RF-powered D2D IT link in (REF ).",
"So, $\\widehat{t_e^{\\rm {D}^*}}\\triangleq \\left[1-\\frac{\\mathcal {Y}_4\\mathrm {W}_0\\left(\\frac{\\mathcal {Y}_4-1}{\\exp (1)}\\right)}{\\mathrm {W}_0\\left(\\frac{\\mathcal {Y}_4-1}{\\exp (1)}\\right)-\\mathcal {Y}_4+1}\\right]^{-1}$ where $\\mathcal {Y}_4\\triangleq \\sum \\limits _{i=1}^{N/2}\\frac{2\\,\\mathcal {Y}_{3,i}}{N}$ .",
"Fig.",
"REF (b) shows that although the quality of approximation gets degraded with increasing $N$ , it is still very much acceptable as the average percentage error is always less than $1.5\\%$ .",
"Figure: Examples to give graphical insights on optimal MS strategy.Next we present graphical insights on optimal MS.",
"In Fig.",
"REF we have plotted two $N=10$ user deployments with $L=1$ unit and shown optimal MS in each case.",
"User pairing is same as discussed before, i.e., user 1 transmits to 10, 4 transmits to 7, and so on.",
"We note that when transmitter (Tx) and receiver (Rx) are placed almost opposite to each other with the HAP being in the center and far from the Tx, then cellular mode is preferred.",
"Otherwise, mostly ($\\approx 66\\%$ times) D2D is preferred.",
"Figure: Performance enhancement provided by joint optimal MS-TA.Finally via Fig.",
"REF , we show that the joint optimization scheme taking advantage of proximity can achieve spectral efficiency gains by optimal MS and can effectively solve the tradeoff between efficient ET and IT by optimal TA.",
"As a result it provides significant gain in terms of $\\tau _{\\rm {S}}$ over that achieved by benchmark scheme having uniform TA for all three phases (ET, uplink and downlink IT) with all nodes selecting cellular mode.",
"Fig.",
"REF also shows that higher gains are achieved when the average link qualities become poorer due to increased $n$ ."
],
[
"Conclusion",
"We have presented a novel system architecture and transmission protocol for efficient the RF-powered D2D communications.",
"To maximize the sum-throughput of RF-EH small cell OFDMA network, we have derived the joint global optimal MS and TA by resolving the underlying combinatorial issue.",
"Analytical insights on the impact of harvested energy on the optimal decision-making have been provided.",
"We have observed that the jointly optimal MS and TA can provide about $45\\%$ enhancement in achievable sum-throughput.",
"Lastly, we have showed that with our proposed joint MS and TA about $66\\%$ nodes follow D2D mode, and the optimal TA scheme with fixed D2D mode for all nodes very closely follows the sum-throughput performance of the jointly optimal scheme."
],
[
"Acknowledgments",
"This work was supported by the Department of Science and Technology under Grant SB/S3/EECE/0248/2014 along with the 2016 Raman Charpak and 2016-2017 IBM PhD Fellowship programs.",
"In addition, the views of Dr. G. C. Alexandropoulos expressed here are his own and do not represent Huawei's ones.",
"10 E. Hossain and M. Hasan, “5G cellular: Key enabling technologies and research challenges,” IEEE Instrum.",
"Meas.",
"Mag., vol.",
"18, no.",
"3, pp.",
"11–21, Jun.",
"2015.",
"D. Mishra et al., “Smart RF energy harvesting communications: Challenges and opportunities,” IEEE Commun.",
"Mag., vol.",
"53, no.",
"4, pp.",
"70–78, Apr.",
"2015.",
"H. Ju and R. Zhang, “Throughput maximization in wireless powered communication networks,” IEEE Trans.",
"Wireless Commun., vol.",
"13, no.",
"1, pp.",
"418–428, Jan. 2014.",
"H. Chen et al., “Harvest-then-cooperate: Wireless-powered cooperative communications,” IEEE Trans.",
"Signal Process., vol.",
"63, no.",
"7, pp.",
"1700–1711, Apr.",
"2015.",
"P. Grover and A. Sahai, “Shannon meets Tesla: Wireless information and power transfer,” in Proc.",
"IEEE ISIT, Austin, Jun.",
"2010, pp.",
"2363–2367.",
"A. H. Sakr and E. Hossain, “Cognitive and energy harvesting-based D2D communication in cellular networks: Stochastic geometry modeling and analysis,” IEEE Trans.",
"Commun., vol.",
"63, no.",
"5, pp.",
"1867–80, May 2015.",
"S. Gupta et al., “Energy harvesting aided device-to-device communication underlaying the cellular downlink,” IEEE Access, Aug. 2016.",
"R. Atat et al., “Energy harvesting-based D2D-assisted machine-type communications,” IEEE Trans.",
"Commun., vol.",
"65, no.",
"3, pp.",
"1289–1302, Mar.",
"2017.",
"D. Mishra and S. De, “i$^2$ RES: Integrated information relay and energy supply assisted RF harvesting communication,” IEEE Trans.",
"Commun., vol.",
"65, no.",
"3, pp.",
"1274–1288, Mar.",
"2017.",
"D. Feng et al., “Mode switching for energy-efficient device-to-device communications in cellular networks,” IEEE Trans.",
"Wireless Commun., vol.",
"14, no.",
"12, pp.",
"6993–7003, Dec. 2015.",
"D. D. Penda et al., “Energy efficient D2D communications in dynamic TDD systems,” IEEE Trans.",
"Commun., vol.",
"65, no.",
"3, pp.",
"1260–1273, Mar.",
"2017.",
"E. Boshkovska et al., “Practical non-linear energy harvesting model and resource allocation for SWIPT systems,” IEEE Commun.",
"Lett., vol.",
"19, no.",
"12, pp.",
"2082–2085, Dec. 2015.",
"S. Boyd and L. Vandenberghe, Convex Optimization.",
"Cambridge University Press, 2004.",
"E. W. Weisstein, “Lambert W-Function,” From MathWorld.",
"[Online].",
"Available: http://mathworld.wolfram.com/LambertW-Function.html"
]
] | 1709.01755 |
[
[
"Moduli of formal torsors"
],
[
"Abstract We construct the moduli stack of torsors over the formal punctured disk in characteristic p > 0 for a finite group isomorphic to the semidirect product of a p-group and a tame cyclic group.",
"We prove that the stack is a limit of separated Deligne-Mumford stacks with finite and universally injective transition maps."
],
[
"Introduction",
"The main subject of this paper is the moduli space of formal torsors, that is, $G$ -torsors (also called principal $G$ -bundles) over the formal punctured disk $\\operatorname{\\textup {Spec}}k((t))$ for a given finite group (or étale finite group scheme) $G$ and field $k$ .",
"More precisely, we are interested in a space over a field $k$ whose $k$ -points are $G$ -torsors over $\\operatorname{\\textup {Spec}}k((t))$ .",
"Since torsors may have non-trivial automorphisms, this space should actually be a stack in groupoids and it should not be confused with $\\operatorname{\\textup {B}}G=[\\operatorname{\\textup {Spec}}k((t))/G]$ , which is a stack defined over $k((t))$ .",
"The case where the characteristic of $k$ and the order of $G$ are coprime is called tame and the other case is called wild.",
"The two cases are strikingly different: in the tame case the moduli space is expected to be zero-dimensional, while in the wild case it is expected to be infinite-dimensional.",
"An important work on this subject is Harbater's one [5].",
"He constructed the coarse moduli space for pointed formal torsors when $k$ is an algebraically closed field of characteristic $p>0$ and $G$ is a $p$ -group.",
"This coarse moduli space is isomorphic to the inductive limit $\\varinjlim _{n}\\mathbb {A}^{n}$ of affine spaces such that the transition map $\\mathbb {A}^{n}\\rightarrow \\mathbb {A}^{n+1}$ is the composition of the closed embedding $\\mathbb {A}^{n}\\hookrightarrow \\mathbb {A}^{n+1}$ and the Frobenius map of $\\mathbb {A}^{n+1}$ .",
"In particular it is neither a scheme nor an algebraic space, but an ind-scheme.",
"Some of the differences between Harbater's space and our space are explained in REF .",
"As a consequence Harbater shows that there is a bijective correspondence between $G$ -torsors over the affine line $\\mathbb {A}^{1}$ and over $\\operatorname{\\textup {Spec}}k((t))$ .",
"In this direction an important development has been given by Gabber and Katz in [7].",
"Anyway, besides those works, the moduli problem of formal torsors have not been developed since then, as far as the authors know.",
"In recent works [14], [13] of the second author, an unexpected relation of this moduli space to singularities of algebraic varieties was discovered.",
"He has formulated a conjectural generalization of the motivic McKay correspondence by Batyrev [3] and Denef–Loeser [4] to arbitrary characteristics, which relates a motivic integral over the moduli space of formal torsors with a stringy invariant of wild quotient singularities.",
"The motivic integral can be viewed as the motivic counterpart of mass formulas for local Galois representations, see [11], [12].",
"The conjecture has been proved only for the case of the cyclic group of order equal to the characteristic.",
"The first and largest problem for other groups is the construction of the moduli space.",
"From the arithmetic viewpoint, the case where $k$ is finite is the most interesting, which motivates us to remove the “algebraically closed” assumption in Harbater's work.",
"The main resut of this paper is to construct the moduli stack of formal torsors and to show that it is a limit of Deligne-Mumford stacks (DM stacks for short) when $k$ is an arbitrary field of characteristic $p>0$ and $G$ is an étale group scheme over $k$ which is geometrically the semidirect product $H\\rtimes C$ of a $p$ -group $H$ and a cyclic group $C$ of order coprime with $p$ .",
"This is an important step towards the general case, because, if $k$ is algebraically closed, then connected $G$ -torsors over $\\operatorname{\\textup {Spec}}k((t))$ (or equivalently Galois extensions of $k((t))$ with group $G$ ) exist only for semidirect products as before.",
"Moreover any $G^{\\prime }$ -torsor for a general $G^{\\prime }$ is induced by some connected $G$ -torsor along an embedding $G\\hookrightarrow G^{\\prime }$ .",
"To give the precise statement of the result, we introduce some notation.",
"We denote by $\\Delta _{G}$ the category fibered in groupoids over the category of affine $k$ -schemes such that for a $k$ -algebra $B$ , $\\Delta _{G}(\\operatorname{\\textup {Spec}}B)$ is the category of $G$ -torsors over $\\operatorname{\\textup {Spec}}B((t))$ .",
"This category turns out to be too large and we define a certain subcategory $\\Delta _{G}^{*}$ (see REF for the definition), although we do not lose anything by this when $B$ is a field.",
"The following is the precise statement of the main result: Theorem A Let $k$ be a field of positive characteristic $p$ and $G$ be a finite and étale group scheme over $k$ such that $G\\times _{k}\\overline{k}$ is a semidirect product $H\\rtimes C$ of a $p$ -group $H$ and a cyclic group $C$ of rank coprime with $p$ .",
"Then there exists a direct system $\\mathcal {X}_{*}$ of separated DM stacks with finite and universally injective transition maps, with a direct system of finite and étale atlases (see REF for the definition) $X_{n}\\longrightarrow \\mathcal {X}_{n}$ from affine schemes and with an isomorphism $\\varinjlim _{n}\\mathcal {X}_{n}\\simeq \\Delta _{G}^{*}$ .",
"If $G$ is a constant $p$ -group then $\\Delta _{G}=\\Delta _{G}^{*}$ and the stacks $\\mathcal {X}_{n}$ are smooth and integral.",
"More precisely there is a strictly increasing sequence $v\\colon \\mathbb {N}\\longrightarrow \\mathbb {N}$ such that $X_{n}=\\mathbb {A}^{v_{n}}$ , the maps $\\mathbb {A}^{v_{n}}\\longrightarrow \\mathcal {X}_{n}$ are finite and étale of degree $\\sharp $ G and the transition maps $\\mathbb {A}^{v_{n}}\\longrightarrow \\mathbb {A}^{v_{n+1}}$ are composition of the inclusion $\\mathbb {A}^{v_{n}}\\longrightarrow \\mathbb {A}^{v_{n+1}}$ and the Frobenius $\\mathbb {A}^{v_{n+1}}\\longrightarrow \\mathbb {A}^{v_{n+1}}$ .",
"If $G$ is an abelian constant group of order $p^{r}$ then we also have an equivalence $\\left(\\varinjlim _{n}\\mathbb {A}^{v_{n}}\\right)\\times \\operatorname{\\textup {B}}G\\simeq \\Delta _{G}$ and the map from $\\varinjlim _{n}\\mathbb {A}^{v_{n}}$ to the sheaf of isomorphism classes of $\\Delta _{G}$ , which is nothing but the rigidification $\\Delta _{G}\\mathbin {\\!\\!",
"}G$ , is an isomorphism.",
"As a consequence of this theorem the fibered category $\\Delta _{G}^{*}$ is a stack.",
"We now explain the outline of our construction.",
"We first consider the case of a constant group scheme of order $p^{r}$ .",
"Following Harbater's strategy, we prove the theorem in this case by induction.",
"The initial step of induction is the case $G\\simeq (\\mathbb {Z}/p\\mathbb {Z})^{n}$ , where we can explicitly describe the moduli space by the Artin-Schreier theory.",
"Since a general $p$ -group has a central subgroup $H\\subset G$ isomorphic to $(\\mathbb {Z}/p\\mathbb {Z})^{n}$ , we have a natural map $\\Delta _{G}\\longrightarrow \\Delta _{G/H}$ , enabling the induction to work.",
"Next we consider the case of the group scheme $\\mu _{n}$ of $n$ -th roots of unity with $n$ coprime to $p$ .",
"In this case, we have the following explicit description of $\\Delta _{G}^{*}$ also in characteristic zero: Theorem B Let $k$ be a field and $n\\in \\mathbb {N}$ such that $n\\in k^{*}$ .",
"We have an equivalence $\\bigsqcup _{q=0}^{n-1}\\operatorname{\\textup {B}}(\\mu _{n})\\longrightarrow \\Delta _{\\mu _{n}}^{*}$ where the map $\\operatorname{\\textup {B}}(\\mu _{n})\\longrightarrow \\Delta _{\\mu _{n}}^{*}$ in the index $q$ maps the trivial $\\mu _{n}$ -torsor to the $\\mu _{n}$ -torsor $\\frac{k((t))[Y]}{(Y^{n}-t^{q})}\\in \\Delta _{\\mu _{n}}(k)$ .",
"When $G$ is a constant group of the form $H\\rtimes C$ and $k$ contains all $n$ -th roots of unity, then $C\\simeq \\mu _{n}$ and there exists a map $\\Delta _{G}^{*}\\longrightarrow \\Delta _{\\mu _{n}}^{*}$ .",
"Using Theorem REF for $p$ -groups, we show that the fiber products $\\Delta _{G}^{*}\\times _{\\Delta _{\\mu _{n}}^{*}}\\operatorname{\\textup {Spec}}k$ with respect to $n$ maps $\\operatorname{\\textup {Spec}}k\\rightarrow \\Delta _{\\mu _{n}}^{*}$ induced from the equivalence in Theorem REF are limits of DM stacks.",
"Finally, to conclude that $\\Delta _{G}^{*}$ itself is a limit of DM stacks and also to reduce the problem to the case of a constant group, we need a lemma (Lemma REF ) roughly saying that if $\\mathcal {Y}$ is a $G$ -torsor over a stack $\\mathcal {X}$ for a constant group $G$ and $\\mathcal {Y}$ is a limit of DM stacks, then $\\mathcal {X}$ is also a limit of DM stacks.",
"This innocent-looking lemma turns out to be rather hard to prove and we will make full use of 2-categories.",
"The paper is organized as follows.",
"In Section we set up notation and terminology frequently used in the paper.",
"In Section we collect basic results on power series rings, finite and universally injective morphisms and torsors.",
"In Section , after introducing a few notions and proving a few easy results, the rest of the section is devoted to the proof of the lemma mentioned above.",
"Section is the main body of the paper, where we prove Theorems REF and REF .",
"Lastly we include two Appendices about limits of fibered categories and rigidification.",
"We thank Ted Chinburg, Ofer Gabber, David Harbater, Florian Pop, Shuji Saito, Takeshi Saito, Melanie Matchett Wood and Lei Zhang for stimulating discussion and helpful information.",
"The second author was supported by JSPS KAKENHI Grant Numbers JP15K17510 and JP16H06337.",
"Parts of this work were done when the first author was staying as the Osaka University and when the second author was staying at the Max Planck Institute for Mathematics and the Institut des Hautes Ãtudes Scientifiques.",
"We thank hospitality of these institutions."
],
[
"Notation and terminology",
"Given a ring $B$ we denote by $B((t))$ the ring of Laurent series $\\sum _{i=r}^{\\infty }b_{i}t^{i}$ with $b_{i}\\in B$ and $r\\in \\mathbb {Z}$ , that is, the localization $B[[t]]_{t}=B[[t]][t^{-1}]$ of the formal power series ring $B[[t]]$ with coefficients in $B$ .",
"This should not be confused with the fraction field of $B[[t]]$ (when $B$ is a domain).",
"By a fibered category over a ring $B$ we always mean a category fibered in groupoids over the category $\\operatorname{\\textup {Aff}}/B$ of affine $B$ -schemes.",
"Recall that a finite map between fibered categories is by definition affine and therefore represented by finite maps of algebraic spaces.",
"By a vector bundle on a scheme $X$ we always mean a locally free sheaf of finite rank.",
"A vector bundle on a ring $B$ is a vector bundle on $\\operatorname{\\textup {Spec}}B$ or, before sheafification, a projective $B$ -module of finite type.",
"If $\\mathcal {C}$ is a category, $X\\colon \\mathcal {C}\\longrightarrow (\\textup {groups})$ is a functor of groups and $S$ is a set we denote by $X^{(S)}\\colon \\mathcal {C}\\longrightarrow (\\textup {groups})$ the functor so defined: if $c\\in \\mathcal {C}$ then $X^{(S)}(c)$ is the set of functions $u\\colon S\\longrightarrow X(c)$ such that $\\lbrace s\\in S\\ | \\ u(s)\\ne 1_{X(c)}\\rbrace $ is finite.",
"We recall that for a morphism $f\\colon \\mathcal {X}\\rightarrow \\mathcal {Y}$ of fibered categories over a ring $B$ , $f$ is faithful (resp.",
"fully faithful, an equivalence) if and only if for every affine $B$ -scheme $U$ , $f_{U}\\colon \\mathcal {X}(U)\\rightarrow \\mathcal {Y}(U)$ is so (see [9]).",
"A morphism of fibered categories is called a monomorphism if it is fully faithful.",
"We also note that every representable (by algebraic spaces) morphism of stacks is faithful ([9]).",
"A map $f\\colon \\mathcal {Y}\\longrightarrow \\mathcal {X}$ between fibered categories over $\\operatorname{\\textup {Aff}}/k$ is a torsor under a sheaf of groups $\\mathcal {G}$ over $\\operatorname{\\textup {Aff}}/k$ if it is given a 2-Cartesian diagram $ \\begin{tikzpicture}[xscale=1.5,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {Y}}; \\node (A0_1) at (1, 0) {\\operatorname{\\textup {Spec}}k}; \\node (A1_0) at (0, 1) {\\mathcal {X}}; \\node (A1_1) at (1, 1) {\\operatorname{\\textup {B}}\\mathcal {G}}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); \\end{tikzpicture} $ By a stack we mean a stack over the category $\\operatorname{\\textup {Aff}}$ of affine schemes with respect to the fppf topology, unless a different site is specified.",
"We often abbreviate “Deligne-Mumford stack” to “DM stack”."
],
[
"Preliminaries",
"In this section we collect some general results that will be used later."
],
[
"Some results on power series",
"Lemma 2.1 Let $C$ be a ring, $J\\subseteq C$ be an ideal and assume that $C$ is $J$ -adically complete.",
"If $U\\subseteq \\operatorname{\\textup {Spec}}C$ is an open subset containing $\\operatorname{\\textup {Spec}}(C/J)$ then $U=\\operatorname{\\textup {Spec}}C$ .",
"Let $U=\\operatorname{\\textup {Spec}}C-V(I)$ , where $I\\subseteq C$ is an ideal.",
"The condition $\\operatorname{\\textup {Spec}}(C/J)\\subseteq U$ means that $I+J=C$ .",
"In particular there exists $g\\in I$ and $j\\in J$ such that $g=1+j$ .",
"Since $j$ is nilpotent in all the rings $C/J^{n}$ we see that $g$ is invertible in all the rings $C/J^{n}$ , which easily implies that $g$ is invertible in $C$ .",
"Thus $I=C$ .",
"Lemma 2.2 Let $R$ be a ring, $X$ be a quasi-affine scheme formally étale over $R$ , $C$ be an $R$ -algebra and $J$ be an ideal such that $C$ is $J$ -adically complete.",
"Then the projection $C\\longrightarrow C/J^{n}$ induces a bijection $X(C)\\longrightarrow X(C/J^{n})\\text{ for all }n\\in \\mathbb {N}$ Since $X$ is formally étale the projections $C/J^{m}\\longrightarrow C/J^{n}$ for $m\\ge n$ induce bijections $X(C/J^{m})\\longrightarrow X(C/J^{n})$ Thus it is enough to prove that if $Y$ is any quasi-affine scheme over $R$ then the natural map $\\alpha _{Y}\\colon Y(C)\\longrightarrow \\varprojlim _{n\\in N}Y(C/J^{n})$ is bijective.",
"This is clear when $Y$ is affine.",
"Let $B=\\operatorname{\\textup {H}}^{0}(\\mathcal {O}_{Y})$ , so that $Y$ is a quasi-compact open subset of $U=\\operatorname{\\textup {Spec}}B$ .",
"The fact that $\\alpha _{U}$ is an isomorphism tells us that $\\alpha _{Y}$ is injective.",
"To see that it is surjective we have to show that if $B\\longrightarrow C$ is a map such that all $\\operatorname{\\textup {Spec}}C/J^{n}\\longrightarrow \\operatorname{\\textup {Spec}}B$ factors through $Y$ then also $\\phi \\colon \\operatorname{\\textup {Spec}}C\\longrightarrow \\operatorname{\\textup {Spec}}B$ factors through $Y$ .",
"But the first condition implies that $\\phi ^{-1}(Y)$ is an open subset of $\\operatorname{\\textup {Spec}}C$ containing $\\operatorname{\\textup {Spec}}C/J$ .",
"The equality $\\phi ^{-1}(Y)=\\operatorname{\\textup {Spec}}C$ then follows from REF .",
"Corollary 2.3 Let $B$ be a ring, $f\\colon Y\\longrightarrow \\operatorname{\\textup {Spec}}B[[t]]$ an étale map, $\\xi \\colon \\operatorname{\\textup {Spec}}L\\longrightarrow \\operatorname{\\textup {Spec}}B$ a geometric point and assume that the geometric point $\\operatorname{\\textup {Spec}}L\\longrightarrow \\operatorname{\\textup {Spec}}B\\longrightarrow \\operatorname{\\textup {Spec}}B[[t]]$ is in the image on $f$ .",
"Then there exists an étale neighborhood $\\operatorname{\\textup {Spec}}B^{\\prime }\\longrightarrow \\operatorname{\\textup {Spec}}B$ of $\\xi $ such that $\\operatorname{\\textup {Spec}}B^{\\prime }[[t]]\\longrightarrow \\operatorname{\\textup {Spec}}B[[t]]$ factors through $Y\\longrightarrow \\operatorname{\\textup {Spec}}B[[t]]$ .",
"We can assume $Y$ affine, say $Y=\\operatorname{\\textup {Spec}}C$ .",
"Set $B^{\\prime }=C/tC$ , so that the induced map $f_{0}\\colon Y_{0}=\\operatorname{\\textup {Spec}}B^{\\prime }\\longrightarrow \\operatorname{\\textup {Spec}}B$ is étale.",
"By hypothesis the geometric point $\\operatorname{\\textup {Spec}}L\\longrightarrow \\operatorname{\\textup {Spec}}B$ is in the image of $f_{0}$ and therefore factors through $f_{0}$ .",
"Moreover the map $Y_{0}\\longrightarrow Y$ gives an element of $Y(B^{\\prime })$ which, by REF , lifts to an element of $Y(B^{\\prime }[[t]])$ , that is a factorization of $\\operatorname{\\textup {Spec}}B^{\\prime }[[t]]\\longrightarrow \\operatorname{\\textup {Spec}}B[[t]]$ through $Y\\longrightarrow \\operatorname{\\textup {Spec}}B[[t]]$ .",
"Lemma 2.4 Let $R$ be a ring, $S$ be an $R$ -algebra and consider the map $\\omega _{S/R}\\colon R[[t]]\\otimes _{R}S\\rightarrow S[[t]]$ The image of $\\omega _{S/R}$ is the subring of $S[[t]]$ of series $\\sum s_{n}t^{n}$ such that there exists a finitely generated $R$ submodule $M\\subseteq S$ with $s_{n}\\in M$ for all $n\\in \\mathbb {N}$ .",
"If any finitely generated $R$ submodule of $S$ is contained in a finitely presented $R$ submodule of $S$ then $\\omega _{S/R}$ is injective.",
"The claim about the image of $\\omega _{S/R}$ is easy.",
"Given an $R$ -module $M$ we define $M[[t]]$ as the $R$ -module $M^{\\mathbb {N}}$ .",
"Its elements are thought of as series $\\sum _{n}m_{n}t^{n}$ and $M[[t]]$ has a natural structure of $R[[t]]$ -module.",
"This association extends to a functor $\\operatorname{\\textup {Mod}}A\\rightarrow \\operatorname{\\textup {Mod}}A[[t]]$ which is easily seen to be exact.",
"Moreover there is a natural map $\\omega _{M/R}\\colon R[[t]]\\otimes _{R}M\\rightarrow M[[t]]$ Since both functors are right exact and $\\omega _{M/R}$ is an isomorphism if $M$ is a free $R$ -module of finite rank, we can conclude that $\\omega _{M/R}$ is an isomorphism if $M$ is a finitely presented $R$ -module.",
"Let $\\mathcal {P}$ be the set of finitely presented $R$ submodules of $S$ .",
"By hypothesis this is a filtered set.",
"Passing to the limit we see that the map $\\omega _{S/R}\\colon R[[t]]\\otimes _{R}S\\simeq \\varinjlim _{M\\in \\mathcal {P}}(R[[t]]\\otimes _{R}M)\\longrightarrow \\varinjlim _{M\\in \\mathcal {P}}M[[t]]=\\bigcup _{M\\in \\mathcal {P}}M[[t]]\\subseteq S[[t]]$ is injective.",
"Lemma 2.5 Let $N$ be a finite set and denote by $\\underline{N}\\colon \\operatorname{\\textup {Aff}}/\\mathbb {Z}\\longrightarrow (\\textup {Sets})$ the associated constant sheaf.",
"Then the maps $\\underline{N}(B)\\longrightarrow \\underline{N}(B[[t]])\\longrightarrow \\underline{N}(B((t)))$ are bijective.",
"In other words if $B$ is a ring and $B((t))\\simeq C_{1}\\times \\cdots \\times C_{l}$ (resp.",
"$B[[t]]=C_{1}\\times \\cdots \\times C_{l}$ ) then $B\\simeq B_{1}\\times \\cdots \\times B_{l}$ and $C_{j}=B_{j}((t))$ (resp.",
"$C_{j}=B_{j}[[t]]$ ).",
"Notice that $\\underline{N}$ is an affine scheme étale over $\\operatorname{\\textup {Spec}}\\mathbb {Z}$ .",
"Since $A=B[[t]]$ is $t$ -adically complete we obtain that $\\underline{N}(B[[t]])\\longrightarrow \\underline{N}(B[[t]]/tB[[t]])$ is bijective thanks to REF .",
"Since $B\\longrightarrow B[[t]]/tB[[t]]$ is an isomorphism we can conclude that $\\underline{N}(B)\\longrightarrow \\underline{N}(B[[t]])$ is bijective.",
"Let $n$ be the cardinality of $N$ and $C$ be a ring.",
"An element of $\\underline{N}(C)$ is a decomposition of $\\operatorname{\\textup {Spec}}C$ into $n$ -disjoint open and closed subsets.",
"In particular if $n=2$ then $\\underline{N}(C)$ is the set of open and closed subsets of $\\operatorname{\\textup {Spec}}C$ .",
"Taking this into account it is easy to reduce the problem to the case $n=2$ .",
"In this case another way to describe $\\underline{N}$ is $\\underline{N}=\\operatorname{\\textup {Spec}}\\mathbb {Z}[x]/(x^{2}-x)$ , so that $\\underline{N}(C)$ can be identified with the set of idempotents of $C$ .",
"Consider the map $\\alpha _{B}\\colon \\underline{N}(B[[t]])\\longrightarrow \\underline{N}(B((t)))$ , which is injective since $B[[t]]\\longrightarrow B((t))$ is so.",
"If, by contradiction, $\\alpha _{B}$ is not surjective, we can define $k>0$ as the minimum positive number for which there exist a ring $B$ and $a\\in B[[t]]$ such that $a/t^{k}\\in \\underline{N}(B((t)))$ and $a/t^{k}\\notin B[[t]]$ .",
"Let $B,a$ as before and set $a_{0}=a(0)$ .",
"It is easy to check that $a_{0}^{2}=0$ in $B$ .",
"Set $C=B/\\langle a_{0}\\rangle $ .",
"By REF we have that $B[[t]]/a_{0}B[[t]]=C[[t]]$ and that $B((t))/a_{0}B((t))=C((t))$ .",
"Thus we have a commutative diagram $ \\begin{tikzpicture}[xscale=2.7,yscale=-1.2] \\node (A0_0) at (0, 0) {\\underline{N}(B[[t]])}; \\node (A0_1) at (1, 0) {\\underline{N}(B((t)))}; \\node (A1_0) at (0, 1) {\\underline{N}(C[[t]])}; \\node (A1_1) at (1, 1) {\\underline{N}(C((t)))}; (A0_0) edge [->]node [auto] {\\scriptstyle {\\alpha _B}} (A0_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); (A0_1) edge [->]node [auto] {\\scriptstyle {\\beta }} (A1_1); (A1_0) edge [->]node [auto] {\\scriptstyle {\\alpha _C}} (A1_1); \\end{tikzpicture} $ in which the vertical maps are bijective, since the topological space of a spectrum does not change modding out by a nilpotent.",
"By construction $\\beta (a/t^{k})=a^{\\prime }/t^{k-1}$ where $a^{\\prime }=(a-a_{0})/t$ .",
"By minimality of $k$ we must have that $a^{\\prime }/t^{k-1}\\in C[[t]]$ .",
"Since the vertical maps in the above diagram are bijective we can conclude that also $a/t^{k}\\in B[[t]]$ , a contradiction.",
"Lemma 2.6 Let $M,N$ be vector bundles on $B((t))$ .",
"Then the functor $\\operatorname{\\underline{\\textup {Hom}}}_{B((t))}(M,N)\\colon \\operatorname{\\textup {Aff}}/B\\longrightarrow (\\textup {Sets}),\\ C\\longmapsto \\operatorname{\\textup {Hom}}_{C((t))}(M\\otimes C((t)),N\\otimes C((t)))$ is a sheaf in the fpqc topology.",
"Set $H=\\operatorname{\\textup {Hom}}_{B((t))}(M,N)$ , which is a vector bundle over $B((t))$ .",
"Moreover if $C$ is a $B$ -algebra we have $H\\otimes _{B((t))}C((t))\\simeq \\operatorname{\\underline{\\textup {Hom}}}_{B((t))}(M,N)(C)$ because $M$ and $N$ are vector bundles.",
"By REF we have to prove descent on coverings indexed by a finite set and, by REF , it is enough to consider a faithfully flat map $B\\longrightarrow C$ .",
"If $i_{1},i_{2}\\colon C\\longrightarrow C\\otimes _{B}C$ are the two inclusions, descent corresponds to the exactness of the sequence $0\\longrightarrow H\\longrightarrow C((t))\\otimes H{(i_{1}-i_{2})\\otimes \\textup {id}_{H}}(C\\otimes _{B}C)((t))\\otimes H$ Since this sequence is obtained applying $\\otimes _{B((t))}H$ to the exact sequence $0\\longrightarrow B((t))\\longrightarrow C((t)){i_{1}-i_{2}}(C\\otimes _{B}C)((t))$ and $H$ is flat we get the result."
],
[
"Finite and universally injective morphisms",
"Definition 2.7 A map $\\mathcal {X}\\longrightarrow \\mathcal {Y}$ between algebraic stacks is universally injective (resp.",
"universally bijective, a universal homeomorphism) if for all maps $\\mathcal {Y}^{\\prime }\\longrightarrow \\mathcal {Y}$ from an algebraic stack the map $|\\mathcal {X}\\times _{\\mathcal {Y}}\\mathcal {Y}^{\\prime }|\\longrightarrow |\\mathcal {Y}^{\\prime }|$ on topological spaces is injective (resp.",
"bijective, an homeomorphism).",
"Remark 2.8 In order to show that a map $\\mathcal {X}\\longrightarrow \\mathcal {Y}$ is universally injective (resp.",
"universally bijective, a universal homeomorphism) it is enough to test on maps $\\mathcal {Y}^{\\prime }\\longrightarrow \\mathcal {Y}$ where $\\mathcal {Y}^{\\prime }$ is an affine scheme.",
"Indeed injectivity and surjectivity can be tested on the geometric fibers.",
"Moreover if $\\bigsqcup _{i}Y_{i}\\longrightarrow \\mathcal {Y}$ is a smooth surjective map and the $Y_{i}$ are affine then $|\\mathcal {X}|\\longrightarrow |\\mathcal {Y}|$ is open if $|\\mathcal {X}\\times _{\\mathcal {Y}}Y_{i}|\\longrightarrow |Y_{i}|$ is open for all $i$ .",
"In particular if $\\mathcal {X}\\longrightarrow \\mathcal {Y}$ is representable then it is universally injective (resp.",
"bijective, a universal homeomorphism) if and only if it is represented by map of algebraic spaces which are universally injective (resp.",
"universally bijective, universal homeomorphisms) in the usual sense.",
"Proposition 2.9 Let $f\\colon \\mathcal {X}\\longrightarrow \\mathcal {Y}$ be a map of algebraic stacks.",
"Then $f$ is finite and universally injective if and only if it is a composition of a finite universal homeomorphism and a closed immersion.",
"More precisely, if $\\mathcal {I}=\\operatorname{\\textup {Ker}}(\\mathcal {O}_{\\mathcal {Y}}\\longrightarrow f_{*}\\mathcal {O}_{\\mathcal {X}})$ , then $\\mathcal {X}\\longrightarrow \\operatorname{\\textup {Spec}}(\\mathcal {O}_{\\mathcal {Y}}/\\mathcal {I})$ is finite and a universal homeomorphism.",
"The if part in the statement is clear.",
"So assume that $f$ is finite and universally injective and consider the factorization $\\mathcal {X}{g}\\mathcal {Z}=\\operatorname{\\textup {Spec}}(\\mathcal {O}_{\\mathcal {Y}}/\\mathcal {I}){h}\\mathcal {Y}$ .",
"Since $f$ is finite, the map $g$ is finite and surjective.",
"Since $h$ is a monomorphism, given a map $U\\longrightarrow \\mathcal {Z}$ from a scheme we have that $\\mathcal {X}\\times _{\\mathcal {Z}}U\\longrightarrow \\mathcal {X}\\times _{\\mathcal {Y}}U$ is an isomorphism, which implies that $g$ is also universally injective as required.",
"Remark 2.10 The following properties of morphisms of schemes are stable by base change and fpqc local on the base: finite, closed immersion, universally injective, surjective and universal homeomorphism (see [9]).",
"In particular for representable maps of algebraic stacks those properties can be checked on an atlas.",
"Remark 2.11 Let $f\\colon \\mathcal {S}^{\\prime }\\longrightarrow \\mathcal {S}$ be a map of algebraic stacks, $\\mathcal {U},\\mathcal {V}$ and $\\mathcal {U}^{\\prime },\\mathcal {V}^{\\prime }$ algebraic stacks with a map to $\\mathcal {S}$ and $\\mathcal {S}^{\\prime }$ respectively and $u\\colon \\mathcal {U}^{\\prime }\\longrightarrow \\mathcal {U}$ , $v\\colon \\mathcal {V}^{\\prime }\\longrightarrow \\mathcal {V}$ be $\\mathcal {S}$ -maps.",
"If $f,u,v$ are finite and universally injective then so is the induced map $\\mathcal {U}^{\\prime }\\times _{\\mathcal {S}^{\\prime }}\\mathcal {V}^{\\prime }\\longrightarrow \\mathcal {U}\\times _{\\mathcal {S}}\\mathcal {V}$ .",
"The map $(\\mathcal {U}\\times _{\\mathcal {S}}\\times \\mathcal {V})\\times _{\\mathcal {S}}\\mathcal {S}^{\\prime }\\longrightarrow \\mathcal {U}\\times _{\\mathcal {S}}\\mathcal {V}$ is finite and universally injective.",
"The map $\\mathcal {U}^{\\prime }\\longrightarrow \\mathcal {U}\\times _{\\mathcal {S}}\\mathcal {S}^{\\prime }$ is also finite and universally injective because $\\mathcal {U}\\times _{\\mathcal {S}}\\mathcal {S}^{\\prime }\\longrightarrow \\mathcal {U}$ and $\\mathcal {U}^{\\prime }\\longrightarrow \\mathcal {U}$ are so (use [9] for the universal injectivity).",
"Thus we can assume $\\mathcal {S}=\\mathcal {S}^{\\prime }$ and $f=\\textup {id}$ .",
"In this case it is enough to use the factorization $\\mathcal {U}^{\\prime }\\times _{\\mathcal {S}}\\mathcal {V}^{\\prime }\\longrightarrow \\mathcal {U}\\times _{\\mathcal {S}}\\mathcal {V}^{\\prime }\\longrightarrow \\mathcal {U}\\times _{\\mathcal {S}}\\mathcal {V}$ ."
],
[
"Some results on torsors",
"In what follows, actions of groups (or sheaves of groups) are supposed to be right actions.",
"Recall that for a sheaf $\\mathcal {G}$ of groups on a site $\\mathcal {C}$ , $\\operatorname{\\textup {B}}\\mathcal {G}$ denotes the category of $\\mathcal {G}$ -torsors over objects of $\\mathcal {C}$ , and that given a map $\\mathcal {G}\\rightarrow \\mathcal {H}$ of sheaves of groups, then there exists a functor $\\operatorname{\\textup {B}}\\mathcal {G}\\rightarrow \\operatorname{\\textup {B}}\\mathcal {H}$ sending a $\\mathcal {G}$ -torsor $\\mathcal {P}$ to the $\\mathcal {H}$ -torsor $(\\mathcal {P}\\times \\mathcal {H})/\\mathcal {G}$ .",
"Lemma 2.12 Let $\\mathcal {G}$ be a sheaf of groups on a site $\\mathcal {C}$ and $\\mathcal {H}$ be a sheaf of subgroups of the center $Z(\\mathcal {G})$ .",
"Then $\\mathcal {H}$ is normal in $\\mathcal {G}$ , the map $\\mu \\colon \\mathcal {G}\\times \\mathcal {H}\\longrightarrow \\mathcal {G}$ restriction of the multiplication is a morphism of groups and the first diagram $ \\begin{tikzpicture}[xscale=2.1,yscale=-1.2] \\node (A0_0) at (0, 0) {\\operatorname{\\textup {B}}(\\mathcal {G}\\times \\mathcal {H})}; \\node (A0_1) at (1, 0) {\\operatorname{\\textup {B}}(\\mathcal {G})}; \\node (A0_2) at (2, 0) {\\mathcal {G}\\times \\mathcal {H}}; \\node (A0_3) at (3, 0) {\\mathcal {G}}; \\node (A1_0) at (0, 1) {\\operatorname{\\textup {B}}(\\mathcal {G})}; \\node (A1_1) at (1, 1) {\\operatorname{\\textup {B}}(\\mathcal {G}/\\mathcal {H})}; \\node (A1_2) at (2, 1) {\\mathcal {G}}; \\node (A1_3) at (3, 1) {\\mathcal {G}/\\mathcal {H}}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_3) edge [->]node [auto] {\\scriptstyle {}} (A1_3); (A0_2) edge [->]node [auto] {\\scriptstyle {\\operatorname{\\textup {pr}}_1}} (A1_2); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); (A0_2) edge [->]node [auto] {\\scriptstyle {\\mu }} (A0_3); (A1_2) edge [->]node [auto] {\\scriptstyle {}} (A1_3); \\end{tikzpicture} $ induced by the second one is 2-Cartesian.",
"A quasi-inverse $\\operatorname{\\textup {B}}(\\mathcal {G})\\times _{\\operatorname{\\textup {B}}(\\mathcal {G}/\\mathcal {H})}\\operatorname{\\textup {B}}(\\mathcal {G})\\longrightarrow \\operatorname{\\textup {B}}(\\mathcal {G}\\times \\mathcal {H})=\\operatorname{\\textup {B}}(\\mathcal {G})\\times \\operatorname{\\textup {B}}(\\mathcal {H})$ is obtained as follows: given $(\\mathcal {P},\\mathcal {Q},\\lambda )\\in \\operatorname{\\textup {B}}(\\mathcal {G})\\times _{\\operatorname{\\textup {B}}(\\mathcal {G}/\\mathcal {H})}\\operatorname{\\textup {B}}(\\mathcal {G})$ (so that $\\lambda \\colon \\mathcal {P}/\\mathcal {H}\\longrightarrow \\mathcal {Q}/\\mathcal {H}$ is a $\\mathcal {G}/\\mathcal {H}$ -equivariant isomorphism) we associate $(\\mathcal {P},\\mathcal {I}_{\\lambda })$ , where $\\mathcal {I}_{\\lambda }$ is the fiber of $\\lambda $ along the map $\\operatorname{\\underline{\\textup {Iso}}}^{\\mathcal {G}}(\\mathcal {P},\\mathcal {Q})\\longrightarrow \\operatorname{\\underline{\\textup {Iso}}}^{\\mathcal {G}/\\mathcal {H}}(\\mathcal {P}/\\mathcal {H},\\mathcal {Q}/\\mathcal {H})$ and the action of $\\mathcal {H}$ is given by $\\mathcal {H}\\longrightarrow \\mathcal {G}\\longrightarrow \\operatorname{\\underline{\\textup {Aut}}}(\\mathcal {Q})$ .",
"Let $(\\mathcal {P},\\mathcal {Q},\\lambda )\\in \\operatorname{\\textup {B}}(\\mathcal {G})\\times _{\\operatorname{\\textup {B}}(\\mathcal {G}/\\mathcal {H})}\\operatorname{\\textup {B}}(\\mathcal {G})$ over an object $c\\in \\mathcal {C}$ .",
"The composition $\\mathcal {H}\\longrightarrow \\mathcal {G}\\longrightarrow \\operatorname{\\underline{\\textup {Aut}}}(\\mathcal {Q})$ has image in $\\operatorname{\\underline{\\textup {Aut}}}^{\\mathcal {G}}(\\mathcal {Q})$ because $\\mathcal {H}$ is central.",
"Moreover the map $\\operatorname{\\underline{\\textup {Iso}}}^{\\mathcal {G}}(\\mathcal {P},\\mathcal {Q})\\longrightarrow \\operatorname{\\underline{\\textup {Iso}}}^{\\mathcal {G}/\\mathcal {H}}(\\mathcal {P}/\\mathcal {H},\\mathcal {Q}/\\mathcal {H})$ is $\\mathcal {H}$ -equivariant.",
"It is also an $\\mathcal {H}$ -torsor: locally when $\\mathcal {P}$ and $\\mathcal {Q}$ are isomorphic to $\\mathcal {G}$ , the previous map become $\\mathcal {G}\\longrightarrow \\mathcal {G}/\\mathcal {H}$ .",
"Thus $\\mathcal {I}_{\\lambda }$ is an $\\mathcal {H}$ -torsor over $c\\in \\mathcal {C}$ .",
"Thus we have two well defined functors $\\Lambda \\colon \\operatorname{\\textup {B}}(\\mathcal {G})\\times _{\\operatorname{\\textup {B}}(\\mathcal {G}/\\mathcal {H})}\\operatorname{\\textup {B}}(\\mathcal {G})\\longrightarrow \\operatorname{\\textup {B}}(\\mathcal {G})\\times \\operatorname{\\textup {B}}(\\mathcal {H})\\text{ and }\\Delta \\colon \\operatorname{\\textup {B}}(\\mathcal {G})\\times \\operatorname{\\textup {B}}(\\mathcal {H})\\longrightarrow \\operatorname{\\textup {B}}(\\mathcal {G})\\times _{\\operatorname{\\textup {B}}(\\mathcal {G}/\\mathcal {H})}\\operatorname{\\textup {B}}(\\mathcal {G})$ and we must show they are quasi-inverses of each other.",
"Let's consider the composition $\\Lambda \\circ \\Delta $ and $(\\mathcal {P},\\mathcal {E})\\in \\operatorname{\\textup {B}}(\\mathcal {G})\\times \\operatorname{\\textup {B}}(\\mathcal {H})$ over $c\\in \\mathcal {C}$ .",
"We have $\\Delta (\\mathcal {P},\\mathcal {E})=(\\mathcal {P},(\\mathcal {P}\\times \\mathcal {E}\\times \\mathcal {G})/\\mathcal {G}\\times \\mathcal {H},\\lambda )$ where $\\lambda $ is the inverse of $[(\\mathcal {P}\\times \\mathcal {E}\\times \\mathcal {G})/\\mathcal {G}\\times \\mathcal {H}]/\\mathcal {H}\\longrightarrow \\mathcal {P}/\\mathcal {H},\\ (p,e,1)\\longrightarrow p$ We have to give an $\\mathcal {H}$ -equivariant map $\\mathcal {E}\\longrightarrow \\mathcal {I}_{\\lambda }$ .",
"Given a global section $e\\in \\mathcal {E}$ , that is a map $\\mathcal {H}\\longrightarrow \\mathcal {E}$ , we get a $\\mathcal {G}\\times \\mathcal {H}$ equivariant morphism $\\mathcal {P}\\times \\mathcal {H}\\longrightarrow \\mathcal {P}\\times \\mathcal {E}$ and thus a $\\mathcal {G}$ -equivariant morphism $\\delta \\colon \\mathcal {P}\\longrightarrow (\\mathcal {P}\\times \\mathcal {H}\\times \\mathcal {G})/\\mathcal {G}\\times \\mathcal {H}\\longrightarrow (\\mathcal {P}\\times \\mathcal {E}\\times \\mathcal {G})/\\mathcal {G}\\times \\mathcal {H}$ which is easily seen to induce $\\lambda $ .",
"Mapping $e$ to $\\delta $ gives an $\\mathcal {H}$ -equivariant map $\\mathcal {E}\\rightarrow \\mathcal {I}_{\\lambda }$ .",
"There are several conditions that must be checked but they are all elementary and left to the reader.",
"Now consider $\\Delta \\circ \\Lambda $ and $(\\mathcal {P},\\mathcal {Q},\\lambda )\\in \\operatorname{\\textup {B}}(\\mathcal {G})\\times _{\\operatorname{\\textup {B}}(\\mathcal {G}/\\mathcal {H})}\\operatorname{\\textup {B}}(\\mathcal {G})$ over an object $c\\in \\mathcal {C}$ .",
"It is easy to see that $(\\mathcal {P}\\times \\mathcal {I}_{\\lambda }\\times \\mathcal {G})/\\mathcal {G}\\times \\mathcal {H}\\longrightarrow \\mathcal {Q},\\ (p,\\phi ,g)\\mapsto \\phi (p)g$ is a $\\mathcal {G}$ -equivariant morphism and it induces a morphism $\\Delta \\circ \\Lambda (\\mathcal {P},\\mathcal {Q},\\lambda )\\longrightarrow (\\mathcal {P},\\mathcal {Q},\\lambda )$ .",
"Remark 2.13 If $X\\longrightarrow Y$ is integral (e.g.",
"finite) and a universal homeomorphism of schemes and $G$ is an étale group scheme over a field $k$ then $\\operatorname{\\textup {B}}G(Y)\\longrightarrow \\operatorname{\\textup {B}}G(X)$ is an equivalence.",
"Indeed by [1] the fiber product induces an equivalence between the category of schemes étale over $Y$ and the category of schemes étale over $X$ .",
"Lemma 2.14 Let $G$ be a finite group scheme over $k$ of rank $\\operatorname{\\textup {rk}}G$ , $U\\longrightarrow \\mathcal {G}$ a finite, flat and finitely presented map of degree $\\operatorname{\\textup {rk}}G$ and $\\mathcal {G}\\longrightarrow T$ be a map locally equivalent to $\\operatorname{\\textup {B}}G$ , where $U$ , $\\mathcal {G}$ and $T$ are categories fibered in groupoids.",
"If $U\\longrightarrow T$ is faithful then it is an equivalence.",
"By changing the base $T$ we can assume that $T$ is a scheme, $\\mathcal {G}=\\operatorname{\\textup {B}}G\\times T$ and $U$ is an algebraic space.",
"We must prove that if $P\\longrightarrow U$ is a $G$ -torsor and $P\\longrightarrow U\\longrightarrow T$ is a cover of degree $\\operatorname{\\textup {rk}}G$ then $f\\colon U\\longrightarrow T$ is an isomorphism.",
"It follows that $f\\colon U\\longrightarrow T$ is flat, finitely presented and quasi-finite.",
"Moreover $f\\colon U\\longrightarrow \\operatorname{\\textup {B}}G\\times T\\longrightarrow T$ is proper.",
"We can conclude that $f\\colon U\\longrightarrow T$ is finite and flat.",
"Looking at the ranks of the involved maps we see that $f$ must have rank 1."
],
[
"Direct system of Deligne-Mumford\nstacks",
"In this section we discuss some general facts about direct limits of DM stacks.",
"For the general notion of limit see Appendix .",
"By a direct system in this section we always mean a direct system indexed by $\\mathbb {N}$ .",
"Definition 3.1 Let $\\mathcal {X}$ be a category fibered in groupoid over $\\mathbb {Z}$ .",
"A coarse ind-algebraic space for $\\mathcal {X}$ is a map $\\mathcal {X}\\longrightarrow X$ to an ind-algebraic space $X$ which is universal among maps from $\\mathcal {X}$ to an ind-algebraic space and such that, for all algebraically closed field $K$ , the map $\\mathcal {X}(K)/\\simeq \\longrightarrow X(K)$ is bijective.",
"Lemma 3.2 Let $\\mathcal {Z}_{*}$ be a direct system of quasi-compact and quasi-separated algebraic stacks admitting coarse moduli spaces $\\mathcal {Z}_{n}\\longrightarrow \\overline{\\mathcal {Z}_{n}}$ .",
"Then the limit of those maps $\\Delta \\longrightarrow \\overline{\\Delta }$ is a coarse ind-algebraic space.",
"Assume moreover that the transition maps of $\\mathcal {Z}_{*}$ are finite and universally injective.",
"Then for all $n\\in \\mathbb {N}$ and all reduced rings $B$ the functors $\\mathcal {Z}_{n}(B)\\longrightarrow \\mathcal {Z}_{n+1}(B)\\longrightarrow \\Delta (B)$ are fully faithful.",
"In particular the maps $\\overline{\\mathcal {Z}_{n}}\\longrightarrow \\overline{\\mathcal {Z}_{n+1}}$ are universally injective and, if all $\\mathcal {Z}_{m}$ are DM, $\\mathcal {Z}_{n}\\longrightarrow \\Delta $ preserves the geometric stabilizers.",
"The first claim follows easily taking into account that, since $\\mathcal {Z}_{n}$ is quasi-compact and quasi-separated, a functor from $\\mathcal {Z}_{n}$ to an ind-algebraic space factors through an algebraic space and therefore uniquely through $\\overline{\\mathcal {Z}_{n}}$ .",
"It is also easy to reduce the second claim to the case of some $\\mathcal {Z}_{n}$ .",
"Denote by $\\psi \\colon \\mathcal {Z}_{n}\\longrightarrow \\mathcal {Z}_{n+1}$ the transition map.",
"Let $\\xi ,\\eta \\in \\mathcal {Z}_{n}(B)$ and $a\\colon \\psi (\\xi )\\longrightarrow \\psi (\\eta )$ .",
"Set $\\psi (\\eta )=\\zeta \\in \\mathcal {Z}_{n+1}(B)$ .",
"If $W$ is the base change of $\\mathcal {Z}_{n}\\longrightarrow \\mathcal {Z}_{n+1}$ along $\\operatorname{\\textup {Spec}}B{\\zeta }\\mathcal {Z}_{n+1}$ then $\\overline{\\xi }=(\\xi ,\\zeta ,a),\\overline{\\eta }=(\\eta ,\\zeta ,\\textup {id})\\in W(B)$ .",
"A lifting of $a$ to an isomorphism $\\xi \\longrightarrow \\eta $ is exactly an isomorphism $\\overline{\\xi }\\longrightarrow \\overline{\\eta }$ .",
"Such an isomorphism exists and it is unique because, since $W\\longrightarrow \\operatorname{\\textup {Spec}}B$ is an homeomorphism and $B$ is reduced it has at most one section.",
"Applying the above property when $B$ is an algebraically closed field we conclude that $\\overline{\\mathcal {Z}_{n}}\\longrightarrow \\overline{\\mathcal {Z}_{n+1}}$ is universally injective.",
"If all $\\mathcal {Z}_{m}$ are DM then the geometric stabilizers are constant.",
"Since for all algebraically closed field $K$ the functor $\\mathcal {Z}_{n}(K)\\longrightarrow \\mathcal {Z}_{n+1}(K)$ is fully faithful we see that $\\mathcal {Z}_{n}\\longrightarrow \\mathcal {Z}_{n+1}$ is an isomorphism on geometric stabilizers.",
"Definition 3.3 Given a direct system of stacks $\\mathcal {Y}_{*}$ , a direct system of smooth (resp.",
"étale) atlases for $\\mathcal {Y}_{*}$ is a direct system of algebraic spaces $U_{*}$ together with smooth (resp.",
"étale) atlases $U_{i}\\longrightarrow \\mathcal {Y}_{i}$ and 2-Cartesian diagrams $ \\begin{tikzpicture}[xscale=2.1,yscale=-1.2] \\node (A0_0) at (0, 0) {U_i}; \\node (A0_1) at (1, 0) {U_{i+1}}; \\node (A1_0) at (0, 1) {\\mathcal {Y}_i}; \\node (A1_1) at (1, 1) {\\mathcal {Y}_{i+1}}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); \\end{tikzpicture} $ for all $i\\in \\mathbb {N}$ .",
"Lemma 3.4 Let $\\mathcal {Y}_{*}$ be a direct system of stacks and $\\mathcal {X}$ be a quasi-compact and quasi-separated algebraic stack.",
"Then the functor $\\varinjlim _{n}\\operatorname{\\textup {Hom}}(\\mathcal {X},\\mathcal {Y}_{n})\\longrightarrow \\operatorname{\\textup {Hom}}(\\mathcal {X},\\varinjlim _{n}\\mathcal {Y})$ is an equivalence of categories.",
"If the transition maps of $\\mathcal {Y}_{*}$ are faithful (resp.",
"fully faithful) so are the transition maps in the above limit.",
"Denotes by $\\zeta _{\\mathcal {X}}$ the functor in the statement.",
"When $\\mathcal {X}$ is an affine scheme $\\zeta _{\\mathcal {X}}$ is an equivalence thanks to REF .",
"In general there is a smooth atlas $U\\longrightarrow \\mathcal {X}$ from an affine scheme.",
"It is easy to see that the functor $\\zeta _{\\mathcal {X}}$ is faithful.",
"If two morphisms become equal in the limit it is enough to pullback to $U$ and get a finite index for $\\zeta _{U}$ .",
"By descent this index will work in general.",
"The next step is to look at the case when $\\mathcal {X}$ is a quasi-compact scheme.",
"Using the faithfulness just proved and taking a Zariski covering of $\\mathcal {X}$ (here one uses that the intersection of two open quasi-compact subschemes of $\\mathcal {X}$ is again quasi-compact) one proves that $\\zeta _{\\mathcal {X}}$ is an equivalence.",
"Finally using that $\\zeta _{U}$ , $\\zeta _{U\\times _{\\mathcal {X}}U}$ and $\\zeta _{U\\times _{\\mathcal {X}}U\\times _{\\mathcal {X}}U}$ are equivalences and using descent one get that $\\zeta _{\\mathcal {X}}$ is an equivalence.",
"The last statement can be proved directly.",
"Lemma 3.5 Consider a 2-Cartesian diagram $ \\begin{tikzpicture}[xscale=1.5,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {Y}}; \\node (A0_1) at (1, 0) {\\operatorname{\\textup {Spec}}k}; \\node (A1_0) at (0, 1) {\\mathcal {X}}; \\node (A1_1) at (1, 1) {\\operatorname{\\textup {B}}G}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_0) edge [->]node [auto] {\\scriptstyle {F}} (A1_0); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); \\end{tikzpicture} $ where $G$ is a finite group and $\\mathcal {X}$ is a stack over $\\operatorname{\\textup {Aff}}/k$ .",
"Suppose that there exists a direct system $\\mathcal {Y}_{*}$ of DM stacks of finite type over $k$ with finite and universally injective transition maps, affine diagonal, with a direct system of étale atlases $Y_{n}\\longrightarrow \\mathcal {Y}_{n}$ from affine schemes and with an isomorphism $\\varinjlim _{n}\\mathcal {Y}_{n}\\simeq \\mathcal {Y}$ .",
"Then there exists a direct system of DM stacks $\\mathcal {X}_{*}$ of finite type over $k$ with finite and universally injective transition maps, affine diagonal, with a direct system of étale atlases $X_{n}\\longrightarrow \\mathcal {X}_{n}$ from affine schemes and with an isomorphism $\\varinjlim _{n}\\mathcal {X}_{n}\\simeq \\mathcal {X}$ .",
"If all the stacks in $\\mathcal {Y}_{*}$ are separated then the stacks $\\mathcal {X}_{*}$ can also be chosen separated.",
"If $Y_{n}\\longrightarrow \\mathcal {Y}_{n}$ is finite and étale then $X_{n}\\longrightarrow \\mathcal {X}_{n}$ can also be chosen finite and étale.",
"The remaining part of this section is devoted to the proof of the above Lemma.",
"Its outline is as follows.",
"For some data $\\underline{\\omega }$ , we define a stack $\\mathcal {X}_{\\underline{\\omega }}$ , and for a suitable sequence $\\underline{\\omega _{u}}$ , $u\\in \\mathbb {N}$ of such data, we will prove that the sequence $\\mathcal {X}_{\\underline{\\omega _{0}}}\\rightarrow \\mathcal {X}_{\\underline{\\omega _{1}}}\\rightarrow \\cdots $ has the desired property.",
"To do this, we reduce the problem to proving a similar property for the induced sequence $\\mathcal {Z}_{\\underline{\\omega _{u}}}\\cong \\mathcal {X}_{\\underline{\\omega _{u}}}\\times _{\\mathcal {X}}\\mathcal {Y}$ , $u\\in \\mathbb {N}$ .",
"Then we describe $\\mathcal {Z}_{\\underline{\\omega }}$ using fiber products of simple stacks.",
"Once these are done, it is straightforward to see the desired properties of $\\mathcal {Z}_{\\underline{\\omega _{u}}}$ , $u\\in \\mathbb {N}$ .",
"We recall that stacks and more generally category fibered in groupoids form 2-categories; 1-morphisms are base preserving functors between them and 2-morphisms are base preserving natural isomorphisms between functors.",
"In a diagram of stacks, 1-morphisms (functors) are written as normal thin arrows and 2-morphisms as thick arrows.",
"For instance, in the diagram of categories fibered in groupoids ${A[r]^{f}[d]_{h} & B[d]^{g}@2[dl]^{\\lambda }\\\\C[r]_{i} & D}$ $A,B,C$ and $D$ are categories fibered in groupoids, $f$ , $g$ , $h$ and $i$ are functors and $\\lambda $ is a natural isomorphism $g\\circ f\\rightarrow i\\circ h$ .",
"We will also say that $\\lambda $ makes the diagram 2-commutative.",
"For a diagram including several 2-morphisms such as ${A[r]^{f}[d]_{i} & @2[dl]_{\\lambda }B[r]^{g}[d] & @2_{\\lambda ^{\\prime }}[dl]C[d]^{h}\\\\D[r]_{j} & E[r]_{k} & F}$ the induced natural isomorphism means that the induced natural isomorphism of the two outer paths from the upper left to the bottom right; concretely, in the above diagram, it is the natural isomorphism $h\\circ g\\circ f\\rightarrow k\\circ j\\circ i$ induced by $\\lambda $ and $\\lambda ^{\\prime }$ .",
"Let us denote the functor $\\mathcal {X}\\longrightarrow \\operatorname{\\textup {B}}G$ in REF by $Q$ .",
"An object of $\\mathcal {Y}$ over $T\\in \\operatorname{\\textup {Aff}}/k$ is identified with a pair $(\\xi ,c)$ such that $\\xi \\in \\mathcal {X}(T)$ and $c$ is a section of the $G$ -torsor $Q(\\xi )\\longrightarrow T$ , in other words $c\\in Q(\\xi )(T)$ .",
"For each $g\\in G$ , we define an automorphism $\\iota _{g}\\colon \\mathcal {Y}\\rightarrow \\mathcal {Y}$ sending $(x,c)$ to $(x,cg)$ .",
"By construction we have $\\iota _{1}=\\textup {id}$ and $\\iota _{g}\\circ \\iota _{h}=\\iota _{hg}$ and we interpret those maps as a map $\\iota \\colon \\mathcal {Y}\\times G\\longrightarrow \\mathcal {Y}$ .",
"For all $\\xi \\in \\mathcal {X}$ the map $\\iota $ induces the action of $G$ on $Q(\\xi )$ .",
"Definition 3.6 We define a category fibered in groupoids $\\widetilde{\\mathcal {X}}$ as follows.",
"An object of $\\widetilde{\\mathcal {X}}$ over a scheme $T$ is a tuple $(P,\\eta ,\\mu )$ where $P$ is a $G$ -torsor over $T\\in \\operatorname{\\textup {Aff}}/k$ with a $G$ -action $m_{P}\\colon P\\times G\\rightarrow P$ , $\\eta \\colon P\\rightarrow \\mathcal {Y}$ is a morphism and $\\mu $ is a natural isomorphism ${P\\times G[r]^{m_{P}}[d]_{\\eta \\times \\textup {id}_{G}} & @2[dl]_{\\mu }P[d]^{\\eta }\\\\\\mathcal {Y}\\times G[r]_{\\iota } & \\mathcal {Y}}$ such that if $\\mu _{g}$ denotes the natural isomorphism induced from $\\mu $ by composing $P\\simeq P\\times \\lbrace g\\rbrace \\hookrightarrow P\\times G$ and $m_{g}\\colon P\\rightarrow P$ denotes the action of $g$ , then the diagram ${P[r]^{m_{h}}[d]_{\\eta } & @2[dl]_{\\mu _{h}}P[r]^{m_{g}}[d]_{\\eta } & @2[dl]_{\\mu _{g}}P[d]_{\\eta }\\\\\\mathcal {Y}[r]_{\\iota _{h}} & \\mathcal {Y}[r]_{\\iota _{g}} & \\mathcal {Y}}$ induces $\\mu _{hg}$ .",
"A morphism $(P,\\eta ,\\mu )\\rightarrow (P^{\\prime },\\eta ^{\\prime },\\mu ^{\\prime })$ over $T$ is a pair $(\\alpha ,\\beta )$ where $\\alpha \\colon P\\rightarrow P^{\\prime }$ is a $G$ -equivariant isomorphism over $T$ and $\\beta $ is a natural isomorphism ${P[rr]^{\\alpha }[dr]_{\\eta } & & P^{\\prime }[dl]^{\\eta ^{\\prime }}@{<=}[dll(0.6)]_(0.7){\\beta }\\\\& \\mathcal {Y}}$ such that the diagram ${P\\times G[dd]_{\\eta \\times \\textup {id}}[rrr][dr]^{\\alpha \\times \\textup {id}} & & @2[dl]_{\\textup {id}} & [dl]^{\\alpha }P[dd]^{\\eta }\\\\& @2[l]^{\\beta ^{-1}\\times \\textup {id}}P^{\\prime }\\times G[r][dl]^{\\eta ^{\\prime }\\times \\textup {id}} & P^{\\prime }[dr]_{\\eta ^{\\prime }}@2[dl]_{\\mu ^{\\prime }} & @2[l]^(0.4){\\beta }\\\\\\mathcal {Y}\\times G[rrr]_{\\iota } & & & \\mathcal {Y}}$ induces $\\mu $ .",
"There is a functor $\\mathcal {X}\\longrightarrow \\widetilde{\\mathcal {X}}$ : given $\\xi \\in \\mathcal {X}(T)$ one get a $G$ -torsor $Q(\\xi )\\rightarrow T$ and a morphism $\\eta \\colon Q(\\xi )\\rightarrow \\mathcal {Y}$ and using the Cartesian diagram relating $\\mathcal {X}$ and $\\mathcal {Y}$ we also get a natural transformation as above.",
"The following is a generalization of [8] for stacks without geometric properties.",
"Lemma 3.7 The functor $\\mathcal {X}\\longrightarrow \\widetilde{\\mathcal {X}}$ is an equivalence.",
"The forgetful functor $\\widetilde{\\mathcal {X}}\\longrightarrow \\operatorname{\\textup {B}}G$ composed with the functor in the statement is $Q\\colon \\mathcal {X}\\longrightarrow \\operatorname{\\textup {B}}G$ .",
"Since $\\mathcal {X}$ and $\\widetilde{\\mathcal {X}}$ are stacks, it is enough to show that the functor $\\Xi \\colon \\mathcal {Y}\\longrightarrow \\widetilde{\\mathcal {Y}}=\\widetilde{\\mathcal {X}}\\times _{\\operatorname{\\textup {B}}G}\\operatorname{\\textup {Spec}}k$ is an equivalence.",
"An object of the stack $\\widetilde{\\mathcal {Y}}$ can be regarded as a pair $(\\eta ,\\mu )$ such that $\\eta \\colon T\\times G\\rightarrow \\mathcal {Y}$ is a morphism and $\\mu $ is a natural isomorphism as in (REF ) with $m_{P}\\colon P\\times G\\rightarrow P$ replaced with $\\textup {id}_{T}\\times m_{G}\\colon T\\times G\\times G\\rightarrow T\\times G$ , where $m_{G}$ is the multiplication of $G$ .",
"A morphism $(\\eta ,\\mu )\\rightarrow (\\eta ^{\\prime },\\mu ^{\\prime })$ in $\\widetilde{\\mathcal {Y}}(T)$ is a natural isomorphism $\\beta \\colon \\eta \\rightarrow \\eta ^{\\prime }$ satisfying the same compatibility as in (REF ) where $P$ and $P^{\\prime }$ are replaced with $T\\times G$ and $\\alpha $ is replaced with $\\textup {id}_{T\\times G}$ .",
"The functor $\\Xi $ sends an object $\\rho \\in \\mathcal {Y}(T)$ to $(\\widetilde{\\rho }\\colon T\\times G\\rightarrow \\mathcal {Y},\\mu )$ such that $\\widetilde{\\rho }|_{T\\times \\lbrace g\\rbrace }=\\iota _{g}\\circ \\rho $ and $\\mu $ is the canonical natural isomorphism, and a morphism $\\gamma \\colon \\rho \\rightarrow \\rho ^{\\prime }$ to $(\\textup {id}_{T\\times G},\\widetilde{\\gamma })$ where $\\widetilde{\\gamma }|_{T\\times \\lbrace g\\rbrace }=\\iota _{g}(\\gamma )$ .",
"One also get a functor $\\Lambda \\colon \\widetilde{\\mathcal {Y}}\\longrightarrow \\mathcal {Y}$ by composing with the identity of $G$ and it is easy to see that $\\Lambda \\circ \\Xi =\\textup {id}$ .",
"The compatibilities defining the objects of $\\widetilde{\\mathcal {Y}}$ also allow to define an isomorphism $\\Xi \\circ \\Lambda \\simeq \\textup {id}$ .",
"Set $\\delta _{u}\\colon \\mathcal {Y}_{u}\\longrightarrow \\mathcal {Y}$ for the structure maps and $\\delta _{u,v}\\colon \\mathcal {Y}_{u}\\longrightarrow \\mathcal {Y}_{v}$ for the transition maps for all $u\\le v\\in \\mathbb {N}$ .",
"Given $w\\ge v\\ge u\\in \\mathbb {N}$ we denote by $\\mathcal {R}(u,v,w)$ the collection of tuples $(\\omega ,\\omega ^{\\prime },\\theta ,\\theta ^{\\prime })$ forming 2-commutative diagrams: ${\\mathcal {Y}_{u}\\times G[r]^{\\omega }[d]_{\\delta _{u}\\times \\textup {id}} & \\mathcal {Y}_{v}[d]^{\\delta _{v}}@2[dl]_{\\theta }\\\\\\mathcal {Y}\\times G[r]_{\\iota } & \\mathcal {Y}}$ ${\\mathcal {Y}_{v}\\times G[r]^{\\omega ^{\\prime }}[d]_{\\delta _{v}\\times \\textup {id}} & \\mathcal {Y}_{w}[d]^{\\delta _{w}}@2[dl]_{\\theta ^{\\prime }}\\\\\\mathcal {Y}\\times G[r]_{\\iota } & \\mathcal {Y}}$ We also require the existence of a natural isomorphism ${\\mathcal {Y}_{u}\\times G[r]^{\\omega }[d]_{\\delta _{u,v}\\times \\textup {id}} & \\mathcal {Y}_{v}[d]^{\\delta _{v,w}}@2[dl]_{\\zeta }\\\\\\mathcal {Y}_{v}\\times G[r]_{\\omega ^{\\prime }} & \\mathcal {Y}_{w}}$ compatible with $\\theta $ and $\\theta ^{\\prime }$ and, for $g,h\\in G$ , the existence of a natural isomorphism $\\lambda _{h,g}$ ${\\mathcal {Y}_{u}[r]^{\\omega _{h}}[d]_{\\omega _{hg}} & \\mathcal {Y}_{v}[d]^{\\omega ^{\\prime }_{g}}@2[dl]_{\\lambda _{h,g}}\\\\\\mathcal {Y}_{v}[r]_{\\delta _{v,w}} & \\mathcal {Y}_{w}}$ such that the natural isomorphism induced by ${\\mathcal {Y}_{u}[drr]^{\\omega _{h}}[ddd]_{\\delta _{u}}[rrrr]^{\\delta _{v,w}(\\omega _{hg})} & & {}@2[d]^{\\lambda _{h,g}} & & \\mathcal {Y}_{w}[ddd]^{\\delta _{w}}\\\\& & @2[dll]_{\\theta _{h}}\\mathcal {Y}_{v}[urr]^{\\omega ^{\\prime }_{g}}[d]_{\\delta _{v}} & & @2[dll]_{\\theta ^{\\prime }_{g}}\\\\{} & & \\mathcal {Y}[drr]_{\\iota _{g}}@2[d]^{\\text{id}}\\\\\\mathcal {Y}[urr]_{\\iota _{h}}[rrrr]_{\\iota _{hg}} & & {} & & \\mathcal {Y}}$ coincides with the one induced by $\\theta _{hg}$ and $\\delta _{v,w}$ .",
"Since the transition maps of $\\mathcal {Y}_{*}$ are faithful, the functor $\\operatorname{\\textup {Hom}}(\\mathcal {Y}_{u},\\mathcal {Y}_{w})\\rightarrow \\operatorname{\\textup {Hom}}(\\mathcal {Y}_{u},\\mathcal {Y})$ is faithful as well, which means that natural transformations $\\zeta $ and $\\lambda _{g,h}$ are uniquely determined.",
"Definition 3.8 For $\\underline{\\omega }=(\\omega ,\\omega ^{\\prime },\\theta ,\\theta ^{\\prime })\\in \\mathcal {R}(u,v,w)$ , we define the following category fibered in groupoids $\\mathcal {X}_{\\underline{\\omega }}$ as follows.",
"An object of $\\mathcal {X}_{\\underline{\\omega }}$ over $T$ is a triple $(P,\\eta ,\\mu )$ of a $G$ -torsor $P$ over $T$ , $\\eta \\colon P\\longrightarrow \\mathcal {Y}_{u}$ and a natural isomorphism $\\mu $ making the diagram ${P\\times G[r]^{m_{P}}[d]_{\\eta \\times \\textup {id}} & P[d]^{\\delta _{u,v}(\\eta )}@2[dl]_{\\mu }\\\\\\mathcal {Y}_{u}\\times G[r]_{\\omega } & \\mathcal {Y}_{v}}$ 2-commutative such that the diagram ${P[rr]^{m_{h}}[dd]_{\\eta } & & @2[dll]_{\\mu _{h}}P[rr]^{m_{g}}[d]_{\\delta _{u,v}(\\eta )} & & @2[dll]_{\\delta _{v,w}(\\mu _{g})}P[dd]^{\\delta _{u,w}(\\eta )}\\\\& & \\mathcal {Y}_{v}[drr]_{\\omega ^{\\prime }_{g}}@2[d]^{\\lambda _{h,g}}\\\\\\mathcal {Y}_{u}[urr]_{\\omega _{h}}[rrrr]_{\\delta _{v,w}(\\omega _{hg})} & & {} & & \\mathcal {Y}_{w}}$ induces $\\delta _{v,w}(\\mu _{hg})$ .",
"A morphism $(P,\\eta ,\\mu )\\longrightarrow (P^{\\prime },\\eta ^{\\prime },\\mu ^{\\prime })$ in $\\mathcal {X}_{\\underline{\\omega }}(T)$ is a pair $(\\alpha ,\\beta )$ where $\\alpha \\colon P\\longrightarrow P^{\\prime }$ is a $G$ -equivariant isomorphism over $T$ and $\\beta $ is a natural isomorphism making the following diagram 2-commutative ${P[rr]^{\\alpha }[dr]_{\\eta } & & P^{\\prime }[dl]^{\\eta ^{\\prime }}@{<=}[dll(0.6)]_(0.7){\\beta }\\\\& \\mathcal {Y}_{u}}$ such that the diagram ${P\\times G[dd]_{\\eta \\times \\textup {id}}[rrr][dr]^{\\alpha \\times \\textup {id}} & & @2[dl]_{\\mathrm {id}} & [dl]^{\\alpha }P[dd]^{\\delta _{u,v}(\\eta )}\\\\& @2[l]^{\\beta ^{-1}\\times \\textup {id}}P^{\\prime }\\times G[r][dl]^{\\eta ^{\\prime }\\times \\textup {id}} & P^{\\prime }[dr]_{\\delta _{u,v}(\\eta ^{\\prime })}@2[dl]_{\\mu ^{\\prime }} & @2[l]^(0.4){\\delta _{u,v}(\\beta )}\\\\\\mathcal {Y}_{u}\\times G[rrr]_{\\omega } & & & \\mathcal {Y}_{v}}$ induces the natural isomorphism $\\mu $ .",
"By REF for all algebraic stacks $\\mathcal {X}$ the functor $\\varinjlim _{w}\\operatorname{\\textup {Hom}}(\\mathcal {X},\\mathcal {Y}_{w})\\rightarrow \\operatorname{\\textup {Hom}}(\\mathcal {X},\\mathcal {Y})$ is an equivalence.",
"This allows us to choose increasing functions $v,w\\colon \\mathbb {N}\\longrightarrow \\mathbb {N}$ such that $u\\le v(u)\\le w(u)$ and $\\underline{\\omega _{u}}=(\\omega _{u},\\omega _{u}^{\\prime },\\theta _{u},\\theta _{u}^{\\prime })\\in \\mathcal {R}(u,v(u),w(u))$ , so that $\\mathcal {X}_{\\underline{\\omega _{u}}}$ is defined, for all $u\\in \\mathbb {N}$ .",
"Moreover we can assume there exist natural isomorphisms $\\kappa $ and $\\kappa ^{\\prime }$ ${\\mathcal {Y}_{u}\\times G[r]^{\\omega _{u}}[d]_{\\delta _{u,u+1}\\times \\textup {id}} & \\mathcal {Y}_{v(u)}[d]^{\\delta _{v(u),v(u+1)}}@2[dl]_{\\kappa }\\\\\\mathcal {Y}_{u+1}\\times G[r]_{\\omega _{u+1}} & \\mathcal {Y}_{v(u+1)}}$ ${\\mathcal {Y}_{v(u)}\\times G[r]^{\\omega ^{\\prime }_{u}}[d]_{\\delta _{v(u),v(u+1)}\\times \\textup {id}} & \\mathcal {Y}_{w(u)}[d]^{\\delta _{w(u),w(u+1)}}@2[dl]_{\\kappa ^{\\prime }}\\\\\\mathcal {Y}_{v(u+1)}\\times G[r]_{\\omega ^{\\prime }_{u+1}} & \\mathcal {Y}_{w(u+1)}}$ such that $\\delta _{v(u+1)}(\\kappa )$ is induced from $\\theta _{u}$ and $\\theta _{u+1}$ and $\\delta _{w(u+1)}(\\kappa ^{\\prime })$ is induced from $\\theta ^{\\prime }_{u}$ and $\\theta ^{\\prime }_{u+1}$ .",
"Again, since the transition maps of $\\mathcal {Y}_{*}$ are faithful, natural transformations $\\kappa $ and $\\kappa ^{\\prime }$ are uniquely determined.",
"For each $u\\in \\mathbb {N}$ , there exist canonical functors $\\mathcal {X}_{\\underline{\\omega _{u}}}\\rightarrow \\mathcal {X}_{\\underline{\\omega _{u+1}}}$ and $\\mathcal {X}_{\\underline{\\omega _{u}}}\\rightarrow \\widetilde{\\mathcal {X}}$ , which lead to a functor $\\varinjlim _{u}\\mathcal {X}_{\\underline{\\omega _{u}}}\\longrightarrow \\widetilde{\\mathcal {X}}.$ Proposition 3.9 The functor $\\varinjlim _{u}\\mathcal {X}_{\\underline{\\omega _{u}}}\\longrightarrow \\widetilde{\\mathcal {X}}$ is an equivalence.",
"By definition, every object and every morphism of $\\widetilde{\\mathcal {X}}$ come from ones of $\\mathcal {X}_{\\underline{\\omega _{u}}}$ for some $u$ .",
"Namely the above functor is essentially surjective and full.",
"To see the faithfulness, we take objects $(P,\\eta ,\\mu )$ , $(P^{\\prime },\\eta ^{\\prime },\\mu ^{\\prime })$ of $\\mathcal {X}_{\\underline{\\omega _{u}}}(T)$ and their images $(P,\\eta _{\\infty },\\mu _{\\infty })$ , $(P,\\eta ^{\\prime }_{\\infty },\\mu _{\\infty }^{\\prime })$ in $\\widetilde{\\mathcal {X}}(T)$ .",
"The map $\\operatorname{\\textup {Hom}}_{\\mathcal {X}_{\\underline{\\omega _{u}}}(T)}((P,\\eta ,\\mu ),(P^{\\prime },\\eta ^{\\prime },\\mu ^{\\prime }))\\rightarrow \\operatorname{\\textup {Hom}}_{\\widetilde{\\mathcal {X}}(T)}((P,\\eta _{\\infty },\\mu _{\\infty }),(P^{\\prime },\\eta ^{\\prime }_{\\infty },\\mu ^{\\prime }_{\\infty }))$ is compatible to projections to the set $\\operatorname{\\textup {Iso}}_{T}^{G}(P,P^{\\prime })$ of $G$ -equivariant isomorphisms over $T$ .",
"The fibers over $\\alpha \\in \\operatorname{\\textup {Iso}}_{T}^{G}(P,P^{\\prime })$ are respectively identified with subsets of $\\operatorname{\\textup {Hom}}_{\\mathcal {Y}_{u}(P)}(\\eta ,\\eta ^{\\prime }\\circ \\alpha )$ and of $\\operatorname{\\textup {Hom}}_{\\mathcal {Y}(P)}(\\eta _{\\infty },\\eta ^{\\prime }_{\\infty }\\circ \\alpha )$ .",
"Since $\\mathcal {Y}(P)$ is the limit of the categories $\\mathcal {Y}_{u}(P)$ by REF we get the faithfulness.",
"Definition 3.10 For $\\underline{\\omega }=(\\omega ,\\omega ^{\\prime },\\theta ,\\theta ^{\\prime })\\in \\mathcal {R}(u,v,w)$ , we define $\\mathcal {Z}_{\\underline{\\omega }}$ as the stack of pairs $(\\eta ,\\mu )$ where $\\eta \\colon T\\times G\\longrightarrow \\mathcal {Y}_{u}$ is a morphism and $\\mu $ is a natural isomorphism making the diagram ${T\\times G\\times G[d]_{\\eta \\times \\textup {id}}[r]^(0.6){\\textup {id}\\times m_{G}} & T\\times G[d]^{\\delta _{u,v}(\\eta )}@2[dl]_{\\mu }\\\\\\mathcal {Y}_{u}\\times G[r]_{\\omega } & \\mathcal {Y}_{v}}$ 2-commutative and such that ${T\\times G[rr]^{\\textup {id}\\times h}[dd]_{\\eta } & & @2[dll]_{\\mu _{h}}T\\times G[rr]^{\\textup {id}\\times g}[d]_{\\delta _{u,v}(\\eta )} & & @2[dll]_{\\delta _{v,w}(\\mu _{g})}T\\times G[dd]^{\\delta _{u,w}(\\eta )}\\\\{} & & \\mathcal {Y}_{v}[drr]^{\\omega ^{\\prime }_{g}}@2[d]^{\\lambda _{h,g}}\\\\\\mathcal {Y}_{u}[urr]^{\\omega _{h}}[rrrr]_{\\delta _{v,w}(\\omega _{hg})} & & {} & & \\mathcal {Y}_{w}}$ induces $\\delta _{v,w}(\\tau _{hg})$ .",
"A morphism $(\\eta ,\\mu )\\rightarrow (\\eta ^{\\prime },\\mu ^{\\prime })$ in $\\mathcal {Z}_{\\underline{\\omega }}$ is a natural isomorphism $\\beta \\colon \\eta \\rightarrow \\eta ^{\\prime }$ , ${@/_{25pt}/[d]_{\\eta }=\"a\"T\\times G@/^{25pt}/[d]^{\\eta ^{\\prime }}=\"b\"\\\\\\mathcal {Y}_{u}@2\"a\";\"b\"^{\\beta }}$ such that the diagram ${@/_{25pt}/[d]_{\\eta }=\"a\"T\\times G\\times G@/^{25pt}/[d]^{\\eta ^{\\prime }}=\"b\"[rrr]^{\\textup {id}\\times m_{G}} & & @2[dl]_{\\mu ^{\\prime }} & @/_{25pt}/[d]_{\\delta _{u,v}(\\eta ^{\\prime })}=\"c\"T\\times G@/^{25pt}/[d]^{\\delta _{u,v}(\\eta )}=\"d\"\\\\\\mathcal {Y}_{u}\\times G[rrr]_{\\omega }@2\"b\";\"a\"_{\\beta ^{-1}\\times \\textup {id}} & {} & & \\mathcal {Y}_{v}@2\"d\";\"c\"_{\\delta _{u,v}(\\beta )}}$ induces the natural isomorphism $\\mu $ .",
"Lemma 3.11 There exists a natural equivalence $\\mathcal {Z}_{\\underline{\\omega }}\\simeq \\mathcal {X}_{\\underline{\\omega }}\\times _{\\mathcal {X}}\\mathcal {Y}$ , where the fiber product is taken with respect to the functors $\\mathcal {X}_{\\underline{\\omega }}\\rightarrow \\widetilde{\\mathcal {X}}\\simeq \\mathcal {X}$ and $\\mathcal {Y}\\rightarrow \\mathcal {X}$ .",
"Set $\\widetilde{\\mathcal {Z}}_{\\underline{\\omega }}=\\mathcal {X}_{\\underline{\\omega }}\\times _{\\mathcal {X}}\\mathcal {Y}$ .",
"We may identify an object of $\\widetilde{\\mathcal {Z}}_{\\underline{\\omega }}(T)$ with a tuple $(P,\\eta ,\\mu ,s)$ such that $(P,\\eta ,\\mu )$ is an object of $\\mathcal {X}_{\\underline{\\omega }}(T)$ and $s$ is a section of $P\\rightarrow T$ .",
"A morphism $(P,\\eta ,\\mu ,s)\\rightarrow (P^{\\prime },\\eta ^{\\prime },\\mu ^{\\prime },s^{\\prime })$ is identified with a morphism $(\\alpha ,\\beta )\\colon (P,\\eta ,\\mu )\\rightarrow (P^{\\prime },\\eta ^{\\prime },\\mu ^{\\prime })$ in $\\mathcal {X}_{\\underline{\\omega }}$ satisfying $\\alpha \\circ s=s^{\\prime }$ .",
"The section $s$ induces a $G$ -equivariant isomorphism $T\\times G\\longrightarrow P$ .",
"Identifying $P$ with $T\\times G$ through this isomorphism, we see that $\\widetilde{\\mathcal {Z}}_{\\underline{\\omega }}$ is equivalent to $\\mathcal {Z}_{\\underline{\\omega }}$ .",
"Notice that if $\\mathcal {U}\\longrightarrow \\mathcal {V}$ is a $G$ -torsor then $\\mathcal {V}$ has affine diagonal (resp.",
"is separated) if and only if $\\mathcal {U}$ has the same property.",
"The “only if” part is clear.",
"The “if” part follows because $\\operatorname{\\textup {B}}G$ is separated, descent and the 2-Cartesian diagrams $ \\begin{tikzpicture}[xscale=1.8,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {U}}; \\node (A0_1) at (1, 0) {\\mathcal {U}\\times \\mathcal {U}}; \\node (A0_2) at (2, 0) {\\operatorname{\\textup {Spec}}k}; \\node (A1_0) at (0, 1) {\\mathcal {V}}; \\node (A1_1) at (1, 1) {\\mathcal {V}\\times _{\\operatorname{\\textup {B}}G}\\mathcal {V}}; \\node (A1_2) at (2, 1) {\\operatorname{\\textup {B}}G}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_1) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A0_2) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A0_2); \\end{tikzpicture} $ From this remark and the above lemma, the proof of REF reduces to: Lemma 3.12 The stacks $\\mathcal {Z}_{\\underline{\\omega _{*}}}$ form a direct system of DM stacks of finite type over $k$ with affine diagonal, with finite and universally injective transition maps, and with a direct system of étale atlases $Z_{*}\\longrightarrow \\mathcal {Z}_{\\underline{\\omega _{*}}}$ from affine schemes.",
"Moreover if all $\\mathcal {Y}_{*}$ are separated so are the $\\mathcal {Z}_{\\underline{\\omega _{*}}}$ and if $Y_{n}\\longrightarrow \\mathcal {Y}_{n}$ is finite and étale then $Z_{*}\\longrightarrow \\mathcal {Z}_{\\underline{\\omega _{*}}}$ can be chosen to be finite and étale.",
"To prove this lemma, we will describe $\\mathcal {Z}_{\\underline{\\omega }}$ by using fiber products of simpler stacks.",
"In what follows, for a stack $\\mathcal {K}$ and a finite set $I$ , we denote by $\\mathcal {K}^{I}$ the product $\\prod _{i\\in I}\\mathcal {K}$ and identify its objects over a scheme $T$ with the morphisms $T\\times I=\\sqcup _{i}T\\rightarrow \\mathcal {K}$ .",
"Let $\\mathcal {W}_{\\underline{\\omega }}$ be the stack of pairs $(\\eta ,\\mu )$ where $\\eta \\colon T\\times G\\longrightarrow \\mathcal {Y}_{u}$ and $\\mu $ is a natural isomorphism as in (REF ), but not necessarily satisfying the compatibility imposed on objects of $\\mathcal {Z}_{\\underline{\\omega }}$ .",
"Remark 3.13 Let $F,G\\colon \\mathcal {W}_{1}\\longrightarrow \\mathcal {W}_{0}$ be two maps of stacks and denote by $\\mathcal {W}_{2}$ the stack of pairs $(w,\\zeta )$ were $w\\in \\mathcal {W}_{1}(T)$ and $\\zeta \\colon G(w)\\longrightarrow F(w)$ is an isomorphism in $\\mathcal {W}_{0}(T)$ .",
"Then there is a 2-Cartesian diagram $ \\begin{tikzpicture}[xscale=2,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {W}_2}; \\node (A0_1) at (1, 0) {\\mathcal {W}_1}; \\node (A1_0) at (0, 1) {\\mathcal {W}_1}; \\node (A1_1) at (1, 1) {\\mathcal {W}_1\\times \\mathcal {W}_0}; (A0_0) edge [->]node [auto] {\\scriptstyle {p}} (A0_1); (A1_0) edge [->]node [auto] {\\scriptstyle {\\Gamma _F}} (A1_1); (A0_1) edge [->]node [auto] {\\scriptstyle {\\Gamma _G}} (A1_1); (A0_0) edge [->]node [auto] {\\scriptstyle {p}} (A1_0); \\end{tikzpicture} $ where $\\Gamma _{*}$ denotes the graph and $p$ the projection.",
"Notice that the sheaf of isomorphisms of an object of a fiber product can be expressed as fiber products of the sheaves of isomorphisms of its factors.",
"In particular, if $\\mathcal {W}_{0},\\mathcal {W}_{1}$ have affine diagonal, then $\\mathcal {W}_{2}$ has affine diagonal.",
"If $\\mathcal {W}_{1}$ has affine diagonal and $F$ is affine then $p$ is also affine.",
"This is because the graph $\\Gamma _{F}$ can be factors as the diagonal $\\mathcal {W}_{1}\\longrightarrow \\mathcal {W}_{1}\\times \\mathcal {W}_{1}$ followed by $\\textup {id}\\times F\\colon \\mathcal {W}_{1}\\times \\mathcal {W}_{1}\\longrightarrow \\mathcal {W}_{1}\\times \\mathcal {W}_{0}$ .",
"This remark particularly gives: Lemma 3.14 Let $\\Phi \\colon \\mathcal {Y}_{u}^{G}\\rightarrow \\mathcal {Y}_{v}^{G\\times G}$ be the morphism sending $\\eta \\colon T\\times G\\rightarrow \\mathcal {Y}_{u}$ to the composition $\\Phi (\\eta )\\colon T\\times G\\times G\\xrightarrow{}T\\times G\\xrightarrow{}\\mathcal {Y}_{u}\\xrightarrow{}\\mathcal {Y}_{v}$ and let $\\Psi \\colon \\mathcal {Y}_{u}^{G}\\rightarrow \\mathcal {Y}_{v}^{G\\times G}$ be the morphism sending $\\eta \\colon T\\times G\\rightarrow \\mathcal {Y}_{u}$ to the composition $\\Psi (\\eta )\\colon T\\times G\\times G\\xrightarrow{}\\mathcal {Y}_{u}\\times G\\xrightarrow{}\\mathcal {Y}_{v}.$ Let $\\Gamma _{\\Phi },\\Gamma _{\\Psi }\\colon \\mathcal {Y}_{u}^{G}\\rightarrow \\mathcal {Y}_{u}^{G}\\times \\mathcal {Y}_{v}^{G\\times G}$ be their respective graph morphisms.",
"Then $\\mathcal {W}_{\\underline{\\omega }}\\simeq \\mathcal {Y}_{u}^{G}\\times _{\\Gamma _{\\Phi },\\mathcal {Y}_{u}^{G}\\times \\mathcal {Y}_{v}^{G\\times G},\\Gamma _{\\Psi }}\\mathcal {Y}_{u}^{G}.$ Let $(\\eta ,\\mu )\\in \\mathcal {W}_{\\underline{\\omega }}(T)$ .",
"In the two diagrams, ${T\\times G\\times G\\times G[rr]^{\\textup {id}_{T}\\times m_{G}\\times \\textup {id}_{G}}[d]_{\\eta } & & @2[dll]_{\\mu \\times \\textup {id}}T\\times G\\times G[rr]^{\\textup {id}_{T}\\times m_{G}}[d]_{\\delta _{u,v}(\\eta )} & & @2[dll]_{\\delta _{v,w}(\\mu )}T\\times G[d]^{\\delta _{u,w}(\\eta )}\\\\\\mathcal {Y}_{u}\\times G\\times G[rr]_{\\omega \\times \\textup {id}_{G}} & & \\mathcal {Y}_{v}\\times G[rr]_{\\omega ^{\\prime }} & & \\mathcal {Y}_{w}}$ and ${T\\times G\\times G\\times G[rr]^{\\textup {id}_{T\\times G}\\times m_{G}}[d]_{\\eta } & & @2[dll]_{\\textrm {canonical}}T\\times G\\times G[rr]^{\\textup {id}_{T}\\times m_{G}}[d]_{\\delta _{u,v}(\\eta )} & & @2[dll]_{\\delta _{v,w}(\\mu )}T\\times G[d]^{\\delta _{u,w}(\\eta )}\\\\\\mathcal {Y}_{u}\\times G\\times G[rr]_{\\delta _{u,v}\\times m_{G}} & & \\mathcal {Y}_{v}\\times G[rr]_{\\omega ^{\\prime }} & & \\mathcal {Y}_{w}}$ the paths from $T\\times G\\times G\\times G$ to $\\mathcal {Y}_{w}$ through the upper right corner are identical; we denote this morphism $T\\times G\\times G\\times G\\rightarrow \\mathcal {Y}_{w}$ by $r(\\eta )$ .",
"As for the paths through the left bottom corner, there is a natural isomorphism between them given by $\\zeta $ (REF ) and $\\lambda _{h,g}$ (REF ).",
"We identify the two lower paths through this natural isomorphism and denote it by $s(\\eta )$ .",
"We denote the natural isomorphism $r(\\eta )\\rightarrow s(\\eta )$ induced from the former diagram by $\\alpha (\\mu )$ and the one from the latter diagram by $\\beta (\\eta )$ .",
"The compatibility (REF ) is nothing but $\\alpha (\\mu )=\\beta (\\mu )$ .",
"For a stack $\\mathcal {K}$ , we denote by $I(\\mathcal {K})$ its inertia stack.",
"An object of $I(\\mathcal {K})$ is a pair $(x,\\alpha )$ where $x$ is an object of $\\mathcal {K}$ and $\\alpha $ is an automorphism of $x$ .",
"There is an equivalence $I(\\mathcal {K})\\simeq \\mathcal {K}\\times _{\\Delta ,\\mathcal {K}\\times \\mathcal {K},\\Delta }\\mathcal {K}$ .",
"We have the forgetting morphism $I(\\mathcal {K})\\rightarrow \\mathcal {K}$ , which has the section $\\mathcal {K}\\rightarrow I(\\mathcal {K})$ , $x\\mapsto (x,\\textup {id})$ .",
"If $\\mathcal {K}$ is a DM stack of finite type with finite diagonal, then $I(\\mathcal {K})\\rightarrow \\mathcal {K}$ is finite and unramified and $\\mathcal {K}\\rightarrow I(\\mathcal {K})$ is a closed immersion.",
"Lemma 3.15 Let $\\Theta ,\\Lambda \\colon \\mathcal {W}_{\\underline{\\omega }}\\rightarrow I(\\mathcal {Y}_{w}^{G\\times G\\times G})$ be the functors sending an object $(\\eta ,\\mu )$ of $\\mathcal {W}_{\\underline{\\omega }}$ to $(r(\\eta ),\\beta (\\mu )^{-1}\\circ \\alpha (\\mu ))$ and to $(r(\\eta ),\\textup {id})$ respectively.",
"Consider also the functor $\\mathcal {Z}_{\\underline{\\omega }}\\rightarrow \\mathcal {Y}_{w}^{G\\times G\\times G}$ sending $(\\eta ,\\mu )$ to $r(\\eta )$ .",
"Then $\\mathcal {Z}_{\\underline{\\omega }}\\times _{\\mathcal {Y}_{w}^{G\\times G\\times G}}I(\\mathcal {Y}_{w}^{G\\times G\\times G})\\simeq \\mathcal {W}_{\\underline{\\omega }}\\times _{\\Gamma _{\\Theta },\\mathcal {W}_{\\underline{\\omega }}\\times I(\\mathcal {Y}_{w}^{G\\times G\\times G}),\\Gamma _{\\Lambda }}\\mathcal {W}_{\\underline{\\omega }}.$ From REF , the right hand side is regarded as the stack of pairs $((\\eta ,\\mu ),\\epsilon )$ such that $(\\eta ,\\mu )$ is an object of $\\mathcal {W}_{\\underline{\\omega }}$ and $\\epsilon $ is a natural isomorphism $\\Theta ((\\eta ,\\mu ))\\rightarrow \\Lambda ((\\eta ,\\mu ))$ .",
"Thus $\\epsilon $ is an isomorphism $r(\\eta )\\rightarrow r(\\eta )$ making the diagram ${r(\\eta )[d]_{\\beta (\\mu )^{-1}\\circ \\alpha (\\mu )}[r]^{\\epsilon } & r(\\eta )[d]^{\\textup {id}}\\\\r(\\eta )[r]_{\\epsilon } & r(\\eta )}$ commutative.",
"Therefore $\\beta (\\mu )^{-1}\\circ \\alpha (\\mu )=\\textup {id}$ , equivalently, the compatibility (REF ) holds, and $\\epsilon $ can be an arbitrary automorphism of $r(\\eta )$ .",
"This shows the equivalence of the lemma.",
"Lemma 3.16 The stack $\\mathcal {Z}_{\\underline{\\omega }}$ is a DM stack of finite type with affine diagonal and it is separated if all the $\\mathcal {Y}_{*}$ are separated.",
"Moreover, for functions $v,w\\colon \\mathbb {N}\\rightarrow \\mathbb {N}$ and $\\underline{\\omega _{u}}\\in \\mathcal {R}(u,v(u),w(u))$ , $u\\in \\mathbb {N}$ , the morphism $\\mathcal {Z}_{\\underline{\\omega _{u}}}\\rightarrow \\mathcal {Z}_{\\underline{\\omega _{u+1}}}$ is finite and universally injective.",
"If $\\mathcal {U}$ , $\\mathcal {V}$ and $\\mathcal {W}$ are DM stacks of finite type with affine (resp.",
"finite) diagonals, then so is $\\mathcal {U}\\times _{\\mathcal {W}}\\mathcal {V}$ .",
"Indeed, $\\mathcal {U}\\times \\mathcal {V}$ is a DM stack of finite type with affine (resp.",
"finite) diagonal and $\\mathcal {U}\\times _{\\mathcal {W}}\\mathcal {V}\\rightarrow \\mathcal {U}\\times \\mathcal {V}$ is an affine (resp.",
"finite) morphism since it is a base change of the diagonal $\\mathcal {W}\\rightarrow \\mathcal {W}\\times \\mathcal {W}$ .",
"Hence $\\mathcal {U}\\times _{\\mathcal {W}}\\mathcal {V}$ is a DM stack of finite type with affine (resp.",
"finite) diagonal.",
"From REF and REF , $\\mathcal {Z}_{\\underline{\\omega }}\\times _{\\mathcal {Y}_{w}^{G\\times G\\times G}}I(\\mathcal {Y}_{w}^{G\\times G\\times G})$ is a DM stack of finite type with affine (resp.",
"finite, provided that all $\\mathcal {Y}_{*}$ are separated) diagonal.",
"Since the section $\\mathcal {Y}_{w}^{G\\times G\\times G}\\rightarrow I(\\mathcal {Y}_{w}^{G\\times G\\times G})$ is a closed immersion, the same conclusion holds for $\\mathcal {Z}_{\\underline{\\omega }}$ .",
"From REF , we can conclude that the morphism $I(\\mathcal {Y}_{w(u)}^{G\\times G\\times G})\\rightarrow I(\\mathcal {Y}_{w(u+1)}^{G\\times G\\times G})$ is finite and universally injective.",
"From REF , REF and REF , $\\mathcal {Z}_{\\underline{\\omega _{u}}}\\times _{\\mathcal {Y}_{w(u)}^{G\\times G\\times G}}I(\\mathcal {Y}_{w(u)}^{G\\times G\\times G})\\rightarrow \\mathcal {Z}_{\\underline{\\omega _{u+1}}}\\times _{\\mathcal {Y}_{w(u+1)}^{G\\times G\\times G}}I(\\mathcal {Y}_{w(u+1)}^{G\\times G\\times G})$ is finite and universally injective, and so is the composition $\\mathcal {Z}_{\\underline{\\omega _{u}}}\\rightarrow \\mathcal {Z}_{\\underline{\\omega _{u}}}\\times _{\\mathcal {Y}_{w(u)}^{G\\times G\\times G}}I(\\mathcal {Y}_{w(u)}^{G\\times G\\times G})\\rightarrow \\mathcal {Z}_{\\underline{\\omega _{u+1}}}\\times _{\\mathcal {Y}_{w(u+1)}^{G\\times G\\times G}}I(\\mathcal {Y}_{w(u+1)}^{G\\times G\\times G}).$ The morphism $\\mathcal {Z}_{\\underline{\\omega _{u}}}\\rightarrow \\mathcal {Z}_{\\underline{\\omega _{u+1}}}$ factorizes this and hence is finite and universally injective.",
"If $\\mathcal {U}\\rightarrow \\mathcal {V}$ is a map of stacks and $\\mathcal {V}$ has affine diagonal then $\\mathcal {U}\\times _{\\mathcal {V}}\\mathcal {U}\\rightarrow \\mathcal {U}\\times \\mathcal {U}$ is affine, because it is the base change of $\\Delta \\colon \\mathcal {V}\\rightarrow \\mathcal {V}\\times \\mathcal {V}$ along $\\mathcal {U}\\times \\mathcal {U}\\rightarrow \\mathcal {V}\\times \\mathcal {V}$ .",
"Therefore the morphism $\\mathcal {Z}_{\\underline{\\omega _{u}}}\\rightarrow \\mathcal {W}_{\\underline{\\omega _{u}}}\\times \\mathcal {W}_{\\underline{\\omega _{u}}}\\rightarrow (\\mathcal {Y}_{u}^{G}\\times \\mathcal {Y}_{u}^{G})\\times (\\mathcal {Y}_{u}^{G}\\times \\mathcal {Y}_{u}^{G})$ induced from equivalences in REF and REF is affine.",
"Pulling back a direct system of étale atlases for $(\\mathcal {Y}_{u}^{G}\\times \\mathcal {Y}_{u}^{G})\\times (\\mathcal {Y}_{u}^{G}\\times \\mathcal {Y}_{u}^{G})$ to $\\mathcal {Z}_{\\underline{\\omega _{u}}}$ , we obtain a system of atlases as in REF , which completes the proof of REF and the one of REF ."
],
[
"The stack of formal $G$ -torsors",
"We fix a field $k$ and an étale group scheme $G$ over $k$ .",
"In this section we will introduce and study the stack of formal $G$ -torsors.",
"Definition 4.1 We denote by $\\Delta _{G}$ the category fibered in groupoids over $\\operatorname{\\textup {Aff}}/k$ whose objects $B$ are $\\Delta _{G}(B)=\\operatorname{\\textup {B}}G(B((t)))$ .",
"Unfortunately, as we will see, $\\Delta _{G}$ is too big and we need to add some further requirements on the torsors considered.",
"Definition 4.2 A vector bundle $M$ on $B((t))$ is free locally on $B$ if there exists an fpqc covering $\\lbrace B\\longrightarrow B_{i}\\rbrace _{i\\in I}$ such that $M\\otimes _{B((t))}B_{i}((t))$ is a free $B_{i}((t))$ -module.",
"We define by $\\Delta _{G}^{*}$ the full subfibered category of $\\Delta _{G}$ of $G$ -torsors ${A}$ over $B((t))$ such that for all field extensions $L/k$ , all one dimensional representations $V$ of $G_{L}=G\\times _{k}L$ and all $B\\otimes _{k}L$ -algebras $R$ the $R((t))$ -module $[({A}\\otimes _{B((t))}R((t)))\\otimes _{L}V]^{G_{L}}$ is a vector bundle which is free locally on $R$ .",
"Remark 4.3 Notice that the $R((t))$ -module $[({A}\\otimes _{B((t))}R((t)))\\otimes V]^{G_{L}}$ is automatically a line bundle and the formation of invariants commutes with arbitrary base changes.",
"More generally if $H$ is a finite group scheme over a field $F$ , $C\\longrightarrow D$ is a map of $F$ -algebras, $W\\in \\operatorname{\\textup {Rep}}_{F}H$ and ${B}$ is an $H$ -torsor over $C$ then $({B}\\otimes W)^{H}\\otimes _{C}D\\simeq [({B}\\otimes _{C}D)\\otimes W]^{H}$ and it is a vector bundle of rank $\\dim _{k}W$ .",
"Indeed the problem is fpqc local on $C$ , so that one can assume that ${B}\\simeq C[H]$ and use the canonical isomorphism $U\\simeq (U\\otimes C[H])^{H}$ which holds for all $C$ -modules $U$ with a coaction of $C[H]$ (see [6]).",
"Let $L_{0}/k$ be a finite field extension such that $G_{0}=G\\times _{k}L_{0}$ is constant and contains all the $\\sharp G_{0}$ -roots of unity, $B$ a $k$ -algebra and ${A}\\in \\Delta _{G}(B)$ .",
"Set also $B_{0}=B\\otimes _{k}L_{0}$ .",
"We claim that ${A}\\in \\Delta _{G}^{*}(B)$ if and only if for all one dimensional representations $W$ of $G_{0}$ the $B_{0}((t))$ -module $[({A}\\otimes _{B((t))}B_{0}((t)))\\otimes W)^{G_{0}}$ is free locally on $B_{0}$ .",
"One easily see that we must show that if $L/L_{0}$ is a field extension all one dimensional representations $V$ of $G_{L}=G\\times _{k}L$ comes from $G_{0}$ .",
"Since $G_{L}$ is constant the representation $V$ is determined by a character $G(L)\\longrightarrow L^{*}$ .",
"Since $G_{0}$ is also constant and $L_{0}$ has all the $\\sharp G_{0}$ -roots of unity there is a character $G(L)\\simeq G_{0}(L_{0})\\longrightarrow L_{0}^{*}\\subseteq L^{*}$ inducing the given one.",
"Remark 4.4 If $\\operatorname{\\textup {char}}k=p$ and $G$ is a $p$ -group then $\\Delta _{G}=\\Delta _{G}^{*}$ .",
"A one dimensional representation of a $p$ -group is trivial: the image of a map $G\\longrightarrow \\mathbb {G}_{m}$ would be a diagonalizable étale $p$ -group scheme and thus trivial.",
"Moreover if ${A}\\in \\Delta _{G}(B)$ and $V$ is the trivial $B$ -representation of $G$ then $({A}\\otimes V)^{G}\\simeq {A}^{G}\\simeq B((t))$ is free.",
"Remark 4.5 By construction we have that if $k^{\\prime }/k$ is a field extension then $\\Delta _{G}\\times _{k}k^{\\prime }\\simeq \\Delta _{G\\times _{k}k^{\\prime }}$ and $\\Delta _{G}^{*}\\times _{k}k^{\\prime }\\simeq \\Delta _{G\\times _{k}k^{\\prime }}^{*}$ .",
"Corollary 4.6 The fiber category $\\Delta _{G}$ is a pre-stack in the fpqc topology.",
"Let $D_{1},D_{2}\\in \\Delta _{G}(B)=\\operatorname{\\textup {B}}G(B((t)))$ .",
"We must show that $I=\\operatorname{\\underline{\\textup {Iso}}}_{\\Delta _{G}}(D_{1},D_{2})\\colon \\operatorname{\\textup {Aff}}/B\\longrightarrow (\\textup {Sets})$ is an fpqc sheaf.",
"By REF , $\\operatorname{\\underline{\\textup {Hom}}}_{B((t))}(D_{1},D_{2})$ is a sheaf, so that in particular $I$ is separated.",
"Thus we must show that if $B\\longrightarrow C$ is an fpqc covering, $\\phi \\colon D_{1}\\longrightarrow D_{2}$ is a map and $\\phi \\otimes C((t))$ is a $G$ -equivariant morphisms of $C((t))$ -algebras, then $\\phi $ is also $G$ -equivariant.",
"But this is obvious since $D_{i}$ is a subset of $D_{i}\\otimes C((t))$ .",
"Definition 4.7 If $\\mathcal {X}$ is a category fibered in groupoids over $\\mathbb {F}_{p}$ then its Frobenius $F_{\\mathcal {X}}\\colon \\mathcal {X}\\longrightarrow \\mathcal {X}$ is the functor mapping $\\xi \\in \\mathcal {X}(B)$ to $F_{B}^{*}(\\xi )\\in \\mathcal {X}(B)$ , where $F_{B}\\colon B\\longrightarrow B$ is the absolute Frobenius.",
"The Frobenius is $\\mathbb {F}_{p}$ -linear, natural in $\\mathcal {X}$ and coincides with the usual Frobenius if $\\mathcal {X}$ is a scheme.",
"A category fibered in groupoid $\\mathcal {X}$ over $\\mathbb {F}_{p}$ is called perfect if the Frobenius $F_{\\mathcal {X}}\\colon \\mathcal {X}\\longrightarrow \\mathcal {X}$ is an equivalence.",
"Example 4.8 As a consequence of REF DM stacks étale over a perfect field are perfect.",
"Proposition 4.9 If $k$ is perfect the fiber category $\\Delta _{G}$ is perfect.",
"By REF the functor $\\operatorname{\\textup {B}}G(B((t)))\\longrightarrow \\operatorname{\\textup {B}}G(B((t)))$ induced by the Frobenius $F_{B}\\colon B\\longrightarrow B$ is an equivalence: the $p$ -th powers of elements in $B((t))$ are in the image of $F_{B}\\colon B((t))\\longrightarrow B((t))$ and therefore the spectrum of this map is integral and a universal homeomorphism.",
"Another example of a perfect object that will be used later is the following: Definition 4.10 If $X$ is a functor $\\operatorname{\\textup {Aff}}/\\mathbb {F}_{p}\\longrightarrow (\\textup {Sets})$ we denote by $X^{\\infty }$ the direct limit of the direct system of Frobenius morphisms $X{F}X{F}\\cdots $ .",
"Notice that if $X$ is a $k$ -pre-sheaf then $X^{\\infty }$ does not necessarily have a $k$ -structure unless $k$ is perfect.",
"Proposition 4.11 Let $H$ be a central subgroup of $G$ .",
"Then the equivalence $\\operatorname{\\textup {B}}G\\times \\operatorname{\\textup {B}}H\\longrightarrow \\operatorname{\\textup {B}}G\\times _{\\operatorname{\\textup {B}}(G/H)}\\operatorname{\\textup {B}}G$ of REF induces an equivalence $\\Delta _{G}\\times \\Delta _{H}\\longrightarrow \\Delta _{G}\\times _{\\Delta _{G/H}}\\Delta _{G}$ .",
"If $\\mathcal {X}$ is a fibered category with a map $\\mathcal {X}\\longrightarrow \\Delta _{G}$ then we have a 2-Cartesian diagram $ \\begin{tikzpicture}[xscale=2.1,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {X}\\times \\Delta _H}; \\node (A0_1) at (1, 0) {\\Delta _G}; \\node (A1_0) at (0, 1) {\\mathcal {X}}; \\node (A1_1) at (1, 1) {\\Delta _{G/H}}; (A0_0) edge [->]node [auto] {\\scriptstyle {\\alpha }} (A0_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_0) edge [->]node [auto] {\\scriptstyle {\\operatorname{\\textup {pr}}_1}} (A1_0); \\end{tikzpicture} $ where $\\alpha $ is given by $\\mathcal {X}\\times \\Delta _{H}\\longrightarrow \\Delta _{G}\\times \\Delta _{H}=\\Delta _{G\\times H}\\longrightarrow \\Delta _{G}$ and the last map is induced by the multiplication $G\\times H\\longrightarrow G$ .",
"The first claim is clear.",
"For the other we have the following Cartesian diagrams $ \\begin{tikzpicture}[xscale=2.1,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {X}\\times \\Delta _H}; \\node (A0_1) at (1, 0) {\\Delta _G\\times \\Delta _H}; \\node (A0_2) at (2, 0) {\\Delta _G}; \\node (A1_0) at (0, 1) {\\mathcal {X}}; \\node (A1_1) at (1, 1) {\\Delta _G}; \\node (A1_2) at (2, 1) {\\Delta _{G/H}}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A0_2); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_1) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A0_2) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A0_0) edge [->]node [auto] {\\scriptstyle {\\operatorname{\\textup {pr}}_1}} (A1_0); (A0_1) edge [->]node [auto] {\\scriptstyle {\\operatorname{\\textup {pr}}_1}} (A1_1); \\end{tikzpicture} $"
],
[
"The group $G=\\protect \\mathbb {Z}/p\\protect \\mathbb {Z}$ in characteristic {{formula:86cf87b1-c831-43fe-8fe9-abf7eda42b5d}} .",
"In this section we consider $G=\\mathbb {Z}/p\\mathbb {Z}$ over $k=\\mathbb {F}_{p}$ .",
"Notice that in this case $\\Delta _{\\mathbb {Z}/p\\mathbb {Z}}=\\Delta _{\\mathbb {Z}/p\\mathbb {Z}}^{*}$ by REF .",
"Let $C$ be an $\\mathbb {F}_{p}$ -algebra.",
"By Artin-Schreier a $\\mathbb {Z}/p\\mathbb {Z}$ -torsor over $C$ is of the form $C[X]/(X^{p}-X-c)$ , where $c\\in C$ and the action is induced by $X\\longmapsto X+f$ for $f\\in \\mathbb {F}_{p}$ .",
"Lemma 4.12 Let $c,d\\in C$ .",
"Then $ \\begin{tikzpicture}[xscale=5.5,yscale=-0.7] \\node (A0_0) at (0, 0) {\\lbrace u\\in C\\ | \\ u^{p}-u+c=d\\rbrace }; \\node (A0_1) at (1, 0) {\\operatorname{\\textup {Iso}}_{C}^{\\mathbb {Z}/p\\mathbb {Z}}(\\frac{C[X]}{(X^{p}-X-c)},\\frac{C[X]}{(X^{p}-X-d)})}; \\node (A1_0) at (0, 1) {u}; \\node (A1_1) at (1, 1) {(X\\longmapsto X-u)}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A1_0) edge [serif cm->]node [auto] {\\scriptstyle {}} (A1_1); \\end{tikzpicture} $ is bijective.",
"The map in the statement is well defined and it induces a morphism $\\operatorname{\\textup {Spec}}(C[X]/(X^{p}-X-(d-c)))\\longrightarrow \\operatorname{\\underline{\\textup {Iso}}}^{\\mathbb {Z}/p\\mathbb {Z}}(C[X]/(X^{p}-X-c),C[X]/(X^{p}-X-d))=I$ The group $\\mathbb {Z}/p\\mathbb {Z}$ acts on both sides and the map is equivariant.",
"Since both sides are $\\mathbb {Z}/p\\mathbb {Z}$ -torsors it follows that the above map is an isomorphism.",
"Notation 4.13 If $C$ is an $\\mathbb {F}_{p}$ -algebra we identify $(\\operatorname{\\textup {B}}\\mathbb {Z}/p\\mathbb {Z})(C)$ with the category whose objects are elements of $C$ and a morphism $c{u}d$ is an element $u\\in C$ such that $u^{p}-u+c=d$ .",
"Composition is given by the sum, identities correspond to $0\\in C$ and the inverse of $u\\in C$ is $-u$ .",
"In particular we see that if $c\\in (\\operatorname{\\textup {B}}\\mathbb {Z}/p\\mathbb {Z})(C)$ then $c\\simeq c^{p}$ .",
"Lemma 4.14 Any element $b\\in tB[[t]]$ is of the form $u^{p}-u$ for a unique element $u\\in tB[[t]]$ .",
"Let $b,u\\in tB[[t]]$ and $b_{s},u_{s}$ for $s\\in \\mathbb {N}$ their coefficients, so that $b_{0}=u_{0}=0$ .",
"We extend the symbol $b_{s},u_{s}$ for $s\\in \\mathbb {Q}$ by setting $b_{s}=u_{s}=0$ if $s\\notin \\mathbb {N}$ .",
"The equation $u^{p}-u=b$ translates in $b_{s}=u_{s/p}^{p}-u_{s}$ for all $s\\in \\mathbb {N}$ .",
"A simple computation shows that, given $b$ , the only solution of the system is $u_{s}=-\\sum _{n\\in \\mathbb {N}}b_{s/p^{n}}^{p^{n}}$ Notation 4.15 In what follows we set $S=\\lbrace n\\ge 1\\ | \\ p\\nmid n\\rbrace $ and $\\mathbb {A}^{(S)}\\colon \\operatorname{\\textup {Aff}}/\\mathbb {F}_{p}\\longrightarrow (\\textup {Sets})$ where $\\mathbb {A}^{(S)}(B)$ is the set of maps $b\\colon S\\longrightarrow B$ such that $\\lbrace s\\in S\\ | \\ b_{s}\\ne 0\\rbrace $ is finite.",
"Given $k\\in \\mathbb {N}$ we set $\\phi _{k}\\colon \\mathbb {A}^{(S)}\\longrightarrow \\Delta _{\\mathbb {Z}/p\\mathbb {Z}},\\ \\phi _{k}(b)=\\sum _{s\\in S}b_{s}t^{-sp^{k}}\\in B((t))=\\Delta _{\\mathbb {Z}/p\\mathbb {Z}}(B)$ and $\\psi _{k}\\colon \\mathbb {A}^{(S)}\\times \\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})\\longrightarrow \\Delta _{\\mathbb {Z}/p\\mathbb {Z}}$ , $\\psi _{k}(b,b_{0})=\\phi _{k}(b)+b_{0}$ .",
"Notice that for all $b\\in \\mathbb {A}^{(S)}(B)$ and $b_{0}\\in B$ there is a natural map $\\psi _{k+1}\\circ (F_{\\mathbb {A}^{(S)}}\\times \\textup {id}_{\\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})})(b,b_{0}){-\\phi _{k}(b)}\\psi _{k}(b,b_{0})$ which therefore induces a functor $(\\mathbb {A}^{(S)})^{\\infty }\\times \\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})\\longrightarrow \\Delta _{\\mathbb {Z}/p\\mathbb {Z}}$ .",
"Theorem 4.16 The functor $(\\mathbb {A}^{(S)})^{\\infty }\\times \\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})\\longrightarrow \\Delta _{\\mathbb {Z}/p\\mathbb {Z}}$ is an equivalence of fibered categories.",
"Essential surjectivity.",
"Let $b(t)=\\sum _{j}b_{j}t^{j}\\in \\Delta _{\\mathbb {Z}/pZ}(B)$ .",
"By REF and the definition of the map in the statement we can assume that $b_{j}=0$ for $j>0$ .",
"Let $k\\in \\mathbb {N}$ be a sufficiently large positive integer such that every $j<0$ with $b_{j}\\ne 0$ is written as $-j=p^{k-m(j)}s(j)$ for some $m(j)\\ge 0$ and $s(j)\\in S$ .",
"Then $b_{j}t^{j}\\simeq (b_{j}t^{j})^{p^{m(j)}}=b_{j}^{p^{m(j)}}t^{-p^{k}s(j)}$ if $b_{j}\\ne 0$ .",
"We see therefore that, up to change $b$ with an isomorphic element, $b$ can be written as $\\psi _{k}(c)$ for some $c\\in (\\mathbb {A}^{(S)}\\times \\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z}))(B)$ .",
"Faithfulness.",
"Let $([b,k],b_{0}),([c,k],c_{0})\\in (\\mathbb {A}^{(S)})^{\\infty }(B)\\times \\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})(B)$ and $u,v\\colon (b,b_{0})\\longrightarrow (c,c_{0})$ two morphisms in $\\mathbb {A}^{(S)}\\times \\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})$ , that is $b=c$ and $u^{p}-u=v^{p}-v=c_{0}-b_{0}$ with $u,v\\in B$ .",
"If $\\psi _{k}(u)=\\psi _{k}(v)$ then $u=v$ by definition of $\\psi _{k}$ as desired.",
"Fullness.",
"Let $([b,k],b_{0}),([c,k^{\\prime }],c_{0})\\in (\\mathbb {A}^{(S)})^{\\infty }(B)\\times \\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})(B)$ and let $u\\colon \\psi _{k}(b,b_{0})\\longrightarrow \\psi _{k^{\\prime }}(c,c_{0})$ be a map in $\\Delta _{\\mathbb {Z}/p\\mathbb {Z}}$ .",
"We want to lift this morphism to $(\\mathbb {A}^{(S)})^{\\infty }\\times \\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})$ .",
"We can assume $k=k^{\\prime }$ .",
"The element $u=\\sum _{q}u_{q}t^{q}\\in B((t))$ can be written as $u=u_{-}+u_{+}$ , where $u_{-}=\\sum _{q<0}u_{q}t^{q}$ and $u_{+}=\\sum _{q\\ge 0}u_{q}t^{q}$ .",
"In particular we obtain that $u_{-}^{p}-u_{-}=\\phi _{k}(c)-\\phi _{k}(b)$ and $u_{+}^{p}-u_{+}=c_{0}-b_{0}$ .",
"By REF it follows that $u_{+}\\in B$ .",
"It suffices to show that $u_{-}=0$ .",
"To see this, we first show that $c-b$ is nilpotent.",
"We have $\\phi _{k}(c)\\simeq \\phi _{k}(b)$ and, applying $F_{B}$ to both side we get $\\phi _{0}(b)\\simeq \\phi _{0}(b)^{p^{k}}=F_{B}^{k}(\\phi _{k}(b))\\simeq F_{B}^{k}(\\phi _{k}(c))=\\phi _{0}(c)^{p^{k}}\\simeq \\phi _{0}(c)$ .",
"Thus there exists $v=\\sum _{q<0}v_{q}t^{q}\\in B((t))$ such that $\\phi _{0}(c)-\\phi _{0}(b)=\\sum _{s\\in S}(c_{s}-b_{s})t^{-s}=v^{p}-v=\\sum _{q<0}(v_{q/p}^{p}-v_{q})t^{q}$ where we set $v_{l}=0$ if $l\\in \\mathbb {Q}-\\mathbb {Z}_{<0}$ .",
"In particular for $s\\in S$ and $l\\in \\mathbb {N}$ we obtain $b_{s}-c_{s}=v_{-s}$ and $v_{-sp^{l}}=v_{-s}^{p^{l}}$ .",
"Since $v_{-sp^{l}}=0$ for $l\\gg 0$ we see that $b_{s}-c_{s}$ is nilpotent.",
"This means that there exists $j>0$ such that $F_{\\mathbb {A}^{(S)}}^{j}(b)=F_{\\mathbb {A}^{(S)}}^{j}(c)$ .",
"Thus, up to replace $k$ by $k+j$ , we can assume $b=c$ , so that $u_{-}^{p}=u_{-}$ .",
"If $u_{-}=\\sum _{q<0}u_{q}t^{q}$ and we put $u_{q}=0$ for $q\\notin \\mathbb {Z}$ then we have $u_{q}=(u_{q/p^{l}})^{p^{l}}$ for every $q\\in \\mathbb {Q}$ with $q<0$ and $l\\in \\mathbb {N}$ .",
"For each $q$ , taking a sufficiently large $l$ with $q/p^{l}\\notin \\mathbb {Z}$ , we see $u_{q}=0$ as desired.",
"Remark 4.17 The addition $\\mathbb {Z}/p\\mathbb {Z}\\times \\mathbb {Z}/p\\mathbb {Z}\\longrightarrow \\mathbb {Z}/p\\mathbb {Z}$ induces maps $\\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})\\times \\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})\\longrightarrow \\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})$ and $\\Delta _{\\mathbb {Z}/p\\mathbb {Z}}\\times \\Delta _{\\mathbb {Z}/p\\mathbb {Z}}\\longrightarrow \\Delta _{\\mathbb {Z}/p\\mathbb {Z}}$ .",
"The ind-scheme $(\\mathbb {A}^{(S)})^{\\infty }$ also has a natural group structure by addition.",
"Notice that the functor in the last theorem preserves the induced “group structure” on both sides.",
"This is because the maps $\\psi _{k}$ preserve the sum and the Frobenius of $\\mathbb {A}^{(S)}$ is a group homomorphism.",
"In particular the induced map from $(\\mathbb {A}^{(S)})^{\\infty }$ to the coarse ind-algebraic space of $\\Delta _{\\mathbb {Z}/p\\mathbb {Z}}$ is an isomorphism of sheaves of groups.",
"Remark 4.18 If $B$ is an $\\mathbb {F}_{p}$ -algebra, $G$ is any constant $p$ -group and $H$ is a central subgroup consisting of elements of order at most $p$ then any map $\\operatorname{\\textup {Spec}}B\\longrightarrow \\Delta _{G/H}$ lifts to a map $\\operatorname{\\textup {Spec}}B\\longrightarrow \\Delta _{G}$ .",
"More generally any $G/H$ -torsor over $B$ extends to a $G$ -torsor.",
"This follows from the fact that there is an exact sequence of sets $\\operatorname{\\textup {H}}^{1}(B,G)\\longrightarrow \\operatorname{\\textup {H}}^{1}(B,G/H)\\longrightarrow \\operatorname{\\textup {H}}^{2}(B,H)=0$ .",
"The last vanishing follows because $H\\simeq (\\mathbb {Z}/p\\mathbb {Z})^{r}$ for some $r$ and using the Artin-Schreier sequence.",
"Corollary 4.19 If $G$ is an étale $p$ -group scheme over a field $k$ then $\\Delta _{G}=\\Delta _{G}^{*}$ is a stack in the fpqc topology.",
"If $B$ is a $k$ -algebra and $A/k$ is a finite $k$ -algebra then $(B\\otimes _{k}A)((t))\\simeq B((t))\\otimes _{k}A$ by REF .",
"Therefore $\\Delta _{G}$ satisfies descent along coverings of the form $B\\longrightarrow B\\otimes _{k}A$ .",
"This implies that it is enough to show that $\\Delta _{G}\\times _{k}L\\simeq \\Delta _{G\\times _{k}L}$ is a stack, where $L/k$ is a finite field extension such that $G\\times _{k}L$ is constant.",
"Again using base change, we can assume $k=\\mathbb {F}_{p}$ and $G$ a constant $p$ -group.",
"If $\\sharp G=p^{l}$ we proceed by induction on $l$ .",
"If $l=1$ then $\\Delta _{\\mathbb {Z}/p\\mathbb {Z}}\\simeq (\\mathbb {A}^{(S)})^{\\infty }\\times \\operatorname{\\textup {B}}(\\mathbb {Z}/p\\mathbb {Z})$ which is a product of stacks.",
"For a general $G$ let $H$ a non-trivial central subgroup.",
"By induction $\\Delta _{G/H}$ is a stack and it is enough to show that all base change of $\\Delta _{G}\\longrightarrow \\Delta _{G/H}$ along a map $\\operatorname{\\textup {Spec}}B\\longrightarrow \\Delta _{G/H}$ is a stack.",
"This fiber product is $\\operatorname{\\textup {Spec}}B\\times \\Delta _{H}$ thanks to REF and REF , which is a stack by inductive hypothesis."
],
[
"Tame cyclic case",
"Let $k$ be a field and $n\\in \\mathbb {N}$ such that $n\\in k^{*}$ .",
"The aim of this section is to prove Theorem REF .",
"Set $G=\\mu _{n}$ , the group of $n$ -th roots of unity, which is a finite and étale group scheme over $k$ .",
"In particular $\\Delta _{G}(B)$ can be seen as the category of pairs $(\\mathcal {L},\\sigma )$ where $\\mathcal {L}$ is an invertible sheaf over $B((t))$ and $\\sigma \\colon \\mathcal {L}^{\\otimes n}\\longrightarrow B((t))$ is an isomorphism.",
"When $\\mathcal {L}=B((t))$ the isomorphism $\\sigma $ will often be thought of as an element $\\sigma \\in B((t))^{*}$ .",
"Lemma 4.20 The subfibered category $\\Delta _{G}^{*}$ of $\\Delta _{G}$ is the subfibered category of pairs $(\\mathcal {L},\\sigma )\\in \\Delta _{G}(B)$ such that $\\mathcal {L}$ is free locally on $B$ .",
"The $G$ -torsor corresponding to $(\\mathcal {L},\\sigma )\\in \\Delta _{G}(B)$ is ${A}=\\bigoplus _{m\\in \\mathbb {Z}/n\\mathbb {Z}}\\mathcal {L}^{m}$ where the multiplication is induced by $\\sigma $ .",
"The one dimensional representations of $G$ over $k$ are of the form $V_{m}$ where $V_{m}=k$ with the action of $G=\\mu _{n}$ via the character $m\\in \\mathbb {Z}/m\\mathbb {Z}\\simeq \\operatorname{\\textup {Hom}}(\\mu _{n},\\mathbb {G}_{m})$ .",
"In particular $({A}\\otimes V_{m})^{\\mu _{n}}\\simeq \\mathcal {L}^{-m}$ .",
"This easily implies the result.",
"Lemma 4.21 For a $k$ -algebra $B$ , we have $\\mu _{n}(B)=\\lbrace b\\in B^{*}\\mid b^{n}=1\\rbrace =\\lbrace b\\in B((t))^{*}\\mid b^{n}=1\\rbrace =\\mu _{n}(B((t))).$ Let $L$ and $R$ denote the left and right sides respectively.",
"Obviously $L\\subset R$ .",
"It is also easy to see $R\\cap B[[t]]=L$ .",
"Thus it suffices to show that $R\\subset B[[t]]$ .",
"Conversely, we suppose that it was not the case.",
"We define the naive order $\\operatorname{\\textup {ord}}_{naive}(a)$ of $a=\\sum _{i\\in \\mathbb {Z}}a_{i}t^{i}\\in B[[t]]$ as $\\min \\lbrace i\\mid a_{i}\\ne 0\\rbrace $ .",
"Elements outside $B[[t]]$ have negative naive orders and choose an element $c=\\sum _{i\\in \\mathbb {Z}}c_{i}t^{i}\\in R\\setminus B[[t]]$ such that $\\operatorname{\\textup {ord}}_{naive}(c)$ attains the maximum, say $i_{0}<0$ .",
"Taking derivatives of $c^{n}=1$ , we get $ncc^{\\prime }=0$ with $c^{\\prime }$ the derivative of $c$ .",
"Since $nc$ is invertible, $c^{\\prime }=0$ .",
"If $\\operatorname{\\textup {char}}k=0$ it immediately follows that $c\\in B$ .",
"So assume $\\operatorname{\\textup {char}}k=p>0$ .",
"In this case $c_{i}=0$ for all $i$ with $p\\nmid i$ .",
"This means that $c$ is in the image of the injective $B$ -algebra homomorphism $f\\colon B[[t]]\\rightarrow B[[t]]$ , $t\\mapsto t^{p}$ .",
"Let $d$ be the unique preimage of $c$ under $f$ , which is explicitly given by $d=\\sum _{i\\in \\mathbb {Z}}c_{pi}t^{i}$ .",
"In particular, $\\operatorname{\\textup {ord}}_{naive}(d)=\\operatorname{\\textup {ord}}_{naive}(c)/p>\\operatorname{\\textup {ord}}_{naive}(c).$ Since $f(d^{n})=c^{n}=1$ and $f$ is injective, we have $d^{n}=1$ and $d\\in R\\setminus B[[t]]$ .",
"This contradicts the way of choosing $c$ .",
"We have proved the lemma.",
"We first define the functor $\\psi \\colon \\bigsqcup _{q=0}^{n-1}\\operatorname{\\textup {B}}(G)\\longrightarrow \\Delta _{G}^{*}$ .",
"An object of $\\bigsqcup _{q=0}^{n-1}\\operatorname{\\textup {B}}(G)$ over a $k$ -algebra $A$ is a factorization $A=\\prod _{q}A_{q}$ plus a tuple $(L_{q},\\xi _{q})_{q}$ where $L_{q}$ is an $A_{q}$ -module and $\\xi _{q}\\colon L_{q}^{\\otimes n}\\longrightarrow A_{q}$ is an isomorphism.",
"A morphism $(A=\\prod _{q}A_{q},L_{q},\\xi _{q})\\longrightarrow (A=\\prod _{q}A^{\\prime }_{q},L^{\\prime }_{q},\\xi ^{\\prime }_{q})$ exists if and only if $A_{q}\\simeq A_{q^{\\prime }}$ as $A$ -algebras (so that such isomorphism is unique) and in this case is a collection of isomorphisms $L_{q}\\longrightarrow L^{\\prime }_{q}$ compatible with the maps $\\xi _{q}$ and $\\xi ^{\\prime }_{q}$ .",
"To such an object we associate the invertible $A((t))=\\prod _{q}A_{q}((t))$ -module $L=\\prod _{q}(L_{q}\\otimes _{A_{q}}A_{q}((t)))$ together with the map $L^{\\otimes n}\\simeq \\prod _{q}(L_{q}^{\\otimes n}\\otimes _{A_{q}}A_{q}((t)))\\longrightarrow \\prod _{q}A_{q}((t))=A((t)),\\ (x_{q}\\otimes 1)_{q}\\mapsto (\\xi _{q}(x_{q})t^{q})_{q}$ It is easy to see that the functor $\\psi $ on $\\operatorname{\\textup {B}}G$ in the index $q$ is the one in the statement.",
"We are going to show that $\\psi $ is an equivalence.",
"Since $\\Delta _{G}^{*}$ is a prestack it will be enough to show that $\\psi $ is an epimorphism and that it is fully faithful.",
"$\\psi $ epimorphism.",
"Let $\\chi \\in \\Delta _{G}^{*}(B)$ .",
"We can assume that $\\chi =(B((t)),b)$ .",
"We have $(B((t)),b)\\simeq (B((t)),b^{\\prime })$ if and only if there exists $u\\in B((t))^{*}$ such that $u^{n}b=b^{\\prime }$ .",
"For $c\\in B((t))^{*}$ we define $\\operatorname{\\textup {ord}}c\\colon \\operatorname{\\textup {Spec}}B\\rightarrow \\mathbb {Z}$ as follows: if $x\\in \\operatorname{\\textup {Spec}}B$ is a point with the residue field $\\kappa $ and $c_{x}\\in \\kappa ((t))$ is the induced power series, then $(\\operatorname{\\textup {ord}}c)(x):=\\operatorname{\\textup {ord}}c_{x}$ .",
"This function is upper semicontinuous.",
"From the additivity of orders, $\\operatorname{\\textup {ord}}b+\\operatorname{\\textup {ord}}(b^{-1})$ is constant zero.",
"Since $\\operatorname{\\textup {ord}}b$ and $\\operatorname{\\textup {ord}}(b^{-1})$ are both upper semicontinuous, they are in fact locally constant.",
"Thus we may suppose that $\\operatorname{\\textup {ord}}b$ is constant, equivalently that if $b_{j}$ are coefficients of $b$ , then for some $i$ , $b_{i}$ is a unit and $b_{j}$ are nilpotents for $j<i$ .",
"Thus we can write $b=b_{-}+t^{i}b_{+}$ with $b_{t}\\in B[[t]]^{*}$ and $b_{-}\\in B((t))$ nilpotent.",
"Set $\\omega =b/(t^{i}b_{+})\\in B((t))$ , $A=B((t))[Y]/(Y^{n}-\\omega )$ and $C=B((t))/(b_{-})$ .",
"We have that $\\omega =1$ in $C$ and therefore that $A\\otimes _{B((t))}C$ has a section.",
"Since $A/B((t))$ is étale and $B((t))\\longrightarrow C$ is surjective with nilpotent kernel the section extends, that is $\\omega $ is an $n$ -th power.",
"Thus we can assume $b_{-}=0$ .",
"Since $B[[t]][Y]/(Y^{n}-b_{+})$ is étale over $B[[t]]$ , by REF we can assume there exists $\\hat{b}\\in B[[t]]^{*}$ such that $\\hat{b}^{n}=b_{+}$ .",
"In conclusion we reduce to the case $b=t^{i}$ and, multiplying by a power of $t^{n}$ , we can finally assume $0\\le i<n$ .",
"$\\psi $ fully faithful.",
"If $(\\mathcal {L},\\sigma )\\in \\Delta _{G}(B)$ then, by Lemma REF , its automorphisms are canonically isomorphic to $\\mu _{n}(B((t)))=\\mu _{n}(B)$ .",
"This easily implies that the restriction of $\\psi $ on each component is fully faithful.",
"Given two objects $\\alpha ,\\beta \\in \\bigsqcup _{q=0}^{n-1}\\operatorname{\\textup {B}}(G)$ and an isomorphism $\\psi (\\alpha )\\longrightarrow \\psi (\\beta )$ of their images the problem of finding an isomorphism $\\alpha \\longrightarrow \\beta $ inducing the given one is local and easily reducible to the following claim: if $(B((t)),t^{q})\\simeq (B((t)),t^{q^{\\prime }})$ then $q\\equiv q^{\\prime }\\mod {n}$ .",
"But the first condition means that there exists $u\\in L((t))^{*}$ such that $u^{n}t^{q}=t^{q^{\\prime }}$ and, applying $\\operatorname{\\textup {ord}}$ , we get the result."
],
[
"General $p$ -groups",
"In this section we consider the case of a constant $p$ -group $G$ over a field $k$ of characteristic $p$ and the aim is to prove Theorem REF in this case.",
"Notice that in this case $\\Delta _{G}=\\Delta _{G}^{*}$ by REF .",
"We setup the following notation for this section.",
"All groups considered in this section are constant.",
"Notation 4.22 We set $S=\\lbrace n\\ge 1\\ | \\ p\\nmid n\\rbrace $ and, given a finite dimensional $\\mathbb {F}_{p}$ -vector space $H$ (regarded as an abelian $p$ -group), we set $X_{H}=(\\mathbb {A}^{(S)})^{\\infty }\\otimes _{\\mathbb {F}_{p}}H\\colon \\operatorname{\\textup {Aff}}/\\mathbb {F}_{p}\\longrightarrow (\\textup {Sets})$ (that is $X_{H}=[(\\mathbb {A}^{(S)})^{\\infty }]^{n}$ if $\\dim _{\\mathbb {F}_{p}}H=n$ after the choice of a basis of $H$ ).",
"The functor $X_{H}$ is a sheaf of abelian groups and we have an isomorphism $X_{H}\\times \\operatorname{\\textup {B}}H\\longrightarrow \\Delta _{H}$ thanks to REF .",
"We also set, for $m\\in \\mathbb {N}$ , $S_{m}=\\lbrace n\\in S\\ | \\ n\\le m\\rbrace $ , $X_{H,m}=\\mathbb {A}^{(S_{m})}\\otimes _{\\mathbb {F}_{p}}H$ , which is a sheaf of abelian groups.",
"In particular $X_{H}=\\varinjlim _{m}X_{H,m}$ as sheaves of abelian groups, where the transition map $X_{H,m}\\longrightarrow X_{H,m+1}$ is the composition of the inclusion $\\mathbb {A}^{(S_{m})}\\otimes _{\\mathbb {F}_{p}}H\\longrightarrow \\mathbb {A}^{(S_{m+1})}\\otimes _{\\mathbb {F}_{p}}H$ and the Frobenius of $\\mathbb {A}^{(S_{m+1})}\\otimes _{\\mathbb {F}_{p}}H$ .",
"We finally set $\\Delta _{H,m}=X_{H,m}\\times \\operatorname{\\textup {B}}(H)$ , so that we have an equivalence $\\varinjlim _{m}(\\Delta _{H,m})\\simeq \\Delta _{H}$ .",
"Proposition 4.23 Let $G$ be a $p$ -group and $H$ be a central subgroup which is an $\\mathbb {F}_{p}$ -vector space.",
"Then $H$ is naturally a subgroup sheaf of the inertia stack of $\\Delta _{G}$ (see Appendix for the inertia stack as a group sheaf) and the quotient map $\\Delta _{G}\\longrightarrow \\Delta _{G/H}$ is the composition of the rigidification $\\Delta _{G}\\longrightarrow \\Delta _{G}\\mathbin {\\!\\!",
"}H$ and an $X_{H}$ -torsor $\\Delta _{G}\\mathbin {\\!\\!",
"}H\\longrightarrow \\Delta _{G/H}$ which is induced by $\\Delta _{G}\\times \\Delta _{H}\\longrightarrow \\Delta _{G}$ (after rigidification).",
"The subgroup $H$ acts on any $G$ -torsor because its central.",
"Moreover the functor $\\Delta _{G}\\longrightarrow \\Delta _{G/H}$ sends isomorphisms coming from $H$ to the identity and therefore factors through the rigidification $\\Delta _{G}\\mathbin {\\!\\!",
"}H$ .",
"Rigidifying both sides of $\\Delta _{G}\\times \\Delta _{H}\\longrightarrow \\Delta _{G}$ we get a map $(\\Delta _{G}\\mathbin {\\!\\!",
"}H)\\times X_{H}\\longrightarrow \\Delta _{G}\\mathbin {\\!\\!",
"}H$ over $\\Delta _{G/H}$ .",
"Using REF we can deduce that $\\Delta _{G}\\mathbin {\\!\\!",
"}H\\longrightarrow \\Delta _{G/H}$ is an $X_{H}$ -torsor.",
"Lemma 4.24 Let $G$ be a $p$ -group and $H$ be a central subgroup which is an $\\mathbb {F}_{p}$ -vector space.",
"Let also $\\mathcal {Y}_{*}$ be a direct system of quasi-separated stacks over $\\mathbb {N}$ with a direct system of smooth (étale) atlases $U_{*}$ made of quasi-compact schemes, $\\varinjlim _{n}\\mathcal {Y}_{n}\\longrightarrow \\Delta _{G/H}$ a map and $\\varinjlim _{n}U_{n}\\longrightarrow \\Delta _{G}$ a lifting.",
"Then there exists a strictly increasing map $q\\colon \\mathbb {N}\\longrightarrow \\mathbb {N}$ , a direct system of quasi-separated stacks $\\mathcal {Z}_{*}$ with a direct system of smooth (étale) atlases $U_{*}\\times X_{H,q_{*}}$ (where the transition morphisms $U_{i}\\times X_{H,q_{i}}\\longrightarrow U_{i+1}\\times X_{H,q_{i+1}}$ is the product of the given map $U_{i}\\longrightarrow U_{i+1}$ and the map $X_{H,q_{i}}\\longrightarrow X_{H,q_{i+1}}$ of REF ), compatible maps $\\mathcal {Z}_{i}\\longrightarrow \\mathcal {Y}_{i}$ induced by the projection $U_{i}\\times X_{H,q_{i}}\\longrightarrow U_{i}$ and which are a composition of an $H$ -gerbe $\\mathcal {Z}_{i}\\longrightarrow \\mathcal {Z}_{i}\\mathbin {\\!\\!",
"}H$ and a $X_{H,q_{i}}$ -torsor $\\mathcal {Z}_{i}\\mathbin {\\!\\!",
"}H\\longrightarrow \\mathcal {Y}_{i}$ and an equivalence $\\varinjlim _{n}\\mathcal {Z}_{n}\\simeq (\\varinjlim _{n}\\mathcal {Y}_{n})\\times _{\\Delta _{G/H}}\\Delta _{G}$ Moreover there is a factorization $U_{i}\\times X_{H,q_{i}}\\longrightarrow U_{i}\\times _{\\mathcal {Y}_{i}}\\mathcal {Z}_{i}\\longrightarrow \\mathcal {Z}_{i}$ where the first arrow is an $H$ -torsor.",
"Consider one index $i\\in \\mathbb {N}$ and the Cartesian diagrams $ \\begin{tikzpicture}[xscale=2.1,yscale=-1.2] \\node (A0_0) at (0, 0) {P^{\\prime }_{U,i}}; \\node (A0_1) at (1, 0) {P^{\\prime }_i}; \\node (A0_2) at (2, 0) {\\Delta _G}; \\node (A1_0) at (0, 1) {P_{U,i}}; \\node (A1_1) at (1, 1) {P_i}; \\node (A1_2) at (2, 1) {\\Delta _{G}\\mathbin {\\!\\!",
"}H}; \\node (A2_0) at (0, 2) {U_i}; \\node (A2_1) at (1, 2) {\\mathcal {Y}_i}; \\node (A2_2) at (2, 2) {\\Delta _{G/H}}; (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A2_1) edge [->]node [auto] {\\scriptstyle {}} (A2_2); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A0_2); (A0_2) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A1_1) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A2_0); (A1_1) edge [->]node [auto] {\\scriptstyle {}} (A2_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); (A2_0) edge [->]node [auto] {\\scriptstyle {}} (A2_1); (A1_2) edge [->]node [auto] {\\scriptstyle {}} (A2_2); \\end{tikzpicture} $ Set also $R_{i}=U_{i}\\times _{\\mathcal {Y}_{i}}U_{i}$ , which is a quasi-compact algebraic space.",
"By REF $P_{i}\\longrightarrow \\mathcal {Y}_{i}$ is a $X_{H}$ -torsor, $P^{\\prime }_{i}\\longrightarrow P_{i}$ an $H$ -gerbe.",
"Moreover the lifting $U_{i}\\longrightarrow \\Delta _{G}$ gives an isomorphism $P^{\\prime }_{U,i}\\simeq U_{i}\\times \\Delta _{H}$ and $P_{U,i}\\simeq U_{i}\\times X_{H}$ by REF .",
"Thus the $X_{H}$ -torsor $P_{i}\\longrightarrow \\mathcal {Y}_{i}$ , by descent along $U_{i}\\longrightarrow \\mathcal {Y}_{i}$ is completely determined by the identification $R_{i}\\times X_{H}\\simeq _{R_{i}}R_{i}\\times X_{H}$ , which consists of an element $\\omega _{i}\\in X_{H}(R_{i})$ satisfying the cocycle condition on $U_{i}\\times _{\\mathcal {Y}_{i}}U_{i}\\times _{\\mathcal {Y}_{i}}U_{i}$ .",
"The given equivalence $P_{i}\\simeq (P_{i+1})_{|\\mathcal {Y}_{i}}$ of $X_{H}$ -torsors over $\\mathcal {Y}_{i}$ is completely determined by its pullback on $U_{i}$ , which is given by multiplication by $\\gamma _{i}\\in X_{H}(U_{i})$ .",
"The compatibility this element has to satisfy is expressed by $(\\omega _{i+1})_{|R_{i}}(s_{i}^{*}\\gamma _{i})=(t_{i}^{*}\\gamma _{i})\\omega _{i}\\text{ in }X_{H}(R_{i})$ where $s_{i},t_{i}\\colon R_{i}\\longrightarrow U_{i}$ are the two projections.",
"Since all $R_{i}$ are quasi-compact and $X_{H}\\simeq \\varinjlim _{j}X_{H,j}$ we can find an increasing sequence of natural numbers $q\\colon \\mathbb {N}\\longrightarrow \\mathbb {N}$ and elements $e_{q_{i}}\\in X_{H,q_{i}}(R_{i})$ , $f_{q_{i}}\\in X_{H,q_{i+1}}(U_{i})$ such that: the element $e_{q_{i}}$ is mapped to $\\omega _{i}$ under the map $X_{H,q_{i}}(R_{i})\\longrightarrow X_{H}(R_{i})$ and it satisfies the cocycle condition in $X_{H,q_{i}}(U_{i}\\times _{\\mathcal {Y}_{i}}U_{i}\\times _{\\mathcal {Y}_{i}}U_{i})$ ; the element $f_{q_{i}}$ is mapped to $\\gamma _{i}$ under the map $X_{H,q_{i+1}}(U_{i})\\longrightarrow X_{H}(U_{i})$ and, if $\\overline{e_{q_{i}}}$ is the image of $e_{q_{i}}$ under the map $X_{H,q_{i}}(R_{i})\\longrightarrow X_{H,q_{i+1}}(R_{i})$ , it satisfies $(e_{q_{i+1}})_{|R_{i}}(s_{i}^{*}f_{q_{i}})=(t_{i}^{*}f_{q_{i}})\\overline{e_{q_{i}}}\\text{ in }X_{H,q_{i+1}}(R_{i})$ The data of $1)$ determine $X_{H,q_{i}}$ -torsors $Q_{i}\\longrightarrow \\mathcal {Y}_{i}$ with a map $U_{i}\\longrightarrow Q_{i}$ over $\\mathcal {Y}_{i}$ and together with an $X_{H,q_{i}}$ -equivariant map $Q_{i}\\longrightarrow P_{i}$ such that $U_{i}\\longrightarrow Q_{i}\\longrightarrow P_{i}$ is the given map.",
"The data of $2)$ determine an $X_{H,q_{i}}$ -equivariant map $Q_{i}\\longrightarrow (Q_{i+1})_{|\\mathcal {Y}_{i}}$ inducing the equivalence $P_{i}\\simeq (P_{i+1})_{|\\mathcal {Y}_{i}}$ .",
"Consider also the $H$ -gerbe $Q_{i}^{\\prime }\\longrightarrow Q_{i}$ pullback of $P_{i}^{\\prime }\\longrightarrow P_{i}$ along $Q_{i}\\longrightarrow P_{i}$ .",
"We set $\\mathcal {Z}_{i}=Q_{i}^{\\prime }$ .",
"We have Cartesian diagrams $ \\begin{tikzpicture}[xscale=1.5,yscale=-1.2] \\node (A0_0) at (0, 0) {Q_i^{\\prime }}; \\node (A0_1) at (1, 0) {P_i^{\\prime }}; \\node (A0_2) at (2, 0) {P_{i+1}^{\\prime }}; \\node (A0_3) at (3, 0) {Q_i^{\\prime }}; \\node (A0_4) at (4, 0) {Q_{i+1}^{\\prime }}; \\node (A0_5) at (5, 0) {P_{i+1}^{\\prime }}; \\node (A1_0) at (0, 1) {Q_i}; \\node (A1_1) at (1, 1) {P_i}; \\node (A1_2) at (2, 1) {P_{i+1}}; \\node (A1_3) at (3, 1) {Q_i}; \\node (A1_4) at (4, 1) {Q_{i+1}}; \\node (A1_5) at (5, 1) {P_{i+1}}; (A1_4) edge [->]node [auto] {\\scriptstyle {}} (A1_5); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A0_2); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_3) edge [->]node [auto] {\\scriptstyle {}} (A1_3); (A0_2) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A0_4) edge [->]node [auto] {\\scriptstyle {}} (A0_5); (A0_3) edge [->]node [auto] {\\scriptstyle {}} (A0_4); (A0_4) edge [->]node [auto] {\\scriptstyle {}} (A1_4); (A1_1) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A0_5) edge [->]node [auto] {\\scriptstyle {}} (A1_5); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_3) edge [->]node [auto] {\\scriptstyle {}} (A1_4); \\end{tikzpicture} $ Notice that, if $\\mathcal {M}$ is a stack over $\\mathcal {Y}_{i}$ , then $\\mathcal {M}\\times _{\\mathcal {Y}_{i+1}}U_{i+1}\\simeq \\mathcal {M}\\times _{\\mathcal {Y}_{i}}U_{i}$ because $U_{*}$ is a direct system of atlases.",
"Pulling back along $U_{i+1}\\longrightarrow \\mathcal {Y}_{i+1}$ the above diagrams, we obtain the bottom rows of the following diagrams.",
"$ \\begin{tikzpicture}[xscale=3.2,yscale=-1.2] \\node (A0_0) at (0, 0) { X_{H,q_i}\\times U_{i}}; \\node (A0_1) at (1, 0) { X_{H}\\times U_{i}}; \\node (A0_2) at (2, 0) { X_{H}\\times U_{i+1}}; \\node (A1_0) at (0, 1) {Q_i^{\\prime }\\times _{\\mathcal {Y}_{i+1}}U_{i+1}}; \\node (A1_1) at (1, 1) { \\Delta _{H}\\times U_{i}}; \\node (A1_2) at (2, 1) { \\Delta _{H}\\times U_{i+1}}; \\node (A2_0) at (0, 2) { X_{H,q_i}\\times U_{i}}; \\node (A2_1) at (1, 2) { X_{H}\\times U_{i}}; \\node (A2_2) at (2, 2) { X_{H}\\times U_{i+1}}; \\node (A3_0) at (0, 3) { X_{H,q_i}\\times U_{i}}; \\node (A3_1) at (1, 3) { X_{H,q_{i+1}}\\times U_{i+1}}; \\node (A3_2) at (2, 3) { X_{H}\\times U_{i+1}}; \\node (A4_0) at (0, 4) {Q_i^{\\prime }\\times _{\\mathcal {Y}_{i+1}}U_{i+1}}; \\node (A4_1) at (1, 4) {Q_{i+1}^{\\prime }\\times _{\\mathcal {Y}_{i+1}}U_{i+1}}; \\node (A4_2) at (2, 4) { \\Delta _{H}\\times U_{i+1}}; \\node (A5_0) at (0, 5) {X_{H,q_i}\\times U_{i}}; \\node (A5_1) at (1, 5) { X_{H,q_{i+1}}\\times U_{i+1}}; \\node (A5_2) at (2, 5) {X_{H}\\times U_{i+1}}; (A5_0) edge [->]node [auto] {\\scriptstyle {}} (A5_1); (A2_1) edge [->]node [auto] {\\scriptstyle {}} (A2_2); (A0_2) edge [->]node [auto] {\\scriptstyle {\\alpha _{i+1}}} (A1_2); (A4_0) edge [->]node [auto] {\\scriptstyle {}} (A5_0); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A2_0); (A1_1) edge [->]node [auto] {\\scriptstyle {}} (A2_1); (A1_2) edge [->]node [auto] {\\scriptstyle {}} (A2_2); (A2_0) edge [->]node [auto] {\\scriptstyle {}} (A2_1); (A3_0) edge [->]node [auto] {\\scriptstyle {}} (A3_1); (A0_1) edge [->]node [auto] {\\scriptstyle {\\alpha _{i}}} (A1_1); (A4_2) edge [->]node [auto] {\\scriptstyle {}} (A5_2); (A4_0) edge [->]node [auto] {\\scriptstyle {}} (A4_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A3_1) edge [->]node [auto] {\\scriptstyle {\\beta _{i+1}}} (A4_1); (A1_1) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A4_1) edge [->]node [auto] {\\scriptstyle {}} (A5_1); (A0_0) edge [->]node [auto] {\\scriptstyle {\\beta _i}} (A1_0); (A4_1) edge [->]node [auto] {\\scriptstyle {}} (A4_2); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A3_0) edge [->]node [auto] {\\scriptstyle {\\beta _i}} (A4_0); (A3_2) edge [->]node [auto] {\\scriptstyle {\\alpha _{i+1}}} (A4_2); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A0_2); (A3_1) edge [->]node [auto] {\\scriptstyle {}} (A3_2); (A5_1) edge [->]node [auto] {\\scriptstyle {}} (A5_2); \\end{tikzpicture} $ The top rows of the above diagrams is instead obtained using REF , where the map $\\alpha _{i}$ are induced by the map $X_{H}\\longrightarrow X_{H}\\times \\operatorname{\\textup {B}}H=\\Delta _{H}$ and the $\\beta _{i}$ are induced by the $\\alpha _{i}$ .",
"Since $Q_{i}^{\\prime }\\times _{\\mathcal {Y}_{i+1}}U_{i+1}\\simeq Q_{i}^{\\prime }\\times _{\\mathcal {Y}_{i}}U_{i}$ we see that the atlases $U_{i}\\times X_{H,q_{i}}{\\beta _{i}}Q_{i}^{\\prime }\\times _{\\mathcal {Y}_{i}}U_{i}\\longrightarrow Q_{i}^{\\prime }=\\mathcal {Z}_{i}$ define a direct system of smooth (resp.",
"étale) atlases satisfying the requests of the statement.",
"Let us show the last equivalence in the statement.",
"By REF the map $\\varinjlim _{n}(U_{n}\\times _{\\mathcal {Y}_{n}}P_{n})=\\varinjlim _{n}(U_{n}\\times X_{H})\\longrightarrow \\varinjlim _{n}P_{n}$ is a smooth atlas.",
"The map $\\varinjlim _{n}Q_{n}\\longrightarrow \\varinjlim _{n}P_{n}$ is therefore an equivalence because its base change along the above atlas is $\\varinjlim _{n}(U_{n}\\times X_{H,q_{n}})\\longrightarrow \\varinjlim _{n}(U_{n}\\times X_{H})$ , which is an isomorphism.",
"Here we have used REF .",
"Using again this we see that the map $\\varinjlim _{n}\\mathcal {Z}_{n}=\\varinjlim _{n}(Q_{n}\\times _{P_{n}}P_{n}^{\\prime })\\longrightarrow \\varinjlim _{n}P_{n}^{\\prime }=\\varinjlim _{n}(\\mathcal {Y}_{n}\\times _{\\Delta _{G/H}}\\Delta _{G})$ is an equivalence as well.",
"Lemma 4.25 Let $G$ be a $p$ -group, $H$ be a central subgroup which is an $\\mathbb {F}_{p}$ -vector space and $X\\longrightarrow Y$ be a finite, finitely presented and universally injective map of affine schemes.",
"Then a 2-commutative diagram $ \\begin{tikzpicture}[xscale=2.1,yscale=-1.2] \\node (A0_0) at (0, 0) {X}; \\node (A0_1) at (1, 0) {\\Delta _G}; \\node (A1_0) at (0, 1) {Y}; \\node (A1_1) at (1, 1) {\\Delta _{G/H}}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A1_0) edge [->,dashed]node [auto] {\\scriptstyle {}} (A0_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); \\end{tikzpicture} $ always admits a dashed map.",
"Set $X=\\operatorname{\\textup {Spec}}B$ and $Y=\\operatorname{\\textup {Spec}}C$ and consider the induced map $C\\longrightarrow B$ .",
"Since $\\operatorname{\\textup {H}}^{2}(B((t)),H)=\\operatorname{\\textup {H}}^{2}(C((t)),H)=0$ by the Artin-Schreier sequence, we have a commutative diagram $ \\begin{tikzpicture}[xscale=3.3,yscale=-1.2] \\node (A0_0) at (0, 0) {\\operatorname{\\textup {H}}^1(C((t)),H)}; \\node (A0_1) at (1, 0) {\\operatorname{\\textup {H}}^1(C((t)),G)}; \\node (A0_2) at (2, 0) {\\operatorname{\\textup {H}}^1(C((t)),G/H)}; \\node (A0_3) at (3, 0) {0}; \\node (A1_0) at (0, 1) {\\operatorname{\\textup {H}}^1(B((t)),H)}; \\node (A1_1) at (1, 1) {\\operatorname{\\textup {H}}^1(B((t)),G)}; \\node (A1_2) at (2, 1) {\\operatorname{\\textup {H}}^1(B((t)),G/H)}; \\node (A1_3) at (3, 1) {0}; (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A0_2); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_1) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A0_2) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A1_2) edge [->]node [auto] {\\scriptstyle {}} (A1_3); (A0_2) edge [->]node [auto] {\\scriptstyle {}} (A0_3); (A0_0) edge [->]node [auto] {\\scriptstyle {\\alpha }} (A1_0); \\end{tikzpicture} $ with exact rows.",
"By hypothesis there are $u\\in \\operatorname{\\textup {H}}^{1}(B((t)),G)$ and $v\\in \\operatorname{\\textup {H}}^{1}(C((t)),G/H)$ which agree in $\\operatorname{\\textup {H}}^{1}(B((t)),G/H)$ .",
"We can find a common lifting in $\\operatorname{\\textup {H}}^{1}(C((t)),G)$ by proving that the map $\\alpha $ is surjective.",
"By REF we have that $\\operatorname{\\textup {Spec}}B((t))\\longrightarrow \\operatorname{\\textup {Spec}}C((t))$ is the base change of $\\operatorname{\\textup {Spec}}B\\longrightarrow \\operatorname{\\textup {Spec}}C$ and therefore is finite and universally injective.",
"Let $D$ be the image of $C((t))\\longrightarrow B((t))$ .",
"The map $\\operatorname{\\textup {H}}^{1}(C((t)),H)\\longrightarrow \\operatorname{\\textup {H}}^{1}(D,H)$ is surjective because $H\\simeq (\\mathbb {Z}/p\\mathbb {Z})^{l}$ for some $l$ and using the description of $\\mathbb {Z}/p\\mathbb {Z}$ -torsors in REF .",
"By REF the map $\\operatorname{\\textup {Spec}}B((t))\\longrightarrow \\operatorname{\\textup {Spec}}D$ is a finite universal homeomorphism.",
"Thus $\\operatorname{\\textup {H}}^{1}(D,H)\\longrightarrow \\operatorname{\\textup {H}}^{1}(B((t)),H)$ is bijective by REF .",
"Since $p$ -groups have non trivial center we can find a sequence of quotients $G=G_{l}\\longrightarrow G_{l-1}\\longrightarrow G_{l-2}\\longrightarrow \\cdots \\longrightarrow G_{0}\\longrightarrow G_{-1}=0$ where $\\operatorname{\\textup {Ker}}(G_{u}\\longrightarrow G_{u-1})$ is central in $G_{u}$ and an $\\mathbb {F}_{p}$ -vector space.",
"We proceed by induction on $l$ .",
"In the base case $l=0$ , so that $G=G_{0}$ is an $\\mathbb {F}_{p}$ -vector space, following REF it is enough to set $\\mathcal {X}_{n}=X_{G,n}\\times \\operatorname{\\textup {B}}G$ .",
"Consider now the inductive step and set $H=\\operatorname{\\textup {Ker}}(G=G_{l}\\longrightarrow G_{l-1})$ .",
"Of course we can assume $H\\ne 0$ , so that we can use the inductive hypothesis on $G/H$ , obtaining a direct system $\\mathcal {Y}_{*}$ and a map $\\overline{v}\\colon \\mathbb {N}\\longrightarrow \\mathbb {N}$ with a direct system of atlases $\\mathbb {A}^{\\overline{v}_{*}}$ .",
"The first result follows applying REF with $U_{n}=\\mathbb {A}^{\\overline{v}_{n}}$ .",
"We just have to prove the existence of a lifting $\\varinjlim _{n}U_{n}\\longrightarrow \\Delta _{G}$ .",
"Since the schemes $U_{n}$ are affine one always find a lifting $U_{n}\\longrightarrow \\Delta _{G}$ thanks to REF .",
"Thanks to REF any lifting $U_{n}\\longrightarrow \\Delta _{G}$ always extends to a lifting $U_{n+1}\\longrightarrow \\Delta _{G}$ .",
"Assume now $G$ abelian and set $X_{G}=X_{G/H}\\times X_{H}$ .",
"By induction we can assume $\\Delta _{G/H}=X_{G/H}\\times \\operatorname{\\textup {B}}(G/H)$ .",
"By REF and REF there is a lifting $X_{G/H}\\longrightarrow \\Delta _{G}$ of the given map $X_{G/H}\\longrightarrow \\Delta _{G/H}$ .",
"In particular, using REF , we obtain a map $X_{G}=X_{G/H}\\times X_{H}\\longrightarrow X_{G/H}\\times \\Delta _{H}\\longrightarrow \\Delta _{G}$ , which is finite and étale of degree $\\sharp G$ .",
"Since $G$ is abelian and using REF we have $(\\operatorname{\\underline{\\textup {Aut}}}_{\\Delta _{G}}P)(B)=G(B((t)))=G(B)$ for all $P\\in \\Delta _{G}(B)$ .",
"It follows that the rigidification $F=\\Delta _{G}\\mathbin {\\!\\!",
"}G$ is the sheaf of isomorphism classes of $\\Delta _{G}$ and that $\\Delta _{G}\\longrightarrow F$ is a gerbe locally $\\operatorname{\\textup {B}}G$ .",
"Since $X_{G}\\longrightarrow \\Delta _{G}$ and, thanks to REF , $\\mathbb {A}^{v}=\\varinjlim _{n}\\mathbb {A}^{v_{n}}\\longrightarrow \\Delta _{G}$ are finite and étale of degree $\\sharp G$ , by REF we can conclude that $X_{G}\\longrightarrow F$ and $\\mathbb {A}^{v}\\longrightarrow F$ are isomorphisms.",
"Since a gerbe having a section is trivial we get our result.",
"With notation and hypothesis from Theorem REF set $\\mathbb {A}^{v}=\\varinjlim \\mathbb {A}^{v_{*}}$ , $\\overline{\\Delta _{G}}$ for the coarse ind-algebraic space of $\\Delta _{G}$ and consider the induced map $\\mathbb {A}^{v}\\longrightarrow \\overline{\\Delta _{G}}$ .",
"We want to show that when $G$ is non-abelian this map is not an isomorphism in general.",
"The key point is the following Lemma.",
"Lemma 4.26 If $K$ is an algebraically closed field and $P\\in \\Delta _{G}(K)$ then $H=\\operatorname{\\underline{\\textup {Aut}}}_{\\Delta _{G}}(P)$ is (non canonically) a subgroup of $G$ and the fiber of $\\mathbb {A}^{v}(K)\\longrightarrow \\overline{\\Delta _{G}}(K)\\simeq \\Delta _{G}(K)/\\simeq $ over $P$ has cardinality $\\sharp G/\\sharp H$ .",
"There exist $n\\in \\mathbb {N}$ and $P_{n}\\in \\mathcal {X}_{n}(K)$ inducing $P\\in \\Delta _{G}(K)$ .",
"By REF we have $\\operatorname{\\underline{\\textup {Aut}}}_{\\mathcal {X}_{n}}(P_{n})=H$ and, since $\\mathcal {X}_{n}$ is a quasi-separated DM stack, it follows that $H$ is a finite and constant group scheme.",
"Moreover the map $H(K)=\\operatorname{\\textup {Aut}}_{K((t))}^{G}(P)\\longrightarrow \\operatorname{\\textup {Aut}}_{\\overline{K((t))}}^{G}(P\\times \\overline{K((t))})\\simeq G$ is injective (the last isomorphism depends on the choice of a section in $P(\\overline{K((t))})$ .",
"Thanks to REF we have 2-Cartesian diagrams $ \\begin{tikzpicture}[xscale=1.5,yscale=-1.2] \\node (A0_0) at (0, 0) {Y}; \\node (A0_1) at (1, 0) {W}; \\node (A0_2) at (2, 0) {V}; \\node (A0_3) at (3, 0) {\\mathbb {A}^{v_n}}; \\node (A0_4) at (4, 0) {\\mathbb {A}^v}; \\node (A1_0) at (0, 1) {\\operatorname{\\textup {Spec}}K}; \\node (A1_1) at (1, 1) {\\operatorname{\\textup {B}}H}; \\node (A1_2) at (2, 1) {U}; \\node (A1_3) at (3, 1) {\\mathcal {X}_n}; \\node (A1_4) at (4, 1) {\\Delta _G}; \\node (A2_2) at (2, 2) {\\operatorname{\\textup {Spec}}K}; \\node (A2_3) at (3, 2) {\\overline{\\mathcal {X}}_n}; (A1_3) edge [->]node [auto] {\\scriptstyle {}} (A1_4); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_3) edge [->]node [auto] {\\scriptstyle {}} (A1_3); (A1_1) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A2_2) edge [->]node [auto] {\\scriptstyle {}} (A2_3); (A0_3) edge [->]node [auto] {\\scriptstyle {}} (A0_4); (A0_4) edge [->]node [auto] {\\scriptstyle {}} (A1_4); (A0_2) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); (A1_3) edge [->]node [auto] {\\scriptstyle {}} (A2_3); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A0_2); (A1_2) edge [->]node [auto] {\\scriptstyle {}} (A1_3); (A1_2) edge [->]node [auto] {\\scriptstyle {}} (A2_2); (A0_2) edge [->]node [auto] {\\scriptstyle {}} (A0_3); \\end{tikzpicture} $ If $F\\subseteq \\mathbb {A}^{v}(K)$ is the fiber we are looking for then we get an induced map $V(K)\\longrightarrow F$ which is easily seen to be surjective.",
"From REF it follows that $\\mathbb {A}^{v_{n}}(K)\\longrightarrow \\mathbb {A}^{v}(K)$ is injective, which implies that $V(K)\\longrightarrow F$ is bijective.",
"From [9] one get that $\\operatorname{\\textup {B}}H$ is the reduction of $U$ .",
"Thus $W(K)=V(K)$ .",
"Notice that, since $\\overline{\\mathcal {X}_{n}}$ has schematically representable diagonal, $V$ , $W$ and $Y$ are all schemes.",
"Since the vertical maps in the top row are finite and étale of degree $\\sharp G$ and $Y\\longrightarrow W$ is an $H$ -torsor we conclude that $\\sharp Y=G$ and $\\sharp W=\\sharp Y/\\sharp H$ as required.",
"We see that if $G$ is not abelian and $P$ is a Galois extension of $K((t))$ with group $G$ , where $K$ is an algebraically closed field, then the map $\\mathbb {A}^{v}\\longrightarrow \\overline{\\Delta _{G}}$ is not injective.",
"Indeed the fiber of $\\mathbb {A}^{v}(K)\\longrightarrow \\overline{\\Delta _{G}}(K)$ over $P$ has cardinality $\\sharp G/\\sharp Z(G)$ because $\\operatorname{\\textup {Aut}}_{K((t))}^{G}(P)=Z(G)$ .",
"Remark 4.27 The moduli functor $F^{\\prime }$ described in [5] is very similar to the sheaf of isomorphism classes of $\\Delta _{G}$ but with some differences.",
"Firstly $F^{\\prime }$ maps pointed connected affine schemes to sets, while we look at the category of all (non-pointed) affine schemes, which is standard in modern moduli theory.",
"Secondly, for a connected and pointed affine scheme $\\operatorname{\\textup {Spec}}B$ , he defines $F^{\\prime }(\\operatorname{\\textup {Spec}}B)$ as the set of equivalence classes of pointed $G$ -torsors on $B\\otimes _{k}k((t))$ rather than $B((t))$ as in our case.",
"Two covers are defined to be equivalent if they agree after a finite étale pullback of $B\\otimes _{k}k[[t]]$ .",
"This equivalence relation plays the role of “killing terms of positive degrees”, while the same role is played by REF in our setting.",
"This is better understood in the case $G=\\mathbb {Z}/p\\mathbb {Z}$ where one can show that $F^{\\prime }$ is exactly the sheaf $\\overline{\\Delta }$ of isomorphism classes of $\\Delta _{\\mathbb {Z}/p\\mathbb {Z}}$ (one can ignore base points here because $\\mathbb {Z}/p\\mathbb {Z}$ is abelian).",
"A map $\\alpha \\colon F^{\\prime }\\longrightarrow \\overline{\\Delta }$ is well defined because if two torsors $P,Q$ over $B\\otimes k((t))$ become isomorphic after an étale cover of $B\\otimes k[[t]],$ by REF $P\\times B((t)),Q\\times B((t))$ become isomorphic over $C((t))$ , where $C/B$ is an étale covering.",
"The surjectivity of $\\alpha $ is easy: from the descritption of $\\Delta _{\\mathbb {Z}/p\\mathbb {Z}}$ a torsor in $\\overline{\\Delta }(B)$ is given by an element $b\\in B((t))$ with zero positive part and, therefore, belonging to $B\\otimes k((t))\\subseteq B((t))$ (see also REF ).",
"For the injectivity take $e\\in B\\otimes k((t))$ defining a torsor over $B\\otimes k((t))$ which become trivial in $\\overline{\\Delta }(B)$ .",
"Write $e=e_{-}+e_{+}$ as usual.",
"Since $e_{+}\\in B\\otimes k[[t]]$ (which means that its associated torsor extends to $B\\otimes k[[t]]$ ) one has $e=e_{-}$ in $F^{\\prime }(B)$ .",
"Using the same notation and strategy of REF , in particular of the essential surjectivity, one can assume $e=\\phi _{k}(\\underline{b})$ for $\\underline{b}\\in \\mathbb {A}^{(S)}$ .",
"Since $e=0$ in $\\overline{\\Delta }=(\\mathbb {A}^{(S)})^{\\infty }$ it follows that the coefficients of $b$ and $e$ are nilpotent.",
"Since $e^{p}=e$ in $F^{\\prime }(B)$ it follows that $e=0$ in $F^{\\prime }(B)$ ."
],
[
"Semidirect products",
"The aim of this section is to complete the proof of Theorem REF .",
"So let $k$ be a field of positive characteristic $p$ and $G$ be a finite and étale group scheme over $k$ such that $G\\times _{k}\\overline{k}$ is a semidirect product of a $p$ -group and cyclic group of rank coprime with $p$ .",
"Extending the base field by a Galois extension and using REF and REF we can assume that $G$ is constant, say $G=H\\rtimes C_{n}$ where $H$ is a $p$ -group and $C_{n}$ is a cyclic group of order $n$ coprime with $p$ .",
"We can moreover assume that the base field $k$ has all $n$ -roots of unity, so that $C_{n}\\simeq \\mu _{n}$ as group schemes.",
"More precisely we assume that $C_{n}=\\mu _{n}(k)\\subseteq k^{*}$ is the group of $n$ -th roots of unity of $k$ .",
"Lemma 4.28 An object $P\\in \\Delta _{G}$ lies in $\\Delta _{G}^{*}$ if and only if $P/H\\in \\Delta _{C_{n}}^{*}$ .",
"By REF it is enough to test $P$ with one dimensional representations of $G$ .",
"The result then follows from the fact that the characters of $G$ are the characters of $C_{n}$ and that, if $V$ is a representation of $C_{n}$ and $M$ a representation of $G$ over a ring $B((t))$ then $(M\\otimes V)^{G}\\simeq (M^{H}\\otimes V)^{C_{n}}$ .",
"Indeed if $\\phi \\colon \\operatorname{\\textup {B}}_{B((t))}G\\longrightarrow \\operatorname{\\textup {B}}_{B((t))}C_{n}$ is the induced map and $\\pi \\colon \\operatorname{\\textup {B}}_{B((t))}C_{n}\\longrightarrow \\operatorname{\\textup {Spec}}B((t))$ is the structure map then $M\\in \\operatorname{\\textup {QCoh}}(\\operatorname{\\textup {B}}_{B((t))}G)$ , $V\\in \\operatorname{\\textup {Vect}}(\\operatorname{\\textup {B}}_{B((t))}C_{n})$ and, using the projection formula, $(M\\otimes V)^{G}\\simeq (\\pi \\phi )_{*}(M\\otimes \\phi ^{*}V)\\simeq \\pi _{*}\\phi _{*}(M\\otimes \\phi ^{*}V)\\simeq \\pi _{*}(\\phi _{*}M\\otimes V)\\simeq (M^{H}\\otimes V)^{C_{n}}$ Consider $\\operatorname{\\textup {Spec}}k\\longrightarrow \\Delta _{C_{n}}^{*}$ given by $(k((t)),t^{q})$ as in Theorem REF and denote by $\\mathcal {Z}_{G,q}$ the fiber product $\\operatorname{\\textup {Spec}}k\\times _{\\Delta _{C_{n}}^{*}}\\Delta _{G}^{*}$ , which is the fibered category of pairs $(P,\\delta )$ where $P\\in \\Delta _{G}(B)$ and $\\delta \\colon P\\longrightarrow \\operatorname{\\textup {Spec}}(B((t))[X]/(X^{n}-t^{q}))$ is a $G$ -equivariant map.",
"Proposition 4.29 Let $d=(n,q)$ and $G_{d}=H\\rtimes C_{n/d}<G$ .",
"Then the functor $\\mathcal {Z}_{G_{d},q/d}\\longrightarrow \\mathcal {Z}_{G,q}$ induced by $\\Delta _{G_{d}}\\longrightarrow \\Delta _{G}$ and $\\Delta _{C_{n/d}}\\longrightarrow \\Delta _{C_{n}}$ is an equivalence.",
"Set $Q_{n,q}=\\operatorname{\\textup {Spec}}(B((t))[X]/(X^{n}-t^{q}))$ .",
"The map $X\\longmapsto X$ induces a map $Q_{n/d,q/d}\\longrightarrow Q_{n,q}$ which is $C_{n/d}$ -equivariant, that is $Q_{n,q}$ is the $C_{n}$ -torsor induced by the $C_{n/d}$ -torsor $Q_{n/d,q/d}$ .",
"We obtain a quasi-inverse $\\mathcal {Z}_{G,q}\\longrightarrow \\mathcal {Z}_{G_{d},q/d}$ mapping $P{\\phi }Q_{n,q}$ to the fiber product $P\\times _{Q_{n,q}}Q_{n/d,q/d}\\longrightarrow Q_{n/d,q/d}$ .",
"Remark 4.30 If $(n,q)=1$ we have an isomorphism of $B$ -algebras $B((s))\\longrightarrow B((t))[X](X^{n}-t^{q})$ such that the $C_{n}$ -action induced on the left is $s\\longmapsto \\xi ^{\\beta }s$ for $\\xi \\in C_{n}$ , where $\\beta \\in \\mathbb {Z}/n\\mathbb {Z}$ is the inverse of $q$ .",
"Indeed write $\\beta q=1+\\alpha n$ for some $\\alpha ,\\beta \\in \\mathbb {N}$ .",
"We have $(X^{\\beta }/t^{\\alpha })^{n}=t$ in $B((t))[X]/(X^{n}-t^{q})$ and isomorphisms $ \\begin{tikzpicture}[xscale=3.9,yscale=-0.7]\\node (A0_0) at (0.25, 0) {s};\\node (A0_1) at (1, 0) {X};\\node (A0_2) at (2, 0) {X^{\\beta }/t^{\\alpha }};\\node (A1_0) at (0.25, 1) {B((s))};\\node (A1_1) at (1, 1) {B((t))[X]/(X^n-t)};\\node (A1_2) at (2, 1) {B((t))[X]/(X^n-t^q)};(A0_0) edge [|->]node [auto] {\\scriptstyle {}} (A0_1);(A1_0) edge [->]node [auto] {\\scriptstyle {\\simeq }} (A1_1);(A0_1) edge [|->]node [auto] {\\scriptstyle {}} (A0_2);(A1_1) edge [->]node [auto] {\\scriptstyle {\\simeq }} (A1_2); \\end{tikzpicture} $ Definition 4.31 Given a $p$ -group $H$ and an autoequivalence $\\phi \\colon \\Delta _{H}\\longrightarrow \\Delta _{H}$ we define $\\mathcal {Z}_{\\phi }$ as the stack of pairs $(P,u)$ where $P\\in \\Delta _{H}$ and $u\\colon P\\longrightarrow \\phi (P)$ is an isomorphism in $\\Delta _{H}$ .",
"There are two natural autoequivalences of $\\Delta _{H}$ : $\\phi _{\\psi }\\colon \\Delta _{H}\\longrightarrow \\Delta _{H}$ obtained composing by an isomorphism $\\psi \\colon H\\longrightarrow H$ ; $\\phi _{\\xi }\\colon \\Delta _{H}\\longrightarrow \\Delta _{H}$ induced by a $n$ -th root of unity $\\xi $ using the Cartesian diagram $ \\begin{tikzpicture}[xscale=2.7,yscale=-0.6] \\node (A0_0) at (0, 0) {\\phi _\\xi (P)}; \\node (A0_1) at (1, 0) {P}; \\node (A2_0) at (0, 2) {\\operatorname{\\textup {Spec}}B((t))}; \\node (A2_1) at (1, 2) {\\operatorname{\\textup {Spec}}B((t))}; \\node (A3_0) at (0, 3) {\\xi t}; \\node (A3_1) at (1, 3) {t}; (A3_1) edge [|->]node [auto] {\\scriptstyle {}} (A3_0); (A2_0) edge [->]node [auto] {\\scriptstyle {}} (A2_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A2_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A2_0); \\end{tikzpicture} $ Proposition 4.32 Assume $(q,n)=1$ and let $\\zeta \\in C_{n}$ be a primitive $n$ -th root of unity and $\\psi \\colon H\\longrightarrow H$ be the automorphism image of $\\zeta $ under $C_{n}\\longrightarrow \\operatorname{\\textup {Aut}}(H)$ .",
"Set $\\xi =\\zeta ^{\\beta }$ where $\\beta \\in \\mathbb {Z}/n\\mathbb {Z}$ is the inverse of $q$ and $\\phi =\\phi _{\\psi }\\circ \\phi _{\\xi }\\colon \\Delta _{H}\\longrightarrow \\Delta _{H}$ .",
"Then $\\mathcal {Z}_{G,q}$ is an open and closed substack of $\\mathcal {Z}_{\\phi }$ .",
"Let $P\\in \\Delta _{H}$ .",
"We have $\\phi (P)=\\phi _{\\xi }(P)$ as schemes.",
"Thus an isomorphism $u\\colon P\\longrightarrow \\phi (P)$ , via $\\phi (P)=\\phi _{\\xi }(P)\\longrightarrow P$ , corresponds to an isomorphism $v\\colon P\\longrightarrow P$ .",
"The morphism $u$ is over $\\operatorname{\\textup {Spec}}B((t))$ if and only if the following diagram commutes $ \\begin{tikzpicture}[xscale=2.7,yscale=-0.6] \\node (A0_0) at (0, 0) {P}; \\node (A0_1) at (1, 0) {P}; \\node (A2_0) at (0, 2) {\\operatorname{\\textup {Spec}}B((t))}; \\node (A2_1) at (1, 2) {\\operatorname{\\textup {Spec}}B((t))}; \\node (A3_0) at (0, 3) {\\xi t}; \\node (A3_1) at (1, 3) {t}; (A3_1) edge [|->]node [auto] {\\scriptstyle {}} (A3_0); (A2_0) edge [->]node [auto] {\\scriptstyle {}} (A2_1); (A0_0) edge [->]node [auto] {\\scriptstyle {v}} (A0_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A2_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A2_0); \\end{tikzpicture} $ Finally, by going through the definitions, we see that $u$ is $H$ -equivariant if and only if $\\psi (h)=vhv^{-1}$ in $\\operatorname{\\textup {Aut}}(P)$ for all $h\\in H$ .",
"We identify $\\mathcal {Z}_{\\phi }$ with the stack of pairs $(P,v)$ as above.",
"Set $S_{B}=\\operatorname{\\textup {Spec}}B((s))$ with the $C_{n}$ -action given by $s\\mapsto \\lambda ^{\\beta }s$ for $\\lambda \\in C_{n}$ .",
"By REF $S_{B}$ is isomorphic to $\\operatorname{\\textup {Spec}}B((t))[X]/(X^{n}-t^{q})$ and therefore, by construction, an object $(P,\\delta )\\in \\mathcal {Z}_{G,q}(B)$ is a $G$ -torsor over $B((t))$ with a $G$ -equivariant map $\\delta \\colon P\\longrightarrow S_{B}$ .",
"In particular $P{\\delta }S_{B}$ is an $H$ -torsor.",
"Set $g_{0}=(1,\\zeta )\\in H\\rtimes C_{n}=G$ , so that $\\psi (h)=g_{0}hg_{0}^{-1}$ in $G$ .",
"Since $\\delta $ is $G$ -equivariant it follows that $(P,g_{0})\\in \\mathcal {Z}_{\\phi }$ .",
"We therefore get a map $\\mathcal {Z}_{G,q}\\longrightarrow \\mathcal {Z}_{\\phi }$ .",
"This map is fully faithful.",
"Indeed if $(P,\\delta ),(P^{\\prime },\\delta ^{\\prime })\\in \\mathcal {Z}_{G,q}(B)$ the map on isomorphisms is $ \\begin{tikzpicture}[xscale=1.5,yscale=-1.2] \\node (A0_0) at (0, 0) {\\operatorname{\\textup {Iso}}_{\\mathcal {Z}_{G,q}}((P,\\delta ),(P^{\\prime },\\delta ^{\\prime }))=\\operatorname{\\textup {Iso}}_{S_{B}}^{G}(P,P^{\\prime })}; \\node (A1_0) at (0, 1) {\\operatorname{\\textup {Iso}}_{\\mathcal {Z}_{\\phi }}((P,g_{0}),(P^{\\prime },g_{0}))=\\lbrace \\omega \\in \\operatorname{\\textup {Iso}}_{S_{B}}^{H}(P,P^{\\prime })\\ | \\ g_{0}\\omega =\\omega g_{0}\\rbrace }; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); \\end{tikzpicture} $ We are going to show that $\\mathcal {Z}_{G,q}=\\overline{\\mathcal {Z}}_{\\phi }\\subseteq \\mathcal {Z}_{\\phi }$ , where $\\overline{\\mathcal {Z}}_{\\phi }$ is the substack of pairs $(P,v)$ where $v^{n}=\\textup {id}$ .",
"Since $g_{0}^{n}=1$ we have the inclusion $\\subseteq $ .",
"Now let $(P,v)\\in \\overline{\\mathcal {Z}}_{\\phi }$ .",
"Since $v^{n}=1$ we get the map $G=H\\rtimes C_{n}\\longrightarrow \\operatorname{\\textup {Aut}}(P)$ sending $(h,\\zeta ^{m})$ to $hv^{m}$ , defining a $G$ -action on $P$ .",
"By construction the map $\\delta \\colon P\\longrightarrow S_{B}$ is $G$ -equivariant.",
"Since $P/H\\simeq S_{B}$ it remains to show that $G$ acts freely on $P$ .",
"If $p(hv^{l})=p$ for some $p\\in P$ , then $\\delta (ph)\\zeta ^{l}=\\delta (p)\\zeta ^{l}=\\delta (p)$ and therefore $n\\mid l$ .",
"Finally $ph=p$ implies $h=1$ in $H\\subseteq \\operatorname{\\textup {Aut}}(P)$ because $P$ is an $H$ -torsor.",
"We now show that $\\overline{\\mathcal {Z}}_{\\phi }$ is open and closed in $\\mathcal {Z}_{\\phi }$ .",
"If $(P,v)\\in \\mathcal {Z}_{\\phi }(B)$ we have that $v^{n}\\in \\operatorname{\\textup {Aut}}_{B((t))}^{H}(P)$ because $\\psi (h)=vhv^{-1}$ and $\\psi $ has order $n$ .",
"The group scheme $\\operatorname{\\underline{\\textup {Aut}}}_{B((t))}^{H}(P)\\longrightarrow \\operatorname{\\textup {Spec}}B((t))$ is finite and étale, thus the locus $W$ in $\\operatorname{\\textup {Spec}}B((t))$ where $v^{n}=\\textup {id}$ is open and closed in $\\operatorname{\\textup {Spec}}B((t))$ .",
"By REF there is an open and closed subset $\\widetilde{W}$ in $\\operatorname{\\textup {Spec}}B$ inducing $W$ .",
"By construction the base change of $\\overline{\\mathcal {Z}}_{\\phi }\\longrightarrow \\mathcal {Z}_{\\phi }$ along $(P,v)\\colon \\operatorname{\\textup {Spec}}B\\longrightarrow \\mathcal {Z}_{\\phi }$ is $\\widetilde{W}$ , which ends the proof.",
"Proposition 4.33 If $\\phi \\colon \\Delta _{H}\\longrightarrow \\Delta _{H}$ is an equivalence then there exists a direct system of separated DM stacks $\\mathcal {Z}_{*}$ with finite and universally injective transition maps, with a direct system of finite and étale atlases $Z_{n}\\longrightarrow \\mathcal {Z}_{n}$ of degree $(\\sharp H)^{2}$ from affine schemes and an equivalence $\\varinjlim _{n}\\mathcal {Z}_{n}\\simeq \\mathcal {Z}_{\\phi }$ .",
"Consider a direct system of DM stacks $\\mathcal {Y}_{*}$ as in Theorem REF for the $p$ -group $H$ .",
"Denote by $\\Gamma _{\\phi }\\colon \\Delta _{H}\\longrightarrow \\Delta _{H}\\times \\Delta _{H}$ be the graph of $\\phi $ and by $\\gamma _{u,v}\\colon \\mathcal {Y}_{u}\\longrightarrow \\mathcal {Y}_{v}$ the transition maps.",
"By REF $\\mathcal {Z}_{\\phi }$ is the fiber product of $\\Gamma _{\\phi }$ and the diagonal of $\\Delta _{H}$ .",
"There exist an increasing function $\\delta \\colon \\mathbb {N}\\longrightarrow \\mathbb {N}$ and 2-commutative diagrams $ \\begin{tikzpicture}[xscale=1.5,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {Y}_n}; \\node (A0_1) at (1, 0) {\\mathcal {Y}_{n+1}}; \\node (A0_2) at (2, 0) {\\Delta _H}; \\node (A1_0) at (0, 1) {\\mathcal {Y}_{\\delta _n}}; \\node (A1_1) at (1, 1) {\\mathcal {Y}_{\\delta _{n+1}}}; \\node (A1_2) at (2, 1) {\\Delta _H}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A0_2); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_1) edge [->]node [auto] {\\scriptstyle {}} (A1_2); (A0_2) edge [->]node [auto] {\\scriptstyle {\\phi }} (A1_2); (A0_0) edge [->]node [auto] {\\scriptstyle {\\phi _n}} (A1_0); (A0_1) edge [->]node [auto] {\\scriptstyle {\\phi _{n+1}}} (A1_1); \\end{tikzpicture} $ Similarly $\\mathcal {Y}_{n}\\longrightarrow \\mathcal {Y}_{n}\\times \\mathcal {Y}_{n}{\\gamma _{n,\\delta _{n}}\\times \\phi _{n}}\\mathcal {Y}_{\\delta _{n}}\\times \\mathcal {Y}_{\\delta _{n}}$ and $\\mathcal {Y}_{n}\\longrightarrow \\mathcal {Y}_{n}\\times \\mathcal {Y}_{n}{\\gamma _{n,\\delta _{n}}\\times \\gamma _{n,\\delta _{n}}}\\mathcal {Y}_{\\delta _{n}}\\times \\mathcal {Y}_{\\delta _{n}}$ approximate $\\Gamma _{\\phi }$ and the diagonal of $\\Delta _{H}$ respectively.",
"By REF it follows that the fiber product $\\mathcal {Z}_{n}=\\mathcal {Y}_{n}\\times _{\\mathcal {Y}_{\\delta _{n}}\\times \\mathcal {Y}_{\\delta _{n}}}\\mathcal {Y}_{n}$ of the two maps form a direct system of separated DM stacks whose limit is $\\mathcal {Z}_{\\phi }$ .",
"By REF the transition maps are finite and universally injective.",
"Let $Y_{n}\\longrightarrow \\mathcal {Y}_{n}$ be the finite and étale atlases of degree $\\sharp H$ given in Theorem REF .",
"The induced map $Z_{n}=Y_{n}\\times _{\\mathcal {Y}_{\\delta _{n}}\\times \\mathcal {Y}_{\\delta _{n}}}Y_{n}\\longrightarrow \\mathcal {Z}_{n}$ is finite and étale of degree $(\\sharp H)^{2}$ .",
"Since $\\mathcal {Y}_{\\delta _{n}}\\times \\mathcal {Y}_{\\delta _{n}}$ has affine diagonal it follows that $Z_{n}$ is affine.",
"Finally, using the usual properties of fiber products and the fact that $Y_{n}\\longrightarrow \\mathcal {Y}_{n}$ is a direct system of atlases, we see that the maps $Z_{n}\\longrightarrow Z_{n+1}\\times _{\\mathcal {Z}_{n+1}}\\mathcal {Z}_{n}$ are isomorphisms.",
"Recall that $G=H\\rtimes C_{n}$ .",
"Consider $\\pi \\colon \\Delta _{G}^{*}\\longrightarrow \\Delta _{C_{n}}^{*}$ and the decomposition $\\Delta _{C_{n}}^{*}=\\bigsqcup _{q=1}^{n}\\operatorname{\\textup {B}}G_{q}$ , where $G_{q}=C_{n}$ , of Theorem REF .",
"The map $\\bigsqcup _{q=1}^{n}(\\operatorname{\\textup {B}}G_{q}\\times _{\\Delta _{C_{n}}^{*}}\\Delta _{G}^{*})\\longrightarrow \\Delta _{G}^{*}$ is well defined and an equivalence because given $k$ -algebras $A_{1}$ and $A_{2}$ the map $\\Delta _{G}(A_{1}\\times A_{2})\\longrightarrow \\Delta _{G}(A_{1})\\times \\Delta _{G}(A_{2})$ is an equivalence.",
"Thus in the statement of the theorem $\\Delta _{G}^{*}$ can be replaced by $\\widetilde{\\mathcal {Z}}_{G,q}=(\\operatorname{\\textup {B}}G_{q}\\times _{\\Delta _{C_{n}}^{*}}\\Delta _{G}^{*})$ .",
"We must show that $\\widetilde{\\mathcal {Z}}_{G,q}$ is a stack in the fpqc topology if $\\mathcal {Z}_{G,q}$ is so.",
"Let $B$ be a ring, $\\mathcal {U}=\\lbrace B\\longrightarrow B_{i}\\rbrace _{i\\in I}$ a covering and $\\xi \\in \\widetilde{\\mathcal {Z}}_{G,q}(\\mathcal {U})$ be a descent datum.",
"Given a $B$ -scheme $Y$ we denote by $\\mathcal {U}_{Y}=\\mathcal {U}\\times _{B}Y$ and by $\\xi _{Y}\\in \\widetilde{\\mathcal {Z}}_{G,q}(\\mathcal {U}_{Y})$ the pullback.",
"Denote by $r\\colon \\widetilde{\\mathcal {Z}}_{G,q}\\longrightarrow \\operatorname{\\textup {B}}C_{n}$ the structure map.",
"The descent datum $r(\\xi )$ yields a $C_{n}$ -torsor $F\\longrightarrow \\operatorname{\\textup {Spec}}B$ .",
"Let $Y\\longrightarrow \\operatorname{\\textup {Spec}}B$ a $B$ -scheme with a factorization $Y\\longrightarrow F$ .",
"This factorization is a trivialization of the $C_{n}$ -torsor over $Y$ and therefore it induces a descent datum of $(\\widetilde{\\mathcal {Z}}_{G,q}\\times _{\\operatorname{\\textup {B}}C_{n}}\\operatorname{\\textup {Spec}}k)(\\mathcal {U}_{Y})=\\mathcal {Z}_{G,q}(\\mathcal {U}_{Y})$ which is therefore effective, yielding $\\eta _{Y}\\in \\mathcal {Z}_{G,q}(Y)$ and $\\widetilde{\\eta }_{Y}\\in \\widetilde{\\mathcal {Z}}_{G,q}(Y)$ .",
"By construction $\\widetilde{\\eta }_{Y}\\in \\widetilde{\\mathcal {Z}}_{G,q}(Y)$ induces the descent datum $\\xi _{Y}\\in \\widetilde{\\mathcal {Z}}_{G,q}(\\mathcal {U}_{Y})$ .",
"In particular we get $\\widetilde{\\eta }_{F}\\in \\widetilde{\\mathcal {Z}}_{G,q}(F)$ .",
"Since $\\widetilde{\\mathcal {Z}}_{G,q}$ is a prestack, the objects $\\widetilde{\\eta }_{F\\times _{B}F}$ obtained using the two projections $F\\times _{B}F\\longrightarrow F$ are isomorphic via a given isomorphism: they both induce $\\xi _{F\\times _{B}F}\\in \\widetilde{\\mathcal {Z}}_{G,q}(\\mathcal {U}_{F\\times _{B}F})$ which does not depend on the projections being a pullback of $\\xi \\in \\widetilde{\\mathcal {Z}}_{G,q}(\\mathcal {U})$ .",
"In conclusion $\\widetilde{\\eta }_{F}$ gives a descent datum for $\\widetilde{\\mathcal {Z}}_{G,q}$ over the covering $F\\longrightarrow \\operatorname{\\textup {Spec}}B$ .",
"In order to get a global object in $\\widetilde{\\mathcal {Z}}_{G,q}(B)$ inducing the given descent datum $\\xi $ it is enough to notice that, by REF , $\\Delta _{G}$ satisfies descent along coverings $U\\longrightarrow \\operatorname{\\textup {Spec}}B$ which are finite, flat and finitely presented.",
"Thanks to REF , it is enough to show that $\\mathcal {Z}_{G,q}$ is a limit as in the statement.",
"Using REF we can further assume $n$ and $q$ coprime and, using REF , we can replace $\\mathcal {Z}_{G,q}$ by $\\mathcal {Z}_{\\phi }$ , where $\\phi =\\phi _{\\psi }\\circ \\phi _{\\xi }$ as in REF .",
"The conclusion now follows from REF ."
],
[
" Limit of fibered categories",
"In this appendix we discuss the notion of inductive limit of stacks.",
"To simplify the exposition and since general colimits were not needed in this paper we will only talk about limit over the natural numbers $\\mathbb {N}$ .",
"General results can be found in [10].",
"A direct system of categories $\\mathcal {C}_{*}$ (indexed by $\\mathbb {N}$ ) is a collection of categories $\\mathcal {C}_{n}$ for $n\\in \\mathbb {N}$ and functors $\\psi _{n}\\colon \\mathcal {C}_{n}\\longrightarrow \\mathcal {C}_{n+1}$ .",
"Given indexes $n<m$ we also set $\\psi _{n,m}\\colon C_{n}{\\psi _{n}}\\mathcal {C}_{n+1}\\longrightarrow \\cdots \\longrightarrow \\mathcal {C}_{m-1}{\\psi _{m-1}}\\mathcal {C}_{m}$ and $\\psi _{n,n}=\\textup {id}_{\\mathcal {C}_{n}}$ .",
"The limit $\\varinjlim _{n\\in \\mathbb {N}}C_{n}$ or $\\mathcal {C}_{\\infty }$ is the category defined as follows.",
"Its objects are pairs $(n,x)$ with $n\\in \\mathbb {N}$ and $x\\in \\mathcal {C}_{n}$ .",
"Given pairs $(n,x)$ and $(m,y)$ we set $\\operatorname{\\textup {Hom}}_{\\mathcal {C}_{\\infty }}((n,x),(m,x))=\\varinjlim _{q>n+m}\\operatorname{\\textup {Hom}}_{\\mathcal {C}_{q}}(\\psi _{n,q}(x),\\psi _{m,q}(y))$ Composition is defined in the obvious way.",
"There are obvious functors $\\mathcal {C}_{n}\\longrightarrow \\mathcal {C}_{\\infty }$ .",
"Given a category $\\mathcal {D}$ we denote by $\\operatorname{\\textup {Hom}}(\\mathcal {C}_{*},\\mathcal {D})$ the category whose objects are collections $(F_{n},\\alpha _{n})$ where $F_{n}\\colon \\mathcal {C}_{n}\\longrightarrow \\mathcal {D}$ are functors and $\\alpha _{n}\\colon F_{n+1}\\circ \\psi _{n}\\longrightarrow F_{n}$ are natural isomorphisms of functors $\\mathcal {C}_{n}\\longrightarrow \\mathcal {D}$ .",
"There is an obvious functor $\\operatorname{\\textup {Hom}}(\\mathcal {C}_{\\infty },\\mathcal {D})\\longrightarrow \\operatorname{\\textup {Hom}}(\\mathcal {C}_{*},\\mathcal {D})$ and we have: Proposition 1.1 [10] The functor $\\operatorname{\\textup {Hom}}(\\mathcal {C}_{\\infty },\\mathcal {D})\\longrightarrow \\operatorname{\\textup {Hom}}(\\mathcal {C}_{*},\\mathcal {D})$ is an equivalence.",
"Let $\\mathcal {S}$ be a category with fiber products.",
"A direct system of fibered categories $\\mathcal {X}_{*}$ over $\\mathcal {S}$ (indexed by $\\mathbb {N}$ ) is a direct system of categories $\\mathcal {X}_{*}$ together with maps $\\mathcal {X}_{n}\\longrightarrow \\mathcal {S}$ making $\\mathcal {X}_{n}$ into a fibered category over $\\mathcal {S}$ and such that the transition maps $\\mathcal {X}_{n}\\longrightarrow \\mathcal {X}_{n+1}$ are maps of fibered categories.",
"Result [10] translates into what follows.",
"The induced functor $\\mathcal {X}_{\\infty }\\longrightarrow \\mathcal {S}$ makes $\\mathcal {X}_{\\infty }$ into a fibered category over $\\mathcal {S}$ and the maps $\\mathcal {X}_{n}\\longrightarrow \\mathcal {X}_{\\infty }$ are map of fibered categories.",
"Given an object $s\\in \\mathcal {S}$ there is an induced direct system of categories $\\mathcal {X}_{*}(s)$ and there is a natural equivalence $\\varinjlim _{n\\in \\mathbb {N}}\\mathcal {X}_{n}(s){\\simeq }\\mathcal {X}_{\\infty }(s)$ In particular if all the $\\mathcal {X}_{n}$ are fibered in sets (resp.",
"groupoids) so is $\\mathcal {X}_{\\infty }$ .",
"If $\\mathcal {Y}$ is another fibered category over $\\mathcal {S}$ denotes by $\\operatorname{\\textup {Hom}}_{\\mathcal {S}}(\\mathcal {X}_{*},\\mathcal {Y})$ the subcategory of $\\operatorname{\\textup {Hom}}(\\mathcal {X}_{*},\\mathcal {Y})$ of objects $(F_{n},\\alpha _{n})$ where $F_{n}$ are base preserving functors and $\\alpha _{n}$ are base preserving natural transformations.",
"Also the arrows in the category $\\operatorname{\\textup {Hom}}_{\\mathcal {S}}(\\mathcal {X}_{*},\\mathcal {Y})$ are required to be base preserving natural transformations.",
"There is an induced functor $\\operatorname{\\textup {Hom}}_{\\mathcal {S}}(\\mathcal {X}_{\\infty },\\mathcal {Y})\\longrightarrow \\operatorname{\\textup {Hom}}_{\\mathcal {S}}(\\mathcal {X}_{*},\\mathcal {Y})$ which is an equivalence of categories.",
"A direct check using the definition of fiber product yields the following.",
"Proposition 1.2 Let $\\mathcal {X}_{*},\\mathcal {Y}_{*}$ and $\\mathcal {Z}_{*}$ be direct system of categories fibered in groupoids over $\\mathcal {S}$ and assume they are given 2-commutative diagrams $ \\begin{tikzpicture}[xscale=1.9,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {X}_n}; \\node (A0_1) at (1, 0) {\\mathcal {X}_{n+1}}; \\node (A0_2) at (2, 0) {\\mathcal {Z}_{n}}; \\node (A0_3) at (3, 0) {\\mathcal {Z}_{n+1}}; \\node (A1_0) at (0, 1) {\\mathcal {Y}_n}; \\node (A1_1) at (1, 1) {\\mathcal {Y}_{n+1}}; \\node (A1_2) at (2, 1) {\\mathcal {Y}_{n}}; \\node (A1_3) at (3, 1) {\\mathcal {Y}_{n+1}}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_1) edge [->]node [auto] {\\scriptstyle {a_{n+1}}} (A1_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_3) edge [->]node [auto] {\\scriptstyle {b_{n+1}}} (A1_3); (A0_2) edge [->]node [auto] {\\scriptstyle {b_n}} (A1_2); (A0_0) edge [->]node [auto] {\\scriptstyle {a_n}} (A1_0); (A0_2) edge [->]node [auto] {\\scriptstyle {}} (A0_3); (A1_2) edge [->]node [auto] {\\scriptstyle {}} (A1_3); \\end{tikzpicture} $ Then the canonical map $\\varinjlim _{n\\in \\mathbb {N}}(\\mathcal {X}_{n}\\times _{\\mathcal {Y}_{n}}\\mathcal {Z}_{n})\\longrightarrow \\varinjlim _{n\\in \\mathbb {N}}\\mathcal {X}_{n}\\times _{{\\displaystyle \\varinjlim _{n\\in \\mathbb {N}}}\\mathcal {Y}_{n}}\\varinjlim _{n\\in \\mathbb {N}}\\mathcal {Z}_{n}$ is an equivalence.",
"Corollary 1.3 Let $\\mathcal {X}_{*}$ and $\\mathcal {Y}_{*}$ be direct systems of categories fibered in groupoids over $\\mathcal {S}$ and assume to have 2-Cartesian diagrams $ \\begin{tikzpicture}[xscale=1.5,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {Y}_n}; \\node (A0_1) at (1, 0) {\\mathcal {Y}_{n+1}}; \\node (A1_0) at (0, 1) {\\mathcal {X}_n}; \\node (A1_1) at (1, 1) {\\mathcal {X}_{n+1}}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); \\end{tikzpicture} $ Then the following diagrams are also 2-Cartesian for all $n\\in \\mathbb {N}$ : $ \\begin{tikzpicture}[xscale=2.0,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {Y}_n}; \\node (A0_1) at (1, 0) {\\varinjlim \\mathcal {Y}_*}; \\node (A1_0) at (0, 1) {\\mathcal {X}_n}; \\node (A1_1) at (1, 1) {\\varinjlim \\mathcal {X}_*}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); \\end{tikzpicture} $ We have maps $\\mathcal {Y}_{n}{\\simeq }\\mathcal {X}_{n}\\times _{\\mathcal {X}_{m}}\\mathcal {Y}_{m}\\longrightarrow \\mathcal {X}_{n}\\times _{\\mathcal {X}_{\\infty }}\\mathcal {Y}_{\\infty }$ for all $m>n$ .",
"Passing to the limit on $m$ we get the result.",
"Lemma 1.4 Assume that $\\mathcal {S}$ has a Grothendieck topology such that for all coverings $\\mathcal {V}=\\lbrace U_{i}\\longrightarrow U\\rbrace _{i\\in I}$ there exists a finite subset $J\\subseteq I$ for which $\\mathcal {V}_{J}=\\lbrace U_{j}\\longrightarrow U\\rbrace _{j\\in J}$ is also a covering.",
"Let also $\\mathcal {Y}$ be a fibered category.",
"Then $\\mathcal {Y}$ is a stack (resp.",
"pre-stack) if and only if given a finite covering $\\mathcal {U}$ of $U\\in \\mathcal {S}$ the functor $\\mathcal {Y}(U)\\longrightarrow \\mathcal {Y}(\\mathcal {U})$ is an equivalence (resp.",
"fully faithful), where $\\mathcal {Y}(\\mathcal {U})$ is the category of descent data of $\\mathcal {Y}$ over $\\mathcal {U}$ .",
"The “only if” part is trivial.",
"We show the “if” part.",
"Let $\\mathcal {V}=\\lbrace U_{i}\\longrightarrow U\\rbrace _{i\\in I}$ be a general covering and consider a finite subset $J\\subseteq I$ for which $\\mathcal {V}_{J}=\\lbrace U_{i}\\longrightarrow U\\rbrace _{i\\in J}$ is also a covering.",
"Thus the composition $\\mathcal {Y}(U)\\longrightarrow \\mathcal {Y}(\\mathcal {V})\\longrightarrow \\mathcal {Y}(\\mathcal {V}_{J})$ is an equivalence (resp.",
"fully faithful) and it is enough to show that $\\mathcal {Y}(\\mathcal {V})\\longrightarrow \\mathcal {Y}(\\mathcal {V}_{J})$ is faithful.",
"This follows because there is a 2-commutative diagram $ \\begin{tikzpicture}[xscale=3.3,yscale=-1.5] \\node (A0_0) at (0, 0) {\\mathcal {Y}(\\mathcal {V})}; \\node (A0_1) at (1, 0) {\\displaystyle \\prod _{i\\in I}\\mathcal {Y}(U_i)}; \\node (A1_0) at (0, 1) {\\mathcal {Y}(\\mathcal {V}_J)}; \\node (A1_1) at (1, 1) {\\displaystyle \\prod _{i\\in I}(\\prod _{j\\in J}\\mathcal {Y}(U_i\\times _U U_j))}; (A0_0) edge [->]node [auto] {\\scriptstyle {a}} (A0_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); (A0_1) edge [->]node [auto] {\\scriptstyle {b}} (A1_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); \\end{tikzpicture} $ where the functors $a$ and $b$ are faithful.",
"Proposition 1.5 In the hypothesis of REF , if $\\mathcal {X}_{*}$ is a direct system of stacks (resp.",
"pre-stacks) over $\\mathcal {S}$ then $\\mathcal {X}_{\\infty }$ is also a stack (resp.",
"pre-stack) over $\\mathcal {S}$ .",
"It is easy to prove descent (resp.",
"descent on morphisms) and its uniqueness along coverings indexed by finite sets.",
"By REF this is enough.",
"Clearly the site $\\mathcal {S}$ we have in mind in the above proposition is a category fibered in groupoids over the category of affine schemes $\\operatorname{\\textup {Aff}}$ with any of the usual topologies, for instance $\\operatorname{\\textup {Aff}}/X$ , the category of affine schemes together with a map to a given scheme $X$ ."
],
[
"Rigidification revisited",
"Rigidification is an operation that allows us to “kill” automorphisms of a given stack by modding out stabilizers by a given subgroup of the inertia.",
"This operation is described in [2] in the context of algebraic stacks, but one can easily see that this is a very general construction.",
"In this appendix we discuss it in its general form.",
"Let $\\mathcal {S}$ be a site, $\\mathcal {X}$ be a stack in groupoids over $\\mathcal {S}$ and denote by $I(\\mathcal {X})\\longrightarrow \\mathcal {X}$ the inertia stack.",
"The inertia stack can be also thought as the sheaf $\\mathcal {X}^{\\operatorname{\\textup {op}}}\\longrightarrow (\\text{Groups})$ mapping $\\xi \\in \\mathcal {X}(U)$ to $\\operatorname{\\textup {Aut}}_{\\mathcal {X}(U)}(\\xi )$ .",
"By a subgroup sheaf of the inertia stack we mean a subgroup sheaf of the previous functor.",
"Notice that given a sheaf $F\\colon \\mathcal {X}^{\\operatorname{\\textup {op}}}\\longrightarrow (\\textup {Sets})$ and an object $\\xi \\in \\mathcal {X}(U)$ one get a sheaf $F_{\\xi }$ on $U$ by composing $(\\mathcal {S}/U)^{\\operatorname{\\textup {op}}}{\\xi }\\mathcal {X}^{\\operatorname{\\textup {op}}}\\longrightarrow (\\textup {Sets})$ , where the first arrow comes from the 2-Yoneda lemma.",
"Concretely one has $F_{\\xi }(V{g}U)=F(g^{*}\\xi )$ .",
"If $f\\colon V\\longrightarrow U$ is any map in $\\mathcal {S}$ there is a canonical isomorphism $F_{\\xi }\\times _{U}V\\simeq F_{f^{*}\\xi }$ .",
"Notice moreover that a subgroup sheaf $\\mathcal {H}$ of $I(\\mathcal {X})$ is automatically normal: if $\\xi \\in \\mathcal {X}(U)$ and $\\omega \\in I(\\mathcal {X})(\\xi )=\\operatorname{\\textup {Aut}}_{\\mathcal {X}(U)}(\\xi )$ then $\\omega $ induces the conjugation $I(\\mathcal {X})(\\xi )\\longrightarrow I(\\mathcal {X})(\\xi )$ and, since $\\mathcal {H}$ is a subsheaf, the subgroup $\\mathcal {H}(\\xi )$ is preserved by the conjugation.",
"We now describe how to rigidify $\\mathcal {X}$ by any subgroup sheaf $\\mathcal {H}$ of the inertia.",
"We define the category $\\widetilde{\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}}$ as follows.",
"The objects are the same as the ones of $\\mathcal {X}$ .",
"Given $\\xi \\in \\mathcal {X}(U)$ and $\\eta \\in \\mathcal {X}(V)$ an arrow $\\xi \\longrightarrow \\eta $ in $\\widetilde{\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}}$ is a pair $(f,\\phi )$ where $f\\colon U\\longrightarrow V$ and $\\phi \\in (\\operatorname{\\underline{\\textup {Iso}}}_{U}(f^{*}\\eta ,\\xi )/\\mathcal {H}_{\\xi })(U)$ .",
"Given $\\zeta \\in \\mathcal {X}(W)$ and arrows $\\xi {(f,\\phi )}\\eta {(g,\\psi )}\\zeta $ we have that $(\\operatorname{\\underline{\\textup {Iso}}}_{U}(f^{*}g^{*}\\zeta ,f^{*}\\eta )/\\mathcal {H}_{f^{*}\\eta })\\simeq (\\operatorname{\\underline{\\textup {Iso}}}_{V}(g^{*}\\zeta ,\\eta )/\\mathcal {H}_{\\eta })\\times _{V}U$ , because the action of $\\mathcal {H}_{*}$ is free.",
"Moreover composition induces a map $(\\operatorname{\\underline{\\textup {Iso}}}_{U}(f^{*}g^{*}\\zeta ,f^{*}\\eta )/\\mathcal {H}_{f^{*}\\eta })\\times (\\operatorname{\\underline{\\textup {Iso}}}_{U}(f^{*}\\eta ,\\xi )/\\mathcal {H}_{\\xi })\\longrightarrow (\\operatorname{\\underline{\\textup {Iso}}}_{U}(f^{*}g^{*}\\zeta ,\\xi )/\\mathcal {H}_{\\xi })$ One set $(g,\\psi )\\circ (f,\\phi )=(gf,\\omega )$ where $\\omega $ is the image of $(f^{*}\\psi ,\\phi )$ under the above map.",
"It is elementary to show that this defines a category $\\widetilde{\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}}$ together with a map $\\widetilde{\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}}\\longrightarrow \\mathcal {S}$ making it into a category fibered in groupoids.",
"The map $\\mathcal {X}\\longrightarrow \\widetilde{\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}}$ is also a map of fibered categories.",
"Definition 2.1 The rigidification $\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}$ of $\\mathcal {X}$ by $\\mathcal {H}$ over the site $\\mathcal {S}$ is a stackification of the category fibered in groupoids $\\widetilde{\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}}$ constructed above.",
"Depending on the chosen foundation and the notion of category used, a stackification does not necessarily exists.",
"The usual workaround is to talk about universes but in our case one can directly construct a stackification $\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}$ .",
"We denote by $\\mathcal {Z}$ the category constructed as follows.",
"Its objects are pairs $(\\mathcal {G}\\longrightarrow U,F)$ where $U\\in \\mathcal {S}$ , $\\mathcal {G}\\longrightarrow U$ is a gerbe and $F\\colon \\mathcal {G}\\longrightarrow \\mathcal {X}$ is a map of fibered categories satisfying the following condition: for all $y\\in \\mathcal {G}$ lying over $V\\in \\mathcal {S}$ the map $\\operatorname{\\textup {Aut}}_{\\mathcal {G}(V)}(y)\\longrightarrow \\operatorname{\\textup {Aut}}_{\\mathcal {X}(V)}(F(y))$ is an isomorphism onto $\\mathcal {H}(F(y))$ .",
"An arrow $(\\mathcal {G}^{\\prime }\\longrightarrow U^{\\prime },F^{\\prime })\\longrightarrow (\\mathcal {G}\\longrightarrow U,F)$ is a triple $(f,\\omega ,\\delta )$ where $ \\begin{tikzpicture}[xscale=1.6,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {G}^{\\prime }}; \\node (A0_1) at (1, 0) {\\mathcal {G}}; \\node (A1_0) at (0, 1) {U^{\\prime }}; \\node (A1_1) at (1, 1) {U}; (A0_0) edge [->]node [auto] {\\scriptstyle {\\omega }} (A0_1); (A1_0) edge [->]node [auto] {\\scriptstyle {f}} (A1_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); \\end{tikzpicture} $ is a 2-Cartesian diagram and $\\delta \\colon F\\circ \\omega \\longrightarrow F^{\\prime }$ is a base preserving natural isomorphism.",
"The class of arrows between two given objects is in a natural way a category rather than a set.",
"On the other hand, since the maps from the gerbes to $\\mathcal {X}$ are faithful by definition, this category is equivalent to a set: between two 1-arrows there exist at most one 2-arrow.",
"In particular $\\mathcal {Z}$ is a 1-category.",
"It is not difficult to show that $\\mathcal {Z}$ is fibered in groupoids over $\\mathcal {S}$ and that it satisfies descent, i.e., it is a stack in groupoids over $\\mathcal {S}$ .",
"There is a functor $\\Delta \\colon \\mathcal {X}\\longrightarrow \\mathcal {Z}$ mapping $\\xi \\in \\mathcal {X}(U)$ to $F_{\\xi }\\colon \\operatorname{\\textup {B}}\\mathcal {H}_{\\xi }\\longrightarrow \\mathcal {X}\\times U\\longrightarrow \\mathcal {X}$ .",
"If $\\psi \\colon \\xi ^{\\prime }\\longrightarrow \\xi $ is an isomorphism in $\\mathcal {X}(U)$ , then $\\Delta (\\psi )=(\\operatorname{\\textup {B}}(c_{\\psi }),\\lambda _{\\psi })$ where $c_{\\psi }\\colon \\mathcal {H}_{\\xi ^{\\prime }}\\longrightarrow \\mathcal {H}_{\\xi }$ is the conjugation by $\\psi $ and $\\lambda _{\\psi }$ is the unique natural transformation $F_{\\xi ^{\\prime }}\\longrightarrow F_{\\xi }\\circ \\operatorname{\\textup {B}}(c_{\\psi })$ that evaluated in $\\mathcal {H}_{\\xi ^{\\prime }}$ yields $\\xi ^{\\prime }{\\psi }\\xi $ .",
"For the existence and uniquness of $\\lambda _{\\psi }$ recall that, by descent, a natural transformation of functors $Q,Q^{\\prime }$ from a stack of torsors to a stack, is the same datum of an isomorphism between the values of $Q$ and $Q^{\\prime }$ on the trivial torsor which is functorial with respect to the automorphisms of the trivial torsor.",
"In our case a natural transformation $F_{\\xi ^{\\prime }}\\longrightarrow F_{\\xi }\\circ \\operatorname{\\textup {B}}(c_{\\psi })$ is an isomorphism $\\omega \\colon \\xi ^{\\prime }\\longrightarrow \\xi $ (the values of the functors on the trivial torsor $\\mathcal {H}_{\\xi ^{\\prime }}$ ) such that $c_{\\psi }(u)=\\omega u\\omega ^{-1}$ for all $\\xi ^{\\prime }{u}\\xi ^{\\prime }\\in \\mathcal {H}_{\\xi ^{\\prime }}$ (that is for all automorphisms of the trivial torsor).",
"Given an object $z=(\\mathcal {G},F)\\in \\mathcal {Z}(U)$ there is a natural isomorphism making the following diagram 2-Cartesian: $ \\begin{tikzpicture}[xscale=1.5,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {G}}; \\node (A0_1) at (1, 0) {\\mathcal {X}}; \\node (A1_0) at (0, 1) {U}; \\node (A1_1) at (1, 1) {\\mathcal {Z}}; (A0_0) edge [->]node [auto] {\\scriptstyle {F}} (A0_1); (A1_0) edge [->]node [auto] {\\scriptstyle {z}} (A1_1); (A0_1) edge [->]node [auto] {\\scriptstyle {\\Delta }} (A1_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); \\end{tikzpicture} $ For this reason we call $\\Delta \\colon \\mathcal {X}\\longrightarrow \\mathcal {Z}$ the universal gerbe.",
"The key point in proving this is that if we have a gerbe $\\mathcal {G}$ over $U$ and a section $x\\in \\mathcal {G}(U)$ then the functor $\\mathcal {G}\\longrightarrow \\operatorname{\\textup {B}}(\\operatorname{\\underline{\\textup {Aut}}}_{\\mathcal {G}}(x)),\\ y\\longmapsto \\operatorname{\\underline{\\textup {Iso}}}_{\\mathcal {G}}(x,y)$ is well defined and an equivalence.",
"In particular if $(\\mathcal {G},F)\\in \\mathcal {Z}(U)$ then $\\operatorname{\\underline{\\textup {Aut}}}_{\\mathcal {G}}(x)\\simeq \\mathcal {H}_{F(x)}$ via $F$ .",
"Proposition 2.2 The functor $\\Delta \\colon \\mathcal {X}\\longrightarrow \\mathcal {Z}$ induces a fully faithful epimorphism $\\widetilde{\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}}\\longrightarrow \\mathcal {Z}$ .",
"In particular $\\mathcal {Z}$ is a rigidification $\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}$ of $\\mathcal {X}$ by $\\mathcal {H}$ .",
"Given a functor $T\\longrightarrow \\mathcal {Z}$ induced by $(\\mathcal {G}\\longrightarrow T,F)\\in \\mathcal {Z}(T)$ then $T\\times _{\\mathcal {Z}}\\mathcal {X}$ is the stack of triples $(f,\\xi ,\\omega )$ where $f\\colon S\\longrightarrow T$ , $\\xi \\in \\mathcal {X}(S)$ and $\\omega $ is an isomorphism $(\\operatorname{\\textup {B}}\\mathcal {H}_{\\xi },F_{\\xi })\\simeq (\\mathcal {G},F)$ .",
"Denote by $\\Delta \\colon \\mathcal {X}\\longrightarrow \\mathcal {Z}$ the functor and let $\\xi ,\\xi ^{\\prime }\\in \\mathcal {X}(U)$ .",
"Since $\\mathcal {X}\\longrightarrow \\mathcal {Z}$ is clearly an epimorphism, we have to prove that $\\operatorname{\\underline{\\textup {Iso}}}_{U}(\\xi ^{\\prime },\\xi )\\longrightarrow \\operatorname{\\underline{\\textup {Iso}}}_{U}(\\Delta (\\xi ^{\\prime }),\\Delta (\\xi ))$ is invariant by the action of $\\mathcal {H}_{\\xi }$ and an $\\mathcal {H}_{\\xi }$ -torsor.",
"Notice that a functor of the form $\\operatorname{\\textup {B}}\\mathcal {H}_{\\xi ^{\\prime }}\\longrightarrow \\operatorname{\\textup {B}}\\mathcal {H}_{\\xi }$ is locally induced by a group homomorphism $\\mathcal {H}_{\\xi ^{\\prime }}\\longrightarrow \\mathcal {H}_{\\xi }$ .",
"Thus it is enough to prove that if $c\\colon \\mathcal {H}_{\\xi ^{\\prime }}\\longrightarrow \\mathcal {H}_{\\xi }$ is an isomorphism of groups and $\\lambda \\colon F_{\\xi ^{\\prime }}\\longrightarrow F_{\\xi }\\circ \\operatorname{\\textup {B}}c$ is an isomorphism then the set $J$ of $\\phi \\colon \\xi ^{\\prime }\\longrightarrow \\xi $ inducing $(\\operatorname{\\textup {B}}(c),\\lambda )\\colon \\Delta (\\xi ^{\\prime })\\longrightarrow \\Delta (\\xi )$ is non empty and $\\mathcal {H}_{\\xi }(U)$ acts transitively of this set.",
"The natural transformation $\\lambda $ evaluated on the trivial torsor $\\mathcal {H}_{\\xi ^{\\prime }}$ yields an isomorphism $\\phi \\colon \\xi ^{\\prime }\\longrightarrow \\xi $ .",
"The fact that $\\lambda $ is a natural transformation implies that $c=c_{\\phi }$ and $\\lambda =\\lambda _{\\phi }$ , that is $\\phi \\in J$ .",
"Now let $\\xi ^{\\prime }{\\psi }\\xi $ be an isomorphism.",
"A natural isomorphism $\\operatorname{\\textup {B}}(c_{\\psi })\\longrightarrow \\operatorname{\\textup {B}}(c_{\\phi })$ is given by $h\\in \\mathcal {H}_{\\xi }(U)$ (more precisely the multiplication $\\mathcal {H}_{\\xi }\\longrightarrow \\mathcal {H}_{\\xi }$ by $h$ ) such that $hc_{\\psi }(\\omega )=c_{\\phi }(\\omega )h$ for all $\\omega \\in \\mathcal {H}_{\\xi ^{\\prime }}(U)$ .",
"Such an $h$ induces a morphism $\\Delta (\\psi )\\longrightarrow \\Delta (\\phi )$ if and only if $h\\psi =\\phi $ .",
"Since this condition implies the previous one we see that $J=\\mathcal {H}_{\\xi }(U)\\phi $ .",
"We denote by $\\operatorname{\\textup {B}}_{\\mathcal {X}}\\mathcal {H}$ the stack of $\\mathcal {H}$ -torsors over $\\mathcal {X}$ (thought of as a site).",
"An object of $\\operatorname{\\textup {B}}_{\\mathcal {X}}\\mathcal {H}$ is by definition an object $\\xi \\in \\mathcal {X}(U)$ together with an $\\mathcal {H}_{|\\mathcal {X}/\\xi }$ -torsor over $\\mathcal {X}/\\xi $ .",
"Since $\\mathcal {X}$ is fibered in groupoids the forgetful functor $\\mathcal {X}/\\xi \\longrightarrow \\mathcal {S}/U$ is an equivalence.",
"Thus an object of $\\operatorname{\\textup {B}}_{\\mathcal {X}}\\mathcal {H}$ is an object $\\xi \\in \\mathcal {X}(U)$ together with a $\\mathcal {H}_{\\xi }$ -torsor over $U$ .",
"Proposition 2.3 We have: given $\\xi ,\\eta \\in \\mathcal {X}(U)$ we have $\\operatorname{\\underline{\\textup {Iso}}}_{U}(\\Delta (\\xi ),\\Delta (\\eta ))\\simeq \\operatorname{\\underline{\\textup {Iso}}}(\\xi ,\\eta )/\\mathcal {H}_{\\eta }$ ; the functor $\\Delta \\colon \\mathcal {X}\\longrightarrow \\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}$ is universal among maps of stacks $F\\colon \\mathcal {X}\\longrightarrow \\mathcal {Y}$ such that, for all $\\xi \\in \\mathcal {X}(U)$ , $\\mathcal {H}_{\\xi }$ lies in the kernel of $\\operatorname{\\underline{\\textup {Aut}}}_{U}(\\xi )\\longrightarrow \\operatorname{\\underline{\\textup {Aut}}}_{U}(F(\\xi ))$ ; if $ \\begin{tikzpicture}[xscale=1.5,yscale=-1.2] \\node (A0_0) at (0, 0) {\\mathcal {Y}}; \\node (A0_1) at (1, 0) {\\mathcal {X}}; \\node (A1_0) at (0, 1) {\\mathcal {R}}; \\node (A1_1) at (1, 1) {\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}}; (A0_0) edge [->]node [auto] {\\scriptstyle {b}} (A0_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A0_1) edge [->]node [auto] {\\scriptstyle {\\Delta }} (A1_1); (A0_0) edge [->]node [auto] {\\scriptstyle {a}} (A1_0); \\end{tikzpicture} $ is a 2-Cartesian diagram of stacks then for all $\\eta \\in \\mathcal {Y}(U)$ the map $\\operatorname{\\textup {Ker}}(\\operatorname{\\underline{\\textup {Aut}}}_{U}(\\eta )\\longrightarrow \\operatorname{\\underline{\\textup {Aut}}}_{U}(a(\\eta )))\\longrightarrow \\operatorname{\\underline{\\textup {Aut}}}_{U}(b(\\eta ))$ is an isomorphism onto $\\mathcal {H}_{b(\\eta )}$ , so that $b^{*}\\mathcal {H}$ is naturally a subgroup sheaf of $I(\\mathcal {Y})$ , and the induced map $\\mathcal {Y}\\mathbin {\\!\\!",
"}b^{*}\\mathcal {H}\\longrightarrow \\mathcal {R}$ is an equivalence; there is an isomorphism $\\mathcal {X}\\times _{\\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}}\\mathcal {X}\\simeq \\operatorname{\\textup {B}}_{\\mathcal {X}}(\\mathcal {H})$ .",
"Point $1)$ follows from REF , while point $2)$ is a direct consequence of the definition of rigidification.",
"Now consider point $3)$ .",
"The kernel in the statement corresponds to the group of automorphisms of the object $\\eta $ in the fiber product $U\\times _{\\mathcal {R}}\\mathcal {Y}$ .",
"Using that $\\mathcal {X}\\longrightarrow \\mathcal {X}\\mathbin {\\!\\!",
"}\\mathcal {H}$ is the universal gerbe we get the the isomorphism in the statement.",
"In particular there is an epimorphism $\\mathcal {Y}\\mathbin {\\!\\!",
"}b^{*}\\mathcal {H}\\longrightarrow \\mathcal {R}$ .",
"This is fully faithful because, given $\\eta ,\\eta ^{\\prime }\\in \\mathcal {Y}(U)$ , by definition of fiber product one get a Cartesian diagram $ \\begin{tikzpicture}[xscale=4.0,yscale=-1.2] \\node (A0_0) at (0, 0) {\\operatorname{\\underline{\\textup {Iso}}}_U(\\eta ^{\\prime },\\eta )}; \\node (A0_1) at (1, 0) {\\operatorname{\\underline{\\textup {Iso}}}_U(b(\\eta ^{\\prime }),b(\\eta ))}; \\node (A1_0) at (0, 1) {\\operatorname{\\underline{\\textup {Iso}}}_U(a(\\eta ^{\\prime }),a(\\eta ))}; \\node (A1_1) at (1, 1) {\\operatorname{\\underline{\\textup {Iso}}}_U(b(\\eta ^{\\prime }),b(\\eta ))/\\mathcal {H}_{b(\\eta )}}; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_0) edge [->]node [auto] {\\scriptstyle {}} (A1_1); \\end{tikzpicture} $ For the last point, denotes by $\\mathcal {Y}$ the fiber product in the statement.",
"It is the stack of triples $(\\xi ,\\xi ^{\\prime },\\phi )$ where $\\xi ,\\xi ^{\\prime }\\in \\mathcal {X}(U)$ and $\\phi \\in \\operatorname{\\underline{\\textup {Iso}}}_{U}(\\xi ^{\\prime },\\xi )/\\mathcal {H}_{\\xi }$ .",
"The functor $\\mathcal {Y}\\longrightarrow \\operatorname{\\textup {B}}_{\\mathcal {X}}\\mathcal {H}$ which maps $(\\xi ,\\xi ^{\\prime },\\phi )$ to $(\\xi ,P_{\\phi })$ , where $P_{\\phi }$ is defined by the Cartesian diagram $ \\begin{tikzpicture}[xscale=2.5,yscale=-1.2] \\node (A0_0) at (0, 0) {P_\\phi }; \\node (A0_1) at (1, 0) {\\operatorname{\\underline{\\textup {Iso}}}_U(\\xi ^{\\prime },\\xi )}; \\node (A1_0) at (0, 1) {U}; \\node (A1_1) at (1, 1) {\\operatorname{\\underline{\\textup {Iso}}}_U(\\xi ^{\\prime },\\xi )/\\mathcal {H}_\\xi }; (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A0_1); (A0_0) edge [->]node [auto] {\\scriptstyle {}} (A1_0); (A0_1) edge [->]node [auto] {\\scriptstyle {}} (A1_1); (A1_0) edge [->]node [auto] {\\scriptstyle {\\phi }} (A1_1); \\end{tikzpicture} $ is an equivalence.",
"This is because the functor $\\mathcal {X}\\longrightarrow \\operatorname{\\textup {B}}_{\\mathcal {X}}\\mathcal {H}$ sending $\\xi $ to $\\xi $ with the trivial torsor is an epimorphism and the base change $\\mathcal {Y}\\times _{\\operatorname{\\textup {B}}_{\\mathcal {X}}\\mathcal {H}}\\mathcal {X}\\longrightarrow \\mathcal {X}$ is an equivalence since $\\mathcal {Y}\\times _{\\operatorname{\\textup {B}}_{\\mathcal {X}}\\mathcal {H}}\\mathcal {X}$ is the stack of triples $(\\xi ,\\xi ^{\\prime },\\psi )$ where $\\xi ,\\xi ^{\\prime }\\in \\mathcal {X}(U)$ and $\\psi \\colon \\xi ^{\\prime }\\longrightarrow \\xi $ is an isomorphism in $\\mathcal {X}(U)$ ."
]
] | 1709.01705 |
[
[
"On the sharp lower bounds of Zagreb indices of graphs with given number\n of cut vertices"
],
[
"Abstract The first Zagreb index of a graph $G$ is the sum of the square of every vertex degree, while the second Zagreb index is the sum of the product of vertex degrees of each edge over all edges.",
"In our work, we solve an open question about Zagreb indices of graphs with given number of cut vertices.",
"The sharp lower bounds are obtained for these indices of graphs in $\\mathbb{V}_{n,k}$, where $\\mathbb{V}_{n, k}$ denotes the set of all $n$-vertex graphs with $k$ cut vertices and at least one cycle.",
"As consequences, those graphs with the smallest Zagreb indices are characterized."
],
[
" Introduction",
"A topological index is a constant which can be describing some properties of a molecular graph, that is, a finite graph represents the carbon-atom skeleton of an organic molecule of a hydrocarbon.",
"During past few decades these have been used for the study of quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR) and for the structural essence of biological and chemical compounds.",
"One of the most well-known topological indices is called Randić index, a moleculor quantity of branching index [1].",
"It is known as the classical Randić connectivity index, which is the most useful structural descriptor in QSPR and QSAR, see [2], [3], [4], [5].",
"Many mathematicians focus considerable interests in the structural and applied issues of Randić connectivity index, see [6], [7], [8], [9].",
"Based on these perfect considerations, Zagreb indices[10] are introduced as expressing formulas for the total $\\pi $ -electron energy of conjugated molecules below.",
"$ \\nonumber M_1(G) = \\sum _{u \\in V(G)} d(u)^2~\\text{ and}~ M_2(G) = \\sum _{uv \\in E(G)} d(u)d(v),$ where $G$ is a (molecular) graph, $uv$ is a bond between two atoms $u$ and $v$ , and $d(u)$ (or $d(v)$ , respectively) is the number of atoms that are connected with $u$ (or $v$ , respectively).",
"Zagreb indices are employing as molecular descriptors in QSPR and QSAR, see [11], [12].",
"In the interdisplinary of mathemactics, chemistry and physics, it is not surprising that there are numerous studies of properties of the Zagreb indices of molecular graphs [13], [14], [15], [16], [11], [17], [18], [19], [20].",
"In [21], [22], some bounds of (chemical) trees on Zagreb indices are studied and surveyed.",
"Hou et al.",
"[23] found sharp bounds for Zagreb indices of maximal outerplanar graphs.",
"Li and Zhou [11] investigated the maximum and minimum Zagreb indices of graphs with connectivity at most $k$ .",
"The upper bounds on Zagreb indices of trees in terms of domination number is studied by Borovićanin et al. [24].",
"In many mathematical literatures [25], the maximum and minimum Zagreb indices of trees with a given number of vertices of maximum degree are explored.",
"Xu and Hua [26] provided a unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs.",
"Sharp upper and lower bounds of these indices about $k$ -trees are introduced by Wang and Wei [27].",
"Liu and Zhang provided several sharp upper bounds for multiplicative Zagreb indices in terms of graph parameters such as the order, size and radius [28].",
"The bounds for the moments and the probability generating function of multiplicative Zagreb indices in a randomly chosen molecular graph with tree structure.",
"Zhao and Li [29] investigated the upper bounds of Zagreb indices, and proposed an open question: Question 1.1 [29] How can we determine the lower bound for the first and the second Zagreb indices of n-vertex connected graphs with k cut vertices?",
"What is the characterization of the corresponding extremal graphs?",
"In the view of above results and open problem, we proceed to investigate Zagreb indices of graphs with given number of cut vertices in this paper.",
"It is known that there are many results about Zagreb indices on the graph without cycles.",
"We consider the set of all $n$ -vertex graphs with $k$ cut vertices and at least one cycle, denoted by $\\mathbb {V}_{n, k}$ .",
"In addition, the minimum values of $M_1(G) $ and $M_2(G) $ of graphs with given number of cut vertices are provided.",
"Furthermore, we characterize graphs with the smallest Zagreb indices in $\\mathbb {V}_{n,k}$ ."
],
[
"Preliminary",
"In this section, we provide some important statements, and introduce several graph transformations.",
"These are significant in the following section.",
"Let $G = (V, E)$ be a simple connected graph of $n$ vertices and $m$ edges, where $V = V (G)$ is a vertex set and $E = E(G)$ is an edge set.",
"If $v \\in V(G)$ , then $N(v)$ is the neighborhood of $v$ , that is, $N_{G}(v)= \\left\\lbrace u|\\; uv\\in E(G)\\right\\rbrace $ and the degree of $v$ is $d_{G}(v)= \\left| N_G(v)\\right|$ .",
"A pendent vertex is the vertex of degree 1 and a supporting vertex is the vertex adjacent to at least 1 pendent vertex.",
"A pendent edge is incident to a supporting vertex and a pendent vertex.",
"Given sets $S \\subseteq V(G)$ and $F \\subseteq E(G)$ , denote by $G[S]$ the subgraph of $G$ induced by $S$ , $G - S$ the subgraph induced by $V(G) - S$ and $G - F$ the subgraph of G obtained by deleting $F$ .",
"A vertex $u$ (or an edge $e$ , respectively) is said to be a cut vetex (or cut edge, respectively) of a connected graph $G$ , if $G- v$ (or $G -e$ ) has at least two components.",
"A graph G is called 2-connected if there does not exist a vertex whose removal disconnects the graph.",
"A block is a connected graph which does not have any cut vertex.",
"In particular, $K_2$ is a trivial block, and the endblock contains at most one cut vertex.",
"Let $P_n$ , $S_n$ and $C_n$ be a path, a star and a cycle on $n$ vertices, respectively.",
"Let $T$ be a tree, and $C_m$ be a cycle of $G$ .",
"If $G$ contains $T$ as its subgraph via attaching some vertex of $T$ to some vertex of $C_m$ , then we say tree $T$ is a pendent tree of $G$ .",
"Especially, replacing $T$ by $P_{|T|}$ , and choosing its pendent vertex to attach some vertex of $C_m$ , we call path $P_{|T|}$ is a pendent path of $G$ .",
"In this exposition we may use the notations and terminology of (chemical) graph theory (see [30], [11]).",
"We start with an elementary lemma below.",
"Lemma 2.1 Let $G$ be a graph.",
"If $uv \\in E(G)$ , then $M_i(G-uv) < M_i(G)$ with $i = 1, 2$ .",
"Besides the above lemma, we provide an useful tool about maximal 2-connected block on Zagreb indices.",
"Lemma 2.2 Let $G\\in \\mathbb {V}_n^k$ be a graph with the smallest Zagreb indices and $D$ a maximal 2-connected block of $G$ with $i=1, 2$ .",
"If $|D| \\ge 3$ , then $D$ is a cycle.",
"Proof.",
"If $|D| = 3$ , then $D$ is a cycle.",
"Otherwise, we prove the case of $|D| \\ge 4$ by a contradiction.",
"Assume that $D$ is a connected graph without cut vertices and $D$ is not a cycle.",
"Then there exists an edge $uv$ in $D$ such that $D- uv$ has no cut vertices.",
"Obviously, $G- uv \\in \\mathbb {V}_n^k$ .",
"By Lemma REF , $M_i(G-uv) < M_i(G)$ , which is contradicted to the choice of $G$ .",
"The four crucial operations on graphs are given as follows.",
"Operation I.",
"As shown in Fig.1, let $H_1$ be a connected graph with $d_{H_1}(v)\\ge 3$ and $d_{H_1}(v_1)=1$ , and $u_1u_2$ belong to a cycle of $H_1.$ If $H_2=H_1-\\lbrace u_1u_2,v_1v\\rbrace +\\lbrace u_1v_1, u_2v_1\\rbrace ,$ we say that $H_2$ is obtained from $H_1$ by Operation I.",
"(263.7,36.4)(287.7,40.4) (263.7,36.4)(286.2,29.4) $u_2$$u_1$$v_1$$v$$v_2$$v_{\\ell }$$u_1$$u_2$$v$(76.2,34.9)(98.7,27.9) (76.2,34.9)(83.2,49.4) (76.2,34.9)(100.2,38.9) $v_1$$v_{\\ell }$$v_2$*(239.7,36.9)(264.2,36.9)(264.2,54.9)(239.7,54.9)(239.7,36.9)(264.2,36.9)(264.2,54.9)(239.7,54.9) *(52.2,34.9)(76.7,34.9)(76.7,52.9)(52.2,52.9)(52.2,34.9)(76.7,34.9)(76.7,52.9)(52.2,52.9) $H_1$$H_2$Fig.1 The graphs using in Operation I and Lemma REF .Based on the above operation, we obtain a lemma below.",
"Lemma 2.3 If $H_2$ is obtained from $H_1$ by Operation I as shown in Fig.1.",
"Then $M_{i}(H_2)<M_{i}(H_1)$ for $i=1,2.$ Proof.",
"Let $v$ be a vertex of $H_1$ with $d_{H_1}(v)\\ge 3$ and containing at least one pendent vertex $v_1$ , and $u_1u_2$ be an edge of some cycle in $H_1$ with $d_{H_1}(u_1),d_{H_1}(u_2)\\ge 2.$ The neighbors of $v$ are marked as $v_1,v_2,\\ldots ,v_{\\ell }$ for $\\ell \\ge 3$ (see Fig.1).",
"If $v$ doesn't belong to any cycle of $H_1$ .",
"Then $H_2$ denotes the graph obtained from $H_1$ by deleting two edges $vv_1,u_1u_2$ and adding edges $u_1v_1,u_2v_1.$ Note that the function $f(x,y)\\triangleq xy-x-y+3,$ for $(x,y)\\in [2,+\\infty )\\times [2,+\\infty ),$ is more than zero.",
"We now deduce that $\\begin{split}M_1(H_1)-M_1(H_2)&=(d_{H_1}(v))^2+(d_{H_1}(v_1))^2-(d_{H_2}(v))^2-(d_{H_2}(v_1))^2\\\\&=(d_{H_1}(v)+d_{H_2}(v))-(d_{H_1}(v_1)+d_{H_2}(v_1))\\\\&\\ge 5-3=2>0.\\end{split}$ In terms of the property of $f(x,y),$ for $M_2$ , we arrive at $\\begin{split}&M_2(H_1)-M_2(H_2)\\\\&=\\sum \\limits _{j=1}^{\\ell }d_{H_1}(v)d_{H_1}(v_j)+d_{H_3}(u_)d_{H_1}(u_2)\\\\&\\ -\\sum \\limits _{j=2}^{\\ell }d_{H_2}(v)d_{H_2}(v_j)-d_{H_2}(u_1)d_{H_2}(v_1)-d_{H_2}(u_2)d_{H_2}(v_1)\\\\&=\\sum \\limits _{j=2}^{\\ell }d_{H_1}(v_j)+d_{H_1}(u_1)d_{H_1}(u_2)+d_{H_1}(v)-d_{H_2}(u_1)-d_{H_2}(u_2)\\\\&>d_{H_1}(u_1)d_{H_1}(u_2)-d_{H_1}(u_1)-d_{H_1}(u_2)+3\\\\&=f(d_{H_1}(u_1),d_{H_1}(u_2))>0.\\end{split}$ The special case $v$ belongs to some cycles of $H_1$ should be discussed.",
"If $v_1$ is the unique pendent vertex of $H_1$ .",
"Then there are nothing to do.",
"If $H_1$ has another pendent vertex, marked as $w_1$ , and $H_2=H_1-vv_1+v_1w_1.$ Then the conclusion is also verified.",
"The proof precess of the case is similar with the above argument, so it is omitted.",
"Hence, the proof is finished.",
"Operation II.",
"As shown in Fig.",
"2, let $H_3$ be a graph with $d_{H_3}(v)\\ge 3,$ and $w_1w_2$ be an edge included in some cycle of $H_3.$ If $H_4=H_3-\\lbrace vv_2,u_{21}v_2,w_1w_2\\rbrace +\\lbrace v_2w_1,v_2w_2,u_{21}u_{\\ell t_{\\ell }}\\rbrace $ for some $\\ell $ , we say that $H_4$ is obtained from $H_3$ by Operation II.",
"(244.9,33.9)(259.9,33.9) (259.9,33.9)(278.4,33.9) (278.4,33.9)(278.4,33.9) (278.4,33.9)(312.9,55.9) (278.4,33.9)(313.9,34.4) $w_2$$w_1$$v_1$$v_2$$v_{\\ell }$$u_{\\ell t_{\\ell }}$$u_{21}$$u_{2t_2}$$v$$w_1$$w_2$(58.4,31.9)(73.4,31.9) $v_1$(73.4,31.9)(91.9,31.9) $u_{21}$$v_2$$v$$u_{2t_2}$(91.9,31.9)(142.4,32.4) (91.9,31.9)(129.4,48.4) (91.9,31.9)(122.9,60.4) $v_{\\ell }$$u_{\\ell 1}$$u_{\\ell t_{\\ell }}$(313.9,34.4)(330.9,34.9) (330.9,34.9)(352.9,34.9) $H_3$$H_4$(200.9,9.4)(248.9,9.4)(248.9,58.4)(200.9,58.4) (11.9,9.9)(62.9,9.9)(62.9,56.9)(11.9,56.9) Fig.2 The graphs using in Operation II and Lemma REF .Lemma 2.4 If $H_4$ is obtained from $H_3$ by Operation II as shown in Fig.2.",
"Then $M_{i}(H_4)<M_{i}(H_3)$ for $i=1,2.$ Proof.",
"Let $v\\in V(H_3)$ with $d_{H_3}(v)\\ge 3$ and $w_1w_2$ be an edge of some cycle in $H_3$ .",
"Its neighbors are labeled as $v_1,v_2,\\ldots ,v_{\\ell }(\\ell \\ge 3).$ If there is at least one pendent vertex of $v$ , then this case can be reduced to Operation I.",
"So we may assume that $v$ only possesses pendent paths whose length are no less than 2.",
"If $v$ has an unique pendent path, there are nothing to do.",
"If $v$ contains at least two pendent paths, e.g., $P_2(v_2 u_{21} \\ldots u_{2t_2})$ and $P_\\ell (v_\\ell \\ldots u_{\\ell 1}u_{\\ell t_{\\ell }})$ with $t_2,t_{\\ell }\\ge 1$ .",
"Let $H_4=H_3-\\lbrace vv_2,u_{21}v_2,w_1w_2\\rbrace +\\lbrace v_2w_1,v_2w_2,u_{21}u_{\\ell t_{\\ell }}\\rbrace .$ Observe that the function $g(x,y)\\triangleq xy-2x-2y+5,$ for $(x,y)\\in [2,+\\infty )\\times [2,+\\infty ),$ is more than zero.",
"We now deduce that $\\begin{split}M_1(H_3)-M_1(H_4)&=(d_{H_3}(v))^2+(d_{H_3}(u_{\\ell }t_{\\ell }))^2-(d_{H_4}(v))^2-(d_{H_4}(u_{\\ell }t_{\\ell }))^2\\\\&=(d_{H_3}(v)+d_{H_4}(v))-(d_{H_3}(u_{\\ell }t_{\\ell })+d_{H_4}(u_{\\ell }t_{\\ell }))\\\\&\\ge 5-3=2>0.\\end{split}$ Since $d_{H_3}(w_1),d_{H_3}(w_2)\\ge 2.$ For $M_2$ , in terms of the property of $g(x,y),$ we get $\\begin{split}&M_2(H_3)-M_2(H_4)\\\\&=\\sum \\limits _{j=1}^{\\ell }d_{H_3}(v)d_{H_3}(v_j)+d_{H_3}(w_1)d_{H_3}(w_2)+d_{H_3}(u_{21})d_{H_3}(v_2)+d_{H_3}(u_{\\ell (t_{\\ell }-1)})d_{H_3}(u_{\\ell t_{\\ell }})\\\\&\\quad -\\sum \\limits _{j=1,j\\ne 2}^{\\ell }d_{H_4}(v)d_{H_4}(v_j)-d_{H_4}(v_2)d_{H_4}(w_1)-d_{H_4}(v_2)d_{H_4}(w_2)\\\\&\\quad -d_{H_4}(u_{\\ell (t_{\\ell }-1)})d_{H_4}(u_{\\ell t_{\\ell }})-d_{H_4}(u_{21 })d_{H_4}(u_{\\ell t_{\\ell }})\\\\&=\\sum \\limits _{j=1,j\\ne 2}^{\\ell }d_{H_3}(v_j)+d_{H_3}(w_1)d_{H_3}(w_2)+2d_{H_3}(v)-2d_{H_3}(w_1)-2d_{H_3}(w_2)-2\\\\&\\ge d_{H_3}(w_1)d_{H_3}(w_2)-2d_{H_3}(w_1)-2d_{H_3}(w_2)+6\\\\&=g(d_{H_3}(w_1),d_{H_3}(w_2))+1>0.\\end{split}$ Hence, the conclusion is verified.",
"Operation III.",
"As shown in Fig.",
"3, let $G_0$ be a connected graph with $|G_0|\\ge 2$ and having two vertices $u$ and $w$ , and $G_1$ be the graph which contains a cycle $C_1.$ Let $H_5$ be a graph on order $n(\\ge 6)$ obtained from $G_0$ by identifying some vertex of $C_1$ with vertex $u$ and some vertex of $C_2$ with vertex $w$ , respectively.",
"If $H_6$ denote the new graph from $H_5-\\lbrace v_1w,v_0v_2,u_1u_2\\rbrace +\\lbrace u_1v_0,u_2v_1\\rbrace ,$ we say that $H_6$ is obtained from $H_5$ by Operation III.",
"(241.2,40.4)(242,19.9)(264.9,19.9)(265.7,40.4)(241.2,40.4)(242,61)(264.9,61)(265.7,40.4) $u$$w$$v_1$$v_3$$v_0$$u_1$$u_2$$v_{\\ell }$$v_2$$u_1$$u_2$$u$$v_{\\ell }$$v_3$(64.2,39.4)(65,17.5)(88.9,17.5)(89.7,39.4)(64.2,39.4)(65,61.3)(88.9,61.3)(89.7,39.4) (75.2,33.4)(89.7,38.3) (75.2,46.4)(89.7,38.3) $v_2$$v_1$$w$$v_0$(265.7,39.9)(290.2,39.9) (252.7,47.4)(265.7,39.9) (252.7,34.4)(265.7,39.9) $H_5$$H_6$$C_1$$C_2$$C_1$Fig.3 The graphs using in Operation III and Lemma REF .Lemma 2.5 If $H_6$ is obtained from $H_5$ by Operation III as shown in Fig.3.",
"Then $M_{i}(H_6)<M_{i}(H_5)$ for $i=1,2.$ Proof.",
"Let $H_5$ be the graph shown in Fig.3, $u$ and $w$ be two cut-vertex of $H_5.$ $C_1$ and $C_2$ are its two cycles, where $C_2$ is an endblock.",
"$v_1w,v_0v_2,v_2w\\in E(C_2)$ and $u_1u_2\\in E(C_1)$ with with $d_{H_5}(u_1),d_{H_5}(u_2)\\ge 2.$ Let $H_6=H_5-\\lbrace v_2v_0,u_1u_2,v_1w\\rbrace +\\lbrace u_1v_0,u_2v_1\\rbrace .$ We can obtain that $\\begin{split}M_2(H_5)-M_2(H_6)&=(d_{H_5}(w))^2+(d_{H_5}(v_2))^2-(d_{H_6}(w))^2-(d_{H_6}(v_2))^2\\\\&=(d_{H_5}(w)+d_{H_6}(w))-(d_{H_5}(v_2)+d_{H_6}(v_2))\\\\&\\ge 5-3=2>0.\\end{split}$ Similarly, For $M_2$ , we can deduce that $\\begin{split}&M_2(H_5)-M_2(H_6)\\\\&=\\sum \\limits _{j=1}^{t}d_{H_5}(w)d_{H_5}(v_j)+d_{H_5}(v_2)d_{H_5}(v_0)+d_{H_5}(u_1)d_{H_5}(u_2)\\\\&\\quad -\\sum \\limits _{j=2}^{t}d_{H_6}(w)d_{H_6}(v_j)-d_{H_6}(u_1)d_{H_6}(v_0)-d_{H_6}(u_2)d_{H_6}(v_1)\\\\&=\\sum \\limits _{j=3}^{t}d_{H_5}(v_j)+3d_{H_5}(w)+d_{H_5}(u_1)d_{H_5}(u_2)-d_{H_5}(u_1)-d_{H_5}(u_2)\\\\&\\ge d_{H_5}(u_1)d_{H_5}(u_2)-d_{H_5}(u_1)-d_{H_5}(u_2)+11\\\\&=f(d_{H_5}(u_1),d_{H_5}(u_2))+8>0.\\end{split}$ Therefore, the proof is finished.",
"Operation IV.",
"As shown in Fig.",
"4, let $G_0$ be a connected graph having a vertex $v$ , and $G_1$ be a graph which contains a cycle $C_1.$ $H_7$ denotes the graph by attaching some vertex of $C_1$ and $C_2$ to the vertex $v$ , respectively.",
"Clearly, $C_2$ is an endblock of $H_7.$ If $H_8=H_7-\\lbrace vv_2,v_0v_1\\rbrace +v_0v_2$ , we say that $H_8$ is obtained from $H_7$ by Operation IV.",
"(40.2,50.4)(3.2,12.4)(73.2,9.4)(40.2,50.9) (39.7,50.9)(91.7,107.4)(88.7,-4.6)(40.2,51.4) (39.2,50.9)(-21.3,6.9)(-2.8,112.4)(39.7,50.9) $G_0$$v_1$$v_0$$v_2$$v$(175.7,48.9)(124.2,6.9)(125.2,98.4)(165.7,67.4) $G_0$(175.7,48.4)(138.7,10.4)(208.7,7.4)(175.7,48.9) $v_2$(181.7,66.4)(227.2,104.9)(224.2,-7.1)(175.7,48.9) $v$$v_1$$v_0$(165.7,67.4)(181.7,66.4) (190.7,64.4)(175.7,48.9) $H_7$$H_8$$C_2$$C_1$(39.7,50.9)(36.7,39.4) (39.7,50.9)(43.2,39.4) (175.7,48.9)(172.2,37.9) (175.7,48.9)(178.2,36.9) Fig.4 The graphs using in Operation IV and Lemma REF .Lemma 2.6 If $H_8$ is obtained from $H_7$ by Operation IV as shown in Fig.4.",
"Then $M_{i}(H_8)<M_{i}(H_7)$ for $i=1,2.$ Proof.",
"As shown in Fig.4, two cycles $C_1$ and $C_2$ of $H_7$ share a common vertex $v$ with $d_{H_7}(v)\\ge 4$ whose neighbors are labeled as $v_1,v_2,\\ldots v_t.$ Obviously, $t\\ge 4.$ In addition, $C_2$ is an endblock of $H_7$ .",
"Let $H_8$ denote the new graph obtained from $H_7$ by deleting edges $vv_2,v_0v_1$ and linking $v_2$ to $v_0$ .",
"We will deduce the relations of the two graphs $H_7$ and $H_8$ in terms of $M_1$ and $M_2,$ respectively.",
"$\\begin{split}M_1(H_7)-M_1(H_8)&=(d_{H_7}(v))^2+(d_{H_7}(v_1))^2-(d_{H_8}(v))^2-(d_{H_8}(v_1))^2\\\\&=(d_{H_7}(v)+d_{H_8}(v))+(d_{H_7}(v_1)+d_{H_8}(v_1))\\\\&\\ge 7+3=10>0,\\end{split}$ $\\begin{split}M_2(H_7)-M_2(H_8)&=\\sum \\limits _{j=1}^{t}d_{H_7}(v)d_{H_7}(v_{j})+d_{H_7}(v_0)d_{H_7}(v_1)\\\\&\\quad -\\sum \\limits _{j=3}^{t}d_{H_8}(v)d_{H_8}(v_{j})-d_{H_8}(v)d_{H_8}(v_{1})-d_{H_8}(v_0)d_{H_8}(v_2)\\\\&=\\sum \\limits _{j=3}^{t}d_{H_7}(v_{j})+d_{H_7}(v)+(d_{H_7}(v)-2)d_{H_7}(v_2)+5>0.\\end{split}$ Together Eq.REF with Eq.REF , the conclusion is verified.",
"Let $H$ be a connected graph with $|E(H)|-|V(H)|\\ge 0$ and $u,v\\in V(H)$ contained in a cycle of $H$ .",
"Denote by $H(a,b)$ the graph formed from $H$ by attaching two paths $P_a$ and $P_b$ to $u$ and $v$ , respectively.",
"Lemma 2.7 For $d_{H(a,b)}(u),d_{H(a,b)}(v)\\ge 3$ , we have $M_i(H(a,b))\\ge M_i(H(1,a+b-1))$ for $i=1,2.$ Proof.",
"Since $u,v$ belong to some cycle of $H,$ we have $d_{H(a,b)}(u),d_{H(a,b)}(v)\\ge 3.$ Without loss of generality, assume that $d_{H(a,b)}(u)\\ge d_{H(a,b)}(v).$ We now label all vertices of the two paths $P_a$ and $P_b$ as $uu_1u_2\\ldots u_{a-1}$ and $vv_1v_2\\ldots v_{b-1},$ respectively.",
"Suppose that, besides $u_1$ , the other neighbors of $u$ are $w_1,w_2,\\ldots ,w_t$ with $t\\ge 2.$ $H(1,a+b-1)$ is the graph formed from $H(a,b)$ by deleting edge $uu_1$ and connecting $u_1$ with $v_{b-1}$ .",
"For short, we mark $H(a,b)$ and $H(1,a+b-1)$ as $H_0$ and $H^{\\prime }_0,$ respectively.",
"We first consider $M_1$ and deduce that $\\begin{split}M_1(H_0)-M_1(H^{\\prime }_0)&=(d_{H_0}(u))^2+(d_{H_0})(v_{b-1}))^2-(d_{H^{\\prime }_0}(u))^2-(d_{H^{\\prime }_0}(v_{b-1}))^2\\\\&=d_{H_0}(u)+d_{H^{\\prime }_0}(u)+3>0.\\end{split}$ Similarly, for $M_2$ , we get that $\\begin{split}&M_2(H_0)-M_2(H^{\\prime }_0)\\\\=&\\sum \\limits _{j=1}^{t}d_{H_0}(u)d_{H_0}(w_j)+d_{H_0}(u)d_{H_0}(u_1)+d_{H_0}(v_{b-2})d_{H_0}(v_{b-1})\\\\&-\\sum \\limits _{j=1}^{t}d_{H^{\\prime }_0}(u)d_{H^{\\prime }_0}(w_t)-d_{H^{\\prime }_0}(v_{b-2})d_{H^{\\prime }_0}(v_{b-1})-d_{H^{\\prime }_0}(v_{b-1})d_{H^{\\prime }_0}(u_1)\\\\=&\\sum \\limits _{j=1}^{t}d_{H_0}(w_j)+d_{H_0}(u_1)(d_{H_0}(u)-d_{H^{\\prime }_0}(v_{b-1}))-d_{H_0}(v_{b-2})\\\\\\ge & d_{H_0}(u)+d_{H_0}(u_1)-d_{H_0}(v)>0.\\end{split}$ Therefore, we complete the proof.",
"Especially, the two vertices $u$ and $v$ are identified in $H(a,b)$ .",
"Then, use the similar way of Lemma REF , we also got a new graph $H(2,a+b-2)$ such that $M_i(H(a,b))\\ge M_i(H(a^{\\prime },b^{\\prime }))$ with $a^{\\prime }=2,b^{\\prime }=a+b-2$ for $i=1,2.$ Obviously, $P_{a^{\\prime }}=uu_1$ and $u_1$ is a pendant.",
"Hence, from Lemma REF , we deduce that there exists $H^{\\prime }$ with $|H^{\\prime }|=|H|+1$ .",
"(It is obtained from $H$ by subdividing its one edge $w_1w_2$ included in some cycle and marking the vertex as $u_1$ .)",
"such that $M_i(H(a,b))\\ge M_i(H^{\\prime }(1,a+b-2))$ for $i=1,2.$ We list the result as follows.",
"Corollary 2.8 If two vertices $u$ and $v$ are identified in $H(a,b)$ .",
"Then there exists a graph $H^{\\prime }$ on order $|H|+1$ such that $M_i(H(a,b))\\ge M_i(H^{\\prime }(1,a+b-2))$ for $i=1,2.$"
],
[
"Main results",
"In this section, we provide the lowest bounds on Zagreb indices of graphs in $\\mathbb {V}_n^k$ .",
"The corresponding graphs are characterized as well.",
"Theorem 3.1 Let $G$ be a graph in $\\mathbb {V}_n^k$ .",
"Then (i) $M_1(G)\\ge 4n+2$ , the equality holds if and only if $G \\cong C_{n,k}$ , (ii) $M_2(G)\\ge 4n+4$ , the equality holds if and only if $G \\cong C_{n,k}$ .",
"Proof.",
"Choose a graph $G\\in \\mathbb {V}_n^k$ such that $G$ has the minimal value of $M_i$ with $i=1, 2$ .",
"Let $B$ be a cut vertex set of size $k$ in $G$ .",
"$G$ can be divided into $s$ blocks via the $k$ cut vertices, and they are denoted by $D_1, D_2, \\cdots , D_s$ .",
"Clearly, $|D_j|=2$ or $|D_j|\\ge 3$ for some $j$ .",
"We start with a claim.",
"Claim 1 $G$ has only one pendent tree.",
"In fact, the tree is a path.",
"Proof.",
"Since $G$ is the graph for which $M_i(G)$ has the minimum for $i=1,2$ in all connected graphs possessing $k$ cut vertices.",
"We claim that $G$ includes at least a pendent tree.",
"If not, we will get a new graph $G^{\\prime }$ from $G$ , and by Lemma REF Lemma REF and $M_i(G^{\\prime })$ is less than $M_i(G)$ .",
"We get a contradiction.",
"In addition, every pendent tree of $G$ must be a path.",
"If not, from Lemma REF , there exists a new graph $G^{\\prime \\prime }$ such that $M_i(G^{\\prime \\prime })<M_i(G)$ , which contradicts with the choice of $G.$ If $G$ includes at least two pendent paths.",
"By means of Lemma REF and Corollary REF , there is a graph $G_1$ for which $M_i(G_1)<M_i(G)$ .",
"This is a contradiction.",
"Note that the number $|B|$ is not changed during these operations.",
"Thus, we complete the proof of this claim.",
"According to Lemma REF and Claim 1, we know that $G$ is a block graph and its blocks consists of cycle and $K_2$ , and $G$ has a unique pendent path, marked as $X(P)$ .",
"If $G$ just contains one cycle, then there is nothing to do.",
"We now suppose that $G$ possesses at least two cycles.",
"We now claim that all endblocks of $G$ are cycles except for $K_2$ of $X(P)$ .",
"Otherwise, $G$ has no less than two pendent paths which contradicts with Claim 1.",
"Case 1.",
"$G$ just includes two endblocks.",
"According the above argument, we can deduce that the two endblocks of $G$ are one cycle $C_1$ and $K_2.$ From the assumption, $G$ contains another cycle $C_2$ .",
"In terms of Lemma REF and Lemma REF , there is a graph $G^{\\prime }$ for which $M_i(G^{\\prime })<M_i(G)$ for $i=1,2.$ Case 2.",
"The number of endblocks in $G$ is more than two.",
"By means of the assumption, $G$ includes at least two cycles endblocks, e.g., $C_3$ and $C_4$ .",
"We will get a new graph $G\"$ obtained from $G$ such that $M_i(G^{\\prime })<M_i(G)$ for $i=1,2$ through Lemma REF and Lemma REF .",
"By combining Case 1 and Case 2, we deduce a contradiction with the choice of $G$ .",
"Hence, $G$ just possesses unique cycle $C_5$ .",
"Since $G$ belongs to $\\mathbb {V}_n^k$ , we can deduce that $C_5\\cong C_{n-k}$ and $X(P)\\cong P_{k+2}.$ Therefore, $G\\cong C_{n,k}.$ By direct calculation, We arrive at $M_1(C_{n,k})=4n+2$ , $M_2(C_{n,k})=4n+4.$ We hence complete the proof.",
"Acknowledgments This work was partial supported by National Natural Science Foundation of China (Grant Nos.11401348 and 11561032), and supported by Postdoctoral Science Foundation of China and the China Scholarship Council."
]
] | 1709.01542 |
[
[
"Monodromic Dark Energy"
],
[
"Abstract Since the discovery of the accelerated expansion of the Universe, the constraints on the equation of state $w_\\text{DE}$ of dark energy, the stress-energy component responsible for the acceleration, have tightened significantly.",
"These constraints generally assume an equation of state that is slowly varying in time.",
"We argue that there is good theoretical motivation to consider \"monodromic\" scenarios with periodic modulations of the dark energy potential.",
"We provide a simple parametrization of such models, and show that these leave room for significant, periodic departures of $w_\\text{DE}$ from -1.",
"Moreover, simple models with non-standard kinetic term result in interesting large-scale structure phenomenology beyond that of standard slow-roll dark energy.",
"All these scenarios are best constrained in a dedicated search, as current analyses average over relatively wide redshift ranges."
],
[
"Introduction",
"Since the conclusive detection in 1998 [1], [2], overwhelming evidence has accumulated that points towards an accelerated expansion of the Universe which, in the context of General Relativity (GR) implies the existence of a form of stress energy with negative pressure, dark energy [3].",
"In particular, the dark energy equation of state $w_\\text{DE} = p/\\rho $ is now constrained to be close to $-1$ [4], [5], [6], [7] (see [8] for a recent review).",
"A cosmological constant is the simplest possibility, which is so far broadly consistent with all observational constraints coming from geometric and large-scale growth probes.",
"However, the value of the cosmological constant is, from a high-energy physics perspective, highly fine-tuned (see [9] for a review).",
"An alternative is to make the cosmological constant dynamical, by promoting it to the potential energy $V(\\phi )$ of a scalar field $\\phi $ , often dubbed quintessence (see [10] for a comprehensive, and [11] for a brief overview).",
"If the potential is sufficiently flat, for example $V(\\phi ) \\propto (\\phi /\\phi _0)^{-\\alpha }$ with $\\alpha \\ll 1$ , the field rolls slowly and is thus potential energy dominated, yielding a stress-energy contribution with equation of state close to $-1$ [12], [13], [14].",
"Moreover, during the matter-dominated epoch of the Universe, the field follows a tracking solution and thus reduces the need for fine-tuning of the model parameters.",
"An approximately flat potential can be realized in a technically natural way by introducing a shift symmetry $\\phi \\rightarrow \\phi +c$ , which is then weakly broken.",
"In the well-known explicit constructions of inflation in the context of string theory [15], [16], known as axion monodromy, nonperturbative effects lead to periodic modulations of the potential.",
"These are responsible for an array of interesting signatures [17], [18], [19].",
"Motivated by this fact, we study in this paper the phenomenological consequences of a dark energy potential of the form $V(\\phi ) = C \\left(\\frac{\\phi }{\\phi _0}\\right)^{-\\alpha } \\left[1 - A \\sin (\\nu \\phi )\\right]\\,,$ where $A$ is the amplitude of the periodic modulation (with $|A|\\ll 1$ ), while $\\nu $ is the frequency in field space.In axion monodromy, the smooth component of the potential is linear.",
"We have generalized this to a power-law here, though the phenomonological consequences are not sensitive to the shape of the smooth potential.",
"Apart from the shape of the smooth part of the potential (power-law vs. exponential), this is precisely the potential studied in [20].",
"It is different from the Pseudo-Nambu-Goldstone scenario of [21] (see also [22], [23]), where $V(\\phi ) = V_\\star [1 + \\cos (\\phi /f)]$ is a non-monotonic potential, as Eq.",
"(REF ) allows for the field to pass through multiple oscillations driven by the potential (several oscillations can however happen in the latter scenario if multiple decay constants $f_i$ are involved [23]).",
"Further, this is a different physical setup than oscillations of a $U(1)$ quintessence field in a power-law potential [24], [25], [26], [27], which have been shown to be unstable and hence not suitable for explaining a sustained period of acceleration [28].",
"On the other hand, the scenario considered here has a wide range of parameter space which is stable (we discuss theoretical constraints in Sec. ).",
"Moreover, the observable oscillatory features can have periods that are naturally much smaller than a Hubble time.",
"Previous studies of oscillatory potentials [29], [30], [31], [32], and constraints using observational data [33], [34], [32], [35], typically focused on periods of order a Hubble time.",
"The window of rapidly oscillating dark energy model space thus remains largely unexplored.",
"Since the assumption of a smooth evolution of the dark energy density is built into almost all observational constraints published so far, this leaves open the possibility of significant periodic deviations from an equation of state $w_\\text{DE} \\sim -1$ .",
"Moreover, a canonical scalar field is not the only possible option.",
"Models with non-standard kinetic term, referred to as k-essence, can exhibit similar tracking behavior [36], [37].",
"Indeed, non-standard kinetic terms also appear in the context of string theory in form of the Dirac-Born-Infeld action [38], [39].",
"In these models, the sound speed is naturally small, so that perturbations in the dark energy density are not negligible.",
"We will see that this leads to even more interesting signatures in large-scale structure (LSS): unlike the case of k-essence with a smoothly varying equation of state, which is observationally difficult to distinguish from quintessence, a periodic modulation of the form Eq.",
"(REF ) produces signatures in the growth of structure which are accessible to galaxy redshift as well as weak lensing surveys.",
"Again, this is a phenomenological window which currently is almost entirely unexplored.",
"Throughout, we will assume a spatially flat background, and assume $\\Omega _{m0} = 0.27 = 1 - \\Omega _{{\\rm DE},0}$ as fiducial values.",
"We will also choose $\\alpha =0.2$ as default, leading to a time-averaged equation of state of $\\bar{w}_\\text{DE} \\approx -0.9$ .",
"The outline is as follows: Sec.",
"– present the dark energy models.",
"Sec.",
"discusses theoretical constraints on the viable model space.",
"We then derive the observable signatures of these models in Sec.",
", and conclude in Sec.",
"."
],
[
"Monodromic quintessence",
"Consider the following action for a canonical scalar field $\\phi $ minimally coupled to gravity: $S = \\int d^4x \\sqrt{-g} \\left[ \\frac{1}{2} M_\\text{Pl}^2 R + \\frac{1}{2} \\nabla _\\mu \\phi \\nabla ^\\mu \\phi + V(\\phi ) + \\mathcal {L}_m \\right]\\,,$ where $V(\\phi )$ is given by Eq.",
"(REF ), $\\mathcal {L}_m$ is the matter action, and we assume no coupling between $\\phi $ and other species apart from gravity.",
"Again, this is essentially identical to the model proposed in [20].",
"The quintessence contribution to the stress-energy tensor is of the perfect-fluid form, $(T^Q)^\\mu _{\\ \\nu } = \\text{diag}(-\\rho _Q, p_Q,p_Q,p_Q)$ .",
"We consider a matter sector that consists of cold pressureless matter with equation of state $w_m=0$ .",
"Restricting to a spatially homogeneous setting, in which the metric becomes of the Friedmann-Robertson-Walker (FRW) form, Eq.",
"(REF ) leads to the equation of motion $\\ddot{\\phi }+ 3 H \\dot{\\phi }+ V_{,\\phi } =\\:& 0\\,,$ where here and throughout, dots indicate derivatives with respect to time $t$ .",
"We will present numerical integration results of Eq.",
"(REF ) below.",
"First, we begin with the case without oscillations.",
"Upon setting $A=0$ in Eq.",
"(REF ), we recover the well-known power-law potential first considered by [14].",
"A power-law ansatz $\\bar{\\phi }(t) = \\tilde{\\phi }_0 t^p$ , with $\\tilde{\\phi }_0 = \\phi _0/t_0^p$ , in matter domination where $H(t) = 2/(3t)$ yields $p = \\frac{2}{2+\\alpha } \\quad \\mbox{and}\\quad p^2 + p = \\alpha C \\tilde{\\phi }_0^{-\\alpha -2}\\,.$ The equation of state and energy density of this tracking solution are $w_Q =\\:& \\frac{p_Q}{\\rho _Q} = \\frac{\\dot{\\phi }^2/2 - V(\\phi )}{\\dot{\\phi }^2/2 + V(\\phi )} \\nonumber \\\\\\stackrel{\\text{tracking}}{=}\\:& \\frac{-1 + \\alpha ^4/4 + \\alpha ^3/16}{1 + \\alpha /2+\\alpha ^2/4 + \\alpha ^3/16}\\,.$ Figure: Evolution of the energy density (ρ Q \\rho _Q, top) and equation of state (w Q w_Q, bottom) of the monodromic quintessence model.",
"The frequency ν\\nu is given in units of M Pl -1 M_\\text{Pl}^{-1}.We see that $w_Q$ is constant during matter domination, and that $w_Q$ approaches $-1$ as $\\alpha \\rightarrow 0$ .",
"In the following, we will choose $\\alpha =0.2$ as fiducial value, leading to $w_Q=-0.89$ on the tracking solution.",
"In the presence of oscillations, Eq.",
"(REF ) is no longer solvable in closed form.",
"However, if we perturb around the tracking solution $\\bar{\\phi }$ by writing $\\phi (t) = \\bar{\\phi }(t) + \\varphi (t)\\,,$ and work to linear order in the amplitude of oscillations $A$ , we obtain in the limit $\\alpha \\rightarrow 0$ during matter domination: $\\varphi (t) =\\:& \\frac{\\text{const}}{t} \\exp \\left[\\pm i \\omega t \\right]\\,,\\quad \\mbox{with} \\nonumber \\\\\\omega =\\:& \\sqrt{C A \\nu ^2 \\sin [\\nu \\bar{\\phi }]}\\,.$ Thus, if the tracking solution is initialized with a small positive $\\bar{\\phi }$ , $\\varphi $ oscillates around $\\bar{\\phi }$ with a frequency $\\omega $ .",
"If $\\nu \\gg 1/M_\\text{Pl}$ , the field can oscillate many times within a Hubble time.",
"We next turn to the full numerical solution of the system.",
"First, the evolution of the quintessence energy density can be obtained from the continuity equation, $\\dot{\\rho }_Q + 3H (1+w_Q) \\rho _Q = 0\\,,$ while the Hubble rate is determined from the Friedmann equation: $H^2 = \\frac{1}{3M_\\text{Pl}^2} \\left[\\rho _m + \\rho _Q \\right]\\,,$ where $\\rho _m(t) \\propto a^{-3}(t)$ and we have assumed a flat background.",
"Fig.",
"REF shows the evolution of the quintessence equation of state and energy density both with and without oscillations.",
"For this, we have integrated Eq.",
"(REF ) together with Eq.",
"(REF ), using initial conditions at a scale factor $a \\approx 10^{-4}$ given by the tracking solution Eq.",
"(REF ).",
"We adjust the initial value of $(\\phi /\\phi _0)$ [where $C$ is determined from Eq.",
"(REF )] in order to obtain the desired value of $\\Omega _{{\\rm DE,}0} = \\rho _Q(t_0)/(3M_\\text{Pl}^2 H_0^2) = 0.73$ .",
"We show results for two different choices of the oscillation frequency $\\nu $ in field space, both with amplitude $A=0.05$ .",
"Note that the oscillations in the energy density are substantially smaller than those in $w_Q$ , as $\\rho _Q$ is given by a time integral over $w_Q$ [Eq.",
"(REF )].",
"We also show an example with a slightly larger amplitude $A=0.1$ .",
"In this case, the quintessence field becomes trapped in a local minimum of the potential, leading to $w_Q=-1$ (see, e.g.",
"[40]).",
"Indeed, we see that if $| A \\phi _0 \\nu | \\gtrsim 1\\,,$ the potential Eq.",
"(REF ) is no longer monotonic.",
"Since quintessence cannot cross the phantom divide to $w_Q < -1$ , the evolution of the field stops once it has reached this value.",
"Since, on the other hand, the average dark energy equation of state is observationally constrained to $\\bar{w}_\\text{DE} \\lesssim -0.9$ , this leaves a limited parameter space of viable monodromic quintessence models, with $|A| \\lesssim 0.05$ .",
"Note that, while the mass of the field $\\phi $ is in general larger than $H$ in the monodromic quintessence scenario, the perturbations to the energy density of quintessence remain negligible.",
"This is because the sound speed, defined in Eq.",
"(REF ) below, is identical to unity in this model, so that pressure perturbations prevent subhorizon quintessence perturbations from growing.",
"This is qualitatively different in the alternative scenario which we turn to next, which also has a significantly enlarged observationally allowed parameter space."
],
[
"Monodromic k-essence",
"We now generalize the action Eq.",
"(REF ) to a non-standard kinetic term, commonly referred to as k-essence, and consider the following action: $S = \\int d^4x \\sqrt{-g} \\left[ \\frac{1}{2} M_\\text{Pl}^2 R + p(\\phi ,X) + \\mathcal {L}_m \\right]\\,,$ where the kinetic term is defined through $X \\equiv - \\frac{1}{2} \\Lambda ^{-4}\\nabla _\\mu \\phi \\nabla ^\\mu \\phi \\,,$ such that $X>0$ .",
"We have introduced the scale $\\Lambda ^{-4}$ to make $X$ dimensionless.",
"At the background level, $X = \\Lambda ^{-4}\\dot{\\phi }^2/2$ .",
"While this scale is arbitrary, since it can be absorbed by a field redefinition, we choose the most natural value of $\\Lambda = \\sqrt{H_0 M_\\text{Pl}}$ .",
"The stress-energy tensor for this field is still of the perfect-fluid form, with pressure and energy density given by $p_K =\\:& p(\\phi ,X) \\nonumber \\\\\\rho _K =\\:& 2 X p_{,X}(\\phi ,X) - p(\\phi ,X)\\,.$ This immediately yields the equation of state $w_K$ .",
"Interestingly, the non-standard kinetic term yields a sound speed of the field, defined as the derivative of $p_K$ with respect to $\\rho _K$ at fixed field value, which in general is different from unity: $c_s^2 \\equiv \\frac{\\partial p_K}{\\partial \\rho _K}\\Big |_{\\phi }= \\frac{p_{K,X}}{\\rho _{K,X}} = \\frac{p_{,X}}{p_{,X} + 2 X p_{,XX}}\\,.$ To avoid tachyonic behavior, we require $p_{,XX}>0$ .",
"The equation of motion of $\\phi $ at the background level can be compactly phrased in terms of $X = \\Lambda ^{-4}\\dot{\\phi }^2/2$ , yielding $(p_{,X} + 2 X p_{,XX}) \\dot{X} + \\sqrt{2X} (2 X p_{,X\\phi } - p_{,\\phi }) + 6 H X p_{,X} = 0\\,.$ Following [36], we choose $p(\\phi ,X) = V(\\phi ) \\left[- X + X^2 \\right]\\,,$ where $V(\\phi )$ is given by Eq.",
"(REF ).",
"Note that $V(\\phi )$ is not a potential here, but determines the amplitude of the kinetic term.",
"The negative pressure is obtained through the non-canonical kinetic term.",
"Any function of the form $K(\\phi ) X + L(\\phi ) X^2$ (with $L \\ne 0$ ) can be brought into this form through a field redefinition [36].",
"Inserting Eq.",
"(REF ) into Eqs.",
"(REF )–(REF ), we obtain $w(X) = \\frac{1-X}{1-3X}\\quad \\mbox{and}\\quad c_s^2(X) = \\frac{1-2X}{1-6X}\\,.$ A cosmological constant behavior is attained in the limit $X\\rightarrow 1/2$ .",
"In close analogy to the power-law quintessence case, the k-essence model in the absence of oscillations ($A=0$ ) admits a scaling solution in matter domination, where $X = \\bar{X}$ is constant and $\\phi = \\sqrt{2\\bar{X}}\\Lambda ^2 t + \\text{const}$ .",
"This scaling solution has $\\bar{X}(\\alpha ) = \\frac{4-\\alpha }{8-3\\alpha }\\,.$ As in the quintessence case, setting $\\alpha =0$ yields a cosmological constant with $\\bar{X}=1/2$ and $w_K=-1$ ; our fiducial choice in the following, $\\alpha =0.2$ , yields $w_K=-0.9$ .",
"Figure: Evolution of the energy density (ρ K \\rho _K, top), equation of state (w K w_K, middle), and sound speed squared (bottom) of the monodromic k-essence model.The frequency ν\\nu is given in units of M Pl -1 M_\\text{Pl}^{-1}.Before turning to the results from the numerical integration of the full equation of motion Eq.",
"(REF ), let us again derive a simpler equation at linear order in the amplitude $A$ , by expanding around the scaling background.",
"Writing $\\phi = \\sqrt{2\\bar{X}}\\Lambda ^2 t + \\varphi (t)\\,,$ and taking the limit $\\alpha \\rightarrow 0$ , we obtain $\\ddot{\\varphi }(t) + \\frac{2}{t} \\dot{\\varphi }(t) + \\frac{1}{8} A \\,\\Lambda ^2 \\nu \\cos \\left(\\Lambda ^2 \\nu t \\right) = 0\\,.$ This equation can be straightforwardly solved to yield, in the limit of large $\\nu t$ , $\\varphi (t) = - A \\frac{\\Lambda ^2}{8\\nu } \\cos \\left(\\Lambda ^2\\nu t\\right)\\,.$ Correspondingly, all of $\\rho _K$ , $X$ , $w_K$ and $c_s^2$ oscillate around the values given by the scaling solution, with approximately $\\frac{\\Lambda ^2 \\nu }{2\\pi H} \\approx \\frac{\\nu M_\\text{Pl}}{2\\pi }$ oscillations per Hubble time.",
"While the amplitude of oscillations in $\\rho _K$ , and hence the Hubble rate $H$ , are controlled by the field amplitude and are hence of order $A/\\nu $ , the oscillations in $w_K$ and $c_s^2$ are proportional to $\\dot{\\varphi }$ , which oscillates with amplitude $A$ .",
"This will become relevant in the observable effects considered in Sec. .",
"As shown in [41], no pathologies arise in k-essence when crossing the phantom divide $w_K=-1$ , as long as the sound speed vanishes at the crossing, and as long as gradient instabilities are countered by higher-derivative terms neglected in the action Eq.",
"(REF ).",
"The former is precisely what happens in monodromic k-essence, as is clear from Eq.",
"(REF ) (see also Fig.",
"REF ).",
"We will discuss in Sec.",
"whether brief episodes of tachyonic behavior can be allowed.",
"Note that, by appropriate choice of $\\alpha $ and $A$ , one can ensure that $c_s^2 > 0$ always.",
"This however restricts the allowed range of oscillation amplitudes significantly.",
"Fig.",
"REF shows the evolution of the k-essence energy density and equation of state as well as sound speed squared obtained from a numerical integration of Eq.",
"(REF ) [which is quite close to the result of the analytical approximation Eq.",
"(REF )].",
"The initial conditions are again taken from the tracking solution at $a \\approx 10^{-4}$ , and the constant $C$ in $V(\\phi )$ is adjusted to obtain the desired value of $\\Omega _{{\\rm DE,}0}=0.73$ .",
"Qualitatively, the behavior is similar to the monodromic quintessence case.",
"However, there are several important differences.",
"First, the oscillations are not damped [compare Eq.",
"(REF ) with Eq.",
"(REF )].",
"Second, the k-essence field can cross the phantom divide, allowing for significantly larger oscillations even for an average equation of state that is close to $-1$ .",
"Recall that $V(\\phi )$ is not a potential in this model, and so the field does not get stuck even if $V$ is non-monotonic.",
"Third, the k-essence model exhibits an oscillatory sound speed with $|c_s^2| \\ll 1$ , while $c_s^2=1$ always holds for quintessence."
],
[
"Theoretical constraints",
"We now briefly review theoretical constraints on the parameter space of the monodromic dark energy models considered here.",
"We focus on constraints which are independent of the microscopic physics that leads to the potential Eq.",
"(REF ), and discuss the latter at the end of this section.",
"In the quintessence case, we have already found the requirement $| A \\phi _0 \\nu | < 1$ in order to ensure a rolling field; otherwise, the field gets trapped in a local minimum, leading to an effective cosmological constant, which is uninteresting phenomenologically.",
"No such constraint exists for the k-essence case.",
"However, unlike the monodromic quintessence case where $c_s=1$ , in the k-essence case we have to confront the issue of gradient instabilities [42].",
"Within the comoving sound horizon of the k-essence field, defined as $R_s(a) \\equiv \\frac{|c_s(a)|}{a H} \\approx 150 \\,h^{-1}\\,{\\rm Mpc}\\left(\\frac{|c_s(a)|}{0.05} \\right)\\,,$ an additional effective pressure force becomes relevant in the dynamics of the k-essence fluid.",
"In particular, it sources a relative velocity divergence $\\theta _{mK} \\equiv \\partial _{x,i} (v_K^i - v_m^i)$ between k-essence and matter, where $\\partial _x$ denotes a derivative with respect to comoving coordinates, whose evolution equation is given by $\\dot{\\theta }_{mK} + H \\theta _{mK} + \\frac{c_s^2}{1+w_K} a^{-1} \\nabla ^2_x \\delta _K =\\:& 0\\,,$ where $\\delta _K$ is the fractional energy density perturbation in the k-essence component and we have assumed $|c_s^2| \\ll 1$ .",
"If $c_s^2 < 0$ , this leads to an exponential growth instability in the dark energy component, known as gradient instability, as $\\delta _K$ is itself sourced by $- \\theta _{mK}$ .",
"We can formally integrate Eq.",
"(REF ) to yield a stability constraint given by $\\int _0^{\\ln a} d\\ln a^{\\prime } H^{-1}(a^{\\prime }) D_K(a^{\\prime }) \\frac{c_s^2(a^{\\prime })}{1+w_K(a^{\\prime })} > 0\\,,$ where $D_K(a) = \\delta _K(k,a)/\\delta _K(k,1)$ is the k-essence growth factor which is in general scale dependent and a complicated function of time.",
"Let us first consider large-scale perturbations whose time scale $1/\\omega = 1/k$ is longer than one oscillation period of the field.",
"From Eq.",
"(REF ), this implies $\\frac{k}{a H} \\lesssim \\frac{\\nu M_\\text{Pl}}{2\\pi }\\,.$ For these perturbations, the episodes of tachyonic behavior are too short to allow instabilities to grow.",
"We have verified this by evaluating Eq.",
"(REF ) for the k-essence model introduced in the previous section, using the result for the k-essence perturbation obtained from Eq.",
"(REF ) below, and find that the stability constraint is satisfied for a wide range of parameter space, including the regime where $A$ is of order 1.",
"Fundamentally, this is because $c_s$ oscillates around a positive value in these models, and the combination $c_s^2/(1+w_K)$ is in fact always positive.",
"This argument no longer applies to small-scale perturbations with $k/aH \\gg \\nu M_\\text{Pl}/2\\pi $ .",
"These perturbations can in principle grow to become nonlinear within a single epoch of $c_s^2 < 0$ .",
"As argued in [41], this instability can be countered by adding higher-derivative terms to the action Eq.",
"(REF ), for example $\\bar{\\Lambda }^{-2} (\\Box \\phi )^2$ .",
"These higher-derivative terms also control the cutoff of the effective description; that is, the model no longer provides controlled predictions in the effective field theory sense on spatial scales that are smaller than $\\bar{\\Lambda }^{-1}$ .",
"Since the episodes of tachyonic behavior are only brief, of order $(\\nu M_\\text{Pl})^{-1}$ Hubble times, the requirement on the scale $\\bar{\\Lambda }$ associated with the higher-derivative term relaxes by a factor of $(\\nu M_\\text{Pl})^{-1}$ .",
"Nevertheless, the k-essence model considered here either requires a very low cutoff (compared to local measurements of gravity), or significant fine-tuning between the natural scale $\\Lambda = M_\\text{Pl}^2 H_0^2$ and the much higher scale $\\bar{\\Lambda }$ of the higher-derivative terms.",
"The nontrivial sound speed in the k-essence model is induced by the non-canonical kinetic term, $\\propto X + X^2$ .",
"This raises the question of whether higher-order terms in this expansion should be included, especially as $X\\approx 1/2$ is not small.",
"However, it turns out that any k-essence model which admits a tracking solution with $X = $ const can be approximated by the form Eq.",
"(REF ) as long as $X$ does not deviate strongly from its value on the tracking position.",
"To see this, one can simply insert a general Lagrangian of the form $p(\\phi ,X) = V(\\phi ) K(X)$ into Eq.",
"(REF ) and impose the tracking ansatz.",
"Expanding $K(X)$ around the tracking point $X_{\\rm tr}$ , one finds that higher-order corrections to Eq.",
"(REF ) scale as powers of $(X-X_{\\rm tr})$ .",
"Now, even in the model with the strongest oscillations considered here, $A=0.5$ , we find that $|X-X_{\\rm tr}| < 0.1$ always.",
"Thus, as long as the smooth component of $|1+w|$ is in the range allowed by observations, Eq.",
"(REF ) is expected to represent the entire class of tracking k-essence models with an oscillatory potential.",
"Of course, these statements are specific to the class of k-essence models which feature a tracking solution.",
"Finally, we turn to somewhat more model-dependent constraints.",
"First, we have neglected all higher-derivative operators in the dark energy Lagrangians.",
"As discussed above, these higher-derivative operators are enhanced by $\\omega /H$ in the monodromic case, compared to slowly-rolling scenarios, where $\\omega $ is the oscillation frequency of the field in the cosmological solution, given by Eq.",
"(REF ) in case of quintessence and $\\omega /H \\approx \\nu M_\\text{Pl}$ in the k-essence case.",
"For the models considered here, this enhancement is as large as an order of magnitude.",
"Whether such an enhancement makes higher-derivative operators relevant, depends on the microscopic physics responsible for the monodromic potential.",
"In the context of inflation from axion monodromy realized in string theory, which of course happens at an energy scale much higher than dark energy, Ref.",
"[18] found that higher-derivative terms remain suppressed as long as $\\nu M_\\text{Pl}$ is smaller than several thousand, and the oscillation amplitude is of order unity or less."
],
[
"Observables",
"We now consider the observable signatures of the monodromic quintessence and k-essence models.",
"The simplest observables are distances, which are probed by standard candles such as type Ia supernovae, and standard rulers such as the baryon acoustic oscillation (BAO) feature in galaxy clustering.",
"Given the assumed flat geometry, cosmological distances satify the simple relations $\\chi (z) =\\:& (1+z) d_A(z) = \\frac{d_L(z)}{1+z}= \\int _0^z \\frac{dz^{\\prime }}{H(z^{\\prime })}\\,,$ where $\\chi $ is the comoving distance, $d_A$ is the angular diameter distance while $d_L$ is the luminosity distance.",
"The distances correspond to one integral over $H \\propto \\rho _{\\rm DE}$ ; thus, the signature of oscillations is even weaker in the distances than in the density, as can be seen in Fig.",
"REF (see also [31]).",
"Here and throughout, we show the ratio of predicted observables in a monodromic model with $A>0$ to the corresponding model with $A=0$ .",
"This is because we are interested in the oscillatory features as a signal, while smoothly varying changes in the observables could be explained by any standard dark energy model, for which the standard parametrizations through, for example, $w_0,w_a$ [Eq.",
"(REF )], are sufficient.",
"Crucially, by sufficiently fine binning in redshift, standard candles can also probe the derivative ${\\rm d}\\chi /{\\rm d}z = 1/H(z)$ .",
"Similarly, the BAO feature along the line-of-sight direction, as well as Alcock-Paczyński (AP) distortions, probe $1/H(z)$ .",
"As shown in Fig.",
"REF , the oscillations in $H(z)$ are significantly stronger than those in the distance.",
"Moreover, the monodromic k-essence model shows much stronger oscillatory features than the quintessence case, a consequence of the limit on the amplitude of modulations in the quintessence case discussed at the end of Sec. .",
"The signatures of monodromic k-essence are already accessible to current probes, as we will discuss in Sec. .",
"Note however that when the radial BAO or AP scale are averaged over a wide redshift range, the oscillation signal is strongly diluted again.",
"Thus, a dedicated analysis is necessary to obtain optimal constraints on monodromic dark energy models.",
"Figure: Background observables in monodromic dark energy models: comoving distance χ(z)\\chi (z) and Hubble expansion rate H(z)H(z).",
"In all cases, we show the ratio of oscillating models (A>0A > 0) to the corresponding model without oscillations (A=0A=0).Figure: Linear growth-of-structure observables in monodromic dark energy models: linear matter growth factor D(z)D(z), and growth rate f(z)dlnD/dlnaf(z) d\\ln D/d\\ln a.",
"In all cases, we show the ratio of oscillating models (A>0A > 0) to the corresponding model without oscillations (A=0A=0).",
"The dotted lines show the predictions in the k-essence case when neglecting dark energy perturbations.The same holds for the monodromy signatures in the growth of structure which we consider now, beginning with the quintessence case.",
"Since the sound horizon of the field is equal to the comoving horizon, fractional perturbations in the dark energy density are less than $10^{-4}$ and can be neglected, as is usually done in studies of the growth of structure in dark energy cosmologies.",
"The growth of matter perturbations at linear order is governed by the linearized Euler-Poisson system: $\\dot{\\delta }_m + a^{-1} \\theta =\\:& 0 \\nonumber \\\\\\dot{\\theta }+ H \\theta - \\frac{3}{2} a H^2 \\Omega _m(t) \\delta _m =\\:& 0\\,.$ These can be combined into a single equation for the linear growth factor $\\ddot{D} + 2 H \\dot{D} - \\frac{3}{2} \\Omega _{m0} H_0^2 a^{-3} D = 0\\,.$ This relation holds in all models of dark energy where perturbations in the dark energy density can be neglected.",
"We see that the growth factor corresponds to two integrals over the energy density and Hubble rate, and thus expect small imprints of oscillations in the growth factor.",
"However, the large-scale clustering of galaxies also receives contributions from the velocity, induced by redshift-space distortions (RSD) [43].",
"From Eq.",
"(REF ) we have $\\theta = - a \\dot{\\delta }_m = -a H f \\delta _m$ , where $f=d\\ln D/d\\ln a$ is the linear growth rate.",
"As seen in Fig.",
"REF , the signatures in the growth rate are comparable to those in the Hubble rate.",
"The monodromic k-essence model has an even more interesting phenomenology for LSS.",
"Due to the nontrivial sound speed in this model, the comoving sound horizon $R_s$ [Eq.",
"(REF )] is much smaller than $(aH)^{-1}$ .",
"Within the sound horizon, perturbations in the dark energy are suppressed, although the oscillations in the sound speed might lead to interesting behavior even in this regime (see Sec. ).",
"In the following, we consider perturbations which are much larger than $R_s$ , where the field perturbations are unsuppressed.",
"Equivalently, our predictions assume the limit $c_s^2 \\rightarrow 0$ , which Ref.",
"[41] argue should be used for any healthy model which cross the phantom divide $w_K=-1$ .",
"Then, the dark energy comoves with matter, and the linearized Euler-Poisson system becomes [44], [45] $\\dot{\\delta }_m + a^{-1} \\theta =\\:& 0 \\nonumber \\\\\\dot{\\delta }_K - 3 H w_K \\delta _K + a^{-1} (1+w_K) \\theta =\\:& 0 \\nonumber \\\\\\dot{\\theta }+ H \\theta - \\frac{3}{2} H^2 a \\left[ \\Omega _m(t) \\delta _m + \\frac{\\rho _K(t)}{3M_\\text{Pl}H^2} \\delta _K\\right] =\\:& 0\\,,$ where $\\theta $ is the velocity divergence, while $\\delta _K$ is the fractional perturbation in the dark energy density, as in Sec. .",
"Figure: Growth factor of the total stress-energy perturbation δ tot \\delta _{\\rm tot} defined in Eq.",
"(), which determines gravitational lensing.",
"For quintessence, this is the same as the matter growth displayed in Fig. .",
"Shown is the ratio of oscillating models (A>0A > 0) to the corresponding model without oscillations (A=0A=0).",
"The dotted lines show the predictions in the k-essence case when neglecting dark energy perturbations.The resulting growth factor and growth rate are shown in Fig.",
"REF .We integrate Eq.",
"(REF ) assuming adiabatic initial conditions for matter and k-essence, which corresponds to the fastest growing mode.",
"As in the case of the background observables, the signal of oscillations is much stronger in the k-essence case due to the larger amplitudes of the potential modulation that is possible.",
"They can easily reach 5–10% and are thus already accessible to existing data sets, as we will discuss in Sec.",
"; however, again a dedicated analysis should be performed to search for these particular rapidly varying signatures.",
"Moreover, the dotted lines in Fig.",
"REF show the result obtained when neglecting the dark energy density perturbations in the growth factor and growth rate, equivalent to integrating Eq.",
"(REF ) instead of Eq.",
"(REF ).",
"Clearly, the effects are not negligible.",
"The monodromic k-essence case thus offers an avenue to detect dark energy perturbations, which are challenging to detect for smooth equations of state (e.g., [46]).",
"So far, we have considered the growth of the matter sector which (on the large cales considered here) includes cold dark matter and baryons.",
"However, large-scale strucutre also offers the opportunity to measure the total density perturbation $\\delta _{\\rm tot} \\equiv \\Omega _m(t) \\delta _m + \\frac{\\rho _K(t)}{3M_\\text{Pl}H^2} \\delta _K = \\left(\\frac{3}{2} a^2 H^2\\right)^{-1} \\nabla ^2_x\\Phi \\,,$ which sources the gravitational potential $\\Phi $ .",
"Here, we again consider scales larger than $R_s$ , so that pressure perturbations in the dark energy can be neglected.",
"In both models considered, as in almost all dark energy models, there is no anisotropic stress.",
"This means that the two spacetime potentials are equal, and $\\delta _{\\rm tot}$ can be probed directly through gravitational lensing (see, e.g.",
"[11]).",
"The prediction for the growth factor of $\\delta _{\\rm tot}$ is shown in Fig.",
"REF .",
"In the quintessence case, it is identical to the matter growth.",
"However, for k-essence, we see significantly stronger features than in the matter growth factor, which go up to a $\\gtrsim 10\\%$ change in the growth factor in one of the models considered.",
"This strong signature exists because the dark energy density perturbation now contributes directly to the observable, rather than only through its gravitational backreaction on the matter growth; moreover, the oscillations in the growth of the dark energy perturbations are much stronger due to the effect of the oscillating equation of state $w_K$ [second line in Eq.",
"(REF )].",
"Note that, while they show order unity oscillations, the fractional dark energy perturbations remain small at all times (less than 10% of the fractional matter density perturbations).",
"This holds in the monodromic k-essence scenario as long as $A(1+\\bar{w}_K) \\ll 1$ .",
"Such a correlated, but different modulation in the growth of matter and gravitational lensing is a telltale signature of dark energy perturbations."
],
[
"Conclusions",
"The main aim of this paper is to point out that there exists a significant theoretically motivated parameter space of dark energy models, characterized by periodically modulated potentials such as Eq.",
"(REF ), which live outside the standard parametrizations of the equation of state such as [47], [48] $w_{\\rm DE}(a) = w_0 + w_a (a-a_0)\\,.$ That is, these models show significant, observationally detectable features that are not captured by $(w_0,w_a)$ , and require a dedicated search.",
"This is true in particular of models with a non-standard kinetic term, which are not restricted to small oscillations as in the case of quintessence.",
"This search involves scanning the parameter space for periodic modulations with frequency $\\nu $ and amplitude $A$ [and in general, a phase $\\phi $ which we have set to zero in Eq.",
"(REF )] around a slow-rolling background.",
"The most promising observables to search for monodromic dark energy are $(i)$ the expansion rate $H$ ; $(ii)$ the growth rate of structure $f$ ; and $(iii)$ the amplitude of gravitational lensing, as a function of redshift.",
"Hubble rate $H$ : oscillations in the Hubble rate can be probed by taking the derivative of the distance-redshift relation observed with standard candles such as type-Ia Supernovae, for example JLA [4] and Supercal [49].",
"Note that the uncertainties involved in the distance ladder are not crucial here, since one is looking for time-dependent features in the Hubble rate.",
"Alternatively, the BAO feature and AP distortions in the two-point function of galaxies and other tracers allow for a measurement of the Hubble rate on a comoving scale of $\\sim 110 \\,h^{-1}\\,{\\rm Mpc}$ in case of the BAO feature, and a wider range of scales for AP distortions.",
"This corresponds to averaging the quantity shown in Fig.",
"REF over a redshift window of $\\Delta z \\sim 0.03-0.05$ , depending on redshift.",
"Current constraints from BOSS are at the level of approximately 2% [6], which could clearly constrain some of the models presented here in a dedicated analysis.",
"Growth rate $f$ : the growth rate is observable through redshift-space distortions.",
"Current constraints on $f \\sigma _8$ are at the level of $\\sim 10\\%$ [50], [51], [52], [53], [54].",
"In addition, distance indicators such as Supernovae and disk galaxies through the Tully-Fisher relation yield comparable constraints at $z \\lesssim 0.05$ [55], [56], [57], which could be combined with the results from galaxy clustering at higher redshifts.",
"While the constraints on the growth rate are not as precise as those on the Hubble rate, they can be extended to smaller spatial scales and thus are able to constrain higher frequencies of oscillations.",
"For example, a growth rate measurement on comoving scales of $\\sim 20 \\,h^{-1}\\,{\\rm Mpc}$ corresponds to a redshift interval of $\\Delta z \\lesssim 0.01$ .",
"Amplitude of gravitational lensing $\\delta _{\\rm tot}$ : as we have seen in Fig.",
"REF , the monodromic k-essence scenario leads to strong oscillatory features in the total energy density perturbation which sources gravitational lensing.",
"Gravitational lensing observables are projected quantities, and the line-of-sight integration strongly suppresses such features in correlations involving lensing alone, such as cosmic shear.",
"However, galaxy-galaxy lensing, the cross-correlation between lensing and source counts in a narrow redshift range, is projected over a much narrower redshift range, and could show observable oscillatory signatures.",
"Galaxy-galaxy lensing has now been measured at high significance in SDSS [58], CFHTLenS [59], CFHT imaging of stripe 82 [60], in KiDS [61], and DES [62].",
"The DES year-1 constraints on the amplitude of galaxy-galaxy lensing are better than 10%.This was estimated from the error on the linear bias $b_\\times $ in Fig.",
"14 of [62].",
"Hence, this is a further promising probe of the new dark energy phenomenology introduced here.",
"Moreover, combining galaxy-galaxy lensing with probes of $H(z)$ and $f(z)$ will allow for constraints on the speed of sound of dark energy.",
"Intriguingly, the reconstructed equation of state from a combination of the most recent cosmological data sets derived by [63] shows oscillatory features with an amplitude of $\\Delta w_\\text{DE} \\approx 0.6$ ; similar evidence was previously found from the combination of BOSS BAO measurements in [64].",
"In the context of the monodromic models introduced here, this could only be explained by the k-essence scenario.",
"In this context, it would be worthwhile to derive the Bayesian evidence for this scenario, which introduces three parameters in addition to the dark energy density.",
"On the other hand, a monodromic k-essence scenario with such a large oscillation amplitude might be in tension with existing galaxy-galaxy lensing data.",
"Beyond the set of predictions based on linear perturbation theory derived in this paper, interesting phenomenology of monodromic dark energy is expected in the nonlinear regime, in particular for k-essence around the scale of the sound horizon.",
"For example, unusual dynamical effects could appear when the time scale of the oscillations in the dark energy density becomes comparable to the dynamical time of massive halos.",
"Given this potential source of rich phenomenology in the large-scale structure of the Universe, there is strong motivation to look more deeply into the theoretical constraints outlined in Sec.",
"; in particular, the issue of gradient instabilities and whether there are stable monodromic k-essence-type scenarios which do not suffer from a low cutoff or fine-tuning.",
"A promising approach is through the effective field theory of dark energy [41], [65], [66].",
"However, the oscillation period $1/\\nu $ in field space adds a new scale, and time translation is no longer weakly broken in monodromic models; rather, it is replaced by a weakly-broken discrete symmetry $\\phi \\rightarrow \\phi + 2\\pi /\\nu $ , analogously to the case of axion monodromy inflation (see the discussion in [67]).",
"Further, it would be interesting to find microsopic scenarios which lead to a potential of the form Eq.",
"(REF ), as well as to study natural values for and limits on the period and amplitude of oscillations.",
"We leave these interesting questions as open topics for future work.",
"I am indebted to Eiichiro Komatsu and Masahiro Takada for helpful comments on the draft, and to Eric Linder for discussions and pointing out valuable references.",
"I further thank Alex Vikman for dicussions on gradient instabilities.",
"Finally, I thank the Wissenschaftskolleg zu Berlin, where this paper was completed, for hospitality.",
"This work was supported by the Marie Curie Career Integration Grant (FP7-PEOPLE-2013-CIG) “FundPhysicsAndLSS,” and Starting Grant (ERC-2015-STG 678652) “GrInflaGal” from the European Research Council."
]
] | 1709.01544 |
[
[
"Gate induced monolayer behavior in twisted bilayer black phosphorus"
],
[
"Abstract Optical and electronic properties of black phosphorus strongly depend on the number of layers and type of stacking.",
"Using first-principles calculations within the framework of density functional theory, we investigate the electronic properties of bilayer black phosphorus with an interlayer twist angle of 90$^\\circ$.",
"These calculations are complemented with a simple $\\vec{k}\\cdot\\vec{p}$ model which is able to capture most of the low energy features and is valid for arbitrary twist angles.",
"The electronic spectrum of 90$^\\circ$ twisted bilayer black phosphorus is found to be x-y isotropic in contrast to the monolayer.",
"However x-y anisotropy, and a partial return to monolayer-like behavior, particularly in the valence band, can be induced by an external out-of-plane electric field.",
"Moreover, the preferred hole effective mass can be rotated by 90$^\\circ$ simply by changing the direction of the applied electric field.",
"In particular, a +0.4 (-0.4) V/{\\AA} out-of-plane electric field results in a $\\sim$60\\% increase in the hole effective mass along the y (x) axis and enhances the $m^*_{y}/m^*_{x}$ ($m^*_{x}/m^*_{y}$) ratio as much as by a factor of 40.",
"Our DFT and $\\vec{k}\\cdot\\vec{p}$ simulations clearly indicate that the twist angle in combination with an appropriate gate voltage is a novel way to tune the electronic and optical properties of bilayer phosphorus and it gives us a new degree of freedom to engineer the properties of black phosphorus based devices."
],
[
"Introduction",
"Within the family of atomically thin two dimensional (2D) materials, black phosphorus occupies a special status because of its buckled structure [1], its highly anisotropic transport properties and its intermediate size direct size bandgap [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].",
"Experimental and theoretical studies showed that few-layer black phosphorus displays high and anisotropic carrier mobility [14], [15], excellent electron-channel contacts [1], and electronic and optical properties that are tuned by the stacking order[4], [16], [17], in-plane strain[2], and external electric field [2], [7], [18], [19], [20].",
"Meanwhile, high performance field-effect transistors based on black phosphorus have been successfully fabricated with a carrier mobility up to 1000 cm$^{2}$ /V/s [1] and an ON/OFF ratio up to 10$^4$ at room temperature [21].",
"Moreover, black phosphorus has also been implemented into various electronic device applications including gas sensors [22], $p-n$ junctions [23], and solar cells [24].",
"The reported results show that black phosphorus has potential to be adopted in future technological applications.",
"First principles calculations have recently indicated that the distinct potential of this material might be substantially enhanced.",
"For instance, by demonstrating that the optical and electronic properties of bilayer black phosphorous are strongly influenced by the type of stacking.",
"In particular, tunable band gap [24], [25], [26], [2], carrier effective masses along different crystallographic directions[26], [25], [4], and high solar power conversion efficiency have been distinctly demonstrated [24].",
"Moreover, considerable change in carrier mobilities [27], [24] and a continuous transition from a normal insulator to a topological insulator and eventually to a metal as a function of external electric field applied along the out-of-plane direction have been substantiated as well [27].",
"In the present work, the electronic properties of bilayer black phosphorus with an interlayer twist angle of 90$^\\circ $ are systematically investigated by using first-principles calculations within the framework of density functional theory.",
"These results are complemented with an analytic $\\vec{k}\\cdot \\vec{p}$ model that is applicable for arbitrary twist angle.",
"First, the optimum stacking formation map is determined by considering a 20$\\times $ 20 grid (in step of 0.25 Å) of different possible arrangements.",
"Then, the electronic properties including effective masses along crystallographic directions (few meV cohesive energy difference) are systematically investigated under the effect of applied out-of-plane external electric field for both minimum and non-minimum energy stackings."
],
[
"Computational Method",
"The simulations are performed within the framework of density functional theory (DFT) as implemented in the Vienna $Ab$ -initio Simulation Package (VASP) [28], [29], [30], [31].",
"The generalized gradient approximation (GGA) formalism [32] is employed for the exchange-correlation potential.",
"The projector augmented wave (PAW) method [33], [34] and a plane-wave basis set with an energy cutoff of 400 eV are used in the calculations.",
"A regular 1$\\times $ 1$\\times $ 1 $k$ -mesh within the Monkhorst-Pack scheme [35] is adopted for Brillouin-zone integration.",
"We employ a 5$\\times $ 7 supercell resulting in a nearly commensurate lattice with a lattice mismatch of $<$ 1% between top and bottom layers.",
"A vacuum spacing of at least 20 Å is introduced between isolated bilayers to prevent spurious interaction.",
"By using the conjugate gradient method, atomic positions and lattice constants are optimized until the Hellmann-Feynman forces are less than 0.01 eV/Å and pressure on the supercells is decreased to values less than 1 kB.",
"The van der Waals interaction between individual layers are taken into account[36], [37] for a correct description of the structural and electronic properties.",
"In order to investigate the equilibrium bilayer structure, formation energies of all the possible non-symmetric structures are calculated.",
"The structural and electronic properties are obtained in the presence of a perpendicular uniform electric field, ranging from -0.4 to 0.4 V/Å in steps of 0.1 V/Å.",
"The influence of the applied electric field on the effective masses of carriers are investigated using the following formula for the mass tensor, $m^{*}_{i,j}=\\hbar ^{2}\\left(\\frac{\\partial ^{2}E}{\\partial k_i\\partial k_j}\\right)^{-1}.$ where $\\hbar $ is the reduced Planck constant, $k_i$ ($k_j$ ) is the wavevector along the $i$ ($j$ ) direction, and $E$ is the energy eigenvalue.",
"Here, the $i$ and $j$ directions are used for the $\\Gamma $ -X and $\\Gamma $ -Y directions in the first Brillouin zone of bilayer black phosphorus.",
"In order to obtain a reliable numerical estimate of the second derivative of the band energies, we use a dense two-dimensional $k$ -point grid centered at the $\\Gamma $ point.",
"Numerical second derivatives are obtained by using four point forward and backward differences."
],
[
"DFT results for 90$^\\circ $ twisted angle ",
"First of all, we investigate the structural and electronic properties of 90$^\\circ $ twisted bilayer black phosphorus at zero electric field.",
"Figure REF shows the variation of the formation energy difference between 90$^\\circ $ twisted and AB stacked bilayer black phosphorous as a function of stacking of the rotated bilayer structure.",
"Here, all the possible stacking arrangements are considered and the atomic positions are only relaxed along the $z$ -direction.",
"In the rest of this work, unless otherwise stated, we focus on the lowest energy structure and relax the atomic positions in the in-plane directions as well.",
"From Fig.",
"REF we note that the energy difference between different stacking types of 90$^\\circ $ twisted bilayer black phosphorus is at most 0.2 meV/atom.",
"This means that layers can easily slide over each other.",
"The van der Waals corrected interlayer separation of the 90$^\\circ $ twisted bilayer black phosphorus is 3.32 Å for the lowest energy structure, which is about 0.15 Å larger than that of AB stacked bilayer[4].",
"The P-P bond lengths show only subtle changes in twisted bilayer as compared to the naturally stacked bilayer.",
"Figure: (a) Top and side views of 90 ∘ ^\\circ twisted bilayer black phosphorus, and (b) potential energy surface with respect to naturally stacked bilayer as afunction of lateral displacements of layers with respect to each other.Figure: Band structure of 90 ∘ ^\\circ twisted bilayer black phosphorus for different values of the out-of-plane electric field.Figure REF displays the band structure of twisted bilayer black phosphorus as a function of applied electric field strength.",
"In contrast to the AB stacked bilayer, 90$^\\circ $ twisted bilayer has an x-y isotropic electronic structure at zero applied field, meaning that the energy bands in the $\\Gamma $ -X and $\\Gamma $ -Y directions are symmetric.",
"While the conduction band minimum (CBM) is singly degenerate, the valence band maximum (VBM) is found to be almost doubly degenerate with an energy splitting of 0.1 meV.",
"Figure: (b)-(f) The decomposed charge densities corresponding to the VBM of the 90 ∘ ^\\circ twisted bilayer phosphorus at the Γ\\Gamma point for different values of the applied electric field.",
"In (a), we also present the charge of VBM of the AB stacked bilayer black phosphorus.For a better insight, Fig.",
"REF depicts the absolute square of the wavefunctions corresponding to the top of the valence band.",
"Naturally stacked (i.e.",
"AB stacked) and 90$^\\circ $ twisted bilayer black phosphorus display very different spatial characters at the VBM.",
"In the case of AB stacked bilayer (Fig.",
"REF a), the VBM wavefunction is equally localized over the two layers.",
"In the 90$^\\circ $ twisted bilayer (Fig.",
"REF b-f), however, the wavefunction is mainly localized either on the bottom or top layer.",
"According to our calculations, the top of the valence band is mostly located on the bottom layer for zero field.",
"The VBM is almost doubly degenerate with the other state 0.1 meV lower in energy.",
"In that state, the wavefunction is mostly localized over the top layer.",
"This spatial character of wavefunction especially implies the lack of a significant hybridization between states of individual layers near the VBM.",
"Whereas, (not shown), the states of individual layers around CBM are strongly coupled, resulting in a wavefunction that is extended over both layers.",
"Next, we turn to investigate the response of the electronic structure of 90$^\\circ $ twisted bilayer black phosphorus to a static out-of-plane electric field.",
"The electronic properties of black phosphorus are very sensitive to in-plane/out-plane strain[2], [20] and out-of-plane applied electric field[27], [18].",
"For all applied electric field values, the nature of the band gap remains direct at the $\\Gamma $ -point, see Fig.",
"REF .",
"In contrast to the zero field case, the applied electric field restores the x-y anisotropic band structure of naturally stacked bilayer.",
"In addition, the electric field lifts the degeneracy at the top of the valence band.",
"Positive (negative) electric field (which is perpendicular to bilayer) shifts the bands along the $\\Gamma $ -Y ($\\Gamma $ -X)-directions towards lower energies.",
"The amount of splitting of doubly degenerate bands with electric field is found to be 0.1 eV for 0.2 V/Å and becomes 0.2 eV for 0.4 V/Å.",
"After the degeneracy is lifted at the VBM, the state becomes localized on the layer with the higher potential.",
"The applied field fully decouples the states of the two individual layers at the VBM, which can be easily seen by inspecting the charge density plots in Fig.",
"REF .",
"For high fields, the VBM is completely localized over either top or bottom layer depending on the sign or direction of the electric field.",
"By shifting constituent layers on top of each other, we can tune the interaction strength between the layers in AB stacked bilayer black phosphorus.",
"This can be understood by calculating the charge density.",
"Distinct from 90$^\\circ $ twisted bilayer, different stacking types in the non-rotated bilayer result in different $\\pi $ -$\\pi $ interaction distances and strengths, allowing the band gap to be tuned by 0.2-0.3 eV [4].",
"However, we find the band gap of 90$^\\circ $ twisted bilayer is almost insensitive to the precise stacking, retaining a value of $\\sim $ 0.58 eV for different stacking types.",
"For some high energy stacking types, we predict that the energy splitting between the almost degenerate VBM states becomes 3 meV due to the reduction in symmetry and the change in interaction strength.",
"Figure: Direction dependent (a) hole and (b) electron masses in monolayer, AB stacked and 90 ∘ ^\\circ twisted bilayer black phosphorus.",
"Each data point represents the endpoint of a vector whose amplitude corresponds to the effective massin units of m 0 m_0 and the direction of this vector corresponds to thedirection in k space along which the mass is calculated.Figure: Direction-dependent (a) hole and (b) electron effective masses.",
"Each data point represents the end point of a vector whose amplitude corresponds to the effective mass in units of m 0 m_0 and the direction of this vector corresponds to the direction in k space along which the mass is calculated.",
"Variation of (c) hole and (d) electron effective masses along the Γ\\Gamma -X (blue solid dot symbols) and Γ\\Gamma -Y (red solid triangle symbols) directions as a function of out of plane electric field.",
"Parameter m * m^* is in units of m 0 m_0,|c||c| is in units of m 0 m_0Å/V.The band edge effective masses along all reciprocal space directions (rather than just the $\\Gamma $ -X and $\\Gamma $ -Y directions) are displayed in Fig.",
"REF for AB stacked and 90$^\\circ $ twisted bilayer black phosphorus.",
"Notice that, these two different stacking types exhibit quite different electronic and transport properties.",
"First of all, due to its anisotropic band structure, the AB stacked black phosphorus has an apparent anisotropic hole and electron effective masses.",
"Rotation of top layer with respect to bottom layer by 90$^\\circ $ removes the anisotropy between the $x$ and $y$ directions.",
"While the hole effective mass is 0.2 $m_0$ along the $\\Gamma $ -Y direction (armchair direction) in AB-stacked bilayer, it becomes 3.16 $m_0$ along the same direction in the 90$^\\circ $ twisted bilayer.",
"For comparison, we also show the effective masses for monolayer phosphorus which exhibits the strongest anisotropy.",
"We next investigate how an out-of-plane electric field changes the effective masses in 90$^\\circ $ twisted bilayer, see Fig.",
"REF .",
"At zero field, the hole and electron effective masses are respectively 3.16 $m_0$ and 0.095 $m_0$ , along both the $\\Gamma $ -X and $\\Gamma $ -Y directions.",
"For holes, the application of a small positive (negative) electric field results in a sudden reduction of the hole effective mass along the the $\\Gamma $ -Y ($\\Gamma $ -X) direction, due to the lifting of the double degeneracy at the VBM and the partial return to monolayer behavior evinced by the wavefunction's localization on a single layer (Fig.",
"REF c-f).",
"For example, a +(-)0.1 V/Å electric field results in a 0.2 $m_0$ effective mass along $\\Gamma $ -Y ($\\Gamma $ -X).",
"Hence the preferred direction of hole transport can switched by 90$^\\circ $ when changing the direction of the electric field.",
"In addition, the larger hole effective mass can be increased by up to 60% by varying the applied field from 0 V/Å to 0.4 V/Å.",
"This behavior of larger hole effective mass is well described by the linear dependence $m^*(E)$ =$m^*$ (0)+$cE$ where $m^*(E)$ and $m^*(0)$ are the hole effective masses in the presence and absence of the electric field, and fitting constant $c$ is given in Fig.",
"REF (a).",
"The electron effective mass (Fig.",
"REF (b,d)) also becomes anisotropic between $\\Gamma $ -X and $\\Gamma $ -Y directions when a finite electric field is applied, but the effect is weaker than for the holes, and does not display the discontinuity at zero electric field, due to the strong interlayer hybridization for this band.",
"Indeed, the electron effective mass fall within 0.21$\\pm $ 0.04 $m_0$ for the entire range of tested electric field strengths.",
"Similar to the hole case, the variation of effective mass with electric field is well described by a linear fit (Fig.",
"REF (b)).",
"We also note that the interlayer separation of 90$^\\circ $ twisted bilayer black phosphorus increases from 3.3 Å at zero electric field to 3.4 Å for a field strength of $\\pm $ 0.4 V/Å.",
"In this work, we employed semi-local functionals in order to investigate the electronic properties of twisted bilayer black phosphorus.",
"We did not consider GW corrections.",
"These corrections will likely change the results in a quantitative level, but we do not expect any change in a qualitative level.",
"GW corrections will significantly enlarge band gap values.",
"This is because of the fact we previously showed that many body effects are significant for black phosphorus.",
"For instance, the band gap for monolayer black phosphorus is about 0.9 eV for PBE functional.",
"It becomes 2.3 eV when we include GW corrections[2].",
"Regardless of computational method (PBE, hybrid or GW) used in the calculations, the evolution of band gaps as a function of number of layers has almost the same trend[2].",
"Similarly, band dispersions are not affected much.",
"Therefore, we believe that our calculated trends are correct."
],
[
"$\\vec{k}\\cdot \\vec{p}$ approach for arbitrary twist angle",
"We will now describe a simple $\\vec{k}\\cdot \\vec{p}$ Hamiltonian which both reproduces the above described bandstructure features, particularly those near the band edges, and is able to describe arbitrarily misaligned twisted black phosphorous bilayers and their optical absorption properties.",
"The $\\vec{k}\\cdot \\vec{p}$ Hamiltonian is written as, $H=\\begin{pmatrix} H_{TT} & H_{TB}\\\\H_{TB}^\\dagger & H_{BB} \\end{pmatrix},$ in a basis $(\\psi ^\\text{T}_\\text{C},\\psi ^\\text{T}_\\text{V},\\psi ^\\text{B}_\\text{C},\\psi ^\\text{B}_\\text{V})$ of $\\Gamma $ -point conduction/valence (C/V) band wavefunctions on the top/bottom (T/B) layers.",
"For the intralayer terms [38], [39] we use, $H_{\\text{TT/BB}}=\\begin{pmatrix}\\epsilon ^0_\\text{C}+u_\\text{T/B} + \\alpha ^x_\\text{C}\\hat{p}_{ x}^2 + \\alpha ^y_\\text{C}\\hat{p}_{ y}^2&\\gamma _{\\text{ML}}\\hat{p}_{ x} \\\\\\gamma _{\\text{ML}}\\hat{p}_{ x}&\\epsilon ^0_\\text{V}+u_\\text{T/B} + \\alpha ^x_\\text{V}\\hat{p}_{ x}^2 + \\alpha ^y_\\text{V}\\hat{p}_{ y}^2\\end{pmatrix},\\nonumber $ where $\\hat{\\vec{p}}=(-i\\,\\hbar \\partial _x,-i\\,\\hbar \\partial _y)$ for the top layer, $\\hat{\\vec{p}}=(i\\,\\hbar \\partial _y,-i\\,\\hbar \\partial _x)$ for the 90$^\\circ $ rotated bottom layer, and $u_\\text{T}=-u_\\text{B}= eEd/2$ for an out-of-plane electric field.",
"For the interlayer hops we use $H_\\text{T/B}=\\begin{pmatrix}\\gamma _\\text{C} & 0 \\\\0 & \\gamma _\\text{V} \\sum _{\\delta \\!\\vec{g} =(\\pm \\delta \\!g,0),(0,\\pm \\delta \\!g )} e^{i\\delta \\!\\vec{g} \\cdot \\vec{r}}\\\\\\end{pmatrix},$ where $\\delta \\!g=\\frac{2\\pi }{b}-\\frac{2\\pi }{a}$ , $a/b$ are the $x/y$ -directions lattice constants of monolayer black phosphorus, and $\\gamma _\\text{C/V}$ are hopping integrals.",
"To obtain this Hamiltonian we approximated the interlayer hopping matrix elements using the overlap of the wavefunctions, $\\langle \\psi ^\\text{T}_\\text{C/V}|H|\\psi ^\\text{B}_\\text{C/V} \\rangle \\sim \\langle \\psi ^\\text{T}_\\text{C/V}|\\psi ^\\text{B}_\\text{C/V} \\rangle $ , and expanded the $\\Gamma $ -point wavefunctions in the shortest few plane waves consistent with the translational and point-group symmetry of monolayer black phosphorus.",
"We also neglected an interlayer valence-to-conduction band coupling due to its negligible affect on the band structure.",
"The periodicity, $2\\pi /\\delta \\!g $ , which enters Hamiltonian (REF ) is that of the moiré pattern formed from the two black phosphorous layers, rather than the twice larger periodicity of the commensurate unit cell displayed in Fig.",
"REF .",
"Consequently, this Hamiltonian describes a twisted bilayer with a general incommensurate ratio between lattice constants $a$ and $b$ , rather than the exact commensurate ratio $a/b=5/7$ used in the DFT calculations.",
"Nevertheless, we fold the band structure into the smaller Brillouin zone of the commensurate unit cell to match the DFT bands, and fit all free parameters to obtain a good match (Fig.",
"REF caption).",
"For zero applied electric field, Hamiltonian (REF ) posses a $C_{4v}$ symmetry in which both the $\\pi /4$ -rotation and reflection in the $x\\pm y=0$ planes are combined with the exchange of the top and bottom layers.",
"Using this symmetry we find that the pair of doubly degenerate VBM states at the $\\Gamma $ -point belong to the two-dimensional E irreducible representation, while the singly degenerate CBM state belongs to the one dimensional B$_1$ irreducible representation (a \"bonding” state with the corresponding \"anti-bonding” state higher in energy by $2\\gamma _\\text{C}$ ).",
"The application of an electric field (Fig.",
"REF , right panel) breaks the degeneracy of the VBM states so that the top of the valence band resembles that of the single isolated monolayer.",
"Note that the above discussed $C_{4v}$ symmetry is slightly broken by the commensuration of the lattices displayed in Fig.",
"REF , which accounts for the slight lifting of the VBM degeneracy in the DFT calculated bandstructure.",
"We can also adapt Hamiltonian (REF ) to the study of bilayer black phosphorous with an $\\textit {arbitrary twist angle}$ , $\\theta $ , by simply using the appropriately rotated momentum $\\hat{\\vec{p}}=(-i\\,\\hbar \\partial _x\\cos (\\theta )+i\\,\\hbar \\partial _y\\sin (\\theta ) , -i\\,\\hbar \\partial _x\\sin (\\theta )-i\\,\\hbar \\partial _y\\cos (\\theta ) )$ in $H_{BB}$ and neglecting the interlayer $\\gamma _V$ hop.",
"This approach is valid for wavevectors $k< \\text{min}\\lbrace \\delta \\!g/2 , \\theta \\pi /a \\rbrace $ where both the $\\gamma _V$ coupling and the superlattice effects, expected for an almost aligned ($\\theta \\ll 1)$ bilayer, are irrelevant.",
"Dispersion surfaces calculated using this method are displayed in Fig.",
"REF for different choices of $\\theta $ .",
"In this simple model varying $\\theta $ breaks the four-fold rotational symmetry seen in the bandstructure but does not change the energies of the $\\Gamma $ -point band energies.",
"Nevertheless the $\\Gamma $ -point degeneracy of the VBM is not protected by symmetry $\\theta \\ne 90^\\circ $ , and would be slightly lifted if the neglected $\\gamma _V$ hop or superlattice effects were included.",
"As an example of the effect of controllably breaking the $C_{4v}$ symmetry of Hamiltonian (REF ), either by applying an electric field, or using an interlayer rotation $\\theta \\ne 90^\\circ $ , we now calculate the optical absorption of linearly polarized light (Fig.",
"REF ).",
"Here we use the matrix element for optical transition $\\sim \\langle \\psi _i|v_{j}|\\psi _f\\rangle $ with velocity operators $v_{j=x/y}=\\partial H/\\partial p{_j}$ , introduce a phenomenological energy broadening $\\eta =10\\,$ meV, and neglect many-body effects such as the formation of excitons.",
"For small interlayer twist angles (e.g.",
"$\\theta =10^\\circ $ , right panel) the optical absorption resembles that of two isolated monolayers, displaying a strong preference to absorb light polarized in the $x$ -direction, and step-like increases of the absorption when the excitation frequency allows electronic transitions between the VBM on either layer (which are split by the electric field) and the CBM.",
"In contrast, a $90^\\circ $ twist between the layers (left panel) and zero applied electric field produces an isotropic optical absorption.",
"However splitting the degeneracy of the VBM bands on the two layers produces a controllable linear dichroism in which the electric field direction selects the preferred polarization direction for absorption.",
"We note that for intermediate twist angles (e.g.",
"$\\theta =45^\\circ $ , center panel) the absorption resembles a combination of the two extremes.",
"Figure: The bandstructure of 90 ∘ ^\\circ twisted black phosphorus calculated using either DFT (red), or the k →·p →\\vec{k} \\cdot \\vec{p} model (black) for zero and finite electric fields.",
"The values for the fitted parameters are:γ ML =4.5\\gamma _\\text{ML}=4.5\\,eV, γ C =0.39\\gamma _\\text{C}=0.39\\,eV, γ V =0.07\\gamma _\\text{V}=0.07\\,eV, ϵ C 0 -ϵ V 0 =0.99\\epsilon ^0_\\text{C}-\\epsilon ^0_\\text{V}=0.99\\,eV, α C x =1.2\\alpha ^x_\\text{C}=1.2\\,eVÅ 2 ^{2}, α C y =2.7\\alpha ^y_\\text{C}=2.7\\,eVÅ 2 ^{2}, α V x =-5.9\\alpha ^x_\\text{V}=-5.9\\,eVÅ 2 ^{2}, α V y =-2.0\\alpha ^y_\\text{V}=-2.0\\,eVÅ 2 ^{2}, d=0.54Åd=0.54\\,\\text{Å}.Figure: The low-energy dispersion surfaces of bilayer black phosphorus for different interlayer twist angles, θ\\theta , (top row), and the corresponding contour plots of the CBM and VBM bands (middle and bottom rows respectively).",
"Parameters as Fig.",
"except here γ V =0\\gamma _\\text{V}=0.Figure: The optical absorption, gg, of bilayer black phosphorous as a function of the applied electric field, EE, and excitation energy, ω\\omega , calculated for light polarized in the x/y-directions (orange/blue surfaces) and various interlayer twist angle.",
"Band parameters are as per Fig.",
"."
],
[
"Conclusions",
"Using first principles and $\\vec{k}\\cdot \\vec{p}$ based calculations, we investigated the structural, electronic and transport properties of 90$^\\circ $ twisted bilayer black phosphorus.",
"Even though twisted bilayer phosphorus displays isotropic electronic and transport properties, an out-of-plane electric field is able to create significant anisotropy in these properties.",
"We demonstrated that the hole effective mass increases from 3.16 $m_0$ to 5 $m_0$ when the field is raised from 0 V/Å to 0.4 V/Å, corresponding to a 60% increase.",
"The hole effective mass approaches the value of the monolayer limit as the applied field increases and bilayer black phosphorus starts to display monolayer behavior for holes.",
"The states near the VBM are localized over either top or bottom layer depending on the direction of the electric field.",
"In addition, we predicted that the degeneracy of the highest occupied valence bands on the two layers splits by the applied electric field and this produces a controllable linear dichroism in which the electric field direction selects the preferred polarization direction for absorption.",
"In summary, our calculations show that twisting combined with an appropriate gate voltage gives us a new degree of freedom in manipulating the electronic, transport, optical and even thermoelectric properties of few-layer black phosphorus.",
"For instance, when the applied field is zero, the isotropic electronic and thermal properties give rise to isotropic thermoelectric properties in 90$^\\circ $ twisted bilayer black phosphorus.",
"However, by the help of an out-of-plane applied electric field, we can tune the ratio of the Seebeck coefficients along the $\\mathbf {x}$ and $\\mathbf {y}$ directions.",
"Similarly, the optical response can be changed from isotropic to anisotropic, which can be used in conceptually new device designs."
],
[
"acknowledgement",
"This work was supported by the bilateral project between the The Scientific and Technological Research Council of Turkey (TUBITAK) and FWO-Flanders, Flemish Science Foundation (FWO-Vl) and the Methusalem foundation of the Flemish government.",
"Computational resources were provided by TUBITAK ULAKBIM, High Performance and Grid Computing Center (TRGrid e-Infrastructure), and HPC infrastructure of the University of Antwerp (CalcUA) a division of the Flemish Supercomputer Center (VSC), which is funded by the Hercules foundation.",
"We acknowledge the support from TUBITAK (Grant No.",
"115F024), ERC Synergy grant Hetero2D and the EU Graphene Flagship Project.",
"We also thank Vladimir Fal'ko for helpful discussions."
]
] | 1709.01765 |
[
[
"Gammapy - A prototype for the CTA science tools"
],
[
"Abstract Gammapy is a Python package for high-level gamma-ray data analysis built on Numpy, Scipy and Astropy.",
"It enables us to analyze gamma-ray data and to create sky images, spectra and lightcurves, from event lists and instrument response information, and to determine the position, morphology and spectra of gamma-ray sources.",
"So far Gammapy has mostly been used to analyze data from H.E.S.S.",
"and Fermi-LAT, and is now being used for the simulation and analysis of observations from the Cherenkov Telescope Array (CTA).",
"We have proposed Gammapy as a prototype for the CTA science tools.",
"This contribution gives an overview of the Gammapy package and project and shows an analysis application example with simulated CTA data."
],
[
"Introduction",
"The Cherenkov Telescope Array (CTA) will observe the sky in very-high-energy (VHE, E > 20$\\,$ GeV) gamma-ray light soon.",
"CTA will consist of large telescope arrays at two sites, one in the southern (Chile) and one in the northern (La Palma) hemisphere.",
"It will perform surveys of large parts of the sky, targeted observations on Galactic and extra-galactic sources, and more specialized analyses like a measurement of charged cosmic rays, constraints on the intergalactic medium opacity for gamma-rays and a search for dark matter.",
"Compared to current Cherenkov telescope arrays such as H.E.S.S., VERITAS or MAGIC, CTA will have a much improved detection area, angular and energy resolution, improved signal/background classification and sensitivity.",
"CTA is expected to operate for thirty years, and all astronomers will have access to CTA high-level data, as well as CTA science tools (ST) software.",
"The ST can be used for example to generate sky images and to measure source properties such as morphology, spectra and light curves, using event lists as well as instrument response function (IRF) and auxiliary information as input.",
"Gammapy is a prototype for the CTA ST, built on the scientific Python stack and Astropy [1], optionally using Sherpa [2], [3], [4] or other packages for modeling and fitting (see Figure REF ).",
"A 2-dimensional analysis of the sky images for source detection and morphology fitting, followed by spectral analysis, was first implemented.",
"A 3-dimensional analysis with a simultaneous spatial and spectral model of the gamma-ray emission, as well as background (called “cube analysis” in the following) is under development.",
"A first study comparing spectra obtained with the classical 1D analysis and the 3D cube analysis using point source observations with H.E.S.S.",
"is presented in [5].",
"Further developments and verification using data from existing Cherenkov telescope arrays such as H.E.S.S.",
"and MAGIC, as well as simulated CTA data is ongoing.",
"Gammapy is now used for scientific studies with existing ground-based gamma-ray telescopes [6], [7], the Fermi-LAT space telescope [8], as well as for CTA [9], [10], [11].",
"Figure: The Gammapy software stack.",
"Required dependencies (Python, Numpy and Astropy)are illustrated with solid arrows, optional dependencies (Scipy and Sherpa) withdashed arrows.In this writeup we focus on the software and technical aspects of Gammapy.We start with a brief overview of the context in Section , followed by a description of the Gammapy package in Section , the Gammapy project in Section and finally our conclusions concerning Gammapy as as CTA science tool prototype in Section ."
],
[
"Context",
"Prior to Gammapy, in 2011/2012, a collection of Python scripts for the analysis of IACT data was released as a first test of open-source VHE data analysis software: “PyFACT: Python and FITS Analysis for Cherenkov Telescopes” [12].",
"This project is not updated anymore, but it was the first implementation of our idea to build a CTA science tools package based on Python and using Numpy and Scipy, during the first development stages of the CTA project.",
"In 2011, the astronomical Python community came together and created the Astropy project and package [1], which is a key factor making Python the most popular language for astronomical research codes (at least according to this informal analysis [13] and survey [14]).",
"Gammapy is an Astropy affiliated package, which means that where possible it uses the Astropy core package instead of duplicating its functionality, as well as having a certain quality standard such as having automated tests and documentation for the available functionality.",
"In recent years, several other packages have adopted the same approach, to build on Python, Numpy and Astropy.",
"To name just a few, there is ctapipe (github.com/cta-observatory/ctapipe), the prototype for the low-level CTA data processing pipeline (up to the creation of lists of events and IRFs); Naima for modeling the non-thermal spectral energy distribution of astrophysical sources [15]; PINT, a new software for high-precision pulsar timing (github.com/nanograv/PINT), and Fermipy (github.com/fermipy/fermipy), a Python package that facilitates analysis of data from the Large Area Telescope (LAT) with the Fermi Science Tools and adds some extra functionality.",
"We note that many other astronomy projects have chosen Python and Astropy as the basis both for their data calibration and reduction pipelines and their science tools.",
"Some prominent examples are the Hubble space telescope (HST) [16], the upcoming James Webb Space Telescope (JWST) [17] and the Chandra X-ray observatory [2], [18].",
"Even projects like LSST that started their analysis software developments before Astropy existed and are based on C++/SWIG are now actively working towards making their software interoperable with Numpy and Astropy, to avoid duplication of code and development efforts, but also to reduce the learning curve for their science tool software (since many astronomers already are using Python, Numpy and Astropy) [19].",
"For current ground-based IACTs, data and software are mostly private to the collaborations operating the telescopes.",
"CTA will be, for the first time in VHE gamma-ray astronomy, operated as an open observatory.",
"This implies fundamentally different requirements for the data formats and software tools.",
"Along this path, the current experiments, H.E.S.S., MAGIC and VERITAS, have started converting their data to FITS format, yet relying on different (some private, some open) analysis tools, and many slightly different ways to store the same information in FITS files appeared.",
"The CTA high-level data model and data format specifications are currently being written and will give a framework to the current experiments to store data.",
"The methods to link events to IRFs still have to be extended to support multiple event types and the IRF and background model formats will have to be extended to be more precise [20].",
"The Gammapy team is participating in and contributing to the effort to prototype and find a good high-level data model and formats for CTA."
],
[
"Gammapy package",
"Gammapy offers the high-level analysis tools to generate science results (images, spectra, light curves and source catalogs) based on input data consisting of reconstructed events (with an arrival direction and an energy) that are classified according to their types (e.g.",
"gamma-like, cosmic-ray-like).",
"In the data processing chain of CTA, the reconstruction and classification of events are realized by another Python package, ctapipegithub.com/cta-observatory/ctapipe.",
"Data are stored in FITS format.",
"The high-level analysis consists of: selection of a data cube (energy and positions) around a sky position from all event lists, computation of the corresponding exposure, estimation the background directly from the data (e.g.",
"with a ring background model [21]) or from a model (e.g.",
"templates built from real data beforehand), creation of sky images (signal, background, significance, etc) and morphology fitting, spectrum measurement with a 1D analysis or with a 3D analysis by adjusting both spectral and spatial shape of gamma-ray sources, computation of light-curves and phasograms, search for transient signals, derivation of a catalog of excess peaks (or a source catalog).",
"For such an analysis, Gammapy is using the IRFs produced by the ctapipe package and processes them precisely to get an accurate estimation of high-level astrophysical quantities.",
"The Gammapy code base is structured into several sub-packages dedicated to specific classes where each of the packages bundle corresponding functionality in a namespace (e.g.",
"data and observation handling in gammapy.data, IRF functionality in gammapy.irf, spectrum estimation and modeling in gammapy.spectrum ...).",
"The Gammapy features are described in detail in the Gammapy documentation (docs.gammapy.org) and many examples given in the tutorial-style Jupyter notebooks, as well as in [22].",
"Figure REF shows one result of the “CTA data analysis with Gammapy” notebook: a significance sky image of the Galactic center region using 1.5 hours of simulated CTA data.",
"The background was estimated using the ring background estimation technique, and peaks above 5 sigma are shown with white circles.",
"For examples of CTA science studies using Gammapy, we refer you to other posters presented at this conference: Galactic survey [10], PeVatrons [11] and extra-galactic sources [9].",
"Several other examples using real data from H.E.S.S.",
"and Fermi-LAT, as well as simulated data for CTA can be found via docs.gammapy.org by following the link to “tutorial notebooks”.",
"The Gammapy Python package is primarily built on Numpy [23], and Astropy [1] as core dependencies.",
"Data is stored in Numpy arrays or objects such as astropy.coordinates.SkyCoord or astropy.table.Table that hold Numpy array data members.",
"Numpy provides many functions for array-oriented computing and numerics, and Astropy provides astronomy-specific functionality.",
"The Astropy functionality most commonly used in Gammapy is astropy.io.fits for FITS data I/O, astropy.table.Table as a container for tabular data (e.g.",
"event lists, but also many other things like spectral points or source catalogs), astropy.wcs.WCS for world coordinate systems mapping pixel to sky coordinates, as well as astropy.coordinates.SkyCoord and astropy.time.Time objects to represent sky coordinates and times.",
"Astropy.coordinates as well as astropy.time are built on ERFA (github.com/liberfa/erfa), the open-source variant of this IAU Standards of Fundamental Astronomy (SOFA) C library (www.iausofa.org).",
"In Gammapy, we use astropy.units.Quantity objects extensively, where a quantity is a Numpy array with a unit attached, supporting arithmetic in computations and making it easier to read and write code that does computations involving physical quantities.",
"As an example, a script that generates a counts image from an event list using Gammapy is shown in Figure REF .",
"The point we want to make here is that it is possible to efficiently work with events and pixels and to implement algorithms from Python, by storing all data in Numpy arrays and processing via calls into existing C extensions in Numpy and Astropy.",
"E.g.",
"here EventList stores the RA and DEC columns from the event list as Numpy arrays, and SkyImage the pixel data as well, and image.fill(events), and all processing happens in existing C extensions implemented or wrapped in Numpy and Astropy.",
"In this example, the read and write methods call into astropy.io.fits which calls into CFITSIO ([24]), and the image.fill(events) method calls into astropy.wcs.WCS and WCSLib ([25]) as well as numpy.histogramdd.",
"Figure: An example script using Gammapy to make a counts image from an event list.",
"Thisis used in Section to explain how Gammapy achievesefficient processing of event and pixel data from Python: all data is stored inNumpy arrays and passed to existing C extensions in Numpy and Astropy.Gammapy aims to be a base package on which other more specialized packages such as Fermipy (github.com/fermipy/fermipy) for Fermi-LAT data analysis or Naima [15] for the modeling of non-thermal spectral energy distributions of astrophysical sources can build.",
"For this reason we avoid introducing new required dependencies besides Numpy and Astropy.",
"That said, Gammapy does import the following optional dependencies to provide extra functionality (sorted in the order of how common their use is within Gammapy).",
"Scipy [26] is used for integration and interpolation, Matplotlib [27] for plotting and Sherpa [2], [3], [4] for modeling and fitting.",
"In addition, the following packages are used at the moment for functionality that we expect to become available in the Astropy core package within the next year: regions (astropy-regions.readthedocs.io) to handle sky and pixel regions, reproject (reproject.readthedocs.io) for reprojecting sky images and cubes and healpy (healpy.readthedocs.io) for HEALPix data handling.",
"Figure: Application example: significance image for the Galactic centre region using1.5 hours of simulated CTA data.",
"White circles are peaks above 5 sigma."
],
[
"Gammapy project",
"In this section we describe the current setup of the Gammapy project.",
"We are using the common tools and services for Python open-source projects for software development, code review, testing, documentation, package distribution and user support.",
"Gammapy is distributed and installed in the usual way for Python packages.",
"Each stable release is uploaded to the Python package index (pypi.python.org), and downloaded and installed by users via pip install gammapy (pip.pypa.io).",
"Binary packages for conda are available via the conda Astropy channel (anaconda.org/astropy/gammapy) for Linux, Mac and Windows, which conda users can install via conda install gammapy -c astropy.",
"Binary packages for the Macports package manager are also available, which users can install via port install gammapy.",
"At this time, Gammapy is also available as a Gentoo Linux package (packages.gentoo.org/packages/dev-python/gammapy) and a Debian Linux package is in preparation.",
"Gammapy development happens on Github (github.com/gammapy/gammapy).",
"We make extensive use of the pull request system to discuss and review code contributions.",
"For testing we use pytest (pytest.org), for continuous integration Travis-CI (Linux and Mac) as well as Appveyor (Windows).",
"For documentation Sphinx (www.sphinx-doc.org), for tutorial-style documentation Jupyter notebooks (jupyter.org) are used.",
"For Gammapy developer team communication we use Slack (gammapy.slack.com).",
"A public mailing list for user support and discussion is available (groups.google.com/forum/#!forum/gammapy).",
"Two face-to-face meetings for Gammapy were organized so far, the first on in June 2016 in Heidelberg as a coding sprint for developers only, the second on in February 2017 in Paris as a workshop for both Gammapy users and developers.",
"Gammapy is under very active development, especially in the area of modeling, and in the implementation of a simple-to-use, high-level end-user interface (either config file driven or command line tools).",
"We will write a paper on Gammapy later this year and are working towards a version 1.0 release."
],
[
"Conclusions",
"In the past two years, we have developed Gammapy as an open-source analysis package for existing gamma-ray telescope and as a prototype for the CTA science tools.",
"Gammapy is a Python package, consisting of functions and classes that can be used as a flexible and extensible toolbox to implement and execute high-level gamma-ray data analyses.",
"We find that the Gammapy approach, to build on the powerful and well-tested Python packages Numpy and Astropy, brings large benefits: a small codebase that is focused on gamma-ray astronomy in a single high-level language is easy to understand and maintain.",
"It is also easy to modify and extend as new use cases arise, which is important for CTA, since it can be expected that the modeling of the instrument, background and astrophysical emission, as well as the analysis method in general (e.g.",
"likelihood or Bayesian statistical methods) will evolve and improve over the next decade.",
"Last but not least, the Gammapy approach is inherently collaborative (contributions from $\\sim $ 30 gamma-ray astronomers so far), sharing development effort as well as know-how with the larger astronomical community, that to a large degree already has adopted Numpy and Astropy as the basis for astronomical analysis codes in the past 5 years."
],
[
"Acknowledgements",
"This work was conducted in the context of the CTA Consortium.",
"We gratefully acknowledge financial support from the agencies and organizations listed here: www.cta-observatory.org/consortium_acknowledgments We would like to thank the Scientific Python and specifically the Astropy community for providing their packages which are invaluable to the development of Gammapy, as well as tools and help with package setup and continuous integration, as well as building of conda packages.",
"We thank the GitHub (github.com) team for providing us with an excellent free development platform, ReadTheDocs (readthedocs.org) for free documentation hosting, Travis (travis-ci.org) and Appveyor (appveyor.com) for free continuous integration testing, and Slack (slack.com) for a free team communication channel."
]
] | 1709.01751 |
[
[
"Global weak solutions of the Teichm\\\"uller harmonic map flow into\n general targets"
],
[
"Abstract We analyse finite-time singularities of the Teichm\\\"uller harmonic map flow -- a natural gradient flow of the harmonic map energy -- and find a canonical way of flowing beyond them in order to construct global solutions in full generality.",
"Moreover, we prove a no-loss-of-topology result at finite time, which completes the proof that this flow decomposes an arbitrary map into a collection of branched minimal immersions connected by curves."
],
[
"Introduction",
"The Teichmüller harmonic map flow is a gradient flow of the harmonic map energy that evolves a given map $u_0:M\\rightarrow N$ from a closed oriented surface $M$ of arbitrary genus $\\gamma \\ge 0$ into a closed target manifold $N$ of arbitrary dimension, and simultaneously evolves the domain metric on $M$ within the class of constant curvature metrics.",
"It tries to evolve $u_0$ to a branched minimal immersion – a critical point of the energy functional in this situation – but in general there is no such immersion homotopic to $u_0$ , so something more complicated must occur.",
"The development of the theory so far has suggested that the flow should instead decompose $u_0$ into a collection of branched minimal immersions from lower genus surfaces.",
"This paper provides the remaining part of the jigsaw in order to prove this in full generality, by analysing the finite-time singularities that may occur, finding a canonical way of flowing beyond them, and analysing their fine structure in order to prove that no topology is lost except by the creation of additional branched minimal immersions and connecting curves.",
"The resulting global generalised solution will have at most finitely many singular times, together, possibly, with singular behaviour at infinite time that was analysed in [14], [16], [19].",
"Consider the harmonic map energy $E(u,g)=\\frac{1}{2}\\int _M\\vert du\\vert _g^2 \\,dv_g$ acting on a sufficiently regular map $u:M\\rightarrow (N,g_N)$ , and a metric $g$ in the space $\\mathcal {M}_c$ of constant (Gauss-)curvature $-1$ , 0 or 1 (depending on the genus) metrics on $M$ with fixed unit area in the case that the curvature is 0.",
"Critical points are weakly conformal harmonic maps $u:(M,g)\\rightarrow (N,g_N)$ , which are then branched minimal immersions [7] (allowing constant maps in addition).",
"The gradient flow, introduced in [14], can be written with respect to a fixed parameter $\\eta >0$ as ${\\frac{\\partial u}{\\partial t}}=\\tau _g(u);\\qquad {\\frac{\\partial g}{\\partial t}}=\\frac{\\eta ^2}{4} Re(P_g(\\Phi (u,g))),$ where $\\tau _g(u)=\\mathop {\\mathrm {tr}}\\nolimits _g (\\nabla _g du)$ denotes the tension field of $u$ , $P_g$ represents the $L^2$ -orthogonal projection from the space of quadratic differentials on $(M,g)$ onto the space ${\\mathcal {H}}(M,g)$ of holomorphic quadratic differentials, and $\\Phi (u,g)$ is the Hopf differential.",
"The flow decreases the energy $E(t):=E(u(t),g(t))$ according to $\\begin{aligned}\\frac{dE}{dt}&=-\\int _M\\left[|\\tau _g(u)|^2+\\left(\\frac{\\eta }{4}\\right)^2 |Re(P_g(\\Phi (u,g)))|^2\\right]\\\\&=-\\Vert \\partial _tu\\Vert _{L^2}^2-\\frac{1}{\\eta ^2}\\Vert \\partial _tg\\Vert _{L^2}^2\\\\&=-\\Vert \\tau _g(u)\\Vert _{L^2}^2-\\frac{\\eta ^2}{32}\\Vert P_g(\\Phi (u,g))\\Vert _{L^2}^2,\\end{aligned}$ where we use that $\\Vert P_g(\\Phi (u,g))\\Vert _{L^2}^2=2\\Vert Re(P_g(\\Phi (u,g))\\Vert _{L^2}^2$ .",
"We refer to [14] for further details.",
"When the genus $\\gamma $ of $M$ is zero, then there are no nonvanishing holomorphic quadratic differentials, so $g$ remains fixed, and we recover the harmonic map flow [5], which has been studied in detail for two-dimensional domains, cf.",
"[21], [22] and the references therein.",
"In the case that $\\gamma =1$ , this flow can be shown to be equivalent to a flow of Ding-Li-Liu [2], as pointed out in [14], and analysed in [2] and [19]."
],
[
"Construction of a global flow",
"In both cases $\\gamma =0$ and $\\gamma =1$ , one obtains global weak solutions starting with any initial map $u_0\\in H^1(M,N)$ and any initial metric $g_0\\in \\mathcal {M}_c$ [21], [2].",
"For $\\gamma \\ge 2$ it was shown in [15] that a weak solution exists on a time interval $[0,T)$ , for some $T\\in (0,\\infty ]$ , and if $T<\\infty $ then the domain must degenerate in the sense that the injectivity radius of $(M,g)$ must approach zero as $t\\uparrow T$ .",
"In all these cases the flow will be smooth away from finitely many times and as time increases to a singular time the map $u$ splits off one or more (but finitely many) nonconstant harmonic 2-spheres, which will then automatically be branched minimal spheres (see e.g.",
"[4] for this latter fact) as bubbling occurs.",
"At each such singular time $\\tau $ , the continuation of this weak solution is constructed by taking a (unique) limit $(u(\\tau ),g(\\tau ))\\in H^1(M,N)\\times \\mathcal {M}_c$ as $t\\uparrow \\tau $ and continuing the flow past the singular time by restarting with $(u(\\tau ),g(\\tau ))$ as new initial data.",
"This process gives a unique flow within the class of weak solutions with non-increasing energy.",
"It was shown in [3] and [22] that for the harmonic map flow, and in particular for the case $\\gamma =0$ above, we have no loss of energy and precise control on the bubble scales at these singular times.",
"A very similar argument establishes the same properties for all genera $\\gamma $ , and the case $\\gamma \\ge 2$ even follows directly from Proposition REF below that we need for other reasons.",
"The upshot of this singularity analysis is that the flow map before a singular time can be reconstructed from the flow map after the singular time together with the branched minimal spheres representing the bubbles.",
"Whenever a global weak solution of (REF ) exists, i.e.",
"when $T=\\infty $ for $\\gamma \\ge 2$ , and in all cases for $\\gamma =0,1$ , then it was shown in [14], [16], [19] (see also [2], [21]) that either the flow subconverges to a branched minimal immersion, or it subconverges to a collection of branched minimal immersions.",
"This collection may consist partly of bubbles, and it may include a limit branched minimal immersion parametrised over the original domain, but in general, for $\\gamma \\ge 2$ , the domain can split into a collection of lower genus closed surfaces, and the map converges to a branched minimal immersion on some or all of these lower genus surfaces.",
"The way the domain surface can split into lower genus surfaces is described by the classical Deligne-Mumford-type description of how hyperbolic surfaces can degenerate, cf.",
"[20].",
"In particular, when the domain splits, the length of the shortest closed geodesic in the domain will shrink to zero and so-called collar regions around such shrinking geodesics, described by the Collar Lemma of Keen-Randol, see e.g.",
"[20], will degenerate.",
"In all cases, if one is careful to capture all bubbles, including those disappearing down any degenerating collars, it was shown in [19] that all energy in the limit is accounted for by branched minimal immersions from closed surfaces.",
"The upshot of this asymptotic analysis is that when a global weak solution exists, for a domain of arbitrary genus, the map $u(t)$ can be reconstructed from the branched minimal immersions we find, connected together with curves.",
"(See [19] for precise statements.)",
"The theory above leaves open the possibility of the flow stopping in finite time in the case $\\gamma \\ge 2$ if it happens that the injectivity radius of the domain converges to zero, i.e.",
"we have collar degeneration as above but in finite time.",
"We showed in [18] that the flow exists and is smooth for all time in the case that the target $(N,g_N)$ has nonpositive sectional curvature, mirroring the seminal work of Eells-Sampson [5] (although the asymptotic behaviour is more elaborate in our situation, with infinite time singularities reflecting the more complicated structure of the space of critical points).",
"However, in the case of general targets, the theory above has the major omission that the existence time $T$ for $\\gamma \\ge 2$ could be finite, and by such time we cannot expect the flow to have decomposed $u(t)$ into branched minimal immersions.",
"In this paper we show how the flow can be continued in a canonical fashion when this domain degeneration occurs, with the continuation being a finite collection of new flows.",
"By repeating this process a finite number of times, we arrive at a global solution that is smooth except at finitely many singular times.",
"Moreover, our analysis of the collar degeneration singularity allows us to account for all `lost topology' at the singular time in terms of branched minimal spheres, some of which may be conventional bubbles, together with connecting curves, despite the tension field diverging to infinity in general.",
"Combined with the earlier work described above, a consequence is that the flow realises the following: Any smooth map $u_0:M\\rightarrow (N,g_N)$ is decomposed by the flow (REF ) into a finite collection of branched minimal immersions $v_i:\\Sigma _i\\rightarrow (N,g_N)$ from closed Riemann surfaces $\\lbrace \\Sigma _i\\rbrace $ of total genus no more than $\\gamma $ .",
"The original $M$ can be reconstructed from the surfaces $\\lbrace \\Sigma _i\\rbrace $ by removing a finite collection of pairs of tiny discs in $\\coprod _i \\Sigma _i$ and gluing in cylinders.",
"The map $u_0$ is homotopic to the corresponding combination of the $\\lbrace v_i\\rbrace $ together with connecting curves on the glued-in cylinders.",
"For other situations in which maps are decomposed into collections of minimal objects, see [11] and [9], for example.",
"In order to make a continuation of the flow, we require the following basic description of the convergence of the flow as we approach a finite-time singularity.",
"This can be applied to a weak solution (including bubbling) by restricting to a short time interval just prior to a time when the injectivity radius drops to zero, thus avoiding the bubbling and allowing us to consider a smooth flow for simplicity.",
"A far more refined description will be required later in order to ensure that the continuation after the singularity properly reflects the flow just before.",
"Theorem 1.1 Let $M$ be any closed oriented surface of genus $\\gamma \\ge 2$ and let $(N,g_N)$ be any smooth closed Riemannian manifold.",
"Let $(u,g)$ be a smooth solution of (REF ) defined on a time interval $[0,T)$ with $T<\\infty $ that is maximal in the sense that $ \\liminf _{t\\uparrow T} \\operatorname{inj}_{g(t)}(M)=0.$ Then the following properties hold: The `pinching set' $F\\subset M$ defined by $F:=\\lbrace p\\in M:\\, \\liminf _{t\\uparrow T}\\operatorname{inj}_{g(t)}(p)=0\\rbrace $ is nonempty and closed, and its complement $\\mathcal {U}:=M\\setminus F$ is nonempty and supports a complete hyperbolic metric $h$ with finite volume and cusp ends, so that $(\\mathcal {U}, h)$ is conformally equivalent to a finite disjoint union of closed Riemann surfaces $M_i$ with finitely many punctures and genus strictly less than that of $M$ , and so that $g(t)\\rightarrow h \\text{ smoothly locally on } \\mathcal {U}\\text{ as }t\\uparrow T.$ The `bubble' set $S:=\\lbrace p\\in \\mathcal {U}\\ :\\ \\exists \\varepsilon >0 \\text{ s.t.",
"}\\limsup _{t\\uparrow T} E(u(t),g(t),V)\\ge \\varepsilon \\text{ for all neighbourhoods } V\\subset M \\text{ of } p\\rbrace $ is a finite set, and there exists a smooth map $\\bar{u}:\\mathcal {U}\\setminus S\\rightarrow N$ , with $\\bar{u}\\in H^1(\\mathcal {U},h,N)$ , such that $u(t)\\rightarrow \\bar{u} \\text{ as }t\\uparrow T$ smoothly locally in $\\mathcal {U}\\setminus S$ and weakly locally in $H^1$ on $\\mathcal {U}$ .",
"Moreover, $\\bar{u}$ extends to a collection of maps $u_i\\in H^1(M_i,N)$ .",
"The convergence of the metric $g(t)$ here should be contrasted with the convergence of a sequence $g(t_n)$ , with $t_n\\uparrow T$ , that could be deduced from the differential geometric form of Deligne-Mumford compactness (see e.g.",
"[20]).",
"Our convergence is uniform in time, and does not require modification by diffeomorphisms.",
"This theorem already tells us enough to be able to define the continuation of the flow beyond time $T$ .",
"We simply take each closed Riemann surface $M_i$ , equip it with a conformal metric $g_i$ in the corresponding space $\\mathcal {M}_c$ of metrics of constant curvature, and restart the flow on each $M_i$ separately with $u_i$ as the initial map.",
"The choice of $g_i$ is uniquely determined when the genus of $M_i$ is at least one, but on the sphere it is initially defined only up to pull-back by Möbius maps.",
"In this case, we must find a way of making a canonical choice of $g_i$ in order to obtain a canonical choice of continuation.",
"We do this by returning to the limit metric $h$ , which induces a smooth conformal complete hyperbolic metric of finite area on the sphere with punctures, and choose the metric $g_i$ to be the limit $g_\\infty $ of the rescaled Ricci flow on the sphere that starts with the metric $h$ , as given by the following theorem which follows immediately from a combination of [24] (see also the simplifications arising from [25]) and [8], [1] (see also [6]).",
"Note that by Gauss-Bonnet, the volume of the metric $h$ must be $2\\pi (n-2)$ , where $n$ is the number of punctures.",
"Theorem 1.2 Suppose $\\lbrace p_1,\\ldots ,p_n\\rbrace \\subset S^2$ is a finite set of points and $h$ is a complete conformal hyperbolic metric on $S^2\\setminus \\lbrace p_1,\\ldots ,p_n\\rbrace $ .",
"Then there exists a unique smooth Ricci flow $g(t)$ on $S^2$ , $t\\in (0,T)$ , $T=\\frac{n-2}{4}$ , i.e.",
"a smooth complete solution of ${\\frac{\\partial g}{\\partial t}}=-2Kg$ with curvature uniformly bounded below and such that $g(t)\\rightarrow h$ smoothly locally on $S^2\\setminus \\lbrace p_1,\\ldots ,p_n\\rbrace $ as $t\\downarrow 0$ .",
"(Here $K$ is the Gauss curvature.)",
"Moreover, there exists a smooth conformal metric $g_\\infty $ on $S^2$ of constant Gauss curvature 1 such that $\\frac{g(t)}{2(T-t)}\\rightarrow g_\\infty $ smoothly as $t\\uparrow T$ .",
"Theorem REF , with the aid of Theorem REF , thus establishes that our flow can be continued canonically beyond the singular time $T$ as a finite collection of flows.",
"The construction does not require us to stop prior to the singular time $T$ and perform surgery.",
"Instead, we flow right to the singular time, and the surgery we do consists of nothing more than adding points to fill in punctures in the domain (the analogue of adding an arbitrary cap in a traditional surgery argument)."
],
[
"No loss of information at finite-time collar degenerations",
"At this stage we have however not yet established a very strong connection between the flow prior to a collar degeneration singularity and the flows after the singularity.",
"We need to relate the topology of $M$ to the topology of the surfaces $M_i$ , and to relate the topology of the map $u(t)$ prior to the singularity to the flow maps afterwards, and most of this paper will be devoted to achieving this.",
"The former issue is dealt with by the following Proposition 1.3 In the setting of Theorem REF , the injectivity radius converges uniformly to a continuous limit: $\\operatorname{inj}_{g(t)}(x)\\rightarrow \\left\\lbrace \\begin{aligned}& \\operatorname{inj}_h(x) &\\quad & \\text{for }x\\in \\mathcal {U}\\\\& 0 & \\quad & \\text{for }x\\in F=M\\setminus \\mathcal {U},\\end{aligned}\\right.$ as $t\\uparrow T$ .",
"Moreover, the set $F$ from (REF ) consists of $k\\in \\lbrace 1,\\ldots ,3(\\gamma -1)\\rbrace $ components $\\lbrace F_j\\rbrace $ , and the total number of punctures in Theorem REF is $2k$ .",
"Furthermore, there exist $\\delta _0\\in (0,\\mathop {\\mathrm {arsinh}}\\nolimits (1))$ and $t_0\\in [0,T)$ such that for every $t\\in [t_0,T)$ there are exactly $k$ simple closed geodesics $\\sigma _j(t)\\subset (M,g(t))$ with length $\\ell _j(t)=L_{g(t)}(\\sigma _j(t))< 2\\delta _0$ and the lengths of these geodesics decay according to $\\ell _j(t)\\le C (T-t)(E(t)-E(T))\\rightarrow 0,\\text{ as }t\\uparrow T,$ for some $C=C(\\eta ,\\gamma )$ .",
"In addition, for every $\\delta \\in (0,\\delta _0]$ and $t\\in [t_0,T)$ the set $\\delta \\text{-thin}(M,g(t))$ consists of the union of the (possibly empty) disjoint cylindrical `subcollar' regions $\\mathcal {C}_j=\\mathcal {C}_j(t,\\delta )$ around $\\sigma _j(t)$ which are isometric to $(-X_j, X_j) \\times S^1 \\text{equipped with the metric }\\rho ^2_{j}(s)(ds^2+d\\theta ^2)$ where $X_j=X_j(t,\\delta )= \\frac{2\\pi }{\\ell _j(t)}\\arccos \\left(\\frac{\\sinh (\\frac{\\ell _j(t)}{2})}{\\sinh \\delta }\\right), \\text{ if } 2\\delta \\ge \\ell _j(t), \\text{ while } X_j=0 \\text{ if } 2\\delta <\\ell _j(t)$ and $\\rho _{j}(s)=\\rho _{\\ell _j(t)}(s)=\\frac{\\ell _j(t)}{2\\pi \\cos (\\frac{\\ell _j(t) s}{2\\pi })},$ and for all $t$ sufficiently large (depending in particular on $\\delta $ ) we have $F_j\\subset \\mathcal {C}_j(t,\\delta )$ .",
"The subcollars $\\mathcal {C}_j$ are subsets of collar neighbourhoods of the collapsing simple closed geodesics described by the Collar lemma (see e.g.",
"[20]).",
"If $\\delta (t)\\downarrow 0$ sufficiently slowly so that $\\delta (t)^{-1}(T-t)(E(t)-E(T))\\rightarrow 0$ as $t\\uparrow T$ , then $X_j(t,\\delta (t))\\rightarrow \\infty $ as $t\\uparrow T$ .",
"This proposition gives us a topological description of how $M$ can be reconstructed from the $M_i$ .",
"We remove $2k$ small discs from the $M_i$ at the punctures described in Theorem REF , and glue in cylinders corresponding to the $k$ collar regions from the proposition.",
"(We see that there will be $2k$ punctures.)",
"The proposition also demonstrates what we must establish in order to relate the flow map before the singularity to the flow maps after the singularity.",
"The continuation of the flow is given in terms of the smooth local limit $\\bar{u}$ on $\\mathcal {U}\\setminus S$ from Theorem REF .",
"Therefore we can potentially lose parts of the map at the points $S$ and parts of the map `at infinity' in $\\mathcal {U}$ .",
"As we shall see in part REF of Theorem REF , the loss of energy at points in $S$ is entirely accounted for in terms of bubbles, i.e.",
"maps $\\omega _i:S^2\\rightarrow N$ that are harmonic and non-constant and are thus themselves branched minimal spheres.",
"On the other hand, we have to be concerned about parts of the map that are lost at infinity in $\\mathcal {U}$ .",
"By Proposition REF , we must specifically be concerned with the restriction of the flow map $u(t)$ to the collar regions $\\mathcal {C}_j$ .",
"If we view these collar regions conformally as the cylinder (REF ) with the flat metric $g_0=ds^2+d\\theta ^2$ , then any fixed length portion of an end of these cylinders will have injectivity radius $\\operatorname{inj}_{g(t)}$ bounded below by a positive number, uniformly as $t\\uparrow T$ , and thus by Proposition REF , it will remain in a compact subset of $\\mathcal {U}$ and the map there will be captured in the limit $\\bar{u}$ .",
"However, this says nothing about what happens away from the ends of the cylinders, and we have to be concerned because the map there need not become harmonic since the tension field is a priori unbounded in $L^2$ .",
"Nevertheless, part REF of Theorem REF will show that near enough the centre of these cylinders – essentially on the $[T-t]^\\frac{1}{2}\\text{-thin}$ part of $(M,g(t))$ – we will be able to describe the map as a collection of bubbles connected together by curves.",
"This leaves the worry that a little outside this thin region (for example where the injectivity radius is of the order of $[T-t]^{\\frac{1}{2}-\\varepsilon }$ ) we might accumulate `unstructured' energy that is lost down the collars in the limit, and is not representing any branched minimal immersion or curve, but instead represents some arbitrary map.",
"Again, this is ruled out in the following Theorem REF , part REF , where we show that all lost energy lives not just on the $[T-t]^{\\frac{1}{2}}\\text{-thin}$ part but even on the $[T-t]\\text{-thin}$ part.",
"Theorem 1.4 In the setting of Theorem REF , we can extract a finite collection of branched minimal spheres at the singular time in order to obtain no loss of energy/topology in the following sense.",
"There exists a sequence $t_n\\uparrow T$ such that $\\left[ \\Vert \\tau _{g}(u)(t_n)\\Vert _{L^2(M,g(t_n))}+\\Vert P_{g}(\\Phi (u,g))(t_n)\\Vert _{L^2(M,g(t_n))}\\right] \\cdot (T-t_n)^\\frac{1}{2}\\rightarrow 0,$ and so that At each $x\\in S$ , finitely many bubbles (i.e.",
"nonconstant harmonic maps $S^2\\rightarrow (N,g_N)$ ) develop as $t_n\\uparrow T$ .",
"All of these bubbles develop at scales of order $o\\left((T-t_n)^\\frac{1}{2}\\right)$ and they account for all of the energy that is lost near $x\\in S$ , as is made precise in part REF of Proposition REF .",
"In particular, if $\\omega _1,\\ldots ,\\omega _{m}$ is the complete list of bubbles developing at points in $S$ then $\\begin{aligned}E_{thick}:=\\lim _{\\delta \\downarrow 0}\\lim _{t\\uparrow T} E(u(t),g(t),\\delta \\text{-thick}(\\mathcal {U},h))&= E(\\bar{u}, h, \\mathcal {U}) + \\sum _{l=1}^{m} E(\\omega _l)\\\\&=\\sum _{i} E(u_i, M_i)+ \\sum _{l=1}^{m} E(\\omega _l).\\end{aligned}$ All the energy $E_{thin}:=E(T)-E_{thick},$ $ E(T):=\\lim _{t\\uparrow T}E(t)$ , lost down the collars concentrates on the $[T-t]\\text{-thin}$ part in the sense that $E_{thin}=\\lim _{t\\uparrow T} E(u(t),g(t),[T-t]\\text{-thin}(M ,g(t))).$ In fact, we have the more refined information that $E_{thin}=\\lim _{K\\rightarrow \\infty }\\liminf _{t\\uparrow T} E\\big (u(t),g(t),[K(T-t)(E(t)-E(T))]\\text{-thin}(M ,g(t))\\big ).$ The restriction of the maps $u(t_n)$ to the $(T-t_n)^{\\frac{1}{2}}\\text{-thin}$ part of the degenerating subcollars $\\mathcal {C}_j$ from Proposition REF has tension $\\Vert \\tau _{g_0}(u(t_n))\\Vert _{L^2}\\rightarrow 0$ as $n\\rightarrow \\infty $ with respect to $g_0=ds^2+d\\theta ^2$ and hence can be assumed to converge to a full bubble branch as explained in Proposition REF below.",
"In the following proposition from [19], we write $a_n\\ll b_n$ , for sequences $a_n$ and $b_n$ , if $a_n< b_n$ for each $n$ and $b_n-a_n\\rightarrow \\infty $ as $n\\rightarrow \\infty $ .",
"Proposition 1.5 (Contents of Theorem 1.9 and Definition 1.10 of [19]) For any sequence of maps $u_n:[-\\hat{X}_n, \\hat{X}_n]\\times S^1\\rightarrow N$ , $\\hat{X}_n\\rightarrow \\infty $ , for which the tension with respect to the flat metric $g_0=ds^2+d\\theta ^2$ satisfies $\\Vert \\tau _{g_0}(u_n)\\Vert _{L^2}\\rightarrow 0$ there exists a subsequence that converges to a full bubble branch in the following sense: There exist a finite number of sequences $s_n^m$ (for $m\\in \\lbrace 0,\\ldots ,\\bar{m}\\rbrace $ , $\\bar{m}\\in {\\mathbb {N}}$ ) with $-\\hat{X}_n=: s_n^0\\ll s_n^1\\ll \\cdots \\ll s_n^{\\bar{m}}:= \\hat{X}_n$ such that The connecting cylinders $(s_n^{m-1}+\\lambda ,s_n^m-\\lambda )\\times S^1$ , $\\lambda $ large, are mapped near curves in the sense that $ \\lim _{\\lambda \\rightarrow \\infty }\\limsup _{n\\rightarrow \\infty }\\sup _{s\\in (s_n^{m-1}+\\lambda ,s_n^m-\\lambda )}\\mathop {{\\mathrm {osc}}}\\limits (u_n;\\lbrace s\\rbrace \\times S^1)=0,$ for each $m\\in \\lbrace 1,\\ldots ,\\bar{m}\\rbrace $ .",
"For each $m\\in \\lbrace 1,\\ldots ,\\bar{m}-1\\rbrace $ (if nonempty) the translated maps $u_n^m(s,\\theta ):=u_n(s+s_n^m,\\theta )$ converge weakly in $H^1$ locally on $(-\\infty ,\\infty )\\times S^1$ to a harmonic map $\\omega ^m$ and strongly in $H_{loc}^1((-\\infty ,\\infty )\\times S^1)$ away from a finite number of points at which bubbles can be extracted in a way that each bubble is counted no more than once, and so that in this convergence of $u_n^m$ to a bubble branch there is no-loss-of-energy on compact subsets of $(-\\infty ,\\infty )\\times S^1$ .",
"Since $(-\\infty ,\\infty )\\times S^1$ is conformally equivalent to the sphere with two points removed, $\\omega ^m$ extends to a harmonic map from $S^2$ .",
"This map can then be considered as a bubble (in particular a branched minimal immersion) if it is nonconstant.",
"If it is constant, then there must be a nonzero number of bubbles developing within.",
"See Theorem 1.5 of [19] for details.",
"Remark 1.6 Proposition REF will give a more general version of part REF of Theorem REF , establishing the no-loss-of-energy property and control on the bubble scales also at finite-time singularities as considered in [15] at which the metrics do not degenerate.",
"As mentioned earlier, the analogue of this result when the underlying surface is $M=S^2$ can already be found in [22] since (REF ) is then just the harmonic map flow.",
"That theorem also elaborates on the sense in which the finite collection of bubbles develop, and the strategy of its proof broadly carries over to our situation here.",
"The key point of Theorem REF is that the degenerating collars, and indeed the whole surface, can be divided up into two regions: First, the cylinders making up $[T-t]\\text{-thin}(M,g(t))$ (and even those making up $[T-t]^{1/2}\\text{-thin}$ ) are sufficiently collapsed that when we rescale, the evolving map $u$ can be seen to have very small tension and can thus be represented in terms of branched minimal spheres.",
"Second, on the remaining $[T-t]\\text{-thick}$ part, the limiting energy is fully accounted for by the energy of the limits $u_i$ and the energy of the bubbles.",
"This latter assertion is not a priori so clear since one might have a part of the flow map drifting down the collar, always living in a region such as where the injectivity radius is of the order of e.g.",
"$[T-t_n]^{1/3}$ .",
"Such a part of the map would have no reason to look harmonic in any way, and might carry some nontrivial topology.",
"This `unstructured' energy could in principle drift down the collar not because energy was flowing around the domain, but because the injectivity radius itself is evolving.",
"The key to ruling out this latter bad behaviour is the following theorem, which gives a more precise description of the convergence of the metric than the one given in Theorem REF and which asserts essentially that by time $t\\in [0,T)$ , the metric $g(t)$ has settled down to its limit $h$ on the $[T-t]\\text{-thick}$ part.",
"As we shall see, this represents an instance of a more general theory from [20] describing the convergence of a general `horizontal curve' of hyperbolic metrics.",
"Theorem 1.7 In the setting of Theorem REF , there exists $\\bar{K}<\\infty $ depending on $\\eta $ and the genus of $M$ (and determined in Lemma REF ) such that the following holds true: The `pinching set' $F\\subset M$ defined in (REF ) can be characterised as $F =\\bigcap _{t<T} \\lbrace p\\in M: \\operatorname{inj}_{g(t)}(p)< \\delta _K(t)\\rbrace ,$ for any $K\\ge \\bar{K}$ , where $\\delta _K(t):=K(T-t)\\left(E(t)-E(T)\\right)\\downarrow 0$ as $t\\uparrow T$ and $E(T):=\\lim _{t\\uparrow T} E(t)$ .",
"Equivalently, we have $\\mathcal {U}:=M\\setminus F=\\bigcup _{t<T}\\, [\\delta _K(t)]\\text{-thick}(M,g(t)).$ In addition to the claims on $\\mathcal {U}$ , $h$ and the convergence $g(t)\\rightarrow h$ made in Theorem REF , for any $K\\ge \\bar{K}$ , $t_0\\in [0,T)$ and $t\\in [t_0,T)$ , we have that for every $l\\in {\\mathbb {N}}$ $\\Vert g(t)-h\\Vert _{C^l([\\delta _K(t_0)]\\text{-thick}(M,g(t_0)),g(t_0))}+\\Vert g(t)-h\\Vert _{C^l([\\delta _{K}(t_0)]\\text{-thick}(\\mathcal {U},h),h)}\\le CK^{-\\frac{1}{2}}\\left[\\frac{\\delta (t)}{\\delta (t_0)}\\right]^\\frac{1}{2},$ where we abbreviate $\\delta (t)=\\delta _1(t)$ and where $C$ depends only on $l$ , the genus of $M$ and $\\eta $ .",
"Furthermore, for $K>0$ sufficiently large (depending on $\\eta $ , the genus of $M$ and an upper bound $E_0$ for the initial energy) and for all $t_0\\in [0,T)$ – or for arbitrary $K>0$ and $t_0\\in [0,T)$ sufficiently large – we have $\\begin{aligned}\\sup _{t\\in [t_0,T)}\\Vert g(t)-h\\Vert _{C^l([K(T-t_0)]\\text{-thick}(M,g(t_0)),g(t_0))}\\qquad \\\\\\qquad +\\sup _{t\\in [t_0,T)}\\Vert g(t)-h\\Vert _{C^l([K(T-t_0)]\\text{-thick}(\\mathcal {U},h),h)}&\\le C\\frac{(E(t_0)-E(T))^\\frac{1}{2}}{K^\\frac{1}{2}}\\rightarrow 0,\\end{aligned}$ as $t_0\\uparrow T$ , where $C$ depends on $l$ , the genus of $M$ and $\\eta $ .",
"Remark 1.8 Although we do not require it here, one should be able to improve the smooth local convergence $u(t)\\rightarrow \\bar{u}$ of Theorem REF to quantitative control on the size of $u(t)-\\bar{u}$ over, say, the $[T-t]^\\frac{1}{2}\\text{-thick}$ part of $(M,g(t))$ , away from $S$ , with respect to an appropriate weighted norm.",
"In summary we obtain that the flow (REF ) decomposes any smooth map $u_0:M\\rightarrow N$ into a collection of branched minimal immersions $v_i:\\Sigma _i\\rightarrow N$ through global solutions that are smooth away from finitely many times as follows: As discussed earlier, at each singular time $t_m$ for which $\\operatorname{inj}_{g(t)}(M)\\nrightarrow 0$ as $t\\uparrow t_m$ , all of the lost energy is accounted for in terms of bubbles $\\omega _m^j:S^2\\rightarrow (N,g_N)$ , which we add to the collection of minimal immersions $v_i$ (adding that same number of copies of $S^2$ to the collection of domains $\\Sigma _i$ ).",
"At singular times for which $\\operatorname{inj}_{g(t)}(M)\\rightarrow 0$ the results discussed above apply and we add both the bubbles developing at the singular points $S\\subset \\mathcal {U}$ and those that are disappearing down one of the degenerating collars to the set of minimal immersions $v_i$ (again adding the corresponding number of $S^2$ 's to the collection of the $\\Sigma _i$ 's) and continue the flow on the closed lower genus surfaces $M_i$ as described above.",
"If the genus of any of the closed surfaces $M_i$ is 0 or 1, then its continuation will be a weak solution that flows forever afterwards according to the theory of the harmonic map flow [21] or the theory in [2].",
"If the genus of any of the surfaces $M_i$ is larger than 1, then the subsequent flow might develop a further finite-time singularity at which a collar degenerates, in which case we repeat the process above to continue the flow further still.",
"Each time we restart the flow after a singularity caused by the degeneration of one or more collars, the genus of the surfaces underlying the continued flows will decrease, so repeating the process finitely many times gives us a global weak solution as desired.",
"As the energy is conformally invariant, the resulting global solution has non-increasing energy and the total number of singular times $t_m$ is a priori bounded in terms of the genus, the initial energy and the target $(N,g_N)$ .",
"We can relate the domain(s) and map(s) before a singular time $t_m$ to the flow(s) after the singular time as explained above and can thus reconstruct the initial map and the initial domain in terms of the map(s) and domain(s) at any time $t\\in (t_m,t_{m+1})$ and the collection of all of the bubbles $v_i$ obtained at the singular times $t_1<..<t_m$ as well as connecting curves on cylinders.",
"We can then apply the asymptotic analysis as discussed above (principally from [19]) to each of the obtained global flows, eventually adding also the bubbles developing at infinite time as well as the limiting maps $u_j^\\infty :M^\\infty _j \\rightarrow N$ obtained at infinite time, which are branched minimal immersions defined on surfaces of total genus no more than $\\gamma $ , to the collection of the $(\\Sigma _i,v_i)$ .",
"This gives the decomposition of the initial map into branched minimal immersions $v_i:\\Sigma _i\\rightarrow N$ described earlier on.",
"This paper is organised as follows.",
"In Section we carry out the analysis of the metric component of the flow, proving part REF of Theorem REF as well as Theorem REF and Proposition REF .",
"The resulting control on the evolution of the metric then allows us to analyse the map component in the subsequent Section .",
"In Section REF we focus on the properties of the map on the non-degenerate part of the surface, stating and proving Proposition REF , which yields both part REF of Theorem REF as well as part REF of Theorem REF .",
"Parts REF and REF of Theorem REF are then proven in Section REF where we analyse the map component on the degenerating part of the surface.",
"Acknowledgements: The second author was supported by EPSRC grant number EP/K00865X/1."
],
[
"Analysis of the metric component",
"In this section we analyse the metric component of the flow, proving first part REF of Theorem REF , then Theorem REF , and finally Proposition REF .",
"This analysis is based on the theory of general horizontal curves we developed in [20], some of which we recall here.",
"A horizontal curve of metrics on a smooth closed oriented surface $M$ of genus at least 2 is a smooth one-parameter family $g(t)$ of hyperbolic metrics on $M$ for $t$ within some interval $I\\subset {\\mathbb {R}}$ so that for each $t\\in I$ , there exists a holomorphic quadratic differential $\\Psi (t)$ such that ${\\frac{\\partial g}{\\partial t}}=Re(\\Psi )$ .",
"This makes $g(t)$ move orthogonally to modifications by diffeomorphisms, as described in [20].",
"An important property of horizontal curves is that we can bound the $C^l$ norm of their velocity, $l\\in {\\mathbb {N}}$ , in terms of a much weaker norm of $\\partial _tg$ and the injectivity radius.",
"In fact, [20] gives that for any $x\\in M$ and $l\\in {\\mathbb {N}}$ $|\\partial _t{g}(t)|_{C^l(g(t))}(x)\\le C[\\operatorname{inj}_{g(t)}(x)]^{-\\frac{1}{2}}\\Vert \\partial _tg (t)\\Vert _{L^2(M,g(t))},$ $C$ depending only on the genus of $M$ and $l$ , where $|\\Omega |_{C^l(g)}(x):=\\sum _{k=0}^l |\\nabla _{g}^{k} \\Omega |_g(x)$ , with $\\nabla _g$ the Levi-Civita connection, or its extension.",
"We furthermore recall that for every point $x\\in M$ the map $t\\mapsto \\operatorname{inj}_{g(t)}(x)$ is locally Lipschitz on the interval $I$ over which $g$ is defined (cf.",
"[20]) and that $\\bigg |\\frac{d}{dt}\\left[\\operatorname{inj}_{g(t)}(x)\\right]^{\\frac{1}{2}}\\bigg |\\le K_0\\Vert \\partial _t g(t)\\Vert _{L^2(M,g(t))}$ holds true for a constant $K_0<\\infty $ that depends only on the genus of $M$ , see [20].",
"These estimates play an important role in the proof of the following convergence result for finite length horizontal curves, proven in [20], which we will use to analyse the metric component of the flow.",
"In order to state this result, we introduce some more notation: If $g(t)$ is defined for $t$ in some interval $[0,T)$ , then we let $\\mathcal {L}(s):=\\int _s^T\\Vert \\partial _tg(t)\\Vert _{L^2(M,g(t))}dt\\in [0,\\infty ]$ denote the length of the restriction of $g$ to the interval $[s,T)$ .",
"Given a tensor $\\Omega $ defined in a neighbourhood of some $K\\subset M$ , we write $\\Vert \\Omega \\Vert _{C^l(K,g)}:= \\sup _K |\\Omega |_{C^l(g)}.$ Theorem 2.1 ([20]) Let $M$ be a closed oriented surface of genus $\\gamma \\ge 2$ , and suppose $g(t)$ is a smooth horizontal curve in $\\mathcal {M}_{-1}$ , for $t\\in [0,T)$ , with finite length $\\mathcal {L}(0)<\\infty $ .",
"Then there exist a nonempty open subset $\\mathcal {U}\\subset M$ , whose complement has $k\\in \\lbrace 0,\\ldots ,3(\\gamma -1)\\rbrace $ components, and a complete hyperbolic metric $h$ on $\\mathcal {U}$ for which $(\\mathcal {U},h)$ is of finite volume and is conformally a finite disjoint union of closed Riemann surfaces (of genus strictly less than that of $M$ if $\\mathcal {U}$ is not the whole of M) with $2k$ punctures, such that $g(t)\\rightarrow h$ smoothly locally on $\\mathcal {U}$ .",
"Moreover, defining $\\mathcal {I}:M\\rightarrow [0,\\infty )$ by $\\mathcal {I}(x)=\\left\\lbrace \\begin{aligned}& \\operatorname{inj}_h(x) &\\quad & \\text{on }\\mathcal {U}\\\\& 0 & \\quad & \\text{on }F=M\\setminus \\mathcal {U},\\end{aligned}\\right.$ we have $\\operatorname{inj}_{g(t)}\\rightarrow \\mathcal {I}$ uniformly on $M$ as $t\\uparrow T$ , and indeed that $\\left\\Vert [\\operatorname{inj}_{g(t)}]^\\frac{1}{2}-\\mathcal {I}^\\frac{1}{2}\\right\\Vert _{C^0}\\le K_0\\mathcal {L}(t)\\rightarrow 0\\qquad \\text{as }t\\uparrow T,$ where $K_0$ is chosen as in (REF ) and depends only on $\\gamma $ .",
"Furthermore, for any $l\\in {\\mathbb {N}}$ and $\\delta >0$ , if we take $t_0\\in [0,T)$ sufficiently large so that $(2K_0 \\mathcal {L}(t_0))^2< \\delta , \\qquad K_0 \\text{ the constant obtained in {(\\ref {est:inj-weaker-copy2})} }$ then $\\delta \\text{-thick}(M,g(s))\\subset \\mathcal {U}$ for every $s\\in [t_0,T)$ , and we have for every $t\\in [t_0,T)$ $\\Vert g(t)-h\\Vert _{C^l(\\delta \\text{-thick}(\\mathcal {U},h),h)}+\\Vert g(t)-h\\Vert _{C^l(\\delta \\text{-thick}(M,g(s)),g(s))}\\le C\\delta ^{-\\frac{1}{2}}\\mathcal {L}(t),$ where $C$ depends only on $l$ and $\\gamma $ .",
"We first apply this result to prove the properties of the metric component claimed in our basic convergence result, i.e.",
"part REF of Theorem REF To this end we first note that for any smooth solution $(u,g)$ of (REF ) defined on $[0,T)$ , $T<\\infty $ , the metric component is by definition a smooth horizontal curve.",
"Furthermore, its length is finite as $\\begin{aligned}\\mathcal {L}(t)^2 & =\\left(\\int _t^T\\Vert \\partial _tg(t)\\Vert _{L^2(M,g(t))}dt\\right)^2\\le (T-t) \\int _t^T\\Vert \\partial _tg(t)\\Vert _{L^2(M,g(t))}^2dt\\\\&\\le \\eta ^2(T-t) \\left(E(t)-E(T)\\right),\\end{aligned}$ by (REF ), where we abbreviate $E(t):=E(u(t),g(t))$ and $E(T):=\\lim _{s\\uparrow T}E(s)$ .",
"In particular, defining $\\bar{K}:= 5K_0^2 \\eta ^2,$ to depend only on $\\eta $ and $\\gamma $ , and defining $\\delta _K(t):=K (T-t)(E(t)-E(T)),$ which we will be considering for $K\\ge \\bar{K}$ and $t\\in [0,T)$ , we have $[K_0\\mathcal {L}(t)]^2 \\le \\frac{1}{5} \\delta _{\\bar{K}}(t)$ for all $t\\in [0,T)$ .",
"We may thus analyse the metric component $g$ of any solution of (REF ) with the above Theorem REF .",
"In the setting of Theorem REF , the assumption (REF ) that the metric component degenerates as $t$ approaches $T$ combined with the uniform convergence of the injectivity radius furthermore guarantees that the pinching set $F$ must be non-empty.",
"Part REF of Theorem REF concerning the local convergence of $g(t)$ to a limit $h$ and the properties of $h$ , $\\mathcal {U}$ and $F$ is thus a direct consequence of Theorem REF and the fact that $\\mathcal {L}(0)<\\infty $ .",
"To prove the refined properties of the metric component stated in Theorem REF and Proposition REF we shall use the following lemma, where $\\bar{K}$ will be chosen as in (REF ) above.",
"Lemma 2.2 Let $(u,g)$ be a smooth solution of (REF ) on $[0,T)$ , $T<\\infty $ , on a surface $M$ of genus $\\gamma \\ge 2$ .",
"Then there exists a constant $\\bar{K}$ depending only on $\\eta $ and $\\gamma $ so that the following holds true.",
"If we define $\\delta _K(t)$ as in (REF ) then for every $t_0\\in [0,T)$ the assumption (REF ) of Theorem REF is satisfied for $t_0$ and any $\\delta >0$ with $\\delta \\ge \\delta _{\\bar{K}}(t_0)$ and thus estimate (REF ) holds true for any $t_0\\in [0,T)$ , $s,t\\in [t_0,T)$ , and any $\\delta >0$ with $\\delta \\ge \\delta _{\\bar{K}}(t_0)$ .",
"Furthermore For every $K\\ge \\bar{K}$ the pinching set $F$ defined in (REF ) can be characterised by (REF ).",
"The metrics $(g(t))_{t\\in [t_0,T)}$ are uniformly equivalent and their injectivity radii are of comparable size at points $x\\in \\delta _{\\bar{K}}(t_0)\\text{-thick}(M,g(t_0))$ in the sense that for every $s,t\\in [t_0,T)$ $g(s)(x)\\le C_1\\cdot g(t)(x) \\text{ and }C_1^{-1}\\cdot h(x)\\le g(t)(x)\\le C_1 \\cdot h(x)$ and $\\operatorname{inj}_{g(s)}(x)\\le C_2\\cdot \\operatorname{inj}_{g(t)}(x)$ where $C_1\\ge 1$ depends only on the genus of $M$ , while $C_2\\ge 1 $ is a universal constant.",
"For every $K\\ge \\bar{K}$ , every $x\\in \\delta _K(t_0)\\text{-thick}(M,g(t_0))$ , every $s,t\\in [t_0,T)$ and every $l\\in {\\mathbb {N}}$ we have $ \\vert \\partial _tg(t)\\vert _{C^l(g(s))}(x)\\le C\\delta _K(t_0)^{-1/2}\\Vert \\partial _tg(t)\\Vert _{L^2(M,g(t))}, $ where $C$ depends only on $l$ and the genus of $M$ .",
"[Proof of Lemma REF ] We first remark that the claims are trivially true if $\\delta _K(t_0)=0$ and hence $g\\vert _{[t_0,T)}$ is constant in time, so we may assume without loss of generality that $\\delta _K(t_0)>0$ .",
"Define $\\bar{K}$ as in (REF ).",
"Then (REF ) implies that $(2K_0\\mathcal {L}(t_0))^2\\le \\frac{4}{5} \\delta _{\\bar{K}}(t_0)<\\delta _{\\bar{K}}(t_0),$ and so (REF ) is satisfied for $\\delta \\ge \\delta _{\\bar{K}}(t_0)$ as claimed in the lemma.",
"To prove part REF of the lemma we combine (REF ) with (REF ) to obtain that $\\operatorname{inj}_{g(t)}(p)\\le (K_0 \\mathcal {L}(t))^2 < \\delta _{\\bar{K}}(t)$ for every $p\\in F$ and every $t\\in [0,T)$ and thus that $F \\subset \\bigcap _{t\\in [0,T)} \\delta _K(t)\\text{-thin}(M,g(t)) \\text{for any } K\\ge \\bar{K}.$ As the reverse inclusion is trivially satisfied this establishes the characterisation (REF ) of the pinching set for each $K\\ge \\bar{K}$ .",
"The proofs of parts REF and REF of the lemma are now based on estimates on the velocity and the injectivity radius that were derived in [20] for general horizontal curves under the same hypothesis that (REF ) holds true: Lemma 3.2 and Remark 3.5 of [20] establish that (REF ) and (REF ) hold true for arbitrary horizontal curves, times $s,t\\in [t_0,T)$ and points $x\\in \\delta \\text{-thick}(M,g(t_0))$ provided $t_0$ and $\\delta $ are so that (REF ) is satisfied.",
"Combined with (REF ) this immediately yields part REF of the lemma.",
"Finally, (REF ), and hence part REF of the lemma follows immediately from [20], with $\\delta $ there equal to $\\delta _{K}(t_0)$ here, because the hypotheses of that lemma are implied by (REF ).",
"Parts REF and REF of Lemma REF will be used in the next section for the fine analysis of the map component, but before that we complete the proofs of Theorem REF and Proposition REF .",
"[Proof of Theorem REF ] We let $\\bar{K}$ be the constant obtained in Lemma REF , i.e.",
"given by (REF ), and set as usual $\\delta _K(t)=K(T-t)(E(t)-E(T))$ .",
"For this choice of $\\bar{K}$ the characterisation (REF ) of the pinching set $F$ has already been proven in Lemma REF and from this lemma we furthermore know that (REF ) holds true for any $t_0$ and any $\\delta \\ge \\delta _{\\bar{K}}(t_0)$ and thus in particular for $\\delta =\\delta _K(t_0)$ , $K\\ge \\bar{K}$ .",
"Hence (REF ) follows from the corresponding estimate (REF ) of Theorem REF and the bound (REF ) on $\\mathcal {L}(t)$ .",
"It remains to prove (REF ).",
"For this we observe that for $K>0$ sufficiently large and for all $t_0\\in [0,T)$ – or for arbitrary $K>0$ and $t_0\\in [0,T)$ sufficiently large – we can be sure that $\\bar{K}(E(t_0)-E(T))\\le K$ and hence by (REF ) that (REF ) is satisfied for $t_0$ and $\\delta =K(T-t_0)$ .",
"This allows us to apply estimate (REF ) of Theorem REF also for such values of $\\delta $ which then gives that $\\begin{aligned}\\sup _{t\\in [t_0,T)}\\Vert g(t)-h\\Vert _{C^l([K(T-t_0)]\\text{-thick}(M,g(t_0)),g(t_0))}&\\le C\\frac{\\mathcal {L}(t_0)}{K^\\frac{1}{2}(T-t_0)^\\frac{1}{2}}\\\\&\\le C\\frac{(E(t_0)-E(T))^\\frac{1}{2}}{K^\\frac{1}{2}}\\rightarrow 0,\\end{aligned}$ as $t_0\\uparrow T$ , using (REF ), as well as that $\\sup _{t\\in [t_0,T)}\\Vert g(t)-h\\Vert _{C^l([K(T-t_0)]\\text{-thick}(\\mathcal {U},h),h)}\\le C\\frac{(E(t_0)-E(T))^\\frac{1}{2}}{K^\\frac{1}{2}}\\rightarrow 0,$ where $C$ depends only on $l$ , $\\eta $ and the genus of $M$ .",
"This completes the proof of Theorem REF .",
"[Proof of Proposition REF ] The uniform convergence of the injectivity radius follows from Theorem REF as $(g(t))_{t\\in [0,T)}, \\, T<\\infty $ , is a horizontal curve of finite length.",
"We furthermore recall that standard results from the theory of hyperbolic surfaces give that for any $\\delta <\\mathop {\\mathrm {arsinh}}\\nolimits (1)$ the $\\delta \\text{-thin}$ part of a hyperbolic surface is given by the union of disjoint subcollar regions around the simple closed geodesics of length $\\ell <2\\delta $ as described in the proposition and refer to the appendix of [18] as well as the references therein for further details.",
"For $K\\ge K_0$ , $K_0$ as in (REF ), we define closed sets $F_K(t):=\\lbrace p: \\operatorname{inj}_{g(t)}(p)\\le (K\\mathcal {L}(t))^2\\rbrace ,$ for $t\\in [0,T)$ .",
"It follows from the slightly stronger result [20] that the sets $F_K(t)$ are nested, becoming only smaller as $t$ increases, and that the pinching set $F$ can be written as $F=\\bigcap _{t\\in [0,T)}F_K(t).$ It is useful for us to appeal to this fact for some $K>K_0$ , and we choose $K=2K_0$ .",
"Thus for $t_0$ sufficiently large, chosen in particular so that $(2K_0\\mathcal {L}(t_0))^2<\\mathop {\\mathrm {arsinh}}\\nolimits (1)$ , the pinching set $F$ has the same number $k\\in \\lbrace 1,...,3(\\gamma -1)\\rbrace $ of connected components as the sets $F_{2K_0}(t) $ , $t\\in [t_0,T)$ , with the connected components of $F_{2K_0}(t) $ being disjoint closed subcollars around geodesics $\\sigma _j(t)$ of length $\\ell _j(t)\\le 2(2K_0\\mathcal {L}(t))^2\\le C(T-t)(E(t)-E(T))$ whose interior is as described in the proposition.",
"In particular given any $\\delta \\in (0,\\mathop {\\mathrm {arsinh}}\\nolimits (1))$ and $t\\in [t_0,T)$ sufficiently large (depending in particular on $\\delta $ ), we know that the connected components $F_j$ of the pinching set are contained in the corresponding subcollar $\\mathcal {C}_j(t,\\delta )$ as claimed in the proposition.",
"It thus remains to show that there exists a number $\\delta _0\\in (0,\\mathop {\\mathrm {arsinh}}\\nolimits (1))$ so that any simple closed geodesic in $(M,g(t))$ , $t\\in [t_0,T)$ , that does not coincide with one of the $\\sigma _j(t)$ must have length at least $2\\delta _0$ .",
"To this end we observe that the characterisation (REF ) this time with $K=K_0$ gives $\\Omega :=(2K_0\\mathcal {L}(t_0))^2\\text{-thick}(M,g(t_0))\\subset M\\setminus F_{K_0}(t_0)\\subset \\mathcal {U}$ and since $\\Omega $ is closed, it is a compact subset of $M$ and hence also of $\\mathcal {U}$ .",
"Therefore over $\\Omega $ the injectivity radius $\\operatorname{inj}_{g(t)}(\\cdot )$ , $t\\in [t_0,T)$ , is bounded uniformly from below by some constant $\\delta _0\\in (0,\\mathop {\\mathrm {arsinh}}\\nolimits (1))$ thanks to (REF ).",
"Consequently, any simple closed geodesic in $(M,g(t))$ that enters $\\Omega $ must have length at least $2\\delta _0$ .",
"The only alternative is that the simple closed geodesic in $(M,g(t))$ is fully contained in one of the $k$ cylinders $\\mathcal {C}_j(t_0,(2K_0\\mathcal {L}(t_0))^2) $ in which case it must be homotopic to $\\sigma _j(t)$ (up to change of orientation) and hence coincide with $\\sigma _j(t)$ ."
],
[
"Analysis of the map component",
"The challenges of analysing the map component are of a different nature depending on whether we consider a region where the metric has already settled down or a region in a collar that will ultimately degenerate.",
"Roughly speaking, on the non-degenerate part of the surface we control the metric but cannot hope to bound the tension while on the degenerating part of the surface the metric is not controlled but the tension tends to zero when computed with respect to the flat metric in collar coordinates along a sequence of times $t_n\\uparrow T$ as considered in Theorem REF .",
"We will analyse the map component separately on these two different regions, with the analysis on the non-degenerate part, and hence the proofs of part REF of Theorem REF and of part REF of Theorem REF , carried out in Section REF .",
"Parts REF and REF of Theorem REF , which concern the part of the map that is lost on degenerating collars, are then proven in Section REF .",
"In both of these sections we use a local energy estimate that is derived in Section REF"
],
[
"Local energy estimates",
"The goal of this section is to prove the following lemma on the evolution of cut-off energies $E_\\varphi (t):=\\frac{1}{2}\\int \\varphi ^2\\vert du(t)\\vert _{g(t)}^2 dv_{g(t)},$ for functions $\\varphi \\in C^\\infty (M,[0,1])$ .",
"Lemma 3.1 (Local energy estimate) Let $(u,g)$ be a smooth solution of (REF ) on a closed surface of genus at least 2, and for an interval $[0,T)$ , $T<\\infty $ , and let $\\varphi \\in C^\\infty (M,[0,1])$ be such that there exists $t_0\\in [0,T)$ and $K\\ge \\bar{K}$ , for $\\bar{K}$ the constant obtained in Lemma REF , so that $ \\mathop {\\mathrm {supp}}\\nolimits (\\varphi )\\subset \\delta _K(t_0)\\text{-thick}(M,g(t_0)),$ where as usual $\\delta _K(t):= K(T-t)(E(t)-E(T)).$ Then the limit $\\lim _{t\\uparrow T} E_\\varphi (t)$ exists and (assuming the flow is not constant in time on $[t_0,T)$ ) for any $t_0\\le t< s<T$ we have $\\begin{aligned}\\vert E_\\varphi (t)-E_\\varphi (s)\\vert &\\le E(t)-E(s)+C\\big [\\delta _K(t_0)^{-\\frac{1}{2}}+\\Vert d\\varphi \\Vert _{L^\\infty (M,g(t_0))}\\big ] (s-t)^{\\frac{1}{2}} (E(t)-E(s))^\\frac{1}{2}\\\\&\\le E(t)-E(T)+C\\big [\\delta _K(t_0)^{-\\frac{1}{2}}+\\Vert d\\varphi \\Vert _{L^\\infty (M,g(t_0))}\\big ] (T-t)^{\\frac{1}{2}} (E(t)-E(T))^\\frac{1}{2}\\end{aligned}$ where $C$ depends only on the coupling constant $\\eta $ , the genus of $M$ and an upper bound $E_0$ for the initial energy.",
"A first step in the proof of Lemma REF is to show the following analogue of well-known local energy estimates for harmonic map flow as found e.g.",
"in [22].",
"Lemma 3.2 Let $(u,g)$ be a (smooth) solution of (REF ) on $[0,T)$ and let $\\varphi \\in C^\\infty (M,[0,1])$ be any given function.",
"Then the evolution of the cut-off energy $E_\\varphi (t)$ defined in (REF ) is controlled by $\\begin{aligned}\\bigg |\\tfrac{d}{dt} E_\\varphi +\\int \\varphi ^2\\vert \\tau _g(u)\\vert ^2 dv_g\\bigg |&\\le 2\\sqrt{2} E(u,g)^{1/2} \\Vert d\\varphi \\Vert _{L^\\infty (M,g)}\\big (\\int \\varphi ^2\\vert \\tau _g(u)\\vert ^2 dv_g\\big )^{1/2}\\\\&\\qquad + \\Vert \\partial _tg\\Vert _{L^\\infty (\\text{supp}(\\varphi ),g)} E_\\varphi .\\end{aligned}$ The equation of the map component can be described by $\\partial _tu-\\Delta _gu=A_g(u)( du, du)=g^{ij}A(u)(\\partial _{x_i} u,\\partial _{x_j} u)\\perp T_uN$ if we view $(N,g_N)$ as a submanifold of Euclidean space using Nash's embedding theorem and denote by $A$ the second fundamental form of $N\\hookrightarrow {\\mathbb {R}}^K$ .",
"We multiply this equation with $\\varphi ^2 \\partial _tu$ and integrate over $(M,g)$ to obtain $0 =\\int \\varphi ^2\\vert \\partial _tu\\vert ^2 dv_g+\\int \\langle \\partial _td u, d u\\rangle _g \\varphi ^2 dv_g+\\partial _tu \\langle du,d(\\varphi ^2)\\rangle _g dv_g.$ We now recall that $\\partial _tg$ is given as the real part of a quadratic differential and thus has zero trace, which implies that $\\frac{d}{dt}dv_g=0$ .",
"As $\\varphi $ is independent of time while $\\partial _tu=\\tau _g(u)$ we thus obtain $\\begin{aligned}\\bigg |\\tfrac{d}{dt} E_\\varphi +\\int \\varphi ^2\\vert \\tau _g(u)\\vert ^2 dv_g\\bigg |&\\le \\tfrac{1}{2}\\tfrac{d}{d\\varepsilon }\\vert _{\\varepsilon =0}\\int \\vert du\\vert _{g(t+\\varepsilon )}^2\\varphi ^2 dv_g + 2\\int \\varphi \\vert d\\varphi \\vert \\cdot \\vert \\tau _g(u)\\vert \\cdot \\vert du\\vert dv_g\\\\&\\le \\Vert \\partial _tg\\Vert _{L^\\infty (\\mathop {\\mathrm {supp}}\\nolimits (\\varphi ),g)}E_\\varphi \\\\& \\quad + 2 \\Vert d\\varphi \\Vert _{L^\\infty (M,g)}(2E(u,g))^{1/2}\\cdot \\bigg (\\int \\varphi ^2\\vert \\tau _g(u)\\vert ^2 dv_g\\bigg )^{1/2}\\end{aligned}$ as claimed.",
"Based on this lemma as well as the control on the metric on the $\\delta _K(t_0)\\text{-thick}$ part of the domain obtained in Lemma REF , we can now prove our main energy estimate.",
"[Proof of Lemma REF ] Given $K\\ge \\bar{K}$ , with $\\bar{K}$ as in Lemma REF , we set as usual $\\delta _K(t):=K(T-t)(E(t)-E(T))$ and consider a cut-off function $\\varphi $ as in the lemma for which (REF ) is satisfied for some $t_0$ .",
"This assumption on the support of $\\varphi $ allows us to bound any $C^l$ norm of $\\partial _tg$ on $\\mathop {\\mathrm {supp}}\\nolimits (\\varphi )$ using estimate (REF ) of Lemma REF , which implies in particular that $\\Vert \\partial _t{g}(t)\\Vert _{L^\\infty (\\mathop {\\mathrm {supp}}\\nolimits \\varphi ,g(t))} \\le C[\\delta _K(t_0)]^{-\\frac{1}{2}}\\Vert \\partial _tg(t)\\Vert _{L^2(M,g(t))}, \\text{ for any } t\\in [t_0,T)$ holds true with a constant $C$ that depends only on the genus.",
"Furthermore, the equivalence (REF ) of the metrics on $\\delta _{\\bar{K}}(t_0)\\text{-thick}(M,g(t_0))$ , and thus in particular on $\\mathop {\\mathrm {supp}}\\nolimits (\\varphi )$ , obtained in the same lemma allows us to bound $\\Vert d\\varphi \\Vert _{L^\\infty (M,g(t))}\\le \\sqrt{C_1} \\Vert d\\varphi \\Vert _{L^\\infty (M,g(t_0))} \\text{ for } t\\in [t_0,T).$ The local energy estimate (REF ) of Lemma REF thus reduces to $\\begin{aligned}\\vert \\tfrac{d}{dt} E_\\varphi \\vert \\le & \\Vert \\tau _g(u)\\Vert _{L^2(M,g)}^2+C\\Vert d\\varphi \\Vert _{L^\\infty (M,g(t_0))}\\Vert \\tau _g(u)\\Vert _{L^2(M,g)}+C\\delta _K(t_0)^{-\\frac{1}{2}} \\Vert \\partial _tg\\Vert _{L^2(M,g)} \\\\\\le &\\big ( -\\tfrac{dE}{dt}\\big ) +C\\cdot \\big [\\Vert d\\varphi \\Vert _{L^\\infty (M,g(t_0))}+\\delta _K(t_0)^{-\\frac{1}{2}}\\big ] \\big (-\\tfrac{dE}{dt}\\big )^\\frac{1}{2}\\end{aligned}$ for $t\\in [t_0,T)$ , where the constant $C$ now depends not only on the genus but also on the coupling constant and an upper bound $E_0$ on $E(0)\\ge E(t)$ and where we used the evolution equation (REF ) of the total energy in the second step.",
"Integrating (REF ) over $[t,s]\\subset [t_0,T)$ yields the claim of Lemma REF .",
"Lemma REF will allow us to determine both the part of the degenerating collars where energy can be lost as well as the scale at which energy concentrates around points in the singular set $S\\subset \\mathcal {U}$ .",
"This will then allow us to capture two collections of bubbles, one developing at the bubble points $S$ , but also an additional collection that are disappearing down the collars.",
"By further analysing what can happen between these bubbles, this will allow us to prove Theorem REF .",
"This bubbling analysis will be carried out along a sequence of times $t_n$ for which (REF ) holds.",
"The existence of such $t_n\\uparrow T$ follows by a standard argument: Integrating (REF ) in time implies that $\\int _0^T \\Vert \\tau _g(u)\\Vert _{L^2}^2 dt\\qquad \\text{ and }\\qquad \\int _0^T \\Vert P_g(\\Phi )\\Vert _{L^2}^2 dt$ are bounded in terms of an upper bound $E_0$ for the initial energy, and (for the second integral) the coupling constant $\\eta $ .",
"Here we suppress the dependence of $\\Phi $ on $u$ and $g$ , of the $L^2$ measure on $g$ , and of $u$ and $g$ on $t$ .",
"These bounds imply that whenever a smooth function $f:[0,T)\\rightarrow [0,\\infty )$ has infinite integral, there exists a sequence of times $t_n\\uparrow T$ such that $\\left[\\Vert \\tau _g(u)\\Vert _{L^2}^2+\\Vert P_g(\\Phi )\\Vert _{L^2}^2\\right](t_n)<f(t_n).$ In particular, we may always choose some sequence $t_n\\uparrow T$ so that (REF ) holds true for $t_n$ (and thus also for any subsequence that we take later)."
],
[
"Analysis of the map component on the non-degenerate part of the surface",
"On compact subsets of $\\mathcal {U}$ we can control the metric component using Lemma REF and may thus think of the evolution equation (REF ) for the map component as a solution of a flow that is akin to the classical harmonic map flow albeit with a (well controlled) time dependent metric.",
"This will allow us to adapt well-known techniques from the theory of the harmonic map flow, in particular from [21] and [22], to analyse the solution on this part of the domain in detail: We prove that as $t\\uparrow T$ energy concentrates only at finitely many points $S$ away from which the maps converge in $C^l$ , for each $l\\in {\\mathbb {N}}$ , and that, along a subsequence of times $t_n\\uparrow T$ as in (REF ), we can extract a finite number of bubbles at each point in $S$ which account for all the energy that is lost near these point.",
"This last part is equivalent to proving that no energy is lost on so-called neck-regions around the bubbles (not to be confused with collar regions around the degenerating geodesics).",
"This fine analysis of the map component on the thick part of the surface applies not only in the case of a finite time degeneration as considered in the present paper but (as a by-product of the following proposition) also gives refined information at singular times as considered in [15] across which the metric remains controlled.",
"Proposition 3.3 (cf.",
"[22]) Let $(u,g)$ be any smooth solution of (REF ) for $t\\in [0,T)$ on a surface of genus at least 2.",
"Let $F$ be the (possibly empty) set given by (REF ) and let $S$ be defined as in (REF ).",
"Then $S$ is a finite set and $u(t)\\text{ converges smoothly locally on } M\\setminus (F\\cup S)$ and weakly locally in $H^1$ on $M\\setminus F$ as $t\\uparrow T$ , to a limit that we denote $u(T)$ .",
"We have no loss of energy at points in $S$ , and the scales of bubbles developing at the points of $ S$ (along a subsequence of times $t_n\\uparrow T$ as in (REF )) are small compared with $(T-t_n)^\\frac{1}{2}$ .",
"Indeed, if $\\omega _1,\\ldots ,\\omega _{m^{\\prime }}$ are the bubbles developing at $x\\in S$ then for every $\\nu >0$ $\\begin{aligned}\\lim _{r\\downarrow 0}\\lim _{t\\uparrow T}E(u(t),g(t),B_{h}(x,r))&= \\lim _{r\\downarrow 0}\\lim _{t\\uparrow T}E(u(t),g(t),B_{g(t)}(x,r))\\\\&=\\lim _{t\\uparrow T}E(u(t),g(t),B_{g(t)}(x,\\nu (T-t)^\\frac{1}{2}))=\\sum _{l=1}^{m^{\\prime }} E(\\omega _l).\\end{aligned}$ In particular, if $\\omega _1,\\ldots ,\\omega _{m^{\\prime \\prime }}$ is the complete list of bubbles developing at points in $S$ and if $\\Omega \\subset \\subset \\mathcal {U}$ is chosen large enough so that $S$ is contained in the interior of $\\Omega $ then $\\begin{aligned}\\lim _{t\\uparrow T} E(u(t),g(t),\\Omega )= E(\\bar{u}, h, \\Omega ) + \\sum _{l=1}^{m^{\\prime \\prime }} E(\\omega _l).\\end{aligned}$ In the setting of Theorem REF , i.e.",
"in case that $\\operatorname{inj}_{g(t)}(M)\\rightarrow 0$ as $t\\uparrow T$ , part REF of the proposition yields the convergence of the maps $u(t)$ on $\\mathcal {U}$ respectively on $\\mathcal {U}\\setminus S$ claimed in part REF of Theorem REF .",
"As the resulting limiting maps can be extended across the punctures to $H^1$ maps from $M_i$ (since their energy is bounded) and as the properties of the metric component claimed in part REF of Theorem REF have already been proven in Section , this then completes the proof of Theorem REF , modulo the proof of Proposition REF .",
"The second part of Proposition REF implies part REF of Theorem REF : The first part of (REF ) follows from (REF ) since $\\delta \\text{-thick}(\\mathcal {U},h)$ is compact for every $\\delta >0$ , while the second part of (REF ) is due to the conformal invariance of the energy.",
"For the proof of Proposition REF we shall use the following standard $\\varepsilon $ -regularity result.",
"Proposition 3.4 There exist constants $\\varepsilon _0>0$ and $C\\in {\\mathbb {R}}$ depending only on the target manifold so that the following holds true.",
"Let $u:B_{g_H}(x,r)\\rightarrow N$ be any smooth map from a ball of radius $r\\in (0,1]$ in the hyperbolic plane $(H,g_{H})$ with energy $E(u,g_H, B_{g_H}(x,r))\\le \\varepsilon _0.$ Then $\\int \\varphi ^2\\big [\\vert \\nabla _{g_H} du\\vert _{g_H}^2+\\vert du\\vert _{g_H}^4 \\big ]\\,dv_{g_H}\\le C\\Vert d\\varphi \\Vert _{L^\\infty (H,g_H)}^2E(u,g_H, B_{g_H}(x,r))+C\\int \\varphi ^2 \\vert \\tau _{g_H}(u)\\vert ^2 dv_{g_H}$ holds true for every function $\\varphi \\in C_c^\\infty (B_{g_H}(x,r),[0,1])$ .",
"Note that the Hessian term $\\vert \\nabla _{g_H} du\\vert _{g_H}^2$ is not referring to the intrinsic Hessian.",
"That term is instead the sum of the corresponding terms for each component of $u$ viewed as a map into Euclidean space, and depends on the isometric embedding of $N$ that we chose.",
"This term can be controlled in terms of the integral of $\\varphi ^2|\\Delta _g u|^2$ and lower order terms simply using integration by parts.",
"This leading order term can be rewritten using (REF ) and the resulting quartic term in $du$ controlled with the Sobolev inequality.",
"The details of a very similar argument can be found in [13].",
"[Proof of Proposition REF ] Part REF of the proposition represents the analogue of Lemma 3.10' of [21] and we shall use properties of horizontal curves from Lemma REF to control the evolution of the metric, see also [15] for a related proof in the non-degenerate case.",
"In the following we shall use several times that for any compact subset $\\Omega \\subset \\mathcal {U}=M\\setminus F$ there exists a number $t_0=t_0(\\Omega )\\in [0,T)$ so that $ \\Omega \\subset \\delta _{2\\bar{K}}(t_0)\\text{-thick}(M,g(t_0)) $ where $\\delta _{K}(t)=K(T-t)(E(t)-E(T))$ and $\\bar{K}$ are as in Lemma REF .",
"Indeed, for solutions of (REF ) which degenerate as described in (REF ), this is a consequence of the uniform convergence of the injectivity radius obtained in Proposition REF , while otherwise $\\operatorname{inj}_{g(t)}(M)$ is bounded away from zero uniformly so (REF ) is trivially satisfied for $t_0$ sufficiently close to $T$ .",
"As a consequence of (REF ) also $\\operatorname{inj}_{g(t_0)}(x)\\ge \\delta _{\\bar{K}}(t_0) \\text{ for all } x\\in M \\text{ with } \\mathop {\\mathrm {dist}}\\nolimits _{g(t_0)}(x,\\Omega )\\le \\delta _{\\bar{K}}(t_0),$ which allows us to apply Lemma REF to control the evolution of the metric as well as Lemma REF to bound the cut-energy on this neighbourhood of $\\Omega $ .",
"We first apply this idea to prove that for any point $p\\in \\mathcal {U}$ for which $\\limsup _{t\\uparrow T} E(u(t),g(t),V)\\ge \\varepsilon _0 \\text{ for every neighbourhood }V\\subset M \\text{ of } p,$ $\\varepsilon _0>0$ the constant obtained in Proposition REF , we also have $\\liminf _{t\\uparrow T} E(u(t),g(t),W)\\ge \\varepsilon _0\\quad \\text{ for every neighbourhood }W\\subset M \\text{ of } p.$ In particular, the set $\\tilde{S}$ of points in $\\mathcal {U}$ for which (REF ) holds is a finite set and we will later see that it agrees with the singular set $S$ defined in (REF ).",
"To show (REF ) for a given $p\\in \\tilde{S}$ we let $t_0\\in [0,T)$ be large enough so that (REF ) holds for $\\Omega =\\lbrace p\\rbrace $ .",
"Given any neighbourhood $W$ of $p$ we then choose $r\\in (0,\\delta _{\\bar{K}}(t_0))$ small enough so that $B_{g(t_0)}(p,r)\\subset W$ and select a cut-off function $\\varphi \\in C_c^\\infty (B_{g(t_0)}(p,r),[0,1])$ with $\\varphi \\equiv 1$ in a neighbourhood $V$ of $p$ .",
"Lemma REF implies that the limit $\\lim _{t\\uparrow T}E_\\varphi (t)$ of the cut-off energy defined in (REF ) exists and thus that, by (REF ), $\\liminf _{t\\uparrow T} E(u(t),g(t),W)\\ge \\lim _{t\\uparrow T}E_\\varphi (t)\\ge \\limsup _{t\\uparrow T} E(u(t),g(t),V)\\ge \\varepsilon _0$ as claimed.",
"Having thus established that there is only a finite subset $\\tilde{S}$ of points in $\\mathcal {U}$ for which (REF ) holds, we now want to prove that $u(t)$ converges smoothly on every compact subset $V$ of $\\mathcal {U}\\setminus \\tilde{S}$ as $t\\uparrow T$ .",
"Given such a compact subset $V$ of $\\mathcal {U}\\setminus \\tilde{S}$ we may choose $r_0\\in (0,1)$ small enough that $ E(u(t),g(t),B_{g(t_0)}(p,r_0))<\\varepsilon _0 \\quad \\text{ for all } t\\in [0,T),\\text{ and all } p\\in V.$ Then choosing $t_0\\in [0,T)$ so that (REF ) holds true for $\\Omega =V$ and reducing $r_0$ if necessary to ensure that $r_0<\\delta _{\\bar{K}}(t_0)$ we know from (REF ) that we can apply both Lemmas REF and REF on balls $B_{g(t_0)}(p,r)$ , $r\\le r_0$ , $p\\in V$ , as they are contained in $\\delta _{\\bar{K}}(t_0)\\text{-thick}(M,g(t_0))$ .",
"We first note that (REF ) from Lemma REF guarantees that for every $t\\in [t_0,T)$ $B_{g(t_0)}(p,\\tfrac{r_0}{C_1})\\subset B_{g(t)}(p,\\tfrac{r_0}{\\sqrt{C_1}})\\subset B_{g(t_0)}(p,r_0).$ We furthermore note that $\\tfrac{r_0}{\\sqrt{C_1}}$ cannot be larger than $\\operatorname{inj}_{g(t)}(p)$ for any $t\\in [t_0,T)$ as otherwise $B_{g(t)}(p,\\tfrac{r_0}{\\sqrt{C_1}})$ , and thus also $ B_{g(t_0)}(p,r_0)$ , would need to contain a curve $\\sigma $ starting and ending in $p$ that is not contractible in $M$ , which would contradict the fact that $r_0<\\operatorname{inj}_{g(t_0)}(p)$ .",
"Hence $B_{g(t)}(p,\\tfrac{r_0}{\\sqrt{C_1}})$ is isometric to a ball in the hyperbolic plane and so the smallness of the energy $E(u(t),g(t),B_{g(t)}(p,\\tfrac{r_0}{\\sqrt{C_1}}))<\\varepsilon _0$ obtained from (REF ) and (REF ) allows us to apply Proposition REF for any $\\varphi \\in C_c^\\infty (B_{g(t_0)}(p,\\tfrac{r_0}{C_1}),[0,1])$ and any time $t\\in [t_0,T)$ .",
"This will be crucial in the proof of the following: Claim: For any $p\\in V$ and $\\varphi \\in C_c^\\infty (B_{g(t_0)}(p,\\tfrac{r_0}{C_1}),[0,1])$ (with $r_0>0$ chosen as above) we have $ \\sup _{t\\in [t_0,T)}\\int \\varphi ^2\\vert \\partial _tu\\vert ^2 dv_g<\\infty .$ In particular there exists a neighbourhood $W$ of $V$ so that $\\sup _{t\\in [t_0,T)}\\Vert u(t)\\Vert _{H^2 (W,g(t))}<\\infty .$ Proof of claim: To prove the first part of the claim, we differentiate (REF ) in time, test with $\\varphi ^2\\partial _tu$ and use that $\\frac{d}{dt}dv_g=0$ to write $\\begin{aligned}&\\tfrac{1}{2}\\tfrac{d}{dt} \\int \\varphi ^2\\vert \\partial _tu\\vert ^2\\,dv_g+\\int \\varphi ^2\\vert d\\partial _tu\\vert ^2 \\, dv_g\\\\&\\qquad =-\\int \\langle d\\partial _tu,d(\\varphi ^2)\\rangle _g \\cdot \\partial _tu \\,dv_g+\\tfrac{d}{d\\varepsilon }\\vert _{\\varepsilon =0}\\int \\Delta _{g(t+\\varepsilon )}u\\cdot \\varphi ^2 \\partial _tu \\,dv_{g(t+\\varepsilon )}\\\\&\\qquad \\qquad +\\int \\partial _t(A_g(u)(du,du))\\cdot \\varphi ^2 \\partial _tu\\,dv_g\\\\&\\qquad \\le \\tfrac{1}{8}\\int \\varphi ^2\\vert d\\partial _tu\\vert ^2 \\, dv_g+C\\Vert d\\varphi \\Vert _{L^\\infty (M,g)}^2\\cdot \\Vert \\partial _tu\\Vert _{L^2(M,g) }^2-\\tfrac{d}{d\\varepsilon }\\vert _{\\varepsilon =0}\\int \\langle du,d(\\varphi ^2\\partial _tu)\\rangle _{g(t+\\varepsilon )} dv_g\\\\&\\qquad \\qquad +C\\Vert \\partial _tg\\Vert _{L^\\infty (\\mathop {\\mathrm {supp}}\\nolimits (\\varphi ),g)}^2 E(u,g) +C\\int \\vert \\partial _tu\\vert ^2 \\vert du\\vert ^2_g\\varphi ^2 dv_g\\\\&\\qquad \\le \\tfrac{1}{4} \\int \\varphi ^2\\vert d\\partial _tu\\vert ^2dv_g+ C\\Vert d\\varphi \\Vert _{L^\\infty (M,g)}^2\\Vert \\partial _tu\\Vert _{L^2(M,g) }^2+C\\Vert \\partial _tg\\Vert _{L^\\infty (\\mathop {\\mathrm {supp}}\\nolimits (\\varphi ),g)}^2\\\\&\\qquad \\qquad +\\hat{C}\\int \\vert \\partial _tu\\vert ^2 \\vert du\\vert _g^2\\varphi ^2 dv_g, \\end{aligned}$ where $C$ and $\\hat{C}$ depend only on a bound $E_0$ on the initial energy and the target manifold, and the value of $\\hat{C}$ is fixed in what follows.",
"To estimate the last term in (REF ) we first apply Proposition REF to get $\\int \\varphi ^2\\vert \\partial _tu\\vert ^2\\vert du\\vert _g^2 dv_g\\le C(\\int \\varphi ^2\\vert \\partial _tu\\vert ^4 dv_g)^{1/2}\\cdot \\big [ \\int \\varphi ^2\\vert \\partial _tu\\vert ^2 dv_g+C\\Vert d\\varphi \\Vert _{L^\\infty (M,g)}^2 \\big ]^{1/2}.$ We then recall that $\\mathop {\\mathrm {supp}}\\nolimits (\\varphi )$ is contained in the ball $B_{g(t)}(p,\\frac{r_0}{\\sqrt{C_1}})$ for every $t\\in [t_0,T)$ and that $\\frac{r_0}{\\sqrt{C_1}}\\le \\min (\\operatorname{inj}_{g(t)}(p),1)$ .",
"We may thus view $(\\mathop {\\mathrm {supp}}\\nolimits (\\varphi ),g(t))$ as a subset of the unit ball in the hyperbolic plane and apply the Sobolev embedding theorem to estimate the first factor in the above inequality by $\\begin{aligned}(\\int \\varphi ^2\\vert \\partial _tu\\vert ^4 dv_g)^{1/2}&=\\Vert \\varphi \\vert \\partial _tu\\vert ^2\\Vert _{L^2} \\le C\\Vert d(\\varphi \\vert \\partial _tu\\vert ^2)\\Vert _{L^1} \\\\&\\le C\\Vert \\partial _tu\\Vert _{L^2(M,g)}\\big (\\int \\vert d\\partial _tu\\vert ^2\\varphi ^2 dv_g\\big )^{1/2}+C\\Vert d\\varphi \\Vert _{L^\\infty (M,g)}\\Vert \\partial _tu\\Vert _{L^2(M,g)}^2.\\end{aligned}$ Combined this allows us to estimate the final term in (REF ) by $\\hat{C}\\int \\vert \\partial _tu\\vert ^2 \\vert du\\vert ^2 \\varphi ^2 dv_g\\le \\frac{1}{4} \\int \\vert d\\partial _tu\\vert ^2 \\varphi ^2 dv_g+C\\Vert \\partial _tu\\Vert _{L^2(M,g)}^2\\big [\\int \\varphi ^2\\vert \\partial _tu\\vert ^2 dv_g+C\\Vert d\\varphi \\Vert _{L^\\infty (M,g)}^2\\big ]$ and thus to reduce (REF ) to $\\begin{aligned}\\frac{d}{dt}\\int \\varphi ^2\\vert \\partial _tu\\vert ^2dv_g +\\int \\varphi ^2\\vert d\\partial _tu\\vert ^2 dv_g&\\le C\\Vert \\partial _tu\\Vert _{L^2(M,g) }^2\\cdot \\big [ \\Vert d\\varphi \\Vert _{L^\\infty (M,g)}^2+\\int \\varphi ^2 \\vert \\partial _tu\\vert ^2dv_g\\big ] \\\\& \\quad +C \\Vert \\partial _tg\\Vert _{L^\\infty (\\mathop {\\mathrm {supp}}\\nolimits (\\varphi ),g)}^2.\\end{aligned}$ Since $\\partial _tg$ is controlled on $\\mathop {\\mathrm {supp}}\\nolimits (\\varphi )\\subset \\delta _{\\bar{K}}(t_0)\\text{-thick}(M,g(t_0))$ by the estimate (REF ) of Lemma REF while estimate (REF ) from the same lemma implies that $\\Vert d\\varphi \\Vert _{L^\\infty (M,g(t))}\\le \\sqrt{C_1} \\Vert d\\varphi \\Vert _{L^\\infty (M,g(t_0))}$ , we thus conclude that $\\begin{aligned}\\frac{d}{dt}\\int \\varphi ^2\\vert \\partial _tu\\vert ^2dv_g\\le &\\, C\\Vert d\\varphi \\Vert _{L^\\infty (M,g(t_0))}^2\\Vert \\partial _tu\\Vert _{L^2(M,g) }^2+C \\Vert \\partial _tu\\Vert _{L^2(M,g)}^2 \\int \\varphi ^2 \\vert \\partial _tu\\vert ^2dv_g\\\\&\\quad +C\\delta _{\\bar{K}}(t_0)^{-1} \\Vert \\partial _tg\\Vert _{L^2(M,g)}^2\\\\\\le & \\,C\\big (-\\tfrac{dE}{dt}\\big )\\int \\varphi ^2 \\vert \\partial _tu\\vert ^2 dv_g+C\\big (-\\tfrac{dE}{dt}\\big ) \\cdot \\big [\\Vert d\\varphi \\Vert _{L^\\infty (M,g(t_0))}^2+\\delta _{\\bar{K}}(t_0)^{-1}\\big ],\\end{aligned}$ by (REF ), where $C$ now depends also on the genus of $M$ and $\\eta $ .",
"Hence (REF ) follows using Gronwall's lemma.",
"The second part of the claim is now an immediate consequence of (REF ) and Proposition REF .",
"Based on the claim we have just proven, we can now establish convergence of $u(t)$ in $C^l(V)$ for every $l\\in {\\mathbb {N}}$ by well-known arguments: First of all, we may reduce the neighbourhood $W$ of $V$ if necessary to ensure that $W\\subset \\delta _{\\bar{K}}(t_0)\\text{-thick}(M,g(t_0))$ , compare (REF ) and (REF ).",
"We then apply the Sobolev embedding theorem to obtain that $\\sup _{t\\in [t_0,T)}\\Vert du(t)\\Vert _{L^p (W,g(t))}<\\infty \\text{ for every } 1\\le p<\\infty .$ The control on the metrics $g(t)$ , $t\\in [t_0,T)$ , obtained in Lemma REF thus allows us to view (REF ) as a uniformly parabolic equation on the fixed surface $(W,g(t_0))$ (for times $t$ in this interval $[t_0,T)$ ) whose right-hand side is in $L^p$ for every $p<\\infty $ .",
"Standard parabolic theory combined with the fact that $u$ is by assumption smooth away from $T$ , implies that $u$ is in the parabolic Sobolev space $W^{2,1;p}(\\tilde{W}\\times [t_0,T))$ for every $p<\\infty $ for a slightly smaller neighbourhood $\\tilde{W}$ of $V$ .",
"In particular $u$ is Hölder continuous with exponent $\\alpha $ for every $\\alpha <1$ on $\\tilde{W} \\times [0,T)$ .",
"Taking covariant derivatives $\\nabla _{g(t)}^l$ of (REF ) allows us to repeat the above argument and obtain that $(x,t)\\mapsto (\\nabla _{g(t)}^l u)(x,t)$ is Hölder continuous on $V\\times [t_0,T)$ for every $l\\in {\\mathbb {N}}$ .",
"As the metrics converge smoothly to $h$ on $V$ , this allows us to conclude that also $u(t)\\rightarrow \\bar{u}$ in $C^l(V,h)$ for every $l\\in {\\mathbb {N}}$ , for some $\\bar{u}$ .",
"As the obtained convergence implies in particular that the set $S$ defined in (REF ), as used in Proposition REF , agrees with the set $\\tilde{S}$ of points satisfying (REF ) considered here, this completes the proof of part REF of Proposition REF .",
"For the proof of part REF of the proposition we closely follow the arguments of [22].",
"Let $p\\in S$ .",
"As above we choose $t_0<T$ so that (REF ) holds true for $\\Omega =\\lbrace p\\rbrace $ which we recall allows us to apply Lemmas REF and REF on balls $B_{g(t_0)}(p,r_0)$ , $r_0\\in (0,\\delta _{\\bar{K}}(t_0))$ since (REF ) ensures that such balls are contained in $\\delta _{\\bar{K}}(t_0)\\text{-thick}(M,g(t_0))$ .",
"We fix such a radius $r_0$ which is small enough so that $B_{g(t_0)}(p,r_0)$ contains no other element of the singular set $S$ .",
"Given any fixed cut-off function $\\psi \\in C_c^\\infty ([0,1),[0,1])$ with $\\psi \\equiv 1$ on $[0, \\frac{1}{2}]$ and with $\\Vert \\psi ^{\\prime }\\Vert _{L^\\infty }\\le 4$ we set $\\varphi _{r}(x):=\\psi \\big (\\tfrac{\\mathop {\\mathrm {dist}}\\nolimits _{g(t_0)}(p,x)^2}{r^2}\\big ), \\quad 0<r<r_0$ and note that $\\Vert d\\varphi _r\\Vert _{L^\\infty (M,g(t_0))}\\le \\frac{C}{r}$ .",
"As $\\mathop {\\mathrm {supp}}\\nolimits (\\varphi _r)\\subset B_{g(t_0)}(p,r_0)$ we can apply Lemma REF to control the associated cut-off energies $E_r(t):=E_{\\varphi _r}(t)$ defined in (REF ) and obtain in particular that $\\lim _{t\\uparrow T} E_r(t)$ exists for every $r\\in (0,r_0)$ .",
"Combined with the local $C^l$ convergence of $u(t)\\rightarrow \\bar{u}$ on $\\mathcal {U}\\setminus S$ and the convergence of the metrics obtained in part REF of Theorem REF this implies that $\\hat{E}_p:=\\lim _{t\\uparrow T} E_r(t)-\\frac{1}{2}\\int \\varphi _r^2\\vert d\\bar{u}\\vert _h^2 dv_h$ is independent of $r\\in (0,r_0)$ .",
"Let now $\\nu >0$ .",
"For $t\\in [t_0,T)$ sufficiently close to $T$ so that $\\nu (T-t)^\\frac{1}{2}<r_0$ we can apply Lemma REF to $s\\mapsto E_{\\nu (T-t)^{1/2}}(s), s\\in [t_0,T)$ , in order to obtain the second inequality of $\\begin{aligned}\\big |E_{\\nu (T-t)^{1/2}}(t)-\\hat{E}_p\\big | & \\le \\big |E_{\\nu (T-t)^{1/2}}(t)-\\lim _{s\\uparrow T}E_{\\nu (T-t)^{1/2}}(s)\\big |+\\frac{1}{2}\\int \\varphi _{\\nu (T-t)^{1/2}}^2\\vert d\\bar{u}\\vert _h^2 dv_h\\\\&\\le E(t)-E(T)+C[\\nu ^{-1}+\\delta _{\\bar{K}}^{-\\frac{1}{2}}(t_0)\\cdot (T-t)^\\frac{1}{2}]\\cdot (E(t)-E(T))^\\frac{1}{2}\\\\&\\qquad + E(\\bar{u}, h, B_{g(t_0)}(p,\\nu (T-t)^{1/2})).\\end{aligned}$ We furthermore note that $B_{g(t_0)}(p,\\nu (T-t)^{1/2}))\\subset B_{h}(p,\\sqrt{C_1}\\nu (T-t)^{1/2})$ , compare (REF ) of Lemma REF , and thus that the last term in (REF ) tends to zero as $t\\uparrow T$ .",
"Passing to the limit $t\\uparrow T$ in (REF ) we thus obtain that also $\\lim _{t\\uparrow T}E_{\\nu (T-t)^{1/2}}(t)=\\hat{E}_p\\text{ for every }\\nu >0.$ Combined with the equivalence (REF ) of the metrics obtained in Lemma REF we therefore get that for any $\\nu >0$ $\\begin{aligned}\\lim _{r\\downarrow 0}\\lim _{t\\uparrow T} E(u(t),g(t), B_{g(t)}(p,r))\\le &\\,\\lim _{r\\downarrow 0}\\lim _{t\\uparrow T} E(u(t),g(t), B_{g(t_0)}(p,\\sqrt{C_1}r))\\le \\lim _{r\\downarrow 0}\\lim _{t\\uparrow T} E_{2\\sqrt{C_1}r}(t)\\\\=&\\,\\hat{E}_p=\\lim _{t\\uparrow T}E_{\\nu C_1^{-1/2} (T-t)^{1/2}}(t)\\\\\\le &\\,\\liminf _{t\\uparrow T} E(u(t),g(t), B_{g(t_0)}(p,\\nu C_1^{-1/2}(T-t)^{1/2}))\\\\\\le &\\,\\liminf _{t\\uparrow T} E(u(t),g(t), B_{g(t)}(p,\\nu (T-t)^{1/2})).\\end{aligned}$ As the `reverse' inequality $\\limsup _{t\\uparrow T} E(u(t),g(t), B_{g(t)}(p,\\nu (T-t)^{1/2}))\\le \\lim _{r\\downarrow 0}\\lim _{t\\uparrow T} E(u(t),g(t), B_{g(t)}(p,r))$ is trivially true, this proves the second equality in (REF ), including the existence of the limits taken, while the first inequality of (REF ) follows directly from the equivalence (REF ) of the metrics $g(t)$ and $h$ obtained in Lemma REF .",
"To establish the final inequality of (REF ) we closely follow [22].",
"Given a sequence of times $t_n\\uparrow T$ as in (REF ) and a point $p\\in S$ we pick local isothermal coordinates centred at $p$ for each of the $g(t_n)$ by identifying $B_{g(t_n)}(p,r_0)$ with the corresponding ball centred at zero of the Poincaré hyperbolic disc, viewed conformally as the unit disc centred at the origin in ${\\mathbb {R}}^2$ , and rescale to obtain a sequence of maps $u_n(x):=u(r_n x,t_n), \\quad r_n:=(T-t_n)^{1/2} $ for which $\\Vert \\tau (u_n)\\Vert _{L^2(\\mathcal {K})}\\rightarrow 0$ for every $\\mathcal {K}\\subset \\subset {\\mathbb {R}}^2$ .",
"Since (REF ) implies that $E(u_n, B(0,\\Lambda )\\setminus B(0,\\lambda ))\\rightarrow 0$ for any $0<\\lambda <\\Lambda $ , a subsequence of the maps $u_n$ converges strongly in $H^1$ away from 0 to a constant map while bubbles $\\lbrace \\omega _j\\rbrace _{j=1}^{m^{\\prime }}$ develop near the origin at scales $\\hat{\\lambda }_n^j\\rightarrow 0, n\\rightarrow \\infty $ .",
"The scales at which the bubbles $\\omega _j$ develop in the original sequence are thus $\\lambda _n^j=r_n\\hat{\\lambda }_n^j=o((T-t_n)^{1/2})$ and the `no-loss-of-energy' result for bubble tree convergence of almost harmonic maps of [3] ensures that all the energy of the $u_n$ is captured by these bubbles i.e.",
"that for every $\\Lambda >0$ we have $\\lim _{n\\rightarrow \\infty } E(u_n,B_\\Lambda (0))=\\sum _{l=1}^{m^{\\prime }} E(\\omega _l).$ Taking the limit $\\Lambda \\downarrow 0$ , and bearing in mind that all but the final equality of (REF ) has already been established, we find that for every $p\\in S$ , we have $\\begin{aligned}\\lim _{r\\downarrow 0}\\lim _{t\\uparrow T} E(u(t),g(t), B_{g(t)}(p,r))= \\sum _{l=1}^{m^{\\prime }} E(\\omega _l),\\end{aligned}$ completing the proof of (REF ).",
"Finally, given any compact subset $\\Omega \\subset \\subset \\mathcal {U}$ which is large enough for $S$ to be contained in the interior of $\\Omega $ we can combine (REF ) with the strong $H^1_{loc}$ convergence of $u(t)\\rightarrow \\bar{u}$ on $\\mathcal {U}\\setminus S$ and the convergence of the metrics to obtain that indeed $\\lim _{t\\uparrow T} E(u(t),g(t),\\Omega )= E(\\bar{u}, h, \\Omega ) + \\sum _{l=1}^{m^{\\prime \\prime }} E(\\omega _l)$ where $\\lbrace \\omega _l\\rbrace _{l=1}^{m^{\\prime \\prime }}$ is the set of all bubbles developing at points in $S$ along a sequence of times $t_n$ as considered in the proposition."
],
[
"All energy lost down collars is represented by bubbles",
"At this point we have a good description of the convergence of $u(t)$ and $g(t)$ locally on $\\mathcal {U}=M\\setminus F$ , with Proposition REF completing the proof of Theorem REF and establishing part REF of Theorem REF .",
"In this section we prove parts REF and REF of Theorem REF , which show that near the centre of degenerating collars, the map is looking like a collection of bubbles, while on larger scales that are nevertheless vanishing scales, where we have no way of showing that the map is becoming harmonic, no energy can be lost.",
"[Proof of part REF of Theorem REF ] As a next step we now prove part REF of Theorem REF which can be seen as quantifying the size of the part of $\\mathcal {U}$ on which the energy has almost reached its limit.",
"As we can only apply the local energy estimate from Lemma REF on regions with sufficiently large injectivity radius, we will obtain the existence of a limit of the energy on the $[T-t]\\text{-thin}$ part by proving that the limit on the $[T-t]\\text{-thick}$ part exists and agrees with $E_{thick}$ and then appealing to the existence of a limit of the total energy $E(t)$ .",
"As above it will be more convenient to work not with energies over given sets, but with cut-off energies $E_\\varphi $ as defined in (REF ).",
"To this end we let $\\delta _K(t)=K(T-t)(E(t)-E(T))$ , $K\\ge \\bar{K}$ , be as in Lemma REF and recall that the characterisation of the pinching set (REF ) implies in particular that for every $t_0\\in [0,T)$ $\\operatorname{inj}_{g(t_0)}(M)< \\delta _K(t_0)$ and thus that $A_{K,t_0}:=\\lbrace x\\in M\\ :\\ \\operatorname{inj}_{g(t_0)}(x)\\le \\delta _{K}(t_0)\\rbrace $ is nonempty.",
"We will always assume that $t_0\\in [0,T)$ is sufficiently large, depending in particular on $K$ , so that $\\delta _K(t_0)\\cdot (\\pi e)<\\mathop {\\mathrm {arsinh}}\\nolimits (1)$ .",
"In this way, not only can we be sure that every point in $A_{K,t_0}$ has injectivity radius less than $\\mathop {\\mathrm {arsinh}}\\nolimits (1)$ , and is thus lying within some collar region around a geodesic of length less than $2\\mathop {\\mathrm {arsinh}}\\nolimits (1)$ , we can also be sure that the 1-fattening of $A_{K,t_0}$ , i.e.",
"$\\lbrace p\\in M\\ |\\ \\mathop {\\mathrm {dist}}\\nolimits _{g(t_0)}(p,A_{K,t_0})<1\\rbrace $ , must lie within $\\delta _{e\\pi K}(t_0)\\text{-thin}(M,g(t_0))$ , and hence also lie within a union of such (pairwise disjoint) collars, since by [20] if $x\\in A_{K,t_0}$ and $y\\in B_{g(t_0)}(x,1)$ lies in the same collar, then $\\operatorname{inj}_{g(t_0)}(y)\\le \\operatorname{inj}_{g(t_0)}(x) \\cdot (\\pi e)\\le \\delta _K(t_0)\\cdot (\\pi e)<\\mathop {\\mathrm {arsinh}}\\nolimits (1)$ , so we cannot escape this collar within a distance 1 of $x$ .",
"In particular, the function $x\\mapsto \\mathop {\\mathrm {dist}}\\nolimits _{g(t_0)}(x,A_{K,t_0})$ is smooth on the 1-fattening of $A_{K,t_0}$ .",
"Given any smooth cut-off function $\\phi :{\\mathbb {R}}\\rightarrow [0,1]$ such that $\\phi (x)=0$ for $x\\le 0$ , $\\phi (x)=1$ for $x\\ge 1$ and $|\\phi ^{\\prime }|\\le 2$ , we can thus define the induced smooth cut-off $\\varphi _{K,t_0}:M\\rightarrow [0,1]$ by $\\varphi _{K,t_0}(x):=\\phi (\\mathop {\\mathrm {dist}}\\nolimits _{g(t_0)}(x,A_{K,t_0})).$ It is immediately apparent that $\\varphi _{K,t_0}\\equiv 0\\qquad \\text{on }\\delta _K(t_0)\\text{-thin}(M,g(t_0)),$ and that the support of $\\varphi _{K,t_0}$ lies within $\\delta _K(t_0)\\text{-thick}(M,g(t_0))$ and hence $\\varphi _{K,t_0}$ has compact support within $\\mathcal {U}$ owing to (REF ).",
"This will shortly allow us to apply Lemma REF to the corresponding local energy $E_{K,t_0}(t):=E_{\\varphi _{K,t_0}}(t)$ that serves as a substitute for the energy of $u(t)$ over $\\delta _K(t_0)\\text{-thick}(M,g(t_0))$ .",
"We also claim that $\\varphi _{K,t_0}\\equiv 1\\qquad \\text{on }\\delta _{e\\pi K}(t_0)\\text{-thick}(M,g(t_0)).$ Indeed, the only way this could fail would be if we could find a point in the 1-fattening of $A_{K,t_0}$ that lies in $\\delta _{e\\pi K}(t_0)\\text{-thick}(M,g(t_0))$ , which we ruled out above.",
"By (REF ), we see that $E_{K,t_0}(t)\\le E\\big (u(t),g(t),\\delta _K(t_0)\\text{-thick}(M,g(t_0))\\big )$ , and so $\\lim _{K\\rightarrow \\infty }\\limsup _{t\\uparrow T}E_{K,t}(t)\\le \\lim _{K\\rightarrow \\infty }\\limsup _{t\\uparrow T} E(u(t),g(t),\\delta _K(t)\\text{-thick}(M,g(t))).$ On the other hand, by (REF ), we see that $E(u(t),g(t),\\delta _{e\\pi K}(t_0)\\text{-thick}(M,g(t_0)))\\le E_{K,t_0}(t)$ , and hence we have the converse inequality $\\lim _{K\\rightarrow \\infty }\\limsup _{t\\uparrow T} E(u(t),g(t),\\delta _{K}(t)\\text{-thick}(M,g(t))\\le \\lim _{K\\rightarrow \\infty }\\limsup _{t\\uparrow T} E_{K,t}(t),$ i.e.",
"we have equality in (REF ) and (REF ).",
"Therefore to prove (REF ), it suffices to show that $E_{thick}=\\lim _{K\\rightarrow \\infty } \\limsup _{t\\uparrow T}E_{K,t}(t).$ We claim first that $E_{thick}=\\limsup _{t_0\\uparrow T}\\lim _{t\\uparrow T} E_{K, t_0}(t)$ where the existence of $\\lim _{t\\uparrow T} E_{K, t_0}(t)$ is guaranteed by Lemma REF .",
"To see (REF ), first recall that for $K$ , $t_0$ as above, the support of $\\varphi _{K,t_0}$ is compact within $\\mathcal {U}$ , and is thus contained within $\\delta \\text{-thick}(\\mathcal {U},h)$ for sufficiently small $\\delta >0$ .",
"By reducing $\\delta $ further, we may assume that all bubble points in $S$ lie within the interior of $\\delta \\text{-thick}(\\mathcal {U},h)$ .",
"Therefore we have $E(u(t),g(t),\\delta \\text{-thick}(\\mathcal {U},h))\\ge E_{K,t_0}(t)$ , and taking the limits $t\\uparrow T$ , $\\delta \\downarrow 0$ and $t_0\\uparrow T$ in that order, we find that $E_{thick}\\ge \\limsup _{t_0\\uparrow T}\\lim _{t\\uparrow T} E_{K, t_0}(t)$ .",
"To see the converse inequality, we observe that by (REF ), for any $\\delta >0$ and $t_0<T$ sufficiently large (depending on $\\delta $ , $K$ etc.)",
"we have $\\varphi _{K,t_0}\\equiv 1$ on $\\delta \\text{-thick}(M,g(t_0))$ , and so $E(u(t),g(t),\\delta \\text{-thick}(\\mathcal {U},h))\\le E_{K,t_0}(t)$ .",
"This time we take limits in the order $t\\uparrow T$ , $t_0\\uparrow T$ and then $\\delta \\downarrow 0$ to give $E_{thick}\\le \\limsup _{t_0\\uparrow T}\\lim _{t\\uparrow T} E_{K, t_0}(t)$ , and hence (REF ).",
"Thus (REF ) would follow if we can prove that as $K\\rightarrow \\infty $ we have $\\limsup _{t_0\\uparrow T}\\vert E_{K,t_0}(t_0)-\\lim _{t\\uparrow T}E_{K,t_0}(t)\\vert \\rightarrow 0.$ But this follows from Lemma REF , which implies that for $t_0\\in [0,T)$ as large as considered above, and every $t\\in [t_0,T)$ , we have $\\vert E_{K,t_0}(t)-E_{K,t_0}(t_0)\\vert \\le E(t_0)-E(T)+\\frac{C}{K^\\frac{1}{2}}+C(T-t_0)^{\\frac{1}{2}}(E(t_0)-E(T))^{\\frac{1}{2}}$ with $C$ depending only on the genus of $M$ , $\\eta $ and an upper bound on the initial energy, which thus yields (REF ) after taking the limits $t\\uparrow T$ , $t_0\\uparrow T$ and $K\\rightarrow \\infty $ , in that order.",
"Now that (REF ) has been proved, we verify that (REF ) follows as a result.",
"In particular, we verify that the limit taken in (REF ) exists.",
"However large we take $K>0$ , for sufficiently large $t<T$ we have $T-t\\ge \\delta _K(t)$ , and hence $E(u(t),g(t),[T-t]\\text{-thin}(M,g(t)))\\ge E(u(t),g(t), \\delta _K(t)\\text{-thin}(M,g(t))).$ Taking a $\\liminf $ as $t\\uparrow T$ and then the limit $K\\rightarrow \\infty $ , and using (REF ) we find that $\\liminf _{t\\uparrow T}E(u(t),g(t),[T-t]\\text{-thin}(M,g(t)))\\ge E_{thin}.$ To obtain the converse inequality, observe that given any $\\delta >0$ , for sufficiently large $t<T$ we have $\\delta \\text{-thin}(\\mathcal {U},h)\\supset [T-t]\\text{-thin}(M,g(t))$ , cf.",
"(REF ), and therefore $E(u(t),g(t),\\delta \\text{-thin}(\\mathcal {U},h))\\ge E(u(t),g(t),[T-t]\\text{-thin}(M,g(t))).$ Provided $\\delta >0$ is sufficiently small (so that the singular set $S$ is in the interior of $\\delta \\text{-thick}(\\mathcal {U},h)$ ), we can then take a limit as $t\\uparrow T$ , followed by a limit as $\\delta \\downarrow 0$ , to give $E_{thin}\\ge \\limsup _{t\\uparrow T}E(u(t),g(t),[T-t]\\text{-thin}(M,g(t))),$ which when combined with (REF ) completes the proof of (REF ) and hence of part REF of the theorem.",
"While part REF of Theorem REF gives good control on where energy can concentrate on the degenerating part of the surface, we currently have no control of what parts of the map are lost down the degenerating parts of the collar at the singular time $T$ .",
"This is addressed by part REF , which we shall now prove.",
"[Proof of part REF of Theorem REF ] Proposition REF tells us that the length $\\ell (t_n)$ of the central geodesic of each degenerating collar is controlled like $\\ell (t_n)=o(T-t_n)$ and hence that the $[T-t_n]\\text{-thin}$ part of such a collar, where all of the lost energy lives, is represented by longer and longer cylinders $\\tilde{\\mathcal {C}}_n:= \\mathcal {C}(t_n, \\delta _n)=(-\\tilde{X}_n,\\tilde{X}_n)\\times S^1$ , $\\delta _n=T-t_n$ , equipped with the corresponding collar metrics $g=\\rho ^2g_0$ .",
"We can indeed consider the maps on the larger subcollars $\\widehat{\\mathcal {C}}_n=(-\\widehat{X}_n,\\widehat{X}_n)$ which correspond to the $[T-t_n]^{\\frac{1}{2}}\\text{-thin}$ parts of the collar, where we note that $1\\ll \\tilde{X}_n \\ll \\widehat{X}_n \\ll X(\\ell _n)$ , compare (REF ).",
"We recall from [18] that $\\rho (y)\\le \\operatorname{inj}_{g(t)}(y)$ as $y$ varies within each collar.",
"Therefore, throughout $\\widehat{\\mathcal {C}}_n$ we have $\\rho \\le (T-t_n)^\\frac{1}{2}$ .",
"By the scaling of the tension field, if we switch from the hyperbolic metric $g_n=g(t_n)$ to the flat cylinder metric $g_0=ds^2+d\\theta ^2$ on each such subcollar, then we can estimate the tension of $u_n:=u(t_n)$ according to $\\Vert \\tau _{g_0}(u_n)\\Vert _{L^2(\\widehat{\\mathcal {C}}_n,g_0)}\\le (\\sup _{\\widehat{\\mathcal {C}}_n}\\rho )\\Vert \\tau _{g_n}(u_n)\\Vert _{L^2(\\widehat{\\mathcal {C}}_n,g_n)}\\le (T-t_n)^\\frac{1}{2}\\Vert \\tau _{g_n}(u_n)\\Vert _{L^2(M,g_n)}\\rightarrow 0$ by (REF ).",
"We can thus view the $u_n$ 's as almost-harmonic maps from longer and longer cylinders $(\\widehat{\\mathcal {C}}_n,g_0)$ and apply Proposition REF to pass to a subsequence that converges to a full bubble branch.",
"It is this estimate (REF ) and the precise information on the degenerate region where energy can concentrate obtained in part REF of Theorem REF that allows us to represent the maps on these parts in terms of branched minimal immersions and curves.",
"We stress that we would not be able to perform this analysis on the whole collar.",
"We also remark that in our situation we obtain the additional information that any bubble obtained in the convergence to a full bubble branch described in Proposition REF will be contained in the $(T-t_n)\\text{-thin}$ part of the surface as we already know that no energy can be lost on $\\lbrace p: \\operatorname{inj}_{g(t)}(p)\\in [(T-t),(T-t)^{1/2}]$ }.",
"MR: Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK PT: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK"
]
] | 1709.01881 |
[
[
"A Compact Kernel Approximation for 3D Action Recognition"
],
[
"Abstract 3D action recognition was shown to benefit from a covariance representation of the input data (joint 3D positions).",
"A kernel machine feed with such feature is an effective paradigm for 3D action recognition, yielding state-of-the-art results.",
"Yet, the whole framework is affected by the well-known scalability issue.",
"In fact, in general, the kernel function has to be evaluated for all pairs of instances inducing a Gram matrix whose complexity is quadratic in the number of samples.",
"In this work we reduce such complexity to be linear by proposing a novel and explicit feature map to approximate the kernel function.",
"This allows to train a linear classifier with an explicit feature encoding, which implicitly implements a Log-Euclidean machine in a scalable fashion.",
"Not only we prove that the proposed approximation is unbiased, but also we work out an explicit strong bound for its variance, attesting a theoretical superiority of our approach with respect to existing ones.",
"Experimentally, we verify that our representation provides a compact encoding and outperforms other approximation schemes on a number of publicly available benchmark datasets for 3D action recognition."
],
[
"Introduction",
"Action recognition is a key research domain in video/image processing and computer vision, being nowadays ubiquitous in human-robot interaction, autonomous driving vehicles, elderly care and video-surveillance to name a few [21].",
"Yet, challenging difficulties arise due to visual ambiguities (illumination variations, texture of clothing, general background noise, view heterogeneity, occlusions).",
"As an effective countermeasure, joint-based skeletal representations (extracted from depth images) are a viable solution.",
"Combined with a skeletal representation, the symmetric and positive definite (SPD) covariance operator scores a sound performance in 3D action recognition [22], [9], [5].",
"Indeed, while properly modeling the skeletal dynamics with a second order statistic, the covariance operator is also naturally able to handle different temporal duration of action instances.",
"This avoids slow pre-processing stages such as time warping or interpolation [20].",
"In addition, the superiority of such representation can be attested by achieving state-of-the-art performance by means of a relative simple classification pipeline [22], [5] where, basicallyFor the sake of precision, let us notice that [22] take advantage of multiple kernel learning in combining several low-level representations and [5] replaces the classical covariance operator with a kernelization., a non-linear Support Vector Machine (SVM) is trained using the Log-Euclidean kernel $K_{\\ell E}(\\mathbf {X},\\mathbf {Y}) = \\exp \\left( - \\dfrac{1}{2 \\sigma ^2} \\Vert \\log \\mathbf {X} - \\log \\mathbf {Y} \\Vert _F^2\\right)$ to compare covariance operators $\\mathbf {X}$ , $\\mathbf {Y}$ .",
"In (REF ), for any SPD matrix $\\mathbf {X}$ , we define $\\log \\mathbf {X} = \\mathbf {U} {\\rm diag} (\\log \\lambda _1, \\dots , \\log \\lambda _d) \\mathbf {U}^\\top ,$ being $\\mathbf {U}$ the matrix of eigenvectors which diagonalizes $\\mathbf {X}$ in terms of the eigenvalues $\\lambda _1 \\ge \\dots \\ge \\lambda _d > 0$ .",
"Very intuitively, for any fixed bandwidth $\\sigma > 0$ , $K_{\\ell E}(\\mathbf {X},\\mathbf {Y})$ is actually computing a radial basis Gaussian function by comparing the covariance operators $\\mathbf {X}$ and $\\mathbf {Y}$ by means of the Frobenius norm $\\Vert \\cdot \\Vert _F$ (after $\\mathbf {X},\\mathbf {Y}$ have been log-projected).",
"Computationally, the latter stage is not problematic (see Section ) and can be performed for each covariance operator before computing the kernel.",
"In addition to its formal properties in Riemannian geometry, this makes (REF ) widely used in practice [9], [22], [5].",
"However, the modern big data regime mines the applicability of such a kernel function.",
"Indeed, since (REF ) has to be computed for every pair of instances in the dataset, the so produced Gram matrix has prohibitive size.",
"So its storage becomes time- and memory-expensive and the related computations (required to train the model) are simply unfeasible.",
"The latter inconvenient can be solved as follows.",
"According to the well established kernel theory [2], the Kernel (REF ) induces an infinite-dimension feature map $\\varphi $ , meaning that $K_{\\ell E}(\\mathbf {X},\\mathbf {Y}) = \\langle \\varphi (\\mathbf {X}), \\varphi (\\mathbf {Y}) \\rangle $ .",
"However, if we are able to obtain an explicit feature map $\\Phi $ such that $K_{\\ell E}(\\mathbf {X},\\mathbf {Y}) \\approx \\langle \\Phi (\\mathbf {X}), \\Phi (\\mathbf {Y}) \\rangle $ , we can directly compute a finite-dimensional feature representation $\\Phi (\\mathbf {X})$ for each action instance separately.",
"Then, with a compact $\\Phi $ , we can train a linear SVM instead of its kernelized version.",
"This is totally feasible and quite efficient even in the big data regime [7].",
"Therefore, the whole pipeline will actually provide a scalable implementation of a Log-Euclidean SVM, whose cost is reduced from quadratic to linear.",
"In our work we specifically tackle the aforementioned issue through the following main contributions.",
"We propose a novel compact and explicit feature map to approximate the Log-Euclidean kernel within a probabilistic framework.",
"We provide a rigorous mathematical formulation, proving that the proposed approximation has null bias and bounded variance.",
"We compare the proposed feature map approximation against alternative approximation schemes, showing the formal superiority of our framework.",
"We experimentally evaluate our method against the very same approximation schemes over six 3D action recognition datasets, confirming with practice our theoretical findings.",
"The rest of the paper is outlined as follows.",
"In Section we review the most relevant related literature.",
"Section proposes the novel approximation and discusses its foundation.",
"We compare it with alternative paradigms in Section .",
"Section draws conclusions and the Appendix reports all proofs of our theoretical results."
],
[
"Related work",
"In this Section, we summarize the most relevant works in both covariance-based 3D action recognition and kernels' approximations.",
"Originally envisaged for image classification and detection tasks, the covariance operator has experienced a growing interest for action recognition, experiencing many different research trends: [9] extends it to the infinite dimensional case, while [10] hierarchically combines it in a temporal pyramid; [22], [12] investigate the conceptual analogy with trial-specific kernel matrices and [5] further proposes a new kernelization as to model arbitrary, non-linear relationships conveyed by the raw data.",
"However, those kernel methods usually do not scale up easily to big datasets due to demanding storage and computational costs.",
"As a solution, the exact kernel representation can be replaced by an approximated, more efficient version.",
"In the literature, this is done according to the following mainstream approaches.",
"The kernel Gram matrix is replaced with a surrogate low-rank version, in order to alleviate both memory and computational costs.",
"Within these methods, [1] applied Cholesky decomposition and [24] exploited Nyström approximation.",
"Instead of the exact kernel function $k$ , an explicit feature map $\\Phi $ is computed, so that the induced linear kernel $\\langle \\Phi (\\mathbf {x}),\\Phi (\\mathbf {y}) \\rangle $ approximates $k(\\mathbf {x},\\mathbf {y})$ .",
"Our work belong to this class of methods.",
"In this context, Rahimi & Recht [17] exploited the formalism of the Fourier Transform to approximate shift invariant kernels $k(\\mathbf {x},\\mathbf {y}) = k(\\mathbf {x}-\\mathbf {y})$ through an expansion of trigonometric functions.",
"Leveraging on a similar idea, Le et al.",
"[13] sped up the computation by exploiting the Walsh-Hadamard transform, downgrading the running cost of [17] from linear to log-linear with respect to the data dimension.",
"Recently, Kar & Karnick [11] proposed an approximated feature maps for dot product kernels $k(\\mathbf {x},\\mathbf {y}) = k(\\langle \\mathbf {x},\\mathbf {y} \\rangle )$ by directly exploiting the MacLaurin expansion of the kernel function.",
"Instead of considering a generic class of kernels, our work specifically focuses on the log-Euclidean one, approximating it through a novel unbiased estimator which ensures a explicit bound for variance (as only provided by [13]) and resulting in a superior classification performance with respect to [17], [13], [11]."
],
[
"The proposed approximated feature map",
"In this Section, we present the main theoretical contribution of this work, namely i) a random, explicit feature map $\\Phi $ such that $\\langle \\Phi (\\mathbf {X}), \\Phi (\\mathbf {Y}) \\rangle \\approx K_{\\ell E}(\\mathbf {X},\\mathbf {Y})$ , ii) the proof of its unbiasedness and iii) a strong theoretical bound on its variance.",
"Construction of the approximated feature map.",
"In order to construct a $\\nu $ dimensional feature map $\\mathbf {X} \\mapsto \\Phi (\\mathbf {X}) = [\\Phi _1(\\mathbf {X}),\\dots ,\\Phi _\\nu (\\mathbf {X})] \\in \\mathbb {R}^{\\nu }$ , for any $d \\times d$ SPD matrix $\\mathbf {X}$ , fix a probability distribution $\\rho $ supported over $\\mathbb {N}$ .",
"Precisely, each component $\\Phi _1, \\dots , \\Phi _\\nu $ of our $\\nu $ -dimensional feature map $\\Phi $ is independently computed according to the following algorithm.",
"[h!]",
"$j = 1,\\dots ,\\nu $ Sample $n$ according to $\\rho $ Sample the $d^n \\times d^n$ matrix $\\mathbf {W}$ of independent Gaussian distributed weights with zero mean and $\\sigma ^2/\\sqrt{n}$ variance Compute $\\log (\\mathbf {X})^{\\otimes n} = \\log \\mathbf {X} \\otimes \\dots \\otimes \\log \\mathbf {X}$ , $n$ times.",
"Assign $\\Phi _j(\\mathbf {X}) = \\dfrac{1}{\\sigma ^{2n}}\\sqrt{ \\dfrac{\\exp (-\\sigma ^{-2})}{\\nu \\rho (n) n!}",
"} {\\rm tr}( \\mathbf {W}^\\top \\log (\\mathbf {X})^{\\otimes n}).",
"\\vspace{-5.69046pt}$ The genesis of (REF ) can be explained by inspecting the feature map $\\varphi $ associated to the kernel $K(x,y)=\\exp \\left(-\\frac{1}{2\\sigma ^2}|x-y|^2\\right)$ , where $x,y \\in \\mathbb {R}$ for simplicity.",
"It results $\\varphi (x) \\propto \\left[ 1, \\sqrt{\\frac{1}{1!\\sigma ^2}}x,\\sqrt{\\frac{1}{2!\\sigma ^4}}x^2,\\sqrt{\\frac{1}{3!\\sigma ^6}}x^3,\\dots \\right].$ Intuitively, we can say that (REF ) approximates the infinite dimensional $\\varphi (x)$ by randomly selecting one of its components: this is the role played by $n \\sim \\rho $ .",
"In addition, we introduce the $\\log $ mapping and replace the exponentiation with a Kronecker product.",
"As a consequence, the random weights $\\mathbf {W}$ ensure that $\\Phi (\\mathbf {X})$ achieves a sound approximation of (REF ), in terms of unbiasedness and rapidly decreasing variance.",
"In the rest of the Section we discuss the theoretical foundation of our analysis, where all proofs have been moved to Appendix for convenience.",
"Unbiased estimation.",
"In order for a statistical estimator to be reliable, we need it to be at least unbiased, i.e., its expected value must be equal to the exact function it is approximating.",
"The unbiasedness of the feature map $\\Phi $ of eq.",
"(REF ) for the Log-Euclidean kernel (REF ) is proved by the following result.",
"Theorem 1 Let $\\rho $ be a generic probability distribution over $\\mathbb {N}$ and consider $\\mathbf {X}$ and $\\mathbf {Y}$ , two generic SPD matrices such that $\\Vert \\log \\mathbf {X} \\Vert _F = \\Vert \\log \\mathbf {Y} \\Vert _{F} = 1.$ Then, $\\langle \\Phi (\\mathbf {X}), \\Phi (\\mathbf {Y}) \\rangle $ is an unbiased estimator of (REF ).",
"That is $\\mathbb {E}[\\langle \\Phi (\\mathbf {X}), \\Phi (\\mathbf {Y}) \\rangle ] = K_{\\ell E} (\\mathbf {X}, \\mathbf {Y}),$ where the expectation is computed over $n$ and $\\mathbf {W}$ which define $\\Phi _j(\\mathbf {X})$ as in (REF ).",
"Once averaging upon all possible realizations of $n$ sampled from $\\rho $ and the Gaussian weights $\\mathbf {W}$ , Theorem REF guarantees that the linear kernel $\\langle \\Phi (\\mathbf {X}), \\Phi (\\mathbf {Y}) \\rangle $ induced by $\\Phi $ is equal to $K_{\\ell E}(\\mathbf {X},\\mathbf {Y})$ .",
"This formalizes the unbiasedness of our approximation.",
"On the assumption $\\Vert \\log \\mathbf {X} \\Vert _F = \\Vert \\log \\mathbf {Y} \\Vert _{F} = 1$ .",
"Under a practical point of view, this assumption may seem unfavorable, but this is not the case.",
"The reason is provided by equation (REF ), which is very convenient to compute the logarithm of a SPD matrix.",
"Since in (REF ), $\\Phi (\\mathbf {X})$ is explicitly dependent on $\\log \\mathbf {X}$ , we can simply use (REF ) and then divide each entry of the obtained matrix by $\\Vert \\log \\mathbf {X} \\Vert _F$ .",
"This is a non-restrictive strategy to satisfy our assumption and actually analogous to require input vectors to have unitary norm, which is very common in machine learning [2].",
"Low-variance.",
"One can note that, in Theorem REF , even by choosing $\\nu = 1$ (a scalar feature map), $\\Phi (\\mathbf {X}) = [\\Phi _1(\\mathbf {X})] \\in \\mathbb {R}$ is unbiased for (REF ).",
"However, since $\\Phi $ is an approximated finite version of the exact infinite feature map associated to (REF ), one would expect the quality of the approximation to be very bad in the scalar case, and to improve as $\\nu $ grows larger.",
"This is indeed true, as proved by the following statement.",
"Theorem 2 The variance of $\\langle \\Phi (\\mathbf {X}), \\Phi (\\mathbf {Y}) \\rangle $ as estimator of (REF ) can be explicitly bounded.",
"Precisely, $\\mathbb {V}_{n,\\mathbf {W}}(K_\\Phi (\\mathbf {X},\\mathbf {Y})) \\le \\dfrac{\\mathcal {C}_\\rho }{\\nu ^3} \\exp \\left(\\dfrac{3 - 2\\sigma ^2}{\\sigma ^4}\\right),$ where $\\Vert \\log \\mathbf {X} \\Vert _F = \\Vert \\log \\mathbf {Y} \\Vert _F = 1$ and the variance is computed over all possible realizations of $n \\sim \\rho $ and $\\mathbf {W}$ , the latter being element-wise sampled from a $\\mathcal {N}(0,\\sigma ^2/\\sqrt{n})$ distribution.",
"Also, $\\mathcal {C}_\\rho \\stackrel{\\rm def}{=} \\sum _{n = 0}^{\\infty } \\frac{1}{\\rho (n) n!",
"}$ , the series being convergent.",
"Let us discuss the bound on the variance provided by Theorem REF .",
"Since the bandwidth $\\sigma $ of the kernel function (REF ) we want to approximate is fixed, the term $\\exp \\hspace{-1.42262pt} \\left(\\hspace{-1.42262pt} \\frac{3 \\hspace{-0.56905pt} - \\hspace{-0.56905pt} 2\\sigma ^2}{\\sigma ^4}\\hspace{-1.42262pt} \\right)$ can be left out from our analysis.",
"The bound in (REF ) is linear in $\\mathcal {C}_\\rho $ and inversely cubic in $\\nu $ .",
"When $\\nu $ grows, the increased dimensionality of our feature map $\\Phi $ makes the variance rapidly vanishing, ensuring that the approximated kernel $K_\\Phi (\\mathbf {X}, \\hspace{-1.42262pt} \\mathbf {Y})= \\langle \\Phi (\\mathbf {X}), \\Phi (\\mathbf {Y}) \\rangle $ converges to the target one, that is $K_{\\ell E}$ .",
"Such trend may be damaged by big values of $\\mathcal {C}_\\rho .$ Since the latter depends on the distribution $\\rho $ , let us fix it to be the geometric distribution $\\mathcal {G}(\\theta )$ with parameter $0 \\le \\theta < 1$ .",
"This yields $\\mathcal {C}_\\rho \\propto \\sum _{n = 0}^\\infty \\dfrac{1}{(1 - \\theta )^n \\cdot n!}",
"= \\exp \\left( \\dfrac{1}{1 - \\theta } \\right).$ There is a tradeoff between a low variance (i.e., $\\mathcal {C}_\\rho $ small) and a reduced computational cost for $\\Phi $ (i.e., $n$ small).",
"Indeed, choosing $\\theta \\approx 1$ makes $\\mathcal {C}_\\rho $ big in (REF ).",
"In this case, the integer $n$ sampled from $\\rho = \\mathcal {G}(\\theta )$ is small with great probability: this leads to a reduced number of Kronecker products to be computed in $\\log (\\mathbf {X})^{\\otimes n}$ .",
"Conversely, when $\\theta \\approx 0$ , despite $n$ and the related computational cost of $\\log (\\mathbf {X})^{\\otimes n}$ are likely to grow, $\\mathcal {C}_\\rho $ is small, ensuring a low variance for the estimator.",
"As a final theoretical result, Theorems REF and REF immediately yield that $\\hspace{-11.38092pt}\\mathbb {P}\\hspace{-1.42262pt}\\left[ \\left| \\hspace{-1.42262pt} K_\\Phi (\\mathbf {X}, \\hspace{-1.42262pt} \\mathbf {Y}) \\hspace{-1.42262pt} - \\hspace{-1.42262pt} K_{\\ell E}(\\mathbf {X}, \\hspace{-1.42262pt} \\mathbf {Y}) \\hspace{-1.42262pt} \\right| \\hspace{-1.42262pt} \\ge \\hspace{-1.42262pt}\\epsilon \\right] \\hspace{-1.42262pt} \\le \\hspace{-1.42262pt} \\frac{\\mathcal {C}_\\rho }{\\nu ^3 \\epsilon ^2}\\hspace{-1.42262pt} \\exp \\hspace{-1.42262pt}\\left(\\hspace{-1.99168pt}\\frac{3\\hspace{-1.42262pt} -\\hspace{-1.42262pt} 2\\sigma ^2}{\\sigma ^4}\\hspace{-1.99168pt}\\right)\\hspace{-1.42262pt}$ for every pairs of unitary Frobenius normed SPD matrices $\\mathbf {X},\\mathbf {Y}$ and $\\epsilon > 0$ , as a straightforward implication of the Chebyshev inequality.",
"This ensures that $K_\\Phi $ differs in module from $K_{\\ell E}$ by more than $\\epsilon $ with a (low) probability $\\mathbb {P}$ , which is inversely cubic and quadratic in $\\nu $ and $\\epsilon $ , respectively.",
"Final remarks.",
"To sum up, we have presented a constructive algorithm to compute a $\\nu $ -dimensional feature map $\\Phi $ whose induced linear kernel is an unbiased estimator of the log-Euclidean one.",
"Additionally, we ensure an explicit bound on the variance which rapidly vanishes as $\\nu $ grows (inversely cubic decrease).",
"This implies that $\\langle \\Phi (\\mathbf {X}), \\Phi (\\mathbf {Y}) \\rangle $ and $K_{\\ell E}(\\mathbf {X},\\mathbf {Y})$ are equal with very high probability, even at low $\\nu $ values.",
"This implements a Log-Euclidean kernel in a scalable manner, downgrading the quadratic cost of computing $K_{\\ell E}(\\mathbf {X},\\mathbf {Y})$ for every $\\mathbf {X},\\mathbf {Y}$ into the linear cost of evaluating the feature map $\\Phi (\\mathbf {X})$ as in (REF ) for every $\\mathbf {X}$ .",
"Practically, this achieve a linear implementation of the log-Euclidean SVM."
],
[
"Results",
"In this Section, we compare our proposed approximated feature map versus the alternative ones by Rahimi & Recht [17], Kar & Karnick [11] and Le et al.",
"[13] (see Section ).",
"Theoretical Comparison.",
"Let us notice that all approaches [17], [11], [13] are applicable also to the log-Euclidean kernel (REF ).",
"Indeed, [17], [13] includes our case of study since $K_{\\ell E}(\\mathbf {X},\\mathbf {Y}) = k(\\log \\mathbf {X} - \\log \\mathbf {Y})$ is logarithmic shift invariant.",
"At the same time, thanks to the assumption $\\Vert \\log \\mathbf {X} \\Vert _F = \\Vert \\log \\mathbf {Y} \\Vert _F = 1$ as in Theorem REF , we obtain $K_{\\ell E}(\\mathbf {X},\\mathbf {Y}) = k(\\langle \\log \\mathbf {X}, \\log \\mathbf {Y} \\rangle )$ (see (REF ) in Appendix ), thus satisfying the hypothesis of Kar & Karnick [11].",
"As we proved in Theorem REF , all works [17], [11], [13] can also guarantee an unbiased estimation for the exact kernel function.",
"Actually, what makes our approach superior is the explicit bound on the variance (see Table REF ).",
"Indeed, [17], [11] are totally lacking in this respect.",
"Moreover, despite an analogous bound is provided in [13], it only ensures a $O(1/\\nu )$ convergence rate for the variance with respect to the feature dimensionality $\\nu $ .",
"Differently, we can guarantee a $O(1/{\\nu ^3})$ trend.",
"This implies that, we achieve a better approximation of the kernel with a lower dimensional feature representation, which ease the training of the linear SVM [7].",
"Table: Theoretical comparison between explicit bounds on variance between the proposed approximation and , , : the quicker the decrease, the better the bound.",
"Here, ν\\nu denotes the dimensionality of the approximated feature vector.Figure: Experimental comparison of our approximation (red curves) against the schemes ofr Rahimi & Recht (pink curves), Kar & Karnick (green curves) and Le et al.",
"(blue curves).",
"Best viewed in colors.Experimental Comparison.",
"We reported here the experimental comparison on 3D action recognition between our proposed approximation and the paradigms of [17], [11], [13].",
"Datasets.",
"For the experiments, we considered UTKinect [23], Florence3D [19], MSR-Action-Pairs (MSR-$pairs$ ) [16], MSR-Action3D [14], [3], HDM-05 [15] and MSRC-Kinect12 [8] datasets.",
"For each, we follow the usual training and testing splits proposed in the literature.",
"For Florence3D and UTKinect, we use the protocols of [20].",
"For MSR-Action3D, we adopte the splits originally proposed by [14].",
"On MSRC-Kinect12, once highly corrupted action instances are removed as in [10], training is performed on odd-index subject, while testing on the even-index ones.",
"On HDM-05, the training exploited all instances of “bd” and “mm” subjects, being “bk”, “dg” and “tr” left out for testing [22], using the 65 action classes protocol of [6].",
"Data preprocessing.",
"As a common pre-processing step, we normalize the data by computing the relative displacements of all joints $x-y-z$ coordinates and the ones of the hip (central) joint, for each timestamp.",
"Results.",
"Figure REF reports the quantitative performance while varying $\\nu $ in the range 10, 20, 50, 100, 200, 500, 1000, 2000, 5000.",
"When comparing with [13], since the data input size must be a multiple of a power of 2, we zero-padded our vectorized representation to match 4096 and (whenever possible) 2048 and 1024 input dimensionality.",
"These cases are then compared with the results related to $\\nu =$ 5000, 2000, 1000 for RGW and [17], [11], respectively.",
"Since all approaches have a random component, we performed ten repetitions for each method and dimensionality setup, averaging the scored classification performances obtained through a linear SVM with $C = 10$ .",
"We employ the publicly available codes for [17], [11], [13].",
"Finally, we also report the classification performance with the exact method obtained by feeding an SVM with the log-Euclidean kernel whose bandwidth $\\sigma $ is chosen via cross validation.",
"Discussion.",
"For large $\\nu $ values, all methods are able to reproduce the performance of the log-Euclidean kernel (black dotted line).",
"Still, in almost all the cases, our approximation is able to outperform the competitors: for instance, we gapped Rahimi and Recht on both MSR-Pairs and MSR-Action3D, while Kar & Karnick scored a much lower performance on HDM-05 and Florence3D.",
"If comparing to Le et al., the performance is actually closer, but this happens for all the methods which are able to cope the performance of the Log-Euclidean kernel with $\\nu \\ge 2000,5000$ .",
"Precisely, the true superiority of our approach is evident in the case of a small $\\nu $ value ($\\nu =$ 10, 20, 50).",
"Indeed, our approximation always provides a much rapid growing accuracy (MSR-Action3D, Florence3D and UTKinect), with only a few cases where the gap is thinner (Kar & Karnick [11] on MSR-$pairs$ and Rahimi & Recth [17] on MSRC-Kinect 12).",
"Therefore, our approach ensures a more descriptive and compact representation, providing a superior classification performance."
],
[
"Conclusions",
"In this work we propose a novel scalable implementation of a Log-Euclidean SVM to perform proficient classification of SPD (covariance) matrices.",
"We achieve a linear complexity by providing an explicit random feature map whose induced linear kernel is an unbiased estimator of the exact kernel function.",
"Our approach proved to be more effective than alternative approximations [17], [11], [13], both theoretically and experimentally.",
"Theoretically, we achieve an explicit bound on the variance on the estimator (such result is totally absent in [17], [11]), which is decreasing with inversely cubic pace versus the inverse linear of [13].",
"Experimentally, through a broad evaluation, we assess the superiority of our representation which is able to provide a superior classification performance at a lower dimensionality."
],
[
"Proofs of all theoretical results",
"In this Appendix we report the formal proofs for both the unbiased approximation (Theorem REF ) and the related rapidly decreasing variance (Theorem REF ).",
"[Proof of Theorem REF ] Use the definition of (REF ) and the linearity of the expectation.",
"We get that $\\mathbb {E}_{n,\\mathbf {W}}\\left[ \\langle \\Phi (\\mathbf {X}), \\Phi (\\mathbf {Y}) \\rangle \\right]$ equals to $\\mathbb {E}_{n}\\left[ \\dfrac{1}{\\sigma ^{4n}} \\dfrac{\\exp (-\\sigma ^{-2})}{\\rho (n) n!}",
"\\mathbb {E}_{\\mathbf {W}} \\left[ {\\rm tr}\\left( \\mathbf {W}^\\top \\log (\\mathbf {X})^{\\otimes n} \\right) {\\rm tr} \\left( \\mathbf {W}^\\top \\log (\\mathbf {Y})^{\\otimes n}\\right) \\right]\\right],$ by simply noticing that the dependence with respect to $\\mathbf {W}$ involves the terms inside the trace operators only.",
"Let us focus on the term ${\\rm tr}\\left( \\mathbf {W}^\\top \\log (\\mathbf {X})^{\\otimes n} \\right)$ .",
"We can expand ${\\rm tr}\\left( \\mathbf {W}^\\top \\log (\\mathbf {X})^{\\otimes n} \\right) = \\sum _{i_1,\\dots ,i_{2n} = 1}^d w_{i_1,\\dots ,i_{2n}} \\log (\\mathbf {X})_{i_1,i_2} \\cdots \\log (\\mathbf {X})_{i_{2n -1},i_{2n}}$ by using the definition of $\\log (\\mathbf {X})^{\\otimes n}$ and the properties of the trace operator.",
"In equation (REF ), we replace the random coefficient $w_{i_1,\\dots ,i_{2n}}$ with $u^{(1)}_{i_1,i_2},\\dots ,u^{(n)}_{i_{2n - 1},i_{2n}}$ independent and identically distributed (i.i.d.)",
"according to a $\\mathcal {N}(0,\\sigma ^2)$ distribution.",
"We can notice that (REF ) can be rewritten as ${\\rm tr}\\left( \\mathbf {W}^\\top \\log (\\mathbf {X})^{\\otimes n} \\right) = \\prod _{\\alpha = 1}^{n} \\sum _{i,j = 1}^d u^{(\\alpha )}_{i,j} \\log (\\mathbf {X})_{ij}.$ Making use of (REF ) in (REF ), we can rewrite $\\mathbb {E}_{n,\\mathbf {W}}\\left[ K_\\Phi (\\mathbf {X},\\mathbf {Y})\\right]$ as $\\mathbb {E}_{n}\\left[ \\dfrac{1}{\\sigma ^{4n}} \\dfrac{\\exp (-\\sigma ^{-2})}{\\rho (n) n!}",
"\\mathbb {E}_{\\mathbf {W}} \\left[ \\left( \\sum _{i,j = 1}^d u^{(1)}_{i,j} \\log (\\mathbf {X})_{ij} \\right)\\left( \\sum _{h,k = 1}^d u^{(1)}_{h,k} \\log (\\mathbf {Y})_{hk} \\right) \\right]^n \\right] $ by also considering the independence of $u^{(\\alpha )}_{i,j}$ are independent.",
"By furthermore using the fact that $\\mathbb {E}_{\\mathbf {W}}\\left[ u^{(1)}_{i,j}u^{(1)}_{h,k} \\right] = 0$ if $i \\ne h$ and $j \\ne k$ and the formula for the variance of a Gaussian distribution, we get $\\mathbb {E}_{n,\\mathbf {W}}\\left[ K_\\Phi (\\mathbf {X},\\mathbf {Y})\\right] = \\mathbb {E}_{n}\\left[ \\dfrac{1}{\\sigma ^{4n}} \\dfrac{\\exp (-\\sigma ^{-2})}{\\rho (n) n!}",
"\\sigma ^{2n} \\left( \\langle \\log (\\mathbf {X}), \\log (\\mathbf {Y}) \\rangle _F \\right)^{n} \\right], $ by introducing the Frobenius inner product $\\langle \\mathbf {A}, \\mathbf {B} \\rangle _F = \\sum _{i,j = 1}^d \\mathbf {A}_{ij} \\mathbf {B}_{ij}$ between matrices $\\mathbf {A}$ and $\\mathbf {B}$ .",
"By expanding the expectation over $\\rho $ , (REF ) becomes $\\mathbb {E}_{n,\\mathbf {W}}\\left[ K_\\Phi (\\mathbf {X},\\mathbf {Y})\\right] &= \\sum _{n = 0}^\\infty \\rho (n) \\dfrac{1}{\\sigma ^{2n}} \\dfrac{\\exp (-\\sigma ^{-2})}{\\rho (n)n!}",
"(\\langle \\log (\\mathbf {X}), \\log (\\mathbf {Y}) \\rangle _F)^n \\nonumber \\\\ &= \\exp \\left(-\\dfrac{1}{\\sigma ^{2}}\\right) \\sum _{n = 0}^\\infty \\left(\\dfrac{\\langle \\log (\\mathbf {X}), \\log (\\mathbf {Y}) \\rangle _F}{\\sigma ^{2}}\\right)^n \\dfrac{1}{n!}.",
"$ The thesis easily comes from (REF ) by using the Taylor expansion for the exponential function and the assumption $\\Vert \\log (\\mathbf {X})\\Vert _F = \\Vert \\log (\\mathbf {Y})\\Vert _F = 1$ .",
"[Proof of Theorem REF ] Due to the independence of the components in $\\Phi $ , by definition of inner product we get $\\mathbb {V}_{n,\\mathbf {W}} \\left[ \\langle \\Phi (\\mathbf {X}),\\Phi (\\mathbf {Y}) \\rangle \\right] = \\nu \\mathbb {V}_{n,\\mathbf {W}} \\left[ \\Phi _1(\\mathbf {X}) \\Phi _1(\\mathbf {Y}) \\right]$ .",
"But then $\\mathbb {V}_{n,\\mathbf {W}} \\left[ \\langle \\Phi (\\mathbf {X}),\\Phi (\\mathbf {Y}) \\rangle \\right] \\le \\nu \\mathbb {E}_{n,\\mathbf {W}} \\left[ \\Phi _1(\\mathbf {X})^2 \\Phi _1(\\mathbf {Y})^2 \\right]$ by definition of variance.",
"Taking advantage of (REF ), yields to the equality between $\\mathbb {V}_{n,\\mathbf {W}} \\left[ K_\\Phi (\\mathbf {X},\\mathbf {Y}) \\right]$ and $\\dfrac{1}{\\nu ^3} \\mathbb {E}_{n,\\mathbf {U}} \\left[ \\dfrac{1}{\\sigma ^{8n}} \\dfrac{\\exp (-2\\sigma ^{-2})}{(\\rho (n) n!",
")^2} \\prod _{\\alpha = 1}^{n} \\left( \\sum _{i,j = 1}^d u^{(\\alpha )}_{i,j} \\log (\\mathbf {X})_{ij} \\right)^{\\hspace{-2.84544pt} 2} \\hspace{-2.84544pt} \\left( \\sum _{h,k = 1}^d u^{(\\alpha )}_{h,k} \\log (\\mathbf {Y})_{hk} \\right)^{\\hspace{-2.84544pt} 2} \\right],$ where $u^{(1)}_{i_1,i_2},\\dots ,u^{(n)}_{i_{2n - 1},i_{2n}}$ are i.i.d.",
"from $\\mathcal {N}(0,\\sigma ^2)$ distribution used to re-parametrize the original weights $\\mathbf {W}$ .",
"Exploit the independence of $u^{(\\alpha )}_{ij}$ to rewrite (REF ) as $\\dfrac{1}{\\nu ^3} \\mathbb {E}_{n} \\left[ \\dfrac{1}{\\sigma ^{8n}} \\dfrac{\\exp (-2\\sigma ^{-2})}{(\\rho (n) n!",
")^2} \\mathbb {E}_{\\mathbf {U}} \\left[ \\left( \\sum _{i,j = 1}^d u^{(1)}_{i,j} \\log (\\mathbf {X})_{ij} \\right)^{\\hspace{-5.0pt}2}\\left( \\sum _{h,k = 1}^d u^{(1)}_{h,k} \\log (\\mathbf {Y})_{hk} \\right)^{\\hspace{-5.0pt} 2} \\right]^{n}\\right].",
"$ By exploiting the zero correlation of the weights in $\\mathbf {U}$ and the formula $\\mathbb {E}[(\\mathcal {N}(0,\\sigma ^2))^4] = 3\\sigma ^4$ [4].",
"Thus, $\\mathbb {V}_{n,\\mathbf {W}} \\left[ K_\\Phi (\\mathbf {X},\\mathbf {Y}) \\right] \\le \\dfrac{1}{\\nu ^3} \\mathbb {E}_{n} \\left[ \\dfrac{1}{\\sigma ^{8n}} \\dfrac{\\exp (-2\\sigma ^{-2})}{(\\rho (n) n!",
")^2} 3^n \\sigma ^{4n} \\left( \\sum _{i,j=1}^d \\log (\\mathbf {X})^2_{ij} \\log (\\mathbf {Y})^2_{ij}\\right)^n \\right].",
"$ Since $\\sum _{i,j=1}^d \\log (\\mathbf {X})^2_{ij} \\log (\\mathbf {Y})^2_{ij} \\le \\left( \\sum _{i,j=1}^d \\log (\\mathbf {X})^2_{ij} \\right) \\left( \\sum _{i,j=1}^d \\log (\\mathbf {Y})^2_{ij} \\right) = 1$ due to the assumption of unitary Frobenius norm for both $ \\log \\mathbf {X}$ and $ \\log \\mathbf {Y}$ , we get $\\mathbb {V}_{n,\\mathbf {W}} \\left[ K_\\Phi (\\mathbf {X},\\mathbf {Y}) \\right] \\le \\dfrac{1}{\\nu ^3} \\mathbb {E}_{n} \\left[ \\dfrac{1}{\\sigma ^{8n}} \\dfrac{\\exp (-2\\sigma ^{-2})}{(\\rho (n) n!",
")^2} 3^n \\sigma ^{4n} \\right].",
"$ We can now expand the expectation over $\\rho $ in (REF ), achieving $\\mathbb {V}_{n,\\mathbf {W}} \\left[ K_\\Phi (\\mathbf {X},\\mathbf {Y}) \\right] \\le \\dfrac{\\exp (-2\\sigma ^{-2})}{\\nu ^3} \\sum _{n = 0}^\\infty \\left(\\dfrac{3}{\\sigma ^4}\\right)^n \\dfrac{1}{n!}",
"\\sum _{n = 0}^\\infty \\dfrac{1}{\\rho (n) n!",
"}, $ since the series of the products is less than the product of the series, provided that both converge.",
"This is actually true since, by exploiting the McLaurin expansion for the exponential function, we easily get $\\sum _{n = 0}^\\infty \\left(\\frac{3}{\\sigma ^4}\\right)^n \\frac{1}{n!}",
"= \\exp \\left( \\frac{3}{\\sigma ^4} \\right)$ .",
"On the other hand, since $\\rho $ is a probability distribution, it must be $\\lim _{n \\rightarrow \\infty } \\frac{\\rho (n+1)}{\\rho (n)} = L$ where $0 < L \\le 1$ , being $\\mathbb {N}$ the support of $\\rho $ and due to $\\sum _{n = 0}^\\infty \\rho (n) = 1$ .",
"Then, since $\\lim _{n \\rightarrow \\infty } \\frac{\\rho (n)}{\\rho (n+1)} = \\frac{1}{L} < \\infty $ and $\\lim _{n \\rightarrow \\infty } \\frac{1}{n +1} = 0$ , by the ration criterion for positive-terms series [18], there must exist a constant $\\mathcal {C}_\\rho > 0$ such that $\\sum _{n = 0}^\\infty \\dfrac{1}{\\rho (n) n!}",
"= \\mathcal {C}_\\rho .$ Therefore, by combining (REF ) in (REF ), we obtain $\\mathbb {V}_{n,\\mathbf {W}} \\left[ K_\\Phi (\\mathbf {X},\\mathbf {Y}) \\right] \\le \\dfrac{\\exp (-2\\sigma ^{-2})}{\\nu ^3} \\exp \\left(\\dfrac{3}{\\sigma ^4}\\right) \\mathcal {C}_\\rho = \\dfrac{\\mathcal {C}_\\rho }{\\nu ^3} \\exp \\left(\\dfrac{3 - 2\\sigma ^2}{\\sigma ^4}\\right), \\nonumber $ which is the thesis."
]
] | 1709.01695 |
[
[
"Polarimetric Study of Near-Earth Asteroid (1566) Icarus"
],
[
"Abstract We conducted a polarimetric observation of the fast-rotating near-Earth asteroid (1566) Icarus at large phase (Sun-asteroid-observer's) angles $\\alpha$= 57 deg--141deg around the 2015 summer solstice.",
"We found that the maximum values of the linear polarization degree are $P_\\mathrm{max}$=7.32$\\pm$0.25 % at phase angles of $\\alpha_\\mathrm{max}$=124$\\pm$8 deg in the $V$-band and $P_\\mathrm{max}$=7.04$\\pm$0.21 % at $\\alpha_\\mathrm{max}$=124$\\pm$6 deg in the $R_\\mathrm{C}$-band.",
"Applying the polarimetric slope-albedo empirical law, we derived a geometric albedo of $p_\\mathrm{V}$=0.25$\\pm$0.02, which is in agreement with that of Q-type taxonomic asteroids.",
"$\\alpha_\\mathrm{max}$ is unambiguously larger than that of Mercury, the Moon, and another near-Earth S-type asteroid (4179) Toutatis but consistent with laboratory samples with hundreds of microns in size.",
"The combination of the maximum polarization degree and the geometric albedo is in accordance with terrestrial rocks with a diameter of several hundreds of micrometers.",
"The photometric function indicates a large macroscopic roughness.",
"We hypothesize that the unique environment (i.e., the small perihelion distance $q$=0.187 au and a short rotational period of $T_\\mathrm{rot}$=2.27 hours) may be attributed to the paucity of small grains on the surface, as indicated on (3200) Phaethon."
],
[
"Introduction",
"The polarimetry of solar system airless bodies (e.g., the Moon, Mercury and asteroids) is a useful diagnostic measure for investigating their surface physical properties, such as the albedo and regolith size.",
"The linear polarization degree $P_\\mathrm {r}$ is defined as $P_\\mathrm {r}=\\frac{I_\\bot -I_\\Vert }{I_\\bot +I_\\Vert },$ where $I_\\bot $ and $I_\\Vert $ denote the intensities of scattered light measured with respect to the scattering plane.",
"In general, $P_\\mathrm {r}$ exhibits a strong dependence on the phase angle (Sun–target–observer's angle, $\\alpha $ ), consisting of a negative branch at $\\alpha \\lesssim $ 20 with a minimum value $P_\\mathrm {min}$ at the phase angle $\\alpha _\\mathrm {min} \\sim $ 10 and a positive branch at $\\alpha \\gtrsim $ 20 with a maximum value $P_\\mathrm {max}$ at $\\alpha _\\mathrm {max}\\sim $ 100 [17].",
"It is well known that the albedo (the $V$ -band geometric albedo, $p_\\mathrm {V}$ , is often referenced) of objects has a strong correlation with the polarization degree (so-called Umov's law) because multiple scattering (which is dependent on the single–scattering albedo) randomizes polarization vectors and eventually weakens the polarization degree of the scattered signal from the bodies.",
"Moreover, it was noted that $P_\\mathrm {max}$ and $\\alpha _\\mathrm {max}$ have a moderate correlation with grain size [4], [12], [56].",
"While intensive polarimetric research on asteroids has been conducted at small phase angles [2], [21], [20], [19], [5], [3], polarimetric studies of asteroids at large $\\alpha $ values remain less common, most likely because of fewer opportunities (limited to near-Earth asteroids, NEAs) and observational difficulty (small solar elongation).",
"Here, we would like to stress the superiority of observatories at middle latitudes ($|l|\\sim $ 45) for NEA observations at large $\\alpha $ values.",
"During the summer solstice, the Sun does not set at latitudes of $|l|>$ 66.6 (a phenomenon called the midnight Sun); in addition, astronomical twilight lasts through the night at observatories at $|l|>$ 48.6, making it difficult to make astronomical observations.",
"When we conduct observations at observatories at longitudes slightly lower than $|l|$ =48.6, we are able to observe NEAs around the Sun as if we were using the Earth as a coronagraph.",
"Taking advantage of this location, we conducted a polarimetric observation of an NEA, (1566) Icarus (=1949 MA), at the Nayoro Observatory ($l$ =+44.4) in Hokkaido, Japan, around the summer solstice in 2015.",
"Icarus is one of the Apollo asteroids that has a small perihelion distance of $q$ =0.187 au.",
"Such asteroids with a small perihelion distance have gained the attention of solar system scientists interested in understanding the mass erosion mechanisms on these bodies [37], [23].",
"Icarus has a diameter of 0.8–1.3 km [62], [43], [28], [49] and a short rotational period of 2.27 hours [45], [18], [27], [65].",
"The asteroid is classified as an S–type asteroid [8] or, more specifically, as a Q–type asteroid [9].",
"Thanks to the favorable location of the observatory, we were able to acquire polarimetric data up to $\\alpha $ =141 and imaging data up to $\\alpha $ =145.",
"The phase angle is overwhelmingly larger than those of previous polarimetric observations at large phase angles [38], [2], [32].",
"We describe our observations in Section 2, the data reduction in Section 3, and the results in Section 4.",
"Finally, we compare our polarimetric results with those of laboratory samples and solar system airless bodies, and we consider the surface regolith properties of the asteroid in Section 5."
],
[
"Observation",
"The journal of our observations is given in Table .",
"We conducted observations for eight nights from UT 2015 June 11 to UT 2015 June 20 using the 1.6 m Pirka telescope at the Nayoro Observatory (14228$^{\\prime }$$58.0^{\\prime \\prime }$ , +4422$^{\\prime }$ 25.1$^{\\prime \\prime }$ , 192.1 m, observatory code number Q33).",
"The observatory has been operated since 2011 by the Faculty of Science, Hokkaido University, Japan.",
"We utilized a Multi-Spectral Imager (MSI) mounted at the $f/$ 12 Cassegrain focus of the Pirka telescope.",
"The combination of the telescope and the instrument enables the acquisition of images that cover a 3.3$\\times $ 3.3 field-of-view (FOV) with a 0.39 pixel resolution [66].",
"We conducted an imaging observation on the first night, June 11, to test the non-sidereal tracking of the fast moving object at a low elevation ($\\sim $ 10).",
"These data were used for the study of photometric functions presented in Section REF .",
"After the second night, we made an imaging polarimetric observation from June 12–20 using a polarimetric module comprising a Wollaston prism and a half-wave plate.",
"To avoid the blending of ordinary and extraordinary rays, a two-slit mask was placed at the focal plane for the polarimetric observation.",
"With the mask, the FOV was subdivided into two adjacent sky areas of 3.3$\\times $ 0.7 each and separated by 1.7 (see Figure REF ).",
"We chose standard Johnson-Cousins $V$ and $R_\\mathrm {C}$ –band filters for this study to examine the wavelength dependence of the polarization degree.",
"We took polarimetric data with exposure times of either 30 seconds or 60 seconds (depending on the apparent magnitude of the asteroid) for a single frame.",
"Between exposures, the half-wave plate was routinely rotated from 0 to 45, from 45 to 22.5, and from 22.5 to 67.5 in sequence to acquire one subset of polarimetric data.",
"Once we acquired the subset of data, the pointing direction of the telescope was shifted by +10 and -10 in turn along the east–west axis (the longer axis of the polarization mask) to acquire the other two subsets.",
"This technique (called dithering) can reduce the effects of pixel-to-pixel inhomogeneity that were not substantially corrected by flat-field correction.",
"Accordingly, each set of data consists of twelve exposures (four exposures with different half-wave plate angles $\\times $ three locations on the CCD chip with the $\\pm $ 10-dithering mode).",
"We took bias frames before and after the asteroid observations in approximately 3 hour intervals.",
"At the end of nightly observation, we obtained dome flat field data at the same focal position of the telescope as that with which we observed the asteroid."
],
[
"DATA ANALYSIS",
"The observed data were analyzed in the same manner as in [40] and [31].",
"The raw observational data were preprocessed using flat images and bias frames by the MSI data reduction package (MSIRED).",
"Cosmic rays on the images were erased using the L.A.Cosmic tool [11].",
"After these processes, we extracted source fluxes on ordinary and extraordinary parts of the images (see Figure 1) using the aperture photometry package in IRAF.",
"The obtained fluxes were used to derive the Stokes parameters after completing the necessary procedures for the Pirka/MSI data (see Appendix A).",
"These procedures contain corrections for polarization efficiencies, the subtraction of instrumental polarization, and the conversion into the standard celestial coordinate system.",
"The linear polarization degree ($P$ ) and the position angle of polarization ($\\theta _\\mathrm {P}$ ) were derived with the following equations: $P = \\sqrt{\\left(q^{\\prime \\prime \\prime }_\\mathrm {pol}\\right)^2+{\\left(u^{\\prime \\prime \\prime }_\\mathrm {pol}\\right)}^2}~,$ and $\\theta _\\mathrm {P}=\\frac{1}{2}\\tan ^{-1}\\left(\\frac{u^{\\prime \\prime \\prime }_\\mathrm {pol}}{q^{\\prime \\prime \\prime }_\\mathrm {pol}}\\right)~,$ where $q^{\\prime \\prime \\prime }_\\mathrm {pol}$ and $u^{\\prime \\prime \\prime }_\\mathrm {pol}$ are the Stokes parameters $Q$ and $U$ , respectively, normalized by $I$ after correcting for instrumental effects.",
"We derived the linear polarization degree with respect to the scattering plane: $P_\\mathrm {r}=P \\cos \\left(2\\theta _\\mathrm {r}\\right)~,$ where $\\theta _\\mathrm {r}$ is given by $\\theta _\\mathrm {r}=\\theta _\\mathrm {P}-\\left(\\phi \\pm 90\\right)~,$ where $\\phi $ is the position angle of the scattering plane on the sky.",
"The polarization degree of each set of four exposures has an error of 0.2–5 % (depending largely on the apparent magnitudes of the nights).",
"We combined these sets of ($q^{\\prime \\prime \\prime }$ , $u^{\\prime \\prime \\prime }$ ) values to obtain nightly averaged $q^{\\prime \\prime \\prime }$ and $u^{\\prime \\prime \\prime }$ values, addressing systematic noise ($\\delta _{q^{\\prime \\prime \\prime }}$ and $\\delta _{u^{\\prime \\prime \\prime }}$ ) and random noise ($\\sigma _{q^{\\prime \\prime \\prime }}$ and $\\sigma _{u^{\\prime \\prime \\prime }}$ ), separately.",
"Regarding systematic noise, we took the arithmetic averages ($\\delta _{\\overline{q}^{\\prime \\prime \\prime }}$ and $\\delta _{\\overline{u}^{\\prime \\prime \\prime }}$ ).",
"Regarding the synthesized random errors, we calculated the variances of the weighted means, given by $\\sigma ^2_{\\overline{q}^{\\prime \\prime \\prime }}=\\frac{1}{\\sum ^n_{i=1}\\left(\\sigma _{q_i^{\\prime \\prime \\prime }}\\right)^{-2}}~~, ~~~~~~~~~\\sigma ^2_{\\overline{u}^{\\prime \\prime \\prime }}=\\frac{1}{\\sum ^n_{i=1}\\left(\\sigma _{u_i^{\\prime \\prime \\prime }}\\right)^{-2}}~~,$ where $\\sigma ^2_{\\overline{q}^{\\prime \\prime \\prime }}$ and $\\sigma ^2_{\\overline{u}^{\\prime \\prime \\prime }}$ are the synthesized random errors for $\\overline{q}^{\\prime \\prime \\prime }$ and $\\overline{u}^{\\prime \\prime \\prime \\prime }$ , respectively.",
"The resultant values for $q^{\\prime \\prime \\prime }$ and $u^{\\prime \\prime \\prime }$ are given by $\\overline{q}^{\\prime \\prime \\prime }= \\sigma ^2_{\\overline{q}^{\\prime \\prime \\prime }} \\sum _{i=1}^n \\frac{q_i^{\\prime \\prime \\prime }}{\\sigma ^2_{q_i^{\\prime \\prime \\prime }}}~~, ~~~~~~~~~\\overline{u}^{\\prime \\prime \\prime }= \\sigma ^2_{\\overline{u}^{\\prime \\prime \\prime }} \\sum _{i=1}^n \\frac{u_i^{\\prime \\prime \\prime }}{\\sigma ^2_{u_i^{\\prime \\prime \\prime }}}~~,$ with total errors of $\\epsilon _{\\overline{q}^{\\prime \\prime \\prime }}=\\sqrt{\\sigma ^2_{\\overline{q}^{\\prime \\prime \\prime }}+\\delta ^2_{\\overline{q}^{\\prime \\prime \\prime }}}~~, ~~~~~~~~~\\epsilon _{\\overline{u}^{\\prime \\prime \\prime }}=\\sqrt{\\sigma ^2_{\\overline{u}^{\\prime \\prime \\prime }}+\\delta ^2_{\\overline{u}^{\\prime \\prime \\prime }}}~~.$ Similarly, using Eqs.",
"(REF )–(REF ), we obtained synthesized $P_\\mathrm {r}$ and $\\theta _\\mathrm {P}$ values, as shown in the following sections."
],
[
"RESULTS",
"We summarize our polarimetric results in Table .",
"We describe our findings below."
],
[
"Phase Angle Dependence and Polarimetric Color",
"Figure REF shows the nightly averaged polarization degrees with respect to the phase angles.",
"At first glance, we noted that the polarization degree increases almost linearly with increasing phase angle at $\\alpha $ =60–100, has a peak $\\alpha \\sim 120$ , and drops at $\\alpha $ =125–140.",
"We also found that the $V$ polarization degrees are higher than the $R_\\mathrm {C}$ polarization degrees regardless of the observed phase angles.",
"The average difference is $\\Delta P_\\mathrm {r}$ =$-0.37$$\\pm $ 0.04%.",
"This trend is similar to that observed for other S-type asteroids [47], [42], [2] and a Q-type asteroid [16].",
"To obtain $\\alpha _\\mathrm {max}$ and $P_\\mathrm {max}$ , we fit our data using the Lumme and Muinonen function [22], [52]: $P_\\mathrm {r}=b \\sin ^{c_1} \\left(\\alpha \\right) \\cos ^{c_2}\\left(\\frac{\\alpha }{2}\\right) \\sin \\left(\\alpha -\\alpha _0\\right),$ where $b$ , $c_{1}$ , $c_{2}$ and $\\alpha _0$ are parameters for fitting.",
"Since our data are not covered at lower phase angles, we fixed the inversion angle $\\alpha _0$ =20 (a typical value for S-type and Q-type asteroids, [3]) and derived the other three parameters by weighting with the square of the errors.",
"We obtained $\\alpha _\\mathrm {max}$ =124$\\pm $ 8 and $P_\\mathrm {max}$ =7.32$\\pm $ 0.25% in the $V$ -band and $\\alpha _\\mathrm {max}$ =124$\\pm $ 6 and $P_\\mathrm {max}$ =7.04$\\pm $ 0.21% in the $R_\\mathrm {C}$ -band.",
"However, we noted that $c_2$ has a negative value, which does not make sense per the original definition of the trigonometric function [52].",
"We discuss this insufficiency and describe the error analysis in Section ."
],
[
"Rotational Variation in $P_\\mathrm {r}$",
"Figure REF shows the polarization degrees with respect to time from UT 2015 June 16 ($\\alpha $ =100.2).",
"We choose the data from this night not only because the sky was clear and stable but also because the time coverage was long enough to see a rotational variability in the polarization degree.",
"We combined each set of data taken at three different positions on the detector (see Section ), excluding several images where field stars overlapped the asteroid.",
"The data cover approximately two rotational periods of the asteroid.",
"From this, we found that the polarization degree was notably constant over the quadrant ($\\sim $ 1/4 because $\\alpha $$\\sim $ 90) of the surface.",
"We determined the upper limit of the rotational variation in $P_\\mathrm {r}$ as 0.3% in the $V$ band and 0.2 % in the $R_\\mathrm {C}$ band with a 1-sigma confidence level.",
"It has been reported that some large asteroids show rotational variations of $P_\\mathrm {r}$ .",
"A notable example is (4) Vesta [13], which showed a 0.1% polarimetric variation, and the maximum of the polarization coincides with the lightcurve minimum, suggesting that albedo variation exists on the surface per the controlled polarization degree and visible magnitude.",
"Similarly, (3) Juno, (9) Metis, and (216) Kleopatra showed rotational variations of 0.15–0.27 %, $\\sim $ 0.1 %, and $\\sim $ 0.2 %, respectively [58], [48], [59].",
"Although the measurement accuracy is too limited to detect such a small variations in the polarization degree, we suggest that Icarus has a quite homogeneous albedo in contrast with these asteroids because our measurement was made at a large phase angle, while these previous detections were made at a small phase angle where the polarization degree itself has small values (1/6$\\sim $ 1/10 of Icarus's $|P_\\mathrm {r}|$ , i.e., 0.5$\\lesssim |P_\\mathrm {r}|\\lesssim $ 1.0%).",
"We will discuss this homogeneity in Section ."
],
[
"Photometric Function and Macroscopic Roughness",
"As a byproduct of our polarimetric observation, we took images without using the polarimetric module.",
"These images were obtained mostly in the $R_\\mathrm {C}$ -band filter when we tested the non-sidereal tracking of the telescope or set the position of the asteroid in the narrow FOV of the polarization mask.",
"Through comparison with the fluxes of field stars with magnitudes listed in the third U.S.",
"Naval Observatory CCD Astrograph Catalog (UCAC3) [67], we derived the $R_\\mathrm {C}$ -band magnitude of Icarus.",
"Applying the $V-R_\\mathrm {C}$ color index of 0.57$\\pm $ 0.08 [18], the magnitude was converted into the $V$ -magnitude.",
"The observed magnitude, $m_\\mathrm {V}$ , was converted into the reduced $V$ -magnitude, $m_\\mathrm {V}(1,1,\\alpha )$ , a magnitude at unit heliocentric and observer's distances that is given by $m_\\mathrm {V}(1,1,\\alpha )=m_\\mathrm {V} - 5~\\log (r_\\mathrm {h} \\Delta )~~,$ where $r_\\mathrm {h}$ and $\\Delta $ are the heliocentric and observer's distances in au.",
"Figure REF is the reduced $V$ -magnitude with respect to the phase angle.",
"In the figure, the magnitude data at $\\alpha >$ 120 were obtained by us, while the data at $\\alpha <$ 110 were obtained from [18] and [65].",
"The phase curve was fitted with the disk-integrated Hapke model [26].",
"$m_\\mathrm {V}(1,1,\\alpha )$ data are converted into the logarithm of $I/F$ (where $F$ is the incidence solar irradiance divided by $\\pi $ , and $I$ is the intensity of reflected light from the asteroid surface) as $-2.5 \\log \\left(\\frac{I}{F}\\right) = m_\\mathrm {V}(1,1,\\alpha ) - m_{V \\odot } -\\frac{5}{2} \\log \\left( \\frac{\\pi }{S}\\right) + m_c ~~,$ where $m_{V \\odot }$ =$-26.74$ [1] is the solar magnitude at 1 au, $S$ is the geometrical cross section of the asteroid in m$^2$ , and $m_c$ = $-5\\log (1.4960 \\times 10^{11})$ = $-55.87$ is a constant to adjust the length unit.",
"The disk-integrated Hapke function is given by $\\nonumber \\frac{I}{F} =\\left[\\left(\\frac{w}{8}\\left[\\left(1+B(\\alpha )\\right)P(\\alpha )-1\\right]+\\frac{r_0}{2}(1-r_0)\\right)\\left(1-\\sin \\left(\\frac{\\alpha }{2}\\right)\\tan \\left(\\frac{\\alpha }{2}\\right)\\ln \\left[\\cot \\left(\\frac{\\alpha }{4}\\right)\\right]\\right) \\right.\\\\\\left.",
"+\\frac{2}{3}r_0^2\\left(\\frac{\\sin (\\alpha )+(\\pi -\\alpha )\\cos (\\alpha )}{\\pi }\\right)\\right]K(\\alpha ,\\bar{\\theta })~~,$ where $w$ is the single-particle scattering albedo.",
"$K(\\alpha ,\\bar{\\theta })$ is a function that characterizes the surface roughness parameterized by $\\bar{\\theta }$ [25].",
"The term $r_0$ is given by $r_0 = \\frac{1-\\sqrt{1-w}}{1+\\sqrt{1-w}}~~.$ The opposition effect term $B(\\alpha )$ is given by $B(\\alpha ) = \\frac{B_0}{1+\\frac{\\tan (\\alpha / 2)}{h}}~~,$ where $B_0$ denotes the amplitude of the opposition effect, and $h$ characterizes the width of the opposition effect.",
"Two parameters of a double Henyey-Greenstein function, $P(\\alpha )$ [41], was employed: $P(\\alpha )=\\frac{(1-c_\\mathrm {HG})(1-b_\\mathrm {HG}^2)}{(1-2~b_\\mathrm {HG} \\cos (\\alpha ) +b_\\mathrm {HG}^2)^{3/2}}+\\frac{c_\\mathrm {HG}(1-b_\\mathrm {HG}^2)}{(1+2~b_\\mathrm {HG} \\cos (\\alpha ) +b_\\mathrm {HG}^2)^{3/2}}~~.$ For the fitting, we fixed opposition parameters as $B_0$ =0.02 and $h$ =0.141 [41] as an analog of S/Q–type asteroid.",
"By changing the initial values of $b_\\mathrm {HG}$ , $c_\\mathrm {HG}$ , $w$ , and $\\bar{\\theta }$ to the range of $0.01 \\le b_\\mathrm {HG} \\le 0.8$ , $0.01 \\le c \\le 0.8$ , $0.1\\le w \\le 0.8$ , and $5\\le \\bar{\\theta } \\le 55$ , we searched for the best-fit parameters.",
"From the fitting, we obtained $b_\\mathrm {HG}$ =0.42$\\pm $ 0.08, $c_\\mathrm {HG}$ =0.41$\\pm $ 0.20, w=0.48$\\pm $ 0.10, and $\\bar{\\theta }$ =48$\\pm $ 6.",
"Although there are large uncertainties in $b_\\mathrm {HG}$ and $c_\\mathrm {HG}$ , we found that $\\bar{\\theta }$ is significantly larger for the 10–30 km-sized S-type asteroids (243) Ida and (951) Gaspra [30], [29].",
"Note that we assumed the diameter of the asteroid to be 1440 m. If we change the assumed size, $w$ would be different, while $\\bar{\\theta }$ is nearly constant.",
"Considering the large uncertainty ($\\sim $ 18%) in the size [24], the fitting provides a reliable result only for $\\bar{\\theta }$ .",
"The large value of $\\bar{\\theta }$ may suggest that there are few small particles equivalent to the wavelength (i.e., micrometer or smaller), resulting in the large macroscopic roughness."
],
[
"Description for Deriving $P_\\mathrm {max}$ and {{formula:f00953f4-f5f4-4105-88ad-bbf7e4821015}} and Their Errors",
"Lumme & Muinonen's equation, Eq.",
"(REF ), has been widely used for the fitting of polarimetric phase curves because it produces several key features of the phase curve, including $P_\\mathrm {r}$ =0 % at $\\alpha $ =0, $\\alpha _0$ and 180, a negative branch at 0$<\\alpha <$$\\alpha _0$ , and a positive branch at $\\alpha >$$\\alpha _0$ .",
"By definition, the power components $c_1$ and $c_2$ should be positive.",
"The equation has fitted the observed polarimetric phase curves with lower phase angle data well in previous studies [52].",
"However, we noted an incompleteness in this function at large phase angles, where we made our observation.",
"We found that the function cannot fit polarimetric data when the phase curve has $\\alpha _\\mathrm {max}\\gtrsim $ 110 .",
"Thus, the derivative $dP(\\alpha )/d\\alpha $ =0 has no root at $\\alpha \\gtrsim $ 110 in the case of $c_2>0$ .",
"Our polarimetric data show $dP(\\alpha )/d\\alpha \\sim $ 0 around $\\alpha $ =120, indicating $\\alpha _\\mathrm {max}\\sim $ 120 .",
"As we mentioned in Section REF , we fit the observed phase profiles with Lumme & Muinonen's equation without any restriction for ranges of $b$ , $c_1$ and $c_2$ and obtained a negative value for $c_2$ .",
"Although $c_2$ is out of the range of the original definition, it gives a reasonable fit to the data, as shown in Figure .",
"We also noted that the best-fit parameters ($b$ , $c_1$ , and $c_2$ ) can be changed for different initial assumptions for $\\alpha _0$ ; however, $\\alpha _\\mathrm {max}$ and $P_\\mathrm {max}$ still remain nearly constant, most likely because our observation covered the range of the maximum where $dP(\\alpha )/d\\alpha $ =0, providing a strict condition for determining reliable estimates for these parameters.",
"We derived the errors for $\\alpha _\\mathrm {max}$ and $P_\\mathrm {max}$ in the following manner: We initially obtained $b$ =0.0482, $c_1$ =0.6521, $c_2$ =$-0.7756$ , and $\\alpha _0$ =15.0 in the $V$ -band and $b$ =0.0418, $c_1$ =0.9149, $c_2$ =$-0.9825$ , and $\\alpha _0$ =15.0 in the $R_\\mathrm {C}$ -band using the trust–region–reflective algorithm with weighting by the inverse squared error.",
"To search for the marginalized 1-sigma errors of $\\alpha _\\mathrm {max}$ and $P_\\mathrm {max}$ , we generated parameter spaces and calculated the chi-square statistic.",
"We thus set the parameter spaces as follows: $ 0.8b_0 < b < 1.2b_0 $ with $ \\Delta b = 0.01 b_0 $ ; $ 0.4c_{1,0} < c_1 < 1.6c_{1,0} $ with $ \\Delta c_1 = 0.01 c_{1,0} $ ; and $ 0.7c_{2,0} < c_2 < 1.3c_{2,0} $ with $ \\Delta c_2 = 0.01 c_{2,0} $ , where the subscript 0 denotes the “best fit” parameter values from the above initial guess and $ \\Delta $ means the bin sizes for the parameter search.",
"As a result, we obtained $\\alpha _\\mathrm {max}$ =124$\\pm $ 8 and $P_\\mathrm {max}$ =7.32$\\pm $ 0.25% in the $V$ -band and $\\alpha _\\mathrm {max}$ =124$\\pm $ 6 and $P_\\mathrm {max}$ =7.04$\\pm $ 0.21% in the $R_\\mathrm {C}$ -band.",
"For confirmation, we fit our data with a simple third order polynomial function and obtained $\\alpha _\\mathrm {max}$ =119$\\pm $ 8 and $P_\\mathrm {max}$ =7.26$\\pm $ 0.28% in the $V$ -band and $\\alpha _\\mathrm {max}$ =122$\\pm $ 6 and $P_\\mathrm {max}$ =7.01$\\pm $ 0.13% in the $R_\\mathrm {C}$ -band.",
"These results are in good agreement with those with Eq.",
"(REF ), ensuring reliability of these derived values."
],
[
"Comparison with Other Airless Bodies in the Solar System",
"One of the unexpected findings from this research is the large $\\alpha _\\mathrm {max}$ values.",
"Taking advantage of the observatory's location and the timing (i.e., the summer solstice), we extended the polarimetric data up to $\\alpha $ =143.",
"The phase angle coverage is overwhelmingly larger than previous polarimetric observations of asteroids, but we marginally detected a drop in polarization beyond $\\alpha _\\mathrm {max}$ .",
"To make it clear how large $\\alpha _\\mathrm {max}$ is, we compared the polarimetric phase curves with those of other solar system airless bodies (Figure REF ).",
"Among them, the Moon and Mercury are observed well around their maximum polarization [34], [14].",
"Both the Moon and Mercury have $\\alpha _\\mathrm {max}\\sim $ 100, which is significantly smaller than Icarus (see Figure REF (a)–(b)).",
"For asteroids, there are four objects in the literature for which polarization data were available at large $\\alpha >$ 100.",
"[38] made a polarimetric observation of an NEA, Toro, and derived $P_\\mathrm {max}$ =8.5$\\pm $ 0.7 % at $\\alpha _\\mathrm {max}$ =110$\\pm $ 10.",
"[32] conducted an observation of another NEA, Toutatis, at $\\alpha $ =74–111 and derived $P_\\mathrm {max}$ =7.0$\\pm $ 0.2% at $\\alpha _\\mathrm {max}$ =107$\\pm $ 10.",
"[2] observed 2000 PN$_9$ at $\\alpha $ =90.7 and 115 and posited that $P_\\mathrm {max}$ =7.7$^{+0.5}_{-0.1}$ % at $\\alpha _\\mathrm {max}$ =103$\\pm $ 12.",
"Among these asteroids, the Toutatis data have good coverage around the maximum phase (Figure REF (c)), showing a clear drop beyond $\\alpha _\\mathrm {max}$ .",
"Once again, our Icarus data clearly show $\\alpha _\\mathrm {max}$ values larger than those for Toutatis.",
"There are several possibilities resulting in the large $\\alpha _\\mathrm {max}$ .",
"Lunar data show a moderate dependence on albedo [39].",
"Thus, smaller albedo values tend to show larger $\\alpha _\\mathrm {max}$ values.",
"Icarus has an albedo typical of stony materials in the solar system (see Section REF ).",
"In addition, lunar data cover an $\\alpha _\\mathrm {max}$ in the range of 92–106, which is much smaller than the Icarus values.",
"Accordingly, the high $\\alpha _\\mathrm {max}$ values cannot be explained by the albedo.",
"[56] examined the size dependence of the polarization properties of laboratory samples and suggested that $\\alpha _\\mathrm {max}$ would increase with increasing size, even up to $\\alpha _\\mathrm {max}\\sim $ 150.",
"Although there may be other factors increasing the $\\alpha _\\mathrm {max}$ values, we hypothesize that one possible explanation for the large $\\alpha _\\mathrm {max}$ of Icarus is that the asteroid could be covered with large grains."
],
[
"Albedo",
"The geometric albedo ($p_\\mathrm {V}$ ) of Icarus was determined by different measurements, but these results do not match well: 0.42 [62], 0.33–0.70 [28], 0.14$^{+0.10}_{-0.06}$ [60], and 0.29$\\pm $ 0.05 [49].",
"We now derive the geometric albedo based on our polarimetric measurement using the so-called slope–albedo law, which is given by $\\log _{10} p_\\mathrm {V} = C_{\\rm 1}~\\log _{10} h_\\mathrm {SLP} + C_2 ~~~~~~,$ where $h_\\mathrm {SLP}$ is the phase slope near the inversion angle (i.e., $dP/d\\alpha $ at $\\alpha =\\alpha _0$ ).",
"$C_{\\rm 1}$ and $C_{\\rm 2}$ are constants that have been determined by several authors.",
"[42] derived $C_{\\rm 1}$ =$-0.98$ and $C_{\\rm 2}$ =$-1.73$ in an early study, and these values were updated to $C_{\\rm 1}$ =$-1.21$ $\\pm $ 0.07 and $C_{\\rm 2}$ =$-1.89$$\\pm $$0.14$ [44] and to $C_{\\rm 1}$ =$-0.80$$\\pm $$0.04$ and $C_{\\rm 2}$ =$-1.47$$\\pm $$0.04$ (when $p_\\mathrm {V}\\ge $ 0.08) [6].",
"We fit our $V$ –polarimetric data at $\\alpha $ =57.2–86.5 constraining the inversion phase angle $\\alpha _0$ =20, which is a typical value for Q-type asteroids [3], and obtained $h_\\mathrm {SLP}$ =$0.0874$$\\pm $$0.0017$ .",
"With $C_{\\rm 1}$ and $C_{\\rm 2}$ as in [44] and [6], we acquired a geometric albedo of $p_\\mathrm {V}$ =$0.25$$\\pm $$0.02$ .",
"This albedo value is consistent with those of Q-type and S-type asteroids [61], [10], [60].",
"Note that the fitted phase angle range is larger than those of previous studies.",
"However, we believe that this range is reasonable for fitting the data not only because our phase curve shows a linear profile at $\\alpha \\lesssim $ 86.5 but also because a study of Itokawa at similar phase angles of $\\alpha $ =41.5–79.2 demonstrated a good match for the albedo ($p_\\mathrm {V}$ =0.24$\\pm $ 0.01 via polarimetry [7] v.s.",
"$p_\\mathrm {V}$ =0.24$\\pm $ 0.02, [33], via remote–sensing observation by the Hayabusa onboard camera).",
"In Section REF , we examined the rotational change in $P$ and found no variability to the accuracy of 0.2–0.3% at $\\alpha $ =100.2.",
"Extrapolating the linear slope to the phase angle (although the phase curve slightly deviated from the line), the upper limit of the polarization variability (0.2–0.3%) is converted into the upper limit of the $h_\\mathrm {SLP}$ variability of $\\sim $ 0.0025.",
"With Eq.",
"(REF ), we put the upper limit of the albedo variation on the quadrant surface at $\\sigma p_\\mathrm {V}$ =0.02.",
"The upper limit would suggest that the surface of the asteroid is quite homogeneous in albedo from a large scale viewpoint (1/4 of surface resolution)."
],
[
"Grain Size Estimate",
"It is known that $P_\\mathrm {max}$ is inversely correlated with the geometric albedo $p_\\mathrm {V}$ (Umov law).",
"$P_\\mathrm {max}$ also depends on the grain size.",
"[56] examined these relationship using lunar soil samples and gave the following equations: $d = 0.03 \\exp (2.9~b)~~,$ and $b=\\log (10^2~A_{\\alpha =5})+a\\log (10~P_\\mathrm {max})~~,$ where $d$ denotes the grain size in .",
"$a$ is 0.795 at 0.43 and 0.845 at 0.65 [56].",
"$A_{\\alpha =5}$ is the albedo at $\\alpha $ =5.",
"Using the phase function we determined in Section REF , we derived $A_{\\alpha =5}$ = 0.215$\\pm $ 0.018 for Icarus.",
"Applying Eq.",
"(REF ) to our polarimetric result, we obtained $d$ =100–130 .",
"In addition, we plotted our data onto the $P_\\mathrm {max}$ –albedo relation for different sizes of laboratory samples (Figure REF ).",
"Similarly, the plot (Figure REF ) shows a trend indicating that Icarus may be covered with particles hundreds of microns in size.",
"This result is consistent with the fact that Icarus has a large macroscopic roughness.",
"The large values of $\\alpha _\\mathrm {max}$ also imply a large particle size.",
"Furthermore, the asteroid exhibits a Q-type spectrum, which is bluer than an S-type spectrum.",
"The blueness can be explained not only by the freshness in terms of the space weathering but also by large grains.",
"It is known that an increase in grain size yields a bluer spectral slope regardless of the types of asteroid [63], [53], [46].",
"Therefore, these optical properties consistently suggest a large grain size on the asteroid.",
"Why is the particle size so large?",
"How did the asteroid lose the small particles from the surface?"
],
[
"Consideration of Mass Ejection around Perihelion",
"Icarus has a critical rotational period (2.273 hours) in which the centrifugal force exceeds the self-gravitational force on the equator.",
"Assuming an Itolawa-like bulk density of $\\sim $ 2000 kg m$^{-3}$ [15], [54] and a spherical body with a 1440-m diameter [24], the ambient gravitational acceleration is approximately 80 micro-G's at the pole and minus 5 micro-G's at the equator, suggesting that granular materials may be ejected from the equatorial region (around a latitude within 30 from the equator) via centrifugal acceleration.",
"In contrast, the rotational axis of Icarus nearly aligns to the ecliptic pole [24], meaning that it is roughly perpendicular to the orbital plane with a moderate inclination to the orbital plane ($i$ =22.3).",
"Under this geometry, the sun shines almost parallel to the polar region.",
"Although regolith grains can remain in the high-latitude region, the oblique sunshine can strip small grains off from the polar region when the asteroid passes through perihelion.",
"Such an idea was suggested for the surface of (3200) Phaethon to explain the dust emission near perihelion [35], [36].",
"The solar radiation pressure is given by $F_\\mathrm {r}$ =$\\beta _\\mathrm {r} F_\\mathrm {g}$ , where $F_\\mathrm {g}=$ 0.169 m s$^{-2}$ is the solar gravity at the perihelion of Icarus ($q$ =0.187 au).",
"$\\beta _\\mathrm {r}$ is the ratio of the solar radiation pressure to the solar gravity, given approximately by $\\beta _\\mathrm {r}$ =1.14/$\\rho _\\mathrm {d} d$ , where $\\rho _\\mathrm {d}$ and $d$ are the particle mass density and diameter, respectively.",
"Thus, the solar radiation pressure exceeds the ambient gravity in the polar region when $d\\lesssim $ 240 (a mass density, $\\rho _\\mathrm {d}$ =1.0 g cm$^{-2}$ , was assumed).",
"Although some cohesive forces, such as van der Waals forces, would work to prevent mass ejection from the surface, we conjecture that the environment of the “fast-rotating” body at a “small solar distance” would be responsible for the paucity of small grains and its unique polarimetric properties.",
"Intriguingly, [50] noted that 2007 MK$_6$ has a strong dynamical connection to Icarus, suggesting that these two asteroids share a common origin.",
"Such groupings of asteroids are also recognized for Phaethon and (155140) 2005 UD [51].",
"It is still unclear if these two bodies were split due to the tidal force of a planet during a close encounter, thermal stress, rotational breakup via YORP acceleration, or other mechanisms.",
"It is important to note that these two bodies have similarities in two aspects: their rapid rotational periods and small perihelion distances.",
"Supposing that these groups of asteroids experienced large-scale splittings that produced their current bodies, they may have had the chance to lose small dust grains during splitting due to strong solar radiation pressure quickly sweeping small dust grains from their orbits before they had the chance to accumulate, producing bodies that lack small dust grains."
],
[
"Summary",
"We made photopolarimetric observations of Icarus at large phase angles $\\alpha $ = 57–141 during its apparition in 2015 and found the following: The combination of the maximum polarization degree and the geometric albedo is in accordance with terrestrial rocks with a diameter of several hundreds of micrometers.",
"The photometric function indicates a large macroscopic roughness.",
"We posit that the unique environment (i.e., the small perihelion distance $q$ =0.187 au and a short rotational period $T_\\mathrm {rot}$ =2.27 hours) may be attributed to the paucity of small grains on the surface, as indicated on Phaethon.",
"The maximum values of the linear polarization degree are $P_\\mathrm {max}$ =7.32$\\pm $ 0.25 % at a phase angle of $\\alpha _\\mathrm {max}$ =124$\\pm $ 8 in the $V$ -band and $P_\\mathrm {max}$ =7.04$\\pm $ 0.21 % at $\\alpha _\\mathrm {max}$ =124$\\pm $ 6 in the $R_\\mathrm {C}$ –band.",
"Applying the polarimetric slope–albedo law, we derived a geometric albedo $p_\\mathrm {V}$ =0.25$\\pm $ 0.02, which is consistent with that of Q-type asteroids.",
"The albedo would be globally constant, showing no significant rotational variation in the polarization degree.",
"$\\alpha _\\mathrm {max}$ is significantly larger than those of Mercury, the Moon and the S–type asteroid Toutatis but consistent with laboratory samples hundreds of microns in size.",
"The $P_\\mathrm {max}$ –albedo relation suggests that Icarus is covered with particles hundreds of microns in size.",
"The photometric function suggest a large macroscopic roughness, supporting the dominance of large grains.",
"To explain the dominance of large grains on the asteroid, we conjecture that a strong radiation pressure around the perihelion passage would strip small grains off of the fast–rotating asteroid."
],
[
"Pirka/MSI Polarimetric Data Analysis Procedures",
"The observed ordinary and extraordinary fluxes at the half-wave plate angle $\\Psi $ in degrees, $I_\\mathrm {o}(\\Psi )$ and $I_\\mathrm {e}(\\Psi )$ , were used to derive $q_\\mathrm {pol}^{\\prime }=\\left(\\frac{R_\\mathrm {q}-1}{R_\\mathrm {q}+1}\\right)\\bigg /p_\\mathrm {eff}~,$ and $u_\\mathrm {pol}^{\\prime }=\\left(\\frac{R_\\mathrm {u}-1}{R_\\mathrm {u}+1}\\right)\\bigg /p_\\mathrm {eff}~,$ where $R_\\mathrm {q}$ and $R_\\mathrm {u}$ are obtained from the observation using the following equations: $R_\\mathrm {q}=\\sqrt{\\frac{I_\\mathrm {e}(0)/I_\\mathrm {o}(0)}{I_\\mathrm {e}(45)/I_\\mathrm {o}(45)}}~,$ and $R_\\mathrm {u}=\\sqrt{\\frac{I_\\mathrm {e}(22.5)/I_\\mathrm {o}(22.5)}{I_\\mathrm {e}(67.5)/I_\\mathrm {o}(67.5)}}~,$ where $p_\\mathrm {eff}$ is a polarization efficiency, which was examined by taking a dome flat image through a pinhole and a Polaroid–like linear polarizer, which produces artificial stars with $P$ =99.97$\\pm $ 0.02 % ($V$ ) and 99.98$\\pm $ 0.01 % ($R_\\mathrm {C}$ ).",
"$p_\\mathrm {eff}$ was measured approximately two months prior to our observation and was determined to be $p_\\mathrm {eff}$ =0.9967$\\pm $ 0.0003 in the $V$ -band and 0.9971$\\pm $ 0.0001 in the $R_\\mathrm {C}$ -band.",
"The instrumental polarization of Pirka/MSI is known to depend on the instrument angle of rotation and can be corrected with the following equation: $\\left( \\begin{array}{r} q^{\\prime \\prime }_\\mathrm {pol} \\\\ u^{\\prime \\prime }_\\mathrm {pol} \\end{array} \\right)= \\left( \\begin{array}{r} q^{\\prime }_\\mathrm {pol} \\\\ u^{\\prime }_\\mathrm {pol} \\end{array} \\right)- \\left( \\begin{array}{rr} \\cos 2\\theta _\\mathrm {rot1} & -\\sin 2\\theta _\\mathrm {rot1} \\\\ \\sin 2\\theta _\\mathrm {rot2} & \\cos 2\\theta _\\mathrm {rot2} \\end{array} \\right)\\left( \\begin{array}{r} q_\\mathrm {inst} \\\\ u_\\mathrm {inst} \\end{array} \\right)~,$ where $\\theta _\\mathrm {rot1}$ denotes the average instrument rotator angle during the exposures with $\\Psi $ =0 and 45, while $\\theta _\\mathrm {rot2}$ denotes the average angle with $\\Psi $ =22.5 and 67.5.",
"$q_\\mathrm {inst}$ and $u_\\mathrm {inst}$ are two components of the Stokes parameters for the instrumental polarization and were determined to be $q_\\mathrm {inst}$ =0.963$\\pm $ 0.029 % in the $V$ -band and 0.703$\\pm $ 0.033 % in the $R_\\mathrm {C}$ -band and $u_\\mathrm {inst}$ =0.453$\\pm $ 0.043 % in the $V$ -band and 0.337$\\pm $ 0.020 % in the $R_\\mathrm {C}$ -band, respectively, by observing the unpolarized stars HD212311 and BD+32 3739 [55].",
"The instrument position angle in celestial coordinates was determined by measuring the polarization position angles of strongly polarized stars for which position angles are reported in [55].",
"The instrument position angle can be corrected using the following equations: $\\left( \\begin{array}{r} q^{\\prime \\prime \\prime }_\\mathrm {pol} \\\\ u^{\\prime \\prime \\prime }_\\mathrm {pol} \\end{array} \\right)= \\left( \\begin{array}{rr} \\cos 2\\theta ^{\\prime }_\\mathrm {off} & \\sin 2\\theta ^{\\prime }_\\mathrm {off} \\\\ -\\sin 2\\theta ^{\\prime }_\\mathrm {off} & \\cos 2\\theta ^{\\prime }_\\mathrm {off} \\end{array} \\right)\\left( \\begin{array}{r} q^{\\prime \\prime }_\\mathrm {pol} \\\\ u^{\\prime \\prime }_\\mathrm {pol} \\end{array} \\right)~~ ,$ and $\\theta ^{\\prime }_\\mathrm {off} = \\theta _\\mathrm {off} - \\theta _\\mathrm {ref}~~,$ where $\\theta _\\mathrm {ref}$ is a given parameter for specifying the position angle of the instrument.",
"Through an observation of strongly polarized stars (HD204827, HD154445, and HD155197) in 2015 May, we derived $\\theta _\\mathrm {off}$ =3.82$\\pm $ 0.38 in the V-band and 3.38$\\pm $ 0.37 in the $R_\\mathrm {C}$ -band.",
"Acknowledgments This research was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (No.",
"2015R1D1A1A01060025).",
"The observations at the Nayoro Observatory were supported by the Optical and Near-infrared Astronomy Inter-University Cooperation Program and Grants-in-Aid for Scientific Research (23340048, 24000004, 24244014, and 24840031) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.",
"We thank the staff members at the Nayoro City Observatory, Y. Murakami, F. Watanabe, Y. Kato, and R. Nagayoshi, for their kind support and Drs.",
"Takashi Ito and Tomoko Arai for their encouragement regarding this work.",
"We also thank the anonymous reviewer for their careful reading of our manuscript and their insightful comments.",
"SH was supported by the Hypervelocity Impact Facility (former name: the Space Plasma Laboratory), ISAS, JAXA.",
"cccrrrrrrrr 11 0pc Observation Journal.",
"Date UT Filter $t_\\mathrm {exp}^{(a)}$ $N^{(b)}$ r$^{(c)}$ $\\Delta ^{(d)}$ $\\alpha ^{(e)}$ $\\theta _\\perp ^{(f)}$ $\\phi ^{(g)}$ mode$^{(h)}$ 2015/06/11 11:23-15:27 R$_\\mathrm {C}$ 60 212 0.928 0.104 145.1 86.3 356.3 Phot 2015/06/12 13:22-16:18 R$_\\mathrm {C}$ 60 92 0.945 0.089 141.3 98.9 8.9 Phot, Pol 2015/06/14 13:48-15:50 V 60 36 0.975 0.064 127.4 144.7 54.7 Phot, Pol 13:10-15:09 R$_\\mathrm {C}$ 60 48 0.975 0.065 127.7 143.7 53.7 Pol 2015/06/15 11:34-15:27 V 30 152 0.989 0.057 116.1 176.0 86.0 Pol 11:42-15:18 R$_\\mathrm {C}$ 30 156 0.989 0.057 116.1 176.0 86.0 Pol 2015/06/16 12:30-17:20 V 30 220 1.005 0.054 100.2 20.6 110.6 Pol 12:22-17:13 R$_\\mathrm {C}$ 30 200 1.005 0.054 100.2 20.5 110.5 Pol 2015/06/17 11:19-12:43 V 30 68 1.018 0.056 86.5 29.8 119.8 Pol 11:08-12:35 R$_\\mathrm {C}$ 30 76 1.018 0.056 86.6 29.8 119.8 Pol 2015/06/19 11:13-11:20 V 30 12 1.046 0.073 64.0 32.2 122.2 Pol 11:02-11:12 R$_\\mathrm {C}$ 30 16 1.046 0.073 64.0 32.2 122.2 Pol 2015/06/20 11:22-11:29 V 30 12 1.060 0.086 57.2 30.1 120.1 Pol 11:12-11:22 R$_\\mathrm {C}$ 30 16 1.060 0.086 57.2 30.1 120.1 Pol (a)Individual effective exposure time in seconds.",
"(b)Number of exposures.",
"(c)Median heliocentric distance in au.",
"(d)Median geocentric distance in au.",
"(e)Median Solar phase angle (Sun–Asteroid–Observer angle) in degrees.",
"(f)Median position angle of normal vector with respect to the scattering plane in degrees.",
"(g)Median position angle of the scattering plane in degrees.",
"(h)Observation mode: Photometry (Phot) or Polarimetry (Pol).",
"ccrrrrrr 8 0pc Degree of linear polarization and position angle of polarization Date Filter $P$$^{(a)}$ $\\epsilon P$$^{(b)}$ $\\theta _\\mathrm {P}^{(c)}$ $\\epsilon \\theta _\\mathrm {P}^{(d)}$ $P_\\mathrm {r}$$^{(e)}$ $\\theta _\\mathrm {r}$$^{(f)}$ 2015/06/12 R$_\\mathrm {C}$ 6.29 0.60 $-82.8$ 2.7 6.28 $-1.7$ 2015/06/14 V 7.14 0.28 $-37.8$ 1.1 7.11 $-2.5$ R$_\\mathrm {C}$ 6.92 0.16 $-36.7$ 0.7 6.92 $-0.4$ 2015/06/15 V 7.26 0.12 $-5.8$ 0.5 7.24 $-1.8$ R$_\\mathrm {C}$ 7.02 0.09 $-5.3$ 0.4 7.01 $-1.3$ 2015/06/16 V 6.77 0.08 18.4 0.3 6.75 $-2.2$ R$_\\mathrm {C}$ 6.33 0.06 18.8 0.3 6.32 $-1.7$ 2015/06/17 V 5.78 0.09 28.2 0.4 5.77 $-1.6$ R$_\\mathrm {C}$ 5.44 0.07 28.2 0.4 5.43 $-1.6$ 2015/06/19 V 4.02 0.16 31.5 1.1 4.02 $-0.7$ R$_\\mathrm {C}$ 3.47 0.20 34.6 1.7 3.46 $2.4$ 2015/06/20 V 3.15 0.16 27.3 1.5 3.13 $-2.8$ R$_\\mathrm {C}$ 2.71 0.13 27.1 1.4 2.69 $-3.0$ (a)Polarization degree in percent.",
"(b)Error of $P$ in percent.",
"(c)Position angle of the strongest electric vector in degrees.",
"(d)Error of $\\theta _P$ in degrees.",
"(e)Polarization degree with respect to the scattering plane in percent.",
"(f)Position angle with respect to the scattering plane in degrees."
]
] | 1709.01603 |
[
[
"Opening the Maslov Box for Traveling Waves in Skew-Gradient Systems"
],
[
"Abstract We obtain geometric insight into the stability of traveling pulses for reaction-diffusion equations with skew-gradient structure.",
"For such systems, a Maslov index of the traveling wave can be defined and related to the eigenvalue equation for the linearization $L$ about the wave.",
"We prove two main results about this index.",
"First, for general skew-gradient systems, it is shown that the Maslov index gives a lower bound on the number of real, unstable eigenvalues of $L$.",
"Second, we show how the Maslov index gives an exact count of all unstable eigenvalues for fast traveling waves in a FitzHugh-Nagumo system.",
"The latter proof involves the Evans function and reveals a new geometric way of understanding algebraic multiplicity of eigenvalues."
],
[
"Introduction",
"The paragon of stability analysis for nonlinear waves is a result that relates spectral information to qualities of the wave itself.",
"In principle, this could explain why some patterns and structures are prevalent in nature, while others are not.",
"The classic example of this is Sturm-Liouville theory, which equates the number of unstable modes of a steady state solution of a scalar reaction-diffusion equation to the number of critical points it has.",
"(See §2.3.2 of [26]).",
"For systems of equations, generalizations of Sturm-Liouville theory lead naturally to the Maslov index, which is a winding number for curves of Lagrangian subspaces.",
"One drawback of the Maslov index as a stability index is that it has typically been applied only in a relatively small class of systems, namely those for which the steady state equation has a Hamiltonian structure.",
"In this work, we show how the Maslov index can give a lower bound on the number of unstable eigenvalues for the linearization about a traveling wave in reaction-diffusion equations with skew-gradient structure.",
"Such systems are necessarily not Hamiltonian.",
"Additionally, we show how the same index gives an exact count of the unstable eigenvalues in a FitzHugh-Nagumo system.",
"The proofs use an adaptation of the “Maslov box\" (see, for example, [25], [20], [6], [22]) and an entirely intersection-based formulation of the Maslov index.",
"The systems of interest are reaction-diffusion equations of the form $u_t=u_{xx}+Q Sf(u),$ where $x,t\\in \\mathbb {R}$ are space and time respectively, $u\\in \\mathbb {R}^n$ , and $f(u)=\\nabla F(u)$ is the gradient of a function $F:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ .",
"The matrix $S\\in \\operatorname{GL}_n(\\mathbb {R})$ is positive and diagonal, and $Q\\in \\operatorname{GL}_{n}(\\mathbb {R})$ has the form $Q=\\mathrm {diag}\\lbrace d_1,\\dots ,d_n\\rbrace ,$ where $d_i=\\pm 1$ for all $i$ .",
"Such systems were dubbed “skew-gradient\" by Yanagida [39], [40].",
"We assume that (REF ) possesses a traveling pulse solution $\\hat{u}$ which depends on one variable $z=x-ct$ .",
"Such solutions have a fixed profile and move with a constant speed $c$ .",
"Without loss of generality, we assume that $c<0,$ meaning that the wave moves to the left.",
"There has been considerable progress in the stability analysis of standing waves of (REF )–see below for more detail–but the known results do not apply to traveling waves.",
"We aim to use the Maslov index to give a systematic treatment of traveling waves in such systems.",
"Written in a moving frame, a traveling pulse of (REF ) is a steady state of the equation $u_t=u_{zz}+cu_z+QSf(u),$ which decays exponentially to a constant state $u_\\infty \\in \\mathbb {R}^n$ as $z\\rightarrow \\pm \\infty $ .",
"For simplicity, we take $u_\\infty =0$ , which means that $f(0)=0$ .",
"We make the further assumption that 0 is a stable equilibrium for the kinetics equation associated with (REF ).",
"More precisely, this means that $\\text{there exists } \\beta <0 \\text{ such that the } n \\text{ eigenvalues } \\nu _i \\text{ of } QS f^{\\prime }(0) \\text{ satisfy } \\mathrm {Re }\\, \\nu _i<\\beta .$ Among systems of the form (REF ) are activator-inhibitor systems, which are known to support pattern formation.",
"Assumption (REF ) is therefore natural, since it is of interest to study the stability of structures which emanate from stable, homogeneous states that are destabilized in the presence of diffusion [32], [35].",
"Since $\\hat{u}_t=0$ , the traveling wave equation is an ODE which can be converted to a first order system by introducing the variable $v=S^{-1}u_z$ : $\\left(\\begin{array}{c}u\\\\v\\end{array}\\right)_z=\\left(\\begin{array}{c}Sv\\\\-cv-Qf(u)\\end{array}\\right).$ This perspective is useful, because it opens up the possibility of analyzing (REF ) using dynamical systems techniques.",
"For example, the traveling wave $\\varphi =(\\hat{u},S^{-1}\\hat{u})$ is seen to be a homoclinic orbit to the fixed point $(0,0)$ .",
"The stable and unstable manifolds of this fixed point–$W^s(0)$ and $W^u(0)$ respectively–will play an important role in our analysis, as will their tangent spaces at 0: $V^s(0)=T_0W^s(0),\\hspace{3.61371pt} V^u(0)=T_0W^u(0).$ Our aim is to use the Maslov index to analyze the stability of $\\hat{u}$ , which is defined as follows.",
"Definition 1 The traveling wave $\\hat{u}(z)$ is asymptotically stable relative to (REF ) if there is a neighborhood $V\\subset BU(\\mathbb {R},\\mathbb {R}^n)$ of $\\hat{u}(z)$ such that if $u(z,t)$ solves (REF ) with $u(z,0)\\in V$ , then $||\\hat{u}(z+k)-u(z,t)||_\\infty \\rightarrow 0 $ as $t\\rightarrow \\infty $ for some $k\\in \\mathbb {R}$ .",
"The stability question immediately leads to the operator $L:=\\partial _z^2+c\\partial _z+QSf^{\\prime }(\\hat{u})$ obtained by linearizing the right-hand side of (REF ) around $\\hat{u}$ .",
"It is known [5], [19] that the nonlinear stability of $\\hat{u}$ (in the sense of Definition REF ) is determined by locating the spectrum of $L$ .",
"Assumption (REF ) guarantees that the essential spectrum of $L$ is contained in the left-half plane.",
"It therefore suffices to determine whether $L$ has any eigenvalues of positive real part.",
"This is the task for the Maslov index.",
"Before discussing that topic further, we briefly review the history of stability in skew-gradient systems.",
"The papers [39], [40] considered standing waves, which are pulses with $c=0$ .",
"An instability criterion for these waves was derived in [40] using an orientation index related to derivatives of the Evans function.",
"Stronger results, akin to those obtained in this work (lower bounds on the Morse index and a stability criterion), were then obtained in [13] using the Maslov index.",
"The strategy in that work was to use the index to aid in the calculation of spectral flow [4] for a family of self-adjoint operators.",
"This calculation relied on a change of variables in the eigenvalue equation that revealed a Hamiltonian structure.",
"A similar change of variables was made in [24] to define and use the Maslov index for standing waves in nonlinear Schrödinger equations.",
"An unstable eigenvalue was shown to exist by means of a shooting argument in the manifold of Lagrangian planes.",
"More precisely, a change in the homotopy class of a loop was observed as a (spectral) parameter varied.",
"The existence of the eigenvalue follows since such a change can only occur at an eigenvalue.",
"It is important to note that in each of the cases mentioned above, the waves considered had zero speed.",
"By contrast, [15] and this work consider traveling waves.",
"This difference is significant, since there is no change of variables that makes the eigenvalue equation for $L$ in (REF ) Hamiltonian.",
"However, there is a symplectic form for which the set of Lagrangian planes is invariant under the eigenvalue equation; hence the Maslov index can be defined.",
"The trade-off is that self-adjointness of $L$ is lost, so that in general the spectrum will not be real.",
"This spurred the authors of [15] to consider the Evans function $D(\\lambda )$ [1], [34], and it was shown that the sign of $D^{\\prime }(0)$ is determined by the parity of the Maslov index.",
"On the other hand, the main results of this work are formulated without reference to the Evans function.",
"Consider $\\mathbb {R}^{2n}$ endowed with a symplectic form $\\omega $ .",
"By symplectic, we mean that $\\omega $ is nondegenerate, skew-symmetric and bilinear.",
"An $n$ -dimensional subspace $V\\subset \\mathbb {R}^{2n}$ is called Lagrangian if $\\omega (v_1,v_2)=0$ for all $v_{1,2}\\in V$ .",
"The collection of all such subspaces is clearly a subset of $\\operatorname{Gr}_n(\\mathbb {R}^{2n})$ , the Grassmannian of all $n$ -dimensional subspaces of $\\mathbb {R}^{2n}$ .",
"In fact, this set is actually a smooth manifold of dimension $n(n+1)/2$ , called the Lagrangian Grassmannian and denoted $\\Lambda (n)$ .",
"It is well-known [3], [29], [30] that $\\pi _1(\\Lambda (n))=\\mathbb {Z}$ for all $n$ , and thus a winding number can be defined for loops in this space.",
"This winding number is the Maslov index.",
"Broadly speaking, the Maslov index counts how many times two paths of Lagrangian subspaces intersect each other.",
"(One of the curves may be fixed, which is the traditional way of defining the index [29].)",
"We will consider paths that encode the left and right boundary data of potential eigenfunctions for $L$ .",
"An intersection therefore corresponds to a function that satisfies both boundary conditions and hence is an eigenfunction.",
"The rest of this paper is organized as follows.",
"In §2, we set up the eigenvalue problem and identify the symplectic structure that makes the analysis possible.",
"In §3, the Maslov index is defined, both for a path of Lagrangian planes and for a pair of curves of Lagrangian planes.",
"This includes a careful consideration of the “crossing form” of [33].",
"In §4, we introduce the “Maslov box\" of [20] and show how the Maslov index can be used to give a lower bound on the number of unstable eigenvalues for $L$ in (REF ).",
"We apply the same framework to a FitzHugh-Nagumo system in §5 and show how the Maslov index gives an exact count of the positive, unstable eigenvalues in this case.",
"Additionally, we prove that any unstable spectrum must be real, from which it follows that the Maslov index detects all unstable eigenvalues.",
"Finally, in §6 we show what the Maslov index reveals about the algebraic multiplicity of eigenvalues.",
"This is accomplished by relating the crossing form to derivatives of the Evans function.",
"In particular, we provide a new geometric interpretation of simplicity of an eigenvalue."
],
[
"Eigenvalue Equation and Symplectic Structure",
"As noted above, the stability of $\\hat{u}$ is assessed by determining the spectrum $\\sigma (L)$ of the operator $L$ in (REF ).",
"First, we say that $\\lambda \\in \\mathbb {C}$ is an eigenvalue for $L$ if there exists a solution $p\\in BU(\\mathbb {R},\\mathbb {C}^n)$ to the equation $Lp=\\lambda p.$ The set of isolated eigenvalues of $L$ of finite multiplicity is denoted $\\sigma _n(L)$ .",
"Comparing with (REF ), setting $p_z=Sq$ converts (REF ) to the first order system $\\left(\\begin{array}{c}p\\\\q\\end{array}\\right)^{\\prime }=\\left(\\begin{array}{c c}0 & S\\\\\\lambda S^{-1}-Qf^{\\prime }(\\hat{u}) & -cI\\end{array} \\right)\\left(\\begin{array}{c}p\\\\q\\end{array}\\right).$ As is commonly done for Evans function analyses (see [1]), we abbreviate (REF ) as $Y^{\\prime }(z)=A(\\lambda ,z)Y(z),$ with $Y\\in \\mathbb {C}^n$ and $A(\\lambda ,z)\\in M_n(\\mathbb {C}^{2n})$ .",
"Assumption (REF ) guarantees that $\\hat{u}$ approaches 0 exponentially, and thus there is a well-defined matrix $A_\\infty (\\lambda )=\\lim \\limits _{z\\rightarrow \\pm \\infty }A(\\lambda ,z),$ and this limit is also achieved exponentially quickly.",
"The eigenvalues of $L$ comprise only part of the spectrum; the rest is the essential spectrum $\\sigma _\\mathrm {ess}(L)$ .",
"For systems of the form (REF ), it is known (Lemma 3.1.10 of [26]) that the essential spectrum is given by $\\sigma _\\mathrm {ess}(L)=\\lbrace \\lambda \\in \\mathbb {C}:A_\\infty (\\lambda ) \\text{ has an eigenvalue }\\mu \\in i\\mathbb {R}\\rbrace .$ We claim that $\\sigma _\\mathrm {ess}(L)$ is contained in the half-plane $H=\\lbrace \\lambda \\in \\mathbb {C}:\\mathrm {Re}\\,\\lambda <\\beta \\rbrace .$ Indeed, a simple calculation using (REF ) shows that the eigenvalues of $A_\\infty (\\lambda )$ are given by $\\mu _j(\\lambda )=\\frac{1}{2}\\left(-c\\pm \\sqrt{c^2+4(\\lambda -\\nu _i)}\\right),$ with $\\nu _i$ from (REF ).",
"We need to show that $A_\\infty (\\lambda )$ has no purely imaginary eigenvalues if $\\mathrm {Re } \\lambda \\ge \\beta $ , which is clearly equivalent to showing that $\\mathrm {Re }\\sqrt{c^2+4(\\lambda -\\nu _i)}\\ne -c$ for such $\\lambda $ .",
"The formula $\\mathrm {Re }\\sqrt{a+bi}=\\frac{1}{\\sqrt{2}}\\sqrt{\\sqrt{a^2+b^2}+a}$ and the fact that $\\mathrm {Re}\\,(c^2+4(\\lambda -\\nu _i))>0$ from (REF ) together imply that $\\mathrm {Re}\\sqrt{c^2+4(\\lambda -\\nu _i)}\\ge \\sqrt{\\mathrm {Re}(c^2+4(\\lambda -\\nu _i))}>\\sqrt{c^2}=-c,$ as desired.",
"This calculation actually proves that $A_\\infty (\\lambda )$ has exactly $n$ eigenvalues of positive real part and $n$ eigenvalues of negative real part for $\\lambda \\in (\\mathbb {C}\\setminus H).$ We label these $\\mu _i(\\lambda )$ in order of increasing real part and observe that $\\mathrm {Re}\\,\\mu _1(\\lambda )\\le \\dots \\le \\mathrm {Re}\\,\\mu _n(\\lambda )<0<-c<\\mathrm {Re}\\,\\mu _{n+1}(\\lambda )\\le \\dots \\le \\mathrm {Re}\\,\\mu _{2n}(\\lambda ).$ Furthermore, one sees from (REF ) that for each $1\\le i\\le n$ we have $\\mu _i(\\lambda )+\\mu _{i+n}(\\lambda )=-c.$ It then follows from, for example, Theorem 3.2 of [34] that (REF ) has exponential dichotomies on $\\mathbb {R}^+$ and $\\mathbb {R}^-$ for $\\lambda \\in \\mathbb {C}\\setminus H$ , allowing us to define $n$ -dimensional vector spaces $\\begin{aligned}E^u(\\lambda ,z) & = \\lbrace \\xi (z)\\in \\mathbb {C}^{2n}:\\xi \\text{ solves } (\\ref {eval eqn matrix}) \\text{ and } \\xi \\rightarrow 0 \\text{ as }z\\rightarrow -\\infty \\rbrace \\\\E^s(\\lambda ,z) & = \\lbrace \\xi (z)\\in \\mathbb {C}^{2n}:\\xi \\text{ solves } (\\ref {eval eqn matrix}) \\text{ and } \\xi \\rightarrow 0 \\text{ as }z\\rightarrow \\infty \\rbrace \\end{aligned}.$ We call these sets the unstable and stable bundles respectively.",
"It is known that $E^{s/u}(\\lambda ,z)$ vary analytically in $\\lambda $ for each $z$ .",
"Moreover, the decay of the solutions in $E^{u/s}(\\lambda ,z)$ is exponential, and any solution of (REF ) that is bounded at $-\\infty $ (resp.",
"$\\infty $ ) must be a member of $E^u(\\lambda ,z)$ (resp.",
"$E^s(\\lambda ,z))$ .",
"It follows that $\\lambda \\in \\mathbb {C}$ is an eigenvalue for $L$ if and only if the intersection $E^u(\\lambda ,z)\\cap E^s(\\lambda ,z)$ is nonempty for some (and hence all) $z\\in \\mathbb {R}$ .",
"The fact that any eigenfunction of $L$ must decay exponentially allows us instead to pose the eigenvalue problem on the Hilbert space $H^1(\\mathbb {R},\\mathbb {C}^n)$ .",
"This will pay dividends later when we consider the FitzHugh-Nagumo system.",
"We now focus our attention on real $\\lambda \\ge \\beta $ .",
"In this case, $E^{s/u}(\\lambda ,z)$ are real vector spaces.",
"To identify the symplectic structure, we introduce the matrix $J=\\left(\\begin{array}{c c}0 & Q\\\\-Q & 0\\end{array}\\right).$ Since $Q^2=I$ and $Q^*=Q$ , it follows that $J^2=-I$ and $J^*=-J$ .",
"We therefore call $J$ a complex structure on $\\mathbb {R}^{2n}$ .",
"If we denote by $\\langle \\cdot ,\\cdot \\rangle $ the standard inner product on $\\mathbb {R}^{2n}$ , then $\\omega (a,b):=\\langle a,Jb\\rangle $ defines a symplectic form on $\\mathbb {R}^{2n}$ , see §1 of [18], for example.",
"The following theorem underpins all of the ensuing analysis.",
"Theorem 2.1 Let $Y_1$ , $Y_2$ be any two solutions of (REF ) for fixed $\\lambda \\in \\mathbb {R}$ .",
"Then $\\frac{d}{dz}\\omega (Y_1,Y_2)=-c\\,\\omega (Y_1,Y_2).$ In particular, if $\\omega (Y_1(z_0),Y_2(z_0))=0$ for some $z_0\\in \\mathbb {R}$ , then $\\omega (Y_1,Y_2)\\equiv 0$ .",
"More generally, the symplectic form $\\Omega :=e^{cz}\\omega $ is constant in $z$ on any two solutions of (REF ).",
"A direct computation gives that $\\begin{aligned}\\frac{d}{dz}\\omega (Y_1,Y_2) & =\\omega (Y_1,A(\\lambda ,z)Y_2)+\\omega (A(\\lambda ,z)Y_1,Y_2)\\\\& = \\langle Y_1,JA(\\lambda ,z)Y_2\\rangle +\\langle A(\\lambda ,z)Y_1,JY_2\\rangle \\\\& = \\langle Y_1,\\left[JA+A^TJ\\right]Y_2\\rangle .\\end{aligned}$ In light of (REF ), we therefore need to show that $JA+A^TJ=-cJ.$ Recalling that $S$ and $Q$ are diagonal and that $(f^{\\prime }(\\hat{u}))^T=F^{\\prime \\prime }(\\hat{u})^T=F^{\\prime \\prime }(\\hat{u})=f^{\\prime }(\\hat{u})$ , we compute $\\begin{aligned}JA+A^TJ & =\\left(\\begin{array}{c c}\\lambda QS^{-1}-f^{\\prime }(\\hat{u}) & -cQ\\\\0 & -QS\\end{array}\\right)+\\left(\\begin{array}{c c}-\\lambda S^{-1}Q+f^{\\prime }(\\hat{u}) & 0\\\\cQ & SQ\\end{array}\\right)\\\\& = -c\\left(\\begin{array}{c c}0 & Q\\\\-Q & 0\\end{array}\\right)=-cJ.\\end{aligned}$ For the second part, we see that $\\frac{d}{dz}\\Omega (Y_1,Y_2)=e^{cz}\\left(c\\omega (Y_1,Y_2)+\\frac{d}{dz}\\omega (Y_1,Y_2)\\right)=0.$ For fixed $\\lambda \\in \\mathbb {R}$ , it is a standard result that (REF ) respects linear independence of solutions.",
"It follows that (REF ) induces a flow on $\\operatorname{Gr}_k(\\mathbb {R}^{2n})$ for any $k$ .",
"The following is then a consequence of the preceding theorem.",
"Corollary 2.1 The set of $\\omega $ -Lagrangian planes $\\Lambda (n)$ is an invariant manifold for the equation induced by (REF ) on $\\operatorname{Gr}_n(\\mathbb {R}^{2n})$ .",
"As explained above, eigenvalues are found by looking for intersections of the sets $E^{s/u}(\\lambda ,z)$ .",
"To make use of Corollary REF , it is therefore critical that the stable and unstable bundles are actually Lagrangian.",
"We show now that this is indeed the case.",
"Theorem 2.2 For all $\\lambda \\in \\mathbb {R}\\cap (\\mathbb {C}-H)$ and $z\\in \\mathbb {R}$ , the subspaces $E^u(\\lambda ,z)$ and $E^s(\\lambda ,z)$ are Lagrangian.",
"First, it is clear that $\\omega $ and $\\Omega $ define the same set of Lagrangian planes.",
"By Theorem REF , we just need to show that $\\Omega (Y_1,Y_2)=0$ for some value of $z$ , given $Y_1,Y_2\\in E^{s/u}(\\lambda ,z)$ .",
"We begin with $E^s(\\lambda ,z)$ .",
"By definition, $Y_1,Y_2\\in E^s(\\lambda ,z)$ decay to 0 as $z\\rightarrow \\infty $ .",
"Since $c<0$ , it is easy to see that $\\Omega (Y_1,Y_2)=\\lim \\limits _{z\\rightarrow \\infty }e^{cz}\\omega (Y_1,Y_2)=0.$ Now consider $Y_1,Y_2\\in E^u(\\lambda ,z)$ .",
"The decay of these solutions at $-\\infty $ will be faster than $e^{-cz}$ , by (REF ) and Theorem 3.1 of [34].",
"It follows that $\\Omega (Y_1,Y_2)=\\lim \\limits _{z\\rightarrow -\\infty }e^{cz}\\omega (Y_1,Y_2)=\\lim \\limits _{z\\rightarrow -\\infty }\\omega (e^{cz}Y_1,Y_2)=0.$ This completes the proof.",
"The result of this section is that the stable and unstable bundles define smooth two-parameter curves in $\\Lambda (n)$ –a lower-dimensional submanifold of $\\operatorname{Gr}_n(\\mathbb {R}^{2n})$ .",
"The way to exploit this fact is through the Maslov index, which we discuss in the next section.",
"We close this section by pointing out that the systems we consider are less general than the “skew-gradient\" systems of [39], [40], [13], which allow for a positive, diagonal matrix $D$ to multiply $u_{xx}$ in (REF ).",
"The reason for this is that the proof of Theorem REF breaks down if we include the matrix $D$ due to the presence of the convective term $cu_z$ .",
"It is quite interesting that we are free to control the coupling of the terms in the reaction (through the matrix $S$ ), but that changing the diffusivities of the reagents ruins the symplectic structure."
],
[
"The Maslov Index",
"As mentioned in the introduction, the fundamental group of $\\Lambda (n)$ is infinite cyclic for all $n\\in \\mathbb {N}$ .",
"The homotopy class of a loop in this space is therefore like a winding number.",
"Intuitively, the duality between winding numbers and intersection numbers should allow us to identify the homotopy class of a loop as an intersection count with a codimension one set in $\\Lambda (n)$ .",
"This is indeed the case, as was shown by Arnol'd [3].",
"In fact, Arnol'd extended this definition to non-closed curves under certain assumptions.",
"These assumptions were relaxed considerably in [33], and the intersection number discussed therein is the Maslov index that we will employ.",
"To start, fix a Lagrangian plane $V\\in \\Lambda (n)$ and define the train of $V$ to be $\\Sigma (V)=\\lbrace V^{\\prime }\\in \\Lambda (n):V\\cap V^{\\prime }\\ne \\lbrace 0\\rbrace \\rbrace .$ There is a natural partition of this set into submanifolds of $\\Lambda (n)$ given by $\\Sigma (V)=\\bigcup _{k=1}^n\\Sigma _k(V),\\hspace{14.45377pt}\\Sigma _k(V)=\\lbrace V^{\\prime }\\in \\Lambda (n):\\dim (V\\cap V^{\\prime })=k \\rbrace .$ In particular, the set $\\Sigma _1(V)$ is dense in $\\Sigma (V)$ , and it is a two-sided, codimension one submanifold of $\\Lambda (n)$ (cf.",
"§2 of [33]).",
"In [3], the Maslov index of a loop $\\alpha $ is defined as the number of signed intersections of $\\alpha $ with $\\Sigma _1(V)$ .",
"A homotopy argument is used to ensure that all intersections with $\\Sigma (V)$ are actually with $\\Sigma _1(V)$ , and hence this definition makes sense.",
"More generally, for a curve $\\gamma :[a,b]\\rightarrow \\Lambda (n)$ , it is shown (§2.2 of [3]) that the same index is well-defined, provided that $\\gamma (a),\\gamma (b)\\notin \\Sigma (V)$ and that all intersections with $\\Sigma (V)$ are one-dimensional and transverse.",
"Both the assumptions of transversality at the endpoints and of only one-dimensional crossings were dispensed of in [33].",
"The key was to make robust the notion of intersections with $\\Sigma (V)$ , which was accomplished through the introduction of the “crossing form.” Now let $\\gamma :[a,b]\\rightarrow \\Lambda (n)$ be a smooth curve.",
"The tangent space to $\\Lambda (n)$ at any point $\\gamma (t)$ can be identified with the space of quadratic forms on $\\gamma (t)$ (cf.",
"§1.6 of [16]).",
"This allows one to define a quadratic form that determines whether $\\gamma (t)$ is transverse to $\\Sigma (V)$ at a given intersection; this quadratic form is the crossing form.",
"Specifically, suppose that $\\gamma (t^*)\\in \\Sigma (V)$ for some $t^*\\in [a,b]$ .",
"It can be checked from (REF ) that the plane $J\\cdot \\gamma (t^*)$ is orthogonal to $\\gamma (t^*)$ , with $J$ as in (REF ).",
"Furthermore, any other Lagrangian plane $W$ transverse to $J\\cdot \\gamma (t^*)$ can be written uniquely as the graph of a linear operator $B_W:\\gamma (t^*)\\rightarrow J\\cdot \\gamma (t^*)$ [16].",
"This includes $\\gamma (t)$ for $|t-t^*|<\\delta \\ll 1.$ Writing $B_{\\gamma (t)}=B(t)$ , it follows that the curve $v+B(t)v\\in \\gamma (t)$ for all $v\\in \\gamma (t^*)$ .",
"The crossing form is then defined by $\\Gamma (\\gamma ,V,t^*)(v)=\\frac{d}{dt}\\omega (v,B(t)v)|_{t=t^*}.$ The form is defined on the intersection $\\gamma (t^*)\\cap V$ .",
"It is shown in Theorem 1.1 of [33] that this definition is independent of the choice $J\\cdot \\gamma (t^*)$ ; any other Lagrangian complement of $\\gamma (t^*)$ would produce the same crossing form.",
"The crossing form is quadratic, so it has a well-defined signature.",
"For a quadratic form $Q$ , we use the notation $\\mathrm {sign}(Q)$ for its signature.",
"We also write $n_+(Q)$ and $n_-(Q)$ for the positive and negative indices of inertia of $Q$ (see page 187 of [36]), so that $\\mathrm {sign}(Q)=n_+(Q)-n_-(Q).$ Roughly speaking, $\\mathrm {sign}(\\Gamma (\\gamma ,V,t^*))$ gives the dimension and the direction of the intersection $\\gamma (t^*)\\cap V$ .",
"Reminiscent of Morse theory, a value $t^*$ such that $\\gamma (t^*)\\cap V\\ne \\lbrace 0\\rbrace $ is called a conjugate point or crossing.",
"A crossing is called regular if the associated form $\\Gamma $ is nondegenerate.",
"One can then define the Maslov index as follows.",
"Definition 2 Let $\\gamma :[a,b]\\rightarrow \\Lambda (n)$ and $V\\in \\Lambda (n)$ such that $\\gamma (t)$ has only regular crossings with the train of $V$ .",
"The Maslov index is then given by $\\mu (\\gamma ,V)=-n_-(\\Gamma (\\gamma ,V,a))+\\sum \\limits _{t^*\\in (a,b)}\\mathrm {sign}\\,\\Gamma (\\gamma ,V,t^*)+n_+(\\Gamma (\\gamma ,V,b)),$ where the sum is taken over all interior conjugate points.",
"Remark 3.1 The reader will notice that the Maslov index defined in [33] has a different endpoint convention than Definition REF .",
"Instead, they take $(1/2)\\mathrm {sign}(\\Gamma )$ as the contribution at both $a$ and $b$ .",
"This is merely convention, provided that one is careful to make sure that the additivity property (see Proposition REF below) holds.",
"Our convention follows [15], [20] to make sure that the Maslov index is always an integer.",
"The convention on the endpoints in (REF ) serves to ensure that the Maslov index has (among others) the following nice properties from §2 of [33].",
"Proposition 3.1 Let $\\gamma :[a,b]\\rightarrow \\Lambda (n)$ be a curve with only regular crossings.",
"Then (Additivity by concatenation) For any $c\\in (a,b)$ , $\\mu (\\gamma ,V)=\\mu (\\gamma |_{[a,c]},V)+\\mu (\\gamma |_{[c,b]},V)$ .",
"(Homotopy invariance) Two paths $\\gamma _{1,2}:[a,b]\\rightarrow \\Lambda (2)$ with $\\gamma _1(a)=\\gamma _2(a)$ and $\\gamma _1(b)=\\gamma _2(b)$ are homotopic with fixed endpoints if and only if $\\mu (\\gamma _1,V)=\\mu (\\gamma _2,V)$ .",
"If $\\dim (\\gamma (t)\\cap V)= \\text{ constant}$ , then $\\mu (\\gamma ,V)=0$ .",
"Up until now, we have considered one curve of Lagrangian planes and seen how to count intersections with a fixed reference plane.",
"Alternatively, one could consider two curves of Lagrangian planes and count how many times they intersect each other.",
"This theory is also developed in [33], see §3.",
"Suppose then that we have two curves $\\gamma _{1,2}:[a,b]\\rightarrow \\Lambda (n)$ .",
"If $\\gamma _1(t^*)\\cap \\gamma _2(t^*)\\ne \\lbrace 0\\rbrace $ for some $t^*\\in [a,b],$ then we can define the relative crossing form $\\Gamma (\\gamma _1,\\gamma _2,t^*)=\\Gamma (\\gamma _1,\\gamma _2(t^*),t^*)-\\Gamma (\\gamma _2,\\gamma _1(t^*),t^*)$ on the intersection $\\gamma _1(t^*)\\cap \\gamma _2(t^*)$ .",
"As before, a crossing is regular if $\\Gamma $ in (REF ) is nondegenerate.",
"For two curves with only regular crossings, define the relative Maslov index to be $\\mu (\\gamma _1,\\gamma _2)=-n_-(\\Gamma (\\gamma _1,\\gamma _2,a))+\\sum \\limits _{t^*\\in (a,b)}\\mathrm {sign}\\,\\Gamma (\\gamma _1,\\gamma _2,t^*)+n_+(\\Gamma (\\gamma _1,\\gamma _2,b)),$ where again the sum is taken over interior intersections of $\\gamma _1$ and $\\gamma _2$ .",
"We point out now that it is easy to show that regular crossings are isolated, so the sums in both (REF ) and (REF ) are finite.",
"Also, it is clear from (REF ) that (REF ) coincides with Definition REF in the case where $\\gamma _2=$ constant.",
"Accordingly, most of the properties of the Maslov index in §2 of [33] carry over to the two-curve case without much trouble.",
"However, in moving from paths to pairs of curves, one must be careful about the homotopy axiom.",
"The following is proved in Corollary 3.3 of [33].",
"Proposition 3.2 Let $\\gamma _1$ and $\\gamma _2$ be curves of Lagrangian planes with common domain $[a,b]$ .",
"If $\\gamma _1(a)\\cap \\gamma _2(a)=\\gamma _1(b)\\cap \\gamma _2(b)=\\lbrace 0\\rbrace $ , then $\\mu (\\gamma _1,\\gamma _2)$ is a homotopy invariant, provided that the homotopy respects the stated condition on the endpoints.",
"We are now ready to specialize to the problem at hand.",
"Recall that for $\\lambda \\ge \\beta $ , $E^u(\\lambda ,z)$ and $E^s(\\lambda ,z)$ are both members of $\\Lambda (n)$ for all $z\\in \\mathbb {R}$ .",
"We will relate the Maslov index to eigenvalues of $L$ by looking for intersections of these subspaces as $z$ and $\\lambda $ vary.",
"Thus there are two practical formulations of (REF ) for our purposes: one for curves parametrized by $z$ and another for curves parametrized by $\\lambda $ .",
"For curves parametrized by $z$ , we have the following formula, which is proved as Theorem 3 of [15].",
"Theorem 3.1 Consider the curve $z\\mapsto E^u(\\lambda ,z)$ , for fixed $\\lambda $ .",
"Assume that for a reference plane $V$ , there exists a value $z=z^*$ such that $E^u(\\lambda ,z^*)\\cap V\\ne \\lbrace 0\\rbrace $ .",
"Then the crossing form for $E^u(\\lambda ,\\cdot )$ with respect to $V$ is given by $\\Gamma (E^u(\\lambda ,\\cdot ),V,z^*)(\\zeta )=\\omega (\\zeta ,A(\\lambda ,z^*)\\zeta ),$ restricted to the intersection $E^u(\\lambda ,z^*)\\cap V$ .",
"We postpone deriving the $\\lambda $ -crossing form until §5.",
"For now, we return to the motivation of this project, which is to use features of the wave $\\hat{u}$ (or $\\varphi $ ) itself to determine its stability.",
"More precisely, we want to associate a Maslov index to $\\varphi $ that we can calculate and use to say something about the unstable spectrum of $L$ .",
"The Maslov index of a homoclinic orbit was defined in a rigorous way in [12].",
"The curve of Lagrangian planes is given by $z\\mapsto E^u(0,z)$ , which can be thought of as the space of solutions to (REF ) with $\\lambda =0$ satisfying the `left' boundary condition.",
"In the spirit of a shooting argument, the natural choice of reference plane is $V^s(0)$ , the stable subspace of the linearization of (REF ) about 0.",
"However, for technical reasons this is untenable.",
"Indeed, by translation invariance, the derivative of the wave $\\varphi ^{\\prime }(z)\\in E^u(0,z).$ Since a homoclinic orbit approaches its end state tangent to the stable manifold, there would be a conjugate point at $+\\infty $ .",
"Moreover, this conjugate point would be irregular since it is reached in infinite time.",
"To address this issue, Chen and Hu (§1 of [12]) instead pulled back $V^s(0)$ slightly along $\\varphi $ and used $E^s(0,\\tau )$ , $\\tau \\gg 1$ as a reference plane.",
"The domain of the curve is truncated as well to $(-\\infty ,\\tau ]$ , which forces a conjugate point at the right end point; $\\varphi ^{\\prime }$ (at least) is in the intersection $E^u(0,\\tau )\\cap E^s(0,\\tau )$ .",
"The only requirement on $\\tau $ is that $V^u(0)\\cap E^s(0,z)=\\lbrace 0\\rbrace \\text{ for all } z\\ge \\tau .$ One then arrives at the following definition.",
"Definition 3 Let $\\tau $ satisfy (REF ).",
"The Maslov index of $\\varphi $ is given by $\\mathrm {Maslov}(\\varphi ):=\\sum _{z^*\\in (-\\infty ,\\tau )}\\mathrm {sign}\\,\\Gamma (E^u,E^s(0,\\tau ),z^*)+n_+(\\Gamma (E^u,E^s(0,\\tau ),\\tau )),$ where the sum is taken over all interior crossings of $E^u(0,z)$ with $\\Sigma $ , the train of $E^s(0,\\tau )$ .",
"It was shown in [12] that this definition is independent of $\\tau $ , as long as (REF ) is satisfied.",
"This is important, because we will have to revise the value $\\tau $ to complete the arguments of the next section.",
"Although we consider several curves in this work, any mention of the Maslov index is referring to $\\mathrm {Maslov}(\\varphi )$ .",
"The value $\\lambda =0$ is special because (REF ) is the equation of variations for (REF ) in that case.",
"Accordingly, one can show (cf.",
"§6 of [15]) that $E^u(0,z)$ is tangent to $W^u(0)$ along $\\varphi $ .",
"The Maslov index can therefore be interpreted as the number of twists $W^u(0)$ makes as $\\varphi $ moves through phase space.",
"In the case $n=1$ , $E^u(0)$ is spanned by the velocity to the wave, and conjugate points correspond to zeros of $\\varphi ^{\\prime }(z)$ (albeit with a rotation of the reference plane).",
"In this way, one sees that the Maslov index can be used to derive Sturm-Liouville theory (see also §1 of [6])."
],
[
"The Maslov Box",
"We will see in this section that the set of positive, real eigenvalues of $L$ is bounded above.",
"Since the spectrum of $L$ in $\\mathbb {C}\\setminus H$ consists of isolated eigenvalues of finite multiplicity (cf.",
"page 172 of [1]), it follows that the quantity $\\operatorname{Mor}(L):= \\text{ the number of real, positive eigenvalues of } L \\text{ counting algebraic multiplicity}$ is well defined.",
"The rest of this section is dedicated to proving Theorem 4.1 $|\\mathrm {Maslov}(\\varphi )|\\le \\operatorname{Mor}(L).$ The strategy of the proof is to consider a contractible loop in $\\Lambda (n)\\times \\Lambda (n)$ (the “Maslov box”) consisting of four different curve segments.",
"Since the total Maslov index must be zero, Proposition REF (i) guarantees that the sum of the constituent Maslov indices is zero.",
"Two of these segments have Maslov index zero, one of them is $\\mathrm {Maslov}(\\varphi )$ , and the final segment is bounded above by $\\operatorname{Mor}(L)$ .",
"This strategy has its roots in [25], [21], [20].",
"In particular, [20] coined the term “Maslov box,\" and that paper encounters many of the same difficulties that arise when considering homoclinic orbits (i.e.",
"curves on infinite intervals).",
"The difference between this paper and [20] is that the latter considered gradient reaction-diffusion equations.",
"In that case, the linearized operator $L$ is self-adjoint, and the Maslov index is computed using spectral flow of unitary matrices.",
"From this point forward, we will think of the stable and unstable bundles as curves in $\\Lambda (n)$ .",
"Likewise, we think of the stable and unstable subspaces of $A_\\infty (\\lambda )$ as points in $\\operatorname{Gr}_n(\\mathbb {R}^{2n})$ .",
"We call these spaces $S(\\lambda )$ and $U(\\lambda )$ respectively.",
"(In particular, $V^s(0)=S(0)$ and $V^u(0)=U(0)$ .)",
"It follows from Lemma 3.2 of [1] that $\\begin{aligned}\\lim \\limits _{z\\rightarrow -\\infty }E^u(\\lambda ,z)=U(\\lambda )\\\\\\lim \\limits _{z\\rightarrow \\infty }E^s(\\lambda ,z)=S(\\lambda ).\\end{aligned}$ In light of Theorem REF , this actually proves that $S(\\lambda ),U(\\lambda )\\in \\Lambda (n),$ since $\\Lambda (n)$ is a closed submanifold of $\\operatorname{Gr}_n(\\mathbb {R}^{2n})$ .",
"In what follows, it will be important to know what happens to $E^u(\\lambda ,z)$ as $z\\rightarrow \\infty $ .",
"First, if $\\lambda \\in \\sigma (L)$ , then $E^u(\\lambda ,z)\\cap E^s(\\lambda ,z)\\ne \\lbrace 0\\rbrace $ , so it must be the case that $\\lim \\limits _{z\\rightarrow \\infty } E^u(\\lambda ,z)\\in \\Sigma (S(\\lambda )),$ the train of $S(\\lambda )$ .",
"On the other hand, if $\\lambda \\notin \\sigma (L)$ , then any solution of (REF ) is unbounded at $+\\infty $ , and it is proved in Lemma 3.7 of [1] that $\\lim \\limits _{z\\rightarrow \\infty }E^u(\\lambda ,z)=U(\\lambda ).$ There are a few facts to be gleaned from this observation.",
"First, if $\\lambda \\notin \\sigma (L)$ , then $z\\mapsto E^u(\\lambda ,z)$ (with domain $\\mathbb {R}$ ) forms a loop in $\\Lambda (n)$ , in which case the Maslov index is independent of the choice of reference plane (§1.5 of [3]).",
"This fact was used in [24], which is the first appearance of the Maslov index for solitary waves (known to the author).",
"Also, it follows from (REF ) and (REF ) that $\\lim \\limits _{z\\rightarrow \\infty } E^u(\\lambda ,z)$ is discontinuous in $\\lambda $ at each eigenvalue of $L$ .",
"Indeed, $U(\\lambda )$ is bounded away from $\\Sigma (S(\\lambda ))$ , since $\\mathbb {R}^{2n}=S(\\lambda )\\oplus U(\\lambda )$ .",
"This is the motivation for using the cutoff $x_\\infty $ (or $\\tau $ in this paper) for the unstable bundle in [20], since the homotopy argument requires a continuous curve.",
"We also have the additional motivation for the cutoff of using $\\mathrm {Maslov}(\\varphi )$ explicitly.",
"Proposition 2.2 of [1] guarantees that one can draw a simple, closed curve in $\\mathbb {C}$ containing $\\sigma (L)\\cap (\\mathbb {C}\\setminus H)$ in its interior.",
"An obvious consequence of this is that the real, unstable spectrum of $L$ is bounded above by a constant $M$ .",
"We will make use of the following, slightly stronger fact.",
"Lemma 4.1 There exists $\\lambda _\\mathrm {max}>M$ such that, for all $z\\in \\mathbb {R}$ , $E^u(\\lambda _\\mathrm {max},z)\\cap S(\\lambda _\\mathrm {max})=\\lbrace 0\\rbrace .$ This is proved in §4.5 of [20], and we refer the reader there for the details.",
"We will instead outline the basic idea, which is straightforward.",
"System (REF ) is a perturbation of the autonomous system $Y^{\\prime }(z)=A_\\infty (\\lambda )Y(z)$ , which also induces an equation on $\\operatorname{Gr}_n(\\mathbb {R}^{2n})$ .",
"In this latter system, $U(\\lambda )$ is an attracting fixed point, as is observed in the proof of Lemma 3.7 in [1].",
"We can therefore find a small ball $B$ around $U(\\lambda )$ in $\\Lambda (n)$ on the boundary of which the vector field points inward.",
"Furthermore, this ball can be taken small enough to be disjoint from $\\Sigma (S(\\lambda ))$ , which is itself closed in $\\Lambda (n)$ .",
"For large enough $\\lambda $ , (REF ) is essentially autonomous, so $B$ will still be positively invariant.",
"Finally, since any $\\lambda >M$ is not an eigenvalue of $L$ , the curve $z\\mapsto E^u(\\lambda ,z)$ will both emanate from and return to $U(\\lambda )$ .",
"It will therefore be trapped in the ball $B$ , and hence there will be no intersections with $S(\\lambda )$ .",
"Now fix the value $\\lambda _\\mathrm {max}$ guaranteed by the preceding lemma.",
"The immediate goal is to set, once and for all, the value $\\tau $ appearing in Definition REF .",
"Since $E^s(\\lambda _\\mathrm {max},z)\\rightarrow S(\\lambda _\\mathrm {max})$ as $z\\rightarrow \\infty $ , it follows from Lemma REF that we can find a value $z=\\tau _\\mathrm {max}$ such that $E^u(\\lambda _\\mathrm {max},z)\\cap E^s(\\lambda _\\mathrm {max},\\zeta )=\\lbrace 0\\rbrace , \\hspace{7.22743pt} \\text{ for all } z\\in \\mathbb {R}\\text{ and for all }\\zeta \\ge \\tau _\\mathrm {max}.$ Similarly, for each $\\lambda \\in [0,\\lambda _\\mathrm {max}]$ we can find $\\tau _\\lambda $ and an open interval $I_\\lambda $ containing $\\lambda $ such that $U(\\lambda )\\cap E^s(\\lambda ,z)=\\lbrace 0\\rbrace ,\\hspace{7.22743pt} \\text{ for all } z\\ge \\tau _\\lambda , \\lambda \\in I_\\lambda .$ (For $\\lambda =\\lambda _\\mathrm {max}$ , the value $\\tau _\\mathrm {max}$ defined above works just fine.)",
"Extracting a finite subcover $\\cup _{k=1}^NI_{\\lambda _k}$ of $[0,\\lambda _\\mathrm {max}]$ , we set $\\tau =\\max \\lbrace \\tau _{\\lambda _1},\\dots ,\\tau _{\\lambda _k}\\rbrace .$ The preceding can be summarized in the following proposition.",
"Proposition 4.1 With $\\tau $ given by (REF ), the following are true.",
"$E^u(\\lambda _\\mathrm {max},z)\\cap E^s(\\lambda _\\mathrm {max},\\tau )=\\lbrace 0\\rbrace $ for all $z\\in (-\\infty ,\\tau ]$ .",
"$U(\\lambda )\\cap E^s(\\lambda ,\\tau )=\\lbrace 0\\rbrace $ for all $\\lambda \\in [0,\\lambda _\\mathrm {max}].$ Now consider the rectangle $Q=[0,\\lambda _\\mathrm {max}]\\times [-\\infty ,\\tau ].$ $Q$ is mapped into $\\Lambda (n)\\times \\Lambda (n)$ by the function $G(\\lambda ,z)=(E^u(\\lambda ,z),E^s(\\lambda ,\\tau )),$ where $G(\\lambda ,-\\infty )$ is defined to be $(U(\\lambda ),E^s(\\lambda ,\\tau ))$ .",
"Notice that $G$ is continuous, see §3 of [1].",
"Since $Q$ is contractible, the image $G(Q)\\subset \\Lambda (2)\\times \\Lambda (2)$ is contractible as well.",
"Let $F:Q\\times [0,1]\\rightarrow Q$ be a deformation retract (page 361 of [31]) of $Q$ onto the point $(0,-\\infty )$ .",
"Composing $F$ and $G$ then gives a deformation retract of $G(Q)$ onto $G(0,-\\infty )=(U(0),E^s(0,\\tau ))$ .",
"In particular, we see that the image of the boundary $\\partial Q$ (with a counterclockwise orientation) under $G$ is homotopic with fixed endpoints to the constant path $(U(0),E^s(0,\\tau ))$ .",
"We will call this (closed) boundary curve $\\alpha $ .",
"Since $U(0)\\cap E^s(0,\\tau )=\\lbrace 0\\rbrace $ by Proposition REF (ii), we see that Proposition REF applies, so $\\mu (\\alpha )=\\mu (U(0),E^s(0,\\tau ))=0.$ The Maslov index in this case is for pairs of Lagrangian planes, since $\\alpha \\subset \\Lambda (n)\\times \\Lambda (n)$ .",
"We can describe the loop $\\alpha $ as the concatenation of four curve segments.",
"(See Figure REF below.)",
"Define: $\\begin{aligned}\\alpha _1 & = (E^u(0,z),E^s(0,\\tau )), \\hspace{7.22743pt} z\\in [-\\infty ,\\tau ]\\\\\\alpha _2 & = (E^u(\\lambda ,\\tau ),E^s(\\lambda ,\\tau )), \\hspace{7.22743pt} \\lambda \\in [0,\\lambda _\\mathrm {max}]\\\\\\alpha _3 & = (E^u(\\lambda _\\mathrm {max},-z),E^s(\\lambda _\\mathrm {max},\\tau )), \\hspace{7.22743pt} z\\in [-\\tau ,\\infty ] \\\\\\alpha _4 & =(U(\\lambda _\\mathrm {max}-\\lambda ),E^s(\\lambda _\\mathrm {max}-\\lambda ,\\tau )), \\hspace{7.22743pt} \\lambda \\in [0,\\lambda _\\mathrm {max}].\\end{aligned}$ Using the notation of [31], page 326, it is clear that $\\alpha =\\alpha _1*\\alpha _2*\\alpha _3*\\alpha _4$ .",
"As explained above, $\\mu (\\alpha )=0$ , since $G(Q)$ is contractible.",
"Proposition REF (i) then asserts that $0=\\mu (\\alpha )=\\mu (\\alpha _1)+\\mu (\\alpha _2)+\\mu (\\alpha _3)+\\mu (\\alpha _4).$ It is a direct consequence of Proposition REF (i) that $\\mu (\\alpha _3)=0$ , since there are no conjugate points.",
"Likewise, Proposition REF (ii) says that $\\mu (\\alpha _4)=0$ .",
"Comparing (REF ) with Definition REF , we see that $\\mu (\\alpha _1)=\\mathrm {Maslov}(\\varphi ).$ Taken together with (REF ), these observations show that $|\\mathrm {Maslov}(\\varphi )|=|\\mu (\\alpha _2)|.$ To prove Theorem REF , it therefore suffices to show that $|\\mu (\\alpha _2)|\\le \\operatorname{Mor}(L).$ Figure: “Maslov Box\": Domain in λz\\lambda z-planeRemark 4.1 Notice in Figure REF the conjugate point in the upper left corner.",
"This crossing corresponds to the translation invariance of (REF ) (i.e.",
"$E^u(0,\\tau )\\cap E^s(0,\\tau )\\ne \\lbrace 0\\rbrace $ ).",
"The contributions of this crossing to $\\mu (\\alpha _1)$ and $\\mu (\\alpha _2)$ can be determined using (REF ).",
"Suppose that $\\lambda ^*$ is a conjugate point for $\\alpha _2$ .",
"By definition, this means that $E^u(\\lambda ^*,\\tau )\\cap E^s(\\lambda ^*,\\tau )\\ne \\lbrace 0\\rbrace .$ But this is precisely the condition that $\\lambda $ be an eigenvalue of $L$ .",
"Furthermore, the dimension of the intersection in (REF ) captures the geometric multiplicity of $\\lambda ^*$ as an eigenvalue.",
"By the triangle inequality, we therefore have $|\\mu (\\alpha _2)|\\le \\sum _{\\lambda ^*\\in [0,\\lambda _{\\mathrm {max}}]}\\dim (E^u(\\lambda ^*,\\tau )\\cap E^s(\\lambda ^*,\\tau )),$ where the sum is taken over all conjugate points.",
"Since $[0,\\lambda _\\mathrm {max}]$ contains all possible real, unstable eigenvalues of $L$ , and the geometric multiplicity of an eigenvalue is no greater than its algebraic multiplicity, we see that $|\\mu (\\alpha _2)|\\le \\operatorname{Mor}(L)$ , proving Theorem REF ."
],
[
"Counting Eigenvalues in a FitzHugh-Nagumo System",
"There are two reasons that the inequality in Theorem REF cannot be improved to equality in general.",
"First, the Maslov index counts signed intersections, so that two different eigenvalues of $L$ might offset in the calculation of $\\mu (\\alpha _2)$ if the crossing forms have different signatures.",
"Second, a given eigenvalue might be deficient (i.e.",
"have lesser geometric than algebraic multiplicity).",
"In this and the next section, we consider traveling waves in a FitzHugh-Nagumo system wherein neither of these potential pitfalls occurs.",
"Additionally, we prove that any unstable spectrum must be real, so that the Maslov index actually counts the total number of unstable eigenvalues.",
"The FitzHugh-Nagumo system is given by $\\begin{aligned}u_t & = u_{xx}+g(u)-v\\\\v_t & = dv_{xx}+\\epsilon (u-\\gamma v),\\end{aligned}$ where $g(u)=u(1-u)(u-a)$ , $0<a<1/2$ and $\\epsilon ,\\gamma >0$ .",
"Typically, $\\epsilon $ is taken to be very small, and (REF ) is studied using techniques of singular perturbation theory.",
"The stability of various traveling and standing fronts and pulses has been studied for the variation of (REF ) in which either $d=0$ or $0<d\\ll 1$ [23], [37], [17], [1], [38].",
"If $d=1$ , one checks that (REF ) is of the form (REF ), with $f(u)=\\left(\\begin{array}{c}g(u)-v\\\\-u+\\gamma v\\end{array}\\right), \\hspace{3.61371pt} Q=\\left(\\begin{array}{c c }1 & 0\\\\0 & -1\\end{array}\\right), \\hspace{3.61371pt} S=\\left(\\begin{array}{c c }1 & 0 \\\\ 0 & \\epsilon \\end{array}\\right).$ The case $d=O(1)$ is considered in [12], in which it is shown that standing waves for (REF ) are stable.",
"In [11], the authors use variational techniques to prove that traveling waves exist as well, but the stability question remains open.",
"For $d=1$ , the same traveling waves are constructed in [14] using geometric singular perturbation theory.",
"(See also §6 of [15].)",
"It falls out of this construction that the wave $\\varphi =(\\hat{u},\\hat{v})$ moves to the left (i.e.",
"$c<0$ ) and is homoclinic to 0 as an orbit $(\\hat{u},\\hat{v},\\hat{u}^{\\prime },\\hat{v}^{\\prime }/\\epsilon )$ in four-dimensional phase space.",
"In an abuse of notation, we will use $\\varphi $ for both the solution $\\varphi =(\\hat{u},\\hat{v})$ of (REF ) and the corresponding homoclinic orbit in phase space.",
"In what follows, we show how the Maslov index provides the framework for showing that these waves are stable.",
"The subsequent calculation of $\\mathrm {Maslov}(\\varphi )$ , which completes the stability proof, is the topic of [14].",
"To start, note that we are concerned with the spectrum of $L_\\epsilon =\\partial _z^2+c\\partial _z+\\left(\\begin{array}{c c}g^{\\prime }(\\hat{u}) & -1\\\\\\epsilon & -\\epsilon \\gamma \\end{array}\\right),$ acting on $BU(\\mathbb {R},\\mathbb {R}^2)$ .",
"The subscript $\\epsilon $ serves both to remind the reader that the operator is $\\epsilon $ -dependent and to distinguish results that are general for (REF ) from those that are specific to (REF ).",
"To apply the methods of this paper, we must verify that condition (REF ) is met.",
"Indeed, a simple calculation gives that $g^{\\prime }(0)=-a$ and the eigenvalues of $QSf^{\\prime }(0)$ are $\\nu _i=\\frac{-(a+\\epsilon \\gamma )\\pm \\sqrt{(a+\\epsilon \\gamma )^2-4\\epsilon (a\\gamma +1)}}{2}.$ For $\\epsilon $ sufficiently small, the $\\nu _i$ are easily seen to be negative and distinct.",
"It then follows from (REF ) that the eigenvalues of $A_\\infty (\\lambda )$ for $\\lambda \\ge 0$ are given by $\\mu _1(\\lambda )<\\mu _2(\\lambda )<0<-c<\\mu _3(\\lambda )<\\mu _4(\\lambda ).$ The benefit of having simple eigenvalues is that we can give analytically varying bases of $E^s(\\lambda ,z)$ and $E^u(\\lambda ,z)$ that separate solutions with different growth rates, see [15] for details.",
"This will be important in §6 when we discuss the symplectic Evans function.",
"We now proceed to show that any unstable eigenvalues of $L_\\epsilon $ must be real.",
"After that, we address the issue of direction of crossings by deriving the $\\lambda $ crossing form and showing that it is positive definite at all conjugate points.",
"Finally, in §6 we show that the algebraic and geometric multiplicities of any unstable eigenvalues of $L_\\epsilon $ are the same.",
"This will prove: Theorem 5.1 $\\mathrm {Maslov}(\\varphi )=\\operatorname{Mor}(L_\\epsilon )=|\\sigma (L_\\epsilon )\\cap \\lbrace \\lambda \\in \\mathbb {C}:\\mathrm {Re}\\,\\lambda \\ge 0\\rbrace |.$"
],
[
"Realness of $\\sigma (L_\\epsilon )$",
"For reference, we write out the eigenvalue problem for $L_\\epsilon $ as a first order system, as in (REF ): $\\left(\\begin{array}{c}p\\\\q\\\\r\\\\s\\end{array} \\right)_z=\\left(\\begin{array}{c c c c}0 & 0 & 1 & 0\\\\0 & 0 & 0 & \\epsilon \\\\\\lambda -g^{\\prime }(\\hat{u}) & 1 & -c & 0\\\\-1 & \\frac{\\lambda }{\\epsilon }+\\gamma & 0 & -c\\end{array}\\right)\\left(\\begin{array}{c}p\\\\q\\\\r\\\\s\\end{array}\\right).$ As in §2, we abbreviate (REF ) as $Y^{\\prime }(z)=A(\\lambda ,z)Y(z).$ The discussion of $\\sigma _\\mathrm {ess}(L)$ from §2 applies to $L_\\epsilon $ as well.",
"However, the upper bound $\\beta $ on the real part of the essential spectrum now depends on $\\epsilon $ .",
"This is not a problem, since $\\epsilon $ is fixed in the stability analysis.",
"However, a few of the results to follow need $\\epsilon $ to be “sufficiently small.\"",
"For completeness, we record the following lemma on $\\sigma _\\mathrm {ess}(L_\\epsilon )$ .",
"Lemma 5.1 For each $\\epsilon >0$ sufficiently small, there exists $\\beta _\\epsilon <0$ such that $\\sigma _\\mathrm {ess}(L_\\epsilon )\\subset H_\\epsilon :=\\lbrace \\lambda \\in \\mathbb {C}:\\mathrm {Re}\\,\\lambda <\\beta _\\epsilon \\rbrace .$ The analysis of $L_\\epsilon $ is complicated by the presence of the $\\partial _z$ term in (REF ).",
"We can sidestep this difficulty by considering instead the operator $L_c:=e^{cz/2}L_\\epsilon e^{-cz/2},$ as is done in [7], [20].",
"It is a routine calculation to see that for $(p, q)^T\\in BU(\\mathbb {R},\\mathbb {C}^2)$ , we have $L_c\\left(\\begin{array}{c}p\\\\q\\end{array}\\right)=\\left(\\begin{array}{c}p_{zz}+\\left(g^{\\prime }(\\hat{u})-\\frac{c^2}{4}\\right)p-q\\\\q_{zz}+\\epsilon p-\\left(\\frac{c^2}{4}+\\epsilon \\gamma \\right)q\\end{array}\\right).$ Furthermore, if $L_\\epsilon P=\\lambda P$ , then $L_c(e^{cz/2}P)=\\lambda e^{cz/2}P$ .",
"This proves that the eigenvalues of $L_\\epsilon $ and $L_c$ are the same, provided that $e^{cz/2}P$ is bounded for a given eigenvector $P$ of $L_\\epsilon $ .",
"This is clearly the case as $z\\rightarrow \\infty $ since $c<0$ .",
"For the other tail, let $\\lambda \\in \\mathbb {C}\\setminus H_\\epsilon $ be an eigenvalue of $L_\\epsilon $ with associated eigenvector $P$ .",
"Since $A_\\infty (\\lambda )$ is hyperbolic with simple eigenvalues, $P$ must decay at least as fast as $e^{\\mu _3(\\lambda )z}$ as $z\\rightarrow -\\infty $ .",
"It follows that $e^{cz/2}P$ is bounded at $-\\infty $ if $\\frac{c}{2}+\\mu _3(\\lambda )>0.$ This is indeed the case, by (REF ).",
"We therefore consider the eigenvalue problem $L_cP=\\lambda P.$ Making the change of variables $\\tilde{q}=\\frac{1}{\\sqrt{\\epsilon }}q$ , we can rewrite (REF ) as (dropping the tildes) $\\left(\\begin{array}{c c}\\partial _z^2+\\left(g^{\\prime }(\\hat{u})-\\frac{c^2}{4}\\right) & -\\sqrt{\\epsilon }\\\\\\sqrt{\\epsilon } & \\partial _z^2-\\left(\\frac{c^2}{4}+\\epsilon \\gamma \\right)\\end{array}\\right)\\left(\\begin{array}{c}p\\\\q\\end{array}\\right)=\\lambda \\left(\\begin{array}{c}p\\\\q\\end{array}\\right).$ $L_c$ is now seen to be of the form $L_c=\\left(\\begin{array}{c c}L_p & -\\sqrt{\\epsilon }\\\\\\sqrt{\\epsilon } & L_q\\end{array}\\right),$ where $L_{p/q}$ are self-adjoint on $H^1(\\mathbb {R})$ .",
"Since any eigenfunction of $L_c$ in $BU(\\mathbb {R},\\mathbb {C}^2)$ is exponentially decaying (provided $\\lambda \\in \\mathbb {C}\\setminus H_\\epsilon $ ), we are free to consider the spectrum of $L_c$ as an operator on the Hilbert space $H^1(\\mathbb {R},\\mathbb {C}^2)$ instead.",
"The payoff of studying $L_c$ instead of $L_\\epsilon $ is the following result, a version of which was proved in Lemma 4.1 of [13].",
"We reproduce the proof here for convenience of the reader.",
"Lemma 5.2 For $\\epsilon >0$ sufficiently small, if $\\lambda \\in \\sigma _{n}(L_c)\\cap (\\mathbb {C}\\setminus H_\\epsilon )$ and $\\mathrm {Re}\\,\\lambda \\ge -\\frac{c^2}{8} $ , then $\\lambda \\in \\mathbb {R}$ .",
"Consequently, the same is true for $L_\\epsilon $ .",
"Let $\\lambda =a+bi$ be an eigenvalue for $L_c$ with corresponding eigenvector $(p, q)^T$ .",
"Assume further that $a\\ge -c^2/8$ .",
"Notice that the second equation in (REF ) can be solved for $q$ , since $L_q+c^2/8$ is negative definite (and hence $\\lambda \\notin \\sigma (L_q))$ .",
"Explicitly, we have $q=-{\\sqrt{\\epsilon }}\\left(L_q-a-bi\\right)^{-1}p$ Next, substitute this expression into the first equation of (REF ) to obtain $L_pp+\\epsilon (L_q-a-bi)^{-1}p=(a+bi)p.$ Taking the $H^1$ pairing $\\langle \\cdot ,\\cdot \\rangle $ with $p$ in (REF ) yields $\\langle L_pp,p\\rangle +\\epsilon \\langle (L_q-a-bi)^{-1}p,p\\rangle =(a+bi)\\langle p,p\\rangle .$ Recalling that $L_p$ is self-adjoint, we extract the imaginary parts of (REF ): $\\epsilon \\, \\mathrm {Im}\\langle \\left(L_q-a-bi\\right)^{-1}p,p \\rangle = b\\langle p,p\\rangle .$ The operator inverse in (REF ) can be decomposed into (self-adjoint) real and imaginary parts as follows: $(L_q-a-bi)^{-1}=\\left((L_q-a)^2+b^2\\right)^{-1}(L_q-a)+ib\\left((L_q-a)^2+b^2\\right)^{-1}.$ Combining (REF ) and (REF ), we arrive at $b\\langle \\left[\\epsilon \\left((L_q-a)^2+b^2\\right)^{-1}-I\\right]p,p\\rangle =0,$ where $I$ denotes the identity operator.",
"For operators on $H^1(\\mathbb {C},\\mathbb {C}^2)$ we write $A<B$ if $(B-A)$ is positive definite.",
"Since $L_q-a<0$ (independently of $\\epsilon $ ) it follows from the inequality $\\left((L_q-a)^2+b^2\\right)^{-1}<(L_q-a)^{-2}$ and the fact that $(L_q-a)^{-2}$ is bounded that $\\epsilon \\left((L_q-a)^2+b^2\\right)^{-1}-I<0$ for $\\epsilon $ small enough.",
"In conjunction with (REF ), this implies that $b=0$ , as desired."
],
[
"Monotonicity of $\\lambda $ -Crossings",
"Recall that conjugate points along $\\alpha _2$ correspond to eigenvalues of $L_\\epsilon $ .",
"We will show below that the crossing form (in $\\lambda $ ) is positive definite at all such crossings.",
"This is the most significant difference between skew-gradient systems and the gradient systems considered in [20], since the crossing form is always positive definite in the latter case (cf.",
"§4.1 and §5.5).",
"Conversely, we rely on the smallness of $\\epsilon $ to get monotonicity of the crossings for $L_\\epsilon $ in (REF ).",
"We stress that the $\\lambda $ crossing form developed in this section would be the same for general systems (REF ).",
"We focus on $L_\\epsilon $ only because we are able to prove that the form is positive definite in this case.",
"To derive the $\\lambda $ crossing form, we first take a closer look at the $z$ crossing form (REF ).",
"Suppose that $z^*$ is a conjugate point for $\\alpha _1$ , and that $\\xi \\in E^u(0,z^*)\\cap E^s(0,\\tau )$ .",
"By virtue of being in $E^u(0,z^*)$ , we know that there exists a solution $u(z)$ of (REF ) such that $u(z)\\in E^u(0,z)$ and $E^u(0,z^*)=\\xi $ .",
"It follows that (REF ) can be rewritten $\\Gamma (E^u(\\lambda ,\\cdot ),E^s(\\lambda ,\\tau ),z^*)(\\xi )=\\omega (\\xi ,A(0,z^*)\\xi )=\\omega (u(z),\\partial _z u(z))|_{z=z^*}.$ In other words, the crossing form simplifies when evaluated on a vector that is part of a solution to a differential equation.",
"Now suppose that $\\lambda =\\lambda ^*$ is a conjugate point for $\\alpha _2$ , with $\\xi \\in E^u(\\lambda ^*,\\tau )\\cap E^s(\\lambda ^*,\\tau )$ .",
"From (REF ), we know that we must evaluate two crossing forms–one where the curve $E^s(\\lambda ,\\tau )$ is frozen at $\\lambda =\\lambda ^*$ and one where $E^u(\\lambda ,\\tau )$ is frozen at $\\lambda =\\lambda ^*$ .",
"To simplify the calculations, we will work with $\\Omega $ instead of $\\omega $ .",
"Since one of these forms is just a scaled version of the other, it is clear that the signatures are the same, and hence the Maslov indices are as well.",
"First consider the curve $\\lambda \\mapsto E^u(\\lambda ,\\tau )$ and reference plane $E^s(\\lambda ^*,\\tau )$ .",
"As in §3, we can write $E^u(\\lambda ,\\tau )$ for $|\\lambda -\\lambda ^*|$ small as the graph of an operator $B_\\lambda :E^u(\\lambda ^*,\\tau )\\rightarrow J\\cdot E^u(\\lambda ^*,\\tau )$ .",
"This, in turn, generates a smooth curve $\\gamma (\\lambda )=(\\xi +B_\\lambda \\xi )\\in E^u(\\lambda ,\\tau )$ with $\\gamma (\\lambda ^*)=\\xi $ .",
"By flowing backwards in $z$ , we obtain a one-parameter family $u(\\lambda ,z)$ of solutions to (REF ) in $E^u(\\lambda ,\\tau )$ , with $u(\\lambda ^*,\\tau )=\\xi $ .",
"The same reasoning as for $z$ then shows that $\\Gamma (E^u(\\cdot ,\\tau ),E^s(\\lambda ^*,\\tau ),\\lambda ^*)(\\xi )=\\Omega (u(\\lambda ,z),\\partial _\\lambda u(\\lambda ,z))|_{\\lambda =\\lambda ^*,z=\\tau }.$ The case where $E^s(\\lambda ,\\tau )$ varies and $E^u(\\lambda ^*,\\tau )$ is fixed is identical.",
"We can generate a smooth family of solutions $v(\\lambda ,z)\\in E^s(\\lambda ,z)$ with $v(\\lambda ^*,\\tau )=\\xi $ .",
"This half of the crossing form is then given by $\\Gamma (E^s(\\cdot ,\\tau ),E^u(\\lambda ^*,\\tau ),\\lambda ^*)(\\xi )=\\Omega (v(\\lambda ,z),\\partial _\\lambda v(\\lambda ,z))|_{\\lambda =\\lambda ^*,z=\\tau }.$ By uniqueness of solutions, we importantly have $u(\\lambda ^*,z)=v(\\lambda ^*,z):=P(z),$ which is a $\\lambda ^*$ -eigenvector of $L_\\epsilon $ .",
"Putting together (REF ), (REF ), and (REF ), we see that $\\begin{aligned}\\Gamma (E^u(\\cdot ,\\tau ),E^s(\\cdot ,\\tau ),\\lambda ^*)(\\xi ) & = \\lbrace \\Omega (u(\\lambda ,z),\\partial _\\lambda u(\\lambda ,z))-\\Omega (v(\\lambda ,z),\\partial _\\lambda v(\\lambda ,z))\\rbrace |_{\\lambda =\\lambda ^*,z=\\tau }\\\\& = \\lbrace \\Omega (u(\\lambda ,z),\\partial _\\lambda u(\\lambda ,z))+\\Omega (\\partial _\\lambda v(\\lambda ,z),v(\\lambda ,z))\\rbrace |_{\\lambda =\\lambda ^*,z=\\tau }\\\\& = \\partial _\\lambda \\Omega (v(\\lambda ,z),u(\\lambda ,z))|_{\\lambda =\\lambda ^*,z=\\tau },\\end{aligned}$ where the last equality follows from (REF ).",
"The expression obtained in (REF ) will be useful in the next section when we relate the crossing form to the Evans function.",
"For now, we compute (REF ) and (REF ) directly.",
"For (REF ), we use the equality of mixed partials and the fact that $u$ solves (REF ) to obtain $(\\partial _\\lambda u(\\lambda ,z))_z=A(\\lambda ,z)\\partial _\\lambda u(\\lambda ,z)+A_\\lambda u(\\lambda ,z),$ where $A_\\lambda :=\\partial _\\lambda A(\\lambda ,z)=\\left(\\begin{array}{c c c c}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\1 & 0 & 0 & 0\\\\0 & \\epsilon ^{-1} & 0 & 0\\end{array}\\right).$ Next, apply $\\omega (u(\\lambda ,z),\\cdot )$ to (REF ) to see that $\\begin{aligned}\\omega (u(\\lambda ,z),A_\\lambda u(\\lambda ,z)) & =\\omega (u(\\lambda ,z),(\\partial _\\lambda u(\\lambda ,z))_z)-\\omega (u(\\lambda ,z),A(\\lambda ,z)\\partial _\\lambda u(\\lambda ,z))\\\\& = \\omega (u(\\lambda ,z),(\\partial _\\lambda u(\\lambda ,z))_z)+\\omega (A(\\lambda ,z)u(\\lambda ,z),\\partial _\\lambda u(\\lambda ,z))+c\\omega (u(\\lambda ,z),\\partial _\\lambda u(\\lambda ,z))\\\\& =\\partial _z\\omega (u(\\lambda ,z),\\partial _\\lambda u(\\lambda ,z))+c\\omega (u(\\lambda ,z),\\partial _\\lambda u(\\lambda ,z)).\\end{aligned}$ The second equality follows from the proof of Theorem REF , specifically (REF ).",
"Applying an integrating factor and using (REF ) and (REF ) then shows that $\\begin{aligned}\\Gamma (E^u(\\cdot ,\\tau ),E^s(\\lambda ^*,\\tau ),\\lambda ^*)(\\xi ) & =\\Omega (u,\\partial _\\lambda u)(\\lambda ^*,\\tau )\\\\& =\\int \\limits _{-\\infty }^{\\tau }\\partial _z\\Omega (u(\\lambda ,z),\\partial _\\lambda u(\\lambda ,z))|_{\\lambda =\\lambda ^*}\\,dz =\\int \\limits _{-\\infty }^{\\tau }e^{cz}\\omega (P,A_\\lambda P)\\,dz.\\end{aligned}$ The preceding calculation makes use of the fact that $u(\\lambda ,z)\\in E^u(\\lambda ,z)$ , and hence it decays faster than $e^{cz}$ as $z\\rightarrow -\\infty $ , by (REF ).",
"The calculation of the crossing form for the stable bundle using the solutions $v(\\lambda ,z)$ is identical until the last step.",
"Indeed, those solutions decay at $+\\infty $ , so an application of the Fundamental Theorem gives $\\begin{aligned}\\Gamma (E^s(\\cdot ,\\tau ),E^u(\\lambda ^*,\\tau ),\\lambda ^*)(\\xi ) & =\\Omega (v,\\partial _\\lambda v)(\\lambda ^*,\\tau )\\\\& =-\\int \\limits _{\\tau }^{\\infty }\\partial _z\\Omega (v,\\partial _\\lambda v)(\\lambda ^*,z)\\,dz =-\\int \\limits _{\\tau }^{\\infty }e^{cz}\\omega (P,A_\\lambda P)\\,dz.\\end{aligned}$ Combining (REF ), (REF ), and (REF ), we see that the relative crossing form is given by $\\Gamma (E^u(\\cdot ,\\tau ),E^s(\\cdot ,\\tau ),\\lambda ^*)(\\xi )=\\int _{-\\infty }^{\\infty }e^{cz}\\,\\omega (P,A_\\lambda P)\\,dz,$ where $P\\in E^u(\\lambda ^*,z)\\cap E^s(\\lambda ^*,z)$ is the $\\lambda ^*$ -eigenfunction of $L_\\epsilon $ satisfying $P(\\tau )=\\xi $ .",
"Writing $P:=(p,q,p_z,q_z/\\epsilon )$ , it is straightforward to calculate from (REF ) that $\\omega (P,A_\\lambda P)=p^2-\\frac{q^2}{\\epsilon }.$ The following theorem shows that $\\Gamma $ is positive definite for each conjugate point of $\\alpha _2$ , which proves that $\\mathrm {Maslov}(\\varphi )$ equals the sum of the geometric multiplicities of all unstable eigenvalues of $L_\\epsilon $ .",
"Theorem 5.2 Let $\\lambda \\in \\sigma (L_\\epsilon )\\cap (\\mathbb {R}^{+}\\cup \\lbrace 0\\rbrace )$ with corresponding eigenvector $P=(p,q)^T$ .",
"Suppose further that $0<\\epsilon <\\frac{c^4}{16}$ .",
"Then $\\int \\limits _{-\\infty }^{\\infty }e^{cz}\\left(p^2-\\frac{q^2}{\\epsilon }\\right)\\,dz>0.$ In other words, the crossing form (REF ) is positive definite for all $\\lambda \\in [0,\\lambda _\\mathrm {max}].$ The proof of this theorem uses the following Poincaré-type inequality.",
"Lemma 5.3 Suppose $h\\in H^1(\\mathbb {R})$ satisfies $\\int \\limits _{-\\infty }^{\\infty }e^{cz}\\left(h^2+(h_z)^2\\right)\\,dz<\\infty .$ Then for all $R\\in \\mathbb {R}$ (including $R=\\infty $ ), we have $\\frac{c^2}{4}\\int \\limits _{-\\infty }^{R}e^{cz}h^2\\,dz\\le \\int \\limits _{-\\infty }^{R}e^{cz}(h_z)^2\\,dz.$ The proof of this inequality is a simple estimate using the fact that (REF ) defines a norm on an exponentially weighted Sobolev space.",
"For more details, we refer the reader to Lemma 4.1 of [27], the source of this result.",
"Written as a system, the eigenvalue equation $L_\\epsilon P=\\lambda P$ is $\\begin{aligned}p_{zz}+cp_z+(f^{\\prime }(\\hat{u})-\\lambda )p-q =0\\\\q_{zz}+cq_z+\\epsilon p-(\\epsilon \\gamma +\\lambda )q =0.\\end{aligned}$ Now, multiply the second equation in (REF ) by $e^{cz}q$ to obtain $\\left(e^{cz}q_z\\right)_zq-(\\epsilon \\gamma +\\lambda )e^{cz}q^2=-e^{cz}\\epsilon pq.$ Since $p,q$ and their derivatives all decay exponentially in both tails, we can integrate (REF ) to obtain (after an integration by parts) $\\int \\limits _{-\\infty }^{\\infty }e^{cz}(q_z)^2\\,dz+(\\epsilon \\gamma +\\lambda )\\int \\limits _{-\\infty }^{\\infty }e^{cz}q^2\\,dz=\\epsilon \\int \\limits _{-\\infty }^{\\infty }e^{cz}pq\\,dz.$ It then follows from (REF ), (REF ), and the Cauchy-Schwarz inequality that $\\begin{aligned}\\frac{c^2}{4\\epsilon }\\int \\limits _{-\\infty }^{\\infty }e^{cz}q^2\\,dz & \\le \\frac{1}{\\epsilon }\\int \\limits _{-\\infty }^{\\infty }e^{cz}(q_z)^2\\,dz\\\\&<\\int \\limits _{-\\infty }^{\\infty }pq\\,dz\\le \\left(\\int \\limits _{-\\infty }^{\\infty }e^{cz}p^2\\,dz\\right)^{1/2}\\left(\\int \\limits _{-\\infty }^{\\infty }e^{cz}q^2\\,dz\\right)^{1/2}.\\end{aligned}$ Dividing the first and last terms in the inequality by $||q||_{1,c}$ (the $e^{cz}$ -weighted $L^2$ norm) and squaring yields $\\frac{c^4}{16\\epsilon }\\int \\limits _{-\\infty }^{\\infty }e^{cz}\\frac{q^2}{\\epsilon }\\,dz<\\int \\limits _{-\\infty }^{\\infty }e^{cz}p^2\\,dz,$ and the result now follows.",
"Remark 5.1 The proof of the preceding theorem uses estimates that are very similar to calculations in [11].",
"However, the objectives of the calculations are very different.",
"In [11], the goal is to establish the existence of a traveling wave using variational techniques.",
"Conversely, we are considering the stability issue, particularly what happens to eigenvalues as the spectral parameter varies."
],
[
"Multiplicity of Eigenvalues: the Evans Function",
"There is one remaining loose end to tie up if we want the Maslov index to give a complete picture of the unstable spectrum of $L_\\epsilon $ , namely, the multiplicity of eigenvalues.",
"In general, for $\\lambda \\in \\sigma _n(L)$ , the geometric multiplicity of $\\lambda $ is given by $\\dim \\ker (L-\\lambda I)$ .",
"For $\\lambda \\in \\mathbb {C}\\setminus H_\\epsilon \\cap \\sigma _n(L_\\epsilon )$ , this number is bounded above by two (or $n$ , in the general setting of (REF )), since $E^u(\\lambda ,z)$ and $E^s(\\lambda ,z)$ are only two-dimensional.",
"Since $\\dim \\ker (L-\\lambda I)=\\dim (E^u(\\lambda ,z)\\cap E^s(\\lambda ,z))$ , it is clear from (REF ) that the dimension of a crossing for $\\alpha _2$ gives the geometric multiplicity of $\\lambda $ .",
"The algebraic multiplicity of $\\lambda $ , on the other hand, is trickier.",
"It is given by $\\dim \\ker (L-\\lambda I)^\\alpha $ , where $\\alpha $ is the ascent of $\\lambda $ , i.e.",
"the smallest $\\alpha $ for which $\\dim \\ker (L-\\lambda I)^\\alpha =\\dim \\ker (L-\\lambda I)^{a+1}$ .",
"See §6.D of [1] for more details.",
"There is nothing obvious about the Maslov index that addresses the algebraic multiplicity of an eigenvalue.",
"In [20], self-adjoint operators are studied, and this issue is moot, since the two multiplicities coincide.",
"However, for our purposes it is not obvious that the two multiplicities are the same.",
"One tool that demonstrably gives information about the algebraic multiplicity of eigenvalues is the Evans function [1], [34].",
"Briefly, the Evans function $D(\\lambda )$ is a Wronskian-type determinant that detects linear dependence between the sets $E^s(\\lambda ,z)$ and $E^u(\\lambda ,z)$ .",
"Thus it is zero if and only if $\\lambda $ is an eigenvalue of $L$ .",
"It is also true (§2.E of [1]) that the order of $\\lambda $ as a root of $D$ is equal to the algebraic multiplicity of $\\lambda $ as an eigenvalue of $L$ .",
"This is the key to relating the Maslov index to algebraic multiplicity, as a symplectic version of the Evans function was developed in [8], [9].",
"The Maslov index in particular was used in Evans function analyses in [10], [15].",
"The Evans function for (REF ) is developed in detail in [15], so we refer the reader there for more background.",
"Since the eigenvalues of $A_\\infty (\\lambda )$ are real and simple for real $\\lambda \\ge 0$ , we can find solutions $u_i(\\lambda ,z)$ , $i=1\\dots 4$ to (REF ) such that $\\begin{aligned}\\lim \\limits _{z\\rightarrow \\infty }e^{-\\mu _i(\\lambda )z}u_i(\\lambda ,z) & =\\eta _i(\\lambda ), \\hspace{14.45377pt}i=1,2\\\\\\lim \\limits _{z\\rightarrow -\\infty }e^{-\\mu _i(\\lambda )z}u_i(\\lambda ,z) & = \\eta _i(\\lambda ), \\hspace{14.45377pt}i=3,4,\\end{aligned}$ where $\\eta _i(\\lambda )$ is a nonzero eigenvector of $A_\\infty (\\lambda )$ corresponding to eigenvalue $\\mu _i(\\lambda )$ .",
"We can then write $\\begin{aligned}E^s(\\lambda ,z)=\\mathrm {sp}\\lbrace u_1(\\lambda ,z),u_2(\\lambda ,z) \\rbrace \\\\E^u(\\lambda ,z)=\\mathrm {sp}\\lbrace u_3(\\lambda ,z),u_4(\\lambda ,z) \\rbrace \\end{aligned}.$ Although the Evans function can be defined without picking bases of the stable and unstable bundles (e.g.",
"[1]), the symplectic structure cannot be exploited without isolating particular solutions.",
"In [15], these bases are used to define the Evans function as follows.",
"Definition 4 The Evans function $D(\\lambda )$ for (REF ) is given by $\\begin{aligned}D(\\lambda ) & =e^{2cz}\\det \\left[u_1(\\lambda ,z),u_2(\\lambda ,z),u_3(\\lambda ,z),u_4(\\lambda ,z)\\right] \\\\&= -\\det \\left[\\begin{array}{c c}\\Omega (u_1(\\lambda ,z),u_3(\\lambda ,z)) & \\Omega (u_1(\\lambda ,z),u_4(\\lambda ,z))\\\\\\Omega (u_2(\\lambda ,z),u_3(\\lambda ,z)) & \\Omega (u_2(\\lambda ,z),u_4(\\lambda ,z))\\end{array}\\right]\\end{aligned}.$ We call the second formulation of $D$ in (REF ) the “symplectic Evans function.\"",
"As mentioned above, $D(\\lambda )=0$ if and only if $\\lambda \\in \\sigma _n(L_\\epsilon )$ , and the order of $\\lambda $ as a root of $D$ is equal to its algebraic multiplicity as an eigenvalue of $L_\\epsilon $ .",
"$D$ is also independent of $z$ , which follows from Theorem REF .",
"The following theorem is the main result of this section.",
"Theorem 6.1 Let $\\lambda ^*\\in \\sigma _n(L_\\epsilon )\\cap (\\mathbb {C}\\setminus H_\\epsilon )$ .",
"Then the geometric and algebraic multiplicities of $\\lambda ^*$ are equal.",
"This is equivalent to $\\lambda =\\lambda ^*$ being a regular conjugate point of $\\alpha _2$ .",
"We prove this separately for $\\lambda $ with geometric multiplicity one and two.",
"Recall that a crossing is regular if the associated crossing form (REF ) is nondegenerate.",
"First, suppose that $\\lambda ^*$ is an eigenvalue of $L_\\epsilon $ with geometric multiplicity one.",
"The goal is to show that $D^{\\prime }(\\lambda ^*)\\ne 0$ .",
"Let $P(z)$ be a corresponding eigenfunction.",
"We can perform a change of basis near $\\lambda =\\lambda ^*$ so that $\\begin{aligned}E^s(\\lambda ,z)=\\mathrm {sp}\\lbrace U(\\lambda ,z),a_s(\\lambda ,z) \\rbrace \\\\E^u(\\lambda ,z)=\\mathrm {sp}\\lbrace V(\\lambda ,z),a_u(\\lambda ,z)\\rbrace \\end{aligned},$ with $U(\\lambda ^*,z)=V(\\lambda ^*,z)=P(z)$ .",
"Doing so changes $D(\\lambda )$ by multiplication with a nonzero analytic function $C(\\lambda )$ (§4.1 of [34]).",
"Since $D(\\lambda ^*)=0$ , we have $\\frac{d}{d\\lambda }\\left[D(\\lambda )C(\\lambda )\\right]|_{\\lambda =\\lambda ^*}=D^{\\prime }(\\lambda ^*)C(\\lambda ^*),$ so making this change of basis does not affect whether or not the derivative of $D$ at $\\lambda ^*$ vanishes.",
"It therefore suffices to consider $\\tilde{D}^{\\prime }(\\lambda ^*)$ , with $\\tilde{D}(\\lambda )=-\\det \\left[\\begin{array}{c c}\\Omega (U(\\lambda ,z),V(\\lambda ,z)) & \\Omega (U(\\lambda ,z),a_u(\\lambda ,z))\\\\\\Omega (a_s(\\lambda ,z),V(\\lambda ,z)) & \\Omega (a_s(\\lambda ,z),a_u(\\lambda ,z))\\end{array}\\right].$ (See Theorem 2 and Corollary 1 of [15] for more details on the derivation of this formula.)",
"The desired derivative is computed using Jacobi's formula (§8.3 of [28]), and an identical calculation is carried out in equation (4.5) of [15].",
"The result is that $D^{\\prime }(\\lambda ^*)=\\Omega (a_s(\\lambda ,z),a_u(\\lambda ,z))\\,\\partial _\\lambda \\Omega (U(\\lambda ,z),V(\\lambda ,z))|_{\\lambda =\\lambda ^*,z=\\tau }.$ Define $\\xi =P(\\tau )$ .",
"Comparing with (REF ), we see that $\\partial _\\lambda \\Omega (U(\\lambda ,z),V(\\lambda ,z))|_{\\lambda =\\lambda ^*,z=\\tau }=-\\Gamma (E^u(\\cdot ,\\tau ),E^s(\\cdot ,\\tau ),\\lambda ^*)(\\xi ),$ which is nonzero, since $\\lambda ^*$ is a regular crossing by Theorem REF .",
"It would follow that $D^{\\prime }(\\lambda ^*)\\ne 0$ , and hence that $\\lambda ^*$ is a simple eigenvalue of $L_\\epsilon $ , if $\\Omega (a_s(\\lambda ^*,z),a_u(\\lambda ^*,z))\\ne 0.$ It turns out that this is equivalent to $\\lambda ^*$ having geometric multiplicity one.",
"Indeed, if $\\Omega (a_s(\\lambda ^*,z),a_u(\\lambda ^*,z))=0$ , then $\\mathrm {sp}\\lbrace a_s(\\lambda ^*,z),a_u(\\lambda ^*,z) \\rbrace $ is a Lagrangian plane.",
"A simple dimension-counting argument (cf.",
"page 85 of [10]) then implies that $E^s(\\lambda ^*,z)=\\mathrm {sp}\\lbrace U,a_s\\rbrace =\\mathrm {sp}\\lbrace V,a_u\\rbrace =E^u(\\lambda ^*,z).$ We now turn to the case where $\\lambda ^*$ is a two-dimensional crossing, meaning that $\\dim (E^u(\\lambda ^*,z)\\cap E^s(\\lambda ^*,z))=2.$ By making another change of basis if necessary, we are free to assume that $\\begin{aligned}u_1(\\lambda ^*,z)=u_4(\\lambda ^*,z)\\\\u_2(\\lambda ^*,z)=u_3(\\lambda ^*,z)\\end{aligned}.$ We then set $\\xi _1=u_1(\\lambda ^*,\\tau )$ and $\\xi _2=u_2(\\lambda ^*,\\tau )$ .",
"Since the algebraic multiplicity of $\\lambda ^*$ is no less than its geometric multiplicity, we know a priori that $D^{\\prime }(\\lambda ^*)=0$ .",
"This is easily verified by applying the product rule to (REF ), which is the zero matrix for $\\lambda =\\lambda ^*$ .",
"What we need to verify is that $D^{\\prime \\prime }(\\lambda ^*)\\ne 0$ , and that this is equivalent to the regularity of the crossing form.",
"To see this, we use (REF ) to write out $D(\\lambda )=\\Omega (u_1(\\lambda ,z),u_4(\\lambda ,z))\\Omega (u_2(\\lambda ,z),u_3(\\lambda ,z))-\\Omega (u_1(\\lambda ,z),u_3(\\lambda ,z))\\Omega (u_2(\\lambda ,z),u_4(\\lambda ,z)).$ Evaluating at $\\lambda =\\lambda ^*$ , each of the four terms in (REF ) is zero, using (REF ) and the fact that $E^{u/s}(\\lambda ,z)$ are Lagrangian planes.",
"As mentioned above, we can see from (REF ) that $D^{\\prime }(\\lambda ^*)=0$ , since the derivative produces a series of four terms, each of which is a product with a factor of zero.",
"Computing $D^{\\prime \\prime }(\\lambda ^*)$ from the general Leibniz rule, we see that the only surviving terms are those for which each factor in (REF ) is differentiated once.",
"Explicitly, we compute that $\\begin{aligned}D^{\\prime \\prime }(\\lambda ^*) = & 2\\left\\lbrace \\partial _\\lambda \\Omega (u_1(\\lambda ,z),u_4(\\lambda ,z))\\partial _\\lambda \\Omega (u_2(\\lambda ,z),u_3(\\lambda ,z))\\right.\\\\& \\left.-\\partial _\\lambda \\Omega (u_1(\\lambda ,z),u_3(\\lambda ,z))\\partial _\\lambda \\Omega (u_2(\\lambda ,z),u_4(\\lambda ,z)) \\right\\rbrace |_{\\lambda =\\lambda ^*,z=\\tau }\\\\& = -2\\det \\left[\\begin{array}{c c}\\partial _\\lambda \\Omega (u_1(\\lambda ,z),u_3(\\lambda ,z)) & \\partial _\\lambda \\Omega (u_1(\\lambda ,z),u_4(\\lambda ,z))\\\\\\partial _\\lambda \\Omega (u_2(\\lambda ,z),u_3(\\lambda ,z)) & \\partial _\\lambda \\Omega (u_2(\\lambda ,z),u_4(\\lambda ,z))\\end{array}\\right]|_{\\lambda =\\lambda ^*,z=\\tau }\\\\&= 2\\det \\left[\\begin{array}{c c}\\partial _\\lambda \\Omega (u_1(\\lambda ,z),u_4(\\lambda ,z)) & \\partial _\\lambda \\Omega (u_1(\\lambda ,z),u_3(\\lambda ,z))\\\\\\partial _\\lambda \\Omega (u_2(\\lambda ,z),u_4(\\lambda ,z)) & \\partial _\\lambda \\Omega (u_2(\\lambda ,z),u_3(\\lambda ,z))\\end{array}\\right]|_{\\lambda =\\lambda ^*,z=\\tau }.\\end{aligned}$ We see from (REF ) that the last matrix (obtained by switching columns and taking a transpose in the previous line) is exactly the matrix of the crossing form $\\Gamma $ in (REF ).",
"To say that $\\Gamma $ is nondegenerate means that the determinant in (REF ) is nonzero, hence $D^{\\prime \\prime }(\\lambda ^*)\\ne 0$ , as desired.",
"Remark 6.1 Although we phrased the preceding theorem for the operator $L_\\epsilon $ , it is clear that the proof generalizes to (REF ).",
"At an $n$ -dimensional crossing $\\lambda ^*$ , the first $(n-1)$ derivatives of $D(\\lambda )$ are forced to vanish.",
"The $n^{\\text{th}}$ derivative will then contain a factor corresponding to the $\\lambda $ -crossing form $\\Gamma $ .",
"The number of zeros of $\\Gamma $ in normal form (page 186 of [36]) counts the discrepancy between the algebraic and geometric multiplicities of $\\lambda ^*$ as an eigenvalue of $L$ .",
"Although algebraic versus geometric multiplicity seems like a picayune detail, it is actually critical in the case $\\lambda =0$ .",
"Indeed, this eigenvalue is always present for traveling waves in autonomous equations.",
"For such waves in semilinear parabolic systems, we have the following well-known result (e.g.",
"[1]).",
"Theorem 6.2 Suppose that the operator $L$ in (REF ) satisfies There exists $\\beta <0$ such that $\\sigma (L)\\setminus \\lbrace 0\\rbrace \\subset \\lbrace \\lambda \\in \\mathbb {C}:\\mathrm {Re }\\lambda <\\beta \\rbrace $ .",
"0 is a simple eigenvalue.",
"Then $\\hat{u}$ is stable in the sense of Definition REF .",
"It is possible that $E^u(0,z)\\cap E^s(0,z)=\\mathrm {sp}\\lbrace \\varphi ^{\\prime }(z) \\rbrace $ is one dimensional, but that $\\lambda =0$ is still not a simple eigenvalue.",
"In [2] (pp.",
"57-60), it is shown that $\\lambda =0$ is simple if and only if the wave is transversely constructed, in the following sense.",
"With the equation $c^{\\prime }=0$ appended to (REF ), $W^{cu}(0)$ and $W^{cs}(0)$ are each $(n+1)$ -dimensional.",
"The wave $\\varphi $ is said to be transversely constructed if $W^{cu}(0)$ and $W^{cs}(0)$ intersect transversely in $\\mathbb {R}^{2n+1}$ , and their (necessarily one-dimensional) intersection is $\\varphi (z)$ .",
"Thus the geometric interpretation of simplicity is that two manifolds intersect transversely in augmented phase space.",
"By contrast, the understanding of simplicity afforded by the symplectic structure requires no variation in $c$ .",
"Instead, we see that $\\lambda =0$ (or any other eigenvalue) is simple if the curve $\\lambda \\mapsto E^u(\\lambda ,\\tau )$ transversely intersects the train of $E^s(0,\\tau )$ for all sufficiently large $\\tau $ .",
"To see this, notice that Theorem REF proves that the eigenvalue is simple if and only if the relative crossing form (REF ) is regular.",
"But if the integral (REF ) is nonzero, then so will be the integral in (REF ) for $\\tau $ large enough.",
"Alternatively, for an eigenvalue $\\lambda ^*$ with geometric multiplicity one, being simple is equivalent to the curves $\\lambda \\mapsto E^u(\\lambda ,\\tau ),E^s(\\lambda ,\\tau )$ intersecting non-tangentially at $\\lambda =\\lambda ^*$ ."
]
] | 1709.01908 |
[
[
"Complexity Classification of Conjugated Clifford Circuits"
],
[
"Abstract Clifford circuits -- i.e.",
"circuits composed of only CNOT, Hadamard, and $\\pi/4$ phase gates -- play a central role in the study of quantum computation.",
"However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable.",
"We show that in contrast, \"conjugated Clifford circuits\" (CCCs) -- where one additionally conjugates every qubit by the same one-qubit gate $U$ -- can perform hard sampling tasks.",
"In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary $U$ can give rise to sampling tasks which cannot be efficiently classically simulated to constant multiplicative error, unless the polynomial hierarchy collapses.",
"Furthermore, by standard techniques, this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture.",
"This work can be seen as progress towards classifying the computational power of all restricted quantum gate sets."
],
[
"Introduction",
"Quantum computers hold the promise of efficiently solving certain problems, such as factoring integers [56], which are believed to be intractable for classical computers.",
"However, experimentally implementing many of these quantum algorithms is very difficult.",
"For instance, the largest number factored to date using Shor's algorithm is 21 [45].",
"These considerations have led to an intense interest in algorithms which can be more easily implemented with near-term quantum devices, as well as a corresponding interest in the difficulty of these computational tasks for classical computers.",
"Some prominent examples of such models are constant-depth quantum circuits [60], [11] and non-adaptive linear optics [1].",
"In many of these constructions, one can show that these “weak\" quantum devices can perform sampling tasks which cannot be efficiently simulated classically, even though they may not be known to be capable of performing difficult decision tasks.",
"Such arguments were first put forth by Bremner, Jozsa, and Shepherd [14] and Aaronson and Arkhipov [1], who showed that exactly simulating the sampling tasks performed by weak devices is impossible assuming the polynomial hierarchy is infinite.",
"The proofs of these results use the fact that the output probabilities of quantum circuits can be very difficult to compute – in fact they can be $\\displaystyle \\mathsf {Gap}\\mathsf {P}$ -hard and therefore $\\displaystyle \\#\\mathsf {P}$ -hard to approximate.",
"In contrast the output probabilities of classical samplers can be approximated in $\\displaystyle \\mathsf {BPP}^\\mathsf {NP}$ by Stockmeyer's approximate counting theorem [59], and therefore lie in the polynomial hierarchy.",
"Hence a classical simulation of these circuits would collapse the polynomial hierarchy to the third level by Toda's Theorem [61].",
"Similar hardness results have been shown for many other models of quantum computation [60], [43], [26], [18], [24], [13], [17], [42].",
"A curious feature of many of these “weak\" models of quantum computation is that they can be implemented using non-universal gate sets.",
"That is, despite being able to perform sampling problems which appear to be outside of $\\displaystyle \\mathsf {BPP}$ , these models are not themselves known to be capable of universal quantum computation.",
"In short these models of quantum computation seem to be “quantum-intermediate\" between $\\displaystyle \\mathsf {BPP}$ and $\\displaystyle \\mathsf {BQP}$ , analogous to the $\\displaystyle \\mathsf {NP}$ -intermediate problems which are guaranteed to exist by Ladner's theorem [40].",
"From the standpoint of computational complexity, it is therefore natural to study this intermediate space between $\\displaystyle \\mathsf {BPP}$ and $\\displaystyle \\mathsf {BQP}$ , and to classify its properties.",
"One natural way to explore the space between $\\displaystyle \\mathsf {BPP}$ and $\\displaystyle \\mathsf {BQP}$ is to classify the power of all possible quantum gate sets over qubits.",
"The Solovay-Kitaev Theorem states that all universal quantum gate sets, i.e.",
"those which densely generate the full unitary group, have equivalent computational power [21].",
"Therefore the interesting gates sets to classify are those which are non-universal.",
"However just because a gate set is non-universal does not imply it is weaker than $\\displaystyle \\mathsf {BQP}$ – in fact some non-universal gates are known to be capable of universality in an “encoded\" sense, and therefore have the same computational power as $\\displaystyle \\mathsf {BQP}$ [36].",
"Other non-universal gate sets are efficiently classically simulable [30], while others seem to lie “between $\\displaystyle \\mathsf {BPP}$ and $\\displaystyle \\mathsf {BQP}$ \" in that they are believed to be neither universal for $\\displaystyle \\mathsf {BQP}$ nor efficiently classically simulable [14].",
"It is a natural open problem to fully classify all restricted gate sets into these categories according to their computational complexity.",
"This is a challenging problem, and to date there has only been partial progress towards this classification.",
"One immediate difficulty in approaching this problem is that there is not a known classification of all possible non-universal gate sets.",
"In particular this would require classifying the discrete subgroups of $\\displaystyle SU(2^n)$ for all $\\displaystyle n\\in \\mathbb {N}$ , which to date has only been solved for $\\displaystyle n\\le 2$ [34].",
"Therefore existing results have characterized the power of modifications of known intermediate gate sets, such as commuting circuits and linear optical elements [7], [18], [51].",
"Others works have classified the classical subsets of gates [6], [31], or else given sufficient criteria for universality so as to rule out the existence of certain intermediate families [57].",
"A complete classification of this space “between $\\displaystyle \\mathsf {BPP}$ and $\\displaystyle \\mathsf {BQP}$ \" would require a major improvement in our understanding of universality as well as the types of computational hardness possible between $\\displaystyle \\mathsf {BPP}$ and $\\displaystyle \\mathsf {BQP}$ .",
"One well-known example of a non-universal family of quantum gates is the Clifford group.",
"Clifford circuits – i.e.",
"circuits composed of merely CNOT, Hadamard and Phase gates – are a discrete subgroup of quantum gates which play an important role in quantum error correction [29], [15], measurement-based quantum computing [53], [54], [9], and randomized benchmarking [44].",
"However a well-known result of Gottesman and Knill states that circuits composed of Clifford elements are efficiently classically simulable [30], [5].",
"That is, suppose one begins in the state $\\displaystyle {\\left|{0}\\right\\rangle }^{\\otimes n}$ , applies polynomially many gates from the set $\\displaystyle \\mathrm {CNOT}, \\mathrm {H}, \\mathrm {S}$ , then measures in the computational basis.",
"Then the Gottesman-Knill theorem states that one can compute the probability a string $\\displaystyle y$ is output by such a circuit in classical polynomial time.",
"One can also sample from the same probability distribution on strings as this circuit as well.",
"A key part of the proof of this result is that the quantum state at intermediate stages of the circuit is always a “stabilizer state\" – i.e.",
"the state is uniquely described by its set of stabilizers in the Pauli group – and therefore has a compact representation.",
"Therefore the Clifford group is incapable of universal quantum computation (assuming $\\displaystyle \\mathsf {BPP}\\ne \\mathsf {BQP}$ ).",
"In this work, we will study the power of a related family of non-universal gates, known as Conjugated Clifford gates, which we introduce below.",
"These gates are non-universal by construction, but not known to be efficiently classically simulable either.",
"Our main result will be to fully classify the computational power of this family of intermediate gate sets."
],
[
"Our results",
"This paper considers a new “weak\" model of quantum computation which we call “conjugated Clifford circuits\" (CCCs).",
"In this model, we consider the power of quantum circuits which begin in the state $\\displaystyle {\\left|{0}\\right\\rangle }^{\\otimes n}$ , and then apply gates from the set $\\displaystyle (U^\\dagger \\otimes U^\\dagger )(\\mathrm {CNOT})(U \\otimes U), U^\\dag HU,U^\\dag SU$ where $\\displaystyle U$ is a fixed one-qubit gate.",
"In other words, we consider the power of Clifford circuits which are conjugated by an identical one-qubit gate $\\displaystyle U$ on each qubit.",
"These gates manifestly perform a discrete subset of unitaries so this gate set is clearly not universal.",
"Although this transformation preserves the non-universality of the Clifford group, it is unclear if it preserves its computational power.",
"The presence of generic conjugating unitaries (even the same $\\displaystyle U$ on each qubit, as in this model) breaks the Gottesman-Knill simulation algorithm [30], as the inputs and outputs of the circuit are not stabilizer states/measurements.",
"Hence the intermediate states of the circuit are no longer efficiently representable by the stabilizer formalism.",
"This, combined with prior results showing hardness for other modified versions of Clifford circuits [37], [38], leads one to suspect that CCCs may not be efficiently classically simulable.",
"However prior to this work no hardness results were known for this model.",
"In this work, we confirm this intuition and provide two results in this direction.",
"First, we provide a complete classification of the power of CCCs according to the choice of $\\displaystyle U$ .",
"We do this by showing that any $\\displaystyle U$ which is not efficiently classically simulable by the Gottesman-Knill theorem suffices to perform hard sampling problems with CCCsMore precisely, we show that any $\\displaystyle U$ that cannot be written as a Clifford times a $\\displaystyle Z$ -rotation suffices to perform hard sampling problems with CCCs.",
"See Theorem REF for the exact statement.. That is, for generic $\\displaystyle U$ , CCCs cannot be efficiently classically simulated to constant multiplicative error by a classical computer unless the polynomial hierarchy collapses.",
"This result can be seen as progress towards classifying the computational complexity of restricted gate sets.",
"Indeed, given a non-universal gate set $\\displaystyle G$ , a natural question is to classify the power of $\\displaystyle G$ when conjugated by the same one-qubit unitariy $\\displaystyle U$ on each qubit, as this transformation preserves non-universality.",
"Our work resolves this question for one of the most prominent examples of non-universal gate sets, namely the Clifford group.",
"As few examples of non-universal gate sets are knownThe only examples to our knowledge are matchgates, Clifford gates, diagonal gates, and subsets thereof., this closes one of the major gaps in our understanding of intermediate gate sets.",
"Of course this does not complete the complexity classification of all gate sets, as there is no known classification of all possible non-universal gate sets.",
"However it does make progress towards this goal.",
"Second, we show that under an additional complexity-theoretic conjecture, classical computers cannot efficiently simulate CCCs to constant error in total variation distance.",
"This is a more experimentally achievable model of error for noisy quantum computations.",
"The proof of this result uses standard techniques introduced by Aaronson and Arkhipov [1], which have also been used in other models [26], [16], [13], [17], [46], [42].",
"This second result is interesting for two reasons.",
"First, it means our results may have relevance to the empirical demonstration of quantum advantage (sometimes referred to as “quantum supremacy\") [52], [13], [4], as our results are robust to noise.",
"Second, from the perspective of computational complexity, it gives yet another conjecture upon which one can base the supremacy of noisy quantum devices.",
"As is the case with other quantum supremacy proposals [1], [26], [16], [46], [42], in order to show that simulation of CCCs to additive error still collapses the polynomial hierarchy, we need an additional conjecture stating that the output probabilities of these circuits are hard to approximate on average.",
"Our conjecture essentially states that for most Clifford circuits $\\displaystyle V$ and most one-qubit unitaries $\\displaystyle U$ , it is $\\displaystyle \\#\\mathsf {P}$ -hard to approximate a constant fraction of the output probabilities of the CCC $\\displaystyle U^{\\otimes n} V (U^\\dagger )^{\\otimes n}$ to constant multiplicative error.",
"We prove that this conjecture is true in the worst case – in fact, for all non-Clifford $\\displaystyle U$ , there exists a $\\displaystyle V$ such that some outputs are $\\displaystyle \\#\\mathsf {P}$ -hard to compute to multiplicative error.",
"However, it remains open to extend this hardness result to the average case, as is the case with other supremacy proposals as well [1], [26], [16], [46], [42].",
"To the best of our knowledge our conjecture is independent of the conjectures used to establish other quantum advantage results such as boson sampling [1], Fourier sampling [26] or IQP [16], [17].",
"Therefore our results can be seen as establishing an alternative basis for belief in the advantage of noisy quantum devices over classical computation.",
"One final motivation for this work is that CCCs might admit a simpler fault-tolerant implementation than universal quantum computing, which we conjecture to be the case.",
"It is well-known that many stabilizer error-correcting codes, such as the 5-qubit and 7-qubit codes [41], [22], [58], admit transversal Clifford operations [29].",
"That is, performing fault-tolerant Clifford operations on the encoded logical qubits can be done in a very simple manner – by simply performing the corresponding Clifford operation on the physical qubits.",
"This is manifestly fault-tolerant, in that an error on one physical qubit does not “spread\" to more than 1 qubit when applying the gate.",
"In contrast, performing non-Clifford operations fault-tolerantly on such codes requires substantially larger (and non-transversal) circuits – and therefore the non-transversal operations are often the most resource intensive.",
"The challenge in fault-tolerantly implementing CCCs therefore lies in performing the initial state preparation and measurement.",
"Initial preparation of non-stabilizer states in these codes is equivalent to the challenge of producing magic states, which are already known to boost Clifford circuits to universality using adaptive Clifford circuits [15], [12] (in contrast our construction would only need non-adaptive Clifford circuits with magic states).",
"Likewise, measuring in a non-Clifford basis would require performing non-Clifford one-qubit gates prior to fault-tolerant measurement in the computational basis.",
"Therefore the state preparation/measurement would be the challenging part of fault-tolerantly implementing CCCs in codes with transversal Cliffords.",
"It remains open if there exists a code with transversal conjugated CliffordsOf course one can always “rotate\" a code with transversal Clifford operations to obtain a code with transversal conjugated Cliffords.",
"If the code previously had logical states $\\displaystyle {\\left|{0}\\right\\rangle }_L,{\\left|{1}\\right\\rangle }_L$ , then by setting the states $\\displaystyle {\\left|{0}\\right\\rangle }^{\\prime }_L = U^\\dagger _L {\\left|{0}\\right\\rangle }_L$ and $\\displaystyle {\\left|{1}\\right\\rangle }^{\\prime }_L = U^\\dagger _L {\\left|{1}\\right\\rangle }_L$ , one obtains a code in which the conjugated Clifford gates (conjugated by $\\displaystyle U$ ) are transversal.",
"However having the ability to efficiently fault-tolerantly prepare $\\displaystyle {\\left|{0}\\right\\rangle }_L$ in the old code does not imply the same ability to prepare $\\displaystyle {\\left|{0}\\right\\rangle }^{\\prime }_L$ in the new code.",
"and easy preparation and measurement in the required basis.",
"Such a code would not be ruled out by the Eastin-Knill Theorem [23], which states that the set of transversal gates must be discrete for all codes which correct arbitrary one qubit errors.",
"Of course this is not the main motivation for exploring the power of this model – which is primarily to classify the space between $\\displaystyle \\mathsf {BPP}$ and $\\displaystyle \\mathsf {BQP}$ – but an easier fault-tolerant implementation could be an unexpected bonus of our results."
],
[
"Proof Techniques",
"To prove these results, we use several different techniques."
],
[
"Proof Techniques: classification of exact sampling hardness",
"To prove exact (or multiplicative) sampling hardness for CCCs for essentially all non-Clifford $\\displaystyle U$ , we use the notion of postselection introduced by Aaronson [2].",
"Postselection is the (non-physical) ability to discard all runs of the computation which do not achieve some particular outcomes.",
"Our proof works by showing that postselecting such circuits allows them to perform universal quantum computation.",
"Hardness then follows from known techniques [2], [14], [1].",
"One technical subtlety that we face in this proof, which is not present in other results, is that our postselected gadgets perform operations which are not closed under inversion.",
"This means one cannot use the Solovay-Kitaev theorem to change quantum gate sets [21].",
"This is a necessary step in the proof that $\\displaystyle \\mathsf {PostBQP}=\\mathsf {PP}$ [2], which is a key part of the hardness proof (see [18]).",
"Fortunately, it turns out that we can get away without inverses due to a recent inverse-free Solovay-Kitaev theorem of Sardharwalla et al.",
"[55], which removes the needs for inverses if the gate set contains the Paulis.",
"Our result would have been much more difficult to obtain without this prior result.",
"To our knowledge this is the first application of their result to structural complexity.",
"A further difficulty in the classification proof is that the postselection gadgets we derive do not work for all non-Clifford $\\displaystyle U$ .",
"In general, most postselection gadgets give rise to non-unitary operations, and for technical reasons we need to work with unitary postselection gadgets to apply the results of [55].",
"Therefore, we instead use several different gadgets which cover different portions of the parameter space of $\\displaystyle U$ 's.",
"Our initial proof of this fact used a total of seven postselection gadgets found by hand.",
"We later simplified this to two postselection gadgets by conducting a brute-force search for suitable gadgets using Christopher Granade and Ben Criger's QuaEC package [28].",
"We include this simplified proof in this writeup.",
"A final difficulty that one often faces with postselected universality proofs is that one must show that the postselection gadgets boost the original gate set to universality.",
"In general this is a nontrivial task; there is no simple test of whether a gate set is universal, though some sufficient (but not necessary) criteria are known [57].",
"Prior gate set classification theorems have solved this universality problem using representation theory [7], [57] or Lie theory [18], [51].",
"However, in our work we are able to make use of a powerful fact: namely that the Clifford group plus any non-Clifford unitary is universal.",
"This follows from results of Nebe, Rains and Sloane [49], [50], [19] classifying the invariants of the Clifford groupHowever we note that in our proofs we will only use the fact that the Clifford group plus any non-Clifford element is universal on a qubit.",
"This version of the theorem admits a direct proof using the representation theory of $\\displaystyle SU(2)$ .. As a result our postselected universality proofs are much simpler than in other gate set classification theorems."
],
[
"Proof techniques: additive error",
"To prove hardness of simulation to additive error, we follow the techniques of [1], [16], [26], [46].",
"In these works, to show hardness of sampling from some probability distribution with additive error, one combines three different ingredients.",
"The first is anti-concentration – showing that for these circuits, the output probabilities in some large set $\\displaystyle T$ are somewhat large.",
"Second, one uses Markov's inequality to argue that, since the simulation error sums to $\\displaystyle \\varepsilon $ , on some other large set of output probabilities $\\displaystyle S$ , the error must be below a constant multiple of the average.",
"If $\\displaystyle S$ and $\\displaystyle T$ are both large, they must have some intersection – and on this intersection $\\displaystyle S\\cap T$ , the imagined classical simulation is not only a simulation to additive error, but also to multiplicative error as well (since the output probability in question is above some minimum).",
"Therefore a simulation to some amount $\\displaystyle \\varepsilon $ of additive error implies a multiplicative simulation to the output probabilities on a constant fraction of the outputs.",
"The impossibility of such a simulation is then obtained by assuming that computing these output probabilities is multiplicatively hard on average.",
"In particular, one assumes that it is a $\\displaystyle \\#\\mathsf {P}$ -hard task to compute the output probability on $\\displaystyle |S\\cap T|/2^n$ -fraction of the outputs.",
"This leads to a collapse of the polynomial hierarchy by known techniques [1], [14].",
"We follow this technique to show hardness of sampling with additive error.",
"In our case, the anticoncentration theorem follows from the fact that the Clifford group is a “2-design\" [62], [64] – i.e.",
"a random Clifford circuit behaves equivalently to a random unitary up to its second moment – and therefore must anticoncentrate, as a random unitary does (the fact that unitary designs anticoncentrate was also shown independently by several groups [33], [42], [35]).",
"This is similar to the hardness results for $\\displaystyle \\mathsf {IQP}$ [16] and $\\displaystyle \\mathsf {DQC}1$ [46], in which the authors also prove their corresponding anticoncentration theorems.",
"In contrast it is open to prove the anticoncentration theorem used for Boson Sampling and Fourier Sampling [1], [26], though these models have other complexity-theoretic advantagesFor instance, for these models it is known to be $\\displaystyle \\#\\mathsf {P}$ -hard to exactly compute most output probabilities of their corresponding circuit.",
"This is a necessary but not sufficient condition for the supremacy conjectures to be true, which require it to be $\\displaystyle \\#\\mathsf {P}$ -hard to approximately compute most output probabilities of their corresponding circuit..",
"Therefore the only assumption needed is the hardness-on-average assumption.",
"We also show that our hardness assumption is true for worst-case inputs.",
"This result follows from combining known facts about $\\displaystyle \\mathsf {BQP}$ with the classification theorem for exact sampling hardness."
],
[
"Relation to other works on modified Clifford circuits",
"While we previously discussed the relation of our results to prior work on gate set classification and sampling problems, here we compare our results to prior work on Clifford circuits.",
"We are not the first to consider the power of modified Clifford circuits.",
"Jozsa and van den Nest [37] and Koh [38], categorized the computational power of a number of modified versions of Clifford circuits.",
"The closest related result is the statement in [37] that if the input state to a Clifford circuit is allowed to be an arbitrary tensor product of one-qubit states, then such circuits cannot be efficiently classically simulated unless the polynomial hierarchy collapses.",
"Their hardness result uses states of the form $\\displaystyle {\\left|{0}\\right\\rangle }^{\\otimes n/2} {\\left|{\\alpha }\\right\\rangle }^{\\otimes n/2}$ , where $\\displaystyle {\\left|{\\alpha }\\right\\rangle } = \\cos (\\pi /8) {\\left|{0}\\right\\rangle }+i\\sin (\\pi /8){\\left|{1}\\right\\rangle }$ is a magic state.",
"They achieve postselected hardness via the use of magic states to perform T gates, using a well-known construction (see e.g.",
"[15]).",
"So in the [37] construction there are different input states on different qubits.",
"In contrast, our result requires the same input state on every qubit – as well as measurement in that basis at the end of the circuit.",
"This ensures our modified circuit can be interpreted as the action of a discrete gate set, and therefore our result has relevance for the classification of the power of non-universal gate sets."
],
[
"Preliminaries",
"We denote the single-qubit Pauli matrices by $\\displaystyle X = \\sigma _x = \\left(\\begin{matrix} 0 & 1 \\\\ 1 & 0 \\end{matrix}\\right)$ , $\\displaystyle Y=\\sigma _y = \\left(\\begin{matrix} 0 & -i \\\\ i & 0 \\end{matrix}\\right)$ , $\\displaystyle Z= \\sigma _z = \\left(\\begin{matrix} 1 & 0 \\\\ 0 & -1 \\end{matrix}\\right)$ , and $\\displaystyle I = \\left(\\begin{matrix} 1 & 0 \\\\ 0 & 1 \\end{matrix}\\right)$ .",
"The $\\displaystyle \\pm 1$ -eigenstates of $\\displaystyle Z$ are denoted by $\\displaystyle {\\left|{0}\\right\\rangle }$ and $\\displaystyle {\\left|{1}\\right\\rangle }$ respectively.",
"The rotation operator about an axis $\\displaystyle t \\in \\lbrace x,y,z\\rbrace $ with an angle $\\displaystyle \\theta \\in [0,2\\pi )$ is $R_t (\\theta ) = e^{-i \\theta \\sigma _t/2} = \\cos (\\theta /2) I - i \\sin (\\theta /2) \\sigma _t.$ We will use the fact that any single-qubit unitary operator $\\displaystyle U$ can be written as $ U = e^{i \\alpha } R_z(\\phi ) R_x(\\theta ) R_z(\\lambda ),$ where $\\displaystyle \\alpha , \\phi , \\theta , \\lambda \\in [0,2\\pi )$ [48].",
"For linear operators $\\displaystyle A$ and $\\displaystyle B$ , we write $\\displaystyle A \\propto B$ to mean that there exists $\\displaystyle \\alpha \\in \\mathbb {C}\\backslash \\lbrace 0\\rbrace $ such that $\\displaystyle A = \\alpha B$ .",
"For linear operators, vectors or complex numbers $\\displaystyle a$ and $\\displaystyle b$ , we write $\\displaystyle a \\sim b$ to mean that $\\displaystyle a$ and $\\displaystyle b$ differ only by a global phase, i.e.",
"there exists $\\displaystyle \\theta \\in [0,2\\pi )$ such that $\\displaystyle a = e^{i\\theta } b$ .",
"For any subset $\\displaystyle S\\subseteq \\mathbb {R}$ and $\\displaystyle k \\in \\mathbb {R}$ , we write $\\displaystyle kS$ to refer to the set $\\displaystyle \\lbrace kn:n\\in S \\rbrace $ .",
"For example, $\\displaystyle k\\mathbb {Z}= \\lbrace kn:n\\in \\mathbb {Z}\\rbrace $ .",
"We denote the set of odd integers by $\\displaystyle \\mathbb {Z}_{odd}$ .",
"We denote the complement of a set $\\displaystyle S$ by $\\displaystyle S^c$ ."
],
[
"Clifford circuits and conjugated Clifford circuits",
"The $\\displaystyle n$ -qubit Pauli group $\\displaystyle \\mathcal {P}_n$ is the set of all operators of the form $\\displaystyle i^k P_1 \\otimes \\ldots \\otimes P_n$ , where $\\displaystyle k \\in \\lbrace 0,1,2,3\\rbrace $ and each $\\displaystyle P_j$ is a Pauli matrix.",
"The $\\displaystyle n$ -qubit Clifford group is the normalizer of $\\displaystyle \\mathcal {P}_n$ in the $\\displaystyle n$ -qubit unitary group $\\displaystyle \\mathcal {U}_n$ , i.e.",
"$\\displaystyle \\mathcal {C}_n = \\lbrace U \\in \\mathcal {U}_n: U \\mathcal {P}_n U^\\dag = \\mathcal {P}_n\\rbrace $ .",
"The elements of the Clifford group, called Clifford operations, have an alternative characterization: an operation is a Clifford operation if and only if it can be written as a circuit comprising the following gates, called basic Clifford gates: Hadamard, $\\displaystyle \\pi /4$ phase, and controlled-NOT gates, whose matrix representations in the computational basis are $H=\\frac{1}{\\sqrt{2}}\\left(\\begin{matrix} 1 & 1 \\\\ 1 & -1 \\end{matrix}\\right),\\quad S=\\left(\\begin{matrix} 1 & 0 \\\\ 0 & i \\end{matrix}\\right),\\quad \\mbox{and}\\quad \\mathrm {CNOT} = \\left(\\begin{matrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\end{matrix}\\right)$ respectively.",
"An example of a non-Clifford gate is the $\\displaystyle T$ gate, whose matrix representation is given by $\\displaystyle T=\\left(\\begin{matrix} 1 & 0 \\\\ 0 & e^{i \\pi /4} \\end{matrix}\\right)$ .",
"We denote the group generated by the single-qubit Clifford gates by $\\displaystyle \\langle S, H \\rangle $ .",
"We will make use of the following fact about Clifford operations.",
"Fact 1 $\\displaystyle R_z(\\phi )$ is a Clifford operation if and only if $\\displaystyle \\phi \\in \\tfrac{\\pi }{2} \\mathbb {Z}$ .",
"A Clifford circuit is a circuit that consists of computational basis states being acted on by the basic Clifford gates, before being measured in the computational basis.",
"Without loss of generality, we may assume that the input to the Clifford circuit is the all-zero state $\\displaystyle {\\left|{0}\\right\\rangle }^{\\otimes n}$ .",
"We define conjugated Clifford circuits (CCCs) similarly to Clifford circuits, except that each basic Clifford gate $\\displaystyle G$ is replaced by a conjugated basic Clifford gate $\\displaystyle (U^{\\otimes k})^ \\dag g U^{\\otimes k}$ , where $\\displaystyle k =1$ when $\\displaystyle g = H, S$ and $\\displaystyle k=2$ when $\\displaystyle g=\\mathrm {CNOT}$ .",
"In other words, Definition 2.1 Let $\\displaystyle U$ be a single-qubit unitary gate.",
"A $\\displaystyle U$ -conjugated Clifford circuit ($\\displaystyle U$ -CCC) on $\\displaystyle n$ qubits is defined to be a quantum circuit with the following structure: Start with $\\displaystyle {\\left|{0}\\right\\rangle }^{\\otimes n}$ .",
"Apply gates from the set $\\displaystyle \\lbrace U^\\dag HU, U^\\dag SU, (U^\\dag \\otimes U^\\dag )\\mathrm {CNOT}(U \\otimes U) \\rbrace $ .",
"Measure each qubit in the computational basis.",
"Because the intermediate $\\displaystyle U$ and $\\displaystyle U^\\dag $ gates cancel, we may equivalently describe a $\\displaystyle U$ -CCC as follows: Start with $\\displaystyle {\\left|{0}\\right\\rangle }^{\\otimes n}$ .",
"Apply $\\displaystyle U^{\\otimes n}$ .",
"Apply gates from the set $\\displaystyle \\lbrace H, S, \\mathrm {CNOT} \\rbrace $ .",
"Apply $\\displaystyle (U^\\dag )^{\\otimes n}$ .",
"Measure each qubit in the computational basis."
],
[
"Notions of classical simulation of quantum computation",
"Let $\\displaystyle \\mathcal {P}= \\lbrace p_z\\rbrace _z$ and $\\displaystyle \\mathcal {Q}= \\lbrace q_z\\rbrace _z$ be (discrete) probability distributions, and let $\\displaystyle \\varepsilon \\ge 0$ .",
"We say that $\\displaystyle \\mathcal {Q}$ is a multiplicative $\\displaystyle \\varepsilon $ -approximation of $\\displaystyle \\mathcal {P}$ if for all $\\displaystyle z$ , $ |p_z - q_z| \\le \\varepsilon p_z.$ We say that $\\displaystyle \\mathcal {Q}$ is an additive $\\displaystyle \\varepsilon $ -approximation of $\\displaystyle \\mathcal {P}$ if $ \\frac{1}{2}\\sum _z|p_z - q_z| \\le \\varepsilon .$ Note that any multiplicative $\\displaystyle \\varepsilon $ -approximation is also an additive $\\displaystyle \\varepsilon /2$ -approximation, since summing Eq.",
"(REF ) over all $\\displaystyle z$ produces Eq.",
"(REF ).",
"Here the factor of 1/2 is present so that $\\displaystyle \\varepsilon $ is the total variation distance between the probability distributions.",
"A weak simulation with multiplicative (additive) error $\\displaystyle \\varepsilon >0$ of a family of quantum circuits is a classical randomized algorithm that samples from a distribution that is a multiplicative (additive) $\\displaystyle \\varepsilon $ -approximation of the output distribution of the circuit.",
"Note that from an experimental perspective, additive error is the more appropriate choice, since the fault-tolerance theorem merely guarantees additive closeness between the ideal and realized output distributions [3].",
"There are of course other notions of simulability of quantum circuits – such as strong simulation where one can compute individual output probabilities.",
"We discuss these further in Section ."
],
[
"Postselection gadgets",
"Our results involve the use of postselection gadgets to simulate unitary operations.",
"In this section, we introduce some terminology to describe these gadgets.",
"Definition 2.2 Let $\\displaystyle U$ be a single-qubit operation.",
"Let $\\displaystyle k,l \\in \\mathbb {Z}^+$ with $\\displaystyle k>l$ .",
"A $\\displaystyle k$ -to-$\\displaystyle l$ $\\displaystyle U$ -CCC postselection gadget $\\displaystyle G$ is a postselected circuit fragment that performs the following procedure on an $\\displaystyle l$ -qubit system: Introduce a set $\\displaystyle T$ of $\\displaystyle (k-l)$ ancilla registers in the state $\\displaystyle {\\left|{a_1\\ldots a_{k-l}}\\right\\rangle }$ , where $\\displaystyle a_1\\ldots a_{k-l} \\in \\lbrace 0,1\\rbrace ^{k-l}$ .",
"Apply $\\displaystyle U^{\\otimes (k-l)}$ to the set $\\displaystyle T$ of registers.",
"Apply a $\\displaystyle k$ -qubit Clifford operation $\\displaystyle \\Gamma $ to both the system and ancilla.",
"Choose a subset $\\displaystyle S$ of $\\displaystyle (k-l)$ registers and apply $\\displaystyle (U^{\\dag })^{\\otimes (k-l)}$ to $\\displaystyle S$ .",
"Postselect on the subset $\\displaystyle S$ of qubits being in the state $\\displaystyle {\\left|{b_1 \\ldots b_{k-l}}\\right\\rangle }$ , where $\\displaystyle b_1\\ldots b_{k-l} \\in \\lbrace 0,1\\rbrace ^{k-l}$ .",
"An example of a $\\displaystyle 4$ -to-$\\displaystyle 1$ $\\displaystyle U$ -CCC postselection gadget is the circuit fragment described by the following diagram: $@*=<0em>@C=1em @R=1em {& @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]&*+<1em,.9em>{\\hphantom{ \\ \\Gamma \\ }} [0,0]=\"i\",[0,0].",
"[3,0]=\"e\",!C *{ \\ \\Gamma \\ },\"e\"+UR;\"e\"+UL **{-};\"e\"+DL **{-};\"e\"+DR **{-};\"e\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{ U^\\dag } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!L!<-.5em,0em>=<0em>{{\\left\\langle {b_1}\\right|}}@{-} [0,-1]\\\\& &*!R!<.5em,0em>=<0em>{{\\left|{a_1}\\right\\rangle }} & *+<.6em>{U} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]&*+<1em,.9em>{\\hphantom{ \\ \\Gamma \\ }} @{-} [0,-1]& *+<.6em>{ U^\\dag } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!L!<-.5em,0em>=<0em>{{\\left\\langle {b_2}\\right|}}@{-} [0,-1]\\\\& & *!R!<.5em,0em>=<0em>{{\\left|{a_2}\\right\\rangle }} & *+<.6em>{U} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]&*+<1em,.9em>{\\hphantom{ \\ \\Gamma \\ }} @{-} [0,-1]& *+<.6em>{ U^\\dag } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!L!<-.5em,0em>=<0em>{{\\left\\langle {b_3}\\right|}}@{-} [0,-1]& \\\\& &*!R!<.5em,0em>=<0em>{{\\left|{a_3}\\right\\rangle }} & *+<.6em>{U} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<1em,.9em>{\\hphantom{ \\ C \\ }} @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]\\\\ \\\\}$ Let $\\displaystyle G$ be a $\\displaystyle U$ -CCC postselection gadget as described in Definition REF .",
"The action $\\displaystyle A(G)$ (also denoted $\\displaystyle A_G$ ) of $\\displaystyle G$ is defined to be the linear operation that it performs, i.e.",
"$A(G) = A_G = {\\left\\langle {b_1\\ldots b_l}\\right|}_S \\left( \\prod _{i \\in S} U_i^\\dag \\right) \\Gamma \\left( \\prod _{i \\in T} U_i \\right) {\\left|{a_1\\ldots a_l}\\right\\rangle }_T ,$ and the normalized action of $\\displaystyle G$ , when it exists, is $ \\tilde{A}_G = \\frac{A_G}{(\\det A_G)^{2^{-l}}}.$ Note that the above normalization is chosen so that $\\displaystyle \\det \\tilde{A}_G = 1$ .",
"We say that a $\\displaystyle U$ -CCC postselection gadget $\\displaystyle G$ is unitary if there exists $\\displaystyle \\alpha \\in \\mathbb {C}\\backslash \\lbrace 0\\rbrace $ and a unitary operator $\\displaystyle U$ such that $\\displaystyle A_G = \\alpha U$ .",
"It is straightforward to check that the following are equivalent conditions for gadget unitarity.",
"Lemma 2.3 A $\\displaystyle U$ -CCC postselection gadget $\\displaystyle G$ is unitary if and only if either one of the following holds: There exists $\\displaystyle \\gamma >0$ such that $\\displaystyle A_G^\\dag A_G = \\gamma I$ , $\\displaystyle \\tilde{A}_G^\\dag \\tilde{A}_G = I$ , i.e.",
"$\\displaystyle \\tilde{A}_G $ is unitary.",
"Similarly, we say that a $\\displaystyle U$ -CCC postselection gadget $\\displaystyle G$ is Clifford if there exists $\\displaystyle \\alpha \\in \\mathbb {C}\\backslash \\lbrace 0\\rbrace $ and a Clifford operator $\\displaystyle U$ such that $\\displaystyle A_G = \\alpha U$ .",
"The following lemma gives a necessary condition for a gadget to be Clifford.",
"Lemma 2.4 If $\\displaystyle G$ is a Clifford $\\displaystyle U$ -CCC postselection gadget, then $ A_G X A_G^\\dag \\propto X \\mbox{ or } A_G X A_G^\\dag \\propto Y \\mbox{ or } A_G X A_G^\\dag \\propto Z ,$ and $ A_G Z A_G^\\dag \\propto X \\mbox{ or } A_G Z A_G^\\dag \\propto Y \\mbox{ or } A_G Z A_G^\\dag \\propto Z.$ If $\\displaystyle G$ is a Clifford $\\displaystyle U$ -CCC postselection gadget, then there exists $\\displaystyle \\alpha \\in \\mathbb {C}\\backslash \\lbrace 0\\rbrace $ and a Clifford operation $\\displaystyle \\Gamma $ such that $\\displaystyle A_G = \\alpha \\Gamma $ .",
"Since $\\displaystyle \\Gamma $ is Clifford, $\\displaystyle \\Gamma X\\Gamma ^\\dag $ is a Pauli operator.",
"But $\\displaystyle \\Gamma X\\Gamma ^\\dag \\lnot \\sim I$ , otherwise, $\\displaystyle X \\sim I$ , which is a contradiction.",
"Hence, $\\displaystyle \\Gamma X\\Gamma ^\\dag \\sim X$ or $\\displaystyle Y$ or $\\displaystyle Z$ , which implies Eq.",
"(REF ).",
"The proof of Eq.",
"(REF ) is similar, with $\\displaystyle X$ replaced with $\\displaystyle Z$ ."
],
[
"Classification results",
"In this section, we classify the hardness of weakly simulating $\\displaystyle U$ -CCCs as we vary $\\displaystyle U$ .",
"As we shall see, it turns out that the classical simulation complexities of the $\\displaystyle U$ -CCCs associated with this notion of simulation are all of the following two types: the $\\displaystyle U$ -CCCs are either efficiently simulable, or are hard to simulate to constant multiplicative error unless the polynomial hierarchy collapses.",
"To facilitate exposition, we will introduce the following terminology to describe these two cases: Let $\\displaystyle \\mathcal {C}$ be a class of quantum circuits.",
"Following the terminology in [38], we say that $\\displaystyle \\mathcal {C}$ is in $\\displaystyle \\mathsf {PWEAK}$ if it is efficiently simulable in the weak sense by a classical computer.",
"We say that $\\displaystyle \\mathcal {C}$ is $\\displaystyle \\mathsf {PH}$ -supreme (or that it exhibits $\\displaystyle \\mathsf {PH}$ -supremacy) if it satisfies the property that if $\\displaystyle \\mathcal {C}$ is efficiently simulable in the weak sense by a classical computer to constant multiplicative error, then the polynomial hierarchy ($\\displaystyle \\mathsf {PH}$ ) collapses.",
"The approach we take to classifying the $\\displaystyle U$ -CCCs is to decompose each $\\displaystyle U$ into the form given by Eq.",
"(REF ), $U = e^{i \\alpha } R_z(\\phi ) R_x(\\theta ) R_z(\\lambda ),$ and study how the classical simulation complexity changes as we vary $\\displaystyle \\alpha , \\phi , \\theta $ and $\\displaystyle \\lambda $ .",
"Two simplifications can immediately be made.",
"First, the outcome probabilities of the $\\displaystyle U$ -CCC are independent of $\\displaystyle \\alpha $ , since $\\displaystyle \\alpha $ appears only in a global phase.",
"Second, the probabilities are also independent of $\\displaystyle \\lambda $ .",
"To see this, note that the outcome probabilities are all of the form: $ |{\\left\\langle {b}\\right|} R_z(-\\lambda )^{\\otimes n} V R_z(\\lambda )^{\\otimes n} {\\left|{0}\\right\\rangle }|^2 = |{\\left\\langle {b}\\right|} V {\\left|{0}\\right\\rangle }|^2 ,$ which is independent of $\\displaystyle \\lambda $ .",
"In the above expression, $\\displaystyle b \\in \\lbrace 0,1\\rbrace ^n$ and $V = R_x(-\\theta )^{\\otimes n} R_z(-\\phi )^{\\otimes n} \\Gamma R_z(\\phi )^{\\otimes n} R_x(\\theta )^{\\otimes n}$ for some Clifford circuit $\\displaystyle \\Gamma $ .",
"The equality follows from the fact that the computational basis states are eigenstates of $\\displaystyle R_z(\\lambda )^{\\otimes n}$ with unit-magnitude eigenvalues.",
"Hence, to complete the classification, it suffices to just restrict our attention to the two-parameter family $\\displaystyle \\lbrace R_z(\\phi ) R_x(\\theta )\\rbrace _{\\phi ,\\theta }$ of unitaries.",
"We first prove the following lemma (see Table REF for a summary): Lemma 3.1 Let $\\displaystyle U = R_z(\\phi ) R_x(\\theta )$ , where $\\displaystyle \\phi , \\theta \\in [0,2\\pi )$ .",
"Then $\\displaystyle U$ -CCCs are in $\\displaystyle \\mathsf {PWEAK}$ , if $\\displaystyle \\phi \\in [0,2\\pi )$ and $\\displaystyle \\theta \\in \\pi \\mathbb {Z}$ , or $\\displaystyle \\phi \\in \\frac{\\pi }{2} \\mathbb {Z}$ and $\\displaystyle \\theta \\in \\frac{\\pi }{2} \\mathbb {Z}$ .",
"$\\displaystyle U$ -CCCs are $\\displaystyle \\mathsf {PH}$ -supreme, if $\\displaystyle \\phi \\notin \\frac{\\pi }{2} \\mathbb {Z}$ and $\\displaystyle \\theta \\in \\frac{\\pi }{2} \\mathbb {Z}_{odd}$ , or $\\displaystyle \\theta \\notin \\frac{\\pi }{2} \\mathbb {Z}$ .",
"Table: Complete complexity classification of U\\displaystyle U-CCCs (where U=R z (φ)R x (θ)\\displaystyle U= R_z(\\phi ) R_x(\\theta )) with respect to weak simulation, as we vary φ\\displaystyle \\phi and θ\\displaystyle \\theta .",
"The roman numerals in parentheses indicate the parts of Lemma that are relevant to the corresponding box.",
"All U\\displaystyle U-CCCs are either in 𝖯𝖶𝖤𝖠𝖪\\displaystyle \\mathsf {PWEAK} (i.e.",
"can be efficiently simulated in the weak sense) or 𝖯𝖧\\displaystyle \\mathsf {PH}-supreme (i.e.",
"cannot be simulated efficiently in the weak sense, unless the polynomial hierarchy collapses.",
")We defer the proof of Lemma REF to Sections REF and REF .",
"Lemma REF allows us to prove our main theorem: Theorem 3.2 Let $\\displaystyle U$ be a single-qubit unitary operator.",
"Consider the following two statements: $\\displaystyle U$ -CCC is in $\\displaystyle \\mathsf {PWEAK}$ .",
"There exists a single-qubit Clifford operator $\\displaystyle \\Gamma \\in \\langle S, H \\rangle $ and $\\displaystyle \\lambda \\in [0,2\\pi )$ such thator alternatively, we could restrict the range of $\\displaystyle \\lambda $ to be in $\\displaystyle [0,\\pi ]$ , since any factor of $\\displaystyle R_z(\\pi /2) \\sim S$ can be absorbed into the Clifford operator $\\displaystyle \\Gamma $ .",
"$ U \\sim \\Gamma R_z(\\lambda ) .$ Then, (B) implies (A).",
"If the polynomial hierarchy is infinite, then (A) implies (B).",
"In other words, if we assume that the polynomial hierarchy is infinite, then $\\displaystyle U$ -CCCs are $\\displaystyle \\mathsf {PH}$ -supreme if and only if they cannot be written in the form $\\displaystyle U \\sim \\Gamma R_z(\\lambda )$ , where $\\displaystyle \\Gamma $ is a Clifford circuit and $\\displaystyle R_z(\\lambda )$ is a $\\displaystyle Z$ -rotation.",
"Since $\\displaystyle R_z(\\lambda ) {\\left|{0}\\right\\rangle } \\sim {\\left|{0}\\right\\rangle }$ , it follows that for any $\\displaystyle \\Gamma $ , $\\displaystyle \\Gamma R_z(\\lambda )$ -CCCs have the same outcome probabilities as $\\displaystyle \\Gamma $ -CCCs.",
"But $\\displaystyle C$ -CCCs are efficiently simulable, by the Gottesman-Knill Theorem, since $\\displaystyle \\Gamma \\in \\langle S, H \\rangle $ .",
"Hence, $\\displaystyle U$ -CCCs are in $\\displaystyle \\mathsf {PWEAK}$ .",
"Let $\\displaystyle U$ be such that $\\displaystyle U$ -CCCs are in $\\displaystyle \\mathsf {PWEAK}$ .",
"Using the decomposition in Eq.",
"(REF ), write $\\displaystyle U = e^{i \\alpha } R_z(\\phi ) R_x(\\theta ) R_z(\\lambda )$ .",
"Since we assumed that the polynomial hierarchy is infinite, Lemma REF implies that $\\displaystyle \\theta \\in \\pi \\mathbb {Z}$ , or $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}$ and $\\displaystyle \\phi \\in \\tfrac{\\pi }{2} \\mathbb {Z}$ .",
"In Case (a), $\\displaystyle \\theta \\in 2\\pi \\mathbb {Z}$ or $\\displaystyle \\pi \\mathbb {Z}_{odd}$ .",
"If $\\displaystyle \\theta \\in 2\\pi \\mathbb {Z}$ , then $U \\sim R_z(\\phi ) R_x(2\\pi \\mathbb {Z}) R_z(\\gamma ) = I. R_z(\\phi +\\gamma ),$ which is of the form given by Eq.",
"(REF ).",
"If $\\displaystyle \\pi \\mathbb {Z}_{odd}$ , then $U \\sim R_z(\\phi ) R_x(\\pi \\mathbb {Z}_{odd}) R_z(\\gamma ) \\sim R_z(\\phi ) X R_z(\\gamma ) = X R_z(\\gamma - \\phi ),$ which is again of the form given by Eq.",
"(REF ).",
"In Case (b), $U &\\in & e^{i\\alpha } R_z(\\pi \\mathbb {Z}/2 ) R_x(\\pi \\mathbb {Z}/2 ) R_z(\\gamma ) \\nonumber \\\\&=& e^{i\\alpha } R_z(\\pi \\mathbb {Z}/2 ) H R_z(\\pi \\mathbb {Z}/2 ) H R_z(\\gamma ) .$ But the elements of $\\displaystyle R_z(\\pi \\mathbb {Z}/2 )$ are of the form $\\displaystyle S^j$ , for $\\displaystyle j \\in \\mathbb {Z}$ , up to a global phase.",
"Therefore, $\\displaystyle R_z(\\pi \\mathbb {Z}/2 ) H R_z(\\pi \\mathbb {Z}/2 ) H $ is Clifford, and $\\displaystyle U$ is of the form Eq.",
"(REF ).",
"Hence, Theorem REF tells us that under the assumption that the polynomial hierarchy is infinite, $\\displaystyle U$ -CCCs can be simulated efficiently (in the weak sense) if and only if $\\displaystyle U \\sim \\Gamma R_z(\\lambda )$ for some single qubit Clifford operator $\\displaystyle \\Gamma $ , i.e.",
"if $\\displaystyle U$ is a Clifford operation times a $\\displaystyle Z$ -rotation."
],
[
"Proofs of efficient classical simulation",
"In this section, we prove Cases (i) and (ii) of Lemma REF ."
],
[
"Proof of Case (i): $\\displaystyle \\phi \\in [0,2\\pi )$ and {{formula:e07379e9-857d-4ef3-b992-1bb6efa00aef}}",
"Theorem 3.3 Let $\\displaystyle U = R_z(\\phi ) R_x(\\theta )$ .",
"If $\\displaystyle \\phi \\in [0,2\\pi )$ and $\\displaystyle \\theta \\in \\pi \\mathbb {Z}$ , then $\\displaystyle U$ -CCCs are in $\\displaystyle \\mathsf {PWEAK}$ .",
"First, we consider the case where $\\displaystyle \\theta \\in 2\\pi \\mathbb {Z}$ .",
"In this case, $\\displaystyle U = R_z(\\phi )$ , and the amplitudes of the $\\displaystyle U$ -CCC can be written as ${\\left\\langle {y}\\right|} R_z(-\\phi )^{\\otimes n} \\Gamma R_z(\\phi )^{\\otimes n} {\\left|{x}\\right\\rangle } \\sim {\\left\\langle {y}\\right|} \\Gamma {\\left|{x}\\right\\rangle }$ for some Clifford operation $\\displaystyle \\Gamma $ and computational basis states $\\displaystyle {\\left|{x}\\right\\rangle }$ and $\\displaystyle {\\left|{y}\\right\\rangle }$ .",
"By the Gottesman-Knill Theorem, these $\\displaystyle U$ -CCCs can be efficiently weakly simulated.",
"Next, we consider the case where $\\displaystyle \\theta \\in \\pi \\mathbb {Z}_{odd}$ .",
"In this case, $\\displaystyle U = R_z(\\phi )R_x(\\pi ) \\sim R_z(\\phi ) X$ , and the amplitudes of the $\\displaystyle U$ -CCC can be written as ${\\left\\langle {y}\\right|} X^{\\otimes n} R_z(-\\phi )^{\\otimes n} \\Gamma R_z(\\phi )^{\\otimes n} X^{\\otimes n} {\\left|{x}\\right\\rangle } \\sim {\\left\\langle {\\bar{y}}\\right|} \\Gamma {\\left|{\\bar{x}}\\right\\rangle }$ for some Clifford operation $\\displaystyle \\Gamma $ and computational basis states $\\displaystyle {\\left|{x}\\right\\rangle }$ and $\\displaystyle {\\left|{y}\\right\\rangle }$ , where $\\displaystyle \\bar{z}$ is the bitwise negation of $\\displaystyle z$ .",
"By the Gottesman-Knill Theorem, these $\\displaystyle U$ -CCCs can be efficiently weakly simulated.",
"Putting the above results together, we get that $\\displaystyle U$ -CCCs are in $\\displaystyle \\mathsf {PWEAK}$ ."
],
[
"Proof of Case (ii): $\\displaystyle \\phi \\in \\tfrac{\\pi }{2} \\mathbb {Z}$ and {{formula:ba4e0907-316c-44b1-96ee-e15e9af576b7}}",
"Theorem 3.4 Let $\\displaystyle U = R_z(\\phi ) R_x(\\theta )$ .",
"If $\\displaystyle \\phi \\in \\tfrac{\\pi }{2} \\mathbb {Z}$ and $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}$ , then $\\displaystyle U$ -CCCs are in $\\displaystyle \\mathsf {PWEAK}$ .",
"The elements of $\\displaystyle R_z(\\tfrac{\\pi }{2} \\mathbb {Z})$ are of the form $\\displaystyle S^j$ , where $\\displaystyle j \\in \\mathbb {Z}$ , up to a global phase.",
"Therefore, $\\displaystyle U = R_z(\\phi ) R_x(\\theta ) = R_z(\\phi ) H R_z(\\theta ) H $ is a Clifford operation, and so, the $\\displaystyle U$ -CCCs consist of only Clifford gates.",
"By the Gottesman-Knill Theorem, these $\\displaystyle U$ -CCCs can be be efficiently (weakly) simulated."
],
[
"Proofs of hardness",
"In this section, we prove Cases (iii) and (iv) of Lemma REF .",
"Our proof uses postselection gadgets, similar to the techniques used in [14], [18].",
"One can also prove hardness using techniques from measurement-based-quantum computing, at least for certain $\\displaystyle U$ .",
"We give such a proof in Appendix for the interested reader; we believe this proof may be more intuitive for those who are familiar with measurement-based quantum computing.",
"We start by proving a lemma that will be useful for the proofs of hardness.",
"Lemma 3.5 (Sufficient condition for $\\displaystyle \\mathsf {PH}$ -supremacy) Let $\\displaystyle U$ be a single-qubit gate.",
"If there exists a unitary non-Clifford $\\displaystyle U$ -CCC postselection gadget $\\displaystyle G$ , then $\\displaystyle U$ -CCCs are $\\displaystyle \\mathsf {PH}$ -supreme.",
"Suppose such a gadget $\\displaystyle G$ exists.",
"Then, since the Clifford group plus any non-Clifford gate is universal [49], [50], [19], the Clifford group plus $\\displaystyle G$ must be universal on a single qubit.",
"Then, by the inverse-free Solovay-Kitaev Theorem of Sardharwalla et al.",
"[55], using polynomially many gates from the set $\\displaystyle G,H,S$ one can compile any desired one-qubit unitary $\\displaystyle V$ to inverse exponential accuracy (since in particular $\\displaystyle \\langle H,S\\rangle $ contains the Paulis).",
"In particular, since any three-qubit unitary can be expressed as a product of a constant number of CNOTs and one-qubit unitaries, one can compile any gate in the set $\\displaystyle \\lbrace $ CCZ, Controlled-H, all one-qubit gates $\\displaystyle \\rbrace $ to inverse exponential accuracy with polynomial overheard.",
"In his proof that $\\displaystyle \\mathsf {PostBQP}=\\mathsf {PP}$ , Aaronson showed that postselected poly-sized circuits of the above gates can compute any language in $\\displaystyle \\mathsf {PP}$ [2].",
"Furthermore, as his postselection succeeds with inverse exponential probability, compiling these gates to inverse exponential accuracy is sufficient for performing arbitrary $\\displaystyle \\mathsf {PP}$ computations.",
"Hence, by using polynomially many gadgets for $\\displaystyle G$ , CNOT, $\\displaystyle H$ and $\\displaystyle S$ , one can compile Aaronson's circuitsMore specifically, we compile the circuit given by $\\displaystyle (U^\\dagger )^{\\otimes n}$ , then Aaronson's circuit, then $\\displaystyle U^{\\otimes n}$ , as we need to cancel the $\\displaystyle U$ 's at the beginning and the $\\displaystyle U^\\dagger $ s at the end in order to perform Aaronson's circuit which starts and measures in the computational basis.",
"However as the $\\displaystyle U,U^\\dagger $ are one-qubit gates, one can cancel them to inverse exponential accuracy using our gates, and hence this construction suffices.",
"for computing $\\displaystyle \\mathsf {PP}$ to inverse exponential accuracy, and hence these circuits can compute $\\displaystyle \\mathsf {PP}$ -hard problems.",
"$\\displaystyle \\mathsf {PH}$ -supremacy then follows from the techniques of [14], [1].",
"Namely, a weak simulation of such circuits with constant multiplicative error would place $\\displaystyle \\mathsf {PP}\\subseteq \\mathsf {BPP}^\\mathsf {NP}\\subseteq \\Delta _3$ by Stockmeyer counting, and hence by Toda's theorem this would result in the collapse of $\\displaystyle \\mathsf {PH}$ to the third level.",
"In fact, by the arguments of Fujii et al.",
"[25], one can collapse $\\displaystyle \\mathsf {PH}$ to the second level as well, by placing $\\displaystyle \\mathsf {coC}_=\\mathsf {P}$ in $\\displaystyle \\mathsf {SBP}$ , and we refer the interested reader to their work for the complete argument."
],
[
"Proof of Case (iii): $\\displaystyle \\phi \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$ and {{formula:c1b0cccf-1c51-4009-b687-e39ed2ec6cda}}",
"Let $\\displaystyle U= R_z(\\phi ) R_x(\\theta )$ .",
"Consider the following $\\displaystyle U$ -CCC postselection gadget: $I(\\phi ,\\theta ) = \\qquad @*=<0em>@C=1em @R=1em {& @{-} [0,-1]& @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]& *+<.6em>{U^\\dag } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& @{-} [0,-1]& {\\left\\langle {0}\\right|} \\\\& *!R!<.5em,0em>=<0em>{{\\left|{0}\\right\\rangle }} & *+<.6em>{U} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]}$ We now prove some properties about $\\displaystyle I(\\phi , \\theta )$ .",
"Theorem 3.6 The action of $\\displaystyle I(\\phi ,\\theta )$ is $ A_{I(\\phi ,\\theta )} = \\left(\\begin{matrix} \\cos ^2 \\tfrac{\\theta }{2} & \\tfrac{i}{2} \\sin \\theta \\ e^{-i \\phi } \\\\ -\\tfrac{i}{2} \\sin \\theta \\ e^{i \\phi } & -\\sin ^2 \\tfrac{\\theta }{2} \\end{matrix}\\right).$ $\\displaystyle I(\\phi ,\\theta )$ is a unitary gadget if and only if $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ .",
"When $\\displaystyle I(\\phi ,\\theta )$ is unitary, $ \\tilde{A}_{I(\\phi ,\\theta )} = \\frac{i}{\\sqrt{2}} \\left(\\begin{matrix} 1 & i (-1)^k e^{-i\\phi } \\\\ -i(-1)^k e^{i\\phi } & -1 \\end{matrix}\\right) ,$ where $\\displaystyle k = \\tfrac{\\theta }{\\pi }- \\tfrac{1}{2}$ .",
"$\\displaystyle I(\\phi ,\\theta )$ is a Clifford gadget if and only if $\\displaystyle \\phi \\in \\tfrac{\\pi }{2} \\mathbb {Z}$ and $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ .",
"$\\displaystyle I(\\phi ,\\theta )$ is a unitary non-Clifford gadget if and only if $\\displaystyle \\phi \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$ and $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ .",
"By direct calculation.",
"By Eq.",
"(REF ), $A_{I(\\phi ,\\theta )}^\\dag A_{I(\\phi ,\\theta )} = \\left(\\begin{matrix} \\cos ^2 \\tfrac{\\theta }{2} & \\tfrac{i}{4} \\sin (2\\theta ) e^{-i \\phi } \\\\ -\\tfrac{i}{4} \\sin (2\\theta ) e^{i \\phi } & \\sin ^2 \\tfrac{\\theta }{2} \\end{matrix}\\right) .$ If $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ , then $\\displaystyle A_{I(\\phi ,\\theta )}^\\dag A_{I(\\phi ,\\theta )} = \\tfrac{1}{2} I$ , which implies that $\\displaystyle I(\\phi ,\\theta )$ is a unitary gadget, by Lemma REF .",
"Conversely, assume that $\\displaystyle I(\\phi ,\\theta )$ is a unitary gadget.",
"Suppose that $\\displaystyle \\theta \\notin \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ .",
"Then $\\displaystyle \\sin (2\\theta ) \\ne 0$ , which implies that $\\displaystyle A_{I(\\phi ,\\theta )}^\\dag A_{I(\\phi ,\\theta )} \\lnot \\propto I$ , which is a contradiction.",
"Hence, $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ .",
"Next, $\\displaystyle k = \\tfrac{\\theta }{\\pi }- \\tfrac{1}{2}$ implies that $\\displaystyle \\theta = \\tfrac{\\pi }{2}(2k+1)$ .",
"Since $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ , it follows that $\\displaystyle k \\in \\mathbb {Z}$ .",
"Then $\\displaystyle \\sin \\theta = (-1)^k$ , $\\displaystyle \\cos ^2 \\tfrac{\\theta }{2} = \\tfrac{1}{2}$ and $\\displaystyle \\sin ^2 \\tfrac{\\theta }{2} = \\tfrac{1}{2}$ .",
"Hence, $ A_{I(\\phi ,\\theta )} = \\left(\\begin{matrix} \\tfrac{1}{2} & \\tfrac{i}{2} (-1)^k e^{-i\\phi } \\\\ -\\tfrac{i}{2} (-1)^k e^{i \\phi } & -\\tfrac{1}{2} \\end{matrix}\\right) .$ Hence, $\\displaystyle \\det A_{I(\\phi ,\\theta )} = -\\tfrac{1}{2}$ .",
"Plugging this and Eq.",
"(REF ) into Eq.",
"(REF ) gives Eq.",
"(REF ).",
"$\\displaystyle (\\Leftarrow )$ Let $\\displaystyle \\phi \\in \\tfrac{\\pi }{2} \\mathbb {Z}$ and $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ .",
"Write $\\displaystyle \\phi = \\tfrac{\\pi }{2} l$ and $\\displaystyle \\theta = \\tfrac{\\pi }{2}(2k+1)$ .",
"Then, by Eq.",
"(REF ), $\\tilde{A}_{I(\\phi ,\\theta )} = \\frac{i}{\\sqrt{2}} \\left(\\begin{matrix} 1 & i^{1+2k+3l} \\\\ i^{3+2k+l} & -1 \\end{matrix}\\right).$ Now, it is straightforward to check that for all $\\displaystyle k,l \\in \\mathbb {Z}$ , $\\displaystyle \\tilde{A}_{I(\\phi ,\\theta )} X \\tilde{A}_{I(\\phi ,\\theta )}^\\dag \\in \\lbrace -X,Z,-Z\\rbrace $ and $\\displaystyle \\tilde{A}_{I(\\phi ,\\theta )} Z \\tilde{A}_{I(\\phi ,\\theta )}^\\dag \\in \\lbrace -Y,X,Y,-X\\rbrace $ .",
"This shows that $\\displaystyle \\tilde{A}_{I(\\phi ,\\theta )}$ maps the Pauli group to itself, under conjugation, which implies that $\\displaystyle \\tilde{A}_{I(\\phi ,\\theta )}$ is Clifford.",
"$\\displaystyle (\\Rightarrow )$ Assume that $\\displaystyle I(\\phi ,\\theta )$ is a Clifford gadget.",
"Suppose that $\\displaystyle \\phi \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$ or $\\displaystyle \\theta \\notin \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ .",
"But $\\displaystyle I(\\phi ,\\theta )$ is unitary, and hence, $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ .",
"So $\\displaystyle \\phi \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$ .",
"By Lemma REF , $\\displaystyle \\tilde{A}_{I(\\phi ,\\theta )} X \\tilde{A}_{I(\\phi ,\\theta )}^\\dag \\sim X$ or $\\displaystyle Y$ or $\\displaystyle Z$ .",
"But, as we compute, $\\tilde{A}_{I(\\phi ,\\theta )} X \\tilde{A}_{I(\\phi ,\\theta )}^\\dag = \\left(\\begin{matrix} (-1)^k \\sin \\phi & -e^{-i\\phi } \\cos \\phi \\\\ -e^{i\\phi } \\cos \\phi & -(-1)^k \\sin \\phi \\end{matrix}\\right) .$ If $\\displaystyle \\tilde{A}_{I(\\phi ,\\theta )} X \\tilde{A}_{I(\\phi ,\\theta )}^\\dag \\sim X$ or $\\displaystyle Y$ , then $\\displaystyle \\sin \\phi =0$ , which is a contradiction, since $\\displaystyle \\phi \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$ .",
"Hence, $\\displaystyle \\tilde{A}_{I(\\phi ,\\theta )} X \\tilde{A}_{I(\\phi ,\\theta )}^\\dag \\sim Z$ , which implies that $\\displaystyle \\cos \\phi = 0$ .",
"But this also contradicts $\\displaystyle \\phi \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$ .",
"Hence, $\\displaystyle \\phi \\in \\tfrac{\\pi }{2} \\mathbb {Z}$ and $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ .",
"Follows from Parts 2 and 3 of Theorem REF .",
"Theorem 3.7 Let $\\displaystyle U = R_z(\\phi ) R_x(\\theta )$ .",
"If $\\displaystyle \\phi \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$ and $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ , then $\\displaystyle U$ -CCCs are $\\displaystyle \\mathsf {PH}$ -supreme.",
"By Theorem REF , when $\\displaystyle \\phi \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$ and $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}_{odd}$ , then $\\displaystyle I(\\phi ,\\theta )$ is a unitary non-Clifford $\\displaystyle U$ -CCC postselection gadget.",
"Hence, by Lemma REF , $\\displaystyle U$ -CCCs are $\\displaystyle \\mathsf {PH}$ -supreme."
],
[
"Proof of Case (iv): $\\displaystyle \\theta \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$",
"Let $\\displaystyle U= R_z(\\phi ) R_x(\\theta )$ .",
"Consider the following $\\displaystyle U$ -CCC postselection gadget: $J(\\phi ,\\theta ) = \\qquad @*=<0em>@C=1em @R=1em {& @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]\\\\& *!R!<.5em,0em>=<0em>{{\\left|{0}\\right\\rangle }} & *+<.6em>{U} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{S} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]& *+<.6em>{U^\\dag } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& @{-} [0,-1]& {\\left\\langle {0}\\right|}}$ We now prove some properties about $\\displaystyle J(\\phi , \\theta )$ .",
"Theorem 3.8 The action of $\\displaystyle J(\\phi ,\\theta )$ is $ A_{J(\\phi ,\\theta )} &=&\\frac{1}{\\sqrt{2}} e^{-i \\frac{\\pi }{4}} \\left(\\begin{matrix} i + \\cos \\theta & 0 \\\\ 0 & 1+ i \\cos \\theta \\end{matrix}\\right) \\nonumber \\\\&=& \\frac{i}{\\sqrt{2}} e^{-i \\tfrac{\\pi }{4}} \\sqrt{1+\\cos ^2 \\theta } \\ S^\\dag R_z(2 \\tan ^{-1}( \\cos \\theta )).$ $\\displaystyle J(\\phi ,\\theta )$ is a unitary gadget for all $\\displaystyle \\theta , \\phi \\in [0,2\\pi )$ .",
"The normalized action is $\\tilde{A}_{J(\\phi ,\\theta )} \\sim S^\\dag R_z(2 \\tan ^{-1}( \\cos \\theta )).$ $\\displaystyle J(\\phi ,\\theta )$ is a Clifford gadget if and only if $\\displaystyle \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}$ .",
"$\\displaystyle J(\\phi ,\\theta )$ is a unitary non-Clifford gadget if and only if $\\displaystyle \\theta \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$ .",
"By direct calculation.",
"The determinant of $\\displaystyle A_{J(\\phi ,\\theta )}$ is $\\det A_{J(\\phi ,\\theta )} = \\tfrac{1}{2} (1+ \\cos ^2 \\theta ) \\ne 0$ for all $\\displaystyle \\theta $ and $\\displaystyle \\phi $ .",
"Hence, $\\displaystyle A_{J(\\phi , \\theta )} \\propto S^\\dag R_z(2 \\tan ^{-1}( \\cos \\theta ))$ for all $\\displaystyle \\theta $ and $\\displaystyle \\phi $ , which implies that $\\displaystyle J(\\phi , \\theta )$ is a unitary gadget for all $\\displaystyle \\theta $ and $\\displaystyle \\phi $ .",
"Hence, $\\tilde{A}_{J(\\phi ,\\theta )} = \\frac{A_{J(\\phi ,\\theta )}}{\\sqrt{\\det A_{J(\\phi ,\\theta )}}} = i e^{-i \\frac{\\pi }{4}} S^\\dag R_z(2 \\tan ^{-1}( \\cos \\theta )).",
"$ $J(\\phi ,\\theta ) \\mbox{ is a Clifford gadget}& \\Leftrightarrow & S^\\dag R_z(2 \\tan ^{-1}( \\cos \\theta )) \\mbox{ is Clifford} \\nonumber \\\\& \\Leftrightarrow & R_z(2 \\tan ^{-1}( \\cos \\theta )) \\mbox{ is Clifford} \\nonumber \\\\& \\Leftrightarrow & 2 \\tan ^{-1}( \\cos \\theta ) \\in \\tfrac{\\pi }{2} \\mathbb {Z}\\quad \\mbox{ by Fact \\ref {fact:RzClifford}} \\nonumber \\\\& \\Leftrightarrow & \\cos \\theta \\in \\lbrace 0,1,-1\\rbrace \\nonumber \\\\& \\Leftrightarrow & \\theta \\in \\tfrac{\\pi }{2} \\mathbb {Z}.$ Follows from Parts 2 and 3 of Theorem REF .",
"Theorem 3.9 Let $\\displaystyle U = R_z(\\phi ) R_x(\\theta )$ .",
"If $\\displaystyle \\theta \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$ , then $\\displaystyle U$ -CCCs are $\\displaystyle \\mathsf {PH}$ -supreme.",
"By Theorem REF , when $\\displaystyle \\theta \\notin \\tfrac{\\pi }{2} \\mathbb {Z}$ , then $\\displaystyle I(\\phi ,\\theta )$ is a unitary non-Clifford $\\displaystyle U$ -CCC postselection gadget.",
"Hence, by Lemma REF , $\\displaystyle U$ -CCCs are $\\displaystyle \\mathsf {PH}$ -supreme."
],
[
"Weak simulation of CCCs with additive error",
"Here we show how to achieve additive hardness of simulating conjugated Clifford circuits, under additional hardness assumptions.",
"Specifically, we will show that under these assumptions, there is no classical randomized algorithm which given a one-qubit unitary $\\displaystyle U$ and a Clifford circuit $\\displaystyle V$ , samples the output distribution of $\\displaystyle V$ conjugated by $\\displaystyle U$ 's up to constant $\\displaystyle \\ell _1$ error.",
"In the following, let $\\displaystyle V$ be a Clifford circuit on $\\displaystyle n$ qubits, $\\displaystyle U$ be a one-qubit unitary which is not a $\\displaystyle Z$ -rotation times a Clifford, and $\\displaystyle y\\in \\lbrace 0,1\\rbrace ^n$ be an $\\displaystyle n$ -bit string.",
"Define $p_{y,U,V}=\\left|{\\left\\langle {y}\\right|} (U^\\dagger )^{\\otimes n} V U^{\\otimes n} {\\left|{0^n}\\right\\rangle }\\right|^2.$ In other words $\\displaystyle p_{y,U,V}$ is the probability of outputting the string $\\displaystyle y$ when applying the circuit $\\displaystyle V$ conjugated by $\\displaystyle U$ 's to the all $\\displaystyle 0$ 's state, and then measuring in the computational basis.",
"Let the corresponding probability distribution on $\\displaystyle y$ 's given $\\displaystyle U$ and $\\displaystyle V$ be denoted $\\displaystyle D(U,V)$ .",
"Theorem 4.1 Assuming that PH is infinite and Conjecture REF , then there is no classical algorithm which given a one-qubit unitary $\\displaystyle U$ and an $\\displaystyle n$ -qubit Clifford circuit $\\displaystyle V$ , outputs a probability distribution which is $\\displaystyle 1/100$ close to $\\displaystyle D(U,V)$ in total variation distance.",
"Conjecture 4.1 For any $\\displaystyle U$ which is not equal to a Z-rotation times a Clifford, it is $\\displaystyle \\#\\mathsf {P}$ -hard to approximate a $\\displaystyle 6/50$ fraction of the $\\displaystyle p_{y,U,V}$ over the choice of $\\displaystyle y,V$ to within multiplicative error $\\displaystyle 1/2+o(1)$ .",
"In order to prove this we'll actually prove a more general theorem described below; the result will then follow from simply setting $\\displaystyle a=c=1/5$ , $\\displaystyle \\varepsilon =1/100$ .",
"One can in general plug in any values they like subject to the constraints; for instance one can strengthen the hardness assumption by assuming computing a smaller fraction of the $\\displaystyle p_{y,U,V}$ is still $\\displaystyle \\#\\mathsf {P}$ -hard to obtain larger allowable error in the simulation.",
"These parameters are similar to those appearing in other hardness conjectures, for example those used for $\\displaystyle \\mathsf {IQP}$ [16].",
"Theorem 4.2 Pick constants $\\displaystyle 0<\\varepsilon ,a,c <1$ such that $\\displaystyle (1-a)^2/2-c>0$ and $\\displaystyle \\frac{2\\varepsilon }{ac}<1$ .",
"Then assuming Conjecture REF , given a one-qubit unitary $\\displaystyle U$ and an $\\displaystyle n$ -qubit Clifford circuit $\\displaystyle V$ , one cannot weakly simulate the distribution $\\displaystyle D(U,V)$ with a randomized classical algorithm with total variation distance error $\\displaystyle \\varepsilon $ , unless the polynomial hierarchy collapses to the third level.",
"Conjecture 4.2 For any $\\displaystyle U$ which is not equal to a Z-rotation times a Clifford, it is $\\displaystyle \\#\\mathsf {P}$ -hard to multiplicatively approximate $\\displaystyle (1-a)^2/2-c$ fraction of the $\\displaystyle p_{y,U,V}$ over the choice of $\\displaystyle (y,V)$ , up to multiplicative error $\\displaystyle \\frac{2\\varepsilon }{ac}+o(1)$ .",
"[Proof of Theorem REF ] Suppose by way of contradiction that there exists a classical poly-time randomized algorithm which given inputs $\\displaystyle U,V$ outputs samples from a distribution $\\displaystyle D^{\\prime }(U,V)$ such that $\\displaystyle \\frac{1}{2}|D(U,V)-D^{\\prime }(U,V)|_1 < \\varepsilon $ .",
"In particular, let $\\displaystyle q_{y,U,V}$ be the probability that $\\displaystyle D^{\\prime }(U,V)$ outputs $\\displaystyle y$ – i.e.",
"the probability that the simulation outputs $\\displaystyle y$ under inputs $\\displaystyle U,V$ .",
"By our simulation assumption, for all $\\displaystyle U,V$ we have that $\\displaystyle \\sum _{y} |q_{y,U,V} - p_{y,U,V}| \\le 2\\varepsilon $ .",
"Therefore by Markov's inequality, given our constant $\\displaystyle 0<c<1$ , we have that for all $\\displaystyle U$ and $\\displaystyle V$ there exists a set $\\displaystyle S^{\\prime }\\subseteq \\lbrace 0,1\\rbrace ^n$ of output strings $\\displaystyle y$ of size $\\displaystyle |S^{\\prime }|/2^n > 1-c$ , such that for all $\\displaystyle y\\in S^{\\prime }$ , $|q_{y,U,V} - p_{y,U,V}|\\le \\frac{2\\varepsilon }{c 2^n}.$ In particular, by averaging over $\\displaystyle V$ 's, we see that for any $\\displaystyle U$ as above, there exists a set $\\displaystyle S \\subset \\lbrace 0,1\\rbrace ^n \\times \\mathcal {C}$ of pairs $\\displaystyle (y,V)$ such that for all $\\displaystyle (y,V)\\in S$ , $\\displaystyle |q_{y,U,V} - p_{y,U,V}|\\le \\frac{2\\varepsilon }{c 2^n}.$ Furthermore $\\displaystyle S$ has measure at least $\\displaystyle (1-c)$ over a uniformly random choice of $\\displaystyle (y,V)$ .",
"We now show the following anticoncentration lemma (similar theorems were shown independently in [33], [42], [35]): Lemma 4.3 For any fixed $\\displaystyle U$ and $\\displaystyle y$ as above, and for any constant $\\displaystyle 0<a<1$ , we have that at least $\\displaystyle \\frac{(1-a)^2}{2}$ fraction of the Clifford circuits $\\displaystyle V$ have the property that $p_{y,U,V} \\ge \\frac{a}{2^n} .",
"$ We will prove Lemma REF shortly.",
"First, we will show why this implies Theorem REF .",
"In particular, by averaging Lemma REF over $\\displaystyle y$ 's, we see that for any $\\displaystyle U$ as above, there exists a set $\\displaystyle T \\subset \\lbrace 0,1\\rbrace ^n \\times \\mathcal {C}$ of pairs $\\displaystyle (y,V)$ such that for all $\\displaystyle (y,V)\\in T$ , $\\displaystyle p_{y,U,V} \\ge \\frac{a}{2^n}$ .",
"Furthermore $\\displaystyle T$ has measure at least $\\displaystyle \\frac{(1-a)^2}{2}$ over a uniformly random choice of $\\displaystyle (y,V)$ .",
"Since we assumed that $\\displaystyle (1-a)^2/2 + (1-c) >1$ , then $\\displaystyle S\\cap T$ must be nonempty, and in particular must contain $\\displaystyle (1-a)^2/2-c$ fraction of the pairs $\\displaystyle (y,V)$ .",
"On this set $\\displaystyle S\\cap T$ , we have that $q_{y,U,V} \\le p_{y,U,V} + \\frac{2\\varepsilon }{c2^n} =p_{y,U,V} + \\frac{2\\varepsilon }{ac}\\frac{a}{2^n} \\le \\left(1 + \\frac{2\\varepsilon }{ac}\\right)p_{y,U,V}, $ and likewise $q_{y,U,V} \\ge p_{y,U,V} - \\frac{2\\varepsilon }{c2^n} =p_{y,U,V} - \\frac{2\\varepsilon }{ac}\\frac{a}{2^n} \\ge \\left(1 - \\frac{2\\varepsilon }{ac}\\right)p_{y,U,V} .",
"$ Since $\\displaystyle 1-\\frac{2\\varepsilon }{ac}>0$ (which we guaranteed by assumption), $\\displaystyle q_{y,U,V}$ is a multiplicative approximation to $\\displaystyle p_{y,U,V}$ with multiplicative error $\\displaystyle \\frac{2\\varepsilon }{ac}$ for $\\displaystyle (y,V)$ in the set $\\displaystyle S\\cap T$ .",
"The set $\\displaystyle S\\cap T$ contains at least $\\displaystyle (1-a)^2/2-c$ fraction of the total pairs $\\displaystyle (y,V)$ .",
"On the other hand, by Conjecture REF we have that computing a $\\displaystyle (1-a)^2/2 -c$ fraction of the $\\displaystyle p_{y,U,V}$ to this level of multiplicative error is a $\\displaystyle \\#\\mathsf {P}$ -hard task.",
"So approximating $\\displaystyle p_{y,U,V}$ to this level of multiplicative error for this fraction of outputs is both $\\displaystyle \\#\\mathsf {P}$ -hard, and achievable by our simulation algorithm.",
"This collapses $\\displaystyle \\mathsf {PH}$ to the third level by known arguments [1], [14].",
"In particular, by applying Stockmeyer's approximate counting algorithm [59] to $\\displaystyle p_{y,U,V}$ , one can multiplicatively approximate $\\displaystyle q_{y,U,V}$ to multiplicative error $\\displaystyle \\frac{1}{\\mathsf {poly}}$ in $\\displaystyle \\mathsf {F}\\mathsf {BPP}^\\mathsf {NP}$ for those elements in $\\displaystyle S\\cap T$ .",
"But since $\\displaystyle q_{y,U,V}$ is a $\\displaystyle \\frac{2\\varepsilon }{ac}$ -approx to $\\displaystyle p_{y,U,V}$ , this is a $\\displaystyle \\frac{2\\varepsilon }{ac}+o(1)$ multiplicative approximation to $\\displaystyle p_{y,U,V}$ in $\\displaystyle S\\cap T$ .",
"Hence a $\\displaystyle \\#\\mathsf {P}$ -hard quantity is in $\\displaystyle \\mathsf {F}\\mathsf {BPP}^\\mathsf {NP}$ .",
"This collapses $\\displaystyle \\mathsf {PH}$ to the third level by Toda's theorem [61].",
"To complete our proof of Theorem REF , we will prove Lemma REF .",
"[Proof of Lemma REF ] To prove this, we will make use of the fact that the Clifford group is an exact 2-designThe Clifford group is also a 3-design, but we will only need the fact it is a 2-design for our proof.",
"[62], [64].",
"The fact that the Clifford group is a 2-design means that for any polynomial $\\displaystyle p$ over the variables $\\displaystyle \\lbrace V_{ij}\\rbrace $ and their complex conjugates, which is of degree at most 2 in the $\\displaystyle V_{ij}$ 's and degree at most 2 in the $\\displaystyle V_{ij}^*$ 's, we have that $\\frac{1}{|C|} \\sum _{V \\in C} p(V,V^*) = \\int p(V,V^*) \\mathrm {d}V, $ where $\\displaystyle C$ denotes the Clifford group and the integral $\\displaystyle \\mathrm {d}V$ is taken over the Haar measure.",
"In other words, the expectation values of low-degree polynomials in the entries of the matrices are exactly identical to the expectation values over the Haar measure.",
"In particular, note that $\\displaystyle p_{y,U,V}$ is a degree-1 polynomial in the entries of $\\displaystyle V$ and their complex conjugates, and $\\displaystyle p_{y,U,V}^2$ is a degree-2 polynomial in these variables.",
"Therefore, since the Clifford group is an exact 2-design, we have that for any $\\displaystyle y$ and $\\displaystyle U$ , $\\frac{1}{|C|} \\sum _{V \\in \\mathcal {C}} p_{y,U,V} = \\int p_{y,U,V} \\mathrm {d}V = \\frac{1}{2^n}$ and $\\frac{1}{|C|} \\sum _{V \\in \\mathcal {C}} p_{y,U,V}^2 = \\int p_{y,U,V}^2 \\mathrm {d}V = \\frac{2}{2^{2n} - 1} \\left(1 - \\frac{1}{2^n}\\right),$ where the values of these integrals over the Haar measure are well known – see for instance Appendix D of [32].",
"Following [16], we now invoke the Paley-Zygmund inequality, which states that: Fact 2 Given a parameter $\\displaystyle 0<a<1$ , and a non-negative random variable $\\displaystyle p$ of finite variance, we have $\\mathrm {Pr}[p\\ge a \\mathbb {E}[p]] \\ge (1-a)^2 \\mathbb {E}[p]^2/\\mathbb {E}[p^2] .$ Applying this inequality to the random variable $\\displaystyle p_{y,U,V}$ over the choice of the Clifford circuit $\\displaystyle V$ , we have that $\\mathrm {Pr}_V\\left[p_{y,U,V}\\ge \\frac{a}{2^n}\\right] \\ge (1-a)^2 \\frac{2^{-2n}}{\\frac{2 - 2^{-n+1}}{2^{2n}-1} } = (1-a)^2 \\frac{1 - 2^{-2n}}{2 - 2^{-n+1}} \\ge \\frac{(1-a)^2}{2}$ which implies the claim.",
"This completes the proof of Theorem REF ."
],
[
"Evidence in favor of hardness conjecture",
"In Section , we saw that by assuming an average case hardness conjecture (namely Conjecture REF ), we could show that a weak simulation of CCCs to additive error would collapse the polynomial hierarchy.",
"A natural question is: what evidence do we have that Conjecture REF is true?",
"In this section, we show that the worst-case version of Conjecture REF is true.",
"In fact, we show that for any $\\displaystyle U\\ne C R_Z(\\theta )$ for a Clifford $\\displaystyle C$ , there exists a Clifford circuit V and an output y such that computing $\\displaystyle p_{y,U,V}$ is $\\displaystyle \\#\\mathsf {P}$ -hard to constant multiplicative error.",
"Therefore certainly some output probabilities of CCCs are $\\displaystyle \\#\\mathsf {P}$ -hard to compute.",
"Conjecture REF is merely conjecturing further that computing a large fraction of such output probabilities is just as hard.",
"Theorem 5.1 (Worst-case version of Conjecture REF ) For any $\\displaystyle U$ which is not equal to a Z-rotation times a Clifford, there exists a Clifford circuit $\\displaystyle V$ and string $\\displaystyle y\\in \\lbrace 0,1\\rbrace ^n$ such that it is $\\displaystyle \\#\\mathsf {P}$ -hard to multiplicatively approximate a $\\displaystyle p_{y,U,V}$ to multiplicative error $\\displaystyle 1/2 -o(1)$ .",
"This follows from combining the ideas from the proof of Lemma REF with previously known facts about $\\displaystyle \\mathsf {BQP}$ .",
"In particular, we will use the following facts: There exists a uniform family of poly-size $\\displaystyle \\mathsf {BQP}$Even $\\displaystyle \\mathsf {IQP}$ suffices here [16].",
"circuits $\\displaystyle C_x$ where $\\displaystyle x\\in \\lbrace 0,1\\rbrace ^n$ using a gate set with algebraic entries such that computing $\\displaystyle |{\\left\\langle {0^n}\\right|}C_x{\\left|{0^n}\\right\\rangle }|^2$ to multiplicative error $\\displaystyle 1/2$ is $\\displaystyle \\#\\mathsf {P}$ -hard [16].",
"For any poly-sized quantum circuit $\\displaystyle C$ over a gate set with algebraic entries, any non-zero output probability has magnitude at least inverse exponential [39].",
"As shown in the proof of Theorem REF , for any $\\displaystyle U$ which is not a Clifford gate times a Z rotation, there is a postselection gadget $\\displaystyle G$ which performs a unitary but non-Clifford one-qubit operation.",
"Furthermore all ancilla qubits in $\\displaystyle G$ begin in the state $\\displaystyle {\\left|{0}\\right\\rangle }$ .",
"From these facts, we can now prove the theorem.",
"Let $\\displaystyle p=|\\langle 0^n|C_x | 0^n \\rangle |^2$ .",
"By Fact 2, the circuit $\\displaystyle C_x$ from Fact 1 either has $\\displaystyle p=0$ or $\\displaystyle p \\ge 2^{-O(n^c)}$ for some constant $\\displaystyle c$ .",
"Now suppose we compile the circuit $\\displaystyle C_x$ from Fact 1 using Clifford gates plus the postselection gadget $\\displaystyle G$ – call this new circuit with postselection $\\displaystyle C^{\\prime }_x$ .",
"By Sardharwalla et al.",
"[55] we can compile this circuit with accuracy $\\displaystyle \\varepsilon =2^{-O(n^c)-100}$ with only polynomial overhead.",
"Let $\\displaystyle \\ell \\in \\lbrace 0,1\\rbrace ^k$ be the string of postselection bits of the circuit $\\displaystyle C^{\\prime }_x$ (which without loss of generality are the last bits of the circuit), and let $\\displaystyle \\alpha $ is the probability that all postselections succeed.",
"Note $\\displaystyle \\alpha $ is a known and easily calculated quantity, since each postselection gadget is unitary so succeeds with a known constant probability.",
"Let $\\displaystyle p^{\\prime } = |\\langle 0^n \\ell |C^{\\prime }_x | 0^{n+k} \\rangle |^2/\\alpha $ .",
"Then we have that: If $\\displaystyle p=0$ then $\\displaystyle p^{\\prime } \\le 2^{-O(n^c)-100}$ .",
"If $\\displaystyle p \\ne 0$ then $\\displaystyle p-2^{-O(n^c)-100} \\le p^{\\prime } \\le p + 2^{-O(n^c)-100} $ .",
"Since $\\displaystyle p\\ge 2^{-O(n^c)}$ , this is a multiplicative approximation to $\\displaystyle p$ with error $\\displaystyle 2^{-100}$ .",
"Now suppose that one can compute $\\displaystyle |\\langle 0^n \\ell |C^{\\prime }_x | 0^{n+k} \\rangle |^2$ to multiplicative error $\\displaystyle \\gamma $ to be chosen shortly.",
"Then immediately one can compute $\\displaystyle p^{\\prime } =|\\langle 0^n \\ell |C^{\\prime }_x | 0^{n+k} \\rangle |^2/\\alpha $ to the same amount of multiplicative error – call this estimate $\\displaystyle p^{\\prime \\prime }$ .",
"By the above argument, if $\\displaystyle p=0$ then $\\displaystyle p^{\\prime \\prime }<2^{-O(n^c)-100}(1+\\gamma )$ .",
"On the other hand if $\\displaystyle p>0$ then $\\displaystyle p^{\\prime }>2^{-O(n^c)}$ , so $\\displaystyle p^{\\prime \\prime }>2^{-O(n^c)}(1-\\gamma )$ .",
"So long as $\\displaystyle \\gamma $ is chosen such that $\\displaystyle 2^{-100}(1+\\gamma )<(1-\\gamma )$ these two cases can be distinguished – which holds in particular if $\\displaystyle \\gamma \\approx 1/2$ .",
"Therefore, if $\\displaystyle p^{\\prime \\prime }<2^{-O(n^c)}$ then we can infer that $\\displaystyle p=0$ .",
"If $\\displaystyle p^{\\prime \\prime }>2^{-O(n^c)}(1-\\gamma )$ , then $\\displaystyle p>0$ so $\\displaystyle p^{\\prime \\prime }$ is a $\\displaystyle \\gamma $ approximation to $\\displaystyle p^{\\prime }$ and hence a $\\displaystyle \\gamma +2^{-100}+\\gamma 2^{-100}$ approximation to $\\displaystyle p$ .",
"In either case we have computed a $\\displaystyle \\gamma +2^{-100}+\\gamma 2^{-100}$ approximation to $\\displaystyle p$ .",
"Therefore, if $\\displaystyle \\gamma =1/2 -2^{-99}$ , then we have computed a 1/2-multiplicative approximation to $\\displaystyle p$ , which is $\\displaystyle \\#\\mathsf {P}$ -hard by Fact 1.",
"Therefore, computing some the probability that the CCC correspoding to $\\displaystyle C^{\\prime }_x$ outputs $\\displaystyle {\\left|{0^{n}\\ell }\\right\\rangle }$ to multiplicative error $\\displaystyle 1/2 - 2^{-99}$ is $\\displaystyle \\#\\mathsf {P}$ -hard.",
"One can similarly improve this hardness to $\\displaystyle 1/2 - o(1)$ .",
"Given that the worst-case version of Conjecture REF is true, a natural question to ask is how difficult it would be to prove the average-case conjecture.",
"To do so would in particular prove quantum advantage over classical computation with realistic error, and merely assuming the polynomial hierarchy is infinite.",
"In some ways this would be stronger evidence for quantum advantage over classical computation than Shor's factoring algorithm, as there are no known negative complexity-theoretic consequences if factoring is contained in $\\displaystyle \\mathsf {P}$ .",
"Unfortunately, recent work has shown that proving Conjecture REF would be a difficult task.",
"Specifically, Aaronson and Chen [4] demonstrated an oracle relative to which $\\displaystyle \\mathsf {PH}$ is infinite, but classical computers can efficiently weakly simulate quantum devices to constant additive error.",
"Therefore, any proof which establishes quantum advantage with additive error under the assumption that $\\displaystyle \\mathsf {PH}$ is infinite must be non-relativizing.",
"In particular this implies any proof of Conjecture REF would require non-relativizing techniques – in other words it could not remain true if one allows for classical oracle class in the circuit.",
"This same barrier holds for proving the similar average-case hardness conjectures to show advantage for Boson Sampling, $\\displaystyle \\mathsf {IQP}$ , $\\displaystyle \\mathsf {DQC}1$ , or Fourier sampling.",
"Therefore any proof of Conjecture REF would require facts specific to the Clifford group.",
"We leave this as an open problem.",
"We also note that it remains open to prove the average-case exact version of Conjecture REF - i.e.",
"whether it is hard to exactly compute a large fraction of $\\displaystyle p_{y,U,C}$ .",
"We believe this may be a more tractable problem to approach than Conjecture REF .",
"However this remains open, as is the analogous average-case exact conjecture corresponding to $\\displaystyle \\mathsf {IQP}$ .",
"We note the corresponding average-case exact conjecture for Boson Sampling and Fourier sampling are known to be true [1], [26], though these models are not known to anticoncentrate."
],
[
"Summary of simulability of CCCs",
"For completeness, in this section we summarize the simulability of $\\displaystyle U$ -CCCs when $\\displaystyle U$ is not a Clifford rotation times a $\\displaystyle Z$ rotation.",
"There are various notions of classical simulation at play here.",
"The results of this paper so far have focused of notions of approximate weak simulation.",
"A weak simulation of a family of quantum circuits is a classical randomized algorithm that samples from the same distribution as the output distribution of the circuit.",
"On the other hand, a strong simulation of a family of quantum circuits is a classical algorithm that computes not only the joint probabilities, but also any marginal probabilities of the outcomes of the measurements in the circuit.",
"Following [38], we can further refine these definitions according to the number of qubits being measured: a strong(1) simulation computes the marginal output probabilities on individual qubits, and a strong($\\displaystyle n$ ) simulation computes the probability of output strings $\\displaystyle y\\in \\lbrace 0,1\\rbrace ^n$ .",
"Similarly, a weak(1) simulation samples from the marginal output probabilities on individual qubits, and a weak($\\displaystyle n$ ) simulation samples from $\\displaystyle p(y_1,\\ldots ,y_n)$ .",
"A weak$\\displaystyle ^+$ simulation samples from the same distribution on all $\\displaystyle n$ output qubits up to constant additive error.",
"Our previous results have shown that efficient weak($\\displaystyle n$ ) simulations (Theorem REF ), weak$\\displaystyle ^+$ simulations (Theorem REF ), and strong($\\displaystyle n$ ) simulations (Theorem REF ) of CCCs are implausible.",
"However it is natural to ask if it is possible to simulate single output probabilities of CCCs.",
"It turns out the answer to this question is yes.",
"This follows immediately from Theorem 5 of [38], which showed more generally that Clifford circuits with product inputs or measurements have an efficient strong(1) and weak(1) simulation.",
"Therefore this completes the complexity classification of the simulability of such circuits.",
"We note that $\\displaystyle \\mathsf {IQP}$ has identical properties in this regard.",
"This emphasizes that the difficulty in simulating CCCs (or $\\displaystyle \\mathsf {IQP}$ circuits) comes from the difficulty of simulating all of the marginal probability distributions contained in the output distribution, where the marginal is taken over a large number of output bits.",
"The probabilities of computing individual output bits of either model are easy for classical computation.",
"This is summarized in Figure REF .",
"Figure: Relationships between different notions of classical simulation and summary of the hardness of simulating CCCs.",
"An arrow from A\\displaystyle A to B\\displaystyle B (A→B\\displaystyle A \\rightarrow B) means that an efficient A\\displaystyle A-simulation of a computational task implies that there is an efficient B\\displaystyle B-simulation for the same task.",
"Note also that an weak(n\\displaystyle n) simulation exists if and only if a weak simulation exists.",
"For a proof of these relationships, see .",
"The two curves indicate the boundary between efficiencies of simulation of U\\displaystyle U-CCCs, where U\\displaystyle U is not a Clifford operation times a Z\\displaystyle Z rotation.",
"“Hard” means that an efficient simulation of U\\displaystyle U-CCCs is not possible, unless 𝖯𝖧\\displaystyle \\mathsf {PH} collapses.",
"“Conjectured hard” means that an efficient simulation of U\\displaystyle U-CCCs is not possible, if we assume Conjecture .",
"“Easy” means that an efficient simulation of U\\displaystyle U-CCCs exists.",
"Note that when U\\displaystyle U is a Clifford operation times a Z\\displaystyle Z rotation, all the above notions become easy."
],
[
"Open Problems",
"Our work leaves open a number of problems.",
"What is the computational complexity of commuting CCCs?",
"In other words, can the gate set $\\displaystyle CZ,S$ conjugated by a one-qubit gate $\\displaystyle U$ ever give rise to quantum advantage?",
"Note that this does not follow from Bremner, Jozsa and Shepherd's results [14], as their hardness proof uses the gate set $\\displaystyle CZ,T$ or $\\displaystyle CCZ,CZ,Z$ conjugated by one-qubit gates.",
"If this is true, it would say that the “intersection\" of CCCs and IQP remains computationally hard.",
"One can also consider the computational power of arbitrary fragments of the Clifford group, which were classified in [31].",
"Perhaps by studying such fragments of the Clifford group one could achieve hardness with lower depth circuits (see additional question below).",
"We showed that Clifford circuits conjugated by tensor-product unitaries are difficult to simulate classically.",
"A natural extension of this question is: suppose your gate set consists of all two-qubit Clifford gates, conjugated by a unitary $\\displaystyle U$ which is not a tensor product of the same one-qubit gate.",
"Can one show that all such circuits are difficult to simulate classically (say exactly)?",
"Such a theorem could be a useful step towards classifying the power of all two-qubit gate sets.",
"Generic Clifford circuits have a depth which is linear in the number of qubits [5].",
"In particular the lowest-depth decomposition for a generic Clifford circuit over $\\displaystyle n$ qubits to date has depth $\\displaystyle 14n-4$ [47].",
"Such depth will be difficult to achieve in near-term quantum devices without error-correction.",
"As a result, others have considered quantum supremacy experiments with lower-depth circuits.",
"For instance, Bremner, Shepherd and Montanaro showed advantage for a restricted version of $\\displaystyle \\mathsf {IQP}$ circuits with depth $\\displaystyle O(\\log n)$ [17] with long-range gates (which becomes depth $\\displaystyle O(n^{1/2}\\log n)$ if one uses SWAP gates to simulate long-range gates using local operations on a square lattice).",
"We leave open the problem of determining if quantum advantage can be achieved with CCCs of lower depth (say $\\displaystyle O(n^{1/2})$ or $\\displaystyle O(n^{1/3})$ ) with local gates only.",
"In order to establish quantum supremacy for CCCs, we conjectured that it is $\\displaystyle \\#\\mathsf {P}$ -hard to approximate a large fraction of the output probabilities of randomly chosen CCCs (Conjecture REF ).",
"Is it also $\\displaystyle \\#\\mathsf {P}$ -hard to exactly compute that large of a fraction of the output probabilities?",
"This is a necessary but not sufficient condition for Conjecture REF to be true, and we believe it may be a more approachable problem."
],
[
"Acknowledgments",
"We thank Scott Aaronson, Mick Bremner, Alex Dalzell, Daniel Gottesman, Aram Harrow, Ashley Montanaro, Ted Yoder and Mithuna Yoganathan for helpful discussions.",
"AB was partially supported by the NSF GRFP under Grant No.",
"1122374, by a Vannevar Bush Fellowship from the US Department of Defense, and by an NSF Waterman award under grant number 1249349.",
"JFF acknowledges support from the Air Force Office of Scientific Research under AOARD grant no.",
"FA2386-15-1-4082.",
"This material is based on research supported in part by the Singapore National Research Foundation under NRF Award No.",
"NRF-NRFF2013-01.",
"DEK is supported by the National Science Scholarship from the Agency for Science, Technology and Research (A*STAR)."
],
[
"Measurement-based Quantum Computing Proof of Multiplicative Hardness for CCCs for certain $\\displaystyle U$ 's",
"We will prove the following theorem using techniques from measurement-based quantum computing (MBQC).",
"Theorem A.1 CCCs with $\\displaystyle U=R_Z(\\theta )H$ cannot be efficiently weakly classically simulated to multiplicative error $\\displaystyle 1/2$ unless the polynomial hierarchy collapses, for any $\\displaystyle \\theta $ which is not an integer multiple of $\\displaystyle \\pi /4$ .",
"This is a weaker version of Lemma REF .",
"We include it for pedagogical reasons, as it provides a different way of understanding the main theorem using MBQC techniques, and it includes a more detailed walkthrough of the hardness construction.",
"Furthermore, it does not rely on the theorem that the Clifford group plus any non-Clifford element is universal; instead one can directly prove postselected universality by finding a qubit rotation by an irrational multiple of $\\displaystyle \\pi $ .",
"As in the proof of Lemma REF , we will first show that CCCs can perform universal quantum computation (i.e., the class $\\displaystyle \\mathsf {BQP}$ ) under postselection.",
"This first step will make extensive use of ideas from Measurement-Based Quantum Computation [9].",
"Next, we will show that these circuits can furthermore perform $\\displaystyle \\mathsf {PostBQP}$ under postselection.",
"This extension requires the inverse-free Solovay-Kitaev theorem of Sardharwalla et al.",
"[55].",
"Theorem A.2 Postselected CCCs can be used to simulate universal quantum computation under the choice of $\\displaystyle U=R_Z(\\theta )H$ , and for any choice of $\\displaystyle \\theta $ other than integer multiples of $\\displaystyle \\pi /4$ .",
"We will first describe the proof without reference to Measurement Based Quantum Computing (MBQC) so as to be understood by the broadest possible audience.",
"We will then summarize the proof in MBQC language for those familiar with the area.",
"Our proof will make use of four gadgets to show that under postselection, we can perform arbitrary 1-qubit gates in this model.",
"For the first gadget, consider the following quantum circuit: $@*=<0em>@C=1em @R=1em {*!R!<.5em,0em>=<0em>{{\\left|{\\psi }\\right\\rangle }} & @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]& *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *=<1.8em,1.4em>{=\"j\",\"j\"-<.778em,.322em>;{\"j\"+<.778em,-.322em> ur,_{}},\"j\"-<0em,.4em>;p+<.5em,.9em> **{-},\"j\"+<2.2em,2.2em>*{},\"j\"-<2.2em,2.2em>*{} } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& {\\left\\langle {0}\\right|} \\\\*!R!<.5em,0em>=<0em>{{\\left|{0}\\right\\rangle }} & *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& {\\left|{\\psi ^{\\prime }}\\right\\rangle }}$ Here the notation $\\displaystyle {\\left\\langle {0}\\right|}$ denotes that we postselect that measurement outcome on obtaining the state $\\displaystyle {\\left|{0}\\right\\rangle }$ , and the two-qubit gate is controlled-Z.",
"This gadget performs teleportation [27].",
"One can easily calculate that $\\displaystyle {\\left|{\\psi ^{\\prime }}\\right\\rangle }=H{\\left|{\\psi }\\right\\rangle }$ – in other words, this gadget performs the $\\displaystyle H$ gate [27], [14].",
"Likewise, if one postselects the first outcome to be $\\displaystyle {\\left|{1}\\right\\rangle }$ , then the gate performed is $\\displaystyle XH$ .",
"By chaining these gadgets together, one can perform any product of these operations.",
"For instance, the following circuit performs $\\displaystyle HXH$ : $@*=<0em>@C=1em @R=1em {*!R!<.5em,0em>=<0em>{{\\left|{\\psi }\\right\\rangle }} & @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]& @{-} [0,-1]& *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *=<1.8em,1.4em>{=\"j\",\"j\"-<.778em,.322em>;{\"j\"+<.778em,-.322em> ur,_{}},\"j\"-<0em,.4em>;p+<.5em,.9em> **{-},\"j\"+<2.2em,2.2em>*{},\"j\"-<2.2em,2.2em>*{} } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& {\\left\\langle {1}\\right|} \\\\*!R!<.5em,0em>=<0em>{{\\left|{0}\\right\\rangle }} & *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]& *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *=<1.8em,1.4em>{=\"j\",\"j\"-<.778em,.322em>;{\"j\"+<.778em,-.322em> ur,_{}},\"j\"-<0em,.4em>;p+<.5em,.9em> **{-},\"j\"+<2.2em,2.2em>*{},\"j\"-<2.2em,2.2em>*{} } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& {\\left\\langle {0}\\right|} \\\\*!R!<.5em,0em>=<0em>{{\\left|{0}\\right\\rangle }} & *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& {\\left|{\\psi ^{\\prime }}\\right\\rangle }}$ The correctness follows from the fact that the order in which quantum measurements are taken is irrelevant.",
"By stringing together $\\displaystyle n$ of these, we can perform $\\displaystyle n$ gates from the set $\\displaystyle \\lbrace H, XH\\rbrace $ .",
"These generate a finite set of one-qubit gates which contain the Paulis.",
"Now clearly circuits composed of these gadgets do not have the form of conjugated Clifford circuits with $\\displaystyle U = R_Z(\\theta ) H$ .",
"But we can easily correct this by inserting $\\displaystyle R_Z(\\theta )$ 's at the beginning of each line, and $\\displaystyle R_Z(-\\theta )$ 's at the end of each line.",
"$@*=<0em>@C=1em @R=1em {*!R!<.5em,0em>=<0em>{{\\left|{\\psi }\\right\\rangle }} & @{-} [0,-1]& *+<.6em>{R_Z(\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]& *+<.6em>{R_Z(-\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *=<1.8em,1.4em>{=\"j\",\"j\"-<.778em,.322em>;{\"j\"+<.778em,-.322em> ur,_{}},\"j\"-<0em,.4em>;p+<.5em,.9em> **{-},\"j\"+<2.2em,2.2em>*{},\"j\"-<2.2em,2.2em>*{} } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& {\\left\\langle {0}\\right|} \\\\*!R!<.5em,0em>=<0em>{{\\left|{0}\\right\\rangle }} & *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{R_Z(\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]& *+<.6em>{R_Z(-\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& {\\left|{\\psi ^{\\prime }}\\right\\rangle }}$ Clearly this is equivalent to our original gadget at the Z rotations commute through and cancel.",
"Now the gadget has the property that Every input line begins with $\\displaystyle R_Z(\\theta )$ , and every output line ends with $\\displaystyle R_Z(-\\theta )$ .",
"Every ancillary input begins with $\\displaystyle {\\left|{0}\\right\\rangle }$ then applies $\\displaystyle R_Z(\\theta )H$ .",
"Every ancillary output applies $\\displaystyle HR_Z(-\\theta )$ and measures in the computational basis.",
"All gates in between are Clifford.",
"When composing such gadgets, the $\\displaystyle R_Z(-\\theta )$ at the end of each output line cancels with the $\\displaystyle R_Z(\\theta )$ at the beginning of each input line.",
"Hence composing gadgets with the above properties will always form a CCC.",
"For instance our prior circuit performing $\\displaystyle HXH$ becomes $@*=<0em>@C=1em @R=1em {*!R!<.5em,0em>=<0em>{{\\left|{\\psi }\\right\\rangle }} & @{-} [0,-1]& *+<.6em>{R_Z(\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]& @{-} [0,-1]& *+<.6em>{R_Z(-\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]&*+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *=<1.8em,1.4em>{=\"j\",\"j\"-<.778em,.322em>;{\"j\"+<.778em,-.322em> ur,_{}},\"j\"-<0em,.4em>;p+<.5em,.9em> **{-},\"j\"+<2.2em,2.2em>*{},\"j\"-<2.2em,2.2em>*{} } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& {\\left\\langle {1}\\right|} \\\\*!R!<.5em,0em>=<0em>{{\\left|{0}\\right\\rangle }} & *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{R_Z(\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]& *+<.6em>{R_Z(-\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *=<1.8em,1.4em>{=\"j\",\"j\"-<.778em,.322em>;{\"j\"+<.778em,-.322em> ur,_{}},\"j\"-<0em,.4em>;p+<.5em,.9em> **{-},\"j\"+<2.2em,2.2em>*{},\"j\"-<2.2em,2.2em>*{} } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& {\\left\\langle {0}\\right|} \\\\*!R!<.5em,0em>=<0em>{{\\left|{0}\\right\\rangle }} & *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{R_Z(\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& @{-} [0,-1]&*!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]&*+<.6em>{R_Z(-\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& {\\left|{\\psi ^{\\prime }}\\right\\rangle }}$ Thus, by simply replacing our input state $\\displaystyle {\\left|{\\psi }\\right\\rangle }$ with the state $\\displaystyle H{\\left|{0}\\right\\rangle }$ , and our output state with a Hadamard followed by measurement, this postselected circuit would be simulating the circuit which starts in the state $\\displaystyle H{\\left|{0}\\right\\rangle }$ , applies $\\displaystyle HXH$ , then applies $\\displaystyle H$ and measures.",
"Furthermore, this state will have the form of a CCC.",
"More generally, by stringing $\\displaystyle n$ such gadgets together to form a CCC, clearly one can simulate any one-qubit quantum circuit where the initial state is $\\displaystyle H{\\left|{0}\\right\\rangle }$ , one performs $\\displaystyle n$ gates from the set $\\displaystyle \\lbrace H,XH\\rbrace $ , and then applies $\\displaystyle H$ and measures.",
"This allows us to simulate one-qubit gates from the set $\\displaystyle \\lbrace H,XH\\rbrace $ with postselected CCC circuits.",
"However, such gates are not universal for a single qubit.",
"In order to show postselected CCCs can perform universal quantum computation, we will need to find a way to simulate all single qubit gates.",
"To do so, we will consider adding features to our gadget.",
"So far the Clifford part of our CCCs are all commuting; let's consider adding a non-commuting one-qubit gate $\\displaystyle X$ to make a new gadget: $@*=<0em>@C=1em @R=1em {*!R!<.5em,0em>=<0em>{{\\left|{\\psi }\\right\\rangle }} & @{-} [0,-1]& *+<.6em>{R_Z(\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]&*+<.6em>{X} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{R_Z(-\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *=<1.8em,1.4em>{=\"j\",\"j\"-<.778em,.322em>;{\"j\"+<.778em,-.322em> ur,_{}},\"j\"-<0em,.4em>;p+<.5em,.9em> **{-},\"j\"+<2.2em,2.2em>*{},\"j\"-<2.2em,2.2em>*{} } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& {\\left\\langle {0}\\right|} \\\\*!R!<.5em,0em>=<0em>{{\\left|{0}\\right\\rangle }} & *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{R_Z(\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]&@{-} [0,-1]& *+<.6em>{R_Z(-\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& {\\left|{\\psi ^{\\prime }}\\right\\rangle }}$ By commuting the $\\displaystyle R_Z(\\theta )$ rightwards on both lines, and noting that $@*=<0em>@C=1em @R=1em {& @{-} [0,-1]& *+<.6em>{R_Z(\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{X} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{R_Z(-\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]&@{-} [0,-1]}$ is equivalent to $@*=<0em>@C=1em @R=1em {& @{-} [0,-1]& *+<.6em>{R_Z(2\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{X} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]&@{-} [0,-1]}$ we can see this performs the same quantum operation as $@*=<0em>@C=1em @R=1em {*!R!<.5em,0em>=<0em>{{\\left|{\\psi }\\right\\rangle }} & @{-} [0,-1]& @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]& *+<.6em>{R_Z(2\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]&*+<.6em>{X} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *=<1.8em,1.4em>{=\"j\",\"j\"-<.778em,.322em>;{\"j\"+<.778em,-.322em> ur,_{}},\"j\"-<0em,.4em>;p+<.5em,.9em> **{-},\"j\"+<2.2em,2.2em>*{},\"j\"-<2.2em,2.2em>*{} } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& {\\left\\langle {0}\\right|} \\\\*!R!<.5em,0em>=<0em>{{\\left|{0}\\right\\rangle }} & *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]&@{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& {\\left|{\\psi ^{\\prime }}\\right\\rangle }}$ which since $\\displaystyle HX=ZH$ , is equivalent to $@*=<0em>@C=1em @R=1em {*!R!<.5em,0em>=<0em>{{\\left|{\\psi }\\right\\rangle }} & @{-} [0,-1]& @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]& *+<.6em>{R_Z(2\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]&*+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *+<.6em>{Z} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *=<1.8em,1.4em>{=\"j\",\"j\"-<.778em,.322em>;{\"j\"+<.778em,-.322em> ur,_{}},\"j\"-<0em,.4em>;p+<.5em,.9em> **{-},\"j\"+<2.2em,2.2em>*{},\"j\"-<2.2em,2.2em>*{} } =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& {\\left\\langle {0}\\right|} \\\\*!R!<.5em,0em>=<0em>{{\\left|{0}\\right\\rangle }} & *+<.6em>{H} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]&@{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& @{-} [0,-1]& {\\left|{\\psi ^{\\prime }}\\right\\rangle }}$ By direct computation, gadget (REF ) (which is equivalent to gadget (REF )) performs the operation $\\displaystyle HR_Z(2\\theta )$ .",
"Let us call this gate $\\displaystyle G_0(\\theta )$ .",
"Likewise, if one postselects on $\\displaystyle {\\left|{1}\\right\\rangle }$ , one obtains the gate $\\displaystyle G_1(\\theta )=XHR_Z(2\\theta )$ .",
"(This gadget is well-known in MBQC; see below).",
"Therefore, by applying our gadgets (REF ) and (REF ), we can create postselected CCCs to simulate the evolution of a one-qubit circuit which evolves by gates in the set $\\displaystyle \\lbrace H,XH,G_0(\\theta ),G_1(\\theta )\\rbrace $ .",
"Intuitively, as long as the choice of $\\displaystyle \\theta $ is not pathological, these gates will generate all one-qubit gates.",
"Therefore we have all one-qubit gates at our disposal via these gadgets.",
"We will prove this statement rigorously in Lemma REF , which we defer to the end of this appendix.",
"In fact, we show that as long as $\\displaystyle \\theta $ is not set to $\\displaystyle k\\pi /4$ for some integer $\\displaystyle k$ , then the set of one qubit gates generated by these gadgets is universal on a qubit.",
"Thus postselected CCC's (where $\\displaystyle \\theta \\ne k\\pi /4$ ) can simulate arbitrary one-qubit operations.",
"To prove that postselected CCC's can perform universal quantum computation, we need to show how to perform an entangling two qubit gate.",
"We can then appeal to the result of Brylinski & Brylinski [8] and Bremner et al.",
"[10] that any entangling two-qubit gate, plus the set of all-one-qubit gates, is universal for quantum computation.",
"But performing entangling two-qubit gates is trivial in our setup, since the Clifford group (and the conjugated Clifford group) contains entangling two-qubit gates.",
"For example, we can easily perform the controlled-Z gate between qubits with the following gadget: $@*=<0em>@C=1em @R=1em {& @{-} [0,-1]& *+<.6em>{R_Z(\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]&*!<0em,.025em>-=-<.2em>{\\bullet }@{-} [1,0] @{-} [0,-1]& *+<.6em>{R_Z(-\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& @{-} [0,-1]\\\\& @{-} [0,-1]& *+<.6em>{R_Z(\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& *!<0em,.025em>-=-<.2em>{\\bullet }@{-} [0,-1]& *+<.6em>{R_Z(-\\theta )} =\"i\",\"i\"+UR;\"i\"+UL **{-};\"i\"+DL **{-};\"i\"+DR **{-};\"i\"+UR **{-},\"i\" @{-} [0,-1]& @{-} [0,-1]}$ This gadget clearly has the correct form, and hence composes with the gadgets (REF ) and REF to form universal quantum circuits.",
"This shows how to simulate $\\displaystyle \\mathsf {BQP}$ with postselected CCCs.",
"We can now recast this proof in the language of Measurement-Based Quantum Computing.",
"Our result essentially follows from that fact that measuring graph states in the bases $\\displaystyle HR_Z(2\\theta )$ and $\\displaystyle H$ , combined with postselection, is universal for quantum computing.",
"More formally, let $\\displaystyle E$ be series of Controlled-Z operations that create a graph state out of $\\displaystyle H^{\\otimes n} {\\left|{0}\\right\\rangle }^{\\otimes n}$ (we will specify the cluster state later).",
"Let $\\displaystyle U=R_Z(\\theta )H$ for some $\\displaystyle \\theta $ to be specified later.",
"Then consider creating the CCC for the Clifford circuit $\\displaystyle C=X^S E$ , where the notation $\\displaystyle X^S$ denotes that we apply an $\\displaystyle X$ gate to some subset $\\displaystyle S\\subseteq [n]$ of the qubits.",
"We have that $H^{\\otimes n} R_Z(-\\theta )^{\\otimes n} X^{S} E R_Z(\\theta )^{\\otimes n} H^{\\otimes n} {\\left|{0}\\right\\rangle }^{\\otimes n}&= H^{\\otimes n} R_Z(-\\theta )^{\\otimes n} X^{S} R_Z(\\theta )^{\\otimes n} E H^{\\otimes n} {\\left|{0}\\right\\rangle }^{\\otimes n}\\\\&= H^{\\otimes n} \\left(XR_Z(2\\theta )\\right)^{S} E H^{\\otimes n} {\\left|{0}\\right\\rangle }^{\\otimes n} \\\\&= H^{\\otimes n} \\left(XR_Z(2\\theta )\\right)^{S} {\\left|{\\mathrm {Cluster}}\\right\\rangle } \\\\&= \\left(\\left(Z H R_Z(2\\theta )\\right)^{S} \\otimes H^{\\bar{S}}\\right) {\\left|{\\mathrm {Cluster}}\\right\\rangle } ,$ where the first equality follows from the fact that $\\displaystyle R_Z$ and $\\displaystyle E$ commute as they are both diagonal in the $\\displaystyle Z$ basis, the second follows from the fact that on the lines without an $\\displaystyle X$ the $\\displaystyle R_Z(\\theta ) $ and the $\\displaystyle R_Z(-\\theta ) $ cancel, and on the lines with an $\\displaystyle X$ we have $\\displaystyle R_Z(-\\theta ) X R_Z(\\theta ) = X R_Z(2\\theta )$ , the third follows from the fact that $\\displaystyle E$ is constructed such that $\\displaystyle E H^{\\otimes n} {\\left|{0}\\right\\rangle }^{\\otimes n} = {\\left|{\\mathrm {Cluster}}\\right\\rangle }$ , and the fourth from the fact that $\\displaystyle HX=ZH$ .",
"Now since we're measuring in the Z basis at the end of the circuit, the last row of $\\displaystyle Z$ 's can be ignored, so the circuit is equivalent to: $ \\left(\\left(H R_Z(2\\theta )\\right)^{S} \\otimes H^{\\bar{S}}\\right) {\\left|{\\mathrm {Cluster}}\\right\\rangle } .",
"$ Now we simply need to show that measurement based quantum computation with postselection on such a state is universal for quantum computing.",
"In other words, we need to show that if we can construct a Cluster state and measure some qubits in the $\\displaystyle H$ basis and others in the $\\displaystyle HR_Z(2\\theta )$ basis, and postselect on the outcomes, then we can perform universal quantum computation.",
"It was previously known to be universal for MBQC if different $\\displaystyle \\theta $ 's occur on each qubit [9].",
"In our setup we do not have this flexibility, but we instead have the additional ability to postselect.",
"Universality of this model follows from the fact that by preparing an appropriate Cluster state (using the standard trick to perform 1-qubit gates with MBQC), this gives us the ability to apply the one-qubit gate $\\displaystyle H R_Z(2\\theta )$ using postselection.",
"Likewise, postselecting on $\\displaystyle {\\left|{1}\\right\\rangle }$ performs the operation $\\displaystyle X H R_Z(2\\theta )$ .",
"As discussed previously, by Lemma REF , as long as $\\displaystyle \\theta $ is not set to $\\displaystyle k\\pi /4$ for some integer $\\displaystyle k$ , this is a universal gate set on a qubit.",
"The addition of entangling two-qubit operations on the Cluster state (namely, controlled-Z) boosts this model to universality.",
"We have now shown that postselected CCCs can perform $\\displaystyle \\mathsf {BQP}$ under postselection.",
"We now extend this to show they can perform $\\displaystyle \\mathsf {PostBQP}=\\mathsf {PP}$ under postselection.",
"This requires using the inverse-free Solovay-Kitaev algorithm of [55].",
"From this, the hardness result follows via known techniques [14], [1].",
"Theorem A.3 Postselected CCCs with $\\displaystyle U=R_Z(\\theta )H$ can decide any language in $\\displaystyle \\mathsf {PostBQP}=\\mathsf {PP}$ , for any choice of $\\displaystyle \\theta $ other than integer multiples of $\\displaystyle \\pi /4$ .",
"To prove this, we will apply Aaronson's result that Postselected $\\displaystyle \\mathsf {BQP}$ circuits, denoted $\\displaystyle \\mathsf {PostBQP}$ , can decide any language in $\\displaystyle \\mathsf {PP}$ .",
"Aaronson's proof works by showing that a particular universal quantum gate set – namely the gate set $\\displaystyle G$ consisting of Toffoli, controlled-Hadamard, and one qubit gates – can decide $\\displaystyle \\mathsf {PP}$ under postselection.",
"We previously showed that our postselected CCCs can perform a different universal quantum gate $\\displaystyle G^{\\prime }$ consisting of controlled-Z, $\\displaystyle HR_Z(2\\theta )$ , $\\displaystyle XHR_Z(2\\theta )$ , $\\displaystyle H$ and $\\displaystyle XH$ .",
"Therefore, in order to show that postselected CCCs can compute $\\displaystyle \\mathsf {PP}$ , we need to show how to simulate Aaronson's gate set $\\displaystyle G$ using our gate set $\\displaystyle G^{\\prime }$ .",
"One difficulty is that we must be extremely accurate in our simulation of these gates.",
"This is because postselected quantum circuits may postselect on exponentially tiny events.",
"Therefore, in order to simulate Aaronson's postselected circuits for $\\displaystyle \\mathsf {PP}$ , we will need to simulate each gate to inverse exponential accuracy.",
"Normally in quantum computing this simulation is handled by the Solovay-Kitaev Theorem, which roughly states that any universal gate set can simulate any other universal gate set to error $\\displaystyle \\varepsilon $ with only $\\displaystyle \\mathrm {polylog}(1/\\varepsilon )$ overhead.",
"Therefore with polynomial overhead, one can obtain inverse exponential accuracy in the simulation.",
"This is why the choice of gate set is irrelevant in the definition of $\\displaystyle \\mathsf {PostBQP}$ .",
"One catch, however, is that the Solovay-Kitaev theorem requires that the gate set is closed under inversion, i.e.",
"for any gate $\\displaystyle g\\in G$ , we have $\\displaystyle g^{-1}\\in G$ as well.",
"This is an essential part of the construction of this theorem (which makes use of group commutators).",
"It is an open problem to remove this requirement [21], [39].",
"As a corollary, it is open whether or not the class $\\displaystyle \\mathsf {PostBQP}$ can still compute all languages in $\\displaystyle \\mathsf {PP}$ if the gate set used is not closed under inversion.",
"It is possible the class could be weaker with non-inversion-closed gate sets.",
"Unfortunately, the gate set $\\displaystyle G^{\\prime }$ we have at our disposal is not closed under inversion.",
"Furthermore, since we obtained the gates using postselection gadgets, it is not clear how to generate the inverses of the gadgets, as postselection is a non-reversible operation.",
"Therefore we cannot appeal to the Solovay-Kitaev theorem to show we can compute languages in $\\displaystyle \\mathsf {PP}$ .",
"Fortunately, however, even though our gate set does not have inverses, it does have a special property – namely, our set of one qubit gates contains the Pauli group.",
"It turns out that recently, [55] proved a Solovay-Kitaev theorem for any set of one qubit gates containing the Paulis, but which is not necessarily closed under inversion.",
"Therefore, by this result, even though our gate set is not closed under inversion, we can still apply any one-qubit gate to inverse exponential accuracy with merely polynomial overhead.",
"So we can apply arbitrary one-qubit gates.",
"It turns out this is sufficient to apply gates from Aaronson's gate set $\\displaystyle G$ consisting of Toffoli, controlled-H and one qubit gates with inverse exponential accuracy.",
"To see this, first not that it is well-known one can construct controlled-V operations for arbitrary one-qubit gates V using a finite circuit of controlled-NOT and one-qubit gates – see [48] for details.",
"Furthermore, it is possible to construct Toffoli using a finite circuit of one qubit gates and controlled-V operations [48].",
"This, together with the fact that controlled-NOT is equal to controlled-Z conjugated by Hadamard on one qubit, shows that each gate in $\\displaystyle G$ has an exact decomposition as a finite number of controlled-Z gates and one-qubit gates.",
"Hence, using controlled-Z gates and one-qubit gates compiled to exponential accuracy, one can obtain circuits from $\\displaystyle G$ with inverse exponential accuracy.",
"Thus, our gate set $\\displaystyle G^{\\prime }$ can efficiently simulate gates from $\\displaystyle G$ , and hence our postselected CCCs can compute all languages in $\\displaystyle \\mathsf {PostBQP}=\\mathsf {PP}$ as well.",
"From this, the hardness result follows via known techniques [14], [1].",
"Corollary A.4 Conjugated Clifford circuits cannot be weakly simulated classically to multiplicative error unless the polynomial hierarchy collapses to the third level, for the choice of $\\displaystyle U=R_Z(\\theta )H$ for any $\\displaystyle \\theta $ which is not an integer multiple of $\\displaystyle \\pi /4$ .",
"To complete our proof, we merely need to show the following lemma: Lemma A.5 So long as $\\displaystyle \\theta $ is not an integer multiple of $\\displaystyle \\pi /4$ , then the gates $\\displaystyle H R_Z(2\\theta )$ and $\\displaystyle X H R_Z(2\\theta )$ are universal on a qubit.",
"Furthermore, so long as $\\displaystyle \\theta $ is not an integer multiple of $\\displaystyle \\pi /4$ , then at least one of these gates is a rotation of the Bloch sphere by an irrational multiple of $\\displaystyle \\pi $ .",
"For convenience of notation, define $\\displaystyle G_0 = -iH R_Z(2\\theta )$ and $\\displaystyle G_1= -X H R_Z(2\\theta )$ .",
"We will actually begin by proving something stronger: namely, that as long as $\\displaystyle \\theta $ is not an integer multiple of $\\displaystyle \\pi /4$ , then one of the rotations $\\displaystyle G_0$ and $\\displaystyle G_1$ is by an irrational multiple of $\\displaystyle \\pi $ .",
"We will prove this by contradiction.",
"Suppose that both $\\displaystyle G_0$ and $\\displaystyle G_1$ are rotations by rational multiples of $\\displaystyle \\pi $ , call their rotation angles $\\displaystyle \\phi _0$ and $\\displaystyle \\phi _1$ , respectively.",
"By direct computation, first eigenvalue of $\\displaystyle G_0$ is given by $\\frac{1}{2\\sqrt{2}} \\left( -2\\sin (\\theta ) - i\\sqrt{6+2\\cos (2\\theta )} \\right).",
"$ Since this must be equal to $\\displaystyle e^{\\pm i\\phi _0/2}$ , and by considering the real part of this equation, we have that $\\cos (\\phi _0/2) = -\\frac{\\sin (\\theta )}{\\sqrt{2}}.$ By an identical argument, for gate $\\displaystyle G_1$ we have that $\\cos (\\phi _1/2) = - \\frac{\\cos (\\theta )}{\\sqrt{2}}.$ Squaring these terms and summing them, we obtain that $\\cos ^2(\\phi _0/2) +\\cos ^2(\\phi _1/2) = \\frac{1}{2}.",
"$ Or, applying the fact $\\displaystyle \\cos ^2 t = \\frac{1+\\cos 2t }{2}$ and simplifying, one can see this is equivalent to $\\cos (\\phi _0) + \\cos (\\phi _1) +\\cos (0)= 0.",
"$ Since we are assuming by way of contradiction that $\\displaystyle G_0,G_1$ are of finite order, we are assuming that $\\displaystyle \\phi _0,\\phi _1$ are rational multiples of $\\displaystyle \\pi $ .",
"Previously, Crosby [20] and Włodarski [63] classified all possible solutions to the equation $\\displaystyle \\cos (\\alpha _1)+\\cos (\\alpha _2)+\\cos (\\alpha _3)$ where each $\\displaystyle \\alpha _i$ are rational multiples of $\\displaystyle \\pi $ .",
"The four possible solution families to this equation (assuming without loss of generality that $\\displaystyle 0\\le \\alpha _i\\le \\pi $ ) are [63] $\\displaystyle \\left\\lbrace \\beta ,\\pi -\\beta ,\\pi /2\\right\\rbrace $ where $\\displaystyle 0\\le \\beta \\le \\pi $ $\\displaystyle \\left\\lbrace \\delta , \\frac{2\\pi }{3}-\\delta ,\\frac{2\\pi }{3}+\\delta \\right\\rbrace $ where $\\displaystyle 0\\le \\delta \\le \\frac{\\pi }{3} $ $\\displaystyle \\left\\lbrace \\frac{2\\pi }{5},\\frac{4\\pi }{5},\\frac{\\pi }{3}\\right\\rbrace $ $\\displaystyle \\left\\lbrace \\frac{\\pi }{5},\\frac{3\\pi }{5},\\frac{2\\pi }{3}\\right\\rbrace .$ Since we have that one of our three angles is $\\displaystyle 0$ , the latter two cases are immediately ruled out, and we must have that the angles $\\displaystyle \\lbrace \\phi _0,\\phi _1\\rbrace $ are either $\\displaystyle \\lbrace \\pi /2,\\pi \\rbrace $ or $\\displaystyle \\lbrace 2\\pi /3,2\\pi /3\\rbrace $ .",
"One can easily see that the first solution corresponds to $\\displaystyle \\theta =k\\pi /2$ for an integer $\\displaystyle k$ , and the second solution corresponds to $\\displaystyle \\theta = k\\pi /4$ for an odd integer $\\displaystyle k$ .",
"Therefore, so long as $\\displaystyle \\theta $ is not an integer multiple of $\\displaystyle \\pi /4$ , we have a contradiction, as there are no further solutions to these equations where the $\\displaystyle \\phi _i$ are rational multiples of $\\displaystyle \\pi $ .",
"So if $\\displaystyle \\theta $ is set to any value other than $\\displaystyle k\\pi /4$ for an integer $\\displaystyle k$ , we have that at least one of the gates $\\displaystyle G_0$ and $\\displaystyle G_1$ is a rotation by an irrational multiple of $\\displaystyle \\pi $ .",
"Now what remains to be shown is that the gates $\\displaystyle G_0$ and $\\displaystyle G_1$ are universal in the general case.",
"This can be shown easily by the classification of continuous subgroups of $\\displaystyle SU(2)$ .",
"The continuous subgroups of $\\displaystyle SU(2)$ are $\\displaystyle U(1)$ (corresponding to all rotations about one axis), $\\displaystyle U(1)\\times \\mathbb {Z}_2$ (corresponding to all rotations about an axis $\\displaystyle a$ , plus a rotation by $\\displaystyle \\pi $ through another axis perpendicular to $\\displaystyle a$ ), and $\\displaystyle SU(2)$ .",
"By our prior result we know that either $\\displaystyle G_0$ or $\\displaystyle G_1$ generates all rotations about its axis of rotation on the Bloch sphere.",
"Therefore, if we can show that neither $\\displaystyle G_0$ nor $\\displaystyle G_1$ are rotations by angle $\\displaystyle \\pi $ we are done, as these then must generate all of $\\displaystyle SU(2)$ .",
"However this follows immediately from equations REF and REF , since these equations imply that we can have either $\\displaystyle \\phi _0=\\pi $ or $\\displaystyle \\phi _1=\\pi $ only when $\\displaystyle \\theta $ is a rational multiple of $\\displaystyle \\pi /2$ .",
"Hence, as long as $\\displaystyle \\theta $ is not a rational multiple of $\\displaystyle \\pi /4$ , neither $\\displaystyle G_0$ nor $\\displaystyle G_1$ is a rotation by $\\displaystyle \\pi $ , and furthermore one is a rotation by an irrational multiple of $\\displaystyle \\pi $ .",
"These gates generate a continuous group which is neither $\\displaystyle U(1)$ nor $\\displaystyle U(1)\\times \\mathbb {Z}_2$ , and therefore by the above observation these generate all of $\\displaystyle SU(2)$ ."
]
] | 1709.01805 |
[
[
"Stratification for multiplicative character sums"
],
[
"Abstract We prove a stratification result for certain families of $n$-dimensional (complete algebraic) multiplicative character sums.",
"The character sums we consider are sums of products of $r$ multiplicative characters evaluated at rational functions, and the families (with $nr$ parameters) are obtained by allowing each of the $r$ rational functions to be replaced by an \"offset\", i.e.",
"a translate, of itself.",
"For very general such families, we show that the stratum of the parameter space on which the character sum has maximum weight $n+j$ has codimension at least $j\\lfloor(r-1)/2(n-1)\\rfloor$ for $1\\le j\\le n-1$ and $\\lceil nr/2\\rceil$ for $j=n$."
],
[
"Introduction",
"In this paper we are interested in multiplicative character sums of the following form: $S:=\\sum _{m\\in \\kappa ^n}\\chi _1(F_1(m))\\chi _2(F_2(m))\\dots \\chi _r(F_r(m)),$ where $\\kappa $ is a finite field, $F_i\\in \\kappa [x_1,\\dots ,x_n]$ , and $\\chi _i:\\kappa ^\\times \\rightarrow \\times $ is a multiplicative character (extended to $\\kappa $ by stipulating $\\chi _i(0)=0$ ), for each $1\\le i\\le r$ .",
"It is reasonable to expect square root cancellation for generic polynomials $F_i$ , namely, that ${S}\\le C(\\#\\kappa )^{n/2}$ for some constant $C=C(n,r,\\lbrace \\deg F_i\\rbrace )$ independent of $\\kappa $ for generic choices of the $F_i$ 's (with respect to the $\\chi _i$ 's).",
"However, character sums of this form seem difficult to deal with, especially if square root cancellation is desired.",
"One can certainly find a multiplicative character $\\chi $ and integers $e_i\\ge 0$ to write $\\chi _i=\\chi ^{e_i}$ , so that $S=\\sum _{m\\in \\kappa ^n}\\chi (F_1(m)^{e_1}F_2(m)^{e_2}\\dots F_r(m)^{e_r})$ .",
"But the square root cancellation result of Katz [10] about sums of the form $\\sum _m \\chi (F(m))$ requires that the homogeneous part of highest degree (the “leading form\") of $F$ defines a nonsingular projective variety, which is obviously not the case for our sums as soon as $r>1$ or some $e_i>1$ .",
"A generalization of Katz's result by Rojas-León [12] allows singular leading forms, but the ability to establish square root cancellation is lost with the presence of a single singular point.",
"A subsequent paper of Rojas-León [13] allows the leading form to be a product of polynomials, but the result applies to additive characters only, and also requires that the factors of the leading form together define a nonsingular variety, among other conditions.",
"The present paper confirms that if the $F_i$ 's are each allowed to vary independently within an “offset family\" (the family of polynomials $F_i(\\,\\,\\cdot \\,+x^{(i)})$ parametrized by the “offset\" $x^{(i)}\\in \\kappa ^n$ ), then for generic members of this family, square root cancellation indeed holds as long as $r\\ge 2n-1$ .",
"In fact we are able to obtain a stratification result in the sense of Fouvry and Katz [3], i.e.",
"to bound the dimensions of the subscheme (the stratum) on which the character sum has maximum weight $n+j$ , for each $1\\le j\\le n$ .",
"Having maximum weight $w$ means being a sum of a bounded number of complex numbers of absolute values $\\le (\\#\\kappa )^{w/2}$ , so maximum weight $n$ leads to square root cancellation.",
"To formulate the precise statement of our results, we first introduce the following Notations, Conventions, and Definitions.",
"If $\\chi $ is a multiplicative character, let $\\operatorname{ord}\\chi $ denote its order.",
"A rational function $F\\in \\kappa (x_1,\\dots ,x_n)$ is called $d$ th-power-free if each irreducible factor of $F$ has multiplicity strictly between $-d$ and $d$ .",
"We think of a rational function $F\\in \\kappa (x_1,\\dots ,x_n)$ as the quotient of two fixed polynomials $G,H\\in \\kappa [x_1,\\dots ,x_n]$ , define its degree $\\deg F$ as $\\max \\lbrace \\deg G,\\deg H\\rbrace $ , and stipulate that $\\chi (F(x))=0$ if $G(x)=0$ or $H(x)=0$ , where $x$ is the $n$ -tuple $(x_1,\\dots ,x_n)$ .",
"Similarly, we use $x^{(i)}$ to denote an $n$ -tuple $(x^{(i)}_1,\\dots ,x^{(i)}_n)$ .",
"For a subscheme $X\\subset \\mathbb {A}^{nr}_{\\kappa }$ , define its degree $\\deg X$ to be the degree of its closure in $\\mathbb {P}^{nr}_{\\kappa }$ .",
"Define the constants $ \\theta _j=\\theta _j(n,r):= {\\left\\lbrace \\begin{array}{ll} ja +\\max \\lbrace 0,b+j-(n-1)\\rbrace & \\text{if }0\\le j\\le n-1,\\\\{nr/2} & \\text{if }j=n,\\end{array}\\right.}",
"$ if we write ${\\frac{r-1}{2}}=(n-1)a+b$ with $a\\in \\mathbb {N}$ and $0\\le b<n-1$ .",
"In particular, $\\theta _0=0$ , $\\theta _1=*{\\frac{r-1}{2(n-1)}}$ , $\\theta _{n-1}={\\frac{r-1}{2}}$ , and in general $\\theta _j\\ge j*{\\frac{r-1}{2(n-1)}}$ if $n\\ge 2$ .",
"A variety in this paper is an integral separated scheme of finite type over a base field, not necessarily algebraically closed.",
"We now state the main theorem of this paper.",
"Theorem 1.1 There exist integers $C,C^{\\prime }\\in \\mathbb {N}$ and a finite set $\\mathcal {S}$ (whose elements are called exceptional primes) that depend on four parameters $n,r,d,D$ such that the following holds.",
"For each $1\\le i\\le r$ , assume that $d_i:=\\operatorname{ord}\\chi _i\\mid d>0$ , let $F_i\\in \\kappa (x_1,\\dots ,x_n)$ be a $d_i$ th-power-free rational function of degree at most $D$ such that $T_{F_i}:=\\lbrace m\\in \\overline{\\kappa }^n\\mid F_i(x)\\equiv F_i(x+m)\\rbrace $ is finite for each $1\\le i\\le r$ , and consider the following family of character sums parametrized by $(x^{(1)},\\dots ,x^{(r)})\\in \\kappa ^{nr}$ : $S(x^{(1)},\\dots ,x^{(r)}):=\\sum _{m\\in \\kappa ^n}\\prod _{i=1}^r\\chi _i(F_i(m+x^{(i)})).$ Then whenever ${\\rm char\\,}\\kappa \\notin \\mathcal {S}$ , there exist subschemes $\\mathbb {A}^{nr}_\\kappa =X_0\\supset X_1\\supset X_2\\supset \\dots \\supset X_n$ , such that the sum of degrees of irreducible components of each $X_j$ is at most $C^{\\prime }$ , and such that $\\operatorname{codim}X_j\\ge \\theta _j$ (i.e.",
"$\\dim X_j\\le nr-\\theta _j$ ) and ${S(x^{(1)},\\dots ,x^{(r)})}\\le C(\\#\\kappa )^{(n+j-1)/2}$ for each $1\\le j\\le n$ and $(x^{(1)},\\dots ,x^{(r)})\\in \\mathbb {A}^{nr}(\\kappa )\\setminus X_j(\\kappa )$ .",
"The theorem says that square root cancellation holds outside of $X_1$ , so $X_1$ is “the stratum of all exceptional (non-generic) parameter values\", and $\\theta _1=*{\\frac{r-1}{2(n-1)}}$ is a lower bound for $\\operatorname{codim}X_1$ .",
"In particular, we need $r\\ge 2n-1$ (i.e.",
"an offset family with at least $(2n-1)n$ parameters) to show that square root cancellation holds for generic parameter values (i.e.",
"$\\operatorname{codim}X_1>0$ ).",
"We shall call a parameter value $(x^{(1)},\\dots ,x^{(r)})$ $j$ -exceptional if it lies in $X_j(\\kappa )$ , so that “exceptional\" is the same as “1-exceptional\".",
"Notice that our assumptions on $F_i$ are very general: they need not actually be polynomials, only rational functions, and no nonsingularity conditions or relations among the $F_i$ 's are assumed.",
"This is due to the generality of the argument: it relies on the general formalism of $\\ell $ -adic sheaves and weights as in Weil II [5] but requires no explicit cohomological computations.",
"In particular, square root cancellation is not established in the usual way by showing that the middle cohomology is pure of weight $n$ and that the higher cohomology groups vanish.",
"An explicit value of the constant $C$ has been obtained by Katz [9] and it does not actually depend on $d$ , but we do not know a procedure to explicitly determine $C^{\\prime }$ and $\\mathcal {S}$ .",
"It is not clear whether one should expect that better $\\theta _j$ 's can be obtained for general $F_i$ 's, but there should certainly be room for improvement if the $F_i$ 's are nice.",
"A naïve linear interpolation between $\\theta _n={\\frac{nr}{2}}$ and $\\theta _0=0$ yields $\\theta _j\\approx \\frac{jr}{2}$ , so that $\\lim _{r\\rightarrow \\infty }\\frac{\\theta _j}{r}=\\frac{j}{2}$ ; this may be a natural goal to aim for.",
"In contrast, with our current $\\theta _j$ 's the limit is $\\frac{j}{2(n-1)}$ ; in the case $n=2$ , this suggests that our result is asymptotically optimal for general $F_i$ , though for specific $F_i$ 's the situation may be better: in fact, if the $F_i$ 's are pairwise non-associate irreducible polynomials and some $\\chi _i$ is nontrivial, then $\\operatorname{codim}X_n=nr+1$ , i.e.",
"there is no $n$ -exceptional parameter value at all.",
"If we are able to obtain a bound on $\\operatorname{codim}X_{n-1}$ for the $T_i$ 's (see below) that is better than $\\theta _{n-1}=s-1$ , a better bound on $\\operatorname{codim}X_1$ for $S$ will follow."
],
[
"Outline of the proof",
"There are three key ingredients of the proof.",
"The first is an elementary transformation which allows us to express the moments over the family of character sums $S$ in terms of $r$ other families of character sums $T_i$ , $1\\le i\\le r$ .",
"It is a special case of Lemma REF .",
"Proposition 1.2 For $s\\in \\mathbb {N}$ , let $M_\\kappa (r,s)$ denote the $2s$ -th moment of the character sum $S(x^{(1)},\\dots ,x^{(r)})$ over the parameter space $\\kappa ^{nr}$ .",
"We have $M_\\kappa (r,s):=\\sum _{x^{(1)},\\dots ,x^{(r)}\\in \\kappa ^n}{S(x^{(1)},\\dots ,x^{(r)})}^{2s}= \\sum _{m^{(1)},\\dots ,m^{(2s)}\\in \\kappa ^n} \\prod _{i=1}^r T_i(m^{(1)},\\dots ,m^{(2s)})$ where $ T_i(m^{(1)},\\dots ,m^{(2s)}):=\\sum _{x\\in \\kappa ^n}\\prod _{j=1}^s \\chi _i(F_i(m^{(j)}+x)) \\prod _{j=s+1}^{2s} \\chi _i^{-1}(F_i(m^{(j)}+x)) =\\sum _{x\\in \\kappa ^n}\\chi _i(F_{{\\bf m}}(x))$ where $ F_{{\\bf m}}(x):=\\prod _{j=1}^s F_i(m^{(j)}+x) \\prod _{j=s+1}^{2s} F_i(m^{(j)}+x)^{-1}.$ Normally, $M_\\kappa (r,s)/(\\#\\kappa )^{nr}$ is what is called the moment, but in this paper we call $M_\\kappa (r,s)$ the moment for simplicity (to avoid the phrase “power sum of absolute values\").",
"With this terminology, the moments over a subscheme (such as $X_j$ ) do not exceed the moment $M_\\kappa (r,s)$ over the whole parameter space.",
"Notice that the $T_i$ 's are families of character sums of the same form as $S$ but with $2sn$ parameters, so whatever stratification result we prove for general $S$ (as in Theorem REF ) can also be applied to the $T_i$ 's, with $r$ replaced by $2s$ .",
"Recall that the family of character sums $S$ has a naturally associated family $S_k$ for each finite extension $k/\\kappa $ , given by $ S_k({\\bf x})=S_k(x^{(1)},\\dots ,x^{(r)}):= \\sum _{m\\in k^n} \\prod _{i=1}^r \\chi _i(\\operatorname{N}_{k/\\kappa }(F_i(m+x^{(i)}))) $ for ${\\bf x}=(x^{(1)},\\dots ,x^{(r)})\\in k^{nr}$ .",
"Let $M_k(r,s):=\\sum _{{\\bf x}\\in k^{nr}}{S_k({\\bf x})}^{2s}$ denote the $2s$ -th moment of $S_k$ .",
"If we replace $\\kappa $ by $k$ and $\\chi _i$ by $\\chi _i\\circ \\operatorname{N}_{k/\\kappa }$ in Proposition REF , we get $M_k(r,s):=\\sum _{m^{(1)},\\dots ,m^{(2s)}\\in k^n} \\prod _{i=1}^r T_{i;k}(m^{(1)},\\dots ,m^{(2s)})$ where $ T_{i;k}(m^{(1)},\\dots ,m^{(2s)})&:=\\sum _{x\\in k^n}\\prod _{j=1}^s \\chi _i\\circ \\operatorname{N}_{k/\\kappa }(F_i(m^{(j)}+x)) \\prod _{j=s+1}^{2s} (\\chi _i\\circ \\operatorname{N}_{k/\\kappa })^{-1}(F_i(m^{(j)}+x))\\\\ &=\\sum _{x\\in \\kappa ^n}\\chi _i\\circ \\operatorname{N}_{k/\\kappa }(F_{{\\bf m}}(x)).$ The second ingredient connects the moments $M_k(r,s)$ over finite extensions of $k/\\kappa $ to the dimensions of the $X_j$ 's.",
"Proposition 1.3 Let $C,C^{\\prime },\\mathcal {S}$ be as in Theorem REF and assume that $\\deg F_i\\le D$ , $\\operatorname{ord}\\chi _i\\mid d>0$ and $\\operatorname{char}\\kappa \\notin \\mathcal {S}$ .",
"If $Y$ be a smooth subvariety of $\\mathbb {A}^{nr}_\\kappa $ on which the families of character sums $S_k$ are a virtual lisse trace function (see Remark REF ), then for each integer $j$ , either ${S_k({\\bf x})}\\le C(\\#k)^{(n+j-1)/2}$ for any finite extension $k/\\kappa $ and ${\\bf x}\\in Y(k)$ , or $\\displaystyle \\limsup _{\\#k\\rightarrow \\infty } \\frac{M_k(r,s)}{(\\#k)^{\\dim Y}(\\#k)^{(n+j)s}}\\ge \\limsup _{\\#k\\rightarrow \\infty } \\frac{\\sum _{{\\bf x}\\in Y(k)}{S_k({\\bf x})}^{2s}}{(\\#k)^{\\dim Y}(\\#k)^{(n+j)s}}\\ge 1$ for all $s\\in \\mathbb {N}$ .",
"There exists a decomposition of $\\mathbb {A}^{nr}_{\\kappa }$ into smooth varieties $Y$ such that the sum of their degrees does not exceed $C^{\\prime }$ and the restrictions of $S_k({\\bf x})$ to each $Y$ is a virtual lisse trace function.",
"Therefore, for $0\\le j\\le n$ we may take $X_j$ to be the union of those $Y$ on which the alternative (2) holds, which implies that $\\dim X_j\\le \\max \\lbrace \\dim Y:Y\\text{ satisfies (2)}\\rbrace \\le \\inf _{s\\in \\mathbb {N}} \\limsup _{\\#k\\rightarrow \\infty }(\\log _{\\#k}M_k(r,s)-(n+j)s).$ Upper bounds on $M_k(r,s)$ for all finite extensions $k/\\kappa $ thus yield upper bounds on $\\dim X_j$ (i.e.",
"lower bounds on $\\operatorname{codim}X_j$ ).",
"Proposition REF (a) follows from Theorem REF , and (b) is shown in §REF using Lemma REF and Lemma REF .",
"The above two ingredients together allow the following bootstrapping process: Starting from bounds on the moments (for all $s$ and all $k/\\kappa $ ), Propsosition REF yields a stratification result (a lower bound on $\\operatorname{codim}X_j$ for each $j$ ).",
"If the bounds are proved for general $S$ , we may also apply them to the $T_i$ 's.",
"A stratification result for the $T_i$ 's in turn yield bounds on the moments of $S$ in the following manner, and the process can then be repeated: write $\\mathbb {A}^{nr}=\\bigcup _{j=0}^n X_j\\setminus X_{j+1}$ (with $X_{n+1}=\\varnothing $ ), apply the respective bounds on $T_{i,k}({\\bf x})$ (in place of $S_k({\\bf x})$ ) for ${\\bf x}\\in X_j(k)\\setminus X_{j+1}(k)\\subset \\mathbb {A}^{nr}(k)\\setminus X_{j+1}(k)$ , and notice that $\\#X_j(k)\\le C^{\\prime }(\\#k)^{\\dim X_j}$ (see Lemma REF ).",
"This way we obtain new bounds on the right-hand side of (REF ) and hence on the left-hand side $M_k(r,s)$ .",
"For details about this process, see §REF .",
"Starting from the initial input below, each time we run the process, the bounds on the $\\operatorname{codim}X_j$ 's will be improved, and they tend to certain limits which we call $\\theta _j$ , and these are the best codimension bounds obtainable by iterated improvement (see §REF ).",
"The initial input to the iterative bootstrapping process is supplied by the following proposition, the last ingredient of the proof: Proposition 1.4 In the setting of Theorem REF : The number of parameter values ${\\bf m}=(m^{(1)},\\dots ,m^{(2s)})\\in k^{n\\cdot 2s}$ such that $F_{{\\bf m}}(x):=\\prod _{j=1}^s F_i(m^{(j)}+x) \\prod _{j=s+1}^{2s} F_i(m^{(j)}+x)^{-1}$ is a perfect $d_i$ th power in $\\overline{k}(x)=\\overline{k}(x_1,\\dots ,x_n)$ , is $O((\\#k)^{ns})$ as $k$ varies over finite extensions of $\\kappa $ .",
"(multivariate Weil bound) If $F_{{\\bf m}}\\in k(x_1,\\dots ,x_n)$ is not a perfect $d_i$ th power in $\\overline{k}(x_1,\\dots ,x_n)$ , then $T_{i,k}(m^{(1)},\\dots ,m^{(2s)}):=\\sum _{x\\in k^n}\\chi _i\\circ \\operatorname{N}_{k/\\kappa }(F_{{\\bf m}}(x))=O((\\#k)^{n-1/2})$ as $k$ varies over finite extensions of $\\kappa $ .",
"Proposition REF can be seen to be equivalent to the equality $\\operatorname{codim}X_n=ns$ for the sums $T_i$ .",
"It was the insight of Michael Larsen that, via the elementary transformation, this rather weak input, the weakest nontrivial bound $O((\\#k)^{n-1/2})$ (maximum weight $2n-1$ ), with square root many exceptions ($\\operatorname{codim}X_n=ns$ ), can be bootstrapped to yield the strongest, square root cancellation bound $O((\\#k)^{n/2})$ (maximum weight $n$ ) for generic parameter values ($\\operatorname{codim}X_1>0$ ).",
"This would not work if the exponent in Proposition REF (a)(2) were $(n+j-1)s$ instead of $(n+j)s$ , so the integrality of the weights is crucial, since it is exactly the integrality that allows the contrast between $(n+j-1)s$ in (1) and $(n+j)s$ in (2) of REF (a).",
"Proposition REF (a) follows from Corollary REF , and (b) is proved in Remark REF .",
"Although we are unable to determine explicitly the subschemes of $\\mathbb {A}^{nr}_\\kappa $ of exceptional parameter values, we obtain uniform bounds on the sums of the degrees of their irreducible components, and hence are able to bound the number of exceptional values in any box in $\\kappa ^{nr}$ , thanks to the following lemma.",
"This is crucial for our intended application in analytic number theory, which will appear in joint work with Lillian Pierce.",
"Lemma 1.5 Let $X\\subset \\mathbb {A}^N_\\kappa $ be a subscheme of codimension $\\theta $ and let $d$ be the sum of the degrees of its irreducible components.",
"If $\\lbrace B_i\\rbrace _{i=1}^N$ are subsets of $\\kappa $ , the “box\" $B:=\\prod _{i=1}^N B_i$ is naturally a subset of $\\mathbb {A}^N(\\kappa )$ .",
"If $1\\le \\#B_1\\le \\#B_2\\le \\dots \\le \\#B_n<\\infty $ , we have $\\#*{X(\\kappa )\\cap \\prod _{i=1}^N B_i}\\le d\\prod _{i=\\theta +1}^N \\#B_i=d(\\#B)\\prod _{i=1}^\\theta (\\#B_i)^{-1}.$ For the proof, see Remark REF .",
"The following is an easy corollary of Theorem REF and Lemma REF with $N=nr$ .",
"Corollary 1.6 In the setting of Theorem REF , if $\\lbrace B_i\\rbrace _{i=1}^n$ are subsets of $\\kappa $ such that $1\\le \\#B_1\\le \\#B_2\\le \\dots \\le \\#B_n<\\infty $ , and let $B:=\\prod _{i=1}^n B_i\\subset \\kappa ^n$ , then $\\#\\lbrace (x^{(1)},\\dots ,x^{(r)})\\in B^r:{S(x^{(1)},\\dots ,x^{(r)})}>C(\\#\\kappa )^{(n+j-1)/2}\\rbrace \\quad \\le \\quad C^{\\prime }(\\#B)^r {\\bf b}^{-\\theta _j},$ where ${\\bf b}^{-\\theta }$ denotes $(\\#B_1\\#B_2\\dots \\#B_{n_0})^{-r}(\\#B_{n_0+1})^{-\\eta }$ if we write $\\theta =n_0 r+\\eta $ with $n_0\\in \\mathbb {N}$ and $0\\le \\eta <r$ , so that ${\\bf b}^{-\\theta _j}=(\\#B_1)^{-\\theta _j}$ for $0\\le j\\le n-1$ , and ${\\bf b}^{-\\theta _n}={\\left\\lbrace \\begin{array}{ll}(\\#B_1\\#B_2\\dots \\#B_{n/2})^{-r}&\\text{if }n\\text{ is even,}\\\\(\\#B_1\\#B_2\\dots \\#B_{(n-1)/2})^{-r}(\\#B_{(n+1)/2})^{-{r/2}}&\\text{if }n\\text{ is odd.}\\end{array}\\right.",
"}$ Now suppose instead that $F_i$ is a $d_i$ th-power-free polynomial in $\\mathbb {Z}[x_1,\\dots ,x_n]$ such that $F_i(x+m)\\lnot \\equiv F_i(x)$ for all $m\\in \\mathbb {Z}^n$ , or equivalently (Lemma REF ), $F_i$ cannot be made independent of $x_1$ by a linear change of coordinates, for each $1\\le i\\le r$ .",
"By Lemma REF , the reductions of $F_i$ modulo almost all (all but finitely many) primes remain $d_i$ th-power-free in $\\mathbb {F}_p[x_1,\\dots ,x_n]$ and satisfy $T_{F_i}=\\lbrace 0\\rbrace $ .",
"Therefore, if $\\chi _i:\\mathbb {F}_p^\\times \\rightarrow \\times $ is a multiplicative character of order dividing $d_i$ for each $1\\le i\\le r$ , $\\lbrace B_i\\rbrace _{i=1}^n$ are subsets of $\\mathbb {F}_p$ such that $1\\le \\#B_1\\le \\#B_2\\le \\dots \\le \\#B_n<\\infty $ , and $B:=\\prod _{i=1}^n B_i$ , then by the above corollary, $\\#\\lbrace (x^{(1)},\\dots ,x^{(r)})\\in B^r:{S(x^{(1)},\\dots ,x^{(r)})}>C(\\#\\kappa )^{(n+j-1)/2}\\rbrace \\quad \\le \\quad C^{\\prime }(\\#B)^r {\\bf b}^{-\\theta _j}$ for almost all primes $p$ (the finitely many primes in $\\mathcal {S}$ also needs to be excluded).",
"A similar result holds when $F_i=G_i/H_i\\in \\mathbb {Q}(x_1,\\dots ,x_n)$ is $d_i$ th-power-free with $\\gcd (G_i,H_i)=1$ and $G_i,H_i\\in \\mathbb {Z}[x_1,\\dots ,x_n]$ are not invariant under any translations."
],
[
"Proof of the Main Theorem",
"This section presents a complete proof of Theorem REF following the outline given in §1.",
"It relies on some additional lemmas stated and proved in §3."
],
[
"Construction of the stratification (the $X_j$ 's)",
"Fix $D, d, n, r\\in \\mathbb {N}$ where $D$ will be an upper bound for all $\\deg F_i$ 's and $d>0$ will be a common multiple of all $d_i=\\operatorname{ord}\\chi _i$ .",
"Let $\\mathcal {P}_0$ be the arithmetic scheme that parametrizes all finite fields $\\kappa $ and pairs of polynomials $(G_1,H_1),\\dots ,(G_r,H_r)$ of degrees $\\le D$ with $H_i\\lnot \\equiv 0$ , which is an open subvariety of the affine space over $\\mathbb {Z}$ of relative dimension $2r{D+n\\atopwithdelims ()n}$ .",
"Let $\\zeta _d$ be a primitive $d$ th root of unity and let $R:=R_d=\\mathbb {Z}[1/d,\\zeta _d]$ .",
"Then $\\mathbb {G}_{{\\rm m},R}\\rightarrow \\mathbb {G}_{{\\rm m},R}$ defined by $x\\mapsto x^d$ is a cyclic étale covering of degree $d$ , hence induces a continuous surjective homomorphism $\\pi _1(\\mathbb {G}_{{\\rm m},R})\\rightarrow \\mathbb {Z}/d\\mathbb {Z}$ .",
"If we let $\\ell $ be a prime dividing $d$ and compose this with a homomorphism $\\mathbb {Z}/d\\mathbb {Z}\\rightarrow \\overline{\\mathbb {Q}_\\ell }^\\times $ sending 1 to $\\zeta _d\\in \\overline{\\mathbb {Q}_\\ell }$ , we get a 1-dimensional continuous $\\overline{\\mathbb {Q}_\\ell }$ -representation of $\\pi _1(\\mathbb {G}_{{\\rm m},R})$ , and hence a pure lisse $\\overline{\\mathbb {Q}_\\ell }$ -sheaf of weight 0 and rank 1 on $\\mathbb {G}_{{\\rm m},R}$ , denoted $\\mathcal {L}_d$ .",
"For every ${p}\\in \\operatorname{Spec}R$ , the trace function of $\\mathcal {L}_d|_{\\mathbb {G}_{{\\rm m},{\\rm k}({p})}}$ is a multiplicative character $\\chi _d$ of degree $d$ of the residue field ${\\rm k}({p})$ .",
"If $\\kappa $ is a finite field that admits multiplicative characters $\\chi _1,\\dots ,\\chi _r$ of orders $d_1,\\dots ,d_r$ respectively, and $d_i\\mid d$ for all $i$ , then $\\kappa $ is a finite extension of ${\\rm k}({p})$ for any ${p}\\in \\operatorname{Spec}R$ lying above $(\\text{char }\\kappa )\\in \\operatorname{Spec}\\mathbb {Z}$ , and $\\chi _d\\circ {\\rm N}_{\\kappa /{\\rm k}({p})}$ is a multiplicative character of order $d$ of $\\kappa $ , so $\\chi _1,\\dots ,\\chi _r$ are all powers of $\\chi _d\\circ {\\rm N}_{\\kappa /{\\rm k}({p})}$ .",
"Let $\\mathcal {P}$ be the disjoint union of $\\mathcal {P}_{d,e_1,\\dots ,e_r}$ over all $0\\le e_i<d$ , where $\\mathcal {P}_{d,e_1,\\dots ,e_r}$ is a copy of $\\mathcal {P}_0\\times _{\\operatorname{Spec}\\mathbb {Z}}\\operatorname{Spec}R_d$ for each $e_1,\\dots ,e_r$ .",
"For each $1\\le i\\le r$ , consider the “translate and evaluate\" maps $g_i$ and $h_i$ which are morphisms $\\mathbb {A}^{n+nr}_\\mathcal {P}=\\mathbb {A}^{n+nr}_\\mathbb {Z}\\times _{\\operatorname{Spec}\\mathbb {Z}}\\mathcal {P}\\rightarrow \\mathbb {A}^1_\\mathbb {Z}$ defined by $((m,x^{(1)},\\dots ,x^{(r)}),(G_1,H_1,\\dots ,G_r,H_r))\\mapsto G_i(m+x^{(i)})\\text{ and }H_i(m+x^{(i)})$ respectively on $\\mathbb {A}_{\\mathcal {P}_{d,e_1,\\dots ,e_r}}^{n+nr}\\subset \\mathbb {A}_{\\mathcal {P}}^{n+nr}$ .",
"Consider $\\mathbb {G}_{{\\rm m},\\mathbb {Z}}\\subset \\mathbb {A}^1_\\mathbb {Z}$ and its inverse images under the evaluation maps, and define $U:=\\bigcap _{i=1}^r g^{-1}_i(\\mathbb {G}_{{\\rm m},\\mathbb {Z}})\\cap h^{-1}_i(\\mathbb {G}_{{\\rm m},\\mathbb {Z}})$ , an open dense subscheme of $\\mathbb {A}^{n+nr}_\\mathcal {P}$ .",
"On the connected component $U_{d,e_1,\\dots ,e_r}:=U\\cap \\mathbb {A}^{n+nr}_{\\mathcal {P}_{d,e_1,\\dots ,e_r}}$ of $U$ , the maps $g_i$ and $h_i$ factor through $\\mathbb {G}_{{\\rm m},R_d}$ , and we define a sheaf $\\mathcal {L}$ on $U$ by specifying $ \\mathcal {L}|_{U_{d,e_1,\\dots ,e_r}}:= \\bigotimes _{i=1}^r g^*_i\\mathcal {L}_d^{\\otimes e_i}\\otimes h^*_i\\mathcal {L}_d^{\\otimes -e_i}.$ Then for any finite field $\\kappa $ and multiplicative characters $\\chi _i:\\kappa ^\\times \\rightarrow \\times $ with $\\operatorname{ord}\\chi _i\\mid d$ and rational functions $F_i\\in \\kappa (x_1,\\dots ,x_n)$ of degrees $\\le D$ , if we write $\\chi _i=\\chi _d^{e_i}$ , then there exists a closed point $P=(F_1,\\dots ,F_r)\\in \\mathcal {P}_{d,e_1,\\dots ,e_r}\\subset \\mathcal {P}$ such that the trace function of $\\mathcal {L}$ on the fiber $U\\cap \\mathbb {A}^{n+nr}_P$ at a point $(m,x^{(1)},\\dots ,x^{(r)})$ equals $\\prod _{i=1}^r \\chi _i(F_i(m+x^{(i)}))$ .",
"If we now consider the projection $\\pi :U\\rightarrow \\mathbb {A}^{nr}_\\mathcal {P}$ , then the trace function of the complex $\\mathcal {K}:=R\\pi _!\\mathcal {L}$ on $\\mathbb {A}^{nr}_P$ gives rise to the family of character sums that we are interested in: $ \\operatorname{Tr}( {\\rm Frob}_k \\mid \\mathcal {K}_{{\\bf x}} ) = S_k(x^{(1)},\\dots ,x^{(r)}) = \\sum _{m\\in k^n} \\prod _{i=1}^r \\chi _i({\\rm N}_{k/\\kappa }(F_i(m+x^{(i)}))) $ for any finite extension $k/\\kappa $ and ${\\bf x}=(x^{(1)},\\dots ,x^{(r)})\\in \\mathbb {A}^{nr}_P(k)\\cong k^{nr}$ .",
"The trace function of $\\mathcal {K}$ is the same as that of the alternating sum of its cohomology sheaves $R^j\\pi _!\\mathcal {L}$ , which are constructible mixed sheaves of integer weights (possibly away from finitely many primes), since $\\mathcal {L}$ is mixed of integer weights and constructible (in fact pure of weight 0 and lisse); see [11], [5], and [4].",
"Mixed sheaves are iterated extensions of pure sheaves, and the trace function of the mixed sheaf is simply the sum of the trace functions of its pure factors.",
"There exists a decomposition of $\\mathbb {A}^{nr}_\\mathcal {P}$ into finitely many (locally closed) subschemes: $\\mathbb {A}^{nr}_\\mathcal {P}=\\bigcup _{X\\in \\mathcal {X}}X$ , such that the restrictions of these constructible pure factors to each $X\\in \\mathcal {X}$ are lisse, so that $S_k({\\bf x})$ is a virtual lisse trace function on each $X$ (see Remark REF ).",
"Moreover, using Lemma REF , we may assume that $\\pi _\\mathcal {P}|_X:X\\rightarrow \\overline{\\pi _\\mathcal {P}(X)}$ , where $\\pi _\\mathcal {P}:\\mathbb {A}^{nr}_\\mathcal {P}\\rightarrow \\mathcal {P}$ is the structural morphism, is smooth for each $X$ if we work away from finitely many primes, so that every fiber of $\\pi _0|_X$ is smooth over the residue field (a finite field).",
"We may also assume that each $X\\in \\mathcal {X}$ is connected.",
"By Lemma REF applied to $\\overline{X}$ , the closure of $X$ in $\\mathbb {P}^{nr}_{\\overline{\\pi _\\mathcal {P}(X)}}$ , the geometric fibers of $\\pi _\\mathcal {P}|_X$ are equidimensional of degree no more than $\\deg X:=\\deg \\overline{X}$ .",
"We then define $C^{\\prime }:=\\sum _{X\\in \\mathcal {X}} \\deg X$ .",
"Once we obtain the uniform stratification, we now work one fiber at a time, i.e.",
"we restrict to a closed point $P\\in \\mathcal {P}$ parametrizing a particular choice of $F_1,\\dots ,F_r,\\chi _1,\\dots ,\\chi _r$ such that $F_i\\in \\kappa (x_1,\\dots ,x_n)$ is $d_i$ th-power-free, where $\\kappa :={\\rm k}(P)$ .",
"For every $X\\in \\mathcal {X}$ , every connected component $Y$ of the fiber $X_P$ of $X$ over $P$ is a smooth variety over $\\kappa $ , and $S_k({\\bf x})$ , the trace function of $\\mathcal {K}$ on $Y$ , is a virtual lisse trace function (see Theorem REF ).",
"Let $X_j$ be the union of all $Y$ on which $S_k({\\bf x})$ satisfies the alternative (2) in Proposition REF (a) (i.e.",
"has maximum weight $\\ge n+j$ ).",
"Then on the other $Y$ , the $S_k({\\bf x})$ satisfies the alternative (1) (i.e.",
"has maximum weight $\\le n+j-1$ ), and the union of these $Y$ contains $\\mathbb {A}^{nr}_P\\setminus X_j$ , so ${S(x^{(1)},\\dots ,x^{(r)})}\\le C(\\#k)^{(n+j-1)/2}\\text{\\qquad for all }(x^{(1)},\\dots ,x^{(r)})\\in \\mathbb {A}^{nr}(\\kappa )\\setminus X_j(\\kappa ),$ where $C$ is the sum of the ranks of the lisse sheaves (which is bounded by the sum of the maximal ranks of the cohomology sheaves $R^j\\pi _!\\mathcal {L}$ , which is bounded by Katz's constant).",
"It is clear the sum of the degrees of the irreducible components of $X_j$ does not exceed $C^{\\prime }$ .",
"We have thus proved Proposition REF (b)."
],
[
"The bootstrapping process",
"The setting of the bootstrapping process is as follows.",
"We have a family $S$ of character sums, and for each $s\\in \\mathbb {N}$ and each $1\\le i\\le r$ we have the family $T_i$ of character sums obtained from the elementary transformation (REF ).",
"For the family $S$ , we consider the filtration $\\mathbb {A}^{nr}=X_0\\supset X_1\\supset X_2\\supset \\dots \\supset X_n$ , where $X_j$ , $1\\le j\\le n$ , is the union of smooth varieties on which the maximum weight of $S$ is at least $n+j$ .",
"The stratification associated to the filtration consists of the $X_j\\setminus X_{j+1}$ (on which $S_k({\\bf x})$ has maximum weight exactly $n+j$ ).",
"Similarly, let $\\mathbb {A}^{n\\cdot 2s}\\supset Y_1\\supset Y_2\\supset \\dots \\supset Y_n$ be the combined stratification of the $T_i$ 's, so that $Y_j$ , $1\\le j\\le n$ , is the union of smooth varieties on which the maximum weight of some $T_i$ is at least $n+j$ .",
"Define $c(r,j)&:=\\operatorname{codim}X_j=nr-\\dim X_j,\\\\\\text{and\\qquad } c^{\\prime }(2s,j)&:=\\operatorname{codim}Y_j=n\\cdot 2s-\\dim Y_j.$ Denote the $2s$ -th moment of $S_k$ by $M_k(r,s)$ , and define $m(r,s):=\\limsup _{\\#k\\rightarrow \\infty }\\,(\\log _{\\#k} M_k(r,s)-nr-ns).$ The bootstrapping process relies on following three inequalities: Lemma 2.1 $ c(r,j) \\ge \\max _{s\\in \\mathbb {N}}\\, (js-{m(r,s)})$ ; $ m(r,s) \\le \\max _{0\\le j\\le n}\\, ( js-c(r,j) )$ ; $ m(r,s) \\le ns-nr/2+\\max _{0\\le j\\le n}\\, (jr/2-c^{\\prime }(2s,j)) $ .",
"Remark 2.2 It can be shown using Theorem REF that we actually have equality in (2).",
"Therefore, $m(r,-)$ can be seen as a “discrete Legendre transform\" of $c(r,-)$ , so $m(r,s)$ is a convex function of $s$ .",
"(We do not know whether $c(r,j)=\\operatorname{codim}X_j$ is a convex function of $j$ , but the bounds we get from inequality (1) will always be convex.)",
"Applying (1) and then (2) (or vice versa) is an (idempotent) closure operator coming from a Galois connection specified by the right-hand sides of both inequalities.",
"Inequality (3) is a version of (2) with the role of $r$ and $2s$ switched (together with $X_j$ and $Y_j$ ).",
"Since the $T_i$ 's are also of the form of $S$ , any universal bound on $c(r,j)=\\operatorname{codim}X_j$ , in the sense that it holds for all sums of the form $S$ in Theorem REF (for fixed $n$ ), also applies to $c^{\\prime }(2s,j)=\\operatorname{codim}Y_j$ if we simply replace $r$ by $2s$ .",
"Thus we can apply (1) and (3) alternately and repeatedly, which is what we refer to as bootstrapping and what we do in the next subsection.",
"The crucial point is that (3) has the power of breaking convexity and idempotency, because $r$ and $2s$ are switched: the bounds on $m(r,s)$ that we get from (1), which are convex in $r$ , are usually not convex in $s$ , and exactly this gives room for improvement.",
"In fact the iterated improvement process goes on forever; see Lemma REF .",
"The limit bound for $c(r,j)=\\operatorname{codim}X_j$ will turn out to be $\\theta _j=\\theta _j(n,r)$ .",
"In reality, we do not actually compute the intermediate bounds we get during the iterative bootstrapping process, but instead use (1) and (3) repeatedly to first obtain the limiting bound on $c(r,n-1)$ , and then show that the bounds on all $c(r,j)$ we get after bootstrapping one more time is the best we can get.",
"For details, see §REF .",
"(1) Since $X_j$ is the union of smooth varieties on which the alternative (2) in Proposition REF (a) holds, and since $\\dim X_j$ is the maximum of the dimensions of these smooth varieties, we have $ \\limsup _{\\#k\\rightarrow \\infty } \\frac{M_k(r,s)}{(\\#k)^{\\dim X}(\\#k)^{(n+j)s}} \\ge \\limsup _{\\#k\\rightarrow \\infty } \\frac{\\sum _{x\\in X_j(k)}{S_k(x)}^{2s}}{(\\#k)^{\\dim X}(\\#k)^{(n+j)s}} \\ge 1>0.",
"$ Taking the logarithm, we see that $ \\limsup _{\\#k\\rightarrow \\infty } \\,(\\log \\#k) *{ \\frac{\\log M_k(r,s)}{\\log \\#k} - (\\dim X_j+(n+j)s)} >-\\infty .",
"$ Since $\\log \\#k\\rightarrow \\infty $ as $\\#k\\rightarrow \\infty $ , we must have $\\limsup _{\\#k\\rightarrow \\infty } *{ \\log _{\\#k}\\,M_k(r,s) - (\\dim X_j+(n+j)s)} \\ge 0 $ and hence $ m(r,s)&=\\limsup _{\\#k\\rightarrow \\infty }\\,(\\log _{\\#k} M_k(r,s)-nr-ns) \\\\&\\ge \\dim X_j+(n+j)s -nr-ns \\\\ & = js - \\operatorname{codim}X_j.",
"$ Thus $c(r,j)=\\operatorname{codim}X_j\\ge js-m(r,s)$ after rearranging, so $c(r,j)\\ge js-{m(r,s)}$ because $c(r,j)$ is an integer.",
"(2) Consider the decomposition $\\mathbb {A}^{nr}=\\bigcup _{0\\le j\\le n} X_j\\setminus X_{j+1}$ , with $X_{n+1}=\\varnothing $ , and recall that $S_k(x)=O((\\#k)^{(n+j)/2})$ as $k$ varies for $x\\in X_j(k)\\setminus X_{j+1}(k)\\subset \\mathbb {A}^{nr}(k)\\setminus X_{j+1}(k)$ .",
"Moreover, $\\#X_j(k)=O((\\#k)^{\\dim X_j})$ as $k$ varies.",
"Therefore $M_k(r,s) & = \\sum _{x\\in \\mathbb {A}^{nr}(k)} {S_k(x)}^{2s}= \\sum _{j=0}^n \\sum _{x\\in X_j(k)\\setminus X_{j+1}(k)} {S_k(x)}^{2s} \\\\& = \\sum _{j=0}^n O((\\#k)^{\\dim X_j})O((\\#k)^{(n+j)s}),$ so $m(r,s)\\le \\max _{0\\le j\\le n}\\,(\\dim X_j+(n+j)s-nr-ns)=\\max _{0\\le j\\le n}\\,(js-\\operatorname{codim}X_j) .$ (3) Consider the decomposition $\\mathbb {A}^{n\\cdot 2s}=\\bigcup _{0\\le j\\le n}Y_j\\setminus Y_{j+1}$ , with $Y_{n+1}=\\varnothing $ , then $T_{i;k}(m)=O((\\#k)^{(n+j)/2})$ as $k$ varies for all $1\\le i\\le r$ and $m\\in Y_j(k)\\setminus Y_{j+1}(k)=\\mathbb {A}^{n\\cdot 2s}(k)\\setminus Y_{j+1}(k)$ .",
"Moreover, $\\#Y_j(k)=O((\\#k)^{\\dim Y_j})$ as $k$ varies.",
"Therefore $M_k(r,s)&=\\sum _{m\\in \\mathbb {A}^{n\\cdot 2s}(k)}\\prod _{i=1}^r T_{i;k}(m) = \\sum _{j=0}^n \\sum _{m\\in Y_j(k)\\setminus Y_{j+1}(k)} \\prod _{i=1}^r T_{i;k}(m)\\\\&= \\sum _{j=0}^n O((\\#k)^{\\dim Y_j}) O((\\#k)^{(n+j)r/2}),$ which yields $m(r,s) &\\le \\max _{0\\le j\\le n}\\,(\\dim Y_j+(n+j)r/2-nr-ns) \\\\& = \\max _{0\\le j\\le n}\\, ( ns-(n-j)r/2-c^{\\prime }(2s,j)) \\\\& = ns-nr/2+\\max _{0\\le j\\le n}\\, (jr/2-c^{\\prime }(2s,j)).$"
],
[
"The initial bound and iterated improvement",
"In this section we aim to obtain initial bounds for the moments $M_k(r,s)$ to start the bootstrapping process.",
"Recall from (REF ) $M_k(r,s):=\\sum _{m^{(1)},\\dots ,m^{(2s)}\\in k^n} \\prod _{i=1}^r T_{i;k}(m^{(1)},\\dots ,m^{(2s)})$ and from Proposition REF the Weil bound $T_{i;k}(m^{(1)},\\dots ,m^{(2s)})=O((\\#k)^{n-1/2})$ for all but $O((\\#k)^{ns})$ parameter values $(m^{(1)},\\dots ,m^{(2s)})$ .",
"We apply the trivial bound $(\\#k)^n$ to these $O((\\#k)^{ns})$ parameter values, which yields $M_k(r,s) & = ((\\#k)^n)^r O((\\#k)^{ns}) + O((\\#k)^{n-1/2})^r ((\\#k)^n)^{2s}\\\\& ={\\left\\lbrace \\begin{array}{ll} O((\\#k)^{ns+nr}) & \\text{if }s\\le r/2n, \\\\O((\\#k)^{n\\cdot 2s+(n-1/2)r}) & \\text{if }s\\ge r/2n.\\end{array}\\right.",
"}$ and therefore $ m(r,s) \\le {\\left\\lbrace \\begin{array}{ll} 0 &\\text{if }s\\le r/2n, \\\\ns-r/2 &\\text{if } s\\ge r/2n.\\end{array}\\right.",
"}$ Taking $s={r/2n}$ in inequality (1) in Lemma REF , we have $c(r,j)\\ge \\max _{s\\in \\mathbb {N}}\\,(js-{m(r,s)})\\ge j{r/2n},$ so $c^{\\prime }(2s,j)\\ge j{s/n}$ .",
"Now take $s\\ge n{r/2}$ , so that $\\max _{0\\le j\\le n}\\,(jr/2-c^{\\prime }(2s,j))$ is achieved at $j=0$ , and hence $m(r,s)\\le ns-nr/2$ by inequality (3).",
"By inequality (1), we then obtain $c(r,j)\\ge {js-(ns-nr/2)}$ .",
"For $j<n$ , this bound is trivial as $-s+nr/2\\le 0$ , but when $j=n$ we do get a nontrivial bound $c(r,n)\\ge {nr/2}$ , so $\\theta _n={nr/2}$ is indeed a lower bound for $\\operatorname{codim}X_n$ , and we have $c^{\\prime }(2s,n)\\ge {n\\cdot 2s/2}=ns$ .",
"We first aim to iteratively improve the bound on $c(r,n-1)$ .",
"This relies on the following lemma: Lemma 2.3 For any function $\\theta $ of the variable $r\\in \\mathbb {N}$ , let $\\theta ^+$ be the function of $r$ defined by $\\theta ^+(r)=\\max _{s\\in \\mathbb {N}}\\,\\min \\lbrace \\,(n-1)s,\\,{r/2}-s+\\theta (2s),\\,-s+r\\,\\rbrace .$ If $\\theta (r)$ is a universal lower bound for $c(r,n-1)$ for all $r$ , then $\\theta ^+(r)$ is also.",
"Suppose that we have a universal bound $c(r,n-1)\\ge \\theta (r)$ , then $c^{\\prime }(2s,n-1)\\ge \\theta (2s)$ .",
"Therefore, by inequality (3) in Lemma REF , $m(r,s) & \\le ns-nr/2+\\max \\lbrace \\,nr/2-ns,\\,(n-1)r/2-\\theta (2s),\\, (n-2)r/2,\\, \\dots ,\\, 0r/2\\,\\rbrace \\\\& = \\max \\lbrace 0,\\, -r/2+ns-\\theta (2s),\\, ns-r\\rbrace $ where we used the bounds $c^{\\prime }(2s,n)\\ge ns$ and the trivial bounds $c^{\\prime }(2s,j)\\ge 0$ for $j<n-1$ .",
"By inequality (1), $ c(r,n-1) & \\ge \\max _{s\\in \\mathbb {N}} \\, ((n-1)s-{m(r,s)}) \\\\& \\ge \\max _{s\\in \\mathbb {N}}\\min \\lbrace \\,(n-1)s, \\,{r/2}-s+\\theta (2s),\\, -s+r\\,\\rbrace =\\theta ^+(r),$ so $\\theta ^+(r)$ is also a universal lower bound for $c(r,n-1)$ .",
"Lemma 2.4 Let $\\theta ^{(0)}(r):=0$ for all $r$ , and define $\\theta ^{(i)}$ inductively by $\\theta ^{(i+1)}=(\\theta ^{(i)})^+$ , so that $\\theta ^{(i+1)}(r)=\\max _{s\\in \\mathbb {N}}\\,\\min \\lbrace \\,(n-1)s,\\,{r/2}-s+\\theta ^{(i)}(2s),\\,-s+r\\,\\rbrace .$ Then $\\theta ^{(i)}(r)\\nearrow \\theta ^{(\\infty )}(r):= {(r-1)/2}={r/2}-1$ as $i\\rightarrow \\infty $ , for any $r\\ge 1$ and $n\\ge 2$ .",
"We prove that $\\theta ^{(i)}(r)\\le {r/2}-1$ inductively.",
"Consider the second term ${r/2}-s+\\theta ^{(i)}(2s)$ in the definition of $\\theta ^{(i+1)}(r)$ .",
"By induction hypothesis, $\\theta ^{(i)}(2s)\\le {2s/2}-1=s-1$ and hence ${r/2}-s+\\theta ^{(i)}(2s)\\le {r/2}-1$ for all $s\\in \\mathbb {N}$ , thus $\\theta ^{(i+1)}(r)\\le {r/2}-1$ .",
"If $\\theta ^{(i)}(r)\\ge \\theta ^{(j)}(r)$ for all $r\\in \\mathbb {N}$ , then $\\theta ^{(i)}(2s)\\ge \\theta ^{(j)}(2s)$ for all $s\\in \\mathbb {N}$ , so from the definition of $\\theta ^{(i+1)}$ it is clear that $\\theta ^{(i+1)}(r)\\ge \\theta ^{(j+1)}(r)$ .",
"Since clearly $\\theta ^{(1)}(r)\\ge \\theta ^{(0)}(r)$ for all $r\\in \\mathbb {N}$ , we see that $\\theta ^{(i+1)}(r)\\ge \\theta ^{(i)}(r)$ for all $i$ by induction.",
"It remains to show that $\\lim _{i\\rightarrow \\infty }\\theta ^{(i)}(r)\\ge {r/2}-1$ .",
"It suffices to deal with the case $n=2$ , since the $\\theta ^{(i)}(r)$ for $n>2$ is no smaller than the $\\theta ^{(i)}(r)$ for $n=2$ , as is clear from the inductive definition.",
"When $n=2$ , we shall show that $\\theta ^{(i)}(r)\\ge *{\\frac{r}{2}*{1-\\frac{1}{i+1}}}$ by induction.",
"(In fact equality holds if $r$ is even.)",
"This inequality clearly holds for $i=0$ .",
"Assuming that it holds for $\\theta ^{(i)}$ , then $\\theta ^{(i+1)}(r)\\ge \\max _{s\\in \\mathbb {N}}\\min \\lbrace \\,s,\\,*{\\frac{r}{2}}-*{\\frac{s}{i+1}},\\,-s+r\\,\\rbrace .$ If we plot $s$ , ${\\frac{r}{2}}-\\frac{s}{i+1}$ and $-s+r$ as functions of $s$ , it is clear that we should look at the intersection of the first two lines, which corresponds to $s=s_0:={\\frac{r}{2}}(1-\\frac{1}{i+2})$ , or rather $s={s_0}$ .",
"Since $s_0= {\\frac{r}{2}}-\\frac{s_0}{i+1}$ , we have ${s_0}=*{*{\\frac{r}{2}}-\\frac{s_0}{i+1}}=*{\\frac{r}{2}}-*{\\frac{s_0}{i+1}}\\le *{\\frac{r}{2}}-*{\\frac{{s_0}}{i+1}}.$ Since $s_0<{\\frac{r}{2}}$ , we have ${s_0}<\\frac{r}{2}$ , so ${s_0}< -{s_0}+r$ .",
"Therefore at $s={s_0}$ , the minimum of three terms is ${s_0}=*{{\\frac{r}{2}}(1-\\frac{1}{i+2})}$ , which is no less than ${\\frac{r}{2}(1-\\frac{1}{i+2})}$ , so $\\theta ^{(i+1)}(r)\\ge {\\frac{r}{2}(1-\\frac{1}{i+2})}$ .",
"Now $\\lim _{i\\rightarrow \\infty }\\theta ^{(i)}(r)\\ge \\theta ^{(r-1)}(r)\\ge *{\\frac{r}{2}(1-\\frac{1}{r})}=*{\\frac{r-1}{2}}.$ The function $\\theta ^{(i+1)}$ is obtained from $\\theta ^{(i)}$ by applying a functional $(\\cdot )^+$ , and $\\theta ^{(\\infty )}$ is a fixed point of this functional.",
"This functional is monotonic, and this lemma shows that $\\theta ^{(\\infty )}$ is the limiting function obtained from applying the functional repeatedly.",
"It is interesting to note that $n$ does not affect the limiting value (though for $n\\ge 3$ the convergence becomes exponential), and that we are unable to improve from ${r/2}-1$ to ${r/2}$ .",
"Since all $\\theta ^{(i)}(r)$ are universal lower bounds for $c(r,n-1)$ , $\\sup _{i\\in \\mathbb {N}}\\theta ^{(i)}(r)={r/2}-1$ is also a universal lower bound for $c(r,n-1)$ .",
"We now use $c(r,n)\\ge {nr/2}$ and $c(r,n-1)\\ge {(r-1)/2}$ to get bounds for all the other $c(r,j)$ ($1\\le j\\le n-2$ ).",
"With this improved bound for $c(r,n-1)$ , the bound for $m(r,s)$ in the proof of Lemma REF becomes $m(r,s) \\le \\max \\lbrace \\,0,(n-1)s-r/2+1,\\,ns-r\\,\\rbrace , $ hence by inequality (1) $ c(r,j) \\ge \\max _{s\\in \\mathbb {N}}\\min \\lbrace \\, js,\\, (j-n+1)s+{r/2}-1,\\, (j-n)s+r\\,\\rbrace .",
"$ Again, we look at $s=s_0:=({\\frac{r}{2}}-1)/(n-1)$ where the first two terms are equal.",
"Clearly, the maximum $\\max _{s\\in \\mathbb {N}}\\min \\lbrace \\, js,\\, (j-n+1)s+{r/2}-1\\,\\rbrace $ is achieved at $s={s_0}$ or $s={s_0}$ if $j<n$ , and hence it is equal to $\\max \\lbrace \\,j{s_0},\\,-(n-j-1){s_0}+{\\frac{r}{2}}-1\\,\\rbrace $ .",
"The third term $(j-n)s+r$ is greater than the first two terms both at ${s_0}$ and at ${s_0}$ , so it does not play a role: indeed, $(j-n){s_0}+r>(j-n+1){s_0}+{\\frac{r}{2}}-1$ because ${s_0}\\le {\\frac{r}{2}}-1<{\\frac{r}{2}}+1=r-({\\frac{r}{2}}+1)$ , so $(j-n){s_0}+r>(j-n+1){s_0}+{\\frac{r}{2}}-1\\ge j{s_0}$ .",
"Writing ${\\frac{r-1}{2}}={\\frac{r}{2}}-1=(n-1)a+b$ with $a\\in \\mathbb {N}$ and $0\\le b<n-1$ , it is then easy to work out $\\theta _j=\\theta _j(n,r)& :=\\max \\lbrace \\,j{s_0},\\,-(n-j-1){s_0}+*{\\frac{r-1}{2}}\\,\\rbrace \\\\& = ja + \\max \\lbrace 0, b+j-(n-1) \\rbrace $ for all $0\\le j\\le n-1$ .",
"Combined with the bound $c(r,n)\\ge \\theta _n:={nr/2}$ which we proved before, this is exactly what is claimed in Theorem REF .",
"If we just apply the bootstrapping process once, we actually already get bounds $\\vartheta _j$ such that $\\lim _{r\\rightarrow \\infty }\\frac{\\vartheta _j}{r}=\\frac{j}{n}$ ; with all this complicated iterated improvement business, we only improve this limit to $\\frac{j}{n-1}$ , and the improvement becomes less and less significant as $n$ increases.",
"However, we really cannot do better than our $\\theta _j$ 's using the bootstrapping method alone: even if we use $c(r,n)=\\theta _n={nr/2}$ and the better bounds $c(r,j)\\ge {r/2}-1\\ge \\theta _j$ for all $1\\le j\\le n-1$ as input, the only effect is to improve (REF ) to $m(r,s)\\le \\max \\lbrace 0,\\,(n-1)s-r/2+1,\\,ns-nr/2\\rbrace ,$ i.e.",
"to replace the third term $ns-r$ by the smaller $ns-nr/2$ .",
"But for $j<n$ , the arguments above has shown that we get the same result even without the third term, so $\\theta _j$ is not improved using this bound for $m(r,s)$ .",
"For $j=n$ , we still get $c(r,n)\\ge \\max _{s\\in \\mathbb {N}}\\min \\lbrace ns,\\, s+{r/2}-1,\\, {nr/2}\\rbrace = {nr/2}=\\theta _n$ ."
],
[
"An elementary transformation",
"Burgess [1] used a transformation to express moments over a complete family of incomplete character sums in terms of an incomplete family of complete character sums.",
"It has since been used as a routine to obtain Burgess type bounds.",
"A simpler form of the transformation appeared already in [2].",
"We generalize this transformation to the situation where the summand is a product of $r$ factors; in our setting, it is used to express the moments of a complete family of complete character sums in terms of $r$ other complete families of complete character sums.",
"Lemma 3.1 Let $R$ be a commutative ring and let $\\sigma _1,\\dots ,\\sigma _s$ be automorphisms of $R$ .",
"Let $M$ and $X$ be sets, and let $f_1,\\dots ,f_r: M\\times X\\rightarrow R$ be functions.",
"Let $S: X^r\\rightarrow R$ be the function defined by $ S(x^{(1)},\\dots ,x^{(r)}):=\\sum _{m\\in M}\\prod _{i=1}^r f_i(m,x^{(i)}).",
"$ Then $ \\sum _{x^{(1)},\\dots ,x^{(r)}\\in X} \\prod _{j=1}^s S(x^{(1)},\\dots ,x^{(r)})^{\\sigma _j}= \\sum _{m^{(1)},\\dots ,m^{(s)}\\in M} \\prod _{i=1}^r T_i(m^{(1)},\\dots ,m^{(s)}), $ where $ T_i(m^{(1)},\\dots ,m^{(s)}):=\\sum _{x\\in X}\\prod _{j=1}^s f_i(m^{(j)},x)^{\\sigma _j}.",
"$ Remark 3.2 In the case $X=M=\\kappa ^n$ , $R=, if we replace $ s$ by $ 2s$ in this lemma, define $ j$ to be the trivial automorphism for $ 1js$ and complex conjugation for $ s+1j2s$, and let $ fi(m,x)=i(Fi(m+x))$, we get Proposition \\ref {ing elem}.",
"If moreover $ S$ is of the more specific form of $ Ti$, this is an equality between the $ 2s$th moment of the $ 2r$-parameter sum and the $ 2r$th moment of the $ 2s$-parameter sum.$ $& \\sum _{x^{(1)},\\dots ,x^{(r)}\\in X} \\prod _{j=1}^s S(m,x^{(1)},\\dots ,x^{(r)})^{\\sigma _j} \\\\=\\, & \\sum _{x^{(1)},\\dots ,x^{(r)}\\in X} \\prod _{j=1}^s \\sum _{m\\in M} \\prod _{i=1}^r f_i(m,x^{(i)})^{\\sigma _j} \\\\=\\, & \\sum _{x^{(1)},\\dots ,x^{(r)}\\in X} \\sum _{m^{(1)},\\dots ,m^{(s)}\\in M} \\prod _{j=1}^s \\prod _{i=1}^r f_i(m^{(j)},x^{(i)})^{\\sigma _j} &&\\text{(distributive law)}\\\\=\\, & \\sum _{m^{(1)},\\dots ,m^{(s)}\\in M} \\sum _{x^{(1)},\\dots ,x^{(r)}\\in X} \\prod _{i=1}^r \\prod _{j=1}^s f_i(m^{(j)},x^{(i)})^{\\sigma _j} \\\\=\\, & \\sum _{m^{(1)},\\dots ,m^{(s)}\\in M} \\prod _{i=1}^r \\sum _{x\\in X} \\prod _{j=1}^s f_i(m^{(j)},x)^{\\sigma _j} &&\\text{(distributive law)}\\\\=\\, & \\sum _{m^{(1)},\\dots ,m^{(s)}\\in M} \\prod _{i=1}^r T_i(m^{(1)},\\dots ,m^{(s)}).$"
],
[
"Geometric connected components",
"In this section we review some facts about geometric connectedness, in preparation for the proof of Theorem REF .",
"If $\\kappa $ is a field, let $\\kappa ^s$ denote its separable algebraic closure and $\\overline{\\kappa }$ its algebraic closure.",
"Let $X$ be a connected scheme of finite type over $\\kappa $ .",
"For any extension $k/\\kappa $ , let $X_k$ denote $X\\times _\\kappa k$ .",
"For any extension $k^{\\prime }/k$ , $X_{k^{\\prime }}\\rightarrow X_k$ induces a surjection $\\pi _0(X_{k^{\\prime }})\\rightarrow \\pi _0(X_k)$ on the sets of connected components.",
"$G:=\\operatorname{Gal}(\\kappa ^s/\\kappa )$ acts on $\\pi _0(X_{\\kappa ^s})$ , which is identified with $\\pi _0(X_{\\overline{\\kappa }})$ via the bijection $\\pi _0(X_{\\overline{\\kappa }})\\rightarrow \\pi _0(X_{\\kappa ^s})$ [15].",
"Let $G_0\\trianglelefteq G$ denote the kernel of the action, and let $k_0$ denote the subfield of $\\kappa ^s$ fixed by $G_0$ .",
"Since $X$ is of finite type over $\\kappa $ , $X_{\\kappa ^s}$ is noetherian, so $\\pi _0(X_{\\kappa ^s})$ is finite.",
"Therefore, the connected components are clopen, $G_0$ is a subgroup of finite index of $G$ , and $k_0/\\kappa $ is a finite extension.",
"We call $k_0$ the splitting field of $X/\\kappa $ , since it is the smallest extension of $\\kappa $ that “splits\" the geometric connected components of $X/\\kappa $ completely.",
"If $f\\in \\kappa [x]$ is an irreducible polynomial and $X=\\operatorname{Spec}\\kappa [x]/(f)$ then $k_0$ is the splitting field of $f$ .",
"The action of $G$ on $\\pi _0(X_{\\kappa ^s})$ is transitive: by [15], the union of each orbit is the inverse image of a closed subset of $X$ under $X_{\\kappa ^s}\\rightarrow X$ .",
"A partition of $\\pi _0(X_{\\kappa ^s})$ into orbits then yields a partition of $X$ into finitely many nonempty disjoint closed subsets.",
"Since $X$ is connected, there can only be one orbit.",
"Lemma 3.3 Let $k$ be an intermediate field of $\\kappa ^s/\\kappa $ .",
"The following are equivalent: every connected component of $X_k$ is geometrically connected; $\\pi _0(X_{\\kappa ^s})\\rightarrow \\pi _0(X_k)$ is injective (hence bijective); $\\operatorname{Gal}(\\kappa ^s/k)$ acts trivially on $\\pi _0(X_{\\kappa ^s})$ ; $\\operatorname{Gal}(\\kappa ^s/k)\\le G_0$ ; $k_0\\subset k$ .",
"If $k/\\kappa $ is Galois, they are also equivalent to some connected component of $X_k$ is geometrically connected; some fiber of $\\pi _0(X_{\\kappa ^s})\\rightarrow \\pi _0(X_k)$ is a singleton; the action of $\\operatorname{Gal}(\\kappa ^s/k)$ on $\\pi _0(X_{\\kappa ^s})$ has a fixed point.",
"Remark 3.4 Since (5)$\\Rightarrow $ (1), that every connected component of $X_{k_0}$ is geometrically connected.",
"Now suppose that $\\kappa $ is a finite field, so any algebraic extension of $\\kappa $ is Galois.",
"If $k_0\\lnot \\subset k$ , no connected component of $X_k$ is geometrically connected, since (6)$\\Rightarrow $ (5); by [15], $X_k$ has no rational points.",
"(1)$\\iff $ (2) and (6)$\\iff $ (7): if $Y\\in \\pi _0(X_k)$ , the inverse image of $Y$ in $\\pi _0(X_{\\kappa ^s})$ consists of the connected components of $Y_{\\kappa ^s}$ , so it is a singleton iff $Y$ is geometrically connected.",
"(2)$\\iff $ (3) and (7)$\\iff $ (8) follow from [15].",
"(3)$\\iff $ (4) by definition of $G_0$ .",
"(4)$\\iff $ (5) by Galois theory.",
"(3)$\\Rightarrow $ (8) is trivial.",
"Now assume that $k/\\kappa $ is Galois, so $H:=\\operatorname{Gal}(\\kappa ^s/k)$ is normal in $G$ .",
"(8)$\\Rightarrow $ (3): if (8) holds, the stabilizer of some element $Y\\in \\pi _0(X_{\\kappa ^s})$ in $G$ contains $H$ .",
"Since the stabilizers of elements in the same $G$ -orbit are conjugate in $G$ , and since $H\\trianglelefteq G$ , we see that $H$ fixes the $G$ -orbit of $Y$ .",
"Since $G$ acts transitively, $H$ fixes $\\pi _0(X_{\\kappa ^s})$ , i.e.",
"(3) holds."
],
[
"Moments of virtual lisse trace functions",
"For an $\\overline{\\mathbb {Q}_\\ell }$ -sheaf $\\mathcal {F}$ on a scheme $X$ over a finite field $\\kappa $ and any finite extension $k/\\kappa $ , let $f_k:X(k)\\rightarrow of $ F$ be defined by$$f_k(x):=\\iota (\\operatorname{Tr}({\\rm Frob}_k \\mid \\mathcal {F}_{\\overline{x}}))$$where $$ is a fixed isomorphism from $Q$ to $ , and $\\overline{x}$ is a geometric point over $x\\in X(k)$ .",
"We call the collection $\\lbrace f_k\\rbrace _{k/\\kappa \\text{ finite}}$ 's for all finite extensions $k/\\kappa $ the trace function of $\\mathcal {F}$ , thought of as a function in variables $k$ and $x$ .",
"All $\\overline{\\mathbb {Q}_\\ell }$ -sheaves appearing in this paper will be pure or mixed with integer weights with respect to any isomorphism $\\overline{\\mathbb {Q}_\\ell }\\rightarrow , but all the arguments go through if we just fix one isomorphism.",
"For simplicity, we shall talk about purity and mixedness without specifying the isomorphism.$ Theorem 3.5 Let $X$ be a smooth variety over a finite field $\\kappa $ , and let $\\lbrace \\mathcal {F}_i\\rbrace _{i=1}^N$ and $\\lbrace \\mathcal {G}_i\\rbrace _{i=1}^{N^{\\prime }}$ be pure lisse $\\overline{\\mathbb {Q}_\\ell }$ -sheaves on $X$ (of integer weights).",
"For every finite extension $k/\\kappa $ , let $(f_i)_k,(g_i)_k:X(k)\\rightarrow denote the trace functions of $ Fi$ and $ Gi$ respectively, and let $ tk=i=1N (fi)k-i=1N' (gi)k$.Then for each integer $ wZ$, either$ (1) ${t_k(x)}\\le C(\\#k)^{w/2}$ for every finite extension $k/\\kappa $ and $x\\in X(k)$ , where $C=\\sum _{i=1}^N\\operatorname{rank}(\\mathcal {F}_i)+\\sum _{i=1}^{N^{\\prime }}\\operatorname{rank}(\\mathcal {G}_i)$ , or (2) $\\displaystyle \\limsup _{\\#k\\rightarrow \\infty }\\frac{\\sum _{x\\in X(k)}{t_k(x)}^{2s}}{(\\#k)^{\\dim X}(\\#k)^{(w+1)s}}\\ge 1$ for all $s\\ge 1$ or $s=0$ , in particular for all $s\\in \\mathbb {N}$ .",
"Remark 3.6 Since the trace functions $t_k(x)$ in the statement of the theorem comes from a formal difference of lisse sheaves, we say that $t_k(x)$ is a “virtual lisse trace function\".",
"The two alternatives (1) and (2) are clearly mutually exclusive since $\\#X(k)\\le (\\deg X)(\\#k)^{\\dim X}$ (see Remark REF ).",
"We call the smallest $w$ that makes (1) true the maximum weight of the virtual trace function $\\lbrace t_k\\rbrace _{k/\\kappa }$ , which is also the largest $w$ such that the irreducible constituents of weight $w$ among the sheaves $\\mathcal {F}_i$ and $\\mathcal {G}_i$ do not all cancel out.",
"The theorem relates the cumulative and the pointwise behavior of a virtual lisse trace function.",
"It shows that, although one cannot expect a trace function with maximum weight $>w$ has magnitude exceeding $(\\#k)^{w+1}$ at every $k$ -point, it indeed has such magnitude on average in terms of its $2s$ -moments ($s\\ge 1$ ), if the variety is smooth and the sheaves are lisse.",
"A result like this may be well-known to experts, but I cannot find a reference.",
"It is easier to prove if the virtual trace function is an actual trace function, so that no cancellation is possible.",
"The $s=0$ case (with the convention $0^0=1$ ) of (2) can alternatively be obtained by applying the theorem to the constantly 1 trace function, or directly from the Lang–Weil bound.",
"This lemma can be extended to the case where all $\\mathcal {F}_i$ , $\\mathcal {G}_i$ are lisse and mixed and $X$ is normal: an irreducible mixed lisse sheaf on a normal variety is pure and remains irreducible when restricted to a dense open smooth subvariety, and moreover its isomorphism class is determined by the restriction [11].",
"We first reduce to the case that $X$ is geometrically connected.",
"Let $k_0$ be the splitting field (see §REF ) of $X/\\kappa $ .",
"Let $\\lbrace X_j\\rbrace _{j=1}^J$ be the connected components of $X\\times _\\kappa k_0$ , and consider the restrictions of $\\mathcal {F}_i$ and $\\mathcal {G}_i$ to the $X_j$ 's.",
"Suppose that the lemma is true for these $X_j$ 's, which are geometrically connected (Remark REF ).",
"If (1) holds for all of the $X_j$ 's, then (1) holds for $X\\times _\\kappa k_0=\\bigcup _{j=1}^J X_j$ , so (1) holds for $X$ if $k_0\\subset k$ .",
"If $k_0\\lnot \\subset k$ , (1) is vacuously true, since in that case $X(k)=\\varnothing $ (Remark REF ).",
"On the other hand, if (2) holds for some $X_j$ , then (2) holds for $X\\times _\\kappa k_0$ since $X_j\\subset X\\times _\\kappa k_0$ , hence it holds for $X$ since finite extensions of $k_0$ are also finite extensions of $\\kappa $ .",
"Thus we may assume that $X$ is geometrically connected.",
"We then have $\\#X(k)=(\\#k)^{\\dim X}+o((\\#k)^{\\dim X})$ as $\\#k\\rightarrow \\infty $ (Lang–Weil), so we can substitute $\\#X(k)$ for $(\\#k)^{\\dim X}$ in the limsup.",
"Since $x\\mapsto x^s$ is convex for $s\\ge 1$ or $s=0$ , by Jensen's inequality, $\\frac{1}{\\#X(k)}\\sum _{x\\in X(k)}*{\\frac{{t_k(x)}^2}{(\\#k)^{w+1}}}^s\\ge \\ *{\\frac{1}{\\#X(k)}\\sum _{x\\in X(k)}\\frac{{t_k(x)}^2}{(\\#k)^{w+1}}}^s_,$ thus we see that the $s=1$ case of (2) implies (2) for arbitrary $s\\ge 1$ or $s=0$ .",
"We now focus on the case $s=1$ .",
"Since the trace functions of a lisse sheaf are the sums of the trace functions of its irreducible constituents (with multiplicities), we may assume that all $\\mathcal {F}_i$ , $\\mathcal {G}_i$ are irreducible.",
"Furthermore, we can assume that no $\\mathcal {F}_i$ is isomorphic to any $\\mathcal {G}_j$ , since isomorphic sheaves give rise to identical trace functions which cancel each other.",
"Let $w_0$ be the maximum weight that appears among the $\\mathcal {F}_i$ and $\\mathcal {G}_i$ .",
"If $w_0\\le w$ , (1) is true, so we assume that $w_0>w$ , and aim to prove the stronger version of (2) with $w+1$ replaced by $w_0$ .",
"For this purpose, those $\\mathcal {F}_i$ and $\\mathcal {G}_i$ with weights $\\le w_0$ become irrelevant, since their contribution to the limsup is zero.",
"(When ${t_k(x)}^2$ is expanded, any term that involves a pure lisse sheaf of weight $<w_0$ contributes at most $C^2 (\\#k)^{w_0/2}(\\#k)^{(w_0-1)/2}=O((\\#k)^{w_0-\\frac{1}{2}})$ , and $\\#X(k)=O((\\#k)^{\\dim X})$ .)",
"Thus we further assume that all $\\mathcal {F}_i$ , $\\mathcal {G}_i$ are of weight $w_0$ .",
"Notice that $f_k:=\\sum _{i=1}^N (f_i)_k$ and $g_k:=\\sum _{i=1}^{N^{\\prime }} (g_i)_k$ are the trace functions of the semisimple lisse sheaves $\\mathcal {F}:=\\bigoplus _{i=1}^N\\mathcal {F}_i$ and $\\mathcal {G}:=\\bigoplus _{i=1}^{N^{\\prime }}\\mathcal {G}_i$ respectively.",
"Since $\\mathcal {F},\\mathcal {G}$ have weight $w_0$ , the trace functions of the duals $\\mathcal {F}^\\vee $ and $\\mathcal {G}^\\vee $ are $\\overline{f_k}/(\\#k)^{w_0}$ and $\\overline{g_k}/(\\#k)^{w_0}$ respectively, so ${t_k}^2/(\\#k)^{w_0}={f_k-g_k}^2/(\\#k)^{w_0}=(f_k\\overline{f_k}-f_k\\overline{g_k}-g_k\\overline{f_k}+g_k\\overline{g_k})/(\\#k)^{w_0}$ is the trace function of $\\mathcal {A}:=\\mathcal {F}\\otimes \\mathcal {F}^\\vee \\oplus \\,\\mathcal {G}\\otimes \\mathcal {G}^\\vee $ minus that of $\\mathcal {B}:=\\mathcal {F}\\otimes \\mathcal {G}^\\vee \\oplus \\,\\mathcal {G}\\otimes \\mathcal {F}^\\vee $ .",
"By the Grothendieck–Lefschetz trace formula, $\\sum _{x\\in X(k)}{t_k(x)}^2/(\\#k)^{w_0}=\\sum _{j=0}^{2\\dim X}\\operatorname{Tr}*{{\\rm Frob}_\\kappa ^{[k:\\kappa ]}\\mid H^j_c(X_{\\overline{\\kappa }},\\mathcal {A})}-\\operatorname{Tr}*{{\\rm Frob}_\\kappa ^{[k:\\kappa ]}\\mid H^j_c(X_{\\overline{\\kappa }},\\mathcal {B})}$ where $X_{\\overline{\\kappa }}:=X\\times _\\kappa \\overline{\\kappa }$ .",
"Since $\\mathcal {A}$ and $\\mathcal {B}$ are pure of weight 0, the eigenvalues of ${\\rm Frob}_\\kappa ^{[k:\\kappa ]}$ acting on both $H^j_c$ have modulus $\\le ((\\#\\kappa )^{j/2})^{[k:\\kappa ]}=(\\#k)^{j/2}$ ([5], [11]), so $H_j^c$ contributes zero to the limsup unless $j=2\\dim X$ .",
"Since $X$ is smooth and geometrically connected, if $\\mathcal {H}$ is a lisse $\\overline{\\mathbb {Q}_\\ell }$ -sheaf on $X$ , Poincaré duality yields $H^{2\\dim X}_c(X_{\\overline{\\kappa }},\\mathcal {H})\\cong H^0(X_{\\overline{\\kappa }},\\mathcal {H}^\\vee )^\\vee (-\\dim X)\\cong ((\\mathcal {H}^\\vee )^{\\pi _1(X_{\\overline{\\kappa }})})^\\vee (-\\dim X)$ as representations of $\\pi _1(X)/\\pi _1(X_{\\overline{\\kappa }})\\cong \\hat{\\mathbb {Z}}=\\overline{{{\\rm Frob}_\\kappa }}$ , so $\\operatorname{Tr}*{{\\rm Frob}_{\\kappa }^{[k:\\kappa ]}\\mid H_c^{2\\dim X}(X_{\\overline{\\kappa }},\\mathcal {H})}=(\\#k)^{\\dim X}\\sum _i\\lambda _i^{-[k:\\kappa ]},$ where the $\\lambda _i$ are the Frobenius eigenvalues (each appearing as many times as its algebraic multiplicity) on the space of geometric invariants $(\\mathcal {H}^\\vee )^{\\pi _1(X_{\\overline{\\kappa }})}$ .",
"Therefore, if the Frobenius eigenvalues on $(\\mathcal {A}^\\vee )^{\\pi _1(X_{\\overline{\\kappa }})}\\cong \\mathcal {A}^{\\pi _1(X_{\\overline{\\kappa }})}$ and $(\\mathcal {B}^\\vee )^{\\pi _1(X_{\\overline{\\kappa }})}\\cong \\mathcal {B}^{\\pi _1(X_{\\overline{\\kappa }})}$ are the multi-sets $A$ and $B$ respectively, we have $\\limsup _{\\#k\\rightarrow \\infty }\\frac{\\sum _{x\\in X(k)}{t_k(x)}^2}{(\\#k)^{\\dim X}(\\#k)^{w_0}}=\\limsup _{\\#k\\rightarrow \\infty }*{\\sum _{\\lambda \\in A}\\lambda ^{-[k:\\kappa ]}-\\sum _{\\lambda \\in B}\\lambda ^{-[k:\\kappa ]}}.$ Notice that all these eigenvalues $\\lambda $ have modulus 1, since $\\mathcal {A}$ and $\\mathcal {B}$ are pure of weight 0.",
"Therefore, by [8], the limsup is at least 1 if $A\\ne B$ .",
"(In fact, Katz showed that the limsup is at least the square root of the cardinality of the symmetric difference $A\\triangle B$ of the multisets $A$ and $B$ .)",
"We now show that $1\\in A$ but $1\\notin B$ .",
"Recall we assumed that the sheaves (representations) $\\mathcal {F}$ and $\\mathcal {G}$ are sums of irreducibles, i.e.",
"semisimple.",
"Since duals, tensor products (over the field $\\overline{\\mathbb {Q}_\\ell }$ of characteristic 0), and quotients of semisimple representations are semisimple, we find that $\\mathcal {A}^{\\pi _1(X_{\\overline{\\kappa }})}$ and $\\mathcal {B}^{\\pi _1(X_{\\overline{\\kappa }})}$ are semisimple $\\pi _1(X)$ -representations, and since the actions of $\\pi _1(X)$ factor through $\\hat{\\mathbb {Z}}=\\overline{{{\\rm Frob}_\\kappa }}$ , they are semisimple as $\\hat{\\mathbb {Z}}$ -representations.",
"Since $\\mathbb {Z}={{\\rm Frob}_\\kappa }$ is dense in $\\hat{\\mathbb {Z}}$ and the action is continuous, $\\hat{\\mathbb {Z}}$ -irreducibles remain irreducible under the $\\mathbb {Z}$ -action, so $\\mathcal {A}^{\\pi _1(X_{\\overline{\\kappa }})}$ and $\\mathcal {B}^{\\pi _1(X_{\\overline{\\kappa }})}$ are semisimple $\\mathbb {Z}$ -representations, which just means that the action of Frob is diagonalizable.",
"Therefore, the (algebraic) multiplicities of the eigenvalue 1 equal the dimensions of the eigenspaces (the geometric multiplicities).",
"But the eigenspaces associated with eigenvalue 1 simply consist of the elements fixed by ${\\rm Frob}_\\kappa $ .",
"Again, since ${{\\rm Frob}_\\kappa }$ is dense in $\\hat{\\mathbb {Z}}$ through which the actions of $\\pi _1(X)$ factor, these eigenspaces are just $\\mathcal {A}^{\\pi _1(X)}$ and $\\mathcal {B}^{\\pi _1(X)}$ .",
"Since $&\\mathcal {A}=\\mathcal {F}\\otimes \\mathcal {F}^\\vee \\oplus \\,\\mathcal {G}\\otimes \\mathcal {G}^\\vee \\cong \\operatorname{Hom}(\\mathcal {F,F})\\oplus \\operatorname{Hom}(\\mathcal {G,G})\\\\\\text{ and \\quad }&\\mathcal {B}=\\mathcal {F}\\otimes \\mathcal {G}^\\vee \\oplus \\,\\mathcal {G}\\otimes \\mathcal {F}^\\vee \\cong \\operatorname{Hom}(\\mathcal {G,F})\\oplus \\operatorname{Hom}(\\mathcal {F,G}),$ we have $\\mathcal {A}^{\\pi _1(X)}\\cong \\operatorname{Hom}_{\\pi _1(X)}(\\mathcal {F,F})\\oplus \\operatorname{Hom}_{\\pi _1(X)}(\\mathcal {G,G})\\ne 0$ (since we assumed that the weight $w_0$ appears in $\\mathcal {F}$ or in $\\mathcal {G}$ , $\\mathcal {F}$ and $\\mathcal {G}$ cannot both be trivial) and $\\mathcal {B}^{\\pi _1(X)}=\\operatorname{Hom}_{\\pi _1(X)}(\\mathcal {G,F})\\oplus \\operatorname{Hom}_{\\pi _1(X)}(\\mathcal {F,G})=0$ (since we assumed that $\\mathcal {F}$ and $\\mathcal {G}$ have no common irreducible constituents).",
"Thus $1\\in A$ but $1\\notin B$ , hence $A\\ne B$ , which completes the proof."
],
[
"The multivariate Weil bound",
"Lemma 3.7 If $k$ is a finite field, $\\chi :k^\\times \\rightarrow \\times $ is a multiplicative character of order $d$ , and $F\\in k(x_1,\\dots ,x_n)$ is not a perfect $d$ th power over $\\overline{k}$ (equivalently, $F$ is not of the form $aG^d$ with $a\\in k^\\times $ and $G\\in k(x_1,\\dots ,x_n)$ ; see Lemma REF ), then $*{\\sum _{x\\in k^n}\\chi (F_(x))}\\le C(\\#k)^{n-1/2},$ where $C$ depends only on $n$ and $\\deg F$ .",
"Remark 3.8 This lemma was proved for polynomial $F$ in [14] without using $\\ell $ -adic sheaves.",
"We use the machinery of Weil II to obtain a proof for rational functions that is more conceptual.",
"Applying the lemma to $\\chi _i\\circ \\operatorname{N}_{k/\\kappa }$ which has order equal to $d_i=\\operatorname{ord}\\chi _i$ , we get Proposition REF (b).",
"Let $d:=\\operatorname{ord}\\chi $ and let $\\ell $ be a prime other than $\\operatorname{char}k$ .",
"Let $\\mathcal {L}_d$ be the lisse $\\overline{\\mathbb {Q}_\\ell }$ -sheaf of weight 0 on $\\mathbb {G}_{{\\rm m},k}$ associated to the $\\ell $ -adic representation $\\pi _1(\\mathbb {G}_{{\\rm m},k})\\rightarrow \\mathbb {Z}/d\\mathbb {Z}\\cong \\mu _d\\subset \\overline{\\mathbb {Q}_\\ell }^\\times $ where the first map is associated to the cyclic étale covering $\\mathbb {G}_{{\\rm m},k}\\rightarrow \\mathbb {G}_{{\\rm m},k}$ defined by $x\\mapsto x^d$ .",
"Let $V$ be the open subvariety of $\\mathbb {A}^n_k$ on which both the numerator and the denominator of $F$ is nonzero, and let $f:V\\rightarrow \\mathbb {G}_{{\\rm m},k}$ be defined by $F$ .",
"By the Grothendieck–Lefschetz trace formula, $\\sum _{x\\in k^n}\\chi (F(x))=\\sum _{j=0}^{2n}\\operatorname{Tr}*{{\\rm Frob}_k\\mid H^j_c(V_{\\overline{k}},f^*\\mathcal {L}_d)}.$ If $F$ is not a perfect $d_i$ th power over $\\overline{k}$ , $f^*\\mathcal {L}_d$ is not geometrically constant (see REF below), and since it is of rank 1, it has no geometric invariants, thus $H^{2n}_c\\cong H^0$ vanishes.",
"Moreover, $H^j_c$ with $j<2n$ has weights $\\le j\\le 2n-1$ [5] since $f^*\\mathcal {L}_d$ is pure of weight 0.",
"Therefore, $ *{\\operatorname{Tr}*{{\\rm Frob}_k\\mid H^j_c(V_{\\overline{k}},f^*\\mathcal {L}_d)}} \\le \\operatorname{rank}(H^j_c(V_{\\overline{k}}))\\cdot (\\#k)^{(2n-1)/2}.",
"$ Since the ranks of the $H^j_c$ are bounded by Katz's constant $C$ which depends only on $n$ and $\\deg F$ , we obtain $*{\\sum _{x\\in k^n}\\chi (F(x))}\\le C(\\#k)^{n-1/2}$ .",
"Lemma 3.9 Let $X$ and $Z$ be connected schemes, let $G$ be a finite group, and let $\\pi _1(X)\\rightarrow \\operatorname{Gal}(Y/X)\\cong G$ be a surjective homomorphism associated to a Galois étale covering $\\varphi :Y\\rightarrow X$ .",
"Let $G\\rightarrow \\operatorname{GL}_N(\\overline{\\mathbb {Q}_\\ell })$ be a faithful representation and let $\\mathcal {L}$ denote the $\\overline{\\mathbb {Q}_\\ell }$ -sheaf associated to its composition with $\\pi _1(X)\\rightarrow G$ .",
"Then for any morphism $f:Z\\rightarrow X$ , $f$ factors through $\\varphi $ iff $f^*\\mathcal {L}$ is constant.",
"Remark 3.10 If $X=Y=\\mathbb {G}_{{\\rm m},\\overline{k}}$ , $\\varphi $ is the $d$ th power map, $Z=V_{\\overline{k}}$ , and $f:Z\\rightarrow X$ is defined by $F$ , then by the lemma we see that $F$ is a perfect $d$ th power in $\\overline{k}(x_1,\\dots ,x_n)\\iff f$ factors through $\\varphi \\iff f^*\\mathcal {L}$ is constant.",
"$(\\Rightarrow )$ Notice that $\\pi (Y)$ is exactly the kernel of $\\pi _1(X)\\rightarrow G$ .",
"If $f$ factors through $\\varphi $ , the representation associated to $f^*\\mathcal {L}$ , which is $\\pi _1(Z)\\rightarrow \\pi _1(X)\\rightarrow G\\rightarrow \\operatorname{GL}_N(\\overline{\\mathbb {Q}_\\ell })$ , factors through $\\pi _1(Y)$ and hence is trivial.",
"$()$ Consider the following commutative diagram $\\begin{tikzcd} Y\\times _X Z {r} [swap]{d} & Y {d} \\\\ Z {r} & X \\end{tikzcd}$ $\\pi _1(Z)$ acts on $Y\\times _X Z$ via the action of $\\pi _1(X)$ on $Y$ .",
"If $f^*\\mathcal {L}$ is trivial, $\\pi _1(Z)\\rightarrow \\pi _1(X)\\rightarrow G\\cong \\operatorname{Gal}(Y/X)$ is trivial because $G\\rightarrow \\operatorname{GL}_N(\\overline{\\mathbb {Q}_\\ell })$ is faithful, so $\\pi _1(Z)$ acts trivially on $Y$ and hence on $Y\\times _X Z$ .",
"Since $Y\\times _X Z\\rightarrow Z$ is an étale covering, it must be an isomorphism on every connected component, so in particular it has a section $Z\\rightarrow Y\\times _X Z$ .",
"Composing this section with $Y\\times _X Z\\rightarrow Y$ yields a lift $Z\\rightarrow Y$ of $f:Z\\rightarrow X$ ."
],
[
"Mutual transversality of subspaces of a vector space",
"Lemma 3.11 If $k$ is a field and $\\lbrace V_j\\rbrace _{j=1}^N$ are $k$ -subspaces of a vector spaces $V$ over $k$ , then there exists a basis $E$ of $V$ and pairwise disjoint subsets $\\lbrace E_j\\rbrace _{j=1}^N$ of $E$ such that $\\bigcap _{j=1}^N V_j=\\operatorname{span}(E\\setminus \\bigcup _{j=1}^N E_j)$ and $V_j\\subset \\operatorname{span}(E\\setminus E_j)$ for $1\\le j\\le N$ .",
"In other words, the $V_j$ 's can each be replaced by a larger subspace such that their intersection remain unchanged, so that they are now determined by the vanishing of respective sets of coordinates that are disjoint from each other.",
"The ability to treat these disjoint coordinates separately is important in the proof of Lemma REF .",
"This lemma follows from Lemma REF and Lemma REF ((1)$\\Rightarrow $ (3)) below.",
"Lemma 3.12 If $\\lbrace V_j\\rbrace _{j=1}^N$ are subspaces of a $k$ -vector space $V$ , then there exist subspaces $\\lbrace W_j\\rbrace _{j=1}^N$ of $V$ such that $V_j\\subset W_j$ and $(\\bigcap _{j=1}^n W_j)+W_{n+1}=V$ for $1\\le n<N$ and $\\bigcap _{j=1}^N V_j=\\bigcap _{j=1}^N W_j$ .",
"This lemma fails if $V_j$ are finite abelian groups instead of vector spaces, which is the main reason why we cannot extend Lemma REF to the situation of an abelian group variety acting on another variety, the original situation being a vector space acting simply transitively on the affine space.",
"We proceed by induction.",
"If $N=0$ , there is nothing to prove (the empty intersection is always $V$ ).",
"If $N>0$ , given $\\lbrace V_j\\rbrace _{j=1}^N$ , apply the induction hypothesis to $\\lbrace V_j\\rbrace _{j=1}^{N-1}$ to get $\\lbrace W_j\\rbrace _{j=1}^{N-1}$ .",
"Let $U$ be a complement of $(\\bigcap _{j=1}^{N-1} W_j)+V_N$ in $V$ , and let $W_N:=V_N+U\\supset V_N$ , then clearly $(\\bigcap _{j=1}^{N-1} W_j)+W_N=V$ .",
"If $A,B,C\\subset V$ are subspaces satisfying $(A+B)\\cap C=\\lbrace 0\\rbrace $ , it is easy to show that $A\\cap (B+C)=A\\cap B$ .",
"Taking $A=\\bigcap _{j=1}^{N-1} W_j$ , $B=V_N$ and $C=U$ , we see that $\\textstyle \\bigcap _{j=1}^N W_j=A\\cap (B+C)=A\\cap B=(\\bigcap _{j=1}^{N-1} W_j)\\cap V_N=(\\bigcap _{j=1}^{N-1} V_j)\\cap V_N=\\bigcap _{j=1}^N V_j.$ Lemma 3.13 If $\\lbrace W_j\\rbrace _{j=1}^N$ are subspaces of a $k$ -vector space $V$ , the following are equivalent: $(\\bigcap _{j=1}^n W_j)+W_{n+1}=V$ for $1\\le n<N$ ; The natural injective linear map $V/\\bigcap _{j=1}^N W_j\\rightarrow \\bigoplus _{j=1}^N V/W_j$ is an isomorphism; There exists a basis $E$ of $V$ and pairwise disjoint subsets $\\lbrace E_j\\rbrace _{j=1}^N$ of $E$ such that $W_j=\\operatorname{span}(E\\setminus E_j)$ for $1\\le j\\le N$ ; There exist linearly independent subspaces $\\lbrace U_j\\rbrace _{j=1}^N$ of $V$ such that $V=(\\bigcap _{i=1}^N W_i)\\oplus \\bigoplus _{i=1}^N U_i$ and $W_j=(\\bigcap _{i=1}^N W_i)\\oplus \\bigoplus _{1\\le i\\le N, i\\ne j}U_i$ for $1\\le j\\le N$ ; $(\\bigcap _{1\\le j\\le N, j\\ne n}W_j)+W_n=V$ for $1\\le n\\le N$ ; In the dual space $V^*$ , the subspaces $\\lbrace W_j^\\perp \\rbrace _{j=1}^N$ are linearly independent.",
"If $\\operatorname{codim}W_j<\\infty $ for all $1\\le j\\le N$ , they are also equivalent to: $\\operatorname{codim}(\\bigcap _{j=1}^N W_j)=\\sum _{j=1}^N\\operatorname{codim}W_j$ .",
"Remark 3.14 If $\\lbrace W_j\\rbrace _{j=1}^N$ satisfy the equivalent conditions listed in this lemma, they are called mutually transverse.",
"The Chinese Remainder Theorem says that comaximal ideals in a $k$ -algebra are mutually transverse.",
"Condition (6) shows that mutual transversality is a notion dual to linear independence.",
"In fact, one way to prove the equivalence is passing to the dual space using the identifications $(\\bigcap _{j=1}^n W_j)^\\perp =\\sum _{j=1}^n W_j^\\perp $ , $(W+W^{\\prime })^\\perp =W^\\perp \\cap W^{\\prime \\perp }$ and $(V/W)^*=W^\\perp $ , and then taking advantage of the familiar equivalent characterizations of linear independence.",
"Although $V$ is finite-dimensional in our intended application, the proof works for any $V$ .",
"If we consider infinitely many subspaces, the obvious generalizations of the conditions in the lemma are no longer equivalent.",
"(1)$\\Rightarrow $ (2): If $A,B\\subset V$ are subspaces, then the natural injective linear map $V/(A\\cap B)\\rightarrow V/A\\oplus V/B$ is an isomorphism iff $V=A+B$ .",
"Thus if (1) holds, then (2) can be obtained by induction.",
"(2)$\\Rightarrow $ (3): Assume (2).",
"For $1\\le j\\le N$ , let $E^{\\prime }_j$ be the image of a basis of $V/W_j$ under the map $V/W_j\\rightarrow \\bigoplus _{j=1}^N V/W_j\\cong V/\\bigcap _{j=1}^N W_j$ , then $\\coprod _{j=1}^N E^{\\prime }_j$ is a basis of $V/\\bigcap _{j=1}^N W_j$ .",
"Let $E_j$ be a lift of $E^{\\prime }_j$ to $V$ , and let $E_0$ be a basis of $\\bigcap _{j=1}^N W_j$ , then $E:=\\coprod _{j=0}^N E_j$ is a basis of $V$ .",
"By definition of $E_j$ , the image of $E_j$ in $V/W_n$ is $\\lbrace 0\\rbrace $ (i.e.",
"$E_j\\subset W_n$ ) if $n\\ne j$ , so $E\\setminus E_n=\\bigcup _{n\\ne j}E_n\\subset W_n$ .",
"Since $E_n$ is a basis both for $V/\\operatorname{span}(E\\setminus E_n)$ (since $E$ is a basis of $V$ ) and for $V/W_n$ (by definition of $E_j$ ), we conclude that $W_n=\\operatorname{span}(E\\setminus E_n)$ .",
"(3)$\\Rightarrow $ (4): Take $U_j=\\operatorname{span}(E_j)$ .",
"(4)$\\Rightarrow $ (5): Assume (4).",
"Then $\\bigcap _{j\\ne n}W_j=(\\bigcap _{i=1}^N W_j)\\oplus U_n$ , so $\\textstyle (\\bigcap _{j\\ne n}W_j)+W_n=(\\bigcap _{i=1}^N W_i)\\oplus U_n+\\bigoplus _{i\\ne n}U_i=V.$ (5)$\\Rightarrow $ (1): Notice that $\\bigcap _{j=1}^n W_j\\supset \\bigcap _{1\\le j\\le N, j\\ne n+1}W_j$ for $1\\le n<N$ .",
"(2)$\\iff $ (6): The dual of the injective linear map $V/\\bigcap _{j=1}^N W_j\\rightarrow \\bigoplus _{j=1}^N V/W_j$ is canonically identified with the natural surjective map $\\bigoplus _{j=1}^N W_j^\\perp \\rightarrow \\sum _{j=1}^N W_j^\\perp $ .",
"(2)$\\iff $ (7): Clear."
],
[
"Bound on the number of perfect powers in certain offset families of rational functions",
"Lemma 3.15 Let $k$ be a field and $k^s$ its separable algebraic closure, so $k^s=\\overline{k}$ .",
"If $A$ is a reduced $k$ -algebra, then $A\\otimes _k k^s$ is reduced.",
"If $F\\in k[x_1,\\dots ,x_n]$ is irreducible, then $F$ is square-free as a polynomial in $k^s[x_1,\\dots ,x_n]$ .",
"If $F,G\\in k[x_1,\\dots ,x_n]$ are non-associate irreducible polynomials, then $F$ and $G$ have no common factors in $k^s[x_1,\\dots ,x_n]$ .",
"(1) This follows from [15].",
"(2) If $F$ is irreducible over $k$ , then $k[x_1,\\dots ,x_n]/(F)$ is reduced, so by (1), $k^s[x_1,\\dots ,x_n]/(F)$ is reduced, so $F$ is square-free over $k^s$ .",
"(3) If $F,G$ are irreducible over $k$ and non-associate, then $k[x_1,\\dots ,x_n]/(FG)$ is reduced, so by (1), $k^s[x_1,\\dots ,x_n]/(FG)$ is reduced, so $F$ and $G$ have no common factors over $k^s$ .",
"Lemma 3.16 Let $\\kappa $ be a finite field and $\\kappa _0$ its prime field.",
"Let $F\\in \\kappa (x_1,\\dots ,x_n)$ be $d$ th-power-free, let $T=T_F:=\\lbrace m\\in \\overline{\\kappa }^n\\mid F(x)\\equiv F(x+m)\\rbrace $ be the $\\kappa _0$ -subspace of $\\overline{\\kappa }^n$ of translations that leave $F$ invariant, and assume that $\\#T<\\infty $ .",
"For any finite extension $k/\\kappa $ , $r\\in \\mathbb {N}$ and $\\lbrace a_i\\rbrace _{i=1}^r\\subset \\mathbb {Z}$ such that $\\gcd (d,a_i)=1$ , let $P$ be the collection of tuples $(m^{(1)},\\dots ,m^{(r)})\\in k^{nr}$ such that the rational function $\\prod _{i=1}^r F(x+m^{(i)})^{a_i}$ is a perfect $d$ th power over $\\overline{\\kappa }$ .",
"Then $\\#P\\le C(\\#k)^{n{r/2}}(\\#T)^{{r/2}}$ , where the constant $C$ only depends on $r$ and the degree of $F$ and not on $k$ .",
"Remark 3.17 We will not try to optimize the constant $C$ .",
"Notice that if $d\\nmid (\\sum a_i)(\\sum b_j)$ (with $b_j$ 's introduced in the proof below), then $\\prod _{i=1}^r F(x+m^{(i)})$ is never a perfect $d$ th power.",
"However, in the case we are interested in (in the corollary that follows), $a_i=\\pm 1$ , and $\\sum a_i=0$ .",
"By Lemma REF , an irreducible polynomial in $\\kappa [x_1,\\dots ,x_n]$ remains square-free over $\\overline{\\kappa }=\\kappa ^s$ , and that different irreducible polynomials remain relatively prime over $\\overline{\\kappa }$ .",
"Since $F\\in \\kappa (x_1,\\dots ,x_n)$ is $d$ th-power-free, if $F=\\prod _{j=1}^N f_j^{b_j}$ is the factorization of $F$ into irreducible factors over $\\overline{\\kappa }$ , we still have $0<{b_j}<d$ , and in particular $d\\nmid b_j$ .",
"For every $i,j$ , the irreducible factor $f_j(x+m^{(i)})$ appears in $F(x+m^{(i)})^{a_i}$ with multiplicity $a_i b_j$ .",
"Since $\\gcd (d,a_i)=1$ and $d\\nmid b_j$ , we have $d\\nmid a_i b_j$ .",
"Thus, in order for $\\prod _{i=1}^r F(x+m^{(i)})^{a_i}$ to be a perfect $d$ th power, $f_j(x+m^{(i)})$ must also appear in $F(x+m^{(i^{\\prime })})$ for some $i^{\\prime }\\ne i$ , so $cf_j(x+m^{(i)})\\equiv f_{j^{\\prime }}(x+m^{(i^{\\prime })})$ (i.e.",
"$f_{j^{\\prime }}(x)\\equiv f_j(x+m^{(i)}-m^{(i^{\\prime })})$ ) for some $1\\le j^{\\prime }\\le N$ and $c\\in \\overline{\\kappa }^\\times $ .",
"Now, for each $j$ and each tuple $m=(m^{(1)},\\dots ,m^{(r)})\\in k^{nr}$ , define an undirected graph $G_{m,j}$ with vertex set $\\lbrace 1,\\dots ,r\\rbrace $ such that there is an edge between $i$ and $i^{\\prime }$ iff $f_{j^{\\prime }}(x)\\equiv cf_j(x+m^{(i)}-m^{(i^{\\prime })})$ or $cf_j(x+m^{(i^{\\prime })}-m^{(i)})$ for some $j^{\\prime }$ and $c$ .",
"If $m\\in P$ , then $G_{m,j}$ has no isolated point by the last paragraph, and it is then easy to see that it has at most ${r/2}$ components.",
"Clearly, the number of undirected graphs on $\\lbrace 1,\\dots ,r\\rbrace $ is $2^{{r+1\\atopwithdelims ()2}}$ .",
"Given $N$ such graphs $G_1,\\dots , G_N$ , we want to bound the number of tuples $m\\in P$ such that $G_{m,j}=G_j$ for all $1\\le j\\le N$ .",
"Let $V_j$ be the $\\kappa _0$ -subspace of $V:=k^n$ of translations that leave $f_j$ invariant, then $\\bigcap _{j=1}^N V_j\\subset T$ .",
"Choose a basis $E$ of $V$ and subsets $\\lbrace E_j\\rbrace _{j=1}^N$ as in Lemma REF , so that $V_j\\subset \\operatorname{span}(E\\setminus E_j)$ for $1\\le j\\le N$ and $\\bigcap _{j=1}^N V_j=\\operatorname{span}(E\\setminus \\bigcup _{j=1}^N E_j)$ , hence $\\sum _{j=1}^N\\#E_j=\\dim V-\\dim \\bigcap _{j=1}^N V_j\\ge \\dim V-\\dim T$ .",
"An edge connecting $i$ and $i^{\\prime }$ in $G_{m,j}$ poses a constraint between the $E_j$ -coordinates of $m^{(i)}$ and $m^{(i^{\\prime })}$ under this basis; more precisely, for each $1\\le j^{\\prime }\\le N$ there are at most two possibilities for the $E_j$ -coordinates of $m^{(i)}-m^{(i^{\\prime })}$ .",
"Indeed, if $f_{j^{\\prime }}(x)\\equiv cf_j(x\\pm (m^{(i)}-m^{(i^{\\prime })}))$ and $f_{j^{\\prime }}(x)\\equiv c^{\\prime }f_j(x\\pm ({m^{\\prime }}^{(i)}-{m^{\\prime }}^{(i^{\\prime })}))$ , then $f_j(x)\\equiv (c^{\\prime }/c)f_j(x\\pm ({m^{\\prime }}^{(i)}-{m^{\\prime }}^{(i^{\\prime })})\\mp (m^{(i)}-{m^{\\prime }}^{(i)}))$ , so $c^{\\prime }=c$ and $({m^{\\prime }}^{(i)}-{m^{\\prime }}^{(i^{\\prime })})\\pm (m^{(i)}-{m^{\\prime }}^{(i)})$ leaves $f_j$ invariant, hence it lies in $V_j\\subset \\operatorname{span}(E\\setminus E_j)$ , so the $E_j$ -coordinates of ${m^{\\prime }}^{(i)}-{m^{\\prime }}^{(i^{\\prime })}$ are $\\pm $ those of $m^{(i)}-{m^{\\prime }}^{(i)}$ .",
"By induction, if $i$ and $i^{\\prime }$ lie in the same component of $G_{m,j}$ , say with distance $D$ , then there are at most $(2N)^D\\le (2N)^r$ possibilities for the $E_j$ -coordinates of $m^{(i)}-m^{(i^{\\prime })}$ .",
"Therefore, if $H\\subset G_{m,j}$ is a connected component, there are at most $(\\#\\kappa _0)^{\\#E_j}(2N)^{r(\\#H-1)}$ possibilities for the $E_j$ -coordinates of the $m^{(i)}$ 's with $i\\in H$ .",
"If $m\\in P$ , $G_{m,j}$ has at most ${r/2}$ components, so there are at most $(\\#\\kappa _0)^{\\#E_j{r/2}}(2N)^{r^2}$ possibilities for all the $E_j$ -coordinates of $m$ , and hence at most $(\\#\\kappa _0)^{\\sum _{j=1}^N \\#E_j{r/2}}(2N)^{r^2 N}&\\le (\\#\\kappa _0)^{(\\dim V-\\dim T){r/2}}(2N)^{r^2 N}\\\\&=(2N)^{r^2 N}((\\#k)^n/\\#T)^{{r/2}}$ possibilities for the $\\bigcup _{j=1}^N E_j$ -coordinates.",
"The possibilities for the $E\\setminus \\bigcup _{j=1}^N E_j$ -coordinates amount to $(\\#T)^r$ .",
"Therefore, if we take $C=2^{{r+1\\atopwithdelims ()2}\\deg F}(2\\deg F)^{r^2\\deg F}$ , then $\\#P\\le C(\\#k)^{n{r/2}}(\\#T)^{{r/2}}$ , since $N\\le \\deg F$ .",
"Corollary 3.18 Fix a $d$ th-power-free rational function $F\\in \\kappa (x_1,\\dots ,x_n)$ satisfying $\\#T_F<\\infty $ , and fix $r\\in \\mathbb {N}$ .",
"For each finite extension $k/\\kappa $ , let $P_k$ be the collection of tuples $(m^{(1)},\\dots ,m^{(2r)})\\in k^{n2r}$ such that $\\prod _{i=1}^r F(x+m^{(i)})\\prod _{i=r+1}^{2r} F(x+m^{(i)})^{-1}$ is a perfect $d$ th power over $\\overline{\\kappa }$ .",
"Then $\\#P_k=O((\\#k)^{nr})$ as $k$ varies.",
"Remark 3.19 In this case, the exponent is sharp: for any bijection $\\varphi :\\lbrace 1,\\dots ,r\\rbrace \\rightarrow \\lbrace r+1,\\dots ,2r\\rbrace $ , if $m^{(r+i)}=m^{(\\varphi (i))}$ for $1\\le i\\le r$ , then $\\prod _{i=1}^r F(x+m^{(i)})\\prod _{i=r+1}^{2r} F(x+m^{(i)})^{-1}=1$ .",
"The number of such tuples $(m^{(1)},\\dots ,m^{(2r)})$ is asymptotic to $r!\\,(\\#k)^{nr}$ as $\\#k\\rightarrow \\infty $ ."
],
[
"Reductions of a polynomial with integer coefficients",
"Lemma 3.20 Let $F\\in \\mathbb {Z}[x_1,\\dots ,x_n]$ be a polynomial, and let $x$ be the row vector $(x_1,\\dots ,x_n)$ of indeterminates.",
"Then the following are equivalent: $F$ is invariant under some nontrivial translation in $\\overline{\\mathbb {Q}}^n$ , i.e.",
"there exists $0\\ne m\\in \\overline{\\mathbb {Q}}^n$ such that $F(x)\\equiv F(x+m)$ ; $F$ is invariant under some nontrivial translation in $\\mathbb {Z}^n$ ; $F$ can be made independent of one of the indeterminates by a linear change of coordinates, i.e.",
"there exists $A\\in \\operatorname{GL}(n,\\mathbb {Z})$ such that $F(xA)\\in \\mathbb {Z}[x_2,\\dots ,x_n]$ ; When viewed as a morphism $\\mathbb {A}^n_\\mathbb {Z}\\rightarrow \\mathbb {A}^1_\\mathbb {Z}$ , $F$ factors through a linear map $\\mathbb {A}^n_\\mathbb {Z}\\rightarrow \\mathbb {A}^{n-1}_\\mathbb {Z}$ , i.e.",
"there exists a integral $n\\times (n-1)$ matrix $B$ and $f\\in \\mathbb {Z}[x_2,\\dots ,x_n]$ such that $F(x)\\equiv f(xB)$ ; For almost all prime numbers $p$ , the reduction of $F$ modulo $p$ is invariant under some nontrivial translation in $\\overline{\\mathbb {F}_p}^n$ .",
"For infinitely many prime numbers $p$ , the reduction of $F$ modulo $p$ is invariant under some nontrivial translation in $\\overline{\\mathbb {F}_p}^n$ .",
"Remark 3.21 If the conditions are violated, (3) or (4) shows that we can reduce to a lower dimension.",
"In fact, if we start with a homogeneous polynomial $F$ we can reduce to a homogeneous polynomial in lower dimension.",
"The lemma can be shown to hold for $F\\in \\mathbb {Q}(x_1,\\dots ,x_n)$ as well.",
"The implication (2)$\\Rightarrow $ (3) fails if $\\mathbb {Z}$ is replaced by a Dedekind domain that is not a PID.",
"(1)$\\Rightarrow $ (2): Assume (1).",
"Let $0\\ne m=(m_1,\\dots ,m_n)\\in \\overline{\\mathbb {Q}}^n$ be such that $F(x)\\equiv F(x+m)$ , we assume without loss of generality that $m_1\\ne 0$ .",
"Now consider $F(x+tm)-F(x)$ as a polynomial in the single indeterminate $t$ .",
"Since $F(x)\\equiv F(x+m)$ , by induction, every $t\\in \\mathbb {Z}$ is a root of $F(x+tm)-F(x)$ , so $F(x)\\equiv F(x+tm)$ since a nonzero polynomial cannot have infinitely many roots.",
"In particular, $F(x)\\equiv F(x+m/m_1)$ , so we may assume that $m_1=1$ by replacing $m$ with $m/m_1$ .",
"Let $E$ be a number field containing all the $m_i$ 's.",
"Since $F$ has coefficients in $\\mathbb {Z}$ , for any $\\sigma \\in \\operatorname{Gal}(E/\\mathbb {Q})$ , we have $F(x)\\equiv F(x+\\sigma (m))$ , hence $F(x)\\equiv F(x+\\operatorname{Tr}_{E/\\mathbb {Q}}(m))$ .",
"Since $m_1=1$ , the first coordinate of $\\operatorname{Tr}_{E/\\mathbb {Q}}(m)$ is $[E:\\mathbb {Q}]\\ne 0$ , so we may assume that $0\\ne m\\in \\mathbb {Q}^n$ by replacing $m$ with $\\operatorname{Tr}_{E/\\mathbb {Q}}(m)$ .",
"Let $d$ be a common denominator of the $m_i$ 's, then $F(x)\\equiv F(x+dm)$ and $dm\\in \\mathbb {Z}^n$ .",
"(2)$\\Rightarrow $ (3): Suppose that $F$ is invariant under $0\\ne m\\in \\mathbb {Z}^n$ .",
"We showed that $F(x)\\equiv F(x+m)\\Rightarrow F(x)\\equiv F(x+tm)$ for all $t\\in \\overline{\\mathbb {Q}}$ , so dividing $m$ by the $\\gcd (m_1,\\dots ,m_n)$ , we may assume that $\\gcd (m_1,\\dots ,m_n)=1$ , which means that $\\mathbb {Z}^n/\\mathbb {Z}\\cdot m$ is torsion free, hence free.",
"Therefore $\\mathbb {Z}^n\\twoheadrightarrow \\mathbb {Z}^n/\\mathbb {Z}\\cdot m$ splits, and if $A$ is the image of the splitting, we have $\\mathbb {Z}^n=\\mathbb {Z}\\cdot m\\oplus A\\cong \\mathbb {Z}\\oplus \\mathbb {Z}^{n-1}\\cong \\mathbb {Z}^n$ , so there exists $A\\in \\operatorname{GL}(n,\\mathbb {Z})$ such that $m=(1,0,\\dots ,0)A$ .",
"We then have $F(xA)\\equiv F(xA+m)\\equiv F((x+(1,0,\\dots ,0))A)$ , so the polynomial $G(x):=F(xA)$ is invariant under translation by $(1,0,\\dots ,0)$ , so $G(x_1,x_2,\\dots ,x_n)-G(0,x_2,\\dots ,x_n)$ regarded as a polynomial in $x_1$ has all integers as its roots, and therefore must be zero.",
"We conclude that $F(xA)\\equiv G(x)\\equiv G(0,x_2,\\dots ,x_n)\\in \\mathbb {Z}[x_2,\\dots ,x_n]$ .",
"(3)$\\Rightarrow $ (4): Suppose that $F(xA)\\equiv f(x_2,\\dots ,x_n)$ for some $f\\in \\mathbb {Z}[x_2,\\dots ,x_n]$ , so $F(x)\\equiv F((xA^{-1})A)\\equiv f((xA^{-1})_2,\\dots ,(xA^{-1})_n)$ , so we can take $B$ to be the last $n-1$ columns of $A^{-1}$ .",
"(4)$\\Rightarrow $ (5): Suppose that there exists an integral $n\\times (n-1)$ matrix $B$ and $f\\in \\mathbb {Z}[x_2,\\dots ,x_n]$ such that $F(x)\\equiv f(xB)$ .",
"Since $B$ is a linear map from $\\mathbb {Q}^n\\rightarrow \\mathbb {Q}^{n-1}$ , the null space of $B$ is nontrivial, so one can find $0\\ne m\\in \\mathbb {Z}^n$ such that $mB=0$ , so $F(x)\\equiv f(xB)\\equiv f(xB+mB)\\equiv f((x+m)B)\\equiv F(x+m)$ .",
"Since $m\\ne 0$ , the reduction of $m$ modulo $p$ is zero only for finitely many $p$ (the reductions actually lie in $(\\mathbb {F}_p)^n$ ).",
"(5)$\\Rightarrow $ (6): Obvious.",
"(6)$\\Rightarrow $ (1): The conditions $F(x)\\equiv F(x+m)$ and $m\\ne 0$ defines a subscheme $S\\subset \\mathbb {A}^n_\\mathbb {Z}$ over $\\operatorname{Spec}\\mathbb {Z}$ , such that the closed points in the geometric fibers $S_{\\overline{\\mathbb {F}_p}}$ or $S_{\\overline{\\mathbb {Q}}}$ correspond to the tuples $0\\ne m\\in \\overline{\\mathbb {F}_p}^n$ or $\\overline{\\mathbb {Q}}^n$ such that $F(x)\\equiv F(x+m)$ .",
"By Chevalley's theorem, the image of the structural morphism $S\\rightarrow \\operatorname{Spec}\\mathbb {Z}$ is constructible, but a constructible subset of $\\operatorname{Spec}\\mathbb {Z}$ either is finite or contains the generic point (and hence is cofinite).",
"Condition (5) says that infinitely many fibers of $S$ are nonempty, hence the image of the structural morphism contains the generic point $\\operatorname{Spec}\\mathbb {Q}$ .",
"Therefore, $S_\\mathbb {Q}$ is a non-empty affine scheme and hence contains a closed point, which gives a nontrivial translation in $\\overline{\\mathbb {Q}}$ under which $F$ is invariant.",
"Lemma 3.22 Let $F\\in \\mathbb {Z}[x_1,\\dots ,x_n]$ be a $d$ th-power-free polynomial.",
"Then the reduction of $F$ modulo $p$ is $d$ th-power-free for almost all primes $p$ .",
"$F$ fails to be $d$ th-power-free if and only if $F$ can be written as $f^d g$ such that $f$ is not constant.",
"The coefficients of $f$ and $g$ can each be encoded in an $N$ -tuple, where $N$ is the number of monomials of degrees $\\le \\deg F$ in $n$ indeterminates.",
"The conditions $f^d g=F$ and that at least one of the nonconstant terms of $f$ is nonzero define a subscheme $S\\subset \\mathbb {A}^{2N}_\\mathbb {Z}$ .",
"If $F$ fails to be $d$ th-power-free modulo infinitely many primes $p$ , then $S_{\\mathbb {F}_p}$ is nonempty for infinitely many primes, hence $S_\\mathbb {Q}$ is nonempty (cf.",
"proof of (6)$\\Rightarrow $ (1) in the previous lemma) and thus $F$ fails to be $d$ th-power-free in $\\overline{\\mathbb {Q}}[x_1,\\dots ,x_n]$ , hence in $\\mathbb {Q}[x_1,\\dots ,x_n]$ (cf.",
"proof of Lemma REF ), hence in $\\mathbb {Z}[x_1,\\dots ,x_n]$ .",
"Combining the previous two lemmas, we get Corollary 3.23 Let $F\\in \\mathbb {Z}[x_1,\\dots ,x_n]$ be a $d$ th-power-free polynomial not invariant under any nontrivial translations (in $\\overline{\\mathbb {Q}}^n$ or in $\\mathbb {Z}^n$ ), then for almost all primes $p$ , the reduction of $F$ modulo $p$ satisfies the hypothesis in Lemma REF and Corollary REF (with $T=\\lbrace 0\\rbrace $ )."
],
[
"Degree of a projective variety and its number of points in a box",
"For applications in analytic number theory, we are interested in bounding the number of points of a quasi-affine variety $X$ in a box with coordinates in a finite field.",
"We obtain below a bound depending only on $\\dim X$ , $\\deg X$ and the lengths of the $(\\dim X)$ longest sides of the box, which is a trivial generalization of what Tao called a Schwarz–Zippel type bound in his blog post [16].",
"Lemma 3.24 Let $k$ be an algebraically closed field, and let $X$ be a closed subvariety of $\\mathbb {P}^n_k$ of codimension $\\theta $ and degree $d$ .",
"If $\\lbrace B_i\\rbrace _{i=1}^n$ are subsets of $k$ , we identify $B=\\prod _{i=1}^n B_i$ with a subset of $\\mathbb {P}^n(k)$ via the inclusions $\\prod _{i=1}^n B_i\\subset k^n=\\mathbb {A}^n(k)\\subset \\mathbb {P}^n(k)$ .",
"If $1\\le \\#B_1\\le \\#B_2\\le \\dots \\le \\#B_n<\\infty $ , we have $\\#*{X(k)\\cap \\prod _{i=1}^n B_i}\\le d\\prod _{i=\\theta +1}^n \\#B_i=d(\\#B)\\prod _{i=1}^\\theta (\\#B_i)^{-1}.$ Remark 3.25 In typical applications in analytic number theory, one usually takes the $B_i$ 's to be intervals in some finite prime field, but it can also be applied with $B_i$ being the whole underlying set of a finite field, for example in Remark REF .",
"If $k$ is not necessarily algebraically closed, and $X$ is instead a (locally closed) subscheme of $\\mathbb {A}^n_k$ whose irreducible components have sum of degrees $d$ , the lemma still holds because we may apply the lemma to the irreducible components of the closure of $X\\times _k\\overline{k}$ in $\\mathbb {P}^n_{\\overline{k}}$ and add up the bounds.",
"This yields Lemma REF .",
"We proceed by induction on $n$ .",
"If $n=0$ , then $D=0$ and $X$ must be the single point in $\\mathbb {A}^0=\\mathbb {P}^0$ , so $d=1$ and both sides of the inequality are 1.",
"If $n>0$ , for each $\\overline{x}\\in B_n$ , let $X_{\\overline{x}}$ be the closed subvariety $X\\cap H_{\\overline{x}}$ , where $H_{\\overline{x}}$ is the hyperplane $\\lbrace x_n=\\overline{x}x_0\\rbrace $ in $\\mathbb {P}^n_k$ (here we use $(x_1,\\dots ,x_n)$ as the coordinates of $\\mathbb {A}^n_k$ and $[x_0:x_1:\\dots :x_n]$ as the homogeneous coordinates of $\\mathbb {P}^n_k$ ).",
"If $X\\subset H_{\\overline{x}}$ for some $\\overline{x}\\in B_n$ , then $X$ has the same degree $d$ as a subvariety in $H_{\\overline{x}}=\\mathbb {P}^{n-1}_k$ .",
"Therefore $\\#*{X(k)\\cap \\prod _{i=1}^n B_i}=\\#*{X_{\\overline{x}}(k)\\cap \\prod _{i=1}^{n-1} B_i}\\le d\\prod _{i=\\theta }^{n-1}\\#B_i\\le d\\prod _{i=\\theta +1}^n\\#B_i,$ where the first inequality is by the induction hypothesis.",
"If $X\\lnot \\subset H_{\\overline{x}}$ for all $\\overline{x}\\in B_n$ , then each $X_{\\overline{x}}$ is a proper closed subset of the irreducible space $X$ , so it has dimension $<\\dim X$ , hence has codimension at least $\\theta $ in $H_{\\overline{x}}=\\mathbb {P}^{n-1}_k$ .Let $Z_1,\\dots ,Z_s$ be the irreducible components of $X_{\\overline{x}}$ .",
"By Theorem I.7.7 in [7], $\\sum _{j=1}^s\\deg Z_j\\le d$ .",
"Therefore $\\#*{X_{\\overline{x}}(k)\\cap \\prod _{i=1}^{n-1} B_i}& \\le \\sum _{j=1}^s\\#*{Z_j(k)\\cap \\prod _{i=1}^{n-1} B_i}\\\\& \\le \\sum _{j=1}^s\\deg Z_j\\prod _{i=\\operatorname{codim}Z_j+1}^{n-1}\\#B_i\\ \\le \\ d\\prod _{i=\\theta +1}^{n-1}\\#B_i$ by the induction hypothesis, and hence $\\#*{X(k)\\cap \\prod _{i=1}^n B_i}=\\sum _{\\overline{x}\\in B_n}\\#*{X_{\\overline{x}}(k)\\cap \\prod _{i=1}^{n-1} B_i}\\le \\#B_n\\cdot d\\prod _{i=\\theta +1}^{n-1}\\#B_i=d\\prod _{i=\\theta +1}^n\\#B_i.$ If we have a connected closed subscheme $X\\subset \\mathbb {P}^n_Y$ smooth over a base scheme $Y$ , i.e.",
"a family of projective schemes parametrized by $Y$ , the following lemma says that all of these schemes (the fibers), possibly base extended to the algebraic closure (the geometric fibers), are equidimensional and have the same dimension and degree, and its degree equals the sum of the degrees of its irreducible components, so if the previous lemma is applied to the irreducible components (which are varieties if equipped the reduced induced scheme structure), uniform bounds are obtained.",
"Lemma 3.26 Let $Y$ be a scheme and let $X$ be a connected closed subscheme of $\\mathbb {P}^n_Y$ smooth over $Y$ .",
"For $y\\in Y$ , let $X_y$ be the fiber of $X$ over $Y$ , and let $X_{\\overline{y}}:=X_y\\times _{{\\rm k}(y)}\\overline{{\\rm k}(y)}$ .",
"Then there exist constants $D,d\\in \\mathbb {N}_{\\ge 0}$ such that each $X_{\\overline{y}}$ is equidimensional of dimension $D$ and degree $d$ .",
"Since $X\\rightarrow Y$ is smooth, it is flat and locally of finite presentation, and each fiber $X_y$ is smooth over ${\\rm k}(y)$ and hence Cohen–Macaulay.",
"Since $X$ is also connected, by [15], $X\\rightarrow Y$ has relative dimension $D$ for some $D$ , i.e.",
"$X_y$ is equidimensional of dimension $D$ for any $y$ .",
"By ibid., Tag 02NK, $X_{\\overline{y}}$ is also equidimensional of dimension $D$ .",
"Since $X\\rightarrow Y$ is flat, $X_y\\subset \\mathbb {P}^n_{{\\rm k}(y)}$ have the same Hilbert polynomial for all $y$ , and hence the same degree $d$ , for all $y$ .",
"Since the Hilbert polynomial does not change under extension of base field, all $X_{\\overline{y}}\\subset \\mathbb {P}^n_{\\overline{{\\rm k}(y)}}$ have the same degree $d$ ."
],
[
"Existence of smooth decompositions",
"The next lemma assures that we can get a decomposition into smooth morphisms for very general morphisms of schemes (away from finitely many primes), and we can then apply the previous lemma to each of these smooth morphisms.",
"Lemma 3.27 Let $X$ be a noetherian scheme and let $\\varphi :X\\rightarrow Y$ be a scheme morphism of finite presentation.",
"Then there exist finitely many locally closed subsets $\\lbrace X_i\\rbrace _{i=1}^N$ of $X$ such that the induced morphisms $\\varphi |_{X_i}:(X_i)_{\\rm red}\\rightarrow \\overline{\\varphi (X_i)}_{\\rm red}$ are smooth for each $i$ , and such that the image of $X\\setminus \\bigcup _{i=1}^N X_i$ in $\\operatorname{Spec}\\mathbb {Z}$ is finite.",
"Remark 3.28 We call such a collection $\\lbrace X_i\\rbrace _{i=1}^N$ a smooth decomposition of $\\varphi $ , or of $X$ relative to $Y$ (or relative to $\\varphi $ ).",
"As easily seen from the proof below, the collection can be made pairwise disjoint, but we do not need that.",
"Using noetherian induction, we need only prove the following: if $\\varphi |_Z:Z\\rightarrow Y$ admits a smooth decomposition for every proper closed subset $Z\\subset X$ (induction hypothesis), then $\\varphi $ also admits a smooth decomposition.",
"(Notice that a closed subscheme $Z$ of a noetherian scheme $X$ is of finite presentation over $X$ , hence over $Y$ .)",
"If $X$ is reducible, its finitely many irreducible components $Z_j$ are proper closed subsets, so by the induction hypothesis each $\\varphi |_{Z_j}$ admits a smooth decomposition, which together yield a smooth decomposition for $\\varphi $ .",
"If $X$ is irreducible, then $X_{\\rm red}$ and $Y_1:=\\overline{\\varphi (X)}_{\\rm red}$ are integral, and the induced morphism $X_{\\rm red}\\rightarrow Y_1$ is still of finite presentation.",
"Therefore, if the function field $K(Y_1)$ is perfect, there exists an open dense subset $X_1\\subset X$ such that $\\varphi |_{X_1}:X_1\\rightarrow Y_1$ is smooth [6].",
"By the induction hypothesis, $\\varphi |_{X\\setminus X_1}$ admits a smooth decomposition, which together with $X_1$ gives a smooth decomposition for $\\varphi $ .",
"If the function field is not perfect, then it has nonzero characteristic, which means that the generic point $\\eta \\in X$ maps to a single closed point in $\\operatorname{Spec}\\mathbb {Z}$ , so the image of $X=\\overline{\\lbrace \\eta \\rbrace }$ in $\\operatorname{Spec}\\mathbb {Z}$ is a single point, and $\\varnothing $ is a smooth decomposition of $\\varphi $ ."
],
[
"Acknowledgements",
"I thank my collaborator Lillian Pierce for raising the original question that led to the present work.",
"I thank my advisor Michael Larsen for his guidance, his original idea from which this paper stemmed, and numerous helpful discussions.",
"I thank Prof. Guocan Feng, Jianxun Hu, Lixin Liu, Zheng-an Yao and especially Yen-Mei Julia Chen, whose reference letters and encouragement five years ago helped me out of the dark times when my applications to PhD programs failed for two consecutive years.",
"This paper is dedicated to them.",
"I thank my collaborator Lillian Pierce for raising the original question that led to the present work.",
"I thank my advisor Michael Larsen for his guidance, his original idea from which this paper stemmed, and numerous helpful discussions.",
"I thank Prof. Guocan Feng, Jianxun Hu, Lixin Liu, Zheng-an Yao and especially Yen-Mei Julia Chen, whose reference letters and encouragement five years ago helped me out of the dark times when my applications to PhD programs failed for two consecutive years.",
"This paper is dedicated to them."
]
] | 1709.01663 |
[
[
"An inner-loop free solution to inverse problems using deep neural\n networks"
],
[
"Abstract We propose a new method that uses deep learning techniques to accelerate the popular alternating direction method of multipliers (ADMM) solution for inverse problems.",
"The ADMM updates consist of a proximity operator, a least squares regression that includes a big matrix inversion, and an explicit solution for updating the dual variables.",
"Typically, inner loops are required to solve the first two sub-minimization problems due to the intractability of the prior and the matrix inversion.",
"To avoid such drawbacks or limitations, we propose an inner-loop free update rule with two pre-trained deep convolutional architectures.",
"More specifically, we learn a conditional denoising auto-encoder which imposes an implicit data-dependent prior/regularization on ground-truth in the first sub-minimization problem.",
"This design follows an empirical Bayesian strategy, leading to so-called amortized inference.",
"For matrix inversion in the second sub-problem, we learn a convolutional neural network to approximate the matrix inversion, i.e., the inverse mapping is learned by feeding the input through the learned forward network.",
"Note that training this neural network does not require ground-truth or measurements, i.e., it is data-independent.",
"Extensive experiments on both synthetic data and real datasets demonstrate the efficiency and accuracy of the proposed method compared with the conventional ADMM solution using inner loops for solving inverse problems."
],
[
"Introduction",
" Most of the inverse problems are formulated directly to the setting of an optimization problem related to the a forward model [24].",
"The forward model maps unknown signals, i.e., the ground-truth, to acquired information about them, which we call data or measurements.",
"This mapping, or forward problem, generally depends on a physical theory that links the ground-truth to the measurements.",
"Solving inverse problems involves learning the inverse mapping from the measurements to the ground-truth.",
"Specifically, it recovers a signal from a small number of degraded or noisy measurements.",
"This is usually ill-posed [25], [24].",
"Recently, deep learning techniques have emerged as excellent models and gained great popularity for their widespread success in allowing for efficient inference techniques on applications include pattern analysis (unsupervised), classification (supervised), computer vision, image processing, etc [6].",
"Exploiting deep neural networks to help solve inverse problems has been explored recently [23], [1] and deep learning based methods have achieved state-of-the-art performance in many challenging inverse problems like super-resolution [3], [23], image reconstruction [19], automatic colorization [12].",
"More specifically, massive datasets currently enables learning end-to-end mappings from the measurement domain to the target image/signal/data domain to help deal with these challenging problems instead of solving the inverse problem by inference.",
"The pairs $\\left\\lbrace {\\mathbf {x}}, {\\mathbf {y}}\\right\\rbrace $ are used to learn the mapping function from ${\\mathbf {y}}$ to ${\\mathbf {x}}$ , where ${\\mathbf {x}}$ is the ground-truth and ${\\mathbf {y}}$ is its corresponding measurement.",
"This mapping function has recently been characterized by using sophisticated networks, e.g., deep neural networks.",
"A strong motivation to use neural networks stems from the universal approximation theorem [5], which states that a feed-forward network with a single hidden layer containing a finite number of neurons can approximate any continuous function on compact subsets of $\\mathbb {R}^{n}$ , under mild assumptions on the activation function.",
"More specifically, in recent work [3], [23], [12], [19], an end-to-end mapping from measurements ${\\mathbf {y}}$ to ground-truth ${\\mathbf {x}}$ was learned from the training data and then applied to the testing data.",
"Thus, the complicated inference scheme needed in the conventional inverse problem solver was replaced by feeding a new measurement through the pre-trained network, which is much more efficient.",
"One main problem for this strategy is that it requires task-specific training of the networks, i.e., different problems require different networks.",
"Thus, it is very expensive to solve diverse sets of problems.",
"To improve the scope of deep neural network models, more recently, in [4], a splitting strategy was proposed to decompose an inverse problem into two optimization problems, where one sub-problem, related to regularization, can be solved efficiently using trained deep neural networks, leading to an alternating direction method of multipliers (ADMM) framework [2], [16].",
"This method involves training a deep convolutional auto-encoder network for low-level image modeling, which explicitly imposes regularization that spans the subspace that the ground-truth images live in.",
"For the sub-problem that requires inverting a big matrix, a conventional gradient descent algorithm was used, leading to an alternating update, iterating between feed-forward propagation through a network and iterative gradient descent.",
"Thus, an inner loop for gradient descent is still necessary in this framework.",
"In this work, we propose an inner-loop free framework, in the sense that no iterative algorithm is required to solve sub-problems, using a splitting strategy for inverse problems.",
"The alternating updates for the two sub-problems were derived by feeding through two pre-trained deep neural networks, i.e., one using an amortized inference based denoising convolutional auto-encoder network for the proximity operation and one using structured convolutional neural networks for the huge matrix inversion related to the forward model.",
"Thus, the computational complexity of each iteration in ADMM is linear with respect to (w.r.t.)",
"the dimensionality of the signals.",
"The network for the proximity operation imposes an implicit prior learned from the training data, including the measurements as well as the ground-truth, leading to amortized inference.",
"The network for matrix inversion is independent from the training data and can be trained from noise, i.e., a random noise image and its output from the forward model.",
"This independence from training data allows the proposed framework to be used to accelerate almost all the existing training data/example free solutions for inverse problems based on a splitting strategy.",
"To make training the networks for the proximity operation easier, three tricks have been employed: the first one is to use a pixel shuffling technique to equalize the dimensionality of the measurements and ground-truth; the second one is to optionally add an adversarial loss borrowed from the GAN (Generative Adversarial Nets) framework [10] for sharp image generation; the last one is to introduce a perceptual measurement loss derived from pre-trained networks, such as AlexNet [11] or VGG-16 Model [22].",
"Arguably, the speed of the proposed algorithm, which we term Inf-ADMM-ADNN (Inner-loop free ADMM with Auxiliary Deep Neural Network), comes from the fact that it uses two auxiliary pre-trained networks to accelerate the updates of ADMM.",
"Contribution The main contribution of this paper is comprised of i) learning an implicit prior/regularizer using a denoising auto-encoder neural network, based on amortized inference; ii) learning the inverse of a big matrix using structured convolutional neural networks, without using training data; iii) each of the above networks can be exploited to accelerate the existing ADMM solver for inverse problems."
],
[
"Linear Inverse Problem",
" Notation: trainable networks by calligraphic font, e.g., $\\mathcal {A}$ , fixed networks by italic font e.g., $A$ .",
"As mentioned in the last section, the low dimensional measurement is denoted as $\\mathbf {y}\\in \\mathbb {R}^{m}$ , which is reduced from high dimensional ground truth $\\mathbf {x}\\in \\mathbb {R}^{n}$ by a linear operator $A$ such that $\\mathbf {y} = A\\mathbf {x}$ .",
"Note that usually $n \\ge m$ , which makes the number of parameters to estimate no smaller than the number of data points in hand.",
"This imposes an ill-posed problem for finding solution $\\mathbf {x}$ on new observation $\\mathbf {y}$ , since $A$ is an underdetermined measurement matrix.",
"For example, in a super-resolution set-up, the matrix $A$ might not be invertible, such as the strided Gaussian convolution in [20], [23].",
"To overcome this difficulty, several computational strategies, including Markov chain Monte Carlo (MCMC) and tailored variable splitting under the ADMM framework, have been proposed and applied to different kinds of priors, e.g., the empirical Gaussian prior [28], [31], the Total Variation prior [21], [29], [30], etc.",
"In this paper, we focus on the popular ADMM framework due to its low computational complexity and recent success in solving large scale optimization problems.",
"More specifically, the optimization problem is formulated as $\\hat{\\mathbf {x}} = \\arg \\min _{\\mathbf {x}, \\mathbf {z}} \\Vert \\mathbf {y} - A\\mathbf {z}\\Vert ^2 + \\lambda \\mathcal {R}(\\mathbf {x}), \\quad s.t.",
"\\quad \\mathbf {z} = \\mathbf {x}$ where the introduced auxiliary variable $\\mathbf {z}$ is constrained to be equal to ${\\mathbf {x}}$ , and $\\mathcal {R}(\\mathbf {x})$ captures the structure promoted by the prior/regularization.",
"If we design the regularization in an empirical Bayesian way, by imposing an implicit data dependent prior on $\\mathbf {x}$ , i.e., $\\mathcal {R}(\\mathbf {x}; \\mathbf {y})$ for amortized inference [23], the augmented Lagrangian for (REF ) is $\\mathcal {L}(\\mathbf {x}, \\mathbf {z}, \\mathbf {u}) = \\Vert \\mathbf {y} - A\\mathbf {z}\\Vert ^2 + \\lambda \\mathcal {R}(\\mathbf {x}; \\mathbf {y}) + \\langle \\mathbf {u}, \\mathbf {x} - \\mathbf {z} \\rangle + \\beta \\Vert \\mathbf {x} - \\mathbf {z}\\Vert ^2$ where $\\mathbf {u}$ is the Lagrange multiplier, and $\\beta > 0$ is the penalty parameter.",
"The usual augmented Lagrange multiplier method is to minimize $\\mathcal {L}$ w.r.t.",
"$\\mathbf {x}$ and $\\mathbf {z}$ simultaneously.",
"This is difficult and does not exploit the fact that the objective function is separable.",
"To remedy this issue, ADMM decomposes the minimization into two subproblems that are minimizations w.r.t.",
"$\\mathbf {x}$ and $\\mathbf {z}$ , respectively.",
"More specifically, the iterations are as follows: $\\mathbf {x}^{k+1} &= \\arg \\min _{\\mathbf {x}} \\beta \\Vert \\mathbf {x} - \\mathbf {z}^k + \\mathbf {u}^k/2\\beta \\Vert ^2 + \\lambda \\mathcal {R}(\\mathbf {x}; \\mathbf {y}) \\\\\\mathbf {z}^{k+1} &= \\arg \\min _{\\mathbf {z}} \\Vert \\mathbf {y} - A\\mathbf {z}\\Vert ^2 + \\beta \\Vert \\mathbf {x}^{k+1} - \\mathbf {z} + \\mathbf {u}^k /2\\beta \\Vert ^2 \\\\\\mathbf {u}^{k+1} &= \\mathbf {u}^k + 2\\beta (\\mathbf {x}^{k+1} - \\mathbf {z}^{k+1}).$ If the prior $\\mathcal {R}$ is appropriately chosen, such as $\\Vert \\mathbf {x}\\Vert _1$ , a closed-form solution for $(\\ref {eq:minx})$ , i.e., a soft thresholding solution is naturally desirable.",
"However, for some more complicated regularizations, e.g., a patch based prior [8], solving (REF ) is nontrivial, and may require iterative methods.",
"To solve (), a matrix inversion is necessary, for which conjugate gradient descent (CG) is usually applied to update $\\mathbf {z}$ [4].",
"Thus, solving (REF ) and () is in general cumbersome.",
"Inner loops are required to solve these two sub-minimization problems due to the intractability of the prior and the inversion, resulting in large computational complexity.",
"To avoid such drawbacks or limitations, we propose an inner loop-free update rule with two pretrained deep convolutional architectures."
],
[
"Amortized inference for $\\mathbf {x}$ using a conditional proximity operator",
"Solving sub-problem (REF ) is equivalent to finding the solution of the proximity operator $\\mathcal {P}_{\\mathcal {R}}(\\mathbf {v};\\mathbf {y})=\\arg \\min _{\\mathbf {x}} \\frac{1}{2}\\Vert \\mathbf {x} - \\mathbf {v}\\Vert ^2 + \\mathcal {R}(\\mathbf {x};\\mathbf {y})$ where we incorporate the constant $\\frac{\\lambda }{2\\beta }$ into $\\mathcal {R}$ without loss of generality.",
"If we impose the first order necessary conditions [17], we have $\\mathbf {x}=\\mathcal {P}_{\\mathcal {R}}(\\mathbf {v};\\mathbf {y}) \\Leftrightarrow 0 \\in \\partial \\mathcal {R}(\\cdot ;\\mathbf {y})(\\mathbf {x}) + \\mathbf {x} - \\mathbf {v} \\Leftrightarrow \\mathbf {v} - \\mathbf {x} \\in \\partial \\mathcal {R}(\\cdot ;\\mathbf {y})(\\mathbf {x})$ where $\\partial \\mathcal {R}(\\cdot ;\\mathbf {y})$ is a partial derivative operator.",
"For notational simplicity, we define another operator $\\mathcal {F} =: \\mathcal {I} + \\partial \\mathcal {R}(\\cdot ;\\mathbf {y})$ .",
"Thus, the last condition in (REF ) indicates that $\\mathbf {x}^{k+1}=\\mathcal {F}^{-1}(\\mathbf {v})$ .",
"Note that the inverse here represents the inverse of an operator, i.e., the inverse function of ${\\mathcal {F}}$ .",
"Thus our objective is to learn such an inverse operator which projects $\\mathbf {v}$ into the prior subspace.",
"For simple priors like $\\Vert \\cdot \\Vert _1$ or $\\Vert \\cdot \\Vert _2^2$ , the projection can be efficiently computed.",
"In this work, we propose an implicit example-based prior, which does not have a truly Bayesian interpretation, but aids in model optimization.",
"In line with this prior, we define the implicit proximity operator $\\mathcal {G}_{\\mathbf {\\theta }}(\\mathbf {x}; \\mathbf {v}, \\mathbf {y})$ parameterized by $\\mathbf {\\theta }$ to approximate unknown $\\mathcal {F}^{-1}$ .",
"More specifically, we propose a neural network architecture referred to as conditional Pixel Shuffling Denoising Auto-Encoders (cPSDAE) as the operator $\\mathcal {G}$ , where pixel shuffling [20] means periodically reordering the pixels in each channel mapping a high resolution image to a low resolution image with scale $r$ and increase the number of channels to $r^2$ (see [20] for more details).",
"This allows us to transform $\\mathbf {v}$ so that it is the same scale as $\\mathbf {y}$ , and concatenate it with ${\\mathbf {y}}$ as the input of cPSDAE easily.",
"The architecture of cPSDAE is shown in Fig.",
"REF (d)."
],
[
"Inversion-free update of $\\mathbf {z}$",
"While it is straightforward to write down the closed-form solution for sub-problem () w.r.t.",
"${\\mathbf {z}}$ as is shown in (REF ), explicitly computing this solution is nontrivial.",
"$\\mathbf {z}^{k+1} = K \\left(A^\\top \\mathbf {y} + \\beta \\mathbf {x}^{k+1} + \\mathbf {u}^k/2 \\right), \\text{ where } K=\\left(A^\\top A + \\beta \\mathbf {I} \\right)^{-1}$ In (REF ), $A^\\top $ is the transpose of the matrix $A$ .",
"As we mentioned, the term $K$ in the right hand side involves an expensive matrix inversion with computational complexity $O(n^3)$ .",
"Under some specific assumptions, e.g., $A$ is a circulant matrix, this matrix inversion can be accelerated with a Fast Fourier transformation, which has a complexity of order $\\mathcal {O}(n\\log n)$ .",
"Usually, the gradient based update has linear complexity in each iteration and thus has an overall complexity of order $\\mathcal {O}( n_{\\textrm {int}} \\log n)$ , where $n_{\\textrm {int}}$ is the number of iterations.",
"In this work, we will learn this matrix inversion explicitly by designing a neural network.",
"Note that $K$ is only dependent on $A$ , and thus can be computed in advance for future use.",
"This problem can be reduced to a smaller scale matrix inversion by applying the Sherman-Morrison-Woodbury formula: $K = \\beta ^{-1} \\left(\\mathbf {I} - A^\\top B A\\right), \\text{ where } B = \\left( \\beta \\mathbf {I} + AA^\\top \\right)^{-1}.$ Therefore, we only need to solve the matrix inversion in dimension $m \\times m$ , i.e., estimating $B$ .",
"We propose an approach to approximate it by a trainable deep convolutional neural network $\\mathcal {C}_{\\mathbf {\\phi }} \\approx B$ parameterized by $\\mathbf {\\phi }$ .",
"Note that $B^{-1}=\\lambda \\mathbf {I} + AA^\\top $ can be considered as a two-layer fully-connected or convolutional network as well, but with a fixed kernel.",
"This inspires us to design two auto-encoders with shared weights, and minimize the sum of two reconstruction losses to learn the inversion $\\mathcal {C}_{\\mathbf {\\phi }}$ : $\\arg \\min _{\\mathbf {\\phi }} \\mathbb {E}_{\\mathbf {\\varepsilon }}\\left[ \\Vert {\\mathbf {\\varepsilon }} - \\mathcal {C}_{\\mathbf {\\phi }} B^{-1}{\\mathbf {\\varepsilon }}\\Vert _2^2 + \\Vert {\\mathbf {\\varepsilon }} - B^{-1} \\mathcal {C}_{\\mathbf {\\phi }} {\\mathbf {\\varepsilon }}\\Vert _2^2 \\right]$ where $\\mathbf {\\varepsilon }$ is sampled from a standard Gaussian distribution.",
"The loss in (REF ) is clearly depicted in Fig.",
"REF (a) with the structure of $B^{-1}$ in Fig.",
"REF (b) and the structure of ${\\mathcal {C}}_{\\mathbf {\\phi }}$ in Fig.",
"REF (c).",
"Since the matrix $B$ is symmetric, we can reparameterize $\\mathcal {C}_{\\mathbf {\\phi }}$ as $\\mathcal {W}_{\\mathbf {\\phi }}\\mathcal {W}_{\\mathbf {\\phi }}^\\top $ , where $\\mathcal {W}_{\\mathbf {\\phi }}$ represents a multi-layer convolutional network and $\\mathcal {W}_{\\mathbf {\\phi }}^\\top $ is a symmetric convolution transpose architecture using shared kernels with $\\mathcal {W}_{\\mathbf {\\phi }}$ , as shown in Fig.",
"REF (c) (the blocks with the same colors share the same network parameters).",
"By plugging the learned $\\mathcal {C}_{\\mathbf {\\phi }}$ in (REF ) , we obtain a reusable deep neural network $\\mathcal {K}_{\\mathbf {\\phi }}=\\beta ^{-1} \\left(\\mathbf {I} - A^\\top \\mathcal {C}_{\\mathbf {\\phi }} A\\right)$ as a surrogate for the exact inverse matrix $K$ .",
"The update of $\\mathbf {z}$ at each iteration can be done by applying the same $\\mathcal {K}_{\\mathbf {\\phi }}$ as follows: $\\mathbf {z}^{k+1} \\leftarrow \\beta ^{-1} \\left(\\mathbf {I} - A^\\top \\mathcal {C}_{\\mathbf {\\phi }} A\\right) \\left(A^\\top \\mathbf {y} + \\beta \\mathbf {x}^{k+1} + \\mathbf {u}^k/2 \\right).$ Figure: Network for updating 𝐳{\\mathbf {z}} (in black): (a) loss function (), (b) structure of B -1 B^{-1}, (c) struture of 𝒞 φ \\cal {C}_{\\mathbf {\\phi }}.Note that the input ϵ\\epsilon is random noise independent from the training data.",
"Network for updating 𝐳{\\mathbf {z}} (in blue): (d) structure of cPSDAE 𝒢 θ (𝐱;𝐱 ˜,𝐲)\\mathcal {G}_{\\mathbf {\\theta }}(\\mathbf {x}; \\tilde{{\\mathbf {x}}}, \\mathbf {y}) (𝐱 ˜\\tilde{{\\mathbf {x}}} plays the same role as 𝐯{\\mathbf {v}} in training), (e) adversarial training for ℛ(𝐱;𝐲) \\mathcal {R}(\\mathbf {x};\\mathbf {y}).",
"Note again that (a)(b)(c) describes the network for inferring 𝐳{\\mathbf {z}}, which is data-independent and (d)(e) describes the network for inferring 𝐱{\\mathbf {x}}, which is data-dependent."
],
[
"Adversarial training of cPSDAE",
"In this section, we will describe the proposed adversarial training scheme for cPSDAE to update $\\mathbf {x}$ .",
"Suppose that we have the paired training dataset $(\\mathbf {x}_i, \\mathbf {y}_i)_{i=1}^N$ , a single cPSDAE with the input pair $(\\tilde{\\mathbf {x}}, \\mathbf {y})$ is trying to minimize the reconstruction error $\\mathcal {L}_r(\\mathcal {G}_{\\mathbf {\\theta }}(\\tilde{\\mathbf {x}},\\mathbf {y}), \\mathbf {x})$ , where $\\tilde{\\mathbf {x}}$ is a corrupted version of $\\mathbf {x}$ , i.e., $\\tilde{{\\mathbf {x}}} = {\\mathbf {x}}+ {\\mathbf {n}}$ where ${\\mathbf {n}}$ is random noise.",
"Notice $\\mathcal {L}_r$ in traditional DAE is commonly defined as $\\ell _2$ loss, however, $\\ell _1$ loss is an alternative in practice.",
"Additionally, we follow the idea in [18], [7] by introducing a discriminator and a comparator to help train the cPSDAE, and find that it can produce sharper or higher quality images than merely optimizing $\\mathcal {G}$ .",
"This will wrap our conditional generative model $\\mathcal {G}_{\\mathbf {\\theta }}$ into the conditional GAN [10] framework with an extra feature matching network (comparator).",
"Recent advances in representation learning problems have shown that the features extracted from well pre-trained neural networks on supervised classification problems can be successfully transferred to others tasks, such as zero-shot learning [14], style transfer learning [9].",
"Thus, we can simply use pre-trained AlexNet [11] or VGG-16 Model [22] on ImageNet as the comparator without fine-tuning in order to extract features that capture complex and perceptually important properties.",
"The feature matching loss $\\mathcal {L}_f(C(\\mathcal {G}_{\\mathbf {\\theta }}(\\tilde{\\mathbf {x}}, \\mathbf {y})), C(\\mathbf {x}))$ is usually the $\\ell _2$ distance of high level image features, where $C$ represents the pre-trained network.",
"Since $C$ is fixed, the gradient of this loss can be back-propagated to $\\mathbf {\\theta }$ .",
"For the adversarial training, the discriminator $\\mathcal {D}_{\\mathbf {\\psi }}$ is a trainable convolutional network.",
"We can keep the standard discriminator loss as in a traditional GAN, and add the generator loss of the GAN to the previously defined DAE loss and comparator loss.",
"Thus, we can write down our two objectives as follows, $\\mathcal {L}_D(\\mathbf {x}, \\mathbf {y}) &= - \\log \\mathcal {D}_{\\mathbf {\\psi }}(\\mathbf {x}) - \\log \\left( 1 - \\mathcal {D}_{\\mathbf {\\psi }}(\\mathcal {G}_{\\mathbf {\\theta }}(\\tilde{\\mathbf {x}}, \\mathbf {y})) \\right) \\\\\\mathcal {L}_G(\\mathbf {x}, \\mathbf {y}) &= \\lambda _r\\Vert \\mathcal {G}_{\\mathbf {\\theta }}(\\tilde{\\mathbf {x}}, \\mathbf {y}) - \\mathbf {x}\\Vert _2^2 + \\lambda _f\\Vert C(\\mathcal {G}_{\\mathbf {\\theta }}(\\tilde{\\mathbf {x}}, \\mathbf {y})) - C(\\mathbf {x})\\Vert _2^2 - \\lambda _a \\log \\mathcal {D}_{\\mathbf {\\psi }}(\\mathcal {G}_{\\mathbf {\\theta }}(\\tilde{\\mathbf {x}}, \\mathbf {y}))$ The optimization involves iteratively updating $\\mathbf {\\psi }$ by minimizing $\\mathcal {L}_D$ keeping $\\mathbf {\\theta }$ fixed, and then updating $\\mathbf {\\theta }$ by minimizing $\\mathcal {L}_G$ keeping $\\mathbf {\\psi }$ fixed.",
"The proposed method, including training and inference has been summarized in Algorithm REF .",
"Note that each update of ${\\mathbf {x}}$ or ${\\mathbf {z}}$ using neural networks in an ADMM iteration has a complexity of linear order w.r.t.",
"the data dimensionality $n$ .",
"Figure: Inner-loop free ADMM with Auxiliary Deep Neural Nets (Inf-ADMM-ADNN)"
],
[
"Discussion",
"A critical point for learning-based methods is whether the method generalizes to other problems.",
"More specifically, how does a method that is trained on a specific dataset perform when applied to another dataset?",
"To what extent can we reuse the trained network without re-training?",
"In the proposed method, two deep neural networks are trained to infer ${\\mathbf {x}}$ and ${\\mathbf {z}}$ .",
"For the network w.r.t.",
"${\\mathbf {z}}$ , the training only requires the forward model $A$ to generate the training pairs ($\\epsilon , A \\epsilon $ ).",
"The trained network for ${\\mathbf {z}}$ can be applied for any other datasets as long as they share the same $A$ .",
"Thus, this network can be adapted easily to accelerate inference for inverse problems without training data.",
"However, for inverse problems that depends on a different $A$ , a re-trained network is required.",
"It is worth mentioning that the forward model $A$ can be easily learned using training dataset $({\\mathbf {x}}, {\\mathbf {y}})$ , leading to a fully blind estimator associated with the inverse problem.",
"An example of learning $\\hat{A}$ can be found in the supplementary materials (see Section 1).",
"For the network w.r.t.",
"${\\mathbf {x}}$ , training requires data pairs $({\\mathbf {x}}_i, {\\mathbf {y}}_i)$ because of the amortized inference.",
"Note that this is different from training a prior for ${\\mathbf {x}}$ only using training data ${\\mathbf {x}}_i$ .",
"Thus, the trained network for ${\\mathbf {x}}$ is confined to the specific tasks constrained by the pairs (${\\mathbf {x}}, {\\mathbf {y}}$ ).",
"To extend the generality of the trained network, the amortized setting can be removed, i.e, the measurements ${\\mathbf {y}}$ is removed from the training, leading to a solution to proximity operator $\\mathcal {P}_{\\mathcal {R}}(\\mathbf {v})=\\arg \\min _{\\mathbf {x}} \\frac{1}{2}\\Vert \\mathbf {x} - \\mathbf {v}\\Vert ^2 + \\mathcal {R({\\mathbf {x}})}$ .",
"This proximity operation can be regarded as a denoiser which projects the noisy version ${\\mathbf {v}}$ of ${\\mathbf {x}}$ into the subspace imposed by ${\\mathcal {R}}({\\mathbf {x}})$ .",
"The trained network (for the proximity operator) can be used as a plug-and-play prior [26] to regularize other inverse problems for datasets that share similar statistical characteristics.",
"However, a significant change in the training dataset, e.g., different modalities like MRI and natural images (e.g., ImageNet [11]), would require re-training.",
"Another interesting point to mention is the scalability of the proposed method to data of different dimensions.",
"The scalability can be adapted using patch-based methods without loss of generality.",
"For example, a neural network is trained for images of size $64 \\times 64$ but the test image is of size $256 \\times 256$ .",
"To use this pre-trained network, the full image can be decomposed as four $64 \\times 64$ images and fed to the network.",
"To overcome the possible blocking artifacts, eight overlapping patches can be drawn from the full image and fed to the network.",
"The output of these eight patches are then averaged (unweighted or weighted) over the overlapping parts.",
"A similar strategy using patch stitching can be exploited to feed small patches to the network for higher dimensional datasets."
],
[
"Experiments",
" In this section, we provide experimental results and analysis on the proposed Inf-ADMM-ADNN and compare the results with a conventional ADMM using inner loops for inverse problems.",
"Experiments on synthetic data have been implemented to show the fast convergence of our method, which comes from the efficient feed-forward propagation through pre-trained neural networks.",
"Real applications using proposed Inf-ADMM-ADNN have been explored, including single image super-resolution, motion deblurring and joint super-resolution and colorization."
],
[
"Synthetic data",
" To evaluate the performance of proposed Inf-ADMM-ADNN, we first test the neural network ${\\mathcal {K}}_{\\mathbf {\\phi }}$ , approximating the matrix inversion on synthetic data.",
"More specifically, we assume that the ground-truth ${\\mathbf {x}}$ is drawn from a Laplace distribution $\\textrm {Laplace}(\\mu ,b)$ , where $\\mu =0$ is the location parameter and $b$ is the scale parameter.",
"The forward model $A$ is a sparse matrix representing convolution with a stride of 4.",
"The architecture of $A$ is available in the supplementary materials (see Section 2).",
"The noise ${\\mathbf {n}}$ is drawn from a standard Gaussian distribution ${\\mathcal {N}}(0,\\sigma ^2)$ .",
"Thus, the observed data is generated as ${\\mathbf {y}}= A {\\mathbf {x}}+{\\mathbf {n}}$ .",
"Following Bayes theorem, the maximum a posterior estimate of ${\\mathbf {x}}$ given ${\\mathbf {y}}$ , i.e., maximizing $p({\\mathbf {x}}|{\\mathbf {y}}) p({\\mathbf {y}}|{\\mathbf {x}}) p({\\mathbf {x}})$ can be equivalently formulated as $\\mathrm {arg}\\min _{{\\mathbf {x}}} \\frac{1}{2 \\sigma ^2}\\Vert {\\mathbf {y}}-A {\\mathbf {x}}\\Vert _2^2 + \\frac{1}{b} \\Vert {\\mathbf {x}}\\Vert _1$ , where $b=1$ and $\\sigma =1$ in this setting.",
"Following (REF ), (), (), this problem is reduced to the following three sub-problems: $\\mathbf {x}^{k+1} &= {\\mathcal {S}}_{\\frac{1}{2\\beta }}({\\mathbf {z}}^{k} - {\\mathbf {u}}^{k} / 2\\beta )\\\\\\mathbf {z}^{k+1} &= \\arg \\min _{\\mathbf {z}} \\Vert \\mathbf {y} - A\\mathbf {z}\\Vert _2^2 + \\beta \\Vert \\mathbf {x}^{k+1} - \\mathbf {z} + \\mathbf {u}^k /2\\beta \\Vert _2^2 \\\\\\mathbf {u}^{k+1} &= \\mathbf {u}^k + 2\\beta (\\mathbf {x}^{k+1} - \\mathbf {z}^{k+1})$ where the soft thresholding operator ${\\mathcal {S}}$ is defined as ${\\mathcal {S}}_{\\kappa }(a) = \\left\\lbrace \\begin{array}{ccc}& 0 & |a| \\le \\kappa \\\\& a - \\textrm {sgn}(a)\\kappa & |a| > \\kappa \\end{array}\\right.$ and sgn($a$ ) extracts the sign of $a$ .",
"The update of ${\\mathbf {x}}^{k+1}$ has a closed-form solution, i.e., soft thresholding of ${\\mathbf {z}}^{k}-{\\mathbf {u}}^{k}/{2 \\beta }$ .",
"The update of ${\\mathbf {z}}^{k+1}$ requires the inversion of a big matrix, which is usually solved using a gradient descent based algorithm.",
"The update of ${\\mathbf {u}}^{k+1}$ is straightforward.",
"Thus, we compare the gradient descent based update, a closed-form solution for matrix inversionNote that this matrix inversion can be explicitly computed due to its small size in this toy experiment.",
"In practice, this matrix is not built explicitly.",
"and the proposed inner-free update using a pre-trained neural network.",
"The evolution of the objective function w.r.t.",
"the number of iterations and the time has been plotted in the left and middle of Figs.",
"REF .",
"While all three methods perform similarly from iteration to iteration (in the left of Figs.",
"REF ), the proposed inner-loop free based and closed-form inversion based methods converge much faster than the gradient based method (in the middle of Figs.",
"REF ).",
"Considering the fact that the closed-form solution, i.e., a direct matrix inversion, is usually not available in practice, the learned neural network allows us to approximate the matrix inversion in a very accurate and efficient way.",
"Figure: Synthetic data: (left) objective v.s.",
"iterations, (middle) objective v.s.",
"time.MNIST dataset: (right) NMSE v.s.",
"iterations for MNIST image 4×4 \\times super-resolution.Figure: Top two rows : (column 1) LR images, (column 2) bicubic interpolation (×4\\times 4), (column 3) results using proposed method (×4\\times 4), (column 4) HR image.",
"Bottom row: (column 1) motion blurred images, (column 2) results using Wiener filter with the best performance by tuning regularization parameter, (column 3) results using proposed method, (column 4) ground-truth."
],
[
"Image super-resolution and motion deblurring",
"In this section, we apply the proposed Inf-ADMM-ADNN to solve the poplar image super-resolution problem.",
"We have tested our algorithm on the MNIST dataset [13] and the 11K images of the Caltech-UCSD Birds-200-2011 (CUB-200-2011) dataset [27].",
"In the first two rows of Fig.",
"REF , high resolution images, as shown in the last column, have been blurred (convolved) using a Gaussian kernel of size $3 \\times 3$ and downsampled every 4 pixels in both vertical and horizontal directions to generate the corresponding low resolution images as shown in the first column.",
"The bicubic interpolation of LR images and results using proposed Inf-ADMM-ADNN on a $20\\%$ held-out test set are displayed in column 2 and 3.",
"Visually, the proposed Inf-ADMM-ADNN gives much better results than the bicubic interpolation, recovering more details including colors and edges.",
"A similar task to super-resolution is motion deblurring, in which the convolution kernel is a directional kernel and there is no downsampling.",
"The motion deblurring results using Inf-ADMM-ADNN are displayed in the bottom of Fig.",
"REF and are compared with the Wiener filtered deblurring result (the performance of Wiener filter has been tuned to the best by adjusting the regularization parameter).",
"Obviously, the Inf-ADMM-ADNN gives visually much better results than the Wiener filter.",
"Due to space limitations, more simulation results are available in supplementary materials (see Section 3.1 and 3.2).",
"To explore the convergence speed w.r.t.",
"the ADMM regularization parameter $\\beta $ , we have plotted the normalized mean square error (NMSE) defined as $\\textrm {NMSE} = {\\Vert \\hat{{\\mathbf {x}}}-{\\mathbf {x}}\\Vert _2^2}/{\\Vert {\\mathbf {x}}\\Vert _2^2}$ , of super-resolved MNIST images w.r.t.",
"ADMM iterations using different values of $\\beta $ in the right of Fig.",
"REF .",
"It is interesting to note that when $\\beta $ is large, e.g., $0.1$ or $0.01$ , the NMSE of ADMM updates converges to a stable value rapidly in a few iterations (less than 10).",
"Reducing the value of $\\beta $ slows down the decay of NMSE over iterations but reaches a lower stable value.",
"When the value of $\\beta $ is small enough, e.g., $\\beta =0.0001, 0.0005, 0.001$ , the NMSE converges to the identical value.",
"This fits well with the claim in Boyd's book [2] that when $\\beta $ is too large it does not put enough emphasis on minimizing the objective function, causing coarser estimation; thus a relatively small $\\beta $ is encouraged in practice.",
"Note that the selection of this regularization parameter is still an open problem."
],
[
"Joint super-resolution and colorization",
"While image super-resolution tries to enhance spatial resolution from spatially degraded images, a related application in the spectral domain exists, i.e., enhancing spectral resolution from a spectrally degraded image.",
"One interesting example is the so-called automatic colorization, i.e., hallucinating a plausible color version of a colorless photograph.",
"To the best knowledge of the authors, this is the first time we can enhance both spectral and spatial resolutions from one single band image.",
"In this section, we have tested the ability to perform joint super-resolution and colorization from one single colorless LR image on the celebA-dataset [15].",
"The LR colorless image, its bicubic interpolation and $\\times 2$ HR image are displayed in the top row of Fig.",
"REF .",
"The ADMM updates in the 1st, 4th and 7th iterations (on held-out test set) are displayed in the bottom row, showing that the updated image evolves towards higher quality.",
"More results are in the supplementary materials (see Section 3.3).",
"Figure: (top left) colorless LR image, (top middle) bicubic interpolation, (top right) HR ground-truth,(bottom left to right) updated image in 1\\mathbf {1}th, 4\\mathbf {4}th and 7\\mathbf {7}th ADMM iteration.Note that the colorless LR images and bicubic interpolations are visually similar but different in detailsnoticed by zooming out."
],
[
"Conclusion",
"In this paper we have proposed an accelerated alternating direction method of multipliers, namely, Inf-ADMM-ADNN to solve inverse problems by using two pre-trained deep neural networks.",
"Each ADMM update consists of feed-forward propagation through these two networks, with a complexity of linear order with respect to the data dimensionality.",
"More specifically, a conditional pixel shuffling denoising auto-encoder has been learned to perform amortized inference for the proximity operator.",
"This auto-encoder leads to an implicit prior learned from training data.",
"A data-independent structured convolutional neural network has been learned from noise to explicitly invert the big matrix associated with the forward model, getting rid of any inner loop in an ADMM update, in contrast to the conventional gradient based method.",
"This network can also be combined with existing proximity operators to accelerate existing ADMM solvers.",
"Experiments and analysis on both synthetic and real dataset demonstrate the efficiency and accuracy of the proposed method.",
"In future work we hope to extend the proposed method to inverse problems related to nonlinear forward models."
],
[
"Acknowledgments",
"The authors would like to thank NVIDIA for the GPU donations."
],
[
"Learning $A$ from training data",
"The objective to estimate $A$ is formulated as $\\mathrm {arg}\\min _{A} \\sum _{i=1}^{N}\\Vert {\\mathbf {y}}_i - A{\\mathbf {x}}_i\\Vert _2^2 + \\lambda \\phi (A)$ where $({{\\mathbf {x}}_i, {\\mathbf {y}}_i})_{i=1:N}$ are the training pairs and $\\phi (A)$ corresponds to a regularization to $A$ .",
"Empirically, when $m$ is large enough, the regularization plays a less important role.",
"The learned and real kernels for $A$ (of size $4 \\times 4$ ) are visually very similar as is shown in Fig.",
"REF .",
"Figure: (left) Ground-truth kernel for AA , (middle) learned kernel for AA, (right) difference of these two."
],
[
"Structure of matrix $A$ in Section 4.1",
"The degradation matrix $A$ in strided convolution can be decomposed as the product of $H$ and $S$ , i.e., $A= SH$ , where $H$ is a square matrix corresponding to 2-D convolution and $S$ represents the regular 2-D downsampling.",
"In general, the blurring matrix $H$ is a block Toeplitz matrix with Toeplitz blocks.",
"If the convolution is implemented with periodic boundary conditions, i.e., the pixels out of an image is padded with periodic extension of itself, the matrix $H$ is a block circulant matrix with circulant blocks (BCCB).",
"Note that for 1-D case, the matrix $B$ reduces to a circulant matrix.",
"For illustration purposes, an example of matrix $B$ for a 1-D case is given as below.",
"$H=$ $\\left[\\begin{array}{ccccccccccccccccccccc}0.5&0.3&0&0&0&0&0&0&0&0&0&0&0&0&0&0.2\\\\0.2&0.5&0.3&0 &0&0&0&0&0&0&0&0&0&0&0&0\\\\0 &0.2&0.5&0.3&0&0&0&0&0&0&0&0&0&0&0&0\\\\0 &0 &0.2&0.5&0.3&0&0&0&0&0&0&0&0&0&0&0\\\\0 &0 &0 &0.2&0.5&0.3&0&0&0&0&0&0&0&0&0&0\\\\0 &0 &0 &0 &0.2&0.5&0.3&0&0&0&0&0&0&0&0&0\\\\0 &0 &0 &0 &0 &0.2&0.5&0.3&0&0&0&0&0&0&0&0\\\\0 &0 &0 &0 &0 &0 &0.2&0.5&0.3&0&0&0&0&0&0&0\\\\0 &0 &0 &0 &0 &0 &0 &0.2&0.5&0.3&0&0&0&0&0&0\\\\0 &0 &0 &0 &0 &0 &0 &0 &0.2&0.5&0.3&0&0&0&0&0\\\\0 &0 &0 &0 &0 &0 &0 &0 &0 &0.2&0.5&0.3&0&0&0&0\\\\0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0.2&0.5&0.3&0&0&0\\\\0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0.2&0.5&0.3&0&0\\\\0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0.2&0.5&0.3&0\\\\0 & 0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0.2&0.5&0.3\\\\0.3 &0 & 0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0.2&0.5\\end{array}\\right]$ An example of matrix $B$ for 2-D convolution of a $9 \\times 9$ kernel with a $16 \\times 16$ image is given in the top of Fig.",
"REF .",
"Clearly, in this huge matrix, a circulant structure is present in the block scale as well as within each block, which clearly demonstrates the self-similar pattern of BCCB matrix.",
"The downsampling matrix $S$ corresponds to downsampling the original signal and its transpose $S^T$ interpolates the decimated signal with zeros.",
"Similarly, a 1-D example of downsampling matrix is shown in (REF ) for an illustrative purpose.",
"An example of matrix $S$ for downsampling a $16 \\times 16$ image to the size of $4 \\times 4$ , i.e., $S \\in \\mathbb {R}^{16 \\times 256}$ , is displayed in the middle of Fig.",
"REF .",
"The resulting degradation matrix $A$ , which is the product of $S$ and $H$ is shown in the bottom of Fig.",
"REF .",
"$S=\\left[\\begin{array}{ccccccccccccccccccccc}1&0&0&0&{|}&0&0&0&0&{|}&0&0&0&0&{|}&0&0&0&0\\\\0&0&0&0&{|}&1&0&0&0&{|}&0&0&0&0&{|}&0&0&0&0\\\\0&0&0&0&{|}&0&0&0&0&{|}&1&0&0&0&{|}&0&0&0&0\\\\0&0&0&0&{|}&0&0&0&0&{|}&0&0&0&0&{|}&1&0&0&0\\end{array}\\right]$"
],
[
"Motion deblurring",
"The motion blurring kernel is shown in Fig.",
"REF .",
"More results of motion deblurring on held-out testing data for CUB dataset are displayed in Fig.",
"REF , REF ."
],
[
"Super-resolution",
"More results of super-resolution on held-out testing data for CUB dataset are displayed in Fig.",
"REF , REF ."
],
[
"Joint super-resolution and colorization",
"More results of joint super-resolution and colorization on held-out testing data for CelebA dataset are displayed in Fig.",
"REF .",
"Figure: 9×99 \\times 9 motion blurring kernel.Figure: motion blurred imagesFigure: deblurred results using Inf-ADMM-ADNNFigure: bicubic interpolationsFigure: HR groundtruthFigure: HR groundtruth"
],
[
"Network for updating ${\\mathbf {x}}$",
"For MNIST dataset, we did not use the pixel shuffling strategy, since each data point is a $28 \\times 28$ grayscale image, which is relatively small.",
"Alternatively, we used a standard denoising auto-encoder with architecture specifications in Table REF .",
"Table: Network Hyper-Parameters of DAE for MNISTFor CUB-200-2011 dataset, we applied a periodical pixel shuffling layer to the input image of size $256 \\times 256 \\times 3$ with the output of size $64 \\times 64 \\times 48$ .",
"Note that we did not use any stride here since we keep the image scale in each layer identical.",
"The architecture of the cPSDAE is given in Table REF .",
"For CelebA dataset, we applied the periodical pixel shuffling layer to the input image of size $128 \\times 128 \\times 3$ with the output of size $32 \\times 32 \\times 48$ , and the rest of setting is the same as CUB-200-2011 dataset, as shown in Table REF .",
"In terms of the discriminator, we fed the pixel shuffled images.",
"The architecture of the disriminator is the same as the one in DCGAN.",
"Table: Network hyper-parameters of cPSDAE for CUB-200-2011Table: Network hyper-parameters of cPSDAE for CelebA"
],
[
"Network for updating ${\\mathbf {z}}$",
"As described in Section 3.2, the neural network to update ${\\mathbf {z}}$ was designed to have symmetric architecture.",
"The details of this architecture is given in Table REF .",
"Note that $W \\times H$ represents the size of the width and height of measurement ${\\mathbf {y}}$ .",
"Table: Symmetric network hyper-parameters for updating 𝐳{\\mathbf {z}}"
]
] | 1709.01841 |
[
[
"An Efficient Loop-free Version of AODVv2"
],
[
"Abstract Ad hoc On Demand distance Vector (AODV) routing protocol is one of the most prominent routing protocol used in Mobile Ad-hoc Networks (MANETs).",
"Due to the mobility of nodes, there exists many revisions as scenarios leading to the loop formation were found.",
"We demonstrate the loop freedom property violation of AODVv2-11, AODVv2-13, and AODVv2-16 through counterexamples.",
"We present our proposed version of AODVv2 precisely which not only ensures loop freedom but also improves the performance."
],
[
"Introduction",
"Mobile Ad-hoc Networks (MANETs) have different applications from military to disastrous situations where there is no network infrastructure and nodes can freely change their locations due to mobility of nodes.",
"Mobility is the main feature of MANETs which makes them powerful and at the same time error prone in practice.",
"The process of the protocol design are not straightforward and simulations are used to validate the protocol.",
"However, all possible scenarios are not covered during simulations.",
"Since there is no base station or fixed network infrastructure, every node acts as a router and keeps the track of the previously seen packets to efficiently forward the received messages to desired destinations.",
"In essence, MANETs need routing protocols in order to provide a way of communication between two indirectly-connected nodes.",
"Ad hoc On Demand Distance Vector (AODV) routing protocol [1] is one of the most popular routing protocol used in MANETs.",
"It has two main versions, each one with several subversions.",
"The AODV specification is given in plain English, no pseudo code or implementation is provided.",
"It brings out lots of ambiguities which may lead to different implementations, or even worse, could cause the violation of important properties of AODV such as loop freedom.",
"For example, while modeling the ADOV v2 (version 11), we confronted some ambiguities that we worked them out through communication with the AODV authors.",
"For instance, when there are more than one unconfirmed routes, which one is going to be used?",
"The answer was the best one as we speculated.",
"Also what happens if it fails to receive an ack from the best one?",
"The answer was it is going to use the second best route and go on till it gets an ack.",
"It was also not clear when rerr messages are sent while dealing with unconfirmed routes, which turns out that they are never going to be sent if the route is unconfirmed.",
"Therefore, it is really necessary to have a precise specification while easy to read and understand.",
"wRebeca modeling language was introduced in [2] for the formal specification and verification of MANET protocols.",
"It not only provides a means to specify a protocol precisely in a Java-like syntax, but also it is supported by a tool to verify given properties, e.g.",
"loop freedom, on the protocol.",
"Besides, adding new features or updating the existing network protocols invalidates all the verifications that have been done on the older versions.",
"Therefore, as the process of developing the AODV protocol is an ongoing one, its verification should be too.",
"In [2], the applicability of wRebeca is shown through the modeling and verification of the AODVv2 (version 11) protocol.",
"In this paper, we focus on several versions of AODVv2 and their shortcomings to assure loop freedom.",
"First, we provide a short introduction of wRebeca and its important aspects in Section , and then we proceed by explaining AODVv2 concisely in Section .",
"We demonstrate the routing table maintenance procedure of its subversions and in their consequent, the scenarios leading to the violation of the loop freedom property of AODVv2-11, AODVv2-13, and AODVv2-16 through counterexamples in Section .",
"Such scenarios, found automatically by our framework, have been communicated with the AODV group to be validated.",
"We explain the reason in the protocol design which leads to the loop freedom violation of AODVv2-16 and present two solutions to amend the protocol in Section .",
"Then, we discuss an excessive restriction, which the protocol applies to ensure its loop freedom, and its consequence on the performance of the protocol.",
"Finally, we present our proposed version of AODVv2-16 which not only ensures loop freedom but also improves the performance.",
"Inspecting all loop scenarios, makes it clear that loop formations are caused by updating the routing table not carefully enough.",
"In fact, there are many factors that must be considered while updating the routing table, such as sequence numbers and route costs.",
"Keeping the routing table loop-free even gets more sophisticated by maintaining more than one route per each destination.",
"For example, in case only one route per each destination exists in the routing table, a new route with a greater sequence number than the existing ones simply replaces it, but now should it be added to the routing table or replace all other routes?",
"As we see in Section , adding such a new route to the routing table may lead to the loop formation.",
"As a matter of fact, the main cause of loop formation in AODVv2 (version 13) and (version 16) was mishandling the situation as a consequence of which a new route with the greater sequence number is added to the routing table.",
"In the all versions of AODV, there is a function which enforces the loop freedom condition through verifying that each incoming route is not a sub-section of any existing routes.",
"Nevertheless, when an incoming route has a greater destination sequence number, this function gets ignored to value its freshness.",
"This kind of avoidance does not cause any problem when there is at most one route to each destination and the incoming route updates the existing one.",
"However, when there are more than one route to each destination, the incoming route does not update all the existing ones which may lead to the loop formation."
],
[
"Actor Model and the wRebeca Language",
"The computational model of actors [3], [4] has been introduced for the purpose of modeling concurrent and distributed systems.",
"Such modeling has become very popular in practice [5], [6], [7]Scala programming language supports actor-models http://www.scala-lang.org.",
"Actors, the primitives of computation, are independent, well encapsulated, and of course run concurrently.",
"Each actor has its own state indicated by its state variables and its encapsulation prohibits other actors to access its state variables directly.",
"Each actor communicates with others only through message passing and owns a mailbox with a unique address to store the received messages.",
"The behavior of an actor is defined in terms of a set of message servers which specify how the actor reacts upon processing each received message.",
"For example, if one actor wants to change the other actor state variable, it should do it through sending an update message to the other actor.",
"The way this message is going to be processed is declared in the corresponding message server of the other actor.",
"In this model, message delivery is guaranteed but is not in-order.",
"This policy implicitly abstracts from the effects of network, i.e., delays over different routing paths, message conflicts, etc., and consequently makes it a suitable modeling framework for concurrent and distributed applications.",
"The modeling language Rebeca [8] provides an operational interpretation of the actor model through a Java-like syntax to fill the gap between formal verification techniques and the real-world software engineering of concurrent and distributed applications.",
"It is empowered through various extensions introduced for different domains such as probabilistic systems [9], real-time systems [10], software product lines [11], and broadcasting environment [12].",
"Mobility is the intrinsic characteristic of the MANETs which affects the correctness of MANET protocols.",
"We extended Rebeca with the concepts of MANETs to model such networks in a more succinct way, so-called wRebeca [2].",
"It is supported by a toolset for efficient verification of wRebeca models regarding the mobility of nodes.",
"wRebeca provides essential primitives for the modeling of MANET protocols, namely unicast, multicast and broadcast communications, abstracting the services of the data link layer.",
"Furthermore, the concepts of connectivity and the underlying topology are considered for actors.",
"The message delivery is guaranteed for the receiving actors connected to a sender and also is in-order as communications are one-hop.",
"Such an extension allows the modeler to setup the initial topology and specify the dynamic aspect of the networks, i.e., how the underlying topology changes through the novel concept of network constraints.",
"A network constraint establishes a set of static (dis)connectivity relations among the nodes.",
"Therefore, a wRebeca model is analyzed for all mobility scenarios respecting the constraints.",
"The wRebeca is reasonably suitable for modeling MANET protocols.",
"In this setting each network node executing an instance of a MANET protocol can be represented through an actor with some state variables and message servers.",
"There is a complete mapping between messages defined by the protocol specification, e.g., an IETF draf, and message servers.",
"The content of the message is passed through the message server arguments.",
"The body of a message server encodes how a received message is going to be processed as defined by the specification.",
"The information required to be maintained by each node is modeled by state variables.",
"Hence, there is a good traceability, from the model back to the protocol.",
"If we find a problem in the model, then we can trace it back into the protocol easier.",
"The faithfulness of the framework to the MANET domain, make it usable for analysis and design of such protocols [13].",
"As mentioned earlier, wRebeca is an extension of Rebeca with a Java-like syntax to easily read and apply.",
"Every wRebeca model consists of two parts: the reactive class declaration part and the main part.",
"Various components of the system are modeled through declaring different reactive classes.",
"Each reactive class has two major parts: one for maintaining its state, which is called statevars, and the other for specifying its reaction upon receiving different messages, i.e., message servers.",
"The body of message servers may consist of conditional, assignment, and communication statements.",
"The syntax of communication statements that worth mentioning are broadcast, multicast, and unicast.",
"Broadcasting a message is like calling a function, by indicating a message server name along with its parameters.",
"Unicasting/Multicasting a message is slightly different since we need to mention the receivers.",
"In addition, in case of unicast the modeler can specify what is going to happen regarding to the success or failure of the communication.",
"The second part of a wRebeca model is the main part which declares the instances of defined reactive classes and their initialization.",
"Furthermore, the modeler can define a set of constraints to restrict the topology changes in network.",
"For instance, if it is known that two nodes would never get connected, i.e, they would never get into each other communication range, the topologies in which these node are connected together can be ruled out from possible topologies by expressing a constraint by which the link between these node is disconnected."
],
[
"Example",
" REF shows the wRebeca specification of the AODV protocol.",
"َFor brevity some parts of the code have been abstracted away.",
"Network nodes running an instance of AODV are modeled by a reactive class, lines (1-60).",
"Each node has a routing table and an IP which is modeled by the state variables, lines (2-5).",
"Every node can receive different routing messages, i.e.",
"$\\it rreq$ , $\\it rrep$ , and $\\it rerr$ .",
"The procedure of handling these messages is modeled through declaring different message servers while each one is responsible for handling a specific message.",
"For example, the message server $\\it rec\\_rreq$ is responsible for handling received $\\it rreq$ messages, lines (21-41).",
"In line 38, an $\\it rreq$ message is broadcast while an $\\it rrep$ is unicast in line 26.",
"Modeler has specified the behavior of the protocol based on the delivery status, lines (28-35).",
"The second part of a Rebeca model is the main part, lines (65-70), where rebecs get instantiated from declared reactive classes, for example $n1$ from $Node$ .",
"The first pair of parentheses specify the initial topology by indicating the name of other rebecs which are initially in the rebec neighborhood.",
"For example, $n2$ is initially connected to $n1$ and $n4$ .",
"The second pair of parentheses, after colon, specify the parameters of the initial message which is going to be processed by the declared initial message server.",
"Each reactive class declaration at least has one message server namely initial which acts like a constructor in object-oriented languages and used for initialization purposes, lines (6-9).",
"For example, initializing state variables, routing table variables and starting a new route discovery by sending a new packet.",
"As mentioned earlier, the main part in wRebeca has another part named network constraints, lines (69-71).",
"This part is used to reduce the domain of the possible topologies.",
"For example, if it is impossible for $n1$ to get out of the communication range of $n2$ and vice versa, modeler can express this situation by declaring a network constraint containing the relation ${\\it con}(n1, n2)$ .",
"$\\square $"
],
[
"Overview of AODVv2 ",
"The AODV protocol is under continuous development and its working group publish a new version at most every 6 months with the aim to improve the protocol and amend its shortcomings.",
"However, all its (sub)versions almost follow the same design concept.",
"More specifically, it uses some specific routing packets, e.g., $\\it rreq$ , $\\it rrep$ , and $\\it rerr$ , but the way these packets are sent and processed differs in every version.",
"In this section, we briefly explain the common procedure of route discovery and maintenance among its variants.",
"The wRebeca specification of its common code between versions 10, 11, 13, and 16 is given in REF .",
"Some parts of the code, abstracted in this specification, e.g., the one commented by “processing code”, vary in different versions.",
"Figure: The AODV specification given in wRebecaIn this protocol, routes are built upon route discovery requests and maintained in nodes routing tables for further use.",
"The routing table contains information about discovered routes and their status: The following information is maintained for each route: SeqNum: destination sequence number route_state: the state of the route to the destination Metric: indicates the cost or quality of the route, e.g., hop count, the length of the path from the node to the destination via the respective next hop NextHop: IP address of the next hop to the destination The routing table of each node is modeled by a set of variables of array type, namely ${\\it dsn}$ , ${\\it rst}$ , ${\\it hops}$ , and ${\\it nhop}$ to denote SeqNum, route_state, Metric, and NextHop, respectively.",
"In addition, ${\\it dip\\_}$ and ${\\it oip\\_}$ denote destination and originator IPs which are used as indexes to retrieve the information of a route to destination/originator in such arrays.",
"For instance, ${\\it rst}[oip\\_]$ denotes the route state to the originator.",
"Whenever a node intends to send a data packet to another, i.e., when it receives a $\\it newpkt$ message, it looks up its routing table to see if it has a valid route to the intended destination, line 12 of $\\it rec\\_newpkt$ .",
"In case it finds a route, it sends the data packet through the next hop specified in that route, otherwise it starts a route discovery by broadcasting a route request, i.e.",
"$\\it rreq$ after increasing its sequence number, lines (14-17) of $\\it rec\\_newpkt$ .",
"Whenever a node receives a new routing packet, $\\it rreq$ , it updates its routing table with new information to keep it up-to-date, abstracted code at line 22 of $\\it rec\\_rreq$ .",
"$\\it rreq$ messages contain route towards a source while $\\it rrep$ messages carry route information towards a destination.",
"Therefore, as an $\\it rreq$ packet proceeds towards the destination, in each node, a backward path, a path to the source from the node, gets constructed.",
"Similarly, a forward path, a path to the destination from the node, is built while $\\it rrep$ packets traverse the constructed backward path from the destination towards the source.",
"Each node upon receiving an $\\it rreq$ message looks up its routing table and if it has a route to the requested destination it would reply through sending an $\\it rrep$ , lines (25-35) of $\\it rec\\_rreq$ otherwise, it resends the $\\it rreq$ message after increasing the hop count if the maximum number of hop count limit is not reached, lines (36-40) of $\\it rec\\_rreq$ .",
"Whenever a node receives an $\\it rrep$ message, it updates its routing table accordingly to construct a forward path, the abstracted code at line 45 of $\\it rec\\_rrep$ .",
"When the $\\it rrep$ reaches the source, abstracted code at line 48 of $\\it rec\\_rrep$ , a bidirectional route has been formed and the data packet can be sent through next-hops on nodes routing tables towards the destination.",
"When a node which is not the source receives a $\\it rec\\_rrep$ message, it unicasts the $\\it rec\\_rrep$ toward the source after increasing the hop count, lines (49-58) of $\\it rec\\_rrep$ .",
"In addition to the $\\it rreq$ and $\\it rrep$ packets, there is an $\\it rerr$ packet which is sent whenever a node fails to send a packet through a valid route, line 33 of $\\it rec\\_rreq$ and line 58 of $\\it rec\\_rrep$ , in order to informs other interested nodes in the broken route about the failure.",
"From version 10, a new ability has been added to the protocol to maintain more than one route to a destination.",
"For each destination, multiple routes may exist with different next-hops, i.e., unconfirmed next-hop, a next-hop which its bidirectionality has not been confirmed yet.",
"Whenever an $\\it rrep$ is going to send a package to an unconfirmed next-hop, it must request an $\\it ack$ from the receiver to become sure about its bidirectionality.",
"This new feature improves the performance since for sending a packet there is no need to wait for a next-hop to get confirmed, and consequently its route to become valid.",
"Although having multiple routes to one destination has its benefits, it can lead to a loop formation when it is used with not required precautions as we are going to explain in the following section."
],
[
"Loop formation Scenarios",
"We explain how different versions of AODVv2 protocol try to prevent loop formation and how they fail to do so through counterexamples which are obtained by our tool.",
"The AODV protocol manuscript has different sections, e.g.",
"initialization, adjacency monitoring, route maintenance and processing received route information [1].",
"For the purpose of loop formation avoidance, we will focus on the processing received route information part of the specification since a loop is formed if and only if preventative measures have not been taken to account while updating the routing tables.",
"Therefore, in the section for the sake of simplicity, we only focus on evaluating received route information and consequently updating the routing tables, abstracted in the specification of REF and commented by “processing code”.",
"For a comprehensive specification, we refer the interested reader to their corresponding IETF drafts."
],
[
"AODVv2-11",
"This version maintains more than one next hop per each destination which increases the probability of packet delivery since if one route gets broken, there may be other routes that can be used as an alternative.https://tools.ietf.org/html/draft-ietf-manet-aodvv2-11 When there are more than one route, the best one would be used.",
"The best route is chosen based on the concept of route state and cost.",
"Route states are determined by the concept of neighbor states of next hops which determines the adjacency states of the node's neighbors, and can have one of the following values: Confirmed: indicates that a bidirectional link to that neighbor exists.",
"This state is achieved either through receiving an rrep message in response to a previously sent rreq message, or an rrep_ack message as a response to a previously sent rrep message (requested an rrep_ack) to that neighbor.",
"Unknown: indicates that the link to that neighbor is currently unknown.",
"Initially, the states of the links to the neighbors are unknown.",
"Blacklisted: indicates that the link to that neighbor is unidirectional.",
"When a node has failed to receive the rrep_ack message in response to its rreq message to that neighbour, the neighbor state is changed to blacklisted.",
"Hence, it stops forwarding any message to it for an amount of time, ResetTime.",
"After reaching the ResetTime, the neighbor's state will be set to unknown.",
"Such information are kept in the neighbor table of each node.",
"Route states, the states of the routes to each destination, are kept in the routing table and can have one of the following values: unconfirmed: when the neighbor state of the next hop is unknown; active: when the link to the next hop has been confirmed, and the route is currently used; idle: when the link to the next hop has been confirmed, but it has not been used in the last active_interval; invalid: when the link to the next hop is broken, i.e., the neighbor state of the next hop is blacklisted.",
"A route is called valid if it is either active or idle.",
"Although there can exist more than one unconfirmed route to each destination, there can be only one valid route to each destination.",
"When a route state to a destination gets changed to valid, all the routes to the same destination are removed from routing table.",
"Every received route message contains a route and consequently is evaluated to check for any improvement.",
"Note that an $\\it rreq$ message contains a route to its source while an $\\it rrep$ message contains a route to its destination.",
"Therefore, as the routes are identified by their destinations, in the former case, the destination of the route is the originator of the message and in the latter, it is the destination of the message.",
"Note that we say a router is better then others if it has either a greater sequence number than others or an equal sequence number while its cost, e.g., hop count, is less than others.",
"The routing table must be updated if one of the following conditions is realized: no route to the destination exists in the routing table: the route is added to the routing table.",
"all the existing routes to the destination are unconfirmed, i.e., their next hops are unconfirmed: the route is added to the routing table.",
"the incoming route is a better route than the existing valid one: if the next hop of the incoming route is confirmed, it updates the existing valid route with the received route, otherwise it adds the received route to the routing table since it may be confirmed in the future and consequently, replaces the existing route.",
"the incoming route is a better route than the existing invalid one: it updates the existing invalid route with the incoming route."
],
[
"Loop Formation Scenario",
"In this version no constrain has been applied to the unconfirmed next-hop of an incoming route prior its addition to the routing table when the route status of the existing routes are unconfirmed (the second case in Section REF ).",
"This lack of restriction easily leads to a looping scenario which is described in the following.",
"Assume that each route entry of the routing table has the following format: $({\\it dest}, {\\it next\\_hop},{\\it hop\\_count}, {\\it seq\\_num}, {\\it route\\_state})$ , where the first element indicates IP of the destination, the second, IP of the next hop, the third, the length of the route to the destination, the forth, the sequence number of the destination, and the last, the route state, respectively.",
"Consider a network of four nodes as shown in REF .",
"Figure: Two possible network topologies for a network of four nodes $n_1$ initiates a route discovery procedure for destination $n_3$ by broadcasting an ${\\it rreq}$ message with the sequence number 2.",
"$n_2$ receives ${\\it rreq}$ message as it is a neighbor of $n_1$ .",
"Since it is the first time that $n_2$ has received an ${\\it rreq}$ message from $n_1$ , the neighbor state of $n_1$ is set to unconfirmed.",
"Therefore, the route state of the received route is unconfirmed, and $n_1$ adds the incoming route $(n_1, n_1, 1, 2, {\\it unconfirmed})$ to its routing table.",
"Since $n_2$ is not the intended destination of the route request, it rebroadcasts an ${\\it rreq}$ message.",
"$n_4$ also receives the ${\\it rreq}$ message sent by $n_1$ (simultaneous with $n_2$ ) and inserts the incoming route $(n_1, n_1, 1, 2, {\\it unconfirmed})$ to its routing table towards $n_1$ similar to $n_2$ .",
"Then, it rebroadcasts the ${\\it rreq}$ message.",
"$n_2$ after receiving the ${\\it rreq}$ message sent by $n_4$ , adds the route $(n_1, n_4, 2, 2,\\\\ {\\it unconfirmed})$ to its routing table since the existing route to $n_1$ , i.e., $(n_1, n_1, 1, 2, {\\it unconfirmed})$ , is unconfirmed.",
"$n_4$ also adds $(n_1, n_2, 2, 2, {\\it unconfirmed})$ to its routing table after processing the message ${\\it rreq}$ sent by $n_2$ .",
"At this point a loop is formed between $n_2$ and $n_4$ .",
"$n_3$ receives the ${\\it rreq}$ message sent by $n_1$ via $n_2$ , and since it is the destination, it sends an ${\\it rrep}$ message towards $n_2$ .",
"Assume that $n_1$ moves out of the communication ranges of $n_2$ and $n_4$ .",
"$n_2$ receives the message ${\\it rrep}$ sent by $n_3$ and as the route state of the routes towards $n_1$ is unconfirmed, it unicasts an ${\\it rrep}$ message one by one to the existing next hops, i.e., $n_1$ and $n_4$ , till it gets an ack.",
"Due to the movement of $n_1$ , it receives no ack from $n_1$ and the route with the next hop $n_1$ is removed from the routing table.",
"However, it receives an ack from $n_4$ .",
"Therefore, the neighbor state of $n_4$ is set to confirmed and subsequently the respective route state towards $n_1$ to valid.",
"$n_4$ by receiving the message ${\\it rrep}$ from $n_2$ unicasts it to its next hops, i.e., $n_1$ and $n_2$ , similar to $n_2$ .",
"Since it only receives an ack from $n_2$ , it updates its routing table by validating $n_2$ as its next hop to $n_1$ , and hence a loop is formed between $n_2$ and $n_4$ over valid routes.",
"After communicating our result on AODVv2-11 to the AODV group, they revised the protocol to restrict the addition of unconfirmed routes when all the existing routes to a destination are unconfirmed.",
"Hence, only the second step of the procedure of Section REF is revised: an incoming route is added to the routing table if all the existing routes to its destination are unconfirmed while the incoming is better the existing ones.https://tools.ietf.org/html/draft-ietf-manet-aodvv2-13"
],
[
"Loop Formation Scenario",
"Although the scenario of Section REF is prohibited, a loop scenario occurs due to resending the ${\\it rreq}$ messages in a network of four nodes with the topologies shown in REF .",
"At first nodes are connected to each other as shown in REF .",
"$n_1$ initiates a route discovery procedure for destination $n_4$ by broadcasting an ${\\it rreq}$ message to $n_3$ with the sequence number 2.",
"$n_3$ inserts the incoming route $(n_1, n_1, 1, 2, {\\it unconfirmed})$ to its routing table and broadcasts an ${\\it rreq}$ message to its neighbors, $n_2$ and $n_4$ .",
"$n_2$ upon receiving the message ${\\it rreq}$ sent by $n_3$ updates its routing table and adds the incoming route $(n_1, n_3, 2, 2, {\\it unconfirmed})$ to its routing table.",
"topology changes at this point and $n_2$ moves into the communication range of $n_1$ , gets connected to $n_1$ , while $n_3$ leaves the communication range of $n_1$ , gets disconnected from $n_1$ , which leads to the network topology shown in REF .",
"$n_1$ , which has not received an ${\\it rrep}$ message yet, resends the message ${\\it rreq}$ after increasing its sequence number to 3 (due to the timeout to receive such a reply).",
"$n_2$ receives the incoming route $(n_1, n_1, 1, 3, {\\it unconfirmed})$ , since it is a better route it would be added to the routing table.",
"Then, $n_2$ broadcasts an ${\\it rreq}$ message to its neighbors, i.e., $n_3$ and $n_4$ .",
"$n_3$ evaluates the received message sent by $n_2$ and adds the incoming route $(n_1, n_2, 2, 3, {\\it unconfirmed})$ to its routing table since the sequence number of the received message is greater than the stored one, i.e, $(n_1, n_1, 1, 2, {\\it unconfirmed})$ .",
"At this point a loop has been formed between nodes $n_2$ and $n_3$ , similar to the step REF of the scenario explained in Section REF for version 11.",
"Therefore, continuing with a scenario similar to the steps REF -REF of the scenario for version 11, a loop is formed between $n_2$ and $n_3$ over valid routes.",
"This loop scenario occurs because the existing unconfirmed route $(n_1, n_1, 1,\\\\ 2, {\\it unconfirmed})$ has not been replaced by the received better route $(n_1, n_2, 2, 3,\\\\ {\\it unconfirmed})$ .",
"Instead, the received new route is only added to the table.",
"We remark that a new route replaces an existing one only when the route state of the existing route is invalid or the route state of the new route is confirmed."
],
[
"AODVv2-16",
"It is the last AODVv2 protocol which applies even more restrictions for updating the routing table to ensure loop freedom.https://tools.ietf.org/html/draft-ietf-manet-aodvv2-16 It maintains at most two routes for each destination while one is (in)valid and the other is unconfirmed.",
"To prevent loops in this version, an incoming route updates the existing route with the same status.",
"In case no route exists with the same status, it will be added to the table.",
"Therefore, the routing table always keeps better routes for each status."
],
[
"Updating the Routing Table",
"The updating procedure has been revised accordingly: no route exists to the destination: the route is added to the routing table.",
"the incoming route is better than the existing one.",
"Two cases can be distinguished: there is only one matching route with the same destination: the route state of the existing route is invalid: the incoming route must replace the existing one; the route state of the incoming route and the existing one are the same: the incoming route should replace the existing one.",
"the route state of the incoming route is unconfirmed and it offers improvement to the existing valid route: the incoming route should be added to the routing table.",
"there are two matching routes with the same destination where one is valid/invalid and the other is unconfirmed: if the incoming route offers improvement to the existing route with the same status, then it should replace it.",
"if the existing route is invalid and the incoming route is valid: the existing route is replaced by the incoming route even if the incoming route does not offer improvement."
],
[
"Loop Formation Scenario",
"The loop scenario is given for a network of four nodes with the network topologies shown in REF with the initial topology illustrated in REF : $n_1$ initiates a route discovery procedure for destination $n_4$ by broadcasting an ${\\it rreq}$ message to $n_3$ with the sequence number 2.",
"$n_3$ inserts the incoming route $(n_1, n_1, 1, 2, {\\it unconfirmed})$ in its routing table and broadcasts an ${\\it rreq}$ message to $n_2$ and $n_4$ .",
"$n_2$ receives the message ${\\it rreq}$ sent by $n_3$ and updates its routing table by inserting the route $(n_1, n_3, 2, 2, {\\it unconfirmed})$ into its routing table.",
"$n_2$ becomes aware that its connectivity to $n_3$ is bidirectional, for example through receiving an ${\\it rrep}\\_{\\it ack}$ from $n_3$ in response of a sent ${\\it rrep}$ message for another route, therefore the neighbor state of $n_3$ is updated to confirmed and route states of all those routes which use $n_3$ as their next hops must be updated to valid.",
"As a result, the route entry $(n_1, n_3, 2, 2, {\\it unconfirmed})$ of $n_2$ 's routing table gets updated to $(n_1, n_3, 2, 2, {\\it valid})$ .",
"the topology changes at this point and $n_3$ moves out of the communication range of $n_1$ while $n_2$ enters the communication range of $n_1$ , which lead to the network topology shown in REF .",
"$n_1$ resends another ${\\it rreq}$ message with the increased sequence number of 3 to $n_2$ (due to the timeout for receiving a reply).",
"$n_2$ processes the received ${\\it rreq}$ message from $n_1$ , since it has the greater sequence number than the stored one, it is used to update the routing table.",
"As the stored route with next hop $n_3$ is valid, the incoming route $(n_1, n_1, 1, 3, {\\it unconfirmed})$ is added to the routing table as a new route.",
"Then, $n_2$ broadcasts the received ${\\it rreq}$ to its neighbors.",
"the topology changes at this point and $n_3$ moves into the communication range of $n_1$ , gets connected to $n_1$ which leads to network topology shown in REF .",
"assume that the connectivity status of $n_3$ to $n_1$ becomes bidirectional, therefore the route entry $(n_1, n_1, 1, 2, {\\it unconfirmed})$ of $n_3$ 's routing table gets updated to $(n_1, n_1, 1, 2, {\\it valid})$ .",
"$n_3$ receives the incoming route $(n_1, n_2, 2, 3, {\\it unconfirmed})$ via the ${\\it rreq}$ message sent by $n_2$ .",
"Since the incoming route has a greater sequence number than the stored one and the stored one is valid, it will be added to the routing table.",
"At this point a loop between $n_2$ and $n_3$ is formed.",
"Again by continuing with a scenario similar to the steps REF -REF of the scenario for version 11, a loop is formed between $n_2$ and $n_3$ over valid routes.",
"Figure: Three possible network topologies for a network of four nodesBy examining the counter example, we realize that a loop is formed as the loop freedom condition is not always considered and consequently, the new route will be added to the routing table when the sequence number is greater than the existing valid one.",
"To amend this situation, we propose two options: The loop freedom condition should be always considered.",
"Therefore, if the new route does not satisfy the loop freedom condition, it must not be used to update the routing table even if it has a greater sequence number.",
"The new route with a greater sequence number will be added to the routing table while all the existing routes are removed from the routing table.",
"These two approaches differs regarding how they prioritize a new route with a greater sequence number and an existing route.",
"The first solution prefers to keep the valid one by ignoring the new route with a greater sequence number while the second one favors the new route with a greater sequence number over existing routes even the valid ones.",
"Nevertheless, we believe that there is a better approach which not only ensures loop freedom but also boosts the performance by maintaining more eligible routes for forwarding a packet to a destination.",
"Irrespective of which solution is being adopted, we demonstrate through an example how the protocol fails to forward a packet while there could have existed a route if the routing table had been updated better.",
"The example is given for a network which consists of seven nodes with the topologies shown in the REF .",
"Figure: Two possible network topologies for a network of seven nodes $n_1$ initiates a route discovery procedure for the destination $n_7$ by broadcasting an ${\\it rreq}$ message with the sequence number 2.",
"$n_2$ , $n_3$ , and $n_4$ update their routing tables upon receiving the ${\\it rreq}$ message sent by $n_1$ , and broadcast the ${\\it rreq}$ message to their neighbors.",
"$n_5$ receives the ${\\it rreq}$ message sent by $n_2$ and after updating its routing table broadcasts it.",
"$n_6$ receives the ${\\it rreq}$ message sent by $n_5$ and adds the route $(n_1, n_5, 3, 2, {\\it unconfirmed})$ to its routing table.",
"Then, it broadcasts the ${\\it rreq}$ message to $n_7$ .",
"assume that the connectivity status of $n_5$ to $n_6$ becomes bidirectional, therefore, the route $(n_1, n_5, 3, 2, {\\it unconfirmed})$ gets updated to $(n_1, n_5, 3, 2, {\\it valid})$ .",
"$n_6$ receives the ${\\it rreq}$ message sent by $n_3$ and since it is a better route and the stored one is a valid one, the incoming route $(n_1, n_3, 2, 2, {\\it unconfirmed})$ is added to the routing table.",
"$n_6$ receives the ${\\it rreq}$ message sent by $n_4$ and since it doe not improve the existing unconfirmed route, it gets discarded.",
"the topology changes at this point as $n_5$ and $n_3$ move out of the communication range of $n_6$ which leads to the network topology shown in REF .",
"$n_7$ receives the ${\\it rreq}$ message sent by $n_6$ and since it is the destination, it replies through sending an ${\\it rrep}$ message to $n_6$ .",
"$n_6$ receives the ${\\it rrep}$ message sent by $n_7$ .",
"To forward its ${\\it rrep}$ message to the originator, i.e.",
"$n_1$ , it has two next hops in its routing table to the destination $n_1$ : $n_3$ and $n_5$ .",
"Both next hops are going to fail to deliver the message since they have got disconnected due to the topology change.",
"Although the route through $n_4$ does exist, it had been ignored.",
"As the number of nodes increases, the chance of having more than one unconfirmed route and consequently, the effect of ignoring them on the performance raises.",
"In the following section, we present a solution which not only satisfies the loop freedom invariant but also improves the performance by preserving multiple routes for each destination."
],
[
"Proposed Procedure for Updating the Routing Table",
"According to the scenarios mentioned for the different versions of AODV, the main reason leading to loop formation is ignoring the loop freedom condition.",
"In the previous subsection we presented two solutions.",
"While these solutions are loop-free, they impose some restrictions which can degrade the performance.",
"Hence, we present the modified version of these approaches while lifting the unnecessary restriction so that it is possible to have multiple routes to the same destination.",
"Although it is possible to have an infinite number of routes, it is more realistic to bound it since there is a trade off between the storage cost and the performance."
],
[
"Solution 1: Preferring Hop count to Freshness",
"In this approach we treat an incoming route with a greater sequence number in a same way we handle an incoming route with an equal sequence number compared to the existing routes.",
"It means that the loop freedom condition is always checked.",
"The procedure of evaluating the incoming route and updating the routing table is modified accordingly: Figure: The first solution for updating the routing table no route exists to the destination: the route is added to the routing table.",
"the sequence number of the incoming route is equal or greater than all the existing routes to the same destination while its cost, e.g., hop count, is equal or less than all the existing routes: the incoming route is added if the bound has not been already reached.",
"otherwise, no change is applied to the table.",
"The precise specification of this procedure is depicted in the REF .",
"This code replaces the abstracted code at line 22 in the specification of REF in the body of the message server handling ${\\it rreq}$ .",
"We remark that ${\\it rreq}$ and ${\\it rrep}$ messages have parametrizes such as the hop count that the message has been relayed from the originator, destination IP, destination sequence number, originator IP (the origin of the message), and sender IP, specified by $\\it hops\\_$ , $\\it dip\\_$ , $\\it dsn\\_$ , $\\it oip\\_$ , and $\\it sip\\_$ , respectively.",
"As the destination of the route in an ${\\it rreq}$ message is its originator, this code uses ${\\it oip\\_}$ in its evaluations.",
"However, the destination of the route in an ${\\it rrep}$ message is identified by ${\\it dip\\_}$ .",
"Therefore, the code replacing the abstracted code at line 45 in the body of the message server handling ${\\it rrep}$ will be the same while ${\\it dip\\_}$ is used in the evaluations.",
"As more than one route is maintained for each destination, the variables $\\it dsn$ , $\\it rst$ , $\\it hops$ , $\\it nhop$ of the specification of REF become two dimensional of size $n\\times n$ , where $n$ is the number of nodes.",
"The second dimension keeps information of the alternative routes via different next hops for each destination.",
"Thus, ${\\it hops}[i][j]$ indicates the hop count of the $j$ -th route to the destination $i$ .",
"Similarly, ${\\it rst}[i][j]$ refers to the state of the $j$ -th route to the destination with IP $i$ which can have the values 0, 1, or 2 to indicate that the route is unconfirmed, valid, or invalid, respectively.",
"Furthermore, a variable ${\\it neigh\\_state}$ is added to the specification to keep the adjacency state of the neighbors, where ${\\it neigh\\_state}[i]={\\it true}$ indicates that the node is adjacent to the node with the IP address $i$ , while ${\\it false}$ indicates that its adjacency status is either unknown or blacklisted (since timing issues are not taken into account, these two statuses are considered the same).",
"Lines 1-12 add the incoming route if no route previously exists.",
"The route state of the incoming route is set in terms of the neighbor state of the sender message, i.e., ${\\it neigh\\_state}[{\\it sip\\_}]$ .",
"Lines 14-22 check whether the incoming route is a better route than the existing ones.",
"If the incoming route is a loop free, in this solution a route which is not older or longer than the existing ones, then the routing table gets updated, lines 24-48.",
"Lines 26-30 check whether there already exists a route from the sender that must be updated or it should be added to the first empty location.",
"If the neighbor state of the sender is confirmed, all the other routes must be cleared by reinitializing corresponding elements to $-1$ , lines 37-43."
],
[
"Solution 2: Preferring Freshness to Hop count",
"In this approach, we favor incoming routes with greater sequence numbers over the existing routes even the valid ones.",
"Since keeping routes with different sequence numbers jeopardizes the satisfaction of the loop freedom property, all the existing routes to the same destination as the incoming route must be removed from the routing table prior to adding the new route to the routing table.",
"no route exists to the destination: the route is added to the routing table.",
"the sequence number of the incoming route is equal to the existing one while its cost, e.g.",
"hop count, is equal or less than the existing one: the incoming route is added if the bound has not been reached already.",
"the sequence number of the incoming route is greater than all the existing routes to the same destination: the incoming route is added to the routing table after removing all the existing routes to the destination.",
"The precise specification of this procedure is depicted in REF .",
"Variables $\\it rst$ and $\\it nhops$ are defined two-dimensional similar to Section REF .",
"However, $\\it dsn$ and $\\it hops$ arrays are defined one-dimensional since in this solution we always keep routes with the greatest destination sequence number and the least hop counts, and hence all the route to the same destination will have the same destination sequence number and hop count.",
"Lines 3-14 adds the incoming route to the routing table when no route exists to the destination.",
"Then in line 18, the loop free condition is checked, in this solution we consider a route loop-free if it has a larger destination sequence number or an equal one while it has a better hop count than the existing ones.",
"If the neighbor state of the sender is confirmed, the route must be added to the table with the valid route state while all the other routes are cleared, lines 22-31.",
"In lines 34-44, the routing table gets updated with the incoming route which has a better hop count.",
"Otherwise, the route has a greater destination sequence number while its hop count is worse than the existing one.",
"Lines 46-54 checks whether there exists a valid route to the destination.",
"If there is no such a route to prevent loop formation, all the other routes must be cleared prior to adding the new route the routing table, Lines 56-66.",
"Figure: The second solution for updating the routing table"
],
[
"Related Work",
"AODV as a routing protocol of MANETs, which is rapidly growing, drew lots of attention to itself.",
"Modeling and verification of AODV protocol has been the main topic of a great deal of studies.",
"Many publications examined loop freedom as the most important property of this protocol through different approaches from extending the existing formal frameworks like SPIN [14], [15] and UPPAAL [16], [17], [18] to proposing new frameworks like CBS# [19], CWS [20], CMN [21], the $\\omega $ -calculus [22], bA$\\pi $ [23], CMAN [24], [25], RBPT [26] and the bpsi-calculi [27], [28] to support the requirements of the new environment, i.e.",
"MANETs, such as modeling the underlying topology, mobility and local broadcast.",
"However, these approaches can not be easily adopted by a user not familiar with formal modeling concepts such as process algebra and timed automata.",
"The Java-like syntax of wRebeca and its inherit friendliness brought up by the actor model, make it a suitable modeling approach that can be used by protocol designers at the early stages of their protocol development.",
"AODV was analyzed in [15] for some special mobility scenarios (as a part of the specification).",
"A scenario leading to a loop was first discovered in [29].",
"In [30], the route discovery procedure of AODV was analyzed and it was shown that in [31] that for all arbitrary number of nodes, the protocol is loop-free.",
"The loop freedom of AODVv2-04 for an arbitrary number of nodes was examined in [32] through an inductive and compositional proof: It provides an inductive invariant and proves that it is held initially and also preserved by every action, either a protocol action or a change in the network, similar to the approach of [31].",
"They have reported two loop-formation scenarios due to inappropriate setting of timing constants and accepting any valid route when the current route is broken without any further evaluation (to ensure loop formation).",
"[33] propose a process algebra for wireless mesh networks, called AWN, which addresses the main challenges of MANETs.",
"It demonstrates the applicability of such framework through model AODV and proving loop freedom condition.",
"There are several studies such as [34], [31], [35], [36] that use AWN to model and analyze different versions of AODV and verify key properties of protocol namely loop freedom.",
"[35] models dynamic MANET on-demand (DYMO) routing protocol (also known as AODVv2) and shows how it solves some problems discovered in AODV and how it fails to address all the shortcomings.",
"[34] points out some ambiguities in the RFC then analyzes different readings of the AODV RFC, and show which interpretations are loop free.",
"AWN is also used in [36] to shown that ambiguities in RFC can lead to loop formation and monotonically increasing sequence numbers, by themselves, do not guarantee loop freedom."
],
[
"Conclusion and Future Work",
"Aodv is a well-known and yet complex routing protocol which its most important property is loop freedom.",
"Many studies showed the violation of this property over different version or even proved its correctness for some versions.",
"Though loop freedom is preserved for some versions, an even a small change led to a loop formation.",
"Therefore, it is desirable to have an ongoing verification in parallel with its design and development.",
"In this paper, we illustrated the loop freedom violation for AODVv2-11,13 and 16 through 3 counterexamples and explained the reasons led to these loop formation scenarios.",
"As the protocol evolves, counterexamples get more complex and harder to guess.",
"Therefore, we need an automated tool to facilities the verification.",
"wRebeca not only makes verification very easy by handling all difficulties of MANETs behind the scene, it results in an accurate specification.",
"Having a precise specification prevents any ambiguity and facilitates the implementation.",
"In addition to reporting the counterexamples, we proposed two solutions to make AODVv2-16 loop free, respecting two different aspects of the performance.",
"One aspect favors new incoming routes over existing routes even the valid ones while the other values validity over freshness.",
"Then we showed how AODVv2-16 fails to recognize and use loop free routes and therefore fails to deliver a packet even if a loop free route does exist.",
"This problem occurs due to over limitation of route maintenance.",
"Finally, we proposed two solutions which are loop free while having better performance.",
"They targeted two different aspects of performance as solutions for AODVv2-16.",
"Based on our verifications for a network of five nodes while considering all possible topologies, by not applying any network constraint, these two solutions are loop free.",
"In addition, the proposed protocols are specified in a wRebeca model which has a Java-like syntax which makes it easy to read and comprehend.",
"We plan to extend our framework to support timed aspects of MANET protocols to analyze real-time behavior of wireless network protocols.",
"This extension enables us to model and verify the AODV protocol while considering its timing parameters."
]
] | 1709.01786 |
[
[
"Efficient Spherical Designs with Good Geometric Properties"
],
[
"Abstract Spherical $t$-designs on $\\mathbb{S}^{d}\\subset\\mathbb{R}^{d+1}$ provide $N$ nodes for an equal weight numerical integration rule which is exact for all spherical polynomials of degree at most $t$.",
"This paper considers the generation of efficient, where $N$ is comparable to $(1+t)^d/d$, spherical $t$-designs with good geometric properties as measured by their mesh ratio, the ratio of the covering radius to the packing radius.",
"Results for $\\mathbb{S}^{2}$ include computed spherical $t$-designs for $t = 1,...,180$ and symmetric (antipodal) $t$-designs for degrees up to $325$, all with low mesh ratios.",
"These point sets provide excellent points for numerical integration on the sphere.",
"The methods can also be used to computationally explore spherical $t$-designs for $d = 3$ and higher."
],
[
"Introduction",
"Consider the $d$ -dimensional unit sphere $\\mathbb {S}^{d} = \\left\\lbrace {x}\\in \\mathbb {R}^{d+1}: | {x}| = 1 \\right\\rbrace $ where the standard Euclidean inner product is ${x}\\cdot {y}= \\sum _{i=1}^{d+1} x_i y_i$ and $| {x}|^2 = {x}\\cdot {x}$ .",
"A numerical integration (quadrature) rule for $\\mathbb {S}^{d}$ is a set of $N$ points ${x}_j \\in \\mathbb {S}^{d}, j = 1,\\ldots ,N$ and associated weights $w_j > 0, j = 1,\\ldots ,N$ such that $Q_N(f) := \\sum _{j=1}^N w_j f({x}_j) \\approx I(f) := \\int _{\\mathbb {S}^{d}} f({x}) d \\sigma _d({x}) .$ Here $\\sigma _d({x})$ is the normalised Lebesgue measure on $\\mathbb {S}^{d}$ with surface area $\\omega _d := \\frac{2 \\pi ^{(d+1)/2}}{\\Gamma ((d+1)/2)},$ where $\\Gamma (\\cdot )$ is the gamma function.",
"Let $\\mathbb {P}_t(\\mathbb {S}^{d})$ denote the set of all spherical polynomials on $\\mathbb {S}^{d}$ of degree at most $t$ .",
"A spherical $t$ -design is a set of $N$ points $X_N=\\lbrace {x}_1,\\ldots ,{x}_N\\rbrace $ on $\\mathbb {S}^{d}$ such that equal weight quadrature using these nodes is exact for all spherical polynomials of degree at most $t$ , that is $\\frac{1}{N} \\sum _{j=1}^N p({x}_j) =\\int _{\\mathbb {S}^{d}} p({x}) \\mbox{d}\\sigma _d({x}),\\quad \\forall p \\in \\mathbb {P}_t(\\mathbb {S}^{d}).$ Spherical $t$ -designs were introduced by Delsarte, Goethals and Seidel [24] who provided several characterizations and established lower bounds on the number of points $N$ required for a spherical $t$ -design.",
"Seymour and Zaslavsky[55] showed that spherical $t$ -designs exist on $\\mathbb {S}^{d}$ for all $N$ sufficiently large.",
"Bondarenko, Radchenko and Viazovska [8] established that there exists a $C_d$ such that spherical $t$ -designs on $\\mathbb {S}^{d}$ exist for all $N \\ge C_d \\; t^d$ , which is the optimal order.",
"The papers [21], [5], [20] provide a sample of many on spherical designs and algebraic combinatorics on spheres.",
"An alternative approach, not investigated in this paper, is to relax the condition $w_j = 1/N$ that the quadrature weights are equal so that $|w_j / (1/N) - 1| \\le \\epsilon $ for $j = 1,\\ldots ,N$ and $0 \\le \\epsilon < 1$ , but keeping the condition that the quadrature rule is exact for polynomials of degree $t$ (see [57], [69] for example).",
"The aim of this paper is not to find spherical $t$ -designs with the minimal number of points, nor to provide proofs that a particular configuration is a spherical $t$ -design.",
"Rather the aim is to find sequences of point sets which are at least computationally spherical $t$ -designs, have a low number of points and are geometrically well-distributed on the sphere.",
"Such point sets provide excellent nodes for numerical integration on the sphere, as well as hyperinterpolation [56], [40], [59] and fully discrete needlet approximation [65].",
"These methods have a requirement that the quadrature rules are exact for certain degree polynomials.",
"More generally, [41] provides a summary of numerical integration on $\\mathbb {S}^{2}$ with geomathematical applications in mind.",
"A spherical harmonic of degree $\\ell $ on $\\mathbb {S}^{d}$ is the restriction to $\\mathbb {S}^{d}$ of a homogeneous and harmonic polynomial of total degree $\\ell $ defined on $\\mathbb {R}^{d+1}$ .",
"Let $\\mathbb {H}_{\\ell }$ denote the set of all spherical harmonics of exact degree $\\ell $ on $\\mathbb {S}^{d}$ .",
"The dimension of the linear space $\\mathbb {H}_{\\ell }$ is $Z(d,\\ell ):=(2\\ell +d-1)\\frac{\\Gamma (\\ell +d-1)}{\\Gamma (d)\\Gamma (\\ell +1)}\\asymp (\\ell +1)^{d-1},$ where $a_{\\ell }\\asymp b_{\\ell }$ means $c\\:b_{\\ell }\\le a_{\\ell }\\le c^{\\prime } \\:b_{\\ell }$ for some positive constants $c$ , $c^{\\prime }$ , and the asymptotic estimate uses [26].",
"Each pair $\\mathbb {H}_{\\ell }$ , $\\mathbb {H}_{\\ell ^{\\prime }}$ for $\\ell \\ne \\ell ^{\\prime }\\ge 0$ is $\\mathbb {L}_{2}$ -orthogonal, $\\mathbb {P}_{L}(\\mathbb {S}^{d})=\\bigoplus _{\\ell =0}^{L} \\mathbb {H}_{\\ell }$ and the infinite direct sum $\\bigoplus _{\\ell =0}^{\\infty } \\mathbb {H}_{\\ell }$ is dense in $\\mathbb {L}_p(\\mathbb {S}^d)$ , $p \\ge 2$ , see e.g.",
"[64].",
"The linear span of $\\mathbb {H}_{\\ell }$ , $\\ell =0,1,\\dots ,L$ , forms the space $\\mathbb {P}_{L}(\\mathbb {S}^{d})$ of spherical polynomials of degree at most $L$ .",
"The dimension of $\\mathbb {P}_{L}(\\mathbb {S}^{d})$ is $D(d, L) := \\mbox{dim}\\; \\mathbb {P}_{L}(\\mathbb {S}^{d}) = \\sum _{\\ell =0}^L Z(d, \\ell ) = Z(d+1,L).$ Let $P^{\\left(\\alpha ,\\beta \\right)}_{\\ell }(z)$ , $-1\\le z\\le 1$ , be the Jacobi polynomial of degree $\\ell $ for $\\alpha ,\\beta >-1$ .",
"The Jacobi polynomials form an orthogonal polynomial system with respect to the Jacobi weight $w_{\\alpha ,\\beta }(z) := (1-z)^{\\alpha }(1+z)^{\\beta }$ , $-1\\le z \\le 1$ .",
"We denote the normalised Legendre (or ultraspherical/Gegenbauer) polynomials by $P^{(d+1)}_{\\ell }(z) := \\frac{P^{\\left(\\frac{d-2}{2},\\frac{d-2}{2}\\right)}_{\\ell }(z)}{P^{\\left(\\frac{d-2}{2},\\frac{d-2}{2}\\right)}_{\\ell }(1)},$ where, from [61], $P_\\ell ^{(\\alpha ,\\beta )}(1) = \\frac{\\Gamma (\\ell +\\alpha +1)}{\\Gamma (\\ell +1)\\Gamma (\\alpha +1)},$ and [61], $\\bigl |P^{(d+1)}_{\\ell }(z)\\bigr |\\le 1, \\qquad -1\\le z \\le 1.$ The derivative of the Jacobi polynomial satisfies [61] $\\frac{\\mbox{d}\\; P_{\\ell }^{(\\alpha ,\\beta )}(z)}{\\mbox{d}\\; z} =\\frac{\\ell +\\alpha +\\beta +1}{2}\\; P_{\\ell -1}^{(\\alpha +1, \\beta +1)}(z),$ so $\\frac{\\mbox{d}\\; P_{\\ell }^{(d+1)}(z)}{\\mbox{d}\\; z} =\\frac{(\\ell +d-1)(\\ell + d/2)}{d} P_{\\ell -1}^{(d+3)}(z).$ Also if $\\ell $ is odd then the polynomials $P_\\ell ^{(d+1)}$ are odd and if $\\ell $ is even the polynomials $P_\\ell ^{(d+1)}$ are even.",
"A zonal function $K:\\mathbb {S}^{d}\\times \\mathbb {S}^{d}\\rightarrow \\mathbb {R}$ depends only on the inner product of the arguments, i.e.",
"$K({x},{y})= \\mathfrak {K}({x}\\cdot {y})$ , ${x},{y}\\in \\mathbb {S}^{d}$ , for some function $\\mathfrak {K}:[-1,1]\\rightarrow \\mathbb {R}$ .",
"Frequent use is made of the zonal function $P^{(d+1)}_{\\ell }({x}\\cdot {y})$ .",
"Let $\\lbrace Y_{\\ell ,k}: k=1,\\dots ,Z(d,\\ell ), \\; \\ell = 0,\\ldots ,L\\rbrace $ be an orthonormal basis for $\\mathbb {P}_L(\\mathbb {S}^{d})$ .",
"The normalised Legendre polynomial $P^{(d+1)}_{\\ell }({x}\\cdot {y})$ satisfies the addition theorem (see [61], [64], [3] for example) $\\sum _{k=1}^{Z(d,\\ell )} Y_{\\ell ,k}({x}) Y_{\\ell ,k}({y}) =Z(d,\\ell ) P^{(d+1)}_{\\ell }({x}\\cdot {y}).$"
],
[
"Number of Points",
"Delsarte, Goethals and Seidel [24] showed that an $N$ point $t$ -design on $\\mathbb {S}^{d}$ has $N\\ge N^*(d,t)$ where $N^*(d,t) := \\left\\lbrace \\begin{array}{ll}2\\binom{d+k}{d} & \\text{ if } t = 2k+1,\\\\[2ex]\\binom{d+k}{d} + \\binom{d+k-1}{d} & \\text{ if } t = 2k.\\end{array}\\right.$ On $\\mathbb {S}^{2}$ $N^*(2,t) := \\left\\lbrace \\begin{array}{ll}\\frac{(t+1)(t+3)}{4} & \\text{ if $t$ odd},\\\\[2ex]\\frac{(t+2)^2}{4} & \\text{ if $t$ even}.\\end{array}\\right.$ Bannai and Damerell [6], [7] showed that tight spherical $t$ -designs which achieve the lower bounds (REF ) cannot exist except for a few special cases (for example except for $t = 1, 2, 3, 5$ on $\\mathbb {S}^{2}$ ) .",
"Yudin [68] improved (except for some small values of $d,t$ , see Table REF ), the lower bounds (REF ), by an exponential factor $(4/e)^{d+1}$ as $t\\rightarrow \\infty $ , so $N \\ge N^+(d,t) $ where $N^+(d,t) :=2 \\frac{\\int _0^1 (1-z^2)^{(d-2)/2}\\;\\mbox{d} z}{\\int _\\gamma ^1 (1-z^2)^{(d-2)/2}\\;\\mbox{d} z} =\\frac{\\sqrt{\\pi }\\Gamma (d/2)/\\Gamma ((d+1)/2)}{\\int _\\gamma ^1 (1-z^2)^{(d-2)/2}\\;\\mbox{d} z},$ and $\\gamma $ is the largest zero of the derivative $\\frac{d P_{t}^{(d+1)}(z)}{d z}$ and hence the largest zero of $P_{t-1}^{(\\alpha +1,\\alpha +1)}(z)$ where $\\alpha = (d-2)/2$ .",
"Bounds [61], [2] on the largest zero of $P_n^{(\\alpha ,\\alpha )}(z)$ are $\\cos \\left(\\frac{j_0(\\nu )}{n+\\alpha +1/2}\\right) \\le \\gamma \\le \\sqrt{\\frac{(n-1)(n+2\\alpha -1)}{(n+\\alpha -3/2)/(n+\\alpha -1/2)}} \\;\\cos \\left(\\frac{\\pi }{n+1}\\right),$ where $j_0(\\nu )$ is the first positive zero of the Bessel function $J_\\nu (x)$ .",
"Numerically there is strong evidence that spherical $t$ -designs with $N = D(2,t) = (t+1)^2$ points exist, [18] and [17] used interval methods to prove existence of spherical $t$ -designs with $N = (t+1)^2$ for all values of $t$ up to 100, but there is no proof yet that spherical $t$ -designs with $N \\le D(2,t)$ points exist for all degrees $t$ .",
"Hardin and Sloane [35],[36] provide tables of designs with modest numbers of points, exploiting icosahedral symmetry.",
"They conjecture that for $d = 2$ spherical $t$ -designs exist with $N = t^2/2 + o(t^2)$ for all $t$ .",
"The numerical experiments reported here and available from [66] strongly support this conjecture.",
"McLaren [46] defined efficiency $E$ for a quadrature rule as the ratio of the number of independent functions for which the rule is exact to the number of arbitrary constants in the rule.",
"For a spherical $t$ -design with $N$ points on $\\mathbb {S}^{d}$ (and equal weights) $E = \\frac{\\mbox{dim} \\; \\mathbb {P}_t(\\mathbb {S}^{d})}{d N} =\\frac{D(d,t)}{d N}.$ In these terms the aim is to find spherical $t$ -designs with $E \\ge 1$ .",
"McLaren [46] exploits symmetry (in particular octahedral and icosahedral) to seek rules with optimal efficiency.",
"The aim here is not to maximise efficiency by finding the minimal number of points for a $t$ -design on $\\mathbb {S}^{d}$ , but rather a sequence of efficient $t$ -designs with $N \\asymp \\frac{D(d,t)}{d} \\asymp \\frac{(1+t)^d}{d}$ .",
"Such efficient $t$ -designs provide a practical tool for numerical integration and approximation."
],
[
"Geometric Quality",
"The Geodesic distance between two points ${x}, {y}\\in \\mathbb {S}^{d}$ is $\\mbox{dist}({x}, {y}) = \\cos ^{-1}({x}\\cdot {y}),$ while the Euclidean distance is $| {x}- {y}| = \\sqrt{2(1-{x}\\cdot {y})} = 2 \\sin (\\mbox{dist}({x},{y})/2).$ The spherical cap with centre ${z}\\in \\mathbb {S}^{d}$ and radius $\\eta \\in [0, \\pi ]$ is $\\mathcal {C}\\left({z};\\eta \\right) =\\left\\lbrace {x}\\in \\mathbb {S}^{d}: \\mbox{dist}({x},{z}) \\le \\eta \\right\\rbrace .$ The separation distance $\\displaystyle \\delta (X_N) = \\min _{i\\ne j} \\; \\mbox{dist}({x}_i,{x}_j)$ is twice the packing radius for spherical caps of the same radius and centers in $X_N$ .",
"The best packing problem (or Tammes problem) has a long history [21], starting with [62], [53].",
"A sequence of point sets $\\lbrace X_N\\rbrace $ with $N \\rightarrow \\infty $ has the optimal order separation if there exists a constant $c_d^{\\textrm {pck}}$ independent of $N$ such that $\\delta (X_N) \\ge c_d^{\\textrm {pck}} \\; N^{-1/d}.$ The separation, and all the zonal functions considered in subsequent sections, are determined by the set of inner products $\\mathcal {A}(X_N) := \\left\\lbrace {x}_i \\cdot {x}_j, i = 1,\\ldots ,N, j = i+1,\\ldots ,N \\right\\rbrace $ which has been widely used in the study of spherical codes, see [21] for example.",
"Then $\\max _{z\\in \\mathcal {A}(X_N)} z = \\cos (\\delta (X_N)).$ Point sets are only considered different if the corresponding sets (REF ) differ, as they are invariant under an orthogonal transformation (rotation) of the point set and permutation (relabelling) of the points.",
"The mesh norm (or fill radius) $h(X_N) = \\max _{{x}\\in \\mathbb {S}^{d}} \\; \\min _{j=1,\\ldots ,N} \\; \\mbox{dist}({x},{x}_j)$ gives the covering radius for covering the sphere with spherical caps of the same radius and centers in $X_N$ .",
"A sequence of point sets $\\lbrace X_N\\rbrace $ with $N\\rightarrow \\infty $ has the optimal order covering if there exists a constant $c_d^{\\textrm {cov}}$ independent of $N$ such that $h(X_N) \\le c_d^{\\textrm {cov}} \\; N^{-1/d}.$ The mesh ratio is $\\rho (X_N) = \\frac{2 h_{X_N}}{\\delta _{X_N}} \\ge 1.$ A common assumption in numerical methods is that the mesh ratio is uniformly bounded, that is the point sets are quasi-uniform.",
"Minimal Riesz $s$ -energy and best packing points can also produce quasi-uniform point sets [23], [34], [10].",
"Yudin [67] showed that a spherical $t$ -design with $N$ points has a covering radius of the optimal order $1/t$ .",
"Reimer extended this to quadrature rules exact for polynomials of degree $t$ with positive weights.",
"Thus a spherical $t$ -design with $N = O(t^d)$ points provides an optimal order covering.",
"The union of two spherical $t$ -designs with $N$ points is a spherical $t$ -design with $2N$ points.",
"A spherical design with arbitrarily small separation can be obtained as one $N$ point set is rotated relative to the other.",
"Thus an assumption on the separation of the points of a spherical design is used to derive results, see [38] for example.",
"This simple argument is not possible if $N$ is less than twice a lower bound (REF ) or (REF ) on the number of points in a spherical $t$ -design.",
"Bondarenko, Radchenko and Viazovska [9] have shown that on $\\mathbb {S}^{d}$ well-separated spherical $t$ -designs exist for $N \\ge c_d^\\prime \\; t^d$ .",
"This combined with Yudin's result on the covering radius of spherical designs mean that there exist spherical $t$ -designs with $N = O(t^d)$ points and uniformly bounded mesh ratio.",
"There are many other “geometric” properties that could be used, for example the spherical cap discrepancy, see [32] for example, (using normalised surface measure so $|\\mathbb {S}^{d}| = 1$ ) $\\sup _{{x}\\in \\mathbb {S}^{d}, \\eta \\in [0,\\pi ]}\\left| |\\mathcal {C}({x},\\eta )| -\\frac{|X_N\\cap \\mathcal {C}({x},\\eta )| }{N}\\right|,$ or a Riesz $s$ -energy, see [12] for example, $E_s(X_N) = \\sum _{1\\le i < j \\le N} \\frac{1}{|{x}_i - {x}_j|^s}.$ In distinguishing between spherical $t$ -designs with the same number $N$ of points we prefer those with lower mesh ratio.",
"Note that some authors, see [34], [10] for example, define the mesh ratio as $\\tilde{\\rho }(X_N) = h(X_N)/\\delta (X_N) \\ge 1/2$ ."
],
[
"Variational Characterizations",
"Delsarte, Goethals and Seidel [24] showed that $X_N = \\lbrace {x}_1, \\ldots , {x}_N\\rbrace \\subset \\mathbb {S}^{d}$ is a spherical $t$ -design if and only if the Weyl sums satisfy $r_{\\ell ,k}(X_N) := \\sum _{j=1}^N Y_{\\ell ,k} ({x}_j) = 0\\qquad k=1, \\ldots , Z(d,\\ell ), \\quad \\ell = 1,\\ldots , t ,$ as the integral of all spherical harmonics of degree $\\ell \\ge 1$ is zero from orthogonality with the constant ($\\ell = 0$ ) polynomial $Y_{0,1} = 1$ which is not included.",
"In matrix form ${r}(X_N) := \\overline{{Y}} {e}= {0}$ where ${e}= (1,\\ldots ,1)^T\\in \\mathbb {R}^{N}$ and $\\overline{{Y}} \\in \\mathbb {R}^{D(d,t)-1 \\times N}$ is the spherical harmonic basis matrix excluding the first row.",
"Let $\\psi _t: [-1, 1] \\rightarrow \\mathbb {R}$ be a polynomial of degree $t\\ge 1$ with $\\psi _t(z) = \\sum _{\\ell =1}^t a_\\ell P^{(d+1)}_{\\ell }(z), \\qquad a_\\ell > 0 \\mbox{ for } \\ell = 1,\\ldots ,t,$ so the generalised Legendre coefficients $a_\\ell $ for degrees $\\ell =1,\\ldots ,t$ are all strictly positive.",
"Clearly any such function $\\psi _t$ can be scaled by an arbitrary positive constant without changing these properties.",
"Consider now an arbitrary set $X_N$ of $N$ points on $\\mathbb {S}^{d}$ .",
"Sloan and Womersley [58] considered the variational form $V_{t,N,\\psi }(X_N) :=\\frac{1}{N^2} \\sum _{i=1}^N \\sum _{j=1}^N \\psi _t ({x}_i \\cdot {x}_j)$ which from (REF ) satisfies $0 \\le V_{t,N,\\psi }(X_N) \\le \\sum _{\\ell =1}^{t} a_\\ell = \\psi _t (1).$ Moreover the average value is $\\overline{V}_{t, N, \\psi } :=\\int _{\\mathbb {S}^{d}} \\cdots \\int _{\\mathbb {S}^{d}}V_{t,N,\\psi } ({x}_1, \\ldots , {x}_N) d\\sigma _d({x}_1) \\cdots d\\sigma _d({x}_N) =\\frac{\\psi _t(1)}{N}.$ As the upper bound and average of $V_{t,N,\\psi }(X_N)$ depend on $\\psi _t(1)$ , we concentrate on functions $\\psi $ for which $\\psi _t(1)$ does not grow rapidly with $t$ .",
"From the addition theorem (REF ), $V_{t,N,\\psi }(X_N)$ is a weighted sum of squares with strictly positive coefficients $V_{t,N,\\psi }(X_N) =\\frac{1}{N^2}\\sum _{\\ell =1}^t \\frac{a_\\ell }{Z(d,\\ell )}\\sum _{k=1}^{Z(d,\\ell )} \\left(r_{\\ell ,k} (X_N)\\right)^2= \\frac{1}{N^2} {r}(X_N)^T\\; {D}\\; {r}(X_N),$ where ${D}$ is the diagonal matrix with strictly positive diagonal elements $\\frac{a_\\ell }{Z(d,\\ell )}$ for $k = 1,\\ldots ,Z(d,\\ell ), \\ell = 1,\\ldots ,t$ .",
"Thus, from (REF ), $X_N$ is a spherical $t$ -design if and only if $V_{t,N,\\psi }(X_N) = 0.$ Moreover, if the global minimum of $V_{t,N,\\psi }(X_N) > 0$ then there are no spherical $t$ -designs on $\\mathbb {S}^{d}$ with $N$ points.",
"Given a polynomial $\\widehat{\\psi }_t(z)$ of degree $t$ and strictly positive Legendre coefficients, the zero order term may need to be removed to get $\\psi _t(z) = \\widehat{\\psi }_t(z) - a_0$ where for $\\mathbb {S}^{d}$ and $\\alpha = (d-2)/2$ , $a_0 = \\int _{-1}^1 \\widehat{\\psi }_t(z) \\left(1-z^2\\right)^\\alpha dz .$ Three examples of polynomials on $[-1, 1]$ with strictly positive Legendre coefficients for $\\mathbb {S}^{d}$ and zero constant term, with $\\alpha = (d-2)/2$ are: Example 1 $\\psi _{1,t}(z) = z^{t-1} + z^{t} - a_0$ where $a_0 = \\frac{\\Gamma (\\alpha +3/2)}{\\sqrt{\\pi }}\\left\\lbrace \\begin{array}{ll}\\frac{\\Gamma (t/2)}{\\Gamma (\\alpha +1+t/2)} & t \\mbox{ odd}, \\\\[2ex]\\frac{\\Gamma ((t+1)/2)}{\\Gamma (\\alpha +3/2+t/2)} & t \\mbox{ even}.\\end{array} \\right.$ For $d = 2$ this simplifies to $a_0 = 1/t$ if $t$ is odd and $a_0 = 1/(t+1)$ if $t$ is even.",
"This function was used by Grabner and Tichy [32] for symmetric point sets where only even values of $t$ need to be considered, as all odd degree polynomials are integrated exactly.",
"Example 2 $\\psi _{2,t}(z) = \\left(\\frac{1+z}{2} \\right)^t - a_0$ where $a_0 = \\frac{2}{\\sqrt{\\pi }} 4^\\alpha \\Gamma (\\alpha +3/2)\\frac{\\Gamma (\\alpha +1+t)}{\\Gamma (2\\alpha +2+t)}.$ For $d = 2$ this simplifies to $a_0 = 1/(1+t)$ .",
"This is a scaled version of the function $(1+z)^t$ used by Cohn and Kumar [20] for which $a_0$ must be scaled by $2^t$ producing more cancellation errors for large $t$ .",
"Example 3 $\\psi _{3,t}(z) = P_t^{(\\alpha +1,\\alpha )}(z) - a_0$ where $a_0$ is given by (REF ).",
"The expansion in terms of Jacobi polynomials in Szegő [61] gives $\\sum _{\\ell =0}^t Z(d,\\ell ) P^{(d+1)}_{\\ell }(z) = \\frac{1}{a_0} P_t^{(\\alpha +1,\\alpha )}(z).$ For $S^2$ this is equivalent to $\\sum _{\\ell =1}^t (2\\ell + 1) P^{(d+1)}_{\\ell }(z) = (t+1) P_t^{(1,0)}(z) - 1$ used in Sloan and Womersley [58]."
],
[
"Quadrature Error",
"The error for numerical integration depends on the smoothness of the integrand.",
"Classical results are based on the error of best approximation of the integrand $f$ by polynomials [51], (see also [41] for more details on $\\mathbb {S}^{2}$ ).",
"For $f\\in C^\\kappa (\\mathbb {S}^{d})$ , there exists a constant $c=c(\\kappa ,f)$ such that the numerical integration error satisfies $\\left|\\int _{\\mathbb {S}^{d}} f({x}) \\mbox{d} \\sigma _d({x}) - \\frac{1}{N} \\sum _{j=1}^N f({x}_j)\\right|\\le c \\; t^{-\\kappa } .$ If $N = O(t^d)$ then the right-hand-side becomes $N^{-\\kappa /d}$ .",
"Thus for functions with reasonable smoothness it pays to increase the degree of precision $t$ .",
"Similar results are presented in [14], building on the work of [39], [37], for functions $f$ in a Sobolev space $\\mathbb {H}^s(\\mathbb {S}^{d})$ , $s > d/2$ .",
"The worst-case-error for equal weight (quasi Monte-Carlo) numerical integration using an arbitrary point set $X_N$ is $WCE(X_N, s, d) :=\\sup _{f\\in \\mathbb {H}^s(\\mathbb {S}^{d}), ||f||_{\\mathbb {H}^s(\\mathbb {S}^{d})\\le 1}} \\ \\left| \\int _{\\mathbb {S}^{d}} f({x}) \\mbox{d} \\sigma ({x}) -\\frac{1}{N} \\sum _{j=1}^N f({x}_j) \\right|.$ From this it immediately follows that the error for numerical integration satisfies $\\left| \\int _{\\mathbb {S}^{d}} f({x}) \\mbox{d} \\sigma _d({x}) -\\frac{1}{N} \\sum _{j=1}^N f({x}_j) \\right| \\le WCE(X_N, s, d) \\ \\Vert f \\Vert _{\\mathbb {H}^s(\\mathbb {S}^{d})}.$ Spherical $t$ -designs $X_N$ with $N = O(t^d)$ points satisfy the optimal order rate of decay of the worst case error, for any $s > d/2$ , namely $WCE(X_N, s, d) = O\\left(N^{-s/d}\\right), \\qquad N \\rightarrow \\infty .$ Thus spherical $t$ -designs with $N = O(t^d)$ points are ideally suited to the numerical integration of smooth functions."
],
[
"Computational Issues",
"The aim is to find a spherical $t$ -design with $N$ points on $\\mathbb {S}^{d}$ by finding a point set $X_N$ achieving the global minimum of zero for the variational function $V_{t, N, \\psi }(X_N)$ .",
"This section considers several computational issues: the evaluation of $V_{t, N, \\psi }(X_N)$ either as a double sum or using its representation (REF ) as a sum of squares; the parametrisation of the point set $X_N$ ; the number of points $N$ as a function of $t$ and $d$ ; the choice of optimization algorithm which requires evaluation of derivatives with respect to the chosen parameters; exploiting the sum of squares structure which requires evaluating the spherical harmonics and their derivatives; and imposing structure on the point set, for example symmetric (antipodal) point sets.",
"An underlying issue is that optimization problems with points on the sphere typically have many different local minima with different characteristics.",
"Here we are seeking both a global minimizer with value 0 and one with good geometric properties as measured by the mesh ratio.",
"The calculations were performed using Matlab, on a Linux computational cluster using nodes with up to 16 cores.",
"In all cases analytic expressions for the derivatives with respect to the chosen parametrisation were used."
],
[
"Evaluating Criteria",
"Although the variational functions are nonnegative, there is significant cancellation between the (constant) diagonal elements $\\psi _t(1)$ and all the off-diagonal elements with varying signs as $V_{t,N,\\psi }(X_N) =\\frac{1}{N} \\psi _t(1) + \\sum _{i=1}^N \\sum _{\\stackrel{j=1}{j\\ne i}}^N \\psi _t({x}_i\\cdot {x}_j).$ Figure: For d=2d = 2, t=30t = 30, a spherical tt-design with N=482N = 482,the functions ψ k,t \\psi _{k,t} and arrays ψ k,t (x i ·x j )\\psi _{k,t}({x}_i\\cdot {x}_j) for k=1,2,3k = 1, 2, 3.Accurate calculation of such sums is difficult, see [42] for example, especially getting reproducible results on multi-core architecture with dynamic scheduling of parallel non-associative floating point operations [25].",
"Example 1 has $\\psi _{1,t}(1) = 2$ and Example 2 has $\\psi _{2,t}(1) = 1$ , both independent of $t$ , while Example 3 has $\\psi _{3,t}(1) = \\frac{\\Gamma (t+\\alpha +2)}{\\Gamma (t+1)\\Gamma (\\alpha +2)} - 1,$ which grows with the degree $t$ (for $d = 2$ , $\\psi _{3,t}(1) = t$ ).",
"These functions are illustrated in Fig.",
"REF .",
"As the variational objectives can be scaled by an arbitrary positive constant, you could instead have used $\\psi _{3,t} \\frac{\\Gamma (t+1)\\Gamma (\\alpha +2)}{\\Gamma (t+\\alpha +2)}$ .",
"Ratios of gamma functions, as in the expressions for $a_0$ , should not be evaluated directly, but rather simplified for small values of $d$ or evaluated using the log-gamma function.",
"The derivatives, essential for large scale non-linear optimization algorithms, are readily calculated using $\\nabla _{{x}_k} V_{t,N,\\psi }(X_N) =2 \\sum _{\\stackrel{i=1}{i\\ne k}}^N \\psi _t^{\\prime }({x}_i \\cdot {x}_k) {x}_i$ and the Jacobian of the (normalised) spherical parametrisation (see Section REF ).",
"Because of the interest in the use of spherical harmonics for the representation of the Earth's gravitational field there has been considerable work, see [43], [44] and [29] for example, on the evaluation of high degree spherical harmonics for $\\mathbb {S}^{2}$ .",
"For $(x, y, z)^T \\in \\mathbb {S}^{2}$ the real spherical harmonics [54] are usually expressed in terms of the coordinates $z = \\cos (\\theta )$ and $\\phi $ .",
"In terms of the coordinates $(x,\\phi _2) = (\\cos (\\phi _1), \\phi _2)$ , see (REF ) below, they are the $Z(2,\\ell ) = 2\\ell +1$ functions $Y_{\\ell ,\\ell +1-k}(x,\\phi _2) & := & \\hat{c}_{\\ell ,k}(1-x^2)^{k/2} S_\\ell ^{(k)}(x) \\sin (k\\phi _2),\\quad k=1,\\ldots ,\\ell ,\\nonumber \\\\Y_{\\ell ,\\ell +1}(x,\\phi _2) & := & \\hat{c}_{\\ell ,0} S_\\ell ^{(0)}(x),\\\\Y_{\\ell ,\\ell +1+k}(x,\\phi _2) & := & \\hat{c}_{\\ell ,k} (1-x^2)^{k/2}S_\\ell ^{(k)}(x) \\cos (k\\phi _2),\\quad k=1,\\ldots ,\\ell .",
"\\nonumber $ where $S_\\ell ^{(k)}(x) = \\sqrt{\\frac{(\\ell -k)!",
"}{(\\ell +k)!}}",
"P_\\ell ^k(x)$ are versions of the Schmidt semi-normalised associated Legendre functions for which stable three-term recurrences exist for high (about 2700) degrees and orders.",
"The normalization constants $\\hat{c}_{\\ell ,0}$ , $\\hat{c}_{\\ell ,k}$ are, for normalised surface measure, $\\hat{c}_{\\ell , 0} = \\sqrt{ 2\\ell +1}, \\quad \\hat{c}_{\\ell ,k} = \\sqrt{2} \\sqrt{2\\ell +1},\\quad k = 1,\\ldots ,\\ell ,$ For $\\mathbb {S}^{2}$ these expressions can be used to directly evaluate the Weyl sums (REF ), and hence their sum of squares, and their derivatives."
],
[
"Spherical Parametrisations",
"There are many ways to organise a spherical parametrisation of $\\mathbb {S}^{d}$ .",
"For $\\phi _i \\in [0, \\pi ]$ for $i = 1,\\ldots ,d-1$ and $\\phi _d \\in [0, 2\\pi )$ define ${x}\\in \\mathbb {S}^{d}$ by $& x_1 = \\cos (\\phi _1) \\\\& x_i = \\prod _{k=1}^{i-1} \\sin (\\phi _k) \\cos (\\phi _i), \\quad i = 2,\\ldots ,d \\\\& x_{d+1} = \\prod _{k=1}^d \\sin (\\phi _k)$ The inverse transformation used is, for $i = 1,\\ldots ,d-1$ $& \\phi _i = \\left\\lbrace \\begin{array}{ll}0 & \\quad \\mbox{if $x_k = 0, \\quad k = i,\\ldots ,d+1$}, \\\\[1ex]\\cos ^{-1}\\left( x_i / \\sqrt{\\sum _{k=i}^{d+1} x_k^2} \\right) & \\quad \\mbox{otherwise}; \\\\[1ex]\\end{array}\\right.",
"\\\\& \\phi _d = \\tan ^{-1} \\left(x_{d+1}/x_d\\right).$ The last component can be calculated using the four quadrant atan2 function and periodicity to get $\\phi _d \\in [0, 2\\pi )$ .",
"Spherical parametrisations introduce potential singularities when $\\phi _i = 0$ or $\\phi _i = \\pi $ for any $i = 1,\\ldots ,d-1$ .",
"As all the functions considered are zonal, they are invariant under an orthogonal transformation (rotation).",
"Thus the point sets are normalised so that the $d+1$ by $N$ matrix ${X}= [{x}_1 \\cdots {x}_N]$ has $& {X}_{i,j} = 0 \\quad \\mbox{for } i = j+1,\\ldots ,d+1, \\quad j = 1,\\ldots ,\\min (d,N) \\\\& {X}_{i,i} \\ge 0 \\quad \\mbox{for } i = 1,\\ldots ,\\min (d,N).$ The first normalised point is ${x}_1 = {e}_1 = (1,0,\\ldots ,0)\\in \\mathbb {R}^{d+1}$ .",
"Such a rotation can easily be calculated using the $QR$ factorization of ${X}$ combined with sign changes to the rows $Q$ .",
"The corresponding normalised spherical parametrisation has $\\Phi _{i,j} = 0 \\quad \\mbox{for } i = j,\\ldots ,d, \\quad j = 1,\\ldots ,\\min (d,N),$ where the $j$ th column of $\\Phi $ corresponds to the point ${x}_j$ , $j = 1,\\ldots ,N$ .",
"The optimisation variables are then $\\Phi _{i,j}, i = 1,\\ldots ,\\min (j-1,d), \\ j = 2,\\ldots , N$ , stored as the vector ${\\phi }\\in \\mathbb {R}^n$ where $n = \\left\\lbrace \\begin{array}{cl}\\frac{N(N-1)}{2} & \\mbox{ for } N \\le d, \\\\[1ex]N d - \\frac{d(d+1)}{2} & \\mbox{ for } N > d,\\end{array}\\right.$ so $& {\\phi }_p = \\Phi _{i,j}, \\quad i = 1,\\ldots ,\\min (j-1,d), \\quad j = 2,\\ldots ,N, \\\\& p = \\left\\lbrace \\begin{array}{cl}\\frac{\\min (j-1,d) (\\min (j-1,d)-1)}{2} + i & \\mbox{ for } j = 2,\\ldots ,\\min (d,N)\\\\\\frac{d(d-1)}{2} + (j-d-1)d + i & \\mbox{ for } j = d+1,\\ldots ,N, \\quad N > d.\\end{array}\\right.$ It is far easier to work with a spherical parametrisation with bound constraints than to impose the quadratic constraints ${x}_j \\cdot {x}_j = 1, j = 1,\\ldots ,N$ , especially for large $N$ .",
"As the optimization criteria have the effect of moving the points apart, the use of the normalised point sets reduces difficulties with singularities at the boundaries corresponding to $\\Phi _{i,j} = 0$ or $\\Phi _{i,j} = \\pi $ , $i = 1,\\ldots ,d-1$ .",
"For $\\mathbb {S}^{2}$ , these normalised point sets may be rotated (the variable components re-ordered) using $Q = \\begin{bmatrix}0 & 1 & 0 \\\\ 0 & 0 & 1\\\\ 1 & 0 & 0\\end{bmatrix}$ to get the commonly [28], [57], [66] used normalization with the first point at the north pole and the second on the prime meridian.",
"A symmetric (or antipodal) point set (${x}\\in X_N \\iff -{x}\\in X_N$ ) must have $N$ even, so can be represented as ${X}= [\\overline{{X}} \\ {-\\overline{{X}}}]$ where the $d+1$ by $N/2$ array of points $\\overline{{X}}$ is normalised as above.",
"If only zonal function functions depending just on the inner products ${x}_i \\cdot {x}_j$ are used then you could use the variables ${Z}_{i,j} = {x}_i \\cdot {x}_j$ , so ${Z}\\in \\mathbb {R}^{N\\times N}, \\quad {Z}^T = {Z}, \\quad {Z}\\succeq 0, \\quad \\mbox{diag}({Z}) = {e}, \\quad \\mbox{rank}({Z}) = d+1.$ where ${e}= (1,\\ldots ,1)^T\\in \\mathbb {R}^N$ and ${Z}\\succeq 0$ indicates ${Z}$ is positive semi-definite.",
"The major difficulties with such a parametrisation are the number $N(N-1)/2$ of variables and the rank condition.",
"Semi-definite programming relaxations (without the rank condition) have been used to get bounds on problems involving points on the sphere (see, for example, [4])."
],
[
"Degrees of Freedom for $\\mathbb {S}^{d}$",
"Using a normalised spherical parametrisation of $N$ points on $\\mathbb {S}^{d} \\subset \\mathbb {R}^{d+1}$ there are $n = Nd - d(d+1)/2$ variables (assuming $N\\ge d$ ).",
"The number of conditions for a $t$ -design is $m = \\sum _{\\ell =1}^t Z(d,\\ell ) = D(d,t) - 1 = Z(d+1,t) - 1.$ Using the simple criterion that the number of variables $n$ is at least the number of conditions $m$ , gives the number of points as $\\widehat{N}(d,t) := \\left\\lceil \\frac{1}{d}\\left(Z(d+1,t) + \\frac{d(d+1)}{2} -1\\right) \\right\\rceil .$ For $\\mathbb {S}^{2}$ there are $n = 2N-3$ variables and $m = (t+1)^2 - 1$ conditions giving $\\widehat{N}(2,t) := \\left\\lceil (t+1)^2)/2 \\right\\rceil + 1.$ Grabner and Sloan [31] obtained separation results for $N$ point spherical $t$ -designs when $N \\le \\tau \\; 2 N^*$ and $\\tau < 1$ .",
"For $d = 2$ , $\\widehat{N}$ is less than twice the lower bound $N^*$ as $\\widehat{N}(2,t) = 2 N^*(2, t) - t,$ but the difference is only a lower order term.",
"The values for $\\widehat{N}(2,t)$ , $N^*(2,t)$ and the Yudin lower bound $N^+(2,t)$ are available in Tables REF – REF .",
"The idea of exploiting symmetry to reduce the number of conditions that a quadrature rule should satisfy at least goes back to Sobolev [60].",
"For a symmetric point set (both ${x}_j, -{x}_j$ in $X_N$ ) then all odd degree polynomials $Y_{\\ell ,k}$ or $P_\\ell ^{(d+1)}$ are automatically integrated exactly by an equal weight quadrature rule.",
"Thus, for $t$ odd, the number of conditions to be satisfied is $m = \\sum _{\\ell =1}^{(t-1)/2} Z(d,2\\ell ) = \\frac{\\Gamma (t+d)}{\\Gamma (d+1)\\Gamma (t)} - 1.$ The number of free variables in a normalised symmetric point set ${X}= [\\overline{{X}} \\quad \\mbox{$-\\overline{{X}}$}]$ (assuming $N/2 \\ge d$ ) is $n = \\left( \\frac{N d}{2} - \\frac{d(d+1)}{2} \\right) .$ Again the simple requirement that $n \\ge m$ gives the number of points as $\\overline{N}(d,t) := 2\\left\\lceil \\frac{1}{d} \\left(\\frac{\\Gamma (t+d)}{\\Gamma (d+1)\\Gamma (t)} - 1 + \\frac{d(d+1)}{2}\\right) \\right\\rceil .$ For $d = 2$ this simplifies, again for $t$ odd, to $\\overline{N}(2,t) := 2 \\left\\lceil \\frac{t^2 + t + 4}{4} \\right\\rceil .",
"$ $\\overline{N}(2,t)$ is slightly less than $\\widehat{N}(2,t)$ , comparable to twice the lower bound $N^*(2,t)$ as $\\overline{N}(2,t) = 2N^*(2,t) -\\mbox{$\\frac{3}{2}$} t +\\left\\lbrace \\begin{array}{ll}\\frac{3}{2} & \\mbox{ if } \\mod {(}t,4) = 1,\\\\[0.5ex]\\frac{1}{2} & \\mbox{ if } \\mod {(}t,4) = 3.\\end{array}\\right.$ However $\\overline{N}(2,t)$ is not less than $\\tau \\; 2 N^*(2,t)$ , $\\tau < 1$ , as required by Grabner and Sloan [31].",
"The leading term of both $\\widehat{N}(d,t)$ and $\\overline{N}(d,t)$ is $D(d,t)/d$ , see Table REF , where $D(d,t)$ defined in (REF ) is the dimension of $\\mathbb {P}_t(\\mathbb {S}^{d})$ .",
"From (REF ), a spherical $t$ -design with $\\widehat{N}(d,t)$ or $\\overline{N}(d,t)$ points has efficiency $E \\approx 1$ .",
"Also the leading term of both $\\overline{N}(d,t)$ and $\\widehat{N}(d,t)$ is $2^d/d$ times the leading term of the lower bound $N^*(d,t)$ .",
"Table: The lower bound N * (d,t)N^*(d,t), the number of points N ¯(d,t)\\overline{N}(d,t) (symmetric point set)and N ^(d,T)\\widehat{N}(d,T) to match the number of conditions and the dimension of ℙ t (𝕊 d )\\mathbb {P}_t(\\mathbb {S}^{d})for d=2,3,4,5d = 2, 3, 4, 5"
],
[
"Optimization Algorithms",
"As with many optimization problems on the sphere there are many distinct (not related by an orthogonal transformation or permutation) points sets giving local minima of the optimization objective.",
"For example, Erber and Hockney [27] and Calef et al [16] studied the minimal energy problem for the sphere and the large number of stable configurations.",
"Gräf and Potts [33] develop optimization methods on general Riemannian manifolds, in particular $\\mathbb {S}^{2}$ , and both Newton-like and conjugate gradients methods.",
"Using a fast method for spherical Fourier coefficients at non-equidistant points they obtain approximate spherical designs for high degrees.",
"While mathematically it is straight forward to conclude that if $V_{t,N,\\psi }(X_N) = 0$ then $X_N$ is a spherical $t$ -design, deciding when a quantity is zero with the limits of standard double precision floating point arithmetic with machine precision $\\epsilon = 2.2\\times 10^{16}$ is less clear (should $10^{-14}$ be regarded as zero?).",
"Extended precision libraries and packages like Maple or Mathematica can help.",
"A point set $X_N$ with $V_{t,N,\\psi }(X_N) \\approx \\epsilon $ does not give a mathematical proof that is $X_N$ is a spherical $t$ -design, but $X_N$ may still be computationally useful in applications.",
"On the other hand showing that the global minimum if $V_{t,N,\\psi }(X_N)$ is strictly positive, so no spherical $t$ -design with $N$ points exist, is an intrinsically hard problem problem.",
"Semi-definite programming [63] provides an approach [50] to the global optimization of polynomial sum of squares for modest degrees.",
"For $d = 2$ a variety of gradient based bound constrained optimization methods, for example the limited memory algorithm [15], [47], were tried both to minimise the variational forms $V_{t,N,\\psi }(X_N)$ .",
"Classically, see [48] for example, methods can exploit the sum of squares structure ${r}(X_N)^T {r}(X_N)$ .",
"In both cases it is important to provide derivatives of the objective with respect to the parameters.",
"Using the normalised spherical parametrisation ${\\phi }$ of $X_N$ , the Jacobian of the residual ${r}({\\phi })$ is ${A}: \\mathbb {R}^{n} \\rightarrow \\mathbb {R}^{m \\times n}$ where $n = dN - d(d+1)/2$ and $m = D(d,t)-1$ ${A}_{i,j}({\\phi }) = \\frac{\\partial r_i({\\phi })}{\\partial \\phi _j},\\qquad i = 1,\\ldots ,m, \\quad j = 1,\\ldots ,n,$ where $i = (\\ell -1) Z(d+1,\\ell -1) + k$ , for $k = 1,\\ldots ,Z(d,\\ell ), \\ell =1,\\ldots ,t$ .",
"For symmetric point sets with $N = \\overline{N}(d,t)$ points, the number of variables $n$ is given by (REF ) and the number of conditions $m$ by (REF ) corresponding to even degree spherical harmonics.",
"The well-known structure of a nonlinear least squares problem, see [48] for example, gives, ignoring the $1/N^2$ scaling in (REF ), $& f({\\phi }) = {r}({\\phi })^T \\; {D}\\; {r}({\\phi }), \\\\& \\nabla f({\\phi }) = 2 {A}({\\phi })^T {D}\\, {r}({\\phi }), \\\\& \\nabla ^2 f({\\phi }) = 2 {A}({\\phi })^T {D}{A}({\\phi }) +2 \\sum _{i=1}^m r_i({\\phi }) D_{ii} \\nabla ^2 r_i({\\phi }).$ If ${\\phi }^*$ has ${r}({\\phi }^*) = {0}$ and ${A}({\\phi }^*)$ has rank $n$ , the Hessian $\\nabla ^2 f({\\phi }^*) = 2 {A}({\\phi }^*)^T {D}{A}({\\phi }^*)$ is positive definite and ${\\phi }^*$ is a strict global minimizer.",
"Here this is only possible when $n = m$ , for example when $d = 2$ and $t$ is odd, see Tables REF , REF , REF , and in the symmetric case when $t \\mod {4} = 3$ , see Tables REF , REF , REF .",
"For $d = 2$ the other values of $t$ have $n = m + 1$ , so there is generically a one parameter family of solutions even when the Jacobian has full rank.",
"When $d = 3$ , the choice $N = \\widehat{N}(3,t)$ gives $n = m$ , $n = m + 1$ or $n = m+3$ depending on the value of $t$ , see Table REF .",
"Thus a Levenberg-Marquadt or trust region method, see [48] for example, in which the search direction satisfies $\\left({A}^T {D}{A}+ \\nu {I}\\right) {d}= {A}^T {D}{r}$ was used.",
"When $n > m$ the Hessian of the variational form $V_{t,N,\\psi }(X_N)$ evaluated using one of the three example functions (REF ), (REF ) or (REF ) will also be singular at the solution.",
"These disadvantages could have been reduced by choosing the number of points $N$ so that $n < m$ , but then there may not be solutions with $V_{t,N,\\psi }(X_N) = 0$ , that is spherical $t$ -designs may not exist for that number of points.",
"Many local solutions were found as well as (computationally) global solutions which differed depending on the starting point and the algorithm parameters (for example the initial Levenberg-Marquadt parameter $\\nu $ , initial trust region, line search parameters etc).",
"Even when $n = m$ there are often multiple spherical designs for same $t$ , $N$ , which are strict global minimisers, but have different inner product sets $\\mathcal {A}(X_N)$ in (REF ) and different mesh ratios."
],
[
"Structure of Point Sets",
"There are a number of issues with the spherical designs studied here.",
"There is no proof that spherical $t$ -designs on $\\mathbb {S}^{d}$ with $N = t^d/d + O(t^{d-1})$ points exist for all $t$ (that is the constant in the Bondarenko et al result [8] is $C_d = 1/d$ (or lower), as suggested by [35] for $\\mathbb {S}^{2}$ ).",
"The point sets are not nested, that is the points of a spherical $t$ -design are not necessarily a subset of the points of a $t^{\\prime }$ -design for some $t^{\\prime } > t$ .",
"The point sets do not lie on bands of equal $\\phi _1$ (latitude on $\\mathbb {S}^{2}$ ) making them less amenable for FFT based methods.",
"The point sets are obtained by extensive calculation, rather than generated by a simple algorithms as for generalized spiral or equal area points on $\\mathbb {S}^{2}$ [52].",
"Once calculated the point sets are easy to use.",
"An example of a point set on $\\mathbb {S}^{2}$ that satisfies the last three issues are the HELAPix points[30], which provide a hierarchical, equal area (so exact for constants), iso-latitude set of points widely used in cosmology."
],
[
"Spherical $t$ -Designs with no Imposed Symmetry for {{formula:fd7a205a-1f3e-41c0-b4c9-2c7e56495d2b}}",
"From Tables REF , REF and REF the variational criteria based on the three functions $\\psi _{1,t}$ , $\\psi _{2,t}$ and $\\psi _{3,t}$ all have values close to the double precision machine precision of $\\epsilon = 2.2\\times 10^{-16}$ for all degrees $t = 1,\\ldots ,180$ .",
"Despite being theoretically non-negative, rounding error sometimes gives negative values, but still close to machine precision.",
"The potential values using $\\psi _{3,t}$ are slightly larger due to the larger value of $\\psi _{3,t}(1)$ .",
"The tables also give the unscaled sum of squares ${r}(X_N)^T{r}(X_N)$ , which is also plotted in Fig.",
"REF .",
"These tables also list both the Delsarte, Goethals and Seidel lower bounds $N^*(2,t)$ and the Yudin lower bound $N^+(2,t)$ , plus the actual number of points $N$ .",
"The number of points $N = \\widehat{N}(2,t)$ , apart from $t = 3, 5, 7, 9, 11, 13, 15$ when $N = \\widehat{N}(2,t)-1$ .",
"There may well be spherical $t$ -designs with smaller values of $N$ and special symmetries, see [35] for example.",
"For all these point sets the mesh ratios $\\rho (X_N)$ are less than $1.81$ , see Fig.",
"REF .",
"All these point sets are available from [66]."
],
[
"Symmetric Spherical $t$ -Designs for {{formula:5d2b5b11-a0e0-4c08-bd21-7dde93f33192}}",
"For $\\mathbb {S}^{2}$ a $t$ -design with a sightly smaller number of points $\\overline{N}(2,t)$ can be found by constraining the point sets to be symmetric (antipodal).",
"A major computational advantage of working with symmetric point sets is the reduction (approximately half), for a given degree $t$ , in the number of optimization variables $n$ and the number of terms $m$ in the Weyl sums.",
"Tables REF , REF and REF list the characteristics of the calculated $t$ -designs for $t = 1,3,5,\\ldots ,325$ , as a symmetric $2k$ -design is automatically a $2k+1$ -design.",
"These tables have $t = \\overline{N}(2,t)$ except for $t = 1, 7, 11$ .",
"These point sets, again available from [66], provide excellent sets of points for numerical integration on $\\mathbb {S}^{2}$ with mesh ratios all less than $1.78$ for degrees up to 325, as illustrated in Fig.",
"REF ."
],
[
"Designs for $d = 3$",
"For $d = 3$ , $Z(3,\\ell ) = (\\ell +1)^2$ , so the dimension of the space of polynomials of degree at most $t$ in $\\mathbb {S}^{3}$ is $D(3,t) = Z(4,t) = (t+1)(t+2)(2t+3)/6$ .",
"Comparing the number of variables with the number of conditions, with no symmetry restrictions, gives $\\widehat{N}(3,t) = \\left\\lceil \\frac{2 t^3 + 9 t^2 + 13 t + 36}{18} \\right\\rceil ,$ while for symmetric spherical designs on $\\mathbb {S}^{3}$ $\\overline{N}(3,t) = 2 \\left\\lceil \\frac{t^3 + 3 t^2 + 2 t + 30}{18} \\right\\rceil .$ The are six regular convex polytopes with $N = 5, 8, 16, 24, 120$ and 600 vertices on $S^3$ [22] (the 5-cell, 16-cell, 8-cell, 24-cell, 600-cell and 120-cell respectively) giving spherical $t$ -designs for for $t = 2, 3, 5, 7, 9, 11$ and 11.",
"The energy of regular sets on $\\mathbb {S}^{3}$ with $N = 2, 3, 4, 5, 6 ,8, 10, 12, 13, 24, 48$ has been studied by [1].",
"The $N = 24$ vertices of the D4 root system [19] provides a one-parameter family of 5-designs on $\\mathbb {S}^{3}$ .",
"The Cartesian coordinates of the regular point sets are known, and these can be numerically verified to be spherical designs.",
"The three variational criteria using (REF ), (REF ) and (REF ) are given for these point sets in Table REF .",
"Fig.",
"REF clearly illustrates the difference between the widely studied [24], [21], [11] inner product set $\\mathcal {A}(X_N$ ) for a regular point set (the 600-cell with $N = 120$ ) and a computed spherical 13-design with $N = 340$ .",
"The results of some initial experiments in minimising the three variational criteria are given in Tables REF and REF .",
"For $d > 2$ , it is more difficult to quickly generate a point set with a good mesh ratio to serve as an initial point for the optimization algorithms.",
"One strategy is to randomly generate starting points, but this both makes the optimization problem harder and tends to produce nearby point sets which are local minimisers and have poor mesh ratios as the random initial points may have small separation [13].",
"Another possibility is the generalisation of equal area points to $d > 2$ by Leopardi [45].",
"For a given $t$ and $N$ there are still many different point sets with objective values close to 0 and different mesh ratios.",
"To fully explore spherical $t$ -designs for $d > 2$ , a stable implementation of the spherical harmonics is needed, so that least squares minimisation can be fully utilised.",
"Figure: Inner product sets 𝒜(X N )\\mathcal {A}(X_N) for600-cell with N=120N = 120 and 13-design with N=340N = 340 on 𝕊 3 \\mathbb {S}^{3}Table: Spherical tt-designs on 𝕊 2 \\mathbb {S}^2 with no symmetry restrictions,N=N ^(2,t)N = \\widehat{N}(2,t) and degrees t=1t = 1 – 60,except t=3,5,7,9,11,13,15t = 3, 5, 7, 9, 11, 13, 15 when N=N ^(2,t)-1N = \\widehat{N}(2,t)-1Table: Spherical tt-designs on 𝕊 2 \\mathbb {S}^2 with no symmetry restrictions, N=N ^(2,t)N = \\widehat{N}(2,t) and degrees t=61t = 61 – 120Table: Spherical tt-designs on 𝕊 2 \\mathbb {S}^2 with no symmetry restrictions, N=N ^(2,t)N = \\widehat{N}(2,t) and degrees t=121t = 121 – 180Table: Symmetric spherical tt-designs on 𝕊 2 \\mathbb {S}^2 with N=N ¯(2,t)N = \\overline{N}(2,t) and odd degrees t=1t = 1 – 109, except for t=1t = 1 when N=N ¯(2,t)-2N = \\overline{N}(2,t)-2 and t=7,11t = 7, 11 when N=N ¯(2,t)+2N = \\overline{N}(2,t)+2Table: Symmetric spherical tt-designs on 𝕊 2 \\mathbb {S}^2 with N=N ¯(2,t)N = \\overline{N}(2,t) and odd degrees t=111t = 111 – 219Table: Symmetric spherical tt-designs on 𝕊 2 \\mathbb {S}^2 with N=N ¯(2,t)N = \\overline{N}(2,t) and odd degrees t=221t = 221 – 325Table: Regular spherical tt-designs on 𝕊 3 \\mathbb {S}^3 for degrees tt = 1, 2, 3, 5 ,7 ,11Table: Computed spherical tt-designs on 𝕊 3 \\mathbb {S}^3 for degrees tt = 1,...,20, with N=N ^(3,t)N = \\widehat{N}(3,t)Table: Computed symmetric spherical tt-designs on 𝕊 3 \\mathbb {S}^3 for degrees tt = 1,3,...,31, with N=N ¯(3,t)N = \\overline{N}(3,t)This research includes extensive computations using the Linux computational cluster Katana supported by the Faculty of Science, UNSW Sydney."
]
] | 1709.01624 |
[
[
"Pulsations and Period Variations of the ${\\delta}$ Scuti Star AN Lyncis\n in a possible three-body system"
],
[
"Abstract Observations for the $\\delta$ Scuti star AN Lyn have been made between 2008 and 2016 with the 85-cm telescope at Xinglong station of National Astronomical Observatories of China, the 84-cm telescope at SPM Observatory of Mexico and the Nanshan One meter Wide field Telescope of Xinjiang Observatory of China.",
"Data in $V$ in 50 nights and in $R$ in 34 nights are obtained in total.",
"The bi-site observations from both Xinglong Station and SPM Observatory in 2014 are analyzed with Fourier Decomposition to detect pulsation frequencies.",
"Two independent frequencies are resolved, including one non-radial mode.",
"A number of stellar model tracks are constructed with the MESA code and the fit of frequencies leads to the best-fit model with the stellar mass of $M = 1.70 \\pm 0.05~\\mathrm{M_{\\odot}}$ , the metallicity abundance of $Z = 0.020 \\pm 0.001$, the age of $1.33 \\pm 0.01$ billion years and the period change rate $1/P\\cdot \\mathrm{d}P/\\mathrm{d}t =1.06 \\times 10^{-9} ~\\mathrm{yr^{-1}} $, locating the star at the evolutionary stage close to the terminal age main sequence (TAMS).",
"The O-C diagram provides the period change rate of $1/P \\cdot \\mathrm{d}P /\\mathrm{d}t =4.5(8)\\times10^{-7}~\\mathrm{yr^{-1}}$.",
"However, the period change rate calculated from the models is smaller in two orders than the one derived from the O-C diagram.",
"Together with the sinusoidal function signature, the period variations are regarded to be dominated by the light-travel time effect of the orbital motion of a three-body system with two low-luminosity components, rather than the stellar evolutionary effect."
],
[
"Introduction",
"$\\delta $ Scuti stars are pulsating variables locating in the classical Cepheid instability strip, on the main sequence or evolving from the main sequence to the giant branch [9], [10].",
"The general consensus shows that most (possibly all) $\\delta $ Scuti stars evolve in the main-sequence or the immediate post-main-sequence stages [2], [6], [7].",
"The amplitudes of pulsations in $\\delta $ scuti stars are from mmag up to tenths of a magnitude [23], [24], [39].",
"Their pulsation frequencies from ground-based observations are believed to be greater than 5 $\\mathrm {d}^{-1}$ but smaller than 50 $\\mathrm {d}^{-1}$ [3], which is a good criterion for the variable classification.",
"$\\delta $ Scuti pulsators are divided into three groups based on the amplitude.",
"The medium amplitude pulsators with visual amplitude between $0.^{\\mathrm {m}}1$ and $0.^{\\mathrm {m}}3$ are usually multiperiodic and the percentage of them is less than 5% [29], [30].",
"As one of the medium amplitude minority, AN Lyn is an interesting target of investigation for the pulsations and period variations.",
"AN Lyn ($\\mathrm {\\alpha _{2000}=09^h14^m28^s}$ , $\\mathrm {\\delta _{2000}=42^\\circ 46^{\\prime }38^{\\prime \\prime }}$ , 10$^{\\mathrm {m}}$ .58 in $V$ , A7IV-V) is a multiperiodic $\\delta $ Scuti star with the principal period of $\\mathrm {0^{d}.09827}$ and the visual amplitude of $\\mathrm {\\sim 0^m.18}$ [35].",
"It has been observed from 1980 [34] to 2016 with a large amount of data accumulated.",
"The distance was estimated as 529 parsec by [27] using the $uvby-\\beta $ photometry.",
"A more convincing result was $720\\pm 160$ parsec, reported by $Gaia$ satellite [12], [13], [1].",
"[30] pointed out that AN Lyn is a nearly cold and evolved $\\delta $ Scuti star with solar metal abundance based on $uvby-\\beta $ photometry, in which the `nearly cold' means that AN Lyn is closer to the red edge of the instability strip than the blue edge.",
"The last report of AN Lyn's stellar parameters is $\\log g=3.8$ and $\\log T_\\mathrm {eff}=3.8$ [27].",
"With Fourier analysis, [31] detected three independent frequencies, $f_1=10.1756,f_2=18.1309$ and $f_3=9.5598$ $\\mathrm {d^{-1}}$ , who denied that AN Lyn was a monoperiodic pulsator by earlier works.",
"However, it is noticed that [35] confirmed the existence of $f_1$ and $f_2$ but not $f_3$ in his work.",
"As far as the O-C analysis, it was hard for [31] to explain the unusual behaviour of AN Lyn.",
"Since then, the binary nature has been gradually deduced from further O-C analysis.",
"The O-C diagram shows a long-term increasing trend plus a fluctuating pattern.",
"The long-term increasing trend is explained as the period increase due to the stellar evolution effect.",
"[16] reported the period change rate $\\mathrm {d}P/\\mathrm {d}t$ as $7.9\\times 10^{-10}$ d d$^{-1}$ .",
"[22] provided the measurement of $2.09\\left(\\pm 0.51\\right)\\times 10^{-11}$ d d$^{-1}$ .",
"[27] reported the latest period change rate of $1.54\\times 10^{-10}$ d d$^{-1}$ .",
"The dramatic inconsistency comes from the under-sampling of the complex shape of the O-C curve.",
"The fluctuating pattern in the O-C diagram is believed to be caused by the binary motion with the orbital period of $\\mathrm {\\sim }$ 20 years.",
"[16] reported the radial velocities of 11.4 km/s in 2003 and 36.8 km/s in 2004, respectively.",
"The difference between these two values is taken as a proof of the binary nature.",
"At the same time, the amplitude change of $f_1$ was detected as sine function with nearly the same period of binary motion [35], [16], [22], [27].",
"However, the reason for the amplitude variations is still an open question.",
"In order to study the pulsations and period changes of AN Lyn in detail and explore its evolutionary status, we made observations for AN Lyn between 2008 to 2016 and constructed theoretical models to constrain the stellar parameters.",
"Section introduces the observations.",
"In Section , frequency analysis is performed with Fourier Decomposition.",
"Section provides constraints from theoretical models and Section gives the O-C analysis.",
"Finally, Section and Section present discussions and conclusions, respectively."
],
[
"Observations",
"AN Lyn had been observed with the 85-cm telescope in Xinglong station of National Astronomical Observatories of China between January of 2008 and January of 2016.",
"During this period, a bi-site observation campaign was made with the 85-cm telescope at Xinglong station in China and the 84-cm telescope at San Pédro Martir (SPM) Observatory in Mexico during March 13 to 20 of 2014.",
"AN Lyn was also monitored from January 21 to 24 of 2016 with the Nanshan One meter Wide field Telescope of Xinjiang Astronomical Observatory of China.",
"In total, data have been collected for AN Lyn for 50 nights in $V$ band and 34 nights in $R$ band, respectively.",
"The data are reduced with the standard procedure of CCD photometry.",
"After bias and flat-field correction, aperture photometry is performed by using the DAOPHOT program of IRAF.",
"Figure REF shows a CCD image of AN Lyn, where a reference star (GSC 02990-00001) and three check stars (TYC 2990-221-1, GSC 02990-00139, GSC 02990-00335) are marked.",
"The light curves were then produced by computing the magnitude differences between AN Lyn and the reference star and verified by the three check stars.",
"Table REF lists the observation runs and the mean photometric precisions of each run.",
"Figure REF shows the detailed distribution of photometric precisions at each night and the differences among three telescopes.",
"One can find that the precisions in more than 90% observation nights are better than $0^\\mathrm {m}.01$ and the typical photometric precision is around $\\sim 0^\\mathrm {m}.006$ .",
"There are 19 nights in which the precisions are better than $0^\\mathrm {m}.004$ .",
"However, the precisions during four nights are worse than $0^\\mathrm {m}.01$ , in which the worst one is $0^\\mathrm {m}.016$ .",
"The precisions differ among the three telescopes.",
"The 84-cm telescope at San Pédro Martir (SPM) Observatory and the Nanshan One-meter Wide field Telescope of Xinjiang Astronomical Observatory perform consistently that their mean photometric precision for all the runs is $0^\\mathrm {m}.004$ whereas the 85-cm telescope in Xinglong station shows a little worse result with the mean photometric precision of $0^\\mathrm {m}.006$ .",
"There is no apparent distinction in different bands.",
"The photometry in $R$ shows a slightly larger error.",
"Figure: Light curves of AN Lyn in VV during the bi-site observation campaign in March of 2014.Figure: Spectral window of the light curves in VV during the bi-site observation campaign in March of 2014.Figure: The amplitude spectrum of the light curves in VV band from the bi-site observations in March of 2014 and the spectra of the light curves after prewhitenings of frequencies.Figure: Same as Figure but in RR band.Figure: Amplitude variations of five frequencies of AN Lyn listed in Table .",
"The circle stands for the amplitude in VV band while the square is the amplitude in RR band.",
"Note that the amplitudes of f 0 f_0 in VV band changed dramatically, especially in 2016.Figure: The `pulsation power' of AN Lyn in different years.",
"Dramatic variations are seen in 2016.Table: Amplitudes of the five frequencies of AN Lyn in VV in six combined observation datasets listed in Table .",
"Ampl stands for the amplitude of each frequency in mag in the six datasets, Error is the uncertainty of the amplitude in mag.",
"The run names are from the first column of Table .Table: Same as Table but in RR band."
],
[
"Frequency Analysis",
"In this subsection, we report the frequency analysis of AN Lyn.",
"The frequencies in $V$ and $R$ band are resolved using the Period04 package [21] in parallel with the $Sigspec$ package [28].",
"To be convincing enough, we accept the frequencies which must appear in two bands from these two independent packages at the same time.",
"Finally, Two independent frequencies, $f_0=10.172(2)$ , $f_2=28.311(2)$ c/d, as well as their harmonics and combinations $2f_0$ , $f_2-f_0$ , $f_2+f_0$ , are resolved and confirmed.",
"Firstly, the data of the bi-site observation campaign for AN Lyn in $V$ in 2014 are used to perform Fourier Decomposition with the software Period04 to extract frequencies of pulsation.",
"We correct the zero-points of the light curves in each night.",
"The zero-points are set to zero by subtracting the mean value of the averaged maximum and minimum magnitudes in each night.",
"Figure REF shows the light curves of AN Lyn in $V$ during the bi-site observation campaign in March of 2014.",
"Figure REF plots the spectral window and Figure REF depicts the power spectra of the light curves and those after prewhitenings of frequencies.",
"The frequencies with signal-noise ratio (S/N) higher than 4 are taken following the criterion of [8].",
"The light curves are then fitted with the following formula, $m=m_0+\\sum a_i \\sin \\left[2\\pi \\left(f_i t + \\phi _i\\right)\\right]$ where $a_i$ is the amplitude, $f_i$ the frequency, $\\phi _i$ the corresponding phase.",
"Table REF lists the results of frequency analysis while Figure REF depicts the fit curves and observed curves in detail.",
"Eight frequencies are resolved with four independent ones identified: $f_0=10.1721\\pm 0.0005$ , $f_1=10.251\\pm 0.006$ , $f_2=28.311\\pm 0.003$ , and $f_3=7.46\\pm 0.01~\\mathrm {d^{-1}}$ .",
"$f_0$ is the most powerful signal, corresponding to the fundamental mode of AN Lyn with $l=0$ and $n=0$ .",
"The existence of $f_1$ causes `beat' phenomenon in light curves since its value is very close to $f_0$ .",
"However, the frequency of $f_2=28.311\\pm 0.003~\\mathrm {d^{-1}}$ should be an independent frequency with higher amplitude and larger S/N than those of the frequency of $18.125 \\pm 0.003~\\mathrm {d^{-1}}$ mentioned in previous works.",
"The latter should be the linear combination of $f_2-f_0$ .",
"Considering the frequency ratio of the fundamental to first overtone mode of 0.778 [9], the first overtone mode is not detected.",
"The other three frequencies, $f_1$ , $f_2$ , $f_3$ , should belong to nonradial modes of AN Lyn.",
"As the frequency $F=0.77\\pm 0.01~\\mathrm {d^{-1}}$ is located in the low frequency region of $0-5~\\mathrm {d^{-1}}$ , we do not take it as an intrinsic frequency of pulsation of AN Lyn.",
"However, $F$ is believed to be caused by the contamination from the variations of either the atmospheric transparency or the instability of the sensitivity of the detector during the nights.",
"Secondly, the data in $R$ band are collected simultaneously and Fourier Decomposition is also performed after the zero-point correction.",
"The power spectrum is shown in Figure REF and the resolved frequencies are listed in Table REF .",
"Figure REF shows the fit curves and the observed data.",
"In order to avoid meaningless low frequencies, we scan the frequency in the range of 5 c/d to 50 c/d [3].",
"We find $f_0$ , $2f_0$ , $f_1$ , $f_2$ , $f_2-f_0$ , $f_2+f_0$ from the Fourier Decomposition in $R$ band while $f_3$ doesn't emerge, which is in general consistent with the results in $V$ band.",
"The disappeared frequency $f_3$ is worth suspecting.",
"Considering both the absence in $R$ and the relatively low S/N in $V$ with the value of 4.1, which is close to the lower limit of Breger's criterion, this frequency hence isn't considered as an real frequency.",
"Finally, Sigspec is implemented to the data in $V$ and $R$ in order to check the frequencies extracted from previous processes.",
"This package is based on an analytical solution of the probability that a Discrete Fourier Transform (DFT) peak of a given amplitude does not arise from white noise in a non-equally spaced data set [28].",
"The statistical estimator called spectral significance ($sig$ ) are used to evaluate the reliabilities of peaks.",
"$Sigspec$ derives the frequencies by iteratively prewhitening the most significant frequency until the estimator `$sig$ ' is less than 5.",
"We find that the frequencies extracted by period04 show relative large significances hence they emerged at the first several frequencies from Sigspec.",
"However, the frequency $f_1=10.251$ d$^{-1}$ doesn't appear from the Sigspec analysis.",
"Considering that $f_1-f_0$ is smaller than the frequency resolution $1/T$ where $T$ is the observation duration, we don't accept $f_1$ as a convincing frequency.",
"Several frequencies with $sig>5$ are extracted from Sigspec.",
"However, they are not be accepted because of their absence in the results of the Period04 analysis.",
"Although there is not any apparent photometric precision difference between $V$ and $R$ , one can find that the frequencies resolved from the data in $V$ in general show higher signal to noise ratios and lower errors than those from the data in $R$ .",
"Hence the former is used in the following analysis.",
"In short, based on the data in $V$ and $R$ and the independent results from Period04 and Sigspec, five pulsation frequencies are resolved and two frequencies are identified as independent frequencies, which are $f_0=10.1721\\pm 0.0005$ , $f_2=28.311\\pm 0.003$ $\\mathrm {d^{-1}}$ , respectively."
],
[
"Amplitude Variations",
"In order to study the amplitude variations from 8-year observations, 5 frequencies listed in Table REF in $V$ are used to fit light curves of AN Lyn with six datasets presented in this paper.",
"We use the frequencies and combinations listed in Table REF to fit the light curves.",
"We select the runs with enough data and then the fitting is implemented in the runs respectively.",
"The fitting curves and observed data are shown in Figure REF for the light curves in $V$ and in Figure REF for the light curves in $R$ , respectively.",
"Table REF and Table REF list the amplitudes with errors of the 5 frequencies in the six datasets.",
"Figure REF shows the amplitude variations of the independent frequencies and their linear combinations.",
"One can find that only the amplitude of the main frequency $f_0$ in $V$ decreased dramatically from $0^\\mathrm {m}.97$ in 2008 to $0^\\mathrm {m}.76$ in 2016 while the amplitudes of the other frequencies kept relatively stable.",
"The variations of $f_0$ 's amplitude follow the previous prediction from [22], which is mentioned in Section REF .",
"Following [11], the `pulsation power' of AN Lyn can be calculated in $\\mathrm {mmag^2 \\cdot s^{-1}}$ by $E=\\sum f_i \\times A^2_i$ in which $f_i$ the resolved frequency including the independent one and the linear combination, $A_i$ the amplitude.",
"Figure REF plots the variations of `pulsation power', showing that AN Lyn's pulsation power in $V$ band had been changing from 2008 to 2016, synchronizing with the changes of the amplitude of the dominant frequency as shown in Figure REF .",
"Figure: Schematic relation among the value of χ\\chi , the stellar mass MM and the metal abundance ZZ.",
"The color stands for the value of χ -0.5 \\left| \\chi ^{-0.5} \\right|.",
"The lighter the color is, the better the calculated frequencies of the model fit the observed frequencies.Figure: Match between theoretical models and observations with ZZ of 0.020.",
"Solid lines show the evolutional tracks in different initial masses.",
"The thick areas mean that the star satisfies logT∈3.7,3.9\\log T \\in \\left[3.7,3.9\\right] and logg∈3.7,3.9\\log g \\in \\left[3.7,3.9\\right].",
"The squares show the models with f 0 ∈[10.0721,10.2721]f_0 \\in [10.0721,10.2721].",
"The cross indicates the best-fit model with the minimum value of χ 2 \\chi ^2."
],
[
"Constraints from theoretical models",
"We use Modules of Experiments in Stellar Astrophysics (MESA) to calculate stellar models.",
"MESA is a group of source-open, powerful, efficient, thread-safe libraries for a wide range of applications in computational stellar astrophysics [25], [26].",
"MESA can simulate 1-D stellar evolution with a wide range of parameters from very-low mass to massive stars.",
"The parameters we adjusted are the stellar mass $M$ and the metal abundances $Z$ .",
"The mass range is from $1.30\\mathrm {~}{M_{\\odot }}$ to $2.50\\mathrm {~}{M_{\\odot }}$ with the step of $0.05\\mathrm {~}{M_{\\odot }}$ .",
"The range of metal abundances is $Z\\in [0.015,0.025]$ with the step of $0.001$ .",
"275 evolutional tracks with different $M$ and $Z$ are calculated.",
"The simulations are made from the pre-main-sequence to the moment when $\\log T_{\\mathrm {eff}}$ smaller than 3.7, where the star is after the end of post-main sequence.",
"Then 275 evolutional tracks are calculated and the frequencies of the eigen modes are computed.",
"The frequencies in a small range of $f^{\\mathrm {obs}}_i\\pm 0.1~\\mathrm {d^{-1}}$ of different modes with degree $l \\le 3$ are calculated, since the observable modes are believed to have $l \\le 3$ .",
"The coefficients $n=0$ and $l=0$ are set for $f_0$ since it should be the fundamental frequency of AN Lyn.",
"The best-fit model is determined by the minimum value of $\\chi ^2$ $\\chi ^2=\\frac{1}{2}\\sum _{i=0}^1 \\left( f^{\\mathrm {cal}}_i-f^{\\mathrm {obs}}_i\\right)^2$ in which $f^{\\mathrm {cal}}_i$ is the theoretically calculated frequency while $f^{\\mathrm {obs}}_i$ the observed one.",
"Figure REF shows the relation among $\\left| \\chi ^{-0.5} \\right|$ , $M$ and $Z$ .",
"Finally, a model with $M$ of $1.70~\\mathrm {M_{\\odot }}$ and $Z$ of $0.020$ is taken as the best-fit model due to the minimum $\\chi ^2$ value.",
"Figure REF shows the theoretical evolutionary tracks where the best-fit model is marked with the cross on the H-R diagram.",
"The best-fit model has the parameters as: $Z = 0.020 \\pm 0.001$ , $M = 1.70 \\pm 0.05~\\mathrm {M_{\\odot }}$ , $\\log T_{\\mathrm {eff}} = 3.8201 \\pm 0.0002$ , $\\log g = 3.890 \\pm 0.005$ , $\\log L = 1.0126 \\pm 0.0003$ , age = $(1.33 \\pm 0.01) \\times 10^9 ~\\mathrm {yr}$ , $1/P\\cdot \\mathrm {d}P/\\mathrm {d}t =2.142 \\times 10^{-9} ~\\mathrm {yr^{-1}} $ with $\\chi ^2$ of $0.00005 ~\\mathrm {d^{−2}}$ .",
"Table REF lists the observed and corresponding theoretical frequencies of the best-fit model.",
"The typical difference between the theoretical and the observed frequency is $0.001~\\mathrm {d^{-1}}$ .",
"Figure: O-C diagram of AN Lyn.",
"(a): fitted by Equation .",
"The dashed line is the parabola function while the solid line concerns both the parabola and the sine function.",
"(b): fitted by Equation .",
"The dashed line shows the trend of long-period orbital motion while the solid line plots the whole motion caused by the two components.Table: Frequencies observed and calculated with the MESA code for the best-fit model of AN Lyn.",
"δf=f cal -f obs (d -1 )\\delta f = f^{\\mathrm {cal}}-f^{\\mathrm {obs}}(\\mathrm {d}^{-1}).",
"[27] gave the parameters of AN Lyn as $\\log g=3.8\\pm 0.1$ and $\\log T_{\\mathrm {eff}}=3.8\\pm 0.1$ , which are consistent with the corresponding values of the best-fit model within the range of three times of the uncertainties.",
"As for the period change rate, there is an apparent conflict between the theoretical and the observed value.",
"One can find that the prediction from the model is $1/P\\cdot \\mathrm {d}P/\\mathrm {d}t =1.06 \\times 10^{-9} ~\\mathrm {yr^{-1}} $ while the measurement from the O-C diagram mentioned in next Section is $4.5(8)\\times 10^{-7}~\\mathrm {yr^{-1}}$ , much larger by a factor of 424.",
"The distance can be calculated based on the predicted $\\log L$ .",
"The result, 453 pc, is not consistent with the photometry estimation and the parallax measurement mentioned in section .",
"The discussion is made in section REF in which the reason will be attributed to the components in the three-body system."
],
[
"O-C Analysis",
"O-C stands for O[bserved] minus C[alculated] and is a widely used method when investigating the cyclic phenomena [33].",
"We collect 238 times of maximum light of AN Lyn from literature, including 127 ones from [16], 108 from [22] and 3 from [27].",
"We derive 124 times of maximum light from our observations, in which 64 ones in $V$ band and 35 in $R$ band determined from the data with the 85-cm telescope of Xinglong station of China, 9 in $V$ band and 9 in $R$ band with the 84-cm telescope of the SPM observatory of Mexico, 4 in $V$ band and 3 in $R$ band with the Nanshan One meter Wide field Telescope of Xinjiang Observatory of China.",
"The times of maximum light are determined with the three-order polynomial fitting of the light curves.",
"The errors are estimated with the Monte-Carlo simulation.",
"Table REF lists the newly-determined times of maximum light.",
"Figure REF displays the O-C diagram.",
"The ephemeris was calculated following [27] $HJD_\\mathrm {{max}}=2444291.02077+0^\\mathrm {d}.09827447\\times E$ A parabolic fit is made for the O-C data in order to deduce the period variation rate ($1/P \\cdot \\mathrm {d} P / \\mathrm {d} t$ ).",
"An undulatory shape of distribution of the residual emerges after removing the parabolic trend.",
"Hence it is reasonable to fit the O-C data by including both the parabolic and the sinusoidal function as follows, $ O-C = \\delta T_0 + \\delta P E+\\frac{1}{2}\\frac{\\mathrm {d} P}{\\mathrm {d} t} P E^2 + A \\sin \\left[ 2\\pi \\left( \\frac{E}{\\Pi } + \\phi \\right) \\right] \\\\$ where $O-C$ means the value calculated by Equation REF .",
"$\\delta T_0$ means the correction of the first time of maximum light while $\\delta P$ the correction of the pulsation period.",
"$A$ , $\\Pi $ and $\\phi $ stand for the amplitude, the period and the phase of the sinusoidal function, respectively.",
"The fit is shown in Figure REF (a).",
"Three groups of points are not used in the fit because of their apparent deviations, marked with the black cross in Figure REF .",
"Table REF lists the coefficients of Equation REF .",
"One can deduce that the period change rate $1/P \\cdot \\mathrm {d}P /\\mathrm {d}t =4.5(8)\\times 10^{-7}~\\mathrm {yr^{-1}}$ and the orbital period of binary is $23.0\\pm 0.3 $ yr. Table: Coefficients of Equation .Concerning the theoretical prediction from the model in Section , the observation-determined period change rate is around 424 times larger than the theoretical value.",
"Such a dramatic contradiction indicates an unusual cause of period change, other than the evolutionary effect.",
"We guess that it could be caused by the light-travel time effect in a three-body system.",
"The undulatory shape in the O-C diagram could be the evidence of a short-axis component (the second component, $M_2$ ) while the parabola trend reflects a long-axis massive component (the third component, $M_3$ ).",
"We try to do Fourier Decomposition of the O-C data and fit it by using the two orbital frequencies, $O-C=Z+\\sum _{i=2,3} A_i \\sin \\left[ 2\\pi \\left( \\frac{E}{\\Pi _i} + \\phi _i \\right) \\right]$ in which $A_i$ is the amplitude of the orbital motion in days and $\\Pi _i$ the period, $Z$ the zero-point.",
"The result shows that there are two periods in the O-C data: $\\Pi _2=24\\pm 2 ~\\mathrm {yr}$ , $\\Pi _3=66\\pm 5 ~\\mathrm {yr}$ .",
"The fit is shown in Figure REF (b).",
"The three groups of discarded data points marked with cross in Figure REF show an unexpectable discrepancy with the fitted trend, which is the reason for them to be moved out in our analysis.",
"[17], [18], [19], [20] reported the data and [22] used them for O-C analysis firstly.",
"However, by collecting more data, [27] presented a more convincing result, showing a systematic deviation of these three groups of data points from the fitted curve on the O-C diagram.",
"Our observations provide six years of extension of data point distribution, supporting the fit of [27] rather than that of [22]."
],
[
"Amplitude Variations",
"Amplitude variations have been detected for almost all types of pulsating variable stars, such as the high-amplitude $\\delta $ Scuti stars [36], [38], $\\gamma $ Dor stars [32], pulsating white dwarfs [11], [14], [5] etc.",
"The mechanisms that are responsible for the amplitude variations may be attributed to either intrinsic or external reasons.",
"The latter includes the tidal effect due to a component star, like an interesting example given by [4].",
"In this assumption, amplitude variations should be synchronized with the orbital motion.",
"We collect the amplitude and period variation data of $f_0$ from [22] and added the data mentioned in Section REF .",
"The period in each epoch is calculated from the O-C's slopes extracted parabolic part.",
"Figure REF shows the amplitude, period deviation ($\\Delta P=P-0^\\mathrm {d}.09827447$ ) and O-C fluctuations of AN Lyn.",
"The period of $f_0$ amplitude variation is $22.4\\pm 1.1~\\mathrm {yr}$ and the one of $\\Delta P$ is $22.9\\pm 1.5~\\mathrm {yr}$ , while the period of the O-C data is $23.0\\pm 0.3~\\mathrm {yr}$ .",
"The accordance of these three values of period is a strong indication that the companion star can change the pulsation amplitude of AN Lyn.",
"The maximum value of orbital radial velocity is only $2.4~\\mathrm {km/s}$ calculated with Doppler effect $\\Delta P_{\\mathrm {max}}/P=v/c$ .",
"Nevertheless, two radial velocities provided by [16] varied very much (11.4 km/s in 2003 and 36.8 km/s in 2004), which might be due to the zero-point shift of the instrument, rather than the motion of AN Lyn due to binarity.",
"Figure: Amplitude and period deviation of f 0 f_0 and O-C fluctuation.",
"Top panel: Circle points stand for the amplitude data collected from while the triangles the data from our observations.",
"The period of amplitude variation is 22.4±1.1 yr 22.4\\pm 1.1~\\mathrm {yr}.",
"Middle panel: Period deviation of f 0 f_0 defined as ΔP=P-0 d .09827447\\Delta P=P-0^\\mathrm {d}.09827447 with periodicity of 22.9±1.5 yr 22.9\\pm 1.5~\\mathrm {yr}.",
"Bottom panel: O-C fluctuation with the parabola function extracted from Equation with the period of 23.0±0.3 yr 23.0\\pm 0.3~\\mathrm {yr}."
],
[
"Three-body Hypothesis",
"As illustrated in Section and Section , the contradiction of period change rate from the O-C observations and the theoretical predictions implied the possibility of the existence of three components of the AN Lyn system.",
"Based on the orbital period and the lower limit of semi-axis deduced by the O-C's amplitudes, the lower limit of the two components' masses can be estimated by the Kepler third law: $4\\pi ^2 \\frac{a_2^3}{T_2^2}=\\mathrm {G}\\frac{m_2^3}{\\left( m_1+m_2 \\right)^2}$ $4\\pi ^2 \\frac{a_3^3}{T_3^2}=\\mathrm {G}\\frac{m_3^3}{\\left( m_1+m_2+m_3 \\right)^2}$ where $m_1$ , $m_2$ and $m_3$ are the masses of AN Lyn and the two companions, $a_{2,3}$ and $T_{2,3}$ are the semi-axes and the periods of the two orbits, respectively.",
"Due to the stability of this system, the axis of the orbit of the third component should be long enough so that the gravity force on the third component equals approximately to the synthesised force of gravity from both AN Lyn and the second component.",
"To be quantitative, the ratio of $a_3$ and $a_2$ should be larger than or equal to 3 [15].",
"The parameters of the three-body system are listed in Table REF .",
"The existence of two components with minimum masses of $\\sim 0.4 \\mathrm {M_{\\odot }}$ and $\\sim 0.5 \\mathrm {M_{\\odot }}$ respectively implies that the evolutionary history of the AN Lyn system is mysterious.",
"[16] displayed a high resolution spectrum of AN Lyn in the region of H$\\beta $ .",
"It seems that there isn't any spectral line from other stars, which indicates that the luminosities of the two components should be much lower than that of AN Lyn, implying that they are compact stars or red dwarf.",
"However, the axis of the third component's orbit $a_3$ depends on the initial value of period in Equation REF .",
"A wrong period value would enlarge variations in the O-C diagram, and increase $a_3$ , making the parameters of the third component uncertain.",
"We fit the O-C data calculated by different initial linear ephemerides and deduce the lower limit of the third component's mass displayed in Figure REF .",
"One can find that the minimum value of mass of the third component is $\\sim 0.5\\mathrm {M_{\\odot }}$ , corresponding to the initial ephemeris in Equation REF .",
"The stellar model given in section reveals the luminosity of AN Lyn and the distance can be calculated based on the luminosity and the visual magnitude.",
"The result is 453 pc, which is narrowly consistent with $Gaia$ 's observation within the range of three times of uncertainty.",
"The huge discrepancy may be posed by the influence of the light emitted from other two components in the three-body system.",
"The asteroseismic analysis only gives the luminosity of AN Lyn.",
"However, the visual magnitude from the photometry is the sum of three components' brightnesses.",
"The visual magnitude and the absolute magnitude are mismatched so that the system is brighter than that AN Lyn should be.",
"Finally, the distance is shorter than that from the the parallax observation."
],
[
"Conclusions",
"After 8-year observations, the $\\delta $ Scuti star AN Lyn are investigated and analysed further.",
"Two independent frequencies are resolved from a one-week bi-site observation campaign, including one non-radial mode.",
"Amplitude variations of the three frequencies and their linear combinations are detected.",
"The amplitude of the main frequency in $V$ has been changing with time while the other frequencies keep their amplitudes stable.",
"Stellar models are constructed and the theoretical frequencies are fitted with the two observed frequencies.",
"The minimum value of $\\chi ^2$ gives the best-fit model, showing that AN Lyn locates near the terminal age main sequence on the H-R diagram with $M$ of $1.70\\pm 0.05~\\mathrm {M_{\\odot }}$ , $Z$ of $0.020\\pm 0.001$ and age of $1.33 \\pm 0.01$ billion years.",
"O-C diagram of AN Lyn is made with the new observation data together with those in the literature.",
"Since the determined period change rate is much larger than the theoretical prediction due to the evolutionary effect, the variations in the O-C diagram show the evidence of the light-travel time effect in a three-body system with two low-luminosity components, with masses larger than $0.4\\mathrm {M_{\\odot }}$ and $0.5\\mathrm {M_{\\odot }}$ , respectively.",
"Future observation is needed to confirm this hypothesis, including both high-quality time-series photometry and radial velocity observations.",
"Spectroscopic observations in the ultraviolet and infrared bands are also necessary in order to confirm the type of the components."
],
[
"Acknowledgements",
"JNF acknowledges the support from both the NSFC grant 11673003 and the National Basic Research Program of China (973 Program 2014CB845700).",
"JZL acknowledges the support from the CAS `Light of West China' program (2015-XBQN-A-02).",
"LFM acknowledges the financial support from the UNAM under grant PAPIIT IN 105115.",
"This work has made use of data from the European Space Agency (ESA) mission $Gaia$ (https://www.cosmos.esa.int/gaia), processed by the $Gaia$ Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).",
"Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the $Gaia$ Multilateral Agreement."
]
] | 1709.01689 |
[
[
"Motivic multiplicative McKay correspondence for surfaces"
],
[
"Abstract We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead of using cohomology, we deal with the Chow motives.",
"More precisely, we prove that for any smooth proper two-dimensional orbifold with projective coarse moduli space, there is an isomorphism of algebra objects, in the category of complex Chow motives, between the motive of the minimal resolution and the orbifold motive.",
"In particular, the complex Chow ring (resp.",
"Grothendieck ring, cohomology ring, topological K-theory) of the minimal resolution is isomorphic to the complex orbifold Chow ring (resp.",
"Grothendieck ring, cohomology ring, topological K-theory) of the orbifold surface.",
"This confirms the two-dimensional Motivic Crepant Resolution Conjecture."
],
[
"Introduction",
"Finite subgroups of $\\mathop {\\rm SL}\\nolimits _{2}($ are classically studied by Klein [26] and Du Val [15].",
"A complete classification (up to conjugacy) is available : cyclic, binary dihedral, binary tetrahedral, binary octahedral and binary icosahedral.",
"The last three types correspond to the groups of symmetries of Platonic solidsi.e.",
"regular polyhedrons in ${\\bf R}^{3}$ .",
"as the names indicate.",
"Let $G\\subset \\mathop {\\rm SL}\\nolimits _{2}($ be such a (non-trivial) finite subgroup acting naturally on the vector space $V\\mathrel {:=}{2}$ .",
"The quotient $X\\mathrel {:=}V/G$ has a unique rational double pointSuch (isolated) surface singularities are also known as Klein, Du Val, Gorenstein, canonical, simple or A-D-E singularities according to different points of view.. Let $f: Y\\rightarrow X$ be the minimal resolution of singularities: ${& V[d]^{\\pi }\\\\Y[r]^{f} & X}$ which is a crepant resolution, that is, $K_{Y}=f^{*}K_{X}$ .",
"The exceptional divisor, denoted by $E$ , consists of a union of $(-2)$ -curvesi.e.",
"smooth rational curve with self-intersection number equal to $-2$ .",
"meeting transversally.",
"The classical McKay correspondence ([28], cf.",
"also [30]) establishes a bijection between the set $\\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G)$ of non-trivial irreducible representations of $G$ on the one hand and the set $\\mathop {\\rm {Irr}}\\nolimits (E)$ of irreducible components of $E$ on the other hand : $\\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G)&\\simeq & \\mathop {\\rm {Irr}}\\nolimits (E)\\\\\\rho &\\mapsto & E_{\\rho }.$ Thus $E=\\bigcup _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G)}E_{\\rho }$ .",
"Moreover, this bijection respects the `incidence relations' : precisely, for any $\\rho _{1}\\ne \\rho _{2}\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G)$ , the intersection number $(E_{\\rho _{1}}\\cdot E_{\\rho _{2}})$ , which is 0 or 1, is equal to the multiplicity of $\\rho _{2}$ in $\\rho _{1}\\otimes V$ (hence is also equal to the multiplicity of $\\rho _{1}$ in $\\rho _{2}\\otimes V$ ), where $V$ is the 2-dimensional natural representation via $G\\subset \\mathop {\\rm SL}\\nolimits (V)$ .",
"All these informations can be encoded into Dynkin diagrams of A-D-E type, which is on the one hand the dual graph of the exceptional divisor $E$ and on the other hand the McKay graph of the non-trivial irreducible representations of $G$ , with respect to the preferred representation $V$ .",
"Apart from the original observation of McKay, there are many approaches to construct this correspondence geometrically and to extend it to higher dimensions : K-theory of sheaves [21], $G$ -Hilbert schemes [29], [24], [23], [22], motivic integration [4], [5], [13], [14], [35], [27] and derived categories [8] etc.",
"We refer the reader to Reid's note of his Bourbaki talk [30] for more details and history.",
"Following Reid [29], one can recast the above McKay correspondence (the bijection) as follows: the isomorphism classes of irreducible representations index a basis of the homology of the resolution $Y$.",
"This is of course equivalent to say that the conjugacy classes of $G$ index a basis of the cohomology of $Y$.",
"The starting point of this paper is that the quotient $X=V/G$ is the coarse moduli space of a smooth orbifold/Deligne–Mumford stack ${\\mathcal {X}}\\mathrel {:=}[V/G]$ , and that the (co)homology of the coarse moduli space $|I{\\mathcal {X}}|$ of its inertia stack $I{\\mathcal {X}}$ has a basis indexed by the conjugacy classes of $G$ .",
"Thus Reid's McKay correspondence can be stated as an isomorphism of vector spaces: $H^{*}(Y)\\simeq H^{*}(|I{\\mathcal {X}}|).$ Chen and Ruan defined in [11] the orbifold cohomology and the orbifold product (i.e.",
"Chen–Ruan cohomology) for any smooth orbifold.",
"See Definition REF for a down-to-earth construction in the global quotient case.",
"By definition, the orbifold cohomology ring $H^{*}_{orb}([V/G])$ has $H^{*}(|I{\\mathcal {X}}|)$ as the underlying vector space.",
"Therefore it is natural to ask whether there is a multiplicative isomorphism (of algebras) $H^{*}(Y)\\simeq H^{*}_{orb}([V/G]).$ None of the aforementioned beautiful theories (K-theory, $G$ -Hilbert schemes, motivic integration and derived categories) produces an isomorphism which respects the multiplicative structures.",
"Nevertheless, the existence of such an isomorphism of algebras is known.",
"For example, it is a baby case of the result of Ginzburg–Kaledin [20] on symplectic resolutions of symplectic quotient singularities.",
"An explicit isomorphism between the equivariant orbifold quantum cohomology of $[V/G]$ and the equivariant cohomology of its minimal resolution is proposed by Bryan–Graber–Pandharipande in [10], which is verified for the ${2}/{{\\bf Z}_{3}}$ case (see also the related work [7], [9]).",
"We will use the same formula to construct our multiplicative isomorphism.",
"This isomorphism fits perfectly into Ruan's following more general Cohomological Crepant Resolution Conjecture (CCRC) : Conjecture 1.1 (CCRC [31]) Let $M$ be a smooth projective variety endowed with a faithful action of a finite group $G$ .",
"Assume that the quotient $X\\mathrel {:=}M/G$ is Gorenstein, then for any crepant resolution $Y\\rightarrow X$ , there is an isomorphism of graded $-algebras:\\begin{equation}H^{*}_{qc}(Y, \\simeq H^{*}_{orb}\\left([M/G], ).\\right.More generally, given a smooth proper orbifold {\\mathcal {X}} with underlying singular variety X being Gorenstein, then for any crepant resolution Y\\rightarrow X, we have an isomorphism of graded -algebras:H^{*}_{qc}(Y, \\simeq H^{*}_{orb}\\left({\\mathcal {X}}, ).\\right.\\end{equation}$ Here the left hand side is the quantum corrected cohomology algebra, whose underlying graded vector space is just $H^{*}(Y, $ , endowed with the cup product with quantum corrections related to Gromov–Witten invariants with curve classes contracted by the crepant resolution, as defined in [31].",
"Since we only consider in this paper the two-dimensional situation, the Gromov–Witten invariants always vanish hence there are no quantum corrections involved.",
"See Lemma REF for this vanishing.",
"Conjecture REF suggests that one should consider the existence of such multiplicative McKay correspondence in the global situation (instead of a quotient of a vector space by a finite group), that is, a Gorenstein quotient of a surface by a finite group action, or even more generally a two-dimensional proper Gorenstein orbifold.",
"Our following main result confirms this, which also pushes the (surface) McKay correspondence to the motivic level: Theorem 1.2 (Motivic multiplicative global McKay correspondence) Let ${\\mathcal {X}}$ be a smooth proper two-dimensional Deligne–Mumford stack with isolated stacky points.",
"Assume that ${\\mathcal {X}}$ has projective coarse moduli space $X$ with Gorenstein singularities.",
"Let $Y\\rightarrow X$ be the minimal resolution.",
"Then we have an isomorphism of algebra objects in the category $\\mathop {\\rm {CHM}}\\nolimits _{ of Chow motives with complex coefficients:\\begin{equation}Y)_{\\simeq {orb}\\left({\\mathcal {X}}\\right)_{.",
"}In particular, one has an isomorphism of -algebras:\\begin{eqnarray*}\\mathop {\\rm CH}\\nolimits ^{*}(Y)_{ &\\simeq & \\mathop {\\rm CH}\\nolimits ^{*}_{orb}\\left({\\mathcal {X}}\\right)_{;\\\\H^{*}(Y, &\\simeq & H^{*}_{orb}\\left({\\mathcal {X}}, );\\\\K_{0}(Y)_{ &\\simeq & K_{orb}\\left({\\mathcal {X}}\\right)_{;\\\\K^{top}(Y)_{ &\\simeq & K^{top}_{orb}\\left({\\mathcal {X}}\\right)_{.",
"}}}This result also confirms the 2-dimensional case of the so-called \\emph {Motivic HyperKähler Resolution Conjecture} studied in \\cite {MHRCKummer} and \\cite {MHRCK3}.",
"See §\\ref {subsect:Motives} for the basics of Chow motives.",
"}As the definitions of the orbifold theories are particularly explicit and elementary for the global quotient stacks (\\textit {cf.",
"}~§\\ref {subsect:OrbQuo}), we deliberately treat the global quotient case (§\\ref {sect:ProofQuotient}) and the general case (§\\ref {sect:ProofGeneral}) separately.\\right.\\textbf {Convention :} All Chow rings and K-theories are with rational coefficients unless otherwise stated.",
"\\mathop {\\rm {CHM}}\\nolimits is the category of Chow motives with rational coefficients and \\mathop {\\rm {SmProj}}\\nolimits ^{op}\\rightarrow \\mathop {\\rm {CHM}}\\nolimits is the (contra-variant) functor that associates a smooth projective variety its Chow motive (§\\ref {subsect:Motives}).",
"An \\emph {orbifold} means a separated Deligne--Mumford stack of finite type with trivial stabilizer at the generic point.",
"We work over an algebraically closed field of characteristic zero.",
"}\\textbf {Acknowledgement :} The authors want to thank Samuel Boissière, Cédric Bonnafé, Philippe Caldero, Jérôme Germoni and Dmitry Kaledin for interesting discussions and also the referee for his or her very helpful suggestions.",
"The most part of the paper is prepared when Lie Fu is staying with his family at the Hausdorff Institute of Mathematics for the 2017 trimester program on K-theory.",
"He thanks Bonn University for providing the perfect working condition in HIM and the relaxing living style in such a beautiful city.",
"}\\end{eqnarray*}\\section {Crepant resolution conjecture}Let us give the construction of the \\emph {orbifold Chow motive} (as an algebra object) and the \\emph {orbifold Chow ring}.",
"we will first give the down-to-earth definition for an orbifold which is a global Gorenstein quotient by a finite group\\,; then we invoke the techniques in \\cite {MR2450211} to give the construction in the general setting of Deligne--Mumford stacks.",
"We refer to our previous work \\cite {MHRCKummer} (joint with Charles Vial), \\cite {MHRCK3} as well as the original sources (for cohomology and Chow rings) \\cite {MR2104605}, \\cite {MR1971293}, \\cite {MR2450211}, \\cite {MR2285746} for the history and more details.",
"For the convenience of the reader, we start with a reminder on the basic notions of Chow motives.",
"}\\subsection {The category of Chow motives}The idea of (pure) \\emph {motives}, proposed initially by Grothendieck, is to construct a universal cohomology theory X\\mapsto X) for smooth projective varieties.",
"His construction uses directly the algebraic cycles on the varieties together with some natural categorical operations.",
"On the one hand, motives behave just like the classically considered Weil cohomology theories\\,; on the other hand, they no longer take values in the category of vector spaces but in some additive idempotent-complete rigid symmetric monoïdal category.",
"Although the construction works for any adequate equivalence relation on algebraic cycles, we use throughout this paper the finest one, namely the \\emph {rational} equivalence, so that our results will hold for Chow groups and imply the other analogous ones, on cohomology for instance, by applying appropriate realization functors.\\end{equation}Fix a base field k. The category of \\emph {Chow motives} over the field k with rational coefficients, denoted by \\mathop {\\rm {CHM}}\\nolimits , is defined as follows (\\textit {cf.",
"}~\\cite {MR2115000} for a more detailed treatment).",
"An object, called a \\emph {Chow motive}, is a triple (X, p, n), where n is an integer, X is a smooth projective variety over k and p\\in \\mathop {\\rm CH}\\nolimits ^{\\dim X}(X\\times X) is a projector, that is, p\\circ p=p as correspondences.",
"Morphisms between two objects (X, p, n) to (Y, q, m) form the following {\\bf Q}-vector subspace of \\mathop {\\rm CH}\\nolimits ^{m-n+\\dim X}(X\\times Y)\\,:\\mathop {\\rm Hom}\\nolimits _{\\mathop {\\rm {CHM}}\\nolimits }\\left((X, p, n), (Y, q, m)\\right):=q\\circ \\mathop {\\rm CH}\\nolimits ^{m-n+\\dim X}(X\\times Y)\\circ p.The composition of morphisms are defined by the composition of correspondences.We have the following naturally defined contra-variant functor from the category of smooth projective varieties to the category of Chow motives\\,:\\begin{eqnarray*} \\mathop {\\rm {SmProj}}\\nolimits ^{op}&\\rightarrow & \\mathop {\\rm {CHM}}\\nolimits \\\\X&\\mapsto & (X,\\Delta _{X}, 0)\\\\(f\\colon X\\rightarrow Y) &\\mapsto & {}^{t}\\Gamma _{f}\\in \\mathop {\\rm CH}\\nolimits ^{\\dim X}(Y\\times X)\\end{eqnarray*}where {}^{t}\\Gamma _{f} is the transpose of the graph of f. The image X) is called the Chow motive of X.",
"}The category $ CHM$ is additive with direct sum induced by the disjoint union of varieties.",
"By construction, $ CHM$ is idempotent complete (\\textit {i.e.",
"}~pseudo-abelian)\\,: for any motive $ M$ and any projector of it, that is, $ EndCHM(M)$ such that $ =$, we have $ MIm()Im(idM-)$.",
"As an example, let us recall the definition of the so-called \\emph {reduced motive}: for a smooth projective variety $ X$ together with a chosen $ k$-rational point $ x$, the composition of $ Speckx X$ with the structure morphism $ XSpeck$ is identity, hence defines a projector $$ of $ X)$.",
"The reduced motive of the pointed variety $ (X, x)$, denoted by $ (X)$, is by definition $ Im(idX)-)$.",
"One can show that the isomorphism class of $ (X)$ is independent of the choice of the point $ x$ (\\textit {cf.",
"}~\\cite [Exemple 4.1.2.1]{MR2115000}).$ There is also a natural symmetric monoïdal structure on $\\mathop {\\rm {CHM}}\\nolimits $ , compatible with the additive structure, given by $(X, p, n)\\otimes (Y, q, m):=(X\\times Y, p\\times q, n+m).$ Hence the Künneth formula: $X\\times Y)\\cong X)\\otimes Y)$ holds for any smooth projective varieties $X$ and $Y$ .",
"Moreover, this tensor category is rigid, with the following duality functor $(X, p, n)^{\\vee }:=(X, {}^{t}p, \\dim X-n).$ Given an integer $n$ , the motive $(\\mathop {\\rm Spec}\\nolimits k, \\Delta _{\\mathop {\\rm Spec}\\nolimits k}, n)$ is called the $n$ -th Tate motive and is denoted by $(n)$ .",
"They are the tensor invertible objects.",
"For any motive $M$ , the tensor product $M\\otimes (n)$ is denoted by $M(n)$ and called the $n$ -th Tate twist of $M$ .",
"In particular, we have the Poicaré duality: $X)^{\\vee }\\cong X)(\\dim X)$ for any smooth projective variety $X$ .",
"The Lefschetz motive is $\\mathop {{L}}\\nolimits :=(-1)$ .",
"One checks that the reduced motive of $\\mathop {\\bf P}\\nolimits ^1$ is isomorphic to $\\mathop {{L}}\\nolimits $ .",
"The functor $ is considered as a cohomology theory and it is universal in the sense that any Weil cohomology theory must factorize through $ .",
"We can extend the notion of Chow groups from varieties to motives by defining for any integer $i$ and any Chow motive $M$ , $\\mathop {\\rm CH}\\nolimits ^{i}(M):=\\mathop {\\rm Hom}\\nolimits _{\\mathop {\\rm {CHM}}\\nolimits }\\left((-i), M\\right),$ Hence the Chow groups of a smooth projective variety $X$ is recovered as $\\mathop {\\rm CH}\\nolimits ^{i}(X)=\\mathop {\\rm CH}\\nolimits ^{i}\\left(X)\\right).$ In all the above constructions, one can replace for the coefficient field the rational numbers by the complex numbers and obtain the category of complex Chow motives $\\mathop {\\rm {CHM}}\\nolimits _{.\\subsection {Orbifold theory : global quotient case}}Let $ M$ be a smooth projective variety and $ G$ be a finite group acting faithfully on $ M$.",
"Assume that $ G$ preserves locally the canonical bundle\\,: for any $ xM$ fixed by $ gG$, the differential $ DgSL(TxM)$.",
"This amounts to require that the quotient $ X:=M/G$ has only Gorenstein singularities.",
"Denote by $ Mg={xM gx=x}$ the fixed locus of $ gG$, $ Mg,h =MgMh$ (with the reduced scheme structure) and $ X:=[M/G]$ the quotient smooth Deligne--Mumford stack.$ Definition 1.3 (Orbifold theories) We define an auxiliary algebra object $M, G)$ in $\\mathop {\\rm {CHM}}\\nolimits $ with $G$ -action, and the orbifold motive $[M/G])$ will be its subalgebra of invariants.",
"The definitions for Chow rings, cohomology and K-theory are similar.",
"For any $g\\in G$ , the age function, denoted by $\\mathop {\\rm {age}}\\nolimits (g)$ , is a ${\\bf Z}$ -valued locally constant function on $M^{g}$ , whose value on a connected component $Z$ is $\\mathop {\\rm {age}}\\nolimits (g)|_{Z}\\mathrel {:=}\\sum _{j=0}^{r-1} \\frac{j}{r} \\mathop {\\rm rank}\\nolimits (W_{j}),$ where $r$ is the order of $g$ , $W_{j}$ is the eigen-subbundle of the restricted tangent bundle $TM|_{Z}$ , for the natural automorphism induced by $g$ , with eigenvalue $e^{\\frac{2\\pi i}{r}j}$ .",
"The age function is invariant under conjugacy.",
"We endow the direct sums $M, G)\\mathrel {:=}\\bigoplus _{g\\in G} M^{g})(-\\mathop {\\rm {age}}\\nolimits (g))$ $\\mathop {\\rm CH}\\nolimits ^{*}(M, G)\\mathrel {:=}\\bigoplus _{g\\in G} \\mathop {\\rm CH}\\nolimits ^{*-\\mathop {\\rm {age}}\\nolimits (g)}(M^{g})$ $H^{*}(M, G)\\mathrel {:=}\\bigoplus _{g\\in G} H^{*-2\\mathop {\\rm {age}}\\nolimits (g)}(M^{g})$ $K(M, G)\\mathrel {:=}\\bigoplus _{g\\in G} K_{0}(M^{g})_{{\\bf Q}}$ $K^{top}(M, G)\\mathrel {:=}\\bigoplus _{g\\in G} K^{top}(M^{g})_{{\\bf Q}}$ with the natural $G$ -action induced by the following action: for any $g, h\\in G$ , $h: M^{g}&\\xrightarrow{}& M^{hgh^{-1}}\\\\x&\\mapsto & hx.$ For any $g\\in G$ , define $V_{g}\\mathrel {:=}\\sum _{j=0}^{r-1}\\frac{j}{r}[W_{j}]\\in K_{0}(M^{g})_{{\\bf Q}},$ whose virtual rank is $\\mathop {\\rm {age}}\\nolimits (g)$ , where $r$ and $W_{j}$ 's are as in $(1^{\\circ })$ .",
"For any $g_{1}, g_{2}\\in G$ , let $g_{3}=g_{2}^{-1}g_{1}^{-1}$ , we define the (virtual class of ) the obstruction bundle on the fixed locus $M^{\\langle g_{1}, g_{2} \\rangle }$ by $F_{g_{1}, g_{2}}\\mathrel {:=}\\left.V_{g_{1}}\\right|_{M^{<g_{1}, g_{2}>}}+ \\left.V_{g_{2}}\\right|_{M^{<g_{1}, g_{2}>}}+ \\left.V_{g_{3}}\\right|_{M^{<g_{1}, g_{2}>}}+TM^{<g_{1},g_{2}>}-\\left.TM\\right|_{M^{<g_{1},g_{2}>}} \\in K_{0}\\left(M^{<g_{1}, g_{2}>}\\right)_{{\\bf Q}}.$ The orbifold product $\\star _{orb}$ is defined as follows: given $g, h\\in G$ , let $\\iota : M^{<g,h>}\\hookrightarrow M$ be the natural inclusion.",
"For cohomology: $\\star _{orb}: H^{i-2\\mathop {\\rm {age}}\\nolimits (g)}(M^{g}) \\times H^{j-2\\mathop {\\rm {age}}\\nolimits (h)}(M^{h}) &\\rightarrow & H^{i+j-2\\mathop {\\rm {age}}\\nolimits (gh)}(M^{gh})\\\\(\\alpha , \\beta )&\\mapsto & \\iota _{*}\\left(\\alpha |_{M^{<g,h>}}\\smile \\beta |_{M^{<g,h>}}\\smile c_{top}(F_{g,h})\\right)$ For Chow groups: $\\star _{orb}: \\mathop {\\rm CH}\\nolimits ^{i-\\mathop {\\rm {age}}\\nolimits (g)}(M^{g}) \\times \\mathop {\\rm CH}\\nolimits ^{j-\\mathop {\\rm {age}}\\nolimits (h)}(M^{h}) &\\rightarrow & \\mathop {\\rm CH}\\nolimits ^{i+j-\\mathop {\\rm {age}}\\nolimits (gh)}(M^{gh})\\\\(\\alpha , \\beta )&\\mapsto & \\iota _{*}\\left(\\alpha |_{M^{<g,h>}}\\cdot \\beta |_{M^{<g,h>}}\\cdot c_{top}(F_{g,h})\\right)$ For K-theory: $\\star _{orb}: K_{0}(M^{g})_{{\\bf Q}} \\times K_{0}(M^{h})_{{\\bf Q}} &\\rightarrow & K_{0}(M^{gh})_{{\\bf Q}}\\\\(\\alpha , \\beta )&\\mapsto & \\iota _{!",
"}\\left(\\alpha |_{M^{<g,h>}}\\cdot \\beta |_{M^{<g,h>}}\\cdot \\lambda _{-1}(F_{g,h}^{\\vee })\\right)$ where $\\lambda _{-1}$ is obtained from the Lambda operation $\\lambda _{t}\\colon K_{0}(M^{<g,h>})\\rightarrow K_{0}(M^{<g,h>})\\llbracket t\\rrbracket $ by evaluating $t=-1$ (cf. [34]).",
"The definition for topological K-theory is similar.",
"For motives: $\\star _{orb}: M^{g})(-\\mathop {\\rm {age}}\\nolimits (g))\\otimes M^{h})(-\\mathop {\\rm {age}}\\nolimits (h))\\rightarrow M^{gh})(-\\mathop {\\rm {age}}\\nolimits (gh))$ is determined by the correspondence $\\delta _{*}(c_{top}(F_{g,h}))\\in \\mathop {\\rm CH}\\nolimits ^{\\dim M^{g}+\\dim M^{h}+\\mathop {\\rm {age}}\\nolimits (g)+\\mathop {\\rm {age}}\\nolimits (h)-\\mathop {\\rm {age}}\\nolimits (gh)}(M^{g}\\times M^{h}\\times M^{gh}),$ where $\\delta : M^{<g, h>}\\rightarrow M^{g}\\times M^{h}\\times M^{gh}$ is the natural morphism sending $x$ to $(x,x,x)$ .",
"Finally, we take the subalgebra of invariants whose existence is guaranteed by the idempotent completeness of $\\mathop {\\rm {CHM}}\\nolimits $ (see §) : ${orb}\\left([M/G]\\right)\\mathrel {:=}(M, G^{G};$ $\\mathop {\\rm CH}\\nolimits ^{*}_{orb}\\left([M/G]\\right)\\mathrel {:=}\\left(\\mathop {\\rm CH}\\nolimits ^{*}(M, G), \\star _{orb}\\right)^{G};$ and similarly $H^{*}_{orb}\\left([M/G]\\right)\\mathrel {:=}\\left(H^{*}(M, G), \\star _{orb}\\right)^{G};$ $K_{orb}\\left([M/G]\\right)\\mathrel {:=}\\left(K(M, G), \\star _{orb}\\right)^{G};$ $K^{top}_{orb}\\left([M/G]\\right)\\mathrel {:=}\\left(K^{top}(M, G), \\star _{orb}\\right)^{G}.$ These are commutative ${\\bf Q}$ -algebras and depend only on the stack $[M/G]$ (not the presentation)."
],
[
"Orbifold theory : general case",
"Let ${\\mathcal {X}}$ be a smooth proper orbifold with projective coarse moduli space $X$ with Gorenstein singularities.",
"Recall that under the Gorenstein assumption, the age function takes values in integers.",
"Define the orbifold Chow motive and orbifold Chow group as follows: ${orb}({\\mathcal {X}}):=I{\\mathcal {X}})(-\\mathop {\\rm {age}}\\nolimits ):=\\oplus _{i}I{\\mathcal {X}}_{i})(-\\mathop {\\rm {age}}\\nolimits _{i}),$ $\\mathop {\\rm CH}\\nolimits ^{*}_{orb}({\\mathcal {X}}):= \\mathop {\\rm CH}\\nolimits ^{*-\\mathop {\\rm {age}}\\nolimits }(I{\\mathcal {X}}):=\\oplus _{i}\\mathop {\\rm CH}\\nolimits ^{*-\\mathop {\\rm {age}}\\nolimits _{i}}(I{\\mathcal {X}}_{i})\\,;$ where the theory of Chow ring (with rational coefficients) as well as the intersection theory of a stack is the one developed by Vistoli in [33] ; the theory of Chow motives for smooth proper Deligne–Mumford stacks is the so-called DMC motivesDMC stands for Deligne–Mumford Chow.",
"developed by Behrend–Manin in [6] and reviewed in Toën [32], which is proven in [32] to be equivalent to the usual category of Chow motives ; $I{\\mathcal {X}}=\\coprod _{i} I{\\mathcal {X}}_{i}$ is the decomposition into connected components while the age function $\\mathop {\\rm {age}}\\nolimits $ is the locally constant function whose value on $I{\\mathcal {X}}_{i}$ is $\\mathop {\\rm {age}}\\nolimits _{i}$ which is Chen–Ruan's degree shifting number defined in [11].",
"Let us also point out that Toën's second construction in [32] of Chow motives of Delign–Mumford stacks is very close to the orbifold Chow motive defined above with the only difference being the age-shifting.",
"Now the key point is to put a product structure on ${orb}({\\mathcal {X}})$ and $\\mathop {\\rm CH}\\nolimits ^{*}_{orb}({\\mathcal {X}})$ .",
"Consider the moduli space $K_{0,3}({\\mathcal {X}}, 0)$ , constructed by Abramovich–Vistoli [2], of 3-pointed twisted stable maps of genus zero with trivial curve class.",
"It comes equipped with a virtual fundamental class $[K_{0,3}({\\mathcal {X}}, 0)]^{vir}\\in \\mathop {\\rm CH}\\nolimits _{\\dim X}\\left(K_{0,3}({\\mathcal {X}}, 0)\\right)$ together with three (proper) evaluation maps: $e_{i}: K_{0,3}({\\mathcal {X}}, 0)\\rightarrow I{\\mathcal {X}}$ with target being the inertia stack ([1]).",
"Note that in general, the evaluation morphism has target in a different stack, the rigidified cyclotomic inertial stack ([1]).",
"However, in the smooth orbifold case, one can prove that the evaluation morphisms of the degree 0 twisted stable maps land in the inertial stack [16].",
"Pushing forward the virtual fundamental class gives the class $\\gamma :=(e_{1}, e_{2}, \\check{e}_{3})_{*}\\left([K_{0,3}({\\mathcal {X}}, 0)]^{vir}\\right)\\in \\mathop {\\rm CH}\\nolimits _{\\dim X}(I{\\mathcal {X}}^{3}),$ where $\\check{e}_{3}$ is the composition of the evaluation map $e_{3}$ and the involution $I{\\mathcal {X}}\\rightarrow I{\\mathcal {X}}$ inverting the group element (cf.",
"[1]); and we are using again Vistoli's Chow groups ([33]).",
"The orbifold product for the orbifold Chow ring is defined as the action of the correspondence $\\gamma $ : $\\mathop {\\rm CH}\\nolimits ^{*}_{orb}({\\mathcal {X}}) \\times & \\mathop {\\rm CH}\\nolimits ^{*}_{orb}({\\mathcal {X}})\\rightarrow & \\mathop {\\rm CH}\\nolimits ^{*}_{orb}({\\mathcal {X}})\\\\\\parallel &&\\parallel \\\\\\mathop {\\rm CH}\\nolimits ^{*-\\mathop {\\rm {age}}\\nolimits }(I{\\mathcal {X}})\\times & \\mathop {\\rm CH}\\nolimits ^{*-\\mathop {\\rm {age}}\\nolimits }(I{\\mathcal {X}}) \\rightarrow &\\mathop {\\rm CH}\\nolimits ^{*-\\mathop {\\rm {age}}\\nolimits }(I{\\mathcal {X}})\\\\(\\alpha ,\\beta )&\\mapsto & \\mathop {\\rm pr}\\nolimits _{3,*}\\left(\\mathop {\\rm pr}\\nolimits _{1}^{*}(\\alpha )\\cdot \\mathop {\\rm pr}\\nolimits _{2}^{*}(\\beta )\\cdot \\gamma \\right)$ It can be checked (cf.",
"[1]) that the age shifting makes the above orbifold product additive with respect to the degrees (otherwise, it is not!).",
"Similarly, we can define the multiplicative structure on ${orb}({\\mathcal {X}})$ to be $\\gamma \\in \\mathop {\\rm CH}\\nolimits _{\\dim X}(I{\\mathcal {X}}^{3})&=&\\mathop {\\rm Hom}\\nolimits _{\\mathop {\\rm {CHM}}\\nolimits }\\left(I{\\mathcal {X}})(-\\mathop {\\rm {age}}\\nolimits )\\otimes I{\\mathcal {X}})(-\\mathop {\\rm {age}}\\nolimits ), I{\\mathcal {X}})(-\\mathop {\\rm {age}}\\nolimits )\\right)\\\\&=&\\mathop {\\rm Hom}\\nolimits _{\\mathop {\\rm {CHM}}\\nolimits }\\left({orb}({\\mathcal {X}})\\otimes {orb}({\\mathcal {X}}), {orb}({\\mathcal {X}})\\right).$ Thanks to [1], this product structure is associative.",
"On the other hand, when ${\\mathcal {X}}$ is a finite group global quotient stack, the main result of [25] implies that the elementary construction in § actually recovers the above abstract construction."
],
[
"Crepant resolution conjectures",
"With orbifold theories being defined, we can speculate that a motivic or K-theoretic version of the Crepant Resolution Conjecture REF should hold.",
"But the problem is that in the definition of the quantum corrections, there is the subtle convergence property which is difficult to make sense in general for Chow groups / motives or for K-theory.",
"Therefore, we will look at some cases that these quantum corrections actually vanish a priori :"
],
[
"The first one is when the resolution $Y$ is holomorphic symplectic, which implies that all (Chow-theoretic, K-theoretic or cohomological) Gromov–Witten invariants vanish (see the proof of [18]).",
"In this case, we indeed have the following Motivic Hyper-Kähler Resolution Conjecture (MHRC), proposed in [19]: Conjecture 1.4 (MHRC [19], [18]) Let $M$ be a smooth projective holomorphic symplectic variety endowed with a faithful symplectic action of a finite group $G$ .",
"If the quotient $X\\mathrel {:=}M/G$ has a crepant resolution $Y\\rightarrow X$ , then there is an isomorphism of algebra object in the category $\\mathop {\\rm {CHM}}\\nolimits _{ of complex Chow motives:Y)\\simeq {orb}([M/G]).In particular, we have an isomorphism of graded -algebras:\\begin{equation*}\\mathop {\\rm CH}\\nolimits ^{*}(Y)_{\\simeq \\mathop {\\rm CH}\\nolimits ^{*}_{orb}\\left([M/G]\\right)_{.",
"}}Thanks to the orbifold Chern character isomorphism constructed by Jarvis--Kaufmann--Kimura in \\cite {MR2285746}, MHRC also implies the K-theoretic Hyper-Kähler Resolution Conjecture of \\textit {loc.cit.",
"}~.Conjecture \\ref {conj:MHRC} is proven in our joint work with Charles Vial \\cite {MHRCKummer} for Hilbert schemes of abelian varieties and generalized Kummer varieties and in \\cite {MHRCK3} for Hilbert schemes of K3 surfaces.\\end{equation*}\\paragraph {\\textbf {Case 2: Surface minimal resolution}}The second one is the main purpose of the article, namely the surface case, \\textit {i.e.",
"}~\\dim (Y)=2.",
"In this case, the vanishing of quantum corrections is explained in the following lemma.",
"}$ Lemma 1.5 Let $X$ be a surface with Du Val singularities and $\\pi : Y\\rightarrow X$ be the minimal resolution.",
"Then the virtual fundamental class of $\\overline{M_{0,3}}\\left(Y,\\beta \\right)$ is rationally equivalent to zero for any curve class $\\beta $ which is contracted by $\\pi $ .",
"Consider the forgetful-stabilization morphism $f: \\overline{M_{0,3}}\\left(Y,\\beta \\right)\\rightarrow \\overline{M_{0,0}}\\left(Y,\\beta \\right).$ By the general theory, the virtual fundamental class of $\\overline{M_{0,3}}\\left(Y,\\beta \\right)$ is the pull-back of the virtual fundamental class of $\\overline{M_{0,0}}\\left(Y,\\beta \\right)$ .",
"However, the virtual dimension of $\\overline{M_{0,0}}\\left(Y,\\beta \\right)$ is $(\\beta \\cdot K_{Y})+(\\dim Y-3)=-1$ since $\\pi $ is crepant.",
"Therefore, both moduli spaces have zero virtual fundamental class in Chow group, cohomology or K-theory.",
"Thanks to the vanishing of quantum corrections, the motivic version of the Crepant Resolution Conjecture REF for surfaces is exactly the content of our main Theorem REF .",
"See the precise statement in Introduction.",
"We will first give the proof for stacks which are finite group global quotients in §, then the proof in the general case in §."
],
[
"Proof of Theorem ",
"In this section, we show Theorem REF in the following setting: $S$ is a smooth projective surface, $G$ is a finite group acting faithfully on $S$ such that the canonical bundle is locally preserved (Gorenstein condition), $X\\mathrel {:=}S/G$ is the quotient surface (with Du Val singularities) and $Y\\rightarrow X$ is the minimal (crepant) resolution.",
"Recall that $\\mathop {{L}}\\nolimits \\mathrel {:=}(-1)$ is the Lefschetz motive in $\\mathop {\\rm {CHM}}\\nolimits $ (§).",
"For any $x\\in S$ , let $G_{x}\\mathrel {:=}\\lbrace g\\in G ~\\vert ~ gx=x\\rbrace $ be the stabilizer.",
"Let $\\mathop {\\rm {Irr}}\\nolimits (G_{x})$ be the set of isomorphism classes of irreducible representations of $G_{x}$ and $\\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ be that of non-trivial ones.",
"We remark that by assumption, there are only finitely many points of $S$ with non-trivial stabilizer."
],
[
"Resolution side",
"We first compute the Chow motive algebra (or Chow ring) of the minimal resolution $Y$ .",
"For any $x\\in S$ , we denote by $\\overline{x}$ its image in $S/G$ .",
"The Chow motive of $Y$ has the following decomposition in $\\mathop {\\rm {CHM}}\\nolimits $ : $(Y\\simeq S)^{G}\\oplus \\bigoplus _{\\overline{x}\\in S/G}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{\\overline{x}, \\rho }\\simeq \\left(S)\\oplus \\bigoplus _{x\\in S}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }\\right)^{G},$ where $\\mathop {{L}}\\nolimits _{\\overline{x}, \\rho }$ and $\\mathop {{L}}\\nolimits _{x,\\rho }$ are both the Lefschetz motive $\\mathop {{L}}\\nolimits $ corresponding to the irreducible component of the exceptional divisor over $\\overline{x}$ , indexed by the non-trivial irreducible representation $\\rho $ of $G_{x}$ via the classical McKay correspondence.",
"The second isomorphism in (REF ) being just a trick of reindexing, let us explain a bit more on the first one.",
"Let $f:Y\\rightarrow S/G$ be the minimal resolution of singularities.",
"By the classical McKay correspondence, over each singular point $\\overline{x}\\in S/G$ , the exceptional divisor $E_{\\overline{x}}:=f^{-1}(\\overline{x})$ is a union (with A-D-E configuration) of smooth rational curves $\\cup _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}E_{\\overline{x},\\rho }$ .",
"As $f$ is obviously a semi-small morphism, we can invoke the motivic decomposition of De Cataldo–Migliorini [12], with the stratification being $S/G=(S/G)_{reg}\\cup \\mathop {\\rm Sing}\\nolimits (S/G)$ , to obtain directly the first isomorphism in (REF ).",
"It is then not hard to follow the proof in loc.cit.",
"to see that the first isomorphism in (REF ) is induced by the pull-back $f^{*}={}^{t}\\Gamma _{f}:S)^{G}=S/G)\\rightarrow Y)$ together with the push-forward along the inclusions $\\mathop {{L}}\\nolimits =\\widetilde{ (E_{\\overline{x}, \\rho })\\xrightarrow{} Y), where \\widetilde{ is the reduced motive (see §\\ref {subsect:Motives}).",
"We remark that the inverse of the isomorphism (\\ref {eqn:decompRes}) is more complicated to describe and involves the inverse of the intersection matrix (\\textit {cf.",
"}~the definition of \\Gamma ^{\\prime } in the end of \\cite [§2]{MR2067464}).",
"}}$ The product structure of $Y)$ is determined as follows via the above decomposition (REF ), which also uses the classical McKay correspondence.",
"Let $i_{x}:\\lbrace x\\rbrace \\hookrightarrow S$ be the natural inclusion.",
"$S)\\otimes S)\\xrightarrow{} S)$ is the usual product induced by the small diagonal of $S^{3}$ .",
"For any $x$ with non-trivial stabilizer $G_{x}$ and any $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , $S)\\otimes \\mathop {{L}}\\nolimits _{x,\\rho }\\xrightarrow{} \\mathop {{L}}\\nolimits _{x,\\rho } $ is determined by the class $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(S)=\\mathop {\\rm Hom}\\nolimits (S)\\otimes \\mathop {{L}}\\nolimits , \\mathop {{L}}\\nolimits )$ .",
"For any $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ as above, $\\mathop {{L}}\\nolimits _{x,\\rho }\\otimes \\mathop {{L}}\\nolimits _{x,\\rho }\\xrightarrow{} S),$ is determined by $-2x\\in \\mathop {\\rm CH}\\nolimits ^{2}(S)$ .",
"The reason is that each component of the exceptional divisor is a smooth rational curve of self-intersection number equal to $-2$ .",
"For any $\\rho _{1}\\ne \\rho _{2}\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , If they are adjacent, that is, $\\rho _{1}$ appears (with multiplicity 1) in the $G_{x}$ -module $\\rho _{2}\\otimes T_{x}S$ , then by the classical McKay correspondence, the components in the exceptional divisor over $\\overline{x}$ indexed by $\\rho _{1}$ and by $\\rho _{2}$ intersect transversally at one point.",
"Therefore $\\mathop {{L}}\\nolimits _{x,\\rho _{1}}\\otimes \\mathop {{L}}\\nolimits _{x,\\rho _{2}}\\xrightarrow{} S),$ is determined by $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(S)$ .",
"If they are not adjacent, then again the classical McKay correspondence tells us that the two components indexed by $\\rho _{1}$ and $\\rho _{2}$ of the exceptional divisor do not intersect ; hence $\\mathop {{L}}\\nolimits _{x,\\rho _{1}}\\otimes \\mathop {{L}}\\nolimits _{x,\\rho _{2}}\\xrightarrow{} S)$ is the zero map.",
"The other multiplication maps are zero.",
"The $G$ -action on (REF ) is as follows: The $G$ -action of $S)$ is induced by the original action on $S$ .",
"For any $h\\in G$ , it maps for any $x\\in S$ and $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , the Lefschetz motive $\\mathop {{L}}\\nolimits _{x, \\rho }$ isomorphically to $\\mathop {{L}}\\nolimits _{hx, h\\rho }$ , where $h\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{hx})$ is the representation which makes the following diagram commutes: ${G_{x} [rr]_{\\simeq }^{g\\mapsto hgh^{-1}}[dr]_{\\rho } && G_{hx} [dl]^{h\\rho }\\\\&V_{\\rho }&}$"
],
[
"Orbifold side",
"Now we compute the orbifold Chow motive algebra of the quotient stack $[S/G]$ .",
"The computation is quite straight-forward.",
"Here $\\mathop {{L}}\\nolimits \\mathrel {:=}(-1)$ is the Lefschetz motive.",
"First of all, it is easy to see that $\\mathop {\\rm {age}}\\nolimits (g)=1$ for any element $g\\ne \\mathop {\\rm id}\\nolimits $ of $G$ , and $\\mathop {\\rm {age}}\\nolimits (\\mathop {\\rm id}\\nolimits )=0$ .",
"By Definition REF , $S, G)=S)\\oplus \\bigoplus _{\\begin{array}{c}g\\in G\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\bigoplus _{x\\in S^{g}}\\mathop {{L}}\\nolimits _{x,g}=S)\\oplus \\bigoplus _{x\\in S}\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g},$ where $\\mathop {{L}}\\nolimits _{x,g}$ is the Lefschetz motive $(-1)$ indexed by the fixed point $x$ of $g$ .",
"Lemma 2.1 (Obstruction class) For any $g, h\\in G$ different from $\\mathop {\\rm id}\\nolimits $ , the obstruction class is $c_{g,h}={{\\left\\lbrace \\begin{array}{ll}1 ~~~\\text{ if }~~~ g=h^{-1}\\\\ 0 ~~~\\text{ if }~~~ g\\ne h^{-1}\\end{array}\\right.",
"}}$ For any $g\\ne \\mathop {\\rm id}\\nolimits $ and any $x\\in S^{g}$ , the action of $g$ on $T_{x}S$ is diagonalizable with a pair of conjugate eigenvalues, therefore $V_{g}$ in Definition REF is a trivial vector bundle of rank one on $S^{g}$ .",
"Hence for any $g, h\\in G$ different from $\\mathop {\\rm id}\\nolimits $ and $x\\in S$ fixed by $g$ and $h$ , the dimension of the fiber of the obstruction bundle $F_{g,h}$ at $x$ is $\\dim F_{g,h}(x)=\\dim V_{g}(x)+\\dim V_{h}(x)+\\dim V_{{(gh)}^{-1}}(x)-\\dim T_{x}S,$ which is 1 if $g\\ne h^{-1}$ and is 0 if $g=h^{-1}$ .",
"The computation of $c_{g,h}$ follows.",
"Once the obstruction classes are computed, we can write down explicitly the orbifold product from Definition REF , which is summarized in the following proposition.",
"Proposition 2.2 The orbifold product on $S, G)$ is given as follows via the decomposition (REF ): $S)\\otimes S)&\\xrightarrow{}& S);\\\\S)\\otimes \\mathop {{L}}\\nolimits _{x,g}&\\xrightarrow{}& \\mathop {{L}}\\nolimits _{x,g} ~~\\forall gx=x;\\\\\\mathop {{L}}\\nolimits _{x,g}\\otimes \\mathop {{L}}\\nolimits _{x,g^{-1}}&\\xrightarrow{}& S).\\\\$ where the first morphism is the usual product given by small diagonal; the second and the third morphisms are given by the class $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(S)$ and $i_{x}: \\lbrace x\\rbrace \\hookrightarrow S$ is the natural inclusion; all the other possible maps are zero.",
"The $G$ -action on (REF ) is as follows by Definition REF : The $G$ -action on $S)$ is the original action.",
"For any $h\\in G$ , it maps for any $x\\in S$ and $g\\ne \\mathop {\\rm id}\\nolimits \\in G_{x}$ , the Lefschetz motive $\\mathop {{L}}\\nolimits _{x,g}$ isomorphically to $\\mathop {{L}}\\nolimits _{hx, hgh^{-1}}$ ."
],
[
"The multiplicative correspondence",
"With both sides of the correspondence computed, we can give the multiplicative McKay correspondence morphism, which is in the category $\\mathop {\\rm {CHM}}\\nolimits _{ of complex Chow motives.",
"Consider the morphism}\\begin{equation}\\Phi : S)\\oplus \\bigoplus _{x\\in S}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho } \\rightarrow S)\\oplus \\bigoplus _{x\\in S}\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g},\\end{equation}which is given by the following `block diagonal matrix^{\\prime }:\\begin{itemize}\\item \\mathop {\\rm id}\\nolimits : S)\\rightarrow S);\\item For each x\\in S (with non-trivial stabilizer G_{x}), the morphism \\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }\\rightarrow \\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g} is the `matrix^{\\prime } with coefficient\\frac{1}{\\sqrt{|G_{x}|}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\cdot \\chi _{\\rho }(g) at place (\\rho , g)\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})\\times (G_{x}\\backslash \\lbrace \\mathop {\\rm id}\\nolimits \\rbrace ), where \\chi denotes the character, \\rho _{0} is the natural 2-dimensional representation T_{x}S of G_{x}.",
"Note that \\rho _{0}(g) has determinant 1, hence its trace \\chi _{\\rho _{0}}(g) is a real number.\\item The other morphisms are zero.\\end{itemize}$ To conclude the main theorem, one has to show three things: $(i)$ $\\Phi $ is compatible with the $G$ -action; $(ii)$ $\\Phi $ is multiplicative and $(iii)$ $\\Phi $ induces an isomorphism $\\Phi ^{G}$ of complex Chow motives on $G$ -invariants.",
"Lemma 2.3 $\\Phi $ is $G$ -equivariant.",
"The $G$ -action on the first direct summand $S)$ is by definition the same, hence is preserved by $\\Phi |_{S)}=\\mathop {\\rm id}\\nolimits $ .",
"For the other direct summands, since it is a matrix computation, we can treat the Lefschetz motives as 1-dimensional vector spaces: let $E_{x,\\rho }$ be the `generator' of $\\mathop {{L}}\\nolimits _{x,\\rho }$ and $e_{x, g}$ be the `generator' of $\\mathop {{L}}\\nolimits _{x,g}$ .",
"Then the $G$ -actions computed in the previous subsections say that for any $x$ and any $h\\in G_{x}$ , $h.",
"E_{x,\\rho }=E_{hx, h\\rho }~~~~~~\\text{ and }~~~~~~~~h.e_{x,g}=e_{hx, hgh^{-1}},$ where $h\\rho $ is defined in (REF ).",
"Therefore $&&\\Phi (h.E_{x, \\rho })\\\\&=&\\Phi (E_{hx, h\\rho })\\\\&=&\\frac{1}{\\sqrt{|G_{hx}|}}\\sum _{g\\in G_{hx}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\,\\chi _{h\\rho }(g)\\, e_{hx,g}\\\\&=& \\frac{1}{\\sqrt{|G_{x}|}}\\sum _{g\\in G_{x}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\,\\chi _{h\\rho }(hgh^{-1})\\, e_{hx,hgh^{-1}}\\\\&=& \\frac{1}{\\sqrt{|G_{x}|}}\\sum _{g\\in G_{x}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\,\\chi _{\\rho }(g)\\, e_{hx,hgh^{-1}}\\\\&=&\\frac{1}{\\sqrt{|G_{x}|}}\\sum _{g\\in G_{x}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\,\\chi _{\\rho }(g)\\, h.e_{x,g}\\\\&=&h.\\Phi (E_{x, \\rho }),$ where the third equality is a change of variable: replace $g$ by $hgh^{-1}$ , the fourth equality follows from the definition of $h\\rho $ in (REF ) Proposition 2.4 (Multiplicativity) $\\Phi $ preserves the multiplication, i.e.",
"$\\Phi $ is a morphism of algebra objects in $\\mathop {\\rm {CHM}}\\nolimits _{.",
"}{\\begin{xmlelement*}{proof}The cases of multiplying S) with itself or with a Lefschetz motive \\mathop {{L}}\\nolimits _{x,\\rho } are all obviously preserved by \\Phi .",
"We only need to show that for any x\\in S with non-trivial stabilizer G_{x}, the morphism \\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }\\rightarrow \\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g} given by the matrix with coefficient \\frac{1}{\\sqrt{|G_{x}|}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\cdot \\chi _{\\rho }(g) at place (\\rho , g) is multiplicative (note that the result of the multiplication could go outside of these direct sums to S)).",
"Since this is just a matrix computation, let us treat Lefschetz motives as 1-dimensional vector spaces (or equivalently, we are looking at the corresponding multiplicativity of the realization of \\Phi for Chow rings): let E_{x,\\rho } be the `generator^{\\prime } of \\mathop {{L}}\\nolimits _{x,\\rho } and e_{x, g} be the `generator^{\\prime } of \\mathop {{L}}\\nolimits _{x,g}.",
"Then the computations of the products in the previous two subsections say that:\\begin{eqnarray}E_{x,\\rho _{1}}\\cdot E_{x,\\rho _{2}}&=&{\\left\\lbrace \\begin{array}{ll}-2x ~~~\\text{ if } \\rho _{1}=\\rho _{2};\\\\x ~~~~\\text{ if } \\rho _{1}, \\rho _{2} \\text{ are adjacent};\\\\0 ~~~~\\text{ if } \\rho _{1},\\rho _{2} \\text{ are not adjacent};\\\\\\end{array}\\right.",
"}\\\\e_{x,g}\\cdot e_{x,h}&=&{\\left\\lbrace \\begin{array}{ll}x ~~~\\text{ if } g=h^{-1};\\\\0 ~~~~\\text{ if } g\\ne h^{-1};\\\\\\end{array}\\right.",
"}\\end{eqnarray}\\end{xmlelement*}}$ Therefore for any $\\rho _{1}, \\rho _{2}\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , we have $&&\\Phi (E_{x,\\rho _{1}})\\cdot \\Phi (E_{x,\\rho _{2}})\\\\&=&\\frac{1}{|G_{x}|}\\sum _{g\\in G_{x}}\\sum _{h\\in G_{x}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\sqrt{\\chi _{\\rho _{0}}(h)-2}\\,\\chi _{\\rho _{1}}(g)\\chi _{\\rho _{2}}(h)\\, e_{x,g}\\cdot e_{x,h}\\\\&=&\\frac{1}{|G_{x}|}\\sum _{g\\in G_{x}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\sqrt{\\chi _{\\rho _{0}}(g^{-1})-2}\\,\\chi _{\\rho _{1}}(g)\\chi _{\\rho _{2}}(g^{-1})\\cdot x\\\\&=&\\frac{1}{|G_{x}|}\\sum _{g\\in G_{x}}(\\chi _{\\rho _{0}}(g)-2)\\,\\chi _{\\rho _{1}}(g)\\overline{\\chi _{\\rho _{2}}(g)}\\cdot x\\\\&=&\\frac{1}{|G_{x}|}\\left(\\sum _{g\\in G_{x}}\\chi _{\\rho _{0}\\otimes \\rho _{1}}(g)\\overline{\\chi _{\\rho _{2}}(g)}-2\\sum _{g\\in G_{x}}\\chi _{\\rho _{1}}(g)\\overline{\\chi _{\\rho _{2}}(g)}\\right)\\cdot x\\\\&=& \\left(\\langle \\rho _{0}\\otimes \\rho _{1}, \\rho _{2}\\rangle - 2\\langle \\rho _{1}, \\rho _{2}\\rangle \\right)\\cdot x\\\\&=& \\Phi \\left(E_{x,\\rho _{1}}\\cdot E_{x,\\rho _{2}}\\right)$ where the first equality is the definition of $\\Phi $ (and we add the non-existent $e_{x,1}$ with coefficient 0), the second equality uses () the orthogonality among $e_{x,g}$ 's (i.e.",
"$\\mathop {{L}}\\nolimits _{x,g}$ 's), the third equality uses the fact that $\\chi _{\\rho _{0}}$ takes real value; the last equality uses all three cases of ().",
"Proposition 2.5 (Additive isomorphism) Taking $G$ -invariants on both sides of (), $\\Phi ^{G}$ is an isomorphism of complex Chow motives between $Y)$ and ${orb}([S/G])$ .",
"We should prove the following morphism is an isomorphism: $\\Phi ^{G}: S)^{G}\\oplus \\left(\\bigoplus _{x\\in S}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }\\right)^{G} \\rightarrow S)^{G}\\oplus \\left(\\bigoplus _{x\\in S}\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G}.$ Since $\\Phi $ is given by `block diagonal matrix', it amounts to show that for each $x\\in S$ (with $G_{x}$ non trivial), the following is an isomorphism : $\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x,\\rho }\\rightarrow \\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G_{x}}.$ which is equivalent to say that the following square matrix is non-degenerate : $\\left(\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\cdot \\chi _{\\rho }(g)\\right)_{(\\rho , [g])},$ where $\\rho $ runs over the set $\\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ of isomorphism classes of non-trivial irreducible representations and $[g]$ runs over the set of conjugacy classes of $G_{x}$ different from $\\mathop {\\rm id}\\nolimits $ .",
"As this is about a matrix, it is enough to look at the realization of (REF ): $\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}E_{x,\\rho }\\rightarrow \\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}e_{x,g}\\right)^{G_{x}},$ where both sides come equipped with non-degenerate quadratic forms given by intersection numbers and degrees of the orbifold product respectively.",
"More precisely, by () and (): $(E_{x,\\rho _{1}}\\cdot E_{x,\\rho _{2}})&=&{\\left\\lbrace \\begin{array}{ll}-2 ~~~\\text{ if } \\rho _{1}=\\rho _{2};\\\\1 ~~~~\\text{ if } \\rho _{1}, \\rho _{2} \\text{ are adjacent};\\\\0 ~~~~\\text{ if } \\rho _{1},\\rho _{2} \\text{ are not adjacent};\\\\\\end{array}\\right.",
"}\\\\(e_{x,g}\\cdot e_{x,h})&=&{\\left\\lbrace \\begin{array}{ll}1 ~~~\\text{ if } g=h^{-1};\\\\0 ~~~~\\text{ if } g\\ne h^{-1};\\\\\\end{array}\\right.",
"}$ which are both clearly non-degenerate.",
"Now Proposition REF shows that our matrix (REF ) respects the non-degenerate quadratic forms on both sides, therefore it is non-degenerate.",
"Let us note here also an elementary proof which does not use the orbifold product.",
"We first remark that for any $g\\ne \\mathop {\\rm id}\\nolimits $ , $\\rho _{0}(g)\\in \\mathop {\\rm SL}\\nolimits _{2}($ which is of finite order and different from the identity, hence its trace $\\chi _{{0}}(g)\\ne 2$ .",
"Therefore the nondegeneracy of the matrix (REF ) is equivalent to the nondegeneracy of the matrix $\\left(\\chi _{\\rho }(g)\\right)_{(\\rho , [g])},$ which is obtained from the character table of the finite group $G_{x}$ by removing the first row (corresponding to the trivial representation) and the first column (corresponding to $\\mathop {\\rm id}\\nolimits \\in G_{x}$ ).",
"The nondegeneracy of this matrix is a completely general fact, which holds for all finite groups.",
"We will give a proof in Lemma REF at the end of this section.",
"The combination of Lemma REF , Proposition REF and Proposition REF proves the isomorphism of algebra objects () in the main Theorem REF in the global quotient case.",
"For the isomorphisms for the Chow rings and cohomology rings, it is enough to apply realization functors.",
"For the isomorphisms for the K-theory and topological K-theory, it suffices to invoke the construction of orbifold Chern characters in [25] which induce isomorphisms of algebras from (orbifold) K-theory to (orbifold) Chow ring as well as from (orbifold) topological K-theory to (orbifold) cohomology ring.",
"The proof of Theorem REF in the global quotient case is now complete.",
"$\\square $ The following lemma is used in the second proof of Proposition REF .",
"The elegant proof below is due to Cédric Bonnafé.",
"We thank him for allowing us to use it.",
"Recall that for a finite group $G$ , its character table is a square matrix whose rows are indexed by isomorphism classes of irreducible complex representations of $G$ and columns are indexed by conjugacy classes of $G$ .",
"Lemma 2.6 Let $G$ be any finite group.",
"Then the matrix obtained from the character table by removing the first row corresponding to the trivial representation and the first column corresponding to the identity element, is non-degenerate.",
"Denote by $$ the trivial representation and by $\\rho _{1}, \\cdots , \\rho _{n}$ the set of isomorphism classes of non-trivial representations of $G$ .",
"Suppose we have a linear combination $\\sum _{i=1}^{n}c_{i}\\chi _{\\rho _{i}}$ , with $c_{i}\\in , which vanishes for all non-identity conjugacy class, hence for all non-identity elements of $ G$:\\begin{equation}\\sum _{i=1}^{n}c_{i}\\chi _{\\rho _{i}}(g)=0, ~~~~~~~\\forall g\\ne \\mathop {\\rm id}\\nolimits \\in G.\\end{equation}Set $$c_{0}\\mathrel {:=}-\\frac{1}{|G|}\\sum _{i=1}^{n}c_{i}\\dim (\\rho _{i}),$$ and denote by $ reg$ be the character of the regular representation, then (\\ref {eqn:LinDep}) implies that the following linear combination vanishes for all $ gG$: $$c_{0}\\chi _{reg}+\\sum _{i=1}^{n}c_{i}\\chi _{\\rho _{i}}=0.$$If $ c00$, it contradicts to the fact that the trivial representation should appear (with multiplicity 1) in the regular representation.\\\\Hence we have $ c0=0$.",
"Then by the linear independency among the characters of irreducible representations, we must have $ c1==cn=0$.$ Proof of Theorem REF : general orbifold case In this section, we give the proof of Theorem REF in the full generality.",
"As the proof goes essentially in the same way as the global quotient case in §, we will focus on the different aspects of the proof and refer to the arguments in § whenever possible.",
"Recall the setting: ${\\mathcal {X}}$ is a two-dimensional Deligne–Mumford stack with only finitely many points with non-trivial stabilizers ; $X$ is the underlying (projective) singular surface with only Du Val singularities and $Y\\rightarrow X$ is the minimal resolution.",
"For each $x\\in X$ , denote by $G_{x}$ its stabilizer, which is contained in $\\mathop {\\rm SL}\\nolimits _{2}$ .",
"Throughout this section, Chow groups of stacks are as in [33] and Chow motives of stacks or singular ${\\bf Q}$ -varieties are as in [32].",
"Resolution side Similar to (REF ), we have the following decomposition given by the classical McKay correspondence (see Introduction): $Y)\\simeq X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }$ and the multiplication is the following: $X)\\otimes X)\\xrightarrow{} X)$ is the usual intersection product.",
"For any $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , $X)\\otimes \\mathop {{L}}\\nolimits _{x,\\rho }\\xrightarrow{} \\mathop {{L}}\\nolimits _{x,\\rho } $ is given by the class $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(X)=\\mathop {\\rm Hom}\\nolimits (X)\\otimes \\mathop {{L}}\\nolimits , \\mathop {{L}}\\nolimits )$ .",
"For any $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , $\\mathop {{L}}\\nolimits _{x,\\rho }\\otimes \\mathop {{L}}\\nolimits _{x,\\rho }\\xrightarrow{} X),$ is determined by $-2x\\in \\mathop {\\rm CH}\\nolimits ^{2}(X)$ .",
"For any $\\rho _{1}\\ne \\rho _{2}\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , If they are adjacent, that is, $\\rho _{1}$ appears (with multiplicity 1) in the $G_{x}$ -module $\\rho _{2}\\otimes {2}$ , where 2 is such that ${2}/G_{x}$ is the singularity type of $x$ , then $\\mathop {{L}}\\nolimits _{x,\\rho _{1}}\\otimes \\mathop {{L}}\\nolimits _{x,\\rho _{2}}\\xrightarrow{} X),$ is determined by $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(X)$ .",
"If they are not adjacent, then $\\mathop {{L}}\\nolimits _{x,\\rho _{1}}\\otimes \\mathop {{L}}\\nolimits _{x,\\rho _{2}}\\xrightarrow{} X)$ is the zero map.",
"The other multiplication maps are zero.",
"Orbifold side Similar to (REF ), we have ${\\mathcal {X}})=X)\\oplus \\bigoplus _{x\\in X}\\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G_{x}},$ where the action of $G_{x}$ is by conjugacy.",
"Note that degree 0 twisted stable maps with 3 marked points to $\\mathcal {X}$ are either untwisted stable maps to $\\mathcal {X}$ or a twisted map to one of the stacky points of $\\mathcal {X}$ .",
"In the latter case, the irreducible components of the moduli space around these twisted stable maps and the obstruction bundle are the same as those of the twisted stable maps to the orbifold $[\\mathbb {C}^2/G]$ .",
"It is then clear that the orbifold product can be described as if $\\mathcal {X}$ is a global quotient.",
"Therefore the orbifold product on ${\\mathcal {X}})$ is given by the following, via (REF ): $X)\\otimes X) \\xrightarrow{} X)$ is the usual intersection product.",
"For all $g\\in G_{x}$ , $X)\\otimes \\mathop {{L}}\\nolimits _{x,g} \\xrightarrow{} \\mathop {{L}}\\nolimits _{x,g}$ determined by the class of $x\\in X$ .",
"For all $g\\in G_{x}$ , $\\mathop {{L}}\\nolimits _{x,g}\\otimes \\mathop {{L}}\\nolimits _{x,g^{-1}} \\xrightarrow{} X)$ determined by the class of $x\\in X$ .",
"The other multiplication maps are zero.",
"The multiplicative isomorphism Similar to (), we define $\\phi : X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho } \\rightarrow X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g},$ which is given by the following `block matrix': $\\mathop {\\rm id}\\nolimits : X)\\rightarrow X)$ ; For each $x\\in X$ (with non-trivial stabilizer $G_{x}$ ), the morphism $\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }\\rightarrow \\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}$ is the `matrix' with coefficient $\\frac{1}{\\sqrt{|G_{x}|}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\cdot \\chi _{\\rho }(g)$ at place $(\\rho , g)\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})\\times (G_{x}\\backslash \\lbrace \\mathop {\\rm id}\\nolimits \\rbrace )$ , where $\\chi $ denotes the character, $\\rho _{0}$ is the natural 2-dimensional representation 2 of $G_{x}$ such that ${2}/G_{x}$ is the singularity type of $x$ .",
"Note that $\\rho _{0}(g)$ has determinant 1, hence its trace $\\chi _{\\rho _{0}}(g)$ is a real number.",
"The other morphisms are zero.",
"To conclude Theorem REF , on the one hand, the same proof as in Proposition REF shows that $\\phi $ is multiplicative.",
"On the other hand, one sees immediately that $\\phi $ factorizes through $X)\\oplus \\bigoplus _{x\\in X}\\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G_{x}}.$ It is thus enough to show that the following induced map is an (additive) isomorphism: $\\psi : X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho } \\rightarrow X)\\oplus \\bigoplus _{x\\in X}\\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G_{x}},$ However this follows from the proof of Proposition REF , where one shows that (REF ) is an isomorphism.",
"The proof of Theorem REF is complete.",
"$\\Box $"
],
[
"Proof of Theorem ",
"In this section, we show Theorem REF in the following setting: $S$ is a smooth projective surface, $G$ is a finite group acting faithfully on $S$ such that the canonical bundle is locally preserved (Gorenstein condition), $X\\mathrel {:=}S/G$ is the quotient surface (with Du Val singularities) and $Y\\rightarrow X$ is the minimal (crepant) resolution.",
"Recall that $\\mathop {{L}}\\nolimits \\mathrel {:=}(-1)$ is the Lefschetz motive in $\\mathop {\\rm {CHM}}\\nolimits $ (§).",
"For any $x\\in S$ , let $G_{x}\\mathrel {:=}\\lbrace g\\in G ~\\vert ~ gx=x\\rbrace $ be the stabilizer.",
"Let $\\mathop {\\rm {Irr}}\\nolimits (G_{x})$ be the set of isomorphism classes of irreducible representations of $G_{x}$ and $\\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ be that of non-trivial ones.",
"We remark that by assumption, there are only finitely many points of $S$ with non-trivial stabilizer."
],
[
"Resolution side",
"We first compute the Chow motive algebra (or Chow ring) of the minimal resolution $Y$ .",
"For any $x\\in S$ , we denote by $\\overline{x}$ its image in $S/G$ .",
"The Chow motive of $Y$ has the following decomposition in $\\mathop {\\rm {CHM}}\\nolimits $ : $(Y\\simeq S)^{G}\\oplus \\bigoplus _{\\overline{x}\\in S/G}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{\\overline{x}, \\rho }\\simeq \\left(S)\\oplus \\bigoplus _{x\\in S}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }\\right)^{G},$ where $\\mathop {{L}}\\nolimits _{\\overline{x}, \\rho }$ and $\\mathop {{L}}\\nolimits _{x,\\rho }$ are both the Lefschetz motive $\\mathop {{L}}\\nolimits $ corresponding to the irreducible component of the exceptional divisor over $\\overline{x}$ , indexed by the non-trivial irreducible representation $\\rho $ of $G_{x}$ via the classical McKay correspondence.",
"The second isomorphism in (REF ) being just a trick of reindexing, let us explain a bit more on the first one.",
"Let $f:Y\\rightarrow S/G$ be the minimal resolution of singularities.",
"By the classical McKay correspondence, over each singular point $\\overline{x}\\in S/G$ , the exceptional divisor $E_{\\overline{x}}:=f^{-1}(\\overline{x})$ is a union (with A-D-E configuration) of smooth rational curves $\\cup _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}E_{\\overline{x},\\rho }$ .",
"As $f$ is obviously a semi-small morphism, we can invoke the motivic decomposition of De Cataldo–Migliorini [12], with the stratification being $S/G=(S/G)_{reg}\\cup \\mathop {\\rm Sing}\\nolimits (S/G)$ , to obtain directly the first isomorphism in (REF ).",
"It is then not hard to follow the proof in loc.cit.",
"to see that the first isomorphism in (REF ) is induced by the pull-back $f^{*}={}^{t}\\Gamma _{f}:S)^{G}=S/G)\\rightarrow Y)$ together with the push-forward along the inclusions $\\mathop {{L}}\\nolimits =\\widetilde{ (E_{\\overline{x}, \\rho })\\xrightarrow{} Y), where \\widetilde{ is the reduced motive (see §\\ref {subsect:Motives}).",
"We remark that the inverse of the isomorphism (\\ref {eqn:decompRes}) is more complicated to describe and involves the inverse of the intersection matrix (\\textit {cf.",
"}~the definition of \\Gamma ^{\\prime } in the end of \\cite [§2]{MR2067464}).",
"}}$ The product structure of $Y)$ is determined as follows via the above decomposition (REF ), which also uses the classical McKay correspondence.",
"Let $i_{x}:\\lbrace x\\rbrace \\hookrightarrow S$ be the natural inclusion.",
"$S)\\otimes S)\\xrightarrow{} S)$ is the usual product induced by the small diagonal of $S^{3}$ .",
"For any $x$ with non-trivial stabilizer $G_{x}$ and any $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , $S)\\otimes \\mathop {{L}}\\nolimits _{x,\\rho }\\xrightarrow{} \\mathop {{L}}\\nolimits _{x,\\rho } $ is determined by the class $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(S)=\\mathop {\\rm Hom}\\nolimits (S)\\otimes \\mathop {{L}}\\nolimits , \\mathop {{L}}\\nolimits )$ .",
"For any $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ as above, $\\mathop {{L}}\\nolimits _{x,\\rho }\\otimes \\mathop {{L}}\\nolimits _{x,\\rho }\\xrightarrow{} S),$ is determined by $-2x\\in \\mathop {\\rm CH}\\nolimits ^{2}(S)$ .",
"The reason is that each component of the exceptional divisor is a smooth rational curve of self-intersection number equal to $-2$ .",
"For any $\\rho _{1}\\ne \\rho _{2}\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , If they are adjacent, that is, $\\rho _{1}$ appears (with multiplicity 1) in the $G_{x}$ -module $\\rho _{2}\\otimes T_{x}S$ , then by the classical McKay correspondence, the components in the exceptional divisor over $\\overline{x}$ indexed by $\\rho _{1}$ and by $\\rho _{2}$ intersect transversally at one point.",
"Therefore $\\mathop {{L}}\\nolimits _{x,\\rho _{1}}\\otimes \\mathop {{L}}\\nolimits _{x,\\rho _{2}}\\xrightarrow{} S),$ is determined by $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(S)$ .",
"If they are not adjacent, then again the classical McKay correspondence tells us that the two components indexed by $\\rho _{1}$ and $\\rho _{2}$ of the exceptional divisor do not intersect ; hence $\\mathop {{L}}\\nolimits _{x,\\rho _{1}}\\otimes \\mathop {{L}}\\nolimits _{x,\\rho _{2}}\\xrightarrow{} S)$ is the zero map.",
"The other multiplication maps are zero.",
"The $G$ -action on (REF ) is as follows: The $G$ -action of $S)$ is induced by the original action on $S$ .",
"For any $h\\in G$ , it maps for any $x\\in S$ and $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , the Lefschetz motive $\\mathop {{L}}\\nolimits _{x, \\rho }$ isomorphically to $\\mathop {{L}}\\nolimits _{hx, h\\rho }$ , where $h\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{hx})$ is the representation which makes the following diagram commutes: ${G_{x} [rr]_{\\simeq }^{g\\mapsto hgh^{-1}}[dr]_{\\rho } && G_{hx} [dl]^{h\\rho }\\\\&V_{\\rho }&}$"
],
[
"Orbifold side",
"Now we compute the orbifold Chow motive algebra of the quotient stack $[S/G]$ .",
"The computation is quite straight-forward.",
"Here $\\mathop {{L}}\\nolimits \\mathrel {:=}(-1)$ is the Lefschetz motive.",
"First of all, it is easy to see that $\\mathop {\\rm {age}}\\nolimits (g)=1$ for any element $g\\ne \\mathop {\\rm id}\\nolimits $ of $G$ , and $\\mathop {\\rm {age}}\\nolimits (\\mathop {\\rm id}\\nolimits )=0$ .",
"By Definition REF , $S, G)=S)\\oplus \\bigoplus _{\\begin{array}{c}g\\in G\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\bigoplus _{x\\in S^{g}}\\mathop {{L}}\\nolimits _{x,g}=S)\\oplus \\bigoplus _{x\\in S}\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g},$ where $\\mathop {{L}}\\nolimits _{x,g}$ is the Lefschetz motive $(-1)$ indexed by the fixed point $x$ of $g$ .",
"Lemma 2.1 (Obstruction class) For any $g, h\\in G$ different from $\\mathop {\\rm id}\\nolimits $ , the obstruction class is $c_{g,h}={{\\left\\lbrace \\begin{array}{ll}1 ~~~\\text{ if }~~~ g=h^{-1}\\\\ 0 ~~~\\text{ if }~~~ g\\ne h^{-1}\\end{array}\\right.",
"}}$ For any $g\\ne \\mathop {\\rm id}\\nolimits $ and any $x\\in S^{g}$ , the action of $g$ on $T_{x}S$ is diagonalizable with a pair of conjugate eigenvalues, therefore $V_{g}$ in Definition REF is a trivial vector bundle of rank one on $S^{g}$ .",
"Hence for any $g, h\\in G$ different from $\\mathop {\\rm id}\\nolimits $ and $x\\in S$ fixed by $g$ and $h$ , the dimension of the fiber of the obstruction bundle $F_{g,h}$ at $x$ is $\\dim F_{g,h}(x)=\\dim V_{g}(x)+\\dim V_{h}(x)+\\dim V_{{(gh)}^{-1}}(x)-\\dim T_{x}S,$ which is 1 if $g\\ne h^{-1}$ and is 0 if $g=h^{-1}$ .",
"The computation of $c_{g,h}$ follows.",
"Once the obstruction classes are computed, we can write down explicitly the orbifold product from Definition REF , which is summarized in the following proposition.",
"Proposition 2.2 The orbifold product on $S, G)$ is given as follows via the decomposition (REF ): $S)\\otimes S)&\\xrightarrow{}& S);\\\\S)\\otimes \\mathop {{L}}\\nolimits _{x,g}&\\xrightarrow{}& \\mathop {{L}}\\nolimits _{x,g} ~~\\forall gx=x;\\\\\\mathop {{L}}\\nolimits _{x,g}\\otimes \\mathop {{L}}\\nolimits _{x,g^{-1}}&\\xrightarrow{}& S).\\\\$ where the first morphism is the usual product given by small diagonal; the second and the third morphisms are given by the class $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(S)$ and $i_{x}: \\lbrace x\\rbrace \\hookrightarrow S$ is the natural inclusion; all the other possible maps are zero.",
"The $G$ -action on (REF ) is as follows by Definition REF : The $G$ -action on $S)$ is the original action.",
"For any $h\\in G$ , it maps for any $x\\in S$ and $g\\ne \\mathop {\\rm id}\\nolimits \\in G_{x}$ , the Lefschetz motive $\\mathop {{L}}\\nolimits _{x,g}$ isomorphically to $\\mathop {{L}}\\nolimits _{hx, hgh^{-1}}$ ."
],
[
"The multiplicative correspondence",
"With both sides of the correspondence computed, we can give the multiplicative McKay correspondence morphism, which is in the category $\\mathop {\\rm {CHM}}\\nolimits _{ of complex Chow motives.",
"Consider the morphism}\\begin{equation}\\Phi : S)\\oplus \\bigoplus _{x\\in S}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho } \\rightarrow S)\\oplus \\bigoplus _{x\\in S}\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g},\\end{equation}which is given by the following `block diagonal matrix^{\\prime }:\\begin{itemize}\\item \\mathop {\\rm id}\\nolimits : S)\\rightarrow S);\\item For each x\\in S (with non-trivial stabilizer G_{x}), the morphism \\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }\\rightarrow \\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g} is the `matrix^{\\prime } with coefficient\\frac{1}{\\sqrt{|G_{x}|}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\cdot \\chi _{\\rho }(g) at place (\\rho , g)\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})\\times (G_{x}\\backslash \\lbrace \\mathop {\\rm id}\\nolimits \\rbrace ), where \\chi denotes the character, \\rho _{0} is the natural 2-dimensional representation T_{x}S of G_{x}.",
"Note that \\rho _{0}(g) has determinant 1, hence its trace \\chi _{\\rho _{0}}(g) is a real number.\\item The other morphisms are zero.\\end{itemize}$ To conclude the main theorem, one has to show three things: $(i)$ $\\Phi $ is compatible with the $G$ -action; $(ii)$ $\\Phi $ is multiplicative and $(iii)$ $\\Phi $ induces an isomorphism $\\Phi ^{G}$ of complex Chow motives on $G$ -invariants.",
"Lemma 2.3 $\\Phi $ is $G$ -equivariant.",
"The $G$ -action on the first direct summand $S)$ is by definition the same, hence is preserved by $\\Phi |_{S)}=\\mathop {\\rm id}\\nolimits $ .",
"For the other direct summands, since it is a matrix computation, we can treat the Lefschetz motives as 1-dimensional vector spaces: let $E_{x,\\rho }$ be the `generator' of $\\mathop {{L}}\\nolimits _{x,\\rho }$ and $e_{x, g}$ be the `generator' of $\\mathop {{L}}\\nolimits _{x,g}$ .",
"Then the $G$ -actions computed in the previous subsections say that for any $x$ and any $h\\in G_{x}$ , $h.",
"E_{x,\\rho }=E_{hx, h\\rho }~~~~~~\\text{ and }~~~~~~~~h.e_{x,g}=e_{hx, hgh^{-1}},$ where $h\\rho $ is defined in (REF ).",
"Therefore $&&\\Phi (h.E_{x, \\rho })\\\\&=&\\Phi (E_{hx, h\\rho })\\\\&=&\\frac{1}{\\sqrt{|G_{hx}|}}\\sum _{g\\in G_{hx}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\,\\chi _{h\\rho }(g)\\, e_{hx,g}\\\\&=& \\frac{1}{\\sqrt{|G_{x}|}}\\sum _{g\\in G_{x}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\,\\chi _{h\\rho }(hgh^{-1})\\, e_{hx,hgh^{-1}}\\\\&=& \\frac{1}{\\sqrt{|G_{x}|}}\\sum _{g\\in G_{x}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\,\\chi _{\\rho }(g)\\, e_{hx,hgh^{-1}}\\\\&=&\\frac{1}{\\sqrt{|G_{x}|}}\\sum _{g\\in G_{x}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\,\\chi _{\\rho }(g)\\, h.e_{x,g}\\\\&=&h.\\Phi (E_{x, \\rho }),$ where the third equality is a change of variable: replace $g$ by $hgh^{-1}$ , the fourth equality follows from the definition of $h\\rho $ in (REF ) Proposition 2.4 (Multiplicativity) $\\Phi $ preserves the multiplication, i.e.",
"$\\Phi $ is a morphism of algebra objects in $\\mathop {\\rm {CHM}}\\nolimits _{.",
"}{\\begin{xmlelement*}{proof}The cases of multiplying S) with itself or with a Lefschetz motive \\mathop {{L}}\\nolimits _{x,\\rho } are all obviously preserved by \\Phi .",
"We only need to show that for any x\\in S with non-trivial stabilizer G_{x}, the morphism \\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }\\rightarrow \\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g} given by the matrix with coefficient \\frac{1}{\\sqrt{|G_{x}|}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\cdot \\chi _{\\rho }(g) at place (\\rho , g) is multiplicative (note that the result of the multiplication could go outside of these direct sums to S)).",
"Since this is just a matrix computation, let us treat Lefschetz motives as 1-dimensional vector spaces (or equivalently, we are looking at the corresponding multiplicativity of the realization of \\Phi for Chow rings): let E_{x,\\rho } be the `generator^{\\prime } of \\mathop {{L}}\\nolimits _{x,\\rho } and e_{x, g} be the `generator^{\\prime } of \\mathop {{L}}\\nolimits _{x,g}.",
"Then the computations of the products in the previous two subsections say that:\\begin{eqnarray}E_{x,\\rho _{1}}\\cdot E_{x,\\rho _{2}}&=&{\\left\\lbrace \\begin{array}{ll}-2x ~~~\\text{ if } \\rho _{1}=\\rho _{2};\\\\x ~~~~\\text{ if } \\rho _{1}, \\rho _{2} \\text{ are adjacent};\\\\0 ~~~~\\text{ if } \\rho _{1},\\rho _{2} \\text{ are not adjacent};\\\\\\end{array}\\right.",
"}\\\\e_{x,g}\\cdot e_{x,h}&=&{\\left\\lbrace \\begin{array}{ll}x ~~~\\text{ if } g=h^{-1};\\\\0 ~~~~\\text{ if } g\\ne h^{-1};\\\\\\end{array}\\right.",
"}\\end{eqnarray}\\end{xmlelement*}}$ Therefore for any $\\rho _{1}, \\rho _{2}\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , we have $&&\\Phi (E_{x,\\rho _{1}})\\cdot \\Phi (E_{x,\\rho _{2}})\\\\&=&\\frac{1}{|G_{x}|}\\sum _{g\\in G_{x}}\\sum _{h\\in G_{x}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\sqrt{\\chi _{\\rho _{0}}(h)-2}\\,\\chi _{\\rho _{1}}(g)\\chi _{\\rho _{2}}(h)\\, e_{x,g}\\cdot e_{x,h}\\\\&=&\\frac{1}{|G_{x}|}\\sum _{g\\in G_{x}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\sqrt{\\chi _{\\rho _{0}}(g^{-1})-2}\\,\\chi _{\\rho _{1}}(g)\\chi _{\\rho _{2}}(g^{-1})\\cdot x\\\\&=&\\frac{1}{|G_{x}|}\\sum _{g\\in G_{x}}(\\chi _{\\rho _{0}}(g)-2)\\,\\chi _{\\rho _{1}}(g)\\overline{\\chi _{\\rho _{2}}(g)}\\cdot x\\\\&=&\\frac{1}{|G_{x}|}\\left(\\sum _{g\\in G_{x}}\\chi _{\\rho _{0}\\otimes \\rho _{1}}(g)\\overline{\\chi _{\\rho _{2}}(g)}-2\\sum _{g\\in G_{x}}\\chi _{\\rho _{1}}(g)\\overline{\\chi _{\\rho _{2}}(g)}\\right)\\cdot x\\\\&=& \\left(\\langle \\rho _{0}\\otimes \\rho _{1}, \\rho _{2}\\rangle - 2\\langle \\rho _{1}, \\rho _{2}\\rangle \\right)\\cdot x\\\\&=& \\Phi \\left(E_{x,\\rho _{1}}\\cdot E_{x,\\rho _{2}}\\right)$ where the first equality is the definition of $\\Phi $ (and we add the non-existent $e_{x,1}$ with coefficient 0), the second equality uses () the orthogonality among $e_{x,g}$ 's (i.e.",
"$\\mathop {{L}}\\nolimits _{x,g}$ 's), the third equality uses the fact that $\\chi _{\\rho _{0}}$ takes real value; the last equality uses all three cases of ().",
"Proposition 2.5 (Additive isomorphism) Taking $G$ -invariants on both sides of (), $\\Phi ^{G}$ is an isomorphism of complex Chow motives between $Y)$ and ${orb}([S/G])$ .",
"We should prove the following morphism is an isomorphism: $\\Phi ^{G}: S)^{G}\\oplus \\left(\\bigoplus _{x\\in S}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }\\right)^{G} \\rightarrow S)^{G}\\oplus \\left(\\bigoplus _{x\\in S}\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G}.$ Since $\\Phi $ is given by `block diagonal matrix', it amounts to show that for each $x\\in S$ (with $G_{x}$ non trivial), the following is an isomorphism : $\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x,\\rho }\\rightarrow \\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G_{x}}.$ which is equivalent to say that the following square matrix is non-degenerate : $\\left(\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\cdot \\chi _{\\rho }(g)\\right)_{(\\rho , [g])},$ where $\\rho $ runs over the set $\\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ of isomorphism classes of non-trivial irreducible representations and $[g]$ runs over the set of conjugacy classes of $G_{x}$ different from $\\mathop {\\rm id}\\nolimits $ .",
"As this is about a matrix, it is enough to look at the realization of (REF ): $\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}E_{x,\\rho }\\rightarrow \\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}e_{x,g}\\right)^{G_{x}},$ where both sides come equipped with non-degenerate quadratic forms given by intersection numbers and degrees of the orbifold product respectively.",
"More precisely, by () and (): $(E_{x,\\rho _{1}}\\cdot E_{x,\\rho _{2}})&=&{\\left\\lbrace \\begin{array}{ll}-2 ~~~\\text{ if } \\rho _{1}=\\rho _{2};\\\\1 ~~~~\\text{ if } \\rho _{1}, \\rho _{2} \\text{ are adjacent};\\\\0 ~~~~\\text{ if } \\rho _{1},\\rho _{2} \\text{ are not adjacent};\\\\\\end{array}\\right.",
"}\\\\(e_{x,g}\\cdot e_{x,h})&=&{\\left\\lbrace \\begin{array}{ll}1 ~~~\\text{ if } g=h^{-1};\\\\0 ~~~~\\text{ if } g\\ne h^{-1};\\\\\\end{array}\\right.",
"}$ which are both clearly non-degenerate.",
"Now Proposition REF shows that our matrix (REF ) respects the non-degenerate quadratic forms on both sides, therefore it is non-degenerate.",
"Let us note here also an elementary proof which does not use the orbifold product.",
"We first remark that for any $g\\ne \\mathop {\\rm id}\\nolimits $ , $\\rho _{0}(g)\\in \\mathop {\\rm SL}\\nolimits _{2}($ which is of finite order and different from the identity, hence its trace $\\chi _{{0}}(g)\\ne 2$ .",
"Therefore the nondegeneracy of the matrix (REF ) is equivalent to the nondegeneracy of the matrix $\\left(\\chi _{\\rho }(g)\\right)_{(\\rho , [g])},$ which is obtained from the character table of the finite group $G_{x}$ by removing the first row (corresponding to the trivial representation) and the first column (corresponding to $\\mathop {\\rm id}\\nolimits \\in G_{x}$ ).",
"The nondegeneracy of this matrix is a completely general fact, which holds for all finite groups.",
"We will give a proof in Lemma REF at the end of this section.",
"The combination of Lemma REF , Proposition REF and Proposition REF proves the isomorphism of algebra objects () in the main Theorem REF in the global quotient case.",
"For the isomorphisms for the Chow rings and cohomology rings, it is enough to apply realization functors.",
"For the isomorphisms for the K-theory and topological K-theory, it suffices to invoke the construction of orbifold Chern characters in [25] which induce isomorphisms of algebras from (orbifold) K-theory to (orbifold) Chow ring as well as from (orbifold) topological K-theory to (orbifold) cohomology ring.",
"The proof of Theorem REF in the global quotient case is now complete.",
"$\\square $ The following lemma is used in the second proof of Proposition REF .",
"The elegant proof below is due to Cédric Bonnafé.",
"We thank him for allowing us to use it.",
"Recall that for a finite group $G$ , its character table is a square matrix whose rows are indexed by isomorphism classes of irreducible complex representations of $G$ and columns are indexed by conjugacy classes of $G$ .",
"Lemma 2.6 Let $G$ be any finite group.",
"Then the matrix obtained from the character table by removing the first row corresponding to the trivial representation and the first column corresponding to the identity element, is non-degenerate.",
"Denote by $$ the trivial representation and by $\\rho _{1}, \\cdots , \\rho _{n}$ the set of isomorphism classes of non-trivial representations of $G$ .",
"Suppose we have a linear combination $\\sum _{i=1}^{n}c_{i}\\chi _{\\rho _{i}}$ , with $c_{i}\\in , which vanishes for all non-identity conjugacy class, hence for all non-identity elements of $ G$:\\begin{equation}\\sum _{i=1}^{n}c_{i}\\chi _{\\rho _{i}}(g)=0, ~~~~~~~\\forall g\\ne \\mathop {\\rm id}\\nolimits \\in G.\\end{equation}Set $$c_{0}\\mathrel {:=}-\\frac{1}{|G|}\\sum _{i=1}^{n}c_{i}\\dim (\\rho _{i}),$$ and denote by $ reg$ be the character of the regular representation, then (\\ref {eqn:LinDep}) implies that the following linear combination vanishes for all $ gG$: $$c_{0}\\chi _{reg}+\\sum _{i=1}^{n}c_{i}\\chi _{\\rho _{i}}=0.$$If $ c00$, it contradicts to the fact that the trivial representation should appear (with multiplicity 1) in the regular representation.\\\\Hence we have $ c0=0$.",
"Then by the linear independency among the characters of irreducible representations, we must have $ c1==cn=0$.$ Proof of Theorem REF : general orbifold case In this section, we give the proof of Theorem REF in the full generality.",
"As the proof goes essentially in the same way as the global quotient case in §, we will focus on the different aspects of the proof and refer to the arguments in § whenever possible.",
"Recall the setting: ${\\mathcal {X}}$ is a two-dimensional Deligne–Mumford stack with only finitely many points with non-trivial stabilizers ; $X$ is the underlying (projective) singular surface with only Du Val singularities and $Y\\rightarrow X$ is the minimal resolution.",
"For each $x\\in X$ , denote by $G_{x}$ its stabilizer, which is contained in $\\mathop {\\rm SL}\\nolimits _{2}$ .",
"Throughout this section, Chow groups of stacks are as in [33] and Chow motives of stacks or singular ${\\bf Q}$ -varieties are as in [32].",
"Resolution side Similar to (REF ), we have the following decomposition given by the classical McKay correspondence (see Introduction): $Y)\\simeq X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }$ and the multiplication is the following: $X)\\otimes X)\\xrightarrow{} X)$ is the usual intersection product.",
"For any $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , $X)\\otimes \\mathop {{L}}\\nolimits _{x,\\rho }\\xrightarrow{} \\mathop {{L}}\\nolimits _{x,\\rho } $ is given by the class $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(X)=\\mathop {\\rm Hom}\\nolimits (X)\\otimes \\mathop {{L}}\\nolimits , \\mathop {{L}}\\nolimits )$ .",
"For any $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , $\\mathop {{L}}\\nolimits _{x,\\rho }\\otimes \\mathop {{L}}\\nolimits _{x,\\rho }\\xrightarrow{} X),$ is determined by $-2x\\in \\mathop {\\rm CH}\\nolimits ^{2}(X)$ .",
"For any $\\rho _{1}\\ne \\rho _{2}\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , If they are adjacent, that is, $\\rho _{1}$ appears (with multiplicity 1) in the $G_{x}$ -module $\\rho _{2}\\otimes {2}$ , where 2 is such that ${2}/G_{x}$ is the singularity type of $x$ , then $\\mathop {{L}}\\nolimits _{x,\\rho _{1}}\\otimes \\mathop {{L}}\\nolimits _{x,\\rho _{2}}\\xrightarrow{} X),$ is determined by $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(X)$ .",
"If they are not adjacent, then $\\mathop {{L}}\\nolimits _{x,\\rho _{1}}\\otimes \\mathop {{L}}\\nolimits _{x,\\rho _{2}}\\xrightarrow{} X)$ is the zero map.",
"The other multiplication maps are zero.",
"Orbifold side Similar to (REF ), we have ${\\mathcal {X}})=X)\\oplus \\bigoplus _{x\\in X}\\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G_{x}},$ where the action of $G_{x}$ is by conjugacy.",
"Note that degree 0 twisted stable maps with 3 marked points to $\\mathcal {X}$ are either untwisted stable maps to $\\mathcal {X}$ or a twisted map to one of the stacky points of $\\mathcal {X}$ .",
"In the latter case, the irreducible components of the moduli space around these twisted stable maps and the obstruction bundle are the same as those of the twisted stable maps to the orbifold $[\\mathbb {C}^2/G]$ .",
"It is then clear that the orbifold product can be described as if $\\mathcal {X}$ is a global quotient.",
"Therefore the orbifold product on ${\\mathcal {X}})$ is given by the following, via (REF ): $X)\\otimes X) \\xrightarrow{} X)$ is the usual intersection product.",
"For all $g\\in G_{x}$ , $X)\\otimes \\mathop {{L}}\\nolimits _{x,g} \\xrightarrow{} \\mathop {{L}}\\nolimits _{x,g}$ determined by the class of $x\\in X$ .",
"For all $g\\in G_{x}$ , $\\mathop {{L}}\\nolimits _{x,g}\\otimes \\mathop {{L}}\\nolimits _{x,g^{-1}} \\xrightarrow{} X)$ determined by the class of $x\\in X$ .",
"The other multiplication maps are zero.",
"The multiplicative isomorphism Similar to (), we define $\\phi : X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho } \\rightarrow X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g},$ which is given by the following `block matrix': $\\mathop {\\rm id}\\nolimits : X)\\rightarrow X)$ ; For each $x\\in X$ (with non-trivial stabilizer $G_{x}$ ), the morphism $\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }\\rightarrow \\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}$ is the `matrix' with coefficient $\\frac{1}{\\sqrt{|G_{x}|}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\cdot \\chi _{\\rho }(g)$ at place $(\\rho , g)\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})\\times (G_{x}\\backslash \\lbrace \\mathop {\\rm id}\\nolimits \\rbrace )$ , where $\\chi $ denotes the character, $\\rho _{0}$ is the natural 2-dimensional representation 2 of $G_{x}$ such that ${2}/G_{x}$ is the singularity type of $x$ .",
"Note that $\\rho _{0}(g)$ has determinant 1, hence its trace $\\chi _{\\rho _{0}}(g)$ is a real number.",
"The other morphisms are zero.",
"To conclude Theorem REF , on the one hand, the same proof as in Proposition REF shows that $\\phi $ is multiplicative.",
"On the other hand, one sees immediately that $\\phi $ factorizes through $X)\\oplus \\bigoplus _{x\\in X}\\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G_{x}}.$ It is thus enough to show that the following induced map is an (additive) isomorphism: $\\psi : X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho } \\rightarrow X)\\oplus \\bigoplus _{x\\in X}\\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G_{x}},$ However this follows from the proof of Proposition REF , where one shows that (REF ) is an isomorphism.",
"The proof of Theorem REF is complete.",
"$\\Box $"
],
[
"Proof of Theorem ",
"In this section, we give the proof of Theorem REF in the full generality.",
"As the proof goes essentially in the same way as the global quotient case in §, we will focus on the different aspects of the proof and refer to the arguments in § whenever possible.",
"Recall the setting: ${\\mathcal {X}}$ is a two-dimensional Deligne–Mumford stack with only finitely many points with non-trivial stabilizers ; $X$ is the underlying (projective) singular surface with only Du Val singularities and $Y\\rightarrow X$ is the minimal resolution.",
"For each $x\\in X$ , denote by $G_{x}$ its stabilizer, which is contained in $\\mathop {\\rm SL}\\nolimits _{2}$ .",
"Throughout this section, Chow groups of stacks are as in [33] and Chow motives of stacks or singular ${\\bf Q}$ -varieties are as in [32]."
],
[
"Resolution side",
"Similar to (REF ), we have the following decomposition given by the classical McKay correspondence (see Introduction): $Y)\\simeq X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }$ and the multiplication is the following: $X)\\otimes X)\\xrightarrow{} X)$ is the usual intersection product.",
"For any $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , $X)\\otimes \\mathop {{L}}\\nolimits _{x,\\rho }\\xrightarrow{} \\mathop {{L}}\\nolimits _{x,\\rho } $ is given by the class $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(X)=\\mathop {\\rm Hom}\\nolimits (X)\\otimes \\mathop {{L}}\\nolimits , \\mathop {{L}}\\nolimits )$ .",
"For any $\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , $\\mathop {{L}}\\nolimits _{x,\\rho }\\otimes \\mathop {{L}}\\nolimits _{x,\\rho }\\xrightarrow{} X),$ is determined by $-2x\\in \\mathop {\\rm CH}\\nolimits ^{2}(X)$ .",
"For any $\\rho _{1}\\ne \\rho _{2}\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})$ , If they are adjacent, that is, $\\rho _{1}$ appears (with multiplicity 1) in the $G_{x}$ -module $\\rho _{2}\\otimes {2}$ , where 2 is such that ${2}/G_{x}$ is the singularity type of $x$ , then $\\mathop {{L}}\\nolimits _{x,\\rho _{1}}\\otimes \\mathop {{L}}\\nolimits _{x,\\rho _{2}}\\xrightarrow{} X),$ is determined by $x\\in \\mathop {\\rm CH}\\nolimits ^{2}(X)$ .",
"If they are not adjacent, then $\\mathop {{L}}\\nolimits _{x,\\rho _{1}}\\otimes \\mathop {{L}}\\nolimits _{x,\\rho _{2}}\\xrightarrow{} X)$ is the zero map.",
"The other multiplication maps are zero."
],
[
"Orbifold side",
"Similar to (REF ), we have ${\\mathcal {X}})=X)\\oplus \\bigoplus _{x\\in X}\\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G_{x}},$ where the action of $G_{x}$ is by conjugacy.",
"Note that degree 0 twisted stable maps with 3 marked points to $\\mathcal {X}$ are either untwisted stable maps to $\\mathcal {X}$ or a twisted map to one of the stacky points of $\\mathcal {X}$ .",
"In the latter case, the irreducible components of the moduli space around these twisted stable maps and the obstruction bundle are the same as those of the twisted stable maps to the orbifold $[\\mathbb {C}^2/G]$ .",
"It is then clear that the orbifold product can be described as if $\\mathcal {X}$ is a global quotient.",
"Therefore the orbifold product on ${\\mathcal {X}})$ is given by the following, via (REF ): $X)\\otimes X) \\xrightarrow{} X)$ is the usual intersection product.",
"For all $g\\in G_{x}$ , $X)\\otimes \\mathop {{L}}\\nolimits _{x,g} \\xrightarrow{} \\mathop {{L}}\\nolimits _{x,g}$ determined by the class of $x\\in X$ .",
"For all $g\\in G_{x}$ , $\\mathop {{L}}\\nolimits _{x,g}\\otimes \\mathop {{L}}\\nolimits _{x,g^{-1}} \\xrightarrow{} X)$ determined by the class of $x\\in X$ .",
"The other multiplication maps are zero."
],
[
"The multiplicative isomorphism",
"Similar to (), we define $\\phi : X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho } \\rightarrow X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g},$ which is given by the following `block matrix': $\\mathop {\\rm id}\\nolimits : X)\\rightarrow X)$ ; For each $x\\in X$ (with non-trivial stabilizer $G_{x}$ ), the morphism $\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho }\\rightarrow \\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}$ is the `matrix' with coefficient $\\frac{1}{\\sqrt{|G_{x}|}}\\sqrt{\\chi _{\\rho _{0}}(g)-2}\\cdot \\chi _{\\rho }(g)$ at place $(\\rho , g)\\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})\\times (G_{x}\\backslash \\lbrace \\mathop {\\rm id}\\nolimits \\rbrace )$ , where $\\chi $ denotes the character, $\\rho _{0}$ is the natural 2-dimensional representation 2 of $G_{x}$ such that ${2}/G_{x}$ is the singularity type of $x$ .",
"Note that $\\rho _{0}(g)$ has determinant 1, hence its trace $\\chi _{\\rho _{0}}(g)$ is a real number.",
"The other morphisms are zero.",
"To conclude Theorem REF , on the one hand, the same proof as in Proposition REF shows that $\\phi $ is multiplicative.",
"On the other hand, one sees immediately that $\\phi $ factorizes through $X)\\oplus \\bigoplus _{x\\in X}\\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G_{x}}.$ It is thus enough to show that the following induced map is an (additive) isomorphism: $\\psi : X)\\oplus \\bigoplus _{x\\in X}\\bigoplus _{\\rho \\in \\mathop {\\rm {Irr}}\\nolimits ^{\\prime }(G_{x})}\\mathop {{L}}\\nolimits _{x, \\rho } \\rightarrow X)\\oplus \\bigoplus _{x\\in X}\\left(\\bigoplus _{\\begin{array}{c}g\\in G_{x}\\\\g\\ne \\mathop {\\rm id}\\nolimits \\end{array}}\\mathop {{L}}\\nolimits _{x,g}\\right)^{G_{x}},$ However this follows from the proof of Proposition REF , where one shows that (REF ) is an isomorphism.",
"The proof of Theorem REF is complete.",
"$\\Box $"
]
] | 1709.01714 |
[
[
"Theory of a Quantum Scanning Microscope for Cold Atoms"
],
[
"Abstract We propose and analyze a scanning microscope to monitor `live' the quantum dynamics of cold atoms in a Cavity QED setup.",
"The microscope measures the atomic density with subwavelength resolution via dispersive couplings to a cavity and homodyne detection within the framework of continuous measurement theory.",
"We analyze two modes of operation.",
"First, for a fixed focal point the microscope records the wave packet dynamics of atoms with time resolution set by the cavity lifetime.",
"Second, a spatial scan of the microscope acts to map out the spatial density of stationary quantum states.",
"Remarkably, in the latter case, for a good cavity limit, the microscope becomes an effective quantum non-demolition (QND) device, such that the spatial distribution of motional eigenstates can be measured back-action free in single scans, as an emergent QND measurement."
],
[
"Quantum non-Demolition vs. ",
"In this section we summarize the concept of emergent quantum non-demolition (QND) measurements in a more formal way.",
"A familiar definition of a QND measurement [3], [45] considers an observable ${\\hat{A}}$ to be QND, if it commutes with the system Hamiltonian $[\\hat{H}, {\\hat{A}}]=0$ .",
"Such a QND observable can be continuously measured with an arbitrary high Signal-to-Noise Ratio (SNR) [3], [45].",
"In general, for a system observable $\\mathcal {\\hat{O}}$ , which does not commute with the Hamiltonian, $[\\hat{H}, \\mathcal {\\hat{O}}]\\ne 0$ , we define as emergent QND observable OeQND n |nn|O|nn|, with $\\left|n\\right\\rangle $ the energy eigenstates.",
"Measurement of $\\hat{\\mathcal {O}}_{\\rm eQND}$ provides the same information as of $\\hat{\\mathcal {O}}$ for energy eigenstates, but in a non-destructive way.",
"This enables studying properties of energy eigenstates of various quantum systems with very high precision and from different perspectives provided by the corresponding observables $\\mathcal {\\hat{O}}$ .",
"The emergent QND measurement in context of the quantum scanning microscope, as discussed in the main text, considers the eQND observable defined from the $\\delta $ -like probe $f(\\hat{z})$ , where $\\hat{z}$ is the position operator.",
"This allows in particular to map out atomic densities of energy eigenstates for harmonic oscillator and Freidel oscillations for many-body systems in a single scan with high SNR, as illustrated in Figs.",
"1 and 4 of the main text."
],
[
"Engineering of The Sub-wavelength Focusing Function $\\phi _{z_0}(z)$",
"Here we discuss in detail the realization of the focusing function $\\phi _{z_0}(z)$ [c.f.",
"Eq.",
"(1) of the main text], showing that subwavelength resolution can be achieved along with negligible additional forces on the atom."
],
[
"Sub-wavelength spin structure with negligible non-adiabatic potential",
"The atomic internal levels for implementing the focusing function, shown in Fig.",
"2a of the main text, consists of a $\\Lambda $ -system formed by $|g\\rangle ,|r\\rangle ,|e\\rangle $ , described by the Hamiltonian $\\begin{split}\\hat{H}_{ \\mathrm {a} } \\!= &\\!-\\!\\!\\hbar \\left(\\Delta _e\\!+\\!",
"i\\frac{\\Gamma _e}{2}\\right)\\!\\hat{\\sigma }_{ee}+ \\frac{\\hbar }{2} \\left[ \\Omega _0(z)\\hat{\\sigma }_{eg} \\!+\\!",
"\\Omega _1(z)\\hat{\\sigma }_{er} \\!+\\!",
"\\mathrm {H.c.} \\right],\\end{split}$ where $\\Gamma _e$ is the decay rate of the excited state, and we assume Raman resonance $\\Delta _r=0$ .",
"Diagonalizing $\\hat{H}_{ \\mathrm {a} }$ gives the eigenstates $\\begin{split}\\left|D(z)\\right\\rangle &= \\sin \\theta (z)\\left|g\\right\\rangle -\\cos \\theta (z) \\left|r\\right\\rangle ,\\\\\\left|+(z)\\right\\rangle &=\\cos \\chi (z) \\left|e\\right\\rangle + \\sin \\chi (z) [\\cos \\theta (z) \\left|g\\right\\rangle +\\sin \\theta (z)\\left|r\\right\\rangle ],\\\\\\left|-(z)\\right\\rangle &=\\sin \\chi (z) \\left|e\\right\\rangle - \\cos \\chi (z)[\\cos \\theta (z)\\left|g\\right\\rangle +\\sin \\theta (z)\\left|r\\right\\rangle ],\\end{split}$ with the corresponding eigenenergies $E_{D}=0$ and $E_{\\pm } (z)=-({\\hbar }/2)\\lbrace \\tilde{\\Delta }_{e}\\mp [\\Omega _0^2(z)+\\Omega _1^2(z)+\\tilde{\\Delta }_e^2]^{1/2}\\rbrace $ , where $\\tilde{\\Delta }_e=\\Delta _e+i\\Gamma _e/2$ , and the mixing angles defined by $\\theta (z)=\\arctan [\\Omega _1(z)/\\Omega _0(z)]$ and $\\chi (z)=-({1}/{2})\\arctan [\\sqrt{\\Omega _0^2(z)+\\Omega _1^2(z)}/\\tilde{\\Delta }_e]$ .",
"We note that the dark state $\\left|D(z)\\right\\rangle $ is decoupled from the dissipative excited state $\\left|e\\right\\rangle $ , and its spin structure is varying in space controlled by the Rabi frequency configuration.",
"We are interested in the regime where $\\textrm {Re}[E_{\\pm }(z)]$ is much larger than the other energy scales in the model.",
"In the spirit of the Born-Oppenheimer (BO) approximation, we study the slow dynamics by assuming the atomic internal state remains in $\\left|D(z)\\right\\rangle $ adiabatically.",
"This allows us to design the desired sub-wavelength spin structure $|\\langle r| D(z)\\rangle |^2=\\cos ^2\\theta $ .",
"Such a spatially varying internal spin is, however, necessarily accompanied by non-adiabatic corrections to the atomic external motion [42], [43] $V_{ na} (z)= \\langle D(z)|\\frac{\\hat{p}_z^2}{2m}\\left|D(z)\\right\\rangle =\\frac{\\hbar ^2}{2m}[\\partial _z\\theta (z)]^2.$ We now show that $|\\langle r|D(z)\\rangle |^2$ can be made nano-scale with negligible $V_{na}(z)$ .",
"We consider the Rabi frequencies $\\begin{split}\\Omega _{0}(z)&=\\epsilon \\Omega _c,\\\\\\Omega _1(z)& = \\Omega _c [ 1 + \\beta - \\cos k_1 (z-z_0) ],\\end{split}$ where $\\Omega _c$ is a large reference frequency (assumed real positive) and $0<\\epsilon \\sim \\beta \\ll 1$ .",
"Physically, $\\Omega _1(z)$ can be realized, e.g., by super-imposing three phase-coherent laser beams where the first two lasers form the standing wave $\\Omega _c\\cos k_1(z-z_0)$ , and the third propagates perpendicularly, to provide the offset $\\Omega _c(1+\\beta )$ .",
"For Rabi frequencies in Eq.",
"(REF ), the resolution $\\sigma $ , quantified by the full width at half maximum (FWHM) of $|\\langle r|D(z)\\rangle |^2$ , is given in the limit $\\epsilon \\ll 1$ by $\\sigma =\\frac{\\sqrt{2}\\lambda _1}{\\pi }\\Big (\\sqrt{\\epsilon ^2+2\\beta ^2}-\\beta \\Big )^{1/2},$ with $\\lambda _1=2\\pi /k_1$ .",
"The non-adiabatic potential is $V_{na}(z)=\\frac{\\hbar ^2k^2_1}{2m \\epsilon ^2}\\left(\\frac{\\sin k_1(z-z_0)}{1+[1+\\beta -\\cos k_1(z-z_0)]^2/\\epsilon ^2}\\right)^2.$ Importantly, $V_{na}(z)$ decreases rapidly by increasing the ratio $\\beta /\\epsilon $ , as shown in Fig.",
"REF .",
"Physically, increasing $\\beta /\\epsilon $ reduces the maximal population transfer onto the state $\\left|r\\right\\rangle $ during the adiabatic motion, $|\\langle r|D(z)\\rangle |^2_{\\rm max} = (1+\\beta ^2/\\epsilon ^2)^{-1}$ , thus suppressing the corresponding non-adiabatic potential.",
"This sub-wavelength spin structure, with negligible $V_{na}(z)$ , is exploited to realize the focusing function $\\phi _{z_0}(z)$ , as we show below."
],
[
"Sub-wavelength focusing function $\\phi _{z_0}(z)$",
"Being a part of the $\\Lambda $ -system, the level $\\left|r\\right\\rangle $ is also coupled to another level $\\left|t\\right\\rangle $ through a cavity mode $\\hat{c}$ , resulting in an effective dispersive coupling $\\hat{H}_{{\\rm ac}}=\\hbar g(\\hat{z})^{2}\\hat{c}^{\\dag }\\hat{c}/\\Delta _{t}$ between $\\left|r\\right\\rangle $ and the cavity mode (the effects of the spontaneous decay of the state $\\left|t\\right\\rangle $ will be discussed below).",
"After projecting onto the dark state $\\left|D(z)\\right\\rangle $ in the BO approach, one obtains the desired sub-wavelength atom-cavity coupling $\\hat{H}_{{\\rm coup}}=\\frac{\\hbar g^{2}(z)}{\\Delta _{t}}|\\langle r|D(z)\\rangle |^{2}\\hat{c}^{\\dag }\\hat{c}\\equiv \\phi _{z_{0}}(z)\\hat{c}^{\\dag }\\hat{c},$ with the spatial resolution given by Eq.",
"(REF ) [notice that $g(z)$ varies slowly on the scale $\\sigma $ ].",
"As mentioned in the main text, it is convenient to write the focusing function in the form $\\phi _{z_{0}}(z)\\equiv {\\cal A}f_{z_{0}}(z)$ , where $\\mathcal {A}$ has the dimension of energy and $f_{z_{0}}(z)$ is dimensionless and normalized.",
"We choose the normalization $\\int dzf_{z_{0}}(z)=\\ell _{0}$ with $\\ell _{0}$ being the characteristic length scale of the system under measurement, such that the matrix elements of $f_{z_{0}}(\\hat{z})$ are of order 1.",
"Note that through this coupling, the stationary coherent field inside the driven cavity exerts a force on the atom, $V_{{\\rm OL}}(z)=\\hbar g^{2}(z)|\\alpha |^{2}|\\langle r|D(z)\\rangle |^{2}/\\Delta _{t}$ , where $\\alpha =\\sqrt{\\kappa }\\mathcal {E}(i\\delta -\\kappa /2)^{-1}$ is the amplitude of the stationary field, $\\delta $ and $\\mathcal {E}$ are the detuning and the strength of the cavity driving laser, respectively.",
"This force can be compensated by simply detuning the Raman resonance with the offset $\\Delta _{r}=g^{2}(z_{0})|\\alpha |^{2}/\\Delta _{t}$ (c.f.",
"Fig.",
"2a in the main text), which results in nearly perfect compensation of $V_{{\\rm OL}}(z)$ for the dark state $\\left|D(z)\\right\\rangle $ .",
"We also note that the focusing function $\\phi _{z_{0}}(z)$ in by Eq.",
"(REF ) has a periodic set of peaks separated by $\\lambda _{1}=2\\pi /k_{1}$ .",
"To design a single-peak $\\phi _{z_{0}}(z)$ one can simply choose a spatially dependent $\\Omega _{0}(z)$ which is tightly ($\\sim \\lambda _{1}$ ) focused at position $z_{0}$ .",
"Figure: The resolution σ\\sigma and the maximum of the non-adiabatic potentialV na (z)V_{na}(z) [c.f., Eq.",
"(), in unit of the recoil energyE r =ℏ 2 k 1 2 /2mE_{r}=\\hbar ^{2}k_{1}^{2}/2m], vs. β/ϵ\\beta /\\epsilon , for thelaser configuration Eq. ().",
"Also shown isthe maximum overlap between D(z)\\left|D(z)\\right\\rangle and r\\left|r\\right\\rangle .",
"Parameters:ϵ=0.1\\epsilon =0.1.",
"We note that V na (z)V_{na}(z) is strongly suppressedfor β/ϵ≫1\\beta /\\epsilon \\gg 1."
],
[
"Spontaneous emission",
"Here we discuss the spontaneous emission of the dark state originated from the spontaneous decay of the states $\\left|e\\right\\rangle $ and $\\left|t\\right\\rangle $ entering the construction of the focusing function (see Fig.",
"2a of the main text).",
"A more formal derivation of the same results using the stochastic master equation including the atomic internal states and the associated spontaneous decay terms will be presented in a follow-up paper [50].",
"The spontaneous decay rate of the state $\\left|e\\right\\rangle $ enters through the residual coupling between the dark state $\\left|D(z)\\right\\rangle $ and the bright states $\\left|\\pm (z)\\right\\rangle $ in the $\\Lambda $ -configuration, due to the atomic kinetic term.",
"As shown in [42], the corresponding decay rate of $\\left|D(z)\\right\\rangle $ scales with the Rabi frequencies as $\\propto [\\Omega _{1}^{2}(z)+\\Omega _{0}^{2}(z)]^{-1}$ , and can be strongly suppressed by choosing large Rabi frequencies.",
"The spontaneous decay of $\\left|D(z)\\right\\rangle $ due to virtual population of the state $\\left|t\\right\\rangle $ (resulting from $\\hat{H}_{{\\rm coup}}$ ) is the dominant decay channel, and the corresponding decay rate can be calculated as $\\gamma _{D}(z)=\\frac{g^{2}(z)}{\\Delta _{t}^{2}}\\left(\\frac{4\\mathcal {E}^{2}}{\\kappa }\\right)\\Gamma _{t}|\\langle r|D(z)\\rangle |^{2}=\\gamma _{{\\rm sp}}f_{z_{0}}(z),$ where $4\\mathcal {E}^{2}/\\kappa $ is the mean photon number in the driven cavity, $\\Gamma _{t}$ is the spontaneous emission rate of the state $\\left|t\\right\\rangle $ , and we introduce the average spontaneous decay rate $\\gamma _{{\\rm sp}}=\\frac{1}{\\ell _{0}}\\int dz\\gamma _{D}(z)=4\\mathcal {A}\\frac{\\mathcal {E}^{2}\\Gamma _{t}}{\\hbar \\kappa \\Delta _{t}}.$ This has to be compared with the measurement strength $\\gamma =[4\\mathcal {{\\cal A}E}/(\\hbar \\kappa )]^{2}$ , with the result $\\begin{split}\\frac{\\gamma }{\\gamma _{{\\rm sp}}}=\\frac{4\\mathcal {{\\cal A}}\\Delta _{t}}{\\hbar \\kappa \\Gamma _{t}}&=\\frac{4}{\\kappa \\Gamma _{t}\\ell _{0}}\\int dzg^{2}(z)|\\langle r|D(z)\\rangle |^{2}\\\\&\\simeq 4\\mathcal {C}\\frac{\\sigma }{\\ell _{0}}|\\langle r|D(z)\\rangle |_{{\\rm max}}^{2},\\end{split}$ where $\\mathcal {C}\\equiv g^{2}(z_{0})/(\\kappa \\Gamma _{t})$ is the cavity cooperativity, and we use the approximation $\\int dzg^{2}(z)|\\langle r|D(z)\\rangle |^{2}\\simeq \\sigma g^{2}(z_{0})|\\langle r|D(z)\\rangle |_{{\\rm max}}^{2}$ .",
"An implementation of the proposed microscope would require $\\gamma \\gg \\gamma _{{\\rm sp}}$ , so that large SNR can be achieved during the time when spontaneous emission is still negligible.",
"This condition can be met with today's high-Q optical cavities, as shown below.",
"In this section we show that the proposed microscope can be implemented in the state-of-the-art experiments involving cold atoms/trapped ions and optical cavities, and discuss typical experimental parameters.",
"First, as discussed in Appendix REF , a prerequisite for the operation of the microscope is $\\gamma \\gg \\gamma _{{\\rm sp}}$ .",
"The cooperativity $\\mathcal {C}$ of high-Q optical cavity can exceed 100 in state-of-the-art experiments [51].",
"To make an estimation we choose $\\mathcal {C}=150$ , $\\sigma =0.3\\ell _{0}$ and $|\\langle r|D(z)\\rangle |_{{\\rm max}}^{2}=0.4$ (which suffices to render $V_{na}$ being negligible, c.f.",
"Fig.",
"REF ), yielding $\\gamma /\\gamma _{{\\rm sp}}\\simeq 75$ .",
"Such a high ratio guarantees that spontaneous emission is indeed negligible for the observation of the key predictions in the main text: for $\\gamma T\\simeq 75$ with $T$ being the total measurement time, one gets ${\\rm SNR}\\gg 1$ (c.f.",
"Fig.",
"4a of the main text) for a single scan of atomic motional eigenstates in the QND mode of the microscope.",
"Second, the two operation modes of the microscope require either $\\omega \\ll \\kappa $ or $\\omega \\gg \\kappa $ .",
"While the first region $\\omega \\ll \\kappa $ us naturally obtained using cavities with a sufficiently large linewidth, the second condition is also realistic.",
"For example, Ref.",
"[25] reports a coupled BEC-cavity setup with $\\kappa \\simeq 2\\pi \\times 4.5{\\rm kHz}$ which is far smaller than the recoil energy of light-mass alkalies (e.g., $E_{r}\\simeq 2\\pi \\times 60{\\rm {kHz}}$ for $^{7}$ Li at the D2 line).",
"Trapped ions provides another platform for reaching the good cavity limit, due to their large oscillation frequency ($\\sim $ MHz)."
],
[
"Perturbative Elimination of the Cavity Field",
"In this section we consider the relation between the homodyne current $I(t)$ and the localised microscope probe $f_{z_0}(\\hat{z})$ .",
"To obtain the connection we eliminate the cavity field from the stochastic dynamics described by the Eq.",
"(3) in the main text.",
"The aim is to derive effective stochastic master equations (5) and (6) in the main text, with corresponding photocurrents in `bad' and `good' cavity limits respectively.",
"To make the discussion more general, first, we consider an arbitrary system which is coupled to the cavity field $\\hat{c}$ via interaction $\\hat{H}_{{\\rm int}}=\\hbar \\varepsilon \\hat{f}(\\hat{c}+\\hat{c}^{\\dagger })$ where $\\hat{f}$ is a system operator and the coupling is assumed to be weak compared to the cavity decay rate $\\varepsilon \\ll \\kappa $ .",
"This model is related to the atomic system from the main text coupled to a driven cavity via the interaction Eq.",
"(1) linearised around the steady state intracavity field.",
"Transforming to an interaction picture with respect to the system Hamiltonian $\\hat{H}_{{\\rm sys}}$ we obtain the following SME describing the dynamics of the full setup under continuous homodyne monitoring of the cavity output field: dc=-i[f(t)(c+c)+cc,c]dt+D[c]c dt +H[ce-i]c dW(t), where $\\delta $ is the cavity detuning, $\\phi $ is the homodyne angle, and $\\hat{f}(t)=e^{i\\hat{H}_{{\\rm sys}}t}\\hat{f}e^{-i\\hat{H}_{{\\rm sys}}t}$ .",
"The corresponding homodyne current reads: $dX_{\\phi }(t) \\equiv I(t)dt=\\sqrt{\\kappa }\\left\\langle \\hat{c} e^{-i\\phi } + \\hat{c}^{\\dagger }e^{i\\phi }\\right\\rangle _{c}dt+dW(t)$ where $\\langle \\dots \\rangle _{c}\\equiv \\mathrm {Tr}\\lbrace \\ldots \\,\\rho _c(t)\\rbrace $ refers to an expectation value with respect to the conditional density matrix.",
"We eliminate the cavity field along the lines of [52].",
"First, we simply trace out the cavity dynamics from the SME () to obtain a stochastic equation for the system density matrix only ($\\tilde{\\rho }_{c}={\\rm Tr}_{T}\\rho _c$ ): dc =-i[f(t),+]dt+(e-i+ei)dW(t) where we define operators $\\hat{\\eta }={\\rm Tr}_{T}\\lbrace \\hat{c}\\rho _c\\rbrace $ and $\\hat{\\mu }=\\hat{\\eta }-\\langle \\hat{c}\\rangle \\tilde{\\rho }_{c}$ such that $\\langle \\hat{c}\\rangle ={\\rm Tr}_{S}\\,\\hat{\\eta }$ , and operations ${\\rm Tr}_{T}$ and ${\\rm Tr}_{S}$ stand for the partial traces over states of the cavity ($T$ for transducer) and the system respectively.",
"We derive an effective equation for $\\tilde{\\rho }_{c}$ up to the second order in the perturbation $\\varepsilon $ for the deterministic term and up to a linear stochastic term: $d\\tilde{\\rho }_{c}=O(\\varepsilon ^{2})dt+O(\\varepsilon )dW(t)$ .",
"This restricts equations for the operators $\\hat{\\eta }$ and $\\hat{\\mu }$ to $d\\hat{\\eta }(d\\hat{\\mu })=O(\\varepsilon )dt+O(1)dW(t)$ .",
"The equation of motion for $\\eta $ operator reads d =TrT{c dc} =-iTrT{ c[f(t)(c+c),c]} dt-(2-i) dt + TrT{ c(c-c) c e-i+c(c-c) cei} dW(t) -if(t)cdt-(2-i)dt, where in the first deterministic term and in the stochastic term we used the fact that $\\rho _c=\\tilde{\\rho }_{c}\\otimes \\rho _{T}$ to zeroth order in $\\varepsilon $ and that the unperturbed cavity is in a vacuum steady state such that $\\langle \\hat{c}\\hat{c}^{\\dagger }\\rangle =1$ , $\\langle \\hat{c}\\hat{c}\\rangle =\\langle \\hat{c}^{\\dagger }\\hat{c}\\rangle =0$ .",
"Next, for the cavity mean field we have: dc =TrSd -if(t)dt-(2-i)cdt This equation is first order in $\\varepsilon $ which means, to define operator $\\hat{\\mu }$ , we need to know $\\tilde{\\rho }_{c}$ to the zeroth order in $\\varepsilon $ .",
"It is constant in this approximation ($d\\tilde{\\rho }_{c}=0$ ) and we obtain an equation for the $\\hat{\\mu }$ operator using Itô rule: d =d-{ (dc)c+cdc+(dc)dc} -i{ f(t)-f(t)} cdt-(2-i) dt Plugging the solutions of the Eqs.",
"() and () into the equation of motion for the system density operator () we recover an effective equation with the necessary precision in $\\varepsilon $ .",
"There are two cases to consider.",
"`Bad' cavity — If the free evolution of the system can be neglected on a time scale of the cavity decay $1/\\kappa $ (for our harmonic oscillator $\\kappa \\gg \\omega $ ) we obtain: $\\hat{\\eta }\\simeq -i\\varepsilon \\frac{\\hat{f}(t)}{\\kappa /2-i\\delta }\\tilde{\\rho }_{c},\\qquad \\hat{\\mu }\\simeq -i\\varepsilon \\frac{\\hat{f}(t)-\\langle \\hat{f}(t)\\rangle }{\\kappa /2-i\\delta }\\tilde{\\rho }_{c}.$ Substituting these expressions into the Eq.",
"() and restoring the Schrödinger picture we arrive at the effective SME: $d\\tilde{\\rho }_{c}=-\\frac{i}{\\hbar }[\\hat{H}_{{\\rm eff}},\\tilde{\\rho }_{c}]dt+\\gamma \\mathcal {D}[\\hat{f}]\\tilde{\\rho }_{c}dt+\\sqrt{\\gamma }\\mathcal {H}[\\hat{f}]\\tilde{\\rho }_{c}dW(t)$ where $\\hat{H}_{{\\rm eff}}=\\hat{H}_{{\\rm sys}}+(\\hbar \\, \\delta \\,\\varepsilon ^{2}\\hat{f}^{2})/\\lbrace (\\kappa /2)^{2}+\\delta ^{2}\\rbrace $ and $\\gamma =(\\varepsilon ^{2}\\kappa )/\\lbrace (\\kappa /2)^{2}+\\delta ^{2}\\rbrace $ .",
"The homodyne phase is chosen to maximize the signal in the photocurrent ($\\phi =-{\\pi }/{2}+{\\rm arctan}\\lbrace {2\\delta }/{\\kappa }\\rbrace $ ) such that $dX_{\\phi }(t)\\equiv I(t)dt=2\\sqrt{\\gamma }\\langle \\hat{f}\\rangle _{c}dt+dW(t)$ which is obtained by substituting solution of Eq.",
"() into Eq.",
"(REF ).",
"In the main text we consider the cavity driven by a coherent field $\\mathcal {E}$ such that the coupling coefficient is given by $\\varepsilon =({\\mathcal {AE}}/{\\hbar })\\lbrace {\\kappa }/(\\kappa ^2/4+\\delta ^{2})\\rbrace ^{1/2}$ .",
"Defining $\\hat{f}=f_{z_0}(\\hat{z})$ and setting zero detuning $\\delta =0$ the equations (REF ) and (REF ) become Eqs.",
"(4) and (5) in the main text.",
"Ensemble averaging the stochastic dynamics (REF ) over individual trajectories results in the corresponding ME for the unconditional density matrix $\\tilde{\\rho }(t)=\\left\\langle \\tilde{\\rho }_{c}(t)\\right\\rangle _{{\\rm st}}$ : $\\frac{d\\tilde{\\rho }}{dt}=-\\frac{i}{\\hbar }[\\hat{H}_{{\\rm eff}},\\tilde{\\rho }]+\\gamma \\mathcal {D}[\\hat{f}]\\tilde{\\rho }.$ Here we used the non-anticipating property of the stochastic differential equation in Itô form $\\left\\langle \\dots dW(t)\\right\\rangle _{{\\rm st}}=0$ .",
"`Good' cavity — In the case of harmonic oscillator $\\hat{H}_{{\\rm sys}}=\\hbar \\omega (\\hat{a}^{\\dagger }\\hat{a}+1/2)$ the system coupling operator (localised probe $f_{z_0}(\\hat{z})$ ) in the interaction picture reads $\\hat{f}(t)=\\sum _{\\ell }\\hat{f}^{(\\ell )}e^{-i\\ell \\omega t}$ , where $\\hat{f}^{(\\ell )}=\\sum _{n}f_{n,n+l}|n\\rangle \\langle n+\\ell |$ with $f_{mn}=\\langle m|f_{z_0}(\\hat{z})|n\\rangle $ .",
"This allows one to integrate Eq.",
"() and () assuming slow time dependence of $\\tilde{\\rho }_{c}$ as follows: -if()e-it/2-i(+)c -i{ f()-f()} e-it/2-i(+)c Substituting the results into the Eq.",
"(), keeping only non-rotating deterministic terms due to $\\kappa \\ll \\omega $ in the `good' cavity limit (secular approximation), and transforming back to the Schrödinger picture we obtain: dc =-i[Heff,c]dt+2(/2)2+(+)2D[f()]cdt +H[-ie-i/2-i(+)f()]cdW(t), where $\\hat{H}_{{\\rm eff}}=\\hat{H}_{{\\rm sys}}+\\sum _{\\ell }\\frac{\\hbar \\varepsilon ^{2}(\\delta +\\omega \\ell )}{(\\kappa /2)^{2}\\!+\\!",
"(\\delta \\!+\\!\\omega \\ell )^{2}}\\left(\\!\\hat{f}^{(\\ell )}\\hat{f}^{(\\ell )\\dagger }\\!-\\hat{f}^{(\\ell )\\dagger }\\hat{f}^{(\\ell )}\\!\\right).$ To enhance the signal from the QND observable $\\hat{f}^{(0)}$ we choose the cavity detuning $\\delta =0$ and the homodyne angle $\\phi =-{\\pi }/{2}$ .",
"Then, by filtering out sidebands with $\\ell \\ne 0$ from the signal, we obtain a homodyne current (again using Eqs.",
"(REF ) and ()): $dX_{\\phi }(t) \\equiv I(t)dt=2\\sqrt{\\gamma }\\langle \\hat{f}^{(0)}\\rangle _{c}dt+dW(t)$ with $\\gamma =4\\varepsilon ^{2}/\\kappa $ and $\\varepsilon $ defined above.",
"This gives expression for the photocurrent preceding Eq.",
"(6) in the main text.",
"Discarding the sidebands from the photocurrent leads to averaging the effective SME () over corresponding unobserved measurements.",
"This results in dropping stochastic terms with $\\ell \\ne 0$ from the equation and yields the SME (6) in the paper.",
"In the `good' cavity limit $\\kappa \\ll \\omega $ , the additional part in the Hamiltonian $\\hat{H}_{\\rm eff}$ is much smaller than $\\hat{H}_{\\rm sys}$ and can be neglected."
],
[
"Scanning Many-body Systems and the Friedel Oscillation",
"Here we extend the scanning measurement to the many-body case and provide the details on scanning Friedel oscillations discussed in the main text.",
"To derivation the SME describing the scan of a many-body system, we decompose the focusing function, $\\phi _{z_0}(z)=\\mathcal {A}f_{z_0}(z)$ [c.f., Eq.",
"(1) of the main text], in terms of many-body eigenstates, $\\sum _{i=1}^{N}f_{z_{0}}(\\hat{z}_{i})\\rightarrow \\hat{f}_{z_{0}}=\\sum _{\\vec{\\nu },\\vec{\\nu }^{\\prime }}f_{\\vec{\\nu }\\vec{\\nu }^{\\prime }}\\left|\\vec{\\nu }\\right\\rangle \\langle \\vec{\\nu }^{\\prime }|,$ where $\\vec{\\nu }$ is the set of quantum numbers specifying the many-body state $\\left|\\vec{\\nu }\\right\\rangle $ with eigenenergy $E_{\\vec{\\nu }}$ , and $f_{\\vec{\\nu }\\vec{\\nu }^{\\prime }}=\\langle \\vec{\\nu }|\\sum _{i}f_{z_{0}}(\\hat{z}_{i})\\left|\\vec{\\nu }^{\\prime }\\right\\rangle $ are the matrix elements (Note, being a single-particle operator, $\\hat{f}_{z_{0}}$ generates only single-particle transitions).",
"Let us now define a set $\\left\\lbrace \\Delta E_{j}\\right\\rbrace $ of difference between the eigenenergies, $\\Delta E_{j}=E_{\\vec{\\nu }}-E_{\\vec{\\nu }^{\\prime }}$ , for all pairs of eigenstates appearing in Eq.",
"(REF ), and define the associated operators $\\quad \\hat{f}^{(\\Delta E_{j})}=f_{\\vec{\\nu }\\vec{\\nu }^{\\prime }}\\left|\\vec{\\nu }\\right\\rangle \\langle \\vec{\\nu }^{\\prime }|,$ so that $\\hat{f}_{z_{0}}=\\sum _{j}\\hat{f}^{(\\Delta E_{j})}$ .",
"Note that here we assume all $\\Delta E_{j}$ being different [except for $\\Delta E_{j}=0$ corresponding to diagonal contributions of (REF )], as in the example of fermions in a box considered below.",
"In situations where there are (quasi-)degenerate energy differences $\\Delta E_{j}$ , like atoms in a harmonic trap, the definition of the operators $\\hat{f}^{(\\Delta E_{j})}$ should include the summation over the pairs of states with (quasi) degenerate $\\Delta E_{j}$ .",
"The operators $\\hat{f}^{(\\Delta E_{j})}$ are generalizations of $\\hat{f}^{(\\ell )}$ in the single-particle case in Appendix , and provide a `spectral decomposition' of $\\hat{f}_{z_{0}}$ : In the interaction picture with respect to the Hamiltonian of the system, they evolve as $\\hat{f}^{(\\Delta E_{j})}(t)=\\hat{f}^{(\\Delta E_{j})}\\exp (-i\\Delta E_{j}t/\\hbar )$ .",
"Let $\\Delta E$ be a typical level spacing between physically relevant states such that $\\Delta E_{j}\\ge \\Delta E$ .",
"In the good cavity regime $\\kappa \\le \\Delta E$ , these fast rotating terms with $\\Delta E_{j}\\ne 0$ will be suppressed due to the finite time resolution $\\kappa ^{-1}$ of cavity, similar to the single atom case.",
"We eliminate the cavity field in the same fashion as the `good cavity' case in Appendix .",
"The dispersive cavity-atom coupling defines the small coeficient $\\varepsilon =({\\mathcal {AE}}/{\\hbar })\\lbrace {\\kappa }/(\\kappa ^2/4+\\delta ^{2})\\rbrace ^{1/2}$ .",
"Assuming $\\varepsilon \\ll \\kappa $ allows for eliminating the cavity in an expansion of $\\varepsilon /\\kappa $ .",
"Accurate to $O(\\varepsilon ^2)$ in the deterministic term and $O(\\varepsilon )$ in the stochastic term, we arrive at the SME for the conditional density matrix of the atomic system $d\\tilde{\\rho }_{c} & = & -\\frac{i}{\\hbar }[\\hat{H}_{\\rm eff},\\tilde{\\rho }_c]dt+\\gamma \\mathcal {D}[\\hat{f}^{(0)}]\\tilde{\\rho }_{c}dt+\\sqrt{\\gamma }\\mathcal {H}[\\hat{f}^{(0)}]\\tilde{\\rho }_{c}dW(t)\\nonumber \\\\& & +\\sum _{\\Delta E_{j}\\ne 0}\\gamma _{j}\\mathcal {D}[\\hat{f}^{(\\Delta E_{j})}]\\tilde{\\rho }_{c}dt,$ where we have assumed a resonant cavity driving $\\delta =0$ , the homodyne angle $\\phi =-\\pi /2$ .",
"In Eq.",
"(REF ), $\\hat{f}^{(0)}=\\hat{f}^{(\\Delta E_{j}=0)}=\\sum _{\\vec{\\nu }}f_{\\vec{\\nu }\\vec{\\nu }}\\left|\\vec{\\nu }\\right\\rangle \\langle \\vec{\\nu }|$ is the QND observable which measures the local density for an arbitrary eigenstate, with a rate $\\gamma =[4\\mathcal {A}\\mathcal {E}/(\\hbar \\kappa )]^{2}$ .",
"Analogous to the single-particle case [c.f.",
"Eq.",
"()], the last term of Eq.",
"(REF ) describes the suppressed dissipation channels, with the rates $\\gamma _{j}=\\gamma [1+4\\Delta E_{j}^2/\\kappa ^{2}]^{-1}$ .",
"Finally, the Hamiltonian $\\hat{H}_{\\rm eff}=\\hat{H}_{\\rm sys}+\\hbar \\varepsilon ^2\\sum _{\\Delta E_j \\ne 0}\\Delta E_j[(\\kappa /2)^2+\\Delta E_j^2]^{-1}[\\hat{f}^{(\\Delta E_j)}\\hat{f}^{(\\Delta E_j)\\dag }-\\hat{f}^{(\\Delta E_j)\\dag }\\hat{f}^{(\\Delta E_j)}]$ .",
"The second term comes from adiabatic elimination of the cavity, and describes cavity-mediated interactions between particles.",
"Due to the energy hierarchy $\\varepsilon \\ll \\kappa \\le \\Delta E$ , this term is far smaller than $\\hat{H}_{\\rm sys}$ and only weakly disturbs the eigenspectrum of the system.",
"We will neglect this tiny correction in the following discussion.",
"The associated expression for the homodyne current reads $I(t)=2\\sqrt{\\gamma }{\\rm Tr}[\\hat{f}^{(0)}\\tilde{\\rho }_{c}(t)]+\\xi (t).$ We now apply the above analysis to a simple example of a non-interacting Fermi sea, where the presence of a single impurity causes the Friedel oscillation.",
"Consider $N$ fermions in a one-dimensional box of length $L\\gg \\sigma $ , $-L/2\\le z\\le L/2$ , with a point-like impurity at the origin described by the potential $V_{imp}(z)=U\\delta (z)$ .",
"Assuming zero boundary conditions at $z=\\pm L/2$ and taking the limit $U\\rightarrow \\infty $ to simplify anlytical expressions, the single-particle wave functions read $\\begin{split}\\psi _{n}^{(o)}(z) & =\\sqrt{\\frac{2}{L}}\\sin \\left(\\frac{2\\pi n}{L}z\\right),\\\\\\quad \\psi _{n}^{(e)}(z) & =\\sqrt{\\frac{2}{L}}\\sin \\left(\\frac{2\\pi n}{L}\\left|z\\right|\\right),\\end{split}$ where $n=1,\\,2,$$\\ldots $ for both $\\psi _{n}^{(o)}(z)$ (odd parity) and $\\psi _{n}^{(e)}(z)$ (even parity).",
"The corresponding eigenenergies are $\\epsilon _{n}^{(o/e)}=[2\\pi ^{2}\\hbar ^{2}/(mL^{2})]n^{2}$ .",
"The particle density for the ground state is (we assume even $N$ for simplicity) n(z) =n=1N/2[n(o)(z)2+n(e)(z)2] =nF+1L{ 1-[2(N+1)z/L](2z/L)} , where $n_{F}=N/L$ is the average fermionic density.",
"In the vicinity of the impurity, $\\left|z\\right|\\ll L/2\\pi $ , $n(z)$ has the form of Friedel oscillations, $n(z)\\approx n_{F}-\\frac{\\sin (2k_{F}z)}{2\\pi z}=n_{F}\\left[1-\\frac{\\sin (2k_{F}z)}{2k_{F}z}\\right],$ with $k_{F}=\\pi n_{F}$ the Fermi wave vector, and we omit terms $\\sim L^{-1}$ .",
"Note that for $z\\sim L/2\\pi $ the “finite-size” oscillations in $n(z)$ , Eq.",
"(), have the amplitude $\\sim L^{-1}$ that vanishes in the thermodynamic limit with the fixed density $n_{F}$ , in contrast to the Friedel oscillations Eq.",
"(REF ).",
"For this case it is convenient to classify the many-body states in terms of occupations of single-particle states and to use the language of second quantization.",
"We introduce the destruction (thus the associated creation) operators as $\\hat{b}_{n,L(R)}=\\frac{1}{\\sqrt{2}}\\int _{-L/2}^{L/2} dz [\\psi _n^{(o)*}(z)\\mp \\psi _n^{(e)*}(z)]\\hat{\\psi }(z)$ [with $\\hat{\\psi }(z)$ the fermi field operator], which correspond to the left(right) single-particle eigenmodes.",
"The focusing function $f_{z_0}(\\hat{z})$ has zero matrix elements between left and right eigenmodes, $\\langle m,L|f_{z_0}(\\hat{z})|n,R\\rangle = 0$ .",
"Using these bases and for simplicity defining the single-particle quantum number $\\nu \\equiv \\lbrace n,L(R)\\rbrace $ , Eqs.",
"(REF ) and (REF ) can be expressed explicitly: the QND observable $\\hat{f}^{(0)}$ becomes $\\hat{f}^{(0)}=\\sum _{\\nu }f_{\\nu \\nu }b^\\dag _{\\nu }b_{\\nu }$ whereas the last term of Eq.",
"(REF ) (the suppressed dissipations channels) reads $\\sum _{\\nu \\ne \\nu ^{\\prime }}\\gamma _{\\nu \\nu ^{\\prime }}\\mathcal {D}[\\hat{b}^\\dag _\\nu \\hat{b}_{\\nu ^{\\prime }}]$ with the corresponding rates $\\gamma _{\\nu \\nu ^{\\prime }}=\\gamma f_{\\nu \\nu ^{\\prime }}^2[1+4(\\epsilon _\\nu -\\epsilon _{\\nu ^{\\prime }})^2/\\kappa ^2]^{-1}$ , where $f_{\\nu \\nu ^{\\prime }}=\\langle \\nu |f_{z_0}(\\hat{z})| \\nu ^{\\prime }\\rangle $ is the single-particle matrix element and $\\epsilon _\\nu =[2 \\pi ^{2} \\hbar ^{2}/(mL^{2})]n^{2}$ .",
"By truncating to a suitable number of fermi orbitals, Eqs.",
"(REF ) and (REF ) can then be integrated straightforwardly.",
"To resolve the Friedel oscillations in the scan, their period has to be larger than the focusing region $\\sigma $ , $\\pi /k_{F}>\\sigma $ .",
"This condition puts an upper bound on the density of fermions and, therefore, on their total number, $N<L/\\sigma $ , which corresponds to having not more than one fermion per length $\\sigma $ .",
"The gap to the first excited state (the level spacing) in this case can be estimated as $\\Delta E\\sim \\hbar ^{2}/(m\\sigma ^{2})$ , and the condition for the non-demolition scan reads $\\kappa \\le \\hbar ^{2}/(m\\sigma ^{2})$ .",
"For the simulation shown in Fig.",
"4b of the main text, we consider $N=16$ fermions, scanned by a microscope with resolution $\\sigma =0.01L$ and cavity linewidth $\\kappa =4\\pi ^2\\hbar ^2/(mL^2)$ .",
"The dimensionless measurement strength is $\\gamma T=400$ with $T$ being the total scanning time.",
"The filter integration time for post-processing is chosen as $\\tau =\\sigma T/L=0.01T$ ."
]
] | 1709.01530 |
[
[
"An Implicit Discrete Unified Gas-Kinetic Scheme for Simulations of\n Steady Flow in All Flow Regimes"
],
[
"Abstract This paper presents an implicit method for the discrete unified gas-kinetic scheme (DUGKS) to speed up the simulations of the steady flows in all flow regimes.",
"The DUGKS is a multi-scale scheme finite volume method (FVM) for all flow regimes because of its ability in recovering the Navier-Stokes solution in the continuum regime and the free transport mechanism in rarefied flow, which couples particle transport and collision in the flux evaluation at cell interfaces.",
"In this paper the predicted iterations are constructed to update the macroscopic variables and the gas distribution functions in discrete microscopic velocity space.",
"The lower-upper symmetric Gauss-Seidel (LU-SGS) factorization is applied to solve the implicit equations.",
"The fast convergence of implicit discrete unified gas-kinetic scheme (IDUGKS) can be achieved through the adoption of a numerical time step with large CFL number.",
"Some numerical test cases, including the Couette flow, the lid-driven cavity flows under different Knudsen number and the hypersonic flow in transition flow regime around a circular cylinder, have been performed to validate this proposed IDUGKS.",
"The computational efficiency of the IDUGKS to simulate the steady flows in all flow regimes can be improved by one or two orders of magnitude in comparison with the explicit DUGKS."
],
[
"Introduction",
"The simulations for flows over a wide range of Knudsen numbers becomes a challenging issue for numerical modelling.",
"Different flow physics in large variation of temporal and spatial scales cause a difficulty.",
"In the case that the particle mean free path is of the same order as or even larger than representative physical length scale, gas can never be modeled as continuum [1].",
"The particle based methods may solve this kind of problem.",
"But in continuum flow regime, its computational cost becomes unaffordable [2].",
"To provide delicate numerical dissipation for particle collisions in flows especially for non-equilibrium phenomenon in shock-wave structure, the gas-kinetic scheme (GKS) is proposed by Xu et al [3].",
"The GKS is able to give a more exact description for highly non-equilibrium flows than that of Navier-Stokes equation [4].",
"Several studies have been done on implicit GKS for a better computation efficiency.",
"Chit et al [5] applied approximate factorization and alternating direction-implicit (AF-ADI) method on GKS.",
"With a large Courant-Friedrichs-Levy (CFL) number, a fast convergence is achieved in simulation of the invicid compressible flows on structured grid.",
"Compared to explicit GKS, this method even obtains a better accuracy.",
"Li et al [6] proposed an implicit GKS based on matrix free Lower-Upper Symmetric Gauss Seidel (LU-SGS) time marching scheme for simulation of hypersonic inviscid flows on unstructured mesh.",
"This method can be easily implemented on an hybrid unstructured mesh and the good robustness of this method is achieved.",
"The issue is that the GKS is only valid for continuum flows.",
"Many kinetic schemes, such as the discrete ordinate method (DOM), can obtain accurate solutions in kinetic regimes but fail to simulate continuum flows efficiently because of the use of temporal and spatial scales on the order of particle collision time and mean free path [2].",
"Many other asymptotic preserving kinetic schemes successfully extend their validation to continuum invicid flows.",
"But they still cannot capture mass and momentum transport in boundary layer [7].",
"The unified gas kinetic scheme (UGKS) is a finite volume method (FVM) proposed to include all simulation scales from Navier-Stokes solution to kinetic regime.",
"Based on the Boltzmann BGK model [8], it couples particle transport and collision process.",
"In the reconstruction of the gas distribution function at the cell interface, the integral solution of the Boltzmann BGK model is applied.",
"So, the numerical time step is not limited to the particle relaxation time [2].",
"In the update of the flow field, the UGKS has to compute the flux of macroscopic variables with moments for gas distribution function, which introduces additional computational cost in comparison with DOM.",
"To reduce the computational cost and simplify computational process, the DUGKS for simulation of the flows with all Knudsen numbers is proposed [9].",
"Different from the UGKS, the implicit treatment of the collision term is removed and the transformation of gas distribution function is employing with collision.",
"In the flux evaluation, the gas distribution function is reconstructed at the cell interface along the characteristic line.",
"So, the multi-scale dynamics in flows is described but the formulation of the numerical method is greatly simplified [10].",
"For UGKS, the matrix-free LU-SGS method for both continuum and rarified flow simulations has been constructed by Zhu et al (2016) [11].",
"In their work, both macroscopic and microscopic governing equations are implicitly coupled.",
"To treat the collision term in an implicit way, the gas distribution function for equilibrium state is predicted by updating macroscopic variables implicitly.",
"The governing equation for the gas distribution function is fully discretized in an implicit form.",
"Finally, both of implicit governing equations of macroscopic variables and gas distribution function are solved iteratively based on the LU-SGS method.",
"Compared to explicit UGKS, the implicit UGKS has a much faster convergence and the same accuracy in simulation for all flow regimes.",
"But compared to DUGKS, the formulation of flux evaluation of UGKS is still too complicated.",
"To simplify the computational process and improve the computational efficiency, it is necessary to develop an implicit method for DUGKS.",
"This paper is aimed to proposed an implicit method for DUGKS.",
"In explicit DUGKS, the governing equation for macroscopic variables is not required.",
"But in the IDUGKS, the BGK collision term should be treated in an implicit way.",
"As a result, macroscopic variables should be updated implicitly in every implicit time step.",
"Implicit discretization is applied in FVM governing equation for gas distribution function.",
"In our work, the matrix free LU-SGS method is still used for discretizing and solving the linear systems derived from the implicit predicted treatment.",
"The rest of the paper is organized as follows.",
"The classical DUGKS for all Knudsen numbers proposed by Guo et al [9], the matrix-free LU-SGS scheme and the implementation of IDUGKS are described in Section .",
"Then, the numerical validations and the Couette flow in continuum flow regime, several lid-driven cavity flow cases under different Knudsen numbers, the hypersonic circular cylinder case are carried out to show the accuracy and the reliability of the present IDUGKS method in Section .",
"Finally, some remarks concluded from this study are grouped in Section .",
"In two dimensional problems, the DUGKS is based on the Boltzmann BGK model which is written as $\\frac{{\\partial f}}{{\\partial t}} + u\\frac{{\\partial f}}{{\\partial x}} + v\\frac{{\\partial f}}{{\\partial y}} = \\frac{{{f^{eq}} - f}}{\\tau },$ where $f$ and $f^{eq}$ are the gas distribution functions, which are the functions of space $\\left( {x, y} \\right)$ , particle velocity $\\left( {u, v} \\right)$ , time $t$ , and internal variable $\\xi $ .",
"$\\tau $ is the particle collision time.",
"${f^{eq}}$ is the Maxwell distribution function which has the following form ${f^{eq}} = \\rho {\\left( {\\frac{\\lambda }{\\pi }} \\right)^{\\frac{{K + 2}}{2}}}{e^{ - \\lambda \\left( {{{\\left( {u - U} \\right)}^2} + {{\\left( {v - V} \\right)}^2} + {\\xi ^2}} \\right)}},$ where $K$ is the internal freedom degree with $K = 3$ for the 2D diatomic molecule gas flows.",
"The variable $\\lambda = {m \\mathord {\\left\\bad.",
"{\\vphantom{m {\\left( {2RT} \\right)}}} \\right.\\hspace{0.0pt}} {\\left( {2RT} \\right)}}$ , $m$ is the molecular mass, $R$ is the Boltzmann constant, and $T$ is the temperature.",
"$\\rho $ is the density, $U$ and $V$ are the $x$ and $y$ components of the macroscopic velocity in 2D, respectively.",
"Note that, for the gas system with $K$ freedom degree, the square of internal variable $\\xi $ can be taken as ${\\xi ^2} = \\xi _1^2 + \\xi _2^2 + \\cdots + \\xi _K^2 .$ Conservative flow variables can be obtained by moments of gas distribution function with microscopic variables.",
"$\\left( \\begin{array}{c}\\rho \\\\\\rho U\\\\\\rho V\\\\\\rho E\\end{array} \\right) = \\int {\\left( \\begin{array}{c}1\\\\u\\\\v\\\\\\frac{1}{2}\\left( {{u^2} + {v^2} + {\\xi ^2}} \\right)\\end{array} \\right)} fdudvd\\xi ,$ where $\\rho E = \\frac{1}{2}\\rho \\left( {{U^2} + {V^2} + \\frac{{K + 2}}{{2\\lambda }}} \\right)$ is the total energy.",
"In this formulation $\\lambda $ has relation $p = {\\rho \\mathord {\\left\\bad.",
"{\\vphantom{\\rho {2\\lambda }}} \\right.",
"\\hspace{0.0pt}} {2\\lambda }}$ with pressure $p$ and density $\\rho $ [12].",
"The evolution of the gas distribution functions has no relation with internal freedom.",
"So, two reduced distribution functions are constructed by moments with variable $\\begin{aligned}G &= \\int {fd\\xi }, \\\\H &= \\int {{\\xi ^2}fd\\xi },\\end{aligned}$ where $G$ denotes particle density and $H$ reflects internal energy.",
"The evolution equations for this two new distribution functions can be rewritten according to Eq.",
"(REF ) $\\begin{aligned}\\frac{{\\partial G}}{{\\partial t}} + u\\frac{{\\partial G}}{{\\partial x}} + v\\frac{{\\partial G}}{{\\partial y}} &= \\frac{{{G^{eq}} - G}}{\\tau }, \\\\\\frac{{\\partial H}}{{\\partial t}} + u\\frac{{\\partial H}}{{\\partial x}} + v\\frac{{\\partial H}}{{\\partial y}} &= \\frac{{{H^{eq}} - H}}{\\tau },\\end{aligned}$ where ${G^{eq}}$ and ${H^{eq}}$ are reduced equilibrium distribution function for $G$ and $H$ .",
"The forms of governing equations for $G$ and $H$ are the same.",
"So they have the same way in evolution.",
"For simplicity, we use a variable $\\phi $ to represent them.",
"Eq.",
"(REF ) can be rewritten as $\\frac{{\\partial \\phi }}{{\\partial t}} + u\\frac{{\\partial \\phi }}{{\\partial x}} + v\\frac{{\\partial \\phi }}{{\\partial y}} = \\vartheta ,$ where $\\vartheta $ represents the collision term.",
"In the finite volume, Eq.",
"(REF ) can be integrated over a time step, in which the midpoint rule is used for the time integration of the convection term and the trapezoidal rule for the collision term.",
"The discrete form of Eq.",
"(REF ) can be written as $\\left( {{\\phi ^{n + 1}} - \\frac{{\\Delta t}}{2}{\\vartheta ^{n + 1}}} \\right) = \\left( {{\\phi ^n} + \\frac{{\\Delta t}}{2}{\\vartheta ^n}} \\right) - \\frac{{\\Delta t}}{\\Omega }\\sum \\limits _{k = 1}^{Nf} {F_k^{n + \\frac{1}{2}}} ,$ where $F_k^{n + \\frac{1}{2}}$ denotes flux through cell interface $k$ and $Nf$ denotes the number of cell interface of a cell.",
"$\\Omega $ is the volume of the cell.",
"The numerical flux reads $F = \\int {{u_n}\\phi dS}.$ In above equation, ${u_n}$ represents velocity propagate to cell interface.",
"The term $\\left( {{\\phi ^{n + 1}} - \\frac{{\\Delta t}}{2}{\\vartheta ^{n + 1}}} \\right)$ and $\\left( {{\\phi ^n} + \\frac{{\\Delta t}}{2}{\\vartheta ^n}} \\right)$ are substituted by two new distribution functions $\\tilde{\\phi }$ and ${\\tilde{\\phi }^ + }$ [9] $\\begin{aligned}\\tilde{\\phi }&= \\frac{{2\\tau + \\Delta t}}{{2\\tau }}\\phi - \\frac{{\\Delta t}}{{2\\tau }}{\\phi ^{eq}}, \\\\{{\\tilde{\\phi }}^ + } &= \\frac{{2\\tau - \\Delta t}}{{2\\tau + \\Delta t}}\\tilde{\\phi }+ \\frac{{2\\Delta t}}{{2\\tau + \\Delta t}}{\\phi ^{eq}}.\\end{aligned}$ To obtain flux of the cell interface at $n + \\frac{1}{2}$ time step, the distribution function at $n + \\frac{1}{2}$ time step at the cell interface is required.",
"In a half time step, Eq.",
"(REF ) is integrated and a relation is obtained [9] $\\bar{\\phi }\\left( {{{\\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over x} }_{cf}}, {t_{n + \\frac{1}{2}}}\\right.",
"= {\\bar{\\phi }^ + }\\left( {\\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over x} - \\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over u\\right.",
"\\frac{{\\Delta t}}{2}, {t_n}$ ), where $\\bar{\\phi }$ and ${\\bar{\\phi }^ + }$ are two reduced distribution function which reads $\\begin{aligned}\\bar{\\phi }&= \\phi - \\frac{{\\frac{{\\Delta t}}{2}}}{2}\\vartheta ,\\\\{{\\bar{\\phi }}^ + } &= \\phi + \\frac{{\\frac{{\\Delta t}}{2}}}{2}\\vartheta = \\frac{{2\\tau - \\frac{{\\Delta t}}{2}}}{{2\\tau + \\frac{{\\Delta t}}{2}}}\\bar{\\phi }+ \\frac{{\\Delta t}}{{2\\tau + \\frac{{\\Delta t}}{2}}}{\\phi ^{eq}}.\\end{aligned}$ Now, $\\bar{\\phi }$ is the distribution function in Eq.",
"(REF ) at half time step.",
"Thee distribution of equilibrium state can be obtained by macroscopic variables.",
"They are computed via integration in microscopic velocity space with $\\bar{\\phi }$ .",
"The distribution function ${\\bar{\\phi }^ + }\\left( {\\mathord {{\\scriptscriptstyle \\rightharpoonup }} \\over x} - \\mathord {{\\scriptscriptstyle \\rightharpoonup }} \\over u\\right.",
"\\frac{{\\Delta t}}{2}, {t_n}$ )$ is obtained along characteristic line,\\begin{equation}{\\bar{\\phi }^ + }\\left( {{{\\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over x} }_{cf}} - \\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over u\\right.",
"\\frac{{\\Delta t}}{2}, {t_n}\\end{equation} = {\\bar{\\phi }^ + }\\left( {{{\\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over x} }_{cf}}, {t_n}\\right.",
"- u{\\frac{{\\Delta t}}{2}}\\frac{{\\partial {{\\bar{\\phi }}^ + }}}{{\\partial x}} - v{\\frac{{\\Delta t}}{2}}\\frac{{\\partial {{\\bar{\\phi }}^ + }}}{{\\partial y}}.$ In above equation, the spatial derivatives $\\frac{{\\partial {{\\bar{\\phi }}^ + }}}{{\\partial x}}$ and $\\frac{{\\partial {{\\bar{\\phi }}^ + }}}{{\\partial y}}$ , the value at the cell interface ${\\bar{\\phi }^ + }\\left( {{{\\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over x} }_{cf}}, {t_n}\\right.",
"$ are obtained by the least square method.",
"Taking the simple schematic grid shown in Fig.",
"REF , the cells in reconstruction are presented.",
"With the cells adjacent to the cell $({x_0}, {y_0})$ as well as values of them, the fitting formulation is applied, ${\\bar{\\phi }^ + } = \\bar{\\phi }_0^ + + \\varphi \\left[ {\\frac{{\\partial {{\\bar{\\phi }}^ + }}}{{\\partial x}}\\left( {x - {x_0}} \\right) + \\frac{{\\partial {{\\bar{\\phi }}^ + }}}{{\\partial y}}\\left( {y - {y_0}} \\right)} \\right] .$ To determine the spatial derivative terms in above equation, the least-squares regression equations can be reconstructed as $\\tiny \\sum \\limits _{j = 1}^N {\\left[ \\begin{array}{c}{\\left( {{x_j} - {x_0}} \\right)^2}\\\\\\left( {{x_j} - {x_0}} \\right)\\left( {{y_j} - {y_0}} \\right)\\end{array} \\right.}",
"\\left.",
"\\begin{array}{c}\\left( {{x_j} - {x_0}} \\right)\\left( {{y_j} - {y_0}} \\right)\\\\{\\left( {{y_j} - {y_0}} \\right)^2}\\end{array} \\right]\\left[ \\begin{array}{l}\\frac{{\\partial {{\\bar{\\phi }}^ + }}}{{\\partial x}}\\\\\\frac{{\\partial {{\\bar{\\phi }}^ + }}}{{\\partial y}}\\end{array} \\right] = \\sum \\limits _{j = 1}^N {\\left[ \\begin{array}{l}\\left( {{x_j} - {x_0}} \\right)\\left( {\\bar{\\phi }_j^ + - \\bar{\\phi }_0^ + } \\right)\\\\\\left( {{y_j} - {y_0}} \\right)\\left( {\\bar{\\phi }_j^ + - \\bar{\\phi }_0^ + } \\right)\\end{array} \\right]} ,$ where $N$ denotes the total number of cells adjacent to the cell.",
"The $\\varphi $ is the limiter.",
"In incompressible flows, $\\varphi = 1$ .",
"In compressible flow simulation, the Vankatacrishnan limiter is applied [13].",
"$\\varphi = \\left\\lbrace \\begin{array}{*{20}{c}}\\theta \\left( {\\frac{{\\max \\left( {\\bar{\\phi }_j^ + } \\right) - \\bar{\\phi }_0^ + }}{{\\bar{\\phi }_{cf}^ + - \\bar{\\phi }_0^ + }}} \\right),&\\bar{\\phi }_{cf}^ + > \\bar{\\phi }_0^ + ,\\\\\\theta \\left( {\\frac{{\\min \\left( {\\bar{\\phi }_j^ + } \\right) - \\bar{\\phi }_0^ + }}{{\\bar{\\phi }_{cf}^ + - \\bar{\\phi }_0^ + }}} \\right),&\\bar{\\phi }_{cf}^ + < \\bar{\\phi }_0^ + ,\\\\1&\\bar{\\phi }_{cf}^ + = \\bar{\\phi }_0^ + ,\\end{array}\\right.$ where function $\\theta $ can be calculated by using the following formulas $\\theta \\left( x \\right) = \\frac{{{x^2} + 2x}}{{{x^2} + x + 2}} .$ In this paper, this limiter is applied in simulation for hypersonic rarified flow around circular cylinder.",
"After ${\\bar{\\phi }^{n + \\frac{1}{2}}}$ at the cell interface is obtained, the original distribution function $\\phi _{cf}^{n + \\frac{1}{2}}$ can be obtained $\\phi _{cf}^{n + \\frac{1}{2}} = \\frac{{2\\tau }}{{2\\tau + \\frac{{\\Delta t}}{2}}}\\bar{\\phi }_{cf}^{n + \\frac{1}{2}} + \\frac{{\\frac{{\\Delta t}}{2}}}{{2\\tau + \\frac{{\\Delta t}}{2}}}\\phi _{cf}^{eq,n + \\frac{1}{2}} .$ At a cell interface $k$ , the numerical flux can be obtained by $F_k^{n + \\frac{1}{2}} = \\int {{u_n}\\phi _{cf}^{n + \\frac{1}{2}}d{S_k}} .$ In gas kinetic theory, collision term in Boltzmann BGK model which contains only one single relaxation time leads to a fixed Prandtl number [12].",
"In this paper, BGK-Shakhov model is applied to overcome this limitation, the reduced distribution function ${G^{eq}}$ and ${H^{eq}}$ for equilibrium state are modified as [14], $\\small \\begin{aligned}G_{\\Pr }^{eq} &= {G^{eq}} + \\left( {1 - \\Pr } \\right)\\frac{{\\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over u} \\cdot \\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over q}{}\\end{aligned}$ 5pRT( $\\scriptscriptstyle \\rightharpoonup $ u 2RT - - 2 )Geq, Heq = Heq + ( 1 - )$\\scriptscriptstyle \\rightharpoonup $ u $\\scriptscriptstyle \\rightharpoonup $ q 5pRT[ ( $\\scriptscriptstyle \\rightharpoonup $ u 2RT - )( K + 3 - ) - 2K ]RTGeq , where $\\alpha $ represents $\\alpha $ -dimensional problem."
],
[
"Matrix-free LU-SGS scheme",
"Now, the distribution function can be updated by FVM scheme ${\\tilde{\\phi }^{n + 1}} = {\\tilde{\\phi }^{ + ,n}} - \\frac{{\\Delta t}}{\\Omega }\\sum \\limits _{k = 1}^{Nf} {F_k^{n + \\frac{1}{2}}}.$ For simulation of steady state, implicit scheme can be constructed by using the backward Euler method at $n + 1$ time step $\\frac{{{{\\tilde{\\phi }}^{n + 1}} - {{\\tilde{\\phi }}^n}}}{{\\Delta t}}\\Omega = {M^n} + \\left( {\\frac{{\\partial M}}{{\\partial \\tilde{\\phi }}}} \\right)\\Delta \\tilde{\\phi }+ {Q^n} + \\left( {\\frac{{\\partial Q}}{{\\partial \\tilde{\\phi }}}} \\right)\\Delta \\tilde{\\phi },$ where $M$ denotes integration of flux around all interfaces and $Q$ denotes source term, they can be written as $\\begin{aligned}M &= - \\sum \\limits _{k = 1}^{Nf} {F_k^{n + \\frac{1}{2}}} ,\\\\Q &= - \\frac{{2\\Delta t}}{{2\\tau + \\Delta t}}\\tilde{\\phi }+ \\frac{{2\\Delta t}}{{2\\tau + \\Delta t}}{\\phi ^{eq}}.\\end{aligned}$ The $M$ in derivative term $\\frac{{\\partial M}}{{\\partial \\tilde{\\phi }}}$ can be rewritten in delta form $M = - \\sum \\limits _{k = 1}^{Nf} {{u_n}{S_k}\\Delta \\tilde{\\phi }_k^{n + 1}}.$ The final convergence solution will not be affected by different algorithms for implicit fluxes.",
"As a result, the distribution function at the cell interface is constructed by upwind scheme [15].",
"For cell i, $\\Delta \\tilde{\\phi }_k^{n + 1} = \\frac{1}{2}\\left( {\\Delta \\tilde{\\phi }_i^{n + 1} + \\Delta \\tilde{\\phi }_k^{n + 1}} \\right) + \\frac{1}{2}sign\\left( {\\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over u} \\cdot {{\\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over S} }_k\\right.$ )( in + 1 - kn + 1 ).",
"In cell $i$ , the final implicit equation is written as $\\left( {\\frac{{{\\Omega _i}}}{{\\Delta {t_{imp}}}} - \\frac{{\\partial M}}{{\\partial {{\\tilde{\\phi }}_i}}} - \\frac{{\\partial {Q}}}{{\\partial {{\\tilde{\\phi }}_i}}}} \\right)\\Delta {\\tilde{\\phi }_i} - \\frac{{\\partial M}}{{\\partial {{\\tilde{\\phi }}_j}}}\\Delta {\\tilde{\\phi }_j} = {M^n} + {Q^n}.$ After iterating over the whole flow field, a system of linear equations $\\Delta \\tilde{\\phi }$ is obtained.",
"This equations can be discretized and solved by using LU-SGS method [16].",
"Different from explicit DUGKS, macroscopic variables and microscopic distribution function should be marching at the same time step so implicit predicted algorithm is applied to governing equation for macroscopic variables.",
"For DUGKS, the scheme for updating of macroscopic variables reads $\\frac{{{W^{n + 1}} - {W^n}}}{{\\Delta t}}\\Omega = - \\sum \\limits _{k = 1}^{Nf} {\\int {\\psi F_k^{n + \\frac{1}{2}}dudv} },$ where $\\psi = {\\left( {1, u, v, \\frac{1}{2}\\left( {{u^2} + {v^2} + {\\xi ^2}} \\right)} \\right)^T}$ is microscopic variables.",
"When updating macroscopic variables the numerical flux at half time step is still used.",
"Eq.",
"(REF ) can be rewritten as implicit form $\\frac{{{W^{n + 1}} - {W^n}}}{{\\Delta t}}\\Omega = M_W^n + \\left( {\\frac{{\\partial {M_W}}}{{\\partial W}}} \\right)\\Delta W ,$ where the term $\\frac{{\\partial {M_W}}}{{\\partial W}}$ is a Jacobian matrix.",
"In this paper, the form in first-order Roe¡¯s scheme [17] in which the Roe¡¯s flux can be linearized is applied in discretization for macroscopic predicted algorithm [18].",
"According to the idea of LU-SGS method, we first split this Jacobian matrix into three parts: the lower triangular matrix, the upper triangular matrix and the diagonal terms.",
"Eq.",
"(REF ) can be rewritten as $\\left( {L + U + D} \\right)\\Delta W = M_W^n ,$ with $\\left\\lbrace \\begin{array}{l}L = \\sum \\limits _{k \\in L\\left( i \\right)} {\\left( {\\frac{1}{2}\\frac{{\\partial F\\left( W \\right)}}{{\\partial W}} \\cdot S - {{\\left( {{\\Lambda _c}} \\right)}_{ik}}} \\right)} ,\\\\U = \\sum \\limits _{k \\in U\\left( i \\right)} {\\left( {\\frac{1}{2}\\frac{{\\partial F\\left( W \\right)}}{{\\partial W}} \\cdot S - {{\\left( {{\\Lambda _c}} \\right)}_{ik}}} \\right)} ,\\\\D = \\left( {\\frac{{{\\Omega _i}}}{{\\Delta {t_{imp}}}} + \\sum \\limits _{k \\in Nf} {{{\\left( {{\\Lambda _c}} \\right)}_{ij}}} } \\right)I ,\\end{array} \\right.$ where $L\\left( i \\right)$ represents $k < i$ , and $U\\left( i \\right)$ represents $k > i$ .",
"${\\Lambda _c}$ has the form [16] ${\\left( {{\\Lambda _c}} \\right)_i} = \\sum \\limits _{k = 1}^{Nf} {\\left( {\\left| {{{\\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over V} }_i} \\cdot {{\\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over n} }_k\\right.}",
"\\right| + {C_i}} {S_k}$ , where ${\\mathord {{\\scriptscriptstyle \\rightharpoonup }}\\over V} _i$$ represents the macroscopic velocity vector in cell $ i$.",
"$$\\scriptscriptstyle \\rightharpoonup $ n k$ represents unit normal vector of the interface with area $ Sk$.",
"Eq.~(\\ref {Eq29}) is discretized and solved by using LU-SGS method.",
"Finally, the distribution function and macroscopic variables are updated in the same implicit time step.\\begin{equation}\\begin{aligned}{{\\tilde{\\phi }}^{n + 1}} &= {{\\tilde{\\phi }}^n} + \\Delta {{\\tilde{\\phi }}_{imp}},\\\\{W^{n + 1}} &= {W^n} + \\Delta {W_{imp}} .\\end{aligned}\\end{equation}$ In above procedure, the inner variable $\\xi $ is continuous.",
"But for $\\alpha $ -dimensional velocities $u$ , $v$ , velocity space is discretized.",
"In this paper, Gauss-Hermite quadrature formula [19] or Newton-Cotes formula [20] is applied in integration under discrete velocity space."
],
[
"Boundary condition",
"In the IDUGKS, ghost cells are employed along boundary of flow field.",
"In evaluation of explicit flux, variables in ghost cells are directly derived from corresponding inner cells as Ref.[12].",
"In prediction step of implicit scheme, to improve convergence efficiency, a governing equation of boundary is included.",
"Boundary condition for implicit iteration is implemented on linearized relation between inner and ghost cells [21].",
"For implicit flux term $M$ and variable $\\Pi $ (distribution function and macroscopic variables) to be updated, the linearized relation reads $\\Delta \\Pi _{ghost}^{n + 1} - \\left( {\\frac{{\\partial M}}{{\\partial \\Pi }}} \\right)_{inner}^n\\Delta \\Pi _{inner}^{n + 1} = 0 .$ On the solid wall, it is not so easy for distribution function to applied Eq.",
"(REF ).",
"In this paper, diffusive reflection boundary condition is applied [22].",
"$\\Delta f_{ghost}^{n + 1} = \\Delta {\\rho _{ghost}}{\\left( {\\frac{{{\\lambda _{wall}}}}{\\pi }} \\right)^{\\frac{{K + 2}}{2}}}{e^{ - {\\lambda _{wall}}\\left[ {{u^2} + {v^2} + {\\xi ^2}} \\right]}},$ where $\\Delta {\\rho _{ghost}} = \\frac{{ - \\int {\\frac{1}{2}\\left( {{u_n} + \\left| {{u_n}} \\right|} \\right)\\Delta {f_{inner}}dudvd\\xi } }}{{\\int {\\frac{1}{2}\\left( {{u_n} - \\left| {{u_n}} \\right|} \\right)} {{\\left( {\\frac{{{\\lambda _{wall}}}}{\\pi }} \\right)}^{\\frac{{K + 2}}{2}}}{e^{ - {\\lambda _{wall}}\\left[ {{u^2} + {v^2} + {\\xi ^2}} \\right]}}dudvd\\xi }} .$ At the beginning of computation at 0 time step, the forward sweep can begin with boundary condition $\\Delta W_{ghost}^{n + 1} = 0$ for macroscopic variables and $\\Delta f_{ghost}^{n + 1} = 0$ for microscopic distribution function.",
"This treatment will not cause any negative defect to accuracy and convergence.",
"The present IDUGKS will be validated by test cases in different flow regimes.",
"First, the test case of Couette flow is carried out to validate that the IDUGKS is a second-order accurate numerical method.",
"Then, the test cases of Lid-driven cavity flow in different Knudsen numbers are conducted to demonstrate that the IDUGKS is capable of simulating flows in all flow regime.",
"Finally, the hypersonic flow in transition flow regime is performed to show the IDUGKS can treat the flow with shock wave.",
"After comparison results in every case, the computational cost will be presented with residual curve.",
"In all test cases the implicit CFL number is chosen as ${10^3}$ .",
"Comparison of computational efficiency measured with wall clock time between implicit and explicit method will be given at the end of this session.",
"Some remarks will also be given."
],
[
"Couette flow",
"The Couette flow is driven by two parallel plates.",
"the top plate is moving in constant velocity and the other is static.",
"The Reynolds number is chosen as 100 and Mach number is $0.1\\sqrt{3} $ .",
"In discrete velocity space, $9 \\times 9$ discrete points are distributed uniformly in $\\left[ { - 2.5, 2.5} \\right] \\times \\left[ { - 2.5, 2.5} \\right]$ .",
"The velocity distributions along y-direction are plotted in Fig.",
"REF on uniform meshes within $\\left[ {0, 1} \\right] \\times \\left[ {0, 1} \\right]$ of $20 \\times 20$ , $40 \\times 40$ , $80 \\times 80$ , $160 \\times 160$ .",
"It can be seen that the numerical results are in excellent agreement with the analytical ones.",
"To test the convergence order of the IDUGKS, the results from simulations on different meshes have been used to compute the L2 errors in velocity field along y-direction.",
"The L2 error is defined by $E(U) = \\frac{{\\sqrt{\\sum _y {|U(y)-U_e(y)|^2}}}}{{\\sqrt{\\sum _y {U_e(y)^2}}}} .$ where $U_e$ is the analytical solution of Couette flow case.",
"The error in L2 norm with respect to mesh size is plotted in Fig.",
"REF , which shows a nearly second-order accuracy of the implicit scheme."
],
[
"Lid-driven cavity flows",
"The first case is incompressible lid-driven cavity flow at Reynolds number 1000 and Mach number $0.1\\sqrt{3} $ .",
"The uniform mesh with $257 \\times 257$ mesh points within $\\left[ {0, 1} \\right] \\times \\left[ {0, 1} \\right]$ is chosen as computational domain.",
"For discrete velocity space, $9 \\times 9$ discrete points are distributed uniformly in $\\left[ { - 2.5, 2.5} \\right] \\times \\left[ { - 2.5, 2.5} \\right]$ .",
"The streamlines are plotted in Fig.",
"REF and comparison result for V velocity profile along line $Y = 0.5$ is presented in Fig.",
"REF .",
"The V velocity profile obtained using IDUGKS is compared with that from explicit DUGKS and U. Ghia et al [23].",
"In continuum regime, the IDUGKS can obtain the same accuracy as the explicit method.",
"Both of them are in good agreement with the previous literature.",
"For a better demonstration for accuracy of the IDUGKS in description of incompressible flow compared to other numerical scheme, a list including the vorticity, the stream function, and $x$ , $y$ location of vortex center are grouped in Table REF .",
"In Table REF , besides results from [23], all the flow properties in incompressible cavity flow are also compared to results obtained from the lattice Boltzmann method (LBM) in work of Zhuo et al [24] and Hou et al [25].",
"Comparison show that the difference between present results and the benchmark results is less than $2\\% $ .",
"With a much larger CFL number than explicit method, IDUGKS requires much fewer iteration step to reach convergence.",
"The residual curves for comparison are plotted in Fig.",
"REF .",
"In this case, the explicit DUGKS requires 255000 iteration step to reach a residual of $5 \\times {10^{ - 8}}$ .",
"To obtain the same convergence criterion the IDUGKS only needs 2040 iteration step.",
"Now, we shift flow regime to slip flow regime.",
"The Knudsen number is set to be $0.075$ .",
"According to definition of Knudsen number $Kn = \\frac{\\bar{l}}{L_{ref}} ,$ where $\\bar{l}$ denotes the mean free path of particles, in this case, ${L_{ref}}$ is the reference length.",
"In the test cases of cavity flow the reference length is the side length of the cavity.",
"The uniform mesh with $60 \\times 60$ mesh cells within $\\left[ {0, 1} \\right] \\times \\left[ {0, 1} \\right]$ is chosen as computational domain.",
"For discrete velocity space, $60 \\times 60$ discrete points are distributed uniformly in $\\left[ { - 2.5, 2.5} \\right] \\times \\left[ { - 2.5, 2.5} \\right]$ .",
"In this case, the Gauss-Hermite quadrature formula is chosen for the integration of distribution function under discrete velocity space.",
"The horizontal velocity profile and vertical profile are given in Fig.",
"REF .",
"In Fig.",
"REF , the velocity profiles obtained from the IDUGKS are compared with UGKS and DSMC presented in Ref. [11].",
"The IDUGKS is able to reach the same accuracy as UGKS and particle based direct simulation Monte Carlo (DSMC).",
"In slip flow regime like this, there is slip velocity on solid wall as shown in Fig.",
"REF .",
"This phenomenon is different from continuum flow in which fluid does not have any motion on solid wall.",
"Existence of slip velocity on wall is consistence with results from DSMC.",
"The residual curves for comparison between the implicit and the explicit DUGKS are shown in Fig.",
"REF .",
"The convergence criterion for residual is also set to be $5 \\times 10^{-8}$ .",
"The explicit DUGKS requires 11900 iteration steps to obtain convergence while just 230 is needed for the IDUGKS.",
"Even though predicted iteration will cost some time, the implicit method still improve convergence efficiency for about 27 times.",
"Next, the lid-driven cavity flow under $Kn = 1.0$ is simulated.",
"This test is to validate the IDUGKS in simulation for rarefied flows.",
"In this paper, Knudsen number takes its effect in computation by relation with Reynolds number $Re$ and Mach number $Ma$ [26] $Kn = \\frac{Ma}{Re} \\sqrt{\\frac{\\gamma \\pi }{2}} ,$ where $\\gamma $ is ratio of specific heats.",
"The computational domain is also set to be $\\left[ {0, 1} \\right] \\times \\left[ {0, 1} \\right]$ and discretized with $61 \\times 61$ mesh points.",
"$60 \\times 60$ discrete velocity points are distributed in $\\left[ { - 2.5, 2.5} \\right] \\times \\left[ { - 2.5, 2.5} \\right]$ space.",
"In DUGKS, in order to avoid divergence, the error from discrete integration is not allow to be greater than $1\\%$ .",
"As a result, the Gauss-Hermite quadrature formula is applied.",
"The velocity profiles in X and Y direction are plotted in Fig.",
"REF and compared with results from Ref.",
"[11].",
"Results from the IDUGKS fit quite well with that from DSMC.",
"In rarefied flow simulation, the IDUGKS is still be able to capture physical details of gas flow.",
"Compared to particle based Lagrangian method, The DUGKS not only does not cause extra computational cost to trace the trajectory of particle motion, but also it can obtain the same accuracy.",
"The convergence history is presented with residual curve in Fig.",
"REF .",
"For the IDUGKS, 190 iteration steps are required while the explicit method uses 9600 iteration steps.",
"In rarefied flow, a much higher efficiency than explicit method is still obtained by the IDUGKS.",
"To verify the IDUGKS for all flow regimes, in final case of lid-driven cavity flow, test for $Kn = 1.0$ is simulated.",
"The physical space of $\\left[ {0, 1} \\right] \\times \\left[ {0, 1} \\right]$ discretized with $61 \\times 61$ mesh points and microscopic velocity space of $\\left[ { - 2.5, 2.5} \\right] \\times \\left[ { - 2.5, 2.5} \\right]$ with $60 \\times 60$ discrete velocity points are chosen.",
"In discrete integration, the Gauss-Hermite quadrature formula is used.",
"Flow structure in this case where $\\bar{l} \\gg L_{ref}$ is described with horizontal and vertical velocity profile along the central line.",
"In kinetic regime, solution of the IDUGKS nicely fits with the one from DSMC and UGKS in Ref.",
"[11].",
"Residual curves are plotted in Fig.",
"REF .",
"In all flow regimes, from continuum flow to kinetic regime, the DUGKS gives the same exact description for flow structure as UGKS and DSMC.",
"But its formulation is much more simple and easy to be implemented.",
"Implementation of implicit scheme greatly reduces iteration steps thus improves efficiency of the original explicit DUGKS.",
"The IDUGKS is a reliable and efficient method for simulation of multi-scale flows."
],
[
"Hypersonic rarefied flow around circular cylinder",
"In this case, supersonic flow passing through a circular cylinder with $Kn = 1.0$ and $Ma = 5.0$ is computed.",
"The physical space is discretized with 4800 mesh cells and microscopic velocity space of $\\left[ { - 15, 15} \\right] \\times \\left[ { - 15, 15} \\right]$ with $90 \\times 90$ discrete velocity points are chosen.",
"In discrete integration, the Gauss-Hermite quadrature formula is used.",
"In reconstruction at the cell interface, the Vankatakrishnan limiter is implemented.",
"The density contour and pressure contour are presented in Fig.",
"REF and Fig.",
"REF .",
"The pressure distribution on the surface of circular cylinder is extracted to compared with results in Ref.",
"[11].",
"Because the flow is symmetric, we just show the part within ${180^o}$ from the leading edge to the trailing edge.",
"Comparison results are presented in Fig.",
"REF .",
"Like lid-driven cavity flow, in rarefied regime, there is slip on solid wall.",
"Velocity vectors near circular cylinder are plotted in Fig.",
"REF .",
"The mach number contour is also presented in Fig.",
"REF .",
"Slip on surface of solid wall is demonstrated with Fig.",
"REF .",
"To give a quantitative description, the shear stress curve on surface of circular cylinder from leading edge to trailing edge is plotted and compared to results in previous literature [11] in Fig.",
"REF .",
"Heat flux distribution on the surface of circular cylinder is extracted to compared with results in Ref.",
"[11].",
"In this paper, BGK-Shakhov model is applied to capture heat transfer.",
"From leading edge to trailing edge, the heat flux curve fits pretty well with previous literature as shown in Fig.",
"REF .",
"Since about ${50^o}$ clockwise from leading edge, the shear stress begins to drop.",
"This result is the same as previous work based on UGKS and DSMC.",
"Residual curves of explicit and implicit DUGKS are plotted in Fig.",
"REF for comparison.",
"For a suitable growth rate of mesh around shock wave, the adaptive mesh refinement (AMR) is applied in this case.",
"The initial mesh with 2400 mesh cells is shown in Fig.",
"REF .",
"Mesh around shock wave is refined using gradient and vorticity criteria.",
"The final hybrid mesh with 4363 cells and pressure contour captured by it are presented in Fig.",
"REF and Fig.",
"REF .",
"In order to further explore the effects of unstructured hybrid mesh on accuracy and convergence, computation results and residual curve under refined mesh are compared with under structured mesh in this paper.",
"Comparisons for pressure, shear stress and heat flux distribution on the surface of circular cylinder on different meshes are presented in Fig.",
"REF , Fig.",
"REF and Fig.",
"REF .",
"To obtain the same accuracy, the number of cells used in refined mesh is fewer than structured grid.",
"For the present two simulations with different kinds of meshes, the residual curves are plotted in Fig.REF .",
"It can be observed that, the two simulations with structured and unstructured refined meshes reach the steady state quickly and only take 2040 and 1750 iteration steps respectively.",
"In all cases, the implicit DUGKS requires much fewer iteration steps than explicit DUGKS.",
"To measure quantitatively the computational efficiency of numerical scheme proposed in this paper, comparison for computational cost between the IDUGKS and the explicit DUGKS is shown in Table REF by wall clock time.",
"For cavity flow, acceleration rates are $89.9$ , $27.4$ , $26.8$ , $28.5$ respectively.",
"In hypersonic circular cylinder case, the acceleration rate becomes $30.7$ .",
"In all, the present IDUGKS solver is applicable for flows from continuum regime to kinetic regime, from incompressible to hypersonic flows.",
"Compared to explicit DUGKS, much lower computation time is required.",
"The IDUGKS proposed in this paper shows excellent accuracy and efficiency."
],
[
"Conclusions",
"In this paper, an implicit DUGKS is constructed for all Knudsen number flows.",
"The physics represented in DUGKS depends on coupling of particle transport and collision in flux evaluation at cell interface.",
"DUGKS can obtain accurate solution for multi-scale flow problems.",
"Since the application of variation of ratio between explicit time step and particle collision time makes DUGKS an asymptotic preserving method, a physical and a numerical time steps are used in the IDUGKS.",
"In this paper, the coupled implicit macroscopic and microscopic iterative equations are applied, which increases the computation efficiency.",
"Two iterative methods, LU-SGS method and point relaxation scheme, are implemented in prediction step.",
"Comparison results for computational cost with explicit DUGKS proves a much better efficiency of the IDUGKS for steady state solution.",
"In test cases, the scheme proposed in this paper is proved to have the same accuracy as DSMC.",
"The present IDUGKS solver can be easily extended to the 3D case.",
"Many technique for improvement, such as parallelization, immersed boundary (IB) method, can be developed.",
"The IDUGKS constructed in this work is promising in its extensive applications."
],
[
"Acknowledgements",
"The work has been supported by the National Natural Science Foundation of China (Grant No.",
"11472219), the 111 Project of China (B17037), as well as the ATCFD Project (2015-F-016)."
]
] | 1709.01819 |
[
[
"A power structure over the Grothendieck ring of geometric dg categories"
],
[
"Abstract We prove the existence of a power structure over the Grothendieck ring of geometric dg categories.",
"We show that a conjecture by Galkin and Shinder (proved recently by Bergh, Gorchinskiy, Larsen, and Lunts) relating the motivic and categorical zeta functions of varieties can be reformulated as a compatibility between the motivic and categorical power structures.",
"Using our power structure we show that the categorical zeta function of a geometric dg category can be expressed as a power with exponent the category itself.",
"We give applications of our results for the generating series associated with Hilbert schemes of points, categorical Adams operations, and series with exponent a linear algebraic group."
],
[
"Introduction",
"A power structure over the Grothendieck ring of varieties $K_0(Var)$ was defined in [17].",
"It has turned out to be an effective tool in expressing certain generating functions associated to varieties, see e.g.",
"[18].",
"Power structures have a deep connection with $\\lambda $ -ring structures.",
"$K_0(Var)$ has a well-known $\\lambda $ -ring structure induced by the symmetric powers of variaties.",
"These imply that the motivic zeta function of a variety has a particularly nice expression as a power with exponent the motive of the variety.",
"The derived category of coherent sheaves on a variety has been proposed as an analogue of the motive of a variety for a long time now [6].",
"In this analogy, semiorthogonal decompositions are the tools for the simplification of this “motive” similar to splitting by projectors in Grothendieck motivic theory.",
"It has since then turned out that it is better to work with dg enhancements of these triangulated categories.",
"In [10] it was shown that there is a motivic measure, i.e.",
"a ring homomorphism $ \\mu \\colon K_0(Var) \\rightarrow K_0(gdg-cat)$ to a certain ring $K_0(gdg-cat)$ , which is the Grothendieck ring of geometric dg categories.",
"On the other hand, symmetric powers in $K_0(gdg-cat)$ do not induce a $\\lambda $ -ring structure, but instead a $2-\\lambda $ -ring structure [16].",
"For example, in a $\\lambda $ -ring $\\mathrm {Sym}^n(1)=1$ for any integer $n$ , whereas in a $2-\\lambda $ -ring $\\mathrm {Sym}^n(1)=p(n)$ , the number of partitions of $n$ .",
"A natural question is: does the $2-\\lambda $ -ring structure on $K_0(gdg-cat)$ imply the existence of a power structure on it?",
"Our first main result is that this is indeed the case.",
"Theorem 1.1 There exists an effective power structure over the Grothendieck ring $K_0(gdg-cat)$ of geometric dg categories.",
"This power structure is defined as a categorical analogue of the power structure on $K_0(Var)$ , and shares many nice properties of it.",
"One of the difficulties with defining this power structure is that, contrary to a the case of varieties, the power of a dg category does not decompose trivially to strata indexed stabilizers under the action of the symmetric group.",
"In particular, this makes it difficult to understand $\\mu (X^n \\setminus \\Delta )$ , where $X$ is a variety and $\\Delta \\subset X^n$ is the big diagonal in the product where at least two coordinates coincide.",
"Our solution is to use the destackification method developed in [1], which realizes $\\mu (X^n \\setminus \\Delta )$ as a semiorthogonal summand in a larger category.",
"The categorical zeta function of an object in $K_0(gdg-cat)$ collects the symmetric powers of the object into a generating function.",
"The ring homomorphism $\\mu $ turns out not to be a $\\lambda $ -ring homomorphism, and this implies that the expression of the categorical zeta function associated to a variety as a power is different from that of the motivic zeta function.",
"Our second main result is that using our power structure on $K_0(gdg-cat)$ , the categorical zeta function of a geometric dg category $\\mathcal {M}$ can be expressed as a power with exponent the category itself.",
"Theorem 1.2 $Z_{cat}(\\mathcal {M},t)=\\prod _{n=1}^{\\infty }\\left( \\frac{1}{1-q^n}\\right)^{[\\mathcal {M}]}.$ Galkin and Shinder in [15] proved a relation between the motivic and the caregorical zeta functions of varieties of dimensions 1 and 2, and they conjectured the same relationship to hold in any dimension.",
"As an immediate corollary of Theorem REF we get Corollary 1.3 The Galkin-Shinder conjecture is true.",
"Namely, for any variety $X$ $ Z_{cat}(\\mu (X),t)=\\prod _{n \\ge 1} \\mu (Z_{mot}(X,t^n)).",
"$ This result implies also a formula for the generating series of the classes of dg categories of the Hilbert scheme of points associated to a fixed smooth projective variety.",
"Furthermore, our result also implies that the dg version of the categorical Heisenberg action of [22] on the derived category of the symmetric orbifold is an irreducible highest weight representation.",
"As the final draft of this paper was being prepared, the preprint [3] of Bergh, Gorchinskiy, Larsen, and Lunts appeared on the arXiv also proving the conjecture of Galkin and Shinder.",
"Our proof seems to be different than theirs although it also uses Bergh’s destackification method as well as several results from [4], [2].",
"The structure of the paper is the following.",
"In Section 2 we summarize the necessary notions and results about pretriangulated dg categories and actions of finite groups on them.",
"In Section 3 we discuss semiorthogonal decomposition and specialize to the case of geometric dg categories.",
"In Section 4, after recalling the definition of a power structure and the main properties of destackifications, we prove Theorem REF .",
"In Section 5 we prove Theorem REF and Corollary REF .",
"The implications for Hilbert scheme of points and for the Heisenberg action on the derived category of the symmetric orbifold are given in Section 6."
],
[
"Acknowledgement",
"The author would like to thank to Jim Bryan, Sabin Cautis, Eugene Gorsky, Sándor Kovács and Roberto Pirisi for helpful comments and discussions."
],
[
"Pretriangulated dg categories",
"The usefulness of dg enhancements of triangulated categories was first observed in [7].",
"We give here a brief summary of these.",
"During the whole paper we fix an algebraically closed field $k$ .",
"All categories and functors that we will consider are assumed to be $k$ -linear.",
"A dg category is a linear category $\\mathcal {V}$ enriched in the monoidal category of complexes of $k$ -vector spaces.",
"This means that for any $A,B \\in \\mathrm {Ob}(\\mathcal {V})$ there is given a complex $\\mathrm {Hom}^{\\bullet }_{\\mathcal {V}}(A,B)$ .",
"We denote by $H^\\bullet (\\mathcal {V})$ the $k$ -linear category with the same objects as $\\mathcal {V}$ and $ \\mathrm {Hom}_{H^\\bullet (\\mathcal {V})}(A,B)= \\bigoplus _i H^i \\mathrm {Hom}_{\\mathcal {V}}(A,B).",
"$ $H^\\bullet (\\mathcal {V})$ is called the graded homotopy category of $\\mathcal {V}$ .",
"Restricting to the 0-th cohomology of the $\\mathrm {Hom}$ complexes one gets the homotopy category $H^0(\\mathcal {V})$ .",
"The tensor product of dg categories $\\mathcal {V}$ and $\\mathcal {W}$ is defined as follows.",
"$\\mathrm {Ob}(\\mathcal {V}\\boxtimes \\mathcal {W}) = \\mathrm {Ob}(\\mathcal {V}) \\times \\mathrm {Ob}(\\mathcal {W})$ ; for $A \\in \\mathrm {Ob}(\\mathcal {V})$ and $B \\in \\mathrm {Ob}(\\mathcal {W})$ the corresponding object is denoted as $A \\otimes B$ ; $\\operatorname{Hom}(A \\otimes B, A^{\\prime } \\otimes B^{\\prime })=\\operatorname{Hom}(A,A^{\\prime }) \\otimes \\operatorname{Hom}(B,B^{\\prime })$ and the composition map is defined by $(f_1\\otimes g_1)(f_2\\otimes g_2)=(-1)^{\\mathrm {deg}(g_1)}\\mathrm {deg}(f_2)f_1f_1 \\otimes g_1 g_2$ .",
"A dg functor between dg categories is a functor preserving the enrichment, i.e.",
"inducing morphisms of Hom-complexes.",
"A dg functor $F \\;:\\; \\mathcal {V}\\rightarrow \\mathcal {W}$ is called a quasi-equivalence if $H^{\\bullet }(F) \\;:\\; H^{\\bullet }(\\mathcal {V}) \\rightarrow H^{\\bullet }(\\mathcal {W})$ is an equivalence of categories.",
"Given dg categories $\\mathcal {V}$ and $\\mathcal {W}$ the set of covariant dg functors $\\mathcal {V}\\rightarrow \\mathcal {W}$ form the objects of a dg category $\\mathrm {Fun}_{dg}(\\mathcal {V},\\mathcal {W})$ .",
"Let $DG(k)=C(k-\\mathrm {mod})$ be the dg category of complexes of $k$ -vector spaces, or dg $k$ -modules.",
"We denote the dg category $\\mathrm {Fun}_{dg}(\\mathcal {V}^{op},DG(k))$ by $\\mathrm {mod}-\\mathcal {V}$ and call it the category of dg $\\mathcal {V}$ -modules.",
"This is a dg category with shift and cone functors, which are inherited from $DG(k)$ .",
"The Yoneda-embedding realizes the original dg category $\\mathcal {V}$ as s full dg subcategory of $\\mathrm {mod}-\\mathcal {V}$ .",
"The pretriangulated hull $\\mathrm {Pre-Tr}(\\mathcal {V})$ of $\\mathcal {V}$ is the smallest full dg subcategory of $\\mathrm {mod}-\\mathcal {V}$ that contains $\\mathcal {V}$ and is closed under isomorphisms, direct sums, shifts, and cones.",
"The perfect hull $\\mathrm {Perf}(\\mathcal {V})$ of $\\mathcal {V}$ is the full dg subcategory of $\\mathrm {mod}-\\mathcal {V}$ consisting of semi-free dg modules that are homotopy equivalent to a direct summand of an object of the category $\\mathrm {Pre-Tr}(\\mathcal {V})$ .",
"The homotopy category $H^0(\\mathrm {Pre-Tr}(\\mathcal {V}))$ is denoted as $\\mathrm {Tr}(\\mathcal {V})$ .",
"A dg category $\\mathcal {V}$ is called pretriangulated, if the embedding $\\mathcal {V}\\rightarrow \\mathrm {Pre-Tr}(\\mathcal {V})$ is a quasi-equivalence.",
"A dg category $\\mathcal {V}$ is called strongly pretriangulated, if the embedding $\\mathcal {V}\\rightarrow \\mathrm {Pre-Tr}(\\mathcal {V})$ is a dg equivalence.",
"A dg category is called perfect, if $\\mathcal {V}\\rightarrow \\mathrm {Perf}(\\mathcal {V})$ is a quasi-equivalence.",
"If $\\mathcal {V}$ is a pretriangulated category, then $H^{0}(\\mathcal {V})$ is naturally a triangulated category.",
"Given a triangulated category $\\mathcal {T}$ , an enhancement of $\\mathcal {T}$ is a pretriangulated category $\\mathcal {V}$ with an equivalence $\\epsilon \\colon H^{0}(\\mathcal {V}) \\rightarrow \\mathcal {T}$ .",
"If $(\\mathcal {V},\\epsilon )$ is an enhancement of a triangulated category $\\mathcal {T}$ then any strict full triangulated subcategory $\\mathcal {S}\\subset \\mathcal {T}$ has an enhancement given by the full dg subcategory of $\\mathcal {V}$ of objects that go to objects of $\\mathcal {S}$ under $\\epsilon $ together with the restriction of $\\epsilon $ to the 0-th cohomology of this subcategory.",
"Example 2.1 If $\\mathcal {V}$ is any dg category, then $\\mathrm {Pre-Tr}(\\mathcal {V})$ is pretriangulated, and hence $\\mathrm {Tr}(\\mathcal {V})$ is triangulated.",
"If $\\mathcal {A}$ is any $k$ -linear abelian category, then $C(\\mathcal {A})$ , the category of complexes over $\\mathcal {A}$ is pretriangulated.",
"Example 2.2 Let $X$ be a scheme of finite type over $k$ .",
"Let $\\mathcal {O}_X-\\mathrm {mod}$ be the abelian category of all sheaves of $\\mathcal {O}_X$ -modules.",
"Let $I(\\mathcal {O}_X-\\mathrm {mod})$ be the full dg subcategory of $C(\\mathcal {O}_X-\\mathrm {mod})$ consisting of h-injective complexes of injective objects.",
"Then $I(\\mathcal {O}_X-\\mathrm {mod})$ is pretriangulated, and the composition $\\epsilon _{I(\\mathcal {O}_X-\\mathrm {mod})} \\colon H^0(I(\\mathcal {O}_X-\\mathrm {mod})) \\rightarrow H^0(C(\\mathcal {O}_X-\\mathrm {mod})) \\rightarrow D(\\mathcal {O}_X-\\mathrm {mod})$ is an equivalence [20].",
"Therefore, $I(\\mathcal {O}_X-\\mathrm {mod})$ is a dg enhancement of $D(\\mathcal {O}_X-\\mathrm {mod})$ .",
"It is known that the subcategory $D_{qc}(X) \\subset D(\\mathcal {O}_X-\\mathrm {mod})$ of complexes with quasi-coherent cohomology is thick.",
"In fact, $D_{qc}(X)$ is known to be equivalent to $D(QCoh(X))$ , where $QCoh(X)$ is the category of quasi-coherent $\\mathcal {O}_X$ -modules.",
"Denote by $I_{qc}(X) \\subset I(\\mathcal {O}_X-\\mathrm {mod})$ the appropriate enhancement of $D_{qc}(X)$ .",
"Similarly, the subcategory $D_{pf}(X) \\subset D_{qc}(X)$ consisting of perfect complexes is thick.",
"Its enhancement is $I_{pf}(X) \\subset I_{qc}(X)$ .",
"When $X$ is smooth, then $D_{pf}(X)=D^b(Coh(X))$ , where $Coh(X)$ is the category of coherent $\\mathcal {O}_X$ -modules.",
"Example 2.3 Throughout the paper we will need to work with stacks and derived categories associated to them.",
"We will restrict our attention to algebraic stacks of Deligne-Mumford type.",
"Most importantly, we will investigate quotient stacks of the form $\\mathcal {X}=[M/G]$ , where $M$ is a scheme and $G$ is a finite group acting on it.",
"In this case a sheaf on $\\mathcal {X}$ is the same as a $G$ -equivariant sheaf on $M$ .",
"The derived categories $D_{qc}(\\mathcal {X})$ and $D_{pf}(\\mathcal {X})$ can be defined analogously to Example REF .",
"These have the enhancements $I_{qc}(\\mathcal {X})$ and $I_{pf}(\\mathcal {X})$ respectively.",
"A good summary about the details of these constructions can be found in [4].",
"Remark 2.4 In the rest of the paper, except when noted otherwise, we will use the perfect version of the derived categories and their dg enhancements both for schemes and for DM stacks.",
"To ease the notation, we will drop the subscript and write only $D(X)=D_{pf}(X)$ and $I(X)=I_{pf}(X)$ (resp., $D(\\mathcal {X})=D_{pf}(\\mathcal {X})$ and $I(\\mathcal {X})=I_{pf}(\\mathcal {X})$ ).",
"The main advantage of working with dg enhancements is that there is a very-well behaving product for them.",
"Definition 2.5 The completed tensor product of two pretriangulated categories $\\mathcal {V}$ and $\\mathcal {W}$ is defined by $ \\mathcal {V}\\;\\widehat{\\boxtimes }\\;\\mathcal {W}= \\mathrm {Perf}(\\mathcal {V}\\boxtimes \\mathcal {W}) $ Again, to ease the notation, we will drop the hat from the notation of the completed tensor product and just write $\\mathcal {V}\\boxtimes \\mathcal {W}$ .",
"Example 2.6 By [10], if $X_1,\\dots ,X_n$ are smooth projective varieties, then $I(X_1) \\boxtimes \\dots \\boxtimes I(X_n)$ is quasi-isomorphic to $I(X_1 \\times \\dots \\times X_n)$ .",
"With exactly the same proof as in [10] one can get the analogous result for DM stacks."
],
[
"2-representations",
"Suppose that $G$ is a finite group such that $|G|$ is invertible in $k$ .",
"A 2-representation of $G$ on a linear category $\\mathcal {V}$ consists of the following data: for each element $g \\in G$ , a linear functor $\\rho (g) \\colon \\mathcal {V}\\rightarrow \\mathcal {V}$ ; for any pair of elements $(g,h)$ of a $G$ an isomorphism of functors; $ \\phi _{g,h}\\colon (\\rho (g) \\circ \\rho (h)) 0359{\\;\\;\\;\\smash{}\\;\\;\\;}{\\cong }{} \\rho (gh) $ an isomorphism of functors $ \\phi _1 \\colon \\rho (1) 0359{\\;\\;\\;\\smash{}\\;\\;\\;}{\\cong }{} \\mathrm {Id}_{\\mathcal {V}};$ such that the following conditions hold: for any $g,h,k \\in G$ we have $ \\phi _{(gh,k)}(\\phi _{g,h} \\circ \\rho (k))=\\phi _{g,hk}(\\rho (g) \\circ \\phi _{h,k}) ;$ $\\phi _{1,g}=\\phi _1 \\circ \\rho (g) \\quad \\textrm {and} \\quad \\phi _{g,1}=\\rho (g) \\circ \\phi _1.$ Definition 2.7 Suppose we are given a 2-representation of $G$ on $\\mathcal {V}$ .",
"A $G$ -equivariant object of $\\mathcal {V}$ is a pair $(A, (\\epsilon _g)_{g \\in G})$ , where $A \\in \\mathrm {Ob}(\\mathcal {V})$ and $(\\epsilon _g)_{g \\in G}$ is a family isomorphisms $ \\epsilon _g\\colon A \\rightarrow \\rho (g)A, $ satisfying the following compatibility conditions: for $g=1$ , $\\epsilon _1= \\phi ^{-1}_{1,A}\\colon A \\mapsto \\rho (1)A;$ for any $g,h \\in G$ the diagram $\\begin{array}{ccc}A & \\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& \\rho (g)(A) \\\\\\bigg \\downarrow \\unknown.",
"{\\scriptstyle {\\epsilon _{gh}}} & & \\bigg \\downarrow \\unknown.",
"{\\scriptstyle {\\rho (g)(\\epsilon _h)}} \\\\\\rho (gh)(A) & \\hspace{-6.375pt}\\xleftarrow{}\\phantom{}\\hspace{-6.375pt}& \\rho (g)(\\rho (h)(A))\\end{array}$ is commutative.",
"A morphism of equivariant objects from $(A, (\\epsilon _g)_{g \\in G})$ and $(B, (\\eta _g)_{g \\in G})$ is a morphism $f\\colon A \\rightarrow B$ compatible with the $G$ -action.",
"That is, a morphism for which all the following diagram is commutative: $\\begin{array}{ccc}A & \\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& \\rho (g)(A) \\\\\\bigg \\downarrow \\unknown.",
"{\\scriptstyle {f}} & & \\bigg \\downarrow \\unknown.",
"{\\scriptstyle {\\rho (g)(f)}} \\\\B & \\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& \\rho (g)(B).\\end{array}$ The category of $G$ -equivariant objects in $\\mathcal {V}$ is denoted as $\\mathcal {V}^G$ .",
"For a pretriangulated category $\\mathcal {V}$ with a $G$ -action the category $\\mathcal {V}^G$ is not necessarily pretriangulated.",
"But for a strongly pretriangulated category $\\mathcal {V}$ , the category $\\mathcal {V}^G$ is always strongly pretriangulated [29].",
"This is the case for the category $I(X)$ associated to scheme.",
"Example 2.8 One can equip every linear category $\\mathcal {V}$ with the trivial $G$ -action.",
"That is, all $\\rho (g)$ and $\\phi _{g,h}$ , as well as $\\phi _1$ are defined to be the identities.",
"In this case a $G$ -equivariant object in $\\mathcal {V}$ is the same as a representation of $G$ in $\\mathcal {V}$ , i.e.",
"an object $V \\in \\mathcal {V}$ and a homomorphism $G \\rightarrow \\operatorname{Aut}_{\\mathcal {V}}(V)$ .",
"Example 2.9 Let $X$ be a scheme of finite type over $k$ equipped with an action of $G$ .",
"Let $I_G(X)$ be the dg category of equivariant objects in $I(X)$ .",
"Then $I_G(X)$ is pretriangulated, and the composition $\\epsilon _{I_G(X)} \\colon H^0(I_G(X)) \\rightarrow H^0(C_G(\\mathcal {O}_X-\\mathrm {mod})) \\rightarrow D_G(X)$ is an equivalence.",
"It is known, that $D_G(X)=D([X/G])$ .",
"Due to [12], $I_G(X)=(I(X))^G$ , the category of $G$ -equivariant objects in $I(X)$ ."
],
[
"Symmetric powers",
"Let $\\mathcal {V}$ be a pretriangulated category.",
"The $n$ -th tensor power of $\\mathcal {V}$ is $\\mathcal {V}^{\\boxtimes n}=\\mathcal {V}\\boxtimes \\cdots \\boxtimes \\mathcal {V}$ where there are $n$ factors in the tensor product.",
"The symmetric group $S_n$ acts on $\\mathcal {V}^{\\boxtimes n}$ by the transformation $ \\sigma (A_1 \\otimes \\cdots \\otimes A_n)=A_{\\sigma ^{-1}(1)} \\otimes \\cdots \\otimes A_{\\sigma ^{-1}(n)},$ on the objects, and similarly on the $\\mathrm {Hom}$ complexes.",
"The $n$ -th symmetric power $\\operatorname{Sym}^n(\\mathcal {V})$ is defined in [16] as the category of $S_n$ -equivariant objects in $\\mathcal {V}^{\\boxtimes n}$ : $\\operatorname{Sym}^n(\\mathcal {V})=(\\mathcal {V}^{\\boxtimes n})^{S_n}.$ Example 2.10 If $X$ is a smooth projective variety, then $\\operatorname{Sym}^n(I(X))$ is the dg category of $S_n$ -equivariant complexes of injective $\\mathcal {O}$ -modules on $X^n$ which are bounded below and have bounded coherent cohomology.",
"$H^0(\\operatorname{Sym}^n(I(X)))=D([X^n/S^n])=D_{S_n}(X^n)$ is the (bounded) derived category of $S_n$ -equivariant complexes of coherent sheaves on $X^n$ ."
],
[
"Semiorthogonal decomposition of triangulated categories",
"Let $\\mathcal {T}$ be a triangulated category.",
"The triangulated envelope of a class of objects $\\mathcal {E}=(E_i)_{i \\in I}$ is the smallest strictly full triangulated subcategory of $\\mathcal {T}$ that contains $\\mathcal {E}$ .",
"The right orthogonal $\\mathcal {E}^\\perp $ is the full subcategory of $\\mathcal {T}$ whose objects $A$ have the property $\\mathrm {Hom}(E_i[n],A)=0$ for all $i$ and $n$ .",
"Similarly, the left orthogonal ${}^\\perp \\mathcal {E}$ is the full subcategory of $\\mathcal {T}$ whose objects $A$ have the property $\\mathrm {Hom}(A,E_i[n])=0$ for all $i$ and $n$ .",
"Let $\\mathcal {A}\\subset \\mathcal {T}$ be a strictly full triangulated subcategory.",
"$\\mathcal {A}$ is called right admissible (resp.",
"left admissible) if for every $A \\in \\mathcal {T}$ there exists an exact triangle $A_{\\mathcal {A}} \\rightarrow A \\rightarrow A_{\\mathcal {A}^\\perp }$ (resp.",
"$A_{{}^{\\perp }\\mathcal {A}} \\rightarrow A \\rightarrow A_{\\mathcal {A}}$ ) with $A_{\\mathcal {A}} \\in {\\mathcal {A}}$ and $A_{\\mathcal {A}^{\\perp }} \\in {\\mathcal {A}^{\\perp }}$ (resp.",
"$A_{{}^{\\perp }\\mathcal {A}} \\in {{}^{\\perp }\\mathcal {A}}$ ).",
"A subcategory is called admissible if it is both left and right admissible.",
"Proposition 3.1 ([10]) Let $\\mathcal {T}$ be a triangulated category, $\\mathcal {A}\\subset \\mathcal {T}$ a strictly full triangulated subcategory.",
"The following conditions are equivalent: $\\mathcal {A}$ is right (resp.",
"left) admissible in $\\mathcal {T}$ ; the embedding functor $i \\colon \\mathcal {A}\\rightarrow \\mathcal {T}$ has a right (resp.",
"left) adjoint $i^{!",
"}$ (resp.",
"$i^{\\ast }$ ).",
"If these hold, then the compositions $i^{!}",
"\\cdot i$ and $i^{\\ast } \\cdot i$ are isomorphic to the identity functor on $\\mathcal {A}$ .",
"A semiorthogonal sequence in $\\mathcal {T}$ is a sequence of admissible triangulated subcategories $\\mathcal {A}_1,\\dots ,\\mathcal {A}_n$ of $\\mathcal {T}$ such that $\\mathcal {A}_j \\subset {}^{\\perp }\\mathcal {A}_i$ for $j > i$ .",
"In addition, a semiorthogonal sequence is said to be full if it generates $\\mathcal {T}$ .",
"In this case we call such a sequence a semiorthogonal decomposition of $\\mathcal {T}$ and denote this as $ \\mathcal {T}= \\langle \\mathcal {A}_1,\\dots ,\\mathcal {A}_n \\rangle .$ Lemma 3.2 ([8]) If $\\mathcal {T}=\\langle \\mathcal {A}, \\mathcal {B}\\rangle $ is a semiorthogonal decomposition, then $\\mathcal {A}$ is left admissible and $\\mathcal {B}$ is right admissible.",
"Conversely, if $\\mathcal {A}\\subset \\mathcal {T}$ is left admissible, then $\\mathcal {T}=\\langle \\mathcal {A}, {{}^{\\perp }\\mathcal {A}} \\rangle $ is a semiorthogonal decomposition, and if $\\mathcal {B}\\subset \\mathcal {T}$ is right admissible, then $\\mathcal {T}=\\langle \\mathcal {B}^{\\perp }, \\mathcal {B}\\rangle $ is a semiorthogonal decomposition.",
"There are some cases when a semiorthogonal decomposition is induced automatically on the category of equivariant objects.",
"Proposition 3.3 ([24]) Let $\\mathcal {T}$ be a triangulated category with a trivial action of a finite group $G$ .",
"If $\\mathcal {T}^G$ is also triangulated, then there is a completely orthogonal decomposition $ \\mathcal {T}^G=\\langle \\mathcal {T}\\otimes V_0, \\mathcal {T}\\otimes V_1, \\dots , \\mathcal {T}\\otimes V_n \\rangle , $ where $V_0,\\dots ,V_n$ is a list of the finite-dimensional irreducible representations of $G$ .",
"Let $G$ be a finite abelian group which acts on a Karoubian linear category $\\mathcal {C}$ .",
"Let $G^{\\vee }=\\operatorname{Hom}(G,k^{\\ast })$ be the dual group to $G$ .",
"$G^{\\vee }$ acts on the category $\\mathcal {C}^G$ by twisting: for any $\\chi \\in G^\\vee $ let $ \\rho (\\chi ) ( (A,(\\epsilon _g)_{g\\in G}) )= (A,(\\epsilon _g)_{g\\in G}) \\otimes \\chi = (A,(\\epsilon _g \\cdot \\chi (g) )_{g\\in G}).",
"$ For $\\chi ,\\psi \\in G^\\vee $ , the equivariant objects $\\rho (\\chi ) \\left(\\rho (\\psi ) ( (A,(\\epsilon _g)_{g\\in G}) )\\right) $ and $\\rho (\\chi \\psi ) ( (A,(\\epsilon _g)_{g\\in G}) )$ are the same.",
"Let the isomorphisms $ \\phi _{\\chi ,\\psi }\\colon (\\rho (\\chi ) \\circ \\rho (\\psi )) 0359{\\;\\;\\;\\smash{}\\;\\;\\;}{\\cong }{} \\rho (\\chi \\psi ) $ be the identities.",
"Theorem 3.4 ([14]) Let $G$ be a finite abelian group.",
"Suppose that $\\mathcal {C}$ is a Karoubian linear category and $G$ acts on $\\mathcal {C}$ .",
"Then $ (\\mathcal {C}^G)^{G^\\vee } \\cong \\mathcal {C}.$ The following is a very important descent result for semiorthogonal decompositions.",
"Theorem 3.5 ([13]) Let $X$ be a quasi-projective variety, $G$ a finite group acting on $X$ , and let $p_2$ and $a$ be the projection and action morphisms from $G\\times X$ to $X$ .",
"Suppose a semiorthogonal decomposition $ D(X) = \\langle \\mathcal {A}_1,\\dots ,\\mathcal {A}_n \\rangle $ is given such that for any $1 \\le i < j \\le n$ and $A_i \\in \\mathcal {A}_i $ , $F_j \\in \\mathcal {A}_j $ one has $\\mathrm {Hom}(p_2^{\\ast } F_j, a^{\\ast } F_i)=0$ .",
"Let $\\mathcal {B}_i= \\lbrace (A, (\\epsilon _g)_{g \\in G}) \\in \\mathcal {T}^G\\;|\\; A \\in \\mathcal {A}_i \\rbrace $ be the full subcategory in $\\mathcal {T}^G$ consisting of objects $(A, (\\epsilon _g)_{g \\in G})$ such that $A \\in \\mathcal {A}_i$ .",
"Then there is a semiorthogonal decomposition $ D(X)^G= \\langle \\mathcal {B}_1,\\dots ,\\mathcal {B}_n \\rangle .$ The same result is true when $X$ is replaced by a gerbe over a quasi-projective variety.",
"Version (2) is not mentioned in [13] but the proof therein gives this stronger version as well.",
"For a scheme $X$ and a closed subscheme $Z \\subset X$ we denote by $D_Z(X)$ the full subcategory of $D(X)$ of complexes whose cohomology sheaves have their support contained in Z.",
"Proposition 3.6 Let $X$ be a quasi-projective variety, $Z \\subset X$ be a closed subvariety and $j\\colon U=X \\setminus Z \\hookrightarrow X$ be the embedding of the complement of $Z$ .",
"Suppose that $D_Z(X) \\subset D(X)$ is a right admissible subcategory.",
"Then there is a semiorthogonal decomposition $ D(X)=\\langle D_Z(X)^\\perp , D_Z(X) \\rangle ; $ there exists a functor $j_{\\ast }\\colon D(U)\\rightarrow D(X)$ right adjoint to $j^{\\ast }\\colon D(X)\\rightarrow D(U)$ ; moreover, $D_Z(X)^\\perp = j_{\\ast }D(U)$ .",
"(1) follows from Lemma REF .",
"To prove (2) and (3) we have to work with the quasi-coherent categories $D_{qc}(X)$ and $D_{qc}(U)$ .",
"It follows from (1) and [13] that there also exists a semiorthogonal decomposition $ D_{qc}(X)=\\langle D_{qc,Z}(X)^\\perp , D_{qc,Z}(X) \\rangle , $ such that $D_{Z}(X)=D_{qc,Z}(X)\\cap D(X)$ , $D_{Z}(X)^\\perp =D_{qc,Z}(X)^\\perp \\cap D(X)$ .",
"Moreover, in the quasi-coherent case there always exists a right adjoint $j_{\\ast }\\colon D_{qc}(U) \\rightarrow D_{qc}(X)$ of the functor $j^{\\ast }\\colon D_{qc}(X) \\rightarrow D_{qc}(U)$ .",
"This functor has $D_{qc,Z}(X)$ as its kernel [28].",
"By adjunction, for any two objects $A \\in D_{qc,Z}(X)$ , $B \\in D_{qc}(U)$ , $ \\operatorname{Hom}(A, j_{\\ast } B)=\\operatorname{Hom}(j^{\\ast }A,B)=0, $ That is, $j_{\\ast }D_{qc}(U) \\subseteq D_{qc,Z}(X)^\\perp $ .",
"Let $F \\in D_{qc,Z}(X)^\\perp $ .",
"Consider the natural morphism $F \\rightarrow j_{\\ast }j^{\\ast }F$ .",
"This is an isomorphism on $U$ .",
"Therefore, the cone $M$ of this morphism is supported set theoretically on $Z$ .",
"It follows that each cohomology sheaf of $M$ is supported set theoretically on $Z$ , i.e.",
"$M \\in D_{qc,Z}(X)$ .",
"Then, we have the following exact triangle in $D_{qc}(X)$ : $ M[-1] \\rightarrow F \\rightarrow j_{\\ast }j^{\\ast }F \\rightarrow M.$ But since $M[-1] \\in D_{qc,Z}(X)$ and $F \\in D_{qc,Z}(X)^\\perp $ the first morphism can only be 0.",
"In particular, $j_{\\ast }j^{\\ast }F=F \\oplus M$ .",
"Composing $j_{\\ast }j^{\\ast }$ with the projection onto the first component yields to the identity.",
"By definition, $j^{\\ast } \\circ j_{\\ast }= id$ as well.",
"By the uniqueness of adjoints, the composition $j_{\\ast } \\circ j^{\\ast }$ , when restricted to $D_{qc,Z}(X)^\\perp $ , has to be the identity too, and $M=0$ .",
"This shows that $D_{qc,Z}(X)^\\perp \\subseteq j_{\\ast }D_{qc}(U)$ .",
"It also follows that perfect complexes are mapped to perfect complexes under $j_{\\ast }$ .",
"Therefore, the adjoint also exists in the perfect case and $D_Z(X)^\\perp = j_{\\ast }D(U)$ .",
"Remark 3.7 We will often identify $D(U)$ with its image under $j_{\\ast }$ in $D(X)$ , and we will just write shortly the semiorthogonal decomposition as $ D(X)=\\langle D(U), D_Z(X) \\rangle .",
"$ The statements of Proposition REF are also valid in the equivariant setting when $X$ is a variety with a $G$ -action, and $Z \\subset X$ is a closed $G$ -invariant subvariety.",
"Even more generally, we can replace $X$ (resp., $Z$ ) with a Deligne-Mumford stack $\\mathcal {X}$ (resp., closed substack $\\mathcal {Z}$ )."
],
[
"The Grothendieck ring of geometric dg categories",
"A pretriangulated dg category is called geometric, if it is a dg enhancement of a semiorthogonal summand in $D(X)$ for some smooth projective variety $X$ .",
"Let $S_0(gdg-cat)$ (resp.",
"$K_0(gdg-cat)$ ) be the Grothendieck semigroup (resp.",
"group) of geometric dg categories with relations coming from semiorthogonal decompositions [10].",
"Namely, $S_0(gdg-cat)$ (resp.",
"$K_0(gdg-cat)$ ) is the free abelian semigroup (resp., group) generated by quasi-equivalence classes of geometric pretriangulated dg categories $\\mathcal {V}$ modulo the relations $ [\\mathcal {V}]=[\\mathcal {A}]+[\\mathcal {B}], $ where $\\mathcal {A}$ and $\\mathcal {B}$ are dg subcategories of $\\mathcal {V}$ ; $H^0(\\mathcal {A})$ and $H^0(\\mathcal {B})$ are admissible subcategories of $\\mathcal {V}$ ; $H^0(\\mathcal {V})=\\langle H^0(\\mathcal {A}), H^0(\\mathcal {B}) \\rangle $ is a semiorthogonal decomposition.",
"The (completed) tensor product makes $S_0(gdg-cat)$ (resp.",
"$K_0(gdg-cat)$ ) into an associative and commutative semiring (resp.",
"ring) (see Example REF ).",
"Theorem 3.8 [10] There exists a ring homomorphism $\\mu \\colon K_0(Var) \\rightarrow K_0(gdg -cat).\\\\$ The homomorphism $\\mu $ associates to the class of a smooth projective variety $X$ the class of the category $I(X)$ .",
"The proof of this theorem is based on the following two facts.",
"By the results of [25], [5] there is a presentation of $K_0(Var)$ as follows.",
"It is generated by the isomorphism classes of smooth projective varieties with the defining set of relations $ [Y]+[Z]=[X]+[E], $ where $Y$ is the blowup of $X$ along $Z$ with exceptional divisor $E$ .",
"In $K_0(gdg -cat)$ exactly the same type of relation is satisfied: $ [I(Y)]+[I(Z)]=[I(X)]+[I(E)].",
"$ Let $K_0(DM)$ be the Grothendieck group of Deligne-Mumford stacks.",
"$K_0(DM)$ is the group freely generated by equivalence classes $[\\mathcal {X}]$ of (generically tame) Deligne-Mumford stacks $\\mathcal {X}$ subject to the scissor relations $ [\\mathcal {X}]=[\\mathcal {X}\\setminus \\mathcal {Z}]+[\\mathcal {Z}], $ where $\\mathcal {Z}\\subset \\mathcal {X}$ is a closed substack.",
"$K_0(DM)$ can be equipped with a product induces by the product of stacks.",
"It follows from the scissor relations that when $Y$ is a variety with a $G$ -action, and $W \\subset Y$ is a closed $G$ -invariant subvariety, then in $K_0(DM)$ $ [[Y/G]]=[[(Y \\setminus W)/G]]+[[W/G]].",
"$ A DM stack is called projective if it admits a closed embedding to a smooth DM stack which is proper over $Spec(k)$ and has a projective coarse moduli space.",
"Smooth projective stacks play a similar role in $K_0(DM)$ as smooth projective varieties in $K_0(Var)$ .",
"Theorem 3.9 There exists a ring homomorphism $\\mu ^{\\prime } \\colon K_0(DM) \\rightarrow K_0(gdg -cat).\\\\$ The homomorphism $\\mu ^{\\prime }$ associates to the class of a smooth projective Deligne-Mumford stack $\\mathcal {X}$ the class of the category $I(\\mathcal {X})$ .",
"In particular, for a smooth projective variety $X$ with the action of a group $G$ $\\mu ^{\\prime }([X/G])=I_G(X).$ We postpone the proof of this theorem until Section REF .",
"By an abuse of notation we will also write $\\mu $ instead of $\\mu ^{\\prime }$ , but it will be always clear from the context which homomorphism $\\mu $ is used out of the two possibilities."
],
[
"Power structure over $K_0(Var)$",
"Definition 4.1 A power structure [17] over a (semi)ring $R$ is a map $(1+tR[[t]]) \\times R \\rightarrow 1+tR[[t]]\\colon (A(t),m) \\mapsto (A(t))^m$ such that: $(A(t))^0=1$ ; $(A(t))^1=A(t)$ ; $(A(t) \\cdot B(t))^m=(A(t))^m \\cdot (B(t))^m$ ; $(A(t))^{m+n}=(A(t))^m\\cdot (A(t))^n$ ; $(A(t))^{mn}=((A(t))^m)^n$ ; $(1+t)^m=1+mt+$ terms of higher degree; $(A(t^k))^m=(A(t))^m\\big |_{t\\mapsto t^k}$ .",
"A power structure over the ring $K_0(Var)$ was defined in [17] by the following formula: $(A(t))^{[M]}= 1 + \\sum _{k=1}^{\\infty } \\left\\lbrace \\sum _{\\underline{k}: \\sum ik_i=k} \\left[ \\left( ( \\prod _i M^{k_i} \\setminus \\Delta ) \\times \\prod _i A_i^{k_i}\\right) / \\prod _i S_{k_i} \\right] \\right\\rbrace \\cdot t^k.$ Here $A(t)=1+[A_1]t+[A_2]t^2+\\dots $ .",
"The coefficients $A_i$ , $i=1,2,\\dots $ and the exponent $M$ are quasiprojective varieties.",
"The index $\\underline{k}$ is taken from the set $\\lbrace k_i \\;\\colon \\; i \\in \\mathbb {Z}_{>0}, k_i \\in \\mathbb {Z}_{\\ge 0} \\rbrace $ .",
"$\\Delta $ is the large diagonal in $M^{\\sum k_i}$ which consists of $(\\sum k_i)$ -tuples of points of $M$ with at least two coinciding one, and the permutation group $S_{k_i}$ acts by permuting the corresponding $k_i$ factors of $(\\prod _i M^{k_i} \\setminus \\Delta ) \\subset \\prod _i M^{k_i}$ and the spaces $A_i$ simultaneously.",
"Geometrically this means that the coefficient of the monomial $t^k$ in the series $(A(t))^{[M]}$ is represented by the set whose element is a finite subset $K$ of points of the variety $M$ with positive multiplicities such that the total number of points of the set $K$ counted with multiplicities is equal to $k$ , and the set $K$ is endowed with a map $\\varphi \\colon K\\rightarrow \\coprod _{i=0}^\\infty A_i$ such that a point with multiplicity $j$ goes to the component $A_j$ .",
"When each $A_i=1=\\lbrace pt \\rbrace $ , then $\\bigcup _{\\underline{k}: \\sum ik_i=k} \\left( \\prod _i M^{k_i} \\setminus \\Delta \\right) / \\prod _i S_{k_i} = \\bigcup _{\\underline{k}: \\sum ik_i=k}S^k_{\\underline{k}}M = S^k M.$ Here $S^k_{\\underline{k}}M$ is the stratum of $S^k M$ where for each $i$ there are exactly $k_i$ points of multiplicity $i$ .",
"It follows from (REF ) and (REF ) that $\\left(\\frac{1}{1-t}\\right)^{[M]}=\\sum _{n=0}^{\\infty } [\\operatorname{Sym}^n(M)]t^n.$ For sequence of integer $\\underline{k}$ such that $\\sum ik_i=k$ , let us denote $M^k_{\\underline{k}}= p^{-1}_k(S^k_{\\underline{k}}M)$ where $p_k \\colon M^k \\rightarrow S^kM$ is the natural projection.",
"For example, $M^k_{(k,0,\\dots ,0)}=M^k \\setminus \\Delta $ ."
],
[
"Destackification",
"Let $\\mathcal {X}$ be an algebraic stack in the sense of Deligne-Mumford.",
"We will need two basic operations, which can be performed on $\\mathcal {X}$ .",
"First, one can take a blow-up along a smooth center.",
"Second, one can form root stacks over it, i.e.",
"a root constructions along smooth divisors.",
"These two operations are called smooth stacky blow-ups.",
"We will restrict our attention to the case of global quotients $\\mathcal {X}=[M/G]$ .",
"Since $G$ is finite, there is a canonical map $\\mathcal {X}\\rightarrow \\mathcal {X}_{cs}=M/G$ to the coarse moduli space.",
"In general, the quotient $\\mathcal {X}_{cs}=M/G$ is not smooth.",
"As a remedy, the concept of destackifications was developed in [1].",
"The following results are valid in the much more general setting of tame algebraic stacks with diagonalizable stabilizers, but, for simplicity, we specialize to the case of DM stacks.",
"Definition 4.2 A destackification of $\\mathcal {X}$ is a map $f\\colon \\mathcal {Y}\\rightarrow \\mathcal {X}$ of algebraic stacks which is a composition of smooth stacky blowups such that the coarse space $\\mathcal {Y}_{cs}$ of $\\mathcal {Y}$ is smooth; the canonical map $\\mathcal {Y}\\rightarrow \\mathcal {Y}_{cs}$ is a (sequence/fiber product of) root stack(s along distinct divisors); the components of the branch divisor in $\\mathcal {Y}_{cs}$ are smooth, and they only have simple normal crossings.",
"A particular case of [1] is Theorem 4.3 If $\\mathcal {X}=[M/G]$ is a global quotient where $G$ is a finite group, then there exists a destackifications $f\\colon \\mathcal {Y}\\rightarrow \\mathcal {X}$ of $\\mathcal {X}$ .",
"For further calculations it is important to investigate how the derived category of a stack behaves under the two stacky blow-up operations.",
"Let $\\mathcal {X}$ be a DM stack and $\\mathcal {Z}\\subset \\mathcal {X}$ an effective divisor.",
"Fix a positive integer $n$ and let $f\\colon \\mathcal {Y}\\rightarrow \\mathcal {X}$ be the $n$ -th root construction of $\\mathcal {X}$ along $\\mathcal {Z}$ .",
"Let $V_k$ be the $k$ -th irreducible representation of the cyclic group $\\mathbb {Z}_n$ .",
"For a triangulated category $\\mathcal {T}$ , let $\\mathcal {T}^{\\mathbb {Z}_n}$ be the category of equivariant objects with respect to the trivial $\\mathbb {Z}_n$ -action, and let $ \\iota _k \\colon \\mathcal {T}\\rightarrow \\mathcal {T}^{\\mathbb {Z}_n}, \\quad F \\mapsto F \\otimes V_k.$ The preimage $\\mathcal {E}$ of $\\mathcal {Z}$ in $\\mathcal {Y}$ as a stack is equivalent to $[\\mathcal {Z}/\\mathbb {Z}_n]$ , i.e.",
"$\\mathcal {Z}$ equipped with a trivial action of $\\mathbb {Z}_n$ .",
"Denote by $i\\colon \\mathcal {E}\\hookrightarrow \\mathcal {Y}$ the embedding of $\\mathcal {E}$ as the ramification divisor.",
"Let $\\mathcal {U}= \\mathcal {Y}\\setminus \\mathcal {E}$ and and let $j \\colon \\mathcal {U}\\hookrightarrow \\mathcal {Y}$ be the embedding morphism.",
"We have the induced functors $\\begin{array}{r c c c c c l}f_k^{\\ast } \\colon & D(\\mathcal {X}) & \\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& D([\\mathcal {X}/\\mathbb {Z}_n]) &\\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& D(\\mathcal {Y}), & \\\\i_{k \\ast } \\colon & D(\\mathcal {Z}) & \\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& D(\\mathcal {E}) &\\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& D(\\mathcal {Y}).",
"& \\\\\\end{array}$ Proposition 4.4 ([19][24], [4]) For each $k \\in \\lbrace 0,\\dots ,n-1\\rbrace $ the functors $f_k^{\\ast }\\colon D(\\mathcal {X}) \\rightarrow D(\\mathcal {Y}),$ $i_{k \\ast } \\colon D(\\mathcal {Z}) \\rightarrow D(\\mathcal {Y})$ are fully faithful and admit left and right adjoints.",
"Moreover, for $k,l \\in \\lbrace 0,\\dots ,n-1\\rbrace $ $(i_{k \\ast }D(\\mathcal {Z}), i_{l \\ast }D(\\mathcal {Z}))$ is semiorthogonal if $k\\ne l,l+1$ ; $(i_{k \\ast }D(\\mathcal {Z}), f_{l \\ast }D(\\mathcal {Z}))$ is semiorthogonal if $k\\ne l$ ; $(f_{k \\ast }D(\\mathcal {Z}), i_{l \\ast }D(\\mathcal {Z}))$ is semiorthogonal if $k\\ne l-1$ .",
"Proposition 4.5 The functor $j^{\\ast } \\colon D(\\mathcal {Y}) \\rightarrow D(\\mathcal {U})$ has a right adjoint $j_{\\ast } \\colon D(\\mathcal {U}) \\rightarrow D(\\mathcal {Y})$ .",
"Moreover, there is a semiorthogonal decomposition $D(\\mathcal {Y})=\\langle j_{\\ast } D(\\mathcal {U}), i_{0 \\ast }D(\\mathcal {Z}), \\dots , i_{(n-1)\\ast }D(\\mathcal {Z}) \\rangle $ The group $\\mathbb {Z}_n$ acts on $\\mathcal {Z}$ trivially.",
"By Proposition REF there is a semiorthogonal decomposition $D(\\mathcal {E})=D([\\mathcal {Z}/\\mathbb {Z}_n])=\\langle \\iota _0 D(\\mathcal {Z}),\\dots , \\iota _{n-1} D(\\mathcal {Z}) \\rangle .$ Suppose that $F \\in D_{\\mathcal {E}}(\\mathcal {Y})$ .",
"That is, each cohomology sheaf of $F$ is supported set-theoretically on the space underlying $\\mathcal {Z}$ .",
"Then it admits a filtration by sheaves supported on the space underlying $\\mathcal {Z}$ scheme-theoretically.",
"This filtration can also be chosen $\\mathbb {Z}_n$ -equivariantly.",
"By (REF ) any $\\mathbb {Z}_n$ -equivariant sheaf scheme-theoretically supported on the space underlying $\\mathcal {Z}$ can be written as the direct sum of sheaves contained in the categories $i_{l \\ast }D(\\mathcal {Z})$ .",
"Therefore $D_{\\mathcal {E}}(\\mathcal {Y})=\\langle i_{0 \\ast }D(\\mathcal {Z}), \\dots , i_{(n-1)\\ast }D(\\mathcal {Z}) \\rangle .$ Due to Proposition REF the subcategory $D_{\\mathcal {E}}(\\mathcal {Y})$ is admissible in $D(\\mathcal {Y})$ .",
"The statement then follows from Proposition REF and the remark after it.",
"Corollary 4.6 In the setting of Proposition REF $\\mu (\\mathcal {U})=\\mu (\\mathcal {X})-\\mu (\\mathcal {Z})=\\mu (\\mathcal {Y})-\\mu (\\mathcal {E}).$ To compute the derived category of the destackification of an orbifold one has to consider also the derived category of an iterated root construction.",
"The details of this are developed in [4].",
"Here we give a brief summary.",
"Suppose that $\\mathcal {X}$ is a DM stack and $\\mathcal {Z}=(\\mathcal {Z}_i)_{i \\in I}$ is a snc divisor on $\\mathcal {X}$ for some index set $I$ .",
"This means that for each subset $J \\subset I$ and each element $i \\in J$ the morphism $ \\cap _{j \\in J} \\mathcal {Z}_j \\rightarrow \\cap _{j \\in J \\setminus \\lbrace i\\rbrace } \\mathcal {Z}_j $ is the inclusion of a smooth, effective Cartier divisor.",
"The divisors $\\mathcal {Z}_i$ are the components of $\\mathcal {Z}$ .",
"Let $\\underline{n}=(n_i)_{i \\in I}$ be a multi-index with positive integer entries.",
"Let $\\mathcal {Y}_i \\rightarrow \\mathcal {X}$ be the the root stack over $\\mathcal {X}$ of degree $n_i$ and branch divisor $\\mathcal {Z}_i$ .",
"The $\\underline{n}$ -th root stack over $\\mathcal {X}$ with branch divisor $\\mathcal {Z}$ is defined as the fiber product $ \\mathcal {Y}= \\mathcal {Y}_1 \\times _{\\mathcal {X}} \\dots \\times _{\\mathcal {X}} \\mathcal {Y}_{|I|} \\rightarrow \\mathcal {X}.$ Let $\\mathbb {Z}_{\\underline{n}}=\\prod _i\\mathbb {Z}_{n_i}$ .",
"For any multi-index $\\underline{k}$ satisfying $\\underline{n} > \\underline{k} \\ge 0$ , let $V_{\\underline{k}}=\\otimes _i V_{k_i}^{n_i}$ , where $V_{k_i}^{n_i}$ is the $k_i$ -th irreducible representation of $\\mathbb {Z}_{n_i}$ .",
"Let moreover $I_{\\underline{k}}$ be the support of $\\underline{k}$ , $\\mathcal {Z}(I_{\\underline{k}})=\\cap _{i \\in I_{\\underline{k}}} \\mathcal {Z}_i$ , and $\\mathcal {E}(I_{\\underline{k}})$ be the preimage of $\\mathcal {Z}(I_{\\underline{k}})$ in $\\mathcal {Y}$ .",
"For any triangulated category $\\mathcal {T}$ let $ \\iota _{\\underline{k}} \\colon \\mathcal {T}\\rightarrow \\mathcal {T}^{\\mathbb {Z}_{\\underline{n}}}, \\quad F \\mapsto F \\otimes V_{\\underline{k}}.$ Again, let $\\begin{array}{r c c c c c l}f_{\\underline{k}}^{\\ast } \\colon & D(\\mathcal {X}) & \\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& D([\\mathcal {X}/\\mathbb {Z}_{\\underline{n}}]) &\\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& D(\\mathcal {Y}), & \\\\i_{\\underline{k} \\ast } \\colon & D(\\mathcal {Z}(I_{\\underline{k}})) & \\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& D(\\mathcal {E}(I_{\\underline{k}})) &\\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& D(\\mathcal {Y}).",
"& \\\\\\end{array}$ Proposition 4.7 ([4]) For each $\\underline{n} > \\underline{k} \\ge 0$ the functors $f_{\\underline{k}}^{\\ast }\\colon D(\\mathcal {X}) \\rightarrow D(\\mathcal {Y}),$ $i_{\\underline{k} \\ast } \\colon D(\\mathcal {Z}(I_{\\underline{k}})) \\rightarrow D(\\mathcal {Y})$ are fully faithful and admit left and right adjoints.",
"Moreover, for $\\underline{n} > \\underline{k},\\underline{l} \\ge 0$ $(i_{\\underline{k} \\ast }D(\\mathcal {Z}(I_{\\underline{k}})), i_{\\underline{l} \\ast }D(\\mathcal {Z}(I_{\\underline{k}})))$ is semiorthogonal if $\\underline{k}\\ne \\underline{l},\\underline{l}+1_0,\\dots ,\\underline{l}+1_{|I|}$ ; $(i_{\\underline{k} \\ast }D(\\mathcal {Z}(I_{\\underline{k}})), f_{\\underline{l} \\ast }D(\\mathcal {X}))$ is semiorthogonal if $\\underline{k}\\ne \\underline{l}$ .",
"Here $1_i$ denotes the vector which is 1 at position $i$ , and zero everywhere else.",
"Let again $\\mathcal {U}= \\mathcal {Y}\\setminus \\mathcal {E}$ and and let $j \\colon \\mathcal {U}\\hookrightarrow \\mathcal {Y}$ be the embedding morphism.",
"Proposition 4.8 The functor $j^{\\ast } \\colon D(\\mathcal {Y}) \\rightarrow D(\\mathcal {U})$ has a right adjoint $j_{\\ast } \\colon D(\\mathcal {U}) \\rightarrow D(\\mathcal {Y})$ .",
"Moreover, there is a semiorthogonal decomposition $D(\\mathcal {Y})=\\langle j_{\\ast } D(\\mathcal {U}), i_{\\underline{k}_1 \\ast }D(\\mathcal {Z}(I_{\\underline{k}_1})), \\dots , i_{\\underline{k}_m\\ast }D(\\mathcal {Z}(I_{\\underline{k}_m})) \\rangle ,$ where the multi-indices $\\underline{k}$ with $\\underline{n} > \\underline{k} \\ge 0$ are arranged into any sequence $\\underline{k}_1,\\dots , \\underline{k}_m$ such that $\\underline{k}_s \\ge \\underline{k}_t$ implies $s \\ge t$ for all $s,t \\in \\lbrace 1,\\dots ,m\\rbrace $ where $m=\\prod _{i \\in I} n_i$ .",
"The statement of the following lemma is encoded in the proof of [4], but it can also be obtained with an induction argument.",
"Lemma 4.9 With the notations of Proposition REF , $D(\\mathcal {E})=\\langle \\iota _{\\underline{k}_1 \\ast }D(\\mathcal {Z}(I_{\\underline{k}_1})), \\dots , \\iota _{\\underline{k}_m\\ast }D(\\mathcal {Z}(I_{\\underline{k}_m})) \\rangle .$ Replace indices everywhere with multi-indices in the proof of Proposition REF and use Proposition REF (resp.",
"Lemma REF ) instead of Proposition REF (resp.",
"equation REF ).",
"Corollary 4.10 In the setting of Proposition REF $\\mu (\\mathcal {U})=\\mu (\\mathcal {X})-\\mu (\\mathcal {Z})=\\mu (\\mathcal {Y})-\\mu (\\mathcal {E}).$ The other stacky blowup operation is the blowup along a smooth center.",
"In this case the open part away from the center does not appear in general as a semiorthogonal summand.",
"But a weaker result is still true in this case.",
"Proposition 4.11 Let $\\mathcal {X}$ be a smooth projective DM stack and $f\\colon \\mathcal {Y}\\rightarrow \\mathcal {X}$ be its blowup along a smooth closed substack $\\mathcal {Z}\\subset \\mathcal {X}$ of codimension $n$ .",
"Let $\\mathcal {E}\\subset \\mathcal {Y}$ be the exceptional divisor.",
"Then, there is a semiorthogonal decomposition $D(\\mathcal {Y})=\\langle f^{\\ast } D(\\mathcal {X}), i_{1 \\ast }D(\\mathcal {Z}), \\dots , i_{(n-1)\\ast }D(\\mathcal {Z}), \\rangle $ where $i_{k \\ast } \\colon D(\\mathcal {Z}) \\rightarrow D(\\mathcal {Y})$ are certain functors.",
"For schemes this result was proved in [27].",
"In the case of a quotient stack $\\mathcal {X}=[M/G]$ with $G$ an affine algebraic group the statement follows from [13].",
"By [21], every (generically tame) smooth projective DM stack is a quotient stack by an affine algebraic group, so the statement is true for all smooth projective DM stack.",
"Corollary 4.12 In the setting of Proposition REF with $\\mathcal {U}= \\mathcal {Y}\\setminus \\mathcal {E}\\cong \\mathcal {X}\\setminus \\mathcal {Z}$ , $\\mu (\\mathcal {U})=\\mu (\\mathcal {X})-\\mu (\\mathcal {Z})=\\mu (\\mathcal {Y})-\\mu (\\mathcal {E}).$ We are now ready to give the In analogy with Theorem REF , the proof follows from the following two statements.",
"By [2] there is a presentation of $K_0(DM)$ as follows.",
"It is generated by the isomorphism classes of smooth projective DM stacks with the defining set of relations $ [\\mathcal {Y}]+[\\mathcal {Z}]=[\\mathcal {X}]+[\\mathcal {E}], $ where $\\mathcal {Y}\\rightarrow \\mathcal {X}$ is a stacky blowup along a smooth center $\\mathcal {Z}$ with exceptional divisor $\\mathcal {E}$ .",
"By Propositions REF and REF the relation $ [I(\\mathcal {Y})]+[I(\\mathcal {Z})]=[I(\\mathcal {X})]+[I(\\mathcal {E})] $ is satisfied for stacky blowups of smooth projective DM stacks along a smooth centers in $K_0(gdg -cat)$ ."
],
[
"Stratification of categorical powers",
"Recall that for sequence of integer $\\underline{k}$ such that $\\sum ik_i=k$ , $M^k_{\\underline{k}}= p^{-1}_k(S^k_{\\underline{k}}M)$ , where $p_k \\colon M^k \\rightarrow S^kM$ is the natural projection and $S^k_{\\underline{k}}M$ is the stratum of $S^k M$ where for each $i$ there are exactly $k_i$ points of multiplicity $i$ .",
"One of our key results is that for smooth projective varieties the strata $M^k_{\\underline{k}}$ have a particularly nice behaviour with respect to the map $\\mu $ .",
"Theorem 4.13 Let $M$ be a smooth projective variety and $\\underline{k}$ a sequence of nonnegative integers such that $\\sum _i ik_i=k$ .",
"$I([M_{\\underline{k}}^k / S_k])$ is geometric, and therefore $ \\mu ([M_{\\underline{k}}^k / S_k])=I([M_{\\underline{k}}^k / S_k]).$ $I(M_{\\underline{k}}^k)$ is geometric, and therefore $ \\mu (M_{\\underline{k}}^k)=I(M_{\\underline{k}}^k ).$ Part (1): Let $D_{\\underline{k}}$ be a connected component of $M^k_{\\underline{k}}$ .",
"$D_{\\underline{k}}$ can also be described as the fixed point set of an element of $S_k$ of conjugacy type described by $\\underline{k}$ , which are not fixed by any other element in the same conjugacy class.",
"Each point in $D_{\\underline{k}}$ has stabilizer $H=\\prod _i (S_i)^{k_i}$ .",
"Let $\\mathcal {X}_{\\underline{k}}=[D_{\\underline{k}} / H]$ .",
"$\\mathcal {X}_{\\underline{k}}$ is a smooth quasi-projective stack which is equivalent to $[M^k_{\\underline{k}}/S_k]$ .",
"Let $\\overline{\\mathcal {X}_{\\underline{k}}}=[\\overline{D_{\\underline{k}}}/H]$ be its closure in $[M^k/H]$ .",
"$\\overline{\\mathcal {X}_{\\underline{k}}}$ is a smooth projective DM stack and $\\mathcal {X}_{\\underline{k}} \\subset \\overline{\\mathcal {X}_{\\underline{k}}}$ is an open substack.",
"Let $f_{\\underline{k}}\\colon \\mathcal {Y}_{\\underline{k}} \\rightarrow \\overline{\\mathcal {X}_{\\underline{k}}}$ be the destackification of $\\overline{\\mathcal {X}_{\\underline{k}}}$ .",
"Then the coarse space $\\mathcal {Y}_{\\underline{k},cs}$ is a smooth variety and the coarse map $\\mathcal {Y}_{\\underline{k}} \\rightarrow \\mathcal {Y}_{\\underline{k},cs}$ is a root stack (possibly precomposed with a gerbe).",
"$\\mathcal {Y}_{\\underline{k}}$ being obtained from $\\overline{\\mathcal {X}_{\\underline{k}}}$ with stacky blowups is itself a smooth projective Deligne-Mumford stack.",
"Thus, $ \\mu (\\mathcal {Y}_{\\underline{k}})=I(\\mathcal {Y}_{\\underline{k}}).$ The coarse space $(\\overline{\\mathcal {X}_{\\underline{k}}})_{cs}=\\overline{D_{\\underline{k}}}/H$ has smooth locus $(\\mathcal {X}_{\\underline{k}})_{cs}=D_{\\underline{k}}/H=D_{\\underline{k}}$ .",
"Therefore, each center of the stacky blowups in the composite mapping $f_{\\underline{k}}$ avoids the preimage of $\\mathcal {X}_{\\underline{k}}$ .",
"Moreover, the substack $\\mathcal {E}_{\\underline{k}}= f^{-1}_{\\underline{k}}(\\overline{\\mathcal {X}_{\\underline{k}}} \\setminus \\mathcal {X}_{\\underline{k}}) \\subset \\mathcal {Y}_{\\underline{k}}$ is also the preimage of $\\mathcal {Y}_{\\underline{k},cs}$ away from the branch divisor under the coarse map $\\mathcal {Y}_{\\underline{k}} \\rightarrow \\mathcal {Y}_{\\underline{k},cs}$ .",
"Let $\\mathcal {U}_{\\underline{k}}= \\mathcal {Y}_{\\underline{k}} \\setminus \\mathcal {E}_{\\underline{k}}$ .",
"Again, since the destackification algorithm removes the stackiness only from $\\mathcal {E}_{\\underline{k}}$ , the gerbe part of the coarse map $\\mathcal {Y}_{\\underline{k}} \\rightarrow \\mathcal {Y}_{\\underline{k},cs}$ is trivial over $\\mathcal {U}_{\\underline{k}}$ .",
"By Proposition REF , there is a semiorthogonal decomposition $D(\\mathcal {Y}_{\\underline{k}})=\\langle D(\\mathcal {U}_{\\underline{k}}), D_{\\mathcal {E}_{\\underline{k}}}(\\mathcal {Y}_{\\underline{k}}) \\rangle $ It follows that $ I(\\mathcal {U}_{\\underline{k}})$ is geometric and $\\mu (\\mathcal {U}_{\\underline{k}})=I(\\mathcal {U}_{\\underline{k}})$ .",
"On the other hand, by Corollaries REF and REF $ \\mu (\\mathcal {U}_{\\underline{k}})= \\mu (\\mathcal {Y}_{\\underline{k}})-\\mu (\\mathcal {E}_{\\underline{k}})= \\mu (\\overline{\\mathcal {X}_{\\underline{k}}})-\\mu (\\overline{\\mathcal {X}_{\\underline{k}}}\\setminus \\mathcal {X}_{\\underline{k}})=\\mu (\\mathcal {X}_{\\underline{k}}).$ The morphism $f_{\\underline{k}}\\vert _{\\mathcal {U}_{\\underline{k}}} \\colon \\mathcal {U}_{\\underline{k}} \\rightarrow \\mathcal {X}_{\\underline{k}}$ is a (composition of) regular covering(s) without branch locus.",
"Thus, $ I({\\mathcal {U}_{\\underline{k}}})=I(\\mathcal {X}_{\\underline{k}})=I([M_{\\underline{k}}^k / S_k]).",
"$ In particular, $I([M_{\\underline{k}}^k / S_k])$ is geometric.",
"Part (2): We continue to use the notations from the proof of Part (1).",
"Step 1: Assume first that $\\underline{k}=(k,0,\\dots ,0)$ .",
"Then, by the definition of a destackification, the coarse map $\\mathcal {Y}_{\\underline{k}} \\rightarrow \\mathcal {Y}_{\\underline{k},cs}$ is an (iterated) root stack and has no gerbe part.",
"It follows that we have the diagram $\\begin{array}{c c c }\\mathcal {U}_{\\underline{k}} & \\hookrightarrow & \\mathcal {Y}_{\\underline{k}} \\\\\\mathrel {\\rotatebox {90}{\\XMLaddatt {origin}{c}\\,=}}& & \\mathrel {\\rotatebox {90}{\\XMLaddatt {origin}{c}\\,=}}\\\\\\end{array}[V_{\\underline{k}} / A] & \\hookrightarrow & [W_{\\underline{k}} / A],$ $where $ $ is the finite abelian covering group of the root stack, and $ Wk$ is the underlying variety of the root stack, and $ Vk$ is the part of $ Wk$ away from the ramification divisor.",
"Applying Theorem \\ref {thm:finabdec} we obtain that\\begin{equation}D(\\mathcal {U}_{\\underline{k}})^{A^{\\vee }}=D(V_{\\underline{k}}) \\quad \\textrm {and} \\quad D(\\mathcal {Y}_{\\underline{k}})^{A^{\\vee }}=D(W_{\\underline{k}}).\\end{equation}$$\\mathcal {Y}_{\\underline{k}}$ was obtained from $\\overline{\\mathcal {X}}_{\\underline{k}}$ with stacky blowup operations, so $\\mathcal {Y}_{\\underline{k}}$ is itself a smooth projective DM stack.",
"Let us investigate the stacky blowup sequence $f_{\\underline{k}}\\colon \\mathcal {Y}_{\\underline{k}} \\rightarrow \\overline{\\mathcal {X}_{\\underline{k}}}$ .",
"$V_{\\underline{k}}$ is obtained from the quasi-projective variety $D_{\\underline{k}}$ with the stacky blowup operations.",
"It follows that $f_{\\underline{k}}$ induces a regular covering map $V_{\\underline{k}} \\rightarrow D_{\\underline{k}}$ with covering group a finite Abelian group $A^{\\prime }$ acting freely on $V_{\\underline{k}}$ .",
"So we get that $D_{\\underline{k}}=V_{\\underline{k}}/A^{\\prime }$ , and $D(D_{\\underline{k}})=D([V_{\\underline{k}}/A^{\\prime }])=D(V_{\\underline{k}})^{A^{\\prime }}.$ The action of $A^{\\prime }$ on $\\mathcal {Y}_{\\underline{k}}$ also preserves the substacks $\\mathcal {U}_{\\underline{k}}$ and $\\mathcal {E}_{\\underline{k}}$ .",
"Theorem REF combined with (REF ), () and (REF ) gives $\\begin{aligned}D([W_{\\underline{k}}/A^{\\prime }])=D(W_{\\underline{k}})^{A^{\\prime }} & =(D(\\mathcal {Y}_{\\underline{k}})^{A^{\\vee }})^{A^{\\prime }} \\\\& = \\langle (D(\\mathcal {U}_{\\underline{k}})^{A^{\\vee }})^{A^{\\prime }}, (D_{\\mathcal {E}_{\\underline{k}}}(\\mathcal {Y}_{\\underline{k}})^{A^{\\vee }})^{A^{\\prime }} \\rangle \\\\&= \\langle D(D_{\\underline{k}}), (D_{\\mathcal {E}_{\\underline{k}}}(\\mathcal {Y}_{\\underline{k}})^{A^{\\vee }})^{A^{\\prime }} \\rangle .\\end{aligned}$ $[W_{\\underline{k}}/A^{\\prime }]$ is a smooth projective Deligne-Mumford stack.",
"It follows from [4] that $I([W_{\\underline{k}}/A^{\\prime }])=I(W_{\\underline{k}})^{A^{\\prime }}$ is geometric, and therefore $I(D_{\\underline{k}})$ is also geometric.",
"$M_{\\underline{k}}^k$ is a disjoint union of a finite number of connected components, each of which is isomorphic $D_{\\underline{k}}$ .",
"We obtain that $I(M_{\\underline{k}}^k)$ is also geometric.",
"Step 2: In general, $D_{\\underline{k}}^k$ is isomorphic to $D^{\\sum _i k_i}_{(\\sum _i k_i,0\\dots ,0)}$ .",
"Again, because $M_{\\underline{k}}^k$ is a disjoint union of a finite number of connected components, each of which is isomorphic $D_{\\underline{k}}$ , this implies that $I(M_{\\underline{k}}^k)$ is geometric.",
"Let $\\mathcal {M}$ be an arbitrary geometric dg category.",
"Then, $H^0(\\mathcal {M}) \\subset D(M)$ is a semiorthogonal summand for some smooth projective variety $M$ .",
"Let $ \\mathcal {M}_{\\underline{k}}^k=\\left\\lbrace F \\in I(M_{\\underline{k}}^k)\\; |\\; F=G\\vert _{M_{\\underline{k}}^k} \\textrm { for some } G \\in \\mathcal {M}^{ k} \\subset I(\\overline{M^k_{\\underline{k}}}) \\right\\rbrace .$ Proposition 4.14 For every integer sequence $\\underline{k}$ such that $\\sum _i ik_i=k$ , the dg category $\\mathcal {M}_{\\underline{k}}^k$ is geometric.",
"For simplicity, we prove the statement first for the sequence $\\underline{k}=(k,0,\\dots ,0)$ .",
"In this case $\\overline{M^k_{\\underline{k}}}=M^k$ .",
"Since $\\mathcal {M}$ is geometric, $H^0(\\mathcal {M}) \\subset D(M)$ is a semiorthogonal summand for some smooth projective variety $M$ .",
"Applying the mutation functors of [9] on a semiorthogonal sequence of $D(M)$ containing $H^0(\\mathcal {M})$ we can assume that there is a semiorthogonal decomposition $ D(M)=\\langle A, \\phi (H^0(\\mathcal {M})) \\rangle , $ where $\\phi (H^0(\\mathcal {M}))$ is equivalent to $H^0(\\mathcal {M})$ .",
"By Lemma REF , $\\phi (H^0(\\mathcal {M}))$ is right admissible in $D(M)$ .",
"For ease, we replace $\\mathcal {M}$ with $\\phi (\\mathcal {M})$ and assume that $H^0(\\mathcal {M})$ is right admissible in $D(M)$ .",
"It is easy to see that in this case $H^0(\\mathcal {M}^k)$ is also right admissible in $D(M^k)$ .",
"By [23] there is a right admissible subcategory $\\widehat{H^0(\\mathcal {M}^{ k})} \\subset D_{qc}(M^k)$ , which restricts to $H^0(\\mathcal {M}^k)$ in $D(M^k)$ .",
"This means that the functor $i\\ \\colon \\widehat{H^0(\\mathcal {M}^{k})} \\hookrightarrow D_{qc}(M^k)$ has a right adjoint $i^{\\ast } \\colon D_{qc}(M^k) \\rightarrow \\widehat{H^0(\\mathcal {M}^{ k})}$ .",
"On the level of quasi-coherent sheaves the restriction functor $j^{\\ast }\\colon D_{qc}(M^k) \\rightarrow D_{qc}(M^n_{(k,0,\\dots ,0)})$ has a right adjoint $j_{\\ast }\\colon D_{qc}(M^n_{(k,0,\\dots ,0)}) \\rightarrow D_{qc}(M^k)$ .",
"Let's denote by $\\widehat{H^0(\\mathcal {M}^k_{(k,0,\\dots ,0)})}$ the image of $\\widehat{H^0(\\mathcal {M}^{ k})}$ under $j^{\\ast }\\colon D_{qc}(M^k) \\rightarrow D_{qc}(M^n_{(k,0,\\dots ,0)})$ .",
"Since everything in the construction is functorial, $\\widehat{H^0(\\mathcal {M}^k_{(k,0,\\dots ,0)})}\\cap D(M^k_{(k,0,\\dots ,0)})=H^0(\\mathcal {M}^k_{(k,0,\\dots ,0)})$ .",
"Since $j^{\\ast } \\circ j_{\\ast }$ is the identity, the composition $\\begin{array}{c c c c c c c}D_{qc}(M^k_{(k,0,\\dots ,0)}) & \\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& D_{qc}(M^k) & \\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& \\widehat{H^0(\\mathcal {M}^{k})} & \\hspace{-6.375pt}\\xrightarrow{}\\phantom{}\\hspace{-6.375pt}& \\widehat{H^0(\\mathcal {M}^n_{(k,0,\\dots ,0)})}\\end{array}$ is right adjoint to the embedding $\\widehat{H^0(\\mathcal {M}^k_{(k,0,\\dots ,0)})} \\hookrightarrow D_{qc}(M^k_{(k,0,\\dots ,0)})$ .",
"This means that $\\widehat{H^0(\\mathcal {M}^k_{(k,0,\\dots ,0)})} \\subset D_{qc}(M^k_{(k,0,\\dots ,0)})$ is right admissible.",
"Moreover, the above composition maps perfect complexes to perfect complexes.",
"So it restricts to a functor $D(M^k_{(k,0,\\dots ,0)}) \\rightarrow H^0(\\mathcal {M}^k_{(k,0,\\dots ,0)})$ , which is right adjoint to the embedding $H^0(\\mathcal {M}^k_{(k,0,\\dots ,0)}) \\hookrightarrow D(M^k_{(k,0,\\dots ,0)})$ .",
"Hence, $H^0(\\mathcal {M}^k_{(k,0,\\dots ,0)})$ is a semiorthogonal summand in $D(M^k_{(k,0,\\dots ,0)})$ whose enhancement is geometric by Theorem REF (2).",
"The proof of the statement for an arbitrary sequence $\\underline{k}$ is the same except that $M^k$ has to be replaced by the smooth projective variety $\\overline{M}^k_{\\underline{k}}$ It is not true in general that $H^0(\\mathcal {M}^k_{\\underline{k}}) \\subset H^0(\\mathcal {M}^k)$ are semiorthogonal summands since it is not true even for $I(M)$ .",
"But we have Proposition 4.15 For any $[\\mathcal {M}] \\in K_0(gdg-cat)$ , the class of $\\mathcal {M}^k$ decomposes as $ [\\mathcal {M}^k]=\\sum _{\\underline{k}: \\sum ik_i=k} [\\mathcal {M}^k_{\\underline{k}}].",
"$ For any variety $M$ there is a decomposition $M^k=\\bigsqcup _{\\underline{k}: \\sum _i ik_i=k}M_{\\underline{k}}^k$ .",
"By Theorem REF for $\\mathcal {M}=I(M)$ this induces the required decomposition $ [I(M^k)]=\\sum _{\\underline{k}: \\sum ik_i=k} [I(M^k_{\\underline{k}})].$ In general, for a geometric dg category $\\mathcal {M}\\subset I(M)$ each $H^0(\\mathcal {M}^k_{\\underline{k}})$ (resp., $H^0(\\mathcal {M}^k)$ ) is a semiorthogonal summand in the corresponding $D(M^k_{\\underline{k}})$ (resp., $D(M^k)$ ).",
"Applying the restriction functor $I(M) \\rightarrow \\mathcal {M}$ on each component on the above sum we obtain the required decomposition of $[\\mathcal {M}^k]$ ."
],
[
"Power structure over $K_0(gdg-cat)$",
"Let $\\mathcal {A}(t)=1+[\\mathcal {A}_1]t+[\\mathcal {A}_2]t^2+\\dots $ , where the coefficients are (represented by) dg categories $\\mathcal {A}_i, i=1,2,\\dots $ .",
"Let further $\\mathcal {M}$ be a pretriangulated category.",
"We want to define $(\\mathcal {A}(t))^{[\\mathcal {M}]}$ in analogy with the formula (REF ).",
"For a general pretriangulated category $\\mathcal {M}$ the difficulty is that for the tensor power $\\mathcal {M}^{k}$ there is no obvious diagonal component, i.e.",
"a semiorthogonal summand in $\\mathcal {M}^{k}$ representing tensors with “coinciding components”.",
"However, for geometric dg categories we can induce such a summand from the decomposition of the underlying space.",
"From now on we will use the convention that a single number $k$ , when appears in the subscript of a category or a space, means the sequence $\\underline{k}=(k,0,\\dots ,0)$ .",
"For example, $M^k_k=M^k_{(k,0,\\dots ,0)}=M^k \\setminus \\Delta $ .",
"Let us define a power structure over the Grothendieck semiring of geometric dg categories $S_0(gdg-cat)$ by the following formula: $(\\mathcal {A}(t))^{[\\mathcal {M}]}= 1 + \\sum _{k=1}^{\\infty } \\left\\lbrace \\sum _{\\underline{k}: \\sum ik_i=k} \\left[ \\left(\\mathcal {M}_{\\sum _i k_i}^{\\sum _i k_i} \\boxtimes \\prod _i \\mathcal {A}_i^{k_i}\\right)^{\\prod _i S_{k_i}} \\right] \\right\\rbrace \\cdot t^k.$ Lemma 4.16 Suppose that, for each integer $i$ , $H^0(\\mathcal {A}_i)$ is a semiorthogonal summand in $D(A_i)$ , where $A_i$ is a smooth projective variety.",
"Then, the coefficient of the monomial $t^k$ in the series $(\\mathcal {A}(t))^{[\\mathcal {M}]}$ is represented by the (prefect hull of the) category with objects (injective complexes of perfect) sheaves $F=F_{k_1} \\otimes \\dots $ satisfying any of the following equivalent descriptions: $F$ is an $(\\prod _i S_{k_i})$ -equivariant sheaf on $(M^{\\sum _i k_i} \\setminus \\Delta ) \\times A_i^{k_i} $ ; $F_{k_i}=p_{k_i}^{\\ast }(G_{k_i})$ , where $G_{k_i}$ is an $S_{k_i}$ -equivariant object of $\\mathcal {M}_{k_i} \\boxtimes \\mathcal {A}_i^{k_i}$ , i.e.",
"an $S_{k_i}$ -equivariant sheaf on $(M^{k_i} \\setminus \\Delta ) \\times A_i^{k_i}$ .",
"$F$ is a sheaf on the space of $(\\prod _i S_{k_i})$ -equivariant maps $(M^{\\sum _i k_i} \\setminus \\Delta ) \\rightarrow A_i^{k_i} $ ; $F_{k_i}=p_{k_i}^{\\ast }(G_{k_i})$ , where $G_{k_i}$ is an $S_{k_i}$ -equivariant object of $\\mathcal {M}_{k_i} \\boxtimes \\mathcal {A}_i^{k_i}$ , i.e.",
"a sheaf on the space of $S_{k_i}$ -equivariant maps $(M^{k_i} \\setminus \\Delta ) \\times A_i^{k_i}$ .",
"$F$ is a sheaf on $((M^{\\sum _i k_i} \\setminus \\Delta ) \\times A_i^{k_i})/\\prod _i S_{k_i} $ ; $F_{k_i}=p_{k_i}^{\\ast }(G_{k_i})$ , where $G_{k_i}$ is an $S_{k_i}$ -equivariant object of $\\mathcal {M}_{k_i} \\boxtimes \\mathcal {A}_i^{k_i}$ , i.e.",
"a sheaf on $((M^{k_i} \\setminus \\Delta ) \\times A_i^{k_i})/S_{k_i}$ .",
"In the first two descriptions $p_{k_i}$ is the composition $ \\prod _i M^{k_i} \\setminus \\Delta \\hookrightarrow \\prod _i M^{k_i} \\rightarrow M^{k_i} ,$ whereas in the third description $p_{k_i}$ is the induced map on the quotients.",
"Description (I) follows directly from the definition (REF ).",
"The group $S_{k_i}$ acts freely on $(M^{k_i} \\setminus \\Delta ) \\times A_i^{k_i}$ .",
"This gives us description (III).",
"Description (II) follows from description (III) taking into account the description of $((M^{\\sum _i k_i} \\setminus \\Delta ) \\times A_i^{k_i})/\\prod _i S_{k_i}$ as distinct points of $M$ with multiplicities and maps from the points to the appropriate space $A_i$ .",
"Theorem 4.17 The equation (REF ) defines a power structure over the Grothendieck semiring $S_0(gdg-cat)$ of geometric dg categories.",
"Properties 1, 2, 6 and 7 from the definition of a power structure are obvious.",
"Properties 3, 4 and 5 follow straightly from the corresponding properties for the power structure on $K_0(Var)$ by analyzing the spaces on which the sheaves under consideration are defined.",
"Proof of Property 3.",
"Let $\\mathcal {B}(t)=\\sum _{i=0}^{\\infty }[\\mathcal {B}_i]t^{i}$ be another series with coefficients from the same semiring, such that, for each integer $i$ , $H^0(\\mathcal {B}_i)$ is a semiorthogonal summand in $D(B_i)$ , where $B_i$ is a smooth projective variety.",
"Let $[\\mathcal {C}_j]=\\sum _{i=0}^j[\\mathcal {A}_i][\\mathcal {B}_{j-i}]=\\sum _{i=0}^j[\\mathcal {A}_i \\boxtimes \\mathcal {B}_{j-i}]$ be the coefficient of $t^k$ in the product $\\mathcal {A}(t)\\cdot \\mathcal {B}(t)$ .",
"Then $H^0(\\mathcal {C}_i)$ is a semiorthogonal summand in $D(C_i)=D(\\coprod _{i=0}^j A_i \\times B_{j-i})$ .",
"The coefficient of the monomial $t^k$ on the left hand side (LHS) of equation (3) is represented by a category whose objects are sheaves on $\\bigsqcup _{\\underline{k}: \\sum ik_i=k} \\left( ( \\prod _i M^{k_i} \\setminus \\Delta ) \\times \\prod _i C_i^{k_i}\\right) / \\prod _i S_{k_i}.$ Each such sheaf can be written as $F=F_{k_1} \\otimes \\dots $ , where $F_{k_i}=p^{\\ast }_{k_i}(G_{k_i})$ for some sheaf $G_{k_i}$ on $(M^{k_i} \\setminus \\Delta )/S_{k_i}$ with values in $\\mathcal {C}_j=\\sum _{i=0}^j \\mathcal {A}_i \\boxtimes \\mathcal {B}_{j-i}$ .",
"The coefficient of the monomial $t^k$ on the right hand side (RHS) of equation (3) is represented by a category whose objects are sheaves on $\\bigsqcup _{\\begin{array}{c}k_1+k_2=k \\\\ \\underline{k_1}: \\sum ik_{1,i}=k_1\\\\ \\underline{k_2}: \\sum ik_{2,i}=k_2 \\end{array}} \\left( ( \\prod _i M^{k_{1,i}} \\setminus \\Delta ) \\times \\prod _i A_i^{k_{1,i}}\\right) / \\prod _i S_{k_{1,i}} \\\\ \\times \\left( ( \\prod _i M^{k_{2,i}} \\setminus \\Delta ) \\times \\prod _i B_i^{k_{2,i}}\\right) / \\prod _i S_{k_{2,i}}.$ Here the sheaves are of the form $F=F_{k_{1,1}} \\otimes \\dots \\otimes F_{k_{2,1}} \\otimes \\dots $ , where $F_{k_{1,i}}=p^{\\ast }_{k_{1,i}}(G_{k_{1,i}})$ for some sheaf $G_{k_{1,i}}$ on $(X^{k_{1,i}} \\setminus \\Delta )/S_{k_{1,i}}$ with values in $\\mathcal {A}_i$ , and $F_{k_{2,i}}=p^{\\ast }_{k_{2,i}}(G_{k_{2,i}})$ for some sheaf $G_{k_{2,i}}$ on $(X^{k_{2,i}} \\setminus \\Delta )/S_{k_{2,i}}$ with values in $\\mathcal {B}_i$ .",
"The fact that the two underlying spaces of the categories on the LHS and on the RHS are canonically isomorphic follows from the proof of Property 3 for the power structure on $K_0(Var)$ in [17].",
"Then, the identity functor for the sheaves on one space to the other is an equivalence of categories between LHS and RHS.",
"Proof of Property 4.",
"An object of the coefficient of the monomial $t^k$ on the LHS is sheaf on $\\bigsqcup _{\\underline{k}: \\sum ik_i=k}\\left( ( \\prod _i (M\\sqcup N)^{k_i} \\setminus \\Delta ) \\times \\prod _i A_i^{k_i}\\right) / \\prod _i S_{k_i},$ where $\\underline{k}$ is a partition of $k$ .",
"An object of the coefficient of the monomial $t^k$ on the RHS is sheaf on $ \\bigsqcup _{\\begin{array}{c}k_1+k_2=k \\\\ \\underline{k_1}: \\sum ik_{1,i}=k_1\\\\ \\underline{k_2}: \\sum ik_{2,i}=k_2 \\end{array}} \\left( ( \\prod _i M^{k_{1,i}} \\setminus \\Delta ) \\times \\prod _i A_i^{k_{1,i}}\\right) / \\prod _i S_{k_{1,i}} \\\\ \\times \\left( ( \\prod _i N^{k_{2,i}} \\setminus \\Delta ) \\times \\prod _i A_i^{k_{2,i}}\\right) / \\prod _i S_{k_{2,i}}.$ Again, the canonical isomorphism of the underlying spaces of the categories on the LHS and on the RHS follows from the proof of Property 4 for the power structure on $K_0(Var)$ in [17].",
"Proof of Property 5.",
"Again, an object of the coefficient of the monomial $t^k$ on the LHS is sheaf on $\\bigsqcup _{\\underline{k}: \\sum ik_i=k}\\left( ( \\prod _i (M\\times N)^{k_i} \\setminus \\Delta ) \\times \\prod _i A_i^{k_i}\\right) / \\prod _i S_{k_i}$ for some partition $\\underline{k}$ of $k$ .",
"An object of the coefficient of the monomial $t^k$ on the RHS is sheaf on $\\bigsqcup _{\\underline{k}: \\sum ik_i=k}\\left( ( \\prod _i N^{k_i} \\setminus \\Delta ) \\times \\prod _i \\left( \\bigsqcup _{\\underline{i}: \\sum ii_i=i}\\left( ( \\prod _i (M\\times N)^{i_i} \\setminus \\Delta ) \\times \\prod _i A_i^{i_i}\\right) / \\prod _i S_{i_i} \\right)^{k_i}\\right) \\\\ {/} \\prod _i S_{k_i}.$ As above, a canonical isomorphism of the underlying spaces of the categories on the LHS and on the RHS is given in the proof of Property 5 for the power structure on $K_0(Var)$ in [17].",
"A more precise version of Theorem REF is Theorem 4.18 There exists a unique power structure over the Grothendieck ring $K_0(gdg-cat)$ which extends the one defined over the semiring $S_0(gdg-cat)$ .",
"Taking into account Theorem REF , the proof of this theorem is the same as that of the analogous [17] for the power structure on $K_0(Var)$ except for replacing varieties by categories.",
"We do not reproduce it here."
],
[
"The categorical zeta function",
"The categorical zeta-function of a geometric dg category $\\mathcal {M}$ was defined in [15] as $ Z_{cat}(\\mathcal {M},t)=\\sum _{n=0}^{\\infty }[\\operatorname{Sym}^n(\\mathcal {M})]t^n \\in 1+ tK_0(gdg-cat)[[t]].$ Let us recall Theorem REF here.",
"Theorem 5.1 $Z_{cat}(\\mathcal {M},t)=\\prod _{n=1}^{\\infty }\\left( \\frac{1}{1-q^n}\\right)^{[\\mathcal {M}]}.$ The expansion $\\prod _{n=1}^{\\infty }\\frac{1}{1-q^n}=\\sum _{k=0}p(k)$ is well known.",
"Here $p(k)$ is the number of partitions of $k$ .",
"Therefore, we need to show that $[\\operatorname{Sym}^k(\\mathcal {M})]=\\sum _{\\underline{k}: \\sum ik_i=k} \\left[ \\left(\\mathcal {M}_{\\sum _i k_i}^{\\sum _i k_i} \\times \\prod _i [p(i)]^{k_i}\\right)^{\\prod _i S_{k_i}} \\right]$ By Proposition REF there is a decomposition $ [\\mathcal {M}^k]=\\sum _{\\underline{k}: \\sum ik_i=k} [\\mathcal {M}_{\\underline{k}}^k] $ of the classes in $K_0(gdg-cat)$ .",
"This gives the decomposition $ [\\operatorname{Sym}^k(\\mathcal {M})]=\\sum _{\\underline{k}: \\sum ik_i=k} [(\\mathcal {M}_{\\underline{k}}^k)^{S_k}] $ for the classes of the categories of $S_n$ -equivariant objects.",
"Hence, to prove (REF ) it is enough to prove the categorical equivalence $ \\left(\\mathcal {M}_{\\sum _i k_i}^{\\sum _i k_i} \\times \\prod _i [p(i)]^{k_i}\\right)^{\\prod _i S_{k_i}}=(\\mathcal {M}_{\\underline{k}}^k)^{S_k}.",
"$ For simplicity, first suppose that $\\mathcal {M}=I(M)$ .",
"We want to show that $ \\left(I\\left(M^{\\sum _i k_i}_{({ \\sum _i k_i},0\\dots ,0)}\\right) \\times \\prod _i [p(i)]^{k_i}\\right)^{\\prod _i S_{k_i}}=\\mu \\left([M_{\\underline{k}}^k/S_k]\\right) $ for every sequence $\\underline{k}$ of $k$ .",
"The stabilizer of each point in $M_{\\underline{k}}^k$ is $H=\\prod _i (S_i)^{k_i}$ .",
"Therefore, the coarse map $[M_{\\underline{k}}^k/S_k] \\rightarrow M_{\\underline{k}}^k/S_k$ is an $H$ -gerbe, i.e.",
"a Zariski locally trivial fibration whose fiber is $BH$ .",
"The coarse spaces $M_{\\underline{k}}^k/S_k$ and $M^{\\sum _i k_i}_{({\\sum _i k_i},0\\dots ,0)}/ \\prod _i S_{k_i}$ are naturally isomorphic.",
"But the stacks $[M_{\\underline{k}}^k/S_k]$ and $[M^{\\sum _i k_i}_{({\\sum _i k_i},0\\dots ,0)}/ \\prod _i S_{k_i}]$ are not equivalent, since the group in the latter case acts freely.",
"To get a stack equivalent to $[M_{\\underline{k}}^k/S_k]$ we have to equip each configuration in $M^{\\sum _i k_i}_{({\\sum _i k_i},0\\dots ,0)}/ \\prod _i S_{k_i}$ with a map to $BH$ such that a point of multiplicity $i$ in the configuration is mapped to $BS_i$ (i.e.",
"locally the map from the configuration is a product of maps from the points of the configuration).",
"Points of multiplicity $i$ in $M^{\\sum _i k_i}_{({\\sum _i k_i},0\\dots ,0)}/ \\prod _i S_{k_i}$ are those which are in the image of the component $M^{\\sum _i k_i}$ .",
"Equivalently, $[M_{\\underline{k}}^k/S_k]$ is isomorphic to the space of $(\\prod _i S_{k_i})$ -equivariant maps $M^{\\sum _i k_i}_{({\\sum _i k_i},0\\dots ,0)} \\rightarrow BH$ .",
"As a consequence, $ \\mu ([M_{\\underline{k}}^k/S_k]) & =I( [M_{\\underline{k}}^k/S_k]) \\\\ & =I\\left({\\left(\\prod _i S_{k_i}\\right) \\textrm {-equivariant maps } M^{\\sum _i k_i}_{({ \\sum _i k_i},0\\dots ,0)} \\rightarrow BH }\\right) \\\\ & =I\\left( \\left[\\left(M^{\\sum _i k_i}_{({ \\sum _i k_i},0\\dots ,0)} \\times BH\\right) /\\prod _i S_{k_i}\\right] \\right), $ where the group $\\prod _i S_{k_i}$ in the last term acts on the two components simultaneously.",
"The stack $BH$ is equivalent to the quotient stack $[\\lbrace pt\\rbrace /H]$ .",
"Applying Proposition REF for $H=\\prod _i (S_i)^{k_i}$ we obtain that $ I\\left(\\left[\\left(M^{\\sum _i k_i}_{({ \\sum _i k_i},0\\dots ,0)} \\times BH\\right) /\\prod _i S_{k_i}\\right]\\right) & =\\left(I\\left(M^{\\sum _i k_i}_{({ \\sum _i k_i},0\\dots ,0)}\\right) \\times I(BH)\\right)^{\\prod _i S_{k_i}} \\\\ &=\\left(I\\left(M^{\\sum _i k_i}_{({ \\sum _i k_i},0\\dots ,0)}\\right) \\times \\prod _i [p(i)]^{k_i}\\right)^{\\prod _i S_{k_i}}.$ In general, exactly the same proof shows the statement for an arbitrary geometric dg category $\\mathcal {M}$ if we replace $I(M^{\\sum _i k_i}_{({ \\sum _i k_i},0\\dots ,0)})$ (resp., $\\mu ([M_{\\underline{k}}^k/S_k])$ ) with $\\mathcal {M}^{\\sum _i k_i}_{({ \\sum _i k_i},0\\dots ,0)}$ (resp., $(\\mathcal {M}_{\\underline{k}}^k)^{S_k}$ ).",
"By comparing Theorem REF and (REF ) we get Corollary REF ."
],
[
"The class of the Hilbert scheme of points",
"Let $R_1$ and $R_2$ be rings with power structures over them.",
"A ring homomorphism $ \\phi \\colon R_1 \\rightarrow R_2$ induces a natural homomorphism $R_1[[t]] \\rightarrow R_2[[t]]$ (also denoted by $\\phi $ ) by $\\phi (\\sum _n a_n t^n)=\\sum _n \\phi (a_n)t^n$ .",
"Lemma 6.1 ([18]) To define a power structure over a ring $R$ it is sufficient to define the series $(1 - t)^{-a} = 1 + at + \\dots $ for each $a \\in R$ , so that $(1 - t)^{-(a+b)} = (1 - t)^{-a}(1 - t)^{-b}$ , $(1 - t)^{-1}= 1 + t + t^2 +\\dots $ .",
"It follows that to specify a homomorphism between rings with power structures it is enough to specify the homomorphism on elements of the form $(1 - t)^{-a}$ , $a \\in R$ .",
"This and Theorem REF give Proposition 6.2 If a ring homomorphism $\\phi \\colon R_1 \\rightarrow R_2$ is such that $\\phi ((1-t)^{-a})=\\prod _{n=1}^{\\infty }(1-t^n)^{-\\phi (a)}$ , then $\\phi ((A(t))^m)=\\prod _{n=1}^{\\infty } (\\phi (A(t^n)))^{\\phi (m)}$ .",
"In particular, for the ring homomorphism $\\mu \\colon K_0(Var) \\rightarrow K_0 (gdg-cat)$ , $ \\mu \\left((A(t))^{[M]}\\right)=\\prod _{n=1}^{\\infty } (\\mu (A(t^n)))^{\\mu (M)}.",
"$ Let $X$ be any quasi-projective variety.",
"Consider the generating series of the motives of Hilbert scheme of points on $X$ : $ H_X(t)=\\sum _{n=0}^{\\infty } [\\mathrm {Hilb}^n(X)] t^n \\; \\in 1+tK_0(Var)[[t]].$ Here $\\mathrm {Hilb}^n(X)$ is the Hilbert scheme of $n$ points on $X$ .",
"Proposition 6.3 ([18]) For a smooth quasi-projective variety $X$ of dimension $d$ , $H_X(t)= \\left(H_{\\mathbb {A}^d,0}(t)\\right)^{[X]},$ where $H_{\\mathbb {A}^d,0}(t)=\\sum _{n=0}^{\\infty }[\\mathrm {Hilb}^n(\\mathbb {A}^d,0)]t^n$ is the generating series of the motives of Hilbert scheme of $n$ points on $\\mathbb {A}^d$ supported at the origin.",
"Propositions REF (2) and REF give Corollary 6.4 For a smooth projective variety $X$ of dimension $d$ , $\\sum _{n=0}^{\\infty } [I(\\mathrm {Hilb}^n(X))] t^n=\\mu (H_X(t))=\\prod _{n=1}^{\\infty }\\mu \\left(H_{\\mathbb {A}^d,0}(t^n)\\right)^{[I(X)]}.",
"$"
],
[
"Character formula of the categorified Heisenberg action",
"Let $X$ be a smooth projective variety.",
"The geometric dg category $ \\mathbb {I}=\\bigoplus _{n \\ge 0} I([X^n/S_n]) $ is an enhancement of $ \\mathbb {D}=\\bigoplus _{n \\ge 0} D([X^n/S_n]).",
"$ Following [22] we define the following functors.",
"For $1 \\le n \\le N$ and $A \\in D(X)$ let $ P_{N,F}^{(n)} \\colon I(([X^{N-n}/S_{N-n}]) \\rightarrow I([X^N/S_N]), \\quad F \\mapsto \\mathrm {Inf}_{S_n \\times S_{N-n}}^{S_N}( A^{\\otimes n} \\otimes F), $ where $ \\mathrm {Inf}_{S_n \\times S_{N-n}}^{S_N} \\colon I([X^{N}/(S_n \\times S_{N-n})]) \\rightarrow I([X^N/S_N]) $ is the adjoint of the forgetful functor.",
"Let moreover $ P_{F}^{(n)}=\\bigoplus _{N \\ge n} P_{N,F}^{n} \\colon \\mathbb {I}\\rightarrow \\mathbb {I}, \\quad \\textrm {for } n\\ge 1, \\quad P_{F}^{(1)}:=Id_{\\mathbb {I}}.",
"$ Finally, let $Q_{F}^{(n)} \\colon \\mathbb {I}\\rightarrow \\mathbb {I}$ be the right adjoint of $P_{F}^{(n)}$ .",
"These functors are the dg enhancements of the derived functors defined in [22] (see also [11]).",
"In fact, the statement and the whole proof of [22] copies word-by-word to the dg enhancements giving Theorem 6.5 The functors $P_{F}^{(n)}$ and $Q_{F}^{(n)}$ make $\\mathbb {I}$ a (categorified) representation of a (categorical) Heisenberg algebra.",
"Theorem REF gives the class in $K_0(gdg-cat)$ of the categorified “character formula” for this representation.",
"Comparing it to the character formula of the classical Fock space representation of the Heisenberg algebra one obtains Corollary 6.6 $\\mathbb {I}$ is a categorified irreducible highest weight representation of the Heisenberg algebra.",
"It is instructive to compare this result to [26]."
]
] | 1709.01678 |
[
[
"Invariant, super and quasi-martingale functions of a Markov process"
],
[
"Abstract We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process.",
"Applications to semi-Dirichlet forms are given.",
"We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same.",
"Finally, using the co-excessive functions, we present a two-step approach to the existence of invariant probability measures."
],
[
"Introduction",
"Let $E$ be a Lusin topological space endowed with the Borel $\\sigma $ -algebra $\\mathcal {B}$ and $X = (\\Omega , \\mathcal {F}, \\mathcal {F}_t, X_t, \\mathbb {P}^x, \\zeta )$ be a right Markov process with state space $E$ , transition function $(P_t)_{t \\ge 0}$ : $P_t u(x) = \\mathbb {E}^x (u(X_t); t < \\zeta )$ , $t \\ge 0$ , $x\\in E$ .",
"One of the fundamental connections between potential theory and Markov processes is the relation between excessive functions and (right-continuous) supermartingales; see e.g.",
"[14], Chapter VI, Section 10, or [19], Proposition 13.7.1 and Theorem 14.7.1.",
"Similar results hold for (sub)martingales, and together stand as a keystone at the foundations of the so called probabilistic potential theory.",
"For completeness, let us give the precise statement; a short proof is included in Appendix.",
"The following assertions are equivalent for a non-negative real-valued $\\mathcal {B}$ -measurable function $u$ and $\\beta \\ge 0$ .",
"i) $(e^{-\\beta t}u(X_t))_{t\\ge 0}$ is a right continuous $\\mathcal {F}_t$ -supermartingale w.r.t.",
"$\\mathbb {P}^x$ for all $x \\in E$ .",
"ii) The function $u$ is $\\beta $ -excessive.",
"Our first aim is to show that this connection can be extended to the space of differences of excessive functions on the one hand, and to quasimartingales on the other hand (cf.",
"Theorem from Section ), with concrete applications to semi-Dirichlet forms (see Theorem REF below).",
"Recall the following famous characterization from [12]: If $u$ is a real-valued $\\mathcal {B}$ -measurable function then $u(X)$ is an $\\mathcal {F}_t$ -semimartingale w.r.t.",
"all $\\mathbb {P}^x$ , $x\\in E$ if and only if $u$ is locally the difference of two finite 1-excessive functions.",
"The main result from Theorem should be regarded as an extension of Proposition and as a refinement of the just mentioned characterization for semimartingales from Remark .",
"However, we stress out that our result is not a consequence of the two previously known results.",
"In Section we focus on a special class of (0-)excessive functions called invariant, which were studied in the literature from several slightly different perspectives.",
"Here, our aim is to provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, and co-invariant functions, showing that in the Markovian (conservative) case they are all the same.",
"The measurable structure of invariant functions is also involved.",
"We give the results in terms of $L^p(E,m)$ -resolvents of operators, where $m$ is assumed sub-invariant, allowing us to drop the strong continuity assumption.",
"In addition, we show that when the resolvent is associated to a right process, then the martingale functions and the co-martingale ones (i.e., martingale w.r.t.",
"to a dual process) also coincide.",
"The last topic where the existence of (co)excessive functions plays a fundamental role is the problem of existence of invariant probability measures for a fixed Markovian transition function $(P_t)_{t\\ge 0}$ on a general measurable space $(E,\\mathcal {B})$ .",
"Recall that the classical approach is to consider the dual semigroup of $(P_t)_{t\\ge 0}$ acting on the space of all probabilities $P(E)$ on $E$ , and to show that it or its integral means, also known as the Krylov-Bogoliubov measures, are relatively compact w.r.t.",
"some convenient topology (metric) on $P(E)$ (e.g.",
"weak topology, (weighted) total variation norm, Wasserstein metric, etc).",
"In essence, there are two kind of conditions which stand behind the success of this approach: some (Feller) regularity of the semigroup $(P_t)_{t\\ge 0}$ (e.g.",
"it maps bounded and continuous (Lipschitz) functions into bounded and continuous (Lipschitz) functions), and the existence of some compact (or small) sets which are infinitely often visited by the process; see e.g.",
"[26], [27], [28], [13], [23], [20], [22].",
"Our last aim is to present (in Section 4) a result from [7], which offers a new (two-step) approach to the existence of invariant measures (see Theorem below).",
"In very few words, our idea was to first fix a convenient auxilliary measure $m$ (with respect to which each $P_t$ respects classes), and then to look at the dual semigroup of $(P_t)_{t\\ge 0}$ acting not on measures as before, but on functions.",
"In this way we can employ some weak $L^1(m)$ -compactness results for the dual semigroup in order to produce a non-zero and non-negative co-excessive function.",
"At this point we would like to mention that most of the announced results, which are going to be presented in the next three sections, are exposed with details in [6], [7], and [5].",
"The authors had the pleasure to be coauthors of Michael Röckner and part of the results presented in this survey paper were obtained jointly.",
"So, let us conclude this introduction with a \"Happy Birthday, Michael!\""
],
[
"Differences of excessive functions and quasimartingales of Markov processes",
"Recall that the purpose of this section is to study those real-valued measurable functions $u$ having the property that $u(X)$ is a $\\mathbb {P}^x$ -quasimartingale for all $x \\in E$ (in short, \"$u(X)$ is a quasimartingale\", or \"$u$ is a quasimartingale function\").",
"At this point we would like to draw the attention to the fact that in the first part of this section we study quasimartingales with respect to $\\mathbb {P}^x$ for all $x \\in E$ , in particular all the inequalities involved are required to hold pointwise for all $x \\in E$ .",
"Later on we shall consider semigroups or resolvents on $L^p$ or Dirichlet spaces with respect to some duality measure, and in these situations we will explicitly mention if the desired properties are required to hold almost everywhere or outside some exceptional sets.",
"For the reader's convenience, let us briefly present some classic facts about quasimartingales in general.",
"Let $(\\Omega , \\mathcal {F}, \\mathcal {F}_t, \\mathbb {P})$ be a filtered probability space satisfying the usual hypotheses.",
"An $\\mathcal {F}_t$ -adapted, right-continuous integrable process $(Z_t)_{t \\ge 0}$ is called $\\mathbb {P}$ -quasimartingale if ${Var}^\\mathbb {P}(Z):= \\mathop {\\sup }\\limits _{\\tau } \\mathbb {E} \\lbrace \\mathop {\\sum }\\limits _{i = 1}^{n} |\\mathbb {E}[Z_{t_i} - Z_{t_{i-1}}|\\mathcal {F}_{t_{i-1}}]| + |Z_{t_n}|\\rbrace < \\infty ,$ where the supremum is taken over all partitions $\\tau : 0 = t_0 \\le t_1 \\le \\ldots \\le t_n < \\infty $ .",
"Quasimartingales played an important role in the development of the theory of semimartingales and stochastic integration, mainly due to M. Rao's theorem according to which any quasimartingale has a unique decomposition as a sum of a local martingale and a predictable process with paths of locally integrable variation.",
"Conversely, one can show that any semimartingale with bounded jumps is locally a quasimartingale.",
"However, to the best of our knowledge, their analytic or potential theoretic aspects have never been investigated or, maybe, brought out to light, before.",
"We return now to the frame given by a Markov process.",
"Further in this section we deal with a right Markov process $X = (\\Omega , \\mathcal {F}, \\mathcal {F}_t, X_t, \\mathbb {P}^x, \\zeta )$ with state space $E$ and transition function $(P_t)_{t \\ge 0}$ .",
"Although we shall not really be concerned with the lifetime formalism, if $X$ has lifetime $\\xi $ and cemetery point $\\Delta $ , then we make the convention $u(\\Delta ) = 0$ for all functions $u: E \\rightarrow [-\\infty , + \\infty ]$ .",
"Recall that for $\\beta \\ge 0$ , a $\\mathcal {B}$ -measurable function $f:E \\rightarrow [0, \\infty ]$ is called $\\beta $ -supermedian if $P_t^\\beta f \\le f$ pointwise on $E$ , $t \\ge 0$ ; $(P_t^\\beta )_{t \\ge 0}$ denotes the $\\beta $ -level of the semigroup of kernels $(P_t)_{t \\ge 0}$ , $P_t^\\beta := e^{-\\beta } P_t$ .",
"If $f$ is $\\beta $ -supermedian and $\\lim \\limits _{t \\rightarrow 0} P_t f = f$ point-wise on $E$ , then it is called $\\beta $ -excessive.",
"It is well known that a $\\mathcal {B}$ -measurable function $f$ is $\\beta $ -excessive if and only if $\\alpha U_{\\alpha +\\beta }f \\le f$ , $\\alpha >0$ , and $\\lim \\limits _{\\alpha \\rightarrow \\infty } \\alpha U_{\\alpha }f = f$ point-wise on $E$ , where $\\mathcal {U} = (U_{\\alpha })_{\\alpha > 0}$ is the resolvent family of the process $X$ , $U_\\alpha := \\int _0^\\infty e^{-\\alpha t} P_t dt$ .",
"The convex cone of all $\\beta $ -excessive functions is denoted by $E(\\mathcal {U}_\\beta )$ ; here $\\mathcal {U}_\\beta $ denotes the $\\beta $ -level of the resolvent $\\mathcal {U}$ , $\\mathcal {U}_\\beta := (U_{\\beta +\\alpha })_{\\alpha > 0}$ ; the fine topology is the coarsest topology on $E$ such that all $\\beta $ -excessive functions are continuous, for some $\\beta > 0$ .",
"If $\\beta = 0$ we drop the index $\\beta $ .",
"Taking into account the strong connection between excessive functions and supermartingales for Markov processes, the following characterization of M. Rao was our source of inspiration: a real-valued process on a filtered probability space $(\\Omega , \\mathcal {F}, \\mathcal {F}_t, \\mathbb {P})$ satisfying the usual hypotheses is a quasimartingale if and only if it is the difference of two positive right-continuous $\\mathcal {F}_t$ -supermartingales; see e.g.",
"[31], page 116.",
"As a first observation, note that if $u(X)$ is a quasimartingale, then the following two conditions for $u$ are necessary: i) $\\mathop {\\sup }\\limits _{t > 0} P_t|u|< \\infty $ and ii) $u$ is finely continuous.",
"Indeed, since for each $x\\in E$ we have that $\\mathop {\\sup }\\limits _{t} P_t|u|(x) = \\mathop {\\sup }\\limits _{t}\\mathbb {E}^x|u(X_t)| \\le {Var}^{\\mathbb {P}^x}(u(X)) < \\infty $ , the first assertion is clear.",
"The second one follows by the result from [9] which is stated in the proof of Proposition in the Appendix at the end of the paper.",
"For a real-valued function $u$ , a partition $\\tau $ of $\\mathbb {R}^+$ , $\\tau : 0 = t_0 \\le t_1 \\le \\ldots \\le t_n < \\infty $ , and $\\alpha >0$ we set $V^\\alpha (u) := \\mathop {\\sup }\\limits _{\\tau }V^\\alpha _{\\tau }(u), \\quad V^\\alpha _{\\tau }(u) := \\mathop {\\sum }\\limits _{i=1}^{n} P^\\alpha _{t_{i-1}} |u - P^\\alpha _{t_i - t_{i-1}}u| + P^\\alpha _{t_n}|u|$ , where the supremum is taken over all finite partitions of $\\mathbb {R}_+$ .",
"A sequence $(\\tau _n)_{n \\ge 1}$ of finite partitions of $\\mathbb {R}_+$ is called admissible if it is increasing, $\\mathop {\\bigcup }\\limits _{k \\ge 1}\\tau _k$ is dense in $\\mathbb {R}_+$ , and if $r \\in \\mathop {\\bigcup }\\limits _{k \\ge 1}\\tau _k$ then $r + \\tau _n \\subset \\mathop {\\bigcup }\\limits _{k \\ge 1}\\tau _k$ for all $n \\ge 1$ .",
"We can state now our first result, it is a version of Theorem 2.6 from [5].",
"Let $u$ be a real-valued $\\mathcal {B}$ -measurable function and $\\beta \\ge 0$ such that $P_t|u| < \\infty $ for all $t$ .",
"Then the following assertions are equivalent.",
"i) $(e^{-\\beta t}u(X_t))_{t\\ge 0}$ is a $\\mathbb {P}^x$ -quasimartingale for all $x \\in E$ .",
"ii) $u$ is finely continuous and $\\mathop {\\sup }\\limits _{n}V^\\beta _{\\tau _n}(u) < \\infty $ for one (hence all) admissible sequence of partitions $(\\tau _n)_n$ .",
"iii) $u$ is a difference of two real-valued $\\beta $ -excessive functions.",
"The key idea behind the previous result is that by the Markov property is not hard to show that for all $x \\in E$ we have ${Var}^{\\mathbb {P}^x}((e^{-\\alpha t}u(X_t)_{t \\ge 0}) = V^\\alpha (u)(x)$ , meaning that assertion i) holds if and only if $V^\\alpha (u)<\\infty $ .",
"But $V^\\alpha (u)$ is a supremum of measurable functions taken over an uncountable set of partitions, hence it may no longer be measurable, which makes it hard to handle in practice.",
"Concerning this measurability issue, Theorem , ii) states that instead of dealing with $V^\\alpha (u)$ , we can work with $\\mathop {\\sup }\\limits _{n}V^\\alpha _{\\tau _n}(u)$ for any admissible sequence of partitions $(\\tau _n)_{n\\ge 1}$ .",
"This subtile aspect was crucial in order to give criteria to check the quasimartingale nature of $u(X)$ ; see also Proposition REF in the next subsection."
],
[
"Criteria for quasimartingale functions",
"In this subsection, still following [5], we provide general conditions for $u$ under which $(e^{-\\beta t}u(X_t))_{t\\ge 0}$ is a quasimartingale, which means that, in particular, $(u(X_t))_{t\\ge 0}$ is a semimartingale.",
"Let us consider that $m$ is a $\\sigma $ -finite sub-invariant measure for $(P_t)_{t\\ge 0}$ so that $(P_t)_{t \\ge 0}$ extends uniquely to a strongly continuous semigroup of contractions on $L^p(m)$ , $1 \\le p < \\infty $ ; $\\mathcal {U}$ may as well be extended to a strongly continuous resolvent family of contractions on $L^p(m)$ , $1 \\le p < \\infty $ .",
"The corresponding generators $({\\sf L}_p, D({\\sf L}_p) \\subset L^p(m))$ are defined by $D({\\sf L}_p) = \\lbrace U_{\\alpha } f : f \\in L^p(m) \\rbrace ,$ ${\\sf L}_p(U_{\\alpha } f) := \\alpha U_{\\alpha } f - f \\quad {\\rm for \\; all} \\; f \\in L^p(m), \\ 1 \\le p < \\infty ,$ with the remark that this definition is independent of $\\alpha > 0$ .",
"The corresponding notations for the dual structure are $\\widehat{P}_t$ and $(\\widehat{\\sf L}_p, D(\\widehat{\\sf L}_p))$ , and note that the adjoint of ${\\sf L}_p$ is $\\widehat{\\sf L}_{p^\\ast }$ ; $\\frac{1}{p} + \\frac{1}{p^\\ast }=1$ .",
"Throughout, we denote the standard $L^p$ -norms by $\\Vert \\cdot \\Vert _p$ , $1 \\le p \\le \\infty $ .",
"We present below the $L^p$ -version of Theorem ; cf.",
"Proposition 4.2 from [5].",
"The following assertions are equivalent for a $\\mathcal {B}$ -measurable function $u \\in \\mathop {\\bigcup }\\limits _{1 \\le p \\le \\infty } L^p(m)$ and $\\beta \\ge 0$ .",
"i) There exists an $m$ -version $\\widetilde{u}$ of $u$ such that $(e^{-\\beta t}\\widetilde{u}(X_t))_{t\\ge 0}$ is a $\\mathbb {P}^x$ -quasimartingale for $x \\in E$ $m$ -a.e.",
"ii) For an admissible sequence of partitions $(\\tau _n)_{n \\ge 1}$ of $\\mathbb {R}_+$ , $\\mathop {\\sup }\\limits _{n} V^\\beta _{\\tau _n}(u) < \\infty $ $m$ -a.e.",
"iii) There exist $u_1, u_2 \\in E(\\mathcal {U}_\\beta )$ finite $m$ -a.e.",
"such that $u = u_1 - u_2$ $m$ -a.e.",
"Under the assumptions of Proposition REF , if $u$ is finely continuous and one of the equivalent assertions is satisfied then all of the statements hold outside an $m$ -polar set, not only $m$ -a.e., since it is known that an $m$ -negligible finely open set is automatically $m$ -polar; if in addition $m$ is a reference measure then the assertions hold everywhere on $E$ .",
"Now, we focus our attention on a class of $\\beta $ -quasimartingale functions which arises as a natural extension of $D({\\sf L}_p)$ .",
"First of all, it is clear that any function $u \\in D({\\sf L}_p)$ , $1 \\le p < \\infty $ , has a representation $u = U_{\\beta } f = U_{\\beta }(f^+) - U_{\\beta }(f^-)$ with $U_{\\beta }(f^{\\pm }) \\in E(\\mathcal {U}_{\\beta }) \\cap L^p(m)$ , hence $u$ has a $\\beta $ -quasimartingale version for all $\\beta > 0$ ; moreover, $\\Vert P_t u - u \\Vert _p = \\left\\Vert \\int _0^t P_s{\\sf L}_p u ds \\right\\Vert _p \\le t \\Vert {\\sf L}_p u \\Vert _p$ .",
"The converse is also true, namely if $1 < p < \\infty $ , $u \\in L^p(m)$ , and $\\Vert P_t u - u \\Vert _p \\le {const} \\cdot t$ , $t \\ge 0$ , then $u \\in D({\\sf L}_p)$ .",
"But this is no longer the case if $p = 1$ (because of the lack of reflexivity of $L^1$ ), i.e.",
"$\\Vert P_t u - u \\Vert _1 \\le {const} \\cdot t$ does not imply $u \\in D({\\sf L}_1)$ .",
"However, it turns out that this last condition on $L^1(m)$ is yet enough to ensure that $u$ is a $\\beta $ -quasimartingale function.",
"In fact, the following general result holds; see [5], Proposition 4.4 and its proof.",
"Let $1 \\le p < \\infty $ and suppose $\\mathcal {A} \\subset \\lbrace u \\in L^{p^{\\ast }}_+(m) : \\Vert u \\Vert _{p^{\\ast }} \\le 1 \\rbrace $ , $\\widehat{P}_s\\mathcal {A} \\subset \\mathcal {A}$ for all $s \\ge 0$ , and $E = \\mathop {\\bigcup }\\limits _{f \\in \\mathcal {A}}{\\rm supp}(f)$ $m$ -a.e.",
"If $u \\in L^p(m)$ satisfies $\\sup \\limits _{f \\in \\mathcal {A}}\\int _E |P_tu - u| f d m \\le {const} \\cdot t$ for all $t \\ge 0$ , then there exists and $m$ -version $\\widetilde{u}$ of $u$ such that $(e^{-\\beta t}\\widetilde{u}(X_t))_{t\\ge 0}$ is a $\\mathbb {P}^x$ -quasimartingale for all $x \\in E$ $m$ -a.e.",
"and every $\\beta > 0$ .",
"We end this subsection with the following criteria which is not given with respect to a duality measure, but in terms of the associated resolvent $\\mathcal {U}$ ; cf.",
"Proposition 4.1 from [5].",
"Let $u$ be a real-valued $\\mathcal {B}$ -measurable finely continuous function.",
"i) Assume there exist a constant $\\alpha \\ge 0$ and a non-negative $\\mathcal {B}$ -measurable function $c$ such that $U_{\\alpha }(|u| + c) < \\infty , \\quad \\mathop {\\lim \\sup }\\limits _{t \\rightarrow \\infty } P_t^{\\alpha }|u| < \\infty , \\quad |P_t u - u| \\le c t, t \\ge 0,$ and the functions $t \\mapsto P_t(|u| + c)(x)$ are Riemann integrable.",
"Then $(e^{-\\alpha t}u(X_t))_{t \\ge 0}$ is a $\\mathbb {P}^x$ -quasimartingale for all $x\\in E$ .",
"ii) Assume there exist a constant $\\alpha \\ge 0$ and a non-negative $\\mathcal {B}$ -measurable function $c$ such that $|P_t u- u| \\le c t, t \\ge 0, \\quad \\mathop {\\sup }\\limits _{t \\in \\mathbb {R}_+} P_t^{\\alpha }(|u| + c) = : b < \\infty .$ Then $(e^{-\\beta t}u(X_t))_{t \\ge 0}$ is a $\\mathbb {P}^x$ -quasimartingale for all $x\\in E$ and $\\beta > \\alpha $ .",
"iii) Assume there exists $x_0 \\in E$ such that for some $\\alpha \\ge 0$ $U_{\\alpha }(|u|)(x_0) < \\infty , \\quad U_{\\alpha }(|P_t u - u|)(x_0) \\le {const} \\cdot t, \\; t \\ge 0.$ Then $(e^{-\\beta t}u(X_t))_{t \\ge 0}$ is a $\\mathbb {P}^x$ -quasimartingale for $\\delta _{x_0}\\circ U_\\beta $ -a.e.",
"$x\\in E$ and $\\beta > \\alpha $ ; if in addition $\\mathcal {U}$ is strong Feller and topologically irreducible then the $\\mathbb {P}^x$ -quasimartingale property holds for all $x\\in E$ ."
],
[
"Applications to semi-Dirichlet forms",
"Assume now that the semigroup $(P_t)_{t\\ge 0}$ is associated to a semi-Dirichlet form $(\\mathcal {E},\\mathcal {F})$ on $L^2(E,m)$ , where $m$ is a $\\sigma $ -finite measure on the Lusin measurable space $(E,{\\mathcal {B}})$ ; as standard references for the theory of (semi-)Dirichlet forms we refer the reader to [25], [24], [18], [29], but also [3], Chapter 7.",
"By Corollary 3.4 from [4] there exists a (larger) Lusin topological space $E_1$ such that $E\\subset E_1$ , $E$ belongs to ${\\mathcal {B}}_1$ (the $\\sigma $ -algebra of all Borel subsets of $E_1$ ), ${\\mathcal {B}}={\\mathcal {B}}_1|_E$ , and $(\\mathcal {E}, \\mathcal {F})$ regarded as a semi-Dirichlet form on $L^2(E_1 , \\overline{m})$ is quasi-regular, where $\\overline{m}$ is the trivial extension of $m$ to $(E_1,{\\mathcal {B}}_1)$ .",
"Consequently, we may consider a right Markov process $X$ with state space $E_1$ which is associated with the semi-Dirichlet form $(\\mathcal {E},\\mathcal {F})$ .",
"If $u \\in \\mathcal {F}$ then $\\widetilde{u}$ denotes a quasi continuous version of $u$ as a function on $E_1$ which always exists and it is uniquely determined quasi everywhere.",
"Following [15], for a closed set $F$ we define $\\mathcal {F}_{b, F}:=\\lbrace v\\in \\mathcal {F} : v \\mbox{ is bounded and } v=0 \\; m\\mbox{-a.e.",
"on } E\\setminus F\\rbrace $ .",
"The next result is a version of Theorem 5.5 from [5], dropping the a priori assumption that the semi-Dirichlet form is quasi-regular.",
"Let $u \\in \\mathcal {F}$ and assume there exist a nest $(F_n)_{n\\ge 1}$ and constants $(c_n)_{n\\ge 1}$ such that $\\mathcal {E}(u,v) \\le c_n \\Vert v\\Vert _\\infty \\;\\; \\mbox{for all} \\; v\\in \\mathcal {F}_{b, F_n}.$ Then $\\widetilde{u}(X)$ is a $\\mathbb {P}^x$ -semimartingale for $x\\in E_1$ quasi everywhere.",
"The previous result has quite a history behind and we take the opportunity to recall some previous achievements on the subject.",
"First of all, without going into details, note that if $E$ is a bounded domain in $\\mathbb {R}^{d}$ (or more generally in an abstract Wiener space) and the condition from Theorem REF holds for $u$ replaced by the canonical projections, then the conclusion is that the underlying Markov process is a semimartingale.",
"In particular, the semimartingale nature of reflected diffusions on general bounded domains can be studied.",
"This problem dates back to the work of [2], where the authors showed that the reflected Brownian motion on a Lipschitz domain in $\\mathbb {R}^d$ is a semimartingale.",
"Later on, this result has been extended to more general domains and diffusions; see [38], [10], [11], and [30].",
"A clarifying result has been obtained in [11], showing that the stationary reflecting Brownian motion on a bounded Euclidian domain is a quasimartingale on each compact time interval if and only if the domain is a strong Caccioppoli set.",
"At this point it is worth to emphasize that in the previous sections we studied quasimartingales on the hole positive real semi-axis, not on finite intervals.",
"This slight difference is a crucial one which makes our approach possible and completely different.",
"A complete study of these problems (including Theorem REF but only in the symmetric case) have been done in a series of papers by M. Fukushima and co-authors (we mention just [15], [16], and [17]), with deep applications to BV functions in both finite and infinite dimensions.",
"All these previous results have been obtained using the same common tools: symmetric Dirichlet forms and Fukushima decomposition.",
"Further applications to the reflection problem in infinite dimensions have been studied in [33] and [34], where non-symmetric situations were also considered.",
"In the case of semi-Dirichlet forms, a Fukushima decomposition is not yet known to hold, unless some additional hypotheses are assumed (see e.g.",
"[29]).",
"Here is where our study developed in the previous sections played its role, allowing us to completely avoid Fukushima decomposition or the existence of the dual process.",
"On brief, the idea of proving Theorem REF is to show that locally, the conditions from Proposition REF are satisfied, so that $u(X)$ is (pre)locally a semimartingale, and hence a global semimartingale.",
"Assume that $(\\mathcal {E}, \\mathcal {F})$ is quasi-regular and that it is local, i.e., $\\mathcal {E}(u,v)=0$ for all $u,v \\in \\mathcal {F}$ with disjoint compact supports.",
"It is well known that the local property is equivalent with the fact that the associated process is a diffusion; see e.g.",
"[25], Chapter V, Theorem 1.5.",
"As in [16], the local property of $\\mathcal {E}$ allows us to extend Theorem REF to the case when $u$ is only locally in the domain of the form, or to even more general situations, as stated in the next result; for details see Subsection 5.1 from [5].",
"Assume that $(\\mathcal {E}, \\mathcal {F})$ is local.",
"Let $u$ be a real-valued $\\mathcal {B}$ -measurable finely continuous function and let $(v_k)_k \\subset \\mathcal {F}$ such that $v_k \\mathop {\\longrightarrow }\\limits _{k \\rightarrow \\infty } u$ point-wise outside an $m$ -polar set and boundedly on each element of a nest $(F_n)_{n \\ge 1}$ .",
"Further, suppose that there exist constants $c_n$ such that $|\\mathcal {E}(v_k, v)| \\le c_n \\Vert v\\Vert _{\\infty } \\;\\; for \\; all \\; v \\in \\mathcal {F}_{b, F_n}.$ Then $u(X)$ is a $\\mathbb {P}^x$ -semimartingale for $x \\in E$ quasi everywhere."
],
[
"Excessive and invariant functions on $L^p$ -spaces",
"Throughout this section $\\mathcal {U}=(U_\\alpha )_{\\alpha > 0}$ is a sub-Markovian resolvent of kernels on $E$ and $m$ is a $\\sigma $ -finite sub-invariant measure, i.e.",
"$m( \\alpha U_\\alpha f ) \\le m(f)$ for all $\\alpha >0$ and non-negative $\\mathcal {B}$ -measurable functions $f$ ; then there exists a second sub-Markovian resolvent of kernels on $E$ denoted by $\\mathcal {\\widehat{U}}=(\\widehat{U}_\\alpha )_{\\alpha >0}$ which is in weak duality with $\\mathcal {U}$ w.r.t.",
"$M$ in the sense that $\\int _E fU_\\alpha g dm = \\int _E g\\widehat{U}_\\alpha f dm$ for all positive $\\mathcal {B}$ -measurable functions $f,g$ and $\\alpha >0$ .",
"Moreover, both resolvents can be extended to contractions on any $L^p(E,m)$ -space for all $1\\le p\\le \\infty $ , and if they are strongly continuous then we keep the same notations for their generators as in Subsection REF .",
"In this part, our attention focuses on a special class of differences of excessive functions (which are in fact harmonic when the resolvent is Markovian).",
"Extending [1], they are defined as follows.",
"A real-valued $\\mathcal {B}$ -measurable function $v \\in \\bigcup _{1 \\le p \\le \\infty } L^p(E, m)$ is called $\\mathcal {U}$ -invariant provided that $U_{\\alpha }(vf) = v U_{\\alpha } f$ $m$ -a.e.",
"for all bounded and $\\mathcal {B}$ -measurable functions $f$ and $\\alpha > 0$ .",
"A set $A \\in \\mathcal {B}$ is called $\\mathcal {U}$ -invariant if $1_A$ is $\\mathcal {U}$ -invariant; the collection of all $\\mathcal {U}$ -invariant sets is a $\\sigma $ -algebra.",
"If $v \\ge 0$ is $\\mathcal {U}$ -invariant, then by [6], Proposition 2.4 there exists $u \\in E(\\mathcal {U})$ such that $u = v$ $m$ -a.e.",
"If $\\alpha U_{\\alpha } 1 = 1$ $m$ -a.e.",
"then for every invariant function $v$ we have that $\\alpha U_{\\alpha } v = v$ $m$ -a.e, which is equivalent (if $\\mathcal {U}$ is strongly continuous) with $v$ being ${\\sf L}_p$ -harmonic, i.e.",
"$v \\in D({\\sf L}_p)$ and ${\\sf L}_pv=0$ .",
"The following result is a straightforward consequence of the duality between $\\mathcal {U}$ and $\\widehat{\\mathcal {U}}$ ; for its proof see Proposition 2.24 and Proposition 2.25 from [6].",
"The following assertions hold.",
"i) A function $u$ is $\\mathcal {U}$ -invariant if and only if it is $\\widehat{\\mathcal {U}}$ -invariant.",
"ii) The set of all $\\mathcal {U}$ -invariant functions from $L^p(E, m)$ is a vector lattice with respect to the point-wise order relation.",
"Let $\\mathcal {I}_p : = \\lbrace u \\in L^p(E, m) : \\alpha U_\\alpha u = u \\; m\\mbox{-a.e.",
"}, \\; \\alpha > 0 \\rbrace .$ The main result here is the next one, and it unifies and extends different more or less known characterizations of invariant functions; cf.",
"Theorem 2.27 and Proposition 2.29 from [6].",
"Let $u \\in L^p(E, m)$ , $1 \\le p < \\infty $ , and consider the following conditions.",
"i) $\\alpha U_{\\alpha }u = u$ $m$ -a.e.",
"for one (and therefore for all) $\\alpha > 0$ .",
"ii) $\\alpha \\widehat{U}_{\\alpha } u = u$ $m$ -a.e., $\\alpha > 0$ .",
"iii) The function $u$ is $\\mathcal {U}$ -invariant.",
"iv) $U_\\alpha u = u U_\\alpha 1$ and $ \\widehat{U}_{\\alpha } u = u \\widehat{U}_{\\alpha } 1$ $m$ -a.e.",
"for one (and therefore for all) $\\alpha > 0$ .",
"v) The function $u$ is measurable w.r.t.",
"the $\\sigma $ -algebra of all $\\mathcal {U}$ -invariant sets.",
"Then $\\mathcal {I}_p$ is a vector lattice w.r.t.",
"the pointwise order relation and i) $\\Leftrightarrow $ ii) $\\Rightarrow $ iii) $\\Leftrightarrow $ iv) $\\Leftrightarrow $ v).",
"If $\\alpha U_\\alpha 1 = 1$ or $\\alpha \\widehat{U}_{\\alpha } 1 = 1$ $m$ -a.e.",
"then assertions i) - v) are equivalent.",
"If $p= \\infty $ and $\\mathcal {U}$ is $m$ -recurrent (i.e.",
"there exists $0\\le f \\in L^1(E,m)$ s.t.",
"$Uf=\\infty $ $m$ -a.e.)",
"then the assertions i)-v) are equivalent.",
"Similar characterizations for invariance as in Theorem , but in the recurrent case and for functions which are bounded or integrable with bounded negative parts were already investigated in [35].",
"Of special interest is the situation when the only invariant functions are the constant ones (irreducibility) because it entails ergodic properties for the semigroup resp.",
"resolvent; see e.g.",
"[36], [1], and [6]."
],
[
"Martingale functions with respect to the dual Markov process",
"Our aim in this subsection is to identify the $\\mathcal {U}$ -invariant functions with martingale functions and co-martingale ones (i.e., martingales w.r.t some dual process); cf.",
"Corollary REF below.",
"The convenient frame is that from [8] and we present it here briefly.",
"Assume that ${\\mathcal {U}}=(U_{\\alpha })_{\\alpha >0}$ is the resolvent of a right process $X$ with state space $E$ and let ${\\mathcal {T}}_0$ be the Lusin topology of $E$ having ${\\mathcal {B}}$ as Borel $\\sigma $ -algebra, and let $m$ be a fixed ${\\mathcal {U}}$ -excessive measure.",
"Then by Corollary 2.4 from [8], and using also the result from [4], the following assertions hold: There exist a larger Lusin measurable space $(\\overline{E}, \\overline{{\\mathcal {B}}})$ , with $E\\subset \\overline{E}$ , $E\\in \\overline{{\\mathcal {B}}}$ , ${\\mathcal {B}}=\\overline{{\\mathcal {B}}}|_{E}$ , and two processes $\\overline{X}$ and $\\widehat{{X}}$ with common state space $\\overline{E}$ , such that $\\overline{X}$ is a right process on $\\overline{E}$ endowed with a convenient Lusin topology having $\\overline{{\\mathcal {B}}}$ as Borel $\\sigma $ -algebra (resp.",
"$\\widehat{{X}}$ is a right process w.r.t.",
"to a second Lusin topology on $\\overline{E}$ , also generating $\\overline{{\\mathcal {B}}}$ ), the restriction of $\\overline{X}$ to $E$ is precisely $X$ , and the resolvents of $\\overline{X}$ and $\\widehat{{X}}$ are in duality with respect to $\\overline{m}$ , where $\\overline{m}$ is the trivial extension of $m$ to $(E_1,{\\mathcal {B}}_1): \\; \\overline{m}(A):=m(A\\cap E), \\; A \\in \\mathcal {B}_1$ .",
"In addition, the $\\alpha $ -excessive functions, $\\alpha >0$ , with respect to $\\widehat{X}$ on $\\overline{E}$ are precisely the unique extensions by continuity in the fine topology generated by $\\widehat{X}$ of the $\\widehat{{\\mathcal {U}}}_{\\alpha }$ -excessive functions.",
"In particular, the set $E$ is dense in $\\overline{E}$ in the fine topology of $\\widehat{X}$ .",
"Note that the strongly continuous resolvent of sub-Markovian contractions induced on $L^p(m)$ , $1\\le p<\\infty $ , by the process $\\overline{X}$ (resp.",
"$\\widehat{{X}}$ ) coincides with ${\\mathcal {U}}$ (resp.",
"$\\widehat{{\\mathcal {U}}}$ ).",
"Let $u$ be function from $L^p(E, m)$ , $1 \\le p < \\infty $ .",
"Then the following assertions are equivalent.",
"i) The process $(u(X_t))_{t\\ge 0}$ is a martingale w.r.t.",
"$\\mathbb {P}^x$ for $m$ -a.e.",
"$x\\in E$ .",
"ii) The process $(u(\\widehat{X}_t))_{t\\ge 0}$ is a martingale w.r.t.",
"$\\widehat{\\mathbb {P}}^x$ for $m$ -a.e.",
"$x\\in E$ .",
"iii) The function $u$ is ${\\sf L}_p$ -harmonic, i.e.",
"$u \\in D({\\sf L}_p)$ and ${\\sf L}_p u=0$ .",
"iv) The function $u$ is ${\\widehat{\\sf L}}_p$ -harmonic, i.e.",
"$u \\in D({\\widehat{\\sf L}}_p)$ and ${\\widehat{\\sf L}}_p u=0$ .",
"The equivalence $iii) \\Longleftrightarrow iv)$ follows by Theorem , $i) \\Longleftrightarrow ii)$ , while the equivalence $i) \\Longleftrightarrow iii)$ is a consequence of Proposition .",
"$\\hfill \\square $ We make the transition to the next (also the last) section of this paper with an application of Theorem to the existence of invariant probability measures for Markov processes.",
"More precisely, assume that $\\mathcal {U}$ is the resolvent of a right Markov process with transition function $(P_t)_{t \\ge 0}$ .",
"As before, $m$ is a $\\sigma $ -finite sub-invariant measure for $\\mathcal {U}$ (and hence for $(P_t)_{t\\ge 0}$ ), while ${\\sf L}_1$ and $\\widehat{{\\sf L}}_1$ stand for the generator, resp.",
"the co-generator on $L^1(E,m)$ .",
"The following assertions are equivalent.",
"i) There exists an invariant probability measure for $(P_t)_{t\\ge 0}$ which is absolutely continuous w.r.t.",
"$m$ .",
"ii) There exists a non-zero element $\\rho \\in D({\\sf L}_1)$ such that ${\\sf L}_1 \\rho = 0$ .",
"It is well known that a probability measure $\\rho \\cdot m$ is invariant w.r.t.",
"$(P_t)_{t \\ge 0}$ is equivalent with the fact that $\\rho \\in D(\\widehat{\\sf L}_1)$ and $\\widehat{{\\sf L}}_1 \\rho =0$ (see also Lemma , ii) from below).",
"Now, the result follows by Theorem .",
"Regarding the previous result, we point out that if $m(E) < \\infty $ and $(P_t)_{t \\ge 0}$ is conservative (i.e.",
"$P_t1=1$ $m$ -a.e.",
"for all $t>0$ ) then it is clear that $m$ itself is invariant, so that Corollary REF has got a point only when $m(E)=\\infty $ .",
"Also, we emphasize that the sub-invariance property of $m$ is an essential assumption.",
"We present a general result on the existence of invariant probability measures in the next section, where we drop the sub-invariance hypothesis."
],
[
"$L^1$ -harmonic functions and invariant probability measures",
"Throughout this subsection $(P_t)_{t \\ge 0}$ is a measurable Markovian transition function on a measurable space $(E, \\mathcal {B})$ and $m$ is an auxiliary measure for $(P_t)_{t\\ge 0}$ , i.e.",
"a finite positive measure such that $m(f) = 0 \\Rightarrow m(P_tf) = 0$ for all $t > 0$ and all positive $\\mathcal {B}$ -measurable functions $f$ .",
"As we previously announced, our final interest concerns the existence of an invariant probability measure for $(P_t)_{t \\ge 0}$ which is absolutely continuous with respect to $m$ .",
"We emphasize once again that in contrast with the previous section, $m$ is not assumed sub-invariant, since otherwise it would be automatically invariant.",
"Also, any invariant measure is clearly auxiliary, but the converse is far from being true.",
"As a matter of fact, the condition on $m$ of being auxiliary is a minimal one: for every finite positive measure $\\mu $ and $\\alpha >0$ one has that $\\mu \\circ U_\\alpha $ is auxiliary; see e.g.",
"[32] and [7].",
"For the first assertion of the next result we refer to [7], Lemma 2.1, while the second one is a simple consequence of the fact that $P_t1=1$ .",
"i) The adjoint semigroup $(P_t^{\\ast })_{t \\ge 0}$ on $(L^{\\infty }(m))^{\\ast }$ maps $L^1(m)$ into itself, and restricted to $L^1(m)$ it becomes a semigroup of positivity preserving operators.",
"ii) A probability measure $\\rho \\cdot m$ is invariant with respect to $(P_t)_{t \\ge 0}$ if and only if $\\rho $ is $m$ -co-excessive, i.e.",
"$P_t^\\ast \\rho \\le \\rho $ $m$ -a.e.",
"for all $t \\ge 0$ .",
"Inspired by well known ergodic properties for semigroups and resolvents (see for example [6]), our idea in order to produce co-excessive functions is to apply (not for $(P_t)_{t \\ge 0}$ but for its adjoint semigroup) a compactness result in $L^1(m)$ due to [21], saying that an $L^1(m)$ -bounded sequence of elements possesses a subsequence whose Cesaro means are almost surely convergent to a limit from $L^1(m)$ .",
"The auxilliary measure $m$ is called almost invariant for $(P_t)_{t \\ge 0}$ if there exist $\\delta \\in [0, 1)$ and a set function $\\phi : \\mathcal {B} \\rightarrow \\mathbb {R}_+$ which is absolutely continuous with respect to $m$ (i.e.",
"$\\mathop {\\lim }\\limits _{m(A) \\rightarrow 0} \\phi (A) = 0$ ) such that $m (P_t1_A) \\le \\delta m(E) + \\phi (A) \\quad \\mbox{for all} \\; t > 0.",
"$ Clearly, any positive finite invariant measure is almost invariant.",
"Here is our last main result, a variant of Theorem 2.4 from [7].",
"The following assertions are equivalent.",
"i) There exists a nonzero positive finite invariant measure for $(P_t)_{t \\ge 0}$ which is absolutely continuous with respect to $m$ .",
"ii) $m$ is almost invariant.",
"The first named author acknowledges support from the Romanian National Authority for Scientific Research, project number PN-III-P4-ID-PCE-2016-0372.",
"The second named author acknowledges support from the Romanian National Authority for Scientific Research, project number PN-II-RU-TE-2014-4-0657."
],
[
"Appendix",
"Proof of Proposition .",
"i) $\\Rightarrow $ ii).",
"If $(e^{-\\beta t}u(X_t))_{t\\ge 0}$ is a right-continuous supermartingale then by taking expectations we get that $e^{-\\beta t} \\mathbb {E}^x u(X_t) \\le \\mathbb {E}^xu(X_0)$ , hence $u$ is $\\beta $ -supermedian.",
"Now, by [3], Corollary 1.3.4, showing that $u \\in E(\\mathcal {U}_\\beta )$ reduces to prove that $u$ is finely continuous, which in turns follows by the well known characterization according to which $u$ is finely continuous if and only if $u(X)$ has right continuous trajectories $\\mathbb {P}^x$ -a.s. for all $x \\in E$ ; see Theorem 4.8 in [9], Chapter II.",
"ii) $\\Rightarrow $ i).",
"Since $u$ is $\\beta $ -supermedian, by the Markov property we have for all $0\\le s \\le t$ $\\mathbb {E}^x[e^{-\\beta (t+s)}u(X_{t+s}) | \\mathcal {F}_s]=e^{-\\beta (t+s)}\\mathbb {E}^{X_s}u(X_t)=e^{-\\beta (t+s)} P_tu(X_s) \\le e^{-\\beta s}u(X_s),$ hence $(e^{-\\beta t}u(X_t))_{t\\ge 0}$ is an $\\mathcal {F}_t$ -supermartingale.",
"The right-continuity of the trajectories follows by the fine continuity of $u$ via the previously mentioned characterization.",
"$\\hfill \\square $"
]
] | 1709.01864 |
[
[
"Improving the Performance of Polythiophene-based Electronic devices by\n Controlling the Band Gap in the Presence of Graphene"
],
[
"Abstract Density functional theory (DFT) and many body perturbation theory at the G$_0$W$_0$ level are employed to study the electronic properties of polythiophene (PT) adsorbed on graphene surface.",
"Analysis of charge density difference shows the substrate-adsorbate interaction leading to a strong physisorption and interfacial electric dipole moment formation.",
"The electrostatic potential displays a -0.19 eV shift in the graphene work function from its initial value of 4.53 eV, as the result of the interaction.",
"The LDA band gap of the polymer does not show any change, however the energy level lineshapes are modified by the orbital hybridization.",
"The interfacial polarization effects on the band gap and levels alignment are investigated within G$_0$W$_0$ level and shows notable reduction of PT band gap compared to that of the isolated chain."
],
[
"INTRODUCTION",
"Polythiophene (PT) and its derivatives are the most important stable and ease of preparation $\\pi $ –conjugated semiconductors with a broad spectrum of applications in organic electronics.",
"The applications range from the organic light-emitting diodes (OLED) and displays to solar cells and sensors [1], [2], [3], [4], [5], [6].",
"However, PT–derivatives based electronic devices suffer from relatively large electronic band gap and low carrier mobility which reduce the corresponding quantum efficiency [7], [8].",
"On the other hand, due to its ballistic charge transport and high electron mobility, graphene expected to be an outstanding new material for the electronic applications.",
"These exceptional features suggest that graphene can be utilized to improve the electronic characteristic and charge transport of polymer–based organic semiconductors.",
"Single and few multilayers graphene are transparent.",
"This property is also of high importance for those devices that the light should enter their active layer.",
"Recently, organic PT-derivatives, poly (3-hexylthiophene (P3HT) and poly(3-octylthiophene) (P3OT) photovoltaic as electron donors and an organic functionalized graphene as electron-accepter has been fabricated [9].",
"It was shown that this composite works well due to the interaction between graphene and polymers [9].",
"Recently, the crystallization of highly regular P3HT on single layer graphene is also reported [10].",
"The P3HT polymer film deposited on graphene reported to have a very different distribution of crystallite orientations.",
"Also much higher degree of $\\pi -\\pi $ stacking perpendicular to the plane of the graphene film, compared to deposition on silicon surface was exhibited [10].",
"Graphene is reported as an ideal template for growing ultra-flat organic films with a face-on orientation [11].",
"These preference features are accounted to obtain higher charge transports and efficiencies from PT derivatives [10].",
"Therefore, this paper presents a theoretical attempt to unveil and understand theoretically the influence of graphene at the electronic properties of PT.",
"To this end, first the electronic structure of isolated graphene and PT are investigated.",
"Next, the adsorption mechanism and the modification of the electronic structure when PT adsorbed on the graphene surface are inspected.",
"In order to obtain a comprehensive view, we use three different approaches: (i) local density approximation (LDA), (ii) the Hubbard corrected LDA+U functional and (iii) the many-body perturbation theory at the $G_0W_0$ level.",
"The outcomes are compared with the previous available theoretical and experimental data.",
"As our main purpose, PT is brought close to the graphene surface in face-on orientation.",
"Then it is inspected that how the electronic states of isolated PT are perturbed by the graphene surface using the three mentioned approaches.",
"Van der Waals (vdW) interaction are considered when the binding energy are calculated by semi-local PBE-GGA functional.",
"Our results show that the most stable geometry occurs in the adsorption distance of about 3.4-3.5 Å.",
"Since a typical adsorbate-substrate spacing in a physisorption is more than 3 Å, the obtained value is an evidence of a physisorption.",
"Charge density analysis shows no charge is transferred between PT and graphene.",
"However, results show a clear charge density distortion close to the graphene surface, which is a sign of electric dipole formation.",
"Previous many body ab initio studies have clarified that the polarization, even for a physisorbed and weakly coupled interaction, can considerably influence the size of the adsorbate gap [12], [13].",
"It is also well known that local (semi-local) nature of density functional theory (DFT) functional cannot contribute for the polarization effect in the interface, due to self-interaction errors.",
"By ignoring the polarization effects, the LDA PT gap is independent of the underlying substrate, and the only connection with the graphene is done through a weak interaction with slightly change in the electronic states through orbital hybridizations.",
"It is well known that the self-interaction error which appears in the occupied states in the standard DFT with local (or semi-local) exchange-correlation functionals over delocalizes the highest occupied state and pushes it up, therefore reduces the band gap [13], [14], [15], [16].",
"Subsequently, when a molecule is brought close to the substrate, local (semi-local) functionals exaggerate the density distribution for the added electron or hole.",
"The consequent of the delocalization error is incorrect convex behavior (instead of piecewise constant) of the energy with respect to the number of electrons .",
"As a result, the derivatives of the energy with respect to fractional charge produce incorrect ionization energy and electron affinity and smaller band gap [17].",
"DFT+U inspired by the Hubbard model as a typical approach may be used to correct the DFT unphysical curvature of the total energy.",
"Our outcomes show that as for many typical conjugated polymers, DFT+U does not improve the band gaps of isolated PT and the adsorbed PT on graphene significantly [18].",
"In order to describe the dynamical polarization effect of the surrounding electrons, we employed many-body perturbation theory implemented through G$_0$ W$_0$ approximation on the top of the DFT calculations.",
"The self-energy $\\Sigma $ is given as the product of the non-interacting single-particle Green function $G_0$ , and the dynamically screened Coulomb interaction $W_0$ , calculated within the plasmon pole approximation.",
"It is well-known that including dynamical electronic correlation in G$_0$ W$_0$ approximation mimics the long-range image potential effects of electrons near the interface [19].",
"Using G$_0$ W$_0$ , we obtained a relatively strong renormalization of the PT band gap, once it physisorbed on the graphene surface."
],
[
"COMPUTATIONAL DETAILS",
"The formation of a regular PT and graphene layer will be considered, when the lateral PT-PT interactions to be less attractive than the corresponding interaction between PT and the graphene surface.",
"To this end, we should perform the calculations in a supercell with the size large enough to prevent the interactions between PT and its images.",
"In addition, the thiophene oligomer in the supercell must arranged in a way that forming the polymer chain, when the cell subjected to the periodic boundary condition (PBC).",
"At the same time, from the repetition of this unit cell in both two coordinates (x, y), the graphene sheet must be constructed.",
"However, the main axes of the graphene lattice do not commensurate at all with the periodicity of PT.",
"Therefore, constructing PT in the face-on orientation is not a trivial commission.",
"In practice, there are many possible on-face arrangement for the combined system, but regarding the computational limitation was explained, we found that the concurrent assembly can be best done when the oligomer is placed along the base-diameter of the monoclinic supercell shown in Fig.",
"REF (a).",
"Figure: (Color online) (a) Top view showing the monoclinic supercell model describing PT polymer adsorbed on top of the graphene monolayer.",
"For clarity the cell is duplicated in both in-plane directions.",
"(b) Top view showing the unit cell vectors of the hexagonal graphene primitive cell 𝐚 1 \\textbf {a}_1 and 𝐚 2 \\textbf {a}_2 (blue arrows), as well as the monoclinic supercell basis vectors 𝐚 1 ' \\textbf {a}^{\\prime }_1 and 𝐚 2 ' \\textbf {a}^{\\prime }_2 (red arrows).",
"(c) Illustration of the supercell Brillouin zone along with the high-symmetry points that are selected to define k-vector paths.This arrangement provides us an opportunity to study and analyze the electronic behaviour of PT adsorbed on graphene surface, at the cost of increasing the number of carbon atoms in the graphene cell up to 72 atoms.",
"If we denote the graphene primitive vectors by $\\textbf {a}_1$ and $\\textbf {a}_2$ ($|\\textbf {a}_1|$ =$|\\textbf {a}_2|$ =a (1.73 Å)), the supercell vectors assembled in this work can be described by $\\textbf {a$$}_1$ ($|\\textbf {a$$}_1|=\\frac{12a}{\\sqrt{3}}=17.15 $ Å) and $\\textbf {a$$}_2$ ($|\\textbf {a$$}_2|$ =$\\frac{9a}{\\sqrt{3}}$ =12.86 Å).",
"A vacuum region of 14 Å along the perpendicular direction is imposed to guarantee a vanishing interaction between periodically repeated images.",
"By this value the total energy versus z–component of the supercell is converged to less than 2 meV.",
"By repetition of the supercell along the two in-plane vectors, both graphene sheet and PT are built, as shown in Fig.",
"REF (b), while the supercell is large enough to prevent the interaction between PT-PT themselves, which are at the distance of 12.8 Å from each other.",
"The first Brillouin zone (BZ) corresponding to the supercell is shown in Fig.",
"REF (c).",
"The two paths through the three symmetric points of the BZ are shown in Fig.",
"REF (c).",
"These points are selected to investigate the electronic behaviours of the isolated components and combined system.",
"Optimization procedure has been performed within LDA frame work by using plane-wave pseudopotential as implemented in the ABINIT code [20].",
"To converge the total energy to within 1 meV a plane-wave cutoff energy of 30 a.u.",
"is required.",
"The Brillouin zone (BZ) are sampled by $4\\times 4\\times 1$ Monkhorst-Pack k-vectors [21].",
"LDA Troullier–-Martins (TM) pseudopotentials are used in our calculations [22].",
"Both volume of the supercell and the position of PT atoms inside it are relaxed within the forces less than 5 meV/Å.",
"The structure of isolated graphene sheet are relaxed within the same criteria.",
"The U correction are adopted to the p–orbitals of all atoms in the cell within the projector augmented-wave (PAW) method.",
"The PAW kinetic energy cut-off is converged by 20 a.u.",
"To determine the U and J parameters, we follow a semi-empirical strategy by looking for the optimum values when the band gap is increased to its experimental value.",
"Figure: (Color online) (a) The optimized geometry model of PT oligomer used in the supercell explained in the text.",
"Atoms in a hexagonal ring are labelled according to parameters shown Table I.Table: Calculated optimized bond lenghts and angles of polythiophene according to atoms labeled in Fig.",
".",
"The previous available LDA predictions are listed for comparison.",
"All lengths are in Å and angles in degree.The G$_0$ W$_0$ approximation [24], [25] provided by the YAMBO code [26] using TM pseudopotentials.",
"The dielectric function is calculated using the plasmon-pole approximation, while the convergence in the self-energy and the dielectric matrix with respect to the number of bands are considered.",
"The calculations were done with up to 200 empty bands corresponding to a maximal band energy of 0.44 a.u.",
"(12 eV), as well as a cut-off for the dielectric matrix up to 2.5 a.u."
],
[
"Isolated polymer",
"Before considering the electronic properties of PT-absorbed on the graphene surface, we present the LDA band structure of isolated PT chain computed in the supercell shown in Fig REF (b).",
"The polymer structure is constructed from the oligomer in the cell, when subjected to the PBC in both two in-plane directions, as shown in Fig.",
"REF (a).",
"The oligomer including four thiophene rings, as shown in Fig.",
"REF , and is oriented parallel to the supercell base-diameter.",
"The LDA optimized structure and geometry parameters are shown in Fig.s REF (b) and REF , respectively.",
"The value of the corresponding parameters are listed in Table REF .",
"The obtained values are also compared with the previous theoretical LDA data, and the agreements support our technical process.",
"By subjecting the supercell to the PBC, in fact we are neglecting the possible torsion angle between oligomers.",
"By this assumption, the alignment of the $\\pi $ –orbitals will be maximized[27], which is an appropriate model for the chains including more than 10 monomers [23].",
"The LDA electronic band structure of the isolated PT are shown in Fig 3.",
"Both the maximum of the valence and the minimum of the conduction bands are happen at the $\\Gamma $ point.",
"These two bands are stemming from the inter-ring $\\pi $ –bonding and $\\pi ^*$ –antibonding states occur at -3.65 and -2.55 eV, respectively, and therefore, a direct gap of 1.10 eV is opened at the $\\Gamma $ point.",
"Depending on the pseudopotential and other calculation parameters given in Sec.",
", this gap is about 0.12-0.26 eV smaller than the previous LDA reported gaps [28], [23].",
"The LDA valence and conduction bandwidth between point $\\Gamma $ to the point A are found relatively small, as shown in Fig.",
"REF , with value of 0.76 and 0.60 eV, respectively, and reflecting the wave functions are localized on individual chain.",
"Figure: (Color online) DFT-LDA (top) and G 0 _0W 0 _0 (down) band structure of polythiophne (PT).",
"The Fermi energies are set to zero.The LDA band gaps obtained in this study and the previous efforts display relatively large deviation from experimental value (2.1 eV) [29].",
"One of the reasons is the spurious self-interaction error (SIE) related to the delocalization error of DFT exchange-correlation (xc) functionals stemming from dominating Coulomb term that pushes electrons apart [17].",
"In some cases, hybrid functionals can partially correct for the SIE by including a fraction of exact exchange term adjusted to cancel the SIE [23].",
"However, using hybrid functionals is not a general solution and for a large number of atoms is computationally quite expensive.",
"Here, we have performed DFT+U as an alternative method to cure the SIE by applying the U correction term only to the p orbitals.",
"To determine parameters U and J, we follow a semi-empirical strategy and controlling the band gap.",
"By increasing U and J to respectively 6 and 0.6 eV, the band gap is raised up and saturated at about 1.40 eV, which is not a significant improvement.",
"Therefore, applying DFT+U, inspired by the Hubbard model, cannot provide a realistic model to correct the PT band gap, as has been proved in past for many conjugated polymers with delocalized interring $\\pi $ electrons [18].",
"On the other hand, it has been shown that many body perturbations theory based on ab initio calculations of the quasiparticle (QP) band structure with the GW approximation overcomes the problematic effect of self-energy on polymer band gap [13], [30], [14], [31].",
"Therefore, we constructed GW calculations in a non-self-consistently manner from the LDA orbitals and eigenvalues, which is referred as the G$_0$ W$_0$ method.",
"The G$_0$ W$_0$ approach is computationally so expensive for the present supercell introduced in Fig.",
"REF (b).",
"The sever limitation is the large number of atoms necessary to study the adsorption of PT on graphene surface.",
"Therefore, we had to calculate electron self-energy by summing over computationally reasonable number of empty states.",
"The initial calculation was performed with 100 number of unoccupied orbitals.",
"The SIE corrected PT gap improved from LDA predicted 1.10 eV to G$_0$ W$_0$ 2.96 eV at point $\\Gamma $ , as seen in Fig.",
"REF .",
"By increasing the number of empty states up to the 200 states, the energy of states are modified, however the initial G$_0$ W$_0$ gap improves only by about 0.03 eV to 2.99 eV.",
"Indeed, compared to LDA, the G$_0$ W$_0$ predict larger bandwidth for the valence and conduction bands with value of 0.96 and 0.74 eV, respectively.",
"Further increasing of the empty states in our G$_0$ W$_0$ calculations imposed computationally sever challenges, especially when the adsorption of PT on graphene is considered.",
"The previous G$_0$ W$_0$ gap computed within the generalized plasmon pole model and by summing self-energy over 1592 empty states within a smaller supercell including two thiophene rings is 3.10 eV, which is in satisfactory agreement with our outcomes driven from 200 empty states [31].",
"Here we point that the band gap of 3.59 eV is also predicted by a previous higher level calculation based on the self-consistent GW calculations [28].",
"Moreover, the computed G$_0$ W$_0$ quasiparticle gaps computed in this and the previous works are about 1 eV larger than the experimental value.",
"The disagreement has been mostly cured by including electron-hole interaction for the optical response, and inter-chain interactions in bulk polymer [28], [31]."
],
[
"Uncovered graphene",
"At first, all carbon atoms in the graphene layer are relaxed to the ground state minimum potential energy point.",
"The LDA predicts C-C bond length of 1.429 Å for graphene.",
"Fig.",
"REF shows the LDA predicted band structure of pristine graphene deduced from the monoclinic supercell illustrated in Fig.",
"REF (a) including 36 primitive cells and 72 carbons.",
"Two of the six Dirac points at the corners of the hexagonal BZ are folded to point $\\Gamma $ inside the BZ shown in Fig.",
"REF (c), so that there are four-fold degeneracy at this point with the energy of -2.13 eV, forming two pairs of touching cones.",
"Figure: (Color online) DFT-LDA band structure of isolated graphene calculated in the supercell described in the text.",
"The Fermi energy is set to zero.",
"The four touching bands forming two Dirac cones at Γ\\Gamma are specified by red colors."
],
[
"PT adsorbed on graphene",
"The optimum distance between PT and graphene plane is determined by the value which minimized the total energy of the compound.",
"PT adsorption energy on graphene surface as a function of adsorption distance, calculated by using different functionals are depicted in Fig.",
"REF .",
"In agreement with the previous report for poly(para-phenylene) (PPP) adsorbed on graphene [13], we also observe that adsorption energy and distance are sensitive to the choice of the applied functional.",
"Moreover, each of LDA and GGA functionals exhibits different picture.",
"According to LDA, the most stable geometry occurs in the adsorption distance of z$_0$ =3.4 Å with binding energy of 0.77 eV.",
"GGA-PBE functional predicts a shallow potential well at 4.2 Å with binding energy of 0.17 eV.",
"Since a typical substrate-adsorbate spacing in a physisorption is more than 3 Å, these results suggest a physisorption of PT on graphene.",
"The GGA-PBE small binding energy is a result of DFT deficient description of the long-range dispersion forces.",
"The calculated binding energy based on LDA may not be reliable, since LDA does not include the long-ranged vdW interactions [32].",
"In fact, the over binding effect observed in the LDA functional compensates this missing term, and LDA benefits from the cancellation of errors [32].",
"For GGA-PBE, one should contribute a small nonlocal vdW dispersion force, which almost is the binding force for the most of the organic materials [33].",
"Including DFT-D2 correction within Grimme empirical scheme [34], as implemented in the ABINIT code, improves PBE binding energy to 1.16 eV at the adsorption distance of z$_0$ = 3.5 Å.",
"Therefore, the contribution of vdW interactions to the binding energy amounts to 1 eV and indicates a strong physisorption.",
"Figure: (Color online) Adsorption energy of PT on graphene surface obtained by using different functionals: GGA-PBE, LDA, and GGA-D2, which the latter includes a semi-empirical long-range dispersion correction, based on the Grimme ' ^\\prime s method, in terms of the adsorption distance.",
"The zero line indicates the two independent components with no binding energyThe orbital overlap and vdW interaction rearrange the charge density, which in turn should influence the valance and conduction energy levels of the polymer, and hence the band gap.",
"Normally, the degree of the substrate-adsorbate vdW interaction depends on the adsorbed distance, the polarizability of the substrate and the number and type of the atoms involved.",
"The reported vdW corrected GGA binding energy of PPP on the graphene surface is about 0.39 eV [13], which is so much smaller than what we obtained here for PT.",
"To get more insight into the PT physisorption mechanism on graphene, we calculated the charge density rearrangement at the substrate-adsorbate interface by computing the in plane-averaged charge density of combined system relative to the charge densities of isolated components: $\\Delta \\rho =\\rho $ (G+PT)-($\\rho $ (PT)+$\\rho $ (G)).",
"Here, $\\rho $ (G+PT) represents the charge density of the combined system, and $\\rho $ (PT), $\\rho $ (G) are the densities of the seperated polymer and uncovered graphene, respectively.",
"The total plane-averaged and difference charge density, as well as the electrostatic potential are represented in Fig.",
"REF .",
"As shown in the top panel of Fig.",
"REF , there is almost no change in the charge density of the polymer, suggesting no charge transfer between graphene and PT.",
"The calculated charge density difference near the graphene surface obtained in this study is about one order of magnitude larger than for the PPP on graphene, reported in Ref.",
"[13].",
"This result indicates the key role of the sulphur atom in this study, and explains the origin of the strong physisorbtion of PT on graphene.",
"By considering $\\Delta \\rho $ , we find electron charge density of graphene is attracted toward PT and makes a region of charge accumulation ($\\Delta \\rho >0$ ) and a region of charge depletion ($\\Delta \\rho >0$ ).",
"Therefore the interaction between polymer and graphene gives rise to an interfacial electric dipole moment formation near the graphene surface in the region between two components.",
"By PT absorption, the Fermi energy of the graphene exhibits a 1.09 eV downward shift, and the interfacial electric dipole moment makes a modification to graphene work function by a -0.19 eV shift down from its initial value of 4.53 eV (4.34 eV).",
"Figure: (Color online) Top panel: plane-averaged total charge density (red solid line) and charge density difference (black dashed line) of the combined system (G+PT) along the perpendicular direction to the graphene plane.",
"Small charge modification are observed in the region between the graphene and polymer.",
"Down panel: electrostatic potential energy along the perpendicular direction to the graphene plane inside the supercell.",
"The work function (φ\\phi ) of the combined system and uncovered graphene (G) are determined from the corresponding Fermi energies.",
".",
"Work function at the bottom ( φ b \\phi _b) and the top sides (φ t \\phi _t) of the cell are illustrated.",
"Black dashed line shows the electrostatic potential energy of the uncovered graphene.",
"The dotted-black line indicates the Fermi energy of the uncovered graphene.",
"The maximum value of the electrostatic potential energy of the combined system is set to zero.LDA band structure of PT at the 3.4 Å above the graphene sheet is shown in Fig.",
"REF .",
"To identify the PT valence and conduction bands within the states of the combined system, the corresponding states of the isolated PT are appended to this figure.",
"We see that according to LDA, PT band gap is slightly renormalized by a value of about 50 meV.",
"The conduction band at point $\\Gamma $ experiences only a small downward shift of 40 meV, which is the result of the weak charge density variation in PT after adsorption.",
"As the result of the orbital hybridization, from the $\\Gamma $ point toward point A, energy of the PT states are influenced by the graphene orbitals.",
"Figure: (Color online) Top panel: LDA-predicted band structure of the combined system at an adsorption height of 3.4 Å.",
"The LDA valence and conduction bands of the isolated PT (blue-dashed lines) are shown to identify the corresponding states of the polymer (red-solid line) in the combined system.",
"Zero energy is set to the Fermi energy of the combined system.",
"Down panel: G 0 _0W 0 _0-predicted band structure of the combined system at an adsorption height of 3.4 Å.",
"The G 0 _0W 0 _0 valence and conduction bands of the isolated PT (blue-dashed line) are shown to identify the corresponding states of the polymer (red-solid line) in the combined system.",
"The G 0 _0W 0 _0 calculations were done in the three points shown by the square mark in each state.",
"Zero energy is set to the energy of the graphene Dirac point (black-solid line at Γ\\Gamma point) in the combined system.3.29 eV.",
"The Fermi energy of the combined system is illustrated in dotted line of the lower panel.",
"The Fermi energy of the uncovered graphene (at -2.13 eV) is not shown for clarity purpose and obtained 3.50 eVWhile many physical effects can influence the energy level of a molecule near a polarizable substrate, LDA (DFT) is not a successful tool to describe them.",
"An important effect in a physisorption is the Coulomb interaction between the surface and the electron or hole corresponding to the occupied or unoccupied states of the adsorbed molecule.",
"The result of this interaction may cause the surface to polarize.",
"The additional interaction between the polarized surface and the molecule influences the energy levels of the molecule and may strongly renormalize its gap.",
"In classical picture, this subject can be described by a point charge q located at the position $z_0$ above a polarizable surface with dielectric $\\epsilon $ filling the half space $z<z_0$ .",
"The electrostatic interaction potential is given by [35] $V=\\frac{qq^\\prime }{4(z-z_0)},$ where the size of the image charge is $q^\\prime =q(1-\\epsilon )/(1+\\epsilon )$ .",
"As a result, the energy of the unoccupied states should shift downward, since they experience an attractive potential according to Eq.",
"REF .",
"For the occupied states the situation is inverse.",
"Since the interaction with the positive image charge reduces the binding of the electron to the molecular core, these states are shifted upward.",
"In contrast to LDA, long-range image static potential effects, for an electron near an interface can be well described by GW approximation.",
"GW studies confirm that the electronic energy levels of a molecule outside a surface would also obey the image potential of Eq.",
"REF , and the response of the surface to an added electron or hole is reproduced within the G$_0$ W$_0$ approach by means the screened Coulomb potential W$_0$ .",
"[13], [12], [16], [15], [36].",
"The interaction with the image charge shift the unoccupied levels downward, whereas the occupied levels upward and therefore, reduces the band gap.",
"The band structure of the combined PT-graphene system at the G$_0$ W$_0$ level and adsorption distance of 3.4 Å is shown in Fig.",
"REF .",
"For clarity, the valence and conduction bands of the combined system are specified in the red-solid lines and black squares.",
"The corresponding isolated PT bands are also integrated to this figure and depicted by blue-dashed lines and red squares.",
"In contrast to the LDA prediction, image charges induced in the graphene layer significantly perturb PT electronic states, so that the PT band gap reduces from 2.99 eV into 1.60 eV ($\\Delta $ E=1.39 eV).",
"Four fold degeneracy of the graphene layer at the $\\Gamma $ point is split by the vdW interaction, so that a gap of 60 meV is opened at this point.",
"Figure: (Color online) The electronic energy-level alignments of PT polymer obtained from DFT-LDA and G 0 _0W 0 _0 calculations near the combined graphene–PT interface in two different adsorption distances (4 and 3.4 Å).",
"The interface position is shown by red rectangular and the energy levels of polymer by blue line segments.",
"The positions of each valence and conduction bands at points \"Γ\"\"\\Gamma \" and \"A\"\"A\" are shown in eV.",
"The vacuum level is set to 0 eV, and distances between line segments are scaled arbitrarily.We also study the dependence of the band alignment at the interface on the adsorption distance .",
"Due to computational cost, our calculations are limited only to two cases with 3.4 Å and 4 Å distances from graphene layer.",
"The results are shown in Fig.",
"REF , which we plot the LDA and G$_0$ W$_0$ energies of the valence and conduction bands of PT at $\\Gamma $ and A points of BZ.",
"Since the LDA respective valence and conduction positions of PT shift slightly in the same direction, the change in the gap is so small (30 meV at $\\Gamma $ point).",
"At the G$_0$ W$_0$ level and from 4 into 3.4 Å, the valence band moves up by 70 meV, whereas the conduction band moves down by 200 meV.",
"Consequently, the band gap is reduced by 270 meV.",
"Therefore, G$_0$ W$_0$ not only can describe the band gap renormalization upon the dynamic polarization of the substrate, but also it can describe the effect of the substrate-adsorbate distance."
],
[
"Conclusion",
"In summary, by local density approximation within DFT frame work and many-body perturbation theory at the G$_0$ W$_0$ level, we investigate electronic properties and possible modifications of the electronic band structure of polythiophene by adsorbtion on graphene surface.",
"The calculated charge density difference predicts the formation of an interface electric dipole near graphene surface, however we found that the charge transfer between them is negligible.",
"Analysis of the electrostatic potential along the direction perpendicular to the graphene plane shows a -0.19 eV shift in the graphene work function from its initial value of 4.53 eV.",
"According to LDA, the adsorption of PT on graphene does not alter the PT band gap, but orbital overlapping of PT and graphene modify the PT energy levels lineshapin far from the $\\Gamma $ point.",
"By reducing the substrate-adsorbate distance, the LDA valence and conduction bands move slightly in the same direction leading to a small change in the gap.",
"However, by the G$_0$ W$_0$ the valence band moves up and the conduction band moves down, results in a significant band reduction effect.",
"According to our finding in this study, we predict a significant change of the polythiophene electronic behaviours in the presence of graphene."
],
[
"Acknowledgement",
"F. Marsusi appreciates the computer assistance provided by Mrs Z. Zeinali in the Department of Energy Engineering and Physics at Amirkabir University of Technology.",
"I. Fedorov gratefully acknowledge the Center for collective use “High Performance Parallel Computing” of the Kemerovo State University for providing the computational facilities."
]
] | 1709.01763 |
[
[
"Characterizations of democratic systems of translates on locally compact\n abelian groups"
],
[
"Abstract We present characterizations of democratic property for systems of translates on a general locally compact abelian group, along a lattice in that group.",
"That way we generalize the results from [11] on systems of integer translates.",
"Furthermore, we investigate the possibilities of more operative characterizations for lattices with torsion group structure, mainly through examples and counterexamples."
],
[
"Introduction",
"The study of greedy approximations in Banach spaces brought the democratic property to the attention of mathematical community.",
"For early results on the greedy algorithm consult the work of V. N. Temlyakov and .",
"We would like to emphasize the theorem of S. V. Konyagin and V. N. Temlyakov (see ), which states that a basis in a Banach space is greedy if and only if it is unconditional and democratic.",
"Recall that conditional bases are often very difficult to construct and, therefore, the democratic property provides an interesting framework to analyze large classes of conditional bases.",
"Furthermore, as already outlined in an article by P. Wojtaszczyk , one can extend the notion of the democratic property to more general systems than bases.",
"In particular, it is natural to analyze the democratic property within the realm of various reproducing function systems, like wavelets, Gabor systems, etc.",
"Within such systems most often we have a core space, which is generated by integer translations of a single function.",
"This was a point of view taken in ; this paper serves as the main motivating point for our paper.",
"Let us be more precise here.",
"A family $\\mathcal {F}$ of nonzero vectors in a Banach space $(X,\\Vert \\cdot \\Vert )$ is called democratic if there exists a constant $D\\in \\langle 0,+\\infty \\rangle $ such that for any two finite subsets $\\Gamma _1$ and $\\Gamma _2$ of $\\mathcal {F}$ with the same cardinalities one has $\\Big \\Vert \\sum _{f\\in \\Gamma _1}\\frac{f}{\\Vert f\\Vert }\\Big \\Vert \\le D \\,\\Big \\Vert \\sum _{f\\in \\Gamma _2}\\frac{f}{\\Vert f\\Vert }\\Big \\Vert .$ If all vectors in $\\mathcal {F}$ have the same norm, then the above inequality becomes $ \\Big \\Vert \\sum _{f\\in \\Gamma _1}f\\Big \\Vert \\le D \\,\\Big \\Vert \\sum _{f\\in \\Gamma _2}f\\Big \\Vert $ and it can be rephrased by saying that the norms of the sums $\\sum _{f\\in \\Gamma }f$ are mutually comparable (up to an absolute constant) for all finite sets $\\Gamma \\subseteq \\mathcal {F}$ of the same size.",
"Many of the common systems for decomposition and reconstruction of functions begin by considering a closed subspace of $\\textup {L}^2(\\mathbb {R})$ generated by integer translates of a single square-integrable function.",
"More precisely, one can take $\\psi \\in \\textup {L}^2(\\mathbb {R})$ , denote $ \\mathcal {F}_\\psi := \\big (\\psi (\\cdot -k)\\big )_{k\\in \\mathbb {Z}},\\quad \\langle \\psi \\rangle := \\overline{\\mathop {\\textup {span}}\\mathcal {F}_\\psi }, $ and ask to characterize various basis-like properties of the system $\\mathcal {F}_\\psi $ for the Hilbert space $\\langle \\psi \\rangle $ .",
"It turns out that all such properties can be characterized in terms of the periodization of $|\\widehat{\\psi }|^2$ , defined by $ p_\\psi (\\xi ) := \\sum _{k\\in \\mathbb {Z}} |\\widehat{\\psi }(\\xi +k)|^2; \\quad \\xi \\in \\mathbb {T}, $ where $\\mathbb {T}=\\mathbb {R}/\\mathbb {Z}$ is a one-dimensional torus, while the Fourier transform of $f\\in \\textup {L}^1(\\mathbb {R})\\cap \\textup {L}^2(\\mathbb {R})$ is normalized as $ \\widehat{f}(\\xi ) := \\int _{\\mathbb {R}} f(x) e^{-2\\pi i x \\xi } dx; \\quad \\xi \\in \\mathbb {R} $ and then extended by continuity to the whole $\\textup {L}^2(\\mathbb {R})$ .",
"These characterizations are consequences of the isometric isomorphism between the shift-invariant space $\\langle \\psi \\rangle $ and the weighted Lebesgue space $\\textup {L}^2(\\mathbb {T},p_\\psi )$ , which maps the translates of $\\psi $ to the exponentials, and in turn enables the usage of many classical results from the Fourier analysis.",
"An interested reader can find more details and numerous references in ; consult also , , , , , , , , , , , , and .",
"Consider the system of translates $\\mathcal {F}_\\psi $ generated by some $\\psi \\in \\textup {L}^2(\\mathbb {R})$ .",
"It is natural to try to characterize all $\\psi $ for which this system is democratic.",
"This task was initiated in .",
"Some characterizing theorems are provided there, as well as numerous necessary or sufficient conditions in various special situations.",
"However, as stated by the authors in , the characterizing condition which is at the same time simple and operative remains unknown.",
"It is perhaps somewhat intriguing why this problem is so difficult, especially in the light of various other basis-like properties that have been successfully treated even on the higher level of generality.",
"We believe that one aspect of the difficulty of this problem is hidden in the fact that the additive group of integers has an “unsuitable” subgroup structure, that is with respect to this characterization problem.",
"It does not contain any non-trivial elements of the finite order.",
"Why is this important?",
"We propose to elevate the problem to the higher level of generality and thus reveal the elements of its complexity more clearly.",
"The problem can be formulated for functions on locally compact abelian groups $G$ translated by elements of a lattice $\\mathcal {L}\\subseteq G$ .",
"As we will see, the democratic property leads to conditions that need to be checked for a very large class of subsets and this problem is quite demanding from the combinatorics point of view.",
"The problem can be radically simplified, but this discussion leads us to several issues related to seemingly unrelated disciplines, like the subgroup structure of the group $G$ , some ergodic properties and some geometric properties (convexity in particular).",
"In short, we prove that the democratic property is a condition that is characterized via an operative integrability condition that needs to be tested on a family of extreme points of a certain convex set.",
"This family is essentially always boxed in between the class of all finite subsets of $G$ (which is usually much larger than the testing family) and the class of all finite subgroups of $G$ (which, unfortunately, is smaller than the testing family).",
"Let us give a few words about the organization of the present paper.",
"In Section we generalize the results from at the level of arbitrary locally compact abelian groups.",
"Theorem REF characterizes democratic systems of translates by testing sizes of the finite sums $\\sum _{k\\in \\Gamma }\\psi (\\cdot -k)$ for all nonempty finite sets $\\Gamma \\subseteq \\mathcal {L}$ .",
"Theorem REF characterizes democratic systems of translates in terms of boundedness and certain density properties of the generalization of the periodization function $p_\\psi $ .",
"It is suspected that there is a lot of redundancy in verifying those conditions.",
"However, no “operative” equivalent conditions are known, even in the case of integer translates, and only a conjecture was stated in the prequel to this paper .",
"Consequently, Section discusses the possible operative conditions for torsion lattices, but mostly through a series of remarks, examples, counterexamples, and numerical data.",
"We also use the opportunity to formulate a couple of open questions."
],
[
"Translates along general lattices",
"In this section we first describe the setting in which we formulate the questions mentioned in the introduction.",
"Then we raise the main results from to the level of abstract abelian groups."
],
[
"Locally compact abelian setting",
"Most of the following material is taken from standard books on harmonic analysis on locally compact groups, such as , , , or , and it has already been adapted to a similar context, for instance in the papers , , , , , and .",
"Whenever we have two subsets $A,B$ of an additively written abelian group $(G,+)$ , let us use the notation $A+B$ for the so-called sumset, defined as $ A+B := \\lbrace x+y \\,:\\, x\\in A,\\, y\\in B\\rbrace .",
"$ If one of the subsets is just a singleton, for instance $A=\\lbrace x\\rbrace $ , then we simply write $x+B$ instead of $\\lbrace x\\rbrace +B$ , and in this case the sumset is just a translate of $B$ by $x$ .",
"Analogously we define the difference set $A-B$ and the multiple sumset $A_1+A_2+\\cdots +A_n$ .",
"The subgroup generated by a set $A\\subseteq G$ will be denoted by $\\langle A\\rangle $ ; it is the smallest subgroup of $G$ that contains $A$ .",
"Let $(G,+)$ be an abelian topological group, which means that the underlying topology makes both the addition $G\\times G\\rightarrow G$ , $(x,y)\\mapsto x+y$ and the inversion $G\\rightarrow G$ , $x\\mapsto -x$ continuous.",
"Assume that the topology of $G$ is Hausdorff and locally compact, i.e., each point of $G$ has a compact neighborhood.",
"The smallest $\\sigma $ -algebra on $G$ containing all open subsets is called the Borel $\\sigma $ -algebra and denoted $\\mathcal {B}(G)$ .",
"There exist a nontrivial Radon measure $\\lambda _G$ on $(G,\\mathcal {B}(G))$ that is invariant under the translations on $G$ , i.e., $ \\lambda _G(x+A) = \\lambda _G(A) $ for all $A\\in \\mathcal {B}(G)$ and $x\\in G$ .",
"Such measure $\\lambda _G$ is unique up to a constant multiple and it is called the Haar measure of $G$ .",
"It will always be understood whenever we suppress it notationally from integrals or function spaces, i.e., $dx$ will have to be interpreted as $d\\lambda _G(x)$ for an appropriate group $G$ and $\\textup {L}^p(G)$ will be an abbreviation for $\\textup {L}^p(G,\\mathcal {B}(G),\\lambda _G)$ .",
"The measure $\\lambda _G$ is finite if and only if $G$ is compact, while in the case of a discrete group $G$ the Haar measure is simply a constant multiple of the counting measure.",
"If $(\\textup {S}^1,\\cdot )$ is the group of unimodular complex numbers, then any continuous homomorphism $\\xi \\colon (G,+)\\rightarrow (\\textup {S}^1,\\cdot )$ is called a (unitary) character of $G$ and the collection of all characters is denoted $\\widehat{G}$ .",
"If we endow $\\widehat{G}$ with the most obvious pointwise binary operation, $ (\\xi + \\zeta )(x) := \\xi (x) \\zeta (x); \\quad x\\in G, $ and with the topology of uniform convergence on compact subsets of $G$ , it also becomes a locally compact Hausdorff topological group.",
"It is called the dual group of $G$ and it possesses its own Haar measure $\\lambda _{\\widehat{G}}$ .",
"The group $\\widehat{G}$ is compact if and only if $G$ is discrete and vice versa.",
"Let $E_G\\colon G\\times \\widehat{G}\\rightarrow \\textup {S}^1$ be the bi-character function, i.e., $E_G(x,\\xi ) = \\xi (x)$ for $x\\in G$ and $\\xi \\in \\widehat{G}$ , $E_G(\\cdot ,\\xi )$ ; $\\xi \\in \\widehat{G}$ are all characters of $G$ , $E_G(x,\\cdot )$ ; $x\\in G$ are all characters of $\\widehat{G}$ .",
"Here we implicitly use the Pontrjagin duality theorem, i.e., that the dual group of $\\widehat{G}$ is canonically isomorphic to $G$ itself.",
"The fundamental algebraic properties of $E_G$ are $E_G(x+y,\\xi ) = E_G(x,\\xi )E_G(y,\\xi ),\\quad E_G(x,\\xi +\\zeta ) = E_G(x,\\xi )E_G(x,\\zeta ),$ and $E_G(-x,\\xi ) = \\overline{E_G(x,\\xi )} = E_G(x,\\xi )^{-1} = E_G(x,-\\xi )$ for any $x,y\\in G$ and any $\\xi ,\\zeta \\in \\widehat{G}$ .",
"The function $E_G$ serves as an analogue of the pure exponentials, since in the classical case $G=\\mathbb {R}^d$ , $\\widehat{G}\\cong \\mathbb {R}^d$ one can take $E_{\\mathbb {R}^d}\\colon \\mathbb {R}^d\\times \\mathbb {R}^d\\rightarrow \\textup {S}^1,\\quad E_{\\mathbb {R}^d}(x,\\xi ) = e^{2\\pi i x\\cdot \\xi },$ where $x\\cdot \\xi $ stands for the standard inner product on $\\mathbb {R}^d$ .",
"The Fourier transform is initially defined for $f\\in \\textup {L}^1(G)\\cap \\textup {L}^2(G)$ by the formula $\\widehat{f}\\colon \\widehat{G}\\rightarrow \\mathbb {C},\\quad \\widehat{f}(\\xi ) := \\int _G f(x) \\overline{E_G(x,\\xi )} dx;\\quad \\xi \\in \\widehat{G}$ and then the isometry $f\\mapsto \\widehat{f}$ extends to a unitary operator from $\\textup {L}^2(G)$ onto $\\textup {L}^2(\\widehat{G})$ if we choose the appropriate normalization of $\\lambda _{\\widehat{G}}$ depending on the normalization of $\\lambda _G$ .",
"If $H$ is any closed subgroup of $G$ , then its orthogonal complement is defined as $ H^\\perp := \\lbrace \\xi \\in \\widehat{G} \\,:\\, E_G(x,\\xi )=1 \\text{ for each } x\\in H\\rbrace $ and it is actually a closed subgroup of $\\widehat{G}$ .",
"Indeed, $H\\mapsto H^\\perp $ is a bijective correspondence between closed subgroups of $G$ and closed subgroups of $\\widehat{G}$ .",
"If $q\\colon G\\rightarrow G/H$ denotes the canonical epimorphism onto the quotient of $G$ by some closed subgroup $H$ , then the maps $\\widehat{G/H} \\rightarrow H^\\perp ,\\quad \\eta \\mapsto \\eta \\circ q$ and $\\widehat{G}/H^\\perp \\rightarrow \\widehat{H},\\quad \\xi +H^\\perp \\mapsto \\xi |_H$ constitute isomorphisms of topological groups; see .",
"Let us explain how one can construct the bi-character function $E_H$ of $H$ from the bi-character function $E_G$ of $G$ .",
"Observe that for any $x\\in H$ and any $\\xi _1,\\xi _2\\in \\widehat{G}$ such that $\\xi _1-\\xi _2\\in H^\\perp $ we have $E_G(x,\\xi _1-\\xi _2)=1$ , so by (REF ) and (REF ) we get $E_G(x,\\xi _1) = E_G(x,\\xi _2)$ .",
"Thus, $E_G\\colon G\\times \\widehat{G}\\rightarrow \\textup {S}^1$ restricted to $H\\times \\widehat{G}$ factors via a bi-homomorphism $H\\times (\\widehat{G}/H^\\perp )\\rightarrow \\textup {S}^1$ , which becomes precisely the bi-character of $H$ after the identification coming from (REF ).",
"In other words, $E_H(x,\\zeta ) = E_G(x,\\xi )$ whenever $x\\in H$ , $\\zeta \\in \\widehat{H}$ , and $\\xi \\in \\widehat{G}$ is any character such that $\\xi |_H=\\zeta $ .",
"Another structural ingredient we need is a lattice $\\mathcal {L}$ in $G$ , or more precisely, a closed subgroup $\\mathcal {L}$ of $G$ such that the relative topology on $\\mathcal {L}$ inherited from $G$ becomes discrete and such that the quotient space $G/\\mathcal {L}$ is compact (i.e., it is a compact Hausdorff topological group).",
"In all that follows we assume that $\\mathcal {L}$ is countably infinite, so that it serves as a reasonable generalization of the integer lattice $\\mathbb {Z}^d$ .",
"Note that in particular $\\mathcal {L}$ cannot be compact, which also forces the whole group $G$ to be noncompact.",
"Since $G/\\mathcal {L}$ is compact, its dual group is discrete, so from (REF ) applied with $H=\\mathcal {L}$ we get $\\widehat{G/\\mathcal {L}}\\cong \\mathcal {L}^\\perp $ and thus we know that $\\mathcal {L}^\\perp $ is discrete too.",
"On the other hand, since $\\mathcal {L}$ is discrete, its dual must be compact, so by applying (REF ) with $H=\\mathcal {L}$ we obtain $\\widehat{G}/\\mathcal {L}^\\perp \\cong \\widehat{\\mathcal {L}}$ and we conclude that $\\mathcal {L}^\\perp $ is cocompact in $\\widehat{G}$ .",
"From these observations we find that $\\mathcal {L}^\\perp $ is a lattice in $\\widehat{G}$ ."
],
[
"Testing sizes of finite sums",
"The translate of $f\\in \\textup {L}^2(G)$ by $y\\in G$ is the function $T_y f\\in \\textup {L}^2(G)$ defined by $ (T_y f)(x) := f(x-y); \\quad x\\in G. $ Now we take a function $\\psi \\in \\textup {L}^2(G)$ such that $\\Vert \\psi \\Vert _{\\textup {L}^2(G)}\\ne 0$ and consider its translates by the elements of $\\mathcal {L}$ , $ \\mathcal {F}_\\psi := \\big (T_k \\psi \\big )_{k\\in \\mathcal {L}}.",
"$ The basic question we want to answer is: When does $\\mathcal {F}_\\psi $ constitute a democratic system in $\\textup {L}^2(G)$ in the sense of the general definition (REF )?",
"The first step towards the resolution is the following straightforward generalization of .",
"Recall that we have assumed $\\mathcal {L}$ to be infinite and countable.",
"The characterization below fails when $\\mathcal {L}$ is finite, but finite democratic families are not particularly interesting and they have already been discussed in .",
"Theorem 1 The system $\\mathcal {F}_\\psi $ is democratic if and only if there exist constants $c_\\psi ,C_\\psi \\in \\langle 0,+\\infty \\rangle $ such that for every nonempty finite set $\\Gamma \\subset \\mathcal {L}$ we have $c_\\psi \\mathop {\\textup {card}}\\Gamma \\le \\Big \\Vert \\sum _{k\\in \\Gamma } T_k\\psi \\Big \\Vert _{\\textup {L}^2(G)}^2 \\le C_\\psi \\mathop {\\textup {card}}\\Gamma .$ The sufficiency is trivial, as one can simply take $D=\\sqrt{C_\\psi /c_\\psi }$ .",
"For the necessity it is enough to show that for each positive integer $n$ there exists a subset $\\Gamma _n\\subset \\mathcal {L}$ of cardinality $n$ such that $\\frac{1}{2}n\\Vert \\psi \\Vert _{\\textup {L}^2(G)}^2 \\le \\Big \\Vert \\sum _{k\\in \\Gamma _n} T_k\\psi \\Big \\Vert _{\\textup {L}^2(G)}^2 \\le \\frac{3}{2}n\\Vert \\psi \\Vert _{\\textup {L}^2(G)}^2.$ This statement is easily established by the induction on $n$ .",
"For the induction basis $n=1$ we can simply take $\\Gamma _1$ to be a singleton containing an arbitrary element of $\\mathcal {L}$ and use the translation invariance of the $\\textup {L}^2$ norm.",
"In the induction step we take $n\\ge 2$ and assume that we have already constructed the set $\\Gamma _{n-1}$ .",
"By a density argument one can first find a compactly supported continuous function $\\varphi $ on $G$ such that $\\Vert \\varphi -\\psi \\Vert _{\\textup {L}^2(G)} < (1/10n^2)\\Vert \\psi \\Vert _{\\textup {L}^2(G)}$ .",
"Denote $K := \\bigcup _{k\\in \\Gamma _{n-1}} (k+\\mathop {\\textup {supp}}\\varphi )$ .",
"Since the set $K-\\mathop {\\textup {supp}}\\varphi $ is compact, while $\\mathcal {L}$ is not, there exists at least one element $m\\in \\mathcal {L}\\setminus (K-\\mathop {\\textup {supp}}\\varphi )$ .",
"Thus $m+\\mathop {\\textup {supp}}\\varphi $ is disjoint from $K$ , i.e., $\\mathop {\\textup {supp}}T_m\\varphi $ and $\\bigcup _{k\\in \\Gamma _{n-1}}\\mathop {\\textup {supp}}T_k\\varphi $ are mutually disjoint sets.",
"If we define $\\Gamma _n := \\Gamma _{n-1}\\cup \\lbrace m\\rbrace $ , we will get a subset of cardinality $n$ and the previously mentioned disjointness of supports will yield $& \\Big \\Vert \\sum _{k\\in \\Gamma _n} T_k\\varphi \\Big \\Vert _{\\textup {L}^2(G)}^2= \\Big \\Vert \\sum _{k\\in \\Gamma _{n-1}} T_k\\varphi + T_m\\varphi \\Big \\Vert _{\\textup {L}^2(G)}^2 \\\\& = \\Big \\Vert \\sum _{k\\in \\Gamma _{n-1}} T_k\\varphi \\Big \\Vert _{\\textup {L}^2(G)}^2 + \\Vert T_m\\varphi \\Vert _{\\textup {L}^2(G)}^2= \\Big \\Vert \\sum _{k\\in \\Gamma _{n-1}} T_k\\varphi \\Big \\Vert _{\\textup {L}^2(G)}^2 + \\Vert \\varphi \\Vert _{\\textup {L}^2(G)}^2.$ Using this equality and estimating $ \\big | \\Vert f\\Vert _{\\textup {L}^2(G)}^2 - \\Vert g\\Vert _{\\textup {L}^2(G)}^2 \\big | \\le \\Vert f-g\\Vert _{\\textup {L}^2(G)} \\big (\\Vert f\\Vert _{\\textup {L}^2(G)}+\\Vert g\\Vert _{\\textup {L}^2(G)}\\big ) $ for any $f,g\\in \\textup {L}^2(G)$ , we see that $ \\Big \\Vert \\sum _{k\\in \\Gamma _n} T_k\\psi \\Big \\Vert _{\\textup {L}^2(G)}^2 \\quad \\text{and}\\quad \\Big \\Vert \\sum _{k\\in \\Gamma _{n-1}} T_k\\psi \\Big \\Vert _{\\textup {L}^2(G)}^2 + \\Vert \\psi \\Vert _{\\textup {L}^2(G)}^2 $ differ by at most $ \\big (n^2+(n-1)^2+1\\big ) \\Vert \\varphi -\\psi \\Vert _{\\textup {L}^2(G)} \\big (\\Vert \\varphi \\Vert _{\\textup {L}^2(G)}+\\Vert \\psi \\Vert _{\\textup {L}^2(G)}\\big ), $ which is, by the choice of $\\varphi $ , less than $\\Vert \\psi \\Vert _{\\textup {L}^2(G)}^2$ times $ \\big (n^2+(n-1)^2+1\\big ) \\frac{1}{10n^2} \\Big (2+\\frac{1}{10n^2}\\Big ) < \\frac{1}{2}.",
"$ Combining this with the induction hypothesis $ \\frac{1}{2}(n-1)\\Vert \\psi \\Vert _{\\textup {L}^2(G)}^2 \\le \\Big \\Vert \\sum _{k\\in \\Gamma _{n-1}} T_k\\psi \\Big \\Vert _{\\textup {L}^2(G)}^2 \\le \\frac{3}{2}(n-1)\\Vert \\psi \\Vert _{\\textup {L}^2(G)}^2 $ gives (REF ) and completes the inductive proof.",
"Characterization from Theorem REF is still not especially operative, since it might be redundant to verify condition (REF ) for all finite subsets $\\Gamma $ ."
],
[
"Testing the periodization function",
"One can also hope to phrase the answer to the main question in terms of the periodization function, which is now defined as $ p_\\psi (\\xi ) := \\sum _{m\\in \\mathcal {L}^\\perp } |\\widehat{\\psi }(\\xi +m)|^2; \\quad \\xi \\in \\widehat{G}.",
"$ Since $p_\\psi (\\xi )$ only depends on the coset of $\\widehat{G}/\\mathcal {L}^\\perp $ to which $\\xi $ belongs, by (REF ) we can view $p_\\psi $ as a function $\\widehat{\\mathcal {L}}\\rightarrow [0,+\\infty ]$ , $\\tau \\mapsto p_\\psi (\\tau ):=p_\\psi (\\xi )$ .",
"Here one has to extend the character $\\tau $ of $\\mathcal {L}$ arbitrarily to a character $\\xi $ of $G$ , which is always possible via the isomorphism (REF ).",
"Moreover, the formula in tells us how to integrate over the quotient group, so we have $& \\int _{\\widehat{G}/\\mathcal {L}^\\perp } p_\\psi (\\xi ) d\\xi = \\int _{\\widehat{G}/\\mathcal {L}^\\perp } \\!\\Big (\\sum _{m\\in \\mathcal {L}^\\perp } |\\widehat{\\psi }(\\xi +m)|^2\\Big ) d\\xi \\\\& = \\int _{\\widehat{G}} |\\widehat{\\psi }(\\zeta )|^2 d\\zeta = \\Vert \\widehat{\\psi }\\Vert _{\\textup {L}^2(\\widehat{G})}^2 = \\Vert \\psi \\Vert _{\\textup {L}^2(G)}^2 < +\\infty .$ Hence, in fact $p_\\psi \\in \\textup {L}^{1}(\\widehat{G}/\\mathcal {L}^\\perp )\\cong \\textup {L}^{1}(\\widehat{\\mathcal {L}})$ and in particular $p_\\psi $ is finite a.e.",
"The bi-character of $\\mathcal {L}$ will be denoted $e\\colon \\mathcal {L}\\times \\widehat{\\mathcal {L}}\\rightarrow \\textup {S}^1$ from now on.",
"The one thing we need to remember from the construction leading to (REF ) is that $e(k,\\tau ) = E_G(k,\\xi )$ whenever $k\\in \\mathcal {L}$ , $\\xi \\in \\widehat{G}$ , and $\\tau =\\xi |_\\mathcal {L}\\in \\widehat{\\mathcal {L}}$ .",
"For example, when $G=\\mathbb {R}^d$ , $\\mathcal {L}=\\mathbb {Z}^d$ , then $\\widehat{\\mathcal {L}}\\cong \\mathbb {T}^d$ and in addition to (REF ) we have $ e\\colon \\mathbb {Z}^d\\times \\mathbb {T}^d\\rightarrow \\textup {S}^1,\\quad e(k,\\tau ) = e^{2\\pi i k\\cdot \\tau }.",
"$ The Haar measure on $\\widehat{\\mathcal {L}}$ is finite because this group is compact.",
"We find convenient to normalize it so that $\\lambda _{\\widehat{\\mathcal {L}}}(\\widehat{\\mathcal {L}})=1$ .",
"Let us agree to denote $g_\\Gamma \\colon \\widehat{\\mathcal {L}}\\rightarrow [0,+\\infty \\rangle ,\\quad g_\\Gamma (\\tau ) := \\frac{1}{\\mathop {\\textup {card}}\\Gamma }\\Big |\\sum _{k\\in \\Gamma }e(k,\\tau )\\Big |^2; \\quad \\tau \\in \\widehat{\\mathcal {L}}$ for every nonempty finite set $\\Gamma \\subset \\mathcal {L}$ .",
"By (REF ) and (REF ) we can also interpret $g_\\Gamma $ as a function on $\\widehat{G}/\\mathcal {L}^\\perp $ .",
"From (REF ) we immediately see that translating $\\Gamma $ does not affect $g_\\Gamma $ .",
"Basic properties of the functions $g_\\Gamma $ are given in the following lemma and its part (b) also motivates their definition.",
"Lemma 2 Let $\\Gamma \\subset \\mathcal {L}$ be an arbitrary nonempty finite set.",
"(a) The function $g_\\Gamma $ defined by (REF ) is nonnegative, even, continuous, and it integrates to 1.",
"Moreover, $g_\\Gamma $ is a generalized trigonometric polynomial (i.e., a finite linear combination of characters $e(k,\\cdot )$ ; $k\\in \\mathcal {L}$ ) with all coefficients in $[0,1]$ , and its expansion is given by $ g_\\Gamma (\\tau ) = \\sum _{k\\in \\mathcal {L}} \\frac{\\mathop {\\textup {card}}(\\Gamma \\cap (k+\\Gamma ))}{\\mathop {\\textup {card}}\\Gamma }\\, e(k,\\tau ); \\quad \\tau \\in \\widehat{\\mathcal {L}}.",
"$ (b) For any $\\psi \\in \\textup {L}^2(G)$ we have $ \\frac{1}{\\mathop {\\textup {card}}\\Gamma }\\Big \\Vert \\sum _{k\\in \\Gamma } T_k\\psi \\Big \\Vert _{\\textup {L}^2(G)}^2= \\int _{\\widehat{\\mathcal {L}}} g_\\Gamma (\\tau ) p_\\psi (\\tau ) d\\tau .",
"$ (a) The first three mentioned properties of $g_\\Gamma $ are obvious.",
"We can use the well-known fact that the characters of $\\widehat{\\mathcal {L}}$ form an orthonormal basis for $\\textup {L}^2(\\widehat{\\mathcal {L}})$ to also conclude $ \\int _{\\widehat{\\mathcal {L}}} g_\\Gamma (\\tau ) d\\tau = \\frac{1}{\\mathop {\\textup {card}}\\Gamma } \\Big \\Vert \\sum _{k\\in \\Gamma }e(k,\\cdot )\\Big \\Vert _{\\textup {L}^2(\\widehat{\\mathcal {L}})}^2 = \\frac{1}{\\mathop {\\textup {card}}\\Gamma } \\sum _{k\\in \\Gamma }1^2 = 1.",
"$ Furthermore, expanding the expression from definition (REF ) and using (REF ) and (REF ) we get $g_\\Gamma (\\tau ) & = \\frac{1}{\\mathop {\\textup {card}}\\Gamma } \\sum _{k_1,k_2\\in \\Gamma } e(k_1,\\tau )\\overline{e(k_2,\\tau )}= \\frac{1}{\\mathop {\\textup {card}}\\Gamma } \\sum _{k_1,k_2\\in \\Gamma } e(k_1-k_2,\\tau ) \\\\& = \\frac{1}{\\mathop {\\textup {card}}\\Gamma } \\,\\sum _{k\\in \\mathcal {L}} \\Big (\\sum _{\\begin{array}{c}k_1,k_2\\in \\Gamma \\\\k_1-k_2=k\\end{array}}1\\Big )\\, e(k,\\tau )= \\sum _{k\\in \\mathcal {L}} \\frac{\\mathop {\\textup {card}}(\\Gamma \\cap (k+\\Gamma ))}{\\mathop {\\textup {card}}\\Gamma }\\, e(k,\\tau ).$ Observe that the summation in $k$ is actually taken over a finite set $\\Gamma -\\Gamma \\subset \\mathcal {L}$ , so $g_\\Gamma $ really is a generalized trigonometric polynomial.",
"(b) It is readily verified that $ (\\widehat{T_y f})(\\xi ) := \\overline{E_G(y,\\xi )} \\widehat{f}(\\xi ); \\quad \\xi \\in \\widehat{G} $ for $f\\in \\textup {L}^2(G)$ by an easy application of formula (REF ) on a dense subspace $\\textup {L}^1(G)\\cap \\textup {L}^2(G)$ .",
"This fact combined with unitarity of the Fourier transform enables us the following computation for a nonempty finite subset $\\Gamma $ of $\\mathcal {L}$ : $& \\Big \\Vert \\sum _{k\\in \\Gamma } T_k\\psi \\Big \\Vert _{\\textup {L}^2(G)}^2 = \\Big \\Vert \\sum _{k\\in \\Gamma } \\widehat{T_k\\psi }\\Big \\Vert _{\\textup {L}^2(\\widehat{G})}^2= \\int _{\\widehat{G}} \\underbrace{\\Big |\\sum _{k\\in \\Gamma }E_G(k,\\xi )\\Big |^2}_{\\text{$\\mathcal {L}^\\perp $-periodic}} |\\widehat{\\psi }(\\xi )|^2 d\\xi \\\\& = \\int _{\\widehat{G}/\\mathcal {L}^\\perp } \\Big |\\sum _{k\\in \\Gamma }E_G(k,\\xi )\\Big |^2 \\Big (\\sum _{m\\in \\mathcal {L}^\\perp }|\\widehat{\\psi }(\\xi +m)|^2\\Big ) d\\xi = \\int _{\\widehat{\\mathcal {L}}} \\Big |\\sum _{k\\in \\Gamma }e(k,\\tau )\\Big |^2 p_\\psi (\\tau ) d\\tau .$ It remains to divide by $\\mathop {\\textup {card}}\\Gamma $ and apply the definition of $g_\\Gamma $ .",
"Now we can characterize democracy in terms of the periodization function $p_\\psi $ .",
"Theorem 3 The system $\\mathcal {F}_\\psi $ is democratic if and only if $p_\\psi $ is essentially bounded (i.e., $p_\\psi \\in \\textup {L}^\\infty (\\widehat{\\mathcal {L}})$ ) and $\\inf _{\\Gamma } \\int _{\\widehat{\\mathcal {L}}} g_\\Gamma (\\tau ) p_\\psi (\\tau ) d\\tau > 0,$ where the infimum is taken over all nonempty finite sets $\\Gamma \\subset \\mathcal {L}$ .",
"Before the proof we will establish two auxiliary result.",
"As the first ingredient we need an appropriate analogue of the classical Dirichlet kernel on $\\mathbb {T}$ , which was used in for the same purpose, and the classical Fejér kernel derived from it.",
"The key concept that we borrow from ergodic theory is a Følner sequence for $\\mathcal {L}$ , which is a sequence $(F_n)_{n=1}^{\\infty }$ of nonempty finite subsets of $\\mathcal {L}$ such that for each $k\\in \\mathcal {L}$ one has $\\lim _{n\\rightarrow \\infty } \\frac{\\mathop {\\textup {card}}(F_n\\triangle (k+F_n))}{\\mathop {\\textup {card}}F_n} = 0.$ Here $\\triangle $ denotes the symmetric difference of sets, i.e., $A\\triangle B:=(A\\setminus B)\\cup (B\\setminus A)$ .",
"It is a well-known fact that every countable discrete abelian group possesses a Følner sequence, which is just one of many equivalent ways of saying that each countable discrete abelian group is amenable; see .",
"Indeed, an obvious choice of the Følner sets for $\\mathbb {Z}$ are discrete intervals $F_n=\\lbrace -N,\\ldots ,N\\rbrace $ .",
"One similarly verifies that every finitely generated abelian group is amenable.",
"Finally, while considering an arbitrary (not necessarily finitely generated) countable discrete abelian group one only needs to observe that the direct limit of an increasing sequence of countable amenable groups is also amenable; see for details.",
"Furthermore, recall that $\\textup {L}^1(\\widehat{\\mathcal {L}})$ is a Banach algebra with respect to the convolution as multiplication, which is in turn defined as $ (f\\ast h)(\\tau ) := \\int _{\\widehat{\\mathcal {L}}} f(\\tau -\\sigma ) h(\\sigma ) d\\sigma .",
"$ This algebra does not have an identity.",
"We say that a sequence $(f_n)_{n=1}^{\\infty }$ is an approximate identity (a notion used for instance in and ) for $\\textup {L}^1(\\widehat{\\mathcal {L}})$ if for every function $h$ in that space we have $\\lim _{n\\rightarrow \\infty } f_n\\ast h = h$ with convergence in the $\\textup {L}^1$ norm.",
"Lemma 4 If $(F_n)_{n=1}^{\\infty }$ is a Følner sequence for $\\mathcal {L}$ , then $(g_{F_n})_{n=1}^{\\infty }$ constitutes an approximate identity for $\\textup {L}^1(\\widehat{\\mathcal {L}})$ .",
"Let us begin by taking a generalized trigonometric polynomial $ q(\\tau ) = \\sum _{k\\in \\Gamma } \\alpha _k e(k,\\tau ) $ for a finite set $\\Gamma \\subset \\mathcal {L}$ and some complex coefficients $(\\alpha _k)_{k\\in \\Gamma }$ .",
"Observe that by part (a) of Lemma REF we have $ (g_{F_n}\\!\\ast q)(\\tau ) = \\sum _{k\\in \\Gamma } \\frac{\\mathop {\\textup {card}}(F_n\\cap (k+F_n))}{\\mathop {\\textup {card}}F_n} \\,\\alpha _k e(k,\\tau ) $ and thus also $ q(\\tau ) - (g_{F_n}\\!\\ast q)(\\tau ) = \\sum _{k\\in \\Gamma } \\frac{\\mathop {\\textup {card}}(F_n\\setminus (k+F_n))}{\\mathop {\\textup {card}}F_n} \\,\\alpha _k e(k,\\tau ), $ so we can estimate $\\Vert g_{F_n}\\!\\ast q - q \\Vert _{\\textup {L}^1(\\widehat{\\mathcal {L}})} \\le \\sum _{k\\in \\Gamma }\\frac{\\mathop {\\textup {card}}(F_n\\setminus (k+F_n))}{\\mathop {\\textup {card}}F_n} |\\alpha _k|.$ For each $k\\in \\Gamma $ the corresponding term in (REF ) converges to 0 by the Følner property (REF ), which implies $\\lim _{n\\rightarrow \\infty } \\Vert g_{F_n}\\!\\ast q - q \\Vert _{\\textup {L}^1(\\widehat{\\mathcal {L}})} = 0.$ Now take an arbitrary $h\\in \\textup {L}^1(\\widehat{\\mathcal {L}})$ and an $\\varepsilon >0$ .",
"By density there exists a generalized trigonometric polynomial $q$ such that $\\Vert q-h\\Vert _{\\textup {L}^1(\\widehat{\\mathcal {L}})}<\\varepsilon /3$ , while by (REF ) there exists a positive integer $n_0$ such that $n\\ge n_0$ implies $\\Vert g_{F_n}\\!\\ast q-q\\Vert _{\\textup {L}^1(\\widehat{\\mathcal {L}})}<\\varepsilon /3$ .",
"Therefore, for each index $n\\ge n_0$ we have $ \\Vert g_{F_n}\\!\\ast h - h \\Vert _{\\textup {L}^1(\\widehat{\\mathcal {L}})}\\le \\Vert g_{F_n} \\Vert _{\\textup {L}^1(\\widehat{\\mathcal {L}})} \\Vert h - q \\Vert _{\\textup {L}^1(\\widehat{\\mathcal {L}})}+ \\Vert g_{F_n}\\!\\ast q - q \\Vert _{\\textup {L}^1(\\widehat{\\mathcal {L}})}+ \\Vert q - h \\Vert _{\\textup {L}^1(\\widehat{\\mathcal {L}})}< \\varepsilon .",
"$ Existence of approximate identities as in Lemma REF at the level of general countable abelian lattices was named the Fejér property in .",
"It was the working assumption in that paper.",
"The following lemma is a certain folklore and it appears (in some form and with a larger constant) in many texts on Banach spaces; for instance see .",
"For completeness we give its elegant self-contained proof.",
"Lemma 5 For any finite system of vectors $(f_k)_{k\\in \\Gamma }$ in a complex Banach space $(X,\\Vert \\cdot \\Vert )$ and any finite system of coefficients $(\\alpha _k)_{k\\in \\Gamma }$ satisfying $|\\alpha _k|\\le 1$ for each $k\\in \\Gamma $ , we have $ \\Big \\Vert \\sum _{k\\in \\Gamma } \\alpha _k f_k\\Big \\Vert \\le \\pi \\max _{\\Gamma ^{\\prime }\\subseteq \\Gamma } \\Big \\Vert \\sum _{k\\in \\Gamma ^{\\prime }} f_k\\Big \\Vert , $ where the maximum is taken over all subsets $\\Gamma ^{\\prime }$ of $\\Gamma $ .",
"By the dual characterization of the norm, $ \\Vert f\\Vert = \\sup _{\\varphi \\in X^\\ast ,\\,\\Vert \\varphi \\Vert \\le 1} |\\varphi (f)|, $ it is enough to show that for each continuous linear functional $\\varphi $ one has $\\sum _{k\\in \\Gamma } |\\varphi (f_k)| \\le \\pi \\max _{\\Gamma ^{\\prime }\\subseteq \\Gamma } \\Big |\\varphi \\Big (\\sum _{k\\in \\Gamma ^{\\prime }}f_k\\Big )\\Big |.$ Indeed, by the assumption on the coefficients $\\alpha _k$ we can then estimate $\\Big \\Vert \\sum _{k\\in \\Gamma } \\alpha _k f_k\\Big \\Vert & = \\sup _{\\varphi \\in X^\\ast ,\\,\\Vert \\varphi \\Vert \\le 1} \\Big |\\sum _{k\\in \\Gamma } \\alpha _k \\varphi (f_k)\\Big |\\le \\sup _{\\varphi \\in X^\\ast ,\\,\\Vert \\varphi \\Vert \\le 1} \\sum _{k\\in \\Gamma } |\\varphi (f_k)| \\\\& \\le \\pi \\sup _{\\varphi \\in X^\\ast ,\\,\\Vert \\varphi \\Vert \\le 1} \\max _{\\Gamma ^{\\prime }\\subseteq \\Gamma } \\Big |\\varphi \\Big (\\sum _{k\\in \\Gamma ^{\\prime }}f_k\\Big )\\Big |\\le \\pi \\max _{\\Gamma ^{\\prime }\\subseteq \\Gamma } \\Big \\Vert \\sum _{k\\in \\Gamma ^{\\prime }} f_k\\Big \\Vert .$ In order to show (REF ), observe that for every $\\beta \\in \\mathbb {C}$ we have the identity $ \\int _{\\mathbb {T}} \\max \\lbrace \\textup {Re}(e^{2\\pi i\\theta }\\beta ), 0 \\rbrace d\\theta = \\int _{-1/4}^{1/4} |\\beta |\\cos (2\\pi \\theta ) d\\theta = \\frac{|\\beta |}{\\pi }.",
"$ Therefore, if for each $\\theta \\in \\mathbb {T}$ we define $ \\Gamma ^{\\prime }_{\\theta } := \\big \\lbrace k\\in \\Gamma \\,:\\, \\textup {Re}\\big (e^{2\\pi i\\theta }\\varphi (f_k)\\big ) \\ge 0\\big \\rbrace , $ then $\\sum _{k\\in \\Gamma } |\\varphi (f_k)|& = \\pi \\int _{\\mathbb {T}} \\sum _{k\\in \\Gamma ^{\\prime }_{\\theta }} \\textup {Re}\\big (e^{2\\pi i\\theta }\\varphi (f_k)\\big ) d\\theta = \\pi \\int _{\\mathbb {T}} \\textup {Re}\\Big (e^{2\\pi i\\theta } \\varphi \\Big (\\sum _{k\\in \\Gamma ^{\\prime }_{\\theta }}f_k\\Big ) \\Big ) d\\theta \\\\& \\le \\pi \\int _{\\mathbb {T}} \\Big | \\varphi \\Big (\\sum _{k\\in \\Gamma ^{\\prime }_{\\theta }}f_k\\Big ) \\Big | d\\theta \\le \\pi \\max _{\\Gamma ^{\\prime }\\subseteq \\Gamma } \\Big |\\varphi \\Big (\\sum _{k\\in \\Gamma ^{\\prime }}f_k\\Big )\\Big |.",
"$ It is interesting to observe that the constant $\\pi $ in Lemma REF is optimal, already in the one-dimensional case $X=\\mathbb {C}$ .",
"Indeed, for a positive integer $n$ take $\\Gamma :=\\lbrace 0,1,\\ldots ,2n-1\\rbrace $ , $f_k:=e^{\\pi i k/n}$ , and $\\alpha _k:=e^{-\\pi i k/n}$ .",
"It can be easily seen that the largest absolute value of the sum over a subset equals $|\\sum _{k=0}^{n-1}f_k|=1/\\sin (\\pi /2n)$ and its ratio to the left hand side $|\\sum _{k=0}^{2n-1} \\alpha _k f_k|=2n$ converges to $1/\\pi $ as $n\\rightarrow \\infty $ .",
"Finally, we are ready to give the proof of the desired characterization of democracy, following the outline from .",
"We will reduce the claim to Theorem REF .",
"By part (b) of Lemma REF we immediately see that the left inequality in (REF ) is equivalent to condition (REF ), so it remains to show that the right inequality is equivalent with essential boundedness of $p_\\psi $ .",
"One implication is trivial, as $p_\\psi \\in \\textup {L}^\\infty (\\widehat{\\mathcal {L}})$ together with part (b) of Lemma REF guarantees $ \\frac{1}{\\mathop {\\textup {card}}\\Gamma } \\Big \\Vert \\sum _{k\\in \\Gamma } T_k\\psi \\Big \\Vert _{\\textup {L}^2(G)}^2\\le \\Vert g_\\Gamma \\Vert _{\\textup {L}^1(\\widehat{\\mathcal {L}})} \\Vert p_\\psi \\Vert _{\\textup {L}^\\infty (\\widehat{\\mathcal {L}})}= \\Vert p_\\psi \\Vert _{\\textup {L}^\\infty (\\widehat{\\mathcal {L}})} < +\\infty $ for every finite subset $\\Gamma $ of $\\mathcal {L}$ .",
"Conversely, suppose that $\\psi $ is such that the right inequality in (REF ) holds with some finite constant $C_\\psi $ .",
"By Lemma REF for any finite set $\\Gamma \\subset \\mathcal {L}$ and any coefficients $(\\alpha _k)_{k\\in \\Gamma }$ satisfying $|\\alpha _k|\\le 1$ we also have $ \\Big \\Vert \\sum _{k\\in \\Gamma } \\alpha _k T_k\\psi \\Big \\Vert _{\\textup {L}^2(G)}^2 \\le C\\mathop {\\textup {card}}\\Gamma , $ where $C:=\\pi ^2 C_\\psi $ .",
"The same computation from the proof of Lemma REF now gives $ \\int _{\\widehat{\\mathcal {L}}} \\frac{1}{\\mathop {\\textup {card}}\\Gamma } \\Big |\\sum _{k\\in \\Gamma }\\alpha _k\\overline{e(k,\\tau )}\\Big |^2 p_\\psi (\\tau ) d\\tau \\le C $ and the particular choice $\\alpha _k = e(k,\\sigma )$ for a fixed $\\sigma \\in \\widehat{\\mathcal {L}}$ simplifies to $(g_\\Gamma \\ast p_\\psi )(\\sigma ) = \\int _{\\widehat{\\mathcal {L}}} g_\\Gamma (\\sigma -\\tau ) p_\\psi (\\tau ) d\\tau \\le C.$ Finally, we take a Følner sequence $(F_n)_{n=1}^{\\infty }$ for $\\mathcal {L}$ and recall that the sequence $(g_{F_n} \\ast p_\\psi )_{n=1}^{\\infty }$ converges to $p_\\psi $ in the $\\textup {L}^1$ norm by Lemma REF .",
"There exist a subsequence $(g_{F_{n_j}} \\ast p_\\psi )_{j=1}^{\\infty }$ that converges a.e., so by taking $\\Gamma =F_{n_j}$ in (REF ) and letting $j\\rightarrow \\infty $ we conclude that $p_\\psi \\le C$ a.e.",
"Theorem REF is still not much more practical than Theorem REF , as condition (REF ) requires computation of a certain integral for each finite subset $\\Gamma $ .",
"However, there is one thing worth noticing that has now become evident: the democratic property of the system $\\mathcal {F}_\\psi $ is characterized only in terms of the lattice and the periodization function.",
"More precisely, once we are given the function $p_\\psi $ on $\\widehat{\\mathcal {L}}$ (rather than $\\widehat{G}/\\mathcal {L}^\\perp $ ), the democracy condition depends only on properties of the lattice $\\mathcal {L}$ (and its bi-character $e$ ), but not on the ambient group $G$ .",
"In particular, this means that the general case is not any more difficult than the special case $G=\\mathcal {L}$ .",
"In this latter case $\\mathcal {L}^\\perp $ is trivial by (REF ) and Theorem REF calls for a characterization in terms of $p_\\psi =|\\widehat{\\psi }|^2$ , which can a priori be an arbitrary nonnegative integrable function on $\\widehat{G}=\\widehat{\\mathcal {L}}$ .",
"Therefore, in the remaining text we mostly work on the lattice $(\\mathcal {L},+)$ and its (compact) dual group $(\\widehat{\\mathcal {L}},+)$ .",
"For any $A\\in \\mathcal {B}(\\widehat{\\mathcal {L}})$ we will simply write its measure as $|A|$ , instead of $\\lambda _{\\widehat{\\mathcal {L}}}(A)$ , in analogy with the usual practice on the torus $\\mathbb {T}$ .",
"Recall that we have chosen the normalization so that $|\\widehat{\\mathcal {L}}|=1$ .",
"We end this section with a sufficient condition from democracy, which is just an adaptation of .",
"Corollary 6 If there exist constants $c,C\\in \\langle 0,+\\infty \\rangle $ and an open neighborhood $U$ of 0 in $\\widehat{\\mathcal {L}}$ such that $ p_\\psi (\\tau ) \\le C \\ \\text{for a.e.",
"}\\ \\tau \\in \\widehat{\\mathcal {L}} \\quad \\text{and}\\quad p_\\psi (\\tau ) \\ge c \\ \\text{for a.e.",
"}\\ \\tau \\in U, $ then $\\mathcal {F}_\\psi $ is a democratic system.",
"This will be a consequence of Theorem REF as soon as we show that the hypothesis $p_\\psi \\ge c$ a.e.",
"on $U$ implies (REF ).",
"By or there exists an open neighborhood $V\\subseteq \\widehat{\\mathcal {L}}$ of 0 such that $V+V\\subseteq U$ and $-V=V$ .",
"Its characteristic function $\\mathbb {1}_V$ vanishes outside $U$ and satisfies $0\\le (\\mathbb {1}_V\\ast \\mathbb {1}_V)(\\tau )\\le |V|$ for each $\\tau \\in \\widehat{\\mathcal {L}}$ .",
"From the properties of the Haar measure we know $|V|>0$ ; see or .",
"Finally, for each nonempty finite $\\Gamma \\subset \\mathcal {L}$ by part (a) of Lemma REF we have $& \\int _{\\widehat{\\mathcal {L}}} g_\\Gamma (\\tau ) p_\\psi (\\tau ) d\\tau \\ge c \\int _{U} g_\\Gamma (\\tau ) d\\tau \\ge \\frac{c}{|V|}\\int _{\\widehat{\\mathcal {L}}} g_\\Gamma (\\tau ) (\\mathbb {1}_V\\ast \\mathbb {1}_V)(\\tau ) d\\tau \\\\& = \\frac{c}{|V|}\\sum _{k\\in \\Gamma -\\Gamma } \\frac{\\mathop {\\textup {card}}(\\Gamma \\cap (k+\\Gamma ))}{\\mathop {\\textup {card}}\\Gamma } \\int _{\\widehat{\\mathcal {L}}}\\int _{\\widehat{\\mathcal {L}}} e(k,\\sigma ) \\overline{e(k,\\sigma -\\tau )} \\mathbb {1}_{V}(\\sigma ) \\mathbb {1}_{V}(\\sigma -\\tau ) d\\sigma d\\tau \\\\& = \\frac{c}{|V|}\\sum _{k\\in \\Gamma -\\Gamma } \\frac{\\mathop {\\textup {card}}(\\Gamma \\cap (k+\\Gamma ))}{\\mathop {\\textup {card}}\\Gamma } \\Big | \\int _{\\widehat{\\mathcal {L}}} e(k,\\sigma ) \\mathbb {1}_{V}(\\sigma ) d\\sigma \\Big |^2\\ge \\frac{c}{|V|} |V|^2 = c|V|,$ where in the last inequality we estimated the sum of nonnegative terms by its single term for $k=0$ .",
"Therefore, $ \\inf _\\Gamma \\int _{\\widehat{\\mathcal {L}}} g_\\Gamma (\\tau ) p_\\psi (\\tau ) d\\tau \\ge c|V| >0, $ as desired.",
"Even though we would like to find more operative equivalent conditions for democratic systems of translates, obtaining these is an open problem even in the particular case when the lattice is $\\mathcal {L}=\\mathbb {Z}$ , as commented in the paper .",
"It was conjectured in that it is sufficient to test the democratic property on the arithmetic progressions $ \\Gamma = \\lbrace 0,d,2d,\\ldots ,(m-1)d\\rbrace $ for positive integers $d$ and $m$ .",
"In the same paper it was also conjectured that the system $\\mathcal {F}_\\psi $ is democratic if and only if $p_\\psi \\in \\textup {L}^\\infty (\\mathbb {T})$ and $ \\inf _{\\begin{array}{c}d\\in \\mathbb {N}\\\\ 0<\\varepsilon <1/2\\end{array}} \\frac{1}{2\\varepsilon }\\int _{\\bigcup _{j=0}^{d-1}\\left[\\frac{j-\\varepsilon }{d},\\frac{j+\\varepsilon }{d}\\right]} p_\\psi (\\tau ) d\\tau > 0.",
"$"
],
[
"Translates along torsion lattices",
"The goal of this section is to investigate further the conditions from Theorems REF and REF from the viewpoint of convex geometry.",
"Some lattices $\\mathcal {L}$ have very different algebraic structure from the integers.",
"We say that $\\mathcal {L}$ is a torsion lattice if $(\\mathcal {L},+)$ is also a torsion group, i.e., each of its elements has finite order.",
"This is clearly equivalent to the fact that there exists a sequence $(\\mathcal {M}_n)_{n=0}^{\\infty }$ of finite subgroups of $\\mathcal {L}$ such that $\\lbrace 0\\rbrace = \\mathcal {M}_0 \\subset \\mathcal {M}_1 \\subset \\mathcal {M}_2 \\subset \\cdots \\quad \\text{and}\\quad \\bigcup _{n=0}^{\\infty } \\mathcal {M}_n = \\mathcal {L}.$ In order to construct such a sequence recursively one has to enumerate the lattice as $\\mathcal {L}=\\lbrace k_i\\rbrace _{i=1}^{\\infty }$ and at the $n$ -th step take the smallest index $i$ such that $k_i\\notin \\mathcal {M}_{n-1}$ , observing that the sumset $\\mathcal {M}_n:=\\mathcal {M}_{n-1}+\\langle \\lbrace k_i\\rbrace \\rangle $ is a finite subgroup.",
"Any such sequence of subgroups is also a Følner sequence, i.e., it satisfies the Følner property (REF ), which is a trivial consequence of the fact that $k+\\mathcal {M}_n=\\mathcal {M}_n$ as soon as $k\\in \\mathcal {M}_n$ .",
"There are two typical examples of discrete abelian torsion groups to keep in mind.",
"Example 7 Let us use the standard notation $\\mathbb {Z}_n$ for the cyclic group of residues modulo $n$ , i.e., $\\mathbb {Z}_n\\cong \\mathbb {Z}/n\\mathbb {Z}$ .",
"Consider the infinite direct sum of cyclic groups $\\mathcal {L} & := \\bigoplus _{i\\in \\mathbb {N}}\\mathbb {Z}_{n_i} = \\mathbb {Z}_{n_1} \\oplus \\mathbb {Z}_{n_2} \\oplus \\mathbb {Z}_{n_3} \\oplus \\cdots \\\\& \\,= \\big \\lbrace (a_j)_{j=1}^{\\infty }\\in {\\textstyle \\prod _{j=1}^{\\infty }\\mathbb {Z}_{n_j}} : \\text{only finitely many }a_j\\text{ are nonzero} \\big \\rbrace , \\nonumber $ where $n_1,n_2,n_3,\\ldots $ is an arbitrary sequence of positive integers.",
"By and its dual group $\\widehat{\\mathcal {L}}$ is isomorphic to $\\prod _{i\\in \\mathbb {N}}\\mathbb {Z}_{n_i}$ , which are the so-called Vilenkin groups.",
"A particular case is $ \\mathbb {Z}_2^\\omega = \\mathbb {Z}_{2} \\oplus \\mathbb {Z}_{2} \\oplus \\mathbb {Z}_{2} \\oplus \\cdots , $ which we simply call the discrete dyadic group and its dual is known as the Cantor dyadic group.",
"The group $\\mathbb {Z}_2^\\omega $ is bijectively mapped to the set of nonnegative integers $\\mathbb {N}_0$ via $ \\mathbb {Z}_2^\\omega \\rightarrow \\mathbb {N}_0,\\quad (b_j)_{j=1}^{\\infty }\\mapsto \\sum _{j=1}^{\\infty } b_j 2^{j-1} $ and its inverse is given by $ \\mathbb {N}_0\\rightarrow \\mathbb {Z}_2^\\omega ,\\quad b\\mapsto \\text{the sequence of binary digits of $b$ viewed from right to left}.",
"$ Transporting the group operation from $\\mathbb {Z}_2^\\omega $ to $\\mathbb {N}_0$ via the above bijection yields a binary operation called the nim-addition, which plays an important role in the theory of impartial combinatorial games; see .",
"It is often convenient to represent the discrete dyadic group on the set of nonnegative integers, whenever we need to list some of its elements, as we do in Table REF below.",
"It is also natural to regard $\\mathcal {L}=\\mathbb {Z}_2^\\omega $ as a lattice in the so-called Walsh field $G=\\mathbb {W}$ , consisting of two-sided sequences of binary digits $(b_j)_{j\\in \\mathbb {Z}}$ such that $b_j=0$ when $j$ is large enough, but which can have infinitely many nonzero digits for negative indices $j$ .",
"In harmonic analysis $\\mathbb {W}$ is commonly considered as a “toy model” for the (nonnegative) reals and $\\mathbb {Z}_2^\\omega $ is then regarded as a toy model for the (nonnegative) integers.",
"Many difficult problems in analysis and combinatorics can be first studied in those simplified models; see and respectively.",
"If $\\sup _{j}n_j<\\infty $ and $N$ is the least common multiple of the numbers $n_1,n_2,\\ldots $ , then a notable property of (REF ) is $Nk=0$ for each $k\\in \\mathcal {L}$ .",
"We say that $\\mathcal {L}$ has a bounded exponent and the smallest such positive integer $N$ is called the exponent of $\\mathcal {L}$ .",
"Conversely, by the first Prüfer theorem (see ) any countable abelian group of bounded exponent is isomorphic to (REF ) for some bounded sequence of positive integers $(n_j)_{j=1}^{\\infty }$ .",
"Example 8 Take a prime number $p$ and consider the set $ \\mathcal {L} := \\Big \\lbrace \\frac{m}{p^n} \\,:\\, n\\in \\mathbb {N}_0,\\, m\\in \\mathbb {Z},\\, 0\\le m<p^n \\Big \\rbrace $ with the binary operation being the addition modulo 1.",
"This group is called the Prüfer $p$ -group and it is sometimes denoted $\\mathbb {Z}(p^\\infty )$ .",
"The dual group $\\widehat{\\mathcal {L}}$ is isomorphic to the group of $p$ -adic integers, see , so it is important in number theory.",
"The only finite subgroups of $\\mathcal {L}$ are $ \\mathcal {M}_n := \\Big \\lbrace \\frac{m}{p^n} \\,:\\, m\\in \\mathbb {Z},\\, 0\\le m<p^n \\Big \\rbrace $ for each nonnegative integer $n$ and observe that $\\mathcal {M}_0\\subseteq \\mathcal {M}_1\\subseteq \\mathcal {M}_2\\subseteq \\cdots $ and $\\mathcal {M}_n \\cong \\mathbb {Z}_{p^n}$ .",
"It can be said that the Prüfer group is a direct limit of the sequence $(\\mathbb {Z}_{p^n})_{n=1}^{\\infty }$ .",
"Actually, the only subgroups of $\\mathcal {L}$ are $\\mathcal {M}_0,\\mathcal {M}_1,\\mathcal {M}_2,\\ldots $ and the whole group itself, which complements the previous example, where the lattice has abundance of finite subgroups."
],
[
"Redundancy of convex combinations",
"Let us begin with the following easy observation.",
"Suppose that we want to be sparing and verify condition (REF ) from Theorem REF by taking infimum only over a certain collection $\\mathcal {G}$ of finite sets $\\Gamma \\subset \\mathcal {L}$ .",
"Whenever we have $\\Gamma _0,\\Gamma _1,\\ldots ,\\Gamma _m\\in \\mathcal {G}$ such that $g_{\\Gamma _0}$ is a convex combination of functions $g_{\\Gamma _1},\\ldots ,g_{\\Gamma _m}$ , then we can freely throw out the set $\\Gamma _0$ from the collection $\\mathcal {G}$ .",
"Indeed, if $g_{\\Gamma _0} = \\sum _{i=1}^{m} \\gamma _i g_{\\Gamma _i}$ for some nonnegative numbers $\\gamma _1,\\ldots ,\\gamma _m$ adding up to 1, then obviously $ \\int _{\\widehat{\\mathcal {L}}} g_{\\Gamma _0}(\\tau ) p_\\psi (\\tau ) d\\tau = \\sum _{i=1}^{m} \\gamma _i \\int _{\\widehat{\\mathcal {L}}} g_{\\Gamma _i}(\\tau ) p_\\psi (\\tau ) d\\tau \\ge \\min _{1\\le i\\le m} \\int _{\\widehat{\\mathcal {L}}} g_{\\Gamma _i}(\\tau ) p_\\psi (\\tau ) d\\tau .",
"$ It might be easier to spot dependencies among the functions $g_{\\Gamma }$ by looking at “sequences” of their Fourier coefficients $v_{\\Gamma }\\in \\ell ^{\\infty }(\\mathcal {L})$ .",
"By part (a) of Lemma REF we know that $v_\\Gamma \\colon \\mathcal {L}\\rightarrow \\mathbb {C}$ is given by $ v_\\Gamma (k) = \\frac{\\mathop {\\textup {card}}(\\Gamma \\cap (k+\\Gamma ))}{\\mathop {\\textup {card}}\\Gamma } $ and we can view $v_\\Gamma $ as an infinite vector of numbers from $[0,1]$ such that all but finitely many if its entries are equal to 0.",
"For each finite set $\\Gamma \\subset \\mathcal {L}$ we write $\\mathbb {1}_\\Gamma $ for the characteristic function of the set $\\Gamma $ , which can also be thought of as an infinite vector with only finitely many nonzero elements.",
"Several examples for the groups $\\mathcal {L}=\\mathbb {Z}$ and $\\mathcal {L}=\\mathbb {Z}_2^\\omega $ are given in Tables REF and REF respectively.",
"For example, on the latter group we have $v_{\\lbrace 0,1,2\\rbrace } & = \\textstyle \\frac{1}{6}v_{\\lbrace 0,1\\rbrace } + \\frac{1}{6}v_{\\lbrace 0,2\\rbrace } + \\frac{1}{6}v_{\\lbrace 0,3\\rbrace } + \\frac{1}{2}v_{\\lbrace 0,1,2,3\\rbrace } \\\\& = \\textstyle \\frac{1}{3}v_{\\lbrace 0\\rbrace } + \\frac{2}{3}v_{\\lbrace 0,1,2,3\\rbrace } .$ Table: A table of vectors 1 Γ \\mathbb {1}_\\Gamma and v Γ v_\\Gamma for several sets Γ⊂ℤ\\Gamma \\subset \\mathbb {Z}.Dots on either side replace sequences of zeros.Table: A table of vectors 1 Γ \\mathbb {1}_\\Gamma and v Γ v_\\Gamma for several sets Γ⊂ℤ 2 ω ≅ℕ 0 \\Gamma \\subset \\mathbb {Z}_2^\\omega \\cong \\mathbb {N}_0.Subgroups are marked with an asterisk.The Fourier transform $\\ell ^2(\\mathcal {L})\\rightarrow \\textup {L}^2(\\widehat{\\mathcal {L}})$ is a linear bijection, so any convex dependence among the vectors $v_\\Gamma $ translates into convex dependence among the functions $g_\\Gamma $ and vice versa.",
"Certain lattices might not even have many such dependencies.",
"This is why we restrict our attention to torsion groups throughout this section.",
"Here is an easy consequence of Theorem REF and the previous observations.",
"Recall that an extreme point of a convex set is any point that cannot be expressed as a nontrivial convex combination of two different points in that set.",
"Corollary 9 Suppose that $\\mathcal {L}$ is a torsion lattice.",
"Let $\\mathcal {K}\\subseteq \\textup {L}^1(\\widehat{\\mathcal {L}})$ be the convex hull of the functions $g_\\Gamma $ as $\\Gamma $ ranges over all nonempty finite subsets of $\\mathcal {L}$ and let $\\mathcal {E}$ be the collection of functions $g_\\Gamma $ that are also extreme points of $\\mathcal {K}$ .",
"The system $\\mathcal {F}_\\psi $ is democratic if and only if $p_\\psi \\in \\textup {L}^\\infty (\\widehat{\\mathcal {L}})$ and $\\inf _{g\\in \\mathcal {E}} \\int _{\\widehat{\\mathcal {L}}} g(\\tau ) p_\\psi (\\tau ) d\\tau > 0.$ Modulo Theorem REF the necessity of obvious, while for the sufficiency we only need to show that condition (REF ) implies condition (REF ).",
"Taking into account the remarks from the beginning of this subsection, it remains to prove that each function $g_\\Gamma $ is a convex combination of the functions from $\\mathcal {E}$ .",
"Take an arbitrary nonempty finite $\\Gamma _0\\subset \\mathcal {L}$ .",
"Choose a finite subgroup $\\mathcal {M}$ of $\\mathcal {L}$ that contains $\\Gamma _0-\\Gamma _0$ ; it could simply be a member of the sequence (REF ).",
"Let $\\mathcal {Q}$ be the set of all generalized trigonometric polynomials $q$ of the form $ q(\\tau ) = \\sum _{k\\in \\mathcal {M}} \\alpha _k e(k,\\tau ) $ for some coefficients $\\alpha _k\\in \\mathbb {R}$ .",
"Furthermore, let $\\mathcal {K}^{\\prime }\\subseteq \\textup {L}^1(\\widehat{\\mathcal {L}})$ be the convex hull of the functions $g_\\Gamma $ that also belong to the set $\\mathcal {Q}$ .",
"At first we claim that $\\mathcal {K}^{\\prime }=\\mathcal {K}\\cap \\mathcal {Q}$ .",
"The inclusion “$\\subseteq $ ” is obvious as $\\mathcal {Q}$ is a convex set, so we only need to show the reverse inclusion “$\\supseteq $ ”.",
"Any $q\\in \\mathcal {K}\\cap \\mathcal {Q}$ is (by the definition of $\\mathcal {K}$ ) a convex combination of some functions $g_{\\Gamma _1},\\ldots ,g_{\\Gamma _m}$ , i.e., there exist $\\gamma _1,\\ldots ,\\gamma _m>0$ such that $\\sum _{i=1}^{m}\\gamma _i=1$ and $q = \\sum _{i=1}^{m} \\gamma _i g_{\\Gamma _i}$ .",
"If we had $g_{\\Gamma _j}\\notin \\mathcal {Q}$ for some index $j$ , then there would exist $k\\in \\mathcal {L}\\setminus \\mathcal {M}$ such that $v_{\\Gamma _j}(k)>0$ .",
"This would imply that the $k$ -th Fourier coefficient of $q$ is $\\sum _{i=1}^{m} \\gamma _i v_{\\Gamma _i}(k) > 0$ and contradict $q\\in \\mathcal {Q}$ .",
"Therefore, $g_{\\Gamma _i}\\in \\mathcal {Q}$ for $i=1,\\ldots ,m$ , which precisely means $q\\in \\mathcal {K}^{\\prime }$ .",
"Since by the construction $g_{\\Gamma _0}\\in \\mathcal {K}\\cap \\mathcal {Q}$ , in particular we conclude $g_{\\Gamma _0}\\in \\mathcal {K}^{\\prime }$ .",
"Observe that $\\mathcal {Q}$ is a finite-dimensional real vector space; it is isomorphic to $\\mathbb {R}^{\\mathop {\\textup {card}}\\mathcal {M}}$ .",
"Moreover, there are only finitely many different function $g_\\Gamma $ belonging to $\\mathcal {Q}$ .",
"Indeed, take one corresponding $\\Gamma $ .",
"By translating it we may assume that $0\\in \\Gamma $ ; the resulting function $g_{\\Gamma }$ will not change.",
"Then we must have $\\Gamma \\subseteq \\mathcal {M}$ , since existence of $k\\in \\Gamma \\setminus \\mathcal {M}$ would imply $ v_\\Gamma (k) = \\frac{\\mathop {\\textup {card}}(\\Gamma \\cap (k+\\Gamma ))}{\\mathop {\\textup {card}}\\Gamma } \\ge \\frac{\\mathop {\\textup {card}}(\\lbrace k\\rbrace )}{\\mathop {\\textup {card}}\\Gamma } > 0, $ so $g_\\Gamma $ could not belong to $\\mathcal {Q}$ .",
"Using Minkowski's theorem (i.e., a finite-dimensional variant of the Krein-Milman theorem) we conclude that each point of $\\mathcal {K}^{\\prime }$ is a convex combination of some functions $g_\\Gamma $ that are also extreme points of $\\mathcal {K}^{\\prime }$ .",
"In particular this holds for $g_{\\Gamma _0}$ .",
"Finally, it remains to show that each function $g_{\\Gamma }$ which is an extreme point of $\\mathcal {K}^{\\prime }$ is also an extreme point of the larger convex set $\\mathcal {K}$ .",
"If the latter was not true, then we could write $g_\\Gamma $ as a convex combination $\\sum _{i=1}^{m} \\gamma _i g_{\\Gamma _i}$ with $m\\ge 2$ and $\\gamma _i>0$ for each index $i$ .",
"As before, from $g_{\\Gamma }\\in \\mathcal {Q}$ we would conclude $g_{\\Gamma _i}\\in \\mathcal {Q}$ for $i=1,\\ldots ,m$ , so that indeed $g_{\\Gamma _i}\\in \\mathcal {K}^{\\prime }$ for each $i$ , which would contradict the fact that $g_{\\Gamma }$ is an extreme point of $\\mathcal {K}^{\\prime }$ .",
"If $\\Gamma _0\\subseteq \\mathcal {M}_N$ for some large enough integer $N$ , where $(\\mathcal {M}_n)_{n=0}^{\\infty }$ is a sequence from (REF ), then from the previous proof we also know that $g_{\\Gamma _0}$ is an extreme point of $\\mathcal {K}$ if and only if $v_{\\Gamma _0}$ is an extreme point of the convex hull of the points $v_{\\Gamma }$ as $\\Gamma $ ranges over nonempty subsets of $\\mathcal {M}_N$ .",
"When $\\mathcal {M}$ is a subgroup of $\\mathcal {L}$ , then we reserve the notation $\\mathcal {M}^\\perp $ for the orthogonal complement of $\\mathcal {M}$ relative to $\\mathcal {L}$ , i.e., $ \\mathcal {M}^\\perp := \\lbrace \\tau \\in \\widehat{\\mathcal {L}} \\,:\\, e(k,\\tau )=1 \\text{ for each } k\\in \\mathcal {M}\\rbrace .",
"$ Since the topology of $\\mathcal {L}$ is discrete, $\\mathcal {M}$ is automatically closed in $\\mathcal {L}$ , so $\\mathcal {M}\\mapsto \\mathcal {M}^\\perp $ is now a bijective correspondence between subgroups of $\\mathcal {L}$ and closed subgroups of $\\widehat{\\mathcal {L}}$ .",
"Moreover, suppose that $\\mathcal {M}$ is finite.",
"From the isomorphism (REF ) we get $\\widehat{\\mathcal {L}}/\\mathcal {M}^\\perp \\cong \\widehat{\\mathcal {M}}$ and, since the dual group of a finite abelian group is isomorphic to itself (see ), we conclude $\\mathop {\\textup {card}}(\\widehat{\\mathcal {L}}/\\mathcal {M}^\\perp ) = \\mathop {\\textup {card}}\\mathcal {M}.$ Consequently, $\\mathcal {M}^\\perp $ is a closed subgroup of $\\widehat{\\mathcal {L}}$ of finite index and each such subgroup must also be open, since its complement is a union of finitely many closed cosets.",
"Conversely, every closed subgroup of $\\widehat{\\mathcal {L}}$ is certainly of the form $\\mathcal {M}^\\perp $ for some subgroup $\\mathcal {M}$ of $\\mathcal {L}$ .",
"If it is also of finite index, then from the same isomorphism (REF ) we conclude that $\\mathcal {M}$ must be finite.",
"Let us summarize by saying that $\\mathcal {M}\\mapsto \\mathcal {M}^\\perp $ is also a bijective correspondence between finite subgroups of $\\mathcal {L}$ and closed subgroups of $\\widehat{\\mathcal {L}}$ of finite index (which are automatically also open).",
"Take a torsion lattice $\\mathcal {L}$ .",
"Orthogonal complements of the terms in (REF ) satisfy $ \\mathcal {L} = \\mathcal {M}_0^\\perp \\supset \\mathcal {M}_1^\\perp \\supset \\mathcal {M}_2^\\perp \\supset \\cdots \\quad \\text{and}\\quad \\bigcap _{n=0}^{\\infty } \\mathcal {M}_n = \\lbrace 0\\rbrace .",
"$ Suppose that we have the periodization function $p_\\psi $ associated with some square-integrable function $\\psi $ for which we want to verify the democratic property of $\\mathcal {F}_\\psi $ .",
"In any real-world application it is likely that $p_\\psi $ will be given to us with a precision to certain scale and we are required to verify conditions $p_\\psi \\le C$ and $\\int _{\\widehat{\\mathcal {L}}} g_\\Gamma (\\tau ) p_\\psi (\\tau ) d\\tau \\ge c$ , where $C$ (resp.",
"$c$ ) is some reasonably large (resp.",
"small) constant.",
"If we only have information about the averages of $p_\\psi $ on the cosets of $\\mathcal {M}_N^\\perp $ (for some large but fixed positive integer $N$ ), then it only makes sense to test the lower bound for $\\Gamma \\subseteq \\mathcal {M}_N$ , as $g_\\Gamma $ is constant on the cosets of $\\mathcal {M}_N^\\perp $ only for such sets $\\Gamma $ .",
"This allows us to preprocess the finite group $\\mathcal {M}_N$ in the search for extreme points.",
"Numerical experimentation in Example REF suggests that this can save the computational time significantly."
],
[
"Subgroups as extreme points and some examples",
"We continue by investigating a special role of finite subgroups of $\\mathcal {L}$ .",
"It is already hinted by the following auxiliary lemma.",
"Lemma 10 (a) If $\\Gamma =m+\\mathcal {M}$ , where $m\\in \\mathcal {L}$ and $\\mathcal {M}$ is a finite subgroup of $\\mathcal {L}$ , then $ g_\\Gamma (\\tau ) = \\sum _{k\\in \\mathcal {M}} e(k,\\tau ); \\quad \\tau \\in \\widehat{\\mathcal {L}} $ and $ g_\\Gamma (\\tau ) = {\\left\\lbrace \\begin{array}{ll} 1/|\\mathcal {M}^\\perp | & \\text{for } \\tau \\in \\mathcal {M}^\\perp , \\\\ 0 & \\text{for } \\tau \\notin \\mathcal {M}^\\perp .",
"\\end{array}\\right.}",
"$ (b) Conversely, if $\\Gamma \\subseteq \\mathcal {L}$ is a nonempty finite set such that all coefficients of $g_\\Gamma $ are either 0 or 1, i.e., $ g_\\Gamma (\\tau ) = \\sum _{k\\in S} e(k,\\tau ) $ for some finite $S\\subseteq \\mathcal {L}$ , then $\\Gamma $ has to be of the form $\\Gamma =m+\\mathcal {M}$ where $m\\in \\mathcal {L}$ and $\\mathcal {M}$ is a finite subgroup of $\\mathcal {L}$ , and indeed $S=\\mathcal {M}$ .",
"(a) The first claim is a consequence of part (a) of Lemma REF , since the cosets $m+\\mathcal {M}$ and $k+m+\\mathcal {M}$ are either equal (when $k\\in \\mathcal {M}$ ) or disjoint (when $k\\notin \\mathcal {M}$ ).",
"For the second claim observe that from (REF ) we obtain $\\mathop {\\textup {card}}\\mathcal {M} = |\\widehat{\\mathcal {L}}|/|\\mathcal {M}^\\perp | = 1/|\\mathcal {M}^\\perp |,$ so for $\\tau \\in \\mathcal {M}^\\perp $ we have $ g_\\mathcal {M}(\\tau ) = \\sum _{k\\in \\mathcal {M}} e(k,\\tau ) = \\sum _{k\\in \\mathcal {M}} 1 = \\mathop {\\textup {card}}\\mathcal {M} = \\frac{1}{|\\mathcal {M}^\\perp |}.",
"$ On the other hand, if $\\tau \\notin \\mathcal {M}^\\perp $ , then there exists $k_0\\in \\mathcal {M}$ such that $e(k_0,\\tau )\\ne 1$ .",
"Therefore, $g_\\mathcal {M}(\\tau ) & = \\sum _{k\\in \\mathcal {M}} e(k_0+k-k_0,\\tau ) = e(k_0,\\tau ) \\sum _{k\\in \\mathcal {M}} e(k-k_0,\\tau ) \\\\& = e(k_0,\\tau ) \\sum _{m\\in \\mathcal {M}-k_0=\\mathcal {M}} e(m,\\tau ) = \\underbrace{e(k_0,\\tau )}_{\\ne 1}g_\\mathcal {M}(\\tau ),$ which implies $g_\\mathcal {M}(\\tau )=0$ .",
"(b) Recall that $e(k,\\cdot )$ ; $k\\in \\mathcal {L}$ form an orthonormal basis for $\\textup {L}^2(\\widehat{\\mathcal {L}})$ , so in particular these functions are linearly independent.",
"Combining the assumption with part (a) of Lemma REF and equaling the coefficients we see that for any $k\\in \\mathcal {L}$ the sets $\\Gamma $ and $k+\\Gamma $ are either equal or disjoint.",
"Fix an arbitrary $m\\in \\Gamma $ and define $\\mathcal {M}:=\\Gamma -m$ .",
"We need to prove that $\\mathcal {M}$ is a subgroup of $\\mathcal {L}$ .",
"Take any $a,b\\in \\mathcal {M}$ and observe that $ m \\in \\Gamma \\cap (m-b+\\mathcal {M}) = \\Gamma \\cap (-b+\\Gamma ).",
"$ Since this intersection is nonempty, we must have $\\Gamma =-b+\\Gamma $ i.e., $\\mathcal {M}=-b+\\mathcal {M}$ , which in particular implies $a-b\\in -b+\\mathcal {M}=\\mathcal {M}$ .",
"We have just shown $\\mathcal {M}-\\mathcal {M}\\subseteq \\mathcal {M}$ , which verifies that $\\mathcal {M}$ is a subgroup.",
"Applying part (a) and using linear independence once again we also conclude $S=\\mathcal {M}$ .",
"In particular we see that the vector of Fourier coefficients $v_\\mathcal {M}$ of a finite subgroup $\\mathcal {M}$ of $\\mathcal {L}$ has only $\\lbrace 0,1\\rbrace $ -entries.",
"Consequently, the corresponding function $g_\\mathcal {M}$ certainly belongs to the set of extreme points $\\mathcal {E}$ of the set $\\mathcal {K}$ described in Corollary REF .",
"Part (b) of Lemma REF characterizes subgroup cosets as the only nonempty finite sets $\\Gamma $ having $\\lbrace 0,1\\rbrace $ -Fourier coefficients.",
"However, we need to emphasize that in general these are not the only extreme points; Example REF will indirectly disprove that fact on the Prüfer 2-group.",
"On a related note, polytopes whose vertices are tuples with all entries from $\\lbrace 0,1\\rbrace $ are called $0/1$ -polytopes.",
"They are combinatorially interesting and extensively studied; see .",
"When $\\Gamma =\\mathcal {M}$ is a finite subgroup of $\\mathcal {L}$ , then from part (a) of the previous lemma we see that the integral in (REF ) becomes a density-type expression for the periodization: $ \\frac{1}{|\\mathcal {M}^\\perp |} \\int _{\\mathcal {M}^\\perp } p_\\psi (\\tau ) d\\tau .",
"$ Example 11 This is a continuation of Example REF ; recall the discrete dyadic group $\\mathbb {Z}_2^\\omega $ introduced there.",
"Take $ \\mathcal {M}_n := \\mathbb {Z}_2 \\oplus \\cdots \\oplus \\mathbb {Z}_2 \\oplus \\lbrace 0\\rbrace \\oplus \\lbrace 0\\rbrace \\oplus \\cdots \\cong \\mathbb {Z}_2^n $ as finite subgroups that exhaust the whole group.",
"Each $\\mathcal {M}_n$ is actually an $n$ -dimensional vector space over $\\mathbb {Z}_2$ .",
"The number of $k$ -dimensional subspaces of $\\mathcal {M}_n$ , $0\\le k\\le n$ , is given by the particular case $q=2$ of the $q$ -binomial coefficient, $ {n \\atopwithdelims ()k}_q := \\frac{(q^n-1)(q^{n-1}-1)\\cdots (q^{n-k+1}-1)}{(q^1-1)(q^2-1)\\cdots (q^k-1)}.",
"$ Consequently, cardinality of the set $\\mathcal {E}_n$ of extreme points $g_\\Gamma \\in \\mathcal {E}$ such that $\\Gamma \\subseteq \\mathcal {M}_n$ is at least $\\sum _{k=0}^{n}{n \\atopwithdelims ()k}_2$ .",
"The last expression defines sequence A006116 in the encyclopedia OEIS (beginning with 1, 2, 5, 16, 67, ...) and its asymptotic behavior is well-known.",
"We can find $ \\liminf _{n\\rightarrow \\infty }\\frac{\\mathop {\\textup {card}}(\\mathcal {E}_n)}{2^{n^2/4}} >0, $ so the number of extreme points in $\\mathcal {M}_n$ grows super-exponentially in $n$ .",
"The actual numerical data are given in Table REF .",
"A lot of torsion already causes that many functions $g_\\Gamma $ coincide, but it is expected that removing convex combinations reduces that number further significantly.",
"Table: Numerical data for the discrete dyadic group.The following lemma will be needed in the next example.",
"Lemma 12 If $\\Gamma \\subset \\mathcal {L}$ is a nonempty finite set and $\\mathcal {M}$ is a finite subgroup of $\\mathcal {L}$ , then $ \\int _{\\mathcal {M}^\\perp }\\!",
"g_\\Gamma (\\tau ) d\\tau = \\frac{\\mathop {\\textup {card}} \\lbrace (k,k^{\\prime })\\in \\Gamma \\times \\Gamma \\,:\\, k-k^{\\prime }\\in \\mathcal {M}\\rbrace }{\\mathop {\\textup {card}}\\mathcal {M} \\mathop {\\textup {card}}\\Gamma }.",
"$ Let us begin by showing $ \\int _{\\mathcal {M}^\\perp }\\!",
"e(k,\\tau ) d\\tau = {\\left\\lbrace \\begin{array}{ll} |\\mathcal {M}^\\perp | & \\text{for } k\\in \\mathcal {M}, \\\\ 0 & \\text{for } k\\notin \\mathcal {M}.",
"\\end{array}\\right.}",
"$ This claim is obvious for $k\\in \\mathcal {M}$ , so take $k\\in \\mathcal {L}\\setminus \\mathcal {M}$ .",
"By $(\\mathcal {M}^\\perp )^\\perp =\\mathcal {M}$ there exists $\\tau _0\\in \\mathcal {M}^\\perp $ such that $e(k,\\tau _0)\\ne 1$ .",
"Now we can write (using translation invariance of the Haar measure on $\\widehat{\\mathcal {L}}$ ): $\\int _{\\mathcal {M}^\\perp }\\!",
"e(k,\\tau ) d\\tau & = \\int _{\\mathcal {M}^\\perp }\\!",
"e(k,\\tau _0+\\tau -\\tau _0) d\\tau = e(k,\\tau _0) \\int _{\\mathcal {M}^\\perp }\\!",
"e(k,\\tau -\\tau _0) d\\tau \\\\& = e(k,\\tau _0) \\int _{-\\tau _0+\\mathcal {M}^\\perp }\\!",
"e(k,\\tau ) d\\tau = \\underbrace{e(k,\\tau _0)}_{\\ne 1} \\int _{\\mathcal {M}^\\perp }\\!",
"e(k,\\tau ) d\\tau ,$ which implies that the above integral is 0, as needed.",
"Applying part (a) of Lemma REF , integrating term-by-term, and using (REF ) we get $ \\int _{\\mathcal {M}^\\perp }\\!",
"g_\\Gamma (\\tau ) d\\tau = \\frac{\\sum _{k\\in \\mathcal {M}}\\mathop {\\textup {card}}(\\Gamma \\cap (k+\\Gamma ))}{\\mathop {\\textup {card}}\\mathcal {M} \\mathop {\\textup {card}}\\Gamma }.",
"$ In order to transform this formula into the desired one it remains to observe that the numerator above equals $ \\mathop {\\textup {card}}\\big \\lbrace (k_1,k_2)\\in \\Gamma \\times \\Gamma \\,:\\, k_1-k_2\\in \\mathcal {M}\\big \\rbrace , $ which is easily seen by double counting.",
"One might get an impression that, in the case of torsion lattices, it is enough to test the democratic property on subgroups.",
"However, this is not the case, as the following example shows.",
"Example 13 This is a continuation of Example REF ; recall the Prüfer 2-group $\\mathbb {Z}(2^\\infty )$ and its subgroups $\\mathcal {M}_n \\cong \\mathbb {Z}_{2^n}$ .",
"For each positive integer $n$ choose $s_n\\in \\mathcal {M}_n\\setminus \\mathcal {M}_{n-1}$ .",
"Then $\\mathcal {M}_{n-1}$ and $s_n+\\mathcal {M}_{n-1}$ are the only two cosets of the smaller subgroup in the larger one.",
"For any positive integer $n$ define $ \\Gamma _n := \\lbrace 0,s_1\\rbrace + \\lbrace 0,s_3\\rbrace + \\cdots + \\lbrace 0,s_{2n-1}\\rbrace .",
"$ Any $k,k^{\\prime }\\in \\Gamma _n$ have unique representations as $k=\\sum _{j=1}^{n}\\alpha _j s_{2j-1}$ , $k^{\\prime }=\\sum _{j=1}^{n}\\alpha ^{\\prime }_j s_{2j-1}$ , where $\\alpha _j,\\alpha ^{\\prime }_j\\in \\lbrace 0,1\\rbrace $ for each index $j=1,\\ldots ,n$ .",
"For a fixed integer $0\\le m\\le 2n-1$ we observe that $ k-k^{\\prime }\\in \\mathcal {M}_m \\Longleftrightarrow \\alpha _j=\\alpha ^{\\prime }_j \\text{ for all indices $j$ such that } j>(m+1)/2.",
"$ Consequently, $ \\mathop {\\textup {card}}\\big \\lbrace (k,k^{\\prime })\\in \\Gamma _n\\times \\Gamma _n \\,:\\, k-k^{\\prime }\\in \\mathcal {M}_m\\big \\rbrace = {\\left\\lbrace \\begin{array}{ll}2^{n+m/2} & \\text{if } 0\\le m\\le 2n-1 \\text{ is even}, \\\\2^{n+(m+1)/2} & \\text{if } 0\\le m\\le 2n-1 \\text{ is odd},\\end{array}\\right.}",
"$ so Lemma REF gives $ \\int _{\\mathcal {M}_m^\\perp }\\!",
"g_{\\Gamma _n}(\\tau ) d\\tau = {\\left\\lbrace \\begin{array}{ll}2^{-m/2} & \\text{if } 0\\le m\\le 2n-1 \\text{ is even}, \\\\2^{-(m-1)/2} & \\text{if } 0\\le m\\le 2n-1 \\text{ is odd},\\end{array}\\right.}",
"$ From this we conclude $\\int _{\\mathcal {M}_{2i-1}^\\perp \\setminus \\mathcal {M}_{2i}^\\perp }\\!",
"g_{\\Gamma _n}(\\tau ) d\\tau = 2^{-i}$ for $i=1,2,\\ldots ,n-1$ .",
"Let us now choose a square-integrable function $\\psi $ such the periodization function $p_\\psi $ is equal to the characteristic function of the set $\\bigcup _{i=0}^{\\infty }(\\mathcal {M}_{2i}^\\perp \\setminus \\mathcal {M}_{2i+1}^\\perp )$ .",
"(For this purpose one can simply take $G$ to also equal $\\mathbb {Z}(2^\\infty )$ .)",
"Using (REF ) we get $ \\int _{\\widehat{\\mathcal {L}}} g_{\\Gamma _n}(\\tau ) p_\\psi (\\tau ) d\\tau = 1 - \\sum _{i=1}^{\\infty } \\int _{\\mathcal {M}_{2i-1}^\\perp \\setminus \\mathcal {M}_{2i}^\\perp } g_{\\Gamma _n}(\\tau ) d\\tau \\le 1 - \\sum _{i=1}^{n-1} 2^{-i} = 2^{-n+1}, $ so by taking $n\\rightarrow \\infty $ we see that condition (REF ) fails and $\\mathcal {F}_\\psi $ cannot be a democratic system.",
"On the other hand, for each nonnegative integer $n$ by part (a) of Lemma REF we have $ \\int _{\\widehat{\\mathcal {L}}} g_{\\mathcal {M}_n}(\\tau ) p_\\psi (\\tau ) d\\tau = \\frac{1}{|\\mathcal {M}_n^\\perp |} \\int _{\\mathcal {M}_n^\\perp } p_\\psi (\\tau ) d\\tau = \\frac{\\sum _{i\\ge n/2}|\\mathcal {M}_{2i}^\\perp \\setminus \\mathcal {M}_{2i+1}^\\perp |}{|\\mathcal {M}_n^\\perp |}= {\\left\\lbrace \\begin{array}{ll}2/3 & \\text{if $n$ is even}, \\\\1/3 & \\text{if $n$ is odd},\\end{array}\\right.}",
"$ so the above quantities, obtained by testing (REF ) on subgroups only, are bounded from below by $1/3$ .",
"The previous example shows that, in general, testing the democratic property on subgroups is not sufficient."
],
[
"Closing remarks",
"Recall that the paper conjectures the sufficiency of testing the democratic property on the finite arithmetic progressions in $\\mathbb {Z}$ .",
"When we pass to the torsion lattice $\\mathcal {L}$ , sufficiently long progressions automatically become finite subgroups.",
"In Example REF we saw that finite subgroups are not enough, but it might still be sufficient to test condition (REF ) for the democratic property of $\\mathcal {F}_\\psi $ by taking only finite sets $\\Gamma $ that are approximate subgroups in an appropriate sense.",
"The notion of an approximate subgroup was defined in several possible ways in the book .",
"When we are given a concrete sequence of exhausting subgroups (REF ), numerical data suggest that the number of extreme points $g_\\Gamma \\in \\mathcal {E}$ coming from $\\Gamma \\subseteq \\mathcal {M}_n$ grows at most like $e^{P(\\log \\mathop {\\textup {card}}(\\mathcal {M}_n))}$ for some polynomial $P$ .",
"Indeed, Example REF gives a lower bound of the form $c^{\\prime } e^{c(\\log \\mathop {\\textup {card}}(\\mathcal {M}_n))^2}$ for the discrete dyadic group $\\mathbb {Z}_2^\\omega $ , but we were not able to establish the upper bound, either for $\\mathbb {Z}_2^\\omega $ , or for any other torsion lattice $\\mathcal {L}$ .",
"We conclude that characterizations of democratic systems of translates still remain without definite answers and we hope that they might attract researchers from various fields.",
"BR15article author=M.",
"Bownik, author=K.",
"A. Ross, title=The structure of translation-invariant spaces on locally compact abelian groups, journal=J.",
"Fourier Anal.",
"Appl., volume=21, year=2015, number=4, pages=849–884, note=, eprint= CP10article author=C.",
"Cabrelli, author=V.",
"Paternostro, title=Shift-invariant spaces on LCA groups, journal=J.",
"Funct.",
"Anal., volume=258, year=2010, number=6, pages=2034–2059, note=, eprint= C13book author=D.",
"L. Cohn, title=Measure theory, series=Birkhäuser Advanced Texts: Basel Textbooks, publisher=Birkhäuser/Springer, New York, volume=, year=2013 C01book author=J.",
"H. Conway, title=On numbers and games, series=, publisher=A K Peters, Ltd., Natick, MA, volume=, year=2001 F95book author=G.",
"B. Folland, title=A course in abstract harmonic analysis, series=Studies in Advanced Mathematics, publisher=CRC Press, Boca Raton, FL, volume=, year=1995 F55article author=E.",
"Følner, title=On groups with full Banach mean value, journal=Math.",
"Scand., volume=3, year=1955, number=, pages=243–254, note=, eprint= F70book author=L.",
"Fuchs, title=Infinite abelian groups.",
"Vol.",
"I., series=Pure and Applied Mathematics, publisher=Academic Press, New York-London, volume=36, year=1970 G05article author=B.",
"Green, title=Finite field models in additive combinatorics, journal=, volume=, date=, number=, pages=1–27, note=, eprint=, book= title=Surveys in combinatorics 2005, series=London Math.",
"Soc.",
"Lecture Note Ser., publisher=Cambridge Univ.",
"Press, Cambridge, volume=327, year=2005 H11book author=C.",
"Heil, title=A basis theory primer.",
"Expanded edition, series=Applied and Numerical Harmonic Analysis, publisher=Birkhäuser/Springer, New York, volume=, year=2011 HP06article author=C.",
"Heil, author=A.",
"M. Powell, title=Gabor Schauder bases and the Balian-Low theorem, journal=J.",
"Math.",
"Phys., volume=47, year=2006, number=11, pages=113506, 21pp, note=, eprint= HNSS13article author=E.",
"Hernández, author=M.",
"Nielsen, author=H.",
"Šikić, author=F.",
"Soria, title=Democratic systems of translates, journal=J.",
"Approx.",
"Theory, volume=171, year=2013, number=, pages=105–127, note=, eprint= HSWW10aarticle author=E.",
"Hernández, author=H.",
"Šikić, author=G.",
"Weiss, author=E.",
"Wilson, title=On the properties of the integer translates of a square integrable function, journal=Contemp.",
"Math., volume=505, year=2010, number=, pages=233–249, note=, eprint= HSWW10article author=E.",
"Hernández, author=H.",
"Šikić, author=G.",
"Weiss, author=E.",
"Wilson, title=Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform, journal=Colloq.",
"Math., volume=118, year=2010, number=1, pages=313–332, note=, eprint= HW96book author=E.",
"Hernández, author=G.",
"Weiss, title=A first course on wavelets, series=Studies in Advanced Mathematics, publisher=CRC Press, Boca Raton, FL, volume=, year=1996 HR79book author=E.",
"Hewitt, author=K.",
"A. Ross, title=Abstract harmonic analysis.",
"Vol.",
"I: Structure of topological groups.",
"Integration theory.",
"Group representations, series=Die Grundlehren der mathematischen Wissenschaften, volume=115, publisher=Springer-Verlag, Berlin-Heidelberg-New York, year=1979 HR70book author=E.",
"Hewitt, author=K.",
"A. Ross, title=Abstract harmonic analysis.",
"Vol.",
"II: Structure and analysis for compact groups.",
"Analysis on locally compact Abelian groups, series=Die Grundlehren der mathematischen Wissenschaften, volume=152, publisher=Springer-Verlag, Berlin-Heidelberg-New York, year=1970 HMW73article author=R.",
"Hunt, author=B.",
"Muckenhoupt, author=R.",
"Wheeden, title=Weighted norm inequalities for the conjugate function and Hilbert transform, journal=Trans.",
"Amer.",
"Math.",
"Soc., volume=176, year=1973, number=, pages=227–251, note=, eprint= KT99article author=S.",
"V. Konyagin, author=V.",
"N. Temlyakov, title=A remark on greedy approximation in Banach spaces, journal=East J.",
"Approx., volume=5, year=1999, number=3, pages=365–379, note=, eprint= N15article author=M.",
"Nielsen, title=On quasi-greedy bases associated with unitary representations of countable groups, journal=Glas.",
"Mat.",
"Ser.",
"III, volume=50, year=2015, number=1, pages=193–205, note=, eprint= NS07article author=M.",
"Nielsen, author=H.",
"Šikić, title=Schauder bases of integer translates, journal=Appl.",
"Comput.",
"Harmon.",
"Anal., volume=23, year=2007, number=2, pages=259–262, note=, eprint= NS14article author=M.",
"Nielsen, author=H.",
"Šikić, title=On stability of Schauder bases of integer translates, journal=J.",
"Funct.",
"Anal., volume=266, year=2014, number=4, pages=2281–2293, note=, eprint= Pa10article author=M.",
"Paluszynski, title=A note on integer translates of a square integrable function on $R$ , journal=Colloq.",
"Math., volume=118, year=2010, number=2, pages=593–597, note=, eprint= P88book author=A.",
"L. T. Paterson, title=Amenability, series=Mathematical Surveys and Monographs, volume=29, publisher=AMS, Providence, RI, year=1988 R62book author=W.",
"Rudin, title=Fourier analysis on groups, series=Interscience Tracts in Pure and Applied Mathematics, volume=12, publisher=Interscience Publishers, John Wiley and Sons, New York-London, year=1962 S13article author=S.",
"Saliani, title=$\\ell ^2$ -linear independence for the system of integer translates of a square integrable function, journal=Proc.",
"Amer.",
"Math.",
"Soc., volume=141, year=2013, number=3, pages=937–941, note=, eprint= SSl12article author=H.",
"Šikić, author=I.",
"Slamić, title=Linear independence and sets of uniqueness, journal=Glas.",
"Mat.",
"Ser.",
"III, volume=47, year=2012, number=2, pages=415–420, note=, eprint= SSW08article author=H.",
"Šikić, author=D.",
"Speegle, author=G.",
"Weiss, title=Structure of the set of dyadic PFW's, journal=, volume=, date=, number=, pages=263–291, note=, eprint=, book= title=Frames and operator theory in analysis and signal processing, series=Contemp.",
"Math., publisher=AMS, Providence, RI, volume=451, year=2008 SW11article author=H.",
"Šikić, author=E.",
"N. Wilson, title=Lattice invariant subspaces and sampling, journal=Appl.",
"Comput.",
"Harmon.",
"Anal., volume=31, year=2011, number=1, pages=26–43, note=, eprint= Sl16article author=I.",
"Slamić, title=$\\ell ^2$ -linear independence for systems generated by dual integrable representations of LCA groups, journal=to appear in Collect.",
"Math., volume=, year=2016, number=, pages=15pp, note=, eprint= Sl14article author=I.",
"Slamić, title=$\\ell ^p$ -linear independence of the system of integer translates, journal=J.",
"Fourier Anal.",
"Appl., volume=20, year=2014, number=4, pages=766–783, note=, eprint= OEISbook author=N.",
"J.",
"A. Sloane (ed.",
"), title=The On-Line Encyclopedia of Integer Sequences (OEIS), series=, volume=, publisher=published electronically, year=, eprint=https://oeis.org TV06book author=T.",
"Tao, author=V.",
"Vu, title=Additive combinatorics, series=Cambridge Studies in Advanced Mathematics, publisher=Cambridge University Press, Cambridge, volume=105, year=2006 T09article author=T.",
"Tao, title=Some notes on amenability, journal=What's new, year=2009, eprint=terrytao.wordpress.com Tem1article author=V.N.",
"Temlyakov, title=Greedy algorithm and m-term trigonometric approximation, journal=Constr.",
"Approx., volume=14, year=1998, number=4, pages=569–587, eprint= Tem2article author=V.",
"N. Temlyakov, title=The best m-term approximation and greedy algorithms, journal=Adv.",
"Comput.",
"Math., volume=8, year=1998, number=3, pages=249–265, eprint= T06book author=C.",
"Thiele, title=Wave packet analysis, series=CBMS Regional Conference Series in Mathematics, volume=105, publisher=AMS, Providence, RI, year=2006 Wojarticle author=P.",
"Wojtaszczyk, title=Greedy algorithm for general biorthogonal systems, journal=J.",
"Approx.",
"Theory, volume=107, year=2000, number=2, pages=293–314, eprint= Z00article author=G.",
"M. Ziegler, title=Lectures on $0/1$ -polytopes, journal=, volume=, date=, number=, pages=1–41, note=, eprint=, book= title=Polytopes — combinatorics and computation, series=Oberwolfach Seminars, publisher=Birkhäuser, Basel, volume=29, year=2000"
]
] | 1709.01747 |
[
[
"Production and sequential decays of charmed hyperons"
],
[
"Abstract We investigate production and decay of the $\\Lambda_c^+ $ hyperon.",
"The production considered is through the $e^+e^-$ annihilation channel, $e^+e^-\\rightarrow\\Lambda_c^+ \\bar{\\Lambda}_c^-$, with summation over the $\\bar{\\Lambda}_c^- $ anti-hyperon spin directions.",
"It is in this situation that the $\\Lambda_c^+ $ decay chain is identified.",
"Two kinds of sequential decays are studied.",
"The first one is the doubly weak decay $B_1\\rightarrow B_2 M_2$, followed by $B_2 \\rightarrow B_3 M_3$.",
"The other one is the mixed weak-electromagnetic decay $B_1\\rightarrow B_2 M_2$, followed by $B_2 \\rightarrow B_3 \\gamma$.",
"In both schemes $B$ denotes baryons and $M$ mesons.",
"We should also mention that the initial state of the $\\Lambda_c^+ $ hyperon is polarized."
],
[
"Introduction",
"We shall investigate properties of certain sequential decays of the $\\Lambda _c^+$ hyperon, but in order to do so we first need to produce them.",
"To this end we consider the reaction $e^+ e^- \\rightarrow \\Lambda _c^+ \\bar{\\Lambda }_c^-$ , which is analyzed in detail in Refs.",
"[1], [2].",
"In order to describe such an annihilation process two hadronic form factors are needed.",
"They can be parametrized by two parameters, $\\alpha $ and $\\Delta \\Phi $ , with $-1\\le \\alpha \\le 1$ .",
"For their precise definitions we refer to Ref. [2].",
"The general cross-section distribution of this annihilation reaction depends on six structure functions which themselves are functions of $\\alpha $ , $\\Delta \\Phi $ , and $\\theta $ , the scattering angle.",
"In our application, however, we sum over the decay products of the anti-hyperon $\\bar{\\Lambda }_c^- $ , but identify the decay chain of the hyperon $\\Lambda _c^+$ , so called single tag events.",
"In this simplified case only two structure functions are relevant, ${\\cal {R}} &=& 1 +\\alpha \\cos ^2\\!\\theta , \\\\{\\cal {S}} &=& \\sqrt{1-\\alpha ^2}\\sin \\theta \\cos \\theta \\sin (\\Delta \\Phi ).",
"$ The scattering distribution function for the $\\Lambda _c^+$ hyperon production becomes, according to Refs.",
"[2], proportional to $W(\\mathbf {n}) = {\\cal {R}}+ {\\cal {S}}\\, \\mathbf {N}\\cdot \\mathbf {n},$ where $\\mathbf {n}$ is the direction of the hyperon spin vector in the hyperon rest system, $\\mathbf {N}$ the normal to the scattering plane, $\\mathbf {N}=\\frac{1}{\\sin \\theta } \\, \\hat{\\mathbf {p}}\\times \\hat{\\mathbf {k}},$ and $\\cos \\theta =\\hat{\\mathbf {p}}\\cdot \\hat{\\mathbf {k}}$ .",
"The momenta $\\mathbf {k}$ and $\\mathbf {p}$ are the relative momenta in the initial and final states, in the center of momentum (c.m.)",
"system.",
"The meaning of the spin vector $ \\mathbf {n}$ is explained in Ref.",
"[3] From Eq.",
"(REF ) we deduce for the spin distribution function, $S(\\mathbf {P}) = 1 + \\mathbf {P} \\cdot \\mathbf {n} $ and $\\mathbf {P}$ the hyperon polarization, $\\mathbf {P} = ( {\\cal {S}}/ {\\cal {R}} ) \\mathbf {N},$ subject to the restriction $|\\mathbf {P}|\\le 1$ .",
"For an unpolarized initial state hyperon $\\mathbf {P}=0$ ."
],
[
"Weak hyperon decays",
"The weak hyperon decay $c\\rightarrow d\\pi $ , of which $\\Lambda \\rightarrow p\\pi ^-$ is an example, is described by two amplitudes, one S-wave and one P-wave amplitude.",
"The decay distribution is commonly described by three parameters, denoted $\\alpha \\beta \\gamma $ .",
"They are not independent but fulfill the relation $\\alpha ^2 + \\beta ^2 +\\gamma ^2=1.$ The parametrization of this hyperon decay is discussed in detail in Ref.",
"[4] and also in Ref.[2].",
"We denote by $G_c(c,d)$ the distribution function describing the weak hyperon decay $c\\rightarrow d\\pi $ , given the spin vectors $\\mathbf {n}_c\\textbf {}$ and $\\mathbf {n}_d$ , $G_c(c, d) = 1+\\alpha _c \\mathbf {n}_c\\cdot \\mathbf {l}_d+\\alpha _c \\mathbf {n}_d\\cdot \\mathbf {l}_d+\\mathbf {n}_c\\cdot \\mathbf {L}_c(\\mathbf {n}_d,\\mathbf {l}_d ),$ with $\\mathbf {L}_c(\\mathbf {n}_d, \\mathbf {l}_d)=\\,\\gamma _c \\mathbf {n}_d+\\bigg [(1-\\gamma _c)\\mathbf {n}_d\\cdot \\mathbf {l}_d\\bigg ]\\, \\mathbf {l}_d+\\beta _c \\mathbf {n}_d\\times \\mathbf {l}_d .$ The vector $\\mathbf {l}_d$ is a unit vector in the direction of motion of the decay baryon $d$ in the rest system of baryon $c$ .",
"The indices on the $\\alpha \\beta \\gamma $ parameters remind us they characterize hyperon $c$ .",
"Since the spin of baryon $d$ is often not measured, the relevant decay density is obtained by averaging over the spin directions $\\mathbf {n}_d$ , $W_c(\\mathbf {n}_c; \\mathbf {l}_d) =&\\, {\\bigg \\langle }G_c(c,d) {\\bigg \\rangle }_d \\nonumber \\\\=&\\, U_c+ \\mathbf {n}_c\\cdot \\mathbf {V}_c,$ with $U_c=1 , \\qquad \\mathbf {V}_c= \\alpha _c \\mathbf {l}_d.",
"$ Hyperons we study are produced in some reaction, and their states are described by some spin distribution function, Eq.",
"(REF ), $S_c(\\mathbf {P}_c)= 1+\\mathbf {P}_c\\cdot \\mathbf {n}_c.",
"$ The final-state distribution in a production reaction followed by decay is obtained by a folding, pertaining to the intermediate and final hyperon spin directions $\\mathbf {n}_c$ and $\\mathbf {n}_d$ , $W_c(\\mathbf {P}_c;\\mathbf {l}_d)= &\\,{\\bigg \\langle } S_c(\\mathbf {P}_c)G_c(c,d) {\\bigg \\rangle }_{cd}\\nonumber \\\\= &\\,1+ \\mathbf {P}_c\\cdot \\mathbf {V}_c,$ where $\\mathbf {V}_c=\\alpha _c \\mathbf {l}_d$ , from Eq.",
"(REF ).",
"The folding over intermediate spin directions follows the prescription of Ref.",
"[1], $\\big {\\langle } 1\\big {\\rangle }_{\\mathbf {n}} =1, \\quad \\big {\\langle } \\mathbf {n} \\big {\\rangle }_{\\mathbf {n}} =0, \\quad \\big {\\langle } \\mathbf {n}\\cdot \\mathbf {k} \\mathbf {n}\\cdot \\mathbf {l} \\big {\\rangle }_{\\mathbf {n}}=\\mathbf {k}\\cdot \\mathbf {l} .$ From Eq.",
"(REF ) it is clear that if the polarization is known the asymmetry parameter $\\alpha _c$ can be measured, but not the $\\beta _c$ or $\\gamma _c$ parameters.",
"For that to be possible we must measure the polarization of the decay baryon $d$ .",
"If hyperon $c$ is produced within a $c\\bar{c}$ pair in $e^+e^-$ annihilation then the polarization can be determined from the cross-section distribution."
],
[
"Electromagnetic hyperon transitions",
"Electromagnetic transitions such as $\\Sigma ^0\\rightarrow \\Lambda \\gamma $ and $\\Xi ^0\\rightarrow \\Lambda \\gamma $ can also be studied in $\\Lambda _c^+$ decays.",
"An electromagnetic transition $c\\rightarrow d\\gamma $ is described by a transition distribution function similar to that of the weak decay, Eq.",
"(REF ).",
"However, the special feature of the electromagnetic interaction is the photon helicity which takes only two values, $ \\lambda _\\gamma =\\pm 1$ .",
"The electromagnetic transition distribution function corresponding to Eq.",
"(REF ) is $G_{\\gamma }(cd; \\lambda _\\gamma )= (1-\\mathbf {n}_{c}\\cdot \\mathbf {l}_d \\mathbf {l}_d\\cdot \\mathbf {n}_{d})-\\lambda _\\gamma (\\mathbf {n}_{c}\\cdot \\mathbf {l}_d-\\mathbf {n}_{d}\\cdot \\mathbf {l}_d),$ where $\\mathbf {l}_d$ is a unit vector in the direction of motion of hyperon $d$ in the rest system of hyperon $c$ .",
"Averaging over photon polarizations the transition distribution takes a very simple form, $G_\\gamma (c,d)= 1-\\mathbf {n}_{c}\\cdot {\\mathbf {l}}_{d} {\\mathbf {l}}_{d}\\cdot \\mathbf {n}_{d}.$ We notice that when both hadron spins are parallel or anti-parallel to the photon momentum, then the transition probability vanishes, a property of angular-momentum conservation.",
"We also notice that expression (REF ) cannot be written in the $\\alpha \\beta \\gamma $ representation of Eq.",
"(REF )."
],
[
"Two-step weak hyperon decay",
"Now, we apply the above technique to hyperons decaying in two steps, such as $b\\rightarrow c\\rightarrow d$ , accompanied by pions.",
"An example of this decay mode is $ \\Lambda _c^+\\rightarrow \\Lambda \\pi ^+$ followed by $\\Lambda \\rightarrow p\\pi ^-$ .",
"We denote by $G_b(b,c)$ the distribution function describing the hyperon decay $b\\rightarrow c\\pi $ pertaining to spin vectors $\\mathbf {n}_b\\textbf {}$ and $\\mathbf {n}_c$ , $G_b(b,c) = 1+\\alpha _b \\mathbf {n}_b\\cdot \\mathbf {l}_c+\\alpha _b \\mathbf {n}_c\\cdot \\mathbf {l}_c+\\mathbf {n}_b\\cdot \\mathbf {L}_b(\\mathbf {n}_c,\\mathbf {l}_c ),$ with $\\mathbf {L}_b(\\mathbf {n}_c, \\mathbf {l}_c)=\\,\\gamma _b \\mathbf {n}_c+\\bigg [(1-\\gamma _b)\\mathbf {n}_c\\cdot \\mathbf {l}_c\\bigg ]\\, \\mathbf {l}_c+\\beta _b \\mathbf {n}_c\\times \\mathbf {l}_c .$ The vector $\\mathbf {l}_c$ is a unit vector in the direction of motion of baryon $c$ in the rest system of baryon $b$ .",
"Folding together the distribution functions $G_b(b,c)$ and $G_c(c,d)$ , averaging over spin vectors $\\mathbf {n}_c$ and $\\mathbf {n}_d$ following the folding prescription (REF ), we get the decay density distribution function $W_b(\\mathbf {n}_b; \\mathbf {l}_c, \\mathbf {l}_d) =&\\,{\\bigg \\langle } G_b(b,c)G_c(c,d) {\\bigg \\rangle }_{cd} \\nonumber \\\\ =&\\, U_b+ \\mathbf {n}_b\\cdot \\mathbf {V}_b, $ with $U_b=&1 +\\alpha _b\\alpha _c \\mathbf {l}_c \\cdot \\mathbf {l}_d, \\\\\\mathbf {V}_b=&\\alpha _b \\mathbf {l}_c +\\alpha _c\\mathbf {L}_b(\\mathbf {l}_d,\\mathbf {l}_c ).$ The result is interesting.",
"In many cases the asymmetry parameter $\\alpha _c$ for the $c$ hyperon and the polarization $\\mathbf {P}_b$ for the initial-state $b$ hyperon are known.",
"Then, just as in the single-step case of Eq.",
"(REF ), the initial state is described by a spin distribution function $S_b(\\mathbf {P}_b)= 1+\\mathbf {P}_b\\cdot \\mathbf {n}_b.",
"$ For the decay distribution of a polarized hyperon, we obtain $W_b(\\mathbf {P}_b; \\mathbf {l}_c, \\mathbf {l}_d) =&\\,{\\bigg \\langle } S_b( \\mathbf {P}_b) G_b(b,c)G_c(c,d) {\\bigg \\rangle }_{bcd} \\nonumber \\\\ =&\\, U_b+ \\mathbf {P}_b\\cdot \\mathbf {V}_b.",
"$ This is equivalent to making the replacement $\\mathbf {n}_b\\rightarrow \\mathbf {P}_b$ in Eq.",
"(REF ).",
"We conclude that by determining $U_b$ and $\\mathbf {V}_b$ of Eqs.",
"(REF ) and (), we should be able to determine all three decay parameters $\\alpha _b$ , $\\beta _b$ , and $\\gamma _b$ , for the $b$ hyperon, and $\\alpha _c$ for the $c$ hyperon.",
"It is now clear how to get the cross-section distribution for production of $\\Lambda ^+_c$ in $e^+e^-$ annihilation and its subsequent decay $\\Lambda ^+_c\\rightarrow \\Lambda \\pi ^+ $ and $\\Lambda \\rightarrow p \\pi ^-$ .",
"Starting from the expressions for the scattering distribution function, Eq.",
"(REF ), and the polarization, Eq.",
"(REF ).",
"we obtain $\\textrm {d}\\sigma \\propto \\bigg [ {\\cal {R}} U_{\\Lambda _c} + {\\cal {S}}\\mathbf {N}\\cdot \\mathbf {V}_{\\Lambda _c} \\bigg ] \\textrm {d}\\Omega _{\\Lambda _c}\\textrm {d}\\Omega _{\\Lambda } \\textrm {d}\\Omega _p , $ with $\\mathbf {N}$ , Eq.",
"(REF ), the normal to the scattering plane.",
"The functions $ \\cal {R}$ and $ \\cal {S}$ are defined in Eqs.",
"(REF ) and () and depend among other things on the $\\Lambda ^+_c$ scattering angle $\\theta $ (=$\\theta _{\\Lambda _c}$ ).",
"In Eqs.",
"(REF ) and () indices are interpreted as; $b=\\Lambda ^+_c$ , $c=\\Lambda $ , $d=p$ .",
"When integrating over the decay angles $\\Omega _{\\Lambda }$ and $\\Omega _p $ in Eq.",
"(REF ) we observe that the term involving the polarization $\\mathbf {N}\\cdot \\mathbf {V}_{\\Lambda _c}$ vanishes, as does the term involving the angular dependent part of $U_{\\Lambda _c}$ .",
"This results is the cross-section distribution $\\textrm {d}\\sigma \\propto \\bigg [1 +\\alpha \\cos ^2\\!\\theta _{\\Lambda _c}\\bigg ] \\textrm {d}\\Omega _{\\Lambda _c} ,$ describing the annihilation reaction $e^+ e^- \\rightarrow \\Lambda _c^+ \\bar{\\Lambda }_c^-$ .",
"It is more interesting to perform a partial integration.",
"Let us integrate over the angles $\\Omega _{\\Lambda }$ and $\\Omega _p $ keeping $\\cos \\theta _{\\Lambda p}$ of $\\cos \\theta _{\\Lambda p}=\\mathbf {l}_\\Lambda \\cdot \\mathbf {l}_p$ constant.",
"Also in this case does the contribution involving the polarization vanish.",
"We are left with $\\textrm {d}\\sigma \\propto \\bigg [1 +\\alpha \\cos ^2\\!\\theta _{\\Lambda _c}\\bigg ]\\bigg [1+\\alpha _{\\Lambda _c}\\alpha _\\Lambda \\cos \\theta _{\\Lambda p} \\bigg ]\\textrm {d}(\\cos \\theta _{\\Lambda _c}) \\textrm {d}(\\cos \\theta _{\\Lambda p}) .",
"$ The cross-section distribution of Eq.",
"(REF ) applies also to the decay chain, $\\Lambda ^+_c\\rightarrow \\Sigma ^+ \\pi ^0 $ and $\\Sigma ^+\\rightarrow p \\pi ^0 $ , with the corresponding identification of indices $b,$ $c$ , and $d$ ."
],
[
"Differential distributions",
"The cross-section distribution (REF ) is a function of two unit vectors $\\mathbf {l}_1=\\mathbf {l}_{\\Lambda }$ , the direction of motion of the Lambda hyperon in the rest system of the charmed-Lambda hyperon, and $\\mathbf {l}_2=\\mathbf {l}_{p}$ the direction of motion of the proton in the rest system of the Lambda hyperon.",
"In order to handle these vectors we need a common coordinate system which we define as follows.",
"The scattering plane of the reaction $e^+e^-\\rightarrow \\Lambda _c\\bar{\\Lambda }_c$ is spanned by the unit vectors $\\hat{\\mathbf {p}}=\\mathbf {l}_{\\Lambda _c}$ and $\\hat{\\mathbf {k}}=\\mathbf {l}_{e^+}$ , as measured in the c.m.",
"system.",
"We assume the scattering to be to the left, with scattering angle $\\theta \\ge 0$ .",
"If the scattering is to the right we rotate such an event $180^{\\circ }$ around the $\\mathbf {k}$ -axis, so that the scattering appears to be to the left.",
"The scattering plane makes up the $xz$ -plane, with the $y$ -axis along the normal to the scattering plane.",
"We choose a right-handed coordinate system with basis vectors $\\mathbf {e}_z &=& \\hat{\\mathbf {p}}, \\\\\\mathbf {e}_y &=& \\frac{1}{\\sin \\theta } ( \\hat{\\mathbf {p}}\\times \\hat{\\mathbf {k}} ) , \\\\\\mathbf {e}_x &=& \\frac{1}{\\sin \\theta } (\\hat{\\mathbf {p}}\\times \\hat{\\mathbf {k}} )\\times \\hat{\\mathbf {p}}.$ Expressed in terms of them the initial-state momentum $\\hat{\\mathbf {k}}= \\sin \\theta \\, \\mathbf {e}_x +\\cos \\theta \\, \\mathbf {e}_z .$ This coordinate system is used for defining the directional angles of the Lambda and the proton.",
"The directional angles of the Lambda hyperon in the charmed-Lambda hyperon rest system are, $\\mathbf {l}_1=(\\cos \\phi _1 \\sin \\theta _1, \\sin \\phi _1 \\sin \\theta _1, \\cos \\theta _1),$ whereas the directional angles of the proton in the Lambda hyperon rest system are $\\mathbf {l}_2=(\\cos \\phi _2 \\sin \\theta _2, \\sin \\phi _2 \\sin \\theta _2, \\cos \\theta _2).$ An event of the reaction $e^+e^-\\rightarrow \\bar{\\Lambda }_c \\Lambda _c(\\rightarrow \\Lambda (\\rightarrow p\\pi )\\pi )$ is specified by the five dimensional vector ${\\xi }=(\\theta ,\\Omega _{1},\\Omega _{2})$ , and the differential-cross-section distribution as summarized by Eq.",
"(REF ) reads, ${\\textrm {d}\\sigma }\\propto {\\cal {W}}({\\xi })\\ {\\textrm {d}\\!\\cos \\theta \\ \\textrm {d}\\Omega _{1}\\textrm {d}\\Omega _{2}}.$ At the moment, we are not interested in absolute normalizations.",
"The differential-distribution function ${\\cal {W}}({\\xi })$ is obtained from Eqs.",
"(REF , ,REF , , REF ) and can be expressed as, $\\begin{split}{\\cal {W}}({\\xi })=&\\ {\\cal {F}}_0({\\xi })+{{{\\alpha }}}{\\cal {F}}_1({\\xi })+\\alpha _1\\alpha _2\\ {\\bigg (} {\\cal {F}}_2({\\xi })+{{{\\alpha }}}{\\cal {F}}_3({\\xi })\\bigg )\\\\+&\\ \\sqrt{1-{{\\alpha }}^2}\\cos ({{\\Delta \\Phi }})\\, \\bigg ( {\\cal {F}}_7({\\xi })+ {{\\alpha _{1}}}{\\cal {F}}_4 ({\\xi }) + \\beta _1 {\\cal {F}}_6({\\xi })\\\\+&\\ \\gamma _1 \\left( {\\cal {F}}_5({\\xi })- {\\cal {F}}_7({\\xi })\\right)\\bigg ) , \\end{split}$ using a set of eight angular functions ${\\cal {F}}_k({\\xi })$ defined as: ${\\cal {F}}_0({\\xi }) =&1, \\nonumber \\\\{\\cal {F}}_1({\\xi }) =&{\\cos ^2\\!\\theta },\\nonumber \\\\{\\cal {F}}_2({\\xi }) =&\\sin \\theta _1\\sin \\theta _2\\cos (\\phi _1-\\phi _2)+\\cos \\theta _1\\cos \\theta _2 ,\\nonumber \\\\{\\cal {F}}_3({\\xi }) =& {\\cos ^2\\!\\theta }\\ {\\cal {F}}_2({\\xi }), \\nonumber \\\\{\\cal {F}}_4({\\xi }) =& \\sin \\theta \\cos \\theta \\sin \\theta _1 \\sin \\phi _1 ,\\nonumber \\\\{\\cal {F}}_5({\\xi }) =& \\sin \\theta \\cos \\theta \\sin \\theta _2 \\sin \\phi _2 , \\nonumber \\\\{\\cal {F}}_6({\\xi }) =&\\sin \\theta \\cos \\theta \\ (\\cos \\theta _2 \\sin \\theta _1\\cos \\phi _1 \\nonumber \\\\&\\qquad \\qquad \\qquad -\\cos \\theta _1 \\sin \\theta _2\\cos \\phi _2 ) , \\nonumber \\\\{\\cal {F}}_7({\\xi }) =&\\sin \\theta \\cos \\theta \\sin \\theta _1\\sin \\phi _1 \\ {\\cal {F}}_2({\\xi }).$ The differential distribution of Eq.",
"(REF ) involves two parameters related to the $e^+e^-\\rightarrow \\Lambda _c\\bar{\\Lambda }_c$ reaction that can be determined by data: the ratio of form factors $\\alpha $ , and the relative phase of form factors $\\Delta \\Phi $ .",
"In addition, the distribution function ${\\cal {W}}({\\xi })$ depends on the weak-decay parameters $\\alpha _1\\beta _1\\gamma _1$ of the charmed-hyperon decay $\\Lambda _c\\rightarrow \\Lambda \\pi $ , and on the weak-decay parameters $\\alpha _2\\beta _2\\gamma _2$ of the hyperon decay $\\Lambda \\rightarrow p\\pi ^-$ .",
"However, the dependency on $\\beta _2$ and $\\gamma _2$ drops out.",
"Similarly, integrating over $\\textrm {d}\\Omega _{2}$ we get ${\\textrm {d}\\sigma }\\propto \\bigg [ 1+&\\alpha \\, {\\cos ^2\\!\\theta } \\nonumber \\\\+&\\alpha _1\\sqrt{1-{{\\alpha }}^2}\\cos ({{\\Delta \\Phi }})\\sin \\theta \\cos \\theta \\sin \\theta _1 \\sin \\phi _1 \\bigg ]\\, \\textrm {d}\\Omega \\, \\textrm {d}\\Omega _{1} ,$ where now the dependency on $\\beta _1$ and $\\gamma _1$ also drops out.",
"The last term in this equation originates with the scalar $\\mathbf {P}_{\\Lambda _c}\\cdot \\mathbf {N}$ .",
"The charmed-hypern polarization vanishes at $\\theta =0^\\circ $ , $90^\\circ $ and $180^\\circ $ .",
"The distributions presented here will hopefully be of value in the analysis of BESIII data."
],
[
"Mixed weak-electromagnetic hyperon decay",
"Now, we extend the formalism to hyperons decaying in two steps, with one being electromagnetic.",
"An example of such a decay chain is $ \\Lambda _c^+\\rightarrow \\Sigma ^0\\pi ^+$ followed by $\\Sigma ^0\\rightarrow \\Lambda \\gamma $ .",
"As before we employ indices $b$ , $c$ , and $d$ for variables belonging to $\\Lambda _c^+$ , $\\Sigma ^0$ , and $\\Lambda $ .",
"The distribution functions for the weak and electromagnetic transitions are given in Eqs.",
"(REF ) and (REF ), $G_b(b, c) &= 1+\\alpha _b \\mathbf {n}_b\\cdot \\mathbf {l}_c+\\alpha _b \\mathbf {n}_c\\cdot \\mathbf {l}_c+\\mathbf {n}_b\\cdot \\mathbf {L}_b(\\mathbf {n}_c,\\mathbf {l}_c ), \\\\G_\\gamma (c,d) &= 1-\\mathbf {n}_c\\cdot \\mathbf {l}_d \\mathbf {l}_d\\cdot \\mathbf {n}_d.$ Performing a folding of the product of the distribution functions $G_b(b,c)$ and $G_\\gamma (c,d)$ , i.e.",
"averaging over spin vectors $\\mathbf {n}_c$ and $\\mathbf {n}_d$ following the folding prescription (REF ), we get $W_b(\\mathbf {n}_b; \\mathbf {l}_c, \\mathbf {l}_d) =&\\,{\\bigg \\langle } G_b(b,c)G_\\gamma (c,d) {\\bigg \\rangle }_{cd} \\nonumber \\\\ =&\\, U_b+ \\mathbf {n}_b\\cdot \\mathbf {V}_b, $ with $U_b=1 , \\qquad \\mathbf {V}_b= \\alpha _b \\mathbf {l}_c.",
"$ These expressions for $U_b$ and $\\mathbf {V}_b$ are noteworthy.",
"They are in fact the same as those of a one-step $b\\rightarrow c\\pi $ decay, Eq.",
"(REF ).",
"Hence, the electromagnetic decay does not add any structure, Eqs.",
"(REF ) are independent of $\\mathbf {l}_d$ .",
"The initial state spin distribution function for hyperon $b$ produced in $e^+e^-$ annihilation is as above, Eq.",
"(REF ), $S_b(\\mathbf {P}_b)= 1+\\mathbf {P}_b\\cdot \\mathbf {n}_b, $ Folding this distribution function with the decay distribution function of Eq.",
"(REF ), we obtain $W_b(\\mathbf {P}_b; \\mathbf {l}_c, \\mathbf {l}_d) =&\\,{\\bigg \\langle } S_b( \\mathbf {P}_b) G_b(b,c)G_\\gamma (c,d) {\\bigg \\rangle }_{bcd} \\nonumber \\\\ =&\\, U_b+ \\mathbf {P}_b\\cdot \\mathbf {V}_b.",
"$ As noted earlier this is equivalent to making the replacement $\\mathbf {n}_b\\rightarrow \\mathbf {P}_b$ in Eq.",
"(REF ).",
"We also notice if we manage to determine $U_b$ and $\\mathbf {V}_b$ of Eqs.",
"(REF ), the only parameter that can be fixed is $\\alpha _b$ , a meager return.",
"The expression for the cross-section distribution for $\\Lambda ^+_c$ production and subsequent decays $\\Lambda ^+_c\\rightarrow \\Sigma ^0 \\pi ^+ $ and $\\Sigma ^0\\rightarrow \\Lambda \\gamma $ is $\\textrm {d}\\sigma \\propto \\bigg [ {\\cal {R}} U_{\\Lambda _c}+ {\\cal {S}} \\mathbf {N}\\cdot \\mathbf {V}_{\\Lambda _c}\\bigg ] \\textrm {d}\\Omega _{\\Lambda _c}\\textrm {d}\\Omega _{\\Sigma } \\textrm {d}\\Omega _{\\Lambda } ,$ with $\\mathbf {N}$ , Eq.",
"(REF ), the normal to the scattering plane, and $U_{\\Lambda _c}=1$ , $\\mathbf {V}_{\\Lambda _c}=\\alpha _{\\Lambda _c } \\mathbf {l}_{\\Sigma }$ , from Eq.",
"(REF ).",
"The functions $ \\cal {R}$ and $ \\cal {S}$ are defined in Eqs.",
"(REF ) and () and depend among other things on the $\\Lambda ^+_c$ scattering angle $\\theta $ .",
"Finally, we mention that it is possible to expand the $\\Lambda ^+_c$ decay chain by adding the decay $\\Lambda \\rightarrow p\\pi ^-$ ."
],
[
"Acknowledgments",
"Thanks to Andrzej Kupsc and Karin Schönning for valuable discussions and suggestions.",
"In this Appendix we detail the angular integration leading to Eq.",
"(REF ).",
"Consider two unit vectors $\\mathbf {l}_c$ and $\\mathbf {l}_d$ .",
"We want to integrate over the angles $\\Omega _c$ and $\\Omega _d$ keeping $\\cos \\theta _{cd}=\\mathbf {l}_c\\cdot \\mathbf {l}_d$ fixed.",
"To this end we put the vectors in the $xy$ -plane of the coordinate system O', $\\mathbf {l}_c&=& (1,0,0),\\\\\\mathbf {l}_d&=& (\\cos \\theta _{cd},\\sin \\theta _{cd},0), \\\\\\mathbf {l}_c\\times \\mathbf {l}_d&=& (0,0,\\sin \\theta _{cd}).$ We then rotate the coordinate system O' with respect to the space fixed coordinate system O, where the normal to the scattering plane is along the $Z$ -direction.",
"The rotation matrix which transforms the column vector $\\bar{r}_b$ in O' into the column vector $\\bar{r}_s$ in O is the matrix $R^{-1}(\\alpha , \\beta , \\gamma )=\\left( \\begin{array}{lll} \\cos \\alpha \\cos \\beta \\cos \\gamma - \\sin \\alpha \\sin \\gamma & \\cos \\gamma \\cos \\beta \\sin \\alpha +\\sin \\gamma \\cos \\alpha & -\\sin \\beta \\cos \\gamma \\\\-\\sin \\gamma \\cos \\beta \\cos \\alpha -\\cos \\gamma \\sin \\alpha & -\\sin \\gamma \\cos \\beta \\sin \\alpha +\\cos \\gamma \\cos \\alpha & \\sin \\beta \\sin \\gamma \\\\\\cos \\alpha \\sin \\beta & \\sin \\gamma \\sin \\beta & \\cos \\beta \\end{array} \\right).$ with $\\bar{r}_s=R^{-1}(\\alpha , \\beta , \\gamma )\\bar{r}_b$ and $\\alpha \\beta \\gamma $ the Euler angles.",
"The angular integrations can be expressed in terms of the Euler angles, as $\\textrm {d}\\Omega _c \\, \\textrm {d}\\Omega _d=\\textrm {d}(\\cos \\theta _{cd})\\,\\textrm {d}\\alpha \\, \\textrm {d}(\\cos \\beta )\\, \\textrm {d}\\gamma .$ The expression to be integrated, Eq.",
"(), reads $U_b+ \\mathbf {P}\\cdot \\mathbf {V}_b =&1 +\\alpha _b\\alpha _c \\mathbf {l}_c \\cdot \\mathbf {l}_d+\\alpha _b\\mathbf {P}\\cdot \\mathbf {l}_c +\\gamma _b\\mathbf {P}\\cdot \\mathbf {l}_d\\nonumber \\\\&+(1-\\gamma _b)\\mathbf {P}\\cdot \\mathbf {l}_c\\mathbf {l}_d\\cdot \\mathbf {l}_c+\\beta _b \\mathbf {P}\\cdot ( \\mathbf {l}_d\\times \\mathbf {l}_c) ,$ with $\\mathbf {P}$ along the $Z$ -direction.",
"Now, we note that terms proportional to $\\mathbf {P}\\cdot \\mathbf {l}_c$ or $\\mathbf {P}\\cdot \\mathbf {l}_d$ vanish upon integration over angles $\\alpha $ or $\\gamma $ .",
"Therefore, $& \\int \\textrm {d}\\Omega _c\\textrm {d}\\Omega _d\\,(U_b+ \\mathbf {P}\\cdot \\mathbf {V}_b) = \\nonumber \\\\& \\qquad =4\\pi ^2 \\int \\textrm {d}(\\cos \\theta _{cd})\\textrm {d}(\\cos \\beta ) \\,\\bigg (1 +\\alpha _b\\alpha _c \\cos \\theta _{cd} \\nonumber \\\\& \\hspace{113.81102pt}-\\beta _b\\alpha _c P\\sin \\theta _{cd}\\cos \\beta \\bigg )&\\nonumber \\\\&\\qquad =8\\pi ^2\\int \\textrm {d}(\\cos \\theta _{cd})\\, \\bigg (1 +\\alpha _b\\alpha _c \\cos \\theta _{cd} \\bigg ).$ This result leads to Eq.",
"(REF )."
]
] | 1709.01803 |
[
[
"Graphical criteria for positive solutions to linear systems"
],
[
"Abstract We study linear systems of equations with coefficients in a generic partially ordered ring $R$ and a unique solution, and seek conditions for the solution to be nonnegative, that is, every component of the solution is a quotient of two nonnegative elements in $R$.",
"The requirement of a nonnegative solution arises typically in applications, such as in biology and ecology, where quantities of interest are concentrations and abundances.",
"We provide novel conditions on a labeled multidigraph associated with the linear system that guarantee the solution to be nonnegative.",
"Furthermore, we study a generalization of the first class of linear systems, where the coefficient matrix has a specific block form and provide analogous conditions for nonnegativity of the solution, similarly based on a labeled multidigraph.",
"The latter scenario arises naturally in chemical reaction network theory, when studying full or partial parameterizations of the positive part of the steady state variety of a polynomial dynamical system in the concentrations of the molecular species."
],
[
"Introduction",
"A classical problem in applied mathematics is to determine the solutions to a linear system of equations.",
"In applications, it is often the case that only positive or nonnegative real solutions to the system are meaningful, and criteria to assert positivity and nonnegativity of the solutions have thus been developed [1], [13], [2], [18].",
"Nonnegativity is required, for example, in the case of equilibria concentrations of molecular species in biochemistry [3], [4], [9], species abundances at steady state in ecology [14], stationary distributions of Markov chains in probability theory [16], and in Birch's theorem for maximum likelihood estimation in statistics [17].",
"Also in economics and game theory are equilibria often required to be nonnegative [12], [8].",
"Many of these situations arise from considering dynamical systems where the state variables are restricted to the positive or nonnegative orthant.",
"We consider a linear system of equations $Ax+b=0$ , with $\\det A\\ne 0$ and where the coefficients of $A,b$ are in a generic partially ordered ring $R$ .",
"By adding an extra row to $A$ such that the column sums are zero, we might associate in a natural way a Laplacian $L$ with the linear system and the corresponding (so-called) labeled canonical multidigraph.",
"The components of the solution to the linear system are rational functions on the entries of $A$ and $b$ .",
"Their numerator and denominator can, by means of the Matrix-Tree Theorem [24], be expressed as polynomials on the labels of the rooted spanning trees of the canonical multidigraph, and in fact, of any labeled multidigraph with Laplacian $L$ .",
"If the multidigraph is what we call a P-graph (Definition REF ), then we show that the solution to $Ax+b=0$ is nonnegative.",
"These conditions are readily fulfilled if the off-diagonal elements of $L$ (not only $A$ ) are nonnegative.",
"In the applications motivating this work, the P-graph condition is hardly met.",
"This happens for example in the study of biochemical reaction networks, where the system of interest arises from computing the equilibrium points of a dynamical system constrained to certain invariant linear varieties.",
"However, in many instances the solution is nevertheless nonnegative.",
"In the second part of the paper we explore this other scenario.",
"We consider systems of a specific block form, compatible with the application setting, and derive conditions on another (related) multidigraph that ensure nonnegativity of the solution.",
"In this situation, the solution is expressed as a rational function in the labels of rooted spanning forests of the multidigraph.",
"The second scenario is an extension of the generic case given in the first part of the paper.",
"Typically in applications, the entries of $A,b$ depend on parameters and inputs that cannot be fixed beforehand, but must be treated as “unknown” or symbolic variables.",
"Our approach accommodates this since solutions are given as rational functions in these entries.",
"Alternatively, one might view the parameters as functions, and apply the results for the ring of real valued functions.",
"In this way, we can study the nonnegativity of solutions without fixing parameters and inputs.",
"One natural application of our results, which motivated this work, is within biochemical reaction network theory and concerns the parameterization of the positive part of the algebraic variety of steady states.",
"The concentrations of the molecular species in the reactions evolve according to a non-linear ODE system of equations and the steady states are found by equating the equations of the ODE system to zero [4].",
"Our approach is tailored to obtain a full or partial parameterization of the set of positive steady states, possibly constrained to linear invariant varieties, in terms of the parameters of the system and some variables.",
"These parameterizations can be obtained for a large class of non-linear reaction networks.",
"The results given here generalise and complement earlier strategies in finding these parameterizations by means of linear elimination [10], [23], [5], [4].",
"We give an example towards the end of the paper.",
"The paper is organized as follows.",
"In Section we introduce notation, background material and present the solution to a linear system in terms of the spanning trees of an associated multidigraph.",
"In Section we give conditions in terms of the associated multidigraph to decide the nonnegativity of a solution.",
"In Section we develop theory for the second scenario and provide examples.",
"Finally, in Section , we prove the two main results of Section ."
],
[
"Preliminaries",
"Let ${\\mathbb {R}}_{\\ge 0}$ and ${\\mathbb {R}}_{>0}$ be the sets of nonnegative and positive real numbers, respectively.",
"For a ring $R$ , let $R^n$ be the $n$ -tuples of elements in $R$ and $R^{n\\times m}$ be the $n$ times $m$ matrices with entries in $R$ .",
"If $R$ is a partially ordered ring, then the notions of positive, negative, nonpositive and nonnegative elements are well defined [22].",
"These elements form the sets $R_{>0}, R_{<0}, R_{\\le 0}, R_{\\ge 0}$ , respectively.",
"Some examples of $R$ are $\\mathbb {Q}$ , ${\\mathbb {R}}$ and the real functions defined on a domain $\\Omega $ ordered by pointwise comparison.",
"The support of an element $x\\in R^n$ is defined as $\\operatorname{supp}(x)=\\lbrace i\\mid x_i\\ne 0\\rbrace .$ The cardinality of a finite set $F$ is denoted by $|F|$ , the power set by $\\mathcal {P}(F)$ and the disjoint union of two sets $E,F$ by $E\\sqcup F$ .",
"For finite pairwise disjoint sets $F_1,\\dots ,F_k$ , we define the following set of unordered $k$ -tuples of $F_1\\cup \\dots \\cup F_k$ : $F_1\\odot \\cdots \\odot F_k=\\underset{i\\in \\lbrace 1,\\dots ,k\\rbrace }{{\\odot }}\\!\\!F_i = \\Big \\lbrace \\lbrace w_1,\\dots ,w_k\\rbrace \\,\\mid \\,w_j\\in F_j\\text{ for }j=1,\\dots ,k\\Big \\rbrace .$ The $k$ -tuples of $F_1\\odot \\cdots \\odot F_k$ have $k$ elements."
],
[
"Multidigraphs and the Matrix-Tree Theorem",
"In this subsection we introduce notation and concepts related to algebraic graph theory.",
"A multidigraph $\\mathcal {G}$ is a pair of finite sets $(\\mathcal {N}, \\mathcal {E})$ , called the set of nodes and the set of edges, respectively, together with two functions, $s\\colon \\mathcal {E} \\rightarrow \\mathcal {N}$ and $t\\colon \\mathcal {E} \\rightarrow \\mathcal {N}$ , called the source and the target function, respectively.",
"The function $s$ (resp.",
"$t$ ) assigns to each edge the source (resp.",
"target) node of the edge.",
"An edge $e$ is a self-edge if $t(e)=s(e)$ , and two edges $e_1,e_2$ are parallel edges if $t(e_1)=t(e_2)$ and $s(e_1)=s(e_2)$ .",
"A cycle is a closed directed path with no repeated nodes.",
"A tree $\\tau $ is a directed subgraph of $\\mathcal {G}$ such that the underlying undirected graph is connected and acyclic.",
"A tree $\\tau $ is rooted at the node $N$ , if $N$ is the only node without outgoing edges.",
"In that case, there is a unique directed path from every node in $\\tau $ to $N$ .",
"A forest $\\zeta $ is a directed subgraph of $\\mathcal {G}$ whose connected components are trees.",
"A tree (resp.",
"forest) is called a spanning tree (resp.",
"spanning forest) if its node set is $\\mathcal {N}$ .",
"For a spanning tree $\\tau $ (resp.",
"a spanning forest $\\zeta $ ) we use $\\tau $ (resp.",
"$\\zeta $ ) to refer to the edge set of the graph and to the graph itself indistinctly, as the node set in this case is $\\mathcal {N}$ .",
"The number of edges of a spanning forest $\\zeta $ is $|\\mathcal {N}|$ minus the number of connected components of $\\zeta $ .",
"If $\\pi \\colon \\mathcal {E}\\rightarrow R$ is a labeling of $\\mathcal {G}$ with values in a ring $R$ , then any sub-multidigraph $\\mathcal {G}^{\\prime }$ of $\\mathcal {G}$ inherits a labeling from $\\mathcal {G}$ .",
"A labeling is extended to $\\mathcal {P}(\\mathcal {E})$ by $\\pi \\colon \\mathcal {P}(\\mathcal {E})\\rightarrow R,\\quad \\pi (\\mathcal {E}^{\\prime })=\\prod _{e\\in \\mathcal {E}^{\\prime }}\\pi (e) \\quad \\text{for}\\quad \\mathcal {E}^{\\prime }\\subseteq \\mathcal {E}.$ In the following we assume that $\\mathcal {G}=(\\mathcal {N}, \\mathcal {E})$ is a (labeled) multidigraph with no self-loops and node set $\\mathcal {N}=\\lbrace 1,\\dots ,m+1\\rbrace $ for some $m\\ge 0$ .",
"For two sets $F,B\\subseteq \\mathcal {N}$ with $|F|=|B|$ , let ${\\Theta _\\mathcal {G}(F,B)}$ be the set of spanning forests of $\\mathcal {G}$ such that: each forest has $|B|$ connected components (trees), each tree contains a node in $F$ and is rooted at a node in $B$ .",
"Each forest $\\zeta \\in \\Theta _\\mathcal {G}(F,B)$ induces a bijection $g_\\zeta \\colon F\\rightarrow B$ with $g_\\zeta (i)=j$ if $j$ is the root of the tree containing $i$ .",
"A pair $(i_1,i_2)\\in F\\times F$ is an inversion in $g_\\zeta $ if $i_1<i_2$ and $g_\\zeta (i_1)>g_\\zeta (i_2)$ .",
"We denote by $I(g_\\zeta )$ the number of inversions in $g_\\zeta $ and define $\\widetilde{\\Upsilon }_\\mathcal {G}(F,B)=\\sum _{\\zeta \\in \\Theta _\\mathcal {G}(F,B)}(-1)^{I(g_\\zeta )}\\pi (\\zeta )\\quad \\text{ and }\\quad \\Upsilon _\\mathcal {G}(F,B)=\\sum _{\\zeta \\in \\Theta _\\mathcal {G}(F,B)}\\pi (\\zeta ),$ where the empty sum is defined as zero.",
"Let $\\mathcal {E}_{ji}=\\lbrace e\\in \\mathcal {E}\\,\\mid \\,s(e)=j,\\ t(e)=i\\rbrace $ be the set of parallel edges with source $j$ and target $i$ .",
"The Laplacian of $\\mathcal {G}$ is the $(m+1)\\times (m+1)$ matrix $L=(L_{ij})$ with $L_{ij}= \\sum \\limits _{e\\in \\mathcal {E}_{ji}}\\pi (e) \\quad \\text{for }i\\ne j, \\quad \\text{ and }\\quad L_{ii}=-\\sum \\limits _{k\\ne i}L_{ki}.$ The column sums of the Laplacian of $\\mathcal {G}$ are zero by construction.",
"Any square matrix $L\\in R^{(m+1)\\times (m+1)}$ with zero column sums can be realized as the Laplacian of a labeled multidigraph.",
"If this is the case we say that $L$ is a Laplacian.",
"The canonical multidigraph with Laplacian $L$ is defined as the labeled multidigraph with node set $\\mathcal {N}=\\lbrace 1,\\dots ,m+1\\rbrace $ and one edge $j\\rightarrow i$ with label $L_{ij}$ for each nonzero entry $L_{ij}\\ne 0$ , for $i\\ne j$ .",
"This multidigraph has neither parallel edges nor self-loops, thus it is a digraph.",
"All other labeled multidigraphs with the same Laplacian can be obtained from the canonical multidigraph by adding self-edges and splitting edges into parallel edges while preserving the label sums.",
"Theorem 1 (All Minors Matrix-Tree Theorem [15]) Let $L$ be the Laplacian of a labeled multidigraph $\\mathcal {G}$ with $m+1$ nodes and let $B,F\\subseteq \\mathcal {N}$ be such that $|B|=|F|$ .",
"Let $L_{(F,B)}$ be the minor obtained from $L$ by removing the rows with index in $F$ and the columns with index in $B$ .",
"Then $L_{(F,B)}=(-1)^{\\epsilon }\\ \\widetilde{\\Upsilon }_\\mathcal {G}(F,B),\\quad \\text{where}\\quad \\epsilon = m+1-|F|+\\sum _{i\\in F}i+\\sum _{j\\in B}j.$ The All Minors Matrix-Tree Theorem is usually stated for digraphs.",
"Using Lemma 1 in [21] it holds also for multidigraphs.",
"When $|B|=|F|=1$ , Theorem REF is the usual Matrix-Tree Theorem extended to multidigraphs [24].",
"Define $\\Theta _\\mathcal {G}(B)=\\bigcup \\limits _{\\begin{subarray}{c}F\\subseteq \\lbrace 1,\\dots ,m+1\\rbrace \\\\ |B|=|F| \\end{subarray}} \\Theta _\\mathcal {G}(F,B) \\quad \\text{ and }\\quad \\Upsilon _\\mathcal {G}(B)=\\sum \\limits _{\\zeta \\in \\Theta _\\mathcal {G}(B)} \\pi (\\zeta ).$ The set $\\Theta _\\mathcal {G}(B)$ consists of spanning forests of $\\mathcal {G}$ with $|B|$ connected components and each component is a tree rooted at a node in $B$ .",
"If $i,j\\in \\mathcal {N}$ , then $\\Theta _\\mathcal {G}(\\lbrace i\\rbrace ,\\lbrace j\\rbrace )$ is the set of spanning trees rooted at $j$ , hence it is independent of $i$ .",
"We denote the set by $\\Theta _\\mathcal {G}(j)$ and let $\\Upsilon _\\mathcal {G}(j)=\\sum _{\\tau \\in \\Theta _\\mathcal {G}(j)}\\pi (\\tau ).$ For $\\tau \\in \\Theta _\\mathcal {G}(j)$ , we have $I(g_\\tau )=0$ .",
"If $\\mathcal {G}$ is not connected, then $\\Theta _\\mathcal {G}(j)=\\emptyset $ and $\\Upsilon _\\mathcal {G}(j)=0$ ."
],
[
"Linear systems",
"Let $R$ be a ring.",
"Consider a linear system $Ax+b=0$ , where $A=(a_{ij})\\in R^{m\\times m}$ is a nonsingular matrix, $b\\in R^m$ is a vector of independent terms, and $x$ is an $m$ -dimensional vector of unknowns.",
"Define the $(m+1)\\times (m+1)$ matrix $L$ by $L=\\left(\\begin{array}{c|c}A & b\\\\ \\hline \\cdots & \\cdot \\end{array} \\right), \\quad \\text{where}\\quad L_{ij}={\\left\\lbrace \\begin{array}{ll}\\quad a_{ij} & \\text{for }i,j\\le m\\\\\\quad b_i & \\text{for }i\\le m \\text{ and }j=m+1\\\\-\\sum \\limits _{k=1}^m a_{kj} & \\text{for } i=m+1\\text{ and }j\\le m \\\\-\\sum \\limits _{k=1}^m b_{k} & \\text{for } i=j=m+1,\\end{array}\\right.",
"}$ such that $L$ has zero column sums.",
"Therefore, $L$ is the Laplacian of a labeled multidigraph $\\mathcal {G}$ with $m+1$ nodes.",
"The solution to the linear system $Ax+b=0$ can be expressed in terms of the labels of the spanning trees of any labeled multidigraph with Laplacian $L$ as follows.",
"Let $A_{i\\rightarrow b}$ be the matrix obtained by replacing the $i$ -th column of $A$ with the vector $b$ , and let $A_{i}| b$ be the matrix obtained by removing the $i$ -th column of $A$ and taking $b$ as the $m$ -th column.",
"Proposition 1 Let $A\\in R^{m\\times m}$ , $b\\in R^m$ and $x=(x_1,\\dots ,x_m)$ .",
"Let $L$ be as in (REF ) and $\\mathcal {G}$ a labeled multidigraph with Laplacian $L$ .",
"Then, $\\det (A)=(-1)^{m}\\Upsilon _\\mathcal {G}(m+1)$ .",
"Further, if $\\det (A)\\ne 0$ , then the solution to the linear system $Ax+b=0$ is $x_i=\\dfrac{\\Upsilon _\\mathcal {G}(i)}{\\Upsilon _\\mathcal {G}(m+1)}\\,\\,\\in \\,\\,\\widehat{R},\\,\\qquad i=1,\\dots ,m,$ where $\\widehat{R}\\supseteq R$ is an extension ring of $R$ for which the above quotient is defined.",
"By Theorem REF we have (A)=L({m+1},{m+1})=(-1)m+1-1+m+1+m+1G(m+1)=(-1)mG(m+1), (Ai| b)=L({m+1},{i})=(-1)m+1-1+m+1+iG(i)=(-1)i+1G(i), as the number of inversions is 0 in these two cases.",
"By Cramer's rule we have $x_i=\\dfrac{\\det (A_{i\\rightarrow -b})}{\\det (A)}=\\dfrac{-\\det (A_{i\\rightarrow b})}{\\det (A)}=\\dfrac{(-1)^{m-i+1}\\det (A_{i}| b)}{\\det (A)}=\\dfrac{(-1)^m\\Upsilon _\\mathcal {G}(i)}{(-1)^m\\Upsilon _\\mathcal {G}(m+1)}.$ (part A)Example ref:split2example (part A) Let $z_1,\\dots ,z_5\\in R\\setminus \\lbrace 0\\rbrace $ .",
"Consider the linear system with $m=3$ , $\\left(\\begin{array}{ccc}-z_2 & \\phantom{-}0 & \\phantom{-}z_4 \\\\-z_1 & -z_3 & \\phantom{-}0 \\\\-z_2 & \\phantom{-}z_3 & -z_4 \\\\\\end{array} \\right)\\hspace{-2.84544pt}\\left(\\begin{matrix}x_1\\\\ x_2\\\\ x_3 \\end{matrix} \\right) +\\left(\\begin{matrix}0\\\\ z_5\\\\ 0 \\end{matrix} \\right)=\\left(\\begin{matrix}0\\\\ 0\\\\ 0 \\end{matrix} \\right).$ The matrix $L$ and the corresponding canonical multidigraph are $L=\\left(\\begin{array}{ccc|c}-z_2 & 0 & \\phantom{-}z_4 & 0\\\\-z_1 & -z_3 & 0 &z_{5} \\\\-z_2 & \\phantom{-}z_3 & -z_4 &0\\\\\\hline z_1+2z_2 & 0 & 0 & -z_{5}\\end{array} \\right)$ [inner sep=1.2pt] v31) [shape=circle] at (-2,0) 3; v11) [shape=circle] at (0,0) 1; v21) [shape=circle] at (2,0) 2.; *1) [shape=circle] at (0,-1.5) 4; [->] (v11) to[out=0,in=180] node[above,sloped]$-z_1$(v21); [->] (v11) to[out=170,in=10] node[above,sloped]$-z_2$(v31); [->] (v11) to[out=270,in=90] node[left]$z_1+2z_2$(*1); [->] (v21) to[out=155,in=25] node[above,sloped]$z_3$(v31); [->] (*1) to[out=20,in=240] node[right]$\\ z_{5}$(v21); [->] (v31) to[out=-10,in=190] node[below,sloped]$z_4$(v11); Therefore, $\\Upsilon _\\mathcal {G}(2)=\\left(z_1+ 2\\,z_{{2}}\\right) z_4 z_5-z_1z_4z_5=2z_2z_4z_5$ and $\\Upsilon _\\mathcal {G}(4)=-\\det (A)=\\left( z_{{1}}+2\\,z_{{2}} \\right)z_3z_4$ .",
"The terms $\\Upsilon _\\mathcal {G}(1)$ and $\\Upsilon _\\mathcal {G}(3)$ are similarly found to obtain the solution $x_1={\\frac{z_{{5}}}{z_{{1}}+2\\,z_{{2}}}},\\qquad x_2={\\frac{ 2\\,z_{{2}} z_5}{\\left( z_{{1}}+2\\,z_{{2}} \\right)z_3 }},\\qquad x_3={\\frac{ z_{{2}}z_5 }{ \\left( z_{{1}}+2\\,z_{{2}}\\right) z_{{4}}}}.$"
],
[
"Positive solution to a linear system",
"Let $R$ be a partially ordered ring.",
"We are interested in conditions that ensure the solution to the linear system $Ax+b=0$ in $\\widehat{R}^m$ is nonnegative (cf.",
"Proposition REF ).",
"If the off-diagonal entries of $L$ are in $R_{\\ge 0}$ , then this is always the case.",
"Indeed, the edge labels of the canonical multidigraph $\\mathcal {G}$ with Laplacian $L$ are in $R_{\\ge 0}$ .",
"Hence by Proposition REF and the definition of $\\Upsilon _\\mathcal {G}(j)$ , the solution is in $\\widehat{R}^m_{\\ge 0}$ .",
"This condition is however not necessary.",
"Consider Example REF with $R={\\mathbb {R}}$ and $z_i\\ge 0$ .",
"Not all off-diagonal entries of $L$ are in $\\mathbb {R}_{\\ge 0}$ , but the solution is nonetheless in $\\mathbb {R}_{\\ge 0}^3$ if $\\det (A)\\ne 0$ .",
"In the following we consider labeled multidigraphs with no zero labels ($\\pi (e)\\ne 0$ for all $e\\in \\mathcal {E}$ ) and such that the label of each edge is either positive or negative.",
"In this case, the labels of the spanning trees are also either nonnegative or nonpositive.",
"Assuming $\\mathcal {G}$ is a P-graph (Definition REF below), we will show that any nonpositive term in $\\Upsilon _\\mathcal {G}(i)$ corresponding to a spanning tree with nonpositive label cancels with a sum of labels of spanning trees with nonnegative labels.",
"Definition REF below guarantees that any nonpositive label of a spanning tree rooted at $i$ cancels out in $\\Upsilon _\\mathcal {G}(i)$ with a sum of labels of spanning trees with nonnegative labels.",
"Further, $\\Upsilon _\\mathcal {G}(i)\\in R_{\\ge 0}$ and the solution to the linear system given in Proposition REF is in $\\widehat{R}^m_{\\ge 0}$ .",
"This is what happens in Example REF .",
"For a multidigraph $\\mathcal {G}=(\\mathcal {N},\\mathcal {E})$ with labeling $\\pi \\colon \\mathcal {E}\\rightarrow R$ , we let $\\mathcal {E}^-=\\big \\lbrace e\\in \\mathcal {E}\\,\\mid \\,\\pi (e)\\in R_{>0}\\big \\rbrace \\quad \\text{and}\\quad \\mathcal {E}^+=\\big \\lbrace e\\in \\mathcal {E}\\,\\mid \\,\\pi (e)\\in R_{<0}\\big \\rbrace $ denote the set of edges with positive and negative labels, respectively.",
"Definition 1 Let $\\mathcal {G}=(\\mathcal {N},\\mathcal {E})$ be a multidigraph with labeling $\\pi \\colon \\mathcal {E}\\rightarrow R$ .",
"Let $\\mu \\colon \\mathcal {E}^-\\rightarrow \\mathcal {P}(\\mathcal {E}^+)$ be a map.",
"The pair $(\\mathcal {G},\\mu )$ is an edge partition if $\\mathcal {E}=\\mathcal {E}^+\\sqcup \\mathcal {E}^-$ .",
"All cycles in $\\mathcal {G}$ contain at most one edge in $\\mathcal {E}^-$ .",
"The map $\\mu $ is such that for every $e\\in \\mathcal {E}^-$ if $e^{\\prime }\\in \\mu (e)$ , then $s(e)=s(e^{\\prime })$ , if $e^{\\prime }\\in \\mu (e)$ , then every cycle containing $e^{\\prime }$ contains $t(e)$ , if $e\\ne e^{\\prime }$ , then $\\mu (e)\\cap \\mu (e^{\\prime })=\\emptyset $ .",
"We say $\\mathcal {G}$ a P-graph if there is an edge partition $(\\mathcal {G},\\mu )$ such that for every $e\\in \\mathcal {E}^-$ , $\\pi (e)+\\sum \\limits _{e^{\\prime }\\in \\mu (e)}\\pi (e^{\\prime })\\in R_{\\ge 0}$ .",
"In this case we say that the map $\\mu $ is associated with the P-graph $\\mathcal {G}$ .",
"Defintion REF (REF ) is equivalent to the condition that any path from $t(e^{\\prime })$ to $s(e^{\\prime })=s(e)$ contains $t(e)$ .",
"If the labels of a labeled multidigraph $\\mathcal {G}$ are all positive, then $\\mathcal {G}$ is trivially a P-graph.",
"Note that there does not necessarily exist a P-graph with a given Laplacian.",
"(part B)Example ref:split2example (part B) Consider the multidigraph $\\mathcal {G}$ with Laplacian $L$ in Example REF .",
"Since there is only one edge in $\\mathcal {E}^+$ with source 1 but two edges in $\\mathcal {E}^-$ with the same source, condition ([cond4Pgraph]iv) is not satisfied for any choice of $\\mu $ .",
"Hence $\\mathcal {G}$ is not a P-graph.",
"The following multidigraph is a P-graph with Laplacian $L$ .",
"An associated map $\\mu $ is given.",
"[inner sep=1.2pt] (-2.5,-2) rectangle (2.5,1); v31) [shape=circle] at (-2,0) 3; v11) [shape=circle] at (0,0) 1; v21) [shape=circle] at (2,0) 2; *1) [shape=circle] at (0,-1.7) 4; [->] (v11) to[out=0,in=180] node[above,sloped]$-z_1$(v21); [->] (v11) to[out=170,in=10] node[above,sloped]$-z_2$(v31); [->] (v11) to[out=260,in=100] node[left]$z_1$(*1); [->] (v11) to[out=280,in=80] node[right]$2z_2\\ $(*1); [->] (v21) to[out=150,in=30] node[above,sloped]$z_3$(v31); [->] (*1) to[out=30,in=240] node[right]$\\ z_{5}$(v21); [->] (v31) to[out=-10,in=190] node[below,sloped]$z_4$(v11); $\\mu (1{->[-z_1]}2)=\\lbrace 1{->[z_1]} 4 \\rbrace ,$ $\\mu (1{->[-z_2]}3)=\\lbrace 1{->[2z_2]} 4 \\rbrace .$ The entry $z_1+2z_2$ in $L$ is made into two edges in the multidigraph.",
"This is necessary in order to satisfy Definition REF (REF ) and ([cond4Pgraph]iv) simultaneously.",
"For some rings $R$ , such as the ring of real functions over a real domain, it is possible to write any element as the sum of a positive and a negative element.",
"In this case, given a labeled multidigraph with Laplacian $L$ , a new labeled multidigraph with the same Laplacian can be made by splitting each edge (if necessary) into two parallel edges, one with positive label and one with negative label.",
"Hence Definition REF (REF ) can always be fulfilled.",
"The key part of Definition REF is the simultaneous fulfilment of condition (REF ) and ([cond4Pgraph]iv).",
"Remark 1 It follows from Definition REF ([cond4Pgraph]iv), that a necessary condition for a multidigraph to be a P-graph is that the label sums over the edges with the same source are nonnegative, that is, the diagonal entries of the Laplacian are nonpositive.",
"Let $(\\mathcal {G},\\mu )$ be an edge partition and define the set $\\operatorname{im}\\mu $ by $\\operatorname{im}\\mu =\\bigcup _{e\\in \\mathcal {E}^-}\\mu (e)\\subseteq \\mathcal {E}^+.$ By Definition REF (REF ), each edge $e\\in \\operatorname{im}\\mu $ belongs to exactly one set $\\mu (e^{\\prime })$ .",
"We might thus define the “inverse” of $\\mu $ by $\\mu ^{*}\\colon \\operatorname{im}\\mu \\rightarrow \\mathcal {E}^-, \\quad \\mu ^{*}(e^{\\prime })=e\\ \\text{ if }\\ e^{\\prime }\\in \\mu (e).$ We will show that the spanning forests in $\\Theta _\\mathcal {G}(B)$ , $B\\subseteq \\mathcal {N}$ , can be obtained from a smaller set of spanning forests $\\Lambda _\\mathcal {G}(B)\\subseteq \\Theta _\\mathcal {G}(B)$ by replacing edges in $\\mathcal {E}^-$ with edges in $\\operatorname{im}\\mu \\subseteq \\mathcal {E}^+$ .",
"The spanning forests of $\\Lambda _\\mathcal {G}(B)$ are characterized by having as many edges as possible in $\\mathcal {E}^-$ , in a sense that will be made precise in Lemma REF .",
"This further allows us to characterize the labels of the spanning forests in $\\Theta _\\mathcal {G}(B)$ in terms of the associated map $\\mu $ , see Lemma REF and Theorem REF below.",
"In the next two lemmas we will make use of the following fact.",
"Given a spanning forest $\\zeta \\in \\Theta _\\mathcal {G}(B)$ , let $\\zeta ^{\\prime }$ be a submultidigraph obtained by replacing some edges of $\\zeta $ by other edges with the same source.",
"We claim that if $\\zeta ^{\\prime }$ does not contain any cycle, then it belongs to $\\Theta _\\mathcal {G}(B)$ .",
"Indeed, if this is so, then $\\zeta ^{\\prime }$ has $|B|$ connected components since it has $m-|B|$ edges and is a spanning forest.",
"Further, by construction, there is not an edge with source in $B$ .",
"Hence each connected component is a tree rooted at a node in $B$ .",
"Recall that we identify a spanning tree $\\zeta $ with its set of edges, and thus $\\zeta \\cap \\operatorname{im}\\mu $ denotes the set of edges of $\\zeta $ in $\\operatorname{im}\\mu $ .",
"Lemma 1 Let $(\\mathcal {G},\\mu )$ be an edge partition, $B\\subseteq \\mathcal {N}$ , $\\zeta \\in \\Theta _\\mathcal {G}(B)$ , and $\\mathcal {E}_\\zeta =\\Big \\lbrace E\\subseteq \\zeta \\cap \\operatorname{im}\\mu \\mid (\\zeta \\setminus E)\\cup \\mu ^{*}(E)\\in \\Theta _\\mathcal {G}(B)\\Big \\rbrace \\, \\subseteq \\mathcal {P}(\\mathcal {E}^+).$ The set $\\mathcal {E}_\\zeta $ is closed under union, that is, if $E_1,E_2\\in \\mathcal {E}_\\zeta $ , then $E_1\\cup E_2\\in \\mathcal {E}_\\zeta $ .",
"Consider distinct $E_1,E_2\\in \\mathcal {E}_\\zeta $ and let $\\zeta _i=(\\zeta \\setminus E_i)\\cup \\mu ^{*}(E_i)\\in \\Theta _\\mathcal {G}(B)$ for $i=1,2$ .",
"We claim that $E_3=E_1\\cup E_2\\in \\mathcal {E}_\\zeta $ , that is, $\\zeta _3=(\\zeta \\setminus E_3)\\cup \\mu ^{*}(E_3)\\in \\Theta _\\mathcal {G}(B)$ .",
"Since the image of $\\mu ^{*}$ belongs to $\\mathcal {E}^-$ , edges with positive label in $\\zeta _3$ are also edges in $\\zeta _1$ and $\\zeta _2$ by construction.",
"Using that a cycle in $\\mathcal {G}$ contains at most one edge in $\\mathcal {E}^-$ by Definition REF (REF ), we conclude that any cycle in $\\zeta _3$ is also a cycle of $\\zeta _1$ or $\\zeta _2$ .",
"However, they do not contain cycles as they are forests.",
"By the argument above, this implies that $\\zeta _3$ is a spanning forest in $\\Theta _\\mathcal {G}(B)$ .",
"Since $\\zeta \\cap \\operatorname{im}\\mu $ is finite, it follows from the lemma that $\\mathcal {E}_\\zeta $ has a unique maximum with respect to inclusion.",
"That is, there is a set $E_\\zeta \\in \\mathcal {E}_\\zeta $ such that for all $E\\in \\mathcal {E}_\\zeta $ it holds $E\\subseteq E_\\zeta $ .",
"Let $(\\mathcal {G},\\mu )$ be an edge partition.",
"For $B\\subseteq \\mathcal {N}$ , define the set $\\Lambda _\\mathcal {G}(B)=\\big \\lbrace \\zeta \\in \\Theta _\\mathcal {G}(B)\\,\\mid \\,\\mathcal {E}_\\zeta =\\emptyset \\big \\rbrace ,$ which consists of the spanning forests that are maximal with respect to the edge replacement operation defined in Lemma REF and thus have the maximal number of negative labels.",
"Note that we suppress the dependence of $\\mu $ in $\\Lambda _\\mathcal {G}(B)$ .",
"We define a surjective map by $\\psi \\colon \\Theta _\\mathcal {G}(B)\\rightarrow \\Lambda _\\mathcal {G}(B),\\quad \\psi (\\zeta )=\\big (\\zeta \\setminus E_\\zeta \\big )\\cup \\mu ^{*}\\big (E_\\zeta \\big ).$ This gives a partition of $\\Theta _\\mathcal {G}(B)$ , $\\Theta _\\mathcal {G}(B)=\\bigsqcup \\limits _{\\zeta \\in \\Lambda _\\mathcal {G}(B)}\\psi ^{-1}(\\zeta ).$ Lemma 2 Let $(\\mathcal {G},\\mu )$ be an edge partition, $B\\subseteq \\mathcal {N}$ , $\\zeta \\in \\Theta _\\mathcal {G}(B)$ , $e\\in \\zeta \\cap \\mathcal {E}^-$ and $e^{\\prime }\\in \\mu (e)$ .",
"Then $\\zeta ^{\\prime }=(\\zeta \\setminus \\lbrace e\\rbrace )\\cup \\lbrace e^{\\prime }\\rbrace \\in \\Theta _\\mathcal {G}(B).$ By assumption $\\zeta $ does not contain cycles.",
"If $\\zeta ^{\\prime }$ contains a cycle, then it contains $e^{\\prime }$ and $t(e)$ by Definition REF (REF ).",
"It implies that there is a path from $t(e)$ to $s(e^{\\prime })=s(e)$ in $\\zeta $ as well.",
"Since $\\zeta $ contains $e$ , there is also a cycle in $\\zeta $ , and we have reached a contradiction.",
"Hence, $\\zeta ^{\\prime }\\in \\Theta _\\mathcal {G}(B)$ .",
"Lemma 3 With the notation introduced above, it holds that for $\\zeta \\in \\Lambda _\\mathcal {G}(B)$ , $\\psi ^{-1}(\\zeta )=\\left\\lbrace (\\zeta \\cap \\mathcal {E}^+) \\cup E \\,\\mid \\,E\\in \\underset{e\\in \\zeta \\cap \\mathcal {E}^-}{{\\odot }} \\big (\\lbrace e\\rbrace \\cup \\mu (e)\\big ) \\right\\rbrace .$ If $\\psi (\\zeta ^{\\prime })= \\zeta $ , then $\\zeta ^{\\prime } = (\\zeta \\setminus \\mu ^*(E_{\\zeta ^{\\prime }}) )\\cup E_{\\zeta ^{\\prime }}$ by construction.",
"Since $E_{\\zeta ^{\\prime }}\\subseteq \\mathcal {E}^+$ and $\\mu ^*(E_{\\zeta ^{\\prime }})\\subseteq \\mathcal {E}^-$ , we have $\\zeta ^{\\prime }= (\\zeta \\cap \\mathcal {E}^+) \\cup E,\\qquad \\textrm { with }\\quad E=(\\zeta ^{\\prime }\\cap \\mathcal {E}^-) \\cup E_{\\zeta ^{\\prime }}.$ This shows the inclusion $\\subseteq $ by noting that $E_{\\zeta ^{\\prime }}\\subseteq \\operatorname{im}\\mu $ and $\\zeta ^{\\prime }\\cap \\mathcal {E}^- \\subseteq \\zeta \\cap \\mathcal {E}^-$ .",
"To prove the inclusion $\\supseteq $ we proceed as follows.",
"By Lemma REF , the set on the right consists of elements in $\\Theta _\\mathcal {G}(B)$ .",
"For $\\zeta ^{\\prime }= (\\zeta \\cap \\mathcal {E}^+) \\cup E$ , we have $E_{\\zeta ^{\\prime }}=E \\cap \\mathcal {E}^+$ and by the computations above, $\\psi (\\zeta ^{\\prime })=\\zeta $ .",
"Let $(\\mathcal {G},\\mu )$ be an edge partition.",
"Using (REF ) and Lemma REF , we conclude that for $i\\in \\mathcal {N}$ it holds that G(i)=G(i) '-1()(')= G(i)(E+)eE-((e)+e'(e)(e')).",
"We are now in position to prove the main result of the section.",
"Theorem 3 Let $A\\in R^{m\\times m}$ with $\\det (A)\\ne 0$ , $b\\in R^m$ , $x=(x_1,\\dots ,x_m)$ , and $L$ be as in (REF ).",
"If there exists a P-graph with Laplacian $L$ , then each component of the solution to the linear system $Ax+b=0$ is a quotient of two terms in $R_{\\ge 0}$ .",
"Let $\\mathcal {G}$ be a P-graph with Laplacian $L$ and $\\mu $ an associated map.",
"Using () and Definition REF ([cond4Pgraph]iv), $\\Upsilon _\\mathcal {G}(i)\\in R_{\\ge 0}$ for all $i\\in \\mathcal {N}$ , since it is a sum of nonnegative terms.",
"The result follows from Proposition REF .",
"Using () and $\\pi (\\zeta \\cap \\mathcal {E}^+)>0$ for $\\zeta \\in \\Lambda _\\mathcal {G}(i)$ , we might further characterize when $\\Upsilon _\\mathcal {G}(i)$ is different from zero, and, in particular, when $\\det (A)=(-1)^m\\Upsilon _\\mathcal {G}(m+1)$ is different from zero.",
"Proposition 2 Let $A\\in R^{m\\times m}$ , $b\\in R^m$ , $x=(x_1,\\dots ,x_m)$ , and $L$ be as in (REF ).",
"Assume there exists a P-graph $\\mathcal {G}$ with Laplacian $L$ and an associated map $\\mu $ .",
"For $i\\in \\mathcal {N}$ , $\\Upsilon _\\mathcal {G}(i)\\ne 0$ if and only if there exists a spanning tree $\\tau \\in \\Lambda _\\mathcal {G}(i)$ such that $\\pi (e)+\\sum \\limits _{e^{\\prime }\\in \\mu (e)}\\pi (e^{\\prime })\\,\\,\\in \\,\\, R_{>0},\\qquad \\textrm {for all }\\quad e \\in \\tau \\cap \\mathcal {E}^-.$ In particular, $\\det (A)\\ne 0$ if and only if the statement holds for $i=m+1$ .",
"The proposition provides a graphical way to check for zero solutions as well, since $\\Upsilon _\\mathcal {G}(i)\\ne 0$ if and only if the solution to the system $Ax+b=0$ satisfies $x_i\\ne 0$ .",
"The proposition holds for any P-graph and any associated map, and thus holds for either all possible P-graphs and associated maps or none.",
"Further, for $i=m+1$ , the vector $b$ plays no role, and hence in order to check whether a square matrix $A$ has nonzero determinant, one can apply the proposition with arbitrary $b$ , for example $b=0$ .",
"Remark 2 A known criterion for nonnegativity of the solution is the following.",
"If $-A$ is an $M$ -matrix and the entries of $b$ are nonnegative, then the solution is nonnegative, because the inverse of $-A$ has nonnegative entries [11].",
"Example REF is not an $M$ -matrix (and cannot be made one by reordering of columns or rows, see also Remark REF ).",
"Oppositely, for $Ax+b=0$ with $A=\\begin{pmatrix} -4 & \\phantom{-}2 \\\\ \\phantom{-}1& -1\\end{pmatrix},\\quad b=\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix},$ $-A$ is an $M$ -matrix, but there is not a P-graph in this case as Definition REF (iv) fails.",
"Hence the two criteria are complementary.",
"We conclude this section with two observations on the existence of P-graphs.",
"Remark 3 In many applications the rows of the matrix $A$ (and the vector $b$ ) have a natural order that corresponds to the order of the variables $x_1,\\ldots ,x_m$ .",
"This is for example the case if $Ax+b=0$ is the equilibrium equations of a linear dynamical system $\\dot{x}=Ax+b$ .",
"Positivity of a solution to $Ax+b=0$ is independent of the order of the rows, however, the existence of a P-graph is not.",
"As pointed out in Remark REF , there cannot exist a P-graph with Laplacian $L$ unless the diagonal entries of $L$ are nonpositive.",
"This property will generally not be fulfilled if the rows are reordered (without reordering the variables accordingly), as the following example shows with $m=2$ for two orders of the equations: $L= \\left(\\begin{array}{cc|c}-2 & \\phantom{-}1 & \\phantom{-}2 \\\\\\phantom{-}1 & -2 & \\phantom{-}2 \\\\ \\hline \\phantom{-}1 & \\phantom{-}1 & -4\\end{array}\\right), \\qquad \\widehat{L}= \\left(\\begin{array}{cc|c}-1 & \\phantom{-}2 & -2 \\\\\\phantom{-}2 & -1 & -2 \\\\ \\hline -1 & -1 & \\phantom{-}4\\end{array}\\right).$ The canonical multidigraph of the first example is a P-graph as the nondiagonal elements are nonnegative.",
"The second example is obtained from the first by swapping the first two rows, followed by a multiplication with minus one to obtain nonpositive diagonal elements (Remark REF ).",
"However this implies that the third, or $(m+1)$ -th, row also changes sign, causing a positive element in the diagonal.",
"Hence there is not a P-graph with Laplacian $\\widehat{L}$ .",
"In general, a reordering of the equations and/or the variables might be convenient in order to find a P-graph corresponding to the system.",
"Remark 4 Consider a P-graph $\\mathcal {G}$ with associated map $\\mu $ and Laplacian $L$ .",
"By splitting edges with positive labels or merging edges with negative labels, the resulting multidigraph is still a P-graph with Laplacian $L$ .",
"Specifically, if $\\mathcal {G}$ has several parallel edges $e_1,\\dots ,e_\\ell \\in \\mathcal {E}_{ji}$ with negative labels, the multidigraph $\\widehat{\\mathcal {G}}$ with exactly one edge $e$ from $j$ to $i$ and label $\\pi (e_1)+\\dots +\\pi (e_\\ell )$ is also a P-graph.",
"An associated map $\\widehat{\\mu }$ agreeing with $\\mu $ for all edges different from $e$ can be defined as $\\widehat{\\mu }(e)= \\cup _{i=1}^\\ell \\mu (e_i)$ .",
"Here we use that $\\mu (e_i)$ is also a subset of edges in $\\widehat{\\mathcal {G}}$ .",
"The proof is straightforward.",
"Similarly, for any edge $e^{\\prime }\\in \\mathcal {E}_{ji}$ with positive label, the multidigraph $\\widehat{\\mathcal {G}}$ with $\\ell $ edges $e^{\\prime }_1,\\dots ,e^{\\prime }_\\ell $ from $j$ to $i$ fulfilling $\\pi (e^{\\prime }_1)+\\dots + \\pi (e^{\\prime }_\\ell )=\\pi (e^{\\prime })$ is also a P-graph with the same Laplacian.",
"An associated map $\\widehat{\\mu }$ can be defined as $\\widehat{\\mu }(e)=(\\mu (e)\\setminus \\lbrace e^{\\prime }\\rbrace ) \\cup \\lbrace e^{\\prime }_1,\\dots ,e^{\\prime }_\\ell \\rbrace $ if $e^{\\prime }\\in \\mu (e)$ and $\\widehat{\\mu }(e)=\\mu (e)$ otherwise, with the natural identification of edges in $\\mathcal {G}$ and $\\widehat{\\mathcal {G}}$ .",
"Lemma 4 Assume $R$ is totally ordered.",
"Let $\\mathcal {G}=(\\mathcal {N},\\mathcal {E})$ be a P-graph with Laplacian $L$ such that there are two nodes $i,j\\in \\mathcal {N}$ with $\\mathcal {E}_{ji}\\ne \\emptyset $ and $L_{ij}=0$ , that is, the labels of the parallel edges from $j$ to $i$ sum to zero.",
"Then there is a P-graph $\\mathcal {G}^{\\prime }=(\\mathcal {N},\\mathcal {E}^{\\prime })$ with Laplacian $L$ and $\\mathcal {E}^{\\prime }_{ji}=\\emptyset $ .",
"The edges of $\\mathcal {G}^{\\prime }$ and $\\mathcal {G}$ agree after potentially splitting some edges with source $j$ into several parallel edges.",
"Let $\\mu $ be a map associated with the P-graph $\\mathcal {G}$ .",
"Without loss of generality, we can assume that there is only one edge $e^-$ from $j$ to $i$ with negative label (cf.",
"Remark REF ).",
"Consider the multidigraph $\\widehat{\\mathcal {G}}=(\\mathcal {N}, \\widehat{\\mathcal {E}})$ obtained from $\\mathcal {G}$ by removing the edges in $\\mathcal {E}_{ji}$ (that is, $\\widehat{\\mathcal {E}}=\\mathcal {E}\\setminus \\mathcal {E}_{ji}$ ) and let $\\widehat{\\mu }$ be the restriction of $\\mu $ to $\\mathcal {E}$ : $\\widehat{\\mu }(e) = \\mu (e) \\cap \\widehat{\\mathcal {E}}^+= \\mu (e) \\cap (\\mathcal {E}\\setminus \\mathcal {E}_{ji})^+ ,\\qquad e\\in \\widehat{\\mathcal {E}}^-.$ The pair $(\\widehat{\\mathcal {G}}, \\widehat{\\mu })$ is an edge partition.",
"Let E ={eE- (e)+e'(e)(e')R0} {eE- s(e)=j and (e)Eji}, where the inclusion is a consequence of the fact that $\\mathcal {G}$ and $\\widehat{\\mathcal {G}}$ only differ in edges with source $j$ and that $\\mathcal {G}$ is a P-graph.",
"We have $\\sum _{e\\in E\\cup \\mathcal {E}_{ji}^-}\\left(\\pi (e)+\\sum _{e^{\\prime }\\in \\mu (e)}\\pi (e^{\\prime }) \\right) =\\sum _{e\\in E}\\left(\\pi (e)+\\sum _{e^{\\prime }\\in \\widehat{\\mu }(e)}\\pi (e^{\\prime }) \\right) + \\sum _{{e^{\\prime }\\in \\mu (e^-)\\\\ e^{\\prime }\\notin \\mathcal {E}_{ji} }}\\pi (e^{\\prime }) \\\\+\\sum _{e\\in E} \\sum _{e^{\\prime }\\in \\mu (e)\\cap \\mathcal {E}_{ji} }\\pi (e^{\\prime })+ \\left(\\pi (e^-)+\\sum _{e^{\\prime }\\in \\mu (e^-)\\cap \\mathcal {E}_{ji} }\\pi (e^{\\prime }) \\right).$ This whole sum is in $R_{\\ge 0}$ since $\\mathcal {G}$ is a P-graph with associated map $\\mu $ , while the summand of the second row is in $R_{\\le 0}$ because the sum is over labels of edges in $\\mathcal {E}_{ji}$ and $L_{ij}=\\sum _{e\\in \\mathcal {E}_{ji}}\\pi (e)=0$ .",
"We deduce that for $E^\\star =\\lbrace e^{\\prime }\\in \\mu (e^-)\\,\\mid \\,e^{\\prime }\\notin \\mathcal {E}_{ji}\\rbrace $ , $\\sum _{e\\in E}\\left(\\pi (e)+\\sum _{e^{\\prime }\\in \\widehat{\\mu }(e)}\\pi (e^{\\prime })\\right)+ \\sum _{e^{\\prime }\\in E^\\star }\\pi (e^{\\prime }) \\in R_{\\ge 0}.$ The edges in $E^\\star $ belong to $\\widehat{\\mathcal {G}}$ , but not to $\\operatorname{im}\\widehat{\\mu }$ .",
"Roughly speaking, in view of (REF ), we will add some of these edges to $\\operatorname{im}\\widehat{\\mu }$ and modify $\\widehat{\\mathcal {E}}$ by splitting some edges.",
"Let $e\\in E$ .",
"Fix an order of the edges in the set $E^\\star $ , such that $E^\\star =\\lbrace e^{\\prime }_1,\\dots ,e^{\\prime }_\\ell \\rbrace $ and let for $k\\in \\lbrace 1,\\dots ,\\ell \\rbrace $ , $\\alpha _k= \\left(\\pi (e)+\\sum _{e^{\\prime }\\in \\widehat{\\mu }(e)}\\pi (e^{\\prime })\\right)+ \\sum _{i=1}^k\\pi (e^{\\prime }_i).$ Choose the first index $k$ such that $\\alpha _{k}\\ge 0$ .",
"Let $\\beta _1> 0$ and $\\beta _2\\ge 0$ such that $\\alpha _{k-1} + \\beta _1=0$ and $\\beta _1+\\beta _2 = \\pi (e^{\\prime }_{k}).$ If $\\beta _2>0$ , then redefine the multidigraph $\\widehat{\\mathcal {G}}$ by splitting the edge $e^{\\prime }_k$ into two edges $\\bar{e}_1,\\bar{e}_2$ with labels $\\beta _1,\\beta _2$ respectively.",
"If $\\beta _2=0$ , let $\\bar{e}_1=e_k^{\\prime }$ .",
"Recall that $E^\\star \\cap \\operatorname{im}\\widehat{\\mu }=\\emptyset $ .",
"Consider the map $\\widehat{\\mu }^{\\prime }$ given by $\\widehat{\\mu }^{\\prime }(\\bar{e})= {\\left\\lbrace \\begin{array}{ll}\\widehat{\\mu }(\\bar{e}) & \\text{if }\\bar{e}\\ne e \\\\ \\widehat{\\mu }(\\bar{e})\\cup \\lbrace e_1^{\\prime },\\dots ,e_{k-1}^{\\prime },\\bar{e}_1\\rbrace & \\text{if }\\bar{e}= e.\\end{array}\\right.",
"}$ Then $\\widehat{\\mu }^{\\prime }$ fulfils $\\pi (\\bar{e})+\\sum _{e^{\\prime }\\in \\widehat{\\mu }^{\\prime }(\\bar{e})}\\pi (e^{\\prime }) \\quad {\\left\\lbrace \\begin{array}{ll} =\\pi (\\bar{e})+\\sum _{e^{\\prime }\\in \\widehat{\\mu }(\\bar{e})}\\pi (e^{\\prime }) \\in R_{\\ge 0} & \\text{if }\\bar{e}\\notin E \\\\= 0 & \\text{if }\\bar{e}= e \\\\\\notin R_{\\ge 0} & \\text{if }\\bar{e}\\in E\\setminus \\lbrace e\\rbrace .\\end{array}\\right.",
"}$ Further, (REF ) holds with $E$ , $\\widehat{\\mu }$ and $E^\\star $ replaced by $E\\setminus \\lbrace e\\rbrace $ , $\\widehat{\\mu }^{\\prime }$ and $E^\\star \\setminus \\lbrace e_1^{\\prime },\\dots ,e_{k-1}^{\\prime },\\bar{e}_1\\rbrace $ , respectively.",
"By iterating this construction for all edges in $E$ , we obtain a multidigraph $\\widehat{\\mathcal {G}}$ with Laplacian $L$ and a map $\\widehat{\\mu }^{\\prime }$ .",
"$\\widehat{\\mathcal {G}}$ differs from $\\mathcal {G}$ in that $\\mathcal {E}_{ji}=\\emptyset $ and the edges in $E^\\star $ might have been split.",
"In particular both multidigraphs agree on edges with sources different from $j$ .",
"The map $\\widehat{\\mu }^{\\prime }$ fulfils Definition REF ([cond4Pgraph]iv) by construction.",
"All that remains is to show that $(\\widehat{\\mathcal {G}},\\widehat{\\mu }^{\\prime })$ is an edge partition.",
"Conditions (REF ) and (REF ) are readily satisfied.",
"By Remark REF , the pair $(\\widehat{\\mathcal {G}},\\widehat{\\mu }^{\\prime })$ fulfils (REF ) for all edges other than the edges $e^{\\prime }\\in \\mu (e^-)\\cap \\widehat{\\mu }^{\\prime }(e)$ with $e\\in E$ .",
"We only need to prove (REF ) for these edges.",
"Consider a path from $t(e^{\\prime })$ to $j$ in $\\widehat{\\mathcal {G}}$ .",
"We need to show that it contains $t(e)$ .",
"Since there is no edge with source $j$ , the path is also in $\\mathcal {G}$ .",
"Since $\\mathcal {G}$ is a P-graph with associated map $\\mu $ and $t(e^-)=i$ , the path contains $i$ .",
"By considering any edge in $\\mu (e)\\cap \\mathcal {E}_{ji}\\ne \\emptyset $ (cf.",
"()), the subpath from $i$ to $j$ contains $t(e)$ .",
"So condition (REF ) is satisfied.",
"This concludes the proof.",
"The lemma has the consequence that to construct a P-graph corresponding to a Laplacian $L$ , we do not need to consider edges between nodes with zero entry in $L$ ."
],
[
"An extension of the previous statements",
"In this section we consider a generalization of the system studied in Section .",
"The system of interest is a linear square system $Ax+b=0$ in $x=(x_1,\\dots , x_m)$ such that the coefficient matrix $A$ and the vector of independent terms $b$ are of the form $A=\\left(\\begin{array}{ccccc}A_1 & 0 &\\cdots & 0 & 0 \\\\0 & A_2 & \\cdots & 0 & 0 \\\\\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0 &0& \\cdots & A_d & 0 \\\\ \\hline \\multicolumn{5}{c}{A_0} \\end{array}\\right)\\in R^{m\\times m}, \\qquad b= \\left(\\begin{array}{c}b^1 \\\\b^2\\\\\\vdots \\\\b^d \\\\ \\hline b^0 \\end{array}\\right)\\in R^m,$ with $A_0\\in R^{m_0\\times m}$ and $b^0\\in R^{m_0}$ arbitrary, and for $i=1,\\dots ,d$ , $A_i$ is a square matrix of size $m_i$ .",
"$b^i$ is a vector of size $m_i$ and nonzero in at most one entry.",
"We let $\\mathcal {N}=\\lbrace 1,\\ldots ,m+1\\rbrace $ as before and let $\\mathcal {N}_i$ denote the set of indices of the rows corresponding to $A_i$ .",
"Specifically, Ni = {1+ j=1i-1 mj , ... , j=1i mj}, i=1,...,d Let also N0 = Ni=1d Ni = {m-m0+1,...,m+1}.",
"Note that we have $m-m_0=m_1+\\dots +m_d$ .",
"By (ii), we can choose indices $j_1,\\dots ,j_d$ , with $j_i\\in \\mathcal {N}_i$ for all $i=1,\\dots ,d$ , such that $b_j=0$ if $j\\ne j_i$ and $j\\le m-m_0$ .",
"If $b^i$ is the zero vector, then the index $j_i$ is arbitrary (but fixed).",
"Otherwise it is uniquely determined.",
"If $d=0$ , then we are left with a system of the form studied in Section .",
"Hence (REF ) might be considered an extension of the previous case.",
"The linear system has $d+1$ “blocks” or subsystems.",
"For the variables with indices in $\\mathcal {N}_i$ , $i=1,\\ldots ,d$ , there is one subsystem with $|\\mathcal {N}_i|$ linear equations.",
"In addition, there is the subsystem $A_0x+b^0=0$ that might depend on all $m$ variables.",
"Remark 5 If $\\det (A)\\ne 0$ , then we might in principle apply the results of the previous section.",
"However, if the rows $j_1,\\dots ,j_d$ of $A$ are nonnegative and $b^1,\\dots ,b^{d}$ nonpositive, then the conditions of Definition REF will often fail for the examples we have in mind, see Example .",
"In reaction network theory, which is our main source of examples, this type of system arises naturally (after reordering of the equations) and is perhaps the rule rather than the exception.",
"The equations with index different from $j_i$ correspond to equilibrium equations, and those with index equal to one of $j_i$ correspond to conservation relations with $b_{j_i}<0$ .",
"(part A)Example ref:2type (part A) Let $z_1,\\dots , z_5$ be the coordinate functions in ${\\mathbb {R}}^5_{\\ge 0}$ and consider the linear system in $x_1,x_2,x_3$ , -z2x1+z3x2=0, x1+x2-z1=0, z3x2-z4x3+z5=0.",
"Note that $\\det (A)=(z_2+z_3)z_4\\ne 0$ .",
"To apply Theorem REF , we change the sign of the second equation, such that the Laplacian matrix $L$ associated with the system has nonpositive diagonal entries, which is a necessary condition for the existence of a P-graph (see Remark REF ).",
"The matrix $L$ and a multidigraph with Laplacian $L$ are $L=\\left(\\begin{array}{ccc|c}-z_2 & z_3 & 0 &0 \\\\-1 & -1 & 0 & z_1\\\\0 & \\phantom{-}z_3 & -z_4 &z_5\\\\\\hline 1+z_2 & 1-2z_3 & z_4 & -z_1-z_{5}\\end{array} \\right)$ [inner sep=1.2pt] v3) [shape=circle] at (4,0) 3; v1) [shape=circle] at (0,0) 1; v2) [shape=circle] at (2,0) 2; *) [shape=circle] at (2,-1.5) 4; [->] (v1) to[out=10,in=170] node[above,sloped]$-1$(v2); [->] (v2) to[out=190,in=-10] node[below,sloped]$z_3$(v1); [->] (v1) to[out=290,in=160] node[left]$1+z_2$(*); [->] (*) to[out=80,in=280] node[right]$z_1$(v2); [->] (v2) to[out=260,in=100] node[left]1(*); [->] (v2) to[out=230,in=130] node[left]$-2z_3$(*); [->] (v2) to[out=0,in=180] node[above]$z_3$(v3); [->] (v3) to[out=260,in=30] node[above]$z_4$(*); [->] (*) to[out=10,in=280] node[below]$z_5$(v3); Since $\\mu (2{->[-2z_3]}4)$ is necessarily a subset of $\\lbrace 2{->[z_3]}3, 2{->[1]}4\\rbrace $ , Defintion REF ([cond4Pgraph]iv) cannot be satisfied.",
"Hence this multidigraph is not a P-graph.",
"Any multidigraph with Laplacian $L$ will have the same problem, so Theorem REF cannot be applied.",
"If we substitute $z_3$ by a nonnegative real number $\\le 1$ , then the multidigraph is a P-graph.",
"The linear system falls in the setting of the present section with $ A_1= \\left(\\begin{array}{cc}-z_2 & z_3 \\\\1 & 1 \\end{array} \\right), \\quad A_0 = \\left(\\begin{array}{ccc}0 & \\phantom{-}z_3 & -z_4\\end{array} \\right), \\quad b^1 = \\left(\\begin{array}{c}0 \\\\-z_1\\end{array} \\right),\\quad b^0 = \\left(\\begin{array}{c}z_5\\end{array} \\right).$ Definition 2 Let $\\mathcal {G}$ be a labeled multidigraph with $m+1$ nodes and Laplacian $L$ .",
"Then $\\mathcal {G}$ is said to be ${A}$ -compatible if There is not an edge from a node in $\\mathcal {N}_i$ , $i\\ge 0$ , to a node in $\\mathcal {N}_j$ for $i\\ne j$ , $j\\ge 1$ .",
"The $\\ell $ -th row of $L$ agrees with the $\\ell $ -th row of $A|b$ for $\\ell \\notin \\lbrace j_1,\\ldots ,j_d,m+1\\rbrace $ .",
"Furthermore, the Laplacian $L$ is said to be $A$ -compatible, if $\\mathcal {G}$ is $A$ -compatible.",
"The graphical structure of an $A$ -compatible multidigraph is shown in Figure REF .",
"By Definition REF (i), any $A$ -compatible Laplacian has the same block form as $A|b$ in (REF ).",
"In particular, the support of the $j_i$ -th row of $L$ is included in $\\mathcal {N}_i\\cup \\lbrace m+1\\rbrace $ .",
"Figure: The structure of an AA-compatible multidigraph 𝒢\\mathcal {G} with the node m+1m+1 singled out, 𝒩 0 ' =𝒩 0 ∖{m+1}\\mathcal {N}_0^{\\prime }=\\mathcal {N}_0\\setminus \\lbrace m+1\\rbrace .Remark 6 If $d=0$ , then any multidigraph with Laplacian as in (REF ) is $A$ -compatible.",
"Such a multidigraph has only the component with node set $\\mathcal {N}_0$ .",
"Let $F= \\lbrace j_1,\\dots ,j_d,m+1\\rbrace $ and $\\mathcal {N}_{d+1}=\\lbrace m+1\\rbrace ,$ and define $\\mathcal {B}=\\underset{j=1}{\\overset{d+1}{{\\odot }}} \\mathcal {N}_j,\\qquad \\mathcal {B}^k=\\underset{j=1,j\\ne k}{\\overset{d+1}{{\\odot }}} \\mathcal {N}_j,\\quad k=1,\\ldots ,d+1.$ If $B$ is an element in any of the defined sets, then the set $B\\cap \\mathcal {N}_i$ ($i\\ne k$ in the second case) consists of a single element, which we for simplicity denote by $\\beta _i$ , that is, $B\\cap \\mathcal {N}_i=\\lbrace \\beta _i\\rbrace .$ It is easy to show the following lemma using Definition REF (i) (see Figure REF ).",
"Lemma 5 Let $\\mathcal {G}$ be an $A$ -compatible multidigraph.",
"Let $\\zeta $ a spanning forest of $\\mathcal {G}$ and $\\tau $ a connected component of $\\zeta $ .",
"If $\\tau $ contains a node in $\\mathcal {N}_i$ , $i\\ge 0$ , then the root of $\\tau $ is in $\\mathcal {N}_i\\cup \\mathcal {N}_0$ .",
"$\\Theta _{\\mathcal {G}}(F,B)=\\emptyset $ if $B$ contains two elements in $\\mathcal {N}_i$ for some $i>0$ .",
"In particular, if $\\tau $ contains $m+1$ , then the root of $\\tau $ is in $\\mathcal {N}_0$ .",
"We are now ready to state a parallel version of Proposition REF under the assumptions of the current setting.",
"The proof is given in Section REF .",
"Proposition 3 Consider a linear system $Ax+b=0$ as in (REF ) such that $\\det (A)\\ne 0$ , and assume there exists an $A$ -compatible multidigraph $\\mathcal {G}$ .",
"Then, the solution to the linear system is x=k=1d+1 (-bjk) BBk,B( i=1, ikd ajii) G(F,B{}) BB(i=1d ajii)G(F,B), where $b_{j_{d+1}}=-1$ for convenience and $\\ell =1,\\ldots ,m$ .",
"By assuming $d=0$ we retrieve Proposition REF .",
"Several terms in the numerator of (REF ) are readily seen to be zero.",
"Indeed, by Lemma REF (ii), $\\Theta _{\\mathcal {G}}(F,B\\cup \\lbrace \\ell \\rbrace )=\\emptyset $ if $\\ell \\in \\mathcal {N}_i$ for $i\\in \\lbrace 1,\\dots ,d\\rbrace $ and $B\\in \\mathcal {B}^k$ with $k\\in \\lbrace 1,\\dots ,d+1\\rbrace $ , $k\\ne i$ .",
"If the columns of $A_0$ with indices in $\\mathcal {N}_k$ , $k\\in \\lbrace 1,\\dots ,d\\rbrace $ , are zero, then the variables $x_i$ with $i\\in \\mathcal {N}_k$ only appear in the subsystem given by the rows of $A$ with indices in $\\mathcal {N}_k$ and thus this subsystem can be solved independently.",
"Building on the ideas of Section , Proposition REF allows us to study when the solution to the system $Ax+b=0$ is positive.",
"In particular, we give the following characterization.",
"The proof is given in Section REF .",
"Theorem 5 Consider a linear system as in (REF ) such that $\\det (A)\\ne 0$ , the rows $j_1,\\dots ,j_d$ of $A$ are nonnegative and $b_{j_1},\\dots ,b_{j_d}$ are nonpositive.",
"Further, assume there exists an $A$ -compatible P-graph $\\mathcal {G}$ such that (*) for $\\ell \\in \\lbrace 1,\\dots ,m\\rbrace $ and $i\\in \\lbrace 1,\\dots , d\\rbrace $ , any path from $j_i\\in \\mathcal {N}_i$ to $\\ell $ that contains an edge in $\\mathcal {E}^-$ goes through $m+1$ .",
"Then, each component of the solution in (REF ) is the quotient of two terms in $R_{\\ge 0}$ .",
"In particular, if the target of all edges with negative labels is $m+1$ , then condition ([condpossol2]*) is fulfilled, see Example REF for an illustration.",
"If $d=0$ , then any $A$ -compatible multidigraph $\\mathcal {G}$ satisfies ([condpossol2]*), so Theorem REF is a generalization of Theorem REF .",
"Corollary 1 With the hypotheses of Theorem REF , assume there is a path from a node $\\ell \\in \\mathcal {N}_i$ with $i>0$ , to a node $j\\in \\lbrace 1,\\dots ,m\\rbrace $ that contains an edge in $\\mathcal {E}^-$ and that does not go through $m+1$ .",
"Then, the solution to the linear system $Ax+b=0$ fulfils $x_\\ell =0$ .",
"Remark 7 Remark REF could essentially be restated here.",
"There is some flexibility to choose the precise order of the rows of $A$ within each block.",
"All the rows of $A_1,\\dots ,A_d$ can be reordered indiscriminately as well as those of $A_0$ (and $b^0$ ).",
"This would lead to different multidigraphs.",
"(part B)Example ref:2type (part B) Consider the linear system in Example .",
"We have $\\mathcal {N}_1=\\lbrace 1,2\\rbrace $ and $\\mathcal {N}_0=\\lbrace 3,4\\rbrace $ .",
"The following multidigraph is $A$ -compatible [inner sep=1.2pt] v3) [shape=circle] at (4,0) 3; v1) [shape=circle] at (0,0) 1; v2) [shape=circle] at (2,0) 2; *) [shape=circle] at (6,0) 4; [->] (v1) to[out=10,in=170] node[above,sloped]$z_2$(v2); [->] (v2) to[out=190,in=-10] node[below,sloped]$z_3$(v1); [->] (v2) to[out=0,in=180] node[above]$z_3$(v3); [->] (v3) to[out=10,in=170] node[above]$z_4$(*); [->] (*) to[out=190,in=-10] node[below]$z_5$(v3); Indeed, its Laplacian $L$ is $A$ -compatible: $L=\\left(\\begin{array}{rrrr} -z_2 & z_3 & 0&0\\\\z_2 & -2z_3 & 0 & 0\\\\ 0 & z_3 & -z_4&z_5\\\\ 0 & 0 & z_4 & -z_5 \\end{array} \\right).$ The multidigraph is further a P-graph that satisfies ([condpossol2]*) in Theorem REF , since there are no edges with negative label.",
"Thus, we conclude by Theorem REF that the solution to the linear system is nonnegative.",
"Example 1 Consider the following linear system in five variables $x_1,\\dots , x_5$ and assume $z_1,\\dots ,z_9\\in R_{\\ge 0}$ .",
"$\\left(\\begin{array}{ccccc}-z_1& z_2 & 0 & 0 & 0\\\\1 & 1 & 0 & 0 & 0\\\\0&z_2&-z_3-z_4& z_5 & 0\\\\0&0& z_3&-z_5&0\\\\0&0&z_3&z_5&-z_6\\end{array}\\right)\\left(\\begin{array}{c} x_1 \\\\ x_2\\\\ x_3\\\\ x_4\\\\ x_5 \\end{array}\\right) +\\left(\\begin{array}{c} 0 \\\\ -z_7\\\\ z_8\\\\ 0\\\\ z_9 \\end{array}\\right)=\\left(\\begin{array}{c} 0 \\\\ 0\\\\ 0\\\\ 0\\\\ 0\\end{array}\\right).$ This system has the form of (REF ) with $d=1$ , $j_1=2$ , $m_1=2$ and $m_0=3$ .",
"The following is an $A$ -compatible Laplacian $\\left(\\begin{array}{cccccc}-z_1& z_2 & 0 & 0 & 0 & 0\\\\{z_1} & {-2z_2} & {0} & {0} & {0} & {0}\\\\0&z_2&-z_3-z_4& z_5 & 0 & z_8\\\\0&0& z_3&-z_5&0 & 0\\\\0&0&z_3&z_5&-z_6 & z_9\\\\{0}& {0} &{z_4-z_3}& {-z_5} &{z_6}& {-z_8-z_9}\\end{array}\\right).$ The 2nd row of $A|b$ is replaced by another vector with the same support and such that the number of negative entries outside the diagonal are kept as small as possible.",
"An example of a P-graph with Laplacian $L$ and associated map $\\mu $ is: [inner sep=1.2pt] v1) [shape=circle] at (0,0) 1; v2) [shape=circle] at (2,0) 2; v3) [shape=circle] at (4,0) 3; v4) [shape=circle] at (6,0) 4; v5) [shape=circle] at (8,0) 5; *) [shape=circle] at (6,-2) 6; [->] (v1) to[out=10,in=170] node[above,sloped]$z_1$(v2); [->] (v2) to[out=190,in=-10] node[below,sloped]$z_2$(v1); [->] (v2) to[out=0,in=180] node[above]$z_2$(v3); [->] (v3) to[out=-70,in=170] node[left]$-z_3$(*); [->] (v3) to[out=-50,in=150] node[left]$z_4$(*); [->] (*) to[out=130,in=-30] node[right]$z_8$(v3); [->] (v3) to[out=10,in=170] node[above]$z_3$(v4); [->] (v4) to[out=190,in=-10] node[below]$z_5$(v3); [->] (v3) to[out=30,in=150] node[above]$z_3$(v5); [->] (v4) to[out=0,in=180] node[above]$z_5$(v5); [->] (v4) to[out=270,in=90] node[right]$-z_5$(*); [->] (v5) to[out=230,in=40] node[left]$z_6$(*); [->] (*) to[out=20,in=250] node[right]$z_9$(v5); (3->[-z3] 6)={3->[z3] 5} (4->[-z5]6)={4->[z3] 5}.",
"All edges in $\\mathcal {E}^-$ have target node $6=m+1$ .",
"Therefore, by Theorem REF the solution to the linear system is nonnegative.",
"Indeed, the solution is x1= z7z2z2+z1, x2= z1z7z2+z1, x3= z1z2z7+(z1+z2)z8z4 ( z2+z1 ), x4= z3 ( z1z2z7+(z1+z2)z8 ) z4 ( z2+z1 ) z5, x5= 2 z1z2z3z7+(z1+z2)(2 z3z8+z4z9)z6z4 ( z2+z1 ) .",
"Example 2 We consider the following reaction network [6].",
"For convenience we denote the chemical species by $X_1,\\ldots ,X_6$ , and $\\kappa _1,\\dots ,\\kappa _{11}$ denote (unknown) reaction rate constants, one for each of the 11 reactions: X1 + X5 <=>[1][2] X3 ->[3] X1+ X6 X2 + X5 <=>[4][5] X4 ->[6] X2+ X6 X6 ->[7] X5 X1 <=>[8][9] X2 X3 <=>[10][11] X4.",
"Further, let $x_1,\\dots ,x_6$ denote the concentrations (in some unit) of the corresponding species.",
"Assuming mass-action kinetics, the evolution of the concentrations are described by an ODE system of the form, x1 = -1x1x5+(2+3)x3-8x1+9x2 x2 = -4x2x5+(5+6)x4+8x1-9x2 x3 = -1x1x5-(2+3)x3-10x3+11x4 x4 = -4x2x5-(5+6)x4+10x3-11x4 x5 = -1x1x5-4x2x5+2x3+5x4+7x6 x6 = -3x3+6x4-7x6.",
"As is evident from the equations, there are two conserved quantities, ${x}_{1} + {x}_{2} + {x}_{3} + {x}_{4}=T_1$ and $x_3+x_4+ x_5 + x_6 = T_2$ with $T_1,T_2\\in {\\mathbb {R}}_{\\ge 0}$ .",
"Using the theory developed in this section we give a one-dimensional parameterization (in $x_5$ ) of the positive steady state variety constrained to the conservation equation given by $T_1$ , $V_{T_1}=\\Big \\lbrace x\\in {\\mathbb {R}}^6_{>0}\\,|\\, \\dot{x}_i=0, i=1,\\ldots ,6\\Big \\rbrace \\cap \\Big \\lbrace x\\in {\\mathbb {R}}^6_{>0}\\,|\\, {x}_{1} + {x}_{2} + {x}_{3} + {x}_{4}=T_1\\Big \\rbrace .$ Consider the variable $x_5$ as an extra constant of the system.",
"Two equations, for example $\\dot{x}_4=0$ and $\\dot{x}_5=0$ , are redundant at steady state because of the conservation equations, and might be removed.",
"Thus the elements of $V_{T_1}$ fulfil: $\\left(\\begin{array}{ccccc}-(\\kappa _1x_5+\\kappa _8) & \\kappa _9 & \\kappa _2 & 0 & 0\\\\\\kappa _{8}&-(\\kappa _{4}x_{5}+\\kappa _9)&0&\\kappa _{5}+\\kappa _{6}& 0\\\\\\kappa _{1}x_{5}&0& -(\\kappa _{2}+\\kappa _{3}+\\kappa _{10})&\\kappa _{11}&0\\\\1 & 1 & 1 & 1 & 0\\\\0&0&\\kappa _{3}&\\kappa _{6}&-\\kappa _{7}\\end{array}\\right)\\left(\\begin{array}{c} x_1 \\\\ x_2\\\\ x_3\\\\ x_4\\\\ x_6 \\end{array}\\right) =\\left(\\begin{array}{c} 0 \\\\ 0\\\\ 0\\\\ T_1\\\\ 0 \\end{array}\\right).$ This system has the form of (REF ) with $d=1$ and $j_1=4$ .",
"Consider the following $A$ -compatible Laplacian $\\left(\\begin{array}{cccccc}-(\\kappa _1x_5+\\kappa _8) & \\kappa _9 & \\kappa _2 & 0 & 0& 0\\\\\\kappa _{8}&-(\\kappa _{4}x_{5}+\\kappa _9)&0&\\kappa _{5}+\\kappa _{6}& 0 & 0\\\\\\kappa _{1}x_{5}&0& -(\\kappa _{2}+\\kappa _{3}+\\kappa _{10})&\\kappa _{11}&0 & 0\\\\{0} & {\\kappa _{4}x_{5}} & {\\kappa _{10}} & {-(\\kappa _5+2\\kappa _6+\\kappa _{11})} & {0} & {0}\\\\0&0&\\kappa _{3}&\\kappa _{6}&-\\kappa _{7} & 0\\\\{0} &{0} & {0}& {0} &{\\kappa _7} & {0}\\end{array}\\right),$ where the 4-th row (in bold) differs from the one in the matrix of the linear system, and the bottom bold row is the $(m+1)$ -th row.",
"The canonical multidigraph with this Laplacian does not have edges with negative label, hence it is a P-graph and condition ([condpossol2]*) is fulfilled.",
"Therefore, by Theorem REF , the solution to the system is nonnegative: x1 =T1q(x) ( (2+3)411x5+9((2+3)(5+6) +(2+3) 11+(5+6)10)), x2 =T1q(x) ( (5+6)110x5+8((2+3)(5+6) +(2+3) 11+(5+6)10)), x3 =T1 x5q(x) ( 1411x5+19(5+6+11)+4811 ), x4 =T1 x5q(x) ( 1410x5+48(2+3+10)+1910), x6 = T1x57q(x) ( 14(311+610 )x5+139 ( 5+11 ) +48( 26+311) + 6 ( 3+10 ) ( 19+48) ), where $q(x)$ is the following polynomial in $x_5$ , q(x) = 14 ( 10+11 ) x52 + ((2+3)4( 8+11 ) +(5+6)1 ( 9+10 ) + ( 10+11 ) ( 19+48 ) )x5 + ( 8+9)( (2+3) (5+6+ 11) +10(5+6) ).",
"This reaction network is one of many reaction networks that fulfill the hypotheses of Theorem REF .",
"In fact, structural conditions on the reaction network guarantee the hypotheses are fulfilled [20].",
"We conclude with some remarks about how to find an $A$ -compatible multidigraph $\\mathcal {G}$ fulfilling the requirements of Theorem REF .",
"By Remark REF , a necessary condition for an $A$ -compatible P-graph to exist is that the diagonal entries of $A$ with indices different from $j_i$ , $i=1,\\dots , d$ , are nonpositive.",
"Further, consider the subsystem $A^{\\prime }_0x^{\\prime }+b^0=0$ , where $A^{\\prime }_0$ is the square matrix consisting of the last $m_0$ columns of $A_0$ and $x^{\\prime }=(x_{m-m_0+1},\\dots , x_m)$ .",
"Then the submultidigraph of an $A$ -compatible P-graph induced by the set of nodes $\\mathcal {N}_0$ is a P-graph with Laplacian $L$ constructed as in (REF ) for $A_0^{\\prime }$ and $b^0$ , up to indexing of the nodes.",
"Thus, if a P-graph for such a subsystem does not exist (see Section ), then there is not an $A$ -compatible P-graph for the original system.",
"If these necessary conditions for the existence of an $A$ -compatible P-graph are fulfilled, we attempt to find an $A$ -compatible Laplacian $L$ by minimizing the number of negative entries outside the diagonal.",
"The focus is on the undetermined rows $j_1,\\dots ,j_d$ of $L$ .",
"Consider $i\\in \\mathcal {N}_k$ , $k>0$ , and the $i$ -th column sum of $A$ without $a_{j_ki}$ .",
"If this sum is nonpositive for $i\\ne j_k$ , or nonnegative for $i=j_k$ , then a good strategy is to define $L_{j_ki}$ as minus this sum.",
"This gives nonnegative entries in the $j_k$ -th row of $L$ outside the diagonal and a nonpositive entry in the diagonal, while having the $i$ -th entry of row $m+1$ equal to zero."
],
[
" Finding the solution to the linear system (Proposition ",
"We find expressions for $\\det (A)$ and $\\det (A_{\\ell \\rightarrow -b})$ in terms of the spanning forests of $\\mathcal {G}$ and the coefficients in the rows $j_1,\\dots ,j_d$ of $A$ and $b$ .",
"These expressions are found using Theorem REF .",
"The explicit solution to the linear system is subsequently found using Cramer's rule, as in the proof of Proposition REF .",
"This will give Proposition REF .",
"Recall the definition of the sets $F$ , $\\mathcal {B}$ and $\\mathcal {B}^k$ in (REF ) and of $\\beta _i$ in (REF ).",
"We start with an observation about the form of the spanning forests in $\\mathcal {G}$ .",
"By applying Lemma REF repeatedly to a spanning forest $\\zeta \\in \\Theta _{\\mathcal {G}}(F,B)$ with $B=\\widetilde{B}\\cup \\lbrace \\ell \\rbrace $ , $\\widetilde{B}\\in \\mathcal {B}^k$ , $\\ell \\notin \\widetilde{B}$ , for $k\\in \\lbrace 1,\\dots ,d+1\\rbrace $ , $\\ell \\in \\lbrace 1,\\dots ,m+1\\rbrace $ , we obtain g(ji) = {ll i if ki, if k= i, .",
"g(m+1) = {ll m+1 if kd+1, if k= d+1, .",
"for $i=1,\\dots ,d$ .",
"Note that if $k=d+1$ and $\\ell =m+1$ , we obtain the sets $B\\in \\mathcal {B}$ , so the previous display applies to the sets in $\\mathcal {B}$ as well.",
"In particular, the map $g_\\zeta $ is independent of $\\zeta $ and depends only on $F$ and $B$ .",
"Let $L$ be the Laplacian of an $A$ -compatible multidigraph $\\mathcal {G}$ as in Proposition REF .",
"Then $L$ agrees with $A$ on all rows but $j_1,\\dots ,j_d$ , that is, on all rows but the ones with indices in $F\\setminus \\lbrace m+1\\rbrace $ ($A$ is an $m\\times m$ matrix).",
"Therefore, for a set $B\\subseteq \\lbrace 1,\\dots ,m\\rbrace $ with $d$ elements, we have the following equality of minors $A_{(F\\setminus \\lbrace m+1\\rbrace ,B)}=L_{(F,B\\cup \\lbrace m+1\\rbrace )}.$ Lemma 6 With the notation introduced above, for $\\ell =1,\\ldots ,m$ , (A) =(-1)m-dBB(i=1d ajii)G(F,B), (A-b) = (-1)m-d (k=1d (-bjk) BBk,B( i=1, ikd ajii ) G(F,B{}).",
".+BBd+1,B(i=1d ajii ) G(F,B {}) ).",
"The nonzero entries of the $j_i$ -th row of $A$ are in columns with index in $\\mathcal {N}_i$ .",
"To prove (REF ) we consider the Laplace expansion of the determinant of $A$ along the rows $j_1, \\dots ,j_d$ [7], [19].",
"Using (REF ) and Theorem REF we have (A) =BBd+1(-1)i=1d (ji+i)(i=1d ajii )A(F{m+1},B) =(-1)m-dBB(i=1d ajii)G(F,B).",
"By (REF ) with $k=d+1$ and $\\ell =m+1$ , if $\\zeta \\in \\Theta _{\\mathcal {G}}(F,B)$ with $B\\in \\mathcal {B}$ , then $I(g_\\zeta )=0$ and so $\\widetilde{\\Upsilon }_{\\mathcal {G}}(F,B)=\\Upsilon _{\\mathcal {G}}(F,B)$ , see (REF ).",
"This concludes the proof of (REF ).",
"For $B\\subseteq \\lbrace 1,\\dots ,m+1\\rbrace $ , we let $\\varepsilon (\\ell ,B)=\\big |\\lbrace i\\in B\\,\\mid \\,\\ell <i<m+1\\rbrace \\big |,$ and for $B\\in \\mathcal {B}^{k}$ , $k\\in \\lbrace 1,\\ldots ,d+1\\rbrace $ , we define $\\alpha _k(B)=\\sum _{\\begin{subarray}{c}i=1,\\, i\\ne k\\end{subarray}}^d(j_i+\\beta _i)\\quad \\textrm { and }\\quad w_k(B)=\\prod _{\\begin{subarray}{c}i=1, \\, i\\ne k\\end{subarray}}^d a_{j_i\\beta _i}.$ To prove (REF ) we consider the Laplace expansion of the determinant of $A_{\\ell \\rightarrow -b}$ along column $\\ell \\in \\lbrace 1,\\ldots ,m\\rbrace $ .",
"For $i\\le m-m_0$ , we have $b_i=0$ if $i\\ne j_k$ for all $k=1,\\dots ,d$ .",
"It gives $\\det (A_{\\ell \\rightarrow -b})=\\sum _{k=1}^d (-b_{j_k})(-1)^{j_k+\\ell } A_{(\\lbrace j_k\\rbrace ,\\lbrace \\ell \\rbrace )}+\\!\\!\\!\\sum _{k=m-m_0+1}^m(-b_k) (-1)^{k+\\ell }A_{(\\lbrace k\\rbrace ,\\lbrace \\ell \\rbrace )}.$ Fix $k\\in \\lbrace 1,\\dots ,d\\rbrace $ .",
"To compute $A_{(\\lbrace j_k\\rbrace ,\\lbrace \\ell \\rbrace )}$ , we consider the Laplace expansion of the determinant of the submatrix $\\widehat{A}$ of $A$ , given by removing row $j_k$ and column $\\ell $ , along the rows $j_1,\\dots $ , $j_{k-1}$ , $j_{k+1}-1,\\dots ,j_{d}-1$ .",
"These rows correspond to the rows $j_1,\\dots $ , $j_{k-1}$ , $j_{k+1},\\dots ,j_{d}$ of $A$ .",
"Given $j\\in \\lbrace 1,\\dots ,m\\rbrace $ , the $j$ -th column of $\\widehat{A}$ is the $j$ -th column of $A$ if $j<\\ell $ and the $(j+1)$ -th column if $j\\ge \\ell $ .",
"By (REF ) and Theorem REF we have A({jk},{}) = BBk, B(-1)k(B)-(d-k)+(,B) wk(B)A(F{m+1},(B {m+1}){}) = BBk, B wk(B)(-1)-d+k+(,B)+m+1-d-1++jkG(F,B{}).",
"Let $B\\in \\mathcal {B}^{k}$ , $k\\in \\lbrace 1,\\ldots ,d\\rbrace $ and $\\ell \\notin B$ .",
"By Lemma REF (ii), $\\Theta _{\\mathcal {G}}(F,B\\cup \\lbrace \\ell \\rbrace )=\\emptyset $ if $\\ell \\in \\mathcal {N}_i$ , $i\\ne k$ and $i>0$ .",
"By (REF ), we further have - If $\\ell \\in \\mathcal {N}_k$ , then $I(g_\\zeta )=0$ and $\\varepsilon (\\ell , B)=d-k$ .",
"- If $\\ell \\in \\mathcal {N}_0$ , then $\\varepsilon (\\ell , B)=0$ and $I(g_\\zeta )=d-k$ since there are $d-k$ inversions in $g_\\zeta $ : we have $j_i>j_k$ for $i>k$ and $g_\\zeta (j_i)=\\beta _i<\\ell =g_\\zeta (j_k)$ .",
"Therefore, $A_{(\\lbrace j_k\\rbrace ,\\lbrace \\ell \\rbrace )}=(-1)^{\\ell +j_k+m-d} \\sum \\limits _{B\\in \\mathcal {B}^k,\\ell \\notin B}\\omega _k(B)\\Upsilon _{\\mathcal {G}}(F,B\\cup \\lbrace \\ell \\rbrace ).$ Secondly, we find $A_{(\\lbrace k\\rbrace ,\\lbrace \\ell \\rbrace )}$ for $m \\ge k>m-m_0$ similarly to above, by considering the Laplace expansion of the submatrix of $A$ obtained by removing row $k$ and column $\\ell $ , along the rows $j_1, \\dots ,j_d$ .",
"By (REF ) and Theorem REF we obtain A({k},{})=BBd+1,B(-1)d+1(B)+(,B)wd+1(B)A((F{m+1}){k},B{}) =BB,Bwd+1(B)(-1) (,B)+m+1-d-2+k+G(F{k},B{}).",
"By Lemma REF (ii), for $\\ell \\le m-m_0$ , we have $\\Theta _{\\mathcal {G}}(F\\cup \\lbrace k\\rbrace ,B\\cup \\lbrace \\ell \\rbrace )=\\emptyset $ if $B\\in \\mathcal {B}$ , and also $\\Theta _{\\mathcal {G}}(F,B\\cup \\lbrace \\ell \\rbrace )=\\emptyset $ for $B\\in \\mathcal {B}^{d+1}$ .",
"Thus, from above 0 = k=m-m0+1m(-bk) (-1)k+A({k},{}) = BBd+1,B(i=1d ajii ) G(F,B {}).",
"If $\\ell >m-m_0$ , then $\\varepsilon (\\ell , B)=0$ .",
"Using Lemma REF (i), we have for $\\zeta \\in \\Theta _{\\mathcal {G}}(F\\cup \\lbrace k\\rbrace ,B\\cup \\lbrace \\ell \\rbrace )$ that $g_\\zeta (j_i)=\\beta _i$ for all $i\\in \\lbrace 1,\\dots ,d\\rbrace $ , $g_\\zeta (m+1)=m+1$ and $g_\\zeta (k)=\\ell $ .",
"Thus $I(g_\\zeta )=0$ and $A_{(\\lbrace k\\rbrace ,\\lbrace \\ell \\rbrace )}=(-1)^{\\ell +k+m-d+1}\\sum _{B\\in \\mathcal {B},\\ell \\notin B}w_{d+1}(B)\\, \\Upsilon _{\\mathcal {G}}(F\\cup \\lbrace k\\rbrace ,B\\cup \\lbrace \\ell \\rbrace ).$ It only remains to prove that for $\\ell >m-m_0$ , we have $\\sum \\limits _{k=m-m_0+1}^m\\!\\!b_k\\!\\!\\!\\sum \\limits _{B\\in \\mathcal {B}, \\ell \\notin B}\\!\\!\\!w_{d+1}(B) \\Upsilon _{\\mathcal {G}}\\big (F\\cup \\lbrace k\\rbrace ,B \\cup \\lbrace \\ell \\rbrace \\big ) = \\!\\!\\!\\sum \\limits _{B\\in \\mathcal {B}^{d+1},\\ell \\notin B}\\!\\!\\!w_{d+1}(B) \\Upsilon _{\\mathcal {G}}\\big (F,B \\cup \\lbrace \\ell \\rbrace \\big ),$ where the left side is (REF ) summed over $k$ .",
"Note that for $B\\in \\mathcal {B}$ , we have $m+1\\in B$ and hence $B\\setminus \\lbrace m+1\\rbrace \\in \\mathcal {B}^{d+1}$ .",
"Therefore it is sufficient to prove that for $B\\in \\mathcal {B}$ and $\\ell \\notin B$ , we have $\\sum \\limits _{k=m-m_0+1}^mb_k \\Upsilon _{\\mathcal {G}}\\big (F\\cup \\lbrace k\\rbrace ,B \\cup \\lbrace \\ell \\rbrace \\big )= \\Upsilon _{\\mathcal {G}}\\big (F,B \\cup \\lbrace \\ell \\rbrace \\setminus \\lbrace m+1\\rbrace \\big ).$ Let $\\mathcal {E}_{m+1,k}$ be the set of edges in $\\mathcal {G}$ with source $m+1$ and target $k$ .",
"Note that $b_k=\\sum _{e\\in \\mathcal {E}_{m+1, k}}\\pi (e)$ , so it is sufficient to show that $\\bigcup _{k=m-m_0+1}^m \\Big \\lbrace e\\cup \\zeta \\,\\mid \\,e\\in \\mathcal {E}_{m+1,k},\\ \\zeta \\in \\Theta _{\\mathcal {G}}(F\\cup \\lbrace k\\rbrace ,B \\cup \\lbrace \\ell \\rbrace )\\Big \\rbrace =\\Theta _{\\mathcal {G}}\\big (F,B \\cup \\lbrace \\ell \\rbrace \\setminus \\lbrace m+1\\rbrace \\big ),$ as each element $e\\cup \\zeta $ on the left side has label $\\pi (e)$ times the label of the spanning tree $\\zeta \\in \\Theta _{\\mathcal {G}}\\big (F,B \\cup \\lbrace \\ell \\rbrace \\setminus \\lbrace m+1\\rbrace \\big )$ .",
"We will show the equality by proving that the left side is contained in the right side, and vice versa.",
"Consider a spanning forest $\\zeta \\in \\Theta _{\\mathcal {G}}(F\\cup \\lbrace k\\rbrace ,B \\cup \\lbrace \\ell \\rbrace )$ and $e\\in \\mathcal {E}_{m+1,k}$ .",
"Then one connected component of $\\zeta $ is a tree rooted at $m+1$ , and another a tree rooted at $\\ell $ that contains $k$ .",
"In $\\zeta \\cup e$ , these two connected components are merged into a tree rooted at $\\ell $ that contains $m+1$ .",
"Hence the inclusion $\\subseteq $ holds.",
"To prove the other inclusion, we note that a spanning forest $\\zeta \\in \\Theta _{\\mathcal {G}}(F,B \\cup \\lbrace \\ell \\rbrace \\setminus \\lbrace m+1\\rbrace )$ contains exactly one edge $e$ with source $m+1$ .",
"Since $\\mathcal {G}$ is $A$ -compatible, the target of this edge belongs to $\\mathcal {N}_0$ (see Figure REF ), that is, $e\\in \\mathcal {E}_{m+1,k}$ with $m\\ge k>m-m_0$ .",
"One connected component of the subgraph $\\zeta \\setminus \\lbrace e\\rbrace $ is a tree rooted at $m+1$ and another connected component is a tree rooted at $\\ell $ that contains $k$ .",
"So the desired inclusion holds; hence the equality holds.",
"The proof of equation (REF ) now follows by combining the equations (REF )–(REF ).",
"By Cramer's rule, the solution to the linear system is $x_\\ell =\\frac{\\det (A_{\\ell \\rightarrow -b})}{\\det (A)}.$ Now, using Lemma REF , we obtain the expression in the statement of Proposition REF , after combining the two sums of (REF ) into one using $b_{j_{d+1}}=-1$ ."
],
[
"Nonnegativity of the solution (Theorem ",
"The aim of this section is to prove Theorem REF , that is, to prove that the solution to the linear system (REF ) is nonnegative under certain conditions.",
"To do so, we prove that a decomposition, similar to the one in (REF ), holds for $\\Theta _{\\mathcal {G}}(F,B)$ for certain subsets $B\\subseteq \\lbrace 1,\\dots ,m+1\\rbrace $ .",
"Assume the multidigraph $\\mathcal {G}$ is an $A$ -compatible P-graph that satisfies condition ([condpossol2]*) of Theorem REF .",
"In the lemmas below we consider $B=\\widetilde{B}\\cup \\lbrace \\ell \\rbrace $ , where $\\widetilde{B}\\in \\mathcal {B}^{k}$ for $k\\in \\lbrace 1,\\dots , d+1\\rbrace $ and $\\ell \\in \\lbrace 1,\\dots ,m+1\\rbrace , \\ell \\notin \\widetilde{B}$ .",
"Lemma 7 Let $\\zeta \\in \\Theta _\\mathcal {G}(F,B)$ .",
"Any path from $j_k\\in F$ , $k\\in \\lbrace 1,\\dots ,d\\rbrace $ , to $i\\in B$ in $\\zeta $ , does not contain an edge in $\\mathcal {E}^-$ .",
"If $\\zeta ^{\\prime }\\in \\Theta _\\mathcal {G}(B)$ is such that the connected component of $\\zeta ^{\\prime }$ containing $j_k\\in F$ has root $g_\\zeta (j_k)$ for all $k=1,\\dots ,d$ , then also the connected component containing $m+1$ has root $g_\\zeta (m+1)$ .",
"In particular, $\\zeta ^{\\prime }\\in \\Theta _\\mathcal {G}(F,B)$ .",
"(i) By condition ([condpossol2]*), any such path goes through $m+1$ .",
"But this implies that $j_k$ and $m+1$ , which both are in $F$ , also are in the same connected component of $\\zeta $ , contradicting the definition of $\\Theta _{\\mathcal {G}}(F,B)$ .",
"(ii) Let $i\\in B$ be the root of the connected component of $\\zeta ^{\\prime }$ containing $m+1$ .",
"By Lemma REF (i), both $i$ and $g_\\zeta (m+1)$ belong to $\\mathcal {N}_0$ .",
"If $m+1\\in B$ , then necessarily $i=m+1=g_\\zeta (m+1)$ .",
"Otherwise, by our choice of sets $B$ , $g_\\zeta (m+1)$ is the only element both in $B$ and $\\mathcal {N}_0$ .",
"Thus it must hold that $i=g_\\zeta (m+1)$ .",
"Lemma 8 For $\\zeta \\in \\Theta _{\\mathcal {G}}(F,B)$ , we define $\\mathcal {E}^{F}_\\zeta =\\big \\lbrace E\\subseteq \\zeta \\cap \\operatorname{im}\\mu \\,\\mid \\,(\\zeta \\setminus E)\\cup \\mu ^{*}(E)\\in \\Theta _{\\mathcal {G}}(F,B)\\big \\rbrace \\subseteq \\mathcal {P}(\\mathcal {E}^+).$ The set $\\mathcal {E}_\\zeta ^F$ is closed under union, that is, if $E_1,E_2\\in \\mathcal {E}_\\zeta ^F$ , then $E_1\\cup E_2\\in \\mathcal {E}_\\zeta ^F$ .",
"Since $\\Theta _{\\mathcal {G}}(F,B)\\subseteq \\Theta _{\\mathcal {G}}(B)$ , we have $\\mathcal {E}^{F}_\\zeta \\subseteq \\mathcal {E}_\\zeta $ , and since $\\mathcal {E}_\\zeta $ is closed under union by Lemma REF , we have $E_1\\cup E_2\\in \\mathcal {E}_\\zeta $ if $E_1,E_2\\in \\mathcal {E}^{F}_\\zeta $ .",
"Hence $\\zeta _3=(\\zeta \\setminus E_1\\cup E_2)\\cup \\mu ^{*}(E_1\\cup E_2)$ is a spanning forest with $d+1$ connected components, each with a root in $B$ .",
"We show that if $j\\in F$ and $i\\in B$ are in the same connected component of $\\zeta $ , then they are also in the same connected component of $\\zeta _3$ .",
"By Lemma REF (ii), it is enough to show this for $j\\ne m+1$ .",
"Consider the unique path from $j\\ne m+1$ to $i$ (assuming $j\\ne i$ ) in $\\zeta $ .",
"Assume this path contains an edge $e\\in E_1\\cup E_2$ , say $e\\in E_1$ .",
"Then $\\zeta _1=(\\zeta \\setminus E_1)\\cup \\mu ^{*}(E_1)$ , which belongs to $ \\Theta _{\\mathcal {G}}(F,B)$ by hypothesis, has also a path from $j$ to $i$ by (REF ).",
"Since every node different from the root of a rooted tree has exactly one outgoing edge, then $\\mu ^*(e)\\in \\mathcal {E}^-$ must belong to this path because it has source $s(e)$ .",
"But this contradicts Lemma REF (i).",
"Therefore this path does not contain an edge in $E_1\\cup E_2$ and hence it belongs to $\\zeta _3$ .",
"This implies that $j$ and $i$ belong to the same connected component of $\\zeta _3$ .",
"This shows that $\\zeta _3\\in \\Theta _{\\mathcal {G}}(F,B)$ .",
"It is a consequence of the lemma that the set $\\mathcal {E}^{F}_\\zeta $ has a unique maximum with respect to inclusion, which we denote by $E^{F}_\\zeta $ .",
"Define $\\Lambda _\\mathcal {G}(F,B)=\\left\\lbrace \\zeta \\in \\Theta _\\mathcal {G}(F,B)\\,\\mid \\,\\mathcal {E}^{F}_\\zeta =\\emptyset \\right\\rbrace ,$ and a surjective map by $\\psi _F\\colon \\Theta _\\mathcal {G}(F,B)\\rightarrow \\Lambda _\\mathcal {G}(F,B),\\quad \\psi _F(\\zeta )=\\Big (\\zeta \\setminus E^{F}_\\zeta \\Big )\\cup \\mu ^{*}\\Big (E^{F}_\\zeta \\Big ).$ Then we have the following decomposition, analogous to the decomposition in (REF ), $\\Theta _\\mathcal {G}(F,B)=\\bigsqcup \\limits _{\\zeta \\in \\Lambda _\\mathcal {G}(F,B)}\\psi ^{-1}_F(\\zeta ).$ Lemma 9 For $\\zeta \\in \\Lambda _{\\mathcal {G}}(F,B)$ , it holds $\\psi ^{-1}_{F}(\\zeta )=\\left\\lbrace E\\cup (\\zeta \\cap \\mathcal {E}^+) \\,\\mid \\,E\\in \\underset{e\\in \\zeta \\cap \\mathcal {E}^-}{{\\odot }}(\\lbrace e\\rbrace \\cup \\mu (e)) \\right\\rbrace .$ The inclusion $\\subseteq $ is proven analogously to the proof of Lemma REF .",
"By Lemma REF , we know that the set on the right side consists of elements in $\\Theta _{\\mathcal {G}}(B)$ .",
"Further, if $e\\in \\zeta \\cap \\mathcal {E}^-$ and $e^{\\prime }\\in \\mu (e)$ , consider the forest $\\zeta ^{\\prime }=(\\zeta \\setminus \\lbrace e\\rbrace )\\cup \\lbrace e^{\\prime }\\rbrace .$ We will show that $\\zeta ^{\\prime }\\in \\Theta _{\\mathcal {G}}(F,B)$ .",
"Given $j\\in F$ , $j\\ne m+1$ , such that $g_\\zeta (j)=i$ , the path connecting $j$ and $i$ (if any) in $\\zeta $ does not contain $e$ by Lemma REF (i).",
"Thus the path is also in $\\zeta ^{\\prime }$ and $j,i$ are in the same connected component of $\\zeta ^{\\prime }$ .",
"Now by Lemma REF (ii), $\\zeta ^{\\prime }\\in \\Theta _{\\mathcal {G}}(F,B)$ .",
"Thus the set on the right is included in $\\Theta _{\\mathcal {G}}(F,B)$ , and it is straightforward to show that their image by $\\psi _F$ is $\\zeta $ .",
"We proceed analogously to the proof of Theorem REF .",
"By (REF ) and Lemma REF , we obtain the following expression corresponding to (): G(F,B) = G(F,G)(E+)eE-((e)+e'(e)(e')).",
"By Definition REF (iv), we deduce that $\\Upsilon _\\mathcal {G}(F,B)\\in R_{\\ge 0}$ .",
"By Proposition REF , in particular (REF ), and the hypotheses of Theorem REF on the signs of the entries of $A$ and $b$ , we conclude that Theorem REF holds.",
"[Proof of Corollary REF ] By (REF ) and the remark below Proposition REF , it is enough to show that $\\Theta _\\mathcal {G}(F,B\\cup \\lbrace \\ell \\rbrace )=\\emptyset $ for $B\\in \\mathcal {B}^i$ , $i\\in \\lbrace 1,\\ldots ,d\\rbrace $ .",
"Assume thus that there exists $\\zeta \\in \\Theta _\\mathcal {G}(F,B\\cup \\lbrace \\ell \\rbrace )$ with $B\\in \\mathcal {B}^i$ .",
"In particular $g_\\zeta (j_i)=\\ell $ by (REF ) and hence there exists a path from $j_i$ to $\\ell $ in $\\zeta $ .",
"By the structure of an $A$ -compatible multidigraph, all nodes in this path belong to $\\mathcal {N}_i$ .",
"Thus the path can be extended to a path from $j_i$ to $j$ containing a node in $\\mathcal {E}^-$ , but not $m+1$ , contradicting condition ([condpossol2]*).",
"3pt"
]
] | 1709.01700 |
[
[
"User Assignment with Distributed Large Intelligent Surface (LIS) Systems"
],
[
"Abstract In this paper, we consider a wireless communication system where a large intelligent surface (LIS) is deployed comprising a number of small and distributed LIS-Units.",
"Each LIS-Unit has a separate signal process unit (SPU) and is connected to a central process unit (CPU) that coordinates the behaviors of all the LIS-Units.",
"With such a LIS system, we consider the user assignments both for sum-rate and minimal user-rate maximizations.",
"That is, assuming $M$ LIS-Units deployed in the LIS system, the objective is to select $K$ ($K\\!\\leq\\!M$) best LIS-Units to serve $K$ autonomous users simultaneously.",
"Based on the nice property of effective inter-user interference suppression of the LIS-Units, the optimal user assignments can be effectively found through classical linear assignment problems (LAPs) defined on a bipartite graph.",
"To be specific, the optimal user assignment for sum-rate and user-rate maximizations can be solved by linear sum assignment problem (LSAP) and linear bottleneck assignment problem (LBAP), respectively.",
"The elements of the cost matrix are constructed based on the received signal strength (RSS) measured at each of the $M$ LIS-Units for all the $K$ users.",
"Numerical results show that, the proposed user assignments are close to optimal user assignments both under line-of-sight (LoS) and scattering environments."
],
[
"Introduction",
"Large Intelligent Surface (LIS) is a newly proposed wireless communication system [1], [2] that can be seen as an extension of massive MIMO[3], [4], [5] systems, but scales up beyond the traditional antenna-array concept.",
"As envisioned in [1], [2], a LIS allows for an unprecedented focusing of energy in three-dimensional space, remote sensing with extreme precision and unprecedented data-transmissions, which fulfills visions for the 5G communication systems [6] and the concept of Internet of Things [7] where massive connections and various applications are featured.",
"In [1], fundamental limits on the number of independent signal dimensions are derived under the assumption of a single deployed LIS with infinite surface-area.",
"The results reveal that with matched-filtering (MF) applied, the inter-user interference of two users at the LIS is close to a sinc-function, and consequently, as long as the distance between two users are larger than half the wavelength, the inter-user interference is negligible.",
"In practical deployments, compared to a centralized deployment of a single large LIS, a LIS system that comprises a number of small LIS-Units such as in Fig.",
"REF has several advantages.",
"Firstly, the surface-area of each LIS-Unit can be sufficiently small which facilitates flexible deployments and configurations.",
"For instance, LIS units can be added, removed, or replaced without significantly affecting system design.",
"Secondly, each LIS-Unit can have a separate signal process unit (SPU) which makes cable and hardware synchronizations [8] simpler.",
"Thirdly, a distributed LIS-system can provide robust data-transmission and cover a wide area as different LIS-Unit scan be deployed apart from each other.",
"With all these advantages, in this paper we consider optimal user assignments both for sum-rate and minimum user-rate maximizations in a distributed LIS-system that comprises $M$ LIS-Units.",
"The target is to select $K$ ($K\\!\\le \\!M$ ) LIS-Units to serve $K$ autonomous users, with each LIS-Unit serving a user separately and simultaneously.",
"Firstly, following the work in [1] we show that, with a rather small surface-area, each LIS-Unit is effective in inter-user interference suppression.",
"Hence, the achieved user-rate at each LIS-Unit can be evaluated by the received signal strength (RSS) for each of the $K$ users.",
"Secondly, by specifying a cost matrix whose elements are the RSS at each LIS-Unit for each user, we can construct a bipartite graph between different users and LIS-Units.",
"With such a bipartite graph, the optimal user assignment for sum-rate maximization can be transferred to a linear sum assignment problem (LSAP), while the minimum user-rate maximization can be transferred to a linear bottleneck assignment problem (LBAP), respectively.",
"Then, the transferred linear assignment problems (LAPs) are solved through the well-known Kuhn-Munkres algorithm [9], [10], and Threshold algorithm [11], respectively.",
"Both algorithms have time complexities close to $\\mathcal {O}(KM^2)$ and are guaranteed to converge to optimal solutions [12], [13], [14].",
"Lastly, we show through numerical results that, the proposed user assignments solved with the LAPs are close to the optimal schemes both for considered line-of-sight (LOS) and scattering environments.",
"Figure: The diagram of a distributed LIS-communication system.",
"In practical systems, it can also correspond to a centralized deployment of a single large LIS, but with partitioning the LIS into a number of independent segments.We consider the transmission from $K$ autonomous single-antenna users located in a three-dimensional space to a two-dimensional LIS deployed on a plane as depicted in Fig.",
"REF .",
"Expressed in Cartesian coordinates, the center of the $m$ th LIS-Unit is located at $(x_m^{\\mathrm {c}},y_m^{\\mathrm {c}},z_m^{\\mathrm {c}})$ with $z_m^{\\mathrm {c}}\\!=\\!0$ , while users are located at $z\\!>\\!0$ and arbitrary $x$ , $y$ coordinates.",
"For analytical tractability, we assume a perfect LoS propagation and the case in scattering environments is similar.",
"The $k$ th terminal located at $(x_k,y_k,z_k)$ transmits data symbol $a_{k}$ with power $P_k$ , which is assumed to be a Gaussian variable with zero-mean and unit-variance, and independent over index $k$ .",
"Denote $\\lambda $ as the wavelength and consider a narrowband system where the transmit times from users to the LIS are negligible compared to symbol period which yields no temporal interference.",
"Following [1], [2], the effective channel $s_{x_k,\\,y_k,\\,z_k}(x,y)$ for the $k$ th user at position $(x, y, 0)$ at the $m$ th LIS-Unit can be modeled as $ s_{x_k,\\,y_k,\\,z_k}^m(x,y)=\\frac{\\sqrt{ z_k}}{2\\sqrt{\\pi }\\eta ^{\\frac{3}{4}}}\\exp \\!\\left(\\!-\\frac{2\\pi j\\sqrt{\\eta _{k,m}}}{\\lambda }\\right)\\!, $ where the metric $\\eta _{k,m}=(x_k-x)^2+(y_k-y)^2+z_k^2.$ Based on (REF ), the received signal at location $(x, y, 0)$ of the $m$ th LIS-Unit comprising all $K$ users is $ r_m(x,y) = \\sum _{k=0}^{K-1}\\sqrt{P_k}s_{x_k,y_k,z_k}^m(x,y)a_{k} +n_m(x,y),$ where $n_m(x,y)$ is AWGN.",
"Given the received signal (REF ) across the LIS-Unit, the discrete received signal after the MF process corresponding to the $k$ th user equals $ r_{m,k} =\\sum _{\\ell =0}^{K-1}\\sqrt{P_kP_\\ell } \\phi _{k,\\ell }^m a_{\\ell }+w_{m,k},$ where $w_k[m]$ is the effective colored noise after MF which has a zero-mean and satisfies $\\mathbb {E}(w_{m,k}w_{m,k}^{\\rm H})\\!=\\!N_0\\phi _{k,k}$ , and the coefficient $\\phi _{k,\\ell }$ is computed as $ \\phi _{k,\\ell }^m =\\iint \\limits _{(x,\\,y)\\in \\mathcal {S}_m}s_{x_\\ell ,y_\\ell ,z_\\ell }^m(x,y) s^{m,\\ast }_{x_k,y_k,z_k}(x,y)\\mathrm {d}x \\mathrm {d}y.",
"$ The variable $\\phi _{k,\\ell }$ denotes the RSS for the $k$ th user with $\\ell \\!=\\!k$ , and the inter-user interference between the $k$ th and $\\ell $ th users with $\\ell \\!\\ne \\!k$ , respectively."
],
[
"Interference Suppression with the LIS-Unit",
"Next we evaluate the interference suppression property at each of the LIS-Units.",
"As shown in [1], when the surface-area of the LIS is infinitely large, two users can be almost perfectly separated without interfering each other after the MF process.",
"However, in practical deployments the surface-area of each LIS-Unit is limited.",
"Therefore, it is of interest to investigate the interference suppression ability for a LIS-Unit with a finite surface-area.",
"Without loss of generality, we consider two users located in front of a square-shaped LIS-Unit whose center is located at position $x\\!=\\!y\\!=\\!z\\!=\\!0$ .",
"Then, the inter-user interference ($\\ell \\!\\ne \\!k$ ) according to (REF ) equals $ \\phi _{k,\\ell }^m&=&\\int \\limits _{-L/2}^{L/2}\\!\\int \\limits _{-L/2}^{L/2}\\!\\!s_{x_\\ell ,y_\\ell ,z_\\ell }(x,y) s^\\ast _{x_k,y_k,z_k}(x,y)\\mathrm {d}x \\mathrm {d}y \\\\&=&\\int \\limits _{-L/2}^{L/2}\\!\\int \\limits _{-L/2}^{L/2} \\!\\!\\!\\frac{\\sqrt{ z_k z_\\ell }}{4\\pi (\\eta _k\\eta _\\ell )^{\\frac{3}{4}}}\\exp \\!\\left(\\!\\frac{2\\pi j(\\sqrt{\\eta _{k,m}}\\!-\\!\\sqrt{\\eta _{\\ell ,m}})}{\\lambda }\\right)\\!\\mathrm {d}x \\mathrm {d}y.",
"\\\\ $ In [1] we show that, under the condition, $z_k\\!=\\!z_\\ell $ , $L\\!=\\!\\infty $ and $\\lambda $ is sufficient small, $\\phi _{k,\\ell }$ only depends on the distance $d$ between two user that equals $d=\\sqrt{(x_k-x_\\ell )^2+(y_k-y_\\ell )^2+(z_k-z_\\ell )^2}, $ and can be well approximated by a sinc-function.",
"Such a fact leads to an important observation that, as long as two users are located at least $\\lambda /2$ away from each other, $\\phi _{k,\\ell }$ is negligible.",
"However, for a finite $L$ , closed-form expression of (REF ) seems out of reach and we calculate (REF ) through numerical computations.",
"As shown next, we can also see that with a rather small $L$ , the LIS-Unit is still quite efficient in suppressing the inter-user interference.",
"Figure: An LoS scenario where three users communicates to a LIS with separate LIS-Units, with each LIS-Unit serving an individual user simultaneously.In Fig.",
"REF , we evaluate the signal-to-interference ratio (SIR) for a case where $L\\!=\\!0.5$Without explicitly pointed out, the unit of length, wavelength and coordinates are all in meter (m) in the rest of the paper., $\\lambda \\!=\\!0.125$ (corresponding to a carrier-frequency 2.4 GHz) and two users that are uniformly located in front of the LIS-UnitAssuming the LIS-Unit is implemented with discrete antenna-elements according to the sampling theory and the spacing between two adjacent antenna-elements is $\\lambda /2$ , then $L\\!=\\!0.5$ and $L\\!=\\!1$ corresponds to 64 and 256 antenna elements, respectively., with coordinates $-4\\!\\le \\!x, y\\le \\!4$ and $0\\!<\\!z\\le \\!8$ for both users.",
"We test for 1000 realizations of random user locations and report the empirical cumulative probability density function (CDF).",
"As can be seen, in almost 90% of the test cases, the value of 1/SIR is below -20 dB, which shows that the interference from the other user is significantly suppressed.",
"The same results can be seen from Fig.",
"REF where we set a larger $L\\!=\\!1$ , in which case the interference is further reduced and in almost 97% of the test cases, the SIR is below -20 dB.",
"Utilizing the effectiveness of interference suppression with LIS-Unit, we next elaborate on optimal user assignments with the distributed LIS-system for sum-rate maximization and minimum user-rate maximization.",
"Figure: The interference powers normalized by the signal powers, i.e., 1/SIR, are measured with two users in front of a square LIS-Unit with L=0.5L\\!=\\!0.5 whose center is x=y=z=0x\\!=\\!y\\!=\\!z\\!=\\!0.",
"The locations of the two users are drawn from a uniform distribution inside a cube with -4≤x,y≤4-4\\!\\le \\!x, y\\le \\!4 and 0<z≤80\\!<\\!z\\le \\!8.Figure: Repeat of the tests in Fig.",
"with L=1L\\!=\\!1."
],
[
"Optimal user assignments with the LIS",
"We consider a LIS deployment as in Fig.",
"REF , where $M$ small LIS-units are deployed distributively forming a large LIS-system and serve $K$ ($K\\!\\le \\!M$ ) users simultaneously.",
"The target is to find $K$ best LIS-Units that maximize the sum-rate and minimum user-rate, respectively, with each LIS-Unit serving a user separately.",
"Denote a set $\\mathcal {P}$ comprising all the possible assignment schemes, whose cardinality equals $|\\mathcal {P}|=\\frac{M!}{(M-K)!",
"}.$ Each assignment $p\\!\\in \\!\\mathcal {P}$ contains $K$ elements with the $k$ th element $p(k)$ denoting the index of the LIS-Unit which is selected to serve the $k$ th user.",
"The user-rate achieved with the $m$ th LIS-Unit equals $ R_k^{m}=\\log \\left(1+\\frac{\\big (\\phi _{k,k }^{m}\\big )^2}{N_0\\phi _{k,k }^{m}+\\sum \\limits _{\\ell =0,\\ell \\ne k}^{K-1}|\\phi _{k,\\ell }^{m}|^2}\\right)\\!.",
"$ Finding optimal assignments $p^{\\ast }$ can be formulated as the following maximization problems, respectively: $ &&\\text{Sum-rate:\\; \\;\\quad \\quad \\qquad }p^{\\ast }=\\mathop {\\arg \\max }_{p\\in \\mathcal {P}} \\sum _{k=0}^{K-1}R_k^{p(k)}, \\\\ &&\\text{Minimum user-rate:\\; \\;} p^{\\ast }=\\mathop {\\arg \\max }_{p\\in \\mathcal {P}} \\left(\\min _{k}\\left(R_k^{p(k)}\\right)\\!\\right)\\!.",
"\\qquad \\; $ The optimizations (REF ) and () can be solved in a brute-force manner for a small $M$ .",
"However, when $M$ and $K$ are large values, the complexity becomes prohibitive, which is not only because of the cardinality of $\\mathcal {P}$ , but also the computations for evaluating all $\\phi _{k,\\ell }^m$ that needs $MK^2$ operations in (REF ).",
"Therefore, suboptimal user assignment algorithms are needed to simplify the complexities.",
"First, we introduce the RSS based user assignment to reduce of complexity of evaluating $\\phi _{k,\\ell }^m$ ."
],
[
"RSS based user assignment",
"According to the results shown in Sec.",
"II-B, the interference power $|\\phi _{k,\\ell }^m|^2$ when $\\ell \\!\\ne \\!k$ is negligible compared to the signal power $|\\phi _{k,k}|^2$ at each of the LIS-Unit.",
"Therefore, the interference terms $|\\phi _{k,\\ell }^m|^2$ can be ignored in (REF ).",
"Then, maximizing $R_k^m$ is equivalent to maximizing $\\phi _{k,k}^m$ , and the optimization problems can be transferred to the following simplified problems: $ &&\\text{Sum-rate: \\;\\;\\quad \\quad \\qquad } p^{\\ast }=\\mathop {\\arg \\max }_{p\\in \\mathcal {P}} \\sum _{k=0}^{K-1}\\phi _{k,k}^{p(k)}, \\\\ &&\\text{Minimum user-rate: \\;\\;} p^{\\ast }=\\mathop {\\arg \\max }_{p\\in \\mathcal {P}} \\left(\\min _{k}\\left(\\phi _{k,k}^{p(k)}\\right)\\!\\right)\\!.",
"\\qquad \\; $ As only $\\phi _{k,k}^m$ are needed in (REF ) and (), the complexity of evaluating $\\phi _{k,k}^m$ reduces to $MK$ operations.",
"Note that, although (REF ) and () are formulated from the LoS scenario, since $\\phi _{k,k}^m$ denotes the RSS at the LIS-Unit for each user, the optimizations (REF ) and () also valid for scattering scenarios as explained in the next subsection."
],
[
"The validity of RSS based user assignments with reflections",
"In scattering environments, the RSS $\\phi _{k,k}^m$ comprises the signals reflected from different scatters that reach the LIS-Unit.",
"As we are considering a narrowband system, time-differences of signals from different scatters when arriving the LIS-Units are negligible.",
"In Fig.",
"REF , we depict an indoor scenario where besides the LoS signals from the user, there are also signals reflected by a surface that reach the LIS-Unit at the same time.",
"In an ideal case, the reflected signals can be viewed as LoS signals from an image of the user created by the reflecting surface, with additional transmit power loss caused by attenuation[15].",
"An important observation from Fig.",
"REF is that, based on the interference suppressing with the LIS-Unit with MF procedure, the effective channel from the user and the image user are almost orthogonal.",
"In more complex scattering environments with many scattering clusters, signal components from different clusters [16] can still be assumed to be orthogonal which results in coherent RSS addition at the LIS-Units.",
"Therefore, the RSS $\\phi _{k,k}^m$ is still a valid measurement for representing the user-rate that can be achieved in scattering environments.",
"Figure: An indoor scenario where a user communicating to a LIS-Unit with reflections from the wall received by the LIS-Unit."
],
[
"Transferred linear assignment problems",
"With the complexity of evaluating $\\phi _{k,\\ell }^m$ reduced from $MK^2$ to $MK$ , in a second step we reduce the complexity caused by the cardinality of $\\mathcal {P}$ from $M!/(M-K)!$ to $\\mathcal {O}(KM^2)$ .",
"We first introduce a coefficient matrix $W$ such that the element $w_{k,m}\\!=\\!1$ if the $k$ th user is assigned to the $m$ th LIS-Unit; otherwise $ w_{k,m}\\!=\\!0$ .",
"In order to form LAPs, we define a cost matrix $$ with elements $\\varphi _{k,m}= -\\phi _{k,k}^{m}\\le 0.$ Then, the optimization problems (REF ) and () can be equivalently written as linear sum and bottleneck assignment problems, respectively, as: $ &&\\text{LSAP: \\,\\;\\;} \\min \\quad \\sum _{k=0}^{K-1}\\sum _{m=0}^{M-1}w_{k,m}\\varphi _{k,m} \\\\ &&\\text{LBAP:\\quad } \\min \\quad \\max _{k}\\left(\\sum _{m=0}^{M-1}w_{k,m}\\varphi _{k,m}\\right)\\!",
"\\\\&& \\text{\\,s.t.\\;\\;} \\sum _{k=0}^{K-1}w_{k,m}\\le 1,\\quad 0\\le m<M \\\\&& \\qquad \\sum _{m=0}^{M-1}w_{k,m}=1, \\quad 0\\le k<K \\\\&& \\qquad w_{k,m} \\in \\lbrace 0,1\\rbrace , \\quad 0\\le k<K,\\; 0\\le m<M .",
"\\qquad $ The optimizations in (REF ) and () under constraints ()-() can be efficiently solved through graph based algorithms.",
"We first construct a weighted bipartite graph $G\\!=\\!",
"(V,E)$ as depicted in Fig.",
"REF in the following manner.",
"We let $x\\!=\\!",
"[x_0, x_1,\\dots ,x_{k-1}]$ with each vertex $x_k$ representing the $k$ th user, and $y\\!=\\!",
"[y_0, y_1,\\dots ,y_{M-1}]$ with each vertex $y_m$ representing the $m$ th LIS-Unit.",
"Hence, it holds that $x\\cap y\\!=\\!\\emptyset $ , and we set $V\\!=\\!x\\cup y$ and $E\\!\\subseteq \\!x\\times y$ is the set of possible matchings.",
"The weight of each edge $(x_k, y_m)$ in $G$ is $\\varphi _{k,m}$ .",
"Following conventional notations of LAP, we make the following notations.",
"Define labels $\\ell (u)$ for each vertex in graph $G$ , and a feasible labeling $\\ell \\!",
":\\!V\\!\\rightarrow \\!\\mathbb {R}$ satisfies the condition $\\ell (x_k)\\!+\\!\\ell (y_m)\\!\\ge \\!\\varphi _{k,m}$ .",
"The neighborhood of a vertex $x_k$ is the set $N_\\ell (x_k)\\!=\\!\\lbrace y_m:\\ell (x_k)\\!+\\!\\ell (y_m)\\!=\\!\\varphi _{k,m}\\rbrace $ with all vertices $y_m$ that share an edge with $x_k$ , and the neighborhood of a set $\\mathcal {S}$ is $N_\\ell (\\mathcal {S})\\!=\\!\\cup _{x_k\\in \\mathcal {S}}N_\\ell (x_k)$ .",
"Let $\\mathcal {P}$ be a matching of $G$ .",
"A maximum matching is a matching $\\mathcal {P}$ such that any other matching $\\mathcal {P}^{\\prime }{}$ satisfies $|\\mathcal {P}^{\\prime }{}|\\!\\le \\!|\\mathcal {P}|$ .",
"A perfect matching is a matching $\\mathcal {P}$ in which every vertex in $x$ is adjacent to some edge in $\\mathcal {P}$ .",
"A vertex $y_m$ is matched if it is endpoint of edge in $\\mathcal {P}$ , otherwise it is unmatched.",
"Figure: A weighted bipartite graph GG between the users and LIS-Units for solving the linear assignment problems () and (), with the weight of each edge (x k ,y m )(x_k, y_m) equals ϕ k,m \\varphi _{k,m}.With the above notations, we can apply the Kuhn-Munkres algorithm [10], [12], [13] to find a perfect matching $\\mathcal {P}$ , i.e., solutions of $w_{k,m}$ for LSAP (REF ), which is guaranteed to reach a global optimal with a numerical complexity $\\mathcal {O}(KM^2)$ and summarized in Algorithm 1.",
"[ht!]",
"Kuhn-Munkres algorithm for solving (REF ).",
"[1] Initialize $\\ell (y_m)\\!=\\!0, \\forall y_m\\!\\in \\!Y$ , $\\ell (x_k)\\!=\\!\\max \\limits _{m}(\\varphi _{k,m})$ , and $\\mathcal {P}\\!=\\!\\emptyset $ .",
"If $\\mathcal {P}$ is perfect, stop and return $\\mathcal {P}$ ; otherwise select an unmatched vertex $x_k\\!\\in \\!X$ and set $\\mathcal {S}\\!=\\!x_k$ , $\\mathcal {T}\\!=\\!\\emptyset $ .",
"If $N_\\ell (\\mathcal {S})\\!=\\!\\mathcal {T}$ , update labeling according to: $\\Delta =\\min _{x_k\\in \\mathcal {S},\\; y_m\\notin \\mathcal {T}}\\left(\\ell (x_k)+\\ell (y_m)-\\varphi _{k,m} \\right),\\\\\\ell (u)=\\left\\lbrace \\begin{array}{cc}\\ell (u)-\\Delta , &u\\in \\mathcal {S} \\\\ \\ell (u)+\\Delta , &u\\in \\mathcal {T} \\\\ \\ell (u), &\\mathrm {otherwise}.",
"\\end{array}\\right.",
"$ If $N_\\ell (\\mathcal {S})\\!\\ne \\!\\mathcal {T}$ , select $y_m\\!\\in \\!",
"N_\\ell (\\mathcal {S})\\!-\\!\\mathcal {T}$ : If $y_m$ free, $x_k-y_m$ is an augmenting path.",
"Augment $\\mathcal {P}$ and go to Step 2.",
"If $y_m$ is matched to some $x_j$ , extend alternating tree: $\\mathcal {S}\\!=\\!\\mathcal {S}\\!\\cup \\!x_j$ and $\\mathcal {T}\\!=\\!\\mathcal {T}\\!\\cup \\!y_m$ , and then go to Step 3.",
"Further, for LBAP () we can also apply the Threshold Algorithm [12] to find an optimal solution which is summarized in Algorithm 2, and has a similar complexity [14] as Algorithm 1.",
"When $M$ and $K$ are large values, the save of computational costs of Algorithm 1 and 2 compared to the brute-force method is significant.",
"In this section, simulation results are provided for evaluating the optimal user assignments with LSAP (REF ) and LBAP ().",
"The optimal user assignments obtained with brute-force methods to solve the original problem (REF ) and () are also presented as upper bounds.",
"Further, we average the rates obtained with all the possible user assignments with size $|\\mathcal {P}|$ to serves as lower bounds, which reflect the rates achieved with random user assignments.",
"We consider both LoS and scattering scenario, and put a special interested in the rates can be achieved per m$^2$ deployed surface-area of each LIS-Unit and per user.",
"In all the tests, we assume all users transmitting at a transmit power $P\\!=\\!20$ dB and a noise power density $N_0\\!=\\!1$ , if not otherwise explicitly pointed out.",
"We assume that all LIS-Units have identical square shapes with identical surface-area $L\\!\\times \\!L$ m$^2$ each, and are deployed close to each other to form a large LIS-system with centers uniformly distributed along the line $y\\!=\\!z\\!=\\!0$ , and with the middle LIS-Unit centered at location (0,0,0).",
"Further, we assume that the users are uniformly distributed in front of the LIS-Units with coordinates $(x, y, z)$ satisfying $-2\\!\\le \\!x\\!\\le \\!2$ , $-2\\!\\le \\!y\\!\\le \\!2$ , and $0\\!<\\!z\\!\\le \\!4$ .",
"For all the test scenarios, we generate 2000 realizations of random user locations.",
"Note that, although Algorithm 1 and 2 are guaranteed to reach optimal solutions for solving (REF ) and (), there are still rate-losses compared to the brute-force algorithms due to the simplification of using RSS in (REF )-() to replace the capacities in (REF )-().",
"[b] Threshold algorithm for solving ().",
"[1] Initialize $\\varphi ^{\\ast }\\!=\\!\\max \\limits _m(\\min \\limits _k\\varphi _{k,m},\\min \\limits _m\\varphi _{k,m})$ and $\\mathcal {P}\\!=\\!\\emptyset $ .",
"Define a bipartite graph $G(\\varphi ^{\\ast })$ whose edges correspond to $\\varphi _{k,m}\\!\\le \\!\\varphi ^{\\ast }$ .",
"Find a maximum matching $\\mathcal {P}$ in $G(\\varphi ^{\\ast })$ .",
"If $|\\mathcal {P}|\\!=\\!K$ , stop and return $\\mathcal {P}$ ; otherwise go to Step 4.",
"Find a minimal row and column covering of the elements $\\varphi _{k,m}\\!\\le \\!\\varphi ^{\\ast }$ in $\\varphi $ .",
"Set $\\varphi ^{\\ast }$ to the minimum of the remaining elements in $\\varphi $ after removing the rows and columns from the minimal covering, and then go to Step 2."
],
[
"The user assignments in LoS scenario",
"First, we evaluate the user assignment in LoS scenarios, and consider a LIS-system with $M\\!=\\!7$ LIS-Units deployed on a plane.",
"The sum-rate maximized with solving LSAP (REF ) and the brute-force search over (REF ) are presented in Fig.",
"REF , after normalizing by the surface-area of a LIS-Unit and the number of users.",
"As can be seen, the proposed user assignments is quite close to the optimal, and much better than random user assignments.",
"Furthermore, the achieved rate per user first increases when the surface-area increases and then starts to decrease.",
"This is essentially because that, as the surface-area initially increases, the inter-user interference suppression improves which in turn improves the achieved rate of each user.",
"After the inter-user interference suppression becomes perfect, further increasing the surface-area of each LIS-Unit decreases the achieved rate per m$^2$ deployed surface-area as the edge parts of the LIS-Unit provide lower rates compared to the central parts of the LIS-Unit.",
"In Fig.",
"REF , the minimum user-rate maximized with solving LBAP () and the brute-force search over () are presented.",
"As can be seen, the proposed user assignments has $\\sim $ 10% minimum user-rate losses compared to the brute-force search in this case, due to the fact that, the minimum user-rate is more sensitive to the user arrangement than the sum-rate.",
"However, the proposed user assignment has a much lower complexity than the brute-force, and it still significantly outperforms the random user assignments.",
"Figure: Sum-rate maximization in LoS scenario with 6 LIS-Units and 2 users.Figure: Repeat the tests in Fig.",
"for minimum user-rate maximization."
],
[
"The user assignments with reflections",
"Next, to evaluate the user assignment performance in scattering environments, we consider a LIS-system with $M\\!=\\!5$ LIS-Units deployed on the front wall in a hall, and with $N\\!=\\!2$ users randomly located inside the hall.",
"In addition, we consider the wall reflections and create 5 images for each user corresponding to the 5 walls except the front wall where the LIS-Units are deployed according to Fig.",
"REF .",
"The attenuation is assumed to be $-3$ dB for all walls.",
"In Fig.",
"REF , the normalized sum-rates are presented, and as can be seen, the conclusions are similar to those drawn from LoS scenarios except for the cases that $L$ is really small.",
"This is essentially because that, with small $L$ such as $L\\!\\le \\!0.3$ , the inter-user interference suppression is not as good as LoS scenarios due to the presented 5 images for each user.",
"Nevertheless, for relatively large $L$ it can be seen that, the proposed user assignments still work well compared to the brute-force results.",
"Figure: Sum-rate maximization under wall-reflections scenario with 5 LIS-Units and 2 users.Figure: Minimum user-rate maximization with measuring RSS only at the center of each LIS-Unit in a LoS scenario with 5 LIS-Units and 2 users."
],
[
"An extension of the RSS based user arrangements",
"At last we discussion an extension of the proposed RSS based algorithms.",
"Instead of measuring the RSS $\\phi _{k,k}^m$ over the whole LIS-Unit for each user, a reduced complexity RSS measurement is to only measure the RSS at finitely many discrete positions.",
"An ultimate simplified RSS measurement algorithm would be only evaluate the RSS at the center of each LIS-Unit, which we denote as $\\tilde{\\phi }_{k,k}^m$ and is calculated according to (REF ) as $ \\tilde{\\phi }_{k,k}^m=\\frac{z_k}{4\\pi \\eta _k^{\\frac{3}{2}}}\\propto z_k \\eta _{k,m}^{-\\frac{3}{2}}, $ where $\\eta _{k,m}$ is the square of distance between the $k$ th user and the $m$ th LIS-Unit.",
"Clearly, replacing $\\phi _{k,k}^m$ by $\\tilde{\\phi }_{k,k}^m$ would degrade the performance of user assignment.",
"However, under the cases that the users are far away from the LIS-Unit or the surface-area of the LIS-Unit is sufficiently small compared to the distances from the users to the LIS-Unit, $\\tilde{\\phi }_{k,k}^m$ (multiplying with the surface-area of the LIS-Unit) is a good estimate of $\\phi _{k,k}^m$ .",
"Therefore, it is of interest to evaluate the performance degradation in general cases.",
"In Fig.",
"REF , we evaluate the minimum user-rate maximizations with the reduced-complexity RSS measurement.",
"We test a similar case as in Fig.",
"REF with $M\\!=\\!5$ LIS-Units with $L\\!=\\!0.2$ and $N\\!=\\!2$ users.",
"As can be seen, with simplified RSS $\\tilde{\\phi }_{k,k}^m$ , the minimum user-rate is slightly lower then with the original RSS $\\phi _{k,k}^m$ for large transmit power.",
"The sum-rate maximization is also evaluated for the simplified RSS, but as the achieved rates are almost the same with $\\tilde{\\phi }_{k,k}^m$ and $\\phi _{k,k}^m$ , they are not presented.",
"Nevertheless, the results in Fig.",
"REF shows the potential to evaluate the RSS at a number of sampled discrete points on the LIS-Unit (or even only at the center), which can significantly simplify the complexity of computing $\\phi _{k,k}^m$ ."
],
[
"Summary",
"We have considered optimal user assignments for a distributed large intelligent surface (LIS) system with $M$ separate LIS-Units.",
"The objective is to select $K$ best LIS-Units to serve $K$ ($K\\!\\le \\!M$ ) autonomous users simultaneously.",
"By constructing a cost matrix based on the received signal strength (RSS) at each LIS-Unit for each user, we obtain a weighted bipartite graph between the $K$ users and $M$ LIS-Units.",
"Utilizing the effectiveness of LIS-Unit in inter-user interference suppression and with the constructed bipartite graph, the optimal user assignments for sum-rate maximization can be transferred to a linear sum assignment problem (LSAP), and for the minimum user-rate maximization can be transferred to a linear bottleneck assignment problem (LBAP), respectively.",
"The linear assignment problems (LAPs) are then solved through the classical Kuhn-Munkres and Threshold algorithms with time complexities $\\mathcal {O}(KM^2)$ .",
"We show through numerical results that, the proposed user assignments perform close to the optimal assignments both for considered line-of-sight (LoS) and scattering environments."
]
] | 1709.01696 |
[
[
"Some remarks on topological $K$-theory of dg categories"
],
[
"Abstract Using techniques from motivic homotopy theory, we prove a conjecture of Anthony Blanc about semi-topological K-theory of dg categories with finite coefficients.",
"Along the way, we show that the connective semi-topological K-theories defined by Friedlander-Walker and by Blanc agree for quasi-projective complex varieties and we study \\'etale descent of topological K-theory of dg categories."
],
[
"Introduction",
"Blanc defines semi-topological and topological $K$ -theory functors $\\mathrm {K}^{\\mathrm {st}},\\mathrm {K}^\\mathrm {top}:\\mathrm {Cat}_{C}\\rightarrow \\mathcal {S}\\mathrm {p},$ where $\\mathrm {Cat}_{C}$ denotes the $\\infty $ -category of small idempotent complete pretriangulated dg categories over ${C}$By work of L. Cohn , $\\mathrm {Cat}_{C}$ is equivalent to the $\\infty $ -category of small idempotent complete ${C}$ -linear stable $\\infty $ -categories.",
"(${C}$ -linear dg categories in this paper for short) and $\\mathcal {S}\\mathrm {p}$ is the $\\infty $ -category of spectra.",
"When $\\mathcal {C}$ is a ${C}$ -linear dg category, there are natural maps $\\mathrm {K}(\\mathcal {C})\\rightarrow \\mathrm {K}^{\\mathrm {st}}(\\mathcal {C})\\rightarrow \\mathrm {K}^\\mathrm {top}(\\mathcal {C})$ .",
"Moreover, $\\mathrm {K}^\\mathrm {st}(\\mathcal {C})$ is a $\\mathrm {ku}$ -module spectrum, and, by definition, $\\mathrm {K}^\\mathrm {top}(\\mathcal {C})\\simeq \\mathrm {K}^\\mathrm {st}(\\mathcal {C})[\\beta ^{-1}]\\simeq \\mathrm {K}^\\mathrm {st}(\\mathcal {C})\\otimes _{\\mathrm {ku}}\\mathrm {KU},$ where $\\beta \\in \\pi _2\\mathrm {ku}$ is the Bott element.",
"Let $\\mathrm {Sch}_{C}$ denote the category of separated ${C}$ -schemes of finite type.",
"If $F:\\mathrm {Cat}_{C}\\rightarrow \\mathcal {S}\\mathrm {p}$ is a functor and $\\mathcal {C}$ is a ${C}$ -linear dg category, then we define a presheaf $\\underline{\\smash{F}}(\\mathcal {C}):\\mathrm {Sch}_{C}^\\mathrm {op}\\rightarrow \\mathcal {S}\\mathrm {p}$ by the formula $\\underline{\\smash{F}}(\\mathcal {C})(X)\\simeq F(\\mathrm {Perf}_X\\otimes _{C}\\mathcal {C})$ .",
"In other words, $\\underline{\\smash{F}}(\\mathcal {C})$ is the composition of the functor $\\mathrm {Perf}:\\mathrm {Sch}^\\mathrm {op}_{C}\\rightarrow \\mathrm {Cat}_{C}$ , the endofunctor $\\mathrm {Cat}_{C}\\rightarrow \\mathrm {Cat}_{C}\\qquad \\mathcal {D}\\mapsto \\mathcal {D}\\otimes _{C}\\mathcal {C},$ and the functor $F:\\mathrm {Cat}_{C}\\rightarrow \\mathcal {S}\\mathrm {p}$ .",
"In many cases, we will use the restriction of $\\underline{\\smash{F}}(\\mathcal {C})$ to $\\mathrm {Aff}_{C}^\\mathrm {op}$ , $\\mathrm {Sm}_{C}^\\mathrm {op}$ , or $\\mathrm {Aff}_{C}^{\\mathrm {sm},\\mathrm {op}}\\simeq \\mathrm {Sm}_{C}^{\\mathrm {aff},\\mathrm {op}}$ the opposites of the categories of affine ${C}$ -schemes of finite type, smooth separated ${C}$ -schemes of finite type, and smooth affine ${C}$ -schemes, respectively.",
"In this paper, we prove three theorems about semi-topological and topological $K$ -theory of dg categories.",
"First, we prove that $\\mathrm {K}^\\mathrm {sst}(X)\\simeq \\mathrm {K}^{\\mathrm {cn},\\mathrm {st}}(\\mathrm {Perf}_X)$ when $X$ is a quasi-projective complex variety, where $\\mathrm {K}^\\mathrm {sst}(X)$ is the semi-topological $K$ -theory spectrum defined by Friedlander and Walker in and $\\mathrm {K}^{\\mathrm {cn},\\mathrm {st}}(\\mathrm {Perf}_X)$ is the connective version of Blanc's semi-topological $K$ -theory.",
"Second, we prove a conjecture of Blanc, stating that $\\mathrm {K}(\\mathcal {C})/n\\simeq \\mathrm {K}^\\mathrm {st}(\\mathcal {C})/n$ for $n\\geqslant 1$ and any ${C}$ -linear dg category $\\mathcal {C}$ .",
"Third, we prove that $\\underline{\\smash{\\mathrm {K}}}^\\mathrm {top}(\\mathcal {C})$ is ${A}^1$ -invariant and a hypersheaf for the étale topology on $\\mathrm {Sm}_{C}$ for any ${C}$ -linear dg category $\\mathcal {C}$ .",
"Put together, the last two theorems imply that $\\underline{\\smash{\\mathrm {K}}}(\\mathcal {C})/n$ satisfies étale hyperdescent after inverting the Bott element.",
"The first theorem has also been obtained by Blanc and Horel and they also made progress toward the second theorem along the same lines as the argument we give."
],
[
"Acknowledgments.",
"BA would like to thank Tasos Moulinos for patiently explaining Blanc's work on semi-topological $K$ -theory to him on several occasions.",
"Both authors express their gratitude to Anthony Blanc for his email comments on this topic and for looking over a preliminary version of the paper.",
"They also are grateful to a careful referee who made several nice suggestions for improvements.",
"Finally, this paper would probably not be possible without the dogs Boschko and Lima, who created the opportunity for the authors to work together."
],
[
"Comparison of semi-topological $K$ -theories",
"The original definition of semi-topological $K$ -theory is for complex varieties and goes back to work of Friedlander and Walker friedlander-walker,friedlander-walker-comparing.",
"They construct spectra $\\mathrm {K}^\\mathrm {semi}(X)$ and $\\mathrm {K}^\\mathrm {sst}(X)$ when $X$ is quasi-projective and they give a natural map $\\mathrm {K}^\\mathrm {sst}(X)\\rightarrow \\mathrm {K}^\\mathrm {semi}(X)$ .",
"When $X$ is projective and weakly normal, this map is an equivalence by *Theorem 1.4.",
"In their survey , they settle on $\\mathrm {K}^\\mathrm {sst}(X)$ as the `correct' definition of semi-topological $K$ -theory of quasi-projective complex varieties.",
"It is natural to wonder about the relationship between $\\mathrm {K}^{\\mathrm {sst}}(X)$ and $\\mathrm {K}^{\\mathrm {st}}(\\mathrm {Perf}_X)$ .",
"We prove that they are in fact equivalent.",
"Blanc has communicated to us that he was aware of this fact, although it was open at the time of .",
"We recall the definition of semi-topological $K$ -theory of dg categories from .",
"Let $\\mathrm {Sch}_{C}\\rightarrow \\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sch}_{C})$ be the spectral Yoneda functor, where $\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sch}_{C})$ is the stable presentable $\\infty $ -category of presheaves of spectra on $\\mathrm {Sch}_{C}$ .",
"Let $\\mathrm {Sch}_{C}\\rightarrow \\mathcal {S}\\mathrm {p}$ be the composition of $\\mathrm {Sch}_{C}\\rightarrow \\mathcal {S}\\qquad X\\mapsto \\operatorname{Sing}{X({C})}$ with the suspension spectrum functor $\\Sigma ^\\infty _+:\\mathcal {S}\\rightarrow \\mathcal {S}\\mathrm {p}$ , where $\\mathcal {S}$ denotes the $\\infty $ -category of topological spaces.",
"Define the topological realization $\\mathrm {Re}:\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sch}_{C})\\rightarrow \\mathcal {S}\\mathrm {p}$ as the left Kan extension ${\\mathrm {Sch}_{C}[rr]^{X\\mapsto \\Sigma ^\\infty \\operatorname{Sing}{X({C})}_+}[d] && \\mathcal {S}\\mathrm {p}\\\\\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sch}_{C}).",
"@{.>}[urr]^{\\rm Re}&&}$ Given $\\mathcal {C}\\in \\mathrm {Cat}_{C}$ , there is the presheaf $\\underline{\\smash{\\mathrm {K}}}(\\mathcal {C})$ as defined in Section , where $\\mathrm {K}:\\mathrm {Cat}_{C}\\rightarrow \\mathcal {S}\\mathrm {p}$ denotes nonconnective $K$ -theory as defined for example in .",
"Definition 2.1 (Blanc ) The semi-topological $K$ -theory of $\\mathcal {C}$ is the spectrum $\\mathrm {K}^\\mathrm {st}(\\mathcal {C})=\\mathrm {Re}(\\underline{\\smash{\\mathrm {K}}}(\\mathcal {C}))$ .",
"More generally, let $f:X\\rightarrow \\operatorname{Spec}{C}$ be a separated ${C}$ -scheme of finite type and let $\\mathrm {Sch}_X$ be the category of separated $X$ -schemes of finite presentation.",
"There is an adjunction $f^*:\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sch}_{C})\\rightleftarrows \\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sch}_X):f_*$ defined in the usual way.",
"We let $\\underline{\\smash{\\mathrm {K}}}^\\mathrm {st}(\\mathcal {C}):\\mathrm {Sch}^\\mathrm {op}\\rightarrow \\mathcal {S}\\mathrm {p}$ be the presheaf with value at $f:X\\rightarrow {C}$ given by $\\underline{\\smash{\\mathrm {K}}}^\\mathrm {st}(\\mathcal {C})(X)=\\mathrm {Re}(f_*f^*\\underline{\\smash{\\mathrm {K}}}(\\mathcal {C}))$ .",
"In particular, $\\underline{\\smash{\\mathrm {K}}}^\\mathrm {st}(\\mathcal {C})(X)\\simeq \\mathrm {K}^\\mathrm {st}(\\mathrm {Perf}_X\\otimes _{C}\\mathcal {C})$ and $\\underline{\\smash{\\mathrm {K}}}^\\mathrm {st}(\\mathcal {C})(\\operatorname{Spec}{C})\\simeq \\mathrm {K}^\\mathrm {st}(\\mathcal {C})$ .",
"Definition 2.2 (Blanc ) If we apply the same construction with connective $K$ -theory $\\mathrm {K}^\\mathrm {cn}$ we obtain a connective version of semi-topological $K$ -theory, namely $\\mathrm {K}^{\\mathrm {cn},\\mathrm {st}}(\\mathcal {C})=\\mathrm {Re}(\\underline{\\smash{\\mathrm {K}}}^{\\mathrm {cn}}(\\mathcal {C}))$ , where $\\underline{\\smash{\\mathrm {K}}}^\\mathrm {cn}(\\mathcal {C})$ is the presheaf of connective spectra $\\underline{\\smash{\\mathrm {K}}}^\\mathrm {cn}(\\mathcal {C})(X)\\simeq \\mathrm {K}^{\\mathrm {cn}}(\\mathrm {Perf}_X\\otimes _{{C}}\\mathcal {C})$ .",
"This is the theory denoted by $\\underline{\\smash{\\tilde{\\mathrm {K}}}}(\\mathcal {C})$ in Blanc's paper.",
"The following theorem has also been obtained by Blanc and Geoffroy Horel (private communication).",
"Theorem 2.3 If $X$ is a quasi-projective complex variety, then there is a natural equivalence $\\mathrm {K}^{\\mathrm {sst}}(X)\\simeq \\mathrm {K}^{\\mathrm {cn},\\mathrm {st}}(\\mathrm {Perf}_X)$ .",
"To begin, we give the definition of $\\mathrm {K}^{\\mathrm {sst}}(X)$ after *Definition 1.1.",
"Let $\\widetilde{\\mathcal {T}\\mathrm {op}}$ be a small category of topological spaces and continuous maps containing at least the essential image of $r:\\mathrm {Sch}_{C}\\xrightarrow{}\\mathcal {T}\\mathrm {op}$ and the topological simplices $\\Delta ^n_\\mathrm {top}$ .",
"Friedlander and Walker consider the left Kan extension ${\\mathrm {Sch}_{C}^{\\mathrm {op}}[rr]^{\\underline{\\smash{\\mathrm {K}}}^\\mathrm {cn}(X)}[d]_{U\\mapsto U({C})} && \\mathcal {S}\\mathrm {p}\\\\\\widetilde{\\mathcal {T}\\mathrm {op}}^{\\mathrm {op}}[urr]_{r^*\\underline{\\smash{\\mathrm {K}}}^\\mathrm {cn}(X)}.&&}$ By definition, if $Y$ is a topological space, then $r^*\\underline{\\smash{\\mathrm {K}}}^\\mathrm {cn}(X)(Y)\\simeq \\operatornamewithlimits{colim}_{Y\\rightarrow U({C})}\\underline{\\smash{\\mathrm {K}}}^\\mathrm {cn}(X)(U)\\simeq \\operatornamewithlimits{colim}_{Y\\rightarrow U({C})}\\underline{\\smash{\\mathrm {K}}}^\\mathrm {cn}(X\\times _{C}U).$ Evaluating $r^*\\underline{\\smash{\\mathrm {K}}}^\\mathrm {cn}(X)$ at the cosimplicial space $\\Delta ^\\bullet _\\mathrm {top}$ , we obtain a simplicial spectrum $r^*\\underline{\\smash{\\mathrm {K}}}^\\mathrm {cn}(X)(\\Delta ^\\bullet _\\mathrm {top})$ .",
"The semi-topological $K$ -theory of $Y$ is defined to be $\\mathrm {K}^{\\mathrm {sst}}(X)=|\\underline{\\smash{\\mathrm {K}}}^\\mathrm {cn}(X)(\\Delta ^\\bullet _\\mathrm {top})|.$ Note that this process is precisely the composition of the functors $\\widetilde{\\mathrm {Re}}_{\\mathcal {S}\\mathrm {p}}:\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sch}_{C})\\xrightarrow{}\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\widetilde{\\mathcal {T}\\mathrm {op}})\\xrightarrow{}\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\Delta )\\rightarrow \\mathcal {S}\\mathrm {p}$ applied to $\\underline{\\smash{\\mathrm {K}}}^\\mathrm {cn}(X)$ , where $s$ denotes the inclusion of $\\Delta $ into $\\widetilde{\\mathcal {T}\\mathrm {op}}$ classifying the cosimplicial space $\\Delta ^\\bullet _{\\mathrm {top}}$ and the final arrow is geometric realization of a simplicial spectrum.",
"This composition is the stabilization of $\\widetilde{\\mathrm {Re}}:\\mathcal {P}(\\mathrm {Sch}_{C})\\xrightarrow{}\\mathcal {P}(\\widetilde{\\mathcal {T}\\mathrm {op}})\\xrightarrow{}\\mathcal {P}(\\Delta )\\xrightarrow{}\\mathcal {S},.$ where $\\mathcal {P}(\\Delta )$ is the $\\infty $ -category of presheaves of spaces on the simplex category $\\Delta $ , or in other words the $\\infty $ -category of simplicial spaces, and $|-|$ denotes geometric realization.",
"To prove the theorem it suffices to prove that $\\widetilde{\\mathrm {Re}}:\\mathcal {P}(\\mathrm {Sch}_{C})\\rightarrow \\mathcal {S}$ is equivalent to the functor $\\mathcal {P}(\\mathrm {Sch}_{C})\\rightarrow \\mathcal {S}$ obtained via the unstable version of the left Kan extension in (REF ): ${\\mathrm {Sch}_{C}[rr]^{U\\mapsto \\operatorname{Sing}{U({C})}}[d]&&\\mathcal {S}\\\\\\mathcal {P}(\\mathrm {Sch}_{C}).",
"[rru]^{\\mathrm {Re}}&&}$ To prove that $\\widetilde{\\mathrm {Re}}\\simeq \\mathrm {Re}$ , note first that both functors are left adjoints because $\\mathcal {P}(\\mathrm {Sch}_{C})$ is presentable.",
"The only thing to check is that $s_*$ preserves colimits, but this follows because colimits in presheaf categories are computed pointwise (see for example *Corollary 5.1.2.3).",
"Thus, it suffices to prove that the restrictions of $\\widetilde{\\mathrm {Re}}$ and $\\mathrm {Re}$ to $\\mathrm {Sch}_{C}$ are equivalent.",
"On the one hand, we know that $\\mathrm {Re}(U)\\simeq U({C})$ in $\\mathcal {S}$ for $U\\in \\mathrm {Sch}_{C}$ .",
"On the other hand, by definition, $r^*U(\\Delta ^n_\\mathrm {top})\\simeq \\operatornamewithlimits{colim}_{\\Delta ^n_\\mathrm {top}\\rightarrow V({C})}\\operatorname{Hom}_{\\mathrm {Sch}_{C}}(V,U)$ .",
"Using Jouanolou's device , we see that any map $\\Delta ^n_\\mathrm {top}\\rightarrow V({C})$ factors through $W({C})\\rightarrow V({C})$ where $W({C})$ is affine and $\\mathrm {W}({C})\\rightarrow V({C})$ is a vector bundle torsor.",
"Thus, by *Proposition 4.2, $r^*U(\\Delta ^n_\\mathrm {top})\\cong \\operatorname{Hom}_{\\mathcal {T}\\mathrm {op}}(\\Delta ^n_{\\mathrm {top}},U({C}))\\cong \\operatorname{Sing}{U({C})}_n$ and hence $\\widetilde{\\mathrm {Re}}(U)\\simeq \\operatorname{Sing}{U({C})}$ , as desired.",
"Remark 2.4 A theorem of Friedlander and Walker says that when $X$ is smooth and quasi-projective, $\\mathrm {K}^\\mathrm {sst}(X)[\\beta ^{-1}]\\simeq \\mathrm {KU}(X({C}))$ , the complex $K$ -theory spectrum of the space of ${C}$ -points of $X$ (see *Theorem 32).",
"It follows from the theorem that $\\mathrm {K}^\\mathrm {top}(\\mathrm {Perf}_X)=\\mathrm {K}^{\\mathrm {st}}(\\mathrm {Perf}_X)[\\beta ^{-1}]\\simeq \\mathrm {K}^{\\mathrm {sst}}(X)[\\beta ^{-1}]\\simeq \\mathrm {KU}(X({C}))$ when $X$ is smooth and quasi-projective, where $\\mathrm {K}^{\\mathrm {st}}(\\mathrm {Perf}_X)\\simeq \\mathrm {K}^{\\mathrm {cn},\\mathrm {st}}(\\mathrm {Perf}_X)$ because $X$ is smooth and by *Theorem 3.18.",
"This gives a new proof of one of the main theorems of Blanc's paper *Theorem 1.1(b) in the special case of $X$ smooth and quasi-projective.",
"Blanc's theorem says more generally that if $X$ is separated and finite type over ${C}$ , then $\\mathrm {K}^\\mathrm {top}(\\mathrm {Perf}_X)\\simeq \\mathrm {KU}(X({C}))$ .",
"Remark 2.5 It is clear that one could have defined a nonconnective version $\\mathrm {K}^{\\mathrm {nc},\\mathrm {sst}}(X)$ of Friedlander and Walker's $\\mathrm {K}^{\\mathrm {sst}}(X)$ simply by replacing connective $K$ -theory with nonconnective $K$ -theory in *Definition 1.1.",
"If this is done, then the proof of Theorem REF goes through and shows that there are natural equivalences $\\mathrm {K}^{\\mathrm {nc},\\mathrm {sst}}(X)\\simeq \\mathrm {K}^{\\mathrm {st}}(X)$ for quasi-projective complex varieties $X$ ."
],
[
"Blanc's conjecture",
"Let ${\\rm Sch}_{C}$ be a full subcategory closed under taking products with ${A}^1_{C}$ .",
"Let $\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}^{{A}^1}(\\subseteq \\mathcal {P}_{\\mathcal {S}\\mathrm {p}}($ be the full subcategory of ${A}^1$ -invariant presheaves of spectra, i.e., those $F$ such that the pullback maps $F(X)\\rightarrow F(X\\times _{C}{A}^1)$ are equivalences for all $X\\in .The inclusion has a left adjoint,$ LA1:PSp(PSpA1($.",
"A map $ FG$ is an$ A1$-equivalence if $ LA1FLA1G$ is an equivalence.Given a presheaf of spectra $ F$ on $ , we let $\\operatorname{Sing}^{{A}^1}F$ be the presheaf defined by $(\\operatorname{Sing}^{{A}^1}F)(X) = \\operatornamewithlimits{colim}_{\\Delta }F(X\\times _{C}\\Delta ^{\\bullet }_{{C}}),$ where $\\Delta ^\\bullet _{C}$ is the standard cosimplicial affine scheme.",
"It is a standard fact that $\\operatorname{Sing}^{{A}^1}F$ is ${A}^1$ -invariant in the sense that for every $X\\in , thepullback map $ (SingA1F)(X)(SingA1F)(XCA1)$ inducedby the projection $ XCA1CX$ is an equivalence.Moreover, $ FSingA1F$ is an $ A1$-equivalence.",
"It followsthat $ SingA1FLA1F$ for all $ F$ and thatif $ F$ is $ A1$-invariant, then the natural transformation$ FLA1F$ is an equivalence.",
"For proofs of these facts,see~\\cite {morel-voevodsky}*{Section~2.3}.From the natural equivalences $ LA1FSingA1F$, we see that if $ i: is a subcategory (also closed under taking products with ${A}^1_{C}$ ) and if $F$ is a presheaf on $, then $ i*LA1F =LA1i*F$.$ Let $\\mathcal {C}$ be a ${C}$ -linear dg category.",
"Then, $\\mathrm {L}_{{A}^1}\\underline{\\smash{\\mathrm {K}}}(\\mathcal {C})\\simeq \\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})$ , where $\\mathrm {KH}:\\mathrm {Cat}_{C}\\rightarrow \\mathcal {S}\\mathrm {p}$ is the homotopy $K$ -theory of dg categories, as defined for example in .",
"If $F:\\mathrm {Sch}^\\mathrm {op}\\rightarrow \\mathcal {S}\\mathrm {p}$ is a presheaf of spectra, write $F^\\mathrm {st}$ for the presheaf with value at $f:X\\rightarrow \\operatorname{Spec}{C}$ in $\\mathrm {Sch}_{C}$ the spectrum $F^\\mathrm {st}(X)\\simeq \\mathrm {Re}(f_*f^*F)$ .",
"Lemma 3.1 If $F:\\mathrm {Sch}^{\\mathrm {op}}\\rightarrow \\mathcal {S}\\mathrm {p}$ is a presheaf of spectra, then $F^\\mathrm {st}\\simeq (\\mathrm {L}_{{A}^1}F)^\\mathrm {st}\\simeq \\mathrm {L}_{{A}^1}(F^\\mathrm {st}).$ In particular, if $\\mathcal {C}$ is a ${C}$ -linear dg category, then $\\mathrm {K}^{\\mathrm {st}}(\\mathcal {C}) \\simeq \\mathrm {KH}^{\\mathrm {st}}(\\mathcal {C}),$ where $\\mathrm {KH}^\\mathrm {st}(\\mathcal {C})=\\mathrm {Re}(\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C}))$ .",
"Since ${A}^1_{C}\\rightarrow \\operatorname{Spec}{C}$ realizes to an equivalence in $\\mathcal {S}\\mathrm {p}$ , the realization functor $\\mathrm {Re}:\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sch}_{C})\\rightarrow \\mathcal {S}\\mathrm {p}$ factors through the ${A}^1$ -localization $\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sch}_{C})\\rightarrow \\mathcal {P}_{\\mathcal {S}\\mathrm {p}}^{{A}^1}(\\mathrm {Sch}_{C})$ , which is modeled concretely by $\\mathrm {L}_{{A}^1}$ .",
"This proves that $F^\\mathrm {st}\\simeq (\\mathrm {L}_{{A}^1}F)^\\mathrm {st}$ .",
"If we prove that $(\\mathrm {L}_{{A}^1}F)^\\mathrm {st}$ is ${A}^1$ -invariant, then we will have proved that $\\mathrm {L}_{{A}^1}(F^\\mathrm {st})\\simeq F^\\mathrm {st}$ .",
"It is enough to prove that $\\mathrm {st}:\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sch}_{C})\\rightarrow \\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sch}_{C})$ preserves ${A}^1$ -invariant presheaves.",
"Let $G$ be ${A}^1$ -invariant.",
"If $f:X\\rightarrow \\operatorname{Spec}{C}$ , then $f_*f^*G\\simeq g_*g^*G$ where $g:X\\times _{C}{A}^1_{C}\\rightarrow \\operatorname{Spec}{C}$ since $G$ is ${A}^1$ -invariant.",
"Thus, $G^\\mathrm {st}(X)\\simeq \\mathrm {Re}(f_*f^*G)\\simeq \\mathrm {Re}(g_*g^*G)\\simeq G^\\mathrm {st}(X\\times _{C}{A}^1)$ , as desired.",
"The second claim follows from the equivalence $\\underline{\\smash{\\mathrm {K}}}^\\mathrm {st}(\\mathcal {C})\\simeq \\underline{\\smash{\\mathrm {KH}}}^\\mathrm {st}(\\mathcal {C})$ of presheaves evaluated at $\\operatorname{Spec}{C}$ .",
"Write $\\mathcal {S}\\mathrm {p}({C})$ for the $\\infty $ -category of motivic ${P}^1$ -spectra over ${C}$ , $\\underline{\\smash{\\mathrm {Map}}}_{\\mathcal {S}\\mathrm {p}({C})}(-,-)$ for the internal mapping object, and $\\mathrm {Map}_{\\mathcal {S}\\mathrm {p}({C})}(-,-)$ for the (classical) mapping spectrum.",
"A good reference for $\\mathcal {S}\\mathrm {p}({C})$ in the language of $\\infty $ -categories is .",
"Proposition 3.2 Let $\\mathcal {C}$ be a ${C}$ -linear dg category.",
"There is a motivic spectrum $\\mathrm {KGL}(\\mathcal {C})\\in \\mathcal {S}\\mathrm {p}({C})$ such that $\\mathrm {Map}_{\\mathcal {S}\\mathrm {p}({C})}(\\Sigma ^{\\infty }_{{P}^1}X_+,\\mathrm {KGL}(\\mathcal {C})) \\simeq \\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})(X)$ for any $X\\in \\mathrm {Sm}_{C}$ .",
"Below, we use the (nonstandard) notation $T = ({P}^1,\\infty )$ for the based scheme and for the duration of this proof write ${P}^1$ only to denote the unbased scheme.",
"Write $\\mathcal {S}\\mathrm {p}_{S^1}({C})$ for the $\\infty $ -category of ${A}^1$ -invariant, Nisnevich sheaves of spectra on $\\mathrm {Sm}_{C}$ .",
"Note that $\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})\\in \\mathcal {S}\\mathrm {p}_{S^1}({C})$ .",
"Indeed, it is ${A}^1$ -invariant by definition and it is a Nisnevich sheaf, because it is the restriction of a localizing invariant to $\\mathrm {Sm}_{C}$ (see *Theorem 1.1(c)).",
"By for example, we have an equivalence $\\mathcal {S}\\mathrm {p}({C}) \\simeq {\\rm Stab}_{T}(\\mathcal {S}\\mathrm {p}_{S^1}({C})) := \\lim (\\mathcal {S}\\mathrm {p}_{S^1}({C})\\xleftarrow{} \\mathcal {S}\\mathrm {p}_{S^1}({C})\\leftarrow \\cdots ).$ Let $\\beta \\in \\pi _{0}\\mathrm {Map}_{\\mathcal {S}\\mathrm {p}_{S^1}({C})}(T, \\underline{\\smash{\\mathrm {KH}}}(\\mathrm {Perf}_{C}))\\cong \\mathrm {KH}_0(({P}^1,\\infty ))$ be the usual Bott element.",
"Write as well $\\beta :\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C}) \\rightarrow \\mathrm {Map}_{\\mathcal {S}\\mathrm {p}_{S^1}({C})}(T,\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C}))$ for the “multiplication by $\\beta $ ” map, obtained from the $\\underline{\\smash{\\mathrm {KH}}}(\\mathrm {Perf}_{C}))$ -module structure on $\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})$ .",
"Now define $\\mathrm {KGL}(\\mathcal {C})\\in \\mathcal {S}\\mathrm {p}({C})$ to be the “constant” spectrum whose value is $\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})$ and structure maps $\\beta :\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C}) \\rightarrow \\mathrm {Map}_{\\mathcal {S}\\mathrm {p}_{S^1}({C})}(T,\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C}))$ .",
"Since $\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})$ is a localizing invariant and $\\mathrm {Perf}_{{P}^1_X}\\simeq \\mathrm {Perf}_X \\oplus \\mathrm {Perf}_X$ , the projective bundle formula holds in $\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})$ : $\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})({P}^1_X)\\simeq \\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})(X)\\oplus \\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})(X)$ for $X\\in \\mathrm {Sch}_{C}$ .",
"This splitting identifies $\\mathrm {Map}_{\\mathcal {S}\\mathrm {p}_{S^1}({C})}(T,\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C}))$ with $\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})$ via the map $\\beta $ defined above.",
"In particular, we see that $\\mathrm {KGL}(\\mathcal {C})$ is a periodic motivic spectrum and $\\Omega ^{\\infty }_{T}(\\mathrm {KGL}(\\mathcal {C}))\\simeq \\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})$ .",
"It is now immediate that $\\mathrm {Map}_{\\mathcal {S}\\mathrm {p}({C})}(\\Sigma ^{\\infty }_{T}X_+,\\mathrm {KGL}(\\mathcal {C})) \\simeq \\mathrm {Map}_{\\mathcal {S}\\mathrm {p}_{S^1}({C})}(\\Sigma ^{\\infty }_{S^1}X_+,\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})) \\simeq \\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})(X),$ for any $X\\in \\mathrm {Sm}_{{C}}$ .",
"Now, we prove Blanc's conjecture.",
"Blanc has told us that Horel was exploring a similar argument.",
"Theorem 3.3 If $\\mathcal {C}$ is a ${C}$ -linear dg category, then the natural map $\\mathrm {K}(\\mathcal {C})/n\\rightarrow \\mathrm {K}^{\\mathrm {st}}(\\mathcal {C})/n$ is an equivalence for any $n\\geqslant 1$ .",
"By *Theorem 3.18, we may compute the semi-topological $K$ -theory of $\\mathcal {C}$ using only smooth ${C}$ -schemes $\\mathrm {Sm}_{C}\\subseteq \\mathrm {Aff}_{C}$ .",
"In fact, if $F:\\mathrm {Sch}_{C}^\\mathrm {op}\\rightarrow \\mathcal {S}\\mathrm {p}$ is any presheaf of spectra, we can compute $\\mathrm {Re}(F)$ by first restricting $F$ to $\\mathrm {Sm}_{C}$ and then using the realization $\\mathrm {Re}:\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sm}_{C})\\rightarrow \\mathcal {S}\\mathrm {p}$ given as the left Kan extension ${\\mathrm {Sm}_{C}[rr]^{X\\mapsto \\Sigma ^\\infty \\operatorname{Sing}{X({C})}_+}[d] && \\mathcal {S}\\mathrm {p}\\\\\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sm}_{C})[urr]^\\mathrm {Re}.&&}$ Let $\\mathrm {E}$ denote the constant presheaf on $\\mathrm {Sm}_{C}$ with value $\\mathrm {K}(\\mathcal {C})/n$ .",
"There is a natural map $\\mathrm {E}\\rightarrow \\underline{\\smash{\\mathrm {K}}}(\\mathcal {C})/n$ .",
"The topological realization of $\\mathrm {E}$ is $\\mathrm {K}(\\mathcal {C})/n$ since it is a constant sheaf.",
"As topological realization factors through Nisnevich hypersheaves, it suffices to check that $\\mathrm {E}\\rightarrow \\underline{\\smash{\\mathrm {K}}}(\\mathcal {C})/n$ induces an equivalence after Nisnevich hypersheafification.",
"For this, it suffices to see that the natural map $\\mathrm {K}(\\mathcal {C})/n\\rightarrow \\mathrm {K}(R\\otimes _{C}\\mathcal {C})/n$ is an equivalence for every essentially smooth hensel local ring $R$ over ${C}$ .",
"By the proposition above, $\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})$ is represented by a motivic spectrum, $\\mathrm {KGL}(\\mathcal {C})$ .",
"Thus, $\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})/n$ is represented by a motivic spectrum denoted $\\mathrm {KGL}(\\mathcal {C})/n$ .",
"Gabber-Suslin rigidity is valid for $\\mathrm {KGL}(\\mathcal {C})/n$ by *Corollary 0.4.",
"(As noted in loc.",
"cit., the normalization property of that result holds for any motivic spectrum over an algebraically closed field.)",
"In particular, $\\mathrm {KH}(R\\otimes _{C}\\mathcal {C})/n\\rightarrow \\mathrm {KH}(R/\\mathfrak {m}\\otimes _{C}\\mathcal {C})/n\\simeq \\mathrm {K}(\\mathcal {C})/n$ is an equivalence for any essentially smooth hensel local ring $R$ , where $\\mathfrak {m}\\subseteq R$ is the maximal ideal.",
"But by a result of Tabuada *Theorem 1.2(i), whose proof essentially follows the argument of Weibel in the case of associative rings *Proposition 1.6, $\\underline{\\smash{\\mathrm {K}}}(\\mathcal {C})/n$ is ${A}^1$ -homotopy invariant so that $\\underline{\\smash{\\mathrm {K}}}(\\mathcal {C})/n\\simeq \\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})/n$ .",
"It follows that $\\mathrm {K}(R\\otimes _{C}\\mathcal {C})/n\\rightarrow \\mathrm {K}(R/\\mathfrak {m}\\otimes _{C}\\mathcal {C})/n\\simeq \\mathrm {K}(\\mathcal {C})/n$ is an equivalence so that $\\mathrm {E}\\rightarrow \\underline{\\smash{\\mathrm {K}}}(\\mathcal {C})/n$ is an equivalence, which is what we wanted to prove."
],
[
"Descent for topological $K$ -theory of dg catgories",
"In this section we prove the following result.",
"Theorem 4.1 Let $\\mathcal {C}$ be a ${C}$ -linear dg category.",
"The presheaf $\\underline{\\smash{\\mathrm {K}}}^{\\mathrm {top}}(\\mathcal {C}):\\mathrm {Sm}_{C}^\\mathrm {op}\\rightarrow \\mathcal {S}\\mathrm {p}$ satisfies étale hyperdescent.",
"The presheaf $\\underline{\\smash{\\mathrm {K}}}(\\mathcal {C})/n[\\beta ^{-1}]:\\mathrm {Sm}_{C}^\\mathrm {op}\\rightarrow \\mathcal {S}\\mathrm {p}$ satisfies étale hyperdescent.",
"Part (ii) of the theorem is a noncommutative generalization of the main theorem of Thomason .",
"Indeed, if $\\mathcal {C}\\simeq \\mathrm {Perf}_X$ where $X$ is an essentially smooth separated ${C}$ -scheme, then $\\underline{\\smash{\\mathrm {K}}}(\\mathrm {Perf}_X)/n[\\beta ^{-1}]$ is equivalent to the presheaf $Y\\mapsto \\mathrm {K}(Y\\times _{C}X)/n[\\beta ^{-1}],$ which satisfies étale hyperdescent by *Theorem 4.1.",
"In general, we cannot improve the result to semi-topological $K$ -theory.",
"Indeed, it is well-known that $\\underline{\\smash{\\mathrm {K}}}^\\mathrm {st}(\\mathrm {Perf}_X)/n\\simeq \\underline{\\smash{\\mathrm {K}}}(\\mathrm {Perf}_X)/n$ does not satisfy étale descent.",
"The Quillen–Lichtenbaum conjectures (which follow from the, now proved, Bloch–Kato conjecture) give a bound on the failure of étale hyperdescent for $K$ -theory with finite coefficients.",
"For example, if $X$ is an essentially smooth separated ${C}$ -scheme of Krull dimension $d$ , then $\\mathrm {K}(X)/\\ell \\rightarrow \\mathrm {K}(X)/\\ell [\\beta ^{-1}]$ is $2d$ -coconnective.",
"(See *Section 5 for a discussion of the bound $2d$ .)",
"Recall that a map of spectra $M\\rightarrow N$ is $r$ -coconnected if the induced map $\\pi _rM\\rightarrow \\pi _rN$ is an injection and $\\pi _sM\\rightarrow \\pi _sN$ is an isomorphism for $s>r$ .",
"Following the tradition of proposing noncommutative versions of theorems known for ${C}$ -linear dg categories of the form $\\mathrm {Perf}_X$ , we ask the following question.",
"Question 4.2 (Noncommutative Quillen-Lichtenbaum) If $\\mathcal {C}$ is a nice (probably smooth and proper) ${C}$ -linear dg category, is $\\mathrm {K}(\\mathcal {C})/n\\rightarrow \\mathrm {K}^{\\mathrm {top}}(\\mathcal {C})/n$ is $r$ -coconnective for some $r$ .",
"To prove Theorem REF , we make use of the topological realization functor ${\\rm Re}:\\mathcal {S}\\mathrm {p}({C}) \\rightarrow \\mathcal {S}\\mathrm {p}$ , extending the functor of taking complex points of a ${C}$ -scheme.",
"This functor factors through the localization $\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sm}_{C})\\rightarrow \\operatorname{Shv}_{\\mathcal {S}\\mathrm {p}}^{\\mathrm {Nis},{A}^1}(\\mathrm {Sm}_{C})$ , which is equivalent to $\\mathcal {S}\\mathrm {p}_{S^1}({C})$ , the category of motivic $S^1$ -spectra.",
"To see that it factors through the $T$ -stabilization functor $\\mathcal {S}\\mathrm {p}_{S^1}({C})\\rightarrow \\mathcal {S}\\mathrm {p}({C})$ , it is enough to note that the realization of $T$ is $S^2$ , which is already tensor invertible in $\\mathcal {S}\\mathrm {p}$ .",
"Thus, we have a commutative diagram ${\\mathrm {Sm}_{C}[r]^{X\\mapsto \\Sigma ^\\infty _+X({C})}[d] & \\mathcal {S}\\mathrm {p}&\\\\\\mathcal {P}_{\\mathcal {S}\\mathrm {p}}(\\mathrm {Sm}_{C})@{.>}[ur]^{\\rm Re}[r] &\\mathcal {S}\\mathrm {p}_{S^1}({C})@{.>}[u]^{\\rm Re}[r]&\\mathcal {S}\\mathrm {p}({C})@{.>}[ul]_{\\rm Re}}$ of realization functors, and we will abuse notation by not distinguishing them.",
"Lemma 4.3 Let $\\mathcal {C}$ be a ${C}$ -linear dg category.",
"Then, there is an equivalence ${\\rm Re}(\\mathrm {KGL}(\\mathcal {C}))\\simeq \\mathrm {K}^\\mathrm {top}(\\mathcal {C})$ .",
"By definition, $\\mathrm {KGL}(\\mathcal {C})\\in \\lim (\\mathcal {S}\\mathrm {p}_{S^1}({C}) \\xleftarrow{} \\mathcal {S}\\mathrm {p}_{S^1}({C})\\leftarrow \\cdots )$ is the periodic $T$ -spectrum with value $\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})$ and structure maps induced by $\\beta $ : $\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})\\xrightarrow{}\\Omega _T\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})\\xrightarrow{}\\Omega _T^2\\underline{\\smash{\\mathrm {KH}}}(\\mathcal {C})\\xrightarrow{}\\cdots .$ The realization functor $\\mathrm {Re}:\\mathcal {S}\\mathrm {p}({C})\\rightarrow \\mathcal {S}\\mathrm {p}$ factors through the equivalence $\\mathcal {S}\\mathrm {p}_{S^2}\\mathcal {S}\\mathrm {p}\\simeq \\mathcal {S}\\mathrm {p}$ , where $\\mathcal {S}\\mathrm {p}_{S^2}\\mathcal {S}\\mathrm {p}$ is the $\\infty $ -category of $S^2$ -spectra in spectra.",
"The realization functor sends $\\mathrm {KGL}(\\mathcal {C})$ to the $S^2$ -spectrum $\\mathrm {K}^{\\mathrm {st}}(\\mathcal {C})\\xrightarrow{}\\Omega ^2\\mathrm {K}^{\\mathrm {st}}(\\mathcal {C})\\xrightarrow{}\\Omega ^4\\mathrm {K}^{\\mathrm {st}}\\xrightarrow{}\\cdots .$ The underlying spectrum of this $S^2$ -spectrum in spectra is “$\\Omega ^{2\\infty }$ ”, the colimit of the diagram, which is by definition $\\mathrm {K}^{\\mathrm {top}}(\\mathcal {C})$ .",
"Lemma 4.4 If $X\\in \\mathrm {Sm}_{{C}}$ , then ${\\rm Re}(\\underline{\\smash{\\mathrm {Map}}}_{\\mathcal {S}\\mathrm {p}({C})}(\\Sigma ^{\\infty }_{\\mathrm {P}^1}X_+, \\mathrm {KGL}(\\mathcal {C}))) \\simeq \\underline{\\smash{\\mathrm {K}}}^{\\mathrm {top}}(\\mathcal {C})(X)$ .",
"By adjunction, $\\underline{\\smash{\\mathrm {Map}}}_{\\mathcal {S}\\mathrm {p}({C})}(\\Sigma ^\\infty _{{P}^1}X_+,\\mathrm {KGL}(\\mathcal {C}))\\simeq \\mathrm {KGL}(\\mathrm {Perf}_X\\otimes _{C}\\mathcal {C}).$ The claim follows from the fact that ${\\rm Re}(\\mathrm {KGL}(\\mathrm {Perf}_X\\otimes _{C}\\mathcal {C}))\\simeq \\mathrm {K}^\\mathrm {top}(\\mathrm {Perf}_X\\otimes _{C}\\mathcal {C})\\simeq \\underline{\\smash{\\mathrm {K}}}^\\mathrm {top}(\\mathcal {C})(X)$ .",
"Lemma 4.5 If $X\\in \\mathrm {Sm}_{{C}}$ , then $\\underline{\\smash{\\mathrm {K}}}^{\\mathrm {top}}(\\mathcal {C})(X)\\simeq \\mathrm {Map}_{\\mathcal {S}\\mathrm {p}}(\\Sigma ^{\\infty }X({C})_+, \\mathrm {K}^{\\mathrm {top}}(\\mathcal {C}))$ .",
"Because the functor ${\\rm Re}$ is symmetric monoidal, it commutes with internal mapping objects.",
"Since $\\Sigma ^{\\infty }_{{P}^1}X_+$ is dualizable in $\\mathcal {S}\\mathrm {p}({C})$ , the statement of the lemma follows from the previous lemma.",
"[Proof of Theorem REF ] It follows from the equivalence $\\underline{\\smash{\\mathrm {K}}}(\\mathcal {C})/n\\simeq \\underline{\\smash{\\mathrm {K}}}^{\\mathrm {st}}(\\mathcal {C})/n$ of Theorem REF that the second part follows from the first part.",
"On the other hand, Lemma REF shows that $\\underline{\\smash{\\mathrm {K}}}^{\\mathrm {top}}(\\mathcal {C})$ is the restriction of the cohomology theory on spaces represented by $\\mathrm {K}^{\\mathrm {top}}(\\mathcal {C})$ to $\\mathrm {Sch}_{C}$ .",
"It follows that it satisfies étale hyperdescent since any cohomology theory does (see for example *Theorem 5.2).",
"tocsectionReferences blancarticle author=Blanc, Anthony, title=Topological K-theory of complex noncommutative spaces, journal=Compos.",
"Math., volume=152, date=2016, number=3, pages=489–555, issn=0010-437X, bgt1article author=Blumberg, Andrew J., author=Gepner, David, author=Tabuada, Gonçalo, title=A universal characterization of higher algebraic $K$ -theory, journal=Geom.",
"Topol., volume=17, date=2013, number=2, pages=733–838, issn=1465-3060, cohnarticle author = Cohn, Lee, title = Differential Graded Categories are k-linear Stable Infinity Categories, journal = ArXiv e-prints, eprint = http://arxiv.org/abs/1308.2587, year = 2013, dugger-isaksenarticle author=Dugger, Daniel, author=Isaksen, Daniel C., title=Topological hypercovers and $A^1$ -realizations, journal=Math.",
"Z., volume=246, date=2004, number=4, pages=667–689, issn=0025-5874, friedlander-walker-comparingarticle author=Friedlander, Eric M., author=Walker, Mark E., title=Comparing $K$ -theories for complex varieties, journal=Amer.",
"J.",
"Math., volume=123, date=2001, number=5, pages=779–810, issn=0002-9327, friedlander-walkerarticle author=Friedlander, Eric M., author=Walker, Mark E., title=Semi-topological $K$ -theory using function complexes, journal=Topology, volume=41, date=2002, number=3, pages=591–644, issn=0040-9383, friedlander-walker-handbookarticle author=Friedlander, Eric M., author=Walker, Mark E., title=Semi-topological $K$ -theory, conference= title=Handbook of $K$ -theory.",
"Vol.",
"1, 2, , book= publisher=Springer, Berlin, , date=2005, pages=877–924, hornbostel-yagunovarticle author=Hornbostel, Jens, author=Yagunov, Serge, title=Rigidity for Henselian local rings and $\\mathbb {A}^1$ -representable theories, journal=Math.",
"Z., volume=255, date=2007, number=2, pages=437–449, issn=0025-5874, jouanolouarticle author=Jouanolou, J. P., title=Une suite exacte de Mayer-Vietoris en $K$ -théorie algébrique, conference= title=Algebraic $K$ -theory, I: Higher $K$ -theories (Proc.",
"Conf., Battelle Memorial Inst., Seattle, Wash., 1972), , book= publisher=Springer, place=Berlin, , date=1973, pages=293–316.",
"Lecture Notes in Math., Vol.",
"341, httbook author=Lurie, Jacob, title=Higher topos theory, series=Annals of Mathematics Studies, publisher=Princeton University Press, address=Princeton, NJ, date=2009, volume=170, ISBN=978-0-691-14049-0; 0-691-14049-9, morel-voevodskyarticle author=Morel, Fabien, author=Voevodsky, Vladimir, title=${\\bf A}^1$ -homotopy theory of schemes, journal=Inst.",
"Hautes Études Sci.",
"Publ.",
"Math., number=90, date=1999, pages=45–143 (2001), issn=0073-8301, robaloarticle author=Robalo, Marco, title=$K$ -theory and the bridge from motives to noncommutative motives, journal=Adv.",
"Math., volume=269, date=2015, pages=399–550, issn=0001-8708, tabuada-kharticle author=Tabuada, Gonçalo, title=${A}^1$ -homotopy theory of noncommutative motives, journal=J.",
"Noncommut.",
"Geom., volume=9, date=2015, number=3, pages=851–875, issn=1661-6952, tabuada-a1-modlarticle author=Tabuada, Gonçalo, title=${A}^1$ -homotopy invariance of algebraic $K$ -theory with coefficients and du Val singularities, journal=Ann.",
"K-Theory, volume=2, date=2017, number=1, pages=1–25, issn=2379-1683, thomasonarticle author=Thomason, R. W., title=Algebraic $K$ -theory and étale cohomology, journal=Ann.",
"Sci.",
"École Norm.",
"Sup.",
"(4), volume=18, date=1985, number=3, pages=437–552, issn=0012-9593, review=826102, thomason-bottarticle author=Thomason, R. W., title=Bott stability in algebraic $K$ -theory, conference= title=Applications of algebraic $K$ -theory to algebraic geometry and number theory, Part I, II, address=Boulder, Colo., date=1983, , book= series=Contemp.",
"Math., volume=55, publisher=Amer.",
"Math.",
"Soc., Providence, RI, , date=1986, pages=389–406, weibel-kharticle author=Weibel, Charles A., title=Homotopy algebraic $K$ -theory, conference= title=Algebraic $K$ -theory and algebraic number theory (Honolulu, HI, 1987), , book= series=Contemp.",
"Math., volume=83, publisher=Amer.",
"Math.",
"Soc., Providence, RI, , date=1989, pages=461–488,"
]
] | 1709.01587 |
[
[
"Formation of rogue waves from the locally perturbed condensate"
],
[
"Abstract The one-dimensional focusing nonlinear Schrodinger equation (NLSE) on an unstable condensate background is the fundamental physical model, that can be applied to study the development of modulation instability (MI) and formation of rogue waves.",
"The complete integrability of the NLSE via inverse scattering transform enables the decomposition of the initial conditions into elementary nonlinear modes: breathers and continuous spectrum waves.",
"The small localized condensate perturbations (SLCP) that grow as a result of MI have been of fundamental interest in nonlinear physics for many years.",
"Here, we demonstrate that Kuznetsov-Ma and superregular NLSE breathers play the key role in the dynamics of a wide class of SLCP.",
"During the nonlinear stage of MI development, collisions of these breathers lead to the formation of rogue waves.",
"We present new scenarios of rogue wave formation for randomly distributed breathers as well as for artificially prepared initial conditions.",
"For the latter case, we present an analytical description based on the exact expressions found for the space-phase shifts that breathers acquire after collisions with each other.",
"Finally, the presence of Kuznetsov-Ma and superregular breathers in arbitrary-type condensate perturbations is demonstrated by solving the Zakharov-Shabat eigenvalue problem with high numerical accuracy."
],
[
"Uniformization of spectral parameter $\\lambda $",
"In our previous works [15,22,23,37] we use Joukowsky transform (uniformization) of standard spectral parameter $\\lambda $ to simplify analysis of $N$ -breather solutions: $\\lambda = -\\frac{iA}{2} \\biggl (\\xi +\\frac{1}{\\xi }\\biggr ), \\quad \\quad \\quad \\xi = re^{I\\alpha } = e^{z + I\\alpha }\\,.$ In such variables the two-sheeted Riemann $\\lambda $ -surface transforms into the one-sheeted uniformized $\\xi $ -plane.",
"The line of the branchcut $[-iA,iA]$ transforms into a circle of unit radius, while the Riemann sheets become the outer and inner pars of the circle.",
"The purpose of this supplemental paragraph is to illustrate the relation between representations of the breather solutions in $\\lambda $ and $\\xi $ variables.",
"In the work [22] we found, that asymptotics of the general one-breather solution (Eq.",
"(6) with $\\theta _c=0$ ) can be simply expressed using uniformized variables (REF ): $\\psi _1 \\rightarrow A \\exp (\\pm 2i\\alpha )\\,, \\quad \\quad \\quad |x|\\rightarrow \\infty \\,.$ The latter means that single breather changes the phase of the condensate before and after itself on $4\\alpha $ .",
"Now the requirements for breathers which can generate SLCP can be easily obtained.",
"Indeed, the condensate complex phase at infinity should be constant with time for any localized perturbation.",
"Thus, the breather solutions capable to describe the growth of SLCP should have equal condensate phases at $|x|\\rightarrow \\infty $ , or at least have difference in the phases comparable with the relative amplitude of the initial perturbation.",
"Among one-breather solutions of the NLSE, only Kuznetsov-Ma and Peregrine breathers have equal condensate phases at infinity since they correspond to the case $\\alpha = 0$ .",
"For a couple of breathers, the total change of phase before and after them is zero when $\\alpha _1 = -\\alpha _{2} = \\alpha \\,.$ As was found in the work [22], at certain phase synchronisation (close to $\\theta _1+\\theta _2=\\pi $ ) and when the breather eigenvalues located near the branchcut ($r-1=\\tilde{\\varepsilon }\\ll 1$ ), such two-breather (superregular) solution generate SLCP at an initial moment of time.",
"For symmetric one-pair superregular solution (Eq.",
"(7)) the initial (linear) stage of perturbation growth is described by the following expression obtained in [23]: $\\delta \\psi \\approx I\\tilde{\\varepsilon } A\\frac{\\cosh (I\\alpha -2\\gamma t)\\cos (2\\eta x)}{\\cosh ((2A\\tilde{\\varepsilon }\\cos \\alpha )x)},\\\\\\nonumber \\eta =A\\cosh z\\sin \\alpha , \\quad \\quad \\gamma = -A^2\\cosh 2z\\sin 2\\alpha /2.$ Here $\\delta \\psi $ is the SLCP: $\\psi = A + \\delta \\psi $ .",
"The maximum value of the growth rate (defined by $2\\gamma $ ) for the expression (REF ) is achieved at $\\alpha = \\pi /4$ .",
"In this case superregular breather corresponds to the maximum of the MI growth rate (2).",
"Indeed, when $\\alpha = \\pi /4$ ($Im[\\lambda ] = 1/\\sqrt{2}$ ), maximum of the most significant Fourier spectrum sideband mode is located at $k_{max} = \\pm \\sqrt{2}$ (with accuracy $\\sim \\tilde{\\varepsilon }$ ) – see the Fig.1.",
"Note, that the similar result for the Akhmediev breather is well known – see the reference [43].",
"The relation (REF ) defines the family of the following parametric curves $\\mathrm {Re}[\\lambda ] = \\pm \\frac{A\\sin \\alpha }{2}\\biggl (r - \\frac{1}{r}\\biggr )\\,,\\,\\,\\mathrm {Im}[\\lambda ] = \\frac{A\\cos \\alpha }{2}\\biggl (r + \\frac{1}{r}\\biggr )\\,,$ that is illustrated here in Fig.",
"REF .",
"The SLCP can be generated only by eigenvalues located on the grey (long dashed) and orange (short dashed) lines since the distance between the eigenvalues and the branchcut defines the amplitude of the perturbation.",
"Figure: Comparison of ξ\\xi and λ\\lambda parametrizations of spectral parameter.",
"The branch cut and its Joukowsky mapping () are drawn by black solid lines.",
"The pairs of breather eigenvalues (marked by blue points) lie on the rays () (dashed lines, left picture) and on the parametric curves () (dashed lines, right picture)."
],
[
"Breather collisions sinchronization",
"Then we present details of synchronization of breather SLCP development presented by Fig.",
"3 and Fig. 4.",
"First we introduce the breather group and phase velocities, that can be obtained from expressions (5): $V_{gr} = - \\frac{\\mathrm {Re}[\\lambda _n \\cdot \\zeta _n]}{\\mathrm {Re}[\\zeta _n]}\\,, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, V_{ph} = \\mathrm {Re}[\\lambda _n \\cdot \\zeta _n]\\,.$ For the four-breather scenario presented in Fig.",
"3a, we choose the initial moment of time $t_0=0$ and the moment of rogue wave formation $t_{max} = 15$ (with maximum amplitude at $x=0$ ).",
"The problem is to find initial space shifts and phases for each breather.",
"Let us denote breathers in the left superregular pair (see Fig.",
"3a) by indexes 1 (the breather moving to the left) and 2 (the breather moving to the right) and breathers in the right pair by indexes 3 (the breather moving to the left) and 4 (the breather moving to the right).",
"The breathers 2 and 3 have complex conjugated eigenvalues, so they differ only in the direction of the group velocity: $V_{gr_2} = - V_{gr_3} = \\widetilde{V}_{gr}$ .",
"Before the moment $t_0$ , the breather 2 was affected by collisions with the breathers 1 and 4, so the total space shift for the breather 2 can be found as: $x_{0,2} = -\\widetilde{V}_{gr} \\cdot t_{max} - \\Delta x_{0,12} + \\Delta x_{0,42}\\,.$ Here $\\Delta x_{0,12}$ and $ \\Delta x_{0,42}$ are calculated via space-shift formula (9).",
"To obtain further synchronisation inside the left superregular pair, we choose the same space shift for the breather 1: $x_{0,1}=x_{0,2}$ .",
"By analogy we obtain the following space shifts for the breathers 3 and 4: $x_{0,3} = x_{0,4} = \\widetilde{V}_{gr} \\cdot t_{max} - \\Delta x_{0,13} + \\Delta x_{0,43}\\,.$ Synchronization of phases we start from the breathers 2 and 3.",
"According to the Fig.",
"2 the phases of both breathers should be equal to $\\pi $ , but we have to account the phase changing with time $t_{max}$ and phase shifts acquired after collisions with breathers 1 and 4.",
"Denoting the phase velocity for the second and third breather as $V_{ph_2} = V_{ph_3} = \\widetilde{V}_{ph}$ we obtain: $\\theta _2 = \\theta _3 = \\pi + \\widetilde{V}_{ph}\\cdot t_{max} - \\Delta \\theta _{12} + \\Delta \\theta _{42}\\,.$ Here $\\Delta \\theta _{12}$ and $ \\Delta \\theta _{42}$ are calculated via phase-shift formula (8).",
"Now the formation of the rogue wave at $t_{max}$ is obtained and we only need to synchronise formation of small condensate superregular perturbations at $t_0$ .",
"As we discussed in the main text of the paper, superregular synchronization appears at $\\theta _1 + \\theta _2 = \\pi $ (see again the Fig. 2).",
"The breather 1 was affected by collisions with the breathers 3 and 4, while the breather 4 collided with breathers 3 and 1.",
"Thus, the required phase shifts can be found as: $\\theta _1 = \\pi - \\theta _2 - \\Delta \\theta _{13} - \\Delta \\theta _{14}\\,,\\\\\\nonumber \\theta _4 = \\pi - \\theta _3 + \\Delta \\theta _{43} - \\Delta \\theta _{14}\\,.$ Now we discuss synchronization of Kuznetsov-Ma breather scenario of SLCP development, presented by Fig. 4c,d.",
"Our task is to find phases $\\theta _m$ for all $M$ breathers which correspond to minimum amplitude at $t=0$ .",
"A single Kuznetsov-Ma breather has minimum amplitude at $t=0$ when its phase $\\theta $ is equal to zero (like in the Fig. 1a).",
"Thus, for each $m$ -th Kuznetsov-Ma breather we should set zero phase and add phase correction from each of the neighbouring breathers: $\\theta _m = \\sum _{j=1,j\\ne m}^{M} \\Delta \\theta _{mj}$ Where $\\Delta \\theta _{mj}$ is calculated using formulas (8).",
"The combined scenario for superregular and Kuznetsov-Ma breathers presented in Fig.",
"3b can be obtained in similar to the discussed above way, thus we skip its details here."
],
[
"Fourier spectra for SLCP development",
"Here in Fig.",
"REF , REF , REF we present Fourier spectra for all studied in the main text of the paper, scenarios of SLCP development.",
"In general, they are in accordance with Fourier spectra of their elementary building blocks: one-breather Kuznetsov-Ma solution and one-pair superregular breather solution, which are presented in Fig. 1.",
"Namely, the development of SLCP driven by breathers leads to the broadens of the Fourier spectrum.",
"Interactions of several breathers make the Fourier spectrum more complicated.",
"In all cases we can see that superregular breathers with $\\mu \\approx 1/\\sqrt{2}$ are responsible for the fastest perturbation growth.",
"Figure: Fourier spectra for different two-breather collisions, presented in Fig.",
"2.Figure: Fourier spectra for analytical scenarios of rogue wave formation from the condensate locally perturbed by superregular (left) and one Kuznetsov-Ma and one superregular (right) breathers presented in Fig.",
"3.Figure: Fourier spectra for scenarios of rogue wave formation from the condensate locally perturbed by random distributions of superregular (left) and Kuznetsov-Ma (right) breathers presented in Fig.",
"4."
],
[
"Acknowledgments",
"The author is thankful to 1) Prof. V.E.",
"Zakharov for providing the idea to study imaginary and real condensate perturbations separately; 2) Prof. E.A.",
"Kuznetsov for the helpful discussions of continuous spectrum waves; 3) Dr. D.S.",
"Agafontsev for the helpful comments with regard to the Fourier spectrum of superregular breathers and discussion of possible applications of the presented results in the interpretation of numerical simulations with periodically perturbed condensate; and 4) Dr. B. Kibler for the helpful discussions of the broadband random noise SLCP.",
"Numerical code and simulations were performed at the Novosibirsk Supercomputer Center (NSU).",
"The first part of the work was performed with support from the Russian Science Foundation (Grant No.",
"14-22-00174).",
"The study presented in the last part \"Randomly perturbed condensate\" was supported by the RFBR (Grants No.",
"16-31-60086 mol_a_dk and 17-41-543347 r_mol_a)."
]
] | 1709.01913 |
[
[
"Eshelbian dislocation mechanics: $J$-, $M$-, and $L$-integrals of\n straight dislocations"
],
[
"Abstract In this work, using the framework of (three-dimensional) Eshelbian dislocation mechanics, we derive the $J$-, $M$-, and $L$-integrals of a single (edge and screw) dislocation in isotropic elasticity as a limit of the $J$-, $M$-, and $L$-integrals between two straight dislocations as they have recently been derived by Agiasofitou and Lazar [Int.",
"J. Eng.",
"Sci.",
"114 (2017) 16-40].",
"Special attention is focused on the $M$-integral.",
"The $M$-integral of a single dislocation in anisotropic elasticity is also derived.",
"The obtained results reveal the physical interpretation of the $M$-integral (per unit length) of a single dislocation as the total energy of the dislocation which is the sum of the self-energy (per unit length) of the dislocation and the dislocation core energy (per unit length).",
"The latter can be identified with the work produced by the Peach-Koehler force.",
"It is shown that the dislocation core energy (per unit length) is twice the corresponding pre-logarithmic energy factor.",
"This result is valid in isotropic as well as in anisotropic elasticity.",
"The only difference lies on the pre-logarithmic energy factor which is more complex in anisotropic elasticity due to the anisotropic energy coefficient tensor which captures the anisotropy of the material."
],
[
"=1 [pages=1-last]LA-MRC-2017-09.pdf"
]
] | 1709.01836 |
[
[
"Probing Space-time Distortion with Laser Wake Field Acceleration and\n X-ray Free Electron Lasers"
],
[
"Abstract Petawatt class femtosecond lasers and x-ray free electron lasers (XFEL) open up a new page in research fields related to space and vacuum physics.",
"One of fundamental principles can be explored by these new instruments is the equivalence principle, saying that gravitation and acceleration should be treated equivalently.",
"If it is true this must lead to the appearance of the Unruh effect analogous to Hawking's black hole radiation.",
"It says that a detector moving with a constant acceleration $w$ sees a boson's thermal bath with its temperature $T_w=\\hbar w/2\\pi k_B c$.",
"Practical detection of the Unruh effect requires extremely strong acceleration and fast probing sources.",
"Here we demonstrate that x-rays scattering from highly accelerated electrons can be used for such detection.",
"We present two, feasible for the Unruh effect, designs for highly accelerated systems produced in underdense plasma irradiated by high intensity lasers pulses.",
"The first is Thomson scattering of a XFEL pulse from plasma wave excited by an intense laser pulse.",
"Properly chosen observation angles enable us to distinguish the Unruh effect from the normal Doppler shift with a reasonable photon number.",
"The second is a system consisting of electron bunches accelerated by a laser wake-field, as sources and as detectors, which move in a constantly accelerating reference frame (Rindler space) and are probed by x-ray free electron laser pulses.",
"The numbers of photons is shown to be enough to observe reproducible results."
],
[
"Introduction",
"The existence of the Unruh radiation remains a fundamental question in quantum electrodynamics[1].",
"Whether kinematic acceleration, in some sense, is equivalent to the gravitational field with the consequences of this principle and feasibility of Rindler space[2] have yet to be verified experimentally.",
"According to Unruh[1] in a reference frame moving with a constant acceleration $w$ , a detector should experience surrounding vacuum as a boson’s heat bath with its temperature $T_w=\\hbar w/2\\pi k_B c$[1].",
"At the first sight the Unruh effect has no practical meaning: to get a boson gas with the temperature, for example, of the Relic radiation $T_R=2.7 \\ \\ K$[3] the acceleration must exceed $w \\approx 10^{20} g$ , where $g$ is the Earth's gravitational acceleration.",
"However, studying the highly accelerated system in the reality may allow us to understand better hidden properties of vacuum such as meaning of vacuum states, radiation transition times, and so on.",
"Exploring the Unruh hypotheses is one of the important ways to probe the vacuum.",
"Since the Unruh theory has yet to be verified there are several standpoints on it: positive[4] and negative[5].",
"The positive standpoints state that the Unruh effect should exist because the temperature predicted by the theory coincides with that from the Hawking’s theory of radiation of black holes[6].",
"The negative standpoints say that the Rindler space is not congruent physically with Minkowski space and the Unruh effect should not exist[5].",
"Indeed, a non-inertial reference system should be carefully checked on its feasibility to physical processes.",
"For example a reference frame rotating with frequency $\\Omega $ is not feasible for distance $R>c/\\Omega $[7].",
"A reference frame moving with constant acceleration $w$ has also some paradoxes rising the question about its feasibility: (i) an electron moving with constant acceleration radiates while its radiation friction force is zero, the energy conservation requires existence of beginning and an end of the acceleration process[8], (ii) apparently there is no conservative system for several particles in a reference frame moves with constant acceleration $w$ ; (iii) it feels as if there is no fixed photon mode $({\\bf k}, \\omega )$ in this reference frame due to Doppler upshift or downshift.",
"Nevertheless the answers on the feasibility can be found only experimentally.",
"Radiation in a highly accelerated system has another fundamental question which appears recently in the quantum approach for the radiation damping force during interaction of energetic electrons with high power laser pulses[9].",
"For example, Compton scattering in strong laser fields runs mainly as $n\\omega _L+e^- > \\omega _{x-ray}+e^-$ where n is the number of a harmonic, $\\omega _L$ is the laser frequency and $\\omega _{x-ray}$ is the frequency of a scattered photon.",
"This scattering has been considered in the classical approach[9] and in the relativistic quantum approaches[10] with one of results as $n \\approx a_0$ if $a_0 \\gg 1$ , where $a_0$ is normalized vector potential.",
"Unfortunately, so far the QED approach is not available for the problem: the laser field is always considered as the classical field.",
"Therefore, the time of Compton scattering is not known.",
"The duration of scattering cannot be found in the frame of present approaches.",
"This question is critical in numerical investigations of plasma interaction with high power laser pulses.",
"Exploring of Compton scattering in a reference frame moving with constant acceleration could provide us with the answer if the detector velocities in absorption time and emission time were essentially different.",
"Here we propose original designs for systems consisting of highly accelerated electrons which move in wake field of PW-class laser pulses and can be considered as objects in a constantly accelerating reference frame (Rindler space).",
"Femtosecond XFEL pulses are proposed to probe such accelerated systems at extremely high acceleration $w \\approx 2 \\pi c^2 a_0/\\lambda $ at a high temporal resolution.",
"Rindler space[2] is a natural reference frame to investigate a system moving with constant acceleration $w$ for example in $x$ direction.",
"Coordinates in this reference frame link to those in Minkowski space as $ ct=(c^2/w+x^\\prime ) \\sinh (wt^\\prime /c)$ , $ x =(c^2/w+x^\\prime ) \\cosh (wt^\\prime /c)$ with the interval $ds^2=(1+wx^\\prime /c^2 )^2 d(ct^\\prime )^2-({dx^\\prime }^2+dy^2+dz^2) = \\rho ^2 d\\eta ^2 - (d\\rho ^2 + dy^2 + dz^2)$ [$g^{00} = \\rho ^{-2}, g^{ii} = -1, \\sqrt{-g} = \\rho $ ].",
"Unruh used a different form for the metric in [1]: $ds^2 = \\rho d\\eta ^2 - (d\\rho ^2/\\rho + dy^2 + dz^2)$ , $g^{00} = 1/\\rho $ , $g^{11} = -\\rho $ , $\\sqrt{-g} = 1$ .",
"Klein-Gordon equation in a curved space has a following form: $\\sqrt{-1/g} \\ \\ \\partial /{\\partial x^\\mu } \\left[g^{\\mu \\nu } \\sqrt{-g} \\ \\ \\partial / \\partial x^\\nu \\right]\\psi =(m^2 c^2/\\hbar ^2) \\psi $ .",
"[For massless particles, this equation does not contain the Plank constant.]",
"For example, in Rindler coordinates Klein-Gordon equation for a massless particle coincides with an equation for potentials of electro-magnetic field and has a following form: $[\\partial ^2 /\\partial \\eta ^2 - \\rho (\\partial / \\partial \\rho ) \\rho (\\partial / \\partial \\rho ) - \\partial ^2 / \\partial y^2 - \\partial ^2 / \\partial z^2 ] \\psi = 0 $ .",
"One can easily check that a plane wave solution of Klein-Gordon equation in Minkowski space $\\psi \\sim e^{i\\omega (t - x/c)} = e^{i \\rho [\\cosh (\\eta )-\\sinh (\\eta )]}$ obeys Klein-Gordon equation in Rindler space.",
"However, in Rindler space there are other solutions of KG equation in the form of Fulling modes: $\\psi _{FR} \\sim g(\\rho )e^{i(\\varpi \\eta - k_y y-k_z z)}$ where $g(\\rho )$ obeys following equation[11] $\\left[\\rho \\frac{\\partial }{\\partial \\rho } \\rho \\frac{\\partial }{\\partial \\rho } + \\varpi ^2 -{k_y}^2 - {k_z}^2 - \\tilde{m}^2 \\right] g = 0 ,$ These modes can be used for the second quantization in Rindler space similar to that with the plane waves in Minkowski space.",
"It is the reasonable assumption that a detector moving with a constant acceleration should see vacuum as a set of Fulling modes.",
"This is the basis of the Unruh effect.",
"The same results occur while considering Maxwell's equations and Dirac equations.",
"According to Unruh for an observer with constant acceleration, the vacuum state is changed and the detector observes the boson distribution of Rindler particles $\\hat{\\rho } = \\rho _0 e^{-2\\pi \\hat{K}} \\equiv \\rho _0 e^{- \\hat{K}_w/k_B T_w } $ in the vacuum.",
"Where $\\rho _0 = \\prod _j[(1-e^{-2 \\pi \\varpi _j } )]$ , $\\hat{K}_w = \\hbar w \\hat{K} ̂$ , $ \\varpi _j$ is frequency of particle and $\\hat{K}$ is the energy of massless scalar particles.",
"The temperature is called the Unruh temperature given by $T_w=\\hbar w/2\\pi k_B c$ .",
"There are papers exploring the effect[12].",
"Unruh and Wald have shown that the particle bathed in boson's thermal bath is excited absorbing Rindler particles and really emits photons[13].",
"This is called the Unruh radiation.",
"Letaw and Pfautsch considered the excitation of the particle moving circularly with constant velocity[14].",
"Chen and Tajima have shown that the Unruh radiation from electron accelerated with extremely high gradient by the static wave produced by intense laser could be observed[15].",
"Chen and Mourou have recently proposed an interesting application for relativistic plasma mirror stopped abruptly by impinging intense x-ray pulses[16].",
"Also, B. J.",
"B. Crowley et al.",
"showed that the spectral broadening due to acceleration of X-ray Thomson scattering from electrons in plasma accelerated by intense laser could detect the Unruh radiation[17].",
"According to Zel'dovich[18] an electron moving with acceleration $w$ crosses a photon bath with its energy density $W \\sim T^4 \\sim w^4$ .",
"This electron should scatter such photons with the known Thomson cross section $\\sigma = 10 \\alpha ^2/m^2$ [$\\alpha $ is the fine-structure constant] with total power $I \\sim 10 \\alpha ^2 w^4/m^2$ which may exceed the classical radiation of the electron $I \\sim 2 \\alpha w^2/3$ when acceleration $w$ exceeds $\\sqrt{m^2/15 \\alpha }$ .",
"Besides, the electron may annihilate with a positron from the electron-positron bath emitting a characteristic radiation.",
"Ginzburg and Frolov[8] have found an analogy between the Unruh effect and the anomalous Doppler effect.",
"Besides, there are problems related to the Unruh radiation.",
"First of all, the Unruh effect of all particles except photons is purely quantum effect.",
"However, there is no definition of acceleration $w$ in quantum mechanics as discussed by Narozhnii et al.",
"[5] According to Narozhnii et al., there is no conformity of Rindler space-time to Minkowski space-time[5], [11].",
"In a certain boundary condition, the Unruh effect therefore cannot exist[5].",
"Nikishov and Ritus have shown that the excitation of the elementary particle in a constant electric field might be due to pair creation by electric field[19].",
"Another problem is the time of formation of the thermal bath.",
"If an electron appears in the electric field, as acceleration field, instantly what it will see?",
"According to Ginzburg and Frolov[8] the electron will see the thermal bath only after a time far longer than $\\tau = c/w$ .",
"What kind of process results in such thermalization has yet to be explained.",
"Moreover, for photons the Unruh effect contains no Plank constant.",
"According to 'conformity principle' for radiation, saying that any quantum radiation equals to classical one when $\\hbar \\rightarrow 0$ , there should be a classical analog of such radiation.",
"Mechanism of classical analog of Unruh radiation is difficult to determine.",
"Nevertheless, there is a positive point that makes the study of the Unruh radiation interesting.",
"The Unruh radiation has a strong analogy with Hawking radiation[6].",
"Whether the acceleration field is equivalent to the gravitational field is not verified, but the Unruh radiation might give us the answer to that is true or not.",
"An unique complex SACLA consisting of 10 keV/30 fs XFEL systems [20] and coming $\\approx $ 1 PW/25 fs Ti-Sph femtosecond laser with maximum intensity $I =10^{22}$ W/cm$^2$ ($a_0\\approx 100$ ) , which exceeds the similar parameter of J-KAREN with peak intensity of $I =10^{22}$ W/cm$^2$ [21] which already exists, allows developing a new, more adequate approach to explore the Unruh effect.",
"One of the problems in detecting the Unruh effect is the necessity of emitters and detectors in the same Rindler reference frame.",
"For this purpose, it is better to explore the photon scattering rather than photon emission."
],
[
"Thomson scattering from accelerated electrons",
"According to Unruh the emission probability of an electron in Rindler space is given by following equation[1]: $P_j = {\\displaystyle \\sum _{p}}{ \\left| \\int _0^T\\epsilon d \\left( \\frac{\\tau }{T} \\right) \\int _V \\sqrt{-g} d^3 x {\\psi _j}^{\\ast } \\mathinner {\\langle {p}|} A \\mathinner {|{0}\\rangle }_M \\psi _0 \\right|}^2,$ where $\\psi $ is an electron wave function in Rindler space, $A$ is the field operator, $\\mathinner {\\langle {p}|}$ is the field wave function, $\\epsilon $ is a coupling constant, $M$ means the vacuum in Minkowski space.",
"This equation results in Rindler radiation.",
"In the case of photon scattering one can extend the Unruh logic to a photon matrix element: $\\mathinner {\\langle {0}|}a_2 A_{\\mu } (x)A_{\\beta } (x^{\\prime }) {a_1}^{\\dagger } \\mathinner {|{0}\\rangle }_M$ (see Ref.",
"[22], $a, a^{\\dagger }$ are the known operators).",
"It is clear that the result of Fulling-Rindler mode may occur only as a broadening of scattering massless photon similar to the Doppler broadening with the temperature $T_w=\\hbar w/2\\pi k_B c$ .",
"There was a proposal of using Thomson scattering as a detector for the Unruh effect[17] exploiting acceleration of electron quiver motion in a laser pulse, $ w = c \\omega _L a_0$ with zero $v \\times B$ force.",
"To make the story, 3 different laser pulses were considered.",
"1st optical laser pulse should excite a gas-jet to make plasma.",
"2nd optical laser pulse makes acceleration for electrons in the plasma, and 3rd an XFEL pulse coaxial with the 2nd pulse probes accelerated electrons.",
"It was stated that there is the possibility to detect the Unruh effect by obtained spectral broadening.",
"Expected spectrum is given by[17] $S({\\bf k}, \\omega ) = \\sqrt{ \\frac{1}{\\pi (1 - \\cos \\theta ) \\Omega ^2}} \\exp \\left[ - \\frac{1}{\\pi (1 - cos \\theta ) \\Omega ^2} \\omega ^2 \\right],$ where $\\Omega = \\omega _i v_{th} / c$ , $\\omega _i$ is the photon energy of probing laser, $\\theta $ is scattering angle, $v_{th} = \\sqrt{2k_B T_w / m}$ is thermal velocity, $k_B$ is Boltzmann constant, $T_w = T + \\lambda m {w^2} / { 2k_B} q^2 c^2$ is the temperature of electrons in thermal equilibrium, which differs from the Unruh temperature, $T$ is temperature of the plasma, $m$ is electron mass, $c$ is speed of light, $\\omega $ is the change of photon energy of probing laser after scattering, $q$ is a change of wavenumber after scattering and $\\lambda $ is metric parameter, which determines the properties of space time geometry (metric) [17].",
"About 40 photons are expected in a shot of XFEL.",
"However, in this scheme, it is difficult to distinguish whether the expected spectrum broadening occurs due to the Unruh effect or due to the extra Doppler effect.",
"Because of a finite length of probing pulses, electrons are further accelerated and evacuated by the ponederomotive force to vicinity of lower laser field as well as a strong, complicated longitudinal field make this scheme difficult in realization.",
"We consider two schemes that can eliminate extra Doppler shifts and extract the Unruh effect using plasma and intense laser."
],
[
"Thomson scattering from a laser pulse wake wave",
"Electrons in plasma wave exited by a laser pulse in underdense plasma are good candidates for probing the Unruh effect via Thomson scattering.",
"It is well known that a laser pulse propagating in underdense plasma creates a trail of plasma wave in its wake [23].",
"This wave has no group velocity that means plasma electrons in the wave oscillate with plasma frequency, $\\omega _p = \\sqrt{4 \\pi Z N_p e^2 / m}$ , where $N_p$ is the plasma density, $Z$ is the ion charge, and with a limited displacement.",
"Such plasma perturbation propagates with phase velocity $v_p=c\\sqrt{1-4 \\pi ZN_p/a_0N_{cr}}$ where $N_{cr}$ is the critical density for laser pulse with frequency $\\omega _L$ and $a_0$ is its normalized intensity.",
"In contrast to the electric field of a laser pulse, maximal electric field in a wake, and corresponding acceleration, is weaker by factor $\\omega _p / \\omega _0$ [24]: $w = eE_p/m = c \\omega _p a_0 $ [cm/s$^2$ ].",
"Moreover, for large $a_0$ this field strength can be achieved only in the first bucket of plasma wave trail due to the relativistic wave-breaking [25].",
"The maximal acceleration in regular wave buckets therefore is even smaller and is determined by the wave breaking limit field [24]: $w = eE_{WB}/m = c \\omega _p \\sqrt{ 2 a_0} $ [cm/s$^2$ ].",
"However, the number of electrons with identical velocity and moving at identical acceleration in these waves is much larger than that in the case of quiver motion [17].",
"This allows essential increasing of efficiency of Thomson scattering for detection of the Unruh effect.",
"The problem of ponderomotive acceleration also vanishes.",
"In the reference frame moving with the constant group velocity of wave this scheme provide an electron layer with $v=0$ undergoing constant acceleration $w$ .",
"There are two parameters resulting in value of electron acceleration, the laser field strength, $a_0$ , and the plasma density, $N_e$ .",
"However, one can consider three different cases of plasma wave for the Unruh effect detection: (i) a regular wake with minimal possible acceleration, $w = eE_{WB}/m$ , and a large number of identical accelerated structures; (ii) wave breaking limit with far higher acceleration $w = c \\omega _p a_0 $ , but with a single regular accelerated electron group existing for a short time, (iii) relativistically transparent overdense plasma with immense number of accelerated electrons and maximal acceleration.",
"Being based on this, we propose the concept for observation of the Unruh effect by the electron layer of the plasma wave with $k$ vector diagram [26] in Fig.REF .",
"The layer is produced in the plasma and accelerated by wake field.",
"Probing laser is incident from its back and scattered light is detected also backwardly.",
"We use plasma wave similar to flying mirror [25] exploiting incident light from opposite direction to decrease intensity of laser light.",
"Akhiezer and Polovin[27] showed that electrons in plasma wave have acceleration $w(v)$ depending on its velocity.",
"Upon choosing reasonable incident and detection angles, the Doppler shift of electron motion is suppressed and only spectral broadening due to the Unruh effect could be extracted.",
"The Doppler shift would be occurred when we observe the spectrum in the laboratory reference frame.",
"That is shown as [25] $\\varepsilon ^{\\prime } = \\varepsilon _0 \\frac{1 + 2 \\beta \\cos \\theta _D + \\beta ^2}{1 - \\beta ^2} ,$ where $\\varepsilon _0$ and $\\varepsilon ^{\\prime }$ are the photon energy of light before and after scattering respectively.",
"$\\beta $ is denoted as $\\beta = v^2 / c^2$ where $v$ and $c$ are speed of electron layer and light respectively.",
"$\\theta _D$ is the angle between velocity of electron layer and incident light.",
"And the reflection angle $\\theta _R$ is related to the incident angle shown in $k$ vector diagram of Fig.REF as $\\sin \\theta _R = \\frac{\\varepsilon _0}{\\varepsilon ^{\\prime }} \\sin \\theta _D =\\frac{(1 - \\beta ^2)\\sin \\theta _D}{1 + 2 \\beta \\cos \\theta _D + \\beta ^2},$ We can find the angles that the scattering frequency should not be changed due to the Doppler shift from the above equations.",
"This gives $\\theta _D = 174^{\\circ }$ and scattering angle $\\theta _R = 5.4^{\\circ }$ .",
"These angles we use for estimation of spectra of scattered x-rays.",
"When photon energy of probing laser is set to be 1 keV, expected spectral broadening is expected to be $\\Delta \\omega /\\omega _i \\approx 1.4 \\%$ at $a_0 \\approx 100 $ , $N_e \\approx 10^{20}$ cm$^{-3} $ for regular wake, $\\Delta \\omega /\\omega _i \\approx 3 \\% $ at $a_0 \\approx 100 $ , $N_e \\approx 10^{20}$ cm$^{-3} $ for first bucket.",
"$\\omega _p$ raises up to $\\omega _L$ with increase of density.",
"To estimate efficiency of scattering we present the electron density in the form[27] $N_e \\approx \\eta (a_0)N_p,$ where $\\eta (a_0)$ is the compression coefficient depending on the laser intensity.",
"We also define a parameter $\\delta = w_{x-ray}/w_{L}$ as the uniformity parameter, which represents the flatness of the electron layer, where $w_{x-ray}$ and $w_{L}$ are the waists of x-rays and driving laser pulse respectively.",
"We solved the conventional transport equation $dN_{sc}/dx=N_e (d\\sigma / d\\Omega )N_{in}\\delta $ assuming $x=L$ is too small, where $N_{in}$ is the number of incident photons.",
"Finally one can get $N_{sc}$ per 1 shot as $N_{sc}=N_e \\frac{d\\sigma }{d\\Omega }N_{in}L\\delta ,$ where $ d\\sigma / d\\Omega = {r_e}^2 (1 + \\cos ^2 \\theta ) / 2$ is the Thomson scattering cross section with $r_e$ is the classical electron radius and $\\theta $ is a scattering angles, and $L$ is thickness of the layer.",
"We use electrons in the plasma wave with high density.",
"The wave moves with high speed as well as the x-rays.",
"Therefore, the efficient length $L^{\\prime }$ becomes longer.",
"For example in the case of incidence with zero angle the length will be $L^{\\prime }=Lc/(c-v)$ where $v$ is the phase velocity of the plasma wave.",
"During this propagation there will be scattering.",
"If we assume that the wave does not change much in this distance the number of photon drastically increases, at least 10-100 times.",
"In the normal case the angle is not zero and efficient length is shorter.",
"However, the number of photon increases.",
"Results of calculations for spectra and total number of scattered photons and its efficiency are presented in Fig.REF for the maximal and minimal accelerations.",
"The number of incident x-rays is $N_{in} = 10^{12}$ , waists of x-rays $w_{x-ray}=7.5$ $\\mu $ m and laser pulse waist $w_{L}=15$ $\\mu $ m, and thickness $L\\approx 1$ $\\mu $ m. Efficiency is the number of resolvable photons in the total number of photons, meaning that how many photons can be detectable for the Unruh radiation.",
"It is determined by the integral of spectra from resolvable point (1 %, 5 % in the case) to infinity.",
"This number of photons is enough to get reasonable results by 1 shot probing.",
"The case of relativistically under-dense plasma has been included in this calculation.",
"The compression efficiency was calculated using 3D PIC FPLASER3D code[29].",
"One can see that the efficiency of scattering is high enough for verifiable detection of the Unruh effect in both cases.",
"The number of scattering photons exceeds $10^{3}$ even for $a_0 = 100$ and the plasma density $N_p = 10^{20}$ cm$^{-3}$ .",
"We show in Fig.REF that shows the results of 3D PIC simulation when intense laser pulse with $a_0 = 100$ and wavelength of 1 $\\mu $ m irradiated to underdenseplasma (hydrogen gas in this case) with initial electron density of $10^{19}$ cm$^{-3}$ .",
"The critical density is approximately $1.7\\times 10^{21}$ cm$^{-3}$ .",
"We see the electron density of the layer exceeds $10^{20}$ cm$^{-3}$ because of the compression.",
"Expectedly the broadening of the scattered x-rays owing to the Unruh effect is larger for the maximal acceleration.",
"Nevertheless even for $a_0 = 100$ and the plasma density $N_p = 10^{20}$ cm$^{-3}$ it is about $1\\%$ that could be enough for detection depending on the broadening of XFEL pulses.",
"We note that this scheme is free from recoil problem.",
"For co-propagating x-rays and laser pulse recoil effect is small $\\hbar \\omega / 2 \\gamma m c^2 \\approx 10^{-4}$ for $\\gamma = 10$ and 1 keV x-ray.",
"The use of the first bucket of wake field would provide us with the maximal acceleration and give parameters of scattered x-rays good enough for detection of the Unruh effect.",
"However, due to relativistic wave breaking a regular electron layer exists only though for a short time but quite detectable even in the case of high density plasma[30].",
"This requires a tight synchronization of laser pulse and XFEL pulse.",
"Full optical XFEL system[31]can provide such synchronization down to a few fs level."
],
[
"Thomson scattering from electron bunches accelerated by laser wake field",
"The use of Thomson scattering from a laser wake wave demonstrates its serious prospect for detection for the Unruh effect.",
"One contradiction, however, is based on the fact that measurements should include not only source of radiation but also detector in the same reference frame.",
"Here, we propose such a system consisting of a source and detector also with laser wake field acceleration (LWFA) and XFELs.",
"Proposed experimental setup is shown in Fig.REF .",
"Pumping optical laser irradiates a helium gas-jet to create laser wake field.",
"Due to multiple electron self-injection several beams, for example, electron beam 1 and electron beam 2 shown in Fig.REF , are created in an accelerating field $E_P$ and can be used as a source and a detector respectively.",
"Probing XFEL (attosecond harmonics with 1 eV spectrum width) is incident to an accelerated source from its back.",
"Scattered light from the detector is probed and the spectrum is to be obtained.",
"Recently, electron acceleration technology has been developing.",
"LWFA can accelerate electrons with an acceleration gradient of $\\approx 100$ GV/cm.",
"A laser pulse leaves its wake field blowing electrons away in plasma[29].",
"Acceleration field is generated by sparse and dense electrons.",
"Some of blown electrons come back to acceleration field by Coulomb force and built groups of electrons, which is called bunch, are uniformly accelerated with the same high gradient acceleration and velocity.",
"Uniform acceleration means that all electrons are under the same acceleration force in electric field with its gradient equaling to zero and in ideal case lower gamma is better for such purpose because of relativistic effects with higher gamma.",
"What we need is large acceleration not energy.",
"Thus, LWFA is adequate to verify the Unruh effect.",
"Fig.REF shows the typical result of 2D PIC simulation for laser wake field acceleration.",
"Due to multiple electron self-injection[29] or an external injection, the number of bunches may be varied.",
"Accelerating electrical field is the wake field with its strength given by $w = \\frac{e}{m} E_p = c \\omega _p a_0 \\ \\ [\\text{cm/s}^2],$ Again, we consider Thomson scattering from bunches to observe the Unruh effect.",
"Figure.REF shows the experimental process with $k$ vector diagram [26].",
"When we consider Thomson scattering from 1 bunch, scattered spectrum broadens due to velocity of bunches.",
"To eliminate this effect, we take a pair of 2 bunches as source and detector and consider the 2 times Thomson scattering.",
"We use a forward bunch as a source and backward bunch as a detector.",
"Electron beams 1 and 2 are accelerated by different acceleration $w_1$ and $w_2$ that have almost the same value.",
"We could put the source and detector in almost the same accelerating frame.",
"Probing laser is shot from the back side of the source.",
"The source scatters photons to the detector.",
"We observe the spectrum of scattered photons by the detector.",
"Frequency of scattered light by source becomes downshifted due to Doppler effect.",
"And frequency of scattered light by detector becomes upshifted due to Doppler effect.",
"So, observed spectrum of detector should be the same as initial probing pulse ordinary.",
"However, if the Unruh effect exists, accelerated bunches feel the Unruh temperature and the spectrum of scattered light becomes broader.",
"We could extract the Unruh effect felt by each electron beams.",
"Scattering occurs twice by the source and the detector.",
"The frequency of the probing laser becomes shifted in the first scattering by the source.",
"The detector scatters the shifted laser light from the source returning its initial frequency.",
"So, the expected spectrum from the detector, observed in the laboratory frame, should be obtained by convolution of $S({\\bf k},\\nu )$ $\\begin{split}S({\\bf k},\\omega ) & \\approx \\int _{-\\infty }^{\\infty } S({\\bf k},\\nu ) S({\\bf k},\\omega - \\nu ) d\\nu \\\\&= \\sqrt{\\frac{1}{\\pi F(\\theta )}} \\exp [- G(\\theta ) \\omega ^2],\\end{split}$ $ F(\\theta ) = (1-\\cos \\theta _2) {\\Omega _2}^2+(1-\\cos \\theta _1) {\\Omega _1}^2,$ $G(\\theta ) = \\frac{1}{(1-\\cos \\theta _1) {\\Omega _1}^2}- \\cfrac{1}{{(1-\\cos \\theta _1)}^2 {\\Omega _1}^4 \\left[ \\cfrac{1}{(1-\\cos \\theta _1) {\\Omega _1}^2} + \\cfrac{1}{(1-\\cos \\theta _2) {\\Omega _2}^2} \\right]},$ $\\Omega _1 = \\frac{\\omega _i v_{th1}}{c}, \\ \\ \\Omega _2 = \\frac{\\omega _i v_{th2}}{c},$ where $\\theta _1$ and $\\theta _2$ is scattering angle of source and detector, $v_{th1}=\\sqrt{2k_B T_{w1} / m}$ and $v_{th2}=\\sqrt{2k_B T_{w2} / {m}}$ , $T_{w1} = \\hbar {w_1} / 2\\pi k_B c$ and $T_{w2} = \\hbar {w_2} / 2\\pi k_B c$ are the Unruh temperature of beam 1 and 2, respectively.",
"We have to set adequate scattering angles to eliminate the normal Doppler effect of scattered light.",
"That means the scattering angles that we observed spectrum becomes monochromatic.",
"Such scattering angles satisfy the equation shown below.",
"$\\cos (\\pi - \\theta _2) = \\frac{1}{\\beta } \\left[ 1 - \\frac{1-\\beta ^2}{1 - \\cos \\theta _1} \\right],$ where $\\beta = v^2 / c^2$ and $v$ is the velocity of bunches.",
"When $v=0.9c$ , a set of angles satisfying the equation is $\\theta _1=165^\\circ $ , $\\theta _2=172.4^\\circ $ .",
"The angles are described in $k$ vector diagram in Fig.",
"REF The spectral width of the expected spectrum is a critical parameter.",
"Broadening due to the Unruh effect must exceed both a bandwidth of XFEL pulses and a thermal-like Doppler shift owing the energy spread of electrons in the beams.",
"Broadening of XFEL can be solved upon use of harmonics from laser plasma[28] while the problem of low beam energy spread is far from solution.",
"The number of scattered photon per 1 shot from a single pair of source and detector can be estimated only by use of results of 3D PIC simulation shown in Fig.REF .",
"Density of photons from probing XFEL can be evaluated from a simple equation as $N_{in} = I_p \\pi {(w_U/ 2)}^2 \\tau / E_{ph}$ , where $I_p$ , $w_U$ , $\\tau $ and $E_{ph}$ are laser intensity, beam waist, pulse duration, and photon energy respectively.",
"We denote the number of scattered photons from the source as $N_{sc1}$ , the number of scattered photons by the detector as $N_{sc2}$ , and the electron density of bunch as $N_e$ .",
"Since the interaction length $L$ is equal to the size of the bunch, $N_{sc1}$ is given by $N_{sc1} = N_e (d\\sigma / d\\Omega )_1 N_{in} L$ .",
"Then, $N_{sc2}$ becomes $\\begin{split}N_{sc2} &= N_e \\left( \\frac{d\\sigma }{d\\Omega } \\right)_2 N_{sc1} L \\\\&= {N_e}^2 \\left( \\frac{d\\sigma }{d\\Omega } \\right)_2 \\left( \\frac{d\\sigma }{d\\Omega } \\right)_1 L^2 N_{in}, \\\\\\end{split}$ in ideal case where $ (d\\sigma / d\\Omega )_i = {r_e}^2 (1 + \\cos ^2 \\theta _i) / 2$ is the Thomson scattering cross section of each scattering, with $r_e$ is the classical electron radius and $\\theta _i$ is a scattering angles.",
"In this case the efficient length as discussed in the previous section could be considered.",
"Electron beams move with high speed as well as the x-rays.",
"The efficient length $L^{\\prime }$ becomes longer.",
"As we showed, in the case of incidence with zero angle the length will be $L^{\\prime }=Lc/(c-v)$ where $v$ is the phase velocity of the electron beams.",
"Upon setting $N_{in} = 10^{12}$ from the equation, $N_e = 10^{20}$ cm$^{-3}$ and $L = 10$ $\\mu $ m from the results of 3D PIC simulation, one can get $N_{sc2} \\approx 5 \\times 10^{-5}$ in 1 shot.",
"This number means that we can get $10^{-5}$ photons of the spectrum per 1 shot.",
"Since several pairs of bunches can be generated, the number of photons can be increased proportionally.",
"Let us estimate the number of pairs we can use when using SACLA's XFEL that has 10 keV and $\\approx $ 80 $\\mu $ m beam size.",
"Size of the bunch is $\\approx $ 2 $\\mu $ m, 2 bunches are used as a pair.",
"A wavelength of light scattered by a source becomes $\\approx $ 0.8 $\\mu $ m. So, a pair needs $\\approx $ 4.8 $\\mu $ m. The beam size of XFEL is $\\approx $ 80 $\\mu $ m. Therefore, $\\approx $ 30 pairs of bunches can be used in 1 shot probing.",
"When we have 30 pairs of bunches, we could get $N_{sc2} \\approx 1.5\\times 10^{-3}$ .",
"We can obtain sufficient spectrum by probing many times.",
"For example, if we use the laser in 10 Hz repetition rate mode, the expected number of photons is $\\approx 5$ per hour.",
"It seems to be a sufficient number of photons to resolve the scattered spectrum.",
"In this case, the charge of electron beams is smaller and of order nC-pC.",
"Because of low charge of the beams, this experiment would be sensitive to scattering compared to using electrons in plasma wave proposed in previous section.",
"In previous section, the number of electrons in plasma wave is large enough to obtain spectrum and charge is not so important.",
"In using electron beams case, electron charge is important.",
"Small charge electron beams would not scatter the enough number of photons to obtain spectrum while eq.",
"(REF ) is assumed that second scattering scatters all photons from first scattering.",
"To verify the existence of the Unruh radiation, scattered spectrum should have certain width that relates to the acceleration.",
"Fig.REF shows $a_0$ dependency of spectrum width.",
"As seen from this graph, higher acceleration is necessary to observe broader spectrum."
],
[
"Conclusions",
"We have proposed designs for experimental observation effects by spectral shift of Thomson scattering from two systems consisting of highly accelerated electrons.",
"First system is consisted of electrons in the plasma wave excited by an intense laser.",
"Reasonable photon number of Thomson scattering allows us to obtain the spectrum.",
"Choosing certain incident and detection angles enables us to distinguish the Unruh effect from the normal Doppler shift.",
"Second system is consisted of source-detector moving with a constant acceleration.",
"In this case, the laser wake field with the multiple electron self-injection has been considered.",
"Probing the existence of the Unruh effect explores whether acceleration and gravitation are equivalent.",
"Although there are several problems in the existence of the Unruh effect, it should be proved in the experiment.",
"Two proposals enable us to distinguish spectral shift of the Unruh effect from that of the normal Doppler shift.",
"Second proposed experiment has a source and a detector in the reference frame with constant acceleration and is suitable for detecting the effect occurring in the reference frame.",
"The proposed experiment can prove whether the Unruh effect exists or not.",
"New proposals can be realized at SACLA [20] in near future.",
"Lasers with $a_0\\approx 100$ is coming to SACLA exceeding already achieved laser parameter of J-KAREN [21].",
"For now, the maximum intensity of optical laser reached $a_0\\approx 100$ experimentally in the world and is developing to reach higher intensity.",
"In the case of plasma wave the firm detection of the Unruh effect can be performed soon with $a_0 \\approx 100-200$ .",
"Therefore, this proposal could be executed by a combination of XFEL and LWFA with intense optical laser theoretically.",
"The technologies of LWFA and X-ray harmonics are under development but are not yet practical.",
"The systems we proposed are considered to leave only practical problems, such as attosecond harmonics, Laser Wake Field Acceleration, synchronization between PW laser and XFEL and the facility consisting of all these technologies.",
"IMPACT project is running to construct the facility that makes it possible in SACLA.",
"Fundamental problems seem to be solved to verify the Unruh effect.",
"We can expect to execute the proposed experiment at SACLA facility with the PW class laser system and the existing XFEL."
],
[
"Acknowledgement",
"This work was partially supported by IMPACT project.",
"Figure Captions Fig.",
"1.",
"Proposed experimental setup using accelerated electrons of plasma wave with $k$ vector diagram.",
"$k_i$ and $k_r$ is wave vector of incident and scattered light, respectively and $q$ is change of wave vector.",
"The layer is produced in the plasma and accelerated by wake field.",
"Probing laser is incident from its back and scattered light is detected also backwardly.Upon choosing reasonable incident and detection angles, the Doppler shift of electron motion is suppressed and only spectral broadening due to the Unruh effect could be extracted.",
"Fig.",
"2.",
"Dependency of spectrum width and the total number of photons on the laser pulse field for (a) scattering from the first bucket at maximal acceleration and (b) for the wave breaking limit.",
"Solid lines show $\\Delta \\omega / \\omega _i$ for $N_e =$ (1)$10^{22}$ , (2)$10^{21}$ (3)$10^{20}$ cm$^{-3}$ .",
"Dashed lines show the total number of photons for $N_e =$ (4)$10^{22}$ , (5)$10^{21}$ (6)$10^{20}$ cm$^{-3}$ .",
"The number of resolvable photons for (c)the first bucket and (d)the wave breaking limit; the photon energy of probing laser is $\\omega _i=1$ keV.",
"Solid lines show the number of resolvable photons with 1 % resolution for $N_e =$ (1)$10^{22}$ , (2)$10^{21}$ (3)$10^{20}$ cm$^{-3}$ .",
"Dashed lines show the number of resolvable photons with 5 % resolution for $N_e =$ (4)$10^{22}$ , (5)$10^{21}$ (6)$10^{20}$ cm$^{-3}$ .",
"Fig.",
"3.",
"Results of 3D PIC simulation when intense laser pulse with $a_0 = 100$ and wavelength of 1 $\\mu $ m irradiated to underdense plasma (hydrogen gas in this case) with electron density of $N_p = 10^{20}$ cm$^{-3}$ .",
"The critical density is approximately $1.7\\times 10^{21}$ cm$^{-3}$ .",
"We see the electron density of the layer exceeds $10^{20}$ cm$^{-3}$ around $x+ct=55$ [$\\mu $ m] which is focal position of the laser pulse because of the compression.",
"Fig.",
"4.",
"Proposed experimental setup with a source and a detector.",
"$k$ vector diagram is shown.",
"$k_i$ and $k_i$ is wave vector of incident and scattered light, respectively and $q$ is change of wave vector of first scattering and with prime is of second scattering.",
"$E_L$ is electric field of optical laser pulse.",
"$E_P$ is accelerating field of wake field.",
"Beams 1 and 2 are group of electrons used as a source and a detector respectively.",
"Pumping optical laser ( $\\ge 10^{22}$ W/cm$^{-2}$ ) is shot in Helium gas-jet to create Laser Wake Field.",
"Probing XFEL is incident to an accelerated source from its back.",
"Scattered light from the detector is probed and the spectrum is to be obtained.",
"Fig.",
"5.",
"3D simulation for Laser Wake Field Acceleration.",
"Electron density from 3D PIC simulation for LWFA done for the conditions stated below.",
"The plasma channel has the maximum density of $N_{e max} = 4\\times 10^{18}$ cm$^{-3}$ and the minimal density of $N_{e max} = 1.2\\times 10^{18} $ cm$^{-3}$ .",
"The laser intensity is $I = 4 \\times 10^{19}$ W/cm$^{-2}$ , the focus diameter is $D = 18$ $ \\mu $ m and laser pulse duration is 30 fs.",
"One can see multiple self-injection of bunches in the first bucket behind the laser pulse.",
"Some bunches appear in the bucket repeatedly.",
"Fig.",
"6.",
"Proposed experimental process.",
"Forward bunch could be used as source and backward bunch as a detector.",
"Probing laser is irradiated from the back side of the source.",
"Source scatters photons to the detector.",
"We observe the spectrum from the detector.",
"Electron bunches moving with the same acceleration.",
"Fig.",
"7.",
"Dependency of spectrum width and efficiency of number of photons on the laser pulse field.",
"The graph shows the dependency of spectrum width and efficiency on the laser pulse field strength $a_0$ .",
"$\\omega _i$ is the photon energy of probing laser set to be 1 keV.",
"Solid line (1) shows $\\Delta \\omega / \\omega _i$ .",
"Dashed lines show the number of resolvable photons with (2)1 % (3)5 % resolution.",
"The spectral width means the strength of the the Unruh effect.",
"To obtain broader spectral shift, the larger $a_0$ is required."
]
] | 1709.01659 |
[
[
"Redshift and contact forms"
],
[
"Abstract It is shown that the redshift between two Cauchy surfaces in a globally hyperbolic spacetime equals the ratio of the associated contact forms on the space of light rays of that spacetime."
],
[
"Introduction",
"Let $X$ be a spacetime, that is, a connected time-oriented Lorentz manifold [2].",
"The Lorentz scalar product on $X$ will be denoted by $\\langle \\text{ },\\!\\text{ }\\rangle $ and assumed to have signature $(+,-,\\dots ,-)$ with $n\\ge 2$ negative spatial dimensions.",
"Suppose that $n_E$ (`emitter') and $n_R$ (`receiver') are two infinitesimal observers, i.e.",
"future-pointing unit Lorentz length vectors, at events $E, R\\in X$ connected by a null geodesic $\\gamma $ .",
"Then the photon redshift $z=z(n_E,n_R,\\gamma )$ from $n_E$ to $n_R$ along $\\gamma $ is defined by the formula $1+z = \\frac{\\langle n_E,\\dot{\\gamma }(E)\\rangle }{\\langle n_R,\\dot{\\gamma }(R)\\rangle }$ for any affine parametrisation of $\\gamma $ .",
"In other words, $1+z$ is the ratio of the frequencies of any lightlike particle travelling along $\\gamma $ measured by $n_E$ and $n_R$ , see e.g.",
"[11] or [17].",
"If $z>0$ , such particles appear `redder' (having lower frequency) to $n_R$ than to $n_E$ , whence the terminology.",
"Assume now that $X$ is globally hyperbolic [4], [5] and consider its space of light rays $\\mathfrak {N}_X$ .",
"By definition, a point $\\gamma \\in \\mathfrak {N}_X$ is an equivalence class of inextendible future-directed null geodesics up to an orientation preserving affine reparametrisation.",
"A seminal observation of Penrose and Low [18], [13], [14] is that the space $\\mathfrak {N}_X$ has a canonical structure of a contact manifold (see also [16], [12], [1]).",
"A contact form $\\alpha _M$ on $\\mathfrak {N}_X$ defining that contact structure can be associated to any smooth spacelike Cauchy surface $M\\subset X$ .",
"Namely, consider the map $\\iota _M:\\mathfrak {N}_X \\mathrel {\\longrightarrow } T^*M$ taking $\\gamma \\in \\mathfrak {N}_X$ represented by a null geodesic $\\gamma \\subset X$ to the 1-form on $M$ at the point $x=\\gamma \\cap M$ collinear to ${\\langle \\dot{\\gamma }(x),\\cdot \\,\\rangle |}_M$ and having unit length with respect to the induced Riemann metric on $M$ (see formula (REF ) below).",
"This map identifies $\\mathfrak {N}_X$ with the unit cosphere bundle ${\\mathbb {S}}^*M$ of the Riemannian manifold $\\left(M,- {\\langle \\text{ },\\!\\text{ }\\rangle |}_M\\right)$ .",
"Then $\\alpha _M := \\iota _M^*\\,\\lambda _{\\mathrm {can}},$ where $\\lambda _{\\mathrm {can}}=\\sum p_kdq^k$ is the canonical Liouville 1-form on $T^*M$ .",
"Contact forms defining the same contact structure are pointwise proportional.",
"The purpose of the present note is to point out that the ratio of the contact forms on $\\mathfrak {N}_X$ associated to different Cauchy surfaces in $X$ is given by the redshifts between infinitesimal observers having those Cauchy surfaces as their rest spaces.",
"Definition 1.1 Let $M$ and $M^{\\prime }$ be spacelike Cauchy surfaces in $X$ .",
"The redshift from $M$ to $M^{\\prime }$ along $\\gamma \\in {\\mathfrak {N}}_X$ is defined by $z(M,M^{\\prime },\\gamma ):= z\\bigl (n_M(x), n_{M^{\\prime }}(x^{\\prime }), \\gamma \\bigr ),$ where $\\gamma $ is any inextendible null geodesic representing $\\gamma $ , $x=\\gamma \\cap M$ , $x^{\\prime }=\\gamma \\cap M^{\\prime }$ , and $n_M$ and $n_{M^{\\prime }}$ are the future pointing normal unit vector fields on $M$ and $M^{\\prime }$ .",
"Theorem 1.2 Let $M$ and $M^{\\prime }$ be spacelike Cauchy surfaces in $X$ .",
"For every $\\gamma \\in \\mathfrak {N}_X$ , we have $\\frac{\\alpha _{M^{\\prime }}}{\\alpha _{M}}(\\gamma ) =1 + z(M, M^{\\prime }, \\gamma ).$ Remark 1.3 The theorem remains true for partial Cauchy surfaces, i.e.",
"locally closed acausal spacelike hypersurfaces $M,M^{\\prime }\\subset X$ , and for $\\gamma \\in \\mathfrak {N}_X$ corresponding to null geodesics intersecting both $M$ and $M^{\\prime }$ .",
"Remark 1.4 If $M$ and $M^{\\prime }$ are Cauchy surfaces through a point $x\\in X$ such that $n_M(x)=n_{M^{\\prime }}(x)$ , then the theorem shows that the contact forms $\\alpha _M$ and $\\alpha _{M^{\\prime }}$ coincide on the tangent spaces to $\\mathfrak {N}_X$ at all points corresponding to null geodesics passing through $x$ .",
"In other words, an infinitesimal observer at an event $x$ defines a contact form on $T{\\mathfrak {N}}_X$ restricted to the sky ${\\mathfrak {S}}_x\\subset {\\mathfrak {N}}_X$ .",
"The contact geometry of $\\mathfrak {N}_X$ was previously used to recover the causal or, equivalently [15], conformal structure of $X$ , see [14], [16], [10], [7], [8], [9].",
"Theorem REF should make it possible to apply techniques from contact geometry to study the metric structure of a globally hyperbolic spacetime.",
"A token application to the comparison of Liouville and Riemannian volumes on different Cauchy surfaces is given in § below."
],
[
"Proof of Theorem ",
"The key fact is the following basic property of vector fields tangent to variations of pseudo-Riemannian geodesics by curves of the same speed.",
"For Jacobi fields tangent to families of null geodesics in Lorentz manifolds, this computation appears in [18], [16], and [1].",
"Lemma 2.1 Let $\\gamma _s:(a,b)\\rightarrow X$ , $0\\le s <\\varepsilon $ , be a one-parameter family of curves in a pseudo-Riemannian manifold $(X,\\langle \\text{ },\\!\\text{ }\\rangle )$ such that $\\gamma _0$ is a geodesic and $\\langle \\dot{\\gamma }_s, \\dot{\\gamma }_s\\rangle $ is independent of $s$ .",
"If $J(t) := \\left.\\frac{d}{ds}\\right|_{s=0} \\gamma _s(t)$ is the vector field along $\\gamma _0$ tangent to this family, then $\\langle \\dot{\\gamma }_0(t), J(t) \\rangle = \\mathrm {const.",
"}$ Let us show that the $t$ -derivative of this scalar product is zero.",
"Indeed, $\\frac{d}{dt} \\langle \\dot{\\gamma }_0(t), J(t)\\rangle = \\langle \\nabla _t \\dot{\\gamma }_0(t), J(t)\\rangle + \\langle \\dot{\\gamma }_0(t), \\nabla _t J(t)\\rangle ,$ where $\\nabla $ is the pull-back of the Levi-Civita connection of the pseudo-Riemannian metric on $X$ to $(a,b)\\times [0,\\varepsilon )$ by the map $(t,s)\\mapsto \\gamma _s(t)$ .",
"The first term on the right hand side vanishes because the tangent vector of a geodesic is parallel along the geodesic.",
"Note further that $\\nabla _t \\frac{\\partial }{\\partial s} = \\nabla _s \\frac{\\partial }{\\partial t}$ since the Levi-Civita connection has no torsion and $\\left[\\frac{\\partial }{\\partial s},\\frac{\\partial }{\\partial t}\\right]=0$ (see also [17]).",
"Hence, the right hand side of (REF ) is equal to ${\\langle \\dot{\\gamma }_0(t), \\nabla _s \\dot{\\gamma }_s(t)\\Big {|}}_{s=0}\\rangle = \\frac{1}{2}\\left.\\frac{d}{ds}\\right|_{s=0}\\langle \\dot{\\gamma }_s,\\dot{\\gamma }_s\\rangle = 0$ because the speed of $\\gamma _s$ is independent of $s$ by assumption.",
"Remark 2.2 The relevance of torsion in this context is pointed out in the footnote on p. 184 of [18].",
"Let now $M$ be a smooth spacelike Cauchy surface in a spacetime $X$ and $\\gamma $ an inextendible future-directed null geodesic in $X$ intersecting $M$ at the (unique) point $x=\\gamma \\cap M$ .",
"Then $\\iota _M(\\gamma ) = \\frac{{\\langle \\dot{\\gamma }(x),\\cdot \\,\\rangle |}_M}{\\langle \\dot{\\gamma }(x), n_M(x)\\rangle }\\, ,$ where $n_M$ is the future-pointing unit normal vector field on $M$ and $\\gamma \\in \\mathfrak {N}_X$ is the equivalence class of $\\gamma $ .",
"Indeed, since $\\langle \\dot{\\gamma }(x),\\cdot \\,\\rangle $ is a null covector, the Riemannian length of its restriction to $T_xM$ is equal to the Riemannian length of its restriction to the Lorentz normal direction, which is precisely $\\langle \\dot{\\gamma }(x), n_M(x)\\rangle (>0)$ .",
"Thus, if $\\mathbf {v}\\in T_{\\gamma }\\mathfrak {N}_X$ and $v={(\\iota _M)}_*\\mathbf {v}$ , then $\\alpha _M(\\mathbf {v}) = \\lambda _\\mathrm {can}(v) =\\frac{{\\langle \\dot{\\gamma }(x),{(\\pi _M)}_* v\\rangle }}{\\langle \\dot{\\gamma }(x), n_M(x)\\rangle }$ by the definition of the canonical 1-form $\\lambda _\\mathrm {can}$ and formula (REF ), where $\\pi _M:T^*M\\rightarrow M$ denotes the bundle projection.",
"Suppose that $\\gamma _s:(a,b)\\rightarrow X$ , $s\\in [0,\\varepsilon )$ , is a family of null geodesics intersecting $M$ such that the maximal extension of $\\gamma _0$ is $\\gamma $ and the corresponding curve in $\\mathfrak {N}_X$ has tangent vector $\\mathbf {v}$ at $\\gamma $ or, equivalently, ${\\left.\\frac{d}{ds}\\right|}_{s=0}\\iota _M(\\gamma _s) = v$ .",
"Let $x(s)=\\gamma _s\\cap M$ so that $x(0)=x$ .",
"Then ${(\\pi _M)}_* v = {\\left.\\frac{d}{ds}\\right|}_{s=0} x(s)$ because $x(s)=\\pi _M\\circ \\iota _M(\\gamma _s)$ by the definition of $\\iota _M$ .",
"Hence, ${(\\pi _M)}_* v = J(x) + \\tau ^{\\prime }(0)\\dot{\\gamma }(x),$ where $J = \\left.\\frac{d}{ds}\\right|_{s=0} \\gamma _s$ is the Jacobi vector field along $\\gamma _0$ tangent to the family $\\gamma _s$ and $\\tau =\\tau (s)$ is the function defined by $\\gamma _s(\\tau (s))=x(s)$ .",
"Since $\\dot{\\gamma }(x)$ is null, it follows that ${\\langle \\dot{\\gamma }(x),{(\\pi _M)}_* v\\rangle } = {\\langle \\dot{\\gamma }(x),J(x)\\rangle }.$ If $M^{\\prime }$ is another Cauchy surface and $x^{\\prime }=\\gamma \\cap M^{\\prime }$ , we may choose $(a,b)\\subseteq \\mathbb {R}$ so that $\\gamma (a,b)\\ni x, x^{\\prime }$ and a family $\\gamma _s$ as above exists on $(a,b)$ .",
"By formulas (REF ) and (REF ), we obtain $\\alpha _M(\\mathbf {v}) = \\frac{\\langle \\dot{\\gamma }(x),J(x)\\rangle }{\\langle \\dot{\\gamma }(x), n_M(x)\\rangle }\\qquad \\text{and}\\qquad \\alpha _{M^{\\prime }}(\\mathbf {v}) = \\frac{\\langle \\dot{\\gamma }(x^{\\prime }),J(x^{\\prime })\\rangle }{\\langle \\dot{\\gamma }(x^{\\prime }), n_{M^{\\prime }}(x^{\\prime })\\rangle }\\, .$ However, $\\langle \\dot{\\gamma }(x),J(x)\\rangle = \\langle \\dot{\\gamma }(x^{\\prime }),J(x^{\\prime })\\rangle $ by Lemma REF and therefore $\\frac{\\alpha _{M^{\\prime }}(\\mathbf {v})}{\\alpha _M(\\mathbf {v})} = \\frac{\\langle \\dot{\\gamma }(x), n_M(x)\\rangle }{\\langle \\dot{\\gamma }(x^{\\prime }), n_{M^{\\prime }}(x^{\\prime })\\rangle }= 1 + z\\bigl (n_M(x), n_{M^{\\prime }}(x^{\\prime }), \\gamma \\bigr ),$ which proves Theorem REF .",
"Remark 2.3 The proof shows that the ratio $\\frac{\\alpha _{M^{\\prime }}(\\mathbf {v})}{\\alpha _M(\\mathbf {v})}$ , where $\\mathbf {v}$ is a tangent vector to $\\mathfrak {N}_X$ at a point $\\gamma \\in \\mathfrak {N}_X$ , is a positive function depending only on $\\gamma $ .",
"Thus, the contact forms on $\\mathfrak {N}_X$ associated to different Cauchy surfaces in $X$ define the same co-oriented contact structure indeed.",
"This contact structure can also be described as the pull-back of the canonical contact structure on the spherical cotangent bundle $ST^*M$ of a Cauchy surface $M$ by the map $\\rho _M = s_M\\circ \\iota _M$ , where $s_M: T^*M-\\lbrace \\text{zero section}\\rbrace \\rightarrow ST^*M$ is the projection to the spherisation, see [16] and [7]."
],
[
"Liouville measure and Riemannian volume",
"Let $M$ and $M^{\\prime }$ be two spacelike Cauchy surfaces in a globally hyperbolic spacetime $(X, \\langle \\text{ },\\!\\text{ }\\rangle )$ and consider the contact forms $\\alpha _M=\\iota _M^*\\lambda _{\\mathrm {can}}$ and $\\alpha _{M^{\\prime }}=\\iota _{M^{\\prime }}^*\\lambda _{\\mathrm {can}}$ on ${\\mathfrak {N}}_X$ associated to $M$ and $M^{\\prime }$ .",
"Then $\\alpha _{M} = \\bigl (1+z(M,M^{\\prime },\\gamma )\\bigr )^{-1}\\alpha _{M^{\\prime }}$ by Theorem REF and therefore $\\alpha _M\\wedge \\left(d\\alpha _M\\right)^{n-1} = \\bigl (1+z(M,M^{\\prime },\\gamma )\\bigr )^{-n}\\alpha _{M^{\\prime }}\\wedge \\left(d\\alpha _{M^{\\prime }}\\right)^{n-1}$ because $\\alpha \\wedge \\alpha =0$ and $d(f\\alpha )=fd\\alpha + df\\wedge \\alpha $ for any function $f$ and 1-form $\\alpha $ .",
"Recall that the Liouville measure on the unit cosphere bundle of a Riemannian manifold is defined by the non-vanishing $(2n-1)$ -form $\\Omega :=\\lambda _{\\mathrm {can}}\\wedge \\left( d\\lambda _{\\mathrm {can}}\\right)^{n-1}.$ Thus, formula (REF ) may be viewed as a general volume–redshift relation (cf.",
"[11]) for the Liouville measures on the unit cosphere bundles of $M$ and $M^{\\prime }$ with respect to the Riemann metrics $-{\\langle \\text{ },\\!\\text{ }\\rangle |}_M$ and $-{\\langle \\text{ },\\!\\text{ }\\rangle |}_{M^{\\prime }}$ .",
"Indeed, let $\\iota _{M^{\\prime }M} = \\iota _M\\circ (\\iota _{M^{\\prime }})^{-1} : {\\mathbb {S}}^*M^{\\prime }\\overset{\\cong }{\\longrightarrow } {\\mathbb {S}}^*M$ be the map identifying the unit covectors corresponding to the same null geodesic at its intersection points with $M$ and $M^{\\prime }$ .",
"Then (REF ) shows that $(\\iota _{M^{\\prime }M})^*\\Omega _M = \\bigl (1+z(M,M^{\\prime },\\gamma )\\bigr )^{-n}\\Omega _{M^{\\prime }}$ at $\\iota _{M^{\\prime }}(\\gamma )\\in {\\mathbb {S}}^*M^{\\prime }$ .",
"Let $\\mathfrak {L}\\subseteq \\mathfrak {N}_X$ be a (Borel) subset of the space of light rays and denote by $\\mathfrak {L}_x$ the set of null geodesics from $\\mathfrak {L}$ passing through a point $x\\in X$ .",
"Integrating (REF ) over $\\iota _{M^{\\prime }}(\\mathfrak {L})$ , we obtain that $\\int \\limits _{\\iota _M(\\mathfrak {L})} \\Omega _M= \\int \\limits _{\\iota _{M^{\\prime }}(\\mathfrak {L})} \\bigl (1+z(M,M^{\\prime },\\gamma )\\bigr )^{-n} \\Omega _{M^{\\prime }}.$ The Liouville measure is locally the product of the Riemann measure on the base manifold and the surface area measure on the unit sphere in the standard Euclidean space $\\mathbb {R}^n$ , see [3] or [6].",
"Therefore both integrals in (REF ) can be converted to double integrals.",
"Applying this to the left hand side first, we see that $\\int \\limits _{\\iota _M(\\mathfrak {L})} \\Omega _M = \\int \\limits _{M} dV_M(x) \\int \\limits _{\\iota _M(\\mathfrak {L}_x)} d\\omega _x= \\int \\limits _{M} \\omega _M(x,\\mathfrak {L})\\, dV_M(x),$ where $dV_M$ is the Riemann measure on $M$ , $d\\omega _x$ is the surface area measure on the fibre $\\mathbb {S}_x^*M$ , and $\\omega _M(x,\\mathfrak {L}) := \\int \\limits _{\\iota _M(\\mathfrak {L}_x)} d\\omega _x$ is the area of the set $\\iota _M(\\mathfrak {L}_x)$ of unit covectors at $x\\in M$ corresponding to null geodesics from $\\mathfrak {L}$ , i.e.",
"the solid angle spanned by the light rays from $\\mathfrak {L}$ at $x\\in M$ .",
"Now (REF ) takes the form $\\int \\limits _{M} \\omega _M(x,\\mathfrak {L})\\, dV_M(x) =\\int \\limits _{M^{\\prime }} dV_{M^{\\prime }}(x^{\\prime }) \\int \\limits _{\\iota _{M^{\\prime }}(\\mathfrak {L}_{x^{\\prime }})} \\bigl (1+z(M,M^{\\prime },\\gamma )\\bigr )^{-n} d\\omega _{x^{\\prime }}.$ Example 3.1 Assume that the redshift $z(M,M^{\\prime },\\gamma )= z$ is the same for all $\\gamma \\in \\mathfrak {L}$ .",
"Then (REF ) simplifies to $\\int \\limits _{M} \\omega _M(x,\\mathfrak {L})\\, dV_M(x) = \\frac{1}{(1+z)^n}\\int \\limits _{M^{\\prime }} \\omega _{M^{\\prime }}(x^{\\prime },\\mathfrak {L})\\, dV_{M^{\\prime }}(x^{\\prime }).$ Example 3.2 Let $\\mathfrak {L}=\\mathfrak {N}_X$ be the set of all light rays.",
"Then $\\iota _M(\\mathfrak {L}_x) = \\mathbb {S}^*_xM$ for every Cauchy surface $M$ and every point $x\\in M$ .",
"Hence, $\\omega _M(x,\\mathfrak {L}) = \\mathfrak {c}_n,$ where $\\mathfrak {c}_n$ is the area of the standard unit sphere in $\\mathbb {R}^n$ .",
"Therefore (REF ) implies $\\mathfrak {c}_n \\mathrm {Vol}(M) =\\int \\limits _{M^{\\prime }} dV_{M^{\\prime }}(x^{\\prime }) \\int \\limits _{\\mathbb {S}^*_{x^{\\prime }}M^{\\prime }} \\bigl (1+z(M,M^{\\prime },\\gamma )\\bigr )^{-n} d\\omega _{x^{\\prime }}.$ If the redshift is constant as in Example REF , it follows that $\\mathrm {Vol}(M) = \\frac{1}{(1+z)^n} \\mathrm {Vol}(M^{\\prime }).$ More generally, if $\\underline{z}\\le z(M,M^{\\prime },\\gamma ) \\le \\overline{z}$ , then $\\frac{1}{(1+\\overline{z})^n} \\mathrm {Vol}(M^{\\prime }) \\le \\mathrm {Vol}(M)\\le \\frac{1}{(1+\\underline{z})^n} \\mathrm {Vol}(M^{\\prime }).$ Example 3.3 Consider a subset $D\\subseteq M$ and let $\\mathfrak {L}^D = \\lbrace \\gamma \\in \\mathfrak {N}_X\\mid \\gamma \\cap D\\ne \\varnothing \\rbrace $ be the set of all light rays passing through $D$ .",
"Then $\\iota _M(\\mathfrak {L}_x^D) ={\\left\\lbrace \\begin{array}{ll}\\mathbb {S}^*_xM, & x\\in D,\\\\\\varnothing , & x\\in M\\setminus D,\\end{array}\\right.",
"}$ and therefore $\\omega _M(x,\\mathfrak {L}^D) ={\\left\\lbrace \\begin{array}{ll}\\mathfrak {c}_n, & x\\in D,\\\\0, & x\\in M\\setminus D.\\end{array}\\right.",
"}$ Hence, (REF ) gives the following expressions for the volume of $D$ in $M$ : $\\mathrm {Vol}_{M}(D) &=\\frac{1}{\\mathfrak {c}_n}\\int \\limits _{M^{\\prime }} dV_{M^{\\prime }}(x^{\\prime }) \\int \\limits _{\\iota _{M^{\\prime }}(\\mathfrak {L}^D_{x^{\\prime }})} \\bigl (1+z(M,M^{\\prime },\\gamma )\\bigr )^{-n} d\\omega _{x^{\\prime }}\\\\& = \\frac{1}{\\mathfrak {c}_n} \\int \\limits _{\\lbrace \\iota _{M^{\\prime }}(\\gamma ) \\mid \\, \\gamma \\cap D\\ne \\varnothing \\rbrace }\\bigl (1+z(M,M^{\\prime },\\gamma )\\bigr )^{-n}\\Omega _{M^{\\prime }}.$ Thus, the volume of $D\\subseteq M$ can be computed by integrating the redshift factor $\\bigl (1+z(M,M^{\\prime },\\gamma )\\bigr )^{-n}$ with respect to the Liouville measure on $\\mathbb {S}^*M^{\\prime }$ over the subset of all unit covectors on $M^{\\prime }$ corresponding to light rays $\\gamma $ passing through $D$ .",
"For constant redshift, (REF ) reduces to $\\mathrm {Vol}_{M}(D) = \\frac{1}{\\mathfrak {c}_n (1+z)^n}\\int \\limits _{M^{\\prime }} \\omega _{M^{\\prime }}(x^{\\prime },\\mathfrak {L}^D)\\, dV_{M^{\\prime }}(x^{\\prime }).$ Note that if $M$ lies in the past of $M^{\\prime }$ , then $\\omega _{M^{\\prime }}(x^{\\prime },\\mathfrak {L}^D)$ may be interpreted as the solid angle at $x^{\\prime }\\in M^{\\prime }$ subtended by $D\\subseteq M$ .",
"Figure: Cauchy surfaces and light rays (n=2n=2).Example 3.4 Let now $\\mathfrak {L}^{DD^{\\prime }}:= \\mathfrak {L}^D \\cap \\mathfrak {L}^{D^{\\prime }} = \\lbrace \\gamma \\in \\mathfrak {N}_X\\mid \\gamma \\cap D\\ne \\varnothing , \\gamma \\cap D^{\\prime }\\ne \\varnothing \\rbrace $ be the set of all light rays intersecting $D\\subseteq M$ and $D^{\\prime }\\subseteq M^{\\prime }$ .",
"(Example REF is a special case of this situation with $D^{\\prime }=M^{\\prime }$ .)",
"Then $\\iota _M(\\mathfrak {L}_x^{DD^{\\prime }}) ={\\left\\lbrace \\begin{array}{ll}\\iota _M(\\mathfrak {L}_x^{D^{\\prime }}), & x\\in D,\\\\\\varnothing , & x\\in M\\setminus D,\\end{array}\\right.",
"}$ and similarly $\\iota _{M^{\\prime }}(\\mathfrak {L}_{x^{\\prime }}^{DD^{\\prime }}) ={\\left\\lbrace \\begin{array}{ll}\\iota _{M^{\\prime }}(\\mathfrak {L}_{x^{\\prime }}^{D}), & x^{\\prime }\\in D^{\\prime },\\\\\\varnothing , & x^{\\prime }\\in M^{\\prime }\\setminus D^{\\prime }.\\end{array}\\right.",
"}$ Hence, it follows from (REF ) that $\\int \\limits _{D} \\omega _M(x,\\mathfrak {L}^{D^{\\prime }})\\, dV_M(x) =\\int \\limits _{D^{\\prime }} dV_{M^{\\prime }}(x^{\\prime }) \\int \\limits _{\\iota _{M^{\\prime }}(\\mathfrak {L}^{D}_{x^{\\prime }})} \\bigl (1+z(M,M^{\\prime },\\gamma )\\bigr )^{-n} d\\omega _{x^{\\prime }}.$ In the case of constant redshift $z$ , we obtain $\\int \\limits _{D} \\omega _M(x,\\mathfrak {L}^{D^{\\prime }})\\, dV_M(x)= \\frac{1}{(1+z)^n}\\int \\limits _{D^{\\prime }} \\omega _{M^{\\prime }}(x^{\\prime },\\mathfrak {L}^{D})\\, dV_{M^{\\prime }}(x^{\\prime }).$ If $M$ is in the past of $M^{\\prime }$ , then $\\omega _{M^{\\prime }}(x^{\\prime },\\mathfrak {L}^{D})$ is the solid angle subtended by $D$ at $x^{\\prime }$ as in Example REF and $\\omega _M(x,\\mathfrak {L}^{D^{\\prime }})$ is the solid angle at $x\\in M$ spanned by rays emitted from $x$ and received in $D^{\\prime }$ , see Fig.",
"REF ."
]
] | 1709.01741 |
[
[
"Neutrinos from beta processes in a presupernova: probing the isotopic\n evolution of a massive star"
],
[
"Abstract We present a new calculation of the neutrino flux received at Earth from a massive star in the $\\sim 24$ hours of evolution prior to its explosion as a supernova (presupernova).",
"Using the stellar evolution code MESA, the neutrino emissivity in each flavor is calculated at many radial zones and time steps.",
"In addition to thermal processes, neutrino production via beta processes is modeled in detail, using a network of 204 isotopes.",
"We find that the total produced $\\nu_{e}$ flux has a high energy spectrum tail, at $E \\gtrsim 3 - 4$ MeV, which is mostly due to decay and electron capture on isotopes with $A = 50 - 60$.",
"In a tentative window of observability of $E \\gtrsim 0.5$ MeV and $t < 2$ hours pre-collapse, the contribution of beta processes to the $\\nu_{e}$ flux is at the level of $\\sim90\\%$ .",
"For a star at $D=1$ kpc distance, a 17 kt liquid scintillator detector would typically observe several tens of events from a presupernova, of which up to $\\sim 30\\%$ due to beta processes.",
"These processes dominate the signal at a liquid argon detector, thus greatly enhancing its sensitivity to a presupernova."
],
[
"Introduction",
"The advanced evolution of massive stars – that culminates in their collapse, and possible explosion as supernovae – has been observed so far only in the electromagnetic band.",
"Completely different messengers, the neutrinos, dominate the star's energy loss from the core carbon burning phase onward, and, with their fast diffusion time scale, they set the very rapid pace (from months to hours) of the latest stages of nuclear fusion (presupernova).",
"These neutrinos have never been detected; their observation in the future would offer a unique and direct probe of the physical processes that lead to stellar core collapse.",
"In a star's interior, neutrinos are produced via a number of thermal processes – mostly pair production – and via $\\beta $ -processes, i.e., electron/positron captures on nuclei and nuclear decay.",
"The neutrino flux from thermal processes mainly depends on the thermodynamic conditions in the core.",
"The neutrino flux from $\\beta $ reactions have a stronger dependence on the isotopic composition, and thus on the complex network of nuclear reactions that take place in the star.",
"In this respect, the two classes of production, thermal and $\\beta $ , carry complementary information.",
"At this time, the study of the thermal neutrino flux from a presupernova star is fairly mature.",
"Exploratory studies in 2003-2010 [41], [42], [28], [40] showed that they can be detected in the largest neutrino detectors for a star at a distance $D \\mathrel {\\hbox{$<$}\\hbox{$\\sim $}}$ 1$ kpc.",
"Later, detailed descriptions of the thermal processes \\cite {Ratkovic:2003td, Odrzywolek:2007xp, Dutta:2003ny, Misiaszek:2005ax} have been applied to state of the art numerical simulations of stellar evolution, to obtain the time dependent presupernova\\ neutrino\\ flux expected at Earth \\cite {Kato:2015faa,Yoshida:2016imf}.",
"The potential of presupernova\\ neutrinos\\ as an early warning of an imminent nearby supernova\\ was emphasized \\cite {Yoshida:2016imf}.$ For the neutrinos from $\\beta $ processes (henceforth $\\beta $ p), the status is very different.",
"Dedicated studies have developed much more slowly, as it was recognized early on [39], [40] that they required a complex numerical study of realistic stellar models with large nuclear networks.",
"In a recent publication [43], we have approached the challenge of modeling the $\\beta $ p in detail – in addition to the thermal processes – for a realistic, time evolving star simulated with the MESA software instrument [44], [45], [46].",
"The 204 isotope nuclear network of MESA, fully coupled to the hydrodynamics during the entire calculation, made it possible to obtain, for the first time, consistent and detailed emissivities and energy spectra for the $\\beta $ neutrinos, at sample points inside the star at selected times pre-collapse.",
"It was found that $\\beta $ p contribute strongly to the total neutrino emissivities, and even dominate at late times and in the energy window relevant for detection ($E \\mathrel {\\hbox{$>$}\\hbox{$\\sim $}}$ 2$ MeV or so).Using an independent numerical simulation, with a combination of nuclear network and arguments of statistical equilibrium, \\cite {Kato:2017ehj} reached similar conclusions, and calculated the rates of events expected in neutrino\\ detectors as a function of time as well as total numbers of events.$ In this paper, we further extend the study of presupernova neutrinos, with emphasis on a realistic, consistent description of the flux from $\\beta $ p. For two progenitor stars, evolved with MESA, the time-dependent neutrino emissivities for different production processes are integrated over the volume of emission, so to obtain the neutrino luminosities and energy spectra expected at Earth for each neutrino flavor.",
"For several time steps leading to the collapse, the isotopes that dominate the $\\beta $ p emission, for both neutrinos and antineutrinos, are identified.",
"We discuss the prospects of detectability, and how they depend on the distance to the star, ranging from the nearby Betelgeuse to progenitors as far as the horizon of detectability, beyond which no observable signal is expected.",
"In the discussion, the main guaranteed detector backgrounds are taken into account.",
"The paper is structured as follows.",
"In Sec.",
", a concise summary of our simulation is given.",
"Sec.",
"gives the results for the neutrino flux and energy spectrum produced in a presupernova star as a function of the time pre-collapse.",
"Sec.",
"shows the expected neutrino flux at Earth, with a brief discussion of oscillations effects and detectability.",
"A discussion follows in Sec.",
"."
],
[
"Neutrino production and stellar evolution ",
"We simulate the evolution of two stars of initial masses $M=15, 30~\\mathrel {{M_\\odot }}$ (with $\\mathrel {{M_\\odot }}$ the mass of the Sun), from the pre-main sequence phase to core collapse, using MESA r7624 [44], [45], [46], for which inlists and stellar models used are publicly available http://mesastar.org.",
"The MESA runs used here are the same as those in [15], where technical details can be found.",
"Each star is modeled as a single, non-rotating, non-mass losing, solar metallicity object.",
"The calculation stops at the onset of core collapse, which is defined as the time when any part of the star exceeds an infall velocity of 1000 $\\rm {km}\\ \\rm {s}^{-1}$ .",
"We also set the maximum mass of a grid zone to be ${\\Delta {\\mathrm {M}}_{\\rm {max}}}=0.1\\mathrel {{M_\\odot }}$ .",
"The simulations employ a large, in-situ, nuclear reaction network, mesa_204.net, consisting of 204 isotopes up to $^{66}$ Zn, and including all relevant reactions.",
"They also include effects of convective overshoot, semi-convection and thermohaline mixing on the chemical mixing inside the star.",
"In output, MESA gives the time- and space-profiles of the temperature $T$ , matter density $\\rho $ , isotopic composition, and electron fraction, $Y_e$ .",
"These quantities are then used in a separate calculation to derive the neutrino fluxes, as outlined in our previous work [43].",
"For brevity, here only the main elements are summarized.",
"We calculate the spectra for $\\mathrel {{\\nu _e}}$ , $\\mathrel {{\\bar{\\nu }}_e}$ and $\\mathrel {{\\nu _\\mu }},\\mathrel {{\\nu _\\tau }},\\mathrel {{\\bar{\\nu }}_\\mu },\\mathrel {{\\bar{\\nu }}_\\tau }$ (collectively called $\\mathrel {{\\nu _x}}$ and $\\mathrel {{\\bar{\\nu }}_x}$ from here on) resulting from $\\beta $ processes and pair annihilation.",
"Other thermal processes [43] were found to be by far subdominant in the late time neutrino emission from the whole star, and were neglected for simplicity.",
"In the calculation of spectra for the $\\beta $ p, the relevant rates are taken from the nuclear tables of Fuller, Fowler and Newman (FFN) [17], [19], [18], [20], Oda et al.",
"(OEA) [37] and Langanke and Martinez-Pinedo (LMP) [29].",
"For isotopes that appear in multiple tables, the rates of LMP are given precedence, followed by OEA, then finally FFN.",
"This order of precedence is the same as used in MESA [44].",
"As described in FFN [17], [19], [18], [20], the rate of decay from a parent nucleus in the excited state $i$ to a daughter in excited state $j$ is $\\lambda _{ij} = \\log {2}\\frac{f_{ij}(T,\\rho ,\\mu )}{\\langle ft \\rangle _{ij}}.$ Here, $\\langle ft \\rangle _{ij}$ is the comparative half-life, containing all of the nuclear structure information and the weak interaction matrix element.",
"The function $f_{ij}$ is the phase space of the incoming and outgoing electrons or positrons.",
"It uniquely determines the shape of the resulting neutrino spectrum, because the outgoing neutrinos are presumed to be free streaming with no Pauli blocking.",
"Since the shape of the spectrum is entirely determined by the phase space, we can define the spectrum as $\\phi = N \\, f_{ij}(T,\\rho ,\\mu )$ , where $N$ is a normalization factor.",
"We then write the spectra for the $\\beta $ p neutrinos for a single isotope as: $\\phi _{EC,PC} &= & N \\frac{E_{\\nu }^{2}(E_{\\nu } - Q_{ij})^{2}}{1 + \\exp {((E_{\\nu } - Q_{ij} - \\mu _{e})/kT)}} \\Theta (E_{\\nu } - Q_{ij} - m_{e})\\\\\\phi _{\\beta } & = & N \\frac{E_{\\nu }^{2}(Q_{ij} - E_{\\nu })^{2}}{1 + \\exp {((E_{\\nu } - Q_{ij} + \\mu _{e})/kT)}} \\Theta (Q_{ij} - m_{e} - E_{\\nu }),$ where EC (PC) is for electron (positron) capture, and $\\beta $ is for decay.",
"The chemical potential $\\mu _{e}$ is defined including the rest mass such that $\\mu _{e^{-}} = -\\mu _{e^{+}}$ .",
"The parameter $Q_{ij} = M_{p} - M_{d} + E_{i} - E_{j}$ is the $Q$ -value for the transition, where $M_{p,d}$ is the mass of the parent (daughter) and $E_{i,j}$ is the excitation energy.",
"The rates reported in the FFN, OEA, and LMP tables are actually the sum of all possible transitions, so $\\lambda = \\Sigma \\lambda _{ij}$ .",
"So rather than finding individual values for each $Q_{ij}$ , we follow the method of [29] and [43], and instead find an effective $Q$ -value.",
"We calculate the spectrum and its average energy, then adjust the $Q$ -value until the average energy in the rate tables is reproduced.",
"Note that the tabulated average energy is a combined value for both decay and capture, therefore $Q$ is the same for both processes.",
"The parameter $N$ in Eqs.",
"(REF )-() is a normalization factor, defined to reproduce the tabulated rates $\\lambda _i$ for isotope $i$ : $\\lambda _{i} = \\int _{0}^{\\infty } \\phi _{i} dE_{\\nu } \\,\\,\\,\\,\\,\\,\\, i = EC, PC, \\beta ^{\\pm }.$ The total spectrum of neutrinos from $\\beta $ p (comprehensive of both capture and decay processes) is given by the sum over all the isotopes, weighed by their abundances $X_k$ : $\\left(\\frac{dR_{\\beta }}{dE}\\right)_{\\nu _{e},\\bar{\\nu }_{e}} = \\sum _{k} X_{k} \\phi _{k} \\frac{\\rho }{m_{p}A_{k}}.$ Here $m_{p}$ is the mass of the proton, and $A_{k}$ is the atomic number of the isotope $k$ .",
"For neutrinos produced via pair annihilation, the emission rate, differential in the neutrino energy, is $\\left(\\frac{dR}{dE}\\right)_{\\nu _{\\alpha },\\bar{\\nu _{\\alpha }}} = \\int d^{3}{p}_{1}d^{3}{p}_{2}\\left(\\frac{d\\sigma }{dE}\\right)_{\\nu _{\\alpha },\\bar{\\nu _{\\alpha }}} f_{1}f_{2},$ where $f_{i}$ is the Fermi-Dirac distribution function for the electron and positron, and $d\\sigma ~ = \\frac{1}{2\\mathcal {E}_{1}} \\frac{1}{2\\mathcal {E}_{2}} \\frac{1}{(2\\pi )^{2}} \\delta ^{4}(P_{1} + P_{2} - Q_{1} - Q_{2}) \\frac{d^{3}q_{1}}{2 E_{1}} \\frac{d^{3}q_{2}}{2 E_{2}} \\langle | \\mathcal {M} |^{2} \\rangle ~ .$ Here, $$ is the relative velocity of the electron-positron pair, $P_{1,2} = (\\mathcal {E}_{1,2}, \\bf {p}_{1,2})$ is the four-momentum of the electron (positron), and $Q_{1,2} = (E_{1,2}, \\bf {q}_{1,2})$ is the four-momentum of the (anti-)neutrino.",
"The squared matrix element, as given by [33], is $\\langle | \\mathcal {M} |^{2} \\rangle = 8 G_{F}^{2} \\left( \\left( C_{A}^{f} - C_{V}^{f}\\right)^{2}\\left(P_{1}\\cdot Q_{1}\\right) \\left( P_{2}\\cdot Q_{2}\\right)+ \\left( C_{A}^{f} + C_{V}^{f}\\right)^{2}\\left(P_{2}\\cdot Q_{1} \\right) \\left(P_{1}\\cdot Q_{2} \\right)+ m_{e}^{2} \\left( C_{V}^{2} - C_{A}^{2}\\right) \\left(Q_{1}\\cdot Q_{1}\\right)\\right).$ Here, $C_{V}^{e} = 1/2 + 2\\sin ^{2}{\\theta _{W}}$ , $C_{A}^{e} = 1/2$ , and $C_{V,A}^{x} = C_{V,A}^{e} - 1$ .",
"In [43], Eq.",
"(REF ) and (REF ) were used to calculate the spectra for selected times and points inside a star.",
"Here, we integrate over the emission region, to obtain the number luminosity – i.e., the number of neutrinos that leave the star per unit time – and the differential luminosity: $\\frac{dL^{\\nu _{\\alpha }}_{N}}{dE} & = & 4 \\pi \\int {\\left(\\frac{dR}{dE}\\right)_{\\nu _{\\alpha }} r^{2} dr}, \\\\L^{\\nu _{\\alpha }}_{N} & = & \\int {\\frac{dL^{\\nu _{\\alpha }}_{N}}{dE} dE}.$"
],
[
"Results: time profiles and spectra",
"Results were obtained for discrete times (time-to-collapse, $\\tau _{CC}$ ) between the onset of core oxygen burning and the onset of core collapse.",
"An interval of two hours prior to collapse – when the chance for detection is greatest – was mapped in greater detail.",
"Specifically, for the $15\\mathrel {{M_\\odot }}$ (30 $\\mathrel {{M_\\odot }}$ ) model, we took a total of 21 (26) time instants, of which 15 (20) in the final two hours.",
"All the calculated times are shown in Fig.",
"REF , while a subset of seven times is investigated in more detail in other figures and tables.",
"Calculations of numbers of events in detectors use all the calculated times within the last two hours."
],
[
"A neutrino narrative: time-evolving luminosities",
"Let us examine the thermal history of the two progenitors, and how it is reflected in the neutrino luminosity.",
"Fig.",
"REF shows the star's trajectory in the plane of central temperature and central density, $(T_{c},~ \\rho _{c})$ , as the time evolves.",
"It also shows the evolution for the neutrino number luminosities, $L^{\\nu _{\\alpha }}_{N}$ , for different production channels, and the approximate times of ignition of the various fuels.",
"From the figure, it appears that the evolution of the two stars is generally similar, the main difference being that the more massive progenitor evolves faster and is overall brighter in neutrinos.",
"In particular, for the 15 $\\mathrel {{M_\\odot }}$ (30 $\\mathrel {{M_\\odot }}$ ) star the burning stages for the two stars proceed as follows: at $\\tau _{CC}\\approx 10^4$ hrs ($\\tau _{CC}\\approx 10^3$ hrs), oxygen ignition takes place in the core, and proceeds convectively until it ceases at $\\tau _{CC}\\approx 10^3$ hrs ($\\tau _{CC}\\approx 10^2$ hrs).",
"Then, an oxygen shell is ignited and burns until $\\tau _{CC}\\approx 5\\times 10^2$ hrs ($\\tau _{CC}\\approx 10$ hrs).",
"Eventually, silicon burning is ignited in the core and proceeds until $\\tau _{CC}\\approx 10$ hrs ($\\tau _{CC}\\approx 5$ hrs).",
"At that point the star transitions to shell silicon burning, which proceeds until collapse.",
"Interestingly, the 15 $\\mathrel {{M_\\odot }}$ star has an intermediate phase (which is absent in the more massive progenitor) before core silicon burning: a second, off center oxygen burning stage, which lasts until $\\tau _{CC}\\approx 10^2$ hrs.",
"In Fig.",
"REF , we can see how the luminosity of $\\mathrel {{\\nu _e}}$ from $\\beta $ p grows faster than that of thermal processes.",
"For the $15 \\mathrel {{M_\\odot }}$ (30 $\\mathrel {{M_\\odot }}$ ) case, it amounts to $\\sim $ 30$\\%$ ($\\sim $ 10$\\%$ ) of the contribution from pair annihilation at the onset of oxygen burning; it becomes comparable to pair annihilation at $\\tau _{CC}\\approx 6$ min ($\\tau _{CC}\\approx 7$ s), increasing to almost an order of magnitude greater ($\\sim $ 30 times greater) at the onset of core collapse.",
"The luminosity of $\\mathrel {{\\bar{\\nu }}_e}$ from $\\beta $ p follows a more complicated pattern, tracing more closely the phases of stellar evolution.",
"It drops after core oxygen burning ends, and begins increasing again after silicon core ignition.",
"The total $\\mathrel {{\\bar{\\nu }}_e}$ emission is always dominated by pair annihilation, although the disparity decreases as the stars approach core collapse.",
"At the onset of core collapse, the $\\beta $ p contribution is approximately 40$\\%$ ($\\sim 20\\%$ ) of the pair process for the 15 $\\mathrel {{M_\\odot }}$ (30 $\\mathrel {{M_\\odot }}$ model) model.",
"A unique feature of the 15 $\\mathrel {{M_\\odot }}$ model is a short sharp drop in the luminosities of all neutrino species, shortly after shell silicon burning begins, followed by a smooth increase.",
"This peak is absent in the time profiles of the 30 $\\mathrel {{M_\\odot }}$ model, for which the time profiles are smoother.",
"This difference can be traced to differences in the core carbon burning phases of the two stars, which proceed convectively for the $M=15\\mathrel {{M_\\odot }}$ case and radiatively for the $M=30\\mathrel {{M_\\odot }}$ model The dividing line between the two paths is given by the central carbon mass fraction, with critical value X($^{12}$ C)$\\sim $ 20% [53], [51], [54].",
"For the MESA inputs used here, solar metallicity models with Zero Age Main Sequence masses below $\\simeq ~20~\\mathrel {{M_\\odot }}$ have X($^{12}$ C)$\\gtrsim $ 20% and thus undergo convective core Carbon-burning.",
"See, e.g., [47].. For convective core C-burning, efficient neutrino emission decreases the entropy.",
"This entropy loss is missing in the radiative carbon burning case, causing all subsequent burning stages to take place at higher entropy, higher temperatures, and lower densities.",
"In these conditions, density gradients are smaller and extend to larger radii, thus explaining the smoother profiles of the 30 $\\mathrel {{M_\\odot }}$ model.",
"We notice that the neutrino luminosity from pair annihilation increases more slowly in the last few hours of evolution.",
"This can be understood considering that the emissivity for pair annihilation is nearly independent of the density for fixed temperature [23], and therefore directly reflects the moderate increase of the temperature (Fig.",
"REF , right panes) over hour-long periods.",
"Generally, the patterns found here are consistent with those in the recent work by [25].",
"The main difference is in the $\\mathrel {{\\bar{\\nu }}_e}$ luminosity from $\\beta $ p, which in our work is always subdominant, while in Kato et al.",
"it dominates over pair annihilation starting at $\\tau _{CC}\\sim 0.5$ hrs.",
"This discrepancy could be due to the nuclear networks used: in our work, the network mesa_204.net is evolved self-consistently within MESA to obtain mass fractions, and tabulated $\\beta $ p rates from FFN, ODA, and LMP are used (see Sec.",
"and [43]).",
"Instead, Kato et al.",
"calculate mass fractions using nuclear statistical equilibrium, and incorporate many neutron rich isotopes, with rates taken from tables by Tachibana and others [50], [56], [49], [26], [27], which they adapted to the stellar environment of interest (the original tables are for terrestrial conditions)."
],
[
"Neutrino spectra: isotopic contributions",
"Let us now discuss the neutrino energy spectra and the effect of the $\\beta $ p on them.",
"Fig.",
"REF gives the number luminosities, differential in energy, of each neutrino species at seven selected times of the evolution (see Tables REF and REF for exact values).",
"Separate panels show the percentages of the $\\mathrel {{\\nu _e}}$ and $\\mathrel {{\\bar{\\nu }}_e}$ luminosities that originate from $\\beta $ p alone.",
"We observe that the $\\mathrel {{\\nu _e}}$ and $\\mathrel {{\\bar{\\nu }}_e}$ spectra are smooth at all times, as the integration over the emission volume averages out spectral structures due to $\\beta $ p from individual isotopes, that appear at early times and in certain shells [43].",
"The spectra have a maximum at $E \\sim 1-3$ MeV depending on the time.",
"At $E \\mathrel {\\hbox{$>$}\\hbox{$\\sim $}}$ 4$ MeV, the $ e$ spectrum is dominated by the $$p\\ at all the times of interest (fraction of $$p\\ larger than $ 60%$).",
"At all energies, the $$p\\ contribution increases with time, and it exceeds a 90\\% fraction at collapse in the entire energy interval, consistently with fig.",
"\\ref {lumVsTime}.$ The percentage of $\\mathrel {{\\bar{\\nu }}_e}$ from $\\beta $ p is lower, overall.",
"Over time, it increases at low energy ($E \\mathrel {\\hbox{$<$}\\hbox{$\\sim $}}$ 1$ MeV), reaching a $ 50%$ fraction at $ E = 1$ MeV at collapse, and decreases at higher energy.",
"This latter behavior reflects the fact that the electron degeneracy increases with time, thus reducing the phase space for electrons in the final state due to $ -$ decay.",
"The lower number density of positrons (relative to electrons) available for capture also explains the suppression of the $$p\\ $ e$ flux relative to $ e$.$ Figure: Neutrino spectra at selected times pre-collapse for a 15 M ⊙ \\mathrel {{M_\\odot }} star (left) and 30 M ⊙ \\mathrel {{M_\\odot }} star (right).",
"Each set of curves shows times t 1 t_{1} through t c t_{c} (lower to upper curves).",
"The exact values of these times are given in Tables and .",
"The dashing styles in the legend apply to all panels.",
"The first (third) panel shows the differential luminosity for electron (anti-)neutrinos.",
"The second (fourth) panel shows the percentage of that luminosity arising from β\\beta processes.",
"The bottom panel shows the ν x \\nu _{x}/ν ¯ x \\bar{\\nu }_{x} luminosity from pair annihilation.Figure: The time evolution of the neutrino luminosity for a 15 M ⊙ \\mathrel {{M_\\odot }} star (left) and a 30 M ⊙ \\mathrel {{M_\\odot }} star (right), differential in energy, at selected energies.",
"The contributions of the thermal and beta processes are shown separately.",
"Solid lines represent neutrinos while dashed lines show the antineutrino contributions.A complementary view of these results is given in Fig.",
"REF , which shows the time evolution of the neutrino luminosities differential in $E$ , at selected values of $E$ .",
"We see that, for the 15 $\\mathrel {{M_\\odot }}$ model, the neutrino luminosity from $\\beta $ p has a peak at $\\tau _{CC}\\approx 2$ hrs, followed by a minimum and a subsequent fast increase.",
"This the same feature that appears in the total luminosities for the same progenitor (Fig.",
"REF ), and is more pronounced at higher neutrino energy.",
"What can we learn from presupernova neutrinos about the isotopic evolution of a star?",
"To start addressing this question, we investigated what nuclear isotopes contribute the most to the $\\mathrel {{\\nu _e}}$ and $\\mathrel {{\\bar{\\nu }}_e}$ fluxes in the detectable region of the spectrum.",
"This is addressed in Tables REF and REF , where, for selected times in the $\\tau _{CC}\\mathrel {\\hbox{$<$}\\hbox{$\\sim $}}$ 2$ hrs, we list the five strongest contributors to both the total luminosity and the luminosity in the window $ E 2$ MeV (where detectors are most sensitive, see Sec.",
"\\ref {sub:events}).",
"The Tables also give the fraction of the $$p\\ number luminosity that each isotope produces.",
"These tables give us a view into how the isotopic makeup of the star evolves over time.$ Let us first describe results for the 15 $\\mathrel {{M_\\odot }}$ model.",
"In it, silicon shell burning begins at $\\tau _{CC}\\approx 10$ hrs (Sec.",
"REF ).",
"Thus in the last two hours before collapse, the isotopic composition is already heavy.",
"The top five dominant isotopes – for both $\\mathrel {{\\nu _e}}$ and $\\mathrel {{\\bar{\\nu }}_e}$ production – are those with $A\\approx 50-60$ such as iron, manganese, cobalt and chromium.",
"At very late times, $t_{6}$ and $t_{7}$ , photodissociation of nuclei becomes efficient, producing free nucleons.",
"We find that free protons are the strongest contributor to the $\\mathrel {{\\nu _e}}$ luminosity at those times.",
"By summing the contributions listed in Table REF , we see that the five dominant isotopes are producing a large percentage of the luminosity: the $\\mathrel {{\\nu _e}}$ luminosity from the five dominant isotopes is between $\\sim 35 - 45\\%$ for the total energy range, ending with $\\sim 50\\%$ at $t_{c}$ .",
"For $\\mathrel {{\\bar{\\nu }}_e}$ , $78\\%$ of the luminosity is from the top five isotopes at $t_{1}$ .",
"The percentage gradually decreases to $37\\%$ at $t_{c}$ .",
"The results for the 30 $\\mathrel {{M_\\odot }}$ model (Table REF ), reflect its faster evolution.",
"For this star, silicon shell burning begins at $\\tau \\approx 5$ hrs, therefore, it is expected that at $\\tau _{CC}\\sim 2$ hrs, there might still be a contribution from medium-mass nuclei.",
"Indeed, the largest contribution to the $\\mathrel {{\\bar{\\nu }}_e}$ luminosity at $t_{1}$ for the 30 $\\mathrel {{M_\\odot }}$ star is from $^{28}$ Al.",
"Subsequent times show the same pattern as the 15 $\\mathrel {{M_\\odot }}$ model, with mainly isotopes with $A\\approx 50-60$ dominating.",
"We see that free protons appear in the top five isotopes at $t_{3} \\approx 0.05$ hrs ($\\simeq 3$ min) pre-collapse, and are the most dominant contributor from $t_{5}$ on.",
"Free neutrons also appear in the top-five list for $\\mathrel {{\\bar{\\nu }}_e}$ above $E\\ge 2$ MeV at $t_{c}$ .",
"For $\\mathrel {{\\nu _e}}$ , the total contribution of the top-five isotopes is $66\\%$ at $t_{1}$ , drops to about $40\\%$ later, then climbs again to end at $75\\%$ at $t_{c}$ , of which $\\sim 65\\%$ is from free protons.",
"For $\\mathrel {{\\bar{\\nu }}_e}$ , the total fraction is $85\\%$ at $t_{1}$ , and gradually decreases to $40\\%$ at $t_{c}$ .",
"The fact that, in both models, large portions of the $\\mathrel {{\\nu _e}}$ and $\\mathrel {{\\bar{\\nu }}_e}$ luminosities come from a relatively small number of isotopes is promising for future work: it means that efforts to produce more precise neutrino spectra could become more manageable, as they can be targeted to the subset of isotopes identified in Tables REF -REF .",
"Table: List of the five isotopes that most contribute to the produced ν e \\mathrel {{\\nu _e}} (ν ¯ e \\mathrel {{\\bar{\\nu }}_e}) presupernova luminosity in the 15 M ⊙ \\mathrel {{M_\\odot }} model– total and at E≥2E\\ge 2 MeV – at selected times.",
"They are listed in order of decreasing ν e \\mathrel {{\\nu _e}} (ν ¯ e \\mathrel {{\\bar{\\nu }}_e}) luminosity.",
"The number below each isotope is the fraction of the β\\beta p number luminosity produced by that isotope.Table: Same as Table but for the 30 M ⊙ \\mathrel {{M_\\odot }} model."
],
[
"Oscillations of presupernova neutrinos ",
"The flavor composition of the presupernova neutrino flux at Earth differs from the one at production, due to flavor conversion (oscillations).",
"In terms of the original, unoscillated flavor luminosities, $F^0_\\alpha = dL^{\\nu _\\alpha }_{N}/dE$ ($\\alpha = e, \\bar{e}, x$ ), the fluxes of each neutrino species at Earth can be written as $F_e = p F^0_e + (1-p) F^0_x ~, \\hspace{28.45274pt}2 F_x = (1-p) F^0_e + (1+p) F^0_x~,$ where $F_x$ is defined so that the total flux is $F_e + 2 F_x = F^0_e + 2 F^0_x$ , and the geometric factor $(4 \\pi D^2)^{-1}$ , due to the distance $D$ to the star, is omitted for brevity.",
"An expression analogous to eq.",
"(REF ) holds for antineutrinos, with the notation replacements $e \\rightarrow \\bar{e}$ and $p \\rightarrow \\bar{p}$ .",
"The quantities $p$ and $\\bar{p}$ are the $\\mathrel {{\\nu _e}}$ and $\\mathrel {{\\bar{\\nu }}_e}$ survival probabilities.",
"They have been studied extensively for a supernova neutrino burst (see, e.g.",
"[13] and references therein), and at a basic level for presupernova neutrinos [7], [24], [25].",
"Similarly to the burst neutrinos, presupernova neutrinos undergo adiabatic, matter-driven, conversion inside the star.",
"The probabilities $p$ and $\\bar{p}$ are are independent of energy and of time.",
"They are given by the elements of the neutrino mixing matrix, $U_{\\alpha i}$ , in a way that depends on the (still unknown) neutrino mass hierarchy; given the masses $m_i$ ($i=1,2,3$ ), the standard convention defines the normal hierarchy (NH) as $m_3 > m_2$ and vice versa for the inverted hierarchy (IH).",
"For each possibility, we have (see e.g., [31], [25]): $p = {\\left\\lbrace \\begin{array}{ll}|U_{e3}|^2 \\simeq 0.02 & \\text{ NH } \\\\|U_{e2}|^2 \\simeq 0.30 & \\text{ IH }~\\end{array}\\right.}",
"\\hspace{22.76228pt}\\bar{p}= {\\left\\lbrace \\begin{array}{ll}|U_{e1}|^2 \\simeq 0.68 & \\text{ NH } \\\\|U_{e3}|^2 \\simeq 0.02 & \\text{ IH }~.\\end{array}\\right.}",
"$ For simplicity here we do not consider other oscillation effects, namely collective oscillations inside the star and oscillations in the matter of the Earth.",
"The former are expected to be negligible due to the relatively low presupernova neutrino luminosity (compared to the supernova burst), and the latter are suppressed (a $\\sim 1\\%$ effect or less) at the energies of interest here (see e.g., [52]).",
"Eq.",
"(REF ) shows that for the NH the $\\mathrel {{\\nu _e}}$ flux at Earth receives only a very suppressed contribution from the original $\\mathrel {{\\nu _e}}$ .",
"The suppression is weaker for the IH, and therefore – considering that $F^0_x \\ll F^0_e$ – the flux $F_e$ should be much larger in this case.",
"For the $\\mathrel {{\\bar{\\nu }}_e}$ flux, a smaller difference between NH and IH is expected, due to $F^0_x$ and $F^0_{\\bar{e}}$ being comparable (Fig.",
"REF )."
],
[
"Window of observability",
"A detailed discussion of the detectability of presupernova neutrinos is beyond the scope of this paper, and is deferred to future work.",
"Here general considerations are given on the region, in the time and energy domain, where detection might be possible – depending on the distance to the star – and the numbers of events expected in neutrino detectors are given.",
"One can define a conceptual window of observability (WO) as the interval of time and energy where the presupernova flux exceeds all the neutrino fluxes of other origin that are (i) present in a detector at all times, and (ii) indistinguishable from the signal.",
"These fluxes are guaranteed backgrounds, regardless of the details of the detector in use; to them, detector-specific backgrounds will have to be added.",
"Therefore the WO defined here represent a most optimistic, ideal situation.",
"Because observations at neutrino detectors are generally dominated by either $\\mathrel {{\\nu _e}}$ or $\\mathrel {{\\bar{\\nu }}_e}$ , let us discuss the WOs for these two species.",
"In the case of $\\mathrel {{\\nu _e}}$ , the largest competing flux is due to solar neutrinos [8].",
"For $\\mathrel {{\\bar{\\nu }}_e}$ , we consider fluxes from nuclear reactors and from the Earth's natural radioactivity (geoneutrinos) [16].",
"For both $\\mathrel {{\\nu _e}}$ and $\\mathrel {{\\bar{\\nu }}_e}$ , other background fluxes are from atmospheric neutrinos and from the diffuse supernova neutrino background (DSNB, due to all the supernova neutrino bursts in the universe).",
"At the times and energies of interest, however, these are much lower than the solar, reactor, geoneutrinos and presupernova fluxes, and therefore they will be neglected from here on.",
"The reactor neutrino and geoneutrino spectra depend on the location of the detector in relation to working reactors and local geography.",
"The reactor spectrum we use was calculated for the Pyhäsalmi mine in Finland [34], [55], and includes oscillations.",
"The geoneutrino spectrum is generic, and includes vacuum oscillations only, with survival probability at JUNO $\\bar{p}\\simeq 0.55$ for both NH and IH [52] Effects from MSW oscillation are shown to be at the level of 0.3% [52], and therefore can be neglected.",
"Figure: The flux of ν e \\mathrel {{\\nu _e}} (left panels) and ν ¯ e \\mathrel {{\\bar{\\nu }}_e} (right panels) expected at Earth from a 15M ⊙ 15 \\mathrel {{M_\\odot }} star at distance D=1D=1 kpc, calculated at times t 1 t_{1} through t c t_{c} (lower to upper curves).",
"Shown are the cases of normal and inverted mass hierarchy (upper and lower rows respectively).",
"Competing neutrino fluxes from other sources are shown (see legend).",
"Oscillations are included in all cases.Figure: Same as Fig.",
"for a 30 M ⊙ \\mathrel {{M_\\odot }} star.Figs.",
"REF and REF shows the presupernova neutrino signal at Earth for a star at $D=1$ kpc.",
"It appears that, already two hours before collapse, the presupernova $\\mathrel {{\\nu _e}}$ flux emerges above solar neutrinos.",
"The WO becomes wider in energy as the presupernova flux increases with time.",
"An approximate WO is $\\tau _{CC} \\sim 0 - 2 $ hrs, and $E \\sim 1 - 8$ MeV, and it is larger for the IH and for the more massive progenitor, where the presupernova flux is higher.",
"We note that it may be possible to distinguish and subtract solar neutrinos effectively using their arrival direction, e.g., in neutrino-electron scattering events in water Cherenkov detectors [3].",
"With a $\\sim 10^4$ reduction in the solar background, the $\\mathrel {{\\nu _e}}$ WO would extend in energy and time, $\\tau _{CC} \\sim 0 - 24 $ hrs, and $E \\sim 0.5 - 10$ MeV.",
"For the same distance $D=1$ kpc, the WO for $\\mathrel {{\\bar{\\nu }}_e}$ is similar to that of $\\mathrel {{\\nu _e}}$ , but it is overall wider in energy, as the presupernova flux eventually exceeds the geoneutrino one at sub-MeV energy.",
"Approximately, the WO is $\\tau _{CC} \\sim 0 - 2 $ hrs, and $E \\sim 0.5 - 20$ MeV.",
"By increasing the distance $D$ , the WO becomes narrower; unless the background fluxes in Figs.",
"REF and REF are subtracted, it eventually closes completely for $D \\sim 30$ kpc.",
"This maximum distance – which is of the order of the size our galaxy – is independent of the specific detector considered.",
"We will see below that the actual horizon for observation is smaller for realistic detector masses.",
"It is possible that the next supernova in our galaxy will be closer than 1 kpc, thus offering better chances of presupernova neutrino observation.",
"A prime example is the red supergiant Betelgeuse ($\\alpha $ Orionis).",
"Betelgeuse has the largest angular diameter on the sky of any star apart from the Sun, and is the ninth-brightest star in the night sky.",
"As such, it has been well studied.",
"Betelgeuse is estimated to have a mass of 11 - 20 M$_{\\odot }$ [30], [36], [12], [35] ; it lies at a distance of 222$^{+48}_{-34}$ pc [21], [22], and has an age of 8 - 10 Myr, with $<$ 1 Myr of life left until core collapse [12], [22].",
"We find that for $D=200$ pc a presupernova neutrino signal would be practically background-free – in energy windows that are realistic for detection – for several hours, and the WO can extend up to $\\sim 10$ hours."
],
[
"Numbers of events, horizon",
"Let us now briefly discuss expected numbers of events at current and near future detectors of ${\\mathcal {O}}(10)$ kt scale or higher.",
"We consider the three main detection technologies: liquid scintillator (JUNO [5]), water Cherenkov (Super Kamiokande [1]) and liquid argon (DUNE [11]).",
"For each, we consider the dominant detection channel – that will account for the majority of the events in the detector – and the first subdominant process that is sensitive to $\\mathrel {{\\nu _e}}$ .",
"The latter will be especially sensitive to $\\mathrel {{\\nu _e}}$ from the $\\beta $ p. For water Cherenkov and liquid scintillator, the dominant detection process is inverse beta decay (IBD), $\\mathrel {{\\bar{\\nu }}_e}+ p \\rightarrow n + e^+$ , which bears some sensitivity to $\\mathrel {{\\bar{\\nu }}_e}$ from $\\beta $ p. The sensitivity to $\\mathrel {{\\nu _e}}$ from the $\\beta $ p is in the subdominant channel, neutrino-electron elastic scattering (ES), where the contribution of $\\mathrel {{\\nu _e}}$ is enhanced (compared to $\\mathrel {{\\nu _x}}$ ) by the larger cross section.",
"Note that the two channels, IBD and ES, can be distinguished in the detector, at least in part, due to their different final state signatures: neutron capture in coincidence for IBD, and the peaked angular distribution for ES (see, e.g., [10], [6]).",
"In Super Kamiokande, efficient neutron capture will be possible in the upcoming upgrade with Gadolinium addition [9].",
"In liquid scintillator (LS), the main detection processes are the same as in water, with the differences that LS offers little directional sensitivity, but has the advantage of a lower, sub-MeV energy threshold, which can capture most of the presupernova spectrum.",
"In liquid Argon (LAr), the dominant process is $\\mathrel {{\\nu _e}}$ Charged Current scattering on the Argon nucleus.",
"Therefore, LAr is, in principle, extremely sensitive to neutrinos from the $\\beta $ p. However, the relatively high energy threshold ($E_{th}\\sim 5 $ MeV [4]) is a considerable disadvantage compared to LS.",
"Table REF and REF summarize our results for the number of presupernova neutrino events expected above realistic thresholds during the last two hours precollapse.",
"The numbers of background events are not given, because they are affected by large uncertainties on the contributions of detector-specific backgrounds.",
"These ultimately depend on type of search performed, and have not been studied in detail yet for a presupernova signal Most background rejection studies have been performed for type of signals that are either constant in time or very short (e.g., a supernova burst).",
"A presupernova signal is intermediate, rising steadily over a time scale of hours.",
"This feature might require developing different approaches to cut backgrounds..",
"The tables confirm that a large liquid scintillator like JUNO has the best potential, due to its sensitivity at low energy, with $N \\sim 10 - 70$ events (depending on the type of progenitor) recorded from a star at $D=1$ kpc.",
"To these events, the contribution of the $\\beta $ p is at the level of $10-30\\%$ , and is larger for the inverted mass hierarchy, for which the $\\mathrel {{\\nu _e}}$ flux is larger, see Sec.",
"REF .",
"For Betelgeuse, a spectacular signal of more than 200 events in two hours could be seen.",
"One can define (optimistically) the horizon of the detector, $D_h$ , as the distance for which one signal event is expected.",
"We find that JUNO has a horizon $D_h \\sim 2-8$ kpc.",
"Although disadvantaged by the higher energy threshold, SuperKamiokande and DUNE can observe presupernova neutrinos for the closest stars.",
"For the most massive progenitor, SuperKamiokande could reach a horizon $D_h \\sim 1$ kpc; and record $N \\sim 5-60$ events for $D=0.2$ pc.",
"Of these, $\\sim 10-20\\%$ would be from $\\beta $ p. Looking farther in the future, the larger water Cherenkov detector HyperKamiokande [2] – with mass 20 times the mass of SuperKamiokande – might become a reality.",
"Assuming an identical performance as SuperKamiokande, HyperKamiokande will have a statistics of up to thousands of events, and a horizon of $\\sim 4-5$ kpc Due to its mass, HyperKamiokande will have a $\\sim $ 20 times higher level of background than SuperKamiokande, and, probably, a higher energy threshold.",
"Therefore, its performance will be worse, and the figures given here have to be taken as best case scenarios.. At DUNE, a detection is possible only for the closest stars; the number of events varies between $N \\sim 1$ and $ N \\sim 30$ , depending on the parameters, for $D=0.2$ kpc.",
"For the most optimistic scenario (the more massive progenitor and the inverted mass hierarchy), the horizon can reach $D_{h} \\sim 1$ kpc.",
"DUNE will observe a strong component due to $\\beta $ p , at the level of $\\sim 40-80\\%$ of the total signal.",
"Therefore, in principle LAr has the best capability to probe the isotopic evolution of supernova progenitors.",
"Table: Numbers of events expected in the two hours prior to collapse, for a presupernova with progenitor mass M=15M ⊙ M=15\\mathrel {{M_\\odot }}, at distance D=1D=1 kpc and the normal mass hierarchy.",
"The numbers in brackets refer to the inverted mass hierarchy.",
"Different columns give the numbers for different detection channels: the superscripts CCCC and elel refer respectively to the dominant charged current process (inverse beta decay or ν e \\mathrel {{\\nu _e}} absorption on the Ar nucleus) and to neutrino-electron scattering.",
"The subscript β\\beta indicates the contribution of the β\\beta processes to those two channels.",
"The total number of events is given in the last column.",
"The results for Betelgeuse (D=0.2D=0.2 kpc) can be obtained by rescaling by a factor of 25.Table: Same as Tab.",
", for the M=30M ⊙ M=30\\mathrel {{M_\\odot }} progenitor."
],
[
"Discussion",
"We have presented a new calculation of the total neutrino flux from beta processes in a presupernova star, inclusive of time-dependent emissivities and neutrino energy spectra.",
"This is part of a complete and detailed calculation of presupernova neutrino fluxes from most relevant processes – beta and thermal – done using the state of the art stellar evolution code MESA.",
"The beta neutrino flux is strongest in the $\\mathrel {{\\nu _e}}$ channel, where it is comparable to the flux from thermal processes in the few hours pre-collapse, and it even exceeds it in the high energy tail of the spectrum, $E\\mathrel {\\hbox{$>$}\\hbox{$\\sim $}}$ 3$ MeV.",
"This very relevant for current and near future detectors, which are most sensitive above the MeV scale.$ Among the realistic detection technologies, liquid scintillator is best suited to detect presupernova neutrinos.",
"This is due to its lower energy threshold, which allows to capture the bulk of the flux hours or minutes before collapse.",
"In such detector neutrinos from beta processes would contribute up to $\\sim 30\\%$ of the total number of events, for a threshold of $\\sim 0.5$ MeV.",
"The horizon for detection (i.e., the distance from the star where a few events are expected in the detector) is of a few kpc for a 17 kt detector, with tens of events expected for $D\\simeq 1$ kpc.",
"The number of event increases strongly with the mass of the progenitor star; therefore, for medium-high statistics and known $D$ , the presupernova neutrino signal will contribute to establishing the type of progenitor.",
"For high statistics, the time profile of the presupernova signal could provide additional information, e.g., on the time of ignition of the different fuels (fig.",
"REF ).",
"At water Cherenkov and liquid Argon detectors of realistic sizes and thresholds ($E_{th} \\sim 4-5$ MeV), the horizon is generally limited to the closest stars, $D \\sim {\\mathcal {O}}(0.1)$ kpc, but could reach 1 kpc for the most massive progenitors and the inverted neutrino mass hierarchy.",
"For liquid Argon, the contribution of the $\\beta $ neutrinos is strong, and could even dominate the signal.",
"Therefore - at least in principle - liquid Argon detectors offer the possibility of probing the complex nuclear processes in stellar cores.",
"If the high energy tail of a presupernova flux is detected, what nuclei and what processes exactly can we probe?",
"To answer this question, we have identified the isotopes that mostly contribute to the presupernova $\\mathrel {{\\nu _e}}$ flux in the detectable energy window, generally iron, manganese and cobalt isotopes as well as free protons and neutrons.",
"The possibility that neutrino detectors may test the physics of these isotopes is completely novel.",
"In closing, we stress that our calculation used the best available instruments: a state of the art stellar evolution code, combined with the most up-to-date studies of nuclear rates and beta spectra.",
"Still, these instruments are affected by uncertainties, which, naturally, affect the results in this paper.",
"In particular, while total emissivities are relatively robust, it is likely that the highest energy tails of the neutrino spectrum, in the detectable window, are very sensitive to the details of the calculation, i.e., the temperature profile of the star, the nuclear abundances and the quantities in the nuclear tables we have used.",
"Specifically for neutrino spectra, a source of error lies in the single-strength approximation that is adopted here for $\\beta $ p (sec.",
").",
"A recent paper [32], presents an exploratory study of this error and concludes that while the single effective $Q$ -value approach results in the correct emissivity and average energy, the specific energy spectrum could miss important features.",
"A systematic extension of this result to the many isotopes included in MESA would be highly desirable to improve our results.",
"Another interesting addition to the code would be the contribution of neutrino pair production via neutral current de-excitation [32], which is currently omitted in MESA.",
"This de-excitation results in higher energy neutrino spectra than the processes described in this work, and thus makes detections more likely.",
"Until these important improvements become available, our results have to be interpreted conservatively, as a proof of the possibility that current and near future detectors might be able to observe presupernova neutrinos, and therefore offer the first, direct test of the isotopic evolution of a star in the advanced stages of nuclear burning.",
"We thank K. Zuber and Wendell Misch for fruitful discussion.",
"We also acknowledge the National Science Foundation grant number PHY-1205745, and the Department of Energy award DE-SC0015406.",
"This project was also supported by NASA under the Theoretical and Computational Astrophysics Networks (TCAN) grant NNX14AB53G, by NSF under the Software Infrastructure for Sustained Innovation (SI2) grant 1339600, and grant PHY 08-022648 for the Physics Frontier Center \"Joint Institute for Nuclear Astrophysics - Center for the Evolution of the ElementsÓ (JINA-CEE).",
"This research was also partially undertaken at the Kavli Institute for Theoretical Physics which is supported in part by the National Science Foundation under Grant No.",
"NSF PHY-1125915."
]
] | 1709.01877 |
[
[
"Learning Dynamical Systems and Bifurcation via Group Sparsity"
],
[
"Abstract Learning governing equations from a family of data sets which share the same physical laws but differ in bifurcation parameters is challenging.",
"This is due, in part, to the wide range of phenomena that could be represented in the data sets as well as the range of parameter values.",
"On the other hand, it is common to assume only a small number of candidate functions contribute to the observed dynamics.",
"Based on these observations, we propose a group-sparse penalized method for model selection and parameter estimation for such data.",
"We also provide convergence guarantees for our proposed numerical scheme.",
"Various numerical experiments including the 1D logistic equation, the 3D Lorenz sampled from different bifurcation regions, and a switching system provide numerical validation for our method and suggest potential applications to applied dynamical systems."
],
[
"Introduction",
"Nonlinear systems of ordinary differential equations (ODEs) are used to describe countless physical and biological processes.",
"Often, the governing equations that model a system must be derived theoretically or computed to fit a given dataset.",
"Model selection and parameter estimation methods are used to make decisions on the form of the governing equations and the values of the model parameters.",
"One major difficulty is determining the set of appropriate candidate functions to fit to the data, since the task of searching through a large set of potential candidate functions can be computationally intractable.",
"Therefore, it is common to pre-select a small subset of potential candidate functions [9], [25].",
"However, this requires prior knowledge on the system and the potential structure of the governing equations.",
"Another issue involves the estimation of model parameters when using multiple sampling sources, since a `one-size-fit-all' approach may not be possible when the parameters in the governing equations vary over the different sources.",
"In this work, we develop a group-sparse penalized method for model selection and parameter estimation of governing equations where the data is given over multiple sources.",
"Possible applications include parameter analysis through data-driven bifurcation diagrams, analysis of chaotic systems, and parameter estimation from incomplete and non-uniform sources.",
"There have been several recent works utilizing sparse optimization for model selection.",
"These methods were inspired by the regression approach from [6], [47], which used a symbolic regression algorithm to learn physical laws from data by fitting derivatives of the data to candidate functions.",
"In terms of regression approaches, sparsity can be incorporated by adding an $\\ell ^0$ term to the method, which penalizes the number of non-zero candidate functions in the learned model.",
"In some cases, it is possible to relax the $\\ell ^0$ penalty to the $\\ell ^1$ norm and still maintain sparse solutions, see [12], [11], [19], [37], [35].",
"Both the $\\ell ^0$ and $\\ell ^1$ penalties have seen many applications from image processing to data mining, and now in methods for learning governing equations from dynamic data.",
"The key idea of sparse model selection for learning governing equations is to find the best fit of the temporal derivative over a large set of potential candidates by enforcing that the selected model uses only a few terms (i.e.",
"the model should be sparse).",
"This is based on the sparsity-of-effect principle, since one expects that a small number of candidate functions contribute to the observed dynamics.",
"In [8], a sequential least-squares thresholding algorithm was proposed for learning dynamical systems.",
"The algorithm iterates between the least-squares solution and a thresholding step, which is meant to retain only the most meaningful terms in the least-squares approximation.",
"In [51], a joint outlier detection and model selection method was developed for learning governing equations from data with time-intervals of highly corrupted information.",
"A group-sparse penalty is used, namely the $\\ell ^{2,1}$ norm, to couple the intervals of corruption between each variable while also penalizing the size of these intervals.",
"It was also shown in [51] that the separation between the clean data and outliers can be exactly recovered from chaotic systems.",
"An $\\ell ^1$ regularized least-squares approach (related to the LASSO method [49]) is used in [40] to learn nonlinear partial differential equations from spatio-temporal data.",
"The dictionary matrix is constructed from a set of basis functions applied to both the data and its spatial derivatives.",
"In [38], a sequential least-squares method related to [8] was used to learn PDE from spatio-temporal data as well.",
"In [43], an $\\ell ^0$ basis pursuit problem was proposed which used an integrated candidate set to identify governing equations from noisy data.",
"In [44], an $\\ell ^1$ basis pursuit problem was developed for extracting governing equations from under-sampled data.",
"The $\\ell ^1$ basis pursuit approach solves the model selection problem exactly (under certain conditions), when the data is sampled using several proposed strategies.",
"It is important to note that the sparse optimization and data-based methods have seen many applications to various scientific problems over that last few years, see [41], [28], [24], [36], [1], [34], [50], [10], [33], [8], [7], [3], [23], [21], [52], [13], [14], [29], [30], [45], [46], [42], [39].",
"In several of those works, it is noted that multiple sampling sources increase the accuracy of the recovered coefficients, when the coefficients are assumed to remain fixed over all sources.",
"This is, in part, due to the fact that over various sources, the data will exhibit different behaviors which help to distinguish between possible candidates.",
"In this work, we develop a group-sparse model for extracting governing equations from multiple sources, whose parameters may vary between the sources.",
"In particular, we enforce that the learned model is the same between the sources and allow the coefficients to vary for each source.",
"The learned coefficients from group-sparse methods are block-wise sparse in the sense that variables are grouped into subsets which are either simultaneously zero or nonzero.",
"This can be thought of as an `all-or-nothing' penalty within each group.",
"Mathematically, group-sparse methods often use an $\\ell ^{2,0}$ or $\\ell ^{2,1}$ penalty to group together related variables while penalizing the number of active (nonzero) groups [53].",
"To solve the group-sparse optimization problem, several algorithms have been proposed such as hard-iterative thresholding algorithms [22], [4]; simultaneous orthogonal matching pursuits (SOMP) based on correlations [22] or noise stabilization [18], and subspace methods [26].",
"Recovery guarantees for these algorithms were also investigated, for example, using a probability model on the nonzero coefficients [22] or full rank conditions [15].",
"Some applications of group sparsity to data-based learning include microarray data analysis [27], spectrum cartography [2], and source localization [31].",
"Using the group-sparse penalty can lead to a decrease in the number of degrees of freedom in the inverse problem which can potentially increase the accuracy of recovery [17].",
"This is the case for learning governing equations with multiple sources or from different bifurcation regions, as detailed in Section .",
"The paper is outlined as follows.",
"In Section , the framework for sparse model selection is detailed as well as the group-sparse structural condition and problem statement.",
"Convergence conditions related to the numerical solver is discussed in Section .",
"In Section , the numerical method, based on the iterative hard thresholding algorithm, is explained and the computational scheme is presented.",
"Computational results for several dynamical systems are shown in Section .",
"We end with some concluding remarks in Section ."
],
[
"Problem Statement",
"Consider the variable $x(t; \\lambda ^{(i)})\\in \\mathbb {R}^n$ governed by the nonlinear dynamical system: $\\dot{x}(t;\\lambda ^{(i)}) = f(x(t);\\lambda ^{(i)}),\\quad i = 1,\\ldots , m.$ In vector form, $x(t; \\lambda ^{(i)}) = (x_1(t; \\lambda ^{(i)}),\\ldots , x_n(t; \\lambda ^{(i)}))^T$ represents the state of the system at time $t$ , the function $f(x;\\lambda ^{(i)}) = (f_1(x;\\lambda ^{(i)}),\\ldots , f_n(x;\\lambda ^{(i)}))^T$ defines the nonlinear evolution of the system, and $\\lambda ^{(i)}$ is a bifurcation parameter of the system.",
"We would like to learn the function $f$ and the model parameters $\\lambda ^{(i)}$ , when only measurements on $x$ are provided.",
"The parameters $\\lambda ^{(i)}$ correspond to different bifurcation regimes in the observed variables.",
"We have no a priori information on the functions $f$ , the parameters $\\lambda ^{(i)}$ , or the bifurcation region in which the state space $x(t;\\lambda ^{i})$ resides.",
"The velocity $\\dot{x}$ is assumed to be observed or calculated, with sufficient accuracy, from the states $x$ .",
"Since the nonlinear functions $f$ are unknown, we will represent them as a linear combination of a large set of candidate nonlinear functions.",
"This transforms the nonlinear regression problem on $f$ to a linear inverse problem with respect to the coefficients.",
"In the construction below, we develop the proposed approach for polynomial systems; however, it can be easily generalized to other (and possibly redundant) candidate sets.",
"Since we are given the observations of the states $x(t;\\lambda ^{(i)})$ , Equation (REF ) decouples and thus the model selection can be done component-wise.",
"Therefore, we consider the following representation for the nonlinear function $f_j$ : $f_j(x(t);\\lambda ^{(i)}_{j}) = c^{(i)}_{j,0} + \\sum \\limits _{k} c^{(i)}_{j,k}\\ x_k(t;\\lambda ^{(i)}) + \\sum \\limits _{k,l} c^{(i)}_{j,k,l}\\ x_k(t;\\lambda ^{(i)}) \\, x_l(t;\\lambda ^{(i)}) + \\ldots $ where $c^{(i)}_{j} = (c_{j,0} ,c_{j,1},\\ldots , c_{j,n}, c_{j,1,1},\\ldots , c_{j,n,n},\\ldots )$ is the vector of unknown coefficients corresponding to the $j^{th}$ component of Equation (REF ) and the $i^{th}$ source.",
"The parameters $\\lambda ^{(i)}_{j}$ denote the subset of parameters corresponding to $f_j$ .",
"Note that observations $x(t;\\lambda ^{(i)}) $ still depend on the entire system $\\lambda ^{(i)}$ .",
"The model selection problem is to identify the support set of $c^{(i)}_{j}$ , i.e.",
"the indices that correspond to the nonzero values.",
"Since the model is assumed to have the same representation for all $i$ , the support set of $c^{(i)}_{j}$ is also the same for all $i$ and thus we will denote it as $S_j$ .",
"The parameter estimation problem corresponds to learning the nonzero values of $c^{(i)}_{j}$ for each $i$ and $j$ .",
"It is worth noting that if the correct model is identified, then the coefficients $c^{(i)}_{j}$ restricted to the set $S_j$ should be identical to $\\lambda ^{(i)}_{j}$ .",
"To illustrate and clarify the notation, consider Duffing's equation, $\\ddot{u}+\\delta \\dot{u} -\\beta u + u^3=0 $ , which can be written as a nonlinear first-order system: $\\dot{x}_1 &= x_2\\\\\\dot{x}_2 &= \\beta x_1-\\delta x_2 -x_1^3$ The model parameters are $\\lambda = (1,\\beta , -\\delta , -1 )$ , which will lead to different types of observed behaviors.",
"The parameter subsets are $\\lambda ^{(i)}_{1}=1$ (which is the same for all $i$ ) and $\\lambda ^{(i)}_{2}=(\\beta ^{(i)}, -\\delta ^{(i)}, -1 )$ .",
"Consider two states $i\\in \\lbrace 1,2\\rbrace $ .",
"For $i=1$ , if we observe data from the system with $\\beta >0$ and $\\delta >0$ , there would be three equilibrium points: two sinks and a saddle.",
"For $i=2$ , if we observe data from the system with $\\beta <0$ and $\\delta >0$ , there would be one sink (at the origin).",
"The vector $\\lambda $ represents the controls on the behavior of the output and are unknown to the user.",
"Learning the representation of $f_2$ from observations of $x$ and $\\dot{x}$ over these two parameter states, if successful, would yield: $f_2(x; \\lambda ^{(i)}_{2})&= 0 + c^{(i)}_{2,1}\\ x_1(t;\\lambda ^{(i)})+ c^{(i)}_{2,2}\\ x_2(t;\\lambda ^{(i)})+0 +\\ldots + 0+ c^{(i)}_{2,1,1,1}\\ x^3_1(t;\\lambda ^{(i)})+0 +\\ldots $ where the learned coefficient vector is $c^{(i)}_{2} = (0 ,c^{(i)}_{2,1},c^{(i)}_{2,2},0, \\ldots , 0,c^{(i)}_{2,1,1,1} ,0, \\ldots )$ .",
"The support set of $c^{(i)}_{2}$ is the same over all $i$ and, if the representation is accurate, the restriction of $c^{(i)}_{2}$ onto the nonzero values, $(c^{(i)}_{2,1},c^{(i)}_{2,2}, c^{(i)}_{2,1,1,1})$ , should match $\\lambda ^{(i)}_{2}=(\\beta ^{(i)}, -\\delta ^{(i)}, -1 )$ for the two regimes $i\\in \\lbrace 1,2\\rbrace $ .",
"The goal is to recover $c^{(i)}_{j}$ by fitting each $f_j$ to the observed or calculated velocity.",
"The inverse problem can be written as a linear system with unknowns $c^{(i)}_{j}$ as follows.",
"For a fixed parameter state $\\lambda ^{(i)}$ , we denote the observed variables at times $\\lbrace t^{(i)}_1, ... , t^{(i)}_{\\ell _i}\\rbrace $ ($\\ell _i$ is the number of temporal measurements obtained from state $i$ ) as $\\lbrace x(t^{(i)}_1;\\lambda ^{(i)}), \\ x(t^{(i)}_2;\\lambda ^{(i)}),\\ldots ,\\ x(t^{(i)}_{\\ell _i};\\lambda ^{(i)})\\rbrace .$ Similar to [8], [51], [43], [40], [38], [44], the data matrix $X^{(i)}$ , the velocity matrix $V^{(i)}$ , and the dictionary matrix $D^{(i)}$ are defined as: $X^{(i)} &= \\begin{bmatrix}| & | & & | \\\\x^{(i)}_1 & x^{(i)}_2 & \\ldots & x^{(i)}_n \\\\| & | & & | \\\\\\end{bmatrix}_{\\ell _i\\times n}= \\begin{bmatrix}{x}_1(t_1;\\lambda ^{(i)}) & {x}_2(t_1;\\lambda ^{(i)}) &\\ldots & {x}_n(t_1;\\lambda ^{(i)})\\\\{x}_1(t_2;\\lambda ^{(i)}) & {x}_2(t_2;\\lambda ^{(i)})&\\ldots & {x}_n(t_2;\\lambda ^{(i)})\\\\\\vdots &\\vdots &\\ldots & \\vdots \\\\{x}_1(t_{\\ell _i};\\lambda ^{(i)}) &{x}_2(t_{\\ell _i};\\lambda ^{(i)}) &\\ldots & {x}_n(t_{\\ell _i};\\lambda ^{(i)})\\\\\\end{bmatrix}_{\\ell _i\\times n},$ $V ^{(i)} &= \\begin{bmatrix}| & | & & | \\\\\\dot{x}^{(i)}_1 & \\dot{x}^{(i)}_2 & \\ldots & \\dot{x}^{(i)}_n \\\\| & | & & | \\\\\\end{bmatrix}_{\\ell _i\\times n}= \\begin{bmatrix}\\dot{x}_1(t_1;\\lambda ^{(i)}) & \\dot{x}_2(t_1;\\lambda ^{(i)}) &\\ldots & \\dot{x}_n(t_1;\\lambda ^{(i)})\\\\\\dot{x}_1(t_2;\\lambda ^{(i)}) & \\dot{x}_2(t_2;\\lambda ^{(i)})&\\ldots & \\dot{x}_n(t_2;\\lambda ^{(i)})\\\\\\vdots &\\vdots &\\ldots & \\vdots \\\\\\dot{x}_1(t_{\\ell _i};\\lambda ^{(i)}) & \\dot{x}_2(t_{\\ell _i};\\lambda ^{(i)}) &\\ldots & \\dot{x}_n(t_{\\ell _i};\\lambda ^{(i)})\\\\\\end{bmatrix}_{\\ell _i\\times n},$ and $D^{(i)} &= \\left[\\textbf {1}_{\\ell _i,1}, \\ X^{(i)} , \\ (X^{(i)})^2 , \\ (X^{(i)})^3,\\ \\ldots \\right]_{\\ell _i\\times \\overline{n}},$ where $\\overline{n} = {n+p \\atopwithdelims ()n}$ is the total number of monomials up to degree $p$ .",
"Each column of these matrices corresponds to the vectorization of each variable over the temporal measurements.",
"For each $q$ , $(X^{(i)})^q$ denotes the values of all monomials of degree $q$ at times $\\lbrace t^{(i)}_1, ... , t^{(i)}_{\\ell _i}\\rbrace $ .",
"For every index $j\\in \\lbrace 1,\\ldots ,n\\rbrace $ (the components of the system), the problem of finding $f_j(x;\\lambda ^{(i)}_j)$ , for all $ i = 1,\\ldots , m$ , can be reformulated to finding $c_j^{(i)}$ given the data matrix $X^{(i)}$ .",
"In particular, the problem can be stated as: find $c_j^{(i)}\\in \\mathbb {R}^{\\overline{n}}$ such that: $V_j^{(i)} = D^{(i)} c_j^{(i)},\\quad i=1,\\ldots , m,$ where $V_j^{(i)}$ is the $j^{th}$ column of $V^{(i)}$ .",
"Next, define the coefficient matrix by: $C_j = \\begin{bmatrix}| & | & | & | & \\\\c_j^{(1)} & c_j^{(2)} & \\ldots & c_j^{(m)} \\\\| & | & | & | & \\\\\\end{bmatrix}_{\\overline{n}\\times m}$ where each column corresponds to the vector $c_j^{(i)}$ for a fixed $i$ and each row corresponds to the same candidate term in the representation of the $f_j$ 's.",
"Let $D$ be a block diagonal matrix whose diagonal corresponds to the matrices $D^{(i)}$ , for $i = 1,\\ldots , m$ .",
"Let $C^{vec}_j:=[c^{(1)}_j,\\cdots , c^{(m)}_j]$ be the column-based vectorization of the coefficient matrix $C_j$ , and let $V^{vec}_j: =[V^{(1)}_j,\\cdots , V^{(m)}_j] $ be the column-based vectorization of the velocities of component $j^{th}$ along each source.",
"Then the optimization problem can be rewritten as a least-square fitting : ${ \\min \\limits _{C_j} \\Vert DC^{vec}_j -V^{vec}_j \\Vert _2^2 \\left(= \\sum \\limits _{i=1}^m\\Vert D^{(i)} c_j^{(i)} -V_j^{(i)} \\Vert _2^2\\right)}$ This is ill-posed due to errors in the measurements $x$ , errors in approximating the velocity, and issues involving the large set of candidate functions (which would lead to overfitting if one solves Equation (REF ) using the pseudo-inverse, for example).",
"To regularize the problem, we include a penalty on the number of active candidate functions.",
"This will help to prevent overfitting and lead to meaningful result in practice.",
"The assumption on the system is that $c_j^{(i)}$ has the same support set (in $j$ ) for each $i$ , but can differ in value.",
"In other words, we can group each row together to be either zero or nonzero, therefore the number of active (nonzero) rows is sparse.",
"This leads to the following group-sparse optimization problem: $\\boxed{ \\min \\limits _{C_j}\\ \\Vert DC^{vec}_j -V^{vec}_j \\Vert _2^2 + \\gamma \\Vert C_j\\Vert _{2,0} \\quad \\Leftrightarrow \\quad \\min \\limits _{C_j}\\ \\sum \\limits _{i=1}^m\\Vert D^{(i)} c_j^{(i)} -V_j^{(i)} \\Vert _2^2 + \\gamma \\Vert C_j\\Vert _{2,0}}$ where the $\\ell ^{2,0}$ penalty is defined as: $\\Vert A\\Vert _{2,0}: = \\# \\left\\lbrace k: \\left( \\sum _{\\ell } |a_{k,\\, \\ell }|^2 \\right)^{1/2} \\ne 0\\right\\rbrace .$ for any matrix $A=[\\, a_{k,\\, \\ell }\\, ]$ .",
"Although the problem is nonconvex, we can solve it numerically using an iterative hard thresholding algorithm, see Section ."
],
[
"Convergence Guarantees",
"The addition of the $\\ell ^{2,0}$ penalty is to encourage sparse solution from the least-squares minimization.",
"If the matrix $D$ is not full rank or badly conditioned, then Equation (REF ) is not guaranteed to have a unique solution.",
"Indeed, our dictionary matrix, formed by monomials up to possibly high order, will generally not be well-conditioned.",
"In Proposition REF Section , we characterizes properties of the dynamics such that $D$ (or equivalently, each $D^{(i)}$ ) will at least be full rank.",
"In fact, our proposed numerical method is guaranteed to converge to a local minimizer of Equation (REF ) if $D$ is full rank (see Section and Appendix for more details).",
"Proposition 3.1 (General bound) Suppose, for each $i$ , $\\overline{n} \\le \\ell _i$ , there exists a subset $S \\subset [\\ell _i]$ of size $| S | = \\overline{n}$ such that $\\lbrace X^{(i)}(k,-)\\mid k\\in S\\rbrace $ do not belong to a common algebraic hypersurface of degree $\\le p$ (i.e., for any $u_1, u_2, \\dots , u_{\\overline{n}} \\in \\mathbb {R}^n$ , there exists a unique interpolating polynomial $u = P(x)$ of degree $\\le p$ satisfying $u_k = P(X^{(i)}(k,-))$ for $k\\in S$ ).",
"This is a necessary and sufficient condition for the dictionary matrix $D$ to be full rank: for each $D^{(i)}$ , there exists a $\\delta _i > 0$ such that $\\inf _{u} \\frac{\\Vert D^{(i)} u \\Vert _2 }{ \\Vert u \\Vert _2} \\ge \\delta _i.$ In particular, for 1D systems, the conditions above states that the dictionary is full rank if there is a subset of $\\overline{n}$ distinct points $x^{(i)}$ .",
"First consider the one-dimensional case $n=1$ .",
"In this case, $D^{(i)}_{S}$ , the $\\overline{n} \\times \\overline{n}$ dictionary matrix restricted to the rows indexed by $S$ , is a square Vandermonde matrix.",
"Moreover, in this case, the condition that $\\overline{n} = p+1$ points in $S$ admit a unique interpolating polynomial of degree $\\le p$ is equivalent to the condition that the $p+1$ points in $S$ are distinct.",
"It is well-known that a Vandermonde matrix such as $D^{(i)}_{S}$ is invertible if and only if its generating points are distinct; thus, $\\inf _{u} \\frac{\\Vert D^{(i)}_S u \\Vert _2 }{ \\Vert u \\Vert _2} \\ge \\delta ,$ and hence also $\\inf _{u} \\frac{\\Vert D^{(i)} u \\Vert _2 }{ \\Vert u \\Vert _2} \\ge \\delta .$ For the general $n$ -dimensional case, the $\\overline{n} \\times \\overline{n}$ matrix $D^{(i)}_{S}$ is a generalized Vandermonde matrix, and the stated conditions are necessary and sufficient for $D^{(i)}_S$ to be nonsingular, according to Theorem 4.1 in [32].",
"Thus, again, $\\inf _{u} \\frac{\\Vert D^{(i)}_S u \\Vert _2 }{ \\Vert u \\Vert _2} \\ge \\delta ,$ and hence also $\\inf _{u} \\frac{\\Vert D^{(i)} u \\Vert _2 }{ \\Vert u \\Vert _2} \\ge \\delta .$ Figure: An example where the state space quickly approaches a limit cycle, which almost stays on a hypersurface of degree 2.",
"The state space is generated from the Lorenz system, Equation () with μ=7.73\\mu = 7.73 and initialization U 0 =[1,1,2]U_0= [1,1,2], time step dt=0.005dt = 0.005.In Theorem REF , we show that if the matrix $D$ is full rank, our proposed numerical method is guaranteed to converge to a local minimizer of the objective function.",
"Still, the local minimizer we converge to could be far from the global minimizer.",
"An example of when this approach may fail is the case of a limit cycle example (see Figure REF ).",
"The problem is that if we initialize near the limit cycle, the dynamics lie very close to an algebraic hypersurface of degree $p=2$ .",
"This can be avoided if one could sample data away from the limit cycle.",
"Also, our algorithm likely converges to a local minimizer under weaker conditions than those in the above proposition; namely, the convergence requires that $D$ is coercive on sparse subsets.",
"So $D$ can in fact be an underdetermined matrix, where we observe a number of snapshots smaller than the size of the dictionary (see Appendix).",
"It remains an open problem to characterize general conditions on the dynamics under which this weaker condition holds.",
"At least for 1D dynamical systems, we can provide such a characterization.",
"Proposition 3.2 Consider the case $n=1$ , thus $\\overline{n} = p+1$ .",
"Suppose that $\\ell _i \\ge s$ , and at least $s$ of the $\\ell _i$ points in set Equation (REF ) are positive and distinct.",
"Then, restricted to column subsets $S \\subset \\overline{n}$ of size $|S| \\le s$ , the dictionary matrix $D^{(i)}$ is coercive, i.e.",
"there exists a $\\delta > 0$ such that: $\\min _{S: |S| \\le s} \\ \\inf _{u \\in \\mathbb {R}^s}\\ \\frac{\\Vert D^{(i)}_S u \\Vert _2 }{ \\Vert u \\Vert _2} \\ge \\delta .$ Without loss of generality, we assume that $\\ell _i = s$ and that all the points $\\lbrace x(t^{(i)}_k;\\lambda ^{(i)}), k \\in [s]\\rbrace ,$ are distinct and positive.",
"Consider any subset $S \\subset [\\overline{n}]$ of size $|S| \\le s$ ; the corresponding $s \\times s$ submatrix $D^{(i)}_S$ is of the form $\\left[\\begin{array}{cccc}\\end{array}z_1^{a_1} & z_1^{a_2} & \\dots & z_1^{a_s} \\\\z_2^{a_1} & z_2^{a_2} & \\dots & z_2^{a_s} \\\\& & \\dots & \\\\z_s^{a_1} & z_s^{a_2} & \\dots & z_s^{a_s}\\right.$ where $0 \\le a_1 < a_2 < \\dots < a_s$ are integers and $0 < z_1 < z_2 < \\dots < z_s$ .",
"This type of matrix is a so-called generalized Vandermonde matrix and is known to be a totally positive matrix, hence invertible (see, e.g.",
"[16]).",
"Thus, $\\inf _{u \\in \\mathbb {R}^s} \\frac{\\Vert D^{(i)}_S u \\Vert _2 }{ \\Vert u \\Vert _2} \\ge \\delta _S > 0.$ Since there are finitely many such subsets, we take $\\delta $ to be the minimum $\\delta _S$ over all subsets to obtain $\\min _{S: |S| \\le s} \\inf _{u \\in \\mathbb {R}^s} \\frac{\\Vert D^{(i)}_S u \\Vert _2 }{ \\Vert u \\Vert _2} \\ge \\delta _S > 0.$"
],
[
"Numerical Method",
"In this section, we detail the numerical method used in this work to solve Equation (REF ).",
"In particular, we use an iterative thresholding algorithm, which keeps all the coefficients above a certain threshold (determined by the $\\ell ^2$ norm of each row).",
"Each iteration contains one gradient descent step and one reduced least-squares problem.",
"This is similar to [20], which keeps the $s$ -largest rows over each iteration.",
"To the best of our knowledge, the following approach does not appear in the literature, thus for the sake of completeness, we detail the numerical method here and include the corresponding proofs in the Appendix.",
"For the discuss below, we rescale $D$ so that maximum spectral norm (over $i$ ) of the matrices $(D^{(i)})^TD^{(i)}$ is less than or equal to 1.",
"For simplicity, we drop the subscript $j$ in Equation (REF ).",
"Denote the product between $D$ and $A$ by: $D\\star A:=\\left[D^{(1)} A_{- \\,,1},\\, D^{(2)} A_{-,2},\\, \\cdots , D^{(m)} A_{-,m}\\right]\\in \\mathbb {R}^{\\ell _1} \\times \\mathbb {R}^{\\ell _2}\\times \\cdots \\times \\mathbb {R}^{\\ell _m},\\quad \\text{for}\\quad A\\in \\mathbb {R}^{\\overline{n}\\times m}$ and define $V :=\\left[V^{(1)},\\, V^{(2)},\\,\\cdots , V^{(m)}\\right]$ .",
"Then we can replace the least-squares term in Equation (REF ) by: $\\Vert D\\star C-V\\Vert _2^2: = \\sum \\limits _{i=1}^m\\Vert D^{(i)} c^{(i)} -V^{(i)} \\Vert _2^2.$ Let $F$ be the objective function in Equation (REF ): $F(C) := \\Vert D\\star C -V \\Vert _2^2 + \\gamma \\Vert C\\Vert _{2,0}.$ and let $F^*$ be a surrogate function for $F$ : $F^*(C,B) := \\Vert D\\star C-V\\Vert _2^2 -\\Vert D\\star (C -B) \\Vert _2^2 +\\Vert C -B\\Vert _2^2+ \\gamma \\Vert C\\Vert _{2,0}.$ These functions agree when $B=C$ , i.e.",
"$F^*(C,C)=F(C)$ .",
"The numerical scheme is based on minimizing the surrogate function $F^*$ .",
"Equation (REF ) can be simplified to: $F^*(C,B) &= \\Vert C\\Vert _2^2 - 2\\left<C, B + D^T\\star (V- D\\star B)\\right> + \\gamma \\Vert C\\Vert _{2,0} + \\Vert B\\Vert _2^2 + \\Vert V\\Vert _2^2 - \\Vert D\\star B\\Vert _2^2,$ where $\\left<\\cdot ,\\cdot \\right>$ is the summation of the component-wise multiplication of each block.",
"To minimize the surrogate function with respect to $C$ , we only need to minimize the first three terms.",
"The $\\ell ^{2,0}$ penalty is row-separable, thus we can consider the two case for each row of $C$ : either the row is zero or nonzero.",
"Denote the support set by: $S: =\\lbrace k: \\Vert C_{k,-}\\Vert _2\\ne 0\\rbrace $ and let $C_{S}$ be the coefficients restricted onto the support set.",
"Then on rows $S$ , the penalty is constant and the minimizer satisfies: $C_S = \\left(B + D^T\\star (V - D\\star B)\\right)_S.$ To decrease the surrogate function, one chooses between Equation (REF ) or setting the row to zero (this is verified in the Appendix).",
"This process yields: $C = H_{\\sqrt{\\gamma }}\\left(B + D^T\\star (V - D\\star B)\\right),$ where the thresholding function is defined as: $H_{a}(x)={\\left\\lbrace \\begin{array}{ll}&0,\\quad \\text{if} \\quad \\Vert x\\Vert _2 \\le a\\\\& x,\\quad \\text{otherwise}.\\end{array}\\right.",
"}$ and applied row-wise to a matrix.",
"We define an iterative thresholding algorithm using Equations (REF ) and (REF ) by: $C^{k+1} = H_{\\sqrt{\\gamma }}\\left(C^{k} +D^T \\star \\left(V - D\\star C^{k}\\right)\\right).$ Like many proximal descent methods, Equation (REF ) may converge slowly in practice.",
"To adjust the convergence rate, we include an additional step: $\\boxed{ {\\left\\lbrace \\begin{array}{ll}S^{k+1} &= \\text{supp}\\left(H_{\\sqrt{\\gamma }}\\left(C^{k} +D^T \\star \\left(V - D\\star C^{k}\\right)\\right) \\right) \\\\C^{k+1}&= \\underset{C}{\\mathrm {argmin}}\\, \\Vert D\\star C-V \\Vert _2^2 \\quad \\text{s.t.}",
"\\ \\ \\text{supp}(C)\\subset S^{k+1}\\end{array}\\right.",
"}}$ Note that the second step is column-wise separable and the row-support set of each column of $C^{k+1}$ is a subset of $S^{k+1}$ , i.e.",
"$\\text{supp}(c^{(i)})\\subset S^{k+1}$ .",
"Therefore, we can solve each reduced least-squares problem in parallel.",
"Indeed, for each column $i$ , we solve $c^{k+1,i} = \\underset{c}{\\mathrm {argmin}}\\Vert D^{(i)}\\, c-V^{(i)} \\Vert _2^2 \\quad \\text{s.t.}",
"\\ \\ \\text{supp}(c^{(i)})\\subset S^{k+1}.$ A summary of the proposed algorithm is described below.",
"Group Hard-Iterative Thresholding Algorithm for Dynamical Systems Given: initialization matrix ${C}^0, tol $ and parameters $\\gamma $ .",
"$\\Vert C^{k+1}-C^{k}\\Vert _{\\infty }> tol$ for $i = 1$ to $m$ : $\\left(\\widetilde{c^{(i)}}\\right)^ {k+1}= \\left(c^{(i)}\\right)^ {k} - (D^{(i)})^T\\left(D^{(i)}\\left(c^{(i)}\\right)^{k} - V^{(i)}\\right) $ end for $S^{k+1} = \\text{supp}\\left(H_{\\sqrt{\\gamma }} \\left[\\widetilde{c^{(1)}}, \\widetilde{c^{(2)}},\\cdots ,\\widetilde{c^{(m)}}\\right]\\right) $ for $i = 1$ to $m$ : $(c^{(i)})^{k+1}= \\underset{c^{(i)}}{\\mathrm {argmin}}\\Vert D^{(i)} c^{(i)} -V^{(i)} \\Vert _2^2 \\quad \\text{s.t.}",
"\\ \\ \\text{supp}(c^{(i)})\\subset S^{k+1}$ .",
"end for To give an indication of the behavior of the modified scheme for $\\ell ^{2,0}$ regularized least-squares minimization, we have the following theorem.",
"Theorem 4.1 Let $F= \\Vert D\\star C -V \\Vert _2^2 + \\gamma \\Vert C\\Vert _{2,0}$ and let $C^n$ be the sequence generated by Equation (REF ), then $F(C^{n+1}) \\le F(C^{n})$ and there are subsequences that converge to local minimizers.",
"In addition, if $D$ is coercive then the sequence $C^{n}$ converges to a local minimizer.",
"The proof is in the appendix and follows a similar approach to [5].",
"When the sparsity level $s$ can be determined or estimated a priori, one could use a modified method based on the group thresholding method of [20].",
"Specifically, at every iteration, we keep $ks$ indices, where $k>1$ , corresponding to $ks$ -largest rows with respect to the $\\ell ^2$ norm and solve the linear regression on that subset.",
"At the final step, we keep exactly $s$ indices instead of $ks$ and follow the same process.",
"Although similar in nature, the sequential thresholding algorithm found in [8] differs from the proposed algorithm.",
"In particular, the thresholding here is performed on a gradient descent step rather than the psuedo-inverse."
],
[
"Computational Results",
"Logistic Equation.",
"For the first computational text, the proposed model (Equation (REF )) is applied to data generated from the 1D logistic equation $\\dot{x} = f(x) :=\\alpha x(1-x), \\quad t\\in [0,50.0],\\quad x(0) = 0.01,$ where $\\alpha $ is the bifurcation parameter.",
"The simulated data are obtained from two sets, which are generated from Equation (REF ) with $\\alpha = 0.05$ and $\\alpha =0.23$ , respectively.",
"For both data sets, we set the time-step to $dt = 0.005$ .",
"The simulated data and the noisy velocities are plotted in Figure REF .",
"In all of our examples, the velocity data $V^{(i)}$ are approximated from $X^{(i)}$ using the central difference.",
"Figure: Logistic equation: State space (top) and noisy velocity space (bottom) plots for α=0.05\\alpha = 0.05 (left) and α=0.23\\alpha = 0.23 (right).",
"The noise levels are σ noise =0.05%\\sigma _{noise} =0.05\\% and σ noise =0.01%\\sigma _{noise} = 0.01\\%, respectively.",
"The maximal degree of monomials in the dictionary is six.To test the robustness of our proposed model, we re-simulate the data 100 times and compute the probability ($P\\in [0,1]$ ) of recovering the correct terms in the governing equation.",
"We use a fixed thresholding parameter of $\\delta _{thres} = 0.0018$ .",
"With the group-sparsity penalization, our model recovers the correct governing equation with probability $P=1$ , i.e.",
"the method learns both terms: $x$ and $x^2$ .",
"Moreover, the average relative errors between the true coefficients and our approximations are $3.04\\%$ for set 1 and around $0.02\\%$ for set 2.",
"For comparison, if one used the $\\ell _0$ -penalty in place of the $\\ell ^{2,0}$ -penalty in Equation (REF ), then the computed probability that the $\\ell _0$ -penalized method recovers the governing equation reduces dramatically to $P=0.41$ and $P=0.3$ , respectively.",
"Thus, unlike our method, the $\\ell _0$ model will likely misidentify the terms in the governing equation.",
"Figure: Lorenz system: State space plots for different α\\alpha with dt=0.005dt = 0.005.",
"Top: α=-1\\alpha = -1 (left), α=4.7\\alpha =4.7 (middle), α=6.9\\alpha = 6.9 (right), where the left figure is the usual chaos, the middle is chaotic with periodic windows, and the right one is chaotic.",
"Bottom: α=7.075\\alpha = 7.075 (left), α=7.73\\alpha = 7.73 (right), where the dynamics include a pitchfork bifurcation and limit cycles, respectively.Lorenz 3D.",
"We consider the Lorenz system with a single bifurcation parameter $\\alpha $ : ${\\left\\lbrace \\begin{array}{ll}\\dot{x}_1 &= 10(x_2-x_1)\\\\\\dot{x}_2 &= -x_1x_3 + (24-4\\alpha )x_1 +x_1x_2\\\\\\dot{x}_3 & = x_1x_3 - \\dfrac{8}{3}x_3,\\end{array}\\right.",
"}$ see [48].",
"To validate our approach, we use five data sets from different bifurcation regimes associated with $\\alpha = -1$ , $\\alpha = 4.7$ , $\\alpha = 6.9$ , $\\alpha = 7.075$ , and $\\alpha = 7.73$ .",
"For $\\alpha = -1$ , the solution is the usual chaotic system with the initial condition set to $U_0 = [-8,7,27]$ .",
"For $\\alpha = 4.7$ , the initial condition is set to $U_0 =[0,-0.01,9]$ and the system exhibits chaos with periodic windows.",
"For $\\alpha = 6.9$ with $U_0= [1,2,1]$ , the solution exhibits chaos.",
"For $\\alpha = 7.075$ with $U_0=[1,1,2]$ , the solutions undergoes a pitchfork bifurcation.",
"And lastly, for $\\alpha = 7.73$ with $U_0 = [2,1,-5]$ , the system has a limit cycle.",
"The time-step is set to $dt = 0.005$ and the finals times are set to: $T = 7.5$ , $12.5$ , 50, $15.0 $ , and $10.0$ so that the corresponding state spaces exhibit the dynamics as indicated in [48] (see Figure REF ).",
"Noise is added to the velocity, with $\\sigma _{noise} = 0.5\\% $ , see Figure REF .",
"Figure: Noisy velocity space plots corresponding to the data given in Figure with noise level σ noise =0.5%\\sigma _{noise} = 0.5\\%.",
"Top: α=-1\\alpha = -1 (left), α=4.7\\alpha = 4.7 (middle), α=6.9\\alpha = 6.9 (right).",
"Bottom: α=7.075\\alpha = 7.075 (left), α=7.73\\alpha = 7.73 (right).Applying our algorithm to this data with $\\delta _{thres} = 1.7$ yields a recovery rate of $P =0.97$ .",
"Moreover, the recovered coefficients are close to the true values, with relative error smaller than $3\\%$ for $\\alpha = -1$ and less than $0.1\\%$ for the remaining $\\alpha $ 's (see Table ).",
"On the other hand, applying the $\\ell _0$ -penalized model yields less consistent results, for example in the $\\alpha = 7.075$ case the recovery probability is less than $P=0.73$ .",
"tableLorenz system.",
"Recovered coefficients from all five sets for the second component $\\dot{x_2}$ .",
"The true values are highlighted in (red).",
"Table: NO_CAPTION Figure: Switching system: State space plot of the Lorenz system (Equation ()), where the bifurcation parameter α\\alpha switches from -1-1 (blue curve) to 6.66.6 (red curve).Switching Systems.",
"For our last example, we illustrate a potential application of the proposed method to switching systems.",
"Consider the Lorenz system, Equation (REF ), where the parameter $\\alpha $ changes from $-1$ to $6.6$ at some unknown time.",
"The state space of the whole system is plotted in Figure REF .",
"Since the location of the parameter change is unknown (as well as the underlying model), we break the entire trajectory into $M$ sub-trajectories and consider each of these sub-trajectories as our different sources, $X^{(i)}$ .",
"Therefore, this problem fits within our framework– the learned parameters are allowed to vary between each sub-trajectory.",
"In this example we take $M=32$ .",
"The recovered coefficients for the second component $\\dot{y}$ are plotted in Figure REF , where the terms $x$ , $y$ , and $xz$ are correctly identified in all sub-trajectories except the one that contains the switch.",
"Figure: Switching System: Indices of the recovered coefficients for the second component of the switching system from Figure .",
"Our method correctly identifies the terms in the governing equation as well as the location of the switch (the anomalous sub-trajectory at 17)."
],
[
"Conclusion",
"We presented a method for extracting governing equations from multiple sources of data using a group-sparsity constraint as well as developed a new group-thresholding algorithm to solve our proposed optimization problem.",
"Our main contribution is the use of group sparsity for learning physical laws from multiple data sources which are controlled by the same mathematical model with different parameters.",
"We also provide convergence guarantees for the associated regression problem.",
"Lastly, convergence of our algorithm to a local minimizer is detailed."
],
[
"Acknowledgments",
"H.S.",
"acknowledges the support of AFOSR, FA9550-17-1-0125.",
"R.W.",
"and G.T.",
"acknowledge the support of NSF CAREER grant $\\#1255631$ .",
"The authors would like to thank the CNA and NSF for their support of the CNA-KiNet Workshop: “Dynamics and Geometry from High Dimensional Data\" at Carnegie Mellon University in March 2017, where this work was finalized and presented."
],
[
"Appendix",
"Proof of Theorem REF .",
"Part 1: Let $\\widetilde{C}^{k+1}$ be defined as: $\\widetilde{C}^{k+1}:=H_{\\sqrt{\\gamma }}\\left(C^{k} + D^T\\star (V - D\\star C^{k} )\\right),$ then we can show that $F(C^{k+1}) \\le F(C^{k})$ by adapting some of the arguments from [5].",
"The objective function at $C^{k+1}$ can be bounded by: $F(C^{k+1})&=\\Vert D\\star C^{k+1} - V \\Vert _2^2 + \\gamma \\Vert C^{k+1}\\Vert _{2,0}\\\\&=\\Vert D\\star C^{k+1} - V \\Vert _2^2 + + \\gamma \\Vert \\widetilde{C}^{n+1}\\Vert _{2,0}\\\\&\\le \\Vert D\\star \\widetilde{C}^{k+1} - V \\Vert _2^2 + \\gamma \\Vert \\widetilde{C}^{k+1}\\Vert _{2,0}=F(\\widetilde{C}^{k+1})$ where the second line comes from the fact that $C^{k+1}$ and $\\widetilde{C}^{k+1}$ have the same support set (and thus the same value with respect to $\\ell ^{2,0}$ ) and the third line comes from the restricted least-squares update (second step of Equation (REF )).",
"Let $A^{(i)}:=\\textbf {I}- \\left(D^{(i)}\\right)^T D^{(i)}$ and assume that the eigenvalues of $A^{(i)}$ , denoted as $\\lambda ^{(i)}$ , are bounded away from zero and are less than 1, i.e.",
"$\\lambda ^{(i)}\\in [\\overline{\\lambda },1]$ for $\\overline{\\lambda }>0$ .",
"Define the norm with respect to $A$ as $\\Vert - \\Vert _{2,A}$ , then $F(\\widetilde{C}^{k+1})&\\le F(\\widetilde{C}^{k+1})+ \\Vert \\widetilde{C}^{k+1}-C^{k} \\Vert ^2_{2,A^{(i)}}\\\\&=\\Vert D\\star \\widetilde{C}^{k+1} -V \\Vert _2^2 +\\gamma \\, \\Vert \\widetilde{C}^{k+1}\\Vert _{2,0} + \\Vert \\widetilde{C}^{k+1}-C^{k} \\Vert ^2_2 -\\Vert D\\star (\\widetilde{C}^{k+1}-C^{k}) \\Vert ^2_{2} \\\\&=F^*(\\widetilde{C}^{k+1},C^{k})\\\\&=\\underset{C}{\\mathrm {argmin}} \\ F^*(C,C^{k})\\\\&\\le F^*(C^{k},C^{k})\\\\&= F(C^{k}).$ Note that this argument also shows that $F(\\widetilde{C}^{k+1}) \\le F(\\widetilde{C}^{k})$ .",
"This implies that the energy $F$ converges.",
"Part 2: We show convergence to a local minimizer when the dictionary is coercive.",
"To do so, consider the finite sum: $\\sum \\limits _{k=0}^N \\Vert {C}^{k+1}-C^{k} \\Vert _2^2 $ which increases monotonically with respect to $N$ .",
"The sum is bounded by: $\\sum \\limits _{k=0}^N \\Vert C^{k+1}-C^{k} \\Vert _2^2 \\le \\sum \\limits _{k=0}^N \\Vert C^{k+1}-\\widetilde{C}^{k+1} \\Vert _2^2+ \\Vert \\widetilde{C}^{k+1}-C^{k} \\Vert _2^2$ which we will individually bound as follows.",
"The first term is bounded by: $\\sum \\limits _{n=0}^N \\Vert \\widetilde{C}^{k+1}-C^{k} \\Vert _2^2 &= \\sum \\limits _{k=0}^N \\sum \\limits _{i=1}^m \\ \\left\\Vert \\left(\\widetilde{c}^{(i)}\\right)^{k+1}-\\left({c}^{(i)}\\right)^{k} \\right\\Vert ^2_{2}\\\\&\\le \\bar{\\lambda }^{-1}\\, \\sum \\limits _{k=0}^N \\sum \\limits _{i=1}^m \\ \\left\\Vert \\left(\\widetilde{c}^{(i)}\\right)^{k+1}-\\left({c}^{(i)}\\right)^{k} \\right\\Vert ^2_{2,A^{(i)}}\\\\&\\le \\bar{\\lambda }^{-1}\\, \\sum \\limits _{k=0}^N \\left( F(C^{k})-F(C^{k+1})\\right)\\\\&= \\bar{\\lambda }^{-1}\\, \\left( F(C^{0})-F(C^{N+1})\\right)\\\\&\\le \\bar{\\lambda }^{-1}\\, F(C^{0}).$ To bound the second term, consider the norm restricted onto the support set $S^{k+1}$ : $\\sum \\limits _{k=0}^N \\Vert C^{k+1}-\\widetilde{C}^{k+1} \\Vert _2^2 = \\sum \\limits _{k=0}^N \\sum \\limits _{i=1}^m \\ \\left\\Vert \\left({c}^{(i)}\\right)^{k+1}-\\left(\\widetilde{c}^{(i)}\\right)^{k+1} \\right\\Vert ^2_{2}= \\sum \\limits _{k=0}^N \\sum \\limits _{i=1}^m \\ \\left\\Vert \\left({c}^{(i)}\\right)^{k+1}-\\left(\\widetilde{c}^{(i)}\\right)^{k+1} \\right\\Vert ^2_{2 \\, | \\,S^{k+1}}.\\\\$ If the matrix $D$ is coercive over $S^{k+1}$ , with coercivity constant $\\delta >0$ , then $\\sum \\limits _{k=0}^N \\sum \\limits _{i=1}^m \\ \\left\\Vert \\left({c}^{(i)}\\right)^{k+1}-\\left(\\widetilde{c}^{(i)}\\right)^{k+1} \\right\\Vert ^2_{2 \\, | \\,S^{k+1}}& \\le \\delta ^{-1}\\, \\sum \\limits _{k=0}^N \\sum \\limits _{i=1}^m \\ \\left\\Vert D^{(i)}|_{S^{k+1}} \\left( \\left({c}^{(i)}\\right)^{k+1}-\\left(\\widetilde{c}^{(i)}\\right)^{k+1} \\right) \\right\\Vert ^2_{2 \\, | \\,S^{k+1}}\\\\& \\le \\delta ^{-1}\\, \\sum \\limits _{k=0}^N \\left( F(\\widetilde{C}^{k+1} ) - F({C}^{k+1} )\\right)\\\\& \\le \\delta ^{-1}\\, \\sum \\limits _{k=0}^N \\left( F(C^{k} ) - F({C}^{k+1} )\\right)\\\\& \\le \\delta ^{-1}\\,F(C^{0} )$ Combining these two bounds yields: $\\sum \\limits _{k=0}^N \\Vert C^{k+1}-C^{k} \\Vert _2^2 \\le \\sum \\limits _{k=0}^N \\Vert C^{k+1}-\\widetilde{C}^{k+1} \\Vert _2^2+ \\Vert \\widetilde{C}^{k+1}-C^{k} \\Vert _2^2 \\le (\\bar{\\lambda }^{-1}+\\delta ^{-1})\\, F(C^{0}).$ Therefore, for any $\\epsilon >0$ , there exist an $N>0$ such that for all $k>N$ , we have $\\Vert C^{k+1}-C^{k} \\Vert _2 \\le \\epsilon $ .",
"Using this condition and an analogy of Lemma 3.4 from [5] (changing element-wise to row-wise) yields a subsequence $C^k$ converging to a local minimizer."
]
] | 1709.01558 |
[
[
"A magic tilt angle for stabilizing two-dimensional solitons by\n dipole-dipole interactions"
],
[
"Abstract In the framework of the Gross-Pitaevskii equation, we study the formation and stability of effectively two-dimensional solitons in dipolar Bose-Einstein condensates (BECs), with dipole moments polarized at an arbitrary angle $\\theta$ relative to the direction normal to the system's plane.",
"Using numerical methods and the variational approximation, we demonstrate that unstable Townes solitons, created by the contact attractive interaction, may be completely stabilized (with an anisotropic shape) by the dipole-dipole interaction (DDI), in interval $\\theta ^{\\text{cr}}<\\theta \\leq \\pi /2$.",
"The stability boundary, $\\theta ^{\\text{cr}}$, weakly depends on the relative strength of DDI, remaining close to the \"magic angle\", $\\theta_{m}=\\arccos \\left( 1/\\sqrt{3}\\right) $.",
"The results suggest that DDIs provide a generic mechanism for the creation of stable BEC\\ solitons in higher dimensions."
],
[
"Introduction",
"The collisional interaction of matter waves in Bose-Einstein condensates (BECs) resembles nonlinear interaction of optical waves in nonlinear dielectric media [1].",
"If solely the attractive short-range $s$ -wave inter-atomic scattering is present in the BEC, which is tantamount to the Kerr (cubic) nonlinearity in optics in the framework of the man-field approximation, the two- and three-dimensional (2D and 3D) matter-wave solitons are subject to the collapse-driven instability [2], [3], [4].",
"In particular, the well-known instability of 2D Townes solitons [5] is induced by the critical collapse in the same setting [6], [7].",
"Long-range interactions may give rise to effects quite different from those induced by the contact (local) cubic nonlinearity [8].",
"In particular, the experimental realization of BEC in gases of atoms carrying large permanent magnetic moments ($\\sim $ several Bohr magnetons), viz., $^{52}$ Cr [9], $^{164}$ Dy [10], and $^{168}$ Er [11], has drawn a great deal of interest to effects of the dipole-dipole interactions (DDIs), which are intrinsically anisotropic and nonlocal [12], [13].",
"Similar to the situation in nonlocal optical media [14], [15], [16], [17], the long-range nonlocal nonlinearity may play a crucial role in the formation and stabilization of solitons.",
"A wide range of novel solitonic structures were predicted to be supported by the nonlocal nonlinearities, such as discrete solitons [18], [19], [20], [21], azimuthons [22], solitary vortices [23], [24], [25], vector solitons [26], [27], [28], dark-in-bright solitons [29], and other species of self-trapped modes.",
"Even though trapping potentials can be used to stabilize 3D or quasi-2D soliton condensates, dipolar BECs suffer from instabilities against spontaneous excitation of roton and phonon modes at high and low momenta, respectively [30], [31], [34], [33], [32], [35], which manifest themselves at large strengths of DDI [36].",
"For matter waves trapped in a cigar-shaped potential, existence of stable quasi-1D solitons was predicted for combinations of the DDI and local interactions [41], [42], [37], [39], [40], [43], [38].",
"The DDI anisotropy brings the roton instability to trapped dipolar gases in the 2D geometry, and drives the condensates into a biconcave density distribution [44], [45].",
"Stable strongly anisotropic quasi-2D solitons in the condensate with in-plane-oriented dipolar moments have been predicted too [46], [47].",
"In this work, we consider a general setting for the formation of 2D bright solitons supported by the contact interaction and DDI, with the dipoles aligned at an arbitrary tilt angle with respect to the direction normal to system's plane.",
"By reducing the 3D Gross-Pitaevskii equation (GPE) to an effective 2D equation for the “pancake\" geometry, we establish conditions necessary for supporting matter-wave solitons in the dipolar BEC, at different values of the DDI strength, chemical potential, and tilt angle.",
"In addition to the application of the well-known Vakhitov-Kolokolov stability criterion [48], the linear-stability analysis and variational approach are also used for the study of the stability of the 2D dipolar soliton solutions.",
"Starting with a fixed strength of the attractive local interaction, our analysis reveals that the originally unstable 2D Townes solitons may be stabilized with the help of the DDI.",
"It is thus found that 2D solitons are stable if the orientation angle of the dipoles, with respect to the direction normal to the pancake's plane, exceeds a certain critical (“magic\") value, see Eq.",
"(REF ) below, a similar “magic angle\" for the sample's spinning axis being known in the theory of the nuclear magnetic resonance [49], [50].",
"Thus, the DDI in dipolar gases provides a generic mechanism for the soliton formation of stable 2D solitons.",
"The rest of the paper is structured as follows.",
"In Sec.",
"II, we outline the derivation of the effective 2D model for the dipolar BEC polarized at an arbitrary tilt angle, starting from the 3D Gross-Pitaevskii equation.",
"Then, in subsection II.A, numerical solutions for 2D solitons, based on this effective equation, are produced for two different scenarios, which correspond to small and large DDI strengths.",
"In subsection II.B, a variational solution is obtained by minimizing the corresponding Lagrangian, using a 2D asymmetric Gaussian ansatz.",
"The variational approximation (VA) makes it also possible to predict the stability of the solitons on the basis of the Vakhitov-Kolokolov (VK) criterion, which is an essential result, as the stability is the critically important issue for the 2D solitons.",
"Further, in Sec.",
"III, we display a stability map for 2D soliton solutions in the parameter plane of the tilt angle and number of atoms, produced by an accurate numerical solution of the stability-eigenvalue problem for small perturbations.",
"Comparison of the variational and numerical results demonstrates that the VA predicts the “magic angle\", as the stability boundary, quite accurately.",
"In particular, while VA produces the single value of the “magic angle\", given by Eq.",
"(REF ), which does not depend on the relative strength of the DDI, $g_{d}$ , with respect to the local self-attraction, the numerical solution of the stability problem exhibits a very weak dependence on $g_{d}$ .",
"The paper is concluded by Sec.",
"IV.",
"Figure: A dipolar BEC in the 2D “pancake\" geometry, withthe dipole moments 𝐩 ^\\mathbf {\\hat{p}} oriented along the tilt angle defined,θ\\protect \\theta , with respect to the normal axis, 𝐳 ^\\mathbf {\\hat{z}}."
],
[
"The effective 2D model",
"We consider an obliquely polarized dipolar BEC trapped in the pancake-shaped potential, as shown in Fig.",
"1.",
"The oblique orientation of dipole moments is imposed by an external magnetic field, which makes the tilt angle, $\\theta $ , with the direction $\\mathbf {\\hat{z}}$ perpendicular to the pancake's plane.",
"The mean-field dynamics of the BEC at zero temperature is governed by by the GPE, which includes the integral term accounting for the DDI [13]: $i\\hbar \\frac{\\partial \\Psi (\\mathbf {r},t)}{\\partial t}&=&\\left[ -\\frac{\\hbar ^{2}}{2m}\\nabla ^{2}+V(z)+g|\\Psi (\\mathbf {r},t)|^{2}\\right.",
"\\\\&+&\\left.",
"\\left( \\int d^{3}\\mathbf {r}^{\\prime }V_{d}(\\mathbf {r}-\\mathbf {r}^{\\prime })|\\Psi (\\mathbf {r}^{\\prime },t)|^{2}\\right) \\right] \\Psi (\\mathbf {r},t).$ Here, $\\Psi (\\mathbf {r},t)$ is the wave function of condensate, $\\mathbf {r}=(x,y,z)$ is the position vector, $m$ is the atomic mass, and $V(z)=m\\omega _{z}^{2}{z}^{2}/2$ is the confining potential acting in the transverse direction.",
"The anisotropic DDI kernel is $V_{d}(\\mathbf {r})=g_{d}(1-3\\cos ^{2}\\eta )/r^{3}, $ where the DDI strength is $g_{d}=\\mu _{0}\\mu _{m}^{2}/4\\pi $ , with the vacuum permeability $\\mu _{0}$ and magnetic dipole moment $\\mu _{m}$ , while $\\eta $ is the angle between vector $\\mathbf {r}$ and the orientation of dipole moments $\\mathbf {\\hat{p}}$ .",
"Note that this kernel vanishes at $\\eta =\\arccos \\left( 1/\\sqrt{3}\\right) $ , which coincides with the “magic angle\" predicted by the VA as a boundary between stable and unstable solitons, see Eq.",
"(REF ) below.",
"The usual contact interaction is represented in Eq.",
"(REF ) by the local cubic term with coefficient $g=4\\pi \\hbar ^{2}a/m$ , where $a$ is the $s$ -wave scattering length $a$ .",
"The norm of the wave function is fixed by total number of atoms, $N=\\int d^{3}\\mathbf {r}|\\Psi (\\mathbf {r},t)|^{2}$ .",
"The 3D GPE (REF ) can be reduced into an effective 2D equation, provided that the confinement in the $z$ -direction is strong enough.",
"To this end, we assume, as usual, that the 3D wave function is factorized, $\\Psi (\\mathbf {r})=\\psi (\\mathbf {\\rho })\\phi (z)\\exp \\left( -i\\mu t/\\hbar \\right) $ , with transverse coordinates $\\mathbf {\\rho }=(x,y)$ and chemical potential $\\mu $ [51], [52], [53].",
"The transverse wave function is taken as the normalized ground state of the respective trapping potential, $\\phi (z)=\\left( \\pi L_{z}^{2}\\right) ^{-1/4}\\exp \\left( -z^{2}/2L_{z}^{2}\\right) $ , with the characteristic length $L_{z}=\\sqrt{\\hbar /m\\omega _{z}}$ .",
"Then, integrating Eq.",
"(REF ) over the $z$ -coordinate, the factorized ansatz leads one to the following effective 2D equation: $&&\\hspace{-14.45377pt}\\left( \\frac{\\mu }{\\hbar }-\\frac{1}{2}\\omega _{z}\\right)\\hbar \\psi ({\\rho })=-\\frac{\\hbar ^{2}}{2m}\\nabla _{\\perp }^{2}\\psi (\\rho )+\\frac{g}{\\sqrt{2\\pi }L_{z}}|\\psi (\\rho )|^{2}\\psi (\\rho ) \\\\&&\\hspace{-14.45377pt}+\\frac{g_{d}}{L_{z}}\\left[ \\int \\frac{d^{2}\\mathbf {k}_{\\rho }}{(2\\pi )^{2}}\\,n(\\mathbf {k}_{\\rho })V_{2d}\\left( \\frac{\\mathbf {k}_{\\rho }L_{z}}{\\sqrt{2}}\\right) \\,e^{i\\,\\mathbf {k}_{\\rho }\\cdot \\rho }\\right] \\psi (\\rho ), $ where $\\nabla _{\\perp }^{2}\\equiv \\partial ^{2}/\\partial x^{2}+\\partial ^{2}/\\partial y^{2}$ , $n(\\mathbf {k}_{\\rho })\\equiv \\int d^{2}\\rho \\,|\\psi (\\rho )|^{2}\\exp [-i\\,\\mathbf {k}_{\\rho }\\rho ]$ is the Fourier transform of the 2D density, $|\\psi (\\rho )|^{2}$ , and $k_{\\rho }=(k_{x}^{2}+k_{y}^{2})^{1/2}$ .",
"Further, defining that the dipoles are polarized aligned in the $\\left( x,z\\right) $ plane, i.e., $\\mathbf {\\hat{p}}=(\\sin \\theta ,0,\\cos \\theta )$ and $\\cos \\eta =\\mathbf {\\hat{p}}\\cdot \\hat{\\mathbf {r}}$ , in the momentum ($\\mathbf {k}$ -) space, the DDI kernel takes the form of $&&V_{2d}\\left( \\frac{\\mathbf {k}_{\\rho }L_{z}}{\\sqrt{2}}\\right) =-\\frac{2\\sqrt{2\\pi }}{3}(1-3\\cos ^{2}\\theta ) \\\\ \\nonumber &&+\\left[ 1-3\\cos ^{2}\\theta +\\cos (2\\zeta )\\sin ^{2}\\theta \\right] \\pi \\mathbf {k}_{\\rho }L_{z}\\exp \\left( \\frac{\\mathbf {k}_{\\rho }^{2}L_{z}^{2}}{2}\\right)\\\\\\nonumber && \\times \\text{erfc}\\left( \\frac{\\mathbf {k}_{\\rho }L_{z}}{\\sqrt{2}}\\right) ,$ with $\\cos \\zeta \\equiv k_{x}/k_{\\rho }$ and the complementary error function $\\text{erfc}$ in the momentum space.",
"Rescaling Eq.",
"(3) by $\\mu \\rightarrow \\mu /\\hbar \\omega _{z}-1/2$ , $\\nabla _{\\perp }\\rightarrow \\nabla _{\\perp }L_{z}$ , $\\rho \\rightarrow \\rho /L_{z}$ , $\\mathbf {k}_{\\rho }\\rightarrow \\mathbf {k}_{\\rho }L_{z}$ , $\\psi (\\rho )\\rightarrow \\psi (\\rho )\\sqrt{2(2\\pi )^{1/2}|a|L_{z}}$ , $g\\rightarrow g/(4\\pi \\hbar \\omega _{z} |a| L_{z}^2)$ , and $g_{d}\\rightarrow g_{d}/(2\\sqrt{2\\pi }\\hbar \\omega _{z} |a| L_{z}^2)$ , we arrive at the following normalized 2D equation: $\\mu \\psi (\\rho )&=&-\\frac{1}{2}\\nabla _{\\perp }^{2}\\psi (\\rho )+g|\\psi (\\rho )|^{2}\\psi (\\rho ) \\\\&+&g_{d}\\left[ \\int \\frac{d^{2}\\mathbf {k}_{\\rho }}{(2\\pi )^{2}}\\,n(\\mathbf {k}_{\\rho })V_{2d}\\left( \\frac{\\mathbf {k}_{\\rho }}{\\sqrt{2}}\\right) \\,e^{i\\,\\mathbf {k}_{\\rho }\\cdot \\rho }\\right] \\psi (\\rho ).$ According to the rescaling, the norm of the 2D wave function, $N_{2}\\equiv \\int d^{2}\\rho \\,|\\psi ({\\rho })|^{2}$ , is related to the number of atoms: $N=N_{2}\\times (L_{z}/(2\\sqrt{2\\pi }|a|)$ .",
"Our model is based on Eq.",
"(REF ).",
"For example, in the case of the BECs of $^{52}$ Cr atoms, the atomic magnetic moment is $\\mu _{m}=6$ $\\mu _{\\mathrm {Bohr}}$ , and an experimentally relevant trapping frequency, $\\omega _{z}=2\\pi \\times 800~\\text{Hz} $ [54], [55], [56], [57], corresponds to the characteristic transverse length $L_{z}=0.493$ $\\mathrm {\\mu }$ m. With the same trapping frequency, for BECs of $^{168}$ Er atoms, we have $\\mu _{m}=7$ $\\mu _{\\mathrm {Bohr}}$ and $m=2.8\\times 10^{-25}$ gram, which corresponds to a characteristic transverse length $L_{z}=0.274$ $\\mathrm {\\mu }$ m; while for $^{162}$ Dy atoms, we have $\\mu _{m}=10$ $\\mu _{\\mathrm {Bohr}}$ , $m=2.7\\times 10^{-25}$ gram, and $L_{z}=0.279$ $\\mathrm {\\mu }$ m, respectively.",
"Figure: Transverse profiles of 2D solitons in the xx-yy plane normalized to the characteristic transverse length L z L_{z}, by the produced by the numericalsolution of Eq.",
"(), with chemical potential μ=-0.01\\protect \\mu =-0.01, fixed strength of contact attraction, g=-1g=-1, and a small DDI strength, g d =0.1g_{d}=0.1.",
"The tilt angles are (a) θ=0\\protect \\theta =0, (b) θ=0.73\\protect \\theta =0.73 (41.8 ∘ 41.8^{\\circ }), (c) θ=π/3\\protect \\theta =\\protect \\pi /3, and (d) θ=π/2\\protect \\theta =\\protect \\pi /2.Here, the corresponding particle number N 2 N_2 are (a) 7.157.15, (b) 6.236.23, (c) 5.715.71 and (d) 5.345.34, respectively.Figure: The same as in Fig.",
", but for stronger DDI, with g d =1.0g_{d}=1.0.Here, the corresponding particle number N 2 N_2 are (a) 4.694.69, (b) 9.119.11, (c) 4.534.53, and (d) 2.862.86, respectively."
],
[
"2D numerical soliton solutions",
"In the absence of the DDI, $g_{d}=0$ , solutions in the form of isotropic Townes solitons are supported by attractive contact interaction with $g<0$ [6], [7], [58].",
"Then, by fixing the strength of the contact attraction, $g=-1$ , we introduce the DDI in Eq.",
"(REF ) and seek for 2D bright-soliton solutions numerically, by varying the DDI strength, $g_{d}$ , for different values of of the chemical potential, $\\mu $ .",
"The validity of our effective 2D equation for the pancake geometry is ensured by checking that the transverse width of the 2D soliton solutions is larger than the transverse-confinement length, $L_{z}$ in the $z$ -direction.",
"This condition sets a constraint on the available range for the chemical potential, i.e., $|\\mu |/\\hbar \\omega _{z}\\ll 1$ .",
"In our simulations, the 2D effective equations remain valid in the range of $-0.1<\\mu <-0.003.",
"$ The tilt angle of the dipoles in the $\\left( x,z\\right) $ plane was also varied, in the full interval of $0<\\theta <\\pi /2$ .",
"The DDI sign is fixed as $g_{d}>0$ , which corresponds to the natural situation of the repulsion between the dipoles oriented perpendicular to the pancake's plane, $\\theta =0$ .",
"Thus, the DDI is isotropic but repulsive at $\\theta =0$ , being anisotropic at $\\theta \\ne 0$ .",
"Accordingly, the DDI tends to compete with the fixed-strength contact attraction.",
"Numerical solution of Eq.",
"(REF ) produces 2D soliton profiles, typical examples of which are displayed in Figs.",
"REF and REF , for $\\mu =-0.01$ .",
"With the fixed contact-interaction coefficient, $g=-1$ , we find two different scenarios of the evolution of the shape of the 2D solitons.",
"For weak DDI, such as with coefficient $g_{d}=0.1$ , starting with the isotropic profile at $\\theta =0$ [Fig.",
"REF (a)], the transverse widths in $x$ - and $y$ -directions both expand, but at different rates, as the tilt angle increases, see Fig.",
"REF (b-d) for $\\theta =0.73$ ($41.8^{\\circ }$ ) and $\\pi /3$ , respectively.",
"The 2D solitons are wider along the $x$ -direction and narrower along $y$ because the dipoles are tilted in the $\\left( x,z\\right) $ plane.",
"For a larger DDI strength, such as $g_{d}=1.0$ , we still have an isotropic profile at $\\theta =0$ , as shown in Fig.",
"3(a).",
"As the tilt angle increases, the transverse widths in $x$ - and $y$ -directions shrink just slightly, remaining nearly equal at $\\theta =0.73$ ($41.8^{\\circ }$ ), $\\pi /3$ , and $\\pi /2$ , as shown in Figs.",
"3(b-d).",
"Note that, quite naturally, the radius of the isotropic profile, observed at $\\theta =0$ , is smaller in Fig.",
"REF (a) than in Fig.",
"REF (a), as in the latter case the dipole-dipole repulsion is much stronger than the competing contact attraction.",
"Nevertheless, the increase of $\\theta $ makes the expansion of the profiles and the growth of its anisotropy, which are effects of the DDI, more salient in Fig.",
"REF , i.e., when the DDI is weaker.",
"This counter-intuitive evolution of the shape may be explained by the fact that it is shown not for the fixed number of atoms, $N_{2}$ , but for a fixed chemical potential, $\\mu $ .",
"To keep the same $\\mu $ in the case of the stronger DDI competing with the contact self-attraction (in Fig.",
"REF ), the system needs to increase $N_{2}$ , which, in turn, helps the contact interaction to keep the compact nearly-isotropic shape of the soliton.",
"Figure: The scaled 2D particle number, N 2 N_{2} in the solitonsolutions,versus the tilt angle, θ\\protect \\theta , and chemical potential, μ\\protect \\mu , at the fixed strength of the contact interaction, g=-1g=-1, anda small DDI strength, g d =0.1g_{d}=0.1.Figure: The same as in Fig.",
", but for a much strongerDDI, with g d =1.0g_{d}=1.0.",
"In this case, N 2 N_{2} attains ist maximum at θ≈0.73\\protect \\theta \\approx 0.73 (41.8 ∘ 41.8^{\\circ }), irrespective of the value of μ\\protect \\mu .To present a clearer illustration on these trends, we display, in Figs.",
"REF and REF , $N_{2}$ as a function of $\\theta $ and $\\mu $ , for the same small and large strengths of DDI, i.e., $g_{d}=0.1$ and $g_{d}=1.0$ , respectively.",
"In accordance with what is said above, $N_{2}$ decreases monotonously at $g_{d}=0.1$ , as the tilt angle increases from $\\theta =0$ to $\\pi /2$ , at all values of $\\mu $ .",
"However, the stronger DDI strength (with $g_{d}=1.0$ ) produces a completely different picture (also in agreement with the above explanation): as $\\theta $ increases from 0, $N_{2}$ at first increases too, reaching a maximum at $\\theta =\\theta _{0}\\approx 0.73~(\\text{equivalent to }41.8^{\\circ })$ [note that this angle is smaller than the critical (“magic\") one, $\\theta _{m}$ , given below by Eq.",
"(REF ), which is an approximate boundary between the stable and unstable solitons].",
"As mentioned above, the increase of $N_{2}$ is necessary to keep the same value of $\\mu $ while the essentially repulsive DDI competes with the local self-attraction, at $\\theta <\\theta _{0}$ .",
"Then, $N_{2}$ decreases, as $\\theta $ passes $\\theta _{0}$ and approaches $\\pi /2$ .",
"Indeed, in the latter case, the DDI becomes essentially attractive [46], hence the local and nonlocal interaction act together, instead of competing, making it possible to keep the given value of $\\mu $ with a smaller norm.",
"Note that these trends are the same at different values of $\\mu $ , although the corresponding values of $N_{2}$ are, naturally, different.",
"Below, we demonstrate that angle $\\theta _{0}$ can be accurately predicted by the variational approximation, see Fig.",
"REF (a).",
"Figure: Comparison of the norm of the 2D wave function, N 2 N_{2}, asproduced by tne numerical solution and variational approximation (solid anddashed lines, respectively).",
"(a) N 2 (θ)N_{2}(\\protect \\theta ) at fixed μ=-0.01\\protect \\mu =-0.01, for the weak and strong DDI, g d =0.1g_{d}=0.1 and 1 (red and bluelines, respectively).",
"Two other panels display N 2 (μ)N_{2}(\\protect \\mu ) for thesame fixed values of the DDI strength: g d =0.1g_{d}=0.1 (b) and 1.01.0 (c).",
"Inthese panels, fixed values of the tilt angle are θ=0\\protect \\theta =0 (redlines), θ=θ 0 ≈0.73\\protect \\theta = \\protect \\theta _{0}\\approx 0.73 [see Eq.",
"(8); magenta], and θ=π/2\\protect \\theta =\\protect \\pi /2 (blue).",
"In eachpanel we also show, by solid and dashed black horizontal lines, the constantvalue of N 2 N_{2} for the Townes soliton (when the DDI is absent, g d =0g_{d}=0)and its variationally predicted counterpart (see the main text for details).For the condensate of 52 ^{52}Cr atoms, the corresponding total numbers ofatoms, for fixed values of other parameters [see Eq.",
"()],are given on the right vertical axes, as a reference for a possibleexperiment."
],
[
"The variational approximation (VA)",
"In addition to numerical solutions, we have developed the VA, following the lines of Refs.",
"[59], [60] and using the Lagrangian density corresponds to Eq.",
"(REF ) $\\mathcal {L}&=&-\\mu |\\psi |^{2}+\\frac{1}{2}|\\nabla _{\\perp }\\psi |^{2}+\\frac{g}{2}|\\psi |^{4} \\nonumber \\\\&+&\\frac{g_{d}}{2}|\\psi |^{2}\\int d^{2}\\rho ^{\\prime }\\,V_{2d}\\left( \\rho -\\rho ^{\\prime }\\right) |\\psi (\\rho ^{\\prime })|^{2}.",
"$ The corresponding Gaussian ansatz is, naturally, anisotropic: $\\psi _{\\text{ans}}=\\sqrt{\\frac{N_{2}}{\\pi w_{x}w_{y}}}\\exp \\left( -\\frac{x^{2}}{2w_{x}^{2}}-\\frac{y^{2}}{2w_{y}^{2}}\\right) \\,,$ with the 2D norm $N_{2}$ , and different transverse widths in $x$ - and $y$ -directions, $w_{x}$ and $w_{y}$ .",
"Then, the effective Lagrangian $L=\\int dxdy\\,\\mathcal {L}$ is calculated: $L&=&-N_{2}\\mu +\\frac{N_{2}(w_{x}^{2}+w_{y}^{2})}{4w_{x}^{2}w_{y}^{2}} \\\\&+&\\frac{gN_{2}^{2}}{4\\pi w_{x}w_{y}}+\\frac{g_{d}N_{2}^{2}}{8\\pi ^{2}}\\left(f_{1}+f_{2}\\right) , $ were we have introduced the short-hand notation: $&&\\hspace{-14.45377pt}f_{1}=-\\frac{4\\sqrt{2}\\pi ^{3/2}}{3w_{x}w_{y}}\\left(1-3\\cos ^{2}\\theta \\right), \\\\&&\\hspace{-14.45377pt}f_{2}=2\\pi ^{2}\\int _{0}^{\\infty }dk_{\\rho }\\,\\left\\lbrace k_{\\rho }^{2}\\exp \\left( \\frac{2k_{\\rho }^{2}-k_{\\rho }^{2}(w_{x}^{2}+w_{y}^{2})}{4}\\right) \\right.",
"\\\\&&\\hspace{-14.45377pt} \\times \\text{erfc}\\left( \\frac{k_{\\rho }}{\\sqrt{2}}\\right) \\times \\left[ (1-3\\cos ^{2}\\theta )\\,I_{0}\\left( \\frac{k_{\\rho }^{2}(w_{x}^{2}-w_{y}^{2})}{4}\\right) \\right.",
"\\\\&&\\hspace{-14.45377pt}\\left.\\left.",
"-\\left( \\sin ^{2}\\theta \\right) \\,I_{1}\\left( \\frac{k_{\\rho }^{2}(w_{x}^{2}-w_{y}^{2})}{4}\\right) \\right] \\right\\rbrace ,$ with the modified Bessel functions, $I_{0,1}\\left( z\\right) $ .",
"The Euler-Lagrange equations follow from Eq.",
"(REF ) in the form of $\\partial L/\\partial \\left( w_{x, y},N_{2}\\right) =0$ : $\\hspace{-21.68121pt}-\\mu +\\frac{w_{x}^{2}+w_{y}^{2}}{4w_{x}^{2}w_{y}^{2}}+\\frac{gN_{2}}{2\\pi w_{x}w_{y}}+\\frac{g_{d}N_{2}(f_{1}+f_{2})}{4\\pi ^{2}}=0, \\\\\\hspace{-21.68121pt}-\\frac{1}{2w_{x}^{3}}-\\frac{gN_{2}}{4\\pi w_{x}^{2}w_{y}}+\\frac{g_{d}N_{2}}{8\\pi ^{2}}\\left( \\frac{\\partial {f_{1}}}{\\partial {w_{x}}}+\\frac{\\partial {f_{2}}}{\\partial {w_{x}}}\\right) =0, \\\\\\hspace{-21.68121pt}-\\frac{1}{2w_{y}^{3}}-\\frac{gN_{2}}{4\\pi w_{x}w_{y}^{2}}+\\frac{g_{d}N_{2}}{8\\pi ^{2}}\\left( \\frac{\\partial {f_{1}}}{\\partial {w_{y}}}+\\frac{\\partial {f_{2}}}{\\partial {w_{y}}}\\right) =0.",
"$ For small arguments, $0<|z|\\ll \\sqrt{\\alpha +1}$ , the modified Bessel function can be replaced by the first term of its expansion, $I_{\\alpha }(z)\\approx (z/2)^{\\alpha }/\\Gamma (\\alpha +1)$ , where $\\Gamma $ is the Gamma-function.",
"Such an approximation makes it possible to simplify Eqs.",
"(REF )-(15) in the case of $0<k_{\\rho }^{2}(w_{x}^{2}-w_{y}^{2})/4\\ll 1.",
"$ This condition implies that either the soliton is wide in comparison with the characteristic transverse-confinement width, $L_{z}$ (which may be naturally expected from the quasi-2D solitons), i.e., $k_{\\rho }\\ll 1$ , or the profile is an almost symmetric one, with $\\left|w_{x}^{2}-w_{y}^{2}\\right|\\ll w_{x, y}^{2}$ .",
"Further analysis makes it possible to expand, under condition (REF ) and to the first-order in $g_d$ , the VA-predicted 2D norm of the wave function as $&&\\hspace{-14.45377pt}N_{2}(\\mu )=2\\pi -\\frac{g_{d}\\,\\pi ^{3/2}[1+3\\cos (2\\theta )]}{3(1+2\\mu )^{2}\\sqrt{\\frac{-2}{\\mu }-4}}\\times \\\\ \\nonumber &&\\hspace{-14.45377pt}\\left[ (4+2\\mu )(4\\mu -1)\\sqrt{\\frac{-1}{\\mu }-2}+9\\sqrt{2} \\arctan \\sqrt{\\frac{-1}{2\\mu }-1}\\right],$ where $2\\pi $ is the well-known VA prediction for the 2D norm of the Townes solitons [59], which is obviously valid in the limit of $g_{d}=0$ , while the term $\\sim g_{d}$ in Eq.",
"(REF ) is a small correction to it.",
"The correction is a critically important one, as it lifts the degeneracy of the Townes solitons, whose norm does not depend on $\\mu $ [5], [6], [7], and thus makes it possible to check the VK criterion, which states that a necessary condition for the stability of any soliton family supported by self-attractive nonlinearity is $dN_{2}/d\\mu <0$ [48], [61], [62], [63].",
"It originates from the condition that a soliton which may be stable should realize a minimum of the energy for a given value of the norm.",
"Note also that condition $-1/2 < \\mu < 0$ , which is obviously necessary for the validity of Eq.",
"(REF ), definitely holds in the range of $\\mu $ given by Eq.",
"(REF ), dealt with in the present work.",
"Applying the VK criterion to the $N_{2}(\\mu )$ dependence given by Eq.",
"(REF ), we obtain $&&\\hspace{-14.45377pt}\\frac{d\\,N_{2}}{d\\,\\mu }=-\\frac{\\,g_{d}\\pi ^{3/2}\\left( 1-3\\cos ^{2}\\theta \\right) }{\\mu (1+2\\mu )^{3}\\sqrt{\\frac{-2}{\\mu }-4}} \\\\ \\nonumber &&\\hspace{-14.45377pt}\\times \\left[ 2\\mu (4\\mu -13)\\sqrt{\\frac{-1}{\\mu }-2} +3\\sqrt{2}(8\\mu -1)\\tan ^{-1}\\sqrt{\\frac{-1}{2\\mu }-1}\\right].$ It immediately follows from Eq.",
"(REF ) that the VK criterion holds, i.e., the solitons may be stable (in the framework of the VA), if the dipoles are polarized under a sufficiently large angle $\\theta $ with respect to the normal direction, i.e., the polarization is relatively close to the in-plane configuration (cf.",
"Ref.",
"[46]): $\\theta >\\theta _{m}\\equiv \\cos ^{-1}(1/\\sqrt{3})\\approx 0.955~(\\text{tantamount to }54.74^{\\circ }).",
"$ On the other hand, the solitons are predicted to be definitely unstable at $\\theta <\\theta _{m}$ .",
"The same critical (alias “magic\") angle is known, e.g., in the theory of the nuclear magnetic resonance, when a sample is spinning about a fixed axis [49], [50].",
"Note that, in the framework of the approximation based on Eqs.",
"(REF ) and (REF ), at $\\theta =\\theta _{m}$ the 2D norm of the solitons coincides with that of the Townes solitons.",
"In the more general case, we have found the VA-predicted parameters $N_{2}$ and $w_{x,y}$ solving Eqs.",
"(REF )-() numerically.",
"In Fig.. REF we present the comparison of norm of $N_{2}$ , as obtained from the full numerical solution of Eq.",
"(REF ) and its counterpart predicted by the VA (solid and dashed curves, respectively).",
"For the reference, we also show the constant value, $N_{2}^{(T)}\\approx 5.85$ for the Townes solitons ($g_{d}=0$ ), and its above-mentioned VA-predicted counterpart, $N_{2}^{(T)}=2\\pi $ [59].",
"In particular, Fig.",
"REF (a) features the same trends in the dependence $N_{2}(\\theta )$ at fixed $\\mu $ as were identified, and qualitatively explained, above while addressing Figs.",
"REF and REF : in the case of the weak DDI, the dependence is monotonous, while the strong DDI gives rise to a well-pronounced maximum at point (REF ).",
"In Fig.",
"REF , we also depict the 2D norm $N_{2}$ as a function of the chemical potential, $\\mu $ , for (b) weak and (c) strong DDI, i.e., $g_{d}=0.1$ and $1.0$ , respectively, for three fixed tilt angles, namely, $\\theta =0$ (the dipoles polarized perpendicular to the pancake), $\\theta = \\theta _0$ [the special value given by Eq.",
"(REF )], and $\\theta =\\pi /2$ (the in-plane polarization).",
"In particular, it is seen that the slope of the $N_{2}\\left( \\mu \\right) $ dependences, which determines the VK criterion, is definitely positive, slightly or strongly positive (for small or large DDI strength), and slightly negative, for $\\theta =0$ (red curves), $\\theta =\\theta _{0}$ (magenta curves), and $\\theta =\\pi /2$ (blue curves), respectively.",
"These conclusions, which pertain to the weak and strong DDI alike, agree with the prediction of Eq.",
"(REF ), namely, $dN_{2}/d\\mu < 0$ for $\\theta >\\theta _{m}$ , and $dN_{2}/d\\mu > 0$ for $\\theta <\\theta _{m}$ .",
"Lastly, Figs.",
"REF (b,c) also show, as a reference for possible experimental realization, the expected numbers of atoms in the solitons created in the $^{52}$ Cr condensate, transversely trapped under condition (REF ).",
"Figure: The stability map for 2D solitons in the plane of the DDI strength,g d g_{d}, and the tilt angle, θ\\protect \\theta , as produced by the solutionof the eigenvalue problem for small perturbations.",
"The solitons are stableat θ cr <θ≤π/2\\protect \\theta ^{\\text{cr}}<\\protect \\theta \\le \\protect \\pi /2, where θ cr (g d )\\protect \\theta ^{\\text{cr}}(g_{d}) is shown by the blue line.",
"As above, thestrength of the contact interaction is fixed to be g=-1g=-1.",
"Points Aand B correspond, respectively, to the smallest and largestvalues of θ cr \\protect \\theta ^{\\text{cr}}, respectively (the definition of thelargest value excludes the narrow stripe of the quick decrease of θ cr \\protect \\theta ^{\\text{cr}} with the increase of g d g_{d} from 0 to point A).",
"In fact, the difference between the largest and smallest values issmall.",
"The nearly flat shape of the stability boundary roughly agrees withthe analytical prediction given by Eq.",
"()."
],
[
"Stability of the 2D solitons",
"As said above, stability is the critically important issue for 2D solitons, as the usual cubic local self-attraction creates Townes solitons which are subject to the subexponential instability against small perturbations [2], [6], [7].",
"Originally, the perturbations grow with time algebraically, rather than exponentially, but eventually the solitons are quickly destroyed.",
"The subexponential instability implies that, in terms if the above-mentioned VK criterion, the Townes solitons are, formally, neutrally stable, having $dN_{2}/d\\mu =0$ [see the flat black lines in Figs.",
"REF (b,c)].",
"As said above, Eq.",
"(REF ) and Figs.",
"6(b,c) demonstrate that the addition of the DDI to the local self-attraction lifts the degeneracy (the independence of the norm of the Townes solitons on the chemical potential).",
"The resulting sign of the slope, $dN_{2}/d\\mu $ , is the same for the numerical solutions and their counterparts predicted by the variational approximation.",
"The sign is the same too for both the weak and strong DDI ($g_{d}=0.1$ and $g_{d}=1$ ).",
"Equation (REF ) produces an important prediction, that, with the increase of the title angle from $\\theta =0$ to $\\pi /2$ , the slope $dN_{2}/d\\mu $ changes from positive (unstable) to negative (possibly stable) at the “magic angle\" given by Eq.",
"(REF ).",
"Because the VK criterion is only a necessary stability condition, and also because Eq.",
"(REF ) was derived in approximately, under condition Eq.",
"(REF ), it is necessary to develop the consistent linear stability analysis for our numerically generated soliton solutions.",
"To this end, we introduce a perturbed solution as $\\psi (\\rho ,t)=\\lbrace \\psi _{0}(\\rho )+\\epsilon \\left[ p(\\rho )e^{-i\\delta t}+q(\\rho )e^{i\\delta ^{\\ast }t}\\right] \\rbrace e^{-i\\mu t}.$ Here, the asterisk stands for the complex-conjugate value, $\\psi _{0}(\\rho )$ is the unperturbed solution, $\\epsilon $ is an infinitesimal perturbation amplitude, while $p(\\rho )$ and $q(\\rho )$ are eigenmodes of the small perturbation, with the respective eigenvalue $\\delta $ .",
"The instability occurs in the case when $\\delta $ is not real.",
"The unperturbed solution was classified as a stable one if the numerically found instability growth rate, $\\left|\\mathrm {Im}(\\delta )\\right|$ , was smaller than $10^{-7}$ .",
"Results of the stability analysis are summarized in Fig.",
"REF , where the stability map for the soliton solutions is displayed in the plane of the DDI strength, $g_{d}$ , and the tilt angle, $\\theta $ , the stability region being $\\theta ^{\\mathrm {cr}}(g_{d})<\\theta \\le \\pi /2.",
"$ This map is found to be the same, up to the accuracy of the numerically collected data, for the entire interval (REF ) of values of the chemical potential in which the derivation of the effective 2D equation (REF ) is valid.",
"This map shows that the originally unstable Townes solitons, corresponding to $g_{d}=0$ , quickly attains the stability saturation, i.e., expansion of the stability interval (REF ) to its limits, $\\theta ^{\\mathrm {cr}}\\approx \\theta _{m}<\\theta \\le \\pi /2$ , at very small values of $g_{d}$ .",
"At $g_{d}=g_{d}^{\\text{cr}}\\approx 0.059$ , the stability boundary attains its minimum value, $\\theta ^{\\text{cr}}\\approx 0.97$ , as labeled by the point A in Fig.",
"REF .",
"With the increase of $g_{d}$ , the critical tilt slightly increases to $\\theta =1.04$ (tantamount to $59.59^{\\circ }$ ), as labeled by point B, which corresponds to $g_{d}=0.91$ .",
"Comparing these numerically exact results with the analytical prediction given by Eq.",
"(REF ), we conclude that the relative error is limited to $8.2\\%$ , and, although the VA fails to predict the dependence of $\\theta ^{\\text{cr}}$ on $g_{d}$ , the actual dependence is quite weak.",
"Lastly, it is relevant to stress that, setting $d N_2/d\\mu = 0$ to identify the VK-predicted stability boundary, we obtain results, from the full numerical solution, for both weak and strong DDI, with $g_d = 0.1$ and $1.0$ , respectively, which exactly coincide with the stability boundary identified above through the calculation of the linear-stability eigenvalues, i.e., $\\theta ^{\\text{cr}} = 0.97$ and $1.04$ .",
"Before the conclusion, we discuss the possibility to stabilize dipolar BECs with quantum fluctuations.",
"The stability boundary we reveal above is based on the mean-field theory.",
"However, when the quantum fluctuations are taken into consideration, a repulsive, known as Lee-Huang-Yang (LHY), correction may stabilize an attractive Bose gas [65].",
"Recently, experimental observations on stable and ordered arrangement of droplets in an atomic dysprosium BEC illustrated the importance of LHY quantum fluctuations in stabilizing the system against collapse [66], [67].",
"LHY corrections have be shown to stabilize droplets in unstable Bose-Bose mixtures [68], and self-bound filament-like droplets [69].",
"Relations on an arbitrary tilt angle to LHY corrections, and related stability of 2D solitons with DDI interaction deserve further study."
],
[
"Conclusion",
"For the dipolar BEC confined to the pancake geometry, we have investigated the formation and stability of 2D soliton with the atomic magnetic moments polarized in an arbitrary direction.",
"Fixing the strength of contact attractive interaction (which, by itself, would only create unstable Townes solitons), we demonstrate, by means of the numerical methods and VA (variational approximation), combined with the VK (Vakhitov-Kolokolov) criterion, that the 2D solitons can be completely stabilized by the DDI (dipole-dipole interaction) with relative strength $g_{d}$ , which makes the solitons anisotropic.",
"Both the VK criterion and numerically exact linear-stability analysis confirm that, there exists a “magic angle\" of the polarization tilt, $\\theta ^{\\text{cr}}$ , such that the 2D solitons are stable at $\\theta ^{\\mathrm {cr}}(g_{d})<\\theta \\le \\pi /2$ .",
"While the VA predicts $\\theta ^{\\mathrm {cr}}=\\arccos (1/\\sqrt{3})$ which does not depend on $g_{d}$ , the numerically exact results feature a weak dependence of $\\theta ^{\\mathrm {cr}}$ on $g_{d}$ , with the actual values of $\\theta ^{\\mathrm {cr}}$ being quite close to the VA prediction.",
"We also produce physical parameters for experiments in the condensate of $^{52}$ Cr atoms, which should make the creation of the stable 2D solitons possible."
],
[
"ACKNOWLEDGMENTS",
"This work was supported by the Ministry of Science and Technology of Taiwan under Grant Nos.",
"105-2119-M-007-004.",
"The work of Y.L.",
"was supported by Grant No.",
"11575063 from the National Natural Science Foundation of China.",
"The work of B.A.M.",
"was supported, in part, by Grant No.",
"2015616 from the joint program in physics between the Binational (US-Israel) Science Foundation and National Science Foundation (USA)."
]
] | 1709.01646 |
[
[
"Descriptors for Machine Learning of Materials Data"
],
[
"Abstract Descriptors, which are representations of compounds, play an essential role in machine learning of materials data.",
"Although many representations of elements and structures of compounds are known, these representations are difficult to use as descriptors in their unchanged forms.",
"This chapter shows how compounds in a dataset can be represented as descriptors and applied to machine-learning models for materials datasets."
],
[
"Introduction",
"Recent developments of data-centric approaches should accelerate the progress in materials science dramatically.",
"Thanks to the recent advances in computational power and techniques, the results from numerous density functional theory (DFT) calculations with predictive performances have been stored as databases.",
"A combination of such databases and an efficient machine-learning approach should realize prediction and classification models of target physical properties.",
"Consequently, machine-learning techniques are becoming ubiquitous.",
"They are used to explore materials and structures from a huge number of candidates and to extract meaningful information and patterns from existing data.",
"A key factor in controlling the performance of a machine-learning approach is how compounds are represented in a data set.",
"Representations of compounds are called “descriptors” or “features”.",
"To perform machine-learning modeling, available descriptors must be determined according to the evaluation cost of the target property and the extent of the exploration space.",
"Based on these considerations, we aim to select “good” descriptors.",
"Prior or experts' knowledge, including a well-known correlation between the target property and the other properties, can be used to select good descriptors.",
"However, the set of descriptors in many cases is examined by trial-and-error because the predictive performance (i.e., the prediction error and efficiency of the model) strongly depends on the quality and data-size of the target property.",
"Section shows how to prepare descriptors of compounds.",
"Sections and introduce representations of chemical elements (elemental representations) and atomic arrangements (structural representations) required to generate compound descriptors.",
"Sections , , and provide applications of machine-learning models for materials datasets, including the construction of a machine-learning prediction model for the DFT cohesive energy, the construction of the machine-learning interatomic potential (MLIP) for elemental metals, materials discovery of low lattice thermal conductivity (LTC), and materials discovery based on the recommender system approach."
],
[
"Compound descriptors",
"Most candidate descriptors can be classified into three groups.",
"The first is the physical properties of a compound in a library and/or their derivative quantities, which are less available.",
"The second is the physical properties of a compound computed by DFT calculations or their derivative quantities.",
"The third is the properties of elements and the structure of a compound and/or their derivative quantities.",
"Combinations of different groups of descriptors can also be useful.",
"A set of compound descriptors should satisfy the following conditions: (i) the same-dimensional descriptors express compounds with a wide range of chemical compositions.",
"(ii) The same-dimensional descriptors express compounds with a wide range of crystal structures.",
"This is an important feature because crystals are generally composed of unit cells with different numbers of atoms.",
"(iii) A set of descriptors satisfies the translational, rotational, and other invariances for all compounds included in the dataset.",
"Figure: Binary elemental descriptors representing the presence of chemical elements.The number of binary elemental descriptors corresponds to the number of element types included in the training data.Candidates for compound descriptors based on DFT calculations include volume, band gap, cohesive energy, elastic constants, dielectric constants, etc.",
"The electronic structure and phonon properties can also be used as descriptors.",
"Although a few first-principles databases are available, the numbers of compounds and physical properties in the databases remain limited.",
"Nevertheless, when a set of descriptors that can well explain a target property is discovered, a robust prediction model can be derived for the target property.",
"Examples can be found in the literature (e.g., Refs.",
"Fujimura2013AENM:AENM201300060,seko2014machine,PhysRevB.93.115104,PhysRevB.93.054112).",
"Other candidates are simply a binary digit representing the presence of each element in a compound (Fig.",
"REF ) [21].",
"When training data is composed of m kinds of elements, a compound is described by an m-dimensional binary vector with elements of one or zero.",
"As a simple extension, a binary digit can be replaced with the chemical composition.",
"Such an application is shown in Sec.",
".",
"Another useful strategy is to use a set of quantities derived from elemental and structural representations of a compound as descriptors.",
"However, it is difficult to use elemental and structural representations as descriptors in their unchanged forms when the training data and search space cover a wide range of chemical compositions and crystal structures.",
"Consequently, it is essential to consider combined forms as compound descriptors.",
"Figure: Schematic illustration of how to generate compound descriptors.Here we provide compound descriptors derived from elemental and structural representations satisfying the above conditions.",
"These descriptors can be applied not only to crystalline systems but also to molecular systems [17].",
"Figure REF schematically illustrates the procedure to generate such descriptors for compounds.",
"First, the compound is considered to be a collection of atoms, which are described by element types and neighbor environments that are determined by other atoms.",
"Assuming the atoms are represented by $N_{x,{\\rm ele}}$ elemental representations and $N_{x,{\\rm st}}$ structural representations, each atom is described by $N_x = N_{x,{\\rm ele}}+N_{x,{\\rm st}}$ representations.",
"Therefore, compound $\\xi $ is expressed by a collection of atomic representations as a matrix with $(N_a^{(\\xi )},N_x)$ -dimensions, where $N_a^{(\\xi )}$ is the number of atoms in the unit cell of compound $\\xi $ .",
"The representation matrix for compound $\\xi $ , ${X}^{(\\xi )}$ , is written as ${X}^{(\\xi )} =\\begin{pmatrix}x_1^{(\\xi ,1)} & x_2^{(\\xi ,1)} & \\cdots & x_{N_x}^{(\\xi ,1)} \\\\x_1^{(\\xi ,2)} & x_2^{(\\xi ,2)} & \\cdots & x_{N_x}^{(\\xi ,2)} \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\x_1^{(\\xi ,N_a^{(\\xi )})} & x_2^{(\\xi , N_a^{(\\xi )})} & \\cdots & x_{N_x}^{(\\xi ,N_a^{(\\xi )})} \\\\\\end{pmatrix},$ where $x_n^{(\\xi ,i)}$ denotes the $n$ th representation of atom $i$ in compound $\\xi $ .",
"Since the representation matrix is only a representation for the unit cell of compound $\\xi $ , a procedure to transform the representation matrix into a set of descriptors is needed to compare different compounds.",
"One approach for this transformation is to regard the representation matrix as a distribution of data points in an $N_x$ -dimensional space (Fig.",
"REF ).",
"To compare the distributions themselves, representative quantities are subsequently introduced to characterize the distribution as descriptors, such as the mean, standard deviation (SD), skewness, kurtosis, and covariance.",
"The inclusion of the covariance enables the interaction between the element type and crystal structure to be considered.",
"A universal or complete set of representations is ideal because it can derive good machine-learning prediction models for all physical properties.",
"However, finding a universal set of representations is nearly impossible.",
"On the other hand, many elemental and structural representations have been proposed for a long time, not only in literature on the machine learning prediction but also in literature on the standard physics and chemistry.",
"Using these representations, many phenomena in physics and chemistry have been explained.",
"Therefore, it is a good way for generating descriptors to make effective use of the existing representations."
],
[
"Elemental representations",
"The literature contains numerous quantities that can be used as elemental representations.",
"This chapter employs a set of elemental representations composed of the following: (1) atomic number, (2) atomic mass, (3) period and (4) group in the periodic table, (5) first ionization energy, (6) second ionization energy, (7) electron affinity, (8) Pauling electronegativity, (9) Allen electronegativity, (10) van der Waals radius, (11) covalent radius, (12) atomic radius, (13) pseudopotential radius for the s orbital, (14) pseudopotential radius for the p orbital, (15) melting point, (16) boiling point, (17) density, (18) molar volume, (19) heat of fusion, (20) heat of vaporization, (21) thermal conductivity, and (22) specific heat.",
"These representations can be classified into the intrinsic quantities of elements (1)-(7), the heuristic quantities of elements (8)-(14), and the physical properties of elemental substances (15)-(22).",
"Such elemental representations should capture essential information of compounds.",
"Therefore, they should assist in building models with a high predictive performance, as shown in Secs.",
", , and ."
],
[
"Structural representations",
"The literature contains many structural representations that are not intended for machine learning applications.",
"Examples include the simple coordination number, Voronoi polyhedron of a central atom, angular distribution function, and radial distribution function (RDF).",
"Here, we introduce two kinds of pairwise structural representations and two kinds of angular-dependent structural representations (i.e., histogram representations of the partial radial distribution function (PRDF), generalized radial distribution function (GRDF), bond-orientational order parameter (BOP) [23], and angular Fourier series (AFS) [1].",
"Figure: Partial radial distribution functions (PRDFs) and generalized radial distribution functions (GRDFs).The PRDF is a well-established representation for various structures.",
"To transform the PRDF into structural representations applicable to machine learning, a histogram representation of the PRDF is adopted with a given bin width and cutoff radius (Fig.",
"REF ).",
"The number of counts for each bin is used as the structural representation.",
"The GPRF, which is a pairwise representation similar to the PRDF histogram representation, is expressed as ${\\rm GRDF}_n^{(i)} = \\sum _j f_n (r_{ij})$ where $f_n (r_{ij})$ denotes a pairwise function of the distance $r_{ij}$ between atoms $i$ and $j$ .",
"For example, a pairwise Gaussian-type function is expressed as $f_n(r) = \\exp \\left[-p_n (r-q_n)^2\\right]f_c(r)$ where $f_c(r)$ denotes the cutoff function.",
"$p_n$ and $q_n$ are given parameters.",
"The GRDF can be regarded as a generalization of the PRDF histogram because the PRDF histogram is obtained using rectangular functions as pairwise functions.",
"The BOP is also a well-known representation for local structures.",
"The rotationally invariant BOP $Q_l^{(i)}$ for atomic neighborhoods is expressed as $Q_l^{(i)} = \\left[ \\frac{4\\pi }{2l+1} \\sum _{m=-l}^{l} |Q_{lm}^{(i)}|^2 \\right]^{1/2}$ where $Q_{lm}^{(i)}$ corresponds to the average spherical harmonics for neighbors of atom $i$ .",
"The third-order invariant BOP $W_l^{(i)}$ for atomic neighborhoods is expressed by $W_l^{(i)} = \\sum ^{l}_{m_1, m_2, m_3 = -l}\\begin{pmatrix}l & l & l \\\\m_1 & m_2 & m_3 \\\\\\end{pmatrix}Q_{lm_1}^{(i)} Q_{lm_2}^{(i)} Q_{lm_3}^{(i)},$ where the parentheses are the Wigner 3$j$ symbol, satisfying $m_1+m_2+m_3=0$ .",
"A set of both $Q_l^{(i)}$ and $W_l^{(i)}$ up to a given maximum $l$ is used as the structural representations.",
"The AFS is the most general among the four representations.",
"The AFS can include both the radial and angular dependences of an atomic distribution, and is given by ${\\rm AFS}_{n,l}^{(i)} = \\sum _{j,k} f_n(r_{ij})f_n(r_{ik}) \\cos (l \\theta _{ijk})$ where $\\theta _{ijk}$ denotes the bond angle between three atoms."
],
[
"Machine learning of DFT cohesive energy",
"The performances of the descriptors derived from elemental and structural representations have been examined by developing kernel ridge regression (KRR) prediction models for the DFT cohesive energy [17].",
"The dataset is composed of the cohesive energy for 18093 binary and ternary compounds computed by DFT calculations.",
"First, descriptor sets derived only from elemental representations, which are expected to be more dominant than structural representations in the prediction of the cohesive energy, are adopted.",
"Since the elemental representations are incomplete for some of the elements in the dataset, only elemental representations, which are complete for all elements, are considered.",
"The root-mean-square error (RMSE) is estimated for the test data.",
"The test data is comprised of 10% of the randomly selected data.",
"This random selection of the test data is repeated 20 times, and the average RMSE is regarded as the prediction error.",
"Figure: Comparison of the cohesive energy calculated by DFT calculations and that calculated by the KRR prediction model.Only one test dataset is shown.Descriptor sets are composed of (a) the mean of the elemental representation, (b) the means of the elemental and PRDF representations, (c) the means, SDs, and covariances of the elemental and PRDF representations and (d) the means, SDs, and covariances of the elemental and 20 trigonometric GRDF representations.Mean of the PRDF corresponds to the RDF.Structure representations are computed from the optimized structure for each compound.The simplest option is to use only the mean of each elemental representation as a descriptor.",
"The prediction error in this case is 0.249 eV/atom.",
"Figure REF (a) compares the cohesive energy calculated by DFT calculations to that by the KRR model, where only the test data in one of the 20 trials are shown.",
"Numerous data points deviate from the diagonal line, which represents equal DFT and KRR energies.",
"When considering the means, SDs, and covariances of the elemental representations, the prediction model has a slightly smaller prediction error of 0.231 eV/atom.",
"Additionally, skewness and kurtosis are not important descriptors for the prediction.",
"Next, descriptors related to structural representations are introduced.",
"They can be computed from the crystal structure optimized by the DFT calculations or the initial prototype structures.",
"The former is only useful for machine-learning predictions when a target observation is expensive.",
"Since the optimized structure calculation requires the same computational cost as the cohesive energy calculation, the benefit of machine learning is lost when using the optimized structure.",
"The structural representations are computed from the optimized crystal structure only to examine the limitation of the procedure and representations introduced here.",
"KRR models are constructed using many descriptor sets, which are composed of elemental and structural representations.",
"The cutoff radius is set to 6 Å for the PRDF, GRDF, and AFS, while the cutoff radius is set to 1.2 times the nearest-neighbor distance for the BOP.",
"This nearest neighbor definition is common for the BOP.",
"Figure REF compares the DFT and KRR cohesive energies, where the KRR models are constructed by (b) a set of the means of the elemental and PRDF histogram representations and (c) a set of the means, standard deviations, and covariances of the elemental and PRDF histogram representations.",
"When considering the means of the elemental and PRDF representations, the lowest prediction error is as large as 0.166 eV/atom.",
"This means that simply employing the PRDF histogram does not yield a good model for the cohesive energy.",
"However, including the covariances of the elemental and PRDF histogram representations produces a much better prediction model and the prediction error significantly decreases to 0.106 eV/atom.",
"Considering only the means of the GRDFs, prediction models are obtained with errors of 0.149–0.172 eV/atom.",
"These errors are similar to those of prediction models considering the means of the PRDFs.",
"Similar to in the case of the PRDF, the prediction model improves upon considering the SDs and covariances of the elemental and structural representations.",
"The best model shows a prediction error of 0.045 eV/atom, which is about half that of the best PRDF model.",
"This is also approximately equal to the “chemical accuracy” of 43 meV/atom (1 kcal/mol).",
"Figure REF (d) compares the DFT and KRR cohesive energies, where a set of the means, SDs, and covariances of the elemental and trigonometric GRDF representations is adopted.",
"Most of the data are located near the diagonal line.",
"We also obtain the best prediction model with a prediction error of 0.041 eV/atom by considering the means, SDs, and covariances of the elemental, 20 trigonometric GRDF, and 20 BOP representations.",
"Therefore, the present method should be useful to search for compounds with diverse chemical properties and applications from a wide range of chemical and structural spaces without performing exhaustive DFT calculations."
],
[
"Construction of MLIP for elemental metals",
"A wide variety of conventional interatomic potentials (IPs) have been developed based on prior knowledge of chemical bonds in some systems of interest.",
"Examples include Lennard-Jones, embedded atom method (EAM), modified EAM (MEAM), and Tersoff potentials.",
"However, the accuracy and transferability of conventional IPs are often lacking due to the simplicity of their potential forms.",
"On the other hand, the MLIP based on a large dataset obtained by DFT calculations is beneficial to improve the accuracy and transferability.",
"In the MLIP framework, the atomic energy is modeled by descriptors corresponding to structural representations, as shown in Sec.",
".",
"Once the MLIP is established, it has a similar computational cost as conventional IPs.",
"MLIPs have been applied to a wide range of materials, regardless of chemical bonding nature of the materials.",
"Recently, frameworks applicable to periodic systems have been proposed [3], [2], [19].",
"The Lasso regression has been used to derive a sparse representation for the IP.",
"In this section, we demonstrate the applicability of the Lasso regression to derive the IPs of 12 elemental metals (Na, Mg, Ag, Al, Au, Ca, Cu, Ga, In, K, Li, and Zn) [19], [20].",
"The features of linear modeling of the atomic energy and descriptors using the Lasso regression include the following.",
"1) The accuracy and computational cost of the energy calculation can be controlled in a transparent manner.",
"2) A well-optimized sparse representation for the IP, which can accelerate and increase the accuracy of atomistic simulations while decreasing the computational costs, is obtained.",
"3) Information on the forces acting on atoms and stress tensors can be included in the training data in a straightforward manner.",
"4) Regression coefficients are generally determined quickly using the standard least-squares technique.",
"The total energy of a structure can be regarded as the sum of the constituent atomic energies.",
"In the framework of MLIPs with only pairwise descriptors, the atomic energy of atom $i$ is formulated as $E^{(i)} = F\\left(b_1^{(i)}, b_2^{(i)}, \\cdots , b_{n_{\\rm max}}^{(i)} \\right),$ where $b_n^{(i)}$ denotes a pairwise descriptor.",
"Numerous pairwise descriptors are generally used to formulate the MLIP.",
"We use the GRDF expressed by Eqn.",
"(REF ) as the descriptors.",
"For the pairwise function $f_n$ , we introduce Gaussian, cosine, Bessel, Neumann, modified Morlet wavelet, Slater-type orbital, and Gaussian-type orbital functions.",
"Although artificial neural network and Gaussian process black-box models have been used as functions $F$ , we use a polynomial function to construct the MLIPs for the 12 elemental metals.",
"In the approximation considering only the power of $b_n^{(i)}$ , the atomic energy is expressed as $E^{(i)} = w_0 + \\sum _n w_n b_n^{(i)} + \\sum _n w_{n,n} b_n^{(i)}b_n^{(i)} + \\cdots ,$ where $w_0$ , $w_n$ , and $w_{n,n}$ denote the regression coefficients.",
"Practically, the formulation is truncated by the maximum value of power, $p_{\\rm max}$ .",
"The vector ${w}$ composed of all the regression coefficients can be estimated by a regression, which is a machine learning method to estimate the relationship between the predictor and observation variables using a training dataset.",
"For the training data, the energy, forces acting on atoms, and stress tensor computed by DFT calculations can be used as the observations in the regression process since they all are expressed by linear equations with the same regression coefficients [20].",
"A simple procedure to estimate the regression coefficients employs a linear ridge regression [7].",
"This is a shrinkage method where the number of regression coefficients is reduced by imposing a penalty.",
"The ridge coefficients minimize the penalized residual sum of squares and are expressed as $L({w}) = ||{X}{w} - {y}||^2_2 + \\lambda ||{w}||^2_2,$ where ${X}$ and ${y}$ denote the predictor matrix and observation vector, respectively, which correspond to the training data.",
"$\\lambda $ , which is called the regularization parameter, controls the magnitude of the penalty.",
"This is referred to as L2 regularization.",
"The regression coefficients can easily be estimated while avoiding the well-known multicollinearity problem that occurs in the ordinary least-squares method.",
"Although the linear ridge regression is useful to obtain an IP from a given descriptor set, a set of descriptors relevant to the system of interest is generally unknown.",
"Moreover, an MLIP with a small number of descriptors is desirable to decrease the computational cost in atomistic simulations.",
"Therefore, a combination of the Lasso regression [7], [25] and a preparation involving a considerable number of descriptors is used.",
"The Lasso regression provides a solution to the linear regression as well as a sparse representation with a small number of non-zero regression coefficients.",
"The solution is obtained by minimizing the function that includes the L1 norm of regression coefficients and is expressed as $L({w}) = ||{X}{w} - {y}||^2_2 + \\lambda ||{w}||_1.$ Simply adjusting the values of $\\lambda $ for a given training dataset controls the accuracy of the solution.",
"To begin with, training and test datasets are generated from DFT calculations.",
"The test dataset is used to examine the predictive power for structures that are not included in the training dataset.",
"For each elemental metal, 2700 and 300 configurations are generated for the training and test datasets, respectively.",
"The datasets include structures made by isotropic expansions, random expansions, random distortions, and random displacements of ideal face-centered-cubic (fcc), body-centered-cubic (bcc), hexagonal-closed-packed (hcp), simple-cubic (sc), $\\omega $ and $\\beta $ -tin structures, in which the atomic positions and lattice constants are fully optimized.",
"These configurations are made using supercells constructed by the $2\\times 2\\times 2$ , $3\\times 3\\times 3$ , $3\\times 3\\times 3$ , $4\\times 4\\times 4$ , $3\\times 3\\times 3$ and $2\\times 2\\times 2$ expansions of the conventional unit cells for fcc, bcc, hcp, sc, $\\omega $ , and $\\beta $ -tin structures, which are composed of 32, 54, 54, 64, 81 and 32 atoms, respectively.",
"For a total of 3000 configurations for each elemental metal, DFT calculations have been performed using the plane-wave basis projector augmented wave (PAW) method [5] within the Perdew–Burke–Ernzerhof exchange-correlation functional [15] as implemented in the VASP code [12], [11], [13].",
"The cutoff energy is set to 400 eV.",
"The total energies converge to less than 10$^{-3}$ meV/supercell.",
"The atomic positions and lattice constants are optimized for the ideal structures until the residual forces are less than 10$^{-3}$ eV/Å.",
"Table: RMSEs for the test data of linear ridge MLIPs using 240 terms.",
"(Unit: meV/atom)For each MLIP, the RMSE is calculated between the energies for the test data predicted by the DFT calculations and those predicted using the MLIP.",
"This can be regarded as the prediction error of the MLIP.",
"Table REF shows the RMSEs of linear ridge MLIPs with 240 terms for Na and Mg, where the RMSE converges as the number of terms increases.",
"The MLIPs with only pairwise interactions have low predictive powers for both Na and Mg. Increasing pmax improves the predictive power of the MLIPs substantially.",
"Using cosine-type functions with $p_{\\rm max} = 3$ and cutoff radius $R_c = 7.0$ Å, the RMSEs are 1.4 and 1.6 meV/atom for Na and Mg, respectively.",
"By increasing the cutoff radius to $R_c = 9.0$ Å, the RMSE reaches a very small value of 0.4 meV/atom for Na, but the RMSE remains almost unchanged for Mg.",
"The RMSE for Na is not improved, even after considering all combinations of the Gaussian, cosine, Bessel and Neumann descriptor sets.",
"In contrast, the combination of Gaussian, cosine, and Bessel descriptor sets provides the best prediction for Mg with an RMSE of 0.9 meV/atom.",
"Figure: RMSEs for the test data of the linear ridge MLIP using cosine-type and Gaussian-type descriptors with p max =3p_{\\rm max} = 3, R c =7.0R_c = 7.0 Å and λ=0.001\\lambda =0.001 for (a) Na and (b) Mg.RMSEs of the Lasso MLIPs are also shown.The Lasso MLIPs have been constructed using the same dataset.",
"Candidate terms for the Lasso MLIPs are composed of numerous Gaussian, cosine, Bessel, Neumann, polynomial and GTO descriptors.",
"Sparse representations are then extracted from a set of candidate terms by the Lasso regression.",
"Figure REF shows the RMSEs of the Lasso MLIPs for Na and Mg, respectively.",
"The RMSEs of the Lasso MLIP decrease faster than those of the linear ridge MLIPs constructed from a single-type of descriptors.",
"In other words, the Lasso MLIP requires fewer terms than the linear ridge MLIP.",
"For Na, a sparse representation with an RMSE of 1.3 meV/atom is obtained using only 107 terms.",
"This is almost the same accuracy as the linear ridge MLIP with 240 terms based on the cosine descriptors.",
"It is apparent that the Lasso MLIP is more advantageous for Mg than for Na.",
"The obtained sparse representation with 95 terms for Mg has an RMSE of 0.9 meV/atom.",
"This is almost half the terms for the linear ridge MLIP based on the cosine descriptors, which requires 240 terms.",
"Figure: (a) Dependence of RMSEs for the energy and stress tensor of the Lasso MLIP on the number of non-zero regression coefficients for ten elemental metals.",
"Orange open circles and blue open squares show RMSEs for the energy and stress tensor, respectively.",
"(b) Comparison of the energies predicted by the Lasso MLIP and DFT for Al and Zn measured from the energy of the most stable structure.",
"(c) Phonon dispersion relationships for FCC-Al and FCC-Zn.",
"Blue solid and orange broken lines show the phonon dispersion curves obtained by the Lasso MLIP and DFT, respectively.",
"Negative values indicate imaginary modes.Table: RMSEs for the energy, force, and stress tensor of the Lasso MLIPs showing the minimum criterion score.Optimal cutoff radius for each element is also shown.Figure REF (a) shows the dependence of the RMSE for the energy and stress tensor of the Lasso MLIP on the number of non-zero regression coefficients for the other ten elemental metals.",
"The number of selected terms tends to increase as the regularization parameter $\\lambda $ decreases.",
"The RMSEs for the energy and stress tensor tend to decrease.",
"Although multiple MLIPs with the same number of terms are sometimes obtained from different values of $\\lambda $ , only the MLIP with the lowest criterion score with the same number of terms is shown in Fig.",
"REF (a).",
"Table REF shows the RMSEs for the energy, force, and stress tensor of the optimal Lasso MLIP.",
"The MLIPs are obtained with the RMSE for the energy in the range of 0.3–3.5 meV/atom for the ten elemental metals using only 165–288 terms.",
"The RMSEs for the force and stress are within 0.03 eV/Å and 0.15 GPa, respectively.",
"Figure REF (b) compares the energies of the test data predicted by the Lasso MLIP and DFT for Al and Zn.",
"Both the largest and second largest RMSEs for the energy are shown.",
"Regardless of the crystal structure, the DFT and Lasso MLIP energies are similar.",
"In addition, the RMSE is clearly independent of the energy despite the wide range of structures included in both the training and test data.",
"The applicability of the Lasso MLIP to the calculation of the force has been also examined by comparing the phonon dispersion relationships computed by the Lasso MLIP and DFT.",
"The phonon dispersion relationships are calculated by the supercell approach for the fcc structure with the equilibrium lattice constant.",
"The phonon calculations use the phonopy code [27].",
"Figure REF (c) shows the phonon dispersion relationships of the fcc structure for elemental Al and Zn computed by both the Lasso MLIP and DFT.",
"The phonon dispersion relationships calculated by the Lasso MLIP agree well with those calculated by DFT.",
"This demonstrates that the Lasso MLIP is sufficiently accurate to perform atomistic simulations with an accuracy similar to DFT calculations.",
"It is important to use an extended approximation for the atomic energy in transition metals [24].",
"The extended approximation also improves the predictive power for the above elemental metals.",
"The MLIPs are constructed by a second-order polynomial approximation with the AFSs described by Eqn.",
"(REF ) and their cross terms.",
"For elemental Ti, the optimized angular-dependent MLIP is obtained with a prediction error of 0.5 meV/atom (35245 terms), which is much smaller than that of the Lasso MLIP with only the power of pairwise descriptors of 17.0 meV/atom.",
"This finding demonstrates that it is very important to consider angular-dependent descriptors when expressing interatomic interactions of elemental Ti.",
"The angular-dependent MLIP can predict the physical properties much more accurately than existing IPs."
],
[
"Discovery of low lattice thermal conductivity materials",
"Thermoelectric generators are essential to utilize waste heat.",
"The thermoelectric figure of merit should be increased to improve the conversion efficiency.",
"Since the figure of merit is inversely proportional to the thermal conductivity, many works have strived to reduce the thermal conductivity, especially the LTC.",
"To evaluate LTCs with an accuracy comparable to the experimental data, a method that greatly exceeds ordinary DFT calculations is required.",
"Since multiple interactions among phonons, or anharmonic lattice dynamics, must be treated, the computational cost is many orders of magnitudes higher than ordinary DFT calculations of primitive cells.",
"Such expensive calculations are feasible only for a few simple compounds.",
"High-throughput screening of a large DFT database of the LTC is an unrealistic approach unless the exploration space is narrowly confined.",
"Figure: (a) LTC calculated from the first principles calculations for 101 compounds along with volume, VV.",
"(b) Experimental LTC data are shown for comparison when the experimental LTCs are available.Recently, Togo et al.",
"reported a method to systematically obtain the theoretical LTC through first principles anharmonic lattice dynamics calculations [26].",
"Figure REF (a) shows the results of first-principles LTCs for 101 compounds as functions of the crystalline volume per atom, $V$ .",
"PbSe with the rocksalt structure shows the lowest LTC, 0.9 W/mK (at 300 K).",
"Its trend is similar to that in a recent report on low LTC for lead- and tin-chalcogenides.",
"Figure REF (b) compares the computed results with the available experimental data.",
"The satisfactory agreement between the experimental and computed results demonstrates the usefulness of the first-principles LTC data for further studies.",
"A phenomenological relationship has been proposed where $\\log \\kappa _L$ is proportional to $\\log V$ [22].",
"Although a qualitative correlation is observed between our LTC and $V$ , it is difficult to predict the LTC quantitatively or discover new compounds with low LTCs only from the phenomenological relationship.",
"It should be noted that the dependence on $V$ differs remarkably between rocksalt-type and zincblende- or wurtzite-type compounds.",
"However, zincblende- and wurtzite-type compounds show a similar LTC for the same chemical composition.",
"The 101 first-principles LTC data has been used to create a model to predict the LTCs of compounds within a library [21].",
"Firstly, a Gaussian process (GP)-based Bayesian optimization [16] is adopted using two physical quantities as descriptors: $V$ and density, $\\rho $ .",
"These quantities are available in most experimental or computational crystal structure databases.",
"Although a phenomenological relationship is proposed between $\\log \\kappa _L$ and $V$ , the correlation between them is low.",
"Moreover, the correlation between $\\log \\kappa _L$ and $\\rho $ is even worse.",
"We start from an observed data set of five compounds that are randomly chosen from the dataset.",
"The Bayesian optimization searches for the compound with a maximum probability of improvement [9] among the remaining data.",
"That is, the compound with the highest Z-score derived from GP is searched.",
"The compound is included into the observed dataset.",
"Then another compound with the maximum probability of improvement is searched.",
"Both the Bayesian optimization and random searches are repeated 200 times, and the average number of observed compounds required to find the best compound is examined.",
"The average numbers of compounds required for the optimization using the Bayesian optimization and random searches, $N_{\\rm ave}$ , are 11 and 55, respectively.",
"The compound with the lowest LTC among the 101 compounds (i.e., rocksalt PbSe) can be found much more efficiently using a Bayesian optimization with only two variables, $V$ and $\\rho $ .",
"However, using a Bayesian optimization only with these two variables is not a robust method to determine the lowest LTC.",
"As an example, the result of the Bayesian optimization using the dataset after intentionally removing the first and second lowest LTC compounds shows that $N_{\\rm ave}$ is 65 to find LiI using Bayesian optimization only with $V$ and $\\rho $ , which is larger than that of the random search ($N_{\\rm ave} = 50$ ).",
"The delay in the optimization should originate from the fact that LiI is an outlier when the LTC is modeled only with $V$ and $\\rho $ .",
"Such outlier compounds with low LTC are difficult to find only with $V$ and $\\rho $ .",
"To overcome the outlier problem, predictors have been added for constituent chemical elements.",
"There are many choices for such variables.",
"Here, we introduce binary elemental descriptors, which are a set of binary digits representing the presence of chemical elements.",
"Since the 101 LTC data is composed of 34 kinds of elements, there are 34 elemental descriptors.",
"When finding both PbSe and LiI, the compound with the lowest LTC is found with $N_{\\rm ave} = 19$ .",
"The use of binary elemental descriptors improves the robustness of the efficient search.",
"Figure: Relationship between logκ L \\log \\kappa _L and the physical properties derived from the first principles electronic structure and phonon calculations.Correlation coefficient, RR, is shown in each panel.Better correlations with LTC can be found for parameters obtained from the phonon density of states.",
"Figure REF shows the relationships between the LTC and the physical properties.",
"Other than volume and density, the following quantities are obtained by our phonon calculations: mean phonon frequency, maximum phonon frequency, Debye frequency, and Grüneisen parameter.",
"The Debye frequency is determined by fitting the phonon density of states for a range between 0 and 1/4 of the maximum phonon frequency to a quadratic function.",
"The thermodynamic Grüneisen parameter is obtained from the mode-Grüneisen parameters calculated with a quasi-harmonic approximation and mode-heat capacities.",
"The correlation coefficients $R$ between $\\log \\kappa _L$ and these physical properties are shown in the corresponding panels.",
"The present study does not use such phonon parameters as descriptors because a data library for such phonon parameters for a wide range of compounds is unavailable.",
"Hereafter, we show results only with the descriptor set composed of 34 binary elemental descriptors on top of $V$ and $\\rho $ .",
"A GP prediction model has been used to screen for low LTC compounds in a large library of compounds.",
"In the biomedical community, a screening based on a prediction model is called a “virtual screening” [10].",
"For the virtual screening, all 54779 compounds in the Materials Project Database (MPD) library [8], which is composed mostly of crystal structure data available in ICSD [4], are adopted.",
"Most of these compounds have been synthesized experimentally at least once.",
"On the basis of the GP prediction model made by $V$ , $\\rho $ , and the 34 binary elemental descriptors for the 101 LTC data, low-LTC compounds are ranked according to the Z-score of the 54779 compounds.",
"Figure: Dependence of the Z-score on the constituent elements for compounds in the MPD library.Color along the volume and density for each element denote the magnitude of the Z-score.Figure REF shows the distribution of Z-scores for the 54779 compounds along with $V$ and $\\rho $ .",
"The magnitude of the Z-score is plotted in the panels corresponding to the constituent elements.",
"The compounds are widely distributed in $V-\\rho $ space.",
"Thus, it is difficult to identify compounds without performing a Bayesian optimization with elemental descriptors.",
"The widely distributed Z-scores for light elements such as Li, N, O, and F imply that the presence of such light elements has a negligible effect on lowering the LTC.",
"When such light elements form a compound with heavy elements, the compound tends to show a high Z-score.",
"It is also noteworthy that many compounds composed of light elements such as Be and B tend to show a high LTC.",
"Pb, Cs, I, Br, and Cl exhibit special features.",
"Many compounds composed of these elements exhibit high Z-scores.",
"Most compounds showing a positive Z-score are a combination of these five elements.",
"On the other hand, elements in the periodic table neighboring these five elements do not show analogous trends.",
"For example, compounds of Tl and Bi, which neighbor Pb, rarely exhibit high Z-scores.",
"This may sound odd since Bi$_2$ Te$_3$ is a famous thermoelectric compound, and some compounds containing Tl have a low LTC.",
"This may be ascribed to our selection of the training dataset, which is composed only of AB compounds with 34 elements and three kinds of simple crystal structures.",
"In other words, the training dataset is somehow “biased”.",
"Currently, this bias is unavoidable because first-principles LTC calculations are still too expensive to obtain a sufficiently unbiased training dataset with a large enough number of data points to cover the diversity of the chemical compositions and crystal structures.",
"Nevertheless, the usefulness of biased training dataset to find low-LTC materials will be verified in the future.",
"Due to the biased training dataset, all low-LTC materials in the library may not be discovered.",
"However, some of them can be discovered.",
"A ranking of LTCs from the Z-score does not necessarily correspond to the true first-principles ranking.",
"Therefore, a verification process for candidates of low-LTC compounds after the virtual screening is one of the most important steps in “discovering” low-LTC compounds.",
"First principles LTCs have been evaluated for the top eight compounds after the virtual screening.",
"All of them are considered to form ordered structures.",
"However, the LTC calculation is unsuccessful for Pb$_2$ RbBr$_5$ due to the presence of imaginary phonon modes within the supercell used in the present study.",
"All of the top five compounds, PbRbI$_3$ , PbIBr, PbRb$_4$ Br$_6$ , PbICl and PbClBr, show a LTC of $<$ 0.2 W/mK (at 300 K), which is much lower than that of the rocksalt PbSe, [i.e., 0.9 W/mK (at 300 K)].",
"This confirms the powerfulness of the present GP prediction model to efficiently discover low-LTC compounds.",
"The present method should be useful to search for materials in diverse applications where the chemistry of materials must be optimized.",
"Figure: Behavior of the Bayesian optimization for the LTC data to find PbClBr, CuCl, and LiI.Finally, the performance of Bayesian optimization has been examined using the compound descriptors derived from elemental and structural representations for the LTC dataset containing the compounds identified by the virtual screening.",
"GP models are constructed using (1) the means and SDs of the elemental representations and GRDFs and (2) the means and SDs of elemental representations and BOPs.",
"Figure REF shows the behavior of the lowest LTC during Bayesian optimization relative to a random search.",
"The optimization aims to find PbClBr with the lowest LTC.",
"For the GP model with the BOP, the average number of samples required for the optimization, $N_{\\rm ave}$ , is 5.0, which is ten times smaller than that of the random search, $N_{\\rm ave} = 50$ .",
"Hence, the Bayesian optimization more efficiently discovers PbClBr than the random search.",
"To evaluate the ability to find a wide variety of low-LTC compounds, two datasets have been prepared after intentionally removing some low-LTC compounds.",
"In these datasets, CuCl and LiI, which respectively show the 11th-lowest and 12th-lowest LTCs, are solutions of the optimizations.",
"For the GP model with BOPs, the average number of observations required to find CuCl and LiI is $N_{\\rm ave} = 15.1$ and 9.1, respectively.",
"These numbers are much smaller than those of the random search.",
"On the other hand, for the GP model with GRDFs, the average number of observations required to find CuCl and LiI is $N_{\\rm ave} = 40.5$ and 48.6, respectively.",
"The delayed optimization may originate from the fact that both CuCl and LiI are outliers in the model with GRDFs, although the model with GRDFs has a similar RMSE as the model with BOPs.",
"These results indicate that the set of descriptors needs to be optimized by examining the performance of Bayesian optimization for a wide range of compounds to find outlier compounds."
],
[
"Recommender system approach for materials discovery",
"Many atomic structures of inorganic crystals have been collected.",
"Of the few available databases for inorganic crystal structures, the ICSD [4] contains approximately 10$^5$ inorganic crystals, excluding duplicates and incompletes.",
"Although this is a rich heritage of human intellectual activities, it covers a very small portion of possible inorganic crystals.",
"Considering 82 non-radioactive chemical elements, the number of simple chemical compositions up to ternary compounds A$_a$ B$_b$ C$_c$ with integers satisfying $\\max (a, b, c)\\le 15$ is approximately $10^8$ , but increases to approximately $10^{10}$ for quaternary compounds A$_a$ B$_b$ C$_c$ D$_d$ .",
"Although many of these chemical compositions do not form stable crystals, the huge difference between the number of compounds in ICSD and the possible number of compounds implies that many unknown compounds remain.",
"Conventional experiments alone cannot fill this gap.",
"Often, first principles calculations are used as an alternative approach.",
"However, systematic first principles calculations without a priori knowledge of the crystal structures are very expensive.",
"Machine learning is a different approach to consider all chemical combinations.",
"A powerful machine-learning strategy is mandatory to discover new inorganic compounds efficiently.",
"Herein we adopt a recommender system approach to estimate the relevance of the chemical compositions where stable crystals can be formed [i.e., chemically relevant compositions (CRCs)].",
"The compositional similarity is defined using the procedure shown in Sec.",
".",
"A composition is described by a set of 165 descriptors composed of the means, SDs, and covariances of the established elemental representations.",
"The probability for CRCs is subsequently estimated on the basis of a machine-learning two-class classification using the compositional similarity.",
"This approach significantly accelerates the discovery of currently unknown CRCs that are not present in the training database."
]
] | 1709.01666 |
[
[
"Critical fields and fluctuations determined from specific heat and\n magnetoresistance in the same nanogram SmFeAs(O,F) single crystal"
],
[
"Abstract Through a direct comparison of specific heat and magneto-resistance we critically asses the nature of superconducting fluctuations in the same nano-gram crystal of SmFeAs(O, F).",
"We show that although the superconducting fluctuation contribution to conductivity scales well within the 2D-LLL scheme its predictions contrast the inherently 3D nature of SmFeAs(O, F) in the vicinity T_{c}.",
"Furthermore the transition seen in specific heat cannot be satisfactory described either by the LLL or the XY scaling.",
"Additionally we have validated, through comparing Hc2 values obtained from the entropy conservation construction (Hab=-19.5 T/K and Hab=-2.9 T/K), the analysis of fluctuation contribution to conductivity as a reasonable method for estimating the Hc2 slope."
],
[
"Introduction",
"The surprising discovery of superconductivity at 26K in LaFeAsO in 2008 was a beginning of a new era for superconductivity [1].",
"Soon after the initial discovery a great effort was taken to reach higher transition temperatures, resulting in a discovery of dozens of iron based superconductors with new compounds still being synthesized [2].",
"However promising, the abundance of new structures was not followed by the availability of high quality macroscopic samples.",
"In particular crystals of the '1111' family, with the highest $T_{c}=55~K$ , usually grow as flakes of $100-200~\\mu m$ diameter, [3] making it challenging to study their bulk thermodynamics.",
"In the face of such difficulties newly discovered superconductors are traditionally characterized by their transport properties.",
"However the influence of possible filamentary and surface superconductivity together with the defect-induced vortex pinning the resistive transition tends to be difficult to interpret.",
"This is particularly visible when determining the $H{}_{c2}$ slope from the typically smooth and featureless resistive data - depending on the chosen criterion for the transition temperature (10, 50 90% of normal state resistivity) one might obtain$H_{c2}$ slopes differing by more than a factor of 2-3 [4].",
"One debate originating from these issues is the discussion of dimensionality of the superconducting fluctuations in SmFeAs(O, F) with reports of 2D and 3D behaviour [5], [6].",
"In the first work Palecchi et al.",
"reported that the superconducting fluctuation contribution to conductivity could be well parametrized within the 2D-LLL scaling scheme in stark contrast to the second study of Welp et al.",
"who suggested the prevalence of 3D-LLL scaling of the superconducting contribution to specific heat in fields up to 8T.",
"One possible explanation of this apparent discrepancy could be the sample variability.",
"On the other hand the analysis of scaling of the superconducting fluctuation contribution to conductivity suffered from the lack of high quality single crystals and was performed on a polycrystalline sample making a proper analysis of fluctuation conductivity very difficult.",
"Here we measure on the same single crystal of SmFeAs(O, F), both heat capacity and resistivity near the superconducting transition in fields up to 14T applied parallel and perpendicular to the FeAs layers.",
"Analysis of the phenomenology of the resistive transition shows that the low temperature part of the transition is strongly influenced by the vortex dynamics.",
"On the other hand, the onset of the transition can be well accounted for as originating from fluctuation conductivity (with the same Hc2 slope as found in specific heat ) and following the scaling form of the 2D Lowest Landau Level (LLL) theory.",
"The appearance of the specific heat anomaly accompanying the transition also reveals a significant presence of fluctuations, however they cannot be well described neither within the LLL nor the XY scaling schemes.",
"Our analysis shows that despite many similarities with the cuprates the multi-band nature of the iron pnictides makes them even more complex."
],
[
"Experimental",
"SmFeAs(O, F) single crystals were grown under high pressure in a NaCl/KCl flux, typically they grew in the form of $5-10\\:\\mu m$ thick platelets with $100-200~\\mu m$ diameter [3].",
"In the course of this study we have used two crystals of approximately $40\\times 50\\times 5\\mu m^{3}$ and $50\\times 100\\times 10\\mu m^{3}$ .",
"The size of the crystals was estimated from electron microscope images.",
"In order to perform specific heat measurements on such small samples we have employed membrane nano calorimeters [7] combined with the 345 method allowing us to measure specific heat of samples as small as 30um in diameter (corresponding to $\\sim 50~ng$ ) [8].",
"After the specific heat measurements the sample was transferred onto a silicon wafer and subsequently trimmed into a Hall bar and electrically connected using the focused ion beam, as shown in the inset of Fig.REF [9], [10].",
"Measurements were performed in a Quantum Design PPMS cryostat equipped in a 14 Tesla magnet.",
"Figure: Variation of the specific heat anomalyof single crystal SmFeAs(O,F) with the magnetic fields applied alongthe c-axis (upper panel) and parallelto the ab-plane (middle panel).",
"The insetleft of the upper panel exemplifies the procedure used for extractingTc at various fields.",
"The inset of themiddle panel presents the total measured specific heat.",
"The bottompanel shows the field dependence of the difference C(0T)-C(H).C(0T)-C(H)."
],
[
"Specific Heat",
"Reliable measurements of specific heat of the '1111' family or in fact of any crystals that are not of macroscopic size seem to be a formidable task and there are only a few reports of thermodynamic bulk measurements performed on nanogram samples [8], [11], [6].",
"In order to perform these experiments we have employed membrane based nano-calorimeters manufactured by Xensor Integration.",
"These chips although designed for operation in temperatures up to $700\\,K$ turned out to work very reliably at low temperatures down to $1.8\\:K$ .",
"Unlike most nano-calorimeters used in condensed matter physics thermometry in these devices does not relay on resistive thermometers but is based on a set of 6 compensated silicon thermopiles.",
"Such a design proves to be especially useful when employing the 345 method as it allows a direct measurement of the temperature difference between the sample and the chip frame (For design details please refer to Xensor technical note [7]).",
"The superconducting specific heat anomaly amounts to less than 5% of the total specific heat (inset of Fig.REF b).",
"To extract meaningful thermodynamic information regarding the superconducting transition we follow the procedure introduced by Welp.",
"et al.",
"[12], [6] and subtract a linear background.",
"However for clarity we have subtracted the same zero field background line from all curves, explicitly showing the field dependence of the normal state specific heat.",
"(Figure REF , bottom panel).",
"Interestingly the normal state specific heat above the transition is reduced in a magnetic field (Fig.REF c).",
"This can be tentatively ascribed to the modification of the crystal-field split energy levels of the Sm 4f electrons, as no such suppression in observed in the Nd- based counterpart [13].",
"The specific heat near $T_{c}$ is shown in Fig.REF .",
"We have fitted a 'mean field jump' (inset of Fig.REF a) to the transitions assuming entropy conservation as exemplified in the inset of Fig.REF , yielding the following parameters: $T_{c}=50.5~K$ , the upper critical field slopes $H^{\\prime }_{c2}$ , Fig.",
"1 with the field parallel to the c-axis: $\\sim 2.9~T/K$ and $\\sim 19.5~T/K$ parallel to the ab-plane.",
"The estimate of the jump hight yielded $\\Delta C/T=17.7\\:mJ/molK^{2}$ for crystal I and $23.7\\:mJ/molK^{2}$ for crystal II.",
"These values are in fair agreement with data previously reported by Welp et al.",
"who estimated the anisotropy parameter $\\Gamma =\\frac{H_{c2}^{ab}}{H_{c2}^{c}}=8$ with the critical field slope along the c-axis as $-3.5~T/K$ .",
"A qualitative investigation of the shape of the specific heat anomalies reveals a strong superconducting fluctuation contribution with the high temperature fluctuation tail extending almost 5 K above bulk $T_{c}$ , suggesting some similarities to the cuprate superconductors.",
"In the case of the cuprates two scenarios were proposed to describe the behaviour of specific heat in the vicinity of $T_{c}$ in magnetic fields: the 2D and 3D Lowest Landau Level (LLL) theory and the 3D XY model [14], [15], [16], [17].",
"It was shown that the LLL theory should be a valid approximation for describing superconducting fluctuations as soon as the magnetic field becomes strong enough to confine the order parameter to the lowest Landau level, what translates to a criterion $H>H_{LLL}$ with $H_{LLL}\\approx G_{i}H_{c2}(0)$ .",
"The LLL theory predicts that specific heat in the vicinity of $H_{c2}$ should be well described, depending on dimensionality of the fluctuations by[14]: $\\frac{dC}{dT}H^{1/2}=F_{2D}^{C}\\left(\\frac{T-T_{c}(H)}{(TH)^{1/2}}\\right)$ $\\frac{dC}{dT}H^{2/3}=F_{3D}^{C}\\left(\\frac{T-T_{c}(H)}{(TH)^{2/3}}\\right)$ where $F_{2D}^{C}(x)$ and $F_{3D}^{C}(x)$ are scaling functions.",
"On the other end the XY model can be considered a justified description for fields too weak to effectively break the XY symmetry.",
"In this case the specific heat anomaly is expected to follow the scaling relation [16], [17]: $\\left[C(H,T)-C(T,0)\\right]H^{\\frac{\\alpha }{2\\nu }}=G\\left(\\left(\\frac{T}{T_{c}}-1\\right)H^{\\frac{-1}{2\\nu }}\\right)$ where $G(x$ ) is the scaling function and the parameters $\\alpha =-0.007$ and $\\nu =-0.669$ are the critical exponents characteristic for the 3D XY model.",
"Interestingly, although both the past and present analysis produce a similar of value Hc2, specific heat data extended to 14 Tesla reveals that the high field data turns out to be not well described by the 3D Lowest Landau Level (3D-LLL) scaling.",
"A close look at Fig.REF a unveils that although the low field data might suggest an onset of convergence towards 3D-LLL scaling [6], the additional higher field measurements indicate the 'non-convergence' continues.",
"An attempt at describing the data using the 2D-LLL scaling (Fig.REF b) is equally unsuccessful, although it seems to collapse the data slightly better.",
"A representation of the data scaled within the 3D XY scaling [18], [16] framework (Figure REF c) is equally unsatisfying.",
"Suggesting that in fact none of the simple scaling scheme captures all the details of the specific heat anomaly.",
"This is especially clear when comparing available datasets with near-perfect data collapse seen in YBCO [19], [16], [20] or BSCCO [21].",
"Additionally comparing the specific heat anomaly of NdFeAs(O, F) [11] and SmFeAs(O, F) with their hydrogenated counterpart [13] reveals qualitative differences in the shape of the specific heat anomalies.",
"This alone suggest that scaling approaches that proved useful in describing classic and cuprate superconductors are not sufficient to capture the details of the physics of the superconducting transition even within one family ('1111') of the pnictides.",
"Figure: Comparison of 2D (middle panel) and 3D(top panel) LLL-scaling schemes with the critical XY scaling (bottompanel).",
"Best curve collapse for the LLL scaling was obtained for T c =50.2KT_{c}=50.2~Kand H c2 =-3.1T/KH_{c2}=-3.1~T/K and T c =49.7KT_{c}=49.7~K for XY scaling."
],
[
"Activated flux flow",
"In order to perform electric transport measurements the crystals were removed from the calorimetric cell, glued to a silicon substrate and subsequently shaped into a form of a Hall bar and contacted using the FIB technique [9].",
"The measurements were performed using a $1117.77Hz$ excitation with peak current density of $20~A/cm^{2}$ - well in the Ohmic regime [22], in the same magnetic fields as the specific heat measurements.",
"The resulting temperature and magnetic field dependence of the resistive transition is depicted in Fig REF .",
"The most prominent feature is the previously reported broadening of the resistive transition with field applied along the c-axis.",
"Figure: The main panels demonstrate the temperature dependenceof resistivity in magnetic fields along and perpendicular to the c-axis.The dashed lines in the lower panel are best fits to the activatedflux flow model, see text.",
"The inset shows an electron micrographof the sample after preparation for transport measurements.It is worthwhile to investigate to what extent these $\\rho (T,H)$ data can be used to extract the upper critical fields.",
"So far there were two scenarios proposed in order to describe the shape of this transition: the first published study, on polycrystalline samples suggested that the transition width follows the 2D-LLL scaling relation for fields above 8 Tesla [5] .",
"On the other hand later measurements on single crystal SmFeAs(O, F) demonstrated the prevailing influence of vortex dynamic and activated flux flow as determining the broadening of the resistive transition [22].",
"Indeed the basic model of activated flux motion as introduced by Tinkham [23] parametrizes our data surprisingly well in high fields.",
"For fields below 3 Tesla (H║c) there is some discrepancy at low temperature , originating from sample inhomogeneity.",
"This feature becomes unimportant at high fields due to the intrinsic broadening of the transition and suppression of possible filamentary superconductivity.",
"$\\frac{R}{R_{n}}=\\lbrace I_{0}[A(1-t)^{3/2}/2B]\\rbrace ^{-2}$ Tinkhams model (eq.",
"1) describes the phenomenology of the resistive transition with only two material dependant parameters: TC and $A=CJ_{c0}/T_{c}$ where $C$ can be approximated as $C\\approx \\beta \\,8.07\\cdot 10^{-3}\\,T\\,K\\,cm^{2}/A$ , with $\\beta $ being of the order of unity [23].",
"In the case of SmFeAs(O, F) we have obtained the best fits for Tc=54K and the value of $A(H)$ steadily increasing from 120 T and saturating at 380 T for fields above 4 T. The outcome of the fitting procedure is represented by the dashed lines in the bottom panel of Figure REF .",
"Our analysis displays three remarkable facts: (1) The constant $A=\\frac{U_{0}}{2T}$ is proportional to the average vortex activation energy, thus its threefold increase could be thought of as a manifestation of field dependence of the pinning potential as suggested by Lee et al [22] who found a transition between two regimes of $U_{0}(H)$ to occur at 3 T. The saturated high field value of $A(H)$ , yields $J_{c0}=2.5\\cdot 10^{6}A/cm^{2}$ very close to the value found for YBCO [23].",
"(2) The value of $T_{c}$ for which the theory reproduces the data best is $\\sim 54\\,K$ and remains almost the same for all magnetic fields.",
"This remarkable fact was already noticed by Tinkham, originally attributed to the small depression of $T_{c}$ in magnetic field [23].",
"Within our framework the value of $54~K$ is significantly higher than the thermodynamic bulk transition temperature extracted from specific heat of the same crystal, the investigation of the specific heat data suggests a clear physical interpretation for the value of $T_{c}$ used in the activated flux flow model: it is the temperature defining the onset of superconducting fluctuations.",
"Indeed in SmFeAs(O, F) this temperature is about $54-55~K$ and seems to be not influenced by magnetic fields up to 14 T. This opens a question to what degree the thermodynamic values of Hc2 and $T_{c}$ are manifesting themselves in the phenomenology of the resistive transition.",
"In the presence of flux motion the activated flux framework describes most of the shape of the resistive transition remarkably well.",
"However the first 5-10% of the drop in resistance are usually dominated by the presence of superconducting fluctuations above the bulk transition temperature.",
"To the best of our knowledge the superconducting fluctuation conductivity has been addressed twice for the '1111' family of superconductors [24], [5].",
"Pallecchi et al.",
"recognized the shape of the resistive drop as following a 2D-LLL [25], [26] scaling relation.",
"However later it was argued that such a result might have been the effect of using a polycrystalline sample[6].",
"Similarly as in case of specific heat the LLL theory predicts that for sufficiently high fields the superconducting fluctuation contribution to conductivity is expected to follow specific dimension dependant scaling relations [26], [25]: $\\triangle \\sigma (H)=\\left(\\frac{T}{H}\\right)^{\\frac{1}{2}}F_{2D}^{\\sigma }\\left(\\frac{T-T_{c}(H)}{(TH)^{1/2}}\\right)$ $\\triangle \\sigma (H)=\\left(\\frac{T}{H}\\right)^{\\frac{1}{3}}F_{3D}^{\\sigma }\\left(\\frac{T-T_{c}(H)}{(TH)^{2/3}}\\right)$ We have extracted the superconducting fluctuation contribution to conductivity by inverting the resistivity tensor and then subtracting the extrapolated normal state background.",
"To investigate the dimensionality of these fluctuations we have plotted the conductivity data in scaled coordinates.",
"As can be seen in Figure.REF the two 2D-LLL scaling collapses our data set far better then the 3D scheme: in the 2D case above $4\\,T$ (red curve) the collapse is nearly ideal whereas in the 3D case there is a considerable fanning out of the curves both below and above $T_{c}$ .",
"What is worth noting is that the best curve collapse was achieved with $T_{c}=50.2~K$ and $H_{c2}=-3.1~T/K,$ very close the values obtained from the entropy conservation construction done on the raw specific heat data and from specific heat scaling.",
"This comparison established the analysis of the superconducting fluctuation contribution to conductivity as a rather reliable method to establish $H_{c2}$ and$T_{c}$"
],
[
"Sample quality",
"One of the primary concerns when discussing scaling of superconducting fluctuations is the availability of high quality crystals.",
"In particular it is essential for this kind of studies to use single phase crystals, as inhomogeneous substitution throughout the sample could lead to the appearance of several superconducting transitions invalidating the scaling analysis.",
"Proper care of this issue is especially important in materials known to be notoriously problematic to synthesize, such as the 1111 pnictides.",
"Indeed in the case at hand the 'two step' appearance of the resistive transition (Figure REF ) could suggest a presence of a substantial inhomogeneity in the sample.",
"It is instructive to compare this two step behaviour with the appearance of the specific heat anomaly.",
"The specific heat data does not show any signatures of peak doubling which would be necessarily present for a two phase sample with two well defined transition temperatures.",
"Additionally the inspection of magnetoresistance data shows that the first step in resistivity is suppressed by relatively weak fields suggesting that the two step appearance of the resistive transition can be attributed to filamentary/surface superconductivity and can be neglected at high fields where the LLL- fluctuations scaling should be applicable."
],
[
"Conclusions and discussion",
"In this study we have investigated the validity of both XY and LLL scaling schemes in application to a SmFeAs(O, F), a representative of the pnictide superconductor family with the highest $T_{c}$ .",
"The analysis revealed that despite structural similarity to YBCO, SmFeAs(O, F) displays a range of behaviours that cannot be fully accounted for by theoretical approaches developed for the cuprates.",
"This is particularly striking when considering specific heat, in YBCO both LLL and XY scaling are reasonably well describing experimental data and only considerable experimental effort settled the boundaries of applicability of both theories.",
"For SmFeAs(O, F) however none of the approaches provides convincing parametrization of experimental data.",
"This is especially surprising in the case of LLL scaling.",
"Considering the relatively low Tc and its high Ginzburg number one would expect the LLL theory to be an adequate description of the condensate in the vicinity of Hc2 for relatively low fields [14], [15], [6].",
"In this context the presence of 2D conductivity fluctuations is even more puzzling as recent studies of vortex transitions in SmFeAs(O, F) [10] conclusively showed that $\\xi _{c}\\left(T\\right)$ remains larger than the inter-plane distance down to $T*\\approx 0.8T_{c}$ (Fig.REF ) questioning the 2D nature of superconducting fluctuations.",
"Figure: A schematic 'Phase diagram' of SmFeAs(O,F).",
"The insets shows how 2D fluctuations extended in the ab-planecould 'screen' existing 3D fluctuations and lead to the observationof 2D-LLL scaling.This apparent paradox might be explained by invoking the inherently multi-band nature of the pnictide superconductors [27], [28], [29], [30], [31], [32].",
"In the case of weak inter-band interaction the main contribution to fluctuation conductivity would come from the band hosting the order parameter component with the largest $\\xi _{ab}$ , effectively 'shortening' all other conductivity fluctuations, inset Fig.REF .",
"If these were of 2D character one could indeed expect the 2D-LLL to parametrize conductivity well for sufficiently high magnetic fields, Fig.REF .",
"On the other hand specific heat measures the total change of entropy, and thus would pick up contributions from both 2D and 3D fluctuations leading to the breakdown of simple LLL-scaling.",
"In the case of SmFeAs(O, F) it was shown that two gaps open at the same temperature with $\\Delta _{1}(0)=18$ meV and $\\Delta _{2}(0)=6.2$ meV [32].",
"Taking into account the very 2D character of its Fermi surface, the standard BCS expression for the coherence length $\\xi _{0}=\\frac{\\hbar v_{F}}{\\pi }$ suggests that at least one component of the order parameter could indeed be highly two dimensional.",
"In this picture at temperatures above $T_{c}$ the properties of SmFeAs(O, F) are determined by a combination of 2D and 3D fluctuations.",
"On cooling below bulk $T_{c}$ the 3D component begins to dominate most of the phenomenology until $\\xi _{c}\\left(T\\right)$ becomes shorter than the distance between adjacent FeAs layers at which point the dominating component of the order parameter becomes 2D.",
"We would like to thank A. Zheludev, K. Povarov and V. B. Geshkenbein for many enlightening discussions that helped improve this manuscript."
]
] | 1709.01777 |
[
[
"Conditional Generative Adversarial Networks for Speech Enhancement and\n Noise-Robust Speaker Verification"
],
[
"Abstract Improving speech system performance in noisy environments remains a challenging task, and speech enhancement (SE) is one of the effective techniques to solve the problem.",
"Motivated by the promising results of generative adversarial networks (GANs) in a variety of image processing tasks, we explore the potential of conditional GANs (cGANs) for SE, and in particular, we make use of the image processing framework proposed by Isola et al.",
"[1] to learn a mapping from the spectrogram of noisy speech to an enhanced counterpart.",
"The SE cGAN consists of two networks, trained in an adversarial manner: a generator that tries to enhance the input noisy spectrogram, and a discriminator that tries to distinguish between enhanced spectrograms provided by the generator and clean ones from the database using the noisy spectrogram as a condition.",
"We evaluate the performance of the cGAN method in terms of perceptual evaluation of speech quality (PESQ), short-time objective intelligibility (STOI), and equal error rate (EER) of speaker verification (an example application).",
"Experimental results show that the cGAN method overall outperforms the classical short-time spectral amplitude minimum mean square error (STSA-MMSE) SE algorithm, and is comparable to a deep neural network-based SE approach (DNN-SE)."
],
[
"Introduction",
"Dealing with degraded speech signals is a challenging yet important task in many applications, e.g.",
"automatic speaker verification (ASV) [2], speech recognition [3], mobile communications and hearing assistive devices [4], [5], [6].",
"When the receiver is a human user, the objective of SE is to improve quality and intelligibility of noisy speech signals.",
"When it is an automatic speech system, the goal is to improve the noise-robustness of the system, e.g.",
"to reduce the EERs of an ASV system under adverse conditions.",
"In the past, this problem has been tackled with statistical methods like Wiener filter and STSA-MMSE [7].",
"Lately, deep learning methods have been used, such as DNNs [6], [8], deep autoencoders (DAEs) [5], and convolutional neural networks (CNNs) [9].",
"However, to our knowledge, no one has tried to use GANs for SE yet.",
"GANs are a framework recently introduced by Goodfellow et al.",
"[10], which consists of a generative model, or generator (G), and a discriminative model, or discriminator (D), that play a min-max game between each other.",
"In particular, G tries to fool D which is trained to distinguish the output of G from the real data.",
"The architectures used in most of the works today [11] are based on deep convolutional GAN (DCGAN) [12] that successfully tackles training instability issues when GANs are applied to high resolution images.",
"Three key ideas are used to accomplish this goal.",
"First, batch normalization [13] is applied to most of the layers.",
"Then, the networks are designed to have no pooling layers as done in [14].",
"Finally, the training is performed adopting the Adam optimizer [15].",
"So far GANs have been successfully applied to a variety of computer vision and image processing tasks [1], [12], [16], [17].",
"However, their adoption for speech-related tasks is rare with one exception in [18], in which the authors of the report applied a deep visual analogy network [19] as a generator of a GAN for voice conversion, and the results are presented as example audio files without speech quality or intelligibility or other measures.",
"In a related field, for music, the GAN concept was applied to train a recurrent neural network for classical music generation [20].",
"Very recently, a general-purpose cGAN framework called Pix2Pix was proposed for image-to-image translation [1].",
"Motivated by its successful deployment on several tasks, we adapt the framework in this work, aiming to explore the potential of cGANs for SE, as part of the overall goal of investigating the feasibility and performance of GANs for speech processing.",
"Specifically, we use Pix2Pix to learn a mapping between noisy and clean speech spectrograms as well as to learn a loss function for training the mapping."
],
[
"Pix2Pix framework for speech enhancement",
"In GANs, G represents a mapping function from a random noise vector $\\mathbf {z}$ to an output sample $G(\\mathbf {z})$ , ideally indistinguishable from the real data $\\mathbf {x}$ [10].",
"In cGANs, both G and D are conditioned on some extra information $\\mathbf {y}$ [1], and they are trained following a min-max game with the objective: $\\begin{split}L&(D, G) = \\mathbb {E}_{\\mathbf {x},\\mathbf {y} \\sim \\ p_{data}(\\mathbf {x},\\mathbf {y})} [\\log (D(\\mathbf {x}, \\mathbf {y}))] +\\\\ &\\mathbb {E}_{\\mathbf {z} \\sim \\ p_\\mathbf {z}(\\mathbf {z}), \\mathbf {y} \\sim \\ p_{data}(\\mathbf {y})} [\\log (1 - D(G(\\mathbf {z}, \\mathbf {y}),\\mathbf {y}))].\\end{split}$ Pix2Pix differs from other cGAN works, like [21], because it does not use $\\mathbf {z}$ .",
"Isola et al.",
"[1] report that adding a Gaussian noise as an input to G, as done in [22], was not effective.",
"Hence, they introduce noise in the form of dropout, but this technique failed to produce stochastic output.",
"However, we are more interested in an accurate mapping between a noisy spectrogram and a clean one than a cGAN able to capture the full entropy of the distribution it models, so this represents a minor issue.",
"Figure REF shows how the data and the condition are used during training in the particular case of this paper.",
"Figure: Generator (G) and discriminator (D) in the Pix2Pix framework for speech enhancement.",
"G generates an enhanced spectrogram from a noisy input by fooling D, which tries to classify a spectrogram as clean or enhanced, conditioned on the respective noisy spectrogram.In addition to the adversarial loss $L(D, G)$ that is learned from the data, Pix2Pix utilizes also L1 distance between the output of G and the ground truth.",
"The choice of combining different losses, like L2 distance [23] or perceptual losses for a specific task [16], [17], has been shown to be beneficial.",
"In Pix2Pix, L1 distance is preferred to L2 because it encourages less blurring [1] and it tends to generalize better if compared to perceptual losses.",
"Furthermore, G and D, adapted from [12], are a U-Net [24] and a PatchGAN, respectively.",
"Since in image-to-image translation tasks, the input and the output of G share the same structure, G is an encoder-decoder where each feature map of the decoder layers is concatenated with its mirrored counterpart from the encoder to avoid that the innermost layer represents a bottleneck for the information flow.",
"Besides, D is built to model the high frequencies of the data, as the low frequency structure is captured by the L1 loss.",
"This is achieved by considering local image patches.",
"In particular, D is applied convolutionally across the image to classify each patch as real or fake.",
"Then, the obtained scores are averaged together to get a single output.",
"This architecture has the advantage of being smaller and can be applied on images of different sizes [1].",
"When the patch size of D has the same size of the input image, D is equivalent to a classical GAN discriminator.",
"Our Pix2Pix implementation is based on [25], with G that gets a $256 \\times 256$ 1-channel image, while D a $256 \\times 256$ 2-channel image.",
"The main differences with the original framework are the adoption of $5 \\times 5$ filters in the convolutional layers, and the last layer of D which is flattened and fed into a single sigmoid output as in [12]."
],
[
"Preprocessing and training",
"For speech signals with a sample rate of 16 kHz, we computed a time-frequency (T-F) representation using a 512-point short time Fourier transform (STFT) with a hamming window size of 32 ms and a hop size of 16 ms.",
"In this way, the frequency resolution is 16 kHz / 512 = 31.25 Hz per frequency bin.",
"We considered only the 257-point STFT magnitude vectors which cover the positive frequencies due to symmetry.",
"Our generator G accepts $256\\times 256\\times 1$ input, so for training we concatenated all the speech signals and then split the spectrogram every 256 frames, while for testing we zero-padded the spectrogram of each test sample in order to have the number of frames equal to a multiple of 256 and then performed the split accordingly.",
"We also removed the last row of the spectrogram, which is a choice that has a negligible impact since it represents only the highest 31.25 Hz band of the signal, but this allows us to have a power-of-2 input size which makes the design of G and D easier.",
"Before the data are fed to our system, they are also normalized to be in the range $[-1,1]$ .",
"We trained the GANs using stochastic gradient descent (SGD) and adopting the Adam optimizer, for 10 epochs with a batch size of 1 according to [1], updating G twice per each iteration to avoid a fast convergence of D [25].",
"The networks' weights have been initialized from a normal distribution with zero mean and a standard deviation of 0.02 [1].",
"The L1 loss has been added to the GAN loss using a scaling factor of 100 [1].",
"To enhance a speech signal with Pix2Pix, we first compute the T-F representation of it, and then we forward propagate the spectrogram magnitude through G. Finally, we reconstruct the signal with the inverse STFT using the phase of the noisy input."
],
[
"Evaluation metrics",
"The performance of our system is evaluated in terms of PESQ [26] (in particular the wide-band extension [27]), STOI [28], and EER of ASV.",
"PESQ and STOI have been chosen as they are the most used estimators of speech quality and speech intelligibility, respectively.",
"The implementations used in this paper are from [7] for PESQ and from [28] for STOI.",
"Regarding the ASV evaluation, we use the classical Gaussian Mixture Model - Universal Background Model (GMM-UBM) framework [29], which is suitable for short utterances as in this work.",
"We first built a general model, UBM, which is a GMM trained with an expectation-maximization algorithm using a large amount of speech data not belonging to the target speakers.",
"Then, a target speaker model for each specific pass-phrase and each speaker was derived by maximum a posteriori adaptation of the UBM.",
"The approach of adapting UBM is used in order to have a well-trained model for a speaker even when there is no much data available.",
"At this point, for a test utterance we calculate the log likelihood ratio between the claimant speaker model and the UBM.",
"The features extracted from the speech data are 57-dimensional mel-frequency cepstral coefficients (MFCCs), and the GMM mixture number is 512."
],
[
"Baseline methods",
"We compare the results of our approach with other two methods we consider as baselines: STSA-MMSE and an Ideal Ratio Mask (IRM) based DNN-SE algorithms.",
"STSA-MMSE is a statistical-based SE technique, where the a priori signal to noise ratio (SNR) is estimated with the Decision-Directed approach [30] and the noise Power Spectral Density (PSD) is estimated with the noise PSD tracker in [31].",
"The noise PSD estimate is initialized with the first 1000 samples of each utterance, assumed to be a speech-free region.",
"For the DNN-SE algorithm, we use the same procedure and parameters of [6].",
"The IRM is estimated by using a DNN with three hidden layers of 1024 units each, and an output layer with 64 units.",
"The input of the DNN is a 1845-dimensional feature vector, which is a robust representation of a frame that combines MFCCs, amplitude modulation spectrogram, relative spectral transform - perceptual linear prediction (RASTA-PLP), and gammatone filter bank energies, with their delta and double delta for a context of 2 past and 2 future frames.",
"The training label is represented by the IRM, which is computed as in [32] from the T-F representation based on a gammatone filter bank with 64 filters linearly spaced on a Mel frequency scale and with a bandwidth equal to one equivalent rectangular bandwidth [33].",
"The system is trained for 30 epochs with SGD, using the mean square error as error function and a batch size of 1024.",
"In order to enhance a test signal, the DNN provides an estimation of the IRM which is applied to the T-F representation of the noisy signal.",
"Finally, the time domain signal is synthesized."
],
[
"Datasets",
"We use two corpora, TIMIT [34] and RSR2015 [35], as follows: Set 1 (TIMIT) - 4380 utterances of male speakers are used for UBM training.",
"Set 2 (RSR2015) - Text ID from 2 to 30 of sessions 1, 4, and 7 for 50 male speakers (from m051 to m100) are selected to train Pix2Pix and DNN-SE.",
"Set 3 (RSR2015) - Text ID 1 of sessions 1, 4, and 7 for 49 male speakers (from m002 to m050) are used to train the speaker models.",
"Set 4 (RSR2015) - Sessions 2, 3, 5, 6, 8, and 9 of the same text ID and speakers used for training the models, are selected for evaluation.",
"The choice of RSR2015 as the main database for training and testing can be seen as a compromise, because we were interested in the evaluation of an ASV system, which provides another objective measure of the performance, and RSR2015 is widely used for this task.",
"We used 5 different noise types to simulate real-life conditions: Babble, obtained by adding 6 random speech samples from the Librispeech corpus [36]; white Gaussian noise generated in MATLAB; Cantine, recorded by the authors; Market and Airplane, collected by Fondazione Ugo Bordoni (FUB) and available on request from the OCTAVE project [37].",
"Noise data, which were added to the utterances in Set 2, 3, and 4 at different SNR values, used for training and testing are different."
],
[
"Setup",
"Inspired by [2], we investigate two different kinds of Pix2Pix-based SE front-ends: 5 noise specific front-ends (NS-Pix2Pix), each of them trained on only one type of noise, and 1 noise general front-end (NG-Pix2Pix), trained on all types of noise.",
"The same has been done for the DNN-SE front-ends, obtaining 5 noise specific front-ends (NS-DNN) and 1 noise general front-end (NG-DNN).",
"For training, we add noise to clean speech at two different SNRs, 10 and 20 dB.",
"With higher SNR it should be easier to train a G able to capture the underlying structure of the noisy input and generate a clean spectrogram, but a test with lower SNRs for training is worth to explore in the future.",
"For testing, results for 5 different SNR conditions are reported: 0, 5, 10, 15, and 20 dB, as is commonly done for ASV, but an interesting future work is to test on lower SNRs, particularly for intelligibility evaluation.",
"In addition, to find the behavior of the front-ends on noise free conditions, ASV performance on enhanced clean speech data is also reported.",
"In all the tests, the performance of the following front-ends are presented: No enhancement (when no SE algorithm is used on noisy data), STSA-MMSE, NS-DNN, NS-Pix2Pix, NG-DNN, and NG-Pix2Pix.",
"In total, three different tests have been conducted: Test 1 - In the first test, we compute PESQ and STOI for the different front-ends to estimate speech quality and intelligibility.",
"Test 2 - In the second test, the ASV system is trained with enhanced clean speech (except for the No enhancement front-end where clean speech is used) and tested on the 5 types of noise.",
"Test 3 - The last test is performed to evaluate the effects of a multi-condition training on ASV.",
"For No enhancement, STSA-MMSE, NS-DNN, and NS-Pix2Pix the speaker models are built from enhanced clean speech and one kind of enhanced noisy speech, while for NG-DNN and NG-Pix2Pix all kinds of noise are used."
],
[
"Results and Discussion",
"The results of Test 1 are shown in Table REF .",
"It is observed that the average PESQ scores of NS-Pix2Pix and NG-Pix2Pix are always better than the other front-ends.",
"The best performance improvement is achieved between 5 and 15 dB SNR regardless of the noise type.",
"At 20 dB, our approach outperforms the others on Market and White noises, but for Airplane noise STSA-MMSE is the best one, while for Babble and Cantine noises the absence of enhancement is superior indicating that all the SE techniques introduce an amount of distortion surpassing the benefit of noise reduction.",
"At 0 dB, NG-Pix2Pix generally outperforms the noise specific version with an exception (Market noise) and its scores are close to DNN-SE ones.",
"In terms of STOI, Pix2Pix front-ends perform similarly to STSA-MMSE.",
"However, DNN-SE front-ends are superior in almost all the conditions, even though Pix2Pix front-ends achieve the same or very close results in some situations, e.g.",
"low SNRs for Cantine and Market noises.",
"At 20 dB, we observe the same behavior as the PESQ scores, where the evaluation of not enhanced signals gives a better outcome.",
"Table: PESQ and STOI performance for the 5 front-ends: No enhancement (a), STSA-MMSE (b), NS-DNN (c), NS-Pix2Pix (d), NG-DNN (e), NG-Pix2Pix (f).Figure: From left to right: noisy spectrogram (White noise at 0 dB SNR); clean spectrogram; spectrogram of the signal enhanced with NG-Pix2Pix; spectrogram of the signal enhanced with NG-DNN; spectrogram of the signal enhanced with STSA-MMSE.Table: ASV performance in terms of EER on clean speaker modelTable: ASV performance in terms of EER on multi-condition speaker modelThe ASV performances (Tests 2 and 3) are reported in Tables REF and REF , where the results of the baseline systems are from [38].",
"For the clean speaker models, Pix2Pix front-ends generally outperform the baseline methods.",
"One exception is seen for Babble noise, where the NG-DNN front-end gives an EER of 8.73%, marginally better than NS-Pix2Pix (8.76%).",
"At low SNR, DNN-SE front-ends sometimes show better results than Pix2Pix, but overall our approach can be considered superior.",
"On the other hand, the performances of DNN-SE front-ends on multi-condition training are generally better, which presents a substantial improvement if compared with the clean speaker model situation.",
"Our approach is generally better than STSA-MMSE, although the NS-Pix2Pix front-end shows lower performance when it deals with white noise.",
"In general, Pix2Pix can be considered competitive with DNN-SE (better PESQ and EER on the clean speaker models, but worse STOI and EER for multi-condition training) and overall superior to STSA-MMSE.",
"Figure REF shows the spectrograms of a noisy utterance (White noise at 0 dB SNR), together with its clean and enhanced versions with NG-Pix2Pix, NG-DNN, and STSA-MMSE.",
"It is observed that the spectrogram enhanced by the cGAN approach preserves the structure of the original signal better than the other SE techniques, while at the same time more noises remain especially at high frequency regions, as compared with NG-DNN.",
"The spectrogram enhanced by STSA-MMSE is snowy all over the places."
],
[
"Conclusion",
"In this paper we investigated the use of conditional generative adversarial networks (cGANs) for speech enhancement.",
"We adapted the Pix2Pix framework, intended to solve generic image-to-image translation problems, and evaluated the performance of this approach in terms of estimated speech perceptual quality and speech intelligibility, together with equal error rate of a Gaussian Mixture Model - Universal Background Model based speaker verification system.",
"The results we obtained allow us to conclude that cGANs are a promising technique for speech denoising, being globally superior to the classical STSA-MMSE algorithm, and comparable to a DNN-SE algorithm.",
"Future work includes a more extensive evaluation of the framework in more critical SNR situations, and modifications aiming at making it specific for the task.",
"For example, a model with G generating a small size output window from a fixed number of successive frames can be built as it is often done in deep neural networks for speech processing, and a specific perceptual loss to be added to the cGAN loss can be designed."
],
[
"Acknowledgements",
"The authors would like to thank Hong Yu for providing data and speaker verification results for the baseline systems and Morten Kolbæk for his assistance and software used for the speaker verification and DNN speech enhancement baseline systems.",
"This work is partly supported by the Horizon 2020 OCTAVE Project (#647850), funded by the Research European Agency (REA) of the European Commission, and the iSocioBot project, funded by the Danish Council for Independent Research - Technology and Production Sciences (#1335-00162)."
]
] | 1709.01703 |
[
[
"Topology of Force Networks in Granular Media under Impact"
],
[
"Abstract We investigate the evolution of the force network in experimental systems of two-dimensional granular materials under impact.",
"We use the first Betti number, $\\beta_1$, and persistence diagrams, as measures of the topological properties of the force network.",
"We show that the structure of the network has a complex, hysteretic dependence on both the intruder acceleration and the total force response of the granular material.",
"$\\beta_1$ can also distinguish between the nonlinear formation and relaxation of the force network.",
"In addition, using the persistence diagram of the force network, we show that the size of the loops in the force network has a Poisson-like distribution, the characteristic size of which changes over the course of the impact."
],
[
"Introduction",
"When a densely packed granular media is subject to some force, its mechanical response is determined to a large extent by the network of inter-grain forces that spontaneously forms within the granular system.",
"These `force-chains' have been experimentally studied in both two- and three-dimensional systems, for a wide variety of loading forces [1], [2], [3], [4], [5].",
"In particular, Kondic et al.",
"[6] studied the topology of force networks in numerical simulations of a system of isotropically compressed elastic disks, finding that the zeroth Betti number, $\\beta _0$ , of the force networks formed during compression is sensitive to the inter-grain friction and the polydispersity of the granular material being compressed.",
"Other structural properties of granular force networks have been examined using measures from persistent homology [7], [8], [9] and network science [10], [11], [12], [13].",
"Their work leaves open the possibility that similar topological measures may provide insight into other experimental granular systems.",
"Here, we examine the evolution of force network structure in the transient (rather than steady-state) response of a granular material to stress.",
"In particular, experimental studies [14] show that force networks in granular media under impact undergo significant spatiotemporal changes throughout the course of an impact event.",
"These changes are also highly dependent on an “effective Mach number”, $M^\\prime $ , defined as $M^\\prime = t_cv_0/d$ where $t_c$ is the collision time between two grains (in turn a function of grain stiffness and force law), $v_0$ the initial intruder speed, and $d$ the diameter of a grain.",
"To date, these structural changes have not been quantified using topological methods.",
"In this paper, we present topological measures as applied to the spatio-temporal evolution of force networks in a experimental granular system under impact.",
"We show that key topological measures, such as the first Betti number, $\\beta _1$ , as well as persistent homology measures such as the persistence diagram, vary significantly with $M^\\prime $ .",
"These measures also provide insight to the structural differences between the collective stiffening and relaxation of the force network that take place during an impact event."
],
[
"Methods",
"The experimental data comes from a vertical two-dimensional granular bed made up of photoelastic discs confined between two acrylic sheets[15].",
"An intruder is released at varying heights above the granular bed, thus varying the intruder speed $v_0$ at impact.",
"The photoelastic grains are made of materials with different stiffnesses, changing the collision time between grains $t_c$ for different sets of experiments.",
"This provides two ways to tune $M^\\prime $ , as given by Eq.",
"REF .",
"By imaging the impact event at high speed (recording at rates up to 40kHz), with the grains lit from behind with circularly polarized light, we track the motion of the intruder, while simultaneously gaining visual access to the propagation of forces in the granular medium.",
"Further details are given in [15].",
"We focus specifically on results from impacts on the softest particles.",
"In order to extract the force networks from the experimental images, we use an adaptive thresholding method, followed by black-and-white image operations to remove noise in the form of isolated single-pixel flickers (bwmorph).",
"These functions are available in Matlab [16].",
"Note that this method is equivalent to choosing a particular force threshold for the force network, and extracting the corresponding network.",
"For the softest particles in the experimental dataset, this method captures the majority of the visible force network in the granular system, as shown in Fig.",
"REF Figure: The formation of the force network after impact, for M ' =0.6675M^\\prime = 0.6675.",
"Each successive frame is 2 ms later than the preceding frame.",
"A: The experimental images, showing the formation of a dense force network beneath the intruder (black disc at the top of the image).",
"Lighter particles indicate regions of greater stress.",
"B: The force network obtained by the thresholding described in Section 2.",
"The thresholding is able to capture the force network of panel A well."
],
[
"Results",
"The thresholding and noise reduction algorithm was applied to the high-speed videos, beginning with the first frame of impact.",
"We first computed the Betti numbers associated with each thresholded frame of each video using CHOMP [17].",
"We begin chiefly with an analysis of the first Betti number, $\\beta _1$ .",
"Physically, the number of loops (`1-cycles') in the granular force network, as measured by $\\beta _1$ , corresponds directly to the amount of the normal force that the system can support, or how “stiff\" it is.",
"Since we expect from prior studies [14], [15] that the effective stiffness of the granular material is strongly connected with the dynamics of impact, we focus on $\\beta _1$ for the remainder of this paper.",
"Future studies may find it useful to focus on $\\beta _0$ , the number of connected components.",
"The evolution of $\\beta _1$ as a function of time, for several different values of $M^\\prime $ , is shown in Fig.",
"REF A.",
"We note an early period of rapid growth in $\\beta _1$ , corresponding to a rapid stiffening process as a dense force network percolates through the granular bed.",
"After reaching a peak, $\\beta _1$ declines more gradually to a new baseline level, indicating that the mean size of loops in the force network has increased.",
"Figure REF C shows that the maximum number of 1-cycles formed in the force network, which occurs near peak stress of the granular material, is linearly dependent on $M^\\prime $ .",
"In addition, Fig.",
"REF D also reveals that the time taken to reach this maximum stiffness decreases linearly as $M^\\prime $ increases.",
"Further insight into the structure of the force network during impact is provided by Fig.",
"REF B, which shows the spatial density of the loops in the system over the course of an impact, for several different values of $M^\\prime $ .",
"The spatial density of the loops in the force network was determined first by finding the percentage of area involved in the network, which we calculate by finding the percentage of black pixels in each frame of Fig.",
"REF B.",
"We then divide $\\beta _1$ by this measure of the network area to find a spatial density.",
"The spatial density of $\\beta _1$ shows a similar trend to the unnormalized $\\beta _1$ values, suggesting that an increase in $M^\\prime $ drives not only a greater number of loops, but also smaller, more densely packed loops within the granular material.",
"Figure: The first Betti number, β 1 \\beta _1, provides useful insight to the evolution of the network structure during an impact event.",
"A: Evolution of first Betti number, β 1 \\beta _1, as a functionof time after impact during impact events with differentinitial M ' M^\\prime .",
"Each curve rises quickly to a peak valueof β 1 \\beta _1, then decreases slowly to a baseline valueof β 1 \\beta _1, corresponding to changes in the strength of the network over the course of an impact.",
"B: β 1 \\beta _1 normalized by the percentage of area involved in the network (percentage black pixels in each frame of Fig.",
"B), giving a measure of the density of the loops in the system.",
"The network becomes denser as well as more connected as M ' M^\\prime increases.",
"C: The maximum number of loops formed in the force network after impact increases linearly with M ' M^\\prime .",
"D: The time taken for the force network to develop its maximal number of loops decreases with increasing M ' M^\\prime .The nonlinear evolution of the structure of the force network can also be observed in the dependence of $\\beta _1$ on other dynamical variables in the impact.",
"Figure REF shows the relationship between $\\beta _1$ and the intruder acceleration, where downwards has been chosen to be the direction of positive acceleration.",
"Two key features stand out from this result.",
"First, there is significant hysteresis between the initial parts of the curves (upwards arrow), where the force network is building up its contacts and thus becoming mechanically stiffer, and the latter part of the curves (downward arrow), where the force network is relaxing and becoming less stressed.",
"Second, the relaxation branch of the curve seems to follow a similar shape regardless of $M^\\prime $ .",
"In contrast, the initial branch associated with the stiffening network depends heavily on $M^\\prime $ .",
"These two features of the data presented in Fig.",
"REF suggest a complex (and nonlinear) interplay between the force exerted by the intruder on the granular material, and the force evolution in the granular network via the formation of loops.",
"In particular, the hysteresis in the data points to a significantly different physical process underlying the relaxation and stiffening halves of the curve, where the formation of the force network depends more heavily on $M^\\prime $ than its relaxation.",
"Figure: β 1 \\beta _1 as a function of the intruder acceleration, for different M ' M^\\prime .The black arrows mark the directions of increase and decrease as a function of time.",
"β 1 \\beta _1 evolves highly nonlinearly as a function of other system dynamics.",
"In addition, there is a large amount of hysteresis between the stiffening and relaxing branches of the curve.Figure: β 1 \\beta _1 as a function of total photoelastic signal, fordifferent M ' M^\\prime .",
"The total photoelastic signal was obtainedby integrating the intensity of the unthresholded experimentaldata over the granular bed, and corresponds directly to the totalforce on the network.",
"The arrows indicate the evolution of thedata as a function of time.",
"There is significant hysteresisbetween the growth and decay of the force network formed during animpact.The differences between the formation and relaxation of the force network during impact are more clear in Fig.",
"REF , which shows the relationship between $\\beta _1$ and the total photoelastic signal (image intensity summed over each video frame) for different values of $M^\\prime $ .",
"Again, there are significant nonlinearities in the relationship between $\\beta _1$ and the total force on the network, due to the highly nonlinear process by which the network is formed.",
"In addition, we observe again the hysteresis between the formation and decay of the force network, providing further support for the claim that the process by which the granular material stiffens is different from the process by which it relaxes.",
"Figure: β 1 \\beta _1 as a function of total photoelastic intensity,normalized by the maximum values of β 1 \\beta _1 and normalized totalphotoelastic intensity respectively.",
"Different colors indicatedifferent values of M ' M^\\prime .",
"A: the stiffeningbranch of the curves.",
"Varying M ' M^\\prime affects the shape of thecurves.",
"These curves were fitted to quadraticfunctions of the form s 1 x 2 +s 2 x+s 3 s_1 x^2 +s_2 x+s_3, where xx is the normalizedphotoelastic intensity.",
"B: The relaxation branch of thecurves.",
"Again, the shape of the curves shows some variation with M ' M^{\\prime } These curves were fitted to linear functions of theform r 1 x+r 2 r_1 x +r_2.",
"C: r 1 ,r 2 ,s 1 ,s 2 r_1,r_2,s_1,s_2 and s 3 s_3 as functionsof M ' M^\\prime .",
"Circles indicate fitting parameters for therelaxation branch of the curve, and crosses for the stiffeningbranch of the curve.",
"The stiffening and relaxation branches differ by a constant quadratic term, s 1 s_1.Figure REF makes these differences quantitative.",
"The data is first divided between the stiffening and relaxation branches, and then normalized by its maximum values of $\\beta _1$ and total photoelastic intensity, respectively.",
"This normalized data is shown in Fig.",
"REF A and B, which show also that the relaxation and stiffening branches of the hysteresis loop have different dependences on $M^\\prime $ .",
"In particular, during the relaxation of the granular material, $\\beta _1$ shows a linear dependence on the normalized photoelastic intensity, whereas the stiffening of the granular material shows a higher order dependence on normalized photoelastic intensity.",
"This dependence is found by fitting each branch of the loop to a linear function of form $r_1 x +r_2$ for the relaxation branch of the data, and $s_1 x^2+s_2 x +s_3$ for the stiffening branch of the data.",
"Here, $x$ is the normalized photoelastic intensity.",
"We then find the dependence of these fitting parameters on $M^\\prime $ .",
"These results are shown in Fig.",
"REF C. The linear and constant term coefficients ($s_2$ , $s_3$ , $r_1$ , $r_2$ ) for both the normalized stiffening and relaxation processes depend linearly on $M^\\prime $ .",
"We note that these dependencies are very similar, within the scatter of the data.",
"The difference between the processes by which the granular material stiffens and relaxes is encapsulated by $s_1$ , which does not show a dependence on $M^\\prime $ .",
"That is, the difference between the loading and unloading of force chains in the granular materials is affected by the initial loading only in amplitude, and follows a similar pattern for all the impacts we observe.",
"In addition to computing the Betti numbers associated with the thresholded force networks, we also compute the persistence diagrams of the same thresholded force networks.",
"The same thresholding algorithm is used to produce black-and-white images of the force network for each frame.",
"Again using the function bwmorph in Matlab [16], we reduce the force network in each frame of the high-speed video to its skeleton.",
"An example of a point cloud generated by this process is shown in Fig.",
"REF A.",
"The persistence diagram of this point cloud was then computed by constructing its Vietoris-Rips complex, using Perseus [18].",
"An illustration of the Vietoris-Rips complex is shown in Fig.",
"REF B and C. The points in the point cloud are replaced by balls centered at each point, whose diameter $d$ is incremented at each computational timestep (in this case $\\delta d=2\\times 10^{-4}$ pixels).",
"At each timestep, we form a simplex for every set of points whose diameter is at most $d$ .",
"Thus if two balls have pairwise intersections, a line is formed between them, and if three balls have pairwise intersections, a triangle is formed between them.",
"If balls intersect, draw an edge between them.",
"A 1-cycle is “born\" when an empty loop of edges appears (Fig REF B).",
"The 1-cycle later “dies\" when the empty loop is completely filled in by its constituent triangles (Fig REF C) [19].",
"Thus the time between the birth of the 1-cycle and its death (or its lifetime) is correlated to the size of the 1-cycle.",
"For our computations, the total number of iterations was $7\\times 10^5$ .",
"Note that these persistence diagrams are generated by a different method from those which are examined in [6], and thus have a different physical interpretation.",
"Figure: Illustration of the process by which we compute persistence diagrams using the Vietoris-Rips complex.",
"A: Example of a point cloud generated from a frame of a high-speed video by the process described in the text.",
"White indicates the location of a point.",
"B: Illustration of a Vietoris-Rips complex formed from ten points, indicated as white circles with black borders.",
"Each point is replaced a ball whose radius is gradually incremented throughout the computation process, shown in blue.",
"At time t 0 t_0 in the computation, the diameter of the balls is d 0 d_0.",
"We form a simplex for every set of points whose diameter is at most d 0 d_0.",
"In this case, pairwise intersections between balls form lines.",
"An empty loop of edges is formed, and a 1-cycle is “born\".",
"C: The same set of points at a later time in the computation t 1 t_1, when the diameter of the balls has increased to d 1 d_1.",
"Sets of three balls now have pairwise intersections, forming triangles and thus filling in the empty loop of edges formed at t 0 t_0, so that the 1-cycle “dies\".",
"The lifetime of the 1-cycle on the left is thus t 1 -t 0 t_1-t_0.",
"At the same time, the group of four points on the right has formed another empty loop of edges, which will then die at a later time when the four constituent balls have pairwise interactions.Using the computed first-dimensional persistence diagram, we then computed the lifetime of each element on the persistence diagram, defined as $death-birth$ .",
"Since each time is associated with a certain ball diameter, the lifetime of a 1-cycle in the persistence diagram, multiplied by the step size, is a direct measure of the diameter of the 1-cycle.",
"These loop sizes follow an underlying distribution, which also evolves as a function of time after impact.",
"These distributions are displayed in Fig.",
"REF , which shows that a Poisson-like process governs the sizes of the 1-cycles in the network.",
"The change in shape of the distribution with increasing time after impact suggests that the expected value of this random distribution changes as a function of time.",
"Since the distribution is Poisson-like, we choose the following model relation: $P \\sim e^{-x/\\lambda }$ where $x$ is the size of a 1-cycle, and $\\lambda $ can be understood as the characteristic length of the distribution.",
"We can extract the value of $\\lambda $ by fitting an exponential function to each of the curves in Fig.",
"REF .",
"Figure: Log of distribution of persistent lifetimes for M ' =0.6675M^\\prime =0.6675, for different times after impact.",
"The distribution of persistent lifetimes is approximately exponential, with an exponent that changes with increasing time after impact.Figure REF shows the change in $\\lambda $ as a function of time after impact, for two different values of $M^\\prime $ .",
"For longer times after impact, as the intruder settles into the granular bed, Figs.",
"REF A and B look similar.",
"However, for times shortly after impact, the granular force network displays transient responses that vary with $M^{\\prime }$ .",
"For large $M^\\prime $ ($M^\\prime = 0.6675$ ), as illustrated in Fig.",
"REF A, the fluctuations in $\\lambda $ decrease after impact.",
"This supports our earlier results on the decreasing size of the 1-cycles in the force network during the collective stiffening of the granular bed.",
"After $\\lambda $ reaches its minimum value, however, it increases again, at a relatively slower rate, corresponding to the slower relaxation process found earlier in our discussion of Figs.",
"REF and REF .",
"In contrast, when $M^\\prime $ is smaller, $M^\\prime = 0.1683$, as in Fig.",
"REF B, there is no apparent change in the expected size of 1-cycles through the course of the experiment, suggesting that the impact dissipates energy through a chain-like force network, similar to that already existing in the granular material, rather than creating a shock-like network, as in the case for larger $M^\\prime $ .",
"Figure: λ\\lambda as a function of time after impact, for A: M ' =0.6675M^\\prime =0.6675 and B: M ' =0.1683M^\\prime =0.1683.",
"For large M ' M^\\prime , the expectation value of the exponential distribution decreases sharply, then increases to a new baseline value.",
"For small M ' M^\\prime , there is no significant change in the expectation value of the exponential distribution.Figure REF shows that the maximum change in $\\lambda $ varies linearly with $M^\\prime $ .",
"This result suggests (somewhat surprisingly) that there is not necessarily a sudden transition between the low $M^\\prime $ regime, where the impact dissipates energy through chain-like structures in the granular material, and the high $M^\\prime $ regime, where the impact significantly alters the structure of the impact, creating a shock-like network response.",
"Instead, there is a continuous transition where more and more of the network reconfigures over the course of an impact as $M^\\prime $ increases.",
"In addition, the linear relationship between the change in expected 1-cycle size and $M^\\prime $ again points to the fact that the amplitude of the topological response of the force network varies linearly with $M^\\prime $ (as shown in Fig.",
"REF ), despite the fact that the topological response itself is highly nonlinear.",
"Figure: Maximal change in λ\\lambda during the course of an impact as a function of M ' M^\\prime .",
"The change in expected size of 1-cycles during an impact varies continuously and linearly with M ' M^\\prime , suggesting a continuous transition from chain-like to shock-like networks as M ' M^\\prime is increased."
],
[
"Conclusions",
"In this paper, we have shown that simple topological measures such as the first Betti number, $\\beta _1$ , and the persistence diagram, can provide insights to the structural changes that take place in the force network of the granular material over the course of an impact, especially in relation to the “effective Mach number\", $M^\\prime $ .",
"In particular, the amplitude of the topological response depends linearly on the system parameter $M^\\prime $ , as measured by 1) the maximum value of $\\beta _1$ , and 2) the transient decrease in the characteristic size of the 1-cycles in the force network.",
"The response of the network itself, however, is highly nonlinear and displays a complex dependence on both the intruder acceleration and the total photoelastic response.",
"Notably, this response is hysteretic, and can be separated into growth and relaxation branches.",
"The growth and relaxation of the network differ by a constant nonlinear term, suggesting that the hysteresis in the structure of the force network under impact is a function of the properties of the granular material rather than the details of the impact."
],
[
"Acknowledgements",
"This work was supported by NSF grants DMR1206351, DMS grant DMS3530656, by NASA grant NNX15AD38G, and by a DARPA grant.",
"We thank Abe Clark for supplying data and for providing valuable advice.",
"We also thank Alec Peterson for his experimental contributions."
]
] | 1709.01884 |
[
[
"Distant decimals of $\\pi$"
],
[
"Abstract We describe how to compute very far decimals of $$\\pi$$ and how to provide formal guarantees that the decimals we compute are correct.",
"In particular, we report on an experiment where 1 million decimals of $$\\pi$$ and the billionth hexadecimal (without the preceding ones) have been computed in a formally verified way.",
"Three methods have been studied, the first one relying on a spigot formula to obtain at a reasonable cost only one distant digit (more precisely a hexadecimal digit, because the numeration basis is 16) and the other two relying on arithmetic-geometric means.",
"All proofs and computations can be made inside the Coq system.",
"We detail the new formalized material that was necessary for this achievement and the techniques employed to guarantee the accuracy of the computed digits, in spite of the necessity to work with fixed precision numerical computation."
],
[
"Introduction",
"The number $\\pi $ has been exciting the curiosity of mathematicians for centuries.",
"Ingenious formulas to compute this number manually were devised since antiquity with Archimede's exhaustion method and a notable step forward achieved in the eighteenth century, when John Machin devised the famous formula he used to compute one hundred decimals of $\\pi $ .",
"Today, thanks to electronic computers, the representation of $\\pi $ in fractional notation is known up to tens of trillions of decimal digits.",
"Establishing such records raises some questions.",
"How do we know that the digits computed by the record-setting algorithms are correct?",
"The accepted approach is to perform two computations using two different algorithms.",
"In particular, with the help of a spigot formula, it is possible to perform a statistical verification, simply checking that a few randomly spread digits are computed correctly.",
"In this article, we study the best known spigot formula, an algorithm able to compute a faraway digit at a cost that is much lower than computing all the digits up to that position.",
"We also study two algorithms based on arithmetic geometric means, which are based on iterations that double the number of digits known at each step.",
"For these algorithms, we perform all the proofs in real analysis that show that they do converge towards $\\pi $ , giving the rate of convergence, and we then show that all the computations in a framework of fixed precision computations, where computations are only approximated by rational numbers with a fixed denominator, are indeed correct, with a formally proved bound on the difference between the result and $\\pi $ .",
"Last we show how we implement the computations in the framework of our theorem prover.",
"The first algorithm, due to Bailey, Borwein, and Plouffe relies on a formula of the following shape, known as the BBP formula .",
"$\\pi = \\sum _{i=0}^{\\infty } \\frac{1}{16 ^ i} \\left(\\frac{4}{8 i + 1} -\\frac{2}{8 i + 4} - \\frac{1}{8 i + 5} - \\frac{1}{8 i + 6}\\right).",
"$ Because each term of the sum is multiplied by $\\frac{1}{16^i}$ it appears that approximately $n$ terms of the infinite sum are needed to compute the value of the nth hexadecimal digit.",
"Moreover, if we are only interested in the value of the nth digit, the sum of terms can be partitioned in two parts, where the first contains the terms such that $i \\le n$ and the second contains terms that will only contribute when carries need to be propagated.",
"We shall describe how this algorithm is proved correct and what techniques are used to make this algorithm run inside the Coq theorem prover.",
"The second and third algorithms rely on a process known as the arithmetic-geometric mean.",
"This process considers two inputs $a$ and $b$ and successively computes two sequences $a_n$ and $b_n$ such that $a_0 = a$ , $b_0 = b$ , and $a_{n+1} = \\frac{a_n + b_n}{2} \\qquad b_{n+1} = \\sqrt{a_n b_n}$ In the particular case where $a = 1$ and $b = x$ , the values $a_n$ and $b_n$ are functions of $x$ that are easily shown to be continuous and differentiable and it is useful to consider the two functions $y_n (x) = \\frac{a_n(x)}{b_n(x)} \\qquad z_n = \\frac{b^{\\prime }_n(x)}{a^{\\prime }_n(x)}$ A first computation of $\\pi $ is expressed by the following equality: $ \\pi = (2 + \\sqrt{2}) \\prod _{n = 1} ^{\\infty } \\frac{1 + y_n(\\frac{1}{\\sqrt{2}})}{1 + z_n(\\frac{1}{\\sqrt{2}})}.$ Truncations of this infinite product are shown to approximate $\\pi $ with a number of decimals that doubles every time a factor is added.",
"This is the basis for the second algorithm.",
"The third algorithm also uses the arithmetic geometric mean for 1 and $\\frac{1}{\\sqrt{2}}$ , but performs a sum and a single division: $\\pi = \\lim _{n\\rightarrow \\infty } \\frac{4 (a_n(1, \\frac{1}{\\sqrt{2}}))^2}{1 - \\sum _{i=1}^{n-1} 2 ^ {i - 1} (a_{i-1}(1,\\frac{1}{\\sqrt{2}}) -b_{i - 1}(1,\\frac{1}{\\sqrt{2}})) ^ 2}$ It is sensible to use index $n$ in the numerator and $n-1$ in the sum of the denominator, because this gives approximations with comparable precisions of their respective limits.",
"This is the basis for the third algorithm.",
"This third algorithm was introduced in 1976 independently by Brent and Salamin , .",
"It is the one implemented in the mpfr library for high-precision computation to compute $\\pi $ .",
"In this paper, we will recapitulate the mathematical proofs of these algorithms (sections and ), and show what parts of existing libraries of real analysis we were able to reuse and what parts we needed to extend.",
"For each of the algorithms, we study first the mathematical foundations, then we concentrate on implementations where all computations are done with a single-precision fixed-point arithmetic, which amounts to forcing all intermediate results to be rational numbers with a common denominator.",
"This framework imposes that we perform more proofs concerning bounds on accumulated rounding errors.",
"All the work described in this paper was done using the Coq proof assistant .",
"This system provides a library describing the basic definition of real analysis, known as the standard Coq library for reals, where the existence of the type of real numbers as an ordered, archimedian, and complete field with decidable comparison is assumed.",
"This choice of foundation makes that mathematics based on this library is inherently classical, and real numbers are abstract values which cannot be exploited in the programming language that comes in Coq's type theory.",
"The standard Coq library for reals provides notions like convergent sequences, series, power series, integrals, and derivatives.",
"In particular, the sine and cosine functions are defined as power series, $\\pi $ is defined as twice the first positive root of the cosine function, and the library provides a first approximation of $\\frac{\\pi }{2}$ as being between $\\frac{7}{8}$ and $\\frac{7}{4}$ .",
"It also provides a formal description of Machin formulas, relating computation of $\\pi $ to a variety of computations of arctangent at rational arguments, so that it is already possible to compute relatively close approximations of $\\pi $ , as illustrated in .",
"The standard Coq library implements principles that were designed at the end of the 1990s, where values whose existence is questionable should always be guarded by a proof of existence.",
"These principles turned out to be impractical for ambitious formalized mathematics in real analysis, and a new library called Coquelicot was designed to extend the standard Coq library and achieve a more friendly and regular interface for most of the concepts, especially limits, derivatives, and integrals.",
"The developments described in this paper rely on Coquelicot.",
"Many of the intermediate level steps of these proofs are performed automatically.",
"The important parts of our working context in this respect are the Psatz library, especially the psatzl tactic , which solves reliably all questions that can be described as linear arithmetic problems in real numbers and lia , which solves similar problems in integers and natural numbers.",
"Another tool that was used more and more intensively during the development of our formal proofs is the interval tactic , which uses interval arithmetic to prove bounds on mathematical formulas of intermediate complexity.",
"Incidentally, the interval tactic also provides a simple way to prove that $\\pi $ belongs to an interval with rational coefficients.",
"Intensive computations are performed using a library for computing with very large integers, called BigZ .",
"It is quite notable that this library contains an implementation of an optimized algorithm to compute square roots of large integers ."
],
[
"The BBP formula",
"In this section we first recapitulate the main mathematical formula that makes it possible to compute a single hexadecimal at a low cost .",
"Then, we describe an implementation of an algorithm that performs the relevant computation and can be run directly inside the Coq theorem prover."
],
[
"The mathematical Proof",
"We give here a detailed proof of the formula established by David Bayley, Peter Borwein and Simon Plouffe.",
"The level of detail is chosen to mirror the difficulties encountered in the formalization work.",
"$\\pi = \\sum _{i=0}^{\\infty } \\frac{1}{16 ^ i} (\\frac{4}{8 i + 1} -\\frac{2}{8 i + 4} - \\frac{1}{8 i + 5} - \\frac{1}{8 i + 6})$ We first study the properties of the sum $S_k$ for a given $k$ such that $1 < k $ : $S_k = \\sum _{i=0}^{\\infty } \\frac{1}{16 ^ i (8 i + k) }$ By using the notation $\\left[f(x)\\right]_0^{y} = f(y) - f(0)$ and the laws of integration, we get $S_k = \\sqrt{2} ^ k \\sum _{i=0}^{\\infty }\\left[\\frac{x ^ {k + 8i}}{8i+k}\\right]_0^{\\frac{1}{\\sqrt{2}}}= \\sqrt{2} ^ k \\sum _{i=0}^{\\infty } \\int _{0}^{\\frac{1}{\\sqrt{2}}} x ^ {k - 1 + 8i} \\,{\\rm d}x$ Thanks to uniform convergence, the series and the integral can be exchanged and we can then factor out $x^{k-1}$ and recognize a geometric series in $x^8$ .",
"$S_k = \\sqrt{2} ^ k\\int _{0}^{\\frac{1}{\\sqrt{2}}} \\sum _{i=0}^{\\infty } x ^ {k - 1 + 8i} \\,{\\rm d}x =\\sqrt{2} ^ k\\int _{0}^{\\frac{1}{\\sqrt{2}}} \\frac{x ^ {k -1}}{1 - x ^8} \\,{\\rm d}x$ Now replacing the $S_k$ values in the right hand side of (REF ), we get: $S = 4 S_1 -2S_4 -S_5 -S_6 = \\int _{0}^{\\frac{1}{\\sqrt{2}}} \\frac{4\\sqrt{2} - 8 x^3 -4\\sqrt{2}x^4 -8x^5}{ 1 - x^8} \\,{\\rm d}x$ Then, with the variable change $y = \\sqrt{2} x$ and algebraic calculations on the integrand $S=\\int _{0}^1 \\frac{4 -4y}{y^2 -2y +2} + \\frac{4}{1 + (y -1)^2} + 4\\frac{y}{y^2 -2}\\,{\\rm d}y$ We recognize here the respective derivatives of $-2\\ln (y^2 -2y +2)$ , $4 \\arctan (y-1)$ and $2\\ln (2-y^2)$ .",
"Most of these functions have null or compensating values at the bounds of the integral, leaving only one interesting term: $S &=& \\left[-2\\ln (y^2 -2y +2)+4 \\arctan (y-1)+2\\ln (2-y^2)\\right]_0^1\\\\&=& -4 \\arctan (-1) = \\pi $"
],
[
"The formalization of the proof",
"The current version of our formal proof, compatible with Coq version 8.5 and 8.6 is available on the world-wide web .",
"To formalize this proof, we use the Coquelicot library intensively.",
"This library deals with series, power series, integrals and provides some theorems linking these notions that we need for our proof.",
"In Coquelicot, series (named Series) are defined as in standard mathematics as the sum of the terms of an infinite sequence (of type $ nat\\rightarrow R$ in our case) and power series (PSeries) are the series of terms of the form $a_n x^n$ .",
"The beginning of the formalisation follows the proof (steps (REF ) to ()).",
"Then, one of the key arguments of the proof is the exchange of the integral sign and the series allowing the transition from equation () to equation (REF ).",
"The corresponding theorem provided by Coquelicot is the following: Lemma RInt_PSeries (a : nat -> R) (x : R) : Rbar_lt (Rabs x) (CV_radius a) -> RInt (PSeries a) 0 x = PSeries (PS_Int a) x. where (PSeries (PS_Int $a$)) is the series whose (n+1)-th term is $ \\frac{a_n}{n + 1} x^{n+1}$ coming from the equality: $\\int _0^x a_n x^n = \\left[ \\frac{a_n}{n + 1} x^{n+1} \\right]_0^x$ .",
"We use this lemma as a rewriting rule from right to left.",
"Note that the RInt_PSeries theorem assumes that the integrated function is a power series (not a simple series), that is, a series whose terms have the form $a_i x^i$ .",
"In our case, the term of the series is $x^{k-1+8i}$ , that is $x^{k -1} x^{8i}$ .",
"To transform it into an equivalent power series we have first to transform the series $\\sum _i x^{8i}$ into a power series.",
"For that purpose, we define the hole function.",
"Definition hole (n : nat) (a : nat -> R) (i : nat) := if n mod k =? 0 then a (i / n) else 0.",
"and prove the equality given in the following lemma.",
"Lemma fill_holes k a x : k <> 0 -> ex_pseries a (x ^ k) -> PSeries (hole k a) x = Series (fun n => a n * x ^ (k * n)).",
"The premise written in the second line of fill_holes expresses that the series $\\sum _i a_i (x^k)^i$ converges.",
"This equality expresses that the series of term $ a_i (x^k)^i$ is equivalent to the power series which terms are $a_{n/k}$ when n is a multiple of $k$ and 0 otherwise.",
"Then by combining fill_holes with the Coquelicot function (PS_incr_n a n), that shifts the coefficients of the series $ \\sum _{i=0}^{\\infty } a_i x^{n+i}$ to transform it into $\\sum _{i = 0}^{i = n-1} 0.x^{i} + \\sum _{i = n}^{\\infty } a_{i - n} x^i$ that is a power series, we prove the PSeries_hole lemma.",
"Lemma PSeries_hole x a d k : 0 <= x < 1 -> Series (fun i : nat => a * x ^ (d + S n * i)) = PSeries (PS_incr_n (hole (S k) (fun _ : nat => a)) n) x Moreover, the RInt_PSeries theorem contains the hypothesis that the absolute value of the upper bound of the integral, that is $|x|$ , is less than the radius of convergence of the power series associated to $a$ .",
"This is proved in the following lemma.",
"Lemma PS_cv x a : (forall n : nat, 0 <= a n <= 1) -> 0 <= x -> x < 1 -> Rbar_lt (Rabs x) (CV_radius a) It should be noted that in our case $a_n$ is either 1 or 0 and the hypothesis forall n : nat, 0 <= a n <= 1 is easily satisfied.",
"In summary, the first part of the proof is formalized by the Sk_Rint lemma: Lemma Sk_Rint k (a := fun i => / (16 ^ i * (8 * i + k))) : 0 < k -> Series a = sqrt 2 ^ k * RInt (fun x => x ^ (n - 1) / (1 - x ^ 8)) 0 (/ sqrt 2).",
"that computes the value of $S_k$ given by (REF ) from the definition (REF ) of Sk.",
"The remaining of the formalized proof follows closely the mathematical proof described in the previous section.",
"We first perform an integration by substitution (starting from equation (REF )), replacing the variable $x$ by $\\sqrt{2} x$ , by rewriting (from right to left) with the RInt_comp_lin Coquelicot lemma.",
"Lemma RInt_comp_lin f u v a b : RInt (fun y : R => u * f (u * y + v)) a b = RInt f (u * a + v) (u * b + v) This lemma assumes that the substitution function is a linear function, which is the case here.",
"Then we decompose S into three parts (by computation) to obtain equation (REF ), actually decomposed into three integrals that are computed in lemmas RInt_Spart1, RInt_Spart2, and RInt_Spart3 respectively.",
"For instance: Lemma RInt_Spart3 : RInt (fun x => (4 * x) / (x ^ 2 - 2)) 0 1 = 2 * (ln 1 - ln 2).",
"Finally, we obtain the final result, based on the equality $\\arctan 1 = \\frac{\\pi }{4}$ .",
"We now describe how the formula (REF ) can be used to compute a particular decimal of $\\pi $ effectively.",
"This formula is a summation of four terms where each term has the form ${1}/{16 ^ i (8 i + k)}$ for some $k$ .",
"Digits are then expressed in hexadecimal (base 16).",
"Natural numbers strictly less than $2^p$ are used to simulate a modular arithmetic with $p$ bits, where $p$ is the precision of computation.",
"We first explain how the computation of $S_k= \\sum _i{{1}/{16 ^ i (8 i + k)}}$ for a given $k$ is performed.",
"Then, we describe how the four computations are combined to get the final digit.",
"We want to get the digit at position $d$ .",
"The first operation is to scale the sum $S_k$ by a factor $m=16^{d -1}\\, 2^p$ to be able to use integer arithmetic.",
"In what follows, we need that $p$ is greater than four.",
"If we consider $\\lfloor mS_k\\rfloor $ (the integer part of $mS_k$ ), the digit we are looking for is composed of its bits $p$ , $p-1$ , $p-2$ , $p-3$ that can be computed using basic integer operations: $(\\lfloor mS_k\\rfloor \\texttt {\\,mod\\,} 2^p) / 2 ^{p - 4}$ .",
"Using integer arithmetic, we are going to compute an approximation of $\\lfloor mS_k\\rfloor \\texttt {\\,mod\\,} 2^p$ by splitting the sum into three parts $mS_k = \\sum _{0 \\le i < d} \\frac{m}{16^i (8 i + k)} +\\sum _{d \\le i < d + p /4} \\frac{m}{16^i(8 i + k)} +\\sum _{d + p / 4 \\le i} \\frac{m}{16^i (8 i + k)}$ In the first part, the inner term can be rewritten as $\\frac{2^p 16^{d- 1 -i}}{8 i + k}$ where both divisor and dividend are natural numbers.",
"The division can be performed in several stages.",
"To understand this, it is worth comparing the fractional and integer part of $\\frac{16 ^{d-1-i}}{8i+k}$ with the bits of $\\frac{2^p 16 ^ {d-1-i}}{8i+k}$ .",
"For illustration, let us consider the case where $i = 0$ , $k=3$ , $p=4$ , and $d = 2$ .",
"The number we wish to compute is $\\frac{2^416^{2-1}}{3}$ and we only need to know the first 4 bits, that is we need to know this number modulo $2^4$ .",
"The ratio is $85.33\\overline{3}$ , and modulo 16 this is 5.",
"Now, we can look at the number $2^4 \\frac{16}{3}$ .",
"If we note $q$ and $r$ the quotient and the remainder of the division on the left (when viewed as an integer division), we have $2^4 \\frac{16}{3} = 2^4 q + \\frac{2 ^ 4 r}{3}$ Since we eventually want to take this number modulo $2^4$ , the left part of the sum, $2^4 q$ , does not impact the result and we only need to compute $r$ , in other words $16 \\texttt {\\,mod\\,}3$ .",
"In our illustration case, we have $16 \\texttt {\\,mod\\,} 3 = 1$ and $\\frac{2 ^ 4 \\times 1}{3} = 5.333$ , so we do recover the right 4 bits.",
"Also, because we are only interested in bits that are part of the integral part of the result, we can use integer division to perform the last operation.",
"These computations are performed in the following Coq function, that progresses by modifying a state datatype containing the current index and the current sum.",
"In this function, we also take care of keeping the sum under $2^p$ , because we are only concerned with this sum modulo $2^p$ .",
"Inductive NstateF := NStateF (i : nat) (res : nat).",
"Doing an iteration is performed by Definition NiterF k (st : NstateF) := let (i, res) := st in let r := 8 * i + k in let res := res + (2 ^ p * (16 ^ (d - 1 - i) mod r)) / r in let res := if res < 2 ^ p then res else res - 2 ^ p in NStateF (i + 1) res.",
"The summation is performed by $d$ iterations: Definition NiterL k := iter d (NiterF k) (NStateF 0 0).",
"The result of NiterL is a natural number.",
"What we need to prove is that it is a modular result and it is not so far from the real value.",
"As we have turned an exact division into a division over natural numbers, the error is at most 1.",
"After $d$ iterations, it is at most $d$ .",
"This is stated by the following lemma.",
"Lemma sumLE k (f := fun i => ((16 ^ d / 16) * 2 ^ p) / (16 ^ i * (8 * i + k))) : 0 < k -> let (_, res) := NiterL p d k in exists u : nat, 0 <= sum_f_R0 f (d - 1) - res - u * 2 ^ p < d. where sum_f_R0 f n represents the summation $f(0) + f(1) + \\dots + f(n)$ .",
"Let us now turn our attention to the second part of the iteration of formula (REF ).",
"$\\sum _{d \\le i < d + p /4} \\frac{m}{16^i(8 i + k)} = \\sum _{d \\le i < d + p /4} \\frac{2^p16^{d-1-i}}{8i+k}.$ All the terms of this sum are less than $2^p$ .",
"As terms get smaller by a factor of at least 16, we consider only $p/4$ terms.",
"We first build a datatype that contains the current index, the current shift and the current result: Inductive NstateG := NStateG (i : nat) (s : nat) (res : nat).",
"We then define what is a step: Definition NiterG k (st : NstateG) := let (i, s, res) := st in let r := 8 * i + k in let res := res + (s / r) in NStateG (i + 1) (s / 16) res.",
"and we iterate $p/4$ times: Definition NiterR k := iter (p / 4) (NiterG k) (NStateG d (2 ^ (p - 4)) 0).",
"Here we do not need any modulo since the result fits in $p$ bits and as the contribution of each iteration makes an error of at most one unit with the division by $r$ , the total error is then bounded by $p/4$ .",
"This is stated by the following lemma.",
"Lemma sumRE k (f := fun i => ((16 ^ d / 16) * 2 ^ p) / (16 ^ (d + i) * (8 * (d + i) + k))) : 0 < k -> 0 < p / 4 -> let (_, _, s1) := NiterR k in 0 <= sum_f_R0 f (p / 4 - 1) - s1 < p / 4.",
"The last summation is even simpler.",
"We do not need to perform any computation.",
"all the terms are smaller than 1 and quickly decreasing.",
"It is then easy to prove that this summation is strictly smaller than 1.",
"Adding the two computations, we get our approximation.",
"Definition NsumV k := let (_, res1) := NiterL k in let (_, _, res2) := NiterR k in res1 + res2.",
"We know that it is an under approximation and the error is less than $d + p / 4 + 1$ .",
"We are now ready to define our function that extracts the digit: Definition NpiDigit := let delta := d + p / 4 + 1 in if (3 < p) then if 8 * delta < 2 ^ (p - 4) then let Y := 4 * (NsumV 1) + (9 * 2^ p - (2 * NsumV 4 + NsumV 5 + NsumV 6 + 4 * delta)) in let v1 := (Y + 8 * delta) mod 2 ^ p / 2 ^ (p - 4) in let v2 := Y mod 2 ^ p / 2 ^ (p - 4) in if v1 = v2 then Some v2 else None else None else None.",
"This deserves a few comments.",
"In this function, the variable delta represents the error that is done by one application of NsumV.",
"When adding the different sums, we are then going to make an overall error of 8 * delta.",
"Moreover, we know that NsumV is an under approximation.",
"The variable Y computes an under approximation of the result: for those sums that appear negatively, the under approximation is obtained adding delta to the sum before taking the opposite.",
"This explains the fragment ... + 4 * delta that appears on the seventh line.",
"Each of the sums obtained by NsumV actually is a natural number $s$ smaller than $2 ^ p$ , when it is multiplied by a negative coefficient, this should be represented by $2 ^ p - s$ .",
"Accumulating all the compensating instances of $2 ^ p$ leads to the fragment 9 * 2 ^ p - ... that appears on the sixth line.",
"After all these computations, Y + 8 * delta is an over approximation.",
"If both Y and $\\texttt {Y + 8 * delta}$ give the same digit, we are sure that this digit is valid.",
"The correctness of the NpiDigit function is proved with respect to the definition of what is the digit at place $d$ in base $b$ of a real number $r$ , i.e.",
"we take the integer part of $r b^d$ and we take the modulo $b$ : Definition Rdigit (b : nat) (d : nat) (r : R) := (Int_part ((Rabs r) * (b ^ d))) mod b.",
"The correctness is simply stated as Lemma NpiDigit_correct k : NpiDigit = Some k -> Rdigit 16 d PI = k. Note that this is a partial correctness statement.",
"A program that always returns None also satisfies this statement.",
"If we look at the actual program, it is clear that one can precompute a $p$ that fulfills the first two tests, the equality test is another story.",
"A long sequence of 0 (or F) may require a very high precision.",
"This program is executable but almost useless since it is based on a Peano representation of the natural numbers.",
"Our next step was to derive an equivalent program using a more efficient representation of natural numbers, provided by the type BigN .",
"This code also receives some optimizations to implement faster operations of multiplications and divisions by powers of 2 and fast modular exponentiations.",
"Computing within Coq that 2 is the millionth decimal in hexadecimal of $\\pi $ with a precision of 28 bits (27 are required for the first two tests and 28 for the equality test) takes less than 2 minutes.",
"In order to reach the billionth decimal, we implement a very naive parallelization for a machine with at least four cores: each sum is computed on a different core generating a theorem then the final result is computed using these four theorems.",
"With this technique, we get the millionth decimal, 2, in 25 seconds and the billionth decimal, 8, in 19 hours.",
"Note that we could further parallelize inside the individual sums to compute partial sums and then use Coq theorems to glue them together."
],
[
"Algorithms to compute $\\pi $ based on arithmetic geometric means",
"In principle, all the mathematics that we had to describe formally in our study of arithmetic geometric means and the number $\\pi $ are available from the mathematical litterature, essentially from the monograph by J. M. Borwein and P. B. Borwein and the initial papers by R. Brent , E. Salamin .",
"However, we had difficulties using these sources as a reference, because they rely on an extensive mathematical culture from the reader.",
"As a result, we were actually guided by a variety of sources on the world-wide web, including an exam for the selection of French high-school mathematical teachers .",
"It feels useful to repeat these mathematical facts in a first section, hoping that they are exposed at a sufficiently elementary level to be understood by a wider audience.",
"However, some details may still be missing from this exposition and they can be recovered from the formal development itself.",
"This section describes two algorithms, but their mathematical justification has a lot in common.",
"The first algorithm that we present came to us as the object of an exam for high-school teachers , but in reality this algorithm is neither the first one to have been designed by mathematicians, nor the most efficient of the two.",
"However, it is interesting that it brings us good tools to help proving the second one, which is actually more traditional (that second algorithm dates from 1976 , , and it is the one implemented in the mpfr library ) and more efficient (we shall see that it requires much less divisions).",
"In a second part of our study, we concentrate on the accumulation of errors during the computations and show that we can also prove bounds on this.",
"This part of our study is more original, as it is almost never covered in the mathematical litterature, however it re-uses most of the results we exposed in a previous article ."
],
[
"Mathematical basics for arithmetic geometric means",
"Here we enumerate a large collection of steps that make it possible to go from the basic notion of arithmetic-geometric means to the computation of a value of $\\pi $ , together with estimates of the quality of approximations.",
"This is a long section, consisting of many simple facts, but some of the detailed computations are left untold.",
"Explanations given between the formulas should be helpful for the reader to recover most of the steps.",
"However, missing information can be found directly in the actual formal development .",
"As already explained in section , the arithmetic-geometric mean of two numbers $a$ and $b$ is obtained by defining sequences $a_n$ and $b_n$ such that $a_0 = a$ , $b_0 = b$ and $a_{n+1} = \\frac{a_n + b_n}{2} \\qquad b_{n+1} = \\sqrt{a_n b_n}$ A few tests using high precision calculators show that the two sequences $a_n$ and $b_n$ converge rapidly to a common value $M(a,b)$ , with the number of common digits doubling at each iteration.",
"The sequence $a_n$ provides over approximations and the sequence $b_n$ under approximations.",
"Here is an example computation (for each line, we stopped printing values at the first differing digit between $a_n$ and $b_n$ ).",
"Table: NO_CAPTIONThe function $M(a,b)$ also benefits from a scalar multiplication property: $M(ka, kb) = k M(a, b)\\qquad M(a,b) = a M(1, \\frac{b}{a})$ For the sake of computing approximations of $\\pi $ , we will mostly be interested in the sequences $a_n$ and $b_n$ stemming from $a_0=1$ and $b_0 = \\frac{1}{\\sqrt{2}}$ ."
],
[
"Elliptic integrals.",
"We will be interested in complete elliptic integrals of the first kind, noted $K(k)$ .",
"The usual definition of these integrals has the following form $K(k) = \\int _0^{\\frac{\\pi }{2}} \\frac{{\\rm d}\\theta }{\\sqrt{1 - k^2 \\sin ^2 \\theta }}$ But it can be proved that the following equality holds, when setting $a = 1$ and $b = \\sqrt{1- k^2}$ , and using a change of variable (we only use the form $I(a,b)$ ): $K(k) = I(a, b) = \\int _0^{+\\infty } \\frac{{\\rm d}t}{\\sqrt{(a ^ 2 + t ^ 2) (b ^ 2 + t ^ 2)}}$ Note that the integrand in $I$ is symmetric, so that $I(a,b)$ is also half of the integral with infinities as bounds.",
"With the change of variables $s = \\frac{1}{2}(x - \\frac{ab}{x})$ , then reasoning by induction and taking the limit, we also have the following equalities $I(a,b) = I(\\frac{a + b}{2}, \\sqrt{ab})=I(a_n,b_n)=I(M(a,b),M(a,b))=\\frac{\\pi }{2M(a,b)}.$"
],
[
"Equivalence when $x \\rightarrow 0$ and derivatives.",
"Another interesting property for elliptic integrals of the first kind can be obtained by the variable change $u = \\frac{x}{t}$ on the integral on the right-hand side of equation (REF ).",
"$I(1, x) = 2 \\int _0^{\\sqrt{x}}\\frac{{\\rm d}t}{\\sqrt{(1 + t ^ 2)(x + t ^ 2)}}$ Studying this integral when $x$ tends to 0 gives the equivalences for $I$ and $M$ : $I(1,x) \\sim 2 \\ln (\\frac{1}{\\sqrt{x}})\\quad M(1, x) \\sim \\frac{-\\pi }{2 \\ln x} \\qquad \\hbox{when}~x\\rightarrow 0^+.$ For the rest of this section, we will assume that $x$ is a value in the open interval $(0, 1)$ and that $a_0 = 1$ and $b_0 = x$ .",
"Coming back to the sequences $a_n$ and $b_n$ , the following property can be established.",
"$M\\left(a_{n+1}, \\sqrt{a_{n+1}^2 - b_{n+1}^2}\\right) = \\frac{1}{2} M\\left(a_n, \\sqrt{a_n^2 - b_n^2}\\right)$ We can repeat $n$ times and use the fact that $a_0^2- b_0^2 = 1 - x^2$ .",
"$2 ^ n M \\left(a_n, \\sqrt{a_n ^ 2 - b_n ^ 2}\\right) =2 ^ n a_n M\\left(1, \\frac{\\sqrt{a_n ^ 2 - b_n ^ 2}}{a_n}\\right)=M \\left(1, \\sqrt{1 - x ^ 2}\\right)$ Still under the assumption of $a_0 = 1$ and $b_0 = x$ , we define $k_n$ as follows: $k_n(x) = \\frac{\\ln \\left(\\frac{a_n}{\\sqrt{a_n ^ 2 - b_n ^ 2}}\\right)}{2 ^ n}$ Through separate calculation, involving Equation () and the definition of $k$ , we establish the following properties.",
"$\\lim _{n \\rightarrow \\infty } k_n (x) = \\frac{\\pi }{2} \\frac{M(1, x)}{M(1,\\sqrt{1-x^2})}\\qquad k_n^{\\prime } = \\frac{b_n ^ 2}{x (1 - x ^ 2)}$ These derivatives converge uniformly to their limit.",
"Moreover, the sequence of derivatives of $a_n$ is growing and converges uniformly.",
"This guarantees that $x \\mapsto M(1,x)$ is also differentiable and and its derivative is the limit of the derivatives of $a_n$ .",
"We can then obtain the following two equations, the second is our main central formula.",
"$\\left(\\frac{\\pi }{2} \\frac{M(1, x)}{M(1, \\sqrt{1 - x ^ 2})}\\right)^{\\prime }=\\frac{M(1, x) ^ 2}{x (1 - x ^ 2)}\\qquad \\pi = 2 \\sqrt{2} \\frac{M(1, \\frac{1}{\\sqrt{2}})^3}{\\displaystyle \\left(M(1, x)\\right)^{\\prime }(\\frac{1}{\\sqrt{2}})}$ We define the functions $y_n = \\frac{a_n}{b_n}$ and $z_n = \\frac{b^{\\prime }_n}{a^{\\prime }_n}$ .",
"These sequence satisfy $y_0 = \\frac{1}{x} \\qquad y_{n+1} = \\frac{1 + y_n}{2\\sqrt{y_n}} \\qquad z_1 = \\frac{1}{\\sqrt{x}} \\qquad z_{n+1} = \\frac{1 + z_n y_n}{(1 + z_n) \\sqrt{y_n}}$ and the following important chain of comparisons.",
"$y_{n+1} \\le z_{n+1} \\le \\sqrt{y_n}$"
],
[
"Computing with $y_n$ and {{formula:d09e8fc7-c619-4362-97b6-b7e84f9d9012}} (the Borwein algorithm).",
"The first algorithm we will present, proposed by J. M. Borwein and P. B.Borwein, consists in approximating $M$ using the sequences $y_n$ and $z_n$ .",
"The value $M(1, x)^3$ is approximated using $a_n b_n^2$ and $(M(1,x))^{\\prime }$ using $a_n^{\\prime }$ , all values being taken in $\\frac{1}{\\sqrt{2}}$ .",
"From the definition, we can easily derive the following properties: ${1 + y_n} = 2 \\frac{a_{n+1} b_{n+1}^2}{a_{n} b_{n}^2}\\qquad {1 + z_n} = 2 \\frac{a_{n+1}^{\\prime }}{a_n^{\\prime }}$ Repeating the products, we get the following definition of a sequence $\\pi _n$ and the proof of its limit: $\\pi _0 = (2 + \\sqrt{2}) \\qquad \\pi _n = \\pi _0\\prod _{i=1}^{n}\\frac{1 + y_i}{1 + z_i}\\qquad \\lim _{n\\rightarrow \\infty } \\pi _n = \\pi $"
],
[
"Convergence speed.",
"For an arbitrary $x$ in the open interval $(0, 1)$ , using a Taylor expansion of the function $y \\mapsto \\frac{1 + y}{2\\sqrt{y}}$ of order two, and then reasoning by induction, we get the following results: $y_{n+1}(x) - 1 \\le \\frac{(y_n(x) - 1) ^ 2}{8}\\qquad y_{n + 1}(x) \\le 8 \\left(\\frac{(y_1(x) - 1)}{8}\\right)^{2 ^ n}$ For $x = \\frac{1}{\\sqrt{2}}$ , we obtain the following bound: $y_{n+1}(\\frac{1}{\\sqrt{2}}) - 1 \\le 8 \\times 531^{-2^n}$ Using the comparisons of line (REF ) and then reasoning by induction we obtain our final error estimate: $0 \\le \\pi _{p+1} - \\pi \\le \\pi _{p+1} \\left(y_{p+1}(\\frac{1}{\\sqrt{2}}) - 1 \\right)\\le 4 \\pi _0 531 ^{-2 ^ p}$"
],
[
"Computing one million decimals.",
"The first element of the sequence $\\pi _n$ that is close to $\\pi $ with an error smaller than $10 ^ {10 ^ 6}$ is obtained for $n$ satisfying the following comparison.",
"$n \\ge \\frac{\\ln \\left(\\frac{10 ^ 6 \\ln 10 - \\ln (4 \\pi _0)}{\\ln 531}\\right)}{\\ln 2} \\sim 18.5$ For one million hexadecimals, $n$ only needs to be larger than $18.75$ ."
],
[
"Computing with an infinite sum (the Brent-Salamin algorithm).",
"The formula described in this section probably appears in Gauss' work and is repeated by King .",
"It was published and clarified for implementation on modern computers by Brent and Salamin .",
"A good account of the historical aspects is given by Almkvist and Berndt .",
"Our presentation relies on a mathematical exposition given by Gourevitch .",
"In the variant proposed by Brent and Salamin, we compute the right-hand side of the main central formula by computing $a_n^2$ and the ratio $\\frac{b_n^{\\prime }}{b_n}$ .",
"We first introduce a third function $c_n$ .",
"$c_n = \\frac{1}{2}(a_{n-1} - b_{n-1})$ The derivative of function $k_n$ can be expressed with $c_n$ and after combination with equation REF , this gives a formula for the derivative of $\\frac{a_n}{b_n}$ at $\\frac{1}{\\sqrt{2}}$ .",
"$\\left(\\frac{a_n}{b_n}\\right)^{\\prime }(\\frac{1}{\\sqrt{2}}) =\\frac{-2^{n+1} \\sqrt{2} \\, a_n c_n^2}{b_n}$ The derivative of this ratio can be compared to the difference of the ratio of $b_n^{\\prime }$ over $b_n$ at two successive indices, which can be repeated $n$ times.",
"$\\frac{b_{n+1}^{\\prime }}{b_{n+1}}- \\frac{b_n^{\\prime }}{b_n}=\\frac{b_n}{2 a_n}\\left(\\frac{a_n}{b_n}\\right)^{\\prime }\\qquad \\frac{b_{n+1}^{\\prime }}{b_{n+1}} =\\frac{b_1^{\\prime }}{b_1} -\\sqrt{2} \\sum _{k = 1}^{n-1} 2 ^ k c_n^2$ We can then use equations (REF ) and (), where $M^3(1,\\frac{1}{\\sqrt{2}})$ is the limit of $a_n^2 b_n$ to obtain the final definition and limit.",
"$\\pi ^{\\prime }_n = \\frac{4\\, a_n^2}{1 - \\sum _{k=1}^{n-1} 2 ^ {k-1}(a_{k - 1} - b_{k - 1}) ^ 2}\\qquad \\pi = \\lim _{n\\rightarrow \\infty } \\pi ^{\\prime }_n$"
],
[
"Speed of convergence.",
"We can link the Brent-Salamin algorithm with the Borwein algorithm in the following manner: $\\pi ^{\\prime }_n = 2\\sqrt{2}\\,\\frac{a_n^2 b_n}{b_n^{\\prime }} =2\\sqrt{2}\\, \\frac{y_n}{z_n} \\frac{a_n b_n^2}{a_n^{\\prime }} =\\frac{y_n}{z_n} \\pi _n$ Combining bounds (REF ), (REF ), and (REF ) we obtain this first approximation.",
"$| \\pi ^{\\prime }_{n+1} - \\pi | \\le 68 \\times 531 ^ {-2^{n-1}}$ This first approximation is too coarse, as it gives the impression that $\\pi ^{\\prime }_{n+1}$ is needed when $\\pi _{n}$ is enough (the exponent of 2 in bound (REF ) is $n-1$ while it is $n$ in bound (REF )).",
"We can make it better by noting that the difference between $\\pi $ and $\\pi _{n+2}$ is $O(531^{-2^n})$ and the difference between $\\pi _{n+2}$ and $\\pi _{n+1}$ is significantly smaller than $531^{-2^{n-1}}$ , while not being $O(531^{-2^n})$ .",
"$|\\pi ^{\\prime }_{n+1} -\\pi | \\le (132 + 384 \\times 2^n) \\times 531 ^{-2^n}$ For one million decimals of $\\pi $ , we can still use $n=19$ .",
"Each algorithm computes $n$ square roots to compute $\\pi _n$ or $\\pi ^{\\prime }_n$ .",
"However, the first one uses $3n$ division to obtain value $\\pi _n$ , while the second one only performs divisions by 2, which are less costly, and a single full division at the end of the computation to compute $\\pi _n^{\\prime }$ .",
"In our experiments computing these algorithms inside Coq, the second one is twice as fast."
],
[
"Formalization issues for arithmetic geometric means",
"In this section, we describe the parts of our development where we had to proceed differently from the mathematical exposition in section REF .",
"Many difficulties arose from gaps in the existing libraries for real analysis."
],
[
"The arithmetic geometric mean functions.",
"For a given $a_0 = a$ and $b_0 = b$ , the functions $a_n$ and $b_n$ actually are functions of $a$ and $b$ that are defined mutually recursively.",
"Instead of a mutual recursion between two functions, we chose to simply describe a function ag that takes three arguments and returns a pair of two arguments.",
"This can be written in the following manner: Fixpoint ag (a b : R) (n : nat) := match n with 0 => (a, b) | S p => ag ((a + b) / 2) (sqrt (a * b)) p end.",
"This functions takes three arguments, two of which are real numbers, and the third one is a natural number.",
"When the natural number is 0, then the result is the pair of the real numbers, thus expressing that $a_0 = a$ and $b_0 = b$ .",
"When the natural number is the successor of some $p$ , then the two real number arguments are modified in accordance to the arithmetic-geometric mean process, and then the $p$ -th argument of the sequence starting with these new values is computed.",
"As an abbreviation we also use the following definitions, for the special case when the first input is 1.",
"Definition a_ (n : nat) (x : R) := fst (ag 1 x n).",
"Definition b_ (n : nat) (x : R) := snd (ag 1 x n).",
"The function ag_step seems to perform the operation in a different order, but in fact we can really show that $a_{n+1} = \\frac{a_n + b_n}{2}$ and $b_{n+1} = \\sqrt{a_n b_n}$ as expected, thanks to a proof by induction on $n$ .",
"This is expressed with theorems of the following form: Lemma a_step n x : a_ (S n) x = (a_ n x + b_ n x) / 2.",
"Lemma b_step n x : b_ (S n) x = sqrt (a_ n x * b_ n x)."
],
[
"Limits and filters.",
"Cartan proposed in 1937 a general notion that made it possible to develop notions of limits in a uniform way, whether they concern limits of continuous function of limits of sequences.",
"This notion, known as filters is provided in formalized mathematics in Isabelle and more recently in the Coquelicot library .",
"It is also present in a simplified form as convergence nets in Hol-Light .",
"Filters are not real numbers, but objects designed to represent ways to approach a limit.",
"There are many kinds of filters, attached to a wide variety of types, but for our purposes we will mostly be interested in seven kinds of filters.",
"eventually represents the limit towards $\\infty $ , but only for natural numbers, locally $x$ represents a limit approaching a real number $x$ from any side, at_point $x$ represents a limit that is actually not a limit but an exact value: you approach $x$ because you are bound to be exactly $x$ , at_right $x$ represents a limit approaching $x$ from the right, that is, only taking values that are greater than $x$ (and not $x$ itself), at_left $x$ represents a limit approaching $x$ from the left, Rbar_locally p_infty describes a limit going to $+\\infty $ , Rbar_locally m_infty describes a limit going to $-\\infty $ .",
"There is a general notion called filterlim $f$ $F_1$ $F_2$ to express that the value returned by $f$ tends to a value described by the filter $F_2$ when its input is described by $F_1$ .",
"For instance, we constructed formal proofs for the following two theorems: Lemma lim_atan_p_infty : filterlim atan (Rbar_locally p_infty) (at_left (PI / 2)).",
"Lemma lim_atan_m_infty : filterlim atan (Rbar_locally m_infty) (at_right (-PI / 2)).",
"In principle, filters make it possible to avoid the usual $\\varepsilon -\\delta $ proofs of topology and analysis, using faster techniques to relate input and output filters for continuous functions .",
"In practice, for precise proofs like the ones above (which use the at_right and at_left filters), we still need to revert to a traditional $\\varepsilon -\\delta $ framework."
],
[
"Improper integrals.",
"The Coq standard library of real numbers has been providing proper integrals for a long time, more precisely Riemann integrals.",
"The Coquelicot library adds an incomplete treatement of improper integrals on top of this.",
"For improper integrals the bounds are described as limits rather than as direct real numbers.",
"For the needs of this experiment, we need to be able to cut improper integrals into pieces, perform variable changes, and compute the improper integral $\\int _{-\\infty }^{+\\infty } \\frac{{\\rm d}t}{1 + t ^ 2} = \\pi $ The Coquelicot library provides two predicates to describe improper integrals, the first one has the formthe name can be decomposed in is R for Riemann, Int for Integral, and gen for generalized.",
"is_Rint_gen $f$ $B_{1}$ $B_{2}$ $v$ The meaning of this predicate is “the improper integral of function $f$ between bounds $B_1$ and $B_2$ converges and has value $v$ ”.",
"The second predicate is named ex_Rint_gen and it simply takes the same first three arguments as is_Rint_gen, to express that there exists a value $v$ such that is_Rint_gen holds.",
"The Coquelicot library does not provide a functional form, but there is a general tool to construct functions from relations where one argument is uniquely determined by the others, called iota in that library.",
"Concerning elliptic integrals, as a first step we need to express the convergence of the improper integral in equation (REF ).",
"For this we need a general theorem of bounded convergence, which is described formally in our development, because it is not provided by the library.",
"Informally, the statement is that the improper integral of a positive function is guaranteed to converge if that function is bounded above by another function that is known to converge.",
"Here is the formal statement of this theorem: Lemma ex_RInt_gen_bound (g : R -> R) (f : R -> R) F G {PF : ProperFilter F} {PG : ProperFilter G} : filter_Rlt F G -> ex_RInt_gen g F G -> filter_prod F G (fun p => (forall x, fst p < x < snd p -> 0 <= f x <= g x) /\\ ex_RInt f (fst p) (snd p)) -> ex_RInt_gen f F G. This statement exhibits a concept that we needed to devise, the concept of comparison between filters on the real line, which we denote filter_Rlt.",
"This concept will be described in further detail in a later section.",
"Three other lines in this theorem statement deserve more explanations, the lines starting at filter_prod.",
"These lines express that a property must ultimately be satisfied for pairs $p$ of real numbers whose components tend simultaneously to the limits described by the filters F and G, which here also serve as bounds for two generalized Riemann integrals.",
"This property is the conjunction of two facts, first for any argument between the pair of numbers, the function f is non-negative and less than or equal to g at that argument, second the function f is Riemann-integrable between the pair of numbers.",
"Using this theorem of bounded convergence, we can prove that the function $ x \\mapsto \\frac{1}{\\sqrt{(x ^ 2 + a ^ 2)(x ^ 2 + b ^ 2)}}$ is integrable between $-\\infty $ and $+\\infty $ as soon as both $a$ and $b$ are positive, using the function $ x \\mapsto \\frac{1}{m^2 (\\left(\\frac{x}{m}\\right) ^ 2 + 1)} $ as the bounding function, where $m = min(a,b)$ , and then proving that this one is integrable by showing that its integral is related to the arctangent function.",
"Having proved the integrability, we then define a function that returns the following integral value: $\\int _{-\\infty }^{+\\infty } \\frac{{\\rm d} x}{\\sqrt{(x ^ 2 + a ^ 2)(x ^ 2 + b ^ 2)}} $ The definition is done in the following two steps: Definition ellf (a b : R) x := /sqrt ((x ^ 2 + a ^ 2) * (x ^ 2 + b ^ 2)).",
"Definition ell (a b : R) := iota (fun v => is_RInt_gen (ellf a b) (Rbar_locally m_infty) (Rbar_locally p_infty) v).",
"The value of ell $a$ $b$ is properly defined when $a$ and $b$ are positive.",
"This is expressed with the following theorems, and will be guaranteed in all other theorems where ell occurs.",
"Lemma is_RInt_gen_ell a b : 0 < a -> 0 < b -> is_RInt_gen (ellf a b) (Rbar_locally m_infty) (Rbar_locally p_infty) (ell a b).",
"Lemma ell_unique a b v : 0 < a -> 0 < b -> is_RInt_gen (ellf a b) (Rbar_locally m_infty) (Rbar_locally p_infty) v -> v = ell a b."
],
[
"An order on filters.",
"On several occasions, we need to express that the bounds of improper integrals follow the natural order on the real line.",
"However, these bounds may refer to no real point.",
"For instance, there is no real number that corresponds to the limit $0+$ , but it is still clear that this limit represents a place on the real line which is smaller than 1 or $+\\infty $ .",
"This kind of comparison is necessary in the statement of ex_RInt_gen_bound, as stated above, because the comparison between functions would be vacuously true when the bounds of the interval are interchanged.",
"We decided to introduce a new concept, written filter_Rlt $F$ $G$ to express that when $x$ tends to $F$ and $y$ tends to $G$ , we know that ultimately $x < y$ .",
"To be more precise about the definition of filter_Rlt, we need to know more about the nature of filters.",
"Filters simply are sets of sets.",
"Every filter contains the complete set of elements of the type being considered, it is stable by intersection, and it is stable by the operation of taking a superset.",
"Moreover, when a filter does not contain the empty set, it is called a proper filter.",
"For instance, the filter Rbar_locally p_infty contains all intervals of the form $(a,+\\infty )$ and their supersets, the filter locally x contains all open balls centered in x and their supersets, and the filter at_right x contains the intersections of all members of locally x with the interval $({\\tt x}, +\\infty )$ .",
"With two filters $F_1$ and $F_2$ on types $T_1$ and $T_2$ , it is possible to construct a product filter on $T_1 \\times T_2$ , which contains all cartesian products of a set in $F_1$ and a set in $F_2$ and their supersets.",
"This corresponds to pairs of points which tend simultaneously towards the limits described by $F_1$ and $F_2$ .",
"To define a comparison between filters on the real line, we state that $F_1$ is less than $F_2$ if there exists a middle point $m$ , so that the product filter $F_1 \\times F_2$ accepts the set of pairs $v_1,v_2$ such that $v_1 < m < v_2$ .",
"In other words, this means that as $v_1$ tends to $F_1$ and $v_2$ to $F_2$ , it ultimately holds that $v_1 < m < v_2$ .",
"In yet other words, if there exists an $m$ such that the filter $F_1$ contains $(-\\infty , m)$ and $F_2$ contains $(m, +\\infty )$ , then $F_1$ is less than $F_2$ .",
"These are expressed by the following definition and the following theorem: Definition filter_Rlt F1 F2 := exists m, filter_prod F1 F2 (fun p => fst p < m < snd p).",
"Lemma filter_Rlt_witness m (F1 F2 : (R -> Prop) -> Prop) : F1 (Rgt m) -> F2 (Rlt m) -> filter_Rlt F1 F2.",
"We proved a few comparisons between filters, for instance at_right $x$ is smaller than Rbar_locally p_infty for any real $x$ , at_left $a$ is smaller than at_right $b$ if $a\\le b$ , but at_right $c$ is only smaller than at_left $d$ when $c < d$ .",
"We can reproduce for improper integrals the results given by the Chasles relations for proper Riemann integrals.",
"Here is an example of a Chasles relation: if $f$ is integrable between $a$ and $c$ and $a \\le b \\le c$ , then $f$ is integrable between $a$ and $b$ and between $b$ $c$ , and the integrals satisfy the following relation: $ \\int _a^c f(x) \\,{\\rm d} x = \\int _a^b f(x) \\,{\\rm d} x + \\int _b^c f(x)\\,{\\rm d} x$ This theorem is provided in the Coquelicot library for $a$ , $b$ , and $c$ taken as real numbers.",
"With the order of filters, we can simply re-formulate this theorem for $a$ and $c$ being arbitrary filters, and $b$ being a real number between them.",
"This is expressed as follows: Lemma ex_RInt_gen_cut (a : R) (F G: (R -> Prop) -> Prop) {FF : ProperFilter F} {FG : ProperFilter G} (f : R -> R) : filter_Rlt F (at_point a) -> filter_Rlt (at_point a) G -> ex_RInt_gen f F G -> ex_RInt_gen f (at_point a) G. We are still considering whether this theorem should be improved, using the filter locally $a$ instead of at_point $a$ for the intermediate integration bound.",
"The theorem ex_RInt_gen_cut is used three times, once to establish equation (REF ) and twice to establish equation (REF ) at page REF ."
],
[
"From improper to proper integrals.",
"Through variable changes, improper integrals can be transformed into proper integrals and vice-versa.",
"For instance, the change of variable leading to equation (REF ) actually leads to the correspondence.",
"$ \\int _0^{+\\infty } \\frac{{\\rm d}t}{\\sqrt{(a ^ 2 + t ^ 2)(b ^ 2 + t ^2)}} = \\frac{1}{2}\\int _{-\\infty }^{+\\infty } \\frac{{\\rm d} s}{\\sqrt{{((\\frac{a +b}{2}) ^ 2 + s ^ 2)(ab + s ^ 2)}}} $ The lower bounds of the two integrals correspond to each other with respect to the variable change $s = \\frac{1}{2}(t - \\frac{ab}{t})$ , but the first lower bound needs to be considered proper for later uses, while the lower bound for the second integral is necessarily improper.",
"To make it possible to change from one to the other, we establish a theorem that makes it possible to transform a limit bound into a real one.",
"Lemma is_RInt_gen_at_right_at_point (f : R -> R) (a : R) F {FF : ProperFilter F v} : locally a (continuous f) -> is_RInt_gen f (at_right a) F v -> is_RInt_gen f (at_point a) F v. This theorem contains an hypothesis stating that $f$ should be well behaved around the real point being considered, the lower bound.",
"In this case, we use an hypothesis of continuity around this point, but this hypothesis could probably be made weaker."
],
[
"Limit equivalence.",
"Equations (REF .1) and (.2) at page REF rely on the concept of equivalent functions at a limit.",
"For our development, we have not developed a separate concept for this, instead we expressed statements as the ratio between the equivalent functions having limit 1 when the input tends to the limit of interest.",
"For instance equation (.1) is expressed formally using the following lemma: Lemma M1x_at_0 : filterlim (fun x => M 1 x / (- PI / (2 * ln x))) (at_right 0) (locally 1).",
"In this theorem, the fact that $x$ tends to 0 on the right is expressed by using the filter (at_right 0).",
"We did not develop a general library of equivalence, but we still gave ourself a tool following the transitivity of this equivalence relation.",
"This theorem is expressed in the following manner: Lemma equiv_trans F {FF : Filter F} (f g h : R -> R) : F (fun x => g x <> 0) -> F (fun x => h x <> 0) -> filterlim (fun x => f x / g x) F (locally 1) -> filterlim (fun x => g x / h x) F (locally 1) -> filterlim (fun x => f x / h x) F (locally 1).",
"The hypotheses like F (fun x => g x <> 0) express that in the vicinity of the limit denoted by F, the function should be non-zero.",
"The rest of the theorem expresses that if $f$ is equivalent to $g$ and $g$ is equivalent to $h$ , then $f$ is equivalent to $h$ .",
"To perform this proof, we need to leave the realm of filters and fall back on the traditional $\\varepsilon -\\delta $ framework."
],
[
"Uniform convergence and derivatives.",
"During our experiments, we found that the concept of uniform convergence does not fit well in the framework of filters as provided by the Coquelicot library.",
"The sensible approach would be to consider a notion of balls on the space of functions, where a function $g$ is inside the ball centered in $f$ if the value of $g(x)$ is never further from the value of $f(x)$ than the ball radius, for every $x$ in the input type.",
"One would then need to instantiate the general structures of topology provided by Coquelicot to this notion of ball, in particular the structures of UniformSpace and NormedModule.",
"In practice, this does not provide all the tools we need, because we actually want to restrict the concept of uniform convergence to subsets of the whole type.",
"In this case the structure of UniformSpace is still appropriate, but the concept of NormedModule is not, because two functions that differ outside the considered subset may have distance 0 when only considering their values inside the subset.",
"The alternative is provided by a treatment of uniform convergence that was developed in Coq's standard library of real numbers at the end of the 1990's, with a notion denoted CVU $f$ $g$ $c$ $r$, where $f$ is a sequence of functions from $\\mathbb {R}$ to $\\mathbb {R}$ , $g$ is a function from $\\mathbb {R}$ to $\\mathbb {R}$ , $c$ is a number in $\\mathbb {R}$ and $r$ is a positive real number.",
"The meaning is that the sequence of function $f$ converges uniformly towards $g$ inside the ball centered in $c$ of radius $r$ .",
"We needed a formal description of a theorem stating that when the derivatives $f^{\\prime }_n$ of a convergent sequence of functions $f_n$ tend uniformly to a limit function $g^{\\prime }$ , this function $g^{\\prime }$ is the derivative of the limit of the sequence $f_n$ .",
"There is already a similar theorem in Coq's standard library, with the following statement: derivable_pt_lim_CVU : forall fn fn' f g x c r, Boule c r x -> (forall y n, Boule c r y -> derivable_pt_lim (fn n) y (fn' n y)) -> (forall y, Boule c r y -> Un_cv (fun n : nat => fn n y) (f y)) -> CVU fn' g c r -> (forall y : R, Boule c r y -> continuity_pt g y) -> derivable_pt_lim f x (g x) However, this theorem is sometimes impractical to use, because it requires that we already know the limit derivative to be continuous, a condition that can actually be removed.",
"For this reason, we developed a new formal proof for the theorem, with the following statementIt turns out that the theorem derivable_pt_lim_CVU was already introduced by a previous study on the implementation of $\\pi $ in the Coq standard library of real numbers .",
"Lemma CVU_derivable : forall f f' g g' c r, CVU f' g' c r -> (forall x, Boule c r x -> Un_cv (fun n => f n x) (g x)) -> (forall n x, Boule c r x -> derivable_pt_lim (f n) x (f' n x)) -> forall x, Boule c r x -> derivable_pt_lim g x (g' x).",
"In this theorem's statement, the third line expresses that the derivatives f' converge uniformly towards the function g', the fourth line expresses that the functions f converge simply towards the function g inside the ball of center c and radius r, the fifth and sixth line express that the functions f are differentiable everywhere inside the ball and the derivative is f', and the seventh line concludes that the function g is differentiable everywhere inside the ball and the derivative is g'.",
"While most of the theorems we wrote are expressed using concepts from the Coquelicot library, this one is only expressed with concepts coming from Coq's standard library of real numbers, but all these concepts, apart from CVU, have a Coquelicot equivalent (and Coquelicot provides the foreign function interface): Boule c r x is equivalent to Ball c r x in Coquelicot, Un_cv $f$ $l$ is equivalent to filterlim $f$ Eventually (locally $l$ ), and derivable_pt_lim is equivalent to is_derive.",
"We used the theorem CVU_derivable twice in our development, once to establish that function $x \\mapsto M(1, x)$ is differentiable everywhere in the open interval $(0,1)$ and the sequence of derivatives of the $a_n$ functions converges to its derivative, and once to establish that the derivatives of the $k_n$ functions converge to $M^2(1, x) / (x (1 - x ^ 2))$ , as in equation (REF )."
],
[
"Automatic proofs.",
"In this development, we make an extensive use of divisions and square root.",
"To reason about these functions, it is often necessary to show that the argument is non-zero (for division), or positive (for square root).",
"There are very few automatic tools to establish this kind of results in general about real numbers, especially in our case, where we rely on a few transcendental functions.",
"For linear arithmetic formulas, there exists a tool call psatzl R , that is very useful and robust in handling of conjunctions and its use of facts from the current context.",
"Unfortunately, we have many expressions that are not linear.",
"We decided to implement a semi-automatic tactic for the specific purpose of proving that numbers are positive, with the following ordered heuristics: Any positive number is non-zero, all exponentials are positive, $\\pi $ , 1, and 2 are positive, the power, inverse, square root of positive numbers is positive, the product of positive numbers is positive, the sum of an absolute value or a square and a positive number is positive, the sum of two positive numbers are positive, the minimum of two positive numbers is positive, a number recognized by the psatzl R tactic to be positive is positive.",
"This semi-automatic tactic can easily be implemented using Coq's tactic programming language Ltac.",
"We named this tactic lt0 and it is used extensively in our development.",
"Given a function like $x \\mapsto 1 / \\sqrt{(x ^ 2 + a ^ 2)(x ^ 2 + b ^ 2)}$ , the Coquelicot library provides automatic tools (mainly a tactic called auto_derive) to show that this function is differentiable under conditions that are explicitly computed.",
"For this to work, the tool needs to rely on a database of facts concerning all functions involved.",
"In this case, the database must of course contain facts about exponentiation, square roots, and the inverse function.",
"As a result, the tactic auto_derive produces conditions, expressing that $(x ^ 2 + a ^ 2)(x ^ 2 + b ^ 2)$ must be positive and the whole square root expression must be non zero.",
"The tactic auto_derive is used more than 40 times in our development, mostly because there is no automatic tool to show the continuity of functions and we rely on a theorem that states that any differentiable function is continuous, so that we often prove derivability only to prove continuity.",
"When proving that the functions $a_n$ and $b_n$ are differentiable, we need to rely on a more elementary proof tool, called auto_derive_fun.",
"When given a function to derive, which contains functions that are not known in the database, it builds an extra hypothesis, which says that the whole expression is differentiable as soon as the unknown functions are differentiable.",
"This is especially useful in this case, because the proof that $b_n$ is differentiable is done recursively, so that there is no pre-existing theorem stating that $a_n$ and $b_n$ are differentiable when studying the derivative of $b_{n+1}$ .",
"For instance, we can call the following tactic: auto_derive_fun (fun y => sqrt (a_ n y * b_ n y)); intros D. This creates a new hypothesis named D with the following statement: D : forall x : R, ex_derive (fun x0 : R => a_ n x0) x /\\ ex_derive (fun x0 : R => b_ n x0) x /\\ 0 < a_ n x * b_ n x /\\ True -> is_derive (fun x0 : R => sqrt (a_ n x0 * b_ n x0)) x ((1 * Derive (fun x0 : R => a_ n x0) x * b_ n x + a_ n x * (1 * Derive (fun x0 : R => b_ n x0) x)) * / (2 * sqrt (a_ n x * b_ n x))) Another place where automation provides valuable help is when we wish to find good approximations or bounds for values.",
"The interval tactic works on goals consisting of such comparisons and solves them right away, as long as it knows about all the functions involved.",
"Here is an example of a comparison that is easily solved by this tactic: (1 + ((1 + sqrt 2)/(2 * sqrt (sqrt 2)))) / (1 + / sqrt (/ sqrt 2)) < 1 An example of expression where interval fails, is when the expressions being considered are far too large.",
"In our case, we wish to prove that $ 4 \\pi _0 \\frac{1}{531 ^ {2 ^ {19}}} \\le \\frac{1}{10 ^{10 ^ 6 + 4}}$ The numbers being considered are too close to 0 for interval to work.",
"The solution to this problem is to first use monotonicity properties of either the logarithm function (in the current version of our development) or the exponential function (in the first version), thus resorting to symbolic computation before finishing off with the interval tactic.",
"The interval tactic already knows about the $\\pi $ constant, so that it is quite artificial to combine our result from formula (REF ) and this tactic to obtain approximations of $\\pi $ but we can still make this experiment and establish that the member $\\pi _3$ of the sequence is a good enough approximation to know all first 10 digits of $\\pi $ .",
"Here is the statement: Lemma first_computation : 3141592653/10 ^ 9 < agmpi 3 /\\ agmpi 3 + 4 * agmpi 0 * Rpower 531 (- 2 ^ 2) < 3141592654/10 ^ 9.",
"We simply expand fully agmpi, simplify instances of $y_n$ and $z_n$ using the equations (REF ), and then ask the interval tactic to finish the comparisons.",
"We need to instruct the tactic to use 40 bits of precision.",
"This takes some time (about a second for each of the two comparisons), and we conjecture that the expansion of all functions leads to sub-expression duplication, leading also to duplication of work.",
"When aiming for more distant decimals, we will need to apply another solution."
],
[
"Computing large numbers of decimals",
"Theorem provers based on type theory have the advantage that they provide computation capabilities on inductive types.",
"For instance, the Coq system provides a type of integers that supports comfortable computations for integers with size going up to $10 ^{100}$ .",
"Here is an example computation, which feels instantaneous to the user.",
"Compute (2 ^331)= 174980057982640953949800178169409709228253554471456994914 06164851279623993595007385788105416184430592 : Z By their very nature, real numbers cannot be provided as an inductive datatype in type theory.",
"Thus the Compute command will not perform any computation for the similar expression concerning real numbers.",
"The reason is that while some real numbers are defined like integers by applying simple finite operations on basic constants like 0 and 1, other are only obtained by applying a limiting process, which cannot be represented by a finite computation.",
"Thus, it does not make sense to ask to compute an expression like $\\sqrt{2}$ in the real numbers, because there is no way to provide a better representation of this number than its definition.",
"On the other hand, what we usually mean by computing $\\sqrt{2}$ is to provide a suitable approximation of this number.",
"This is supported in the Coq system by the interval tactic, but only when we are in the process of constructing a proof, as in the following example: Lemma anything : 12 / 10 < sqrt 2.",
"Proof.",
"interval_intro (sqrt 2).",
"1 subgoal H : 759250124 * / 536870912 <= sqrt 2 <= 759250125 * / 536870912 ============================ 12 / 10 < sqrt 2 What we see in this dialog is that the system creates a new hypothesis (named H) that provides a new fact giving an approximation of $\\sqrt{2}$ .",
"In this hypothesis, the common numerator appearing in both fractions is actually the number $2 ^{29}$ .",
"Concerning notations, readers will have to know that Coq writes / a for the inverse of a, so that 3 * / 2 is 3 times the inverse of 2.",
"A human mathematician would normally write 3 / 2 and Coq also accepts this syntax.",
"One may argue that 759250124 * / 536870912 is not much better than sqrt 2 to represent that number, and actually this ratio is not exact, but it can be used to help proving that $\\sqrt{2}$ is larger or smaller than another number.",
"Direct computation on the integer datatype can also be used to approximate computations in real numbers.",
"For instance, we can compute the same numerator for an approximation of $\\sqrt{2}$ by computing the integer square root of $2 \\times (2 ^ {29}) ^2$ .",
"Compute (Z.sqrt (2 * (2 ^ 29) ^ 2)).",
"= 759250124: Z This approach of computing integer values for numerators of rational numbers with a fixed denominator is the one we are going to exploit to compute the first million digits of $\\pi $ , using three advantages provided by the Coq system: The Coq system provides an implementation of big integers, which can withstand computations of the order of $10 ^{10 ^{12}}$ .",
"The big integers library already contains an efficient implementation of integer square roots.",
"The Coq system provides a computation methodology where code is compiled into OCaml and then into binary format for fast computation."
],
[
"A framework for high-precision computation",
"If we choose to represent every computation on real numbers by a computation on corresponding approximations of these numbers, we need to express how each operation will be performed and interpreted.",
"We simply provide five values and functions that implement the elementary values of $\\mathbb {R}$ and the elementary operations: multiplication, addition, division, the number 1, and the number 2.",
"We choose to represent the real number $x$ by the integer $\\lfloor m x\\rfloor $ where $m$ is a scaling factor that is mostly fixed for the whole computation.",
"For readability, it is often practical to use a power of 10 as a scaling factor, but in this paper, we will also see that we can benefit from also using scaling factors that are powers of 2 or powers of 16.",
"Actually, it is not even necessary that the scaling factor be any power of a small number, but it turns out that it is the most practical case.",
"Conversely, we shall note $[\\![n]\\!",
"]$ the real value represented by the integer $n$ .",
"Simply, this number is $\\frac{n}{m}$ .",
"When $m$ is the scaling factor, the real number 1 is represented by the integer $m$ and the real number 2 is represented by the number $2\\times m$ .",
"So $[\\![m]\\!]",
"= 1$ , $[\\![2m]\\!]",
"= 2$ .",
"So, we define the following two functions to describe the representations of 1 and 2 with respect to a given scaling factor, in Coq syntax where we use the name magnifier for the scaling factor.",
"Definition h1 (magnifier : bigZ) := magnifier.",
"Definition h2 magnifier := (2 * magnifier)\\end{verbatim} When multiplying two real numbers \\(x\\) and \\(y\\), we need to multiply their representations and take care of the scaling.",
"To understand how to handle the scaling, we should look at the following equality: \\[\\denote{n_1} \\denote{n_2} = \\frac{n_1}{m}\\frac{n_2}{m}\\] To obtain the integer that will represent this result, we need to multiply the product of the represented numbers by \\(m\\) and then take the largest integer below.",
"This is \\[\\lfloor \\frac{n_1 \\times n_2}{m}\\rfloor\\] The combination of the division operation and taking the largest integer below is performed by integer division.",
"So we define our high-precision multiplication as follow.",
"\\begin{verbatim} Definition hmult (magnifier x y : bigZ) := (x * y / magnifier)\\end{verbatim} For division and square root, we reason similarly.",
"For addition, nothing needs to be implemented, we can directly use integer computation.",
"The scaling factor is transmitted naturally (and linearly from the operands to the result).",
"Similarly, multiplication by an integer can be represented directly with integer multiplication, without having to first scale the integer.",
"Here are a few examples.",
"To compute \\(\\frac{1}{3}\\) to a precision of \\(10 ^{-5}\\), we can run the following computation.",
"\\begin{verbatim} Compute let magnifier := (10 ^ 5)hdiv magnifier magnifier (3 * magnifier).",
"= 33333: BigZ.t_ The following illustrates how to compute $\\sqrt{2}$ to the same precision.",
"Compute let magnifier := (10 ^ 5)hsqrt magnifier (2 * magnifier).",
"= 141421: BigZ.t_ In both examples, the real number of interest has the order of magnitude of 1 and is represented by a 5 or 6 digit integer.",
"When we want to compute one million decimals of $\\pi $ we should handle integers whose decimal representation has approximately one million digits.",
"Computation with this kind of numbers takes time.",
"As an example, we propose a computation that handles the 1 million digit representation of $\\sqrt{2}$ and avoids displaying this number (it only checks that the millionth decimal is odd).",
"Time Eval native_compute in BigZ.odd (BigZ.sqrt (2 * 10 ^ (2 * 10 ^ 6))).",
"= true : bool Finished transaction in 91.278 secs (90.218u,0.617s) (successful) This example also illustrates the use of a different evaluation strategy in the Coq system, called native_compute.",
"This evaluation strategy relies on compiling the executed code in OCaml and then on relying on the most efficient variant of the OCaml compiler to produce a code that is executed and whose results are integrated in the memory of the Coq system .",
"This strategy relies on the OCaml compiler and the operating system linker in ways that are more demanding than traditional uses of Coq.",
"Still it is the same compiler that is being used as the one used to compile the Coq system, so that the trusted base is not changed drastically in this new approach.",
"When it comes to time constraints, all scaling factors are not as efficient.",
"In conventional computer arithmetics, it is well-known that multiplications by powers of 2 are less costly, because they can simply be implemented by shifts on the binary representation of numbers.",
"This property is also true for Coq's implementation of big integers.",
"If we compare the computation of $\\sqrt{\\sqrt{2}}$ when the scaling factor is $10 ^{10 ^ 6}$ or $2 ^ {3321929}$ , we get a performance ratio of 1.5, the latter setting is faster even though the scaling factor and the intermediate values are slightly larger.",
"It is also interesting to understand how to stage computations, so that we avoid performing the same computation twice.",
"For this problem, we have to be careful, because values that are precomputed don't have the same size as their original description, and this may not be supported by the native_compute chain of evaluation.",
"Indeed, the following experiment fails.",
"Require Import BigZ.",
"Definition mag := Eval native_compute in (10 ^ (10 ^ 6)) Time Definition z1 := Eval native_compute in let v := mag in (BigZ.sqrt (v * BigZ.sqrt (v * v * 2)))\\end{verbatim} This examples makes Coq fail, because the definition of {\\tt mag} with the pragma {\\tt Eval native\\_compute in} makes that the value \\(10 ^ {10 ^ 6}\\) is precomputed, thus creating a huge object of the Gallina language, which is then passed as a program for the OCaml compiler to compile when constructing {\\tt z1}.",
"The compiler fails because the input program is too large.",
"On the other hand, the following computation succeeds: \\begin{verbatim} Eval native_compute in let v := (10 ^ (10 ^ 6))(BigZ.sqrt (v * BigZ.sqrt (v * v * 2)))."
],
[
"The full approximating algorithm",
"Using all elementary operations described in the previous section, we can describe the recursive algorithm to compute approximations of $\\pi _n$ in the following manner.",
"Fixpoint hpi_rec (magnifier : bigZ) (n : nat) (s2 y z prod : bigZ) {struct n} : bigZ := match n with | 0hmult magnifier (h2 magnifier + s2) prod | S p => let sy := hsqrt magnifier y in let ny := hdiv magnifier (h1 magnifier + y) (2 * sy) in let nz := hdiv magnifier (h1 magnifier + hmult magnifier z y) (hmult magnifier (h1 magnifier + z) sy) in hpi_rec magnifier p s2 ny nz (hmult magnifier prod (hdiv magnifier (h1 magnifier + ny) (h1 magnifier + nz))) end.",
"This function takes as input the scaling factor magnifier, a number of iteration n, the integer s2 representing $\\sqrt{2}$ , the integer y representing $y_p$ for some natural number $p$ larger than 0, the integer z representing $z_p$ , and the integer prod representing the value $\\prod _{i=1}^{p} \\frac{1 + y_i(\\frac{1}{\\sqrt{2}})}{1 +z_i(\\frac{1}{\\sqrt{2}})}$ It computes an integer approximating $\\pi _{n+p} \\times {\\tt magnifier}$ , but not exactly this number.",
"The number s2 is passed as an argument to make sure it is not computed twice, because it is already needed to compute the initial values of y, z, and prod.",
"This recursive function is wrapped in the following functions.",
"Definition hs2 (magnifier : bigZ) := hsqrt magnifier (h2 magnifier).",
"Definition hsyz (magnifier : bigZ) := let hs2 := hs2 magnifier in let hss2 := hsqrt magnifier hs2 in (hs2, (hdiv magnifier (h1 magnifier + hs2) (2 * hss2)), hss2).",
"Definition hpi (magnifier : bigZ) (n : nat) := match n with | 0(h2 magnifier + (hs2 magnifier))| S p => let '(s2, y1 , z1) := hsyz magnifier in hpi_rec magnifier p s2 y1 z1 (hdiv magnifier (h1 magnifier + y1) (h1 magnifier + z1)) end.",
"We can use this function hpi to compute approximations of $\\pi $ at a variety of precisions.",
"Here is a collection of trials performed on a powerful machine.",
"Table: NO_CAPTIONThis table illustrates the advantage there is to compute with a scaling factor that is a power of 2.",
"Each column where the scaling factor is a power of 2 gives an approximation that is slightly more precise than the column to its left, at a fraction of the cost in time.",
"Even if our objective is to obtain decimals of $\\pi $ , it should be efficient to first perform the computations of all the iterations with a magnifier that is a power of 2, only to change the scaling factor at the end of the computation, this is the solution we choose eventually.",
"There remains a question about how much precision is lost when so many computations are performed with elementary operations that each provide only approximations of the mathematical operation.",
"Experimental evidence shows that when computing 17 iterations with a magnifier of $10 ^ {10 ^ 5}$ the last two digits are wrong.",
"The next section shows how we prove bounds on the accumulated error in the concrete computation."
],
[
"Proofs about approximate computations",
"When proving facts about approximate computations, we want to abstract away from the fact that the computations are performed with a datatype that provides fast computation with big integers.",
"What really matters is that we approximate each operation on real numbers by another operation on real numbers and we have a clear description of how the approximation works.",
"In the next section, we describe the abstract setting and the proofs performed in this setting.",
"In a later section, we show how this abstract setting is related to the concrete setting of computing with integers and with the particular datatype of big integers."
],
[
"Abstract reasoning on approximate computations",
"In the case of fixed precision computation as we described in the previous section, we know that all operations are approximated from below by a value which is no further than a fixed allowance $e$ .",
"This does not guarantee that all values are approximated from below, because one of the approximated operations is division, and dividing by an approximation from below may yield an approximation from above.",
"For this reason, most of our formal proofs about approximations are performed in a section where we assume the existence of a collection of functions and their properties.",
"The header of our working section has the following content.",
"Variables (e : R) (r_div : R -> R -> R) (r_sqrt : R -> R) (r_mult : R -> R -> R).",
"Hypothesis ce : 0 < e < /1000.",
"Hypothesis r_mult_spec : forall x y, 0 <= x -> 0 <= y -> x * y - e < r_mult x y <= x * y.",
"In this header, we introduce a constant e, which is used to bound the error made in each elementary operation, we assume that e is positive and suitably small, and then we describe how each rounded operation behaves with respect to the mathematical operation it is supposed to represent.",
"For multiplication, the hypothesis named r_mult_spec describes that the inputs are expected to be positive numbers, and that the result of r_mul x y is smaller than or equal to the product, but the difference is smaller than e in absolute value.",
"We have similar specification hypotheses for the rounded division r_div and the rounded square root r_sqrt.",
"We then use these rounded operations to describe the computations performed in the algorithm.",
"We can now study how the computation of the various sequences of the algorithm are rounded, and how errors accumulate.",
"Considering the sequence $y_n$ , the computation at each step is represented by the following expression.",
"r_div (1 + y) (2 * (r_sqrt y)) In this expression, we have to assume that y comes from a previous computation, and for this reason it is tainted with some error h. The question we wish to address has the following form: if we know that $y_n$ is tainted with an error $h$ that is smaller that a given allowance $e^{\\prime }$ , can we show that $y_{n+1}$ is tainted with an error that is smaller than $f(e^{\\prime })$ for some well-behaved function $f$ ?",
"How much bigger than $e$ must $e^{\\prime }$ be?",
"We were able to answer two questions: if the accumulated error on computing $y_n$ is smaller than $e^{\\prime }$ , then the accumulated error on computing $y_{n+1}$ is also smaller than $e^{\\prime }$ (so for the sequence $y_n$ , the function $f$ is the identity function), the allowance $e^{\\prime }$ needs to be at least $2 e$ (and not more).",
"This is quite surprising.",
"Errors don't really accumulate for this sequence.",
"In retrospect, there are good reasons for this.",
"Rounding errors in the division operation make the result go down, but rounding errors in the square root make the result go up.",
"On the other hand, the input value $y_n$ may be tainted by an error $h$ , but this error is only multiplied by the derivative of the function $y \\mapsto \\frac{1 + y}{2 \\sqrt{y}}$ It happens that this derivative never exceeds $\\frac{1}{14}$ in the region of interest.",
"As an illustration, let's assume $y_n = 1.100$ , we want to compute $y_{n+1}$ , and we are working with three digits of precision.",
"The value of $\\sqrt{1.1}$ is $1.04880\\dots $ but it is rounded down to $1.048$ .",
"$2\\sqrt{1.1}$ is $1.09761\\dots $ but the rounded computation give $2.096$ , $y_{n+1}$ is $1.00113$ .",
"In our computation, we actually compute $(1 + 1.1)/2.096)=1.00190$ .",
"This is an over approximation of $y_{n+1}$ , but this is rounded down to $1.001$ : the last rounding down compensates the over-approximation introduced when dividing by the previously rounded down square root.",
"If our input representation of $y_n$ is an approximation, for example we compute with $1.098$ or $1.102$ , we still obtain $1.001$ .",
"In the end, the lemma we are able to prove has the following statement.",
"Lemma y_error e' y h : e' < /10 -> e <= e' / 2 -> 1 <= y <= 71/50 -> Rabs h < e' -> let y1 := (1 + y)/(2 * sqrt y) in y1 - e' < r_div (1 + (y + h)) (2 * (r_sqrt (y + h))) < y1 + e'.",
"The proof is organized in four parts, where the first part consists in replacing the operations with rounding by expressions where an explicit error ratio is displayed.",
"We basically construct a value e1, taken in the interval $[-\\frac{1}{2},0]$ , so that the following equality holds.",
"r_sqrt (y + h) = sqrt (y + h) + e1 * e' We prefer to define e1 as a ratio between constant bounds, rather than a value in an interval whose bounds are expressed in e', because the automatic tactic interval handles values between numeric constants better.",
"We do the same for the division, introducing a ratio e2.",
"The second part of the proof consists in showing that the propagated error from previous computations has limited impact on the final error.",
"This is stated as follows.",
"set (y2 := (1 + (y + h)) / (2 * sqrt (y + h))).",
"assert (propagated_error : Rabs (y2 - y1) < e' / 14).",
"This step is proved by applying the mean value theorem, using the derivative of the function $y \\mapsto \\frac{1 + y}{2\\sqrt{y}}$ , which was already computed during the proof of convergence of the $y_n$ sequence.",
"The interval tactic is practical here to show the absolute value of the derivative of that function at any point between y and y + h is below $\\frac{1}{14}$ .",
"The mean value theorem makes it possible to factor out the input error in the comparisons, so that we eventually obtain a comparison of an expression with a constant, which we resolve using the interval tactic.",
"The other two parts of the proof are concerned with providing a bound for the impact of the rounding errors introduced by the current computation.",
"Each part is concerned with one direction, and in each case only one of the two possible rounding errors need to be considered.",
"The proof for the lemma y_error is quite long (just under 100 lines), but this is only a preliminary step for the proof of lemma z_error, which shows that the errors accumulated when computing the $z_n$ sequence can also be bounded in a constant fashion.",
"The statement of this lemma has the following shape.",
"Lemma z_error e' y z h h' : e' < /50 -> e <= e' / 4 -> 1 < y < 51/50 -> 1 < z < 6/5 -> Rabs h < e' -> Rabs h' < e' -> let v := (1 + z * y)/((1 + z) * sqrt y) in v - e' < r_div (1 + r_mult (z + h') (y + h)) (r_mult (1 + (z + h')) (r_sqrt (y + h))) < v + e'.",
"In this statement, the fragment r_div (1 + r_mult (z + h') (y + h)) (r_mult (1 + (z + h') (r_sqrt (y + h))) represents the computed expression with rounding operations, using inputs that are tainted by errors h and h', while the fragment (1 + z * y) /((1 + z) * sqrt y) represents the ratio $\\frac{1 + zy}{(1 + z)\\sqrt{y}}$ .",
"This proof is more complex.",
"In this case, we are also able to show that errors do not grow as we compute more elements of the sequence: they stay stable at about 4 times the elementary rounding error introduced by each rounding operation.",
"The proof of this lemma is around 170 lines long.",
"The next step in the computation is to compute the product of ratios $\\prod \\frac{1 + y}{1 + z}$ .",
"For each ratio, we establish a bound on the error as expressed by the following lemma.",
"Lemma quotient_error : forall e' y z h h', e' < / 40 -> Rabs h < e' / 2 -> Rabs h' < e' -> e <= e' / 4 -> 1 < y < 51 / 50 -> 1 < z < 6 / 5 -> Rabs (r_div (1 + (y + h)) (1 + (z + h')) - (1 + y)/(1 + z)) < 13 / 10 * e'.",
"The difference between the second hypothesis (on Rabs h) and the third hypothesis Rabs h' handles the fact that we don't have as precise a bound on error for the computation of $y_n$ and for $z_n$ .",
"The result is that the error on the ratio is bounded at a value just above 5 times the elementary error e. It remains to prove a bound on the error introduced when computing the iterated product.",
"This is done by induction on the number of iterations.",
"The following lemma is used as the induction step: when p represents the product of $k$ terms and v represents one of the ratios, the product of p and v with accumulated errors, adding the error for the rounded multiplication increases by $\\frac{23}{20}$ the error on the ratio.",
"Lemma product_error_step : forall p v e1 e2 h h', 0 <= e1 <= /100 -> 0 <= e2 <= /100 -> e < /5 * e2 -> /2 < p < 921/1000 -> /2 < v <= 1 -> Rabs h < e1 -> Rabs h' < e2 -> Rabs (r_mult (p + h) (v + h') - p * v) < e1 + 23/20 * e2.",
"At this point we write functions rpi_rec and rpi so that they mirror exactly the functions hpi_rec and hpi.",
"The main difference is that rpi_rec manipulates real numbers while hpi_rec manipulates integers.",
"Aside from this, rpi_rec performs a multiplication using r_mult wherever hpi_rec performs a multiplication using hmult.",
"We can now combine all results about the sub-expressions, scale all errors with respect to the elementary error, and obtain a bound on accumulated errors in rpi_rec, as expressed in the following lemma.",
"Lemma rpi_rec_correct (p n : nat) y z prod : (1 <= p)Rabs (y - y_ p (/sqrt 2)) < 2 * e -> Rabs (z - z_ p (/sqrt 2)) < 4 * e -> Rabs (prod - pr p) < 4 * (3/2) * p * e -> Rabs (rpi_rec n y z prod - agmpi (p + n)) < (2 + sqrt 2) * 4 * (3/2) * (p + n) * e + 2 * e. Note that this statement guarantees a bound on errors only if the magnitude of the error e is small enough when compared with the inverse of the number of iterations p + n. In practice, this is not a constraint because we tend to make the error magnitude vanish twice exponentially.",
"In the end, we have to check the approximations for the initial values given as argument to rpi_rec.",
"This yields a satisfying rounding error lemma.",
"Lemma rpi_correct : forall n, (1 <= n)Rabs (rpi n - agmpi n) < (21 * n + 2) * e. In other words, we can guarantee that $\\pi _n$ is computed with an error that grows proportionally to $21 n + 2$ .",
"A similar study for the Brent-Salamin algorithm yields the following error estimate: Lemma rsalamin_correct (n : nat) : 0 <= e <= / 10 ^ (n + 6) / 3 ^ (n + 1) -> Rabs (rsalamin n - salamin_formula (n + 1)) <= (160 * (3 / 2) ^ (n + 1) + 80 * 3 ^ (n + 1) + 100) * e. This error grows exponentially with respect to $n$ , which means that the number of needed extra digits to ensure a given distant decimal is still linear in $n$ .",
"When computing the number of required extra digits for 1 million, we obtain 12 (because $n$ is 19)."
],
[
"From abstract rounding to integer computations",
"In our concrete setting, we don't have the functions r_mult, r_div, and r_sqrt, but functions hmult, hdiv and hsqrt.",
"The type on which these functions operate is bigZ, a type that is designed to make large computations possible inside the Coq system, but that is otherwise not suited to perform intensive proofs.",
"To establish the connection with our proofs of rounded operations, we build a bridge that relies on the better supported type Z.",
"The standard library of reals already provides function INR and IZR to inject natural numbers and integers, respectively, into the type of real numbers.",
"These functions are useful to us, but they must be improved to include the scaling process.",
"We also define functions hR : Z -> R and Rh : R -> Z mapping an integer (respectively a real number) to its representation (respectively to the integer that represents its rounding by default).",
"All these functions are defined in the context of a Coq section where we assume the existence of a scaling factor named magnifier (an integer), and that this scaling factor is larger than 1000, which corresponds to assuming that we perform computations with at least 3 digits of precision.",
"Coming from the type of integers, we can now redefine the functions hmult, hdiv, and hsqrt as in section REF , but with the type Z for inputs and outputs.",
"Definition hR (v : Z) : R := (IZR v /IZR magnifier) Definition RbZ (v : R) : Z := floor v. Definition Rh (v : R) : Z := RbZ( v * IZR magnifier).",
"The abstract functions r_mult, r_div and r_sqrt are then defined by rounding and injecting the result back into the type of real numbers.",
"Definition r_mult (x y : R) : R := hR (Rh (x * y)).",
"The main rounding property can be proved once for all three rounded operations, since it is solely a property of the hR and Rh function.",
"Lemma hR_Rh (v : R) : v - /IZR magnifier < hR (Rh v) <= v. The link to the concrete computing functions is established by the following kind of lemma, the form of which is close to a morphism lemma.",
"Lemma hmult_spec : forall x y : Z, (0 <= x -> 0 <= y -> hR (hmult x y) = r_mult (hR x) (hR y))\\end{verbatim} The hypotheses {\\tt r\\_mult\\_spec}, {\\tt r\\_div\\_spec}, and {\\tt r\\_sqrt\\_spec}, which are necessary for the abstract reasoning in section~\\ref{sec:abstract-approximate}, are then easily obtained by composing a lemma of the form {\\tt hmult\\_spec} with the lemma {\\tt hR\\_Rh}.",
"The complement of the lemma {\\tt hR\\_Rh} is another lemma which expresses that {\\tt Rh} is a left inverse to {\\tt hR}.",
"This lemma is instrumental when showing the correspondence between concrete and abstract algorithms.",
"We now have two views of the algorithm: the algorithm {\\tt hpi} as described in section~\\ref{rounding-big} and the algorithm {\\tt rpi} where the functions {\\tt hmult}, {\\tt hdiv}, {\\tt hsqrt} have been replaced by {\\tt r\\_mult}, {\\tt r\\_div}, {\\tt r\\_sqrt} respectively.",
"We wish to show that these algorithms actually describe the same computation.",
"A new difficulty arises because we need to show that all operations receive and produce non-negative numbers, because these conditions are required by lemmas like {\\tt hmult\\_spec}.",
"This is not as simple as it seems because the result of {\\tt hmult 0 0} is only guaranteed to be larger than {\\tt - e} by the initial specification.",
"The implementation actually satisfies a stronger property.",
"In the end the correspondence lemma has the following form.",
"\\begin{verbatim} Lemma hpi_rpi_rec n p y z prod: (1 <= p)4 * (3/2) * INR (p + n) * /IZR magnifier < /100 -> Rabs (hR y - y_ p (/sqrt 2)) < 2 * /IZR magnifier -> Rabs (hR z - z_ p (/sqrt 2)) < 4 * /IZR magnifier -> Rabs (hR prod - pr p) < 4 * (3/2) * INR p * /IZR magnifier -> hR (hpi_rec n y z prod) = rpi_rec r_div r_sqrt r_mult n (hR y) (hR z) (hR prod).",
"The interesting part of this lemma is the equality stated on the last two lines.",
"The previous lines only state information about the size of the inputs, to help make sure that the intermediate computations never feed a negative number to the operations.",
"This constraint of non-negative operands makes the proof of correspondence tedious, but quite regular.",
"This proof ends up being 120 lines long.In retrospect, it might have been useful to add hypotheses that returned values by all functions were positive, as long as the inputs were.",
"A similar proof is constructed for the main encapsulating function, so that we obtain a lemma of the following shape.",
"Lemma hpi_rpi (n : nat) : 6 * INR n * /IZR magnifier < / 100 -> hR (hpi n) = rpi r_div r_sqrt r_mult n. Lemma integer_pi : forall n, (1 <= n)600 * INR (n + 1) < IZR magnifier < Rpower 531 (2 ^ n)/ 14 -> Rabs (hR (hpi (n + 1)) - PI) < (21 * INR (n + 1) + 3) /IZR magnifier.",
"In the end, we obtain a description of the algorithm based on integers, which can be applied to any number of iterations and any suitable scaling factor.",
"This algorithm can already be used to compute approximations of $\\pi $ inside Coq, but it will not return answers in reasonable time for precisions that go beyond a thousand digits (less than a second for a 7 iterations at 100 digits, 12 seconds for 9 iterations at 500 digits, a minute for 10 iterations at 1000 digits).",
"Concerning the magnitude of the accumulated error, for one million digits the number of iterations is 20, and the error is guaranteed to be smaller than 423."
],
[
"Changing the scaling factor.",
"Although we are culturally attracted by the fractional representation of $\\pi $ in decimal form, it is more efficient to perform most of the costly computations using a scaling factor that is a power of 2.",
"For any two scaling factors $m_1$ and $m_2$ , let us assume that $v_1$ and $v_2$ are linked by the equation $v_2 =\\left\\lfloor \\frac{v_1 \\times m_2}{m_1}\\right\\rfloor .$ If $v_1$ is the representation of a constant $a$ for the scaling factor $m_1$ , then $v_2$ is a reasonably good approximation of $a$ for the scaling factor $m_2$ .",
"This suggests that we could perform all operations with a scaling factor $m_1$ that is a power of 2 and then post-process the result to obtain a representation for the scaling factor $m_2$ .",
"Of course, one more multiplication and one more division need to be performed and a little precision is lost in the process, but the gain in computation time is worth it.",
"The validity of this change in scaling factor is expressed by the following lemma.",
"Lemma change_magnifier : forall m1 m2 x, (0 < m2)(m2 < m1)hR m1 x - /IZR m2 < hR m2 (x * m2/m1) <= hR m1 x.",
"This lemma expresses that the added error for this operation is only one time the inverse of the new scaling factor.",
"In our case, we use this lemma with ${\\tt m1} = 2 ^ {3321942}$ and ${\\tt m2} = 10^{10 ^6+4}$ for instance."
],
[
"Guaranteeing a fixed number of digits.",
"When we want to compute a number $N$ of digits, we don't know in advance whether the digits at position $N+1$ , $N+2$ , ...describe a small number or a large number.",
"If this number is too small or too large we are unable to guarantee the value of the digit at position $N$ .",
"Let's illustrate this problem on a small example.",
"Let's assume we want to compute the integral part of $a$ and we have an approximated value $b$ which is guaranteed to be within $1/4$ of $a$ .",
"Moreover, when computing $b$ with a precision of 2 digits, we know that our computation process may introduce errors of two units in the last place.",
"This means that we actually compute a value $c$ whose distance to $b$ is guaranteed to be smaller than $0.02$ .",
"At the time we discover the result of computing $c$ three cases may occur.",
"if the fractional part of $c$ is smaller than $0.27$ , the integral part of $a$ may be smaller than the integral part of $c$ .",
"For instance, we may have $c = 3.26$ , $b=3.245$ , and $a=2.995$ if the fractional part of $c$ is larger than or equal to $0.73$ , the integral part of $a$ may be larger than the integral part of $c$ .",
"For instance, we may have $c=2.74$ , $b=2.755$ , and $a=3.005$ .",
"if the fractional part of $c$ is larger than or equal to $0.27$ or smaller than $0.73$ , then we now that $a$ , $b$ , and $c$ all share the same integral part.",
"When considering distant decimals, the same problem is transposed through multiplication by a large power of 10.",
"After putting together the error coming from the difference $\\pi _n - \\pi $ , the accumulated rounding errors, and the error coming from the change of scaling factor, this means we need to verify that the last four digits are either larger than 0427 or smaller than 9573.",
"This verification is made in the following definitions, which return a boolean value and a large integer.",
"The meaning of the two values is expressed by the attached lemma.",
"Definition million_digit_pi : bool * Z := let magnifier := (2 ^ 3321942)let n := hpi magnifier 20 in let n' := (n * 10 ^ (10 ^ 6 + 4) / 2 ^ 3321942)let (q, r) := Z.div_eucl n' (10 ^ 4) in ((427 <?",
"r) Lemma pi_osix : fst million_digit_pi = true -> hR (10 ^ (10 ^ 6)) (snd million_digit_pi) < PI < hR (10 ^ (10 ^ 6)) (snd million_digit_pi) + Rpower 10 (-(Rpower 10 6))."
],
[
"Proving the big number computations.",
"The lemma million_digit_pi only states the correctness of computations for computations in the type Z, but this computation is unpractical to perform.",
"The last step is to obtain the same proof for computations on the type bigZ.",
"The library BigZ provides both this type and a coercion function noted [ $\\cdot $ ] so that when x is a big integer of type bigZ, [x] is the corresponding integer of type Z.",
"In what follows, the functions rounding_big.hmult, et cetera operate on numbers of type BigZ, while the functions hmult operate on plain integers.",
"We have the following morphism lemmas: Lemma hmult_morph p x y: [rounding_big.hmult p x y] = hmult [p] [x] [y].",
"Proof.",
"unfold hmult, rounding_big.hmult.",
"rewrite BigZ.spec_div, BigZ.spec_mul; reflexivity.",
"Qed.",
"Lemma hdiv_morph p x y: [rounding_big.hdiv p x y] = hdiv [p] [x] [y].",
"Proof.",
"unfold hdiv, rounding_big.hdiv.",
"rewrite BigZ.spec_div, BigZ.spec_mul; reflexivity.",
"Qed.",
"Using these lemmas, it is fairly routine to prove the correspondence between the algorithms instantiated on both types.",
"Lemma hpi_rec_morph : forall s p n v1 v2 v3, [s] = hsqrt [p] (h2 [p]) -> [rounding_big.hpi_rec p n s v1 v2 v3] = hpi_rec [p] n [s] [v1] [v2] [v3].",
"Lemma hpi_morph : forall p n, [rounding_big.hpi p n]\\end{verbatim} In the end, we have a theorem that expresses the correctness of the computations made with big numbers, with the following statement.",
"\\begin{verbatim} Lemma big_pi_osix : fst rounding_big.million_digit_pi = true -> (IZR [snd rounding_big.million_digit_pi] * Rpower 10 (-(Rpower 10 6)) < PI < IZR [snd rounding_big.million_digit_pi] * Rpower 10 (-(Rpower 10 6)) + Rpower 10 (-(Rpower 10 6)))\\end{verbatim} This statement expresses that the computation returns a boolean value and a large integer.",
"When this boolean value is {\\tt true}, then the large integer is the largest integer \\(n\\) so that \\[\\frac{n}{10^{10^6}} < \\pi.\\] The computation of this value takes approximately 2 hours on a powerful machine.",
"We also implemented similar functions to compute approximations of \\(\\pi\\) using the Brent-Salamin algorithm, and experiments showed the computation is twice as fast.",
"\\section{Related work} Computing approximations of \\(\\pi\\) is a task that is necessary for many projects of formally verified mathematics, but precision beyond tens of digits are practically never required.",
"To our knowledge, this work is the only one addressing explicitly the challenge of computing decimals at position beyond one thousand.",
"Most developments rely on Machin-like formulas to give a computationally relevant definition of \\(\\pi\\).",
"The paper \\cite{BertotAllais14} already provides an overview of methods used to compute \\(\\pi\\) in a variety of provers.",
"In Hol-Light \\cite{hol-light-analysis}, an approximation to the precision of \\(2^{-32}\\) is obtained by approximating \\(\\frac{\\pi}{6}\\) using the intermediate value theorem and a Taylor expansion of the sine function, and the library also provides a description of a variety of Machin-like formulas.",
"In Isabelle/HOL \\cite{Nipkow-Paulson-Wenzel:2002}, one of the Machin-like formulas is provided directly in the basic theory of transcendental functions.",
"Computation of arbitrary mathematical formulas, in the spirit of what is done with the {\\tt interval} tactic, is described in work by H\\\"olzl~\\cite{Hoelzl09}.",
"The HOL Light library contains a formalization of the BBP formula~\\cite{hol-bbp}.",
"Our contribution is to link the formalization of the formula with the actual algorithm that computes the digit.",
"In the Coq system, real numbers can also be approached constructively as in the C-CoRN library \\cite{Cruz-Filipe04}.",
"This was used as the basis for a library providing fairly efficient computation of mathematical functions within the theorem prover \\cite{OConnor-2008,KrebbersSpitters}.",
"Using an advanced Machin-like formula they are capable to compute numbers like \\(\\sqrt{\\pi}\\) at a precision of 500 digits in about 6 seconds (to be compared with less than a second in our case, but our development is not as versatile as theirs).",
"The formalized proof of the Kepler conjecture, under the supervision of T. Hales \\cite{Hales15} also required computing many inequalities between mathematical formulas involving transcendental functions, a task covered more specifically by Solovyev and Hales~\\cite{SolovyevHalesNFM13}, but none of these computations involved precisions in the ranges that we have been studying here.",
"\\section{Conclusion} What we guarantee with our lemmas is that the integer we produce satisfies a property with respect to \\(\\pi\\) and a large power of the base, which is 16 in the case of the the BBP algorithm, and may be any integer in the case of the algebraic-geometric mean algorithms.",
"We do not guarantee that the string produced by the Coq system when printing this large number is correct, but experimental evidence shows that that part of the Coq system (printing large numbers) is correct.",
"That computations can proceed to the end is a nice surprise, because it would be understandable that some parts of the theorem prover have limitations that preclude heavy computing (as is the case when performing computations with natural numbers, which are notoriously naive in their implementation and their space and time complexity).",
"It would be an interesting project to construct a formally verified integer to string converter, but this project is probably not as challenging as what has been presented in this article.",
"The organisation of proofs follows principles that were advocated by Cohen, D\\'en\\`es, M\\\"ortberg, and Siles \\cite{refinement-algebra,refinements-free}, where the algorithm is first studied in a mathematical setting using mathematical objects (in this case real numbers) before being embodied in a more efficient implementation using different data-types.",
"The concrete implementation is then viewed as a refinement of the first algorithm.",
"This approach makes sure that we take advantage of the most comfortable mathematical libraries when performing the most difficult proofs.",
"The refinement approach was used twice: first to establish the correspondence between computations on real numbers and the computations on integers, and second to establish the correspondence between integers and big integers.",
"The first stage does not fit exactly the framework advocated by Cohen and co-authors, because the computations are only approximated and we need to quantify the quality of the approximation.",
"On the other hand, the second stage corresponds quite precisely to what they advocate, and it was a source of great simplification in our formal proof, because the Coq libraries provided too few theorems and tactics to work on the big integers.",
"This experiment also raises the question of {\\em what do we perceive as a formally verified program?}",
"The implementations described in this paper do run and produce output, however they need the whole context of the interactive theorem prover.",
"We experimented with using the extraction facility of the Coq system to produce stand-alone programs that can be compiled with OCaml and run independently.",
"This works, but the resulting program is one order magnitude slower than what runs in the interactive theorem prover.",
"The reason is that the {\\tt BigZ} library exploits an ability to compute directly with machine integers (numbers modulo \\(2^{31}\\)) \\cite{ArmandGregoireSpiwackThery2010}, while the extracted program still views these numbers as records with 31 fields, with no shortcuts to exploit bit-level algorithmics.",
"This raises several questions of trusted base: firstly, the Coq system with the ability to exploit machine integers directly for number computations has a wider trusted base (because the code linking integer computation with machine integer computation needs to be trusted).",
"This first question is handled in another published article by Armand, Gr\\'egoire, Spiwack and Th\\'ery \\cite{ArmandGregoireSpiwackThery2010}.",
"Secondly we also have to trust the implementation of the {\\tt native\\_compute} facility, which generates an {\\tt OCaml} program, calls the OCaml compiler, and then runs and exploits the results of the compiled program.",
"This question is handled in another article by Boespflug, D\\'en\\`es and Gr\\'egoire \\cite{full-throttle}.",
"Thirdly, we could also extract the algorithms as modules to be interfaced with arbitrary libraries for large number computations.",
"We would thus obtain implementations that would be partially verified and whose guarantees would depend on the correct implementation of the large number operations.",
"This is probably the most sensible approach to using formally verified algorithms in the real world.",
"In the direction of formally verified programs, the next stage will be to study how the algorithms studied in this article can be implemented using imperative programming languages, avoiding stack operations and implementing clever memory operations, such as re-using explicitly the space of data that has become useless, instead of relying on a general purpose garbage-collector.",
"Obviously, we would need to interface with a library for large number computations in such a setting.",
"Such libraries already exist, but none of them have been formally verified.",
"We believe that the community of formal verification will produce such a formally verified library for large number computations, probably exploiting the advances provided by the CompCert formally verified compiler \\cite{Leroy-Compcert-CACM} (which provides the precise language for the implementation), and the Why3 tool \\cite{filliatre13esop} to organize proofs of programs with imperative features, based on various forms of Hoare logic.",
"In their current implementation, our algorithms run at speeds that are several orders of magnitude lower than the same algorithms implemented by clever programmers in heavy duty libraries like {\\tt mpfr} \\cite{mpfr}.",
"For now, the algorithms for elementary operations are based on Karatsuba-like divide-and-conquer approaches, with binary tree implementations of large numbers, but it could be interesting to implement fast-Fourier-transform based multiplication as suggested by Sch\\\"onhage and Strassen \\cite{SchoenhageStrassen71} and observe whether this brings a significative improvement in the computation of billions of decimals.",
"In spite of the fun with mathematical curiosities around the \\(\\pi\\) number, the real lesson of this paper is more about the current progress in interactive theorem provers.",
"How much mathematics can be described formally now?",
"How much detail can we give about computations?",
"How reproducible is this experiment?",
"For the question on how much mathematics, it is quite satisfying that real analysis becomes feasible, with concepts such as improper integrals, power series, interchange between limits, with automatic tools to check that mathematical expressions stay within bounds, but also with rigidities coming from the limits of the automatic tool.",
"One of the rigidity that we experienced is the lack of a proper integration of square roots in the automatic tool that deals with equalities in a field.",
"This tool, named {\\tt field}, deals very well with equalities between expressions that contain mostly products, divisions, additions and subtractions, but it won't simplify expressions such as \\(\\sqrt{\\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{2}}} - \\sqrt{\\sqrt{2}}\\).",
"From a human user's perspective, this rigidity is often hard to accept, because once the properties of the square root function are understood, we integrate them directly in our mental calculation process.",
"For the question on how much detail we can give about computations, these experiments show that we can go quite far in the direction of reasoning about computation errors.",
"This is not a novelty, and many other experiments by other authors have been studying how to reason about floating point computations \\cite{BolMel11}.",
"This experiment is slightly different in that it relies more on fixed point computations.",
"For the question on how reproducible is this experiment, we believe that one should distinguish between the task of running the formalized proof and the task of developing it.",
"For the first task, re-running the formal proof, we provide a link to the sources of our developments \\cite{PlouffeAGMSources}, which can be run with Coq version 8.5 and 8.6 and precise versions of the libraries Coquelicot and Interval.",
"For the task of developing the formal proof, this becomes a question at the edge of our scientific expertise, but still a question that is worth asking.",
"In the long run, formally verified mathematics should become practical to a wider audience thanks to the availability of comprehensive and well-documented libraries such as Coquelicot \\cite{BLM15} or mathematical components \\cite{MathematicalComponents}.",
"However, there are some aspects of the work that make reproducibility by less expert users difficult.",
"For instance, it is often difficult to understand the true limits of automatic tools and this form of rigidity may cause users to lose a lot of time, for instance by mistaking a failure to prove a statement with the fact that the statement could be wrong.",
"Another example is illustrated with the use of filters in the Coquelicot library, which requires much more advanced mathematical expertise than what would be expected for an intermediate level library about real analysis.",
"\\bibliographystyle{plain} \\begin{thebibliography}{10} \\bibitem{AlmkvistBerndt88} Gert Almkvist and Bruce Berndt.",
"\\newblock {\\em Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\\pi$, and the Ladies Diary (1988)}, pages 125--150.",
"\\newblock Springer International Publishing, 2016.",
"\\bibitem{CapesAGM95} Anonymous.",
"\\newblock Première composition de mathématiques'', concours externe de recrutement de professeurs certifiés, section mathématiques, 1995.",
"\\bibitem{ArmandGregoireSpiwackThery2010} Micha{\\\"{e}}l Armand, Benjamin Gr{\\'{e}}goire, Arnaud Spiwack, and Laurent Th{\\'{e}}ry.",
"\\newblock Extending {C}oq with {I}mperative {F}eatures and {I}ts {A}pplication to {SAT} {V}erification.",
"\\newblock In {\\em Interactive Theorem Proving, First International Conference, {ITP} 2010}, volume 6172 of {\\em LNCS}, pages 83--98. Springer, 2010.",
"\\bibitem{BBP97} David Bailey, Peter Borwein, and Simon Plouffe.",
"\\newblock On the rapid computation of various polylogarithmic constants.",
"\\newblock {\\em Mathematics of Computation}, 66(218):903--913, 1997.",
"\\bibitem{Bertot:CPP15} Yves Bertot.",
"\\newblock Fixed precision patterns for the formal verification of mathematical constant approximations.",
"\\newblock In {\\em Proceedings of the 2015 Conference on Certified Programs and Proofs}, pages 147--155. ACM, 2015.",
"\\bibitem{BertotAllais14} Yves Bertot and Guillaume Allais.",
"\\newblock {Views of Pi: definition and computation}.",
"\\newblock {\\em {Journal of Formalized Reasoning}}, 7(1):105--129, October 2014.",
"\\bibitem{RacineCarreeGMP} Yves Bertot, Nicolas Magaud, and Paul Zimmermann.",
"\\newblock A proof of {GMP} square root.",
"\\newblock {\\em Journal of Automated Reasoning}, 29(3-4):225--252, 2002.",
"\\bibitem{PlouffeAGMSources} Yves Bertot, Laurence Rideau, and Laurent Th\\'ery.",
"\\newblock Distant decimals of pi, 2017.",
"\\newblock Available at \\url{https://www-sop.inria.fr/marelle/distant-decimals-pi/}.",
"\\bibitem{DBLP:conf/types/Besson06} Fr{\\'e}d{\\'e}ric Besson.",
"\\newblock Fast reflexive arithmetic tactics the linear case and beyond.",
"\\newblock In {\\em Proceedings of the 2006 International Conference on Types for Proofs and Programs}, volume 4502 of {\\em LNCS}, pages 48--62.",
"Springer, 2007.",
"\\bibitem{full-throttle} Mathieu Boespflug, Maxime D{\\'e}n{\\`e}s, and Benjamin Gr{\\'e}goire.",
"\\newblock Full reduction at full throttle.",
"\\newblock In {\\em Certified Programs and Proofs: First International Conference}, volume 7086 of {\\em LNCS}, pages 362--377. Springer, 2011.",
"\\bibitem{BLM15} Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond.",
"\\newblock Coquelicot: A user-friendly library of real analysis for {C}oq.",
"\\newblock {\\em Mathematics in Computer Science}, 9(1):41--62, March 2015.",
"\\bibitem{BolMel11} Sylvie Boldo and Guillaume Melquiond.",
"\\newblock Flocq: A unified library for proving floating-point algorithms in {C}oq.",
"\\newblock In {\\em IEEE Symposium on Computer Arithmetic}, pages 243--252.",
"IEEE Computer Society, 2011.",
"\\bibitem{BorweinAGM} Jonathan~M. Borwein and Peter~B. Borwein.",
"\\newblock {\\em Pi and the AGM: A Study in the Analytic Number Theory and Computational Complexity}.",
"\\newblock Wiley-Interscience, New York, NY, USA, 1987.",
"\\bibitem{Brent76} Richard~P. Brent.",
"\\newblock Fast multiple-precision evaluation of elementary functions.",
"\\newblock {\\em J. ACM}, 23(2):242--251, April 1976.",
"\\bibitem{Filtres-cartan} Henri Cartan.",
"\\newblock Th\\'eorie des filtres.",
"\\newblock {\\em Comptes Rendus de l'Acad\\'emie des Sciences}, 205:595--598, 1937.",
"\\bibitem{refinements-free} Cyril Cohen, Maxime D{\\'e}n{\\`e}s, and Anders M{\\\"o}rtberg.",
"\\newblock Refinements for free!",
"\\newblock In {\\em Certified Programs and Proofs: Third International Conference}, volume 8307 of {\\em LNCS}, pages 147--162. Springer, 2013.",
"\\bibitem{Cruz-Filipe04} Luis Cruz-Filipe.",
"\\newblock {\\em Constructive Real Analysis: a Type-Theoretical Formalization and Applications}.",
"\\newblock PhD thesis, University of Nijmegen, April 2004.",
"\\bibitem{refinement-algebra} Maxime D{\\'{e}}n{\\`{e}}s, Anders M{\\\"{o}}rtberg, and Vincent Siles.",
"\\newblock A refinement-based approach to computational algebra in {C}oq.",
"\\newblock In {\\em Interactive Theorem Proving - Third International Conference, {ITP} 2012}, volume 7406 of {\\em LNCS}, pages 83--98. Springer, 2012.",
"\\bibitem{coq} Coq development team.",
"\\newblock The {C}oq proof assistant, 2016.",
"\\newblock \\url{http://coq.inria.fr}.",
"\\bibitem{filliatre13esop} Jean-Christophe Filli{\\^a}tre and Andrei Paskevich.",
"\\newblock Why3 --- where programs meet provers.",
"\\newblock In {\\em Programming Languages and Systems: 22nd European Symposium on Programming, ESOP 2013}, volume 7792 of {\\em LNCS}, pages 125--128.",
"Springer, 2013.",
"\\bibitem{mpfr} Laurent Fousse, Guillaume Hanrot, Vincent Lef\\`{e}vre, Patrick P{\\'e}lissier, and Paul Zimmermann.",
"\\newblock Mpfr: A multiple-precision binary floating-point library with correct rounding.",
"\\newblock {\\em ACM Trans. Math. Softw.}, 33(2), June 2007.",
"\\bibitem{MathematicalComponents} Georges Gonthier et~al.",
"\\newblock Mathematical components.",
"\\newblock Available at \\url{http://math-comp.github.io/math-comp/}.",
"\\bibitem{Gourevitch99} Boris Gourevitch.",
"\\newblock The world of {P}i, 1999.",
"\\newblock Available at \\url{http://www.pi314.net}, consulted March 2017.",
"\\bibitem{GregoireTheryIJCAR2006} Benjamin Gr{\\'e}goire and Laurent Th{\\'e}ry.",
"\\newblock A purely functional library for modular arithmetic and its application to certifying large prime numbers.",
"\\newblock In {\\em Automated Reasoning: Third International Joint Conference, IJCAR 2006}, volume 4130 of {\\em LNCS}, pages 423--437. Springer, 2006.",
"\\bibitem{Hales15} Thomas~C. Hales et~al.",
"\\newblock A formal proof of the {K}epler conjecture.",
"\\newblock {\\em arXiv}, 1501.02155, 2015.",
"\\bibitem{hol-light-analysis} John Harrison.",
"\\newblock Formalizing basic complex analysis.",
"\\newblock In {\\em From Insight to Proof: Festschrift in Honour of Andrzej Trybulec}, volume 10(23) of {\\em Studies in Logic, Grammar and Rhetoric}, pages 151--165. University of Bia{\\l}ystok, 2007.",
"\\newblock \\url{http://mizar.org/trybulec65/}.",
"\\bibitem{hol-bbp} John Harrison.",
"\\newblock Pi series in {B}ailey/{B}orwein/{P}louffe \\textit{polylogarithmic constants} paper, the {HOL} {L}ight library, 2010.",
"\\newblock Available at \\url{https://github.com/jrh13/hol-light/blob/master/Examples/polylog.ml}.",
"\\bibitem{Harrison98} John Harrison.",
"\\newblock {\\em Theorem Proving with the Real Numbers}.",
"\\newblock Springer Publishing Company, Incorporated, 1st edition, 2011.",
"\\bibitem{Hoelzl09} Johannes H{\\\"o}lzl.",
"\\newblock Proving inequalities over reals with computation in {Isabelle/HOL}.",
"\\newblock In {\\em Proceedings of the {ACM SIGSAM 2009} International Workshop on Programming Languages for Mechanized Mathematics Systems ({PLMMS'09})}, pages 38--45, Munich, 2009.",
"\\bibitem{Isabelle-filters} Johannes H{\\\"o}lzl, Fabian Immler, and Brian Huffman.",
"\\newblock Type classes and filters for mathematical analysis in {Isabelle/HOL}.",
"\\newblock In {\\em Interactive Theorem Proving (ITP 2013)}, volume 7998 of {\\em LNCS}, pages 279--294. Springer, 2013.",
"\\bibitem{King1924} Louis~V. King.",
"\\newblock {\\em On the Direct Numerical Calculation of Elliptic Functions and Integrals}.",
"\\newblock Cambridge University Press, 1924.",
"\\bibitem{KrebbersSpitters} Robbert Krebbers and Bas Spitters.",
"\\newblock Type classes for efficient exact real arithmetic in {C}oq.",
"\\newblock {\\em Logical Methods in Computer Science}, 9(1), 2011.",
"\\bibitem{Leroy-Compcert-CACM} Xavier Leroy.",
"\\newblock Formal verification of a realistic compiler.",
"\\newblock {\\em Commun. ACM}, 52(7):107--115, July 2009.",
"\\bibitem{interval} {\\'{E}}rik Martin{-}Dorel and Guillaume Melquiond.",
"\\newblock Proving tight bounds on univariate expressions with elementary functions in {C}oq.",
"\\newblock {\\em Journal of Automated Reasoning}, 57(3):187--217, 2016.",
"\\bibitem{Nipkow-Paulson-Wenzel:2002} Tobias Nipkow, Markus Wenzel, and Lawrence~C. Paulson.",
"\\newblock {\\em Isabelle/HOL: A Proof Assistant for Higher-order Logic}.",
"\\newblock Springer, Berlin, Heidelberg, 2002.",
"\\bibitem{OConnor-2008} Russell O'Connor.",
"\\newblock Certified exact transcendental real number computation in {C}oq.",
"\\newblock In {\\em Theorem Proving in Higher Order Logics: 21st International Conference, TPHOLs 2008}, volume 5170 of {\\em LNCS}, pages 246--261.",
"Springer, 2008.",
"\\bibitem{Salamin76} Eugene Salamin.",
"\\newblock Computation of \\(\\pi\\) using arithmetic-geometric mean.",
"\\newblock {\\em Mathematics of Computation}, 33(135), 1976.",
"\\bibitem{SchoenhageStrassen71} Arnold Sch{\\\"{o}}nhage and Volker Strassen.",
"\\newblock Schnelle {M}ultiplikation gro{\\ss}er {Z}ahlen.",
"\\newblock {\\em Computing}, 7(3-4):281--292, 1971.",
"\\bibitem{SolovyevHalesNFM13} Alexey Solovyev and Thomas~C. Hales.",
"\\newblock Formal verification of nonlinear inequalities with {T}aylor interval approximations.",
"\\newblock In {\\em NASA Formal Methods: 5th International Symposium, NFM 2013,}, volume 7871 of {\\em LNCS}, pages 383--397. Springer, 2013.",
"\\end{thebibliography} \\end{document}"
]
] | 1709.01743 |
[
[
"Deep Learning Techniques for Music Generation -- A Survey"
],
[
"Abstract This paper is a survey and an analysis of different ways of using deep learning (deep artificial neural networks) to generate musical content.",
"We propose a methodology based on five dimensions for our analysis: Objective - What musical content is to be generated?",
"Examples are: melody, polyphony, accompaniment or counterpoint.",
"- For what destination and for what use?",
"To be performed by a human(s) (in the case of a musical score), or by a machine (in the case of an audio file).",
"Representation - What are the concepts to be manipulated?",
"Examples are: waveform, spectrogram, note, chord, meter and beat.",
"- What format is to be used?",
"Examples are: MIDI, piano roll or text.",
"- How will the representation be encoded?",
"Examples are: scalar, one-hot or many-hot.",
"Architecture - What type(s) of deep neural network is (are) to be used?",
"Examples are: feedforward network, recurrent network, autoencoder or generative adversarial networks.",
"Challenge - What are the limitations and open challenges?",
"Examples are: variability, interactivity and creativity.",
"Strategy - How do we model and control the process of generation?",
"Examples are: single-step feedforward, iterative feedforward, sampling or input manipulation.",
"For each dimension, we conduct a comparative analysis of various models and techniques and we propose some tentative multidimensional typology.",
"This typology is bottom-up, based on the analysis of many existing deep-learning based systems for music generation selected from the relevant literature.",
"These systems are described and are used to exemplify the various choices of objective, representation, architecture, challenge and strategy.",
"The last section includes some discussion and some prospects."
],
[
"Introduction",
"*Chapter Introduction provides motivation for this book.",
"It includes a short summary of the history of computer music and music generation (including previous uses of artificial neural networks as well as other models such as grammars, rules and Markov chains) up to the recent rise of deep learning.",
"It also describes the organization and the public of the book as well as related work.",
"Deep learning has recently become a fast growing domain and is now used routinely for classification and prediction tasks, such as image recognition, voice recognition or translation.",
"It became popular in 2012, when a deep learning architecture significantly outperformed standard techniques relying on handcrafted features in an image classification competition, see more details in Section .",
"We may explain this success and reemergence of artificial neural network techniques by the combination of: availability of massive data; availability of efficient and affordable computing powerNotably, thanks to graphics processing units (GPU), initially designed for video games, which have now one of their biggest markets in data science and deep learning applications.",
"; technical advances, such as: pre-training, which resolved initially inefficient training of neural networks with many layers [80]Although nowadays it has being replaced by other techniques, such as batch normalization [92] and deep residual learning [74].",
"; convolutions, which provide motif translation invariance [111]; LSTM (long short-term memory), which resolved initially inefficient training of recurrent neural networks [83].",
"There is no consensual definition for deep learning.",
"It is a repertoire of machine learning (ML) techniques, based on artificial neural networks.",
"The key aspect and common ground is the term deep.",
"This means that there are multiple layers processing multiple hierarchical levels of abstractions, which are automatically extracted from dataThat said, although deep learning will automatically extract significant features from the data, manual choices of input representation, e.g., spectrum vs raw wave signal for audio, may be very significant for the accuracy of the learning and for the quality of the generated content, see Section REF ..",
"Thus a deep architecture can manage and decompose complex representations in terms of simpler representations.",
"The technical foundation is mostly artificial neural networks, as we will see in Chapter , with many extensions, such as: convolutional networks, recurrent networks, autoencoders, and restricted Boltzmann machines.",
"For more information about the history and various facets of deep learning, see, e.g., a recent comprehensive book on the domain [63].",
"Driving applications of deep learning are traditional machine learning tasksTasks in machine learning are types of problems and may also be described in terms of how the machine learning system should process an example [63].",
"Examples are: classification, regression and anomaly detection.",
": classification (for instance, identification of images) and predictionAs a testimony of the initial DNA of neural networks: linear regression and logistic regression, see Section .",
"(for instance, of the weather) and also more recent ones such as translation.",
"But a growing area of application of deep learning techniques is the generation of content.",
"Content can be of various kinds: images, text and music, the latter being the focus of our analysis.",
"The motivation is in using now widely available various corpora to automatically learn musical styles and to generate new musical content based on this.",
"The first music generated by computer appeared in 1957.",
"It was a 17 seconds long melody named “The Silver Scale” by its author Newman Guttman and was generated by a software for sound synthesis named Music I, developed by Mathews at Bell Laboratories.",
"The same year, “The Illiac Suite” was the first score composed by a computer [78].",
"It was named after the ILLIAC I computer at the University of Illinois at Urbana-Champaign (UIUC) in the United States.",
"The human “meta-composers” were Lejaren A. Hiller and Leonard M. Isaacson, both musicians and scientists.",
"It was an early example of algorithmic composition, making use of stochastic models (Markov chains) for generation as well as rules to filter generated material according to desired properties.",
"In the domain of sound synthesis, a landmark was the release in 1983 by Yamaha of the DX 7 synthesizer, building on groundwork by Chowning on a model of synthesis based on frequency modulation (FM).",
"The same year, the MIDIMusical instrument digital interface, to be introduced in Section REF .",
"interface was launched, as a way to interoperate various software and instruments (including the Yamaha DX 7 synthesizer).",
"Another landmark was the development by Puckette at IRCAM of the Max/MSP real-time interactive processing environment, used for real-time synthesis and for interactive performances.",
"Regarding algorithmic composition, in the early 1960s Iannis Xenakis explored the idea of stochastic compositionOne of the first documented case of stochastic music, long before computers, is the Musikalisches Wurfelspiel (Dice Music), attributed to Wolfgang Amadeus Mozart.",
"It was designed for using dice to generate music by concatenating randomly selected predefined music segments composed in a given style (Austrian waltz in a given key).",
"[209], in his composition named “Atrées” in 1962.",
"The idea involved using computer fast computations to calculate various possibilities from a set of probabilities designed by the composer in order to generate samples of musical pieces to be selected.",
"In another approach following the initial direction of “The Illiac Suite”, grammars and rules were used to specify the style of a given corpus or more generally tonal music theory.",
"An example is the generation in the 1980s by Ebcioğlu's composition software named CHORAL of a four-part chorale in the style of Johann Sebastian Bach, according to over 350 handcrafted rules [42].",
"In the late 1980s David Cope's system named Experiments in Musical Intelligence (EMI) extended that approach with the capacity to learn from a corpus of scores of a composer to create its own grammar and database of rules [27].",
"Since then, computer music has continued developing for the general public, if we consider, for instance, the GarageBand music composition and production application for Apple platforms (computers, tablets and cellphones), as an offspring of the initial Cubase sequencer software, released by Steinberg in 1989.",
"For more details about the history and principles of computer music in general, see, for example, the book by Roads [160].",
"For more details about the history and principles of algorithmic composition, see, for example, [128] and the books by Cope [27] or Dean and McLean [33]."
],
[
"Autonomy versus Assistance",
"When talking about computer-based music generation, there is actually some ambiguity about whether the objective is to design and construct autonomous music-making systems – two recent examples being the deep-learning based Amper™ and Jukedeck systems/companies aimed at the creation of original music for commercials and documentary; or to design and construct computer-based environments to assist human musicians (composers, arrangers, producers, etc.)",
"– two examples being the FlowComposer environment developed at Sony CSL-Paris [153] (introduced in Section REF ) and the OpenMusic environment developed at IRCAM [3].",
"The quest for autonomous music-making systems may be an interesting perspective for exploring the process of compositionAs Richard Feynman coined it: “What I cannot create, I do not understand.” and it also serves as an evaluation method.",
"An example of a musical Turing testInitially codified in 1950 by Alan Turing and named by him the “imitation game” [191], the “Turing test” is a test of the ability for a machine to exhibit intelligent behavior equivalent to (and more precisely, indistinguishable from) the behavior of a human.",
"In his imaginary experimental setting, Turing proposed the test to be a natural language conversation between a human (the evaluator) and a hidden actor (another human or a machine).",
"If the evaluator cannot reliably tell the machine from the human, the machine is said to have passed the test.",
"will be introduced in Section REF .",
"It consists in presenting to various members of the public (from beginners to experts) chorales composed by J. S. Bach or generated by a deep learning system and played by human musiciansThis is to avoid the bias (synthetic flavor) of a computer rendered generated music.. As we will see in the following, deep learning techniques turn out to be very efficient at succeeding in such tests, due to their capacity to learn musical style from a given corpus and to generate new music that fits into this style.",
"That said, we consider that such a test is more a means than an end.",
"A broader perspective is in assisting human musicians during the various steps of music creation: composition, arranging, orchestration, production, etc.",
"Indeed, to compose or to improviseImprovisation is a form of real time composition., a musician rarely creates new music from scratch.",
"S/he reuses and adapts, consciously or unconsciously, features from various music that s/he already knows or has heard, while following some principles and guidelines, such as theories about harmony and scales.",
"A computer-based musician assistant may act during different stages of the composition, to initiate, suggest, provoke and/or complement the inspiration of the human composer.",
"That said, as we will see, the majority of current deep-learning based systems for generating music are still focused on autonomous generation, although more and more systems are addressing the issue of human-level control and interaction."
],
[
"Symbolic versus Sub-Symbolic AI",
"Artificial Intelligence (AI) is often divided into two main streamsWith some precaution, as this division is not that strict.",
": symbolic AI – dealing with high-level symbolic representations (e.g., chords, harmony...) and processes (harmonization, analysis...); and sub-symbolic AI – dealing with low-level representations (e.g., sound, timbre...) and processes (pitch recognition, classification...).",
"Examples of symbolic models used for music are rule-based systems or grammars to represent harmony.",
"Examples of sub-symbolic models used for music are machine learning algorithms for automatically learning musical styles from a corpus of musical pieces.",
"These models can then be used in a generative and interactive manner, to help musicians in creating new music, by taking advantage of this added “intelligent” memory (associative, inductive and generative) to suggest proposals, sketches, extrapolations, mappings, etc.",
"This is now feasible because of the growing availability of music in various forms, e.g., sound, scores and MIDI files, which can be automatically processed by computers.",
"A recent example of an integrated music composition environment is FlowComposer [153], which we will introduce in Section REF .",
"It offers various symbolic and sub-symbolic techniques, e.g., Markov chains for modeling style, a constraint solving module for expressing constraints, a rule-based module to produce harmonic analysis; and an audio mapping module to produce rendering.",
"Another example of an integrated music composition environment is OpenMusic [3].",
"However, a deeper integration of sub-symbolic techniques, such as deep learning, with symbolic techniques, such as constraints and reasoning, is still an open issueThe general objective of integrating sub-symbolic and symbolic levels into a complete AI system is among the “Holy Grails” of AI., although some partial integrations in restricted contexts already exist (see, for example, Markov constraints in [149], [7] and an example of use for FlowComposer in Section REF )."
],
[
"Deep Learning",
"The motivation for using deep learning (and more generally machine learning techniques) to generate musical content is its generality.",
"As opposed to handcrafted models, such as grammar-based [177] or rule-based music generation systems [42], a machine learning-based generation system can be agnostic, as it learns a model from an arbitrary corpus of music.",
"As a result, the same system may be used for various musical genres.",
"Therefore, as more large scale musical datasets are made available, a machine learning-based generation system will be able to automatically learn a musical style from a corpus and to generate new musical content.",
"As stated by Fiebrink and Caramiaux [52], some benefits are it can make creation feasible when the desired application is too complex to be described by analytical formulations or manual brute force design, and learning algorithms are often less brittle than manually designed rule sets and learned rules are more likely to generalize accurately to new contexts in which inputs may change.",
"Moreover, as opposed to structured representations like rules and grammars, deep learning is good at processing raw unstructured data, from which its hierarchy of layers will extract higher level representations adapted to the task."
],
[
"Present and Future",
"As we will see, the research domain in deep learning-based music generation has turned hot recently, building on initial work using artificial neural networks to generate music (e.g., the pioneering experiments by Todd in 1989 [190] and the CONCERT system developed by Mozer in 1994 [139]), while creating an active stream of new ideas and challenges made possible thanks to the progress of deep learning.",
"Let us also note the growing interest by some private big actors of digital media in the computer-aided generation of artistic content, with the creation by Google in June 2016 of the Magenta research project [48] and the creation by Spotify in September 2017 of the Creator Technology Research Lab (CTRL) [176].",
"This is likely to contribute to blurring the line between music creation and music consumption through the personalization of musical content [2]."
],
[
"This Book",
"The lack (to our knowledge) of a comprehensive survey and analysis of this active research domain motivated the writing of this book, built in a bottom-up way from the analysis of numerous recent research works.",
"The objective is to provide a comprehensive description of the issues and techniques for using deep learning to generate music, illustrated through the analysis of various architectures, systems and experiments presented in the literature.",
"We also propose a conceptual framework and typology aimed at a better understanding of the design decisions for current as well as future systems."
],
[
"Other Books and Sources",
"To our knowledge, there are only a few partial attempts at analyzing the use of deep learning for generating music.",
"In [14], a very preliminary version of this work, Briot et al.",
"proposed a first survey of various systems through a multicriteria analysis (considering as dimensions the objective, representation, architecture and strategy).",
"We have extended and consolidated this study by integrating as an additional dimension the challenge (after having analyzed it in [16]).",
"In [65], Graves presented an analysis focusing on recurrent neural networks and text generation.",
"In [90], Humphrey et al.",
"presented another analysis, sharing some issues about music representation (see Section ) but dedicated to music information retrieval (MIR) tasks, such as chord recognition, genre recognition and mood estimation.",
"On MIR applications of deep learning, see also the recent tutorial paper by Choi et al.",
"[21].",
"One could also consult the proceedings of some recently created international workshops on the topic, such as the Workshop on Constructive Machine Learning (CML 2016), held during the 30th Annual Conference on Neural Information Processing Systems (NIPS 2016) [29]; the Workshop on Deep Learning for Music (DLM), held during the International Joint Conference on Neural Networks (IJCNN 2017) [75], followed by a special journal issue [76]; and on the deep challenge of creativity, the related Series of International Conferences on Computational Creativity (ICCC) [186].",
"For a more general survey of computer-based techniques to generate music, the reader can refer to general books such as Roads' book about computer music [160]; Cope's [27], Dean and McLean's [33] and/or Nierhaus' books [144] about algorithmic composition; a recent survey about AI methods in algorithmic composition [51]; and Cope's book about models of musical creativity [28].",
"About machine learning in general, some examples of textbooks are the textbook by Mitchell [132]; a nice introduction and summary by Domingos [39]; and a recent, complete and comprehensive book about deep learning by Goodfellow et al.",
"[63]."
],
[
"Other Models",
"We have to remember that there are various other models and techniques for using computers to generate music, such as rules, grammars, automata, Markov models and graphical models.",
"These models are either manually defined by experts or are automatically learnt from examples by using various machine learning techniques.",
"They will not be addressed in this book as we are concerned here with deep learning techniques.",
"However, in the following section we make a quick comparison of deep learning and Markov models."
],
[
"Deep Learning versus Markov Models",
"Deep learning models are not the only models able to learn musical style from examples.",
"Markov chain models are also widely used, see, for example, [146].",
"A quick comparison (inspired by the analysis of Mozer in [139]Note that he made his analysis in in 1994, long before the deep learning wave.)",
"of the pros (+) and cons (–) of deep neural network models and Markov chain models is as follows: [–] + Markov models are conceptually simple.",
"+ Markov models have a simple implementation and a simple learning algorithm, as the model is a transition probability tableStatistics are collected from the dataset of examples in order to compute the probabilities.. – Neural network models are conceptually simple but the optimized implementations of current deep network architectures may be complex and need a lot of tuning.",
"– Order 1 Markov models (that is, considering only the previous state) do not capture long-term temporal structures.",
"– Order n Markov models (considering n previous states) are possible but require an explosive training set sizeSee the discussion in [139].",
"and can lead to plagiarismBy recopying too long sequences from the corpus.",
"Some promising solution is to consider a variable order Markov model and to constrain the generation (through min order and max order constraints) on some sweet spot between junk and plagiarism [152].. + Neural networks can capture various types of relations, contexts and regularities.",
"+ Deep networks can learn long-term and high-order dependencies.",
"+ Markov models can learn from a few examples.",
"– Neural networks need a lot of examples in order to be able to learn well.",
"– Markov models do not generalize very well.",
"+ Neural networks generalize better through the use of distributed representations [82].",
"+ Markov models are operational models (automata) on which some control on the generation could be attachedExamples are Markov constraints [149] and factor graphs [148].. – Deep networks are generative models with a distributed representation and therefore with no direct control to be attachedThis issue as well as some possible solutions will be discussed in Section REF .. As deep learning implementations are now mature and a large number of examples are available, deep learning-based models are in high demand for their characteristics.",
"That said, other models (such as Markov chains, graphical models, etc.)",
"are still useful and used and the choice of a model and its tuning depends on the characteristics of the problem."
],
[
"Requisites and Roadmap",
"This book does not require prior knowledge about deep learning and neural networks nor music.",
"Chapter Introduction (this chapter) introduces the purpose and rationale of the book.",
"Chapter Method introduces the method of analysis (conceptual framework) and the five dimensions at its basis (objective, representation, architecture, challenge and strategy), dimensions that we discuss within the next four chapters.",
"Chapter Objective concerns the different types of musical content that we want to generate (such as a melody or an accompaniment to an existing melody)Our proposed typology of possible objectives will turn out to be useful for our analysis because, as we will see, different objectives can lead to different architectures and strategies., as well as their expected use (by a human and/or a machine).",
"Chapter Representation provides an analysis of the different types of representation and techniques for encoding musical content (such as notes, durations or chords) for a deep learning architecture.",
"This chapter may be skipped by a reader already expert in computer music, although some of the encoding strategies are specific to neural networks and deep learning architectures.",
"Chapter Architecture summarizes the most common deep learning architectures (such as feedforward, recurrent or autoencoder) used for the generation of music.",
"This includes a short reminder of the very basics of a simple neural network.",
"This chapter may be skipped by a reader already expert in artificial neural networks and deep learning architectures.",
"Chapter Challenge and Strategy provides an analysis of the various challenges that occur when applying deep learning techniques to music generation, as well as various strategies for addressing them.",
"We will ground our study in the analysis of various systems and experiments surveyed from the literature.",
"This chapter is the core of the book.",
"Chapter Analysis summarizes the survey and analysis conducted in Chapter through some tables as a way to identify the design decisions and their interrelations for the different systems surveyedAnd hopefully also for the future ones.",
"If we draw the analogy (at some meta-level) with the expected ability for a model learnt from a corpus by a machine to be able to generalize to future examples (see Section REF ), we hope that the conceptual framework presented in this book, (manually) inducted from a corpus of scientific and technical literature about deep-learning-based music generation systems, will also be able to help in the design and the understanding of future systems.. Chapter Discussion and Conclusion revisits some of the open issues that were touched in during the analysis of challenges and strategies presented in Chapter , before concluding this book.",
"A table of contents, a table of acronyms, a list of references, a glossary and an index complete this book.",
"Supplementary material is provided at the following companion web site: www.briot.info/dlt4mg/"
],
[
"Limits",
"This book does not intend to be a general introduction to deep learning – a recent and broad spectrum book on this topic is [63].",
"We do not intend to get into all technical details of implementation, like engineering and tuning, as well as theoryFor instance, we will not develop the probability theory and information theory frameworks for formalizing and interpreting the behavior of neural networks and deep learning.",
"However, Section REF will introduce the intuition behind the notions of entropy and cross-entropy, used for measuring the progress made during learning., as we wish to focus on the conceptual level, whilst providing a sufficient degree of precision.",
"Also, although having a clear pedagogical objective, we do not provide some end-to-end tutorial with all the steps and details on how to implement and tune a complete deep learning-based music generation system.",
"Last, as this book is about a very active domain and as our survey and analysis is based on existing systems, our analysis is obviously not exhaustive.",
"We have tried to select the most representative proposals and experiments, while new proposals are being presented at the time of our writing.",
"Therefore, we encourage readers and colleagues to provide any feedback and suggestions for improving this survey and analysis which is a still ongoing project."
],
[
"Method",
"*Chapter Method describes the conceptual framework proposed in this book to analyze, classify and compare various deep learning-based music generation systems.",
"The five dimensions considered are: objective, representation, architecture, challenge and strategy.",
"The typologies associated to each dimension have been constructed in a bottom-up way, from the analysis of numerous deep learning-based music generation systems from the literature.",
"In our analysis, we consider five main dimensions to characterize different ways of applying deep learning techniques to generate musical content.",
"This typology is aimed at helping the analysis of the various perspectives (and elements) leading to the design of different deep learning-based music generation systemsIn this book, systems refers to the various proposals (architectures, systems and experiments) about deep learning-based music generation that we have surveyed from the literature.."
],
[
"Dimensions",
"The five dimensions that we consider are as follows.",
"The objectiveWe could have used the term task in place of objective.",
"However, as task is a relatively well-defined and common term in the machine learning community (see Section and [63]), we preferred an alternative term.",
"consists in: The musical nature of the content to be generated.",
"Examples are a melody, a polyphony or an accompaniment; and The destination and use of the content generated.",
"Examples are a musical score to be performed by some human musician(s) or an audio file to be played."
],
[
"Representation",
"The representation is the nature and format of the information (data) used to train and to generate musical content.",
"Examples are signal, transformed signal (e.g., a spectrum, via a Fourier transform), piano roll, MIDI or text."
],
[
"Architecture",
"The architecture is the nature of the assemblage of processing units (the artificial neurons) and their connexions.",
"Examples are a feedforward architecture, a recurrent architecture, an autoencoder architecture and generative adversarial networks."
],
[
"Challenge",
"A challenge is one of the qualities (requirements) that may be desired for music generation.",
"Examples are content variability, interactivity and originality."
],
[
"Strategy",
"The strategy represents the way the architecture will process representations in order to generateNote, that we consider here the strategy relating to the generation phase and not the strategy relating to the training phase, as they could be different.",
"the objective while matching desired requirements.",
"Examples are single-step feedforward, iterative feedforward, decoder feedforward, sampling and input manipulation."
],
[
"Discussion",
"Note that these five dimensions are not orthogonal.",
"The choice of representation is partially determined by the objective and it also constrains the input and output (interfaces) of the architecture.",
"A given type of architecture also usually leads to a default strategy of use, while new strategies may be designed in order to target specific challenges.",
"The exploration of these five different dimensions and of their interplay is actually at the core of our analysisLet us remember that our proposed typology has been constructed in a bottom-up manner from the survey and analysis of numerous systems retrieved from the literature, most of them being very recent.. Each of the first three dimensions (objective, representation and architecture) will be analyzed with its associated typology in a specific chapter, with various illustrative examples and discussion.",
"The challenge and strategy dimensions will be jointly analyzed within the same chapter (Chapter ) in order to jointly illustrate potential issues (challenges) and possible solutions (strategies).",
"As we will see, the same strategy may relate to more than one challenge and vice versa.",
"Last, we do not expect our proposed conceptual framework (and its associated five dimensions and related typologies) to be a final result, but rather a first step towards a better understanding of design decisions and challenges for deep learning-based music generation.",
"In other words, it is likely to be further amended and refined, but we hope that it could help bootstrap what we believe to be a necessary comprehensive study."
],
[
"Objective",
"*Chapter Objective presents the first dimension of the conceptual framework proposed in this book to analyze, classify and compare various deep learning-based music generation systems.",
"This first dimension is the objective, which is the nature of the musical content to be generated.",
"We consider some facets (type – the main one –, destination, use, mode and style) with related taxonomies.",
"For example, main types are: melody, polyphony, multivoice and accompaniment.",
"The first dimension, the objective, is the nature of the musical content to be generated."
],
[
"Facets",
"We may consider five main facets of an objective: Type The musical nature of the generated content.",
"Examples are a melody, a polyphony or an accompaniment.",
"Destination The entity aimed at using (processing) the generated content.",
"Examples are a human musician, a software or an audio system.",
"Use The way the destination entity will process the generated content.",
"Examples are playing an audio file or performing a music score.",
"Mode The way the generation will be conducted, i.e.",
"with some human intervention (interaction) or without any intervention (automation).",
"Style The musical style of the content to be generated.",
"Examples are Johann Sebastian Bach chorales, Wolfgang Amadeus Mozart sonatas, Cole Porter songs or Wayne Shorter music.",
"The style will actually be set though the choice of the dataset of musical examples (corpus) used as the training examples.",
"Main examples of musical types are as follows: Single-voice monophonic melody, abbreviated as Melody It is a sequence of notes for a single instrument or vocal, with at most one note at the same time.",
"An example is the music produced by a monophonic instrument like a fluteAlthough there are non-standard techniques to produce more than one note, the simplest one being to sing simultaneously as playing.",
"There are also non-standard diphonic techniques for voice.. Single-voice polyphony (also named Single-track polyphony), abbreviated as Polyphony It is a sequence of notes for a single instrument, where more than one note can be played at the same time.",
"An example is the music produced by a polyphonic instrument such as a piano or guitar.",
"Multivoice polyphony (also named Multitrack polyphony), abbreviated as Multivoice or Multitrack It is a set of multiple voices/tracks, which is intended for more than one voice or instrument.",
"Examples are: a chorale with soprano, alto, tenor and bass voices or a jazz trio with piano, bass and drums.",
"Accompaniment to a given melody Such as Counterpoint, composed of one or more melodies (voices); or Chord progression, which provides some associated harmony.",
"Association of a melody with a chord progression An example is what is named a lead sheetFigure REF in Chapter Representation will show an example of a lead sheet.",
"and is common in jazz.",
"It may also include lyricsNote that lyrics could be generated too.",
"Although this target is beyond the scope of this book, we will see later in Section REF that, in some systems, music is encoded as a text.",
"Thus, a similar technique could be applied to lyric generation..",
"Note that the type facet is actually the most important facet, as it captures the musical nature of the objective for content generation.",
"In this book, we will frequently identify an objective according to its type, e.g., a melody, as a matter of simplification.",
"The next three facets – destination, use and mode – will turn out important when regarding the dimension of the interaction of human user(s) with the process of content generation."
],
[
"Destination and Use",
"Main examples of destination and use are as follows: Audio system Which will play the generated content, as in the case of the generation of an audio file.",
"Sequencer software Which will process the generated content, as in the case of the generation of a MIDI file.",
"Human(s) Who will perform and interpret the generated content, as in the case of the generation of a music score."
],
[
"Mode",
"There are two main modes of music generation: Autonomous and Automated Without any human intervention; or Interactive (to some degree) With some control interface for the human user(s) to have some interactive control over the process of generation.",
"As deep learning for music generation is recent and basic neural network techniques are non-interactive, the majority of systems that we have analyzed are not yet very interactiveSome examples of interactive systems will be introduced in Section ..",
"Therefore, an important goal appears to be the design of fully interactive support systems for musicians (for composing, analyzing, harmonizing, arranging, producing, mixing, etc.",
"), as pioneered by the FlowComposer prototype [153] to be introduced in Section REF ."
],
[
"Style",
"As stated previously, the musical style of the content to be generated will be governed by the choice of the dataset of musical examples that will be used as training examples.",
"As will be discussed further in Section , we will see that the choice of a dataset, notably properties like coherence, coverage (versus sparsity) and scope (specialized versus large breadth), is actually fundamental for good music generation.",
"*Chapter Representation presents the second dimension of the conceptual framework proposed in this book to analyze, classify and compare various deep learning-based music generation systems.",
"The representation is about the way a musical content is specified and then encoded.",
"A big divide is between an audio or a symbolic representation, the latter type being more frequent as it focuses on the compositional level.",
"Main concepts are: note, duration, rest, chord and meter.",
"Various formats for representation may be used, the most frequent ones being piano roll, MIDI and text.",
"Encoding is the final and lowest-level decision about how concepts or values will be represented and computed by the deep network.",
"This chapter may be skipped by a reader already expert in computer music, although some of the encoding strategies are specific to neural networks and deep learning architectures.",
"The second dimension of our analysis, the representation, is about the way the musical content is represented.",
"The choice of representation and its encoding is tightly connected to the configuration of the input and the output of the architecture, i.e.",
"the number of input and output variables as well as their corresponding types.",
"We will see that, although a deep learning architecture can automatically extract significant features from the data, the choice of representation may be significant for the accuracy of the learning and for the quality of the generated content.",
"For example, in the case of an audio representation, we could use a spectrum representation (computed by a Fourier transform) instead of a raw waveform representation.",
"In the case of a symbolic representation, we could consider (as in most systems) enharmony, i.e.",
"A$\\sharp $ being equivalent to B$\\flat $ and C$\\flat $ being equivalent to B, or instead preserve the distinction in order to keep the harmonic and/or voice leading meaning."
],
[
"Phases and Types of Data",
"Before getting into the choices of representation for the various data to be processed by a deep learning architecture, it is important to identify the two main phases related to the activity of a deep learning architecture: the training phase and the generation phase, as well as the related fourThere may be more types of data depending on the complexity of the architecture, which may include intermediate processing steps.",
"main types of data to be considered: Training phase Training data The set of examples used for training the deep learning system; Validation data (alsoActually, a difference could be made, as will be later explained in Section REF .",
"named Test data) The set of examples used for testing the deep learning systemThe motivation will be introduced in Section REF .. Generation phase Generation (input) data The data that will be used as input for the generation, e.g., a melody for which the system will generate an accompaniment, or a note that will be the first note of a generated melody; Generated (output) data The data produced by the generation, as specified by the objective.",
"Depending on the objectiveAs stated in Section REF , we identify an objective by its type as a matter of simplification., these four types of data may be equal or differentActually, training data and validation data are of the same kind, being both input data of the same architecture.. For instance: in the case of the generation of a melody (for example, in Section REF ), both the training data and the generated data are melodies; whereas in the case of the generation of a counterpoint accompaniment (for example, in Section REF ), the generated data is a set of melodies."
],
[
"Audio versus Symbolic",
"A big divide in terms of the choice of representation (both for input and output) is audio versus symbolic.",
"This also corresponds to the divide between continuous and discrete variables.",
"As we will see, their respective raw material is very different in nature, as are the types of techniques for possible processing and transformation of the initial representationThe initial representation may be transformed, through, e.g., data compression or extraction of higher-level representations, in order to improve learning and/or generation..",
"They in fact correspond to different scientific and technical communities, namely signal processing and knowledge representation.",
"However, the actual processing of these two main types of representation by a deep learning architecture is basically the sameIndeed, at the level of processing by a deep network architecture, the initial distinction between audio and symbolic representation boils down, as only numerical values and operations are considered..",
"Therefore, actual audio and symbolic architectures for music generation may be pretty similar.",
"For example, the WaveNet audio generation architecture (to be introduced in Section REF ) has been transposed into the MidiNet symbolic music generation architecture (in Section REF ).",
"This polymorphism (possibility of multiple representations leading to genericity) is an additional advantage of the deep learning approach.",
"That said, we will focus in this book on symbolic representations and on deep learning techniques for generation of symbolic music.",
"There are various reasons for this choice: the grand majority of the current deep learning systems for music generation are symbolic; we believe that the essence of music (as opposed to soundWithout minimizing the importance of the orchestration and the production.)",
"is in the compositional process, which is exposed via symbolic representations (like musical scores or lead sheets) and is subject to analysis (e.g., harmonic analysis); covering the details and variety of techniques for processing and transforming audio representations (e.g., spectrum, cepstrum, MFCCMel-frequency cepstral coefficients., etc.)",
"would necessitate an additional bookAn example entry point is the recent review by Wyse of audio representations for deep convolutional networks [208].",
"; and as stated previously, independently of considering audio or symbolic music generation, the principles of deep learning architectures as well as the encoding techniques used are actually pretty similar.",
"Last, let us mention a recent deep learning-based architecture which combines audio and symbolic representations.",
"In this proposal from Manzelli et al.",
"[126], a symbolic representation is used as a conditioning inputConditioning will be introduced in Section REF .",
"in addition to the audio representation main input, in order to better guide and structure the generation of (audio) music (see more details in Section REF )."
],
[
"Audio",
"The first type of representation of musical content is audio signal, either in its raw form (waveform) or transformed."
],
[
"Waveform",
"The most direct representation is the raw audio signal: the waveform.",
"The visualization of a waveform is shown in Figure REF and another one with a finer grain resolution is shown in Figure REF .",
"In both figures, the x axis represents time and the y axis represents the amplitude of the signal.",
"Figure: Example of a waveformFigure: Example of a waveform with a fine grain resolution.Excerpt from a waveform visualization (sound of a guitar) by Michael Jancsyreproduced from “https://plot.ly/~michaeljancsy/205.embed”with permission of the authorThe advantage of using a waveform is in considering the raw material untransformed, with its full initial resolution.",
"Architectures that process the raw signal are sometimes named end-to-end architecturesThe term end-to-end emphasizes that a system learns all features from raw unprocessed data – without any pre-processing, transformation of representation, or extraction of features – to produce the final output..",
"The disadvantage is in the computational load: low level raw signal is demanding in terms of both memory and processing."
],
[
"Transformed Representations",
"Using transformed representations of the audio signal usually leads to data compression and higher-level information, but as noted previously, at the cost of losing some information and introducing some bias."
],
[
"Spectrogram",
"A common transformed representation for audio is the spectrum, obtained via a Fourier transformThe objective of the Fourier transform (which could be continuous or discrete) is the decomposition of an arbitrary signal into its elementary components (sinusoidal waveforms).",
"As well as compressing the information, its role is fundamental for musical purposes as it reveals the harmonic components of the signal..",
"Figure REF shows an example of a spectrogram, a visual representation of a spectrum, where the x axis represents time (in seconds), the y axis represents the frequency (in kHz) and the third axis in color represents the intensity of the sound (in dBFSDecibel relative to full scale, a unit of measurement for amplitude levels in digital systems.).",
"Figure: Example of a spectrogram of the spoken words “nineteenth century”.Reproduced from Aquegg's original image at “https://en.wikipedia.org/wiki/Spectrogram”"
],
[
"Chromagram",
"A variation of the spectrogram, discretized onto the tempered scale and independent of the octave, is a chromagram.",
"It is restricted to pitch classesA pitch class (also named a chroma) represents the name of the corresponding note independently of the octave position.",
"Possible pitch classes are C, C$\\sharp $ (or D$\\flat $ ), D, ... A$\\sharp $ (or B$\\flat $ ) and B..",
"The chromagram of the C major scale played on a piano is illustrated in Figure REF .",
"The x axis common to the four subfigures (a to d) represents time (in seconds).",
"The y axis of the score (a) represents the note, the y axis of the chromagrams (b and d) represents the chroma (pitch class) and the y axis of the signal (c) represents the amplitude.",
"For chromagrams (b and d), the third axis in color represents the intensity.",
"Figure: Examples of chromagrams.",
"(a) Musical score of a C-major scale.",
"(b) Chromagram obtained from the score.",
"(c) Audio recording of the C-major scale played on a piano.",
"(d) Chromagram obtained from the audio recording.Reproduced from Meinard Mueller's original image at “https://en.wikipedia.org/wiki/Chroma_feature” under a CC BY-SA 3.0 licenceSymbolic representations are concerned with concepts like notes, duration and chords, which will be introduced in the following sections."
],
[
"Note",
"In a symbolic representation, a note is represented through the following main features, and for each feature there are alternative ways of specifying its value: Pitch – specified by frequency, in Hertz (Hz); vertical position (height) on a score; or pitch notationAlso named international pitch notation or scientific pitch notation., which combines a musical note name, e.g., A, A$\\sharp $ , B, etc.",
"– actually its pitch class – and a number (usually notated in subscript) identifying the pitch class octave which belongs to the $[-1, 9]$ discrete interval.",
"An example is A$_4$ , which corresponds to A440 – with a frequency of 440 Hz – and serves as a general pitch tuning standard.",
"Duration – specified by absolute value, in milliseconds (ms); or relative value, notated as a division or a multiple of a reference note duration, i.e.",
"the whole note .",
"Examples are a quarter noteNamed a crotchet in British English.",
"and an eighth noteNamed a quaver in British English.",
".",
"Dynamics – specified by absolute and quantitative value, in decibels (dB); or qualitative value, an annotation on a score about how to perform the note, which belongs to the discrete set $\\lbrace $$,$ $,$ $,$ ,$ $ ,$ $ }$, from pianissimo to fortissimo.$"
],
[
"Rest",
"Rests are important in music as they represent intervals of silence allowing a pause for breathAs much for appreciation of the music as for respiration by human performer(s)!.",
"A rest can be considered as a special case of a note, with only one feature, its duration, and no pitch or dynamics.",
"The duration of a rest may be specified by absolute value, in milliseconds (ms); or relative value, notated as a division or a multiple of a reference rest duration, the whole rest having the same duration as a whole note .",
"Examples are a quarter rest and an eighth rest , corresponding respectively to a quarter note and an eighth note ."
],
[
"Interval",
"An interval is a relative transition between two notes.",
"Examples are a major third (which includes 4 semitones), a minor third (3 semitones) and a (perfect) fifth (7 semitones).",
"Intervals are the basis of chords (to be introduced in the next section).",
"For instance, the two main chords in classical music are major (with a major third and a fifth) and minor (with a minor third and a fifth).",
"In the pioneering experiments described in [190], Todd discusses an alternative way for representing the pitch of a note.",
"The idea is not to represent it in an absolute way as in Section REF , but in a relative way by specifying the relative transition (measured in semitones), i.e.",
"the interval, between two successive notes.",
"For example, the melody C$_4$ , E$_4$ , G$_4$ would be represented as C$_4$ , +4, +3.",
"In [190], Todd points out as two advantages the fact that there is no fixed bounding of the pitch range and the fact that it is independent of a given key (tonality).",
"However, he also points out that this second advantage may also be a major drawback, because in case of an error in the generation of an interval (resulting in a change of key), the wrong tonality (because of a wrong index) will be maintained in the rest of the melody generated.",
"Another limitation is that this strategy applies only to the specification of a monophonic melody and cannot directly represent a single-voice polyphony, unless separating the parallel intervals into different voices.",
"Because of these pros and cons, an interval-based representation is actually rarely used in deep learning-based music generation systems."
],
[
"Chord",
"A representation of a chord, which is a set of at least 3 notes (a triad)Modern music extends the original major and minor triads into a huge set of richer possibilities (diminished, augmented, dominant 7th, suspended, 9th, 13th, etc.)",
"by adding and/or altering intervals/components., could be implicit and extensional, enumerating the exact notes composing it.",
"This permits the specification of the precise octave as well as the position (voicing) for each note, see an example in Figure REF ; or explicit and intensional, by using a chord symbol combining the pitch class of its root note, e.g., C, and the type, e.g., major, minor, dominant seventh, or diminishedThere are some abbreviated notations, frequent in jazz and popular music, for example C minor = Cmin = Cm = C-; C major seventh = CM7 = Cmaj7 = C$\\Delta $ , etc..",
"Figure: C major chord with an open position/voicing: 1-5-3 (root, 5th and 3rd)We will see that the extensional approach (explicitly listing all component notes) is more common for deep learning-based music generation systems, but there are some examples of systems representing chords explicitly with the intensional approach, as for instance the MidiNet system to be introduced in Section REF ."
],
[
"Rhythm",
"Rhythm is fundamental to music.",
"It conveys the pulsation as well as the stress on specific beats, indispensable for dance!",
"Rhythm introduces pulsation, cycles and thus structure in what would otherwise remain a flat linear sequence of notes."
],
[
"Beat and Meter",
"A beat is the unit of pulsation in music.",
"Beats are grouped into measures, separated by barsAlthough (and because) a bar is actually the graphical entity – the line segment “|” – separating measures, the term bar is also often used, specially in the United States, in place of measure.",
"In this book we will stick to the term measure..",
"The number of beats in a measure as well as the duration between two successive beats constitute the rhythmic signature of a measure and consequently of a piece of musicFor more elaborate music, the meter may change within different portions of the music..",
"This time signature is also often named meter.",
"It is expressed as the fraction $numberOfBeats/BeatDuration$ , where $numberOfBeats$ is the number of beats within a measure; and $beatDuration$ is the duration between two beats.",
"As with the relative duration of a note (see Section REF ) or of a rest, it is expressed as a division of the duration of a whole note .",
"More frequent meters are 2/4, 3/4 and 4/4.",
"For instance, 3/4 means 3 beats per measure, each one with the duration of a quarter note .",
"It is the rhythmic signature of a Waltz.",
"The stress (or accentuation) on some beats or their subdivisions may form the actual style of a rhythm for music as well as for a dance, e.g., ternary jazz versus binary rock."
],
[
"Levels of Rhythm Information",
"We may consider three different levels in terms of the amount and granularity of information about rhythm to be included in a musical representation for a deep learning architecture: None – only notes and their durations are represented, without any explicit representation of measures.",
"This is the case for most systems.",
"Measures – measures are explicitly represented.",
"An example is the system described in Section REFIt is interesting to note that, as pointed out by Sturm et al.",
"in [179], the generated music format also contains bars separating measures and that there is no guarantee that the number of notes in a measure will always fit to a measure.",
"However, errors rarely occur, indicating that this representation is sufficient for the architecture to learn to count, see [60] and Section REF ..",
"Beats – information about meter, beats, etc.",
"is included.",
"An example is the C-RBM system described in Section REF , which allows us to impose a specific meter and beat stress for the music to be generated."
],
[
"Multivoice/Multitrack",
"A multivoice representation, also named multitrack, considers independent various voices, each being a different vocal range (e.g., soprano, alto...) or a different instrument (e.g., piano, bass, drums...).",
"Multivoice music is usually modeled as parallel tracks, each one with a distinct sequence of notesWith possibly simultaneous notes for a given voice, see Section REF ., sharing the same meter but possibly with different strong (stressed) beatsDance music is good at this, by having some syncopated bass and/or guitar not aligned on the strong drum beats, in order to create some bouncing pulse..",
"Note that in some cases, although there are simultaneous notes, the representation will be a single-voice polyphony, as introduced in Section REF .",
"Common examples are polyphonic instruments like a piano or a guitar.",
"Another example is a drum or percussion kit, where each of the various components, e.g., snare, hi-hat, ride cymbal, kick, etc., will usually be considered as a distinct note for the same voice.",
"The different ways to encode single-voice polyphony and multivoice polyphony will be further discussed in Section REF ."
],
[
"Format",
"The format is the language (i.e.",
"grammar and syntax) in which a piece of music is expressed (specified) in order to be interpreted by a computerThe standard format for humans is a musical score.."
],
[
"MIDI",
"Musical Instrument Digital Interface (MIDI) is a technical standard that describes a protocol, a digital interface and connectors for interoperability between various electronic musical instruments, softwares and devices [133].",
"MIDI carries event messages that specify real-time note performance data as well as control data.",
"We only consider here the two most important messages for our concerns: Note on – to indicate that a note is played.",
"It contains a channel number, which indicates the instrument or track, specified by an integer within the set $\\lbrace 0, 1,\\ldots ~, 15\\rbrace $ ; a MIDI note number, which indicates the note pitch, specified by an integer within the set $\\lbrace 0, 1,\\ldots ~, 127\\rbrace $ ; and a velocity, which indicates how loud the note is playedFor a keyboard, it means the speed of pressing down the key and therefore corresponds to the volume., specified by an integer within the set $\\lbrace 0, 1,\\ldots ~, 127\\rbrace $ .",
"An example is “Note on, 0, 60, 50” which means “On channel 1, start playing a middle C with velocity 50”; Note off – to indicate that a note ends.",
"In this situation, the velocity indicates how fast the note is released.",
"An example is “Note off, 0, 60, 20” which means “On channel 1, stop playing a middle C with velocity 20”.",
"Each note event is actually embedded into a track chunk, a data structure containing a delta-time value which specifies the timing information and the event itself.",
"A delta-time value represents the time position of the event and could represent a relative metrical time – the number of ticks from the beginning.",
"A reference, named the division and defined in the file header, specifies the number of ticks per quarter note ; or an absolute time – useful for real performances, not detailed here, see [133].",
"An example of an excerpt from a MIDI file (turned into readable ascii) and its corresponding score are shown in Figures REF and REF .",
"The division has been set to 384, i.e.",
"384 ticks per quarter note (which corresponds to 96 ticks for a sixteenth note ).",
"Figure: Excerpt from a MIDI fileFigure: Score corresponding to the MIDI excerptIn [87], Huang and Hu claim that one drawback of encoding MIDI messages directly is that it does not effectively preserve the notion of multiple notes being played at once through the use of multiple tracks.",
"In their experiment, they concatenate tracks end-to-end and thus posit that it will be difficult for such a model to learn that multiple notes in the same position across different tracks can really be played at the same time.",
"Piano roll, to be introduced in next section, does not have this limitation but at the cost of another limitation."
],
[
"Piano Roll",
"The piano roll representation of a melody (monophonic or polyphonic) is inspired from automated pianos (see Figure REF ).",
"This was a continuous roll of paper with perforations (holes) punched into it.",
"Each perforation represents a piece of note control information, to trigger a given note.",
"The length of the perforation corresponds to the duration of a note.",
"In the other dimension, the localization of a perforation corresponds to its pitch.",
"Figure: Automated piano and piano roll.Reproduced from Yaledmot's post “https://www.youtube.com/watch?v=QrcwR7eijyc” with permission of YouTubeAn example of a modern piano roll representation (for digital music systems) is shown in Figure REF .",
"The x axis represents time and the y axis the pitch.",
"Figure: Example of symbolic piano roll.Reproduced from with permission of Hao Staff Music Publishing (Hong Kong) Co Ltd.There are several music environments using piano roll as a basic visual representation, in place of or in complement to a score, as it is more intuitive than the traditional score notationAnother notation specific to guitar or string instruments is a tablature, in which the six lines represent the chords of a guitar (four lines for a bass) and the note is specified by the number of the fret used to obtain it.. An example is Hao Staff piano roll sheet music [71], shown in Figure REF with the time axis being horizontal rightward and notes represented as green cells.",
"Another example is tabs, where the melody is represented in a piano roll-like format [85], in complement to chords and lyrics.",
"Tabs are used as an input by the MidiNet system, to be introduced in Section REF .",
"The piano roll is one of the most commonly used representations, although it has some limitations.",
"An important one, compared to MIDI representation, is that there is no note off information.",
"As a result, there is no way to distinguish between a long note and a repeated short noteActually, in the original mechanical paper piano roll, the distinction is made: two holes are different from a longer single hole.",
"The end of the hole is the encoding of the end of the note..",
"In Section REF , we will look at different ways to address this limitation.",
"For a more detailed comparison between MIDI and piano roll, see [87] and [200]."
],
[
"Melody",
"A melody can be encoded in a textual representation and processed as a text.",
"A significant example is the ABC notation [202], a de facto standard for folk and traditional musicNote that the ABC notation has been designed independently of computer music and machine learning concerns..",
"Figures REF and REF show the original score and its associated ABC notation for a tune named “A Cup of Tea”, from the repository and discussion platform The Session [99].",
"Figure: Score of “A Cup of Tea” (Traditional).Reproduced from The Session with permission of the managerFigure: ABC notation of “A Cup of Tea”.Reproduced from The Session with permission of the managerThe first six lines are the header and represent metadata: T is the title of the music, M is the meter, L is the default note length, K is the key, etc.",
"The header is followed by the main text representing the melody.",
"Some basic principles of the encoding rules of the ABC notation are as followsPlease refer to [202] for more details.",
": the pitch class of a note is encoded as the letter corresponding to its English notation, e.g., A for A or La; its pitch is encoded as following: A corresponds to A$_4$ , a to an A one octave up and a' to an A two octaves up; the duration of a note is encoded as following: if the default length is marked as 1/8 (i.e.",
"an eighth note , the case for the “A Cup of Tea” example), a corresponds to an eighth note , a/2 to a sixteenth note and a2 to a quarter note Note that rests may be expressed in the ABC notation through the z letter.",
"Their durations are expressed as for notes, e.g., z2 is a double length rest.",
"; and measures are separated by “|” (bars).",
"Note that the ABC notation can only represent monophonic melodies.",
"In order to be processed by a deep learning architecture, the ABC text is usually transformed from a character vocabulary text into a token vocabulary text in order to properly consider concepts which could be noted on more than one character, e.g., g2.",
"Sturm et al.",
"'s experiment, described in Section REF , uses a token-based notation named the folk-rnn notation [179].",
"A tune is enclosed within a “<s>” begin mark and an “<\\s>” end mark.",
"Last, all example melodies are transposed to the same C root base, resulting in the notation of the tune “A Cup of Tea” shown in Figure REF .",
"Figure: Folk-rnn notation of “A Cup of Tea”.Reproduced from with permission of the authors"
],
[
"Chord and Polyphony",
"When represented extensionally, chords are usually encoded with simultaneous notes as a vector.",
"An interesting alternative extensional representation of chords, named Chord2VecChord2Vec is inspired by the Word2Vec model for natural language processing [130]., has recently been proposed in [122]For information, there is another similar model, also named Chord2Vec, proposed in [88].. Rather than thinking of chords (vertically) as vectors, it represents chords (horizontally) as sequences of constituent notes.",
"More precisely, a chord is represented as an arbitrary length-ordered sequence of notes; and chords are separated by a special symbol, as with sentence markers in natural language processing.",
"When using this representation for predicting neighboring chords, a specific compound architecture is used, named RNN Encoder-Decoder which will be described in Section REF .",
"Note that a somewhat similar model is also used for polyphonic music generation by the BachBot system [119] which will be introduced in Section REF .",
"In this model, for each time step, the various notes (ordered in a descending pitch) are represented as a sequence and a special delimiter symbol “|||” indicates the next time frame."
],
[
"Markup Language",
"Let us mention the case of general text-based structured representations based on markup languages (famous examples are HTML and XML).",
"Some markup languages have been designed for music applications, like for instance the open standard MusicXML [62].",
"The motivation is to provide a common format to facilitate the sharing, exchange and storage of scores by musical software systems (such as score editors and sequencers).",
"MusicXML, as well as similar languages, is not intended for direct use by humans because of its verbosity, which is the down side of its richness and effectiveness as an interchange language.",
"Furthermore, it is not very appropriate as a direct representation for machine learning tasks for the same reasons, as its verbosity and richness would create too much overhead as well as bias."
],
[
"Lead Sheet",
"Lead sheets are an important representation format for popular music (jazz, pop, etc.).",
"A lead sheet conveys in upto a few pages the score of a melody and its corresponding chord progression via an intensional notationSee Section REF ..",
"Lyrics may also be added.",
"Some important information for the performer, such as the composer, author, style and tempo, is often also present.",
"An example of lead sheet in shown in Figure REF .",
"Figure: Lead sheet of “Very Late” (Pachet and d'Inverno).Reproduced with permission of the composersParadoxically, few systems and experiments use this rich and concise representation, and most of the time they focus on the notes.",
"Note that Eck and Schmidhuber's Blues generation system, to be introduced in Section REF , outputs a combination of melody and chord progression, although not as an explicit lead sheet.",
"A notable contribution is the systematic encoding of lead sheets done in the Flow Machines project [49], resulting in the Lead Sheet Data Base (LSDB) repository [147], which includes more than 12,000 lead sheets.",
"Note that there are some alternative notations, notably tabs [85], where the melody is represented in a piano roll-like format (see Section REF ) and complemented with the corresponding chords.",
"An example of use of tabs is the MidiNet system to be analyzed in Section REF ."
],
[
"Temporal Scope and Granularity",
"The representation of time is fundamental for musical processes."
],
[
"Temporal Scope",
"An initial design decision concerns the temporal scope of the representation used for the generation data and for the generated data, that is the way the representation will be interpreted by the architecture with respect to time, as illustrated in Figure REF : Global – in this first case, the temporal scope of the representation is the whole musical piece.",
"The deep network architecture (typically a feedforward or an autoencoder architecture, see Sections and ) will process the input and produce the output within a global single stepIn Chapter , we will name it the single-step feedforward strategy, see Section REF ..",
"Examples are the MiniBach and DeepHear systems introduced in Sections REF and REF , respectively.",
"Time step (or time slice) – in this second case, the most frequent one, the temporal scope of the representation is a local time slice of the musical piece, corresponding to a specific temporal moment (time step).",
"The granularity of the processing by the deep network architecture (typically a recurrent network) is a time step and generation is iterativeIn Chapter , we will name it the iterative feedforward strategy, see Section REF ..",
"Note that the time step is usually set to the shortest note duration (see more details in Section REF ), but it may be larger, e.g., set to a measure in the system as discussed in [190].",
"Note step – this third case was proposed by Mozer in [139] in his CONCERT system [139], see Section REF .",
"In this approach there is no fixed time step.",
"The granularity of processing by the deep network architecture is a note.",
"This strategy uses a distributed encoding of duration that allows to process a note of any duration in a single network processing step.",
"Note that, by considering as a single processing step a note rather than a time step, the number of processing steps to be bridged by the network is greatly reduced.",
"The approach proposed later on by Walder in [200] is similar.",
"Figure: Temporal scope for a piano roll-like representationNote that a global temporal scope representation actually also considers time steps (separated by dash lines in Figure REF ).",
"However, although time steps are present at the representation level, they will not be interpreted as distinct processing steps by the neural network architecture.",
"Basically, the encoding of the successive time slices will be concatenated into a global representation considered as a whole by the network, as shown in Figure REF of an example to be introduced in Section REF .",
"Also note that in the case of a global temporal scope the musical content generated has a fixed length (the number of time steps), whereas in the case of a time step or a note step temporal scope the musical content generated has an arbitrary length, because generation is iterative as we will see in Section REF ."
],
[
"Temporal Granularity",
"In the case of a global or a time step temporal scope, the granularity of the time step, corresponding to the granularity of the time discretization, must be defined.",
"There are two main strategies: The most common strategy is to set the time step to a relative duration, the smallest duration of a note in the corpus (training examples/dataset), e.g., a sixteenth note .",
"To be more precise, as stated by Todd in [190], the time step should be the greatest common factor of the durations of all the notes to be learned.",
"This ensures that the duration of every note will be properly represented with a whole number of time steps.",
"One immediate consequence of this “leveling down” is the number of processing steps necessary, independent of the duration of actual notes.",
"Another strategy is to set the time step to a fixed absolute duration, e.g., 10 milliseconds.",
"This strategy permits us to capture expressiveness in the timing of each note during a human performance, as we will see in Section .",
"Note that in the case of a note step temporal scope, there is no uniform discretization of time (no fixed time step) and no need for."
],
[
"Metadata",
"In some systems, additional information from the score may also be explicitly represented and used as metadata, such as note tieA tied note on a music score specifies how a note duration extends across a single measure.",
"In our case, the issue is how to specify that the duration extends across a single time step.",
"Therefore, we consider it as metadata information, as it is specific to the representation and its processing by a neural network architecture., fermata, harmonics, key, meter, and the instrument associated to a voice.",
"This extra information may lead to more accurate learning and generation."
],
[
"Note Hold/Ending",
"An important issue is how to represent if a note is held, i.e.",
"tied to the previous note.",
"This is actually equivalent to the issue of how to represent the ending of a note.",
"In the MIDI representation format, the end of a note is explicitly stated (via a “Note off” eventNote that, in MIDI, a “Note on” message with a null (0) velocity is interpreted as a “Note off” message.).",
"In the piano roll format discussed in Section REF , there is no explicit representation of the ending of a note and, as a result, one cannot distinguish between two repeated quarter notes and a half note .",
"The main possible techniques are to introduce a hold/replay representation, as a dual representation of the sequence of notes.",
"This solution is used, for example, by Mao et al.",
"in their DeepJ system [127] (to be analyzed in Section REF ), by introducing a replay matrix similar to the piano roll-type matrix of notes; to divide the size of the time stepSee Section REF for details of how the value of the time step is defined.",
"by two and always mark a note ending with a special tag, e.g., 0.",
"This solution is used, for example, by Eck and Schmidhüber in [43], and will be analyzed in Section REF ; to divide the size of the time step as before but instead mark a new note beginning.",
"This solution is used by Todd in [190]; or to use a special hold symbol “__” in place of a note to specify when the previous note is held.",
"This solution was proposed by Hadjeres et al.",
"in their DeepBach system [70] to be analyzed in Section REF .",
"Figure: a) Extract from a J. S. Bach chorale and b) its representation using the hold symbol “__”.Reproduced from with permission of the authorsThis last solution considers the hold symbol as a note, see an example in Figure REF .",
"The advantages of the hold symbol technique are it is simple and uniform as the hold symbol is considered as a note; and there is no need to divide the value of the time step by two and mark a note ending or beginning.",
"The authors of DeepBach also emphasize that the good results they obtain using Gibbs sampling rely exclusively on their choice to integrate the hold symbol into the list of notes (see [70] and Section REF ).",
"An important limitation is that the hold symbol only applies to the case of a monophonic melody, that is it cannot directly express held notes in an unambiguous way in the case of a single-voice polyphony.",
"In this case, the single-voice polyphony must be reformulated into a multivoice representation with each voice being a monophonic melody; then a hold symbol is added separately for each voice.",
"Note that in the case of the replay matrix, the additional information (matrix row) is for each possible note and not for each voice.",
"We will discuss in Section REF how to encode a hold symbol."
],
[
"Note Denotation (versus Enharmony)",
"Most systems consider enharmony, i.e.",
"in the tempered system A$\\sharp $ is enharmonically equivalent to (i.e.",
"has the same pitch as) B$\\flat $ , although harmonically and in the composer's intention they are different.",
"An exception is the DeepBach system, described in Section REF , which encodes notes using their real names and not their MIDI note numbers.",
"The authors of DeepBach state that this additional information leads to a more accurate model and better results [70]."
],
[
"Feature Extraction",
"Although deep learning is good at processing raw unstructured data, from which its hierarchy of layers will extract higher-level representations adapted to the task (see Section REF ), some systems include a preliminary step of automatic feature extraction, in order to represent the data in a more compact, characteristic and discriminative form.",
"One motivation could be to gain efficiency and accuracy for the training and for the generation.",
"Moreover, this feature-based representation is also useful for indexing data, in order to control generation through compact labeling (see, for example, the DeepHear system in Section REF ), or for indexing musical units to be queried and concatenated (see Section REF ).",
"The set of features can be defined manually (handcrafted) or automatically (e.g.",
"by an autoencoder, see Section ).",
"In the case of handcrafted features, the bag-of-words (BOW) model is a common strategy for natural language text processing, which may also be applied to other types of data, including musical data, as we will see in Section REF .",
"It consists in transforming the original text (or arbitrary representation) into a “bag of words” (the vocabulary composed of all occurring words, or more generally speaking, all possible tokens); then various measures can be used to characterize the text.",
"The most common is term frequency, i.e.",
"the number of times a term appears in the textNote that this bag-of-words representation is a lossy representation (i.e.",
"without effective means to perfectly reconstruct the original data representation).. Sophisticated methods have been designed for neural network architectures to automatically compute a vector representation which preserves, as much as possible, the relations between the items.",
"Vector representations of texts are named word embeddingsThe term embedding comes from the analogy with mathematical embedding, which is an injective and structure-preserving mapping.",
"Initially used for natural language processing, it is now often used in deep learning as a general term for encoding a given representation into a vector representation.",
"Note that the term embedding, which is an abstract model representation, is often also used (we think, abusively) to define a specific instance of an embedding (which may be better named, for example, a label, see [180] and Section REF ).. A recent reference model for natural language processing (NLP) is the Word2Vec model [130].",
"It has recently been transposed to the Chord2Vec model for the vector encoding of chords, as described in [122] (see Section REF )."
],
[
"Timing",
"If training examples are processed from conventional scores or MIDI-format libraries, there is a good chance that the music is perfectly quantized – i.e., note onsetsAn onset refers to the beginning of a musical note (or sound).",
"are exactly aligned onto the tempo – resulting in a mechanical sound without expressiveness.",
"One approach is to consider symbolic records – in most cases recorded directly in MIDI – from real human performances, with the musician interpreting the tempo.",
"An example of a system for this purpose is Performance RNN [174], which will be analyzed in Section REF .",
"It follows the absolute time duration quantization strategy, presented in Section REF ."
],
[
"Dynamics",
"Another common limitation is that many MIDI-format libraries do not include dynamics (the volume of the sound produced by an instrument), which stays fixed throughout the whole piece.",
"One option is to take into consideration (if present on the score) the annotations made by the composer about the dynamics, from pianissimo to fortissimo , see Section REF .",
"As for tempo expressiveness, addressed in Section REF , another option is to use real human performances, recorded with explicit dynamics variation – the velocity field in MIDI."
],
[
"Audio",
"Note that in the case of an audio representation, expressiveness as well as tempo and dynamics are entangled within the whole representation.",
"Although it is easy to control the global dynamics (global volume), it is less easy to separately control the dynamics of a single instrument or voiceMore generally speaking, audio source separation, often coined as the cocktail party effect, has been known for a long time to be a very difficult problem, see the original article in [19].",
"Interestingly, this problem has been solved in 2015 by deep learning architectures [46], opening up ways for disentangling instruments or voices and their relative dynamics as well as tempo (by using audio time stretching techniques).."
],
[
"Encoding",
"Once the format of a representation has been chosen, the issue still remains of how to encode this representation.",
"The encoding of a representation (of a musical content) consists in the mapping of the representation (composed of a set of variables, e.g., pitch or dynamics) into a set of inputs (also named input nodes or input variables) for the neural network architectureSee Section for more details about the input nodes of a neural network architecture.."
],
[
"Strategies",
"At first, let us consider the three possible types for a variable: Continuous variables – an example is the pitch of a note defined by its frequency in Hertz, that is a real value within the $]0, +\\infty [$ intervalThe notation $]0, +\\infty [$ is for an open interval excluding its endpoints.",
"An alternative notation is $(0, +\\infty )$ ..",
"The straightforward way is to directly encode the variableIn practice, the different variables are also usually scaled and normalized, in order to have similar domains of values ($[0, 1]$ or $[-1, +1]$ ) for all input variables, in order to ease learning convergence.",
"as a scalar whose domain is real values.",
"We call this strategy value encoding.",
"Discrete integer variables – an example is the pitch of a note defined by its MIDI note number, that is an integer value within the $\\lbrace 0, 1,\\ldots ~, 127\\rbrace $ discrete setSee our summary of MIDI specification in Section REF ..",
"The straightforward way is to encode the variable as a real value scalar, by casting the integer into a real.",
"This is another case of value encoding.",
"Boolean (binary) variables – an example is the specification of a note ending (see Section REF ).",
"The straightforward way is to encode the variable as a real value scalar, with two possible values: 1 (for true) and 0 (for false).",
"Categorical variablesIn statistics, a categorical variable is a variable that can take one of a limited – and usually fixed – number of possible values.",
"In computer science it is usually referred as an enumerated type.",
"– an example is a component of a drum kit; an element within a set of possible values: $\\lbrace $ snare, high-hat, kick, middle-tom, ride-cymbal, etc.$\\rbrace $ .",
"The usual strategy is to encode a categorical variable as a vector having as its length the number of possible elements, in other words the cardinality of the set of possible values.",
"Then, in order to represent a given element, the corresponding element of the encoding vector is set to 1 and all other elements to 0.",
"Therefore, this encoding strategy is usually called one-hot encodingThe name comes from digital circuits, one-hot referring to a group of bits among which the only legal (possible) combinations of values are those with a single high (hot!)",
"(1) bit, all the others being low (0)..",
"This frequently used strategy is also often employed for encoding discrete integer variables, such as MIDI note numbers."
],
[
"From One-Hot to Many-Hot and to Multi-One-Hot",
"Note that a one-hot encoding of a note corresponds to a time slice of a piano roll representation (see Figure REF ), with as many lines as there are possible pitches.",
"Note also that while a one-hot encoding of a piano roll representation of a monophonic melody (with one note at a time) is straightforward, a one-hot encoding of a polyphony (with simultaneous notes, as for a guitar playing a chord) is not.",
"One could then consider many-hot encoding – where all elements of the vector corresponding to the notes or to the active components are set to 1; multi-one-hot encoding – where different voices or tracks are considered (for multivoice representation, see Section ) and a one-hot encoding is used for each different voice/track; or multi-many-hot encoding – which is a multivoice representation with simultaneous notes for at least one or all of the voices."
],
[
"Summary",
"The various approaches for encoding are illustrated in Figure REF , showing from left to right a scalar continuous value encoding of A$_4$ (A440), the real number specifying its frequency in Hertz; a scalar discrete integer value encodingNote that, because the processing level of an artificial neural network only considers real values, an integer value will be casted into a real value.",
"Thus, the case of a scalar integer value encoding boils down to the previous case of a scalar continuous value encoding.",
"of A$_4$ , the integer number specifying its MIDI note number; a one-hot encoding of A$_4$ ; a many-hot encoding of a D minor chord (D$_4$ , F$_4$ , A$_4$ ); a multi-one-hot encoding of a first voice with A$_4$ and a second voice with D$_3$ ; and a multi-many-hot encoding of a first voice with a D minor chord (D$_4$ , F$_4$ , A$_4$ ) and a second voice with C$_3$ (corresponding to a minor seventh on bass).",
"Figure: Various types of encoding"
],
[
"Binning",
"In some cases, a continuous variable is transformed into a discrete domain.",
"A common technique, named binning, or also bucketing, consists of dividing the original domain of values into smaller intervalsThis can be automated through a learning process, e.g., by automatic construction of a decision tree., named bins; and replacing each bin (and the values within it) by a value representative, often the central value.",
"Note that this binning technique may also be used to reduce the cardinality of the discrete domain of a variable.",
"An example is the Performance RNN system described in Section REF , for which the initial MIDI set of 127 values for note dynamics is reduced into 32 bins."
],
[
"Pros and Cons",
"In general, value encoding is rarely used except for audio, whereas one-hot encoding is the most common strategy for symbolic representationLet us remind (as pointed out in Section ) that, at the level of the encoding of a representation and its processing by a deep network, the distinction between audio and symbolic representation boils down to nothing, as only numerical values and operations are considered.",
"In fact the general principles of a deep learning architecture are independent of that distinction and this is one of the vectors of the generality of the approach.",
"See also in [126] the example of an architecture (to be introduced in Section REF ) which combines audio and symbolic representations.. A counterexample is the case of the DeepJ symbolic generation system described in Section REF , which is, in part, inspired by the WaveNet audio generation system.",
"DeepJ's authors state that: “We keep track of the dynamics of every note in an N x T dynamics matrix that, for each time step, stores values of each note's dynamics scaled between 0 and 1, where 1 denotes the loudest possible volume.",
"In our preliminary work, we also tried an alternate representation of dynamics as a categorical value with 128 bins as suggested by Wavenet [194].",
"Instead of predicting a scalar value, our model would learn a multinomial distribution of note dynamics.",
"We would then randomly sample dynamics during generation from this multinomial distribution.",
"Contrary to Wavenet's results, our experiments concluded that the scalar representation yielded results that were more harmonious.” [127].",
"The advantage of value encoding is its compact representation, at the cost of sensibility because of numerical operations (approximations).",
"The advantage of one-hot encoding is its robustness (discrete versus analog), at the cost of a high cardinality and therefore a potentially large number of inputs.",
"It is also important to understand that the choice of one-hot encoding at the output of the network architecture is often (albeit not always) associated to a softmax functionIntroduced in Section REF .",
"in order to compute the probabilities of each possible value, for instance the probability of a note being an A, or an A$\\sharp $ , a B, a C, etc.",
"This actually corresponds to a classification task between the possible values of the categorical variable.",
"This will be further analyzed in Section REF ."
],
[
"Chords",
"Two methods of encoding chords, corresponding to the two main alternative representations discussed in Section REF , are implicit and extensional – enumerating the exact notes composing the chord.",
"The natural encoding strategy is many-hot.",
"An example is the RBM-based polyphonic music generation system described in Section REF ; and explicit and intensional – using a chord symbol combining a pitch class and a type (e.g., D minor).",
"The natural encoding strategy is multi-one-hot, with an initial one-hot encoding of the pitch class and a second one-hot encoding of the class type (major, minor, dominant seventh, etc.).",
"An example is the MidiNet systemIn MidiNet, the possible chord types are actually reduced to only major and minor.",
"Thus, a boolean variable can be used in place of one-hot encoding.",
"described in Section REF ."
],
[
"Special Hold and Rest Symbols",
"We have to consider the case of special symbols for hold (“hold previous note”, see Section REF ) and rest (“no note”, see Section REF ) and how they relate to the encoding of actual notes.",
"First, note that there are some rare cases where the rest is actually implicit: in MIDI format – when there is no “active” “Note on”, that is when they all have been “closed” by a corresponding “Note off”; and in one-hot encoding – when all elements of the vector encoding the possible notes are equal to 0 (i.e.",
"a “zero-hot” encoding, meaning that none of the possible notes is currently selected).",
"This is for instance the case in the experiments by Todd (to be described in Section REF )This may appear at first as an economical encoding of a rest, but at the cost of some ambiguity when interpreting probabilities (for each possible note) produced by the softmax output of the network architecture.",
"A vector with low probabilities for each note may be interpreted as a rest or as an equiprobability between notes.",
"See the threshold trick proposed in Section REF in order to discriminate between the two possible interpretations.. Now, let us consider how to encode hold and rest depending on how a note pitch is encoded: value encoding – In this case, one needs to add two extra boolean variables (and their corresponding input nodes) hold and rest.",
"This must be done for each possible independent voice in the case of a polyphony; or one-hot encoding – In that case (the most frequent and manageable strategy), one just needs to extend the vocabulary of the one-hot encoding with two additional possible values: hold and rest.",
"They will be considered at the same level, and of the same nature, as possible notes (e.g., A$_3$ or C$_4$ ) for the input as well as for the output."
],
[
"Drums and Percussion",
"Some systems explicitly consider drums and/or percussion.",
"A drum or percussion kit is usually modeled as a single-track polyphony by considering distinct simultaneous “notes”, each “note” corresponding to a drum or percussion component (e.g., snare, kick, bass tom, hi-hat, ride cymbal, etc.",
"), that is as a many-hot encoding.",
"An example of a system dedicated to rhythm generation is described in Section REF .",
"It follows the single-track polyphony approach.",
"In this system, each of the five components is represented through a binary value, specifying whether or not there is a related event for current time step.",
"Drum events are represented as a binary wordIn this system, encoding is made in text, similar to the format described in Section REF and more precisely following the approach proposed in [22].",
"of length 5, where each binary value corresponds to one of the five drum components; for instance, 10010 represents simultaneous playing of the kick (bass drum) and the high-hat, following a many-hot encoding.",
"Note that this system also includes – as an additional voice/track – a condensed representation of the bass line part and some information representing the meter, see more details in Section REF .",
"The authors [123] argue that this extra explicit information ensures that the network architecture is aware of the beat structure at any given point.",
"Another example is the MusicVAE system (see Section REF ), where nine different drum/percussion components are considered, which gives $2^9$ possible combinations, i.e.",
"$2^9 = 512$ different tokens."
],
[
"Dataset",
"The choice of a dataset is fundamental for good music generation.",
"At first, a dataset should be of sufficient size (i.e.",
"contain a sufficient number of examples) to guarantee accurate learningNeural networks and deep learning architectures need lots of examples to function properly.",
"However, one recent research area is about learning from scarce data.. As noted by Hadjeres in [67]: “I believe that this tradeoff between the size of a dataset and its coherence is one of the major issues when building deep generative models.",
"If the dataset is very heterogeneous, a good generative model should be able to distinguish the different subcategories and manage to generalize well.",
"On the contrary, if there are only slight differences between subcategories, it is important to know if the “averaged model” can produce musically-interesting results.”"
],
[
"Transposition and Alignment",
"A common technique in machine learning is to generate synthetic data as a way to artificially augment the size of the dataset (the number of training examples)This is named dataset augmentation., in order to improve accuracy and generalization of the learnt model (see Section REF ).",
"In the musical domain, a natural and easy way is transposition, i.e.",
"to transpose all examples in all keys.",
"In addition to artificially augmenting the dataset, this provides a key (tonality) invariance of all examples and thus makes the examples more generic.",
"Moreover, this also reduces sparsity in the training data.",
"This transposition technique is, for instance, used in the C-RBM system [109] described in Section REF .",
"An alternative approach is to transpose (align) all examples into a single common key.",
"This has been advocated for the RNN-RBM system [11] to facilitate learning, see Section REF ."
],
[
"Datasets and Libraries",
"A practical issue is the availability of datasets for training systems and also for evaluating and comparing systems and approaches.",
"There are some reference datasets in the image domain (e.g., the MNISTMNIST stands for Modified National Institute of Standards and Technology.",
"dataset about handwritten digits [113]), but none yet in the music domain.",
"However, various datasets or librariesThe difference between a dataset and a library is that a dataset is almost ready for use to train a neural network architecture, as all examples are encoded within a single file and in the same format, although some extra data processing may be needed in order to adapt the format to the encoding of the representation for the architecture or vice-versa; whereas a library is usually composed of a set of files, one for each example.",
"have been made public, with some examples listed below: the Classical piano MIDI database [105]; the JSB Chorales datasetNote that this dataset uses a quarter note quantization, whereas a smaller quantization at the level of a sixteenth note should be used in order to capture the smallest note duration (eighth note), see Section REF .",
"[1]; the LSDB (Lead Sheet Data Base) repository [147], with more than 12,000 lead sheets (including from all jazz and bossa nova song books), developed within the Flow Machines project [49]; the MuseData library, an electronic library of classical music with more than 800 pieces, from CCARH in Stanford University [77]; the MusicNet dataset [188], a collection of 330 freely-licensed classical music recordings together with over 1 million annotated labels (indicating timing and instrumental information); the Nottingham database, a collection of 1,200 folk tunes in the ABC notation [54], each tune consisting of a simple melody on top of chords, in other words an ABC equivalent of a lead sheet; the Session [99], a repository and discussion platform for Celtic music in the ABC notation containing more than 15,000 songs; the Symbolic Music dataset by Walder [201], a huge set of cleaned and preprocessed MIDI files; the TheoryTab database [85], a set of songs represented in a tab format, a combination of a piano roll melody, chords and lyrics, in other words a piano roll equivalent of a lead sheet; the Yamaha e-Piano Competition dataset, in which participants MIDI performance records are made available [210]."
],
[
"Architecture",
"*Chapter Architecture presents the third dimension of the conceptual framework proposed in this book to analyze, classify and compare various deep learning-based music generation systems.",
"The architecture represents the set of computational units (neurons), grouped in layers, and their weighted connexions which process and generate a musical representation.",
"At first, we summarize the history and evolution of artificial neural network architectures.",
"An introduction to artificial neural networks is presented, starting from linear regression up to a typical neural network layer, considered as a basic building block.",
"From it, various architectures are derived and introduced: feedforward (MLP), recurrent (RNN), autoencoder, generative adversarial networks and others.",
"Various ways of composing architectures are also examined.",
"This chapter may be skipped by a reader already expert in neural networks and deep learning architectures.",
"Deep networks are a natural evolution of neural networks, themselves being an evolution of the Perceptron, proposed by Rosenblatt in 1957 [166].",
"Historically speakingSee, for example, [63] for a more detailed analysis of key trends in the history of deep learning., the Perceptron was criticized by Minsky and Papert in 1969 [131] for its inability to classify nonlinearly separable domainsA simple example and a counterexample of linear separability (of a set of four points within a 2-dimensional space and belonging to green cross or red circle classes) are shown in Figure REF .",
"The elements of the two classes are linearly separable if there is at least one straight line separating them.",
"Note that the discrete version of the counterexample corresponds to the case of the exclusive or (XOR) logical operator, which was used as an argument by Minsky and Papert in [131].. Their criticism also served in favoring an alternative approach of Artificial Intelligence, based on symbolic representations and reasoning.",
"Figure: Example and counterexample of linear separabilityNeural networks reappeared in the 1980s, thanks to the idea of hidden layers joint with nonlinear units, to resolve the initial linear separability limitation, and to the backpropagation algorithm, to train such multilayer neural networks [167].",
"In the 1990s, neural networks suffered declining interestMeanwhile, convolutional networks started to gain interest, notably though handwritten digit recognition applications [112].",
"As Goodfellow et al.",
"in [63] put it: “In many ways, they carried the torch for the rest of deep learning and paved the way to the acceptance of neural networks in general.” because of the difficulty in training efficiently neural networks with many layersAnother related limitation, although specific to the case of recurrent networks, was the difficulty in training them efficiently on very long sequences.",
"This was resolved in 1997 by Hochreiter and Schmidhuber with the Long short-term memory (LSTM) architecture [83], presented in Section REF .",
"and due to the competition from support vector machines (SVM) [197], which were efficiently designed to maximize the separation margin and had a solid formal background.",
"An important advance was the invention of the pre-training techniquePre-training consists in prior training in cascade (one layer at a time, also named greedy layer-wise unsupervised training) of each hidden layer [80] [63].",
"It turned out to be a significant improvement for the accurate training of neural networks with several layers [47].",
"That said, pre-training is now rarely used and has been replaced by other more recent techniques, such as batch normalization and deep residual learning.",
"But its underlying techniques are useful for addressing some new concerns like transfer learning, which deals with the issue of reusability (of what has been learnt, see Section ).",
"by Hinton et al.",
"in 2006 [80], which resolved this limitation.",
"In 2012, an image recognition competition (the ImageNet Large Scale Visual Recognition Challenge [168]) was won by a deep neural network algorithm named AlexNetAlexNet was designed by the SuperVision team headed by Hinton and composed of Alex Krizhevsky, Ilya Sutskever and Geoffrey E. Hinton [104].",
"AlexNet is a deep convolutional neural network with 60 million parameters and 650,000 neurons, consisting of five convolutional layers, some followed by max-pooling layers, and three globally-connected layers., with a stunning marginOn the first task, AlexNet won the competition with a 15% error rate whereas other teams did not achieve better than a 26% error rate.",
"over the other algorithms which were using handcrafted features.",
"This striking victory was the event which ended the prevalent opinion that neural networks with many hidden layers could not be efficiently trainedInterestingly, the title of Hinton et al.",
"'s article about pre-training [80] is about “deep belief nets” and does not mention the term “neural nets” because, as Hinton remembers it in [106]: “At that time, there was a strong belief that deep neural networks were no good and could never be trained and that ICML (International Conference on Machine Learning) should not accept papers about neural networks.”."
],
[
"Introduction to Neural Networks",
"The purpose of this section is to review, or to introduce, the basic principles of artificial neural networks.",
"Our objective is to define the key concepts and terminology that we will use when analyzing various music generation systems.",
"Then, we will introduce the concepts and basic principles of various derived architectures, like autoencoders, recurrent networks, RBMs, etc., which are used in musical applications.",
"We will not describe extensively the techniques of neural networks and deep learning, for example covered in the recent book [63].",
"Although bio-inspired (biological neurons), the foundation of neural networks and deep learning is linear regression.",
"In statistics, linear regression is an approach for modeling the (assumed linear) relationship between a scalar variable $\\text{y} \\in {\\rm I\\!R}$ and oneThe case of one explanatory variable is called simple linear regression, otherwise it is named multiple linear regression.",
"or more than one explanatory variable(s) x$_1$ ... x$_n$ , with x$_i \\in {\\rm I\\!R}$ , jointly noted as vector x.",
"A simple example is to predict the value of a house, depending on some factors (e.g., size, height, location...).",
"Equation REF gives the general model of a (multiple) linear regression, where $h(\\text{x}) = b + \\theta _1 \\text{x}_1 + ... + \\theta _n \\text{x}_n = b + \\sum \\limits _{i=1}^{n} \\theta _i \\text{x}_i$ $h$ is the model, also named hypothesis, as this is the hypothetical best model to be discovered, i.e.",
"learnt; $b$ is the biasIt could also be notated as $\\theta _0$ , see Section REF ., representing the offset; and $\\theta _1$ ... $\\theta _n$ are the parameters of the model, the weights, corresponding to the explanatory variables x$_1$ ... x$_n$ ."
],
[
"Notations",
"We will use the following simple notation conventions a constant is in roman (straight) font, e.g., integer 1 and note C$_4$ .",
"a variable of a model is in roman font, e.g., input variable x and output variable y (possibly vectors).",
"a parameter of a model is in italics, e.g., bias $b$ , weight parameter $\\theta _1$ , model function $h$ , number of explanatory variables $n$ and index $i$ of a variable x$_i$ .",
"a probability as well as a probability distribution are in italics and upper case, e.g., probability $P(\\text{note} = \\text{A}_4)$ that the value of variable note is A$_4$ and probability distribution $P(\\text{note})$ of variable note over all possible notes (outcomes)."
],
[
"Model Training",
"The purpose of training a linear regression model is to find the values for each weight $\\theta _i$ and the bias $b$ that best fit the actual training data/examples, i.e.",
"various pairs of values $(\\text{x}, \\text{y})$ .",
"In other words, we want to find the parameters and bias values such that for all values of x, $h(\\text{x})$ is as close as possibleActually, for the neural networks that are more complex (nonlinear models) than linear regression and that will be introduced in Section , the best fit to the training data is not necessarily the best hypothesis because it may have a low generalization, i.e.",
"a low ability to predict yet unseen data.",
"This issue, named overfitting, will be introduced in Section REF .",
"to y, according to some measure named the cost.",
"This measure represents the distance between $h(\\text{x})$ (the prediction, also notated as ŷ) and y (the actual ground value), for all examples.",
"The cost, also named the loss, is usuallyOr also $J(\\theta )$ , $\\mathcal {L}_\\theta $ or $\\mathcal {L}(\\theta )$ .",
"notated $J_\\theta (h)$ and could be measured, for example, by a mean squared error (MSE), which measures the average squared difference, as shown in Equation REF , where $m$ is the number of examples and $(\\text{x}^{(i)}, \\text{y}^{(i)})$ is the $i$ th example pair.",
"$J_\\theta (h) = 1/m \\sum _{i=1}^{m}{(\\text{y}^{(i)} - h(\\text{x}^{(i)}))^2} = 1/m \\sum _{i=1}^{m}{(\\text{y}^{(i)} - \\text{ŷ}^{(i)})^2}$ An example is shown in Figure REF for the case of simple linear regression, i.e.",
"with only one explanatory variable x.",
"Training data are shown as blue solid dots.",
"Once the model has been trained, values of the parameters are adjusted, illustrated by the blue solid bold line which mostly fits the examples.",
"Then, the model can be used for prediction, e.g., to provide a good estimate ŷ of the actual value of y for a given value of x by computing $h(\\text{x})$ (shown in green).",
"Figure: Example of simple linear regression"
],
[
"Gradient Descent Training Algorithm",
"The basic algorithm for training a linear regression model, using the simple gradient descent method, is actually pretty simpleSee, e.g., [142] for more details.",
": initialize each parameter $\\theta _{i}$ and the bias $b$ to a random or some heuristic valuePre-training led to a significant advance, as it improved the initialization of the parameters by using actual training data, via sequential training of the successive layers [47].",
"; compute the values of the model $h$ for all examplesComputing the cost for all examples is the best method but also computationally costly.",
"There are numerous heuristic alternatives to minimize the computational cost, e.g., stochastic gradient descent (SGD), where one example is randomly chosen, and minibatch gradient descent, where a subset of examples is randomly chosen.",
"See, for example, [63] for more details.",
"; compute the cost $J_\\theta (h)$ , e.g., by Equation REF ; compute the gradients $\\frac{\\partial J_\\theta (h)}{\\partial \\theta _i}$ which are the partial derivatives of the cost function $J_\\theta (h)$ with respect to each $\\theta _i$ , as well as to the bias $b$ ; update simultaneouslyA simultaneous update is necessary for the algorithm to behave correctly.",
"all parameters $\\theta _{i}$ and the bias according to the update ruleThe update rule may also be notated as $\\theta := \\theta - \\alpha {\\nabla _\\theta }J_\\theta (h)$ , where ${\\nabla _\\theta }J_\\theta (h)$ is the vector of gradients $\\frac{\\partial J_\\theta (h)}{\\partial \\theta _i}$ .",
"shown in Equation REF , with $\\alpha $ being the learning rate.",
"$\\theta _{i} := \\theta _{i} - \\alpha \\frac{\\partial J_\\theta (h)}{\\partial \\theta _i}$ This represents an update in the opposite direction of the gradients in order to decrease the cost $J_\\theta (h)$ , as illustrated in Figure REF ; and iterate until the error reaches a minimumIf the cost function is convex (the case for linear regression), there is only one global minimum, and thus there is a guarantee of finding the optimal model., or after a certain number of iterations.",
"Figure: Gradient descent"
],
[
"From Model to Architecture",
"Let us now introduce in Figure REF a graphical representation of a linear regression model, as a precursor of a neural network.",
"The architecture represented is actually the computational representation of the modelWe mostly use the term architecture as, in this book, we are concerned with the way to implement and compute a given model and also with the relation between an architecture and a representation..",
"The weighted sum is represented as a computational unitWe use the term node for any component of a neural network, whether it is just an interface (e.g., an input node) or a computational unit (e.g., a weighted sum or a function).",
"We use the term unit only in the case of a computational node.",
"The term neuron is also often used in place of unit, as a way to emphasize the inspiration from biological neural networks., drawn as a squared box with a $\\sum $ , taking its inputs from the x$_i$ nodes, drawn as circles.",
"In the example shown, there are four explanatory variables: x$_1$ , x$_2$ , x$_3$ and x$_4$ .",
"Note that there is some convention of considering the bias as a special case of weight (thus alternatively notated as $\\theta _0$ ) and having a corresponding input node named the bias node, which is implicitHowever, as will be explained later in Section , bias nodes rarely appear in illustrations of non-toy neural networks.",
"and has a constant value notated as $+1$ .",
"This actually corresponds to considering an implicit additional explanatory variable x$_0$ with constant value +1, as shown in Equation REF , alternative formulation of linear regression initially defined in Equation REF .",
"$h(\\text{x}) = \\theta _0 + \\theta _1 \\text{x}_1 + ... + \\theta _n \\text{x}_n = \\sum \\limits _{i=0}^{n} \\theta _i \\text{x}_i$ Figure: Architectural model of linear regression"
],
[
"From Model to Linear Algebra Representation",
"The initial linear regression equation (in Equation REF ) may also be made more compact thanks to a linear algebra notation leading to Equation REF where $h(\\text{x}) = b + \\theta \\text{x}$ $b$ and $h(\\text{x})$ are scalars; $\\theta $ is a row vectorThat is a matrix which has a single row, i.e.",
"a matrix of dimension $1{\\times }n$ .",
"consisting of a single row of $n$ elements: $\\begin{bmatrix} \\theta _1 &\\theta _2 &\\dots &\\theta _n \\end{bmatrix}$ ; x is a column vectorThat is a matrix which has a single column, i.e.",
"a matrix of dimension $n{\\times }1$ .",
"consisting of a single column of $n$ elements: $\\begin{bmatrix} \\text{x}_1\\\\ \\text{x}_2\\\\ \\vdots \\\\ \\text{x}_n \\end{bmatrix}$ ."
],
[
"From Simple to Multivariate Model",
"Linear regression can be generalized to multivariate linear regression, the case when there are multiple variables y$_1$ ... y$_p$ to be predicted, as illustrated in Figure REF with three predicted variables: y$_1$ , y$_2$ and y$_3$ , each subnetwork represented in a different color.",
"Figure: Architectural model of multivariate linear regressionThe corresponding linear algebra equation is Equation REF , where $h(\\text{x}) = b + W \\text{x}$ the $b$ bias vector is a column vector of dimension $p{\\times }1$ , with $b_j$ representing the weight of the connexion between the bias input node and the $j$ th sum operation corresponding to the $j$ th output node; the $W$ weight matrix is a matrix of dimension $p{\\times }n$ , that is with $p$ rows and $n$ columns, with $W_{i,j}$ representing the weight of the connexion between the $j$ th input node and the $i$ th sum operation corresponding to the $i$ th output node; $n$ is the number of input nodes (without considering the bias node); and $p$ is the number of output nodes.",
"For the architecture shown in Figure REF , $n = 4$ (the number of input nodes and of columns of $W$ ) and $p = 3$ (the number of output nodes and of rows of $W$ ).",
"The corresponding $b$ bias vector and $W$ weight matrix are shown in Equations REF and REFIndeed, $b$ and $W$ are generalizations of $b$ and $\\theta $ for the case of univariate linear regression (as shown in Section REF ) to the case of multivariate and thus to multiple rows, each row corresponding to an output node.",
"andBy showing only the connexions to one of the output node, in order to keep readability.",
"in Figure REF .",
"Figure: Architectural model of multivariate linear regression showing the bias and the weights corresponding to the connexions to the third output$b =\\left[\\begin{array}{c}b_1\\\\b_2\\\\b_3\\end{array}\\right]$ $W =\\left[\\begin{array}{cccc}W_{1,1} &W_{1,2} &W_{1,3} &W_{1,4}\\\\W_{2,1} &W_{2,2} &W_{2,3} &W_{2,4}\\\\W_{3,1} &W_{3,2} &W_{3,3} &W_{3, 4}\\end{array}\\right]$"
],
[
"Activation Function",
"Let us now also apply an activation function (AF) to each weighted sum unit, as shown in Figure REF .",
"Figure: Architectural model of multivariate linear regression with activation functionThis activation function allows us to introduce arbitrary nonlinear functions.",
"From an engineering perspective, a nonlinear function is necessary to overcome the linear separability limitation of the single layer Perceptron (see Section ).",
"From a biological inspiration perspective, a nonlinear function can capture the threshold effect for the activation of a neuron through its incoming signals (via its dendrites), determining whether it fires along its output (axone).",
"From a statistical perspective, when the activation function is the sigmoid function, a model corresponds to logistic regression, which models the probability of a certain class or event and thus performs binary classificationFor each output node/variable.",
"See more details in Section REF ..",
"Historically speaking, the sigmoid function (which is used for logistic regression) is the most common.",
"The sigmoid function (usually written $\\sigma $ ) is defined in Equation REF and is shown in Figure REF .",
"It will be further analyzed in Section REF .",
"An alternative is the hyperbolic tangent, often noted tanh, similar to sigmoid but having $[-1, +1]$ as its domain interval ($[0, 1]$ for sigmoid).",
"Tanh is defined in Equation REF and shown in Figure REF .",
"But ReLU is now widely used for its simplicity and effectiveness.",
"ReLU, which stands for rectified linear unit, is defined in Equation REF and is shown in Figure REF .",
"Note that, as some notation convention we use z as the name of the variable of an activation function, as x is usually reserved for input variables.",
"$\\text{sigmoid}(\\text{z}) = \\sigma (\\text{z}) = \\frac{1}{1 + e^{-\\text{z}}}$ Figure: Sigmoid function$\\text{tanh}(\\text{z}) = \\frac{e^\\text{z} - e^{-\\text{z}}}{e^\\text{z} + e^{-\\text{z}}}$ Figure: Tanh function$\\text{ReLU}(\\text{z}) = \\text{max}(0, \\text{z})$ Figure: ReLU function"
],
[
"Basic Building Block",
"The architectural representation (of multivariate linear regression with activation function) shown in Figure REF is an instance (with 4 input nodes and 3 output nodes) of a basic building block of neural networks and deep learning architectures.",
"Although simple, this basic building block is actually a working neural network.",
"It has two layersAlthough, as we will see in Section REF , it will be considered as a single-layer neural network architecture.",
"As it has no hidden layer, it still suffers from the linear separability limitation of the Perceptron.",
": The input layer, on the left of the figure, is composed of the input nodes x$_i$ and the bias node which is an implicit and specific input node with a constant value of 1, therefore usually denoted as $+1$ .",
"The output layer, on the right of the figure, is composed of the output nodes y$_j$ .",
"Training a basic building block is essentially the same as training a linear regression model, which has been described in Section REF ."
],
[
"Feedforward Computation",
"After it has been trained, we can use this basic building block neural network for prediction.",
"Therefore, we simply feedforward the network, i.e.",
"provide input data to the network (feed in) and compute the output values.",
"This corresponds to Equation REF .",
"$\\text{ŷ} = h(\\text{x}) = AF(b + W \\text{x})$ The feedforward computation of the prediction (for the architecture shown in Figure REF ) is illustrated in Equation REF , where $h_j(\\text{x})$ (i.e.",
"ŷ$_j$ ) is the prediction of the $j$ th variable $y_j$ .",
"$\\begin{split}\\text{ŷ} = h(\\text{x}) =h(\\left[\\begin{array}{c}\\text{x}_1\\\\\\text{x}_2\\\\\\text{x}_3\\\\\\text{x}_4\\end{array}\\right])= AF(b + W \\text{x})\\\\=AF(\\left[\\begin{array}{c}b_1\\\\b_2\\\\b_3\\end{array}\\right]+\\left[\\begin{array}{cccc}W_{1,1} &W_{1,2} &W_{1,3} &W_{1,4}\\\\W_{2,1} &W_{2,2} &W_{2,3} &W_{2,4}\\\\W_{3,1} &W_{3,2} &W_{3,3} &W_{3,4}\\end{array}\\right]\\times \\left[\\begin{array}{c}\\text{x}_1\\\\\\text{x}_2\\\\\\text{x}_3\\\\\\text{x}_4\\end{array}\\right])\\\\=AF(\\left[\\begin{array}{c}b_1\\\\b_2\\\\b_3\\end{array}\\right]+\\left[\\begin{array}{c}W_{1,1}\\,\\text{x}_1 + W_{1,2}\\,\\text{x}_2 + W_{1,3}\\,\\text{x}_3 + W_{1,4}\\,\\text{x}_4\\\\W_{2,1}\\,\\text{x}_1 + W_{2,2}\\,\\text{x}_2 + W_{2,3}\\,\\text{x}_3 + W_{2,4}\\,\\text{x}_4\\\\W_{3,1}\\,\\text{x}_1 + W_{3,2}\\,\\text{x}_2 + W_{3,3}\\,\\text{x}_3 + W_{3,4}\\,\\text{x}_4\\end{array}\\right])\\\\=AF(\\left[\\begin{array}{c}b_1 + W_{1,1}\\,\\text{x}_1 + W_{1,2}\\,\\text{x}_2 + W_{1,3}\\,\\text{x}_3 + W_{1,4}\\,\\text{x}_4\\\\b_2 + W_{2,1}\\,\\text{x}_1 + W_{2,2}\\,\\text{x}_2 + W_{2,3}\\,\\text{x}_3 + W_{2,4}\\,\\text{x}_4\\\\b_3 + W_{3,1}\\,\\text{x}_1 + W_{3,2}\\,\\text{x}_2 + W_{3,3}\\,\\text{x}_3 + W_{3,4}\\,\\text{x}_4\\end{array}\\right])\\\\=\\left[\\begin{array}{ccc}h_1(\\text{x})\\\\h_2(\\text{x})\\\\h_3(\\text{x})\\end{array}\\right]=\\left[\\begin{array}{ccc}\\text{ŷ}_1\\\\\\text{ŷ}_2\\\\\\text{ŷ}_3\\end{array}\\right]\\end{split}$"
],
[
"Computing Multiple Input Data Simultaneously",
"Feedforwarding simultaneously a set of examples is easily expressed as a matrix by matrix multiplication, by substituting the single vector example x in Equation REF with a matrix of examples (usually notated as X), leading to Equation REF .",
"Successive columns of the matrix of examples X correspond to the different examples.",
"We use a superscript notation X$^{(k)}$ to denote the $k$ th example, the $k$ th column of the X matrix, to avoid confusion with the subscript notation x$_i$ which is used to denote the $i$ th input variable.",
"Therefore, X$^{(k)}_i$ denotes the $i$ th input value of the $k$ th example.",
"The feedforward computation of a set of examples is illustrated in Equation REF , with predictions $h(\\text{X}^{(k)})$ being successive columns of the resulting output matrix.",
"$h(\\text{X}) = AF(b + W \\text{X})$ $\\begin{split}h(\\text{X})=h(\\left[\\begin{array}{cccc}\\text{X}^{(1)}_1 &\\text{X}^{(2)}_1 &\\hdots &\\text{X}^{(m)}_1\\\\\\text{X}^{(1)}_2 &\\text{X}^{(2)}_2 &\\hdots &\\text{X}^{(m)}_2\\\\\\text{X}^{(1)}_3 &\\text{X}^{(2)}_3 &\\hdots &\\text{X}^{(m)}_3\\\\\\text{X}^{(1)}_4 &\\text{X}^{(2)}_4 &\\hdots &\\text{X}^{(m)}_4\\end{array}\\right])= AF(b + W \\text{X})\\\\=AF(\\left[\\begin{array}{c}b_1\\\\b_2\\\\b_3\\end{array}\\right]+\\left[\\begin{array}{cccc}W_{1,1} &W_{1,2} &W_{1,3} &W_{1,4}\\\\W_{2,1} &W_{2,2} &W_{2,3} &W_{2,4}\\\\W_{3,1} &W_{3,2} &W_{3,3} &W_{3,4}\\end{array}\\right]\\times \\left[\\begin{array}{cccc}\\text{X}^{(1)}_1 &\\text{X}^{(2)}_1 &\\hdots &\\text{X}^{(m)}_1\\\\\\text{X}^{(1)}_2 &\\text{X}^{(2)}_2 &\\hdots &\\text{X}^{(m)}_2\\\\\\text{X}^{(1)}_3 &\\text{X}^{(2)}_3 &\\hdots &\\text{X}^{(m)}_3\\\\\\text{X}^{(1)}_4 &\\text{X}^{(2)}_4 &\\hdots &\\text{X}^{(m)}_4\\end{array}\\right])\\\\=\\left[\\begin{array}{cccc}h_1(\\text{X}^{(1)}) &h_1(\\text{X}^{(2)}) &\\hdots &h_1(\\text{X}^{(m)})\\\\h_2(\\text{X}^{(1)}) &h_2(\\text{X}^{(2)}) &\\hdots &h_2(\\text{X}^{(m)})\\\\h_3(\\text{X}^{(1)}) &h_3(\\text{X}^{(2)}) &\\hdots &h_3(\\text{X}^{(m)})\\end{array}\\right]=\\left[\\begin{array}{cccc}h(\\text{X}^{(1)}) &h(\\text{X}^{(2)}) &\\hdots &h(\\text{X}^{(m)})\\end{array}\\right]\\end{split}$ Note that the main computation taking placeApart from the computation of the $AF$ activation function.",
"In the case of ReLU this is fast.",
"is a product of matrices.",
"This can be computed very efficiently, by using linear algebra vectorized implementation libraries and furthermore with specialized hardware like graphics processing units (GPUs)."
],
[
"Definition",
"Let us now reflect a bit on the meaning of training a model, whether it is a linear regression model (Section REF ) or the basic building block architecture presented in Section .",
"Therefore, let us consider what machine learning actually means.",
"Our starting point is the following concise and general definition of machine learning provided by Mitchell in [132]: “A computer program is said to learn from experience $E$ with respect to some class of tasks $T$ and performance measure $P$ , if its performance at tasks in $T$ , as measured by $P$ , improves with experience $E$ .” At first, note that the word performance actually covers different meanings, specially regarding the computer music context of the book: the execution of (the action to perform) an action, notably an artistic act such as a musician playing a piece of music; a measure (criterium of evaluation) of that action, notably for a computer system its efficiency in performing a task, in terms of time and memoryWith the corresponding analysis measurements, time complexity and space complexity, for the corresponding algorithms.",
"measurements; or a measure of the accuracy in performing a task, i.e.",
"the ability to predict or classify with minimal errors.",
"In the remainder of the book, in order to try to minimize ambiguity, we will use the terms as following: performance as an act by a musician, efficiency as a measure of computational ability, and accuracy as a measure of the quality of a prediction or a classificationIn fact, accuracy may not be a pertinent metric for a classification task with skewed classes, i.e.",
"with one class being vastly more represented in the data than other(s), e.g., in the case of the detection of a rare disease.",
"Therefore a confusion matrix and additional metrics like precision and recall, and possible combinations like F-score, are used (see, e.g., [63] for details).",
"We will not address them in the book, because we are primarily concerned with content generation and not in pattern recognition (classification)..",
"Thus, we could rephrase the definition as: “A computer program is said to learn from experience $E$ with respect to some class of tasks $T$ and accuracy measure $A$ , if its accuracy at tasks in $T$ , as measured by $A$ , improves with experience $E$ .”"
],
[
"Categories",
"We may now consider the three main categories of machine learning with regard to the nature of the experience conveyed by the examples: supervised learning – the dataset is fixed and a correct (expected) answerIt is usually named a label in the case of a classification task and a target in the case of a prediction/regression task.",
"is associated to each example, the general objective being to predict answers for new examples.",
"Examples of tasks are regression (prediction), classification and translation; unsupervised learning – the dataset is fixed and the general objective is in extracting information.",
"Examples of tasks are feature extraction, data compression (both performed by autoencoders, to be introduced in Section ), probability distribution learning (performed by RBMs, to be introduced in Section ), series modeling (performed by recurrent networks, to be introduced in Section ), clustering and anomaly detection; and reinforcement learningTo be introduced in Section .",
"– the experience is incremental through successive actions of an agent within an environment, with some feedback (the reward) providing information about the value of the action, the general objective being to learn a near optimal policy (strategy), i.e.",
"a suite of actions maximizing its cumulated rewards (its gain).",
"Examples of tasks are game playing and robot navigation."
],
[
"Components",
"In his introduction to machine learning [39], Domingos describes machine learning algorithms through three components: representation – the way to represent the model – in our case, a neural network, as it has been introduced and will be further developed in the following sections; evaluation – the way to evaluate and compare models – via a cost function, that will be analyzed in Section REF ; and optimization – the way to identify (search among models for) a best model."
],
[
"Optimization",
"Searching for values (of the parameters of a model) that minimize the cost function is indeed an optimization problem.",
"One of the most simple optimization algorithms is gradient descent, as it has been introduced in Section REF .",
"There are various more sophisticated algorithms, such as stochastic gradient descent (SGD), Nesterov accelerated gradient (NAG), Adagrad, BFGS, etc.",
"(see, for example, [63] for more details).",
"From this basic building block, we will describe in the following sections the main types of deep learning architectures used for music generation (as well as for other purposes): feedforward, autoencoder, restricted Boltzmann machine (RBM), and recurrent (RNN).",
"We will also introduce architectural patterns (see Section REF ) which could be applied to them: convolutional, conditioning, and adversarial."
],
[
"Multilayer Neural Network ",
"A multilayer neural network, also named a feedforward neural network, is an assemblage of successive layers of basic building blocks: the first layer, composed of input nodes, is called the input layer; the last layer, composed of output nodes, is called the output layer; and any layer between the input layer and the output layer is named a hidden layer.",
"An example of a multilayer neural network with two hidden layers is illustrated in Figure REF .",
"The combination of a hidden layer and a nonlinear activation function makes the neural network a universal approximator, able to overcome the linear separability limitationThe universal approximation theorem [86] states that a feedforward network with a single hidden layer containing a finite number of neurons can approximate a wide variety of interesting functions when given appropriate parameters (weights).",
"However, there is no guarantee that the neural network will be able to learn them!.",
"Figure: Example of a feedforward neural network (detailed)"
],
[
"Abstract Representation",
"Note that, in the case of practical (non-toy) illustrations of neural network architectures, in order to simplify the figures, bias nodes are very rarely illustrated.",
"With a similar objective, the sum units and the activation function units are also almost always omitted, resulting in a more abstract view such as that shown in Figure REF .",
"Figure: Example of feedforward neural network (simplified)We can further abstract each layer by representing it as an oblong form (by hiding its nodes)It is sometimes pictured as a rectangle, see Figure REF , or even as a circle, notably in the case of recurrent networks, see Figure REF .",
"as shown in Figure REF .",
"Figure: Example of a feedforward neural network (abstract)"
],
[
"Depth",
"The architecture illustrated in Figure REF is called a 3-layer neural network architecture, also indicating that the depth of the architecture is three.",
"Note that the number of layers (depth) is indeed three and not four, irrespective of the fact that summing up the input layer, the output layer and the two hidden layers gives four and not three.",
"This is because, by convention, only layers with weights (and units) are considered when counting the number of layers in a multilayer neural network; therefore, the input layer is not counted.",
"Indeed, the input layer only acts as an input interface, without any weight or computation.",
"In this book, we will use a superscript (power) notationThe set of compact notations for expressing the dimension of an architecture or a representation will be introduced in Section .",
"to denote the number of layers of a neural network architecture.",
"For instance, the architecture illustrated in Figure REF could be denoted as Feedforward$^3$ .",
"The depth of the first neural network architectures was small.",
"The original Perceptron [166], the ancestor of neural networks, has only an input layer and an output layer without any hidden layer, i.e.",
"it is a single-layer neural network.",
"In the 1980s, conventional neural networks were mostly 2-layer or 3-layer architectures.",
"For modern deep networks, the depth can indeed be very large, deserving the name of deep (or even very deep) networks.",
"Two recent examples, both illustrated in Figure REF , are Figure: (left) GoogLeNet 27-layer deep network architecture.Reproduced from with permission of the authors.",
"(right) ResNet 34-layer deep network architecture.Reproduced from with permission of the authors the 27-layer GoogLeNet architecture [182]; and the 34-layer (up to 152-layer!)",
"ResNet architectureIt introduces the technique of residual learning, reinjecting the input between levels and estimating the residual function $h(\\text{x}) - \\text{x}$ , a technique aimed at very deep networks, see [74] for more details.",
"[74].",
"Note that depth does matter.",
"A recent theorem [44] states that there is a simple radial functionA radial function is a function whose value at each point depends only on the distance between that point and the origin.",
"More precisely, it is radial if and only if it is invariant under all rotations while leaving the origin fixed.",
"on ${\\rm I\\!R}^d$ , expressible by a 3-layer neural network, which cannot be approximated by any 2-layer network to more than a constant accuracy unless its width is exponential in the dimension $d$ .",
"Intuitively, this means that reducing the depth (removing a layer) means exponentially augmenting the width (the number of units) of the layer left.",
"On this issue, the interested reader may also wish to review the analyses in [4] and [193].",
"Note that for both networks pictured in Figure REF , the flow of computation is vertical, upward for GoogLeNet and downward for ResNet.",
"These are different usages than the convention for the flow of computation that we have introduced and used so far, which is horizontal, from left to right.",
"Unfortunately, there is no consensus in the literature about the notation for the flow of computation.",
"Note that in the specific case of recurrent networks, to be introduced in Section , the consensus notation is vertical, upward."
],
[
"Output Activation Function",
"We have seen in Section that, in modern neural networks, the activation function ($AF$ ) chosen for introducing nonlinearity at the output of each hidden layer is often the ReLU function.",
"But the output layer of a neural network has a special status.",
"Basically, there are three main possible types of activation function for the output layer, named in the following, the output activation functionA shorthand for output layer activation function.",
": identity – the case for a prediction (regression) task.",
"It has continuous (real) output values.",
"Therefore, we do not need and we do not want a nonlinear transformation at the last layer; sigmoid – the case of a binary classification task, as in logistic regressionFor details about logistic regression, see, for example, [63] or [73].",
"For this reason, the sigmoid function is also called the logistic function..",
"The sigmoid function (usually written $\\sigma $ ) has been defined in Equation REF and shown in Figure REF .",
"Note its specific shape, which provides a “separation” effect, used for binary decision between two options represented by values 0 and 1; and softmax – the most common approach for a classification task with more than two classes but with only one label to be selectedA very common example is the estimation by a neural network architecture of the next note, modeled as a classification task of a single note label within the set of possible notes.",
"(and where a one-hot encoding is generally used, see Section ).",
"The softmax function actually represents a probability distribution over a discrete output variable with $n$ possible values (in other words, the probability of the occurrence of each possible value $v$ , knowing the input variable x, i.e.",
"$P(\\text{y} = v | \\text{x})$ ).",
"Therefore, softmax ensures that the sum of the probabilities for each possible value is equal to 1.",
"The softmax function is defined in Equation REF and an example of its use is shown in Equation REF .",
"Note that the $\\sigma $ notation is used for the softmax function, as for the sigmoid function, because softmax is actually the generalization of sigmoid to the case of multiple values, being a variadic function, that is one which accepts a variable number of arguments.",
"$\\sigma (\\text{z})_i = \\frac{e^{\\text{z}_i}}{\\sum _{i=1}^n e^{\\text{z}_i}}$ $\\sigma \\left[\\begin{array}{ccc}1.2\\\\0.9\\\\0.4\\end{array}\\right]=\\left[\\begin{array}{ccc}0.46\\\\0.34\\\\0.20\\end{array}\\right]$ For a classification or prediction task, we can simply select the value with the highest probability (i.e.",
"via the argmax function, the indice of the one-hot vector with the highest value).",
"But the distribution produced by the softmax function can also be used as the basis for sampling, in order to add nondeterminism and thus content variability to the generation (this will be detailed in Section )."
],
[
"Cost Function",
"The choice of a cost (loss) function is actually correlated to the choice of the output activation function and to the choice of the encoding of the target y (the true value).",
"Table REFInspired by Ronaghan's concise pedagogical presentation in [165].",
"summarizes the main cases.",
"Table: Relation between output activation function and cost (loss) functionA cross-entropy function measures the difference between two probability distributions, in our case (of a classification task) between the target (true value) distribution (y) and the predicted distribution (ŷ).",
"Note that there are two types of cross-entropy cost functions: binary cross-entropy, when the classification is binary (Boolean), and categorical cross-entropy, when the classification is multiclass with a single label to be selected.",
"In the case of a classification with multiple labels, binary cross-entropy must be chosen joint with sigmoid (because in such cases we want to compare the distributions independently, class per classIn case of multiple labels, the probability of each class is independent from the other class probabilities – the sum is greater than 1.)",
"and the costs for each class are summed up.",
"In the case of multiple simultaneous classifications (multi multiclass single label), each classification is now independent from the other classifications, thus we have two approaches: apply sigmoid and binary cross-entropy for each element and sum up the costs, or apply softmax and categorical cross-entropy independently for each classification and sum up the costs."
],
[
"Interpretation",
"Let us take some examples to illustrate these subtle but important differences, starting with the cases of real and binary values in Figure REF .",
"They also include the basic interpretation of the resultThe interpretation is actually part of the strategy of the generation of music content.",
"It will be explored in Chapter .",
"For instance, sampling from the probability distribution may be used in order to ensure content generation variability, as will be explained in Section ..",
"Figure: Cost functions and interpretation for real and binary values An example of use of the multiclass single label type is a classification among a set of possible notes for a monophonic melody, therefore with only one single possible note choice (single label), as shown in Figure REF .",
"See, for example, the Blues$_C$ system in Section REF .",
"An example of use of the multiclass multilabel type is a classification among a set of possible notes for a single-voice polyphonic melody, therefore with several possible note choices (several labels), as shown in Figure REF .",
"See, for example, the Bi-Axial LSTM system in Section REF .",
"An example of use of the multi multiclass single label type is a multiple classification among a set of possible notes for multivoice monophonic melodies, therefore with only one single possible note choice for each voice, as shown in Figure REF .",
"See, for example, the Blues$_{MC}$ system in Section REF .",
"Another example of use of the multi multiclass single label type is a multiple classification among a set of possible notes for a set of time steps (in a piano roll representation) for a monophonic melody, therefore with only one single possible note choice for each time step.",
"See, for example, the DeepHear$_M$ system in Section REF .",
"An example of use of a multi$^2$ multiclass single label type is a 2-level multiple classification among a set of possible notes for a set of time steps for a multivoice set of monophonic melodies.",
"See, for example, the MiniBach system in Section REF .",
"Figure: Cost function and interpretation for a multiclass single labelFigure: Cost function and interpretation for a multiclass multilabelFigure: Cost function and interpretation for a multi multiclass single labelThe three main interpretations usedIn various systems to be analyzed in Chapter .",
"are argmax (the index of the output vector with the largest value), in the case of a one-hot multiclass single label (in order to select the most likely note), sampling from the probability represented by the output vector, in the case of a one-hot multiclass single label (in order to select a note sorted along its likelihood), and argsortargsort is a numpy library Python function.",
"(the indexes of the output vector sorted according to their diminishing values), in the case of a many-hot multiclass multi label, filtered by some thresholds (in order to select the most likely notes above a probability threshold and under a maximum number of simultaneous notes)."
],
[
"Entropy and Cross-Entropy",
"Mean squared error has been defined in Equation REF in Section REF .",
"Without getting into details about information theory, we now introduce the notion and the formulation of cross-entropyWith some inspiration from Preiswerk's introduction in [156]..",
"The intuition behind information theory is that the information content about an event with a likely (expected) outcome is low, while the information content about an event with an unlikely (unexpected, i.e.",
"a surprise) outcome is high.",
"Let us take the example of a neural network architecture used to estimate the next note of a melody.",
"Suppose that the outcome is $\\text{note} = \\text{B}$ and that it has a probability $P(\\text{note} = \\text{B})$ .",
"We can then introduce the self-information (notated $I$ ) of that event in Equation REF .",
"$I(\\text{note} = \\text{B}) = \\text{log}(1/P(\\text{note} = \\text{B})) = - \\text{log}\\,P(\\text{note} = \\text{B})$ Remember that a probability is by definition within $[0, 1]$ interval.",
"If we look at -log function in Figure REF , we could see that its value is high for a low probability value (unlikely outcome) and its value is null for a probability value equal to 1 (certain outcome), which corresponds to the objective introduced above.",
"Note that the use of a logarithm also makes self-information additive for independent events, i.e.",
"$I(P_1\\,P_2) = I(P_1) + I(P_2)$ .",
"Figure: -log functionThen, let us consider all possible outcomes $\\text{note} = \\text{Note}_i$ , each outcome having $P(\\text{note} = \\text{Note}_i)$ as its associated probability, and $P(\\text{note})$ being the probability distribution for all possible outcomes.",
"The intuition is to define the entropy (notated $H$ ) of the probability distribution for all possible outcomes as the sum of the self-information for each possible outcome, weighted by the probability of the outcome.",
"This leads to Equation REF .",
"$\\begin{split}H(P) = \\sum _{i=0}^{n}{P(\\text{note} = \\text{Note}_i)~I(\\text{note} = \\text{Note}_i)}\\\\= - \\sum _{i=0}^{n}{P(\\text{note} = \\text{Note}_i)\\,\\text{log}\\,P(\\text{note} = \\text{Note}_i)}\\end{split}$ Note that we can further rewrite the definition by using the notion of expectationAn expectation, or expected value, of some function $f(\\text{x})$ with respect to a probability distribution $P(\\text{x})$ , usually notated as $\\mathbb {E}_{\\text{x} \\sim P}[f(\\text{x})]$ , is the average (mean) value that $f$ takes on when x is drawn from $P$ , i.e.",
"$\\mathbb {E}_{\\text{x} \\sim P}\\,[f(\\text{x})] = \\sum _{\\text{x}}^{}{P(\\text{x}) f(\\text{x})}$ (we are here considering the case of discrete variables, which is the case for classification within a set of possible notes)., which leads to Equation REF .",
"$H(P) = \\mathbb {E}_{\\text{note} \\sim P}\\,[I(\\text{note})]= - \\mathbb {E}_{\\text{note} \\sim P}\\,[\\text{log}\\,P(\\text{note})]$ Now, let us introduce in Equation REF the Kullback-Leibler divergence (often abbreviated as KL-divergence, and notated $D_\\text{KL}$ ), as some measureNote that it is not a true distance measure as it not symmetric.",
"of how different are two separate probability distributions $P$ and $Q$ over a same variable (note).",
"$\\begin{split}D_\\text{KL}(P || Q) = \\mathbb {E}_{\\text{note} \\sim P}\\,[\\text{log}\\,\\frac{P(\\text{note})}{Q(\\text{note})}]\\\\= \\mathbb {E}_{\\text{note} \\sim P}\\,[\\text{log}\\,P(\\text{note}) - \\text{log}\\,Q(\\text{note})]\\\\= \\mathbb {E}_{\\text{note} \\sim P}\\,[\\text{log}\\,P(\\text{note})] - \\mathbb {E}_{\\text{note} \\sim P}\\,[\\text{log}\\,Q(\\text{note})]\\end{split}$ $D_\\text{KL}$ may be rewritten as in Equation REFBy using $H(P)$ definition in Equation REF ., where $H(P, Q)$ , named the categorical cross-entropy, is defined in Equation REF .",
"$D_\\text{KL}(P || Q) = - H(P) + H(P, Q)$ $H(P, Q) = - \\mathbb {E}_{\\text{note} \\sim P}\\,[\\text{log}\\,Q(\\text{note})]$ Note that categorical cross-entropy is similar to KL-divergenceAnd, just like KL-divergence, it is not symmetric., while lacking the $H(P)$ term.",
"But minimizing $D_\\text{KL}(P || Q)$ or minimizing $H(P, Q)$ , with respect to $Q$ , are equivalent, because the omitted term $H(P)$ is a constant with respect to $Q$ .",
"Now, rememberSee Section REF .",
"that the objective of the neural network is to predict the ŷ probability distribution, which is an estimation of the y true ground probability distribution, by minimizing the difference between them.",
"This leads to Equations REF and REF .",
"$D_\\text{KL}(\\text{y} || \\text{ŷ})= \\mathbb {E}_{\\text{y}}\\,[\\text{log}\\,\\text{y} - \\text{log}\\,\\text{ŷ}]= \\sum _{i=0}^{n}{\\text{y}_i\\,(\\text{log}\\,\\text{y}_i - \\text{log}\\,\\text{ŷ}_i})$ $H(\\text{y}, \\text{ŷ})= - \\mathbb {E}_{\\text{y}}\\,[\\text{log}\\,\\text{ŷ}]= - \\sum _{i=0}^{n}{\\text{y}_i\\,\\text{log}\\,\\text{ŷ}_i}$ As mentioned above, minimizing $D_\\text{KL}(\\text{y} || \\text{ŷ})$ or minimizing $H(\\text{y}, \\text{ŷ})$ , with respect to ŷ, are equivalent, because the omitted term $H(\\text{y})$ is a constant with respect to ŷ.",
"Last, deriving the binary cross-entropy (that we notate $H_\\text{B}$ ) is easy, as there are only two possible outcomes, which leads to Equation REF .",
"$H_\\text{B}(\\text{y}, \\text{ŷ}) = - (\\text{y}_0\\,\\text{log}\\,\\text{ŷ}_0 + \\,\\text{y}_1\\,\\text{log}\\,\\text{ŷ}_1)$ Because $\\text{y}_1 = 1 - \\text{y}_0$ and $\\text{ŷ}_1 = 1 - \\text{ŷ}_0$ (as the sum of the probabilities of the two possible outcomes is 1), this ends up into Equation REF .",
"$H_\\text{B}(\\text{y}, \\text{ŷ}) = - (\\text{y}\\,\\text{log}\\,\\text{ŷ} + (1 - \\text{y})\\,\\text{log}\\,(1 - \\text{ŷ}))$ More details and principles for the cost functionsThe underlying principle of maximum likelihood estimation, not explained here.",
"can be found, for example, in [63] and [63], respectively.",
"In addition, the information theory foundation of cross-entropy as the number of bits needed for encoding information is introduced, for example, in [37]."
],
[
"Feedforward Propagation",
"Feedforward propagation in a multilayer neural network consists in injecting input dataThe x part of an example, for the generation phase as well as for the training phase.",
"into the input layer and propagating the computation through its successive layers until the output is produced.",
"This can be implemented very efficiently because it consists in a pipelined computation of successive vectorized matrix products (intercalated with $AF$ activation function calls).",
"Each computation from layer $k-1$ to layer $k$ is processed as in Equation REF , which is a generalization of Equation REFFeedforward computation for one layer has been introduced in Section REF ., where $b^{[k]}$ and $W^{[k]}$We use a superscript notation with brackets $^{[k]}$ to denote the $k$ th layer, to avoid confusion with the superscript notation with parentheses $^{(k)}$ to denote the $k$ th example and the subscript notation $_i$ to denote the $i$ th input variable.",
"are respectively the bias and the weight matrix between layer $k-1$ and layer $k$ , and where $\\text{output}^{[0]}$ is the input layer, as shown in Figure REF .",
"$\\text{output}^{[k]} = AF(b^{[k]} + W^{[k]} \\text{output}^{[k-1]})$ Figure: Example of a feedforward neural network (abstract) pipelined computationMultilayer neural networks are therefore often also named feedforward neural networks or multilayer Perceptron (MLP)The original Perceptron was a neural network with no hidden layer, and thus equivalent to our basic building block, with only one output node and with the step function as the activation function..",
"Note that neural networks are deterministic.",
"This means that the same input will deterministically always produce the same output.",
"This is a useful guarantee for prediction and classification purposes but may be a limitation for generating new content.",
"However, this may be compensated by sampling from the resultant probability distribution (see Sections REF and )."
],
[
"Training",
"For the training phaseLet us remember that this a case of supervised learning (see Section )., computing the derivatives becomes a bit more complex than for the basic building block (with no hidden layer) presented in Section REF .",
"Backpropagation is the standard method of estimating the derivatives (gradients) for a multilayer neural network.",
"It is based on the chain rule principle [167], in order to estimate the contribution of each weight to the final prediction error, that is the cost.",
"See, for example, [63] for more details.",
"Note that, in the most common case, the cost function of a multilayer neural network is not convex, meaning that there may be multiple local minima.",
"Gradient descent, as well as other more sophisticated heuristic optimization methods, does not guarantee the global optimum will be reached.",
"But in practice a clever configuration of the model (notably, its hyperparameters, see Section REF ) and well-tuned optimization heuristics, such as stochastic gradient descent (SGD), will lead to accurate solutionsOn this issue, see [24], which shows that 1) local minima are located in a well-defined band, 2) SGD converges to that band, 3) reaching the global minimum becomes harder as the network size increases and 4) in practice this is irrelevant as the global minimum often leads to overfitting (see next section).."
],
[
"Overfitting",
"A fundamental issue for neural networks (and more generally speaking for machine learning algorithms) is their generalization ability, that is their capacity to perform well on yet unseen data.",
"In other words, we do not want a neural network to just perform well on the training dataOtherwise, the best and simpler algorithm would be a memory-based algorithm, which simply memorizes all (x, y) pairs.",
"It has the best fit to the training data but it does not have any generalization ability.",
"but also on future dataFuture data is not yet known but that does not mean that it is any kind of (random) data, otherwise a machine learning algorithm would not be able to learn and generalize well.",
"There is indeed a fundamental assumption of regularity of the data corresponding to a task (e.g., images of human faces, jazz chord progressions, etc.)",
"that neural networks will exploit..",
"This is actually a fundamental dilemma, the two opposing risks being underfitting – when the training error (error measure on the training data) is large; and overfitting – when the generalization error (expected error on yet unseen data) is large.",
"A simple illustrative example of underfit, good fit and overfit models for the same training data (the green solid dots) is shown in Figure REF .",
"Figure: Underfit, good fit and overfit modelsIn order to be able to estimate the potential for generalization, the dataset is actually divided into two portions, with a ratio of approximately 70/30: the training set – which will be used for training the neural network; and the validation set, also named test setActually, a difference could (should) be made, as explained by Hastie et al.",
"in [73]: “It is important to note that there are in fact two separate goals that we might have in mind: Model selection: estimating the performance of different models in order to choose the best one.",
"Model assessment: having chosen a final model, estimating its prediction error (generalization error) on new data.",
"If we are in a data-rich situation, the best approach for both problems is to randomly divide the dataset into three parts: a training set, a validation set, and a test set.",
"The training set is used to fit the models; the validation set is used to estimate prediction error for model selection; the test set is used for assessment of the generalization error of the final chosen model.” However, as a matter of simplification, we will not consider that difference in the book.",
"– which will be used to estimate the capacity of the model for generalization."
],
[
"Regularization",
"There are various techniques to control overfitting, i.e., to improve generalization.",
"They are usually named regularization and some examples of well-known techniques are weight decay (also known as L$^2$), by penalizing over-preponderant weights; dropout, by introducing random disconnections; early stopping, by storing a copy of the model parameters every time the error on the validation set reduces, then terminating after an absence of progress during a pre-specified number of iterations, and returning these parameters; and dataset augmentation, by data synthesis (e.g., by mirroring, translation and rotation for images; by transposition for music, see Section REF ), in order to augment the number of training examples.",
"We will not further detail regularization techniques, see, for example, [63]."
],
[
"Hyperparameters",
"In addition to the parameters of the model, which are the weights of the connexions between nodes, a model also includes hyperparameters, which are parameters at an architectural meta-level, concerning both structure and control.",
"Examples of structural hyperparameters, mainly concerned with the architecture, are number of layers, number of nodes, and nonlinear activation function.",
"Examples of control hyperparameters, mainly concerned with the learning process, are optimization procedure, learning rate, and regularization strategy and associated parameters.",
"Choosing proper values for (tuning) the various hyperparameters is fundamental both for the efficiency and the accuracy of neural networks for a given application.",
"There are two approaches for exploring and tuning hyperparameters: manual tuning or automated tuning – by algorithmic exploration of the multidimensional space of hyperparameters and for each sample evaluating the generalization error.",
"The three main strategies for automated tuning are random search – by defining a distribution for each hyperparameter, sampling configurations, and evaluating them; grid search – as opposed to random search, exploration is systematic on a small set of values for each hyperparameter; and model-based optimization – by building a model of the generalization error and running an optimization algorithm over it.",
"The challenge of automated tuning is its computational cost, although trials may be run in parallel.",
"We will not detail these approaches here; however, further information can be found in [63].",
"Note that this tuning activity is more objective for conventional tasks such as prediction and classification because the evaluation measure is objective, being the error rate for the validation set.",
"When the task is the generation of new musical content, tuning is more subjective because there is no preexisting evaluation measure.",
"It then turns out to be more qualitative, for instance through a manual evaluation of generated music by musicologists.",
"This evaluation issue will be addressed in Section ."
],
[
"Platforms and Libraries",
"Various platformsSee, for example, the survey in [154]., such as CNTK, MXNet, PyTorch and TensorFlow, are available as a foundation for developing and running deep learning systemsThere are also more general libraries for machine learning and data analysis, such as the SciPy library for the Python language, or the language R and its libraries..",
"They include libraries of basic architectures, such as the ones we are presenting in this chapter; components, for example optimization algorithms; runtime interfaces for running models on various hardware, including GPUs or distributed Web runtime facilities; and visualization and debugging facilities.",
"Keras is an example of a higher-level framework to simplify development, with CNTK, TensorFlow and Theano as possible backends.",
"ONNX is an open format for representing deep learning models and was designed to ease the transfer of models between different platforms and tools."
],
[
"Autoencoder",
"An autoencoder is a neural network with one hidden layer and with an additional constraint: the number of output nodes is equal to the number of input nodesThe bias is not counted/considered here as it is an implicit additional input node..",
"The output layer actually mirrors the input layer.",
"It is shown in Figure REF , with its peculiar symmetric diabolo (or sand-timer) shape aspect.",
"Figure: Autoencoder architectureTraining an autoencoder represents a case of unsupervised learning, as the examples do not contain any additional label information (the effective value or class to be predicted).",
"But the trick is that this is implemented using conventional supervised learning techniques, by presenting output data equal to the input dataThis is sometimes called self-supervised learning [110]..",
"In practice, the autoencoder tries to learn the identity function.",
"As the hidden layer usually has fewer nodes than the input layer, the encoder component (shown in yellow in Figure REF ) must compress information while the decoder (shown in purple) has to reconstruct, as accurately as possible, the initial informationCompared to traditional dimension reduction algorithms, such as principal component analysis (PCA), this approach has two advantages: 1) feature extraction is nonlinear (the case of manifold learning, see [63] and Section REF ) and 2) in the case of a sparse autoencoder (see next section), the number of features may be arbitrary (and not necessarily smaller than the number of input parameters)..",
"This forces the autoencoder to discover significant (discriminating) features to encode useful information into the hidden layer nodes (also named the latent variablesIn statistics, latent variables are variables that are not directly observed but are rather inferred (through a mathematical model) from other variables that are observed (directly measured).",
"They can serve to reduce the dimensionality of data.).",
"Therefore, autoencoders may be used to automatically extract high-level features [110].",
"The set of features extracted are often named an embeddingSee the definition of embedding in Section REF .. Once trained, in order to extract features from an input, one just needs to feedforward the input data and gather the activations of the hidden layer (the values of the latent variables).",
"Another interesting use of decoders is the high-level control of content generation.",
"The latent variables of an autoencoder constitute a compact representation of the common features of the learnt examples.",
"By instantiating these latent variables and decoding the embedding, we can generate a new musical content corresponding to the values of the latent variables.",
"We will explore this strategy in Section REF ."
],
[
"Sparse Autoencoder",
"A sparse autoencoder is an autoencoder with a sparsity constraint, such that its hidden layer units are inactive most of the time.",
"The objective is to enforce the specialization of each unit in the hidden layer as a specific feature detector.",
"For instance, a sparse autoencoder with 100 units in its hidden layer and trained on 10$\\times $ 10 pixel images will learn to detect edges at different positions and orientations in images, as shown in Figure REF .",
"When applied to other input domains, such as audio or symbolic music data, this algorithm will learn useful features for those domains too.",
"The sparsity constraint is implemented by adding an additional term to the cost function to be minimized, see more details in [141] or [63].",
"Figure: Visualization of the input image motives that maximally activate each of the hidden units of a sparse autoencoder architecture.Reproduced from with permission of the author"
],
[
"Variational Autoencoder",
"A variational autoencoder (VAE) [102] has the added constraint that the encoded representation, the latent variables, by convention denoted by variable z, follow some prior probability distribution $P(\\text{z})$ .",
"Usually, a Gaussian distributionAlso named normal distribution.",
"is chosen for its generality.",
"This constraint is implemented by adding a specific term to the cost function, by computing the cross-entropy between the values of the latent variables and the prior distributionThe actual implementation is more complex and has some tricks (e.g., the encoder actually generates a mean vector and a standard deviation vector) that we will not detail here.. For more details about VAEs, an example of tutorial could be found in [38] and there is a nice introduction of its application to music in [162].",
"As with an autoencoder, a VAE will learn the identity function, but furthermore the decoder part will learn the relation between a Gaussian distribution of the latent variables and the learnt examples.",
"As a result, sampling from the VAE is immediate, one just needs to sample a value for the latent variables $\\text{z} \\sim P(\\text{z})$ , i.e.",
"z following distribution $P(\\text{z})$ ; input it into the decoder; and feedforward the decoder to generate an output corresponding to the distribution of the examples, following $P(\\text{x} | \\text{z})$ conditional probability distribution learnt by the decoder.",
"This is in contrast to the need for indirect and computationally expensive strategies such as Gibbs sampling for other architectures such as RBM, to be introduced in Section .",
"By construction, a variational autoencoder is representative of the dataset that it has learnt, that is, for any example in the dataset, there is at least one setting of the latent variables which causes the model to generate something very similar to that example [38].",
"A very interesting characteristic of the variational autoencoder architecture for generation purposes – therefore often considered as one type of a class of models named generative models – is in the meaningful exploration of the latent space, as a variational autoencoder is able to learn a “smooth”That is, a small change in the latent space will correspond to a small change in the generated examples, without any discontinuity or jump.",
"For a more detailed discussion about which (and how) interesting effects (smoothness, parsimony and axis-alignment between data and latent variability) a VAE has on the latent representation (the vector of latent variables) learnt, see, e.g., [206].",
"latent space mapping to realistic examples.",
"Note that this general objective is named manifold learning and more generally representation learning [8], that is the learning of a representation capturing the topology of a set of examples.",
"As defined in [63], a manifold is a connected set of points (examples) that can be approximated by a smaller number of dimensions, each one corresponding to a local direction of variation.",
"An intuitive example is a 2D map capturing the topology of cities dispersed on the 3D earth, where a movement on the map corresponds to a movement on the earth.",
"To illustrate the possibilities, let us train a VAE with only two latent variables on the MNIST handwritten digits database dataset [113] (with 60.000 examples, each one being an image of 28$\\times $ 28 pixels).",
"Then, we scan the latent two-dimension plane, sampling latent values for the two latent variables (i.e.",
"sampling points within the 2-dimension latent space) at regular intervals and generating the corresponding artificial digits by decoding the latent pointsAs proposed and implemented in [23]..",
"Figure REF shows examples of artificial digits generated.",
"Figure: Various digits generated by decoding sampled latent points at regular intervals on the MNIST handwritten digits databaseNote that training the VAE has forced it to compress information about the actual examples by splitting (though the encoder) information in two subsets: the specific (discriminative) part, encoded within the latent variables; and the common part, encoded within the weights of the decoder, in order to be able to reconstruct as close as possible each original dataIndeed, there is no magic here, the reversible compression from 28$\\times $ 28 variables to 2 variables must have extracted and stored missing information somewhere..",
"The VAE actually has been forced to find out dimensions of variations for the dataset examples.",
"By looking at the figure, we can guess that the two dimensions could be: from angular to round elements for the 1st variable (z$_1$ , horizontally represented), and the size of the compound element (circle or angle) for the second latent variable (z$_2$ , vertically represented).",
"Note that we cannot expect/force the VAE towards the semantics (meaning) of specific dimensions, as the VAE will automatically extract them (this depends on the dataset as well as on the training configuration), and we can only try to interpret them a posterioriHowever, we will see that we can construct arbitrary characteristic attributes from a subset of examples and impose them on other examples, by doing attribute vector arithmetics, as defined in the immediately following list, and as will be illustrated in Figure REF in Section REF ..",
"Examples of possible dimensions for music generation could be: the number of notesThis will be illustrated in Figure REF in Section REF ., the range (the distance from the lowest to the highest pitch), etc.",
"Once learnt by a VAE, the latent representation (a vector of latent variables) can be used to explore the latent space with various operations to control/vary the generation of content.",
"Some examples of operations on the latent space, as proposed in [162] and [163] for the MusicVAE system described in Section REF , are translation; interpolation; averaging of some points; attribute vector arithmetics, by addition or subtraction of an attribute vector capturing a given characteristicThis attribute vector is computed as the average latent vector for a collection of examples sharing that attribute (characteristic)..",
"Figure REF shows an interesting comparison of melodies resulting from Figure: Comparison of interpolations between the top and the bottom melodies by(left) interpolating in the data (melody) spaceand(right) interpolating in the latent space and decoding it into melodies.Reproduced from with permission of the authors interpolation in the data space, that is the space of representation of melodies; and interpolation in the latent space, which is then decoded into the corresponding melodies.",
"The interpolation in the latent space produces more meaningful and interesting melodies than the interpolation in the data space (which basically just varies the ratio of notes from the two melodies), as can be heard in [161] and [164].",
"More details about these experiments will be provided in Section REF .",
"Variational autoencoders are therefore elegant and promising models, and as a result they are currently among the hot approaches explored for generating content with controlled variations.",
"Application to music generation will be illustrated in Sections REF and REF ."
],
[
"Stacked Autoencoder",
"The idea of a stacked autoencoder is to hierarchically nest successive autoencoders with decreasing numbers of hidden layer units.",
"An example of a 2-layer stacked autoencoderNote that the convention in this case is to count and notate the number of nested autoencoders, i.e.",
"the number of hidden layers.",
"This is different from the depth of the whole architecture, which is double.",
"For instance, a 2-layer stacked autoencoder results in a 4-layer whole architecture, as shown in Figure REF ., i.e.",
"two nested autoencoders that we could notate as Autoencoder$^2$ , is illustrated in Figure REF .",
"Figure: A 2-layer stacked autoencoder architecture, resulting in a 4-layer full architectureThe chain of encoders will increasingly compress data and extract higher-level features.",
"Stacked autoencoders, which are indeed deep networks, are therefore used for feature extraction (an example will be introduced in Section REF ).",
"They are also useful for music generation, as we will see in Section REF .",
"This is because the innermost hidden layer, sometimes named the bottleneck hidden layer, provides a compact and high-level encoding (embedding) as a seed for generation (by the chain of decoders)."
],
[
"Restricted Boltzmann Machine (RBM)",
"A restricted Boltzmann machine (RBM) [81] is a generative stochastic artificial neural network that can learn a probability distribution over its set of inputs.",
"Its name comes from the fact that it is a restricted (constrained) formWhich actually makes RBM practical, as opposed to the general form, which besides its interest suffers from a learning scalability limitation.",
"of a (general) Boltzmann machine [82], named after the Boltzmann distribution in statistical mechanics, which is used in its sampling function.",
"The architectural restrictions of an RBM (see Figure REF ) are that it is organized in layers, just as for a feedforward network or an autoencoder, and more precisely two layers: the visible layer (analog to both the input layer and the output layer of an autoencoder); and the hidden layer (analog to the hidden layer of an autoencoder); as for a standard neural network, there cannot be connections between nodes within the same layer.",
"Figure: Restricted Boltzmann machine (RBM) architectureAn RBM bears some similarity in spirit and objective to an autoencoder.",
"However, there are some important differences: an RBM has no ouput – the input also acts as the output; an RBM is stochastic (and therefore not deterministic, as opposed to a feedforward network or an autoencoder); an RBM is trained in an unsupervised learning manner, with a specific algorithm (named contrastive divergence, see Section REF ), whereas an autoencoder is trained using a standard supervised learning method, with the same data as input and output; and the values manipulated are booleansAlthough there are extensions with multinoulli (categorical) or continuous values, see Section REF .. RBMs became popular after Hinton designed a specific fast learning algorithm for them, named contrastive divergence [79], and used them for pre-training deep neural networks [47] (see Section ).",
"An RBM is an architecture dedicated to learning distributions.",
"Moreover, it can learn efficiently from only a few examples.",
"For musical applications, this is interesting for learning (and generating) chords, as the combinatorial nature of possible notes forming a chord is large and the number of examples is usually small.",
"We will see an example of such an application in Section REF ."
],
[
"Training",
"Training an RBM has some similarity to training an autoencoder, with the practical difference that, because there is no decoder part, the RBM will alternate between two steps: the feedforward step – to encode the input (visible layer) into the hidden layer, by making predictions about hidden layer node activations; and the backward step – to decode/reconstruct the input (visible layer), by making predictions about visible layer node activations.",
"We will not detail here the learning technique behind RBMs, see, for example, [63].",
"Note that the reconstruction process is an example of generative learning (and not discriminative learning, as for training autoencoders which is based on regression)See, for example, a nice introduction to generative learning (and the difference with discriminative learning) in [143].."
],
[
"Sampling",
"After the training phase has been completed, in the generation phase, a sample can be drawn from the model by randomly initializing visible layer vector $v$ (following a standard uniform distribution) and running samplingMore precisely Gibbs sampling (GS), see [107].",
"Sampling will be introduced in Section REF .",
"until convergence.",
"To this end, hidden nodes and visible nodes are alternately updated (as during the training phase).",
"In practice, convergence is reached when the energy stabilizes.",
"The energy of a configuration (the pair of visible and hidden layers) is expressedFor more details, see, for example, [63].",
"in the Equation REF , where $E(\\text{v}, \\text{h}) = -a^\\mathrm {T} \\text{v} - b^\\mathrm {T} \\text{h} -\\text{v}^\\mathrm {T} W \\text{h}$ v and h, respectively, are column vectors representing the visible and the hidden layers; $W$ is the matrix of weights associated with the connections between visible and hidden nodes; $a$ and $b$ , respectively, are column vectors representing the bias weights for visible and hidden nodes, with $a^\\mathrm {T}$ and $b^\\mathrm {T}$ being their respective transpositions into row vectors; and $v^\\mathrm {T}$ is the transposition of v into a row vector."
],
[
"Types of Variables",
"Note that there are actually three possibilities for the nature of RBM variables (units, visible or hidden): Boolean or Bernoulli – this is the case of standard RBMs, in which units (visible and hidden) are Boolean, with a Bernoulli distribution (see [63]); multinoulli – an extension with multinoulli unitsAs explained by Goodfellow et al.",
"in [63]: ““Multinoulli” is a term that was recently coined by Gustavo Lacerdo and popularized by Murphy in [140].",
"The multinoulli distribution is a special case of the multinomial distribution.",
"A multinomial distribution is the distribution over vectors in $\\lbrace 0, .",
".",
".",
", n\\rbrace ^k$ representing how many times each of the $k$ categories is visited when $n$ samples are drawn from a multinoulli distribution.",
"Many texts use the term “multinomial” to refer to multinoulli distributions without clarifying that they refer only to the $n = 1$ case.”, i.e.",
"with more than two possible discrete values; and continuous – another extension with continuous units, taking arbitrary real values (usually within the $[0, 1]$ range).",
"An example is the C-RBM architecture analyzed in Section REF ."
],
[
"Recurrent Neural Network (RNN)",
"A recurrent neural network (RNN) is a feedforward neural network extended with recurrent connexions in order to learn series of items (e.g., a melody as a sequence of notes).",
"The input of the RNN is an element x$_t$This x$_t$ notation – or sometimes s$_t$ to stress the fact that it is a sequence – is very common but unfortunately introduces possible confusion with the notation of x$_i$ as the $i$ th input variable.",
"The context – recurrent versus nonrecurrent network – usually helps to discriminate, as well as the use of the letter $t$ (for time) as the index.",
"An example of an exception is the RNN-RBM system analyzed in Section REF , which uses the x$^{(t)}$ notation.",
"of the sequence, where $t$ represents the index or the time, and the expected output is next element x$_{t+1}$ .",
"In other words the RNN will be trained to predict the next element of a sequence.",
"In order to do so, the output of the hidden layer reenters itself as an additional input (with a specific corresponding weight matrix).",
"This way, the RNN can learn, not only based on the current item but also on its previous own state, and thus, recursively, on the whole of the previous sequence.",
"Therefore, an RNN can learn sequences, notably temporal sequences, as in the case of musical content.",
"Figure: Recurrent neural network (folded)Figure: Recurrent neural network (unfolded)An example of RNN (with two hidden layers) is shown in Figure REF .",
"Recurrent connexions are signaled with a solid square, in order to distinguish them from standard connexionsActually, there are some variations of this basic architecture, depending on the exact nature and location of the recurrent connexions.",
"The most standard case is a recurrent connexion for each hidden unit, as shown in Figure REF .",
"But there are some other cases, see for example in [63].",
"An example of a music generation architecture with recurrent connexions from the output to a special context input will be introduced in Section REF ..",
"The unfolded version of the visual representation is in Figure REF , with a new diagonal axis representing the time dimension, in order to illustrate the previous step value of each layer (in thinner and lighter color).",
"Note that, as for standard connexions (shown in yellow solid lines), recurrent connexions (shown in purple dashed lines) fully connect (with a specific weight matrix) all nodes corresponding to the previous step nodes to the nodes corresponding to the current step, as illustrated in Figure REF .",
"Figure: Standard connexions versus recurrent connexions (unfolded)An RNN can learn a probability distribution over a sequence by being trained to predict the next element at time step $t$ in a sequence as being the conditional probability distribution $P(\\text{s}_t | \\text{s}_{t-1},... , \\text{s}_1)$ , also notated as $P(\\text{s}_t | \\text{s}_{<t})$ , that is the probability distribution $P(\\text{s}_t)$ given all previous elements generated s$_1$ , s$_2$ , ... , s$_{t-1}$ .",
"In summary, recurrent networks (RNNs) are good at learning sequences and therefore are routinely used for natural text processing and for music generation."
],
[
"Visual Representation",
"A more frequent visual representation for an RNN is actually showing the flow upwards and time rightwards, see the folded version (of an RNN with only one hidden layer) in Figure REF and the unfolded version in Figure REF , with h$_t$ being the value of the hidden layer at step $t$ , and x$_t$ and y$_t$ being the values of the input and output at step $t$ .",
"Figure: Recurrent neural network (folded)Figure: Recurrent neural network (unfolded)"
],
[
"Training",
"A recurrent network is not trained in exactly the same manner as a feedforward network.",
"The idea is to present an example element of a sequence (e.g., a note within a melody) as the input x$_t$ and the next element of the sequence (the next note) x$_{t+1}$ as the output y$_t$ .",
"This will train the recurrent network to predict the next element of the sequence.",
"In practice, an RNN is rarely trained element by element but with a sequence as an input and the same sequence shifted left by one step/item as the output.",
"See an example in Figure REFThe end of the sequence is marked by a special symbol..",
"Therefore, the recurrent network will learn to predictPictured as dashed arrows.",
"the next element for all successive elements of the sequence.",
"Figure: Training a recurrent neural networkThe backpropagation algorithm to compute gradients for feedforward networks, introduced in Section REF , has been extended into a backpropagation through time (BPTT) algorithm for recurrent networks.",
"The intuition is in unfolding the RNN through time and considering an ordered sequence of input-output pairs, but with every unfolded copy of the network sharing the same parameters, and then applying the standard backpropagation algorithm.",
"More details may be found, for example, in [63].",
"Note that, a RNN has usually an output layer identical to its input layerRNNs are actually more general and there are actually some rare cases of an RNN with an arbitrary output different from the input (as for a feedforward network).",
"An example is a RNN-based architecture to generate a chord-based accompaniment, to be analyzed in Section REF .",
"As Karpathy puts it in [98]: “Depending on your background you might be wondering: What makes Recurrent Networks so special?",
"A glaring limitation of Vanilla Neural Networks (and also Convolutional Networks) is that their API is too constrained: they accept a fixed-sized vector as input (e.g.",
"an image) and produce a fixed-sized vector as output (e.g.",
"probabilities of different classes).",
"Not only that: These models perform this mapping using a fixed amount of computational steps (e.g.",
"the number of layers in the model).",
"The core reason that recurrent nets are more exciting is that they allow us to operate over sequences of vectors: Sequences in the input, the output, or in the most general case both.”, as a recurrent network predicts the next item, which will be used iteratively as the next input in a recursive way in order to produce a sequence.",
"Note also that training a recurrent network is usually considered as a case of supervised learning as, for each item, the next item is presented as the expected prediction, although it is not an additional label information (effective value or class to be predicted) but only the recurrent information about the next item (intrinsically present within a sequence)."
],
[
"Long Short-Term Memory (LSTM)",
"Recurrent networks suffered from a training problem caused by the difficulty of estimating gradients because in backpropagation through time recurrence brings repetitive multiplications, and could thus lead to over amplify or minimize effectsThis has been coined as the vanishing or exploding gradient problem and also as the challenge of long-term dependencies (see, for example, [63])..",
"This problem has been addressed and resolved by the long short-term memory (LSTM) architecture, proposed by Hochreiter and Schmidhuber in 1997 [83].",
"As the solution has been quite effective, LSTM has become the de facto standard for recurrent networksAlthough, there are a few subsequent but similar proposals, such as gated recurrent units (GRUs).",
"See a comparative analysis of LSTM and GRU in [25]..",
"The idea behind LSTM is to secure information in memory cells, within a blockCells within the same block share input, output and forget gates.",
"Therefore, although each cell might hold a different value in its memory, all cell memories within a block are read, written or erased all at once [83]., protected from the standard data flow of the recurrent network.",
"Decisions about writing to, reading from and forgetting (erasing) the values of cells within a block are performed by the opening or closing of gates and are expressed at a distinct control level (meta-level), while being learnt during the training process.",
"Therefore, each gate is modulated by a weight parameter, and thus is suitable for backpropagation and standard training process.",
"In other words, each LSTM block learns how to maintain its memory as a function of its input in order to minimize loss.",
"See a conceptual view of an LSTM cell in Figure REF .",
"We will not further detail here the inner mechanism of an LSTM cell (and block) because we may consider it here as a black box (please refer to, for example, the original article [83]).",
"Figure: LSTM architecture (conceptual)Note that a more general model of memory with access customized through training has recently been proposed: neural Turing machines (NTM) [66].",
"In this model, memory is global and has read and write operations with differentiable controls, and thus is subject to learning through backpropagation.",
"The memory to be accessed, specified by location or by content, is controlled via an attention mechanism (introduced in next section)."
],
[
"Attention Mechanism",
"The motivation for an attention mechanism has been inspired by the human visual system ability to efficiently track and recognize objects by focusing its attention.",
"It has therefore been first introduced into neural network architectures for image recognition, as for instance for object tracking [35].",
"It has then been adapted to recurrent architectures for natural language processing (and more specifically for translation tasks) and has showed significant improvement for the management of long-term dependencies.",
"The idea of an attention mechanism is to focus at each time step on some specific elements of the input sequence.",
"This is modeled by weighted connexions onto the sequence elements (or onto the sequence of hidden units).",
"Therefore it is differentiable and subject to backpropagation-based learning at a meta-level, as with LSTM gate control described in previous section.",
"For more details, see, for example, [63].",
"Interestingly, a novel architecture for translation of sequences, named Transformer and which is solely based on an attention mechanismThe architecture introduces multi-head attention which allows the model to jointly attend to information from different representation subspaces at different positions [198]., has recently being proposed and shows promising results [198].",
"Its very recent application to music generation will be shortly discussed in Section ."
],
[
"Convolutional Architectural Pattern",
"Convolutional neural network (CNN or ConvNet) architectures for deep learning have become common place for image applications.",
"The concept was originally inspired by both a model of human vision and the convolution mathematical operatorIn mathematics, a convolution is a mathematical operation on two functions sharing the same domain (usually noted $f*g$ ) that produces a third function which is the integral (or the sum in the discrete case – the case of images made of pixels) of the pointwise multiplication of the two functions varying within the domain in an opposing way.",
"In the case of a continuous domain $[low~~high]$ : $(f * g)(\\text{x}) = \\int _{low}^{high} f(\\text{x} - \\text{t})g(\\text{t})d\\text{t}$ In the discrete case: $(f * g)(\\text{n}) = \\sum _{m=low}^{high} f(\\text{n} - \\text{m})g(\\text{m})$ .",
"It has been carefully adapted to neural networks and improved by LeCun, at first for handwritten character and object recognition [111].",
"This resulted in efficient and accurate architectures for pattern recognition, exploiting the spatial local correlation present in natural images."
],
[
"Principles",
"The basic ideaInspired by the nice intuitive explanation provided by Karn in [97].",
"For more technical details see, for example, [117] or [63].",
"is to slide a matrix (named a filter, a kernel or a feature detector) through the entire image (seen as the input matrix); and for each mapping position: compute the dot product of the filter with each mapped portion of the image; and then sum up all elements of the resulting matrix; resulting in a new matrix (composed of the different sums for each sliding/mapping position), named convolved feature, or also feature map.",
"The size of the feature map is controlled by three hyperparameters: depth – the number of filters used; stride – the number of pixels by which we slide the filter matrix over the input matrix; and zero-padding – the padding of the input matrix with zeros around its borderZero-padding allows mapping of the filter up to the borders of the image.",
"It also avoids shrinking the representation, which otherwise would be problematic when using multiple consecutive convolutional layers [63]..",
"Figure: Convolution, filter and feature map.Inspired by Karn's data science blog postAn example is illustrated in Figure REF with some simple settings: depth = 1, stride = 1 and no zero-padding.",
"Various filter matrixes can be used with different objectives, such as detection of different features (e.g., edges or curves) or other operations such as sharpening or blurring.",
"The parameter sharing used by the convolution (because of the shared fixed filter) brings the important property of equivariance to translation, i.e.",
"a motif in an image can be detected independently of its location [63]."
],
[
"Stages",
"A convolution usually consists of three successive stages: a convolution stage, as described in Section REF ; a nonlinear rectification stage, sometimes named detector stage, which applies a nonlinear operation, usually ReLU; and a pooling stage, also named subsampling, to reduce the dimensionality."
],
[
"Pooling",
"The motivation for pooling is to reduce the dimensionality of each feature map while retaining significant information.",
"Operations used for pooling are, for example, max, average and sum.",
"In addition to reducing the dimensionality of data, pooling brings the important property of the invariance to small transformations, distortions and translations in the input image.",
"This provides an overall robustness to the processing [97].",
"Like convolution, pooling has hyperparameters to control the process.",
"A simple example of max pooling with stride = 2 is illustrated in Figure REF .",
"Figure: Pooling.Inspired by Karn's data science blog post"
],
[
"Multilayer Convolutional Architecture",
"A typical example of a convolutional architecture with successive layers – each one including the three stages of convolution, nonlinearity and pooling – is illustrated in Figure REF .",
"The final layer is a fully connected layer, like in standard feedforward networks, and typically ends up in a softmax in order to classify image types.",
"Figure: Convolutional deep neural network architecture.Inspired by Karn's data science blog postNote that a convolution is an architectural pattern, as it may be applied internally to almost any architecture listed."
],
[
"Convolution over Time",
"For musical applications, it could be interesting to apply convolutions to the time dimensionThis approach is actually the basis for time-delay neural networks [108]., in order to model temporally invariant motives.",
"Therefore, the convolution operation will share parameters across time [63], like for RNNsIndeed, RNNs are invariant in time, as remarked in [94]..",
"However, the sharing of parameters is shallow, as it applies only to a small number of temporal neighboring members of the input, in contrast to RNNs that share parameters in a deep way, for all time steps.",
"RNNs are indeed much more frequent than convolutional networks for musical applications.",
"That said, we have noticed the recent occurrence of some convolutional architectures as an alternative to RNN architectures, following the pioneering WaveNet architecture for audio [194], described in Section REF .",
"WaveNet presents a stack of causal convolutional layers, somewhat analogous to recurrent layers.",
"Another example is the C-RBM architecture, described in Section REF .",
"If we now consider the pitch dimension, in most cases pitch intervals are not considered invariants, and thus convolutions should not a priori apply to the pitch dimensionAn exception is Johnson's architecture [94], analyzed in Section REF , which explicitly looks for invariance in pitch (although this seems to be a rare choice) and accordingly uses an RNN over the pitch dimension..",
"This issue of convolution versus recurrence (recurrent networks) for musical applications will be further discussed in Section ."
],
[
"Conditioning Architectural Pattern",
"The idea of a conditioning (sometimes also named conditional) architecture is to parametrize the architecture based on some extra conditioning information, which could be arbitrary, e.g., a class label or data from other modalities.",
"The objective is to have some control over the data generation process.",
"Examples of conditioning information are a bass line or a beat structure in the rhythm generation system to be described in Section REF ; a chord progression in the MidiNet system to be described in Section REF ; some positional constraints on notes in the Anticipation-RNN system to be described in Section REF ; and a musical genre or an instrument in the WaveNet system to be described in Section REF .",
"In practice, the conditioning information is usually fed into the architecture as an additional and specific input layer, shown in purple in Figure REF .",
"Figure: Conditioning architectureThe conditioning layer could be a simple input layer.",
"An example is a tag specifying a musical genre or an instrument in the WaveNet system to be described in Section REF ; or some output of some architecture, being the same architecture, as a way to condition the architecture on some historyThis is close in spirit to a recurrent architecture (RNN).. An example is the MidiNet system to be described in Section REF , in which history information from previous measure(s) is injected back into the architecture; or another architecture.",
"An example is the DeepJ system to be described in Section REF , in which two successive transformation layers of a style tag produce an embedding used as the conditioning input.",
"In the case of conditioning a time-invariant architecture – recurrent or convolutional over time – there are two options global conditioning – if the conditioning input is shared for all time steps; and local conditioning – if the conditioning input is specific to each time step.",
"The WaveNet architecture, which is convolutional over time (see Section REF ), offers the two options, as will be analyzed in Section REF ."
],
[
"Generative Adversarial Networks (GAN) Architectural Pattern",
"A significant conceptual and technical innovation was introduced in 2014 by Goodfellow et al.",
"with the concept of generative adversarial networks (GAN) [64].",
"The idea is to train simultaneously two neural networksIn the original version, two feedforward networks are used.",
"But we will see that other networks may be used, e.g., recurrent networks in the C-RNN-GAN architecture (Section REF ) and convolutional feedforward networks in the MidiNet architecture (Section REF )., as illustrated in Figure REF : Figure: Generative adversarial networks (GAN) architecture.Reproduced from with permission of O'Reilly Media a generative model (or generator) G, whose objective is to transform a random noise vector into a synthetic (faked) sample, which resembles real samples drawn from a distribution of real images; and a discriminative model (or discriminator) D, which estimates the probability that a sample came from the real data rather than from the generator GIn some ways, a GAN represents an automated Turing test setting, with the discriminator being the evaluator and the generator being the hidden actor..",
"This corresponds to a minimax two-player game, with one unique (final) solutionIt corresponds to the Nash equilibrium of the game.",
"In game theory, the intuition of a Nash equilibrium is a solution where no player can benefit by changing strategies while the other players keep theirs unchanged, see, for example, [145].",
": G recovers the training data distribution and D outputs $1/2$ everywhere.",
"The generator is then able to produce user-appealing synthetic samples from noise vectors.",
"The discriminator may then be discarded.",
"The minimax relationship is defined in Equation REF .",
"$\\underset{G}{min}~ \\underset{D}{max}~ V(G, D) = \\mathbb {E}_{\\text{x} \\sim P_{\\text{Data}}}[\\text{log}~D(\\text{x})] + \\mathbb {E}_{\\text{z} \\sim P_{\\text{z}}(\\text{z})}[\\text{log}(1 - D(G(\\text{z})))]$ $D(\\text{x})$ represents the probability that x came from the real data (i.e.",
"the correct estimation by D); and $\\mathbb {E}_{\\text{x} \\sim p_{\\text{Data}}}[\\text{log}~D(\\text{x})]$ is the expectationThe expectation has been introduced in Section REF .",
"of $\\text{log}~D(\\text{x})$ with respect to x being drawn from the real data.",
"It is thus D's objective to estimate correctly real data, that is to maximize the $\\mathbb {E}_{\\text{x} \\sim p_{\\text{Data}}}[\\text{log}~D(\\text{x})]$ term.",
"$D(G(\\text{z}))$ represents the probability that $G(\\text{z})$ came from the real data (i.e.",
"the uncorrect estimation by D); $1 - D(G(\\text{z}))$ represents the probability that $G(\\text{z})$ did not come from the real data, i.e.",
"that it was generated by G (i.e.",
"the correct estimation by D); and $\\mathbb {E}_{\\text{z} \\sim p_{\\text{z}}(\\text{z})}[\\text{log}(1 - D(G(\\text{z})))]$ is the expectation of $\\text{log}(1 - D(G(\\text{z})))$ with respect to $G(\\text{z})$ being produced by G from z random noise.",
"It is thus also D's objective to estimate correctly synthetic data, that is to maximize the $\\mathbb {E}_{\\text{z} \\sim p_{\\text{z}}(\\text{z})}[\\text{log}(1 - D(G(\\text{z})))]$ term.",
"In summary, it is D's objective to estimate correctly both real data and synthetic data and thus to maximize both $\\mathbb {E}_{\\text{x} \\sim p_{\\text{Data}}}[\\text{log}~D(\\text{x})]$ and $\\mathbb {E}_{\\text{z} \\sim p_{\\text{z}}(\\text{z})}[\\text{log}(1 - D(G(\\text{z})))]$ terms, i.e.",
"to maximize $V(G, D)$ .",
"On the opposite side, G's objective is to minimize $V(G, D)$ .",
"Actual training is organized with successive turns between the training of the generator and the training of the discriminator.",
"One of the initial motivations for GAN was for classification tasks to prevent adversaries from manipulating deep networks to force misclassification of inputs (this vulnerability is analyzed in detail in [183]).",
"However, from the perspective of content generation (which is our interest), GAN improves the generation of samples, which become hard to distinguish from the actual corpus examples.",
"To generate music, random noise is used as an input to the generator G, whose goal is to transform random noises into the objective, e.g., melodiesIn that respect, generation from a GAN has some similarity with generation by decoding hidden layer variables of a variational autoencoder (Section REF ), as in both cases generation is done from latent variables.",
"An important difference is that, by construction, a variational autoencoder is representative of the whole dataset that it has learnt, that is, for any example in the dataset, there is at least one setting of the latent variables which causes the model to generate something very similar to that example [38].",
"A GAN does not offer such guarantee and does not offer a smooth generation control interface over the latent space (by, e.g., interpolation or attribute arithmetics, see Section REF ), but it can usually generate better quality (better resolution) images than a variational autoencoder [125].",
"Note that the resolution limitation for a VAE may be a problem too for audio generation of music, but it appears a priori less a direct concern in the case of symbolic generation of music.. An example of the use of GAN for generating music is the MidiNet system, to be described in Section REF ."
],
[
"Challenges",
"Training based on a minimax objective is known to be challenging to optimize [211], with a risk of nonconverging oscillations.",
"Thus, careful selection of the model and its hyperparameters is important [63].",
"There are also some newer techniques, such as feature matchingFeature matching changes the objective for the generator (and accordingly its cost function) to minimize the statistical difference between the features of the real data and the generated samples, see more details in [169]., among others, to improve training [169].",
"A recent proposed alternative both to GANs and to autoencoders is generative latent optimization (GLO) [9].",
"It is an approach to train a generator without the need to learn a discriminator, by learning a mapping from noise vectors to images.",
"GLO can thus be viewed both as an encoder-less autoencoder, and as a discriminator-less GAN.",
"It can also be used, as for a VAE (variational autoencoder) introduced in Section REF , to control generation by exploring the latent space.",
"GLO has been tested on images but not yet on music and needs more evaluation."
],
[
"Reinforcement Learning",
"Reinforcement learning (RL) may appear at first glance to be outside of our interest in deep learning architectures, as it has distinct objectives and models.",
"However, the two approaches have recently been combined.",
"The first move, in 2013, was to use deep learning architectures to efficiently implement reinforcement learning techniques, resulting in deep reinforcement learning [134].",
"The second move, in 2016, is directly related to our concerns, as it explored the use of reinforcement learning to control music generation, resulting in the RL-Tuner architecture [93] to be described in Section REF .",
"Let us start with a reminder of the basic concepts of reinforcement learning, illustrated in Figure REF Figure: Reinforcement learning – conceptual model.Reproduced from with permission of SAGE Publications, Inc./Corwin an agent within an environment sequentially selects and performs actions in an environment; where each action performed brings it to a new state; the agent receives a reward (reinforcement signal), which represents the fitness of the action to the environment (current situation); the objective of the agent being to learn a near optimal policy (sequence of actions) in order to maximize its cumulated rewards (named its gain).",
"Note that the agent does not know beforehand the model of the environment and the reward, thus it needs to balance between exploring to learn more and exploiting (what it has learned) in order to improve its gain – this is the exploration exploitation dilemma.",
"There are many approaches and algorithms for reinforcement learning (for a more detailed presentation, please refer, for example, to [96]).",
"Among them, Q-learning [203] turned out to be a relatively simple and efficient method, and thus is widely used.",
"The name comes from the objective to learn (estimate) the Q function $Q^*(\\text{s}, \\text{a})$ , which represents the expected gain for a given pair $(\\text{s}, \\text{a})$ , where s is a state and a an action, for an agent choosing actions optimally (i.e.",
"by following the optimal policy $\\pi ^*$ ).",
"The agent will manage a table, called the Q-table, with values corresponding to all possible pairs.",
"As the agent explores the environment, the table is incrementally updated, with estimates becoming more accurate.",
"A recent combination of reinforcement learning (more specifically Q-learning) and deep learning, named deep reinforcement learning, has been proposed [134] in order to make learning more efficient.",
"As the Q-table could be hugeBecause of the high combinatorial nature when the number of possible states and possible actions is huge., the idea is to use a deep neural network in order to approximate the expected values of the Q-table through the learning of many replayed experiences.",
"A further optimization, named double Q-learning [196] decouples the action selection from the evaluation, in order to avoid value overestimation.",
"The task of the first network, named the Target Q-Network, is to estimate the gain (Q), while the task of the Q-Network is to select the next action.",
"Reinforcement learning appears to be a promising approach for incremental adaptation of the music to be generated, e.g., based on the feedback from listeners (this issue will be addressed in Section ).",
"Meanwhile, a significant move has been made in using reinforcement learning to inject some control into the generation of music by deep learning architectures, through the reward mechanism, as described in Section REF ."
],
[
"Compound Architectures",
"Often compound architectures are used.",
"Some cases are homogeneous compound architectures, combining various instances of the same architecture, e.g., a stacked autoencoder (see Section REF ), and most cases are heterogeneous compound architectures, combining various types of architectures, e.g., an RNN Encoder-Decoder which combines an RNN and an autoencoder, see Section REF ."
],
[
"Composition Types",
"We will see that, from an architectural point of view, various types of compositionWe are taking inspiration from concepts and terminology in programming languages and software architectures [172], such as refinement, instantiation, nesting and pattern [55].",
"may be used: Composition – at least two architectures, of the same type or of different types, are combined, such as a bidirectional RNN (Section REF ) combining two RNNs, forward and backward in time; and the RNN-RBM architecture (Section REF ) combining an RNN architecture and an RBM architecture.",
"Refinement – one architecture is refined and specialized through some additional constraint(s)Both cases are refinements of the standard autoencoder architecture through additional constraints, in practice adding an extra term onto the cost function., such as a sparse autoencoder architecture (Section REF ); and a variational autoencoder (VAE) architecture (Section REF ).",
"Nested – one architecture is nested into the other one, for example a stacked autoencoder architecture (Section REF ); and the RNN Encoder-Decoder architecture (Section REF ), where two RNN architectures are nested within the encoder and decoder parts of an autoencoder, which we could therefore also notate as Autoencoder(RNN, RNN).",
"Pattern instantiation – an architectural pattern is instantiated onto a given architecture(s), for example the C-RBM architecture (Section REF ) that instantiates the convolutional architectural pattern onto an RBM architecture, which we could notate as Convolutional(RBM); the C-RNN-GAN architecture (Section REF ), where the GAN architectural pattern is instantiated onto an RNN architecture, which we could notate as GAN(RNN, RNN); and the Anticipation-RNN architecture (Section REF ) that instantiates the conditioning architectural pattern onto an RNN with the output of another RNN as the conditioning input, which we could notate as Conditioning(RNN, RNN)."
],
[
"Bidirectional RNN",
"Bidirectional recurrent neural networks (bidirectional RNNs) were introduced by Schuster and Paliwal [171] to handle the case when the prediction depends not only on the previous elements but also on the next elements, as for instance with speech recognition.",
"In practice, a bidirectional RNN combinesSee more details in [171].",
"a first RNN that moves forward through time and begins from the start of the sequence; and a second symmetric RNN that moves backward through time and begins from the end of the sequence.",
"The output y$_t$ of the bidirectional RNN at step $t$ combines the output h$^f_t$ at step $t$ of the hidden layer of the “forward RNN”, and the output h$^b_{N-t+1}$ at step $N-t+1$ of the hidden layer of the “backward RNN”.",
"Figure: Bidirectional RNN architectureAn illustration is in Figure REF .",
"Examples of use are the BLSTM architecture (Section REF ); the C-RNN-GAN architecture (Section REF ) that encapsulates a bidirectional RNN into the discriminator of a GAN; and the MusicVAE architecture (Section REF ) that encapsulates a bidirectional RNN into the encoder of a VAE (variational autoencoder)."
],
[
"RNN Encoder-Decoder",
"The idea of encapsulating two identical recurrent networks (RNNs) into an autoencoder, named the RNN Encoder-DecoderWe could also notate it as Autoencoder(RNN, RNN)., was initially proposed in [20] as a technique to encode a variable length sequence learnt by a recurrent network into another variable length sequence produced by another recurrent networkThis is named sequence-to-sequence learning [181]..",
"The motivation and application target is the translation from one language to another, resulting in sentences of possibly different lengths.",
"The idea is to use a fixed-length vector representation as a pivot representation between an encoder and a decoder architecture, see the illustration in Figure REF .",
"The hidden layer(s) $h^e_t$ of the encoder will act as a memory which Figure: RNN Encoder-Decoder architecture.Inspired from iteratively accumulates information about some input sequence of length $N$ , while reading its successive x$_t$ elementsThe end of the sequence is marked by a special symbol, as when training an RNN, see Section REF ., resulting in a final state h$^e_N$ ; which is passed to the decoder as the summary c of the whole input sequence; and the decoder then iteratively generates the output sequence of length $M$ , by predicting the next item y$_t$ given its hidden state h$^d_t$ and the summary (as a conditioning additional input) cAs noted by Goodfellow et al.",
"in [63], an alternative is to use the summary c only to initialize the initial hidden state of the decoder h$^d_0$ .",
"This is, for instance, the strategy chosen in the GLSR-VAE architecture described in Section REF ..",
"The two components of the RNN Encoder-Decoder are jointly trained to minimize the cross-entropy between input and output.",
"See in Figure REF the example of the Audio Word2Vec architecture for processing audio phonetic structures [26].",
"Figure: RNN Encoder-Decoder audio Word2Vec architecture.Reproduced from with permission of the authorsOne limitation of the RNN Encoder-Decoder approach is the difficulty for the summary to memorize very long sequencesIn text translation applications, sentences have a limited size.. Two possible directions are using an attention mechanismIntroduced in Section REF .",
"; and using a hierarchical model, as proposed in the MusicVAE architecture, to be introduced in Section REF ."
],
[
"Variational RNN Encoder-Decoder",
"An interesting development is a variational version of the RNN Encoder-Decoder, in other words a variational autoencoder (VAE) encapsulating two RNNs.",
"We could notate it as Variational(Autoencoder(RNN, RNN)).",
"The objective is to combine the variational property of the VAE for controlling the generationSee Section REF .",
"; and the sequence generation property of the RNN.",
"Examples of its application to music generation will be introduced in Section REF ."
],
[
"Polyphonic Recurrent Networks",
"The RNN-RBM architecture, to be introduced in Section REF , combines an RBM architecture and a recurrent (LSTM) architecture by coupling them to associate the vertical perspective (simultaneous notes) with the horizontal perspective (temporal sequences of notes) of a polyphony to be generated."
],
[
"Further Compound Architectures",
"It is possible to further combine architectures that are already compound, for example the WaveNet architecture (Section REF ), which is a conditioning convolutional feedforward architecture with some tag as the conditioning input, which we could notate as Conditioning(Convolutional(Feedforward), Tag); and the VRASH architecture (Section REF ), which is a variational autoencoder encapsulating RNNs with the decoder being conditioned on history, which we could notate as Variational(Autoencoder(RNN, Conditioning(RNN, History))).",
"There are also some more specific (ad hoc) compound architectures, for example Johnson's Hexahedria architecture (Section REF ), which combines two layers recurrent on the time dimension with two other layers recurrent on the pitch dimension, as an integrated alternative to the RNN-RBM architecture; and The DeepBach architecture (Section REF ), which combines two feedforward architectures with two recurrent architectures."
],
[
"The Limits of Composition",
"There is a natural tendency to explore possible combinations of different architectures with the hope of combining their respective features and merits.",
"An example of a sophisticated compound architecture is the VRASH architecture (Section REF ), which combines variational autoencoder; recurrent networks; and conditioning (on the decoder).",
"However, note that not all combinations make sense.",
"For instance, recurrence and convolution over the time dimension would compete, as discussed in Section ; and there is no guarantee that combining a maximal variety of types will make a sound and accurate architectureAs in the case of a good cook, whose aim is not to simply mix all possible ingredients but to discover original successful combinations.. We will see in Chapter that an important additional design dimension is the strategy, which governs how an architecture will process representations in order to reach a given objective with some expected properties (the challenges)."
],
[
"Challenge and Strategy",
"*Chapter Challenge and Strategy presents the fourth and the fifth dimensions of the conceptual framework proposed in this book to analyze, classify and compare various deep learning-based music generation systems.",
"We analyze successive limitations and challenges occurring when applying deep learning techniques to music generation.",
"Examples of challenges are: content variability, control, structure, originality and interactivity.",
"For each challenge, we present alternative strategies for addressing it.",
"Examples of strategies are: single-step feedforward, iterative feedforward, decoder feedforward, input manipulation and sampling.",
"The analysis is illustrated by numerous examples of various deep learning-based music generation systems.",
"Each system is summarized along the conceptual framework presented.",
"This chapter is the core of the book.",
"We are now reaching the core of this book.",
"This chapter will analyze in depth how to apply the architectures presented in Chapter to learn and generate music.",
"We will first start with a naive, straightforward strategy, using the basic prediction task of a neural network to generate an accompaniment for a melody.",
"We will see that, although this simple direct strategy does work, it suffers from some limitations.",
"We then will study these limitations, some relatively simple to solve, some more difficult and profound – the challenges.",
"We will analyze various strategiesRemember, and this will be important for the following sections, that, as stated in Chapter , we consider here the strategy related to the generation phase and not the training phase (which could be different).",
"for each challenge, and illustrate them though different systemsAs proposed in Chapter , we use the term systems for various proposals – architectures, models, prototypes, systems and related experiments – for deep learning-based music generation, collected from the related literature.",
"taken from the relevant literature.",
"This also provides an opportunity to study the possible relationships between architectures and strategies."
],
[
"Notations for Architecture and Representation Dimensions",
"At first, let us introduce some compact notations for the dimension of an architecture and for the size of a representation: Architecture-type$^n$ for a $n$ -layer architectureThis notation has actually already been introduced in Section REF ., e.g., Feedforward$^2$ for the 2-layer feedforward architecture of the MiniBach system to be introduced in Section REF , Architecture-type${\\times }n$ for a $n$ -instance compound architecture, e.g., RNN$\\times $ 2 for the double RNN compound architecture of RL-Tuner to be introduced in Section REF , and One-hot${\\times }n$ for a multi-one-hot encoding representation, such as: a $n$ -time steps one-hot encoding, e.g., One-hot$\\times $ 64 for the 64-time steps representation of the DeepHear$_M$ system to be introduced in Section REF , a $n$ -voice one-hot encoding, e.g., One-hot$\\times $ 2 for the melody+chords representation of the Blues$_{MC}$ system to be introduced in Section REF , or a combination of a multi-time steps encoding and a multivoice encoding, e.g., One-hot$\\times $ 64$\\times $ (1+3) for the 64-time steps 1-voice input and 3-voices output representation of the MiniBach system to be introduced in Section REF .",
"An example of a combination of the two notations is LSTM$^2\\times $ 2 for the double 2-layer RNN compound architecture of the Anticipation-RNN system to be introduced in Section REF .",
"The most direct strategy is using the prediction or the classification task of a neural network in order to generate musical content.",
"Let us consider the following objective: for a given melody we want to generate an accompaniment, for example, a counterpoint.",
"We will consider a dataset of examples, each one being a pair $(melody, counterpoint\\,melody(ies))$ .",
"We then train a feedforward neural network architecture in a supervised learning manner on this dataset.",
"Once trained, we can choose an arbitrary melody and feedforward it into the architecture in order to produce a corresponding counterpoint accompaniment, in the style of the dataset.",
"Generation is completed in a single-step of feedforward processing.",
"Therefore, we have named this strategy the single-step feedforward strategy."
],
[
"Example: MiniBach Chorale Counterpoint Accompaniment Symbolic Music Generation System",
"Let us consider the following objective: generating a counterpoint accompaniment to a given melody for a soprano voice, through three matching parts, corresponding to alto, tenor and bass voices.",
"We will use as a corpus the set of J. S. Bach's polyphonic chorales [5].",
"As we want this first introductory system to be simple, we consider only 4 measures long excerpts from the corpus.",
"The dataset is constructed by extracting all possible 4 measures long excerpts from the original 352 chorales, also transposed in all possible keys.",
"Once trained on this dataset, the system may be used to generate three counterpoint voices corresponding to an arbitrary 4 measures long melody provided as an input.",
"Somehow, it does capture the practice of J. S. Bach, who chose various melodies for a soprano and composed the three additional voices melodies (for alto, tenor and bass) in a counterpoint manner.",
"First, we need to decide the input as well as the output representations.",
"We represent four measures of 4/4 music.",
"Both the input and the output representations are symbolic, of the piano roll type, with one-hot encoding for each voice, i.e.",
"a multi-one-hot encoding for the output representation.",
"The three first voices (soprano, alto and tenor) have a scope of 20 possible notes plus an additional token to encode a holdSee Section REF .",
"Note that, as a simplification, MiniBach does not consider rests., while the last voice (bass) has a scope of 27 possible notes plus the hold symbol.",
"Time quantization (the value of the time step) is set at the sixteenth note, which is the minimal note duration used in the corpus.",
"The input representation has a size of 21 possible notes $\\times $ 16 time steps $\\times $ 4 measures, i.e.",
"$21\\times 16\\times 4 = 1,344$ , while the output representation has a size of $(21 + 21 + 28)\\times 16\\times 4 = 4,480$ .",
"The architecture, a feedforward network, is shown in Figure REF .",
"As explained previously and because of the mapping between the representation and the architecture, the input layer has 1,344 nodes and the output layer 4,480.",
"There is a single hidden layer with 200 unitsThis is an arbitrary choice..",
"The nonlinear activation function used for the hidden layer is ReLU.",
"The output layer activation function is sigmoid and the cost function used is binary cross-entropy (this is a case of multi$^2$ multiclass single label, see Section REF ).",
"The detail of the architecture and the encoding is shown in Figure REF .",
"It shows the encoding of successive music time slices into successive one-hot vectors directly mapped to the input nodes (variables).",
"In the figure, each blackened vector element as well as each corresponding blackened input node element illustrate the specific encoding (one-hot vector index) of a specific note time slice, depending of its actual pitch (or a hold in the case of a longer note, shown with a bracket).",
"The dual process happens at the output.",
"Each grey output node element illustrates the chosen note (the one with the highest probability), leading to a corresponding one-hot index, leading ultimately to a sequence of notes for each counterpoint voice.",
"Figure: MiniBach architectureFigure: MiniBach architecture and encodingThe characteristics of this system, named MiniBachMiniBach is actually a strong simplification – but with the same objective, corpus and representation principles – of the DeepBach system to be introduced in Section REF ., are summarized in our multidimensional conceptual framework (as defined in Chapter Method) in Table REF .",
"The notationThese notations, introduced in Section , will be summarized in Section .",
"One-hot$\\times $ 64$\\times $ (1+3) means an encoding with 1 input + 3 output voices, each with 64 (for 4 measures of 16 time steps each) one-hot encodings of notes.",
"The notation Feedforward$^2$ means a 2-layer feedforward architecture (with 1 hidden layer).",
"An example of a chorale counterpoint generated from a soprano melody is shown in Figure REF .",
"Figure: Example of a chorale counterpoint generated by MiniBach from a soprano melodyTable: MiniBach summary"
],
[
"A First Analysis",
"The chorales produced by MiniBach look convincing at first glance.",
"But, independently of a qualitative musical evaluation, where an expert could detect some defects, objective limitations of MiniBach appear: A structural limitation is that the music produced (as well as the input melody) has a fixed size (one cannot produce a longer or shorter piece of music).",
"The same melody will always produce exactly the same accompaniment because of the deterministic nature of a feedforward neural network architecture.",
"The generated accompaniment is produced in a single atomic step, without any possibility of human intervention (i.e.",
"without incrementality and interactivity).",
"Let us now introduce a tentative list of limitations (in most cases, properties not fulfilled) and challengesOur shallow distinction between a limitation and a challenge is as follows: limitations have relatively well-understood solutions, whereas challenges are more profound and still the subject of open research.",
": Ex nihilo generation (vs accompaniment); Length variability (vs fixed length); Content variability (vs determinism); Expressiveness (vs mechanization); Melody-harmony consistency; Control (e.g., tonality conformance, maximum number of repeated notes...); Style transfer; Structure; Originality (vs imitation); Incrementality (vs one-shot generation); Interactivity (vs automation); Adaptability (vs no improvement through usage); and Explainability (vs black box).",
"We will analyze them with possible matching solutions and illustrate them through various examples systems."
],
[
"The MiniBach system is good at generating an accompaniment (a counterpoint composed of three distinct melodies) matching an input melody.",
"This is an example of supervised learning, as training examples include both an input (a melody) and a corresponding output (accompaniment).",
"Now suppose that our objective is to generate a melody on its own – not as an accompaniment of some input melody – while being based on a style learnt from a corpus of melodies.",
"A standard feedforward architecture and its companion single-step feedforward strategy, such as those used in MiniBach (described in Section REF ), are not appropriate for such an objective.",
"Let us introduce some strategies to generate new music content ex nihilo or from minimal seed information, such as a starting note or a high-level description."
],
[
"Decoder Feedforward",
"The first strategy is based on an autoencoder architecture.",
"As explained in Section , through the training phase an autoencoder will specialize its hidden layer into a detector of features characterizing the type of music learnt and its variationsTo enforce this specialization, sparse autoencoders are often used (see Section REF ).. One can then use these features as an input interface to parameterize the generation of musical content.",
"The idea is then to: choose a seed as a vector of values corresponding to the hidden layer units; insert it in the hidden layer; and feedforward it through the decoder.",
"This strategy, that we name decoder feedforward, will produce a new musical content corresponding to the features, in the same format as the training examples.",
"In order to have a minimal and high-level vector of features, a stacked autoencoder (see Section REF ) is often used.",
"The seed is then inserted at the bottleneck hidden layer of the stacked autoencoderIn other words, at the exact middle of the encoder/decoder stack, as shown in Figure REF .",
"and feedforwarded through the chain of decoders.",
"Therefore, a simple seed information can generate an arbitrarily long, although fixed-length, musical content.",
"An example of this strategy is the DeepHear system by Sun [180].",
"The corpus used is 600 measures of Scott Joplin's ragtime music, split into 4 measures long segments.",
"The representation used is piano roll with a multi-one-hot encoding.",
"The quantization (time step) is a sixteenth note, thus the representation includes $4\\times 16 = 64$ time steps (notated as One-hot$\\times $ 64).",
"The number of input nodes is around 5,000, which provides a vocabulary of about 80 possible note values.",
"The architecture is shown in Figure REF and is a 4-layer stacked autoencoder (notated as Autoencoder$^4$ ) with a decreasing number of hidden units, down to 16 units.",
"Figure: DeepHear stacked autoencoder architecture.Extension of a figure reproduced from with permission of the authorAfter a pre-training phaseWe do not detail pre-training here, please refer to, for example, [63]., final training is performed, with each provided example used both as an input and as an output, in the self-supervised learning manner (see Section ) shown in Figure REF .",
"Figure: Training DeepHear.Extension of a figure reproduced from with permission of the authorGeneration is performed by inputing random data as the seed into the 16 bottleneck hidden layer unitsThe units of the hidden layer represent an embedding (see Section REF ), of which an arbitrary instance is named by Sun a label.",
"(shown within a red rectangle) and then by feedforwarding it into the chain of decoders to produce an output (in the same 4 measures long format as the training examples), as shown in Figure REF .",
"We summarize the characteristics of DeepHear$_M$We notate DeepHear$_M$ this DeepHear melody generation system, where $M$ stands for melody, because another experiment with the same DeepHear architecture but with a different objective will be presented later on in Section REF .",
"in Table REF .",
"Figure: Generation in DeepHear.Extension of a figure reproduced from with permission of the authorTable: DeepHear M _M summaryIn [180], Sun remarks that the system produces a certain amount of plagiarism.",
"Some generated music is almost recopied from the corpus.",
"He states that this is because of the small size of the bottleneck hidden layer (only 16 nodes) [180].",
"He measured the similarity (defined as the percentage of notes in a generated piece that are also in one of the training pieces) and found that, on average, it is 59.6%, which is indeed quite high, although it does not prevent most of generated pieces from sounding different."
],
[
"#2 Example: deepAutoController Audio Music Generation System",
"The deepAutoController system, by Sarroff and Casey [170], is similar to DeepHear (see Section REF ) in that it also uses a stacked autoencoder.",
"But the representation is audio, more precisely a spectrum generated by Fourier transform, see [170] for more details.",
"The dataset is composed of 8,000 songs of 10 musical genres, leading to 70,000 frames of magnitude Fourier transformsAs the authors state in [170]: “We chose to use frames of magnitude FFTs (Fast Fourier transforms) for our models because they may be reconstructed exactly into the original time domain signal when the phase information is preserved, the Fourier coefficients are not altered, and appropriate windowing and overlap-add is applied.",
"It was thus easier to subjectively evaluate the quality of reconstructions that had been processed by the autoencoding models.”.",
"The entire data is normalized to the $[0, 1]$ range.",
"The cost function used is mean squared error.",
"The architecture is a 2-layer stacked autoencoder, the bottleneck hidden layer having 256 units and the input and output layers having 1,000 nodes.",
"The authors report that increasing the number of hidden units does not appear to improve the model performance.",
"The system, summarized in Table REF , also provides a user interface, analyzed in Section , to interactively control the generation, e.g., selecting a given input (to be inserted at the bottleneck hidden layer), generating a random input, and controlling (by scaling or muting) the activation of a given unit.",
"Table: deepAutoController summaryAnother strategy is based on sampling.",
"Sampling is the action of generating an element (a sample) from a stochastic model according to a probability distribution."
],
[
"Sampling Basics",
"The main issue for sampling is to ensure that the samples generated match a given distribution.",
"The basic idea is to generate a sequence of sample values in such a way that, as more and more sample values are generated, the distribution of values more closely approximates the target distribution.",
"Sample values are thus produced iteratively, with the distribution of the next sample being dependent only on the current sample value.",
"Each successive sample is generated through a generate-and-test strategy, i.e.",
"by generating a prospective candidate, accepting or rejecting it (based on a defined probability density) and, if needed, regenerating it.",
"Various sampling strategies have been proposed: Metropolis-Hastings algorithm, Gibbs sampling (GS), block Gibbs sampling, etc.",
"Please see, for example, [63] for more details about sampling algorithms."
],
[
"Sampling for Music Generation",
"For musical content, we may consider two different levels of probability distribution (and sampling): item-level or vertical dimension – at the level of a compound musical item, e.g., a chord.",
"In this case, the distribution is about the relations between the components of the chord, i.e.",
"describing the probability of notes to occur together; and sequence-level or horizontal dimension – at the level of a sequence of items, e.g., a melody composed of successive notes.",
"In this case, the distribution is about the sequence of notes, i.e.",
"it describes the probability of the occurrence of a specific note after a given note.",
"An RBM (restricted Boltzmann machine) architecture is generallyA counterexample is the C-RBM convolutional RBM architecture, to be introduced in Section REF , which models both the vertical dimension (simultaneous notes) and the horizontal dimension (sequence of notes) for single-voice polyphonies.",
"used to model the vertical dimension, i.e.",
"which notes should be played together.",
"As noted in Section , an RBM architecture is dedicated to learning distributions and can learn efficiently from few examples.",
"This is particularly interesting for learning and generating chords, as the combinatorial nature of possible notes forming a chord is large and the number of examples is usually small.",
"An example of a sampling strategy applied on an RBM for the horizontal dimension will be presented in Section REF .",
"An RNN (recurrent neural network) architecture is often used for the horizontal dimension, i.e.",
"which note is likely to be played after a given note, as will be described in Section REF .",
"As we will see in Section REF , a sampling strategy may be also added to enforce variability.",
"We will see in Section REF that a compound architecture named RNN-RBM may combine and articulateThis issue of how to articulate vertical and horizontal dimensions, i.e.",
"harmony with melody, will be further analyzed in Section .",
"these two different approaches: an RBM architecture with a sampling strategy for the vertical dimension; and an RNN architecture with an iterative feedforward strategy for the horizontal dimension.",
"An alternative approach is to use sampling as the unique strategy for both dimensions, as witnessed by the DeepBach system to be analyzed in Section REF ."
],
[
"Example: RBM-based Chord Music Generation System",
"In [11], Boulanger-Lewandowski et al.",
"propose to use a restricted Boltzmann machine (RBM) [81] to model polyphonic music.",
"Their objective is actually to improve the transcription of polyphonic music from audio.",
"But prior to that, the authors discuss the generation of samples from the model that has been learnt as a qualitative evaluation and also for music generation [12].",
"In their first experiment, the RBM learns from the corpus the distribution of possible simultaneous notes, i.e.",
"a repertoire of chords.",
"The corpus is the set of J. S. Bach's chorales (as for MiniBach, described in Section REF ).",
"The polyphony (number of simultaneous notes) varies from 0 to 15 and the average polyphony is 3.9.",
"The input representation has 88 binary visible units that span the whole range of piano from A$_0$ to C$_8$ , following a many-hot encoding.",
"The sequences are aligned (transposed) onto a single common tonality (e.g., C major/minor) to ease the learning process.",
"One can sample from the RBM through block Gibbs sampling, by performing alternative steps of sampling the hidden layer nodes (considered as variables) from the visible layer nodes (see Section ).",
"Figure REF shows various examples of samples.",
"The vertical axis represents successive possible notes.",
"Each column represents a specific sample composed of various simultaneous notes, with the name of the chord written below when the analysis is unambiguous.",
"Table REF summarizes this RBM-based chord generation system, which we notate RBM$_C$ (where $C$ stands for chords).",
"Figure: Samples generated by the RBM trained on J. S. Bach chorales.Reproduced from with permission of the authorsTable: RBM C _C summary"
],
[
"Length Variability",
"An important limitation of the single-step feedforward strategy (Section REF ) and of the decoder feedforward strategy (Section REF ) is that the length of the music generated (more precisely the number of times steps or measures) is fixed.",
"It is actually fixed by the architecture, namely the number of nodes of the output layerIn the case of an RBM, the number of nodes of the input layer (which also has the role of an output layer).. To generate a longer (or shorter) piece of music, one needs to reconfigure the architecture and its corresponding representation."
],
[
"Iterative Feedforward",
"The standard solution to this limitation is to use a recurrent neural network (RNN).",
"The typical usage, as initially described for text generation by Graves in [65], is to select some seed information as the first item (e.g., the first note of a melody); feedforward it into the recurrent network in order to produce the next item (e.g., next note); use this next item as the next input to produce the next next item; and repeat this process iteratively until a sequence (e.g., of notes, i.e.",
"a melody) of the desired length is produced.",
"Note the iterative aspect of the generation, processed element by element.",
"Therefore, we name this approach the iterative time step feedforward strategy, abbreviated as the iterative feedforward strategy.",
"Actually, a recursion – current output reenters as the next input – is also often present.",
"However, there are a few rare exceptions, as we will see, e.g., in Sequential (Section REF ) and in BLSTM (Section REF ) architectures, where there is an iteration but no recursion.",
"Note that the iterative feedforward strategy, as the decoder feedforward strategy (Section REF ), is one kind of seed-based generation (see Section ), as the full sequence (e.g., a melody) is generated iteratively from an initial seed item (e.g., a starting note).",
"In [43], Eck and Schmidhuber describe a double experiment undertaken with a recurrent network architecture using LSTMsThis was actually the first experiment in using LSTMs to generate music..",
"In their first experiment, the objective is to learn and generate chord sequences.",
"The format of representation is piano roll, with two types of sequences: melody and chords, although chords are represented as notes.",
"The melodic range as well as the chord vocabulary is strongly constrained, as the corpus consists of 12 measures long blues and is handcrafted (melodies and chords).",
"The 13 possible notes extend from middle C (C$_4$ ) to tenor C (C$_5$ ).",
"The 12 possible chords extend from C to B.",
"A one-hot encoding is used.",
"Time quantization (time step) is set at the eighth note, half of the minimal note duration used in the corpus, which is a quarter note.",
"With 12 measures long music this equates to 96 time steps.",
"An example of chord sequence training example is shown in Figure REF .",
"Figure: A chord training example for blues generation.Reproduced from with permission of the authorsThe architecture for this first experiment is: an input layer with 12 nodes (corresponding to a one-hot encoding of the 12 chord vocabulary), a hidden layer with four LSTM blocks containing two cells eachSee in Section REF for the difference between LSTM cells and blocks.",
"and an output layer with 12 nodes (identical to the input layer).",
"Generation is performed by presenting a seed chord (represented by a note) and by iteratively feedforwarding the network, producing the prediction of the next time step chord, using it as the next input and so on, until a sequence of chords has been generated.",
"The architecture and the iterative generation is illustrated in Figure REF .",
"This system, which we notate Blues$_C$ (where $C$ stands for chords), is summarized in Table REF .",
"Figure: Blues chord generation architectureTable: Blues C _C summary"
],
[
"#2 Example: Blues Melody and Chords Symbolic Music Generation System",
"In Eck and Schmidhuber's second experiment [43], the objective is to simultaneously generate melody and chord sequences.",
"The new architecture is an extension of the previous one: it has an input layer with 25 nodes (corresponding to a one-hot encoding of the 12 chord vocabulary and to a one-hot encoding of the 13 melody note vocabulary), a hidden layer with eight LSTM blocks (four chord blocks and four melody blocks, as we will see below), containing two cells each, and an output layer with 25 nodes (identical to the input layer).",
"The separation between chords and melody is ensured as follows: chord blocks are fully connected to the input nodes and to the output nodes corresponding to chords; melody blocks are fully connected to the input nodes and to the output nodes corresponding to melody; chord blocks have recurrent connections to themselves and to the melody blocks; and melody blocks have recurrent connections only to themselves.",
"Generation is performed by presenting a seed (note and chord) and by recursively feedforwarding it into the network, producing the prediction of the next time step note and chord, and so on, until a sequence of notes with chords is generated.",
"Figure REF shows an example of the melody and chords generated.",
"Table REF summarizes this second system, which we notate Blues$_{MC}$ (where $MC$ stands for melody and chords).",
"Figure: Example of blues generated (excerpt).Reproduced with permission of the authorsTable: Blues MC _{MC} summaryThis second experiment is interesting in that it simultaneously generates melody and chords.",
"Note that in this second architecture, recurrent connexions are asymmetric as the authors wanted to ensure the preponderant role of chords.",
"Chord blocks have recurrent connexions to themselves but also to melody blocks, whereas melody blocks do not have recurrent connexions to chord blocks.",
"This means that chord blocks will receive previous step information about chords and melody, whereas melody blocks cannot use previous step information about chords.",
"This somewhat ad hoc configuration of the recurrent connexions in the architecture is a way to control the interaction between harmony and melody in a master-slave manner.",
"The control of the interaction and consistency between melody and harmony is indeed an effective issue and it will be further addressed in Section where we will analyze alternative approaches."
],
[
"Content Variability",
"A limitation of the iterative feedforward strategy on an RNN, as illustrated by the blues generation experiment described in Section REF , is that generation is deterministic.",
"Indeed, a neural network is deterministicThere are stochastic versions of artificial neural networks – an RBM is an example – but they are not mainstream.. As a consequence, feedforwarding the same input will always produce the same output.",
"As the generation of the next note, the next next note, etc., is deterministic, the same seed note will lead to the same generated series of notesThe actual length of the melody generated depends on the number of iterations..",
"Moreover, as there are only 12 possible input values (the 12 pitch classes), there are only 12 possible melodies."
],
[
"Sampling",
"Fortunately, as we will see, the usual solution is quite simple.",
"The assumption is that the output representation of the melody is one-hot encoded.",
"In other words, the output representation is of a piano roll type, the output activation layer is softmax and generation is modeled as a classification task.",
"See an example in Figure REF , where $P(\\text{x}_t = \\text{C} | \\text{x}_{<t})$ represents the conditional probability for the element (note) x$_t$ at step $t$ to be a C given the previous elements x$_{<t}$ (the melody generated so far).",
"The default deterministic strategy consists in choosing the class (the note) with the highest probability, i.e.",
"$\\text{argmax}\\index {Argmax}_{\\text{x}_t} P(\\text{x}_t | \\text{x}_{<t})$ , that is A$\\flat $ in Figure REF .",
"We can then easily switch to a nondeterministic strategy, by sampling the output which corresponds (through the softmax function) to a probability distribution between possible notes.",
"By sampling a note following the distribution generatedThe chance of sampling a given class/note is its corresponding probability.",
"In the example shown in Figure REF , A$\\flat $ has around one chance in two of being selected and B$\\flat $ one chance in four., we introduce stochasticity in the process and thus variability in the generation.",
"Figure: Sampling the softmax outputCONCERT (an acronym for CONnectionist Composer of ERudite Tunes) developed by Mozer [139] in 1994, was actually one of the first systems for generating music based on recurrent networks (and before LSTM).",
"It is aimed at generating melodies, possibly with some chord progression as an accompaniment.",
"The input and output representation includes three aspects of a note: pitch, duration and harmonic chord accompaniment.",
"The representation of a pitch, named PHCCCH, is inspired by the psychological pitch representation space of Shepard [173], and is based on five dimensions, as illustrated in Figure REF .",
"Figure: CONCERT PHCCCH pitch representation.Inspired by and The three main components are as follows: the pitch height (PH), the (modulo) chroma circle (CC) cartesian coordinates, and the (harmonic) circle of fifths (CH) cartesian coordinates.",
"The motivation is in having a more musically meaningful representation of the pitch by capturing the similarity of octaves and also the harmonic similarity between a note and its fifth.",
"The proximity of two pitches is determined by computing the Euclidean distance between their PHCCCH representations, that distance being invariant under transposition.",
"The encoding of the pitch height is through a scalar variable scaled to range from -9.798 for C$_1$ to +9.798 for C$_5$ .",
"The encoding of the chroma circle and of the circle of fifths is through a six binary value vector, for the reasons detailed in [139].",
"The resulting encoding includes 13 input variables, with some examples shown in Table REF .",
"Note that a rest is encoded as a pitch with a unique code.",
"Table: Examples of PHCCCF pitch representationDurations are considered at a very fine-grain level, each beat (a quarter note) being divided into twelfths, thus having a duration of 12/12.",
"This choice allows to represent binary (two or four divisions) as well as ternary (three divisions) rhythms.",
"In a similar way to the representation of pitch, a duration is represented through a scalar and two circle coordinates, for 1/4 and 1/3 beat cycles, as illustrated in Figure REF , resulting in five dimensions directly encoded through a five binary value vector (see more details in [139]).",
"The temporal scope is a note step, that is the granularity of processing by the architecture is a noteAnd not a fixed time step as for most of recurrent architectures, e.g., in Section REF .",
"The various types of temporal scope have been introduced in Section REF ..",
"Figure: CONCERT duration representation.Inspired by Chords are represented in an extensional way as a triad or a tetrachord, through the root, the third (major or minor) and the fifth (perfect, augmented or diminished), with the possible addition of a seventh component (minor or major).",
"To represent the next note to be predicted, the CONCERT system actually uses both this rich and distributed representation (named next-node-distributed, see Figure REF ) and a more concise and traditional representation (named next-node-local), in order to be more intelligible.",
"The activation function is the sigmoid function rescaled to the $[-1, +1]$ range and the cost function is mean squared error.",
"Figure: CONCERT architecture.Reproduced from with permission of Taylor & Francis (www.tandfonline.com)In the generation phase, the output is interpreted as a probability distribution over a set of possible notes as a basis for deciding the next note in a nondeterministic way, following the sampling strategy.",
"CONCERT has been tested on different examples, notably after training with melodies of J. S. Bach.",
"Figure REF shows an example of a melody generated based on the Bach training set.",
"Although now a bit dated, CONCERT has been a pioneering model and the discussion in the article about representation issues is still relevant.",
"Note also that CONCERT (which is summarized in Table REF ) is representative of the early generation systems, before the advent of deep learning architectures, when representations were designed with rich handcrafted features.",
"One of the benefits of using deep learning architectures is that this kind of rich and deep representation may be automatically extracted and managed by the architecture.",
"Figure: Example of melody generation by CONCERT based on the J. S. Bach training set.Reproduced from with permission of Taylor & Francis (www.tandfonline.com)Table: CONCERT summary"
],
[
"#2 Example: Celtic Melody Symbolic Music Generation System",
"Another representative example is the system by Sturm et al.",
"to generate Celtic music melodies [179].",
"The architecture used is a recurrent network with three hidden layers, which we could notateNote that, as explained in Section REF , we notate the number of hidden layers without considering the input layer.",
"as LSTM$^3$ , with 512 LSTM cells in each layer.",
"The corpus comprises folk and Celtic monophonic melodies retrieved from a repository and discussion platform named The Session [99].",
"Pieces that were too short, too complex (with varying meters) or contained errors were filtered out, leaving a dataset of 23,636 melodies.",
"All melodies are aligned (transposed) onto the single C key.",
"One of the specificities is that the representation chosen is textual, namely the token-based folk-rnn notation, a transformation of the character-based ABC notation (see Section REF ).",
"The number of input and output nodes is equal to the number of tokens in the vocabulary (i.e.",
"with a one-hot encoding), in practice equal to 137.",
"The output of the network is a probability distribution over its vocabulary.",
"Training the recurrent network is done in an iterative way, as the network learns to predict the next item.",
"Once trained, the generation is done iteratively by inputing a random token or a specific token (e.g., corresponding to a specific starting note), feedforwarding it to generate the output, sampling from this probability distribution, and recursively using the selected vocabulary element as a subsequent input, in order to produce a melody element by element.",
"The final step is to decode the folk-rnn representation generated into a MIDI format melody to be played.",
"See in Figure REF for an example of a melody generated.",
"One may also see and listen to results on [178].",
"The results are very convincing, with melodies generated in a clear Celtic style.",
"The system is summarized in Table REF .",
"Figure: Score of “The Mal's Copporim” automatically generated.Reproduced from with permission of the authorsAs observed in [67]: “It is interesting to note that in this approach the bar lines and the repeat bar lines are given explicitly and are to be predicted as well.",
"This can cause some issues, since there is no guarantee that the output sequence of tokens would represent a valid song in ABC format.",
"There could be too many notes in one bar for example, but according to the authors, this rarely occurs.",
"This would tend to show that such an architecture is able to learn to count.”On this issue, see also [60].",
"Table: Celtic system summary"
],
[
"Expressiveness",
"One limitation of most existing systems is that they consider fixed dynamics (amplitude) for all notes as well as an exact quantization (a fixed tempo), which makes the music generated too mechanical, without expressiveness or nuance.",
"A natural approach resides in considering representations recorded from real performances and not simply scores, and therefore with musically grounded (by skilled human musicians) variations of tempo and of dynamics, as discussed in Section .",
"Note that an alternative approach is to automatically augment the generated music information (e.g., a standard MIDI piece) with slight transformations on the amplitude and/or the tempo.",
"An example is the Cyber-João system [30], which performs bossa nova guitar accompaniment with expressiveness, through automatic retrievalBy a mixed use of production rules and case-based reasoning (CBR).",
"and application of rhythmic patternsThese patterns have been manually extracted from a corpus of performances by the guitarist and singer João Gilberto, one of the inventors of the Bossa nova style.",
"One could also consider automatic extraction, as, for example, in [32].. As noted in Section REF , in the case of an audio representation, expressiveness is implicit to the representation.",
"However, it is difficultBut not impossible to achieve, regarding recent achievements made on audio source separation through deep learning techniques, as has been pointed out in Section REF .",
"to separately control the expressiveness (dynamics or tempo) of a single instrument or voice as the representation is global."
],
[
"Example: Performance RNN Piano Polyphony Symbolic Music Generation System",
"In [174], Simon and Oore present their architecture and methodology named Performance RNN.",
"It is an LSTM-based recurrent neural network architecture.",
"One of the specificities is in the dataset characteristics, as the corpus is composed of recorded human performances, with records of exact timing as well as dynamics for each note played.",
"The corpus used is the Yamaha e-Piano Competition dataset, whose participants MIDI performance records are made available to the public [210].",
"It captures more than 1,400 performances by skilled pianists.",
"To create additional training examples, some time stretching (up to 5% faster or slower) as well as some transposition (up to a major third) is applied.",
"The representation is adapted to the objective.",
"At first look, it resembles a piano roll with MIDI note numbers but it is actually a bit different.",
"Each time slice is a multi-one-hot vector of the possible values for each of the following possible events: start of a new note – with 128 possible values (MIDI pitches), end of a note – with 128 possible values (MIDI pitches), time shift – with 100 possible values (from 10 miliseconds to 1 second), and dynamics – with 32 possible values (128 MIDI velocities quantized into 32 binsSee the description of the binning transformation in Section .).",
"An example of a performance representation is shown in Figure REF .",
"Figure: Example of Performance RNN representation.Reproduced from with permission of the authorsSome control is made available to the user, referred to as the temperature, which controls the randomness of the generated events in the following way: a temperature of 1.0 uses the exact distribution predicted, a value smaller than 1.0 reduces the randomness and thus increases the repetition of patterns, and a larger value increases the randomness and decreases the repetition of patterns.",
"Examples are available on the web page [174].",
"Performance RNN is summarized in Table REF .",
"Table: Performance RNN summary"
],
[
"RNN and Iterative Feedforward Revisited",
"As we saw in previous examples, the iterative feedforward strategy is based on the idea of the recurrent neural network (RNN) architecture to iteratively generate successive item of a sequence.",
"It looks like a recurrent neural network architecture and the iterative feedforward strategy are strongly coupled.",
"Indeed, almost all RNN-based systems use an iterative feedforward strategy and recursively reenter the output produced (next time step generated) into the input.",
"But we will introduce in this section some exceptions."
],
[
"#1 Example: Time-Windowed Melody Symbolic Music Generation System",
"The experiments by Todd in [190] were one of the very first attempts (in 1989) at exploring how to use artificial neural networks to generate music.",
"Although the architectures he proposed are not directly used nowadays, his experiments and discussion were pioneering and are still an important source of information.",
"Todd's objective was to generate a monophonic melody in some iterative way.",
"He has experimented with different choices for representing the notes (see Section REF ) and the durations, but finally had decided to use a conventional pitch note representation with a one-hot encoding and a time step temporal scope approach.",
"The time step is set at the duration of an eighth note.",
"In most of experiments, input melodies used for the training are 34 time steps long (that is, four measures and a half long), padded at the end with rests.",
"A note begin is represented with a specific token and is encoded as an additional value encoding node (see Sections REF and REF ).",
"Rests are not encoded explicitly but as the absence of a note, i.e.",
"as the note one-hot encoding being all filled with null values (see Section REF ).",
"The first experiment is what the author named Time-Windowed architecture, where a sliding window of successive time-periods of fixed size is considered.",
"In practice this sliding window of a melody segment is one measure long, i.e.",
"8 time steps.",
"Its representation may be considered as a piano roll, like in the MiniBach architecture (see Section REF ), with the successive one-hot encodings of notes for the 8 successive time steps, notated as One-hot$\\times $ 8.",
"The architecture is a feedforward network (and not an RNN), with a melody segment as its input, the next melody segment as its output and with a single hidden layer.",
"Generation is conducted iteratively (and recursively), melody segment by melody segment.",
"The architecture is illustrated in Figure REF .",
"Figure: Time-Windowed architecture.Inspired from For each time step of the melody segment, the predicted note is the one with the highest probability.",
"Because of the zero-hot encoding of a rest (i.e.",
"as all values being null), there is an ambiguity between the case of every possible note has a low probability and the case of a rest (see Section REF ).",
"For that reason, a probability threshold is introduced, namely 0.5.",
"Thus, the predicted note is the one with the highest probability if it is greater than 0.5 and is a rest otherwise.",
"The network is trained in a supervised way by presenting a melody segment as an input and its corresponding next segment as the output, and repeating this for various segments.",
"Note that, as the architecture is not recurrent, although the network will learn the pairwise correlations between two successive melody segmentsIn that respect, the Time-Windowed model is analog to an order 1 Markov model (considering only the previous state) at the level of a melody measure., there is no explicit memory for learning long term correlations such as in the case of a recurrent network architecture.",
"Thus, although the author does not show a comparison with its next experiment (see next section), the architecture appears to have a low ability to learn long term correlations.",
"The Time-Windowed architecture is summarized in Table REF .",
"Table: Time-Windowed summary"
],
[
"#2 Example: Sequential Melody Symbolic Music Generation System",
"In [190], Todd proposed another architecture, that he named Sequential, as notes are generated in a sequence.",
"It is illustrated in Figure REF .",
"The input layer is divided in two parts, named the context and the plan.",
"The context is the actual memory (of the melody generated so far) and consists in units corresponding to each note (D$_4$ to C$_6$ ), plus a unit about the note begin information (notated as “nb” in Figure REF ).",
"Therefore, it receives information from the output layer which produces next note, with a reentering connexion corresponding to each unitNote that the output layer is isomorphic to the context layer..",
"In addition, as Todd explains it: “A memory of more than just the single previous output (note) is kept by having a self-feedback connection on each individual context unit.”This is a peculiar characteristic of this architecture, as in a standard recurrent network architecture recurrent connexions are encapsulated within the hidden layer (see Figures REF and REF ).",
"The argument by Todd in [190] is that context units are more interpretable than hidden units: “Since the hidden units typically compute some complicated, often uninterpretable function of their inputs, the memory kept in the context units will likely also be uninterpretable.",
"This is in contrast to [this] design, where, as described earlier, each context unit keeps a memory of its corresponding output unit, which is interpretable.” The plan represents a melody (among many) that the network has learnt.",
"Todd has experimented with various encodings, one-hot or distributed (through a many-hot embedding).",
"Figure: Sequential architecture.Inspired from Training is done by selecting a plan (melody) to be learnt.",
"The activations of the context units are initialized to 0 in order to begin with a clean empty context.",
"The network is then feedforwarded and its output, corresponding to the first time step note, is compared to the first time step note of the melody to be learnt, resulting in adjustment of the weights.",
"The output valuesActually, as an optimization, Todd proposes in the following of his description to pass back the target values and not the output values.",
"are passed back to the current context.",
"And then, the network is feedforwarded again, leading to the next time step note, again compared to the melody target, and so on until the last time step of the melody.",
"This process is then repeated for various plans (melodies).",
"Generation of new melodies is conducted by feedforwarding the network with a new plan embedding, corresponding to a new melody (not part of the training plans/melodies).",
"The activations of the context units are initialized to 0 in order to begin with a clean empty context.",
"The generation takes place iteratively, time step after time step.",
"Note that, as opposed to most cases of the iterative feedforward strategy (Section REF ), in which the output is explicitly reentered (recursively) into the input of the architecture, in Todd's Sequential architecture the reentrance is implicit because of the specific nature of the recurrent connexions: the output is reentered into the context units while the input – the plan melody – is constant.",
"After having trained the network on a plan melody, various melodies may be generated by extrapolation by inputing new plans, as shown in Figure REF .",
"A repeat sign : indicates when the network output goes into a fixed loop.",
"Figure: Examples of melodies generated by the Sequential architecture.",
"(o) Original plan melody learnt.",
"(e 1 _1 and e 2 _2) Melodies generated by extrapolating from a new plan melody.Inspired from One could also do interpolation between several (two or more) plans melodies that have been learntNote that this way of doing is actually some precursor of doing interpolation on embeddings of melodies to be generated by combining a decoder feedforward strategy and an iterative feedforward strategy, such as for example in the VRAE or the MusicVAE systems, to be described in Sections REF and REF , respectively..",
"Examples are shown in Figure REF .",
"The Sequential architecture is summarized in Table REF .",
"Figure: Examples of melodies generated by the Sequential architecture.",
"(o A _A and o B _B) Original plan melodies learnt.",
"(i 1 _1 and i 2 _2) Melodies generated by interpolating between o A _A plan and o B _B plan melodies.Inspired from Table: Sequential architecture summary"
],
[
"#3 Example: BLSTM Chord Accompaniment Symbolic Music Generation System",
"The BLSTM (Bidirectional LSTM) chord accompaniment system by Lim et al.",
"[120] is a rare and interesting caseAs noted in Sections REF and REF .",
"of an accompaniment system based on a recurrent architecture.",
"The objective is to generate a progression (sequence) of chords as an accompaniment to a melody (specified symbolically).",
"The corpus is imported from a now defunct lead sheet public data base.",
"The authors selected 2,252 selected lead sheets of various western modern music (rock, pop, country, jazz, folk, R&B, children's song, etc.",
"), all in major key and the majority with a single chord per measure (otherwise only the first chord is considered).",
"This results in a training set of 1,802 songs (making a total of 72,418 measures) and a test set of 450 songs (17,768 measures).",
"All songs are transposed (aligned) to C major key.",
"Desired characteristics are extracted from the original XML files and converted to a CSVCSV stands for Comma-separated values.",
"(spreadsheet) matrix format, as shown in Figure REF .",
"The specificities (simplifications) of the representation are as follows: for the melodyAnd obviously also for the chords., only pitch classes are considered (and octaves are not), resulting in a 12 notes one-hot encoding (named 12-semitone-vector) plus the rest; and for the chords, only their primary triads are considered, with only two types: major and minor, resulting in a 24 chords one-hot encoding.",
"Figure: Example of extracted data from a single measure.Reproduced from under a CC BY 4.0 licenceThe architecture is a bidirectional LSTM with two LSTM layers, each one with 128 units.",
"The motivation is to provide the network with the musical context backward and also forward in time.",
"The time step considered by the architecture is four measures long, as shown in Figure REF .",
"The tanh function is used as the non linear activation function for the hidden layers and softmax is used as the output layer activation function, with categorical cross-entropy as its associated cost function.",
"Figure: BLSTM architecture.Reproduced from under a CC BY 4.0 licenceTraining is done with various four measures long samples as input and their associated four chords as output, generated by sliding a four measures long window over each training song.",
"Generation is done by iteratively feedforwarding successive four measures long melody fragments (time slices) of a song and concatenating the resulting four measures long chord progression fragments.",
"The architecture is peculiar in that, although recurrent, generation is not recursive and the output data has a different nature and structure (chords) than the input data (notes).",
"Furthermore, note that, although the strategy is iterative and the architecture is recurrent, the granularity of each iterative step is quite coarse as it is 4 measures long, as opposed to most of systems based on recurrent architectures and iterative feedforward strategy which consider the time step at the level of the smallest notre duration (see, e.g., the system analyzed in Section REF ).",
"This kind of mixed architecture/strategy between forward/single step and recurrent/iterative may have been motivated by the objective of capturing sufficiently the history of horizontal correlations (between notes of the melody and between chords of the accompaniment) as the LSTM cells focus on capturing the history of vertical correlations (between notes and chords).",
"The system has been evaluated by comparing to some hidden Markov model (HMM) model and to some deep neural network–HMM hybrid model (named DNN-HMM, see details in [199]), both quantitatively (by comparing the accuracies and through confusion matrixes), and qualitatively (through a web-based survey of 25 musically untrained participants).",
"Results are showing a better accuracy and preference for the BLSTM model, see a simple example in Figure REF .",
"The authors note that the evaluation also shows that, when songs are unknown, the preference for the BLSTM model is weaker.",
"They conjecture that this is because BLSTM often generates a more diverse chord sequence than the original.",
"The BLSTM system is summarized in Table REF .",
"Figure: Comparison of generated chord progressions (HMM, DNN-HMM, BLSTM and original).Reproduced from under a CC BY 4.0 licenceTable: BLSTM summary"
],
[
"Summary",
"In summary, we have seen that an RNN architecture is usually coupled to an iterative feedforward strategy, which allows a recursive seed-based variable length generation, as discussed in Section .",
"However, there are some exceptions: the Time-Windowed system by Todd (Section REF ) uses an iterative feedforward strategy on a feedforward architecture in order to generate a melody, and the BLSTM system (Section REF ) uses an iterative feedforward strategy on a recurrent architecture in order to generate a chord accompaniment to a melody.",
"We will see further (with the VRAE system to be described in Section REF ) the use of an RNN Encoder-Decoder compound architecture (Section REF ), as a way to decouple the length of the input sequence with the length of the output sequence, by combining the decoder feedforward strategy with the iterative feedforward strategy.",
"Some other examples of couplings between architectures and strategies, or between challenges, will be discussed in Section .",
"Before that, we will continue to analyze challenges and possible solutions or directions."
],
[
"Melody-Harmony Interaction",
"When the objective is to generate simultaneously a melody with an accompaniment, expressed through some harmony or counterpointHarmony and counterpoint are dual approaches of accompaniment with different focus and priorities.",
"Harmony focuses on the vertical relations between simultaneous notes, as objects on their own (chords), and then considers the horizontal relations between them (e.g., harmonic cadences).",
"Conversely, counterpoint focuses on the horizontal relations between successive notes for each simultaneous melody (a voice), and then considers the vertical relations between their progression (e.g., to avoid parallel fifths).",
"Note that, although their perspectives are different, the analysis and control of relations between vertical and horizontal dimensions are their shared objectives., an issue is the musical consistency between the melody and the harmony.",
"Although a general architecture such as MiniBach (Section REF ) is supposed to have learnt correlations, interactions between vertical and horizontal dimensions are not explicitly considered.",
"We have analyzed in Section REF an example of a specific architecture to generate simultaneously melody and chords, with explicit relations between them (i.e.",
"chords can use previous step information about melody but not the opposite).",
"However, this architecture is a bit ad hoc.",
"In the following sections, we will analyze some more general architectures having in mind interactions between melody and harmony."
],
[
"#1 Example: RNN-RBM Polyphony Symbolic Music Generation System",
"In [11], Boulanger-Lewandowski et al.",
"have associated to the RBM-based architecture introduced in Section REF a recurrent network (RNN) in order to represent the temporal sequence of notes.",
"The idea is to couple the RBM to a deterministic RNN with a single hidden layer, such that the RNN models the temporal sequence to produce successive outputs, corresponding to successive time steps, which are parameters, more precisely the biases, of an RBM that models the conditional probability distribution of the accompaniment notes, i.e.",
"which notes should be played together.",
"In other words, the objective is to combine a horizontal view (temporal sequence) and a vertical view (combination of notes for a particular time step).",
"The resulting architecture named RNN-RBM is shown in Figure REF , and can be interpreted as follows: Figure: RNN-RBM architecture.Reproduced from with permission of the authors the bottom line represents the temporal sequence of RNN hidden units u$^{(0)}$ , u$^{(1)}$ , ..., u$^{(t)}$ , where u$^{(t)}$ notation meansNote that the usual notation would be u$_t$ , as the u$^{(t)}$ notation is usually reserved to index dataset examples ($t$ th example), see Section .",
"the value of the RNN hidden layer u at time (index) $t$ ; and the upper part represents the sequence of each RBM instance at time $t$ , which we could notate RBM$^{(t)}$ , with v$^{(t)}$ its visible layer with $b_\\text{v}^{(t)}$ its bias, h$^{(t)}$ its hidden layer with $b_\\text{h}^{(t)}$ its bias, and $W$ the weight matrix of connexions between the visible layer v$^{(t)}$ and the hidden layer h$^{(t)}$ .",
"There is a specific training algorithm, which we will not detail here, please refer to [11].",
"During generation, each $t$ time step of processing is as follows: compute the biases $b_\\text{v}^{(t)}$ and $b_\\text{h}^{(t)}$ of RBM$^{(t)}$ , via Equations REF and REF respectively, sample from RBM$^{(t)}$ by using block Gibbs sampling to produce v$^{(t)}$ , and feedforward the RNN with v$^{(t)}$ as the input, using the RNN hidden layer value u$^{(t-1)}$ , in order to produce the RNN new hidden layer value u$^{(t)}$ via Equation REF , where $W_{\\text{vu}}$ is the weight matrix and $b_\\text{u}$ the bias for the connexions between the input layer of the RBM and the hidden layer of the RNN; and $W_{\\text{uu}}$ is the weight matrix for the recurrent connexions of the hidden layer of the RNN.",
"$b_\\text{v}^{(t)} = b_\\text{v} + W_{\\text{uv}} \\text{u}^{(t-1)}$ $b_\\text{h}^{(t)} = b_\\text{h} + W_{\\text{uh}} \\text{u}^{(t-1)}$ $\\text{u}^{(t)} = \\text{tanh} (b_\\text{u} + W_{\\text{uu}} \\text{u}^{(t-1)} + W_{\\text{vu}} \\text{v}^{(t)})$ Note that the biases $b_\\text{v}^{(t)}$ and $b_\\text{h}^{(t)}$ of RBM$^{(t)}$ are variable for each time step, in other words they are time dependent, whereas the weight matrix $W$ for the connexions between the visible and the hidden layer of RBM$^{(t)}$ is shared for all time steps (for all RBMs), i.e.",
"it is time independent$W_{\\text{uv}}$ , $W_{\\text{uh}}$ , $W_{\\text{uu}}$ and $W_{\\text{vu}}$ weight matrices are also shared and thus time independent.. Four different corpora have been used in the experiments: classical piano, folk tunes, orchestral classical music and J. S. Bach chorales.",
"Polyphony varies from 0 to 15 simultaneous notes, with an average value of 3.9.",
"A piano roll representation is used with many-hot encoding of 88 units representing pitches from A$_0$ to C$_8$ .",
"Discretization (time step) is a quarter note.",
"All examples are aligned onto a single common tonality: C major or minor.",
"An example of a sample generated in a piano roll representation is shown in Figure REF .",
"The quality of the model has made RNN-RBM, summarized at Table REF , one of the reference architectures for polyphonic music generation.",
"Figure: Example of a sample generated by RNN-RBM trained on J. S. Bach chorales.Reproduced from with permission of the authorsTable: RNN-RBM summaryThere have been a few systems following on and extending the RNN-RBM architecture, but they are not significantly different and furthermore they have not been thoroughly evaluated.",
"However, it is worth mentioning the following: the RNN-DBN architectureThis is apparently the state of the art for the J. S. Bach Chorales dataset in terms of cross-entropy loss., using multiple hidden layers [61]; and the LSTM-RTRBM architecture, using an LSTM instead of an RNN [121]."
],
[
"#2 Example: Hexahedria Polyphony Symbolic Music Generation Architecture",
"The system for polyphonic music proposed by Johnson in his Hexahedria blog [94] is hybrid and original in that it integrates into the same architecture a first part made of two RNNs (actually LSTM) layers, each with 300 hidden units, recurrent over the time dimension, which are in charge of the temporal horizontal aspect, that is the relations between notes in a sequence.",
"Each layer has connections across time steps, while being independent across notes; and a second part made of two other RNN (LSTM) layers, with 100 and 50 hidden units, recurrent over the note dimension, which are in charge of the harmony vertical aspect, that is the relations between simultaneous notes within the same time step.",
"Each layer is independent between time steps but has transversal directed connexions between notes.",
"We can notate this architecture as LSTM$^{2+2}$ in order to highlight the two successive 2-level recurrent layers, recurrent in two different dimensions (time and note).",
"The architecture is actually a kind of integration within a single architectureWe will see in Section REF an alternative architecture, named Bi-Axial LSTM, where each of the 2-level time-recurrent layers is encapsulated into a different architectural module.",
"of the RNN-RBM architecture described in previous section.",
"The main originality is in using recurrent networks not only on the time dimension but also on the note dimension, more precisely on the pitch class dimension.",
"This latter type of recurrence is used to model the occurence of a simultaneous note based on other simultaneous notes.",
"Like for the time relation, which is oriented towards the future, the pitch class relation is oriented towards higher pitch, from C to B.",
"Figure: Hexahedria architecture (folded).Reproduced from with permission of the authorFigure: Hexahedria architecture (unfolded).Reproduced from with permission of the authorThe resulting architecture is shown in its folded form in Figure REF and in its unfolded formOur unfolded pictorial representation of an RNN shown in Figure REF was actually inspired by Johnson's Hexahedria pictorial representation.",
"in Figure REF , with three axes represented: the flow axis, shown horizontally and directed from left to right, represents the flow of (feedforward) computation through the architecture, from the input layer to the output layer; the note axis, shown vertically and directed from top to bottom, represents the connexions between units corresponding to successive notes of each of the two last (note-oriented) recurrent hidden layers; and the time axis, only in the unfolded Figure REF , shown diagonally and directed from top left to bottom right, represents the time steps and the propagation of the memory within a same unit of the two first (time-oriented) recurrent hidden layers.",
"The dataset is constructed by extracting 8 measures long parts from MIDI files from the Classical piano MIDI database [105].",
"The input representation used is piano roll, with the pitch represented as the MIDI note number.",
"More specific information is added: the pitch class, the previous note played (as a way to represent a possible hold), how many times a pitch class has been played in the previous time step and the beat (the position within the measure, assuming a 4/4 time signature).",
"The output representation is also a piano roll, in order to represent the possibility of more than one note at the same time.",
"Generation is done in an iterative way (i.e.",
"following the iterative feedforward strategy), as for most recurrent networks.",
"The system is summarized in Table REF .",
"Table: Hexahedria summary"
],
[
"#3 Example: Bi-Axial LSTM Polyphony Symbolic Music Generation Architecture",
"Johnson recently proposed an evolution of his original Hexahedria architecture, described in Section REF , named Bi-Axial LSTM (or BALSTM) [95].",
"The representation used is piano roll, with note hold and rest tokens added to the vocabulary.",
"Various corpora are used: the JSB Chorales dataset, a corpus of 382 four-part chorales by J. S. Bach [1]; the MuseData library, an electronic classical music library from CCARH in Stanford [77]; the Nottingham database, a collection of 1,200 folk tunes in ABC notation [54]; and the Classical piano MIDI database [105].",
"Each dataset is transposed (aligned) into the key of C major or C minor.",
"The probability of playing a note depends on two types of information: all notes at previous time steps – this is modeled by the time-axis module; and all notes within the current time step that have already been generated (the order being lowest to highest) – this is modeled by the note-axis module.",
"There is an additional front end layer, named “Note Octaves”, which transforms each note into a vector of all its possible corresponding octave notes (i.e.",
"an extensional version of pitch classes).",
"The resulting architecture is illustrated in Figure REFThis figure comes from the description of another system based on the Bi-Axial LSTM architecture, named DeepJ, which will be described in Section REF ..",
"The “x2” represents the fact that each module is stacked twice (i.e.",
"has two layers).",
"The time-axis module is recurrent in time (as for a classical RNN), the LSTM weights being shared across notes in order to gain note transposition invariance.",
"The note-axis moduleNote that, as opposed to Johnson's first architecture (that we refer to as Hexahedria, and which has been introduced in Section REF ), which integrates the 2-level time-recurrent layers with the 2-level note-recurrent layers within a single architecture and therefore notated as LSTM$^{2+2}$ , the Bi-Axial LSTM architecture explicitly separates each 2-level time-recurrent layers into distinct architectural modules and is therefore notated as LSTM$^2\\times $ 2. is recurrent in note.",
"For each note input of the note-axis module, $\\oplus $ represents the concatenation of the corresponding output from the time-axis module with the already predicted lower notes.",
"Sampling (into a binary value, by using a coin flip) is applied to each note output probability in order to compute the final prediction (whether that note is played or not).",
"Figure: Bi-Axial LSTM architecture.Reproduced from with permission of the authorsAs pointed out by Johnson [95], during the training phase, as all the notes at all time steps are known, the training process may be accelerated by processing each layer independently (e.g., on a GPU), by running input through the two time-axis layers in parallel across all notes, and using the two note-axis layers to compute probabilities in parallel across all time steps.",
"The generation phase is sequential for each time step (by following both the iterative feedforward strategy and the sampling strategy).",
"An excerpt of music generated is shown in Figure REF .",
"The Bi-Axial LSTM system, summarized in Table REF , has been evaluated and compared to some other architectures.",
"The author reports noticeably better results with Bi-Axial LSTM, the greatest improvements being on the MuseData [77] and the Classical piano MIDI database [105] datasets, and states in [95] that: “It is likely due to the fact that those datasets contain many more complex musical structures in different keys, which are an ideal case for a translation-invariant architecture.” Note that an extension of the Bi-Axial LSTM architecture with conditioning, named DeepJ, will be introduced in Section REF .",
"Figure: Example of Bi-Axial LSTM generated music (excerpt).Reproduced from Table: Bi-Axial LSTM summary"
],
[
"Control",
"A deep architecture generates musical content matching the style learnt from the corpus.",
"This capacity of induction from a corpus without any explicit modeling or programming is an important ability, as discussed in Chapter and also in [52].",
"However, like a fast car that needs a good steering wheel, control is also needed as musicians usually want to adapt ideas and patterns borrowed from other contexts to their own objective and context, e.g., transposition to another key, minimizing the number of notes, finishing with a given note, etc."
],
[
"Dimensions of Control Strategies",
"Arbitrary control is a difficult issue for deep learning architectures and techniques because neural networks have not been designed to be controlled.",
"In the case of Markov chains, they have an operational model on which one can attach constraints to control the generationTwo examples are Markov constraints [149] and factor graphs [148]..",
"However, neural networks do not offer such an operational entry point and the distributed nature of their representation does not provide a clear relation to the structure of the content generated.",
"Therefore, as we will see, most of strategies for controlling deep learning generation rely on external intervention at various entry points (hooks) and levels: input, output, input and output, and encapsulation/reformulation.",
"Various control strategies can be employed: sampling, conditioning, input manipulation, reinforcement, and unit selection.",
"We will also see that some strategies (such as sampling, see Section REF ) are more bottom-up and others (such as structure imposition, see Section REF , or unit selection, see Section REF ) are more top-down.",
"Lastly, there is also a continuum between partial solutions (such as conditioning/parametrization, see Section REF ) and more general approaches (such as reinforcement, see Section REF )."
],
[
"Sampling",
"Sampling from a stochastic architecture (such as a restricted Boltzmann machine (RBM), see Section REF ), or from a deterministic architecture (in order to introduce variability, see Section REF ), may be an entry point for control if we introduce constraints into the sampling process.",
"This is called constrained sampling, see for example the C-RBM system in Section REF .",
"Constrained sampling is usually implemented by a generate-and-test approach, where valid solutions are picked from a set of random samples generated from the model.",
"But this could be a very costly process and, moreover, with no guarantee of success.",
"A key and difficult issue is therefore how to guide the sampling process in order to fulfill the constraints."
],
[
"Sampling for Iterative Feedforward Generation",
"In the case of an iterative feedforward strategy on a recurrent network, some refinements in the sampling procedure can be made.",
"In Section REF , we introduced the technique of sampling the softmax output of a recurrent network in order to introduce content variability.",
"However, this may sometimes lead to the generation of an unlikely note (with a low probability).",
"Moreover, as noted in [67], generating such a “wrong” note can have a cascading effect on the remaining of the generated sequence.",
"A counter measure consist in adjusting a learnt RNN model (conditional probability distribution $P(\\text{s}_t | \\text{s}_{<t})$ , as defined in Section ) by not considering notes with a probability under a certain threshold.",
"The new model, with a probability distribution $P_{threshold}(\\text{s}_t | \\text{s}_{<t})$ , is defined in Equation REF following [204], where: $P_{threshold}(\\text{s}_t | \\text{s}_{<t}) :={\\left\\lbrace \\begin{array}{ll}0~\\text{if}~P(\\text{s}_t | \\text{s}_{<t}) / \\text{max}_{\\text{s}_t} P(\\text{s}_t | \\text{s}_{<t}) < threshold,\\\\P(\\text{s}_t | \\text{s}_{<t}) / z~\\text{otherwise}.\\end{array}\\right.",
"}$ $\\text{max}_{\\text{s}_t} P(\\text{s}_t | \\text{s}_{<t})$ is the note maximum probability, $threshold$ is the threshold hyperparameter, and $z$ is a normalization constant.",
"A slightly more sophisticated version interpolates between the original distribution $P(\\text{s}_t | \\text{s}_{<t})$ and the $\\text{argmax\\index {Argmax}}_{\\text{s}^{\\prime }_t} P(\\text{s}^{\\prime }_t | \\text{s}_{<t})$ deterministic variantSee Section REF ., with some temperature user control hyperparameter (see more details in [67]).",
"This technique will be further generalized and combined with the conditioning strategy in order to control the generation of notes at specific positions via positional constraints.",
"This will be exemplified by the Anticipation-RNN system to be introduced in Section REF ."
],
[
"Sampling for Incremental Generation",
"In the case of an incremental generation (to be introduced in Section ), the user may select on which part (e.g., a given part of a melody and/or a given voice) sampling will occur (or reoccur), and the interval of possible values on which sampling will occur.",
"In the case of the DeepBach system (to be introduced in Section REF ), this will be the basis for introducing user control on the generation, notably to regenerate only some parts of a music, to restrict note range, and to impose some basic rhythm."
],
[
"Sampling for Variational Decoder Feedforward Generation",
"Another interesting case is the use of sampling for generative models, such as variational autoencoders (VAEs) and generative adversarial networks (GANs), to be introduced in Section REF .",
"We will see that some nice control of the sampling, e.g., to produce an interpolation, averaging or attribute modification, will produce meaningful variations in the content generated by the decoder feedforward strategy.",
"Moreover, as has been discussed in Section REF , a variational autoencoder (VAE) is interesting for its ability for controlling generation over significant dimensions that have been learnt."
],
[
"#1 Example: VRAE Video Game Melody Symbolic Music Generation System",
"In [50], Fabius and van Amersfoort propose the extension of the RNN Encoder-Decoder architecture to the case of a variational autoencoder (VAE), which is therefore named a variational recurrent autoencoder (VRAE).",
"Both the encoder and the decoder encapsulate an RNN (actually an LSTM), as has been explained in Section REF .",
"In terms of strategy, the VRAE combines the iterative feedforward strategy with the decoder feedforward strategy and the sampling strategy.",
"The corpus used in the experiment is a set of MIDI files of eight video game songs from the 1980s and 1990s (Sponge Bob, Super Mario, Tetris...), which are divided into various shorter parts of 50 time steps.",
"A one-hot encoding of 49 possible pitches is used (pitches with too few occurrences of notes were not considered).",
"Experiments have been conducted with 2 or 20 hidden layer units (latent variables).",
"Training takes place as for training recurrent networks, i.e.",
"for each input note presenting the next note as the output.",
"After the training phase, the latent space vector can be sampled and used by the RNN encapsulated within the decoder to generate iteratively a melody.",
"This could be done by random sampling or also by interpolating between the values of the latent variables corresponding to different songs that have been learnt, creating a sort of “medley” of these songs.",
"Figure REF visualizes the organisation of the encoded data in the latent space, each color representing the data points from one song.",
"The result is positive, but the low musical quality of the corpus hampers a careful evaluation.",
"The VRAE system is summarized in Table REF .",
"Figure: Visualization of the VRAE latent space encoded data.Extended from with permission of the authorsTable: VRAE summary"
],
[
"#2 Example: GLSR-VAE Melody Symbolic Music Generation System",
"The architecture proposed by Hadjeres and Nielsen in [69] is based on a variational autoencoder (VAE) architecture (Section REF ), but it proposes an improvement in the control of the variation in the generation, named geodesic latent space regularization (GLSR), with a system named GLSR-VAE.",
"The starting point is that a straight line between two points in the latent space will not necessarily produce the best interpolation in the generated content domain space.",
"The idea is to introduce a regularization to relate variations in the latent space to variations in the attributes of the decoded elements.",
"The details of the definition of the added cost term may be found in [69].",
"The experiment consists in generating chorale melodies in the style of J. S. Bach.",
"The dataset comprises monophonic soprano voices from the J. S. Bach chorales corpus [5].",
"GLSR-VAE shares the principles of representation initiated by the DeepBach system (Section REF ), that is one-hot encoding of a note, with the addition to the vocabulary of the hold symbol “__” and the rest symbol to specify, respectively, a note repetition and a rest (see Section REF ), and using the names of the notes (with no enharmony, e.g., F$\\sharp $ and G$\\flat $ are considered to be different, see Section REF ).",
"Quantization is at the level of a sixteenth note.",
"The latent variable space is set to 12 dimensions (12 latent variables).",
"In the experiments conducted, regularization is executed on a first dimension which has been foundSee Section REF .",
"to represent the number of notes (named z$_1$ ).",
"Figure REF shows the organisation of the encoded data in the latent space, with the number of notes z$_1$ being the abscissa axis, with from left to right an effective progressive increase in the number of notes (shown with scales of colors).",
"Figure REF shows examples of the melodies generated (each 2 measures long, separated by double bar lines) while increasing z$_1$ , showing a progressive correlated densification of the melodies generated.",
"Figure: Visualization of GLSR-VAE latent space encoded data.Reproduced from with permission of the authorsFigure: Examples of 2 measures long melodies (separated by double bar lines) generated by GLSR-VAE.Reproduced from with permission of the authorsGLSR-VAE is summarized in Table REF .",
"More examples of sampling from variational autoencoders will be described in Section REF .",
"Table: GLSR-VAE summary"
],
[
"Sampling for Adversarial Generation",
"Another example of a generative model is a generative adversarial networks (GAN) architecture.",
"In such an architecture, after having trained the generator in an adversarial way, generation of content is done by sampling latent random variables."
],
[
"Example: Mogren's C-RNN-GAN Classical Polyphony Symbolic Music Generation System",
"The objective of Mogren's C-RNN-GAN [135] system is the generation of single voice polyphonic music.",
"The representation chosen is inspired by MIDI and models each musical event (note) via four attributes: duration, pitch, intensity and time elapsed since the previous event, each attribute being encoded as a real value scalar.",
"This allows the representation of simultaneous notes (in practice up to three).",
"The musical genre of the corpus is classical music, retrieved in MIDI format from the Web and contains 3,697 pieces from 160 composers.",
"C-RNN-GAN is based on a generative adversarial networks (GAN) architecture, with both the generator and the discriminator being recurrent networksThis generative GAN architecture encapsulates two recurrent networks, in the same spirit that the generative VRAE variational autoencoder architecture encapsulates two recurrent networks as explained in Section REF ., more precisely each having two LSTM layers with 350 units each.",
"A specificity is that the discriminator (but not the generator) has a bidirectional recurrent architecture, in order to take context from both the past and the future for its decisions.",
"The architecture is shown in Figure REF and summarized in Table REF .",
"Figure: C-RNN-GAN architecture.Reproduced from with permission of the authorsThe discriminator is trained, in parallel to the generator, to classify if a sequence input is coming from the real data.",
"Similar to the case of the encoder part of the RNN Encoder-Decoder, which summarizes a musical sequence into the values of the hidden layer (see Section REF ), the bidirectional RNN decoder part of the C-RNN-GAN summarizes the sequence input into the values of the two hidden layers (forward sequence and backward sequence) and then classifies them.",
"An example of generated music is shown in Figure REF .",
"The author conducted a number of measurements on the generated music.",
"He states that the model trained with feature matchingA regularization technique for improving GANs, see Section REF .",
"achieves a better trade-off between structure and surprise than the other variants.",
"Note that this is consonant with the use of the feature matching regularization technique to control creativity in MidiNet (to be introduced in Section REF ).",
"C-RNN-GAN is summarized in Table REF .",
"Figure: C-RNN-GAN generated example (excerpt).Reproduced from with permission of the authorsTable: C-RNN-GAN summary"
],
[
"Sampling for Other Generation Strategies",
"Sampling may also be combined with other strategies for content generation, as for instance Conditioning, as a way to parametrize generation with constraints, in Section REF , or Input manipulation, as a way to correct the manipulation performed in order to realign the samples with the learnt distribution, in Section REF ."
],
[
"Conditioning",
"The idea of conditioning (sometimes also named conditional architecture) is to condition the architecture on some extra information, which could be arbitrary, e.g., a class label or data from other modalities.",
"Examples are a bass line or a beat structure, in the rhythm generation architecture (Section REF ), a chord progression, in the MidiNet architecture (Section REF ), the previously generated note, in the VRASH architecture (Section REF ), some positional constraints on notes, in the Anticipation-RNN architecture (Section REF ), a musical genre or an instrument, in the WaveNet architecture (Section REF ), and a musical style, in the DeepJ architecture (Section REF ).",
"In practice, the conditioning information is usually fed into the architecture as an additional input layer (for example, see Figure REF ).",
"This distinction between standard input and conditioning input follows a good architectural modularity principleNote that we do not consider conditioning as a strategy because we consider that the essence of conditioning relates to the conditioning architecture.",
"Generation uses a conventional strategy (e.g., single-step feedforward, iterative feedforward...) depending on the type of the architecture (e.g., feedforward, recurrent...)..",
"Conditioning is a way to have some degree of parametrized control over the generation process.",
"The conditioning layer could be a simple input layer.",
"An example is a tag specifying a musical genre or an instrument in the WaveNet system (Section REF ), some output of some architecture, being the same architecture, as a way to condition the architecture on some historyThis is close in spirit to a recurrent architecture (RNN).",
"– an example is the MidiNet system (Section REF ) in which history information from previous measure(s) is injected back into the architecture, or another architecture – examples are the rhythm generation system (Section REF ) in which a feedforward network in charge of the bass line and the metrical structure information produces the conditioning input, and the DeepJ system (Section REF ) in which two successive transformation layers of a style tag produce an embedding used as the conditioning input.",
"If the architecture is time-invariant – i.e.",
"recurrent or convolutional over time –, there are two options global conditioning – if the conditioning input is shared for all time steps, or local conditioning – if the conditioning input is specific to each time step.",
"The WaveNet architecture, which is convolutional over time (see Section REF ), offers the two options, as will be analyzed in Section REF ."
],
[
"#1 Example: Rhythm Symbolic Music Generation System",
"The system proposed by Makris et al.",
"[123] is specific in that it is dedicated to the generation of sequences of rhythm.",
"Another specificity is the possibility to condition the generation relative to some particular information, such as a given beat or bass line.",
"The corpus includes 45 drum and bass patterns, each 16 measures long in 4/4 time signature, from three different rock bands and converted to MIDI.",
"The representation of drums is described in Section REF and summarized as follows.",
"Different drum components (kick, snare, toms, hi-hat, cymbals) are considered as distinct simultaneous voices, following a many-hot approach, and encoded in text as a binary word of length 5, e.g., 10010 represents the simultaneous playing of kick and high-hat.",
"The representation also includes a condensed representation of the bass line part.",
"It captures the voice leading perspective of the bassThe voice leading of the bass has proven a valuable aspect in harmonization systems, see, e.g., [72]., by specifying the pitch difference direction for the bass between two successive time steps.",
"This is represented in a binary word of length 4, the first digit specifying the existence of a bass event (1) or a rest event (0), while the three remaining digits specify the 3 possible directions for voice leading: steady (000), upward (010) and downward (001).",
"Last, the representation includes some additional information representing the metrical structure (the beat structure), also through binary words.",
"See further details in [123].",
"The architecture is a combination of a recurrent network (more precisely, an LSTM) and a feedforward network, representing the conditioning layer.",
"The LSTM (two stacked LSTM layers with 128 or 512 units) is in charge of the drums part, while the feedforward network is in charge of the bass line and the metrical structure information.",
"The outputs of these two networks are then mergedNote that in this system, the conditioning layer is added to the main architecture at its output level and not at its input level.",
"Therefore an additional feedforward merge layer is introduced.",
"We could notate the resulting architecture as Conditioning(Feedforward(LSTM$^2$ ), Feedforward)., resulting in the architecture illustrated in Figure REF .",
"The authors report that the conditioning layer (bass line and beat information) improves the quality of the learning and of the generation.",
"It may also be used in order to mildly influence the generation.",
"More details may be found in the article [123].",
"The architecture is summarized in Table REF .",
"An example of a rhythm pattern generated is shown in Figure REF with in Figure REF the use of a specific and more complex bass line as a conditioning input which produces a rhythm more elaborate.",
"The piano roll like visual representation shows in its five successive lines (downwards) the kick, snare, toms, hi-hat and cymbals components events.",
"Figure: Rhythm generation architecture.Reproduced from with permission of the authorsFigure: Example of a rhythm pattern generated.The five lines of the piano roll correspond (downwards) to:kick, snare, toms, hi-hat and cymbals.Reproduced from with permission of the authorsFigure: Example of a rhythm pattern generated with a specific bass line as the conditioning input.Reproduced from with permission of the authorsTable: Rhythm system summary"
],
[
"#2 Example: WaveNet Speech and Music Audio Generation System",
"WaveNet, by van der Oord et al.",
"[194], is a system for generating raw audio waveforms, quite innovative in that respect.",
"It has been tested in three audio domains: multi-speaker, text-to-speech (TTS) and music.",
"The architecture is based on a convolutional feedforward network with no pooling layer.",
"Convolutions are constrained in order to ensure that the prediction only depends on previous time steps, and are therefore named causal convolutions.",
"The actual implementation is optimized through the use of dilated convolution (also called “à trous”), where the convolution filter is applied over an area larger than its length by skipping input values with a certain step.",
"Incrementally dilated successive convolution layersThe dilation is doubled for every layer up to a limit and then repeated, e.g., 1, 2, 4, ..., 512, 1, 2, 4, ..., 512, ... enable networks to have very large receptive fields with just a few layers while preserving the input resolution throughout the network as well as computational efficiency (see [194] for more details).",
"The architecture is illustrated in Figure REF .",
"Another specificity of WaveNet is in the training/generation asymmetry: during the training phase, predictions for all time steps can be made in parallel, whereas during the generation phase, predictions are sequential (following the iterative feedforward strategy).",
"The WaveNet architecture is made conditioning, as a way to guide the generation, by adding an additional tag as a conditioning input.",
"We could thus notate the architecture as Conditioning(Convolutional(Feedforward), Tag).",
"There are actually two options: global conditioning, if the conditioning input is shared for all time steps; and local conditioning, if the conditioning input is specific to each time step.",
"An example of conditioning for a text-to-speech application domain is to feed in linguistic features from different speakers, e.g., North American or Mandarin Chinese English speakers, in order to generate speech with a specific prosody.",
"Figure: WaveNet architecture.Reproduced from with permission of the authorsThe authors also conducted preliminary work on conditioning models to generate music given a set of tags specifying, for example, genre or instruments.",
"They state (without further details) that their preliminary attempt is promising [194].",
"WaveNet is summarized in Table REF .",
"Last, let us mention, a recent proposal as an offspring from WaveNet, which uses a symbolic representation (associated to the audio input) as the conditioning model/input, in order to better guide and structure the generation of (audio) music (see details in [126]).",
"Table: WaveNet summary"
],
[
"#3 Example: MidiNet Pop Music Melody Symbolic Music Generation System",
"In [212], Yang et al.",
"propose the MidiNet architecture, which is both adversarial and convolutional, for the generation of single or multitrack pop music monophonic melodies.",
"The corpus used is a collection of 1,022 pop music songs from the TheoryTabTabs are piano roll-like leadsheets, including melody, lyrics and notation of chords.",
"online database [85] that provides two channels per tab, one for the melody and the other for the underlying chord progression.",
"This allows two versions of the system: one with only the melody channel and another that additionally uses chords to condition melody generation.",
"After all the preprocessing steps, the dataset is composed of 526 MIDI tabs (representing 4,208 measures).",
"Data augmentation is then performed by circularly shifting all melodies and chords to any of the 12 keys, leading to a final dataset of 50,496 measures of melody and chord pairs for training.",
"The representation is obtained by transforming each channel of MIDI files into a one-hot encoding of 8 measures long piano roll representations, using one of the encodings to represent silence (rest) and neglecting the velocity of the note events.",
"The time step is set at the smallest note, a sixteenth note.",
"All melodies have been transposed in order to fit within the two-octave interval $[$ C$_4,$ B$_5]$However, the authors considered all the 128 MIDI note numbers (corresponding to the $[$ C$_0,$ G$_{10}]$ interval) in a one-hot encoding and state in [212] that: “In doing so, we can detect model collapsing more easily, by checking whether the model generates notes outside these octaves.”.",
"Note that the current representation does not distinguish between a long note and two short repeating notes, and the authors mention considering future extensions in order to emphasize the note onsets.",
"For chords, instead of using a many-hot vector extensional representation of dimension 24 (for the two octaves), the authors state that they found it more efficient to use an intensional representation of dimension 13: 12 for the pitch-class (key) and 1 for the chord type (major or minor).",
"The architectureThe architecture is complex, please see further details in [212].",
"is illustrated in Figures REF and REF .",
"It is composed of a generator and a discriminator, which are both convolutional networks.",
"The generator includes two fully-connected layers (with 1,024 and 512 units respectively) followed by four convolutional layers.",
"Generation takes place iteratively, by sampling one measure after one measure until reaching 8 measures.",
"The generator is conditioned by a module (named Conditioner CNN in Figure REF ) which includes four convolutional layers with a reverse architecture.",
"The conditioning mechanism incorporates history information from previous measures (as a memory mechanism, analog to a RNN), and the chord sequence (only for the generator).",
"The discriminator includes two convolutional layers followed by some fully connected layers and the final output activation function is cross-entropy.",
"The discriminator is also conditioned, but without specific conditioner layers.",
"We could thus notate the architecture as Table: NO_CAPTIONFigure: MidiNet architecture.Reproduced from with permission of the authorsFigure: Architecture of the MidiNet generator.Reproduced from with permission of the authorsThe conditioning information could be only about the previous measure – named “1D conditions” (shown in yellow in Figures REF and REF ); or about various previous measures – named “2D conditions” (shown in blue).",
"Both cases are illustrated in Figure REF .",
"The authors report experiments performed with different variants: melody generation with conditioning on the previous measure (with previous measure as 2D conditions for the generator and as 1D conditions for the discriminatorTo ensure that the discriminator distinguishes between real and generated melodies only from the present measure.",
"); melody generation with conditioning on the previous measure and on the chord sequence (with chord sequence as 1D conditions for the generator, or alternatively also as 2D conditions only for its last convolutional layer in order to highlight the chord condition); and melody generation with conditioning on the previous measure and on the chord sequence in a creative mode (with chord sequence as 2D conditions for all convolutional layers of the generator).",
"For the second variant, which they name stable mode, the authors report that the generation is more chord-dominant and stable, in other words it closely follows the chord progression and seldom generates notes violating chord constraints.",
"For the third variant, named creative modeOn the challenge of creativity, see Section ., the generator sometimes violates the constraint imposed by the chords, to better adhere to the melody of the previous measure.",
"In other words, the creative mode allows a better balance between melody following over chord following.",
"The authors state in [212] that: “Such violations sometimes sound unpleasant, but can be sometimes creative.",
"Unlike the previous two variants, we need to listen to several melodies generated by this model to handpick good ones.",
"However, we believe such a model can still be useful for assisting and inspiring human composers.” MidiNet is summarized in Table REF .",
"Table: MidiNet summary"
],
[
"#4 Example: DeepJ Style-Specific Polyphony Symbolic Music Generation System",
"In [127], Mao et al.",
"propose a system named DeepJ, with the objective of being able to control the style of music generated.",
"In their experiment, they consider 23 styles, each corresponding to a different composer (from Johann Sebastian Bach to Pyotr Ilyich Tchaikovsky) with his/her specific styleIn other words, they identify a style to a composer..",
"They encode the style – or a combination of stylesIn the case of a combination of several styles, the vector must be normalized in order for its sum to be equal to 1.",
"– as a many-hot representation over all possible styles (i.e.",
"composers).",
"Composers are grouped into musical genres.",
"Thus a genre is specified (extensionally) as an equal combination of the styles (composers) of that genre.",
"For example, if the Baroque genre is defined by composers 1 to 4, the Baroque style would be equal to $[0.25, 0.25, 0.25, 0.25, 0, 0,~...]$ .",
"We will see below, when detailing the architecture, that this somewhat simplistic user-defined style encoding will be automatically transformed through the learning phase into an adaptive distributed representation.",
"The foundation of the architecture is the Bi-Axial LSTM architecture proposed by Johnson in [95] (see Section REF ).",
"Music representation is based on piano roll, modeling a note through its MIDI note number, within a truncated range (originally within the $\\lbrace 0, 1,\\ldots ~127\\rbrace $ discrete set, truncated to $\\lbrace 36, 37,\\ldots ~84\\rbrace $ , i.e.",
"four octaves) in order to reduce note input dimensionality.",
"Quantization is 16 time steps per measure, i.e.",
"a time step with the value of a sixteenth note.",
"The representation is similar to that for Bi-Axial LSTM.",
"DeepJ representation uses a replay matrix, dual to the piano roll matrix of notes, in order to distinguish between a held note and a replayed note.",
"DeepJ representation also includes information about dynamics through a scalar variableThe authors comment that they have also tried an alternate representation of dynamics as a categorical value (one-hot encoding) with 128 bins (as in WaveNet, see Section REF ), which is actually the original MIDI discretization.",
"But: “Contrary to WaveNet's results, our experiments concluded that the scalar representation yielded results that were more harmonious.” [127] within the $[0, 1]$ interval.",
"But the main addition is the use of style conditioning, via global conditioningThis means that the conditioning input is shared for all time steps, see Section REF ., as in WaveNet.",
"As has been noted, the user-defined style encoding is too simplistic to be used as it is.",
"Musical styles are not necessarily orthogonal to each other and may share many characteristics.",
"The first transformation layer linearly transforms the user-defined many-hot encoding of the style into a first embedding (a set of hidden/latent variables, pictured as the yellow Embedding box in Figure REF ).",
"The second transformation layer transforms this first embedding in a nonlinear way (through a tanh activationHyperbolic tangent function.)",
"into a second embedding of the style (pictured as the lower yellow Fully-Connected box) to be added as a conditioning input to the time-axis module.",
"A similar transformation and conditioning is performed for the note-axis module.",
"Further details and discussion may be found in [127].",
"DeepJ is summarized in Table REF .",
"Figure: DeepJ architecture.Reproduced from with permission of the authorsThe authors have conducted an initial subjective evaluation with human listeners comparing music generated by DeepJ (an example is shown in Figure REF ) and by Bi-Axial LSTM.",
"They report that DeepJ compositions were usually preferred and they comment that the style conditioning makes generated music more stylistically consistent.",
"They also conducted a second subjective evaluation in order to verify whether DeepJ can generate stylistically distinct music (correctly identified by human listeners).",
"The authors report no statistically significant differences between the classification accuracy for DeepJ music and real composers music.",
"A more objective analysis has also been undertaken by visualizing the style embedding space, shown in Figure REF , with each composer pictured as a dot and each cluster as a color (blue, yellow and red are for baroque, classical and romantic clusters, respectively).",
"The authors found that composers from similar periods do cluster together (same color) and point out the interesting result that Ludwig van Beethoven appears at the limit between the classical and romantic clusters.",
"Figure: Example of baroque music generated by DeepJ.Reproduced from with permission of the authorsFigure: Visualization of DeepJ embedding space.Extended from with permission of the authorsTable: DeepJ summary"
],
[
"#5 Example: Anticipation-RNN Bach Melody Symbolic Music Generation System",
"In [68], Hadjeres and Nielsen propose a system named Anticipation-RNN for generating melodies with unary constraints on notes (to enforce a given note at a given time position to have a given value).",
"The limitation when using a standard iterative feedforward strategy for generation is that enforcing the constraint at time $i$ may retrospectively invalidate the distribution of the previously generated itemsAs the authors put it, imposing a constraint on time index $i$ “twists” the conditional probability distribution $P(\\text{s}_t | \\text{s}_{<t})$ for $t < i$ ., as shown in [151].",
"The idea is then to condition the RNN on information summarizing the set of further (in time) constraints, as a way to anticipate oncoming constraints, in order to generate notes with a correct distribution.",
"Therefore, a second RNN architecture, named Constraint-RNN, is used that functions backward in time.",
"Its outputs are used as additional inputs for the main RNN, which the authors name Token-RNN.",
"The complete architecture is illustrated in Figure REF , with the following notation and meaning: Figure: Anticipation-RNN architecture.Reproduced from with permission of the authors c$_i$ is a positional constraint; and o$_i$ is the output at index $i$ (after $i$ iterations) of Constraint-RNN – it summarizes constraints information from step $i$ to the final step $N$ (the end of the sequence).",
"It will be concatenated (the $\\oplus $ circled plus sign) to input s$_{i-1}$ of Token-RNN in order to predict the next item s$_i$ .",
"Note that Anticipation-RNN is not a symmetric bidirectional recurrent architecture, as in the case of the BLSTM (Section REF ) or the C-RNN-GAN (Section REF ) architectures because what is processed backwards is another sequence (of the constraints associated to the first sequence).",
"Both RNNs (Constraint-RNN and Token-RNN) are implemented as a 2-layer LSTM.",
"The corpus used is the set of soprano voice melodies extracted from the four-voice Chorales of J. S. Bach.",
"Data synthesis is performed by transposing in all keys within the original voice range and by pairing them with some sorted set of constraintsThis is done to reduce the combinatorial explosion, as one does not need to construct all possible pairs $(melody, constraint)$ as long as the coverage is sufficient for good learning.. Anticipation-RNN shares the principles of representation initiated by the DeepBach system to be presented in Section REF , that is one-hot encoding with the addition of the hold symbol “__” and the rest symbol to specify, respectively, a note repetition and a rest, and using the names of the notes with no enharmony.",
"Quantization is at the level of a sixteenth note.",
"Three examples of melodies generated with the same set of positional constraints (each one indicated with a green note within a green rectangle) are shown in Figure REF .",
"The model is indeed able to anticipate each positional constraint by adjusting its direction towards the target (lower-pitched or higher-pitched note).",
"Further details and analysis of the results are provided in [68].",
"Anticipation-RNN is summarized in Table REF .",
"Figure: Examples of melodies generated by Anticipation-RNN.Reproduced from with permission of the authorsTable: Anticipation-RNN summary"
],
[
"#6 Example: VRASH Melody Symbolic Music Generation System",
"The system described by Tikhonov and Yamshchikov in [189], although similar to VRAE (see Section REF ), uses a different representation, separately encoding in a multi-one-hot manner the pitch, the octave and the duration.",
"The training set is composed of various songs (different epochs and genres), derived from MIDI files following filtering and normalization (see the details in [189]).",
"The architecture has four LSTMTo be more precise, a recent evolution named recurrent highway networks (RHNs) [213].",
"layers for the encoder and for the decoder.",
"The authors have experimented with feeding the output of the decoder back into the decoder as a way of including the previously generated note as an additional information (therefore, they have named their final architecture VRASH, for variational recurrent autoencoder supported by history).",
"It is illustrated in Figures REF and REF and summarized in Table REF .",
"In their evaluation, the authors state that the melodies generated are only slightly closer to the corpus (using a cross-entropy measure) than when not adding history information, but that qualitatively the results are better.",
"Figure: VRASH architecture.Reproduced from with permission of the authorsFigure: VRASH architecture with a focus on the decoder.Extended from with permission of the authorsTable: VRASH summary"
],
[
"Input Manipulation",
"The input manipulation strategy was pioneered for images by Deep Dream.",
"The idea is that the initial input content, or a brand new (randomly generated) input content, is incrementally manipulated in order to match a target property.",
"Note that control of the generation is indirect, as it is not applied to the output but to the input, before generation.",
"Examples of target properties are maximizing the similarity to a given target, in order to create a consonant melody, as in DeepHear$_C$ (Section REF ); maximizing the activation of a specific unit, to amplify some visual element associated to this unit, as in Deep Dream (Section REF ); maximizing the content similarity to some initial image and the style similarity to a reference style image, to perform style transfer (Section REF ); and maximizing the similarity of the structure to some reference music, to perform style imposition (Section REF ).",
"Interestingly, this is done by reusing standard training mechanisms, namely backpropagation to compute the gradients, as well as gradient descent (or ascent) to minimize the cost (or to maximize the objective)."
],
[
"#1 Example: DeepHear Ragtime Counterpoint Symbolic Music Generation System",
"In [180], in addition to the generation of melodies (described in Section REF ), Sun proposed to use DeepHear for a different objective: to harmonize a melody, while using the same architecture as well as what has already been learntIt is a simple example of transfer learning (see [63]), using the same domain and the same training but for a different task.. We notate this second experiment DeepHear$_C$ , where $C$ stands for counterpoint, in order to distinguish it from DeepHear$_M$ for melody generation (Section REF ).",
"The idea is to find a label instance of the embedding, i.e.",
"a set of values for the 16 units of the bottleneck hidden layer of the stacked autoencoder, which will result in a decoded output resembling a given melody.",
"Therefore, a simple distance (error) function is defined to represent the distance (similarity) between two melodies (in practice, the number of unmatched notes).",
"Then a gradient descent is conducted on the variables of the embedding, guided by the gradients corresponding to the error function, until a sufficiently similar decoded melody is found.",
"Although this is not a real counterpointAs, for example, in the case of MiniBach (Section REF ) or DeepBach (Section REF ) for real counterpoint generation., but rather the generation of a similar (consonant) melody, the results (tested on ragtime melodies) do produce a naive counterpoint with a ragtime flavor.",
"Note that in DeepHear$_C$ (summarized in Table REF ), the input manipulated is the input of the innermost decoder (the starting point of the chain of decoders) and not the main input of the full architecture.",
"Whereas, in the case of the Deep Dream system to be introduced in Section REF , this is the main input of the full (feedforward) architecture which is manipulated.",
"Table: DeepHear C _C summary"
],
[
"Relation to Variational Autoencoders",
"Note that in the case of the manipulation of the hidden layer units of an autoencoder (or a stacked autoencoder, the case of DeepHear$_C$ ), the input manipulation strategy does have some analogy with variational autoencoders, such as for instance the VRAE system (Section REF ) or the GLSR-VAE system (Section REF ).",
"Indeed in both cases, there is some exploration of possible values for the hidden units in order to generate variations of musical content by the decoder (or the chain of decoders).",
"The important difference is that in the case of a variational autoencoder, the exploration of values is user-directed, although it could be guided by some principle, e.g., geodesic in GLSR-VAE, interpolation or attribute vector arithmetics in MusicVAE (Section REF ), whereas in the case of input manipulation, the exploration of values is automatically guided by the gradient descent (or ascent) mechanism, the user having previously specified a cost function to be minimized (or an objective to be maximized)."
],
[
"#2 Example: Deep Dream Psychedelic Images Generation System",
"Deep Dream, by Mordvintsev et al.",
"[137], has become famous for generating psychedelic versions of standard images.",
"The idea is to use a deep convolutional feedforward neural network architecture (see Figure REF ) and to use it to guide the incremental alteration of an initial input image, in order to maximize the potential occurrence of a specific visual motifTo create a pareidolia effect, where a pareidolia is a psychological phenomenon in which the mind responds to a stimulus, like an image or a sound, by perceiving a familiar pattern where none exists.",
"correlated to the activation of a given unit.",
"Figure: Deep Dream architecture (conceptual)The method is as follows: the network is first trained on a large dataset of images; instead of minimizing the cost function, the objective is to maximize the activation of some specific unit(s) which has (have) been identified to activate for some specific visual feature(s), e.g., a dog's face, see Figure REFInstead of exactly prescribing which feature(s) we want the network to amplify, an alternative is to let the network make that decision, by picking a layer and asking the network to enhance whatever it has detected [137].",
"; an initial image is iteratively slightly altered (e.g., by jitterAdding a small random noise displacement of pixels.",
"), under gradient ascent control, in order to maximize the activation of the specific unit(s).",
"This will favor the emergence of the correlated visual motif (motives), see Figure REF .",
"Figure: Deep Dream.",
"Example of a higher-layer unit maximization transformation.Created by Google's Deep Dream.Original picture: Abbey Road album cover, Beatles, Apple Records (1969).",
"Original photograph by Iain MacmillanNote that the activation maximization of a higher-layer unit(s), as in Figure REF , will favor the emergence in the image of a correlated high-level motif (motives), like a dog's face (see Figure REFAs the authors put it in [137]: “The results are intriguing – even a relatively simple neural network can be used to over-interpret an image, just like as children we enjoyed watching clouds and interpreting the random shapes.",
"This network was trained mostly on images of animals, so naturally it tends to interpret shapes as animals.”); whereas the activation maximization of a lower-layer unit(s), as in Figure REF , will result in texture insertion (see Figure REF ).",
"Figure: Deep Dream architecture focusing on a lower-level unitFigure: Deep Dream.",
"Example of a lower-layer unit maximization transformation.Reproduced from under a CC BY 4.0 licence.",
"Original photograph by Zachi EvenorOne may imagine a direct transposition of the Deep Dream approach to music, by maximizing the activation of a specific nodeParticularly if the role of a node has been identified, through a correlation analysis between node/layer activations and musical motives, as, for example, in Section REF .."
],
[
"#3 Example: Style Transfer Painting Generation System",
"The idea in this approach, named style transfer, pioneered by Gatys et al.",
"[56] and designed for images, is to use a deep convolutional feedforward architecture to independently capture the features of a first image (named the content), and the style (as a correlation between features) of a second image (named the style).",
"Gradient-based learning is then used to guide the incremental modification of an initially random third image, with the double objective of matching both the content and the style descriptions.",
"More precisely, the method is as follows: capture the content information of the first image (the content reference) by feed-forwarding it into the network and by storing units activations for each layer; capture the syle information of the second image (the style reference) by feed-forwarding it into the network and by storing feature spaces, which are correlations between units activations for each layer; and synthesize a hybrid image.",
"The hybrid image is created by generating a random image, defining it as current image, and then iterating the following loop until the two targets (content similarity and style similarity) are reached: capture the contents and the style information of the current image, compute the content cost (distance between reference and current content) and the style cost (distance between reference and current style), compute the corresponding gradients through standard backpropagation, and update the current image guided by the gradients.",
"The architecture and process are summarized in Figure REF (more details may be found in [57]).",
"The content image (on the right) is a photograph of Tübingen's Neckarfront in GermanyThe location of the researchers.",
"(shown in Figure REF ) and the style image (on the left) is the painting “The Starry Night” by Vincent van Gogh (1889).",
"Figure: Style transfer full architecture/process.Extension of a figure reproduced from with permission of the authorsFigure: Tübingen's Neckarfront.",
"Photograph by Andreas Praefcke.Reproduced from with permission of the authorsExamples of transfer for the same content (Tübingen's Neckarfront) and the styles “The Starry Night” by Vincent van Gogh (1889) and “The Shipwreck of the Minotaur” by J. M. W. Turner (1805) are shown in Figures REF and REF , respectively.",
"Figure: Style transfer of “The Starry Night” by Vincent van Gogh (1889) on Tübingen's Neckarfront photograph.Reproduced from with permission of the authorsFigure: Style transfer of “The Shipwreck of the Minotaur” by J. M. W. Turner (1805) on Tübingen's Neckarfront photograph.Reproduced from with permission of the authorsNote that one may balance content and style targetsThrough the $\\alpha $ and $\\beta $ parameters, see at the top of Figure REF the total loss defined as $\\mathcal {L}_{total} = \\alpha \\mathcal {L}_{content} + \\beta \\mathcal {L}_{style}$ .",
"($\\alpha /\\beta $ ratio) in order to favor content or style.",
"In addition, the complexity of the capture may also be adjusted via the number of hidden layers used.",
"These variations are shown in Figure REF : rightwards an increasing $\\alpha /\\beta $ content/style objectives ratio and downwards an increasing number of hidden layers used (from 1 to 5) for capturing the style.",
"The style image is the painting “Composition VII” by Wassily Kandinsky (1913).",
"Figure: Variations on the style transfer of “Composition VII” by Wassily Kandinsky (1913) on Tübingen's Neckarfront photograph.Reproduced from with permission of the authors"
],
[
"Style Transfer vs Transfer Learning",
"Note that although style transfer shares some of the general objectives of transfer learning, it is actually different (in terms of objective and techniques).",
"Transfer learning is about reusing what has been learnt by a neural network architecture for a specific task and applying it to another task and/or domain (e.g., another type of classification).",
"Transfer learning issues will be touched upon in Section ."
],
[
"#4 Example: Music Style Transfer",
"Transposing this style transfer technique to music (music style transfer) is a tempting direction.",
"However, as we will see, the style of a piece of music is more multidimensional and could be related to different types of music representation (composition, performance, sound, etc.",
"), and is thus more difficult to capture via such a simple correlation of activations.",
"Therefore, we will analyze this issue as a specific challenge in Section ."
],
[
"Input Manipulation and Sampling",
"An example of the combination of the input manipulation strategy with the sampling strategy, thus acting both on the input and the outputInterestingly, the input is actually equal to the output because the architecture used is an RBM (see Section ), where the visible layer acts both as input and output., is exemplified in the following section."
],
[
"Example: C-RBM Polyphony Symbolic Music Generation System",
"In the system presented by Lattner et al.",
"in [109], the starting point is to use a restricted Boltzmann machine (RBM) to learn the local structure, seen as the musical texture, of a corpus of musical pieces.",
"The additional idea is to impose, through constraints, a more global structure (form, e.g., AABA, as well as tonality), seen as a structural template inspired by an existing musical piece, on the new piece to be generated.",
"This is called structure impositionThis is an example of score-level composition style transfer (see Section REF )., also earlier coined as templagiarism (short for template plagiarism) by Hofstadter [84].",
"These constraints, concerning structure, tonality and meter, will guide an iterative generation through a search process, manipulating the input, based on gradient descent.",
"The actual objective is the generation of polyphonic music.",
"The representation used is piano roll, with 512 time steps and a range of 64 notes (corresponding to MIDI note numbers 28 to 92).",
"The corpus is Wolfgang Amadeus Mozart's sonatas.",
"Each piece is transposed into all possible keys in order to have sufficient training data for all possible keys (this also helps reduce sparsity in the training data).",
"The architecture is a convolutional restricted Boltzmann machine (C-RBM) [115], i.e.",
"an RBM with convolution, with 512$\\times $ 64 = 32,768 input nodes and 2,048 hidden units.",
"Units have continuous and not Boolean values, as for standard RBMs (see Section REF ).",
"Convolution is only performed on the time dimension, in order to model temporally invariant motives but not pitch invariant motives (there are correlations between notes over the whole pitch range), which would break the notion of tonalityAs the authors state in [109]: “Tonality is another very important higher-order property in music.",
"It describes perceived tonal relations between notes and chords.",
"This information can be used to, for example, determine the key of a piece or a musical section.",
"A key is characterized by the distribution of pitch classes in the musical texture within a (temporal) window of interest.",
"Different window lengths may lead to different key estimations, constituting a hierarchical tonal structure (on a very local level, key estimation is strongly related to chord estimation).”.",
"Training of the C-RBM is undertaken using the RBM-specific contrastive divergence algorithm (see Section , more precisely a more advanced version named persistent contrastive divergence).",
"Generation is performed by sampling with some constraints.",
"Three types of constraints are considered: Self-similarity – the purpose is to specify a global structure (e.g.",
"AABA) in the generated music piece.",
"This is modeled by minimizing the distance (measured through a mean squared error) between the self-similarity matrixes of the reference target and of the intermediate solution.",
"Tonality constraint – the purpose is to specify a key (tonality).",
"To estimate the key in a given temporal window, the distribution of pitch classes in the window is compared with the so-called key profiles of the reference (i.e.",
"paradigmatic relative pitch-class strengths for specific scales and modes [185], in practice the major and minor modes).",
"They are repeated in the time and pitch dimensions of the piano roll matrix, with a modulo octave shift in the pitch dimension.",
"The resulting key estimation vectors are combined (see the article for more details) to obtain an overall key estimation vector.",
"In the same way as for self-similarity, the distance between the target and the intermediate solution key estimations is minimized.",
"Meter constraint – the purpose is to impose a specific meter (also named a time signature, e.g., 3/4, 4/4, see Section REF ) and its related rhythmic pattern (e.g., relatively strong accents on the first and the third beat of a measure in a 4/4 meter).",
"As note intensities are not encoded in the data, only note onsets are considered.",
"The relative occurrence of note onsets within a measure is constrained to follow that of the reference.",
"Generation is performed via constrained sampling (CS), a mechanism used to restrict the set of possible solutions in the sampling process according to some pre-defined constraints.",
"The principles of the process, illustrated in Figure REF , are as follows: A sample is randomly initialized following the standard uniform distribution.",
"A step of constrained sampling (CS) is performed comprising $n$ runs of gradient descent (GD) optimization to impose the high-level structure, and $p$ runs of selective Gibbs sampling (SGS)Selective Gibbs sampling (SGS) is the authors' variant of Gibbs sampling (GS).",
"to selectively realign the sample onto the learnt distribution.",
"A simulated annealing algorithm is applied in order to decrease exploration in relation to a decrease of variance over solutions.",
"Figure: C-RBM architecture.Reproduced from with permission of the authorsThe different steps of constrained sampling are further detailed in [109].",
"Figure REF shows an example of a generated sample in piano roll format.",
"Figure: Piano roll sample generated by C-RBM.Reproduced with permission of the authorsThe results for the C-RBM (summarized in Table REF ) are interesting and promising.",
"One current limitation, stated by the authors, is that constraints only apply to the high-level structure.",
"Initial attempts at imposing low-level structure constraints are challenging because, as constraints are never purely content-invariant, when trying to transfer low-level structure, the template piece can be exactly reconstructed in the GD phase.",
"Therefore, creating constraints for low-level structure would have to be accompanied by an increase in their content invariance.",
"Another issue is convergence and satisfaction of the constraints.",
"As discussed by the authors, their approach is not exact, as opposed to the Markov constraints approach (for Markov chains) proposed in [149].",
"Table: C-RBM summary"
],
[
"Reinforcement",
"The idea of the reinforcement strategy is to reformulate the generation of musical content as a reinforcement learning problem: using the similarity to the output of a recurrent network trained on the dataset as a reward and adding user defined constraints, e.g., some tonality rules according to music theory, as an additional reward.",
"Let us consider the case of a monophonic melody formulated as a reinforcement learning problem: the state represents the musical content (a partial melody) generated so far, and the action represents the selection of the next note to be generated.",
"Let us now consider a recurrent neural network (RNN) trained on the chosen corpus of melodies.",
"Once trained, the RNN will be used as a reference for the reinforcement learning architecture.",
"The reward of the reinforcement learning architecture is defined as a combination of two objectives: adherence to what has been learnt, by measuring the similarity of the action selected, i.e.",
"the next note to be generated, to the note predicted by the recurrent network in a similar state (i.e.",
"the partial melody generated so far); and adherence to user-defined constraints (e.g., consistency with current tonality, avoidance of excessive repetitions, etc.",
"), by measuring how well they are fulfilled.",
"In summary, the reinforcement learning architecture is rewarded to mimic the RNN, while also being rewarded to enforce some user-defined constraints."
],
[
"Example: RL-Tuner Melody Symbolic Music Generation System",
"The reinforcement strategy was pioneered in the RL-Tuner architecture by Jaques et al.",
"[93].",
"This architecture, illustrated in Figure REF , consists in two deep Q network reinforcement learning architecturesAn implementation of the Q-learning reinforcement learning strategy through a deep learning architecture [196].",
"and two recurrent neural network (RNN) architectures.",
"Figure: RL-Tuner architecture.Reproduced from with permission of the authors The initial RNN, named Note RNN, is trained on the dataset of melodies for the task of predicting and generating the next note, following the iterative feedforward strategy.",
"A fixed copy of Note RNN is made, named Reward RNN, which will be used by the reinforcement learning architecture as a reference.",
"The Q Network architecture task is to learn to select the next note (next action a) from the generated (partial) melody so far (current state s).",
"The Q Network is trained in parallel to the other Q Network, named Target Q Network, which estimates the value of the gain (accumulated rewards) and which has been initialized from what Note RNN has learnt.",
"Q Network's reward r combines two rewards, as defined in previous section: adherence to what has been learnt, measured by the probability of Reward RNN to play that note, in practice $\\text{log}\\,P(\\text{a} | \\text{s})$ , the log probability for the next note being a given a melody s; and adherence to music theory constraints, in practiceThis list of musical theory constraints has been selected from [58], see more details in [93].",
": staying in key, beginning and ending with the tonic note, avoiding excessively repeated notes, preferring harmonious intervals, resolving large leaps, avoiding continuously repeating extrema notes, avoiding high auto-correlation, playing motifs, and playing repeated motifs.",
"The total reward $\\text{r}(\\text{s}, \\text{a})$Which means the reward to be received when from state s action a is chosen.",
"is defined by Equation REF , where r$_{MT}$ is the reward concerning music theory and $c$ is a parameter controlling the balance between the two competing constraints.",
"$\\text{r}(\\text{s}, \\text{a}) = \\text{log}\\,P(\\text{a} | \\text{s}) + \\text{r}_{MT}(\\text{a}, \\text{s})/c$ Figure REF shows the evolution during the training phase of the two types of rewards (adherence to Note RNN and to music theory), with three different reinforcement learning algorithms: Q-learning, $\\Psi $ -learning and G-learning (see details in [93]).",
"Figure: Evolution during training of the two types of rewards for the RL-Tuner architecture.Reproduced from with permission of the authorsThe corpus used for the experiments is a set of monophonic melodies extracted from a corpus of 30,000 MIDI songs.",
"The time step is set at a sixteenth note.",
"The one-hot encoding (of dimension 38) considers three octaves of notes plus two special events: note off (encoded as 0) and no note (a rest, encoded as 1).",
"The MIDI note number is translated in order to start the lowest note (C$_3$ ) as a 2 (B$_5$ is encoded as 37) and have special events smoothly integrated within the integer encoding.",
"Note that, as melodies are monophonic, playing a different note implicitly ends the last played note without requiring an explicit note off event, which results in a more compact representation.",
"Note RNN (and its copy Reward RNN) have one LSTM layer with 100 cells.",
"In summary, the reinforcement strategy allows arbitrary user given constraints (control) to be combined with a style learnt by the recurrent network.",
"Note that in the case of RL-Tuner, the reward is known beforehand and dual purpose: handcrafted for the music theory rules and learnt from the dataset by an RNN for the musical style.",
"Therefore, there is an opportunity to add another type of reward, an interactive feedback by the user (see Section ).",
"However, a feedback at the granularity of each note generated may be too demanding and, moreover, not that accurateAs Miles Davis coined it: “If you hit a wrong note, it's the next note that you play that determines if it's good or bad.”.",
"We will discuss in Section the issue of learning from user feedback.",
"RL-Tuner is summarized in Table REF .",
"Table: RL-Tuner summary"
],
[
"Unit Selection",
"The unit selection strategy is about querying successive musical units (e.g., one measure long melodies) from a database and concatenating them in order to generate a sequence according to some user characteristics.",
"Querying is using features which have been automatically extracted by an autoencoder.",
"Concatenation, i.e.",
"“what unit next?”, is controlled by two LSTMs, each one for a different criterium, in order to achieve a balance between direction and transition.",
"This strategy, as opposed to most of the other ones, which are bottom-up, is top-down, as it starts with a structure and fills it."
],
[
"Example: Unit Selection and Concatenation Symbolic Melody Generation System",
"This strategy was pioneered by Bretan et al.",
"[13].",
"The idea is to generate music from a concatenation of musical units, queried from a database.",
"The key process here is unit selection, which is based on two criteria: semantic relevance and concatenation cost.",
"The idea of unit selection to generate sequences was actually inspired by a technique commonly used in text-to-speech (TTS) systems.",
"The objective is to generate melodies.",
"The corpus considered is a dataset of 4,235 lead sheets in various musical styles (jazz, folk, rock...) and 120 jazz solo transcriptions.",
"The granularity of a musical unit is a measure.",
"This means there are roughly 170,000 units in the dataset.",
"The dataset is restricted to a five octaves range (MIDI note numbers 36 to 99) and augmented by transposing each unit in all keys so that all possible pitches are covered.",
"The architecture includes one autoencoder and two LSTM recurrent networks.",
"The first step is feature extraction: 10 features, manually handcrafted, are considered, following a bag-of-words (BOW) approach (see Section REF ), e.g., counts of a certain pitch class, counts of a certain pitch class rhythm tuple, whether the first note is tied to the previous measure, etc.",
"This results in 9,675 actual features.",
"Most of features have integer values, with the exception of rests being represented using a negative pitch value and of some Boolean features.",
"Therefore, each unit is described (indexed) as a feature vector of size 9,675.",
"The autoencoder used has a 2-layer stacked autoencoder architecture, as illustrated in Figure REF .",
"Once trained on the set of feature vectors, in the usual self-supervised way for autoencoders (see Section ), the autoencoder becomes a features extractor encoding a feature vector of size 9,675 into an embedding vector of size 500.",
"Figure: Unit selection indexing architecture.Reproduced from with permission of the authorsThere is one remaining issue for generating a melody: how to select the best (or at least, a very good) candidate from a given (current, named seed by the authors) musical unit as a successor musical unit?",
"Two criteria are considered: Successor semantic relevance – based on a model of transition between units, as learnt by an LSTM recurrent network.",
"In other words, relevance is based on the distance to the (ideal) next unit as predicted by the model.",
"This first LSTM architecture has two hidden layers, each with 128 units.",
"The input and output layers have 512 units (corresponding to the format of the embedding).",
"Concatenation cost – based on another model of transitionAt a more fine-grained note-level transition than the previous model.",
"between the last note of current unit and the first note of the next unit, as learnt by another LSTM recurrent network.",
"This second LSTM architecture is multilayer and its input and output layers have about 3,000 units, corresponding to a multi-one-hot encoding of the characterization of an individual note (as defined by its pitch and its duration).",
"The combination of the two criteria (illustrated in Figure REF , with current (seed) unit in blue and next (candidate) unit in red) is handled by a heuristic-based dynamic ranking process: rank all musical units according to their successor semantic relevance with current musical unitThe initial musical unit of a melody to be generated may be chosen by the user or sorted.",
"; take the top 5% and re-rank them according to the combination of their successor semantic relevance and their concatenation cost; and select the musical unit with the highest combined rank.",
"Figure: Unit selection based on semantic cost.Reproduced from with permission of the authorsThe process is iterated in order to generate successive musical units and thus a melody of arbitrary length.",
"This may at first look like a standard iterative feedforward generation from a recurrent network (see Section REF ), but there are two important differences: the label (instance of the embedding) of the next musical unit is computed through a multicriteria ranking algorithm; and the actual unit is queried from a database with the label as the index.",
"Initial external human evaluation has been conducted by the authors.",
"They found that music generated using one or two measures long units tend to be ranked higher according to naturalness and likeability than four measures long units or note-level generation, with an ideal unit length appearing to be one measure.",
"Note that the unit selection strategy does not directly provide control, but it does provide entry points for control as one may extend the selection framework (currently based on two criteria: successor semantic relevance and concatenation cost) with user defined constraints/criteria.",
"The system is summarized in Table REF .",
"Table: Unit selection summary"
],
[
"Style Transfer",
"In Section REF we introduced style transfer as one example of using the input manipulation strategy to control content generation.",
"The style transfer technique for images (proposed by Gatys et al.",
"[56] and described in Section REF ) is effective and relatively straightforward to apply.",
"However, as opposed to paintings, where the common representation is two-dimensional and uniformly digitalized in terms of pixels, music is a much more complex object with various levels and models of representation (see Chapter and also Section REF ).",
"In their recent analysis, Dai et al.",
"[31] consider three main levels (or dimensions) of representation and associated types of music style transfer: score-level, which they name composition style transfer; sound-level, which they name timbre style transfer; and performance control-level, which they name performance style transfer.",
"They state that music style transfer for each level (namely, composition, timbre and performance style transfer) are very different in nature.",
"They also point out the issue of the interrelation and the entanglement of these different levels (and nature) of representation.",
"Therefore, they point out the need for automated learning of the disentanglementDisentanglement is the objective of separating the different factors governing variability in the data (e.g., in the case of human images, identity of the individual and facial expression, see, for example, [36]).",
"Recent work on disentanglement learning can be found, for example, in [10].",
"Also note that variational autoencoders (VAEs, see Section REF ) are currently among the promising approaches for disentanglement learning because, as Goodfellow et al.",
"put it in [63]: “Training a parametric encoder in combination with the generator network forces the model to learn a predictable coordinate system that the encoder can capture.” of different levels of music representation, in order to ease music style transfer."
],
[
"Composition Style Transfer",
"Style transfer at the composition level means working on symbolic representations.",
"An example is structure imposition, i.e.",
"transferring some existing structure (e.g., an AABA global structure) from an initial composition into another newly generated composition.",
"The C-RBM system, presented in Section REF , implements this kind of structure imposition by considering separately three kinds of structures and associated constraints: global structure (e.g., AABA), tonality and meter (rhythm).",
"One may think that such structure descriptors are too low level to define a style.",
"But they are an interesting first step, as one may consider higher-level style descriptors by aggregating such structure descriptors.",
"Let us imagine, for instance, describing (and later on transferring) the style of a composer like Michel Legrand with his own way of repeating transpositions of motives.",
"Note that in the DeepJ system for controlling the style of the generation (Section REF ) the objective is different, as the style is explicitly specified by the user via a set of musical examples, learnt and applied during generation time through conditioning."
],
[
"Timbre Style Transfer",
"For timbre style transfer, based on audio representations, some researchers have straightforwardly applied Gatys et al.",
"'s technique (Section REF ) to sound (audio), using various kinds of sources (various styles of music as well as speech), as explained in next section."
],
[
"Examples: Audio Timbre Style Transfer Systems",
"Examples of style transfer systems for audio (timbre) are: Ulyanov and Lebedev's system in [192], and Foote et al.",
"'s system in [53].",
"These two systems both use a spectrogram (and not a direct wave signal) as their input representationFor a comparison of various audio representations for audio style transfer, see the recent analysis by Wyse [208]..",
"In [208], Wyse points out two specificities (which he calls “two remarkable aspects”) in the architecture of Ulyanov and Lebedev's system that differentiate it from the image style transfer technique: the network uses only a single layer.",
"Therefore, the only difference between content and style comes from the difference between first-order and second-order (correlations) measures of activation; and the network was not pre-trained and uses random weights.",
"Wyse further adds in [208]: “The blog post claims this unintuitive approach generated results as good as any other, and the sound examples posted are indeed compelling.” We also found convincing the examples of audio style transfer, although not as interesting as painting style transfer, as it sounds similar to some sound modulation/merging of both style and content signals.",
"In their own analysis in [53], Foote et al.",
"summarize the difficulty as follows: “On this level we draw one main conclusion: audio is dissimilar enough from images that we shouldn't expect work in this domain to be as simple as changing 2D convolutions to 1D.” We will try to analyze some possible reasons for this in the next section.",
"Table REF summarizes the main features of these audio (timbre) style transfer systems (which we will reference to as AST in Chapter ).",
"Table: Audio (timbre) style transfer (AST) summary"
],
[
"Limits and Challenges",
"We believe that, in part, the difficulty of directly transposing image style transfer to music comes from the anisotropyIsotropy means invariance of properties regardless of the direction, whereas anisotropy means direction dependence.",
"of global audio music content representation.",
"In the case of a natural image, the correlations between visual elements (pixels) are equivalent whatever the direction (horizontal axis, vertical axis, diagonal axis or any arbitrary direction), i.e.",
"correlations are isotropic.",
"In the case of a global representation of audio data (where the horizontal dimension represents time and the vertical dimension represents the notes), this uniformity no longer holds as horizontal correlations represent temporal correlations and vertical correlations represent harmonic correlations, which are very different in nature (see an illustration in Figure REF ).",
"Figure: Anisotropic music vs an isotropic image.Incorporating Aquegg's original image from “https://en.wikipedia.org/wiki/Spectrogram”and the painting “The Starry Night” by Vincent van Gogh (1889)One direction could be to reformulate the capture of the style information, and therefore the nature of the correlations, in order to take into account the time dimension.",
"Another (also hypothetical) direction could be to use a “time-compressed” representation, by considering the summary learnt by an RNN Encoder-Decoder (see Section REF )."
],
[
"Performance Style Transfer",
"Although it does not directly address performance style transfer, the Performance RNN system described in Section REF provides a background representation for modeling performance (note onsets as well as dynamics).",
"What remains to be undertaken to develop a performance imposition system could be along the following lines: model mappings between performance and other(s) features (e.g., mean duration of notes, modulation, etc.",
"); learn mappings for a given corpus (musician, context, etc.)",
"through correlation analysis, revealing the performance style of a given musician (and corpus); and transpose a mapping to an existing piece in order to transfer the performance style.",
"As noted by Dai et al.",
"in [31], performance style transfer is closely related to expressive performance rendering, see, for instance, the example of the Cyber-João system [30] introduced in Section and a recent system based on deep learning in [124].",
"But it also requires the disentanglement of control (style) and score information (content) as well as the learning of the mappings discussed above.",
"Thus, this is still a direction to be explored."
],
[
"Example: FlowComposer Composition Support Environment",
"A good example of an interactive music composition environment addressing style transfer in different dimensions is the FlowComposer system [150], [153], developed by Pachet et al.",
"during the Flow Machines project [49].",
"Note that it is based on Markov chain models and not (yet) deep learning models.",
"FlowComposer provides possibilities for music style transfer at the following levels: Composition style level – some style transfer may be performed, e.g., automated reharmonization based on a style (corpus) of selected music.",
"See the examples of the automatic reharmonization of Yesterday by John Lennon and Paul McCartney (Figure REF ) in the style of Michel Legrand (Figure REF ) and Bill Evans (Figure REF ).",
"Timbre and performance style levelsJointly, as there is no possibility yet for separating/disentangling these two concerns.",
"– rendering may be done via style transfer, by automated mapping and extrapolation from a library of various instrumental audio performances (through the ReChord component [157]).",
"Figure: Yesterday (Lennon/McCartney) (first 15 measures) – original harmonization.Reproduced from with permission of the authorsFigure: Yesterday (Lennon/McCartney) (first 15 measures) – reharmonization by FlowComposer in the style of Michel Legrand.Reproduced from with permission of the authorsFigure: Yesterday (Lennon/McCartney) (first 15 measures) – reharmonization by FlowComposer in the style of Bill Evans.Reproduced from with permission of the authorsThe FlowComposer control panel includes various fields to select composition style and sliders to set harmonisation conformance, inspiration, average note duration and chord changes, as shown in Figure REF .",
"An example of a lead sheet generated in the style of Bill Evans is shown in Figure REF , with the following user defined characteristics: a 3/4 time signature, a constraint on the first (C7) and last (G7) chords, and a “max order” of four beatsThis very interesting feature controls the maximum amount of successive notes (actually beats) copied from the corpus.",
"It relies on the integration of a new constraint named MaxOrder [152] in the Markov constraints framework [149] underlying FlowComposer.",
"This is one possible way to control originality (see Section ).. Color backgrounds indicate sequences of notes extracted as a whole from a given song in the chosen corpus (here all of Bill Evans' compositions in 3/4).",
"Figure: Flow Composer control panel.Reproduced from with permission of the authorsFigure: Example of a Flow Composer interactively generated lead sheet.Reproduced from with permission of the authors"
],
[
"Structure",
"One challenge is that most existing systems have a tendency to generate music with no clear structure or “sense of direction”Beside the technical improvements brought by LSTMs on the learning of long-term dependencies (see Section REF )..",
"In other words, although the style of the generated music corresponds to the corpus learnt, the music appears to wander without any higher organization, as opposed to human composed music which has some global organization (usually named a form) and identified components, such as an overture, an allegro, an adagio or a finale in classical music; an AABA or an AAB form in jazz; a refrain, a verse or a bridge in song music.",
"Note that there are various possible levels of structure.",
"For instance, an example of a finer-grain structure is at the level of a melodic motif that can be repeated, often being transposed in order to adapt to a new harmonic structure.",
"The reinforcement strategy (used by RL-Tuner in Section REF ) and the structure imposition approach (used by C-RBM in Section REF ) can both enforce (and/or transfer, see Section ) some constraints, possibly high-level, on the generation.",
"About structure imposition, see also a recent proposal combining two graphical models, one for chords and one for melody, for the generation of lead sheets with an imposed structure [148].",
"An alternative top-down approach is followed by the unit selection strategy (see Section REF ), by generating an abstract sequence structure and filling it with musical units, although the structure is not yet very high-level as it effectively stays at the level of a measure.",
"A related challenge is not about the imposition of preexisting high-level structures, but about the capacity for learning high-level structures and, moreover, the capacity for invention (emergence) of high-level structures.",
"Therefore, a natural direction is to explicitly consider and process different levels (hierarchies) of temporality and structure."
],
[
"Example: MusicVAE Multivoice Hierarchical Symbolic Music Generation System",
"In [162], Roberts et al.",
"propose an architecture named MusicVAE, based on a variational recurrent autoencoder (VRAE) with a 2-level hierarchical RNN within the decoder The corpus comprises MIDI files collected from the web, from which three types of musical examples are extracted: monophonic melodiesMusicVAE has since been extended to an arbitrary number of polyphonic tracks, see details in [175]., 2 or 16 measures long; drum patterns, 2 or 16 measures long; and trio sequences with three different voices (melody, bass line and drum pattern), 16 measures long.",
"Encoding of monophonic melodies and bass lines is through tokens representing MIDI events: the 128 “Note on” events corresponding to the 128 possible MIDI note numbers (pitches) of the defined interval, the singleOnly one “Note off” event is needed for all possible pitches as the melody is monophonic.",
"It is used to differentiate a note held from two successive identical notes, see Section REF .",
"“Note off” event and the rest (silence) token.",
"Encoding of drum patterns is done by mapping MIDI standard drum classes through a binning into 9 canonical classes, leading to $2^9 = 512$ categorical tokens representing all possible combinations.",
"Quantization is at the sixteenth note.",
"The architecture follows the principles of a variational autoencoder encapsulating recurrent networks such as VRAE, with two differences: the encoder is a bidirectional recurrent network (see Section REF ) – an LSTM with the input and output layers having 2,048 nodes and a single hidden layer of 512 cells; and the decoder is a hierarchical 2-level recurrent network, composed of a high-level RNN named the conductor – an LSTM with the input and output layers having 512 nodes and a single hidden layer of 1,024 cells – that produces a sequence of embeddings; and a bottom-layer RNN – an LSTM with two hidden layers of 1,024 cells – that uses each embedding as an initial state and also as an additional input concatenated to its previously generated tokenAlong the iterative feedforward strategy.",
"to produce each subsequence.",
"In order to prioritize the conductor RNN over the bottom-layer RNN, its initial state is reinitialized with the decoder generated embedding for each new subsequence.",
"In the case of a multivoice trio (melody, bass and drums), there are three LSTMs, one for each voice.",
"The MusicVAE architecture is illustrated in Figure REF .",
"The authors report that an equivalent “flat” MusicVAE architecture (without hierarchy), although accurate in modeling the style in the case of 2 measures long examples, was inaccurate in the case of 16 measures long examples, with a 27% error increase for the autoencoder reconstruction (0.883 accuracy for the flat architecture and 0.919 accuracy for the hierarchical architecture).",
"Figure: MusicVAE architecture.Reproduced from with permission of the authorsAn example of trio music generated is shown in Figure REF .",
"A preliminary evaluation has been conducted with listeners comparing three versions (flat architecture, hierarchical architecture and real music) for three types of music: melody, trio and drums.",
"The results show a very significant gain with the hierarchical architecture, see more details in [162].",
"Figure: Example of a trio music generated by MusicVAE.Reproduced from with permission of the authorsAn interesting feature of the variational autoencoder architecture is in the capacity for exploring the latent space via various operations such as translation; interpolation – Figure REF in Section REF shows an interesting comparison of melodies resulting from interpolation in the data space (that is the space of representation of melodies) and interpolation in the latent space which is then decoded into the corresponding melodies.",
"One can see (and hear) that the interpolation in the latent space produces much more meaningful and interesting melodies; averaging – Figure REF shows an example of a melody (in the middle of the figure) generated from the combination (averaging) of the latent spaces of two melodies (at the top and bottom of the figure); attribute vector arithmetics, by addition or subtraction of an attribute vector capturing a given characteristic – Figure REF shows an example of a melody (at the bottom of the figure) generated when a “high note density” attribute vector is added to the latent space of an existing melody (at the top of the figure).",
"An attribute vector is computed as the average of the latent vectors for a collection of examples sharing that characteristic (attribute) (e.g., high density of notes, rapid change, high register, etc.).",
"Figure: Example of a melody generated (middle) by MusicVAE by averaging the latent spaces of two melodies (top and bottom).Reproduced from with permission of the authorsFigure: Example of a melody generated (bottom) by MusicVAEby adding a “high note density” attribute vector to the latent space of an existing melody (top).Reproduced from with permission of the authorsFurthermore, Figure REF shows the effect, as a percent change, of modifying individual attributes of 16 measures long melodies by adding (left matrix), or respectively subtracting (right matrix), attribute vectors in the latent space.",
"The vertical axis of each correlation matrix denotes the attribute vector applied and the horizontal axis denotes the attribute measured.",
"These correlation matrixes show that individual attributes can be modified without effecting others, except for the cases when correlations are expected, as for instance between the eighth and sixteenth note syncopations.",
"Figure: Correlation matrices of the effect of adding (left) of subtracting (right) an attribute to other attributes in MusicVAE.Reproduced from with permission of the authorsAudio examples are available in [164] and [161].",
"MusicVAE is summarized in Table REF .",
"Table: MusicVAE summary"
],
[
"Other Temporal Architectural Hierarchies",
"There are some alternative solutions to organize temporal hierarchies within a deep learning architecture.",
"A first example is ClockworkRNN, by Koutník et al.",
"[103].",
"The idea is to partition the hidden recurrent layer into various modules, all fully connected (in a parallel way) to input layer nodes and to output layer nodes, but each module with a different clock rate with interconnections between modules according to their clock rates (neurons of a faster module are fully connected to neurons of a slower module).",
"Another example is SampleRNN, by Mehri et al.",
"[129].",
"It is an extension of WaveNet architecture (Section REF ) inspired from the idea of different clock rates from ClockworkRNN, but with some external modules (full networks) organized via some conditioningConditioning has been introduced in Section ., as opposed to ClockworkRNN's internal modules.",
"Each module is a deep RNN which summarizes the history of its inputs (successive waveform frames) into a conditioning vector for the next module downward which operates on frames of shorter duration.",
"More details of these two architectures may be found in their respective articles, [103] and [129].",
"These examples show that there is an active ongoing research activity to explore various ways to organize temporal hierarchies in deep learning architectures (for audio or symbolic contents), in order to try to better capture longer term structure of music."
],
[
"Originality",
"The issue of the originality of the music generated is not only an artistic issue (creativity) but also an economic one, because it raises the issue of the copyrightOn this issue, see the recent paper by Deltorn [34].. One approach is a posteriori, by ensuring that the generated music is not too similar (e.g., in not having recopied a significant number of notes of a melody) to an existing piece of music.",
"Therefore, existing algorithms to detect similarities in texts may be used.",
"Another approach, more systematic but even more challenging, is a priori, by ensuring that the music generated will not recopy a given portion of music from the training corpusNote that this addresses the issue of significant recopying from the training corpus, but it does not prevent a system from reinventing existing music outside of the training corpus.. A solution for music generation from Markov chains has been proposed [152].",
"It is based on a variable order Markov model and constraints over the order of the generation through some min order and max order constraints, in order to attain some sweet spot between junk and plagiarism.",
"However, there is not yet a solution for deep learning architectures.",
"Let us now analyze some recent directions for favoring originality in the generated musical content."
],
[
"Example: MidiNet Melody Generation System",
"In their description of MidiNet [212] (see Section REF ), the authors discuss two methods to control creativity: restricting the conditioning by inserting the conditioning data only in the intermediate convolution layers of the generator architecture; and decreasing the values of the two control parameters of feature matching regularization, in order to reduce the requirement for the closeness of the distributions of real data and generated distributions of real and generated data.",
"These experiments are interesting but they remain at the level of ad hoc tuning of the hyper-parameters of the architecture."
],
[
"Creative Adversarial Networks",
"Another more systematic and conceptual direction is the concept of creative adversarial networks (CAN) proposed by Elgammal et al.",
"[45], as an extension of the generative adversarial networks (GAN) architecture (introduced in Section )."
],
[
"Creative Adversarial Networks Painting Generation System",
"Elgammal et al.",
"propose in [45] to address the issue of creativity by extending a generative adversarial networks (GAN) architecture into a creative adversarial networks (CAN) architecture to “generate art by learning about styles and deviating from style norms.” [45].",
"Their assumption is that in a standard GAN architecture, the generator objective is to generate images that fool the discriminator and, as a consequence, the generator is trained to be emulative but not creative.",
"In the proposed creative adversarial networks (CAN) (illustrated in Figure REF ), the generator receives from the discriminator not just one but two signals: Figure: Creative adversarial networks (CAN) architecture.Reproduced from with permission of the authors the first signal is analog to the case of the standard GAN (see Equation REF ) and is the discriminator's estimation whether the generated sample is real or faked art; and the second signal is about how easily the discriminator can classify the generated sample into predefined established styles.",
"If the generated sample is style-ambiguous (i.e.",
"the various classes are equiprobable), this means that the sample is difficult to fit within the existing art styles, which may be interpreted as the creation of a new style.",
"These two signals are contradictory forces which push the generator to explore the space for generating items that are close to the distribution of existing art pieces and style-ambiguous.",
"Experiments have been done with paintings from a WikiArt dataset [205].",
"This collection has images of 81,449 paintings from 1,119 artists ranging from the fifteenth century to the twentieth century.",
"It has been tagged with 25 possible painting styles (e.g., cubism, fauvism, high-renaissance, impressionism, pop-art, realism, etc.).",
"Some examples of images generated by CAN are shown in Figure REF .",
"Figure: Examples of images generated by CAN.Reproduced from with permission of the authorsAs the authors discuss, the generated images are not recognized like traditional art, in terms of standard genres (portraits, landscapes, religious paintings, still lifes, etc.",
"), as shown by a preliminary external human evaluation and also a preliminary analysis of their approach.",
"Note that the CAN approach assumes the existence of a prior style classification and reduces the idea of creativity to exploring new styles (which indeed has some grounding in the art history).",
"The necessary prior classification between different styles does have an important role and it will be interesting to experiment with other types of classification, including styles which are automatically constructed.",
"Experimenting with the transposition of the CAN approach to music generation appears as a tempting direction.",
"Note that, as opposed to most of techniques applying control constraints during the generation phase while leaving the training phase untouched (see, e.g., style imposition in C-RBM system in Section REF ), in the CAN approach, the incentive for creativity is applied during the training phase.",
"The important issue of originality (and creativity) will be further discussed in Section ."
],
[
"Incrementality",
"A straightforward use of deep architectures for generation leads to a one-shot generation of a musical content as a whole in the case of a feedforward or autoencoder network architecture, or to an iterative generation of time slices of a musical content in the case of a recurrent network architecture.",
"This is a strong limitation if we compare this to the way a human composer creates and generates music, in most cases very incrementally, though successive refinements of arbitrary parts."
],
[
"Note Instantiation Strategies",
"Let us review how notes are instantiated during generation.",
"There are three main strategies: Single-step feedforward – a feedforward architecture processes in a single processing step a global representation which includes all time steps.",
"An example is MiniBach (Section REF ).",
"Iterative feedforward – a recurrent architecture iteratively processes a local representation corresponding to a single time step.",
"An example is CONCERT (Section REF ).",
"Incremental sampling – a feedforward architecture incrementally processes a global representation which includes all time steps, by incrementally instantiating its variables (each variable corresponding to the possibility of a note at a specific time step).",
"An example is DeepBach (Section REF ).",
"These three strategies are compared and illustrated in Figure REF .",
"The representation is piano roll type with two simultaneous voices (or tracks).",
"The cells in blue are the notes to be played.",
"The rectangles with a thick line labeled as “current” indicate the parts being processed, whereas the parts in light grey indicate the parts already processed.",
"Figure: Note generation/instantiation – three main strategiesIn the case of the incremental sampling strategy (right part of Figure REF ), at each processing step a new cell representing a triplet $(voice, note, time\\,step)$ , labeled as “current”, is randomly chosen and instantiated.",
"Triplets already instantiated are blue-filled if a note is to be played and light grey-filled otherwise.",
"Note that, with this incremental sampling strategy, it is possible to only generate or to regenerate an arbitrary part (slice) of the musical content, for a specific time interval between two time steps and/or for a specific subset of voices/tracks, without the need for regenerating the whole content.",
"In Figure REF , the dashed rectangle indicates a zone selected by the user to perform a selective regenerationWith the single-step feedforward strategy, one could imagine selecting only the desired slice from the regenerated content and “copy/pasting” it into the previously generated content, but with the obvious absence of a guarantee that the old and the new parts will be consistent.."
],
[
"Example: DeepBach Chorale Multivoice Symbolic Music Generation System",
"Hadjeres et al.",
"have proposed the DeepBach architectureThe MiniBach architecture described in Section REF is actually a deterministic single-step feedforward (major) simplification of the DeepBach architecture.",
"for the generation of J. S. Bach chorales [70].",
"The architecture, shown in Figure REF , combines two recurrent networks (LSTMs) and two feedforward networks.",
"As opposed to the standard use of recurrent networks where a single time direction is consideredAn exception is, for example, some bidirectional recurrent architecture used in the BLSTM and in the C-RNN-GAN systems analyzed, respectively, in Sections REF and REF ., DeepBach architecture considers two directions: forward in time and backward in timeThe authors state that this architectural choice somewhat matches the real compositional practice of Bach chorales.",
"Indeed, when reharmonizing a given melody, it is often simpler to start from the cadence and write music backward [70]..",
"Therefore, two recurrent networks (more precisely, LSTMs with 200 cells) are used, one summing up past information and another summing up information coming from the future, together with a nonrecurrent network in charge of notes occurring at the same time.",
"Their three outputs are merged and passed to the input of a final feedforward neural network, with one hidden layer with 200 units.",
"The final output activation function is softmax.",
"Figure: DeepBach architecture.Reproduced from with permission of the authorsThe initial corpus is the set of J. S. Bach's polyphonic (multivoice) chorales [5], where the composer chose various given melodies for a soprano and composed the three additional ones (for alto, tenor and bass) in a counterpoint manner.",
"The initial dataset (352 chorales) is augmented by adding all chorale transpositions which fit within the vocal ranges defined by the initial corpus.",
"This leads to a total corpus of 2,503 chorales.",
"The vocal ranges contain up to 28 different pitches for each voice21 for the soprano, alto and tenor parts and 28 for the bass part..",
"The choice of the representation in DeepBach has some specificities.",
"A hold symbol “__” is used to indicate whether a note is being held (see Section REF ).",
"The authors emphasize in [70] that this representation is well-suited to the sampling method used, more precisely that the fact that they obtain good results using Gibbs sampling relies exclusively on their choice to integrate the hold symbol into the list of notes.",
"Another specificity is that the representation consists in encoding notes using their real names and not their MIDI note numbers (e.g., F$\\sharp $ is considered separately from G$\\flat $ , see Section REF ).",
"Last, the fermata symbol for Bach chorales is explicitly considered as it helps to produce structure and coherent phrases.",
"The first four lines of the example data at top of Figure REF correspond to the four voices.",
"The two bottom lines correspond to metadata (fermata and beat information).",
"Actually this architecture is replicated four times, one for each voice (four in a chorale).",
"Training, as well as generation, is not done in the conventional way for neural networks.",
"The objective is to predict the value of the current note for a given voice (shown in light green with a red “?”, at top center of Figure REF ), using as input information the surrounding contextual notes and their associated metadata, more precisely the three current notes for the three other voices (the thin rectangle in light blue in top center); the six previous notes (the rectangle in light turquoise blue in top left) for all voices; and the six next notes (the rectangle in light grey blue in top right) for all voices.",
"The training set is formed on-line by repeatedly randomly selecting a note in a voice from an example of the corpus and its surrounding context (as previously defined).",
"Generation is performed by incremental sampling, using a pseudo-Gibbs sampling algorithm analog to but computationally simpler than Gibbs sampling algorithmThe difference with Gibbs sampling (based on the non-assumption of compatibility of conditional probability distributions) and the algorithm are detailed and discussed in [70].",
"(see Section REF ), to produce a set of values (each note) of a polyphony, following the distribution that the network has learnt.",
"The algorithm for generation by incremental sampling is shown in Figure REF and has been illustrated in Figure REF .",
"Figure: DeepBach incremental generation/sampling algorithmAn example of a chorale generatedWe will see in Section REF that DeepBach may also be used for a different objective: counterpoint accompaniment.",
"is shown in Figure REF .",
"As opposed to many experiments, a systematic evaluation in a Turing-type test has been conducted (with more than 1,200 human subjects, from experts to novices, via a questionnaire on the WebAn evaluation was also conducted during a live program on a Dutch TV channel.)",
"and the results are very positive, showing a significant difficulty to discriminate between chorales composed by Bach and chorales generated by DeepBach.",
"DeepBach is summarized in Table REF .",
"Figure: Example of a chorale generated by DeepBach.Reproduced from with permission of the authorsTable: DeepBach summary"
],
[
"Interactivity",
"An important issue is that, for most current systems, generation of musical content is an automated and autonomous process.",
"Some interactivity with a human user(s) is fundamental to obtaining a companion system to help humans in their musical tasks (composition, counterpoint, harmonization, analysis, arranging, etc.)",
"in an incremental and interactive manner.",
"An example, already introduced in Section REF , is the FlowComposer prototype [153].",
"A couple of examples of partially interactive incremental systems based on deep network architectures are deepAutoController (Section REF ) and DeepBach (Section REF )."
],
[
"#1 Example: deepAutoController Audio Music Generation System",
"The deepAutoController system [170] introduced in Section REF provides a user interface, shown in Figure REF , to interactively control the generation, for instance by selecting a given input, generating a random input to be feedforwarded into the decoder stack, or controlling (by scaling or muting) the activation of a given unit.",
"Figure: Snapshot of a deepAutoController information window showing hidden units.Reproduced from with permission of the authors"
],
[
"#2 Example: DeepBach Chorale Symbolic Music Generation System",
"The user interface of DeepBach [70] (see Section REF ) is implemented as a plugin for the MuseScore music editor (see Figure REF ).",
"It helps the human user to interactively select and control partial regeneration of chorales.",
"This is made possible by the incremental nature of the generation (see Section ).",
"Moreover, the user can enforce some user-defined constraints, such as freezing a voice (e.g., the soprano) and resampling the other voices in order to reharmonize the fixed melodyIn practice, this means changing from the original objective of generating a 4-voice polyphony from scratch as discussed in Section REF , to generating a 3-voice counterpoint accompaniment for a given melody.",
"; modifying the fermata list in order to impose an end to musical phrases at specific places; restricting the note range for a given voice and a given temporal interval; and imposing a rhythm by restricting the note range to the hold symbol (as it is considered as a note) in specific parts.",
"Figure: DeepBach user interface.Reproduced from with permission of the authors"
],
[
"Interface Definition",
"Let us finally mention, at the junction between control and interactivity, the interesting discussion by Morris et al.",
"in [138] on the issue of what control parameters (for music generation by a Markov chain trained model) should be exposed at the human user level.",
"Some examples of user-level control parameters they have experimented with are as follows major vs minor; following melody vs following chords; and locking a feature (e.g., a chord)."
],
[
"Adaptability",
"One fundamental limitation of current deep learning architectures for the generation of musical content is that they paradoxically do not learn or adapt.",
"Learning is applied during the training phase of the network, but no learning or adaptation occurs during the generation phase.",
"However, one can imagine some feedback from a user, e.g., the composer, producer, listener, about the quality and the adequacy of the generated music.",
"This feedback may be explicit, which puts a task on the user, but it could also be, at least partly, implicit and automated.",
"For instance, the fact that the user quickly stops listening to the music just generated could be interpreted as negative feedback.",
"On the contrary, the fact that the user selects a better rendering after a first quick listen to some initial reproduction could be interpreted as positive feedback.",
"Several approaches are possible.",
"The most straightforward approach, considering the nature of neural networks and supervised learning, would be to add the newly generated musical piece to the training set and eventuallyImmediately, after some time, or after some amount of new feedback, as with a minibatch (see Section REF ).",
"retrain the networkThis could be done in the background..",
"This would reinforce the number of positive examples and gradually update the learnt model and, as a consequence, future generations.",
"However, there is no guarantee that the overall generation quality would improve.",
"This could also lead the model to overfit and loose some generalization.",
"Moreover, there is no direct possibility of negative feedback, as one cannot remove a badly generated example from the dataset because there is almost no chance that it was already present in the dataset.",
"At the junction between adaptability and interactivity, an interesting approach is that of interactive machine learning for music generation, as discussed by Fiebrink and Caramiaux [52].",
"They report on experience with a toolkit they designed, named Wekinator, to allow users to interactively modify the training examples.",
"For instance, they argue in [52] that: “Interactive machine learning can also allow people to build accurate models from very few training examples: by iteratively placing new training examples in areas of the input space that are most needed to improve model accuracy (e.g., near the desired decision boundaries between classes), users can allow complicated concepts to be learned more efficiently than if all training data were representative of future data.” Another approach is to work not on the training dataset but on the generation phase.",
"This leads us back to the issue of control (see Section ), via, for example, a constrained sampling strategy, an input manipulation strategy or, obviously, a reinforcement strategy.",
"The RL-Tuner framework (Section REF ) is an interesting step in this direction.",
"Although the initial motivation for RL-Tuner was to introduce musical constraints on the generation, by encapsulating them into an additional reward, this approach could also be used to introduce user feedback as an additional reward."
],
[
"Explainability",
"A common critique of sub-symbolic approaches of Artificial Intelligence (AI)As opposed to symbolic approaches, see Section REF ., such as neural networks and deep learning, is their black box nature, which makes it difficult to explain and justify their decisions [18].",
"Explainability is indeed a real issue, as we would like to be able to understand and explain what (and how) a deep learning system has learned from a corpus as well as why it ends up generating a given musical content."
],
[
"#1 Example: BachBot Chorale Polyphonic Symbolic Music Generation System",
"Although preliminary, an interesting study conducted with the BachBot system concerns the analysis of the specialization of some of the units (neurons) of the network, through a correlation analysis with some specific motives and progressions.",
"BachBot, by Liang [119], [118], is a system designed to generate chorales in the style of J. S. Bach, an objective shared by DeepBach (Section REF ).",
"All examples from the dataset are aligned onto the same key.",
"The initial representation is piano roll but it is encoded in text, in a similar way to the Celtic melody generation system described in Section REF .",
"One of the specificities of the encoding is the way simultaneous notes are encoded as a sequence of tokens, with a special delimiter symbol “|||” indicating the next time frame, with a constant time step of an eighth note.",
"Actually, a chorale is considered in BachBot as a single-voice polyphony and not as a multivoice polyphony, as for instance in the cases of DeepBach (Section REF ) and MiniBach (Section REF ).",
"Rests are encoded as empty frames.",
"Notes are ordered in a descending pitch and are represented by their MIDI note number, with a boolean indicating if it is tied to a note at the same pitch from previous time stepThis is equivalent to a hold “__” indication.. An example is shown in Figure REF , encoding two successive chords: notes B$_3$ , G$\\sharp _3$ , E$_3$ and B$_2$ , corresponding to a E major with a B as the bass (often notated as E/B in jazz) with the duration of a quarter note, and repeated with a tied note; a fermata, notated as “(.",
")”; and notes A$_3$ , E$_3$ , C$_3$ and A$_2$ , corresponding to a A minor with the duration of an eighth note.",
"Figure: Example of score encoding in BachBot.Reproduced from The architecture is a recurrent network (LSTM).",
"The author used a grid search in order to select the optimal setting for hyperparameters of the architecture (number of layers, number of units, etc.).",
"The selected architecture has three layers and as the author notes in [119]: “Depth matters!",
"Increasing num_layers can yield up to 9% lower validation loss.",
"The best model is 3 layers deep, any further and overfitting occurs.” Generation is done time step by time step, following the iterative feedforward strategy.",
"As for DeepBach (Sections REF and REF ), BachBot may be readapted from the initial 4-voice multivoice chorale generation objective to a melody 3-voice counterpoint accompaniment objective, see details in [119].",
"However, as opposed to DeepBach architecture and representation which stay unchanged, in the case of BachBot both the architecture, the representation temporal scope and the strategy have to be structurally changed from a time step/iterative feedforward generation approach to a global/single-step feedforward generation approach (similar to MiniBach).",
"An interesting preliminary study by the author was the invitation of a musicologist to manually search for possible correlations between unit activation and specific motives and progressions, as shown in Figure REF .",
"Some examples of the correlations foundMore details may be found in [119].",
"are as follows: Neurons 64 and 138 of Layer 1 seem to detect (specifically) perfect cadences (V–I) with root position chords in the tonic key.",
"Neuron 87 of Layer 1 seems to detect an I (first degree) chord on the first downbeat and its reprise four measures later.",
"Neuron 151 of Layer 1 seems to detect A minor cadences that end phrases 2 and 4.",
"Neuron 37 of Layer 2 seems to be looking for I chords: strong peak for a full I and weaker for other similar chords (same bass).",
"Figure: Correlation analysis of BachBot layer/unit activation.Reproduced from BachBot is summarized in Table REF .",
"Table: BachBot summary"
],
[
"#2 Example: deepAutoController Audio Music Generation System",
"In [170], the authors of deepAutoController (Section REF ) discuss the musical effects of different controls over the units of the architecture: “The optimal parameters of the models were mostly inhibitory.",
"Therefore the deactivation of a unit in a hidden layer yields a denser mixture of sounds at the output.",
"Learning to play such an interface may prove difficult for new users, as one typically expects the opposite behavior from a musical synthesizer.",
"$<$ ...$>$ We explored models having non-negative weights by using an asymmetric weight decay as shown in [116].",
"The results are not presented here as they are preliminary.",
"Reconstruction error in such models is worse than without non-negativity constraints.",
"But we find informally that the models are somewhat more intuitive to play as synthesizers.”"
],
[
"Towards Automated Analysis",
"The two previous examples in Sections REF and REF are examples of a preliminary manual correlation analysis.",
"Meanwhile, an active area of research relates to the understanding of the way deep learning architectures work and the explanation of their predictions or decisions via automated analyses.",
"An example of such an approach is using saliency maps with the three following categoriesFollowing Kindermans et al.",
"'s study in [100], actually a critique of the reliability of saliency methods.",
": gradient sensitivity, to estimate how a small change to the input can affect the classification task (see, for example, [6]); signal methods, to isolate input patterns that stimulate neuron activation in higher layers (see, for example, [101]); and attribution methods, to decompose the value at a specific output neuron into contributions from the individual input dimensionsWith an approach analog to reverse correlation, which is used in neurophysiology for studying how sensory neurons add up signals from different sources and sum up stimuli at different times to generate a response (see, for example, [159]).",
"(see, for example, [136]).",
"Note that this type of analysis could also be used with a different objective: to optimize the configuration of the architecture by removing components that are considered to make no contribution and are therefore unnecessary, see, for example, [114] (with its provocative title).",
"Last, let us mention some recent work targeted for image recognition which are showing interesting direction and prospects, like for instance to interactively explore activation atlases of the features the network has learnedUsing a first processing stage inspired by Deep Dream feature visualization by optimization, see Section REF .",
"[17] or to automatically explore incorrect behaviors by generating test counterexamples [155]."
],
[
"Discussion",
"We can observe that the various limitations and challenges that we have analyzed may be partially dependent on one another and, furthermore, conflicting.",
"Thus, resolving one may damper another.",
"For instance, the sampling strategy used by DeepBach (Section REF ) provides incrementality but the length of the generated music is fixed, whereas the iterative feedforward strategy allows variable and unbounded length but incrementality is only forward in time.",
"There is probably no general solution and, as for multicriteria decisions, the selection of architectures and strategies depends on preferences and priorities.",
"Also, as already noted in Section REF , there is no guarantee that combining a variety of different architectures and/or strategies will make a sound and accurate system.",
"As for a good cook, the best outcome is not achieved by simply mixing together all the possible ingredients.",
"Therefore, it is important to continue to deepen our understanding and to explore solutions as well as their possible articulations and combinations.",
"We hope that the survey and analysis conducted in this chapter and in the two next chapters provide a contribution to this understanding."
],
[
"Analysis",
"*Chapter Analysis summarizes the analysis of the various deep learning-based music generation systems considered in this book.",
"For that purpose, various tables are proposed.",
"Correlation tables are also introduced, in order to highlight the relations between the dimensions.",
"We now present a preliminary analysis and summary of the various systems surveyed, following our proposed five dimensions referential, through various tables.",
"This provides material for an analysis of the relations between the different dimensions and the corresponding design decisions."
],
[
"Referencing and Abbreviations",
"At first, we reference in Table REF the various systems that we have analyzed.",
"Then, because of space limitations, we introduce abbreviations for the various possible types for each dimension: objectives in Table REF ; representations in Table REF ; architectures in Table REF ; challenges in Table REF ; and strategies in Table REF .",
"Table: Systems referencingTable: Abbreviations for the types of objectiveTable: Abbreviations for the types of representationTable: Abbreviations for the types of architectureTable: Abbreviations for the types of challengeTable: Abbreviations for the types of strategy"
],
[
"System Analysis",
"We summarize in TablesThis table is split in two because of vertical space limitations.",
"REF and REF how each system is positioned in respect to each of the following four dimensions: objective, representation, architecture and strategy.",
"We then analyze each system in a more detailed manner, dimension by dimension: objective in Table REF ; representation in TablesThis table is split in two because of horizontal space limitations.",
"REF and REF ; architecture and strategy in Table REF ; and challenge in Table REF .",
"For each table, which analyzes each system (line) in respect to the possible types (columns) for a given dimension, the occurrence of an “X” at the crossing of a given line (system) and a given column (type) means that this system does match that given type for that dimension (e.g., follows some representation facet, is based on some type of architecture, fulfills some challenge...).",
"Note that we base this analysis on how each system is presented in the literature referenced, and not as it could be further extended.",
"Furthermore, we use notations such as X$^n$ and X${\\times }n$ (introduced in Section ) to convey additional information about the number of occurrences of a type.",
"Table: Systems summary (1/2)Table: Systems summary (2/2)Table: System ×\\times ObjectiveTable: System ×\\times Representation (1/2)Table: System ×\\times Representation (2/2)Table: System ×\\times Architecture & StrategyNote that, when considering the analysis regarding the challenges in Table REF , we have to keep in mind that the limitations and challenges are not of equal importance and difficulty; and the majority of the systems may be further extended in order to better address some of the challengesFor instance, the RL-Tuner system has the potential for addressing interactivity and adaptability challenges, although, to our knowledge, not yet experimented.. That said, we can see the emergence of some divide between systems using a global versus a time step temporal scope representation (see Section REF ), depending on the following requirements: ex nihilo generation and length variability.",
"This will be further discussed in Section ."
],
[
"Correlation Analysis",
"The last series of tables analyse some correlations between the dimensions: representation with respect to the objective in Table REF ; architecture and strategy with respect to the objective in Table REF ; objective, architecture and strategy with respect to the representation in Table REF ; strategy with respect to the architecture in Table REF ; and architecture and strategy with respect to the challenge in Table REF .",
"The occurrence of an “X” at the crossing of a given line (a given type for the first dimension) and a given column (a given type for the second dimension) means that there is (at least) a systemWithin the set of systems analyzed in the book.",
"which matches both types (for instance, is based on a particular architecture and follows a particular strategy).",
"However, the absence of an “X” does not mean that the two types are incompatible or that such a system does not exist or may not be constructed.",
"Therefore, we add an additional “x” notation to represent an a priori potential compatibility between types and furthermore a possible direction to be explored.",
"Table: Representation ×\\times ObjectiveTable: Architecture & Strategy ×\\times ObjectiveTable: Objective & Architecture & Strategy ×\\times RepresentationTable: Strategy ×\\times ArchitectureTable: Architecture & Strategy ×\\times ChallengeThe analysis of the correlation tables allows us to draw a few first observations about some design decisions and their consequences: audio versus symbolic representation (Table REF ); global versus time step temporal scope representation (Tables REF and REF ).",
"A global temporal scope representation is usually coupled with: a) a feedforward architecture and a single-step feedforward strategy, or b) an autoencoder architecture with a decoder feedforward strategy.",
"A time step temporal representation is usually coupled with a recurrent architecture and an iterative feedforward strategy.",
"However, there are some exceptions (e.g., Time-Windowed in Section REF and BLSTM in Section REF ); and accompaniment objective versus seed-based generation (Table REF ).",
"DeepBach is a notable exception, because thanks to its sampling only strategy, it can generate an accompaniment as well as a complete chorale (see the discussion in Section REF ).",
"Note that in Table REF , the columns corresponding to the expressiveness and explainability challenges are left empty.",
"This is because these challenges are not to be solved with the lone choice of an architecture or a strategy.",
"We do not comment further these tables in this book.",
"We consider them as a first version of analysis tools related to our proposed conceptual framework which could be tried out and improved, for investigating current as well as future systems."
],
[
"Discussion and Conclusion",
"*Chapter Discussion and Conclusion is the last chapter of the book.",
"It revisits some design decision issues introduced during the analysis of challenges and strategies and it discusses some related prospects, before concluding by wrapping up the contributions presented in this book.",
"We now revisit some design decision issues raised through our analysis and discuss related prospects."
],
[
"Global versus Time Step",
"As we have seen in Sections and REF , one important decision is to choose between the two main types of temporal scope representation: global, including all time steps – typically coupled with a feedforward or an autoencoder architecture; and time step, representing a single time step – typically coupled with a recurrent (neural network) (RNN) architectureIn general, the granularity of the time step is set at the level of the smallest notre duration, as discussed in Section REF .",
"Time-Windowed and BLSTM architectures (respectively, Sections REF and REF ) are both peculiar cases of a coarse grained time step (respectively, one and four measures long).",
"Time-Windowed architecture has the additional specificity that it is a feedforward and not a recurrent architecture..",
"The pros and cons are as follows: global [+/–] + allows arbitrary output, e.g., for the objective of generating some accompaniment through the single-step feedforward strategy on a feedforward architecture, as, for example, in the MiniBach system (Section REF ); + supports incremental instantiation (via sampling), as, for example, by the DeepBach system (Section REF ); – does not allow variable length generation; – does not allow seed-based generation for a feedforward architecture; + allows seed-based generation through the decoder feedforward strategy on an autoencoder architecture, as, for example, in the DeepHear system (Section REF ); time step [+/–] +/– supports incremental instantiation, but only forward in time, through the iterative feedforward strategy on a recurrent network architecture; + allows variable length generation, as, for example, in the CONCERT system (Section REF ).",
"Actually some attempt at combining “the best of both worlds” seems to lie in using an RNN Encoder-Decoder architecture, as generation is iterative, which allows variable length content generation, while allowing arbitrary output generation, as the output sequence may have an arbitrary length and contentAs is the case for translation tasks., and allowing the manipulation of a global temporal scope representation (the latent variables).",
"Moreover, in a variational version, such as, for example, VRAE, the latent space could be explored in a disciplined and meaningful manner, as, for example, in the GLSR-VAE and MusicVAE systems (Sections REF and REF ).",
"Note also that an alternative to a recurrent architecture is a convolutional architecture applied over the time dimension, as discussed in the (next) Section ."
],
[
"Convolution versus Recurrent",
"As noted in Section , convolutional architectures, while prevalent for image applications, are more seldom used than recurrent neural network (RNN) architectures in music applications.",
"The few examples of nonrecurrent architectures using convolution on the time dimension that we have encountered and analyzed are WaveNet, a convolutional feedforward architecture for audio (Section REF ); MidiNet, a GAN architecture encapsulating conditional convolutional architectures (Section REF )A comparison between MidiNet and C-RNN-GAN, both using GANs but encapsulating a convolutional network versus a recurrent network, is also interesting.",
"; and C-RBM, a convolutional RBM (Section REF ).",
"Let us try to list and analyze the relative pros of cons of using recurrent architectures or convolutional architectures to model time correlations: recurrent networks are popular and accurate, especially since the arrival of LSTM architectures; convolution should be used a priori only on the time dimension because, as opposed to images where motives are invariant in all dimensions, in music the pitch dimension is a priori not metrically invariantOtherwise this could break the notion of tonality, see the rationale for the C-RBM system in Section REF .",
"However, the system analyzed in Section REF considers recurrence on the pitch class dimension in order to model simultaneous notes (chords).",
"; convolutional networks are typically faster to train and easier to parallelize than recurrent networks [195]; using convolution on the time dimension in place of using a recurrent network implies the multiplication of the number of input variables by the number of time steps considered and thus leads to a significant augmentation of the volume of data to process and of the number of parameters to adjustFor that reason, recurrent networks are still the norm for learning time series of multi-dimensional data like, for example, 2-D images for weather prediction.",
"; sharing weights by convolutions only applies to a small number of temporal neighboring members of the input, in contrast to a recurrent network that shares parameters in a deep way, for all time steps (see Section ); the authors of WaveNet argue that the layers of dilated convolutions allow the receptive field to grow longer in a much cheaper way than using LSTM units; the authors of MidiNet argue that using a conditioning strategy for a convolutional architecture allows the incorporation of information from previous measures into intermediate layers and therefore considers history as a recurrent network would do.",
"A potential output of this initial comparative analysis is that, as there are many systems using nonrecurrent architectures (like feedforward networks or autoencoders), it may be interesting to study whether their extension into a convolutional architecture on the time dimension could bring some effective gain, in terms of efficiency and/or accuracy.",
"Actually, this issue of using convolutional versus recurrent architecture is recently being challenged by the introduction of a novel architecture, named Transformer [198], with an objective similar to that of a RNN Encoder-Decoder architecture (introduced in Section REF ).",
"This architecture does not use convolutions or recurrence and is only based on an attention mechanism (Section REF ).",
"A very recentToo late to analyze it thoroughly in this book.",
"application to music generation, named MusicTransformer has been presented in [89], with apparent very good results about its capacity to model long-term structure."
],
[
"Style Transfer and Transfer Learning",
"Transfer learning is an important issue for deep learning and machine learning in general.",
"As training can be a tedious process, the issue is to be able to reuse, at least partially, what has been learnt in one context and use it in other contexts.",
"Various cases may be considered, e.g., similar source and target domains, similar task, etc.",
"This new research subdomain, named transfer learning, is about methodologies and techniques for the transfer of what has been learnt [63].",
"We have not addressed this important issue in our analysis because it has not yet been specifically addressed for music generation, although we think that it will become an area of investigation.",
"Meanwhile, an example, although still simplistic, is the way the DeepHear architecture and what it has learnt is transfered from the objective of generating a melody to the objective of generating a counterpoint (see Section REF ).",
"Another example is the way the DeepBach architecture allows to adapt the objective from ex nihilo chorale generation to multi-voice accompaniment (see Section REF ).",
"Last, let us remember that style transfer is a very specific case of transfer learning in terms of objective and techniques (see Section REF )"
],
[
"Cooperation",
"All the systems surveyed are basically lone systems (although the architecture may be compound).",
"A more cooperative approach is natural for handling complexity, heterogeneity, scalability and openness, as, for example, pioneered by multi-agent systems [207].",
"An example is the system proposed by Hutchings and McCormack [91].",
"It is composed of two agents: a harmony agent, based on an RNN (LSTM) architecture, in charge of the progression of chords; and a melody agent, based on a rule-based system, in charge of the melody.",
"The two agents work in a cooperative way and alternate between leading and accompanying roles (inspired by, for example, the way musicians function in a jazz band).",
"The authors relate the interesting dynamics between the two agents and also an intereresting balance between harmonic creativity and harmonic consistencyOn this issue, see Section ..",
"This approach appears to be an interesting direction to pursue and extend with more agents and roles."
],
[
"Specialization",
"A general issue is the hyper-specialization of systems designed for a specific objective and/or a specific type of corpus.",
"This is witnessed by the diversity of the architectures and approaches surveyed.",
"Note that this is a known issue for Artificial Intelligence (AI) research in general.",
"There is some tendency towards hyper-specialized systems solving specific problems, especially in the case of competitions organized by conferences or other institutions, with the risk of loosing the initial objective of a general problem solving frameworkAn interesting counterexample is the ongoing research and competition about general game playing [59]..",
"Meanwhile, the general objective of generating interesting musical content is complex and still an opened issue.",
"Thus, we need to work both on general approaches for general problems and specific approaches for specific subproblems, as well as top-down and bottom-up approaches, while not losing interest in how to interpret, generalize and reuse advances and lessons learntWe hope that the survey and analysis conducted in this book will contribute to this research agenda.."
],
[
"Evaluation and Creativity",
"Evaluation of a system generating music mostly consists in a qualitative evaluation of examples of generated musicAbout the possibility for more systematic objective criteria for evaluation, we can for example look at the analysis by Theis et al.",
"for the case of image generation [187].",
"The authors state that an evaluation of image generative models is multicriteria via different possible metrics, such as log-likelihood, Parzen window estimates, or qualitative visual fidelity, and that a good result with respect to one criterion does not necessarily imply a good result with respect to another criterion.. For many experiments, evaluation is only preliminary, and in many cases, only conducted by the designers themselves.",
"There are of course some exceptions, with more systematic and external evaluations (by a more or less expert public).",
"When the corpus is very precise, e.g., in the case of J. S. Bach's chorales for the BachBot or the DeepBach systems (Sections REF and REF ), a Turing test may be conducted: a piece of music being presented to the public who has to guess if it is one of the original pieces or music generated by a computer.",
"But this methodology is more limited when the objective is not to generate music highly conformant to a relatively narrow style (and corpus), as in the case of J. S. Bach choralesBach chorales, and more generally speaking Bach music, are often used for experiments and evaluation, because the corpus is quite homogeneous regarding a given style (e.g., preludes, chorales...) as well as quality.",
"It also fits particularly well with algorithmic composition, of which Bach was somehow a precursor., but to generate more creative music.",
"Moreover, if we consider as a general objective for a system the capacity to assist composers and musiciansAs, for instance, pioneered by the FlowComposer prototype, introduced in Section REF ., rather than to autonomously generate music (see Section REF ), we should maybe consider as an evaluation criteria the satisfaction of the composer (notably, if the assistance of the computer allowed him to compose and create music that he may consider not having been possible otherwise), rather than the satisfaction of the auditors (who remain too often guided by some conformance to a current musical trend).",
"Some fundamental limitation is that there is no clear objective function associated to creativity and to art quality.",
"Therefore, the selection of a musical corpus used as a set of training examples is a first fundamental step and decisionAs noted in Section ..",
"But in order to be able to learn something interesting, a relatively coherent/homogenous corpus needs to be selectedConstructing a corpus with the “best considered” musical pieces, independently of the style (classical, jazz, pop, etc.)",
"– as could do a museum or an exhibition presenting in a single room its best artefacts of different nature and origin –, is not likely to produce interesting results because such a corpus is too much sparse and heterogeneous..",
"This will unfortunately favor the quality (actually, the conformance) of the generated musical content regarding the learnt style, rather than its intrinsic quality (interest).",
"Current experiments and directions to promote creativity rely mostly on constraints to avoid plagiarism and/or heuristics to incentive a generation outside the “comfort zone” that the deep architecture has learned from the corpus, while balancing elements of surprise with predictability/understandability.",
"Such creativity control may be applied during the training phase (the case of the CAN architecture, see Section REF ), or (for most types of control, see Sections and ) during the generation phase.",
"Some alternative (and complementary) direction to better model such an element of surprise could be to include a model of an artificial listener with some model of expectation, e.g., which could evaluate how well a new generated content could be “explained” in terms of references to an existing memory, as proposed, e.g., in [41].",
"Last, some additional fundamental limitation is that current deep learning techniques for learning and generating music are based on artefacts, actual musical data, independently of the processes and the culture that have led to them.",
"If we want to envision more profound systems, it is likely that we will have to incorporate some modeling of the context and the process leading to musical artefacts and not so the artefacts themselves.",
"Indeed, when considering art history, creation takes place within a historico-cultural context with refinementsE.g., the extension of classical harmony based on triads (only root, third and fifth) to extended chords.",
"as well as possible rupturesE.g., movements like dodecaphonism or free jazz.. One possible direction would then be to not just model content generation from a frozen artistic corpus outside of its history, but to try to model a more dynamical process of creation including the historico-cultural contextThe modeling of the context is one of the limitations of current deep learning architectures and is a topic of ongoing research.",
"An illustrating real counterexample is the case of a Chinese woman (chairwoman of China's biggest air conditioners maker) who had found her face displayed in 2018 in the port city of Ningbo on a huge screen that displays images of people caught jaywalking by surveillance cameras.",
"It was then found that the artificial intelligence-backed monitoring system had captured her face from an advertisement on the side of a moving bus [184]., with the history and dynamics maybe addressed by recurrent architectures and/or reinforcement learning, although this appears yet a long way to go."
],
[
"Conclusion",
"The use of deep learning techniques for the creation of musical content, and more generally speaking creative artistic content, is nowadays getting increased attention.",
"This book presented a survey and an analysis of various strategies and techniques for using deep learning to generate musical content.",
"We have proposed a multi-criteria conceptual framework based on five dimensions: objective, representation, architecture, challenge and strategy.",
"We have analyzed and compared various systems and experiments proposed by various researchers in the literature.",
"We hope that the conceptual framework provided in this book will help in understanding the issues and in comparing various approaches for using deep learning for music generation, and therefore contribute to this research agenda."
]
] | 1709.01620 |
[
[
"Effectiveness of Anonymization in Double-Blind Review"
],
[
"Abstract Double-blind review relies on the authors' ability and willingness to effectively anonymize their submissions.",
"We explore anonymization effectiveness at ASE 2016, OOPSLA 2016, and PLDI 2016 by asking reviewers if they can guess author identities.",
"We find that 74%-90% of reviews contain no correct guess and that reviewers who self-identify as experts on a paper's topic are more likely to attempt to guess, but no more likely to guess correctly.",
"We present our findings, summarize the PC chairs' comments about administering double-blind review, discuss the advantages and disadvantages of revealing author identities part of the way through the process, and conclude by advocating for the continued use of double-blind review."
],
[
"Introduction",
"Peer review is a cornerstone of the academic publication process but can be subject to the flaws of the humans who perform it.",
"Evidence suggests that subconscious biases influence one's ability to objectively evaluate work: In a controlled experiment with two disjoint program committees, the ACM International Conference on Web Search and Data Mining (WSDM'17) found that reviewers with author information were $1.76\\times $ more likely to recommend acceptance of papers from famous authors, and $1.67\\times $ more likely recommend acceptance of papers from top institutions [6].",
"A study of three years of the Evolution of Languages conference (2012, 2014, and 2016) found that, when reviewers knew author identities, review scores for papers with male first authors were 19% higher, and for papers with female first authors 4% lower [4].",
"In a medical discipline, US reviewers were more likely to recommend acceptance of papers from US-based institutions [2].",
"These biases can affect anyone, regardless of the evaluator's race and gender [3].",
"Luckily, double-blind review can mitigate these effects [2], [6], [1] and reduce the perception of bias [5], making it a constructive step toward a review system that objectively evaluates papers based strictly on the quality of the work.",
"Three conferences in software engineering and programming languages held in 2016 — the IEEE/ACM International Conference on Automated Software Engineering (ASE), ACM International Conference on Object-Oriented Programming, Systems, Languages, and Applications (OOPSLA), and the ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI) — collected data on anonymization effectiveness, which weSven Apel and Sarfraz Khurshid were the ASE'16 PC chairs, Claire Le Goues and Yuriy Brun were the ASE'16 review process chairs, Yannis Smaragdakis was the OOPSLA'16 PC chair, and Emery Berger was the PLDI'16 PC chair.",
"use to assess the degree to which reviewers were able to successfully deanonymize the papers' authors.",
"We find that anonymization is imperfect but fairly effective: 70%–86% of the reviews were submitted with no author guesses, and 74%–90% of reviews were submitted with no correct guesses.",
"Reviewers who believe themselves to be experts on a paper's topic were more likely to attempt to guess author identities but no more likely to guess correctly.",
"Overall, we strongly support the continued use of double-blind review, finding the extra administrative effort minimal and well-worth the benefits."
],
[
"Methodology",
"The authors submitting to ASE 2016, OOPSLA 2016, and PLDI 2016 were instructed to omit author information from the author block and obscure, to the best of their ability, identifying information in the paper.",
"PLDI authors were also instructed not to advertise their work.",
"ASE desk-rejected submissions that listed author information on the first page, but not those that inadvertently revealed such information in the text.",
"Authors of OOPSLA submissions who revealed author identities were instructed to remove the identities, which they did, and no paper was desk-rejected for this reason.",
"PLDI desk-rejected submissions that revealed author identities in any way.",
"The review forms included optional questions about author identities, the answers to which were only accessible to the PC chairs.",
"The questions asked if the reviewer thought he or she knew the identity of at least one author, and if so, to make a guess and to select what informed the guess.",
"The data considered here refer to the first submitted version of each review.",
"For ASE, author identities were revealed to reviewers immediately after submission of an initial review; for OOPSLA, ahead of the PC meeting; for PLDI, only for accepted papers, after all acceptance decisions were made.",
"Threats to validity.",
"Reviewers were urged to provide a guess if they thought they knew an author.",
"A lack of a guess could signify not following those instructions.",
"However, this risk is small, e.g., OOPSLA PC members were allowed to opt out uniformly and yet 83% of the PC members participated.",
"Asking reviewers if they could guess author identities may have affected their behavior: they may not have thought about it had they not been asked.",
"Data about reviewers' confidence in guesses may affect our conclusions.",
"Reviewers could submit multiple guesses per paper and be considered correct if at least one guess matched, so making many uninformed guesses could be considered correct, but we did not observe this phenomenon.",
"In a form of selection bias, all conferences' review processes were chaired by, and this article is written by, researchers who support double-blind review."
],
[
"Anonymization Effectiveness",
"For the three conferences, 70%–86% of reviews were submitted without guesses, suggesting that reviewers typically did not believe they knew or were not concerned with who wrote most of the papers they reviewed.",
"Figure REF summarizes the number of reviewers, papers, and reviews processed by each conference, and the distributions of author identity guesses.",
"Figure: Guess rate, and correct guess rate, by self-reported reviewer expertisescore (X: expert, Y: knowledgable, Z: informed outsider).When reviewers did guess, they were more likely to be correct (ASE 72% of guesses were correct, OOPSLA 85%, and PLDI 74%).",
"However, 75% of ASE, 50% of OOPSLA, and 44% of PLDI papers had no reviewers correctly guess even one author, and most reviews contained no correct guess (ASE 90%, OOPSLA 74%, PLDI 81%).",
"Are experts more likely to guess and guess correctly?",
"All reviews included a self-reported assessment of reviewer expertise (X for expert, Y for knowledgable, and Z for informed outsider).",
"Figure REF summarizes guess incidence and guess correctness by reviewer expertise.",
"For each conference, X reviewers were statistically significantly more likely to guess than Y and Z reviewers ($p \\le 0.05$ ).",
"But the differences in guess correctness were not significant, except the Z reviewers for PLDI were statistically significantly correct less often than the X and Y reviewers ($p \\le 0.05$ ).",
"We conclude that reviewers who considered themselves experts were more likely to guess author identities, but were no more likely to guess correctly.",
"Are papers frequently poorly anonymized?",
"One possible reason for deanonymization is poor anonymization.",
"Poorly anonymized papers may have more reviewers guess, and also a higher correct guess rate.",
"Figure REF shows the distribution of papers by the number of reviewers who attempted to guess the authors.",
"The largest proportion of papers (26%–30%) had only a single reviewer attempt to guess.",
"Fewer papers had more guesses.",
"The bar shading indicates the fractions of the author identity guesses that are correct; papers with more guesses have lower rates of incorrect guesses.",
"Combining the three conferences' data, the $\\chi ^2$ statistic indicates that the rates of correct guessing for papers with one, two, and three or more guesses are statistically significantly different ($p \\le 0.05$ ).",
"This comparison is also statistically significant for OOPSLA alone, but not for ASE and PLDI.",
"Comparing guess rates (we use one-tailed $z$ tests for all population proportion comparisons) between paper groups directly: For OOPSLA, the rate of correct guessing is statistically significantly different between one-guess papers and each of the other two paper groups.",
"For PLDI, the same is true between one-guess and three-plus-guess paper groups.",
"This evidence suggests that a minority of papers may be easy to unblind.",
"For ASE, only 1.5% of the papers had three or more guesses, while for PLDI, 13% did.",
"However, for PLDI, 40% of all the guesses corresponded to those 13% of the papers, so improving the anonymization of a relatively small number of papers would potentially significantly reduce the number of guesses.",
"Since the three conferences only began using the double-blind review process recently, the occurrences of insufficient anonymization are likely to decrease as authors gain more experience with anonymizing submissions, further increasing double-blind effectiveness.",
"Figure: Distributions of papers by number of guesses.",
"Thebar shading indicates the fraction of the guesses that are correct.Figure: Acceptance rate of papers by reviewer guessing behavior.Are papers with guessed authors more likely to be accepted?",
"We investigated if paper acceptance correlated with either the reviewers' guesses or with correct guesses.",
"Figure REF shows the acceptance rate for each conference for papers without guesses, with at least one correct guess, and with all incorrect guesses.",
"We observed different behavior at the three conferences: ASE submissions were accepted at statistically the same rate regardless of reviewer guessing behavior.",
"Additional data available for ASE shows that for each review's paper rating (strong accept, weak accept, weak reject, strong reject), there is no statistically significant differences in acceptance rates for submissions with different guessing behavior.",
"OOPSLA and PLDI submissions with no guesses were less likely to be accepted ($p\\le 0.05$ ) than those with at least one correct guess.",
"PLDI submissions with no guesses were also less likely to be accepted ($p \\le 0.05$ ) than submissions with all incorrect guesses (for OOPSLA, for the same test, $p = 0.57$ ).",
"One possible explanation is that OOPSLA and PLDI reviewers were more likely to affiliate work they perceived as of higher-quality with known researchers, and thus more willing to guess the authors of submissions they wanted to accept.",
"How do reviewers deanonymize?",
"OOPSLA and PLDI reviewers were asked if the use of citations revealed the authors.",
"Of the reviews with guesses, 37% (11% of all reviews) and 44% (11% of all reviews) said they did, respectively.",
"The ASE reviewers were asked what informed their guesses.",
"The answers were guessing based on paper topic (75 responses); obvious unblinding via reference to previous work, dataset, or source code (31); having previously reviewed or read a draft (21); or having seen a talk (3).",
"The results suggest that some deanonymization may be unavoidable.",
"Some reviewers discovered GitHub repositories or project websites while searching for related work to inform their reviews.",
"Some submissions represented clear extensions of or indicated close familiarity with the authors' prior work.",
"However, there also exist straightforward opportunities to improve anonymization.",
"For example, community familiarity with anonymization, consistent norms, and clear guidelines could address the incidence of direct unblinding.",
"However, multiple times at the PC meetings, the PC chairs heard a PC member remark about having been sure another PC member was a paper author, but being wrong.",
"Reviewers may be overconfident, and sometimes wrong, when they think they know an author through indirect unblinding."
],
[
"PC Chairs' Observations",
"After completing the process, the PC chairs of all three conferences reflected on the successes and challenges of double-blind review.",
"All PC chairs were strongly supportive of continuing to use double-blind review in the future.",
"All felt that double-blind review mitigated effects of (subconscious) bias, which is the primary goal of using double-blind review.",
"Some PC members also felt so, indicating anecdotally that they were more confident that their reviews and decisions had less bias.",
"One PC member remarked that double-blind review is liberating, since it allows for evaluation without concern about the impact on the careers of people they know personally.",
"All PC chairs have arguments in support of their respective decisions on the timing of revealing the authors (i.e., after review submission, before PC meeting, or only for accepted papers).",
"The PLDI PC chair advocated strongly for full double-blind, which enables rejected papers to be anonymously resubmitted to other double-blind venues with common reviewers, addressing one cause of deanonymization.",
"The ASE PC chairs observed that in a couple of cases, revealing author identities helped to better understand a paper's contribution and value.",
"The PLDI PC chair revealed author identities on request, when deemed absolutely necessary to assess the paper.",
"This happened extremely rarely, and could provide the benefit observed by the ASE PC chairs without sacrificing other benefits.",
"That said, one PC member remarked that one benefit of serving on a PC is learning who is working on what; full anonymization eliminates learning the who, though still allows learning the what.",
"Overall, none of the PC chairs felt that the extra administrative burden imposed by double-blind review was large.",
"The ASE PC chairs recruited two review process chairs to assist, and all felt the effort required was reasonable.",
"The OOPSLA PC chair noted the level of effort required to implement double-blind review, including the management of conflicts of interest, was not high.",
"He observed that it was critical to provide clear guidance to the authors on how to anonymize papers.",
"(e.g., http://2016.splashcon.org/track/splash-2016-oopsla#FAQ-on-Double-Blind-Reviewing).",
"PLDI allowed authors to either anonymize artifacts (e.g., source code) or to submit non-anonymized versions to the PC chair, who distributed to reviewers when appropriate, on demand.",
"The PC chair reported that this presented only a trivial additional administrative burden.",
"The primary source of additional administration in double-blind review is conflict of interest management.",
"This task is simplified by conference management software that straightforwardly allows authors and reviewers to declare conflicts based on names and affiliations, and chairs to quickly cross-check declared conflicts.",
"ASE PC chairs worked with the CyberChairPro maintainer to support this task.",
"Neither ASE nor OOPSLA observed unanticipated conflicts discovered when author identities were revealed.",
"The PLDI PC chair managed conflicts of interest more creatively, creating a script that validated author-declared conflicts by emailing PC members lists of potentially-conflicted authors mixed with a random selection of other authors, and asking the PC member to identify conflicts.",
"The PC chair examined asymmetrically declared conflicts and contacted authors regarding their reasoning.",
"This identified erroneous conflicts in rare instances.",
"None of the PC chairs found identifying conflicts overly burdensome.",
"The PLDI PC chair reiterated that the burden of full double-blind reviewing is well worth maintaining the process integrity throughout the entire process, and for future resubmissions."
],
[
"Conclusions",
"Data from ASE 2016, OOPSLA 2016, and PLDI 2016 suggest that, while anonymization is imperfect, it is fairly effective.",
"The PC chairs of all three conferences strongly support the continued use of double-blind review, find it effective at mitigating (both conscious and subconscious) bias in reviewing, and judge the extra administrative burden to be relatively minor and well-worth the benefits.",
"Technological advances and the now developed author instructions reduce the burden.",
"Having a dedicated organizational position to support double-blind review can also help.",
"The ASE and OOPSLA PC chairs point out some benefits of revealing author identities mid-process, while the PLDI PC chair argues some of those benefits can be preserved in a full double-blind review process that only reveals the author identities of accepted papers, while providing significant additional benefits, such as mitigating bias throughout the entire process and preserving author anonymity for rejected paper resubmissions."
],
[
"Acknowledgments",
"Kathryn McKinley suggested an early draft of the reviewer questions used by OOPSLA and PLDI.",
"This work is supported in part by the National Science Foundation under grants CCF-1319688, CCF-1453474, CCF-1563797, CCF-1564162, and CNS-1239498, and by the German Research Foundation (AP 206/6)."
]
] | 1709.01609 |
[
[
"Adaptively time stepping the stochastic Landau-Lifshitz-Gilbert equation\n at nonzero temperature: implementation and validation in MuMax3"
],
[
"Abstract Thermal fluctuations play an increasingly important role in micromagnetic research relevant for various biomedical and other technological applications.",
"Until now, it was deemed necessary to use a time stepping algorithm with a fixed time step in order to perform micromagnetic simulations at nonzero temperatures.",
"However, Berkov and Gorn have shown that the drift term which generally appears when solving stochastic differential equations can only influence the length of the magnetization.",
"This quantity is however fixed in the case of the stochastic Landau-Lifshitz-Gilbert equation.",
"In this paper, we exploit this fact to straightforwardly extend existing high order solvers with an adaptive time stepping algorithm.",
"We implemented the presented methods in the freely available GPU-accelerated micromagnetic software package MuMax3 and used it to extensively validate the presented methods.",
"Next to the advantage of having control over the error tolerance, we report a twenty fold speedup without a loss of accuracy, when using the presented methods as compared to the hereto best practice of using Heun's solver with a small fixed time step."
],
[
"Introduction",
"Micromagnetic simulations of systems at nonzero temperatures are an increasingly important tool to numerically investigate magnetic systems relevant for technological applications.",
"Historically, the foundations for a description of thermal fluctuations in micromagnetic systems were laid by Brown when he investigated the thermal switching of single-domain particles[2], [3].",
"Today, these particles are used in promising biomedical applications such as disease detection and tumor treatment[4], [5], [6], [7].",
"In order for these applications to be successful, a full understanding of the particles' thermal switching is important.",
"For example, many characterization procedures [8], [9], [10], [11], [12], [13], [14] require this knowledge to accurately determine the particles' properties.",
"Also diagnostic particle imaging [15], [16], [17], [18], [19], [20], [21], [22], [23], and therapeutic applications[24], [25] rely on models of the particles' thermal switching.",
"Currently, these models are often based on approximations that not always take into account that, e.g.",
"the magnetization state in large particles can deviate from a uniform magnetization[26], or the particles might interact with each other via the magnetostatic interaction[27].",
"In such cases, the analytical models do not accurately reflect the true magnetization dynamics of the particles, and one has to rely on numerical models, the most accurate of which are based on a micromagnetic approach [28], [29], [30], [31], [32], [33], [34].",
"Next to their relevance in magnetic nanoparticle research, thermal fluctuations also play an important role in (exchange-coupled) continuous magnetic systems.",
"One technologically relevant example is domain wall motion through a magnetic nanostrip, proposed as the operating principle for the racetrack memory[35], [36], [37] and for logic devices[38], [39], [40], [41], [42].",
"Recently, even smaller magnetization structures, i.e.",
"skyrmions, have been proposed in both memory[43] and logic devices[44].",
"As the information carriers in these devices become smaller, the influence of thermal fluctuations further grows in importance: at such small spatial scales, the thermal stability of the bits themselves starts to become a relevant research question[45].",
"At the same time, their thermal depinning becomes an inherently stochastic process[46].",
"When numerically investigating domain wall motion at low driving forces, the dynamics can only be captured by considering the interplay between thermal fluctuations, the disorder energy landscape of the material and the driving forces.",
"The resulting motion is then called domain wall creep[47].",
"Until now, full micromagnetic simulations are still prohibitively expensive in all but the smallest of such systems[48].",
"Thermal fluctuations are also critical to the design of magnetic storage elements.",
"They do not only determine the data retention limit of any magnetic storage system, but can also influence the read and write process of a MRAM cell.",
"So estimation of read and write errors requires stochastic micromagnetic modeling of the spin valve during the application of the spin-torque current.",
"This very challenging because of large time scales that are involved[49].",
"There exist different theoretical approaches, each with their respective advantages and disadvantages, to study thermally induced magnetization dynamics[32], [50].",
"Following Brown[3], it is possible to derive the Fokker-Planck equation describing the time-dependent probability distribution of the magnetization directions of an ensemble of uniformly magnetized magnetic nanoparticles[3], [32].",
"However, only in the simplest cases, e.g.",
"when the particles' anisotropy axes are aligned with the applied field, an analytical solution can be found.",
"In more complex cases approximations have to be introduced and when considering continuous systems consisting of several exchange-coupled finite difference cells, this approach becomes intractable.",
"Alternatively, thermal fluctuations can be included as a stochastic term in the Landau-Lifshitz-Gilbert (LLG) equation, henceforth named stochastic LLG (sLLG) equation by adding a thermal field term to the effective field.",
"This approach was presented by Lyberatos[51] and is based on the fact that finite difference cells can be considered as dipoles comparable to single-domain particles.",
"This method has the advantage that it is a straightforward extension of the LLG equation.",
"On the other hand, in contrast to the Fokker-Planck approach, the resulting sLLG equation has to be solved many times in order to gather enough data to draw conclusions about averaged quantities.",
"Integrating stochastic differential equations (SDE) requires the use of non-Riemann calculus to be able to deal with the discontinuous thermal field term.",
"A full discussion of this topic lies beyond the scope of this article, and for an excellent introduction we refer to Ref. [32].",
"In general, SDE's are not trivial to numerically integrate, and require specialized methods[52], which are only suited to integrate SDE's written in either their Ito or Stratonovich form, as otherwise a drift term might appear in the solution.",
"When considering variable time stepping algorithms, the complexity further increases[53], [54], sometimes even canceling the relative advantage obtained by using such methods.",
"However, Berkov and Gorn have shown that the drift term in the sLLG equation manifests itself only in the length of the magnetization vector[1] which, in contrast to the Landau-Lifshitz-Bloch equation[55], is held constant.",
"As shown in Ref.",
"[1], this can be seen more clearly when writing the sLLG equation in spherical coordinates.",
"In Cartesian coordinates, numerical noise would build up in this direction if it weren't accounted for by renormalizing the magnetization after each time step to ensure that the drift term does not influence the magnetization dynamics.",
"Consequently, the Ito and Stratonovich interpretation are equivalent for integrating the sLLG equation, enabling the use of higher order solvers to integrate the sLLG eqation[56].",
"Despite this result, in literature one often still finds the recommendation to use the second order Heun's solver (here denoted with “RK12”, because it is a second order Runge-Kutta type solver with embedded first order solution) with a very small fixed time step of the order of femtoseconds to simulate micromagnetic systems at finite temperatures[57], [58], [59].",
"In this paper, we present the use of higher order solvers with adaptive time stepping for such simulations and show that this method offers significant advantages."
],
[
"Methods",
"The Landau-Lifshitz-Gilbert (LLG) equation[60], [61] contains a precession and a damping term, and describes the magnetization dynamics at the nanometer length scale and the picosecond timescale.",
"$\\dot{\\mathbf {m}} = -\\frac{\\gamma }{1+\\alpha ^2} \\left( \\mathbf {m}\\times \\mathbf {B}_\\mathrm {eff}+\\alpha \\mathbf {m}\\times \\left( \\mathbf {m}\\times \\mathbf {B}_\\mathrm {eff}\\right) \\right) $ In this equation, $\\gamma $ denotes the gyromagnetic ratio, $\\alpha $ the dimensionless Gilbert damping parameter and $\\mathbf {B}_\\mathrm {eff}$ the effective field.",
"There is more than one way to add thermal fluctuations to the LLG equation.",
"We choose to add a stochastic thermal field $B_\\mathrm {therm}$ as a contribution to the effective field term in both the precession and damping term, although it has been shown that the field contribution could also be omitted in the damping term if the size of the thermal field is rescaled adequately[58].",
"A second common option[32] is to add thermal fluctuation directly as an extra thermal torque term to the LLG equation.",
"The properties of the thermal field $B_\\mathrm {therm}$ were determined by Brown when he investigated the thermal switching of single-domain particles[3].",
"Later, it was realized that this theory was also applicable to micromagnetic simulations as each finite difference cell can be considered as such a particle [51].",
"The thermal field is given by $\\langle \\mathbf {B}_\\mathrm {therm} \\rangle =0\\\\\\langle \\mathbf {B}_{\\mathrm {therm},i}(t)\\mathbf {B}_{\\mathrm {therm},j}(t^{\\prime }) \\rangle =q\\delta (t-t^{\\prime })\\delta _{ij}\\\\q=\\frac{2k_\\mathrm {B}T\\alpha }{M_\\mathrm {s}\\gamma V}$ Here, the operator $\\langle \\cdot \\rangle $ denotes a time average, $\\langle \\cdot \\cdot \\rangle $ a correlation, $\\delta $ the Dirac delta function and the indices $i$ and $j$ run over the $x$ , $y$ and $z$ axes in a Cartesian coordinate system.",
"The thermal field has zero average [Eq.",
"(REF )], is uncorrelated in time and space [Eq.",
"()] and its size $q$ is given by Eq.",
"().",
"In this equation, $k_\\mathrm {B}$ denotes the Boltzmann constant, $T$ the temperature, $M_\\mathrm {s}$ the saturation magnetization, and $V$ the volume on which the thermal fluctuations act, i.e.",
"the volume of a single finite difference cell.",
"Equations (REF ) to () are determined such that the effect of the thermal fluctuations is independent of the spatial discretization used: when splitting up a volume into subvolumes and averaging the thermal fluctuations within those, one will recover the same resulting dynamics as in the undivided volume.",
"The same is also true for the time step $\\Delta t$ : when averaged out over a larger time, thermal fluctuations decrease in strength and again, this proportionality is determined such that the average dynamics do not depend on the time discretization."
],
[
"Implementation in MuMax$^3$ ",
"MuMax$^3$ is a GPU-accelerated micromagnetic software package which numerically solves the LLG equation using a finite-difference discretization[59].",
"The thermal field is included in the effective field as $\\mathbf {B}_\\mathrm {therm} = \\eta \\sqrt{ \\frac{2\\alpha k_\\mathrm {B} T}{M_\\mathrm {s}\\gamma V\\Delta t} }$ where $\\Delta t$ denotes the time step and $\\eta $ is a random vector drawn from a standard normal distribution whose value is redetermined after every time step[57], [59].",
"MuMax$^3$ provides several explicit Runge-Kutta methods to time step the LLG equation, the details of which can be found in Ref. [59].",
"Here, we will only mention the ones relevant for this work.",
"Previously, simulations at nonzero temperatures in MuMax$^3$ were performed with the widely used Heun's method (RK12), using a very small time step of the order of 5 fs.",
"In Section , we will validate our results by comparing them to the solution obtained with this solver when there are no analytical solutions available.",
"For dynamical simulations at zero temperature, the default solver is the Dormand-Prince method (RK45).",
"This solver offers 5th order error convergence and contains an embedded 4th order method to estimate the error.",
"Generally speaking (i.e.",
"when not investigating very fast dynamics, or in the absence of thermal fluctuations), it is not advantageous to implement even higher-order solvers.",
"The reason for this is threefold: 1) For the moderately small torques encountered in typical micromagnetic simulations, the performance of the solver is limited by its stability regime.",
"This means that using even slightly larger time steps will result in much larger errors no matter how small the exerted torques are.",
"2) Higher order methods typically need more intermediate torque evaluations per time step, thus reducing the advantage obtained by taking a larger time step.",
"3) Due to the memory required to store the results of the intermediate torque evaluations, higher-order solvers disproportionately increase the memory consumption compared to the obtained gain in performance."
],
[
"Sixth order Runge-Kutta-Fehlberg solver",
"Due to the large size of the stochastic thermal field, simulations at nonzero temperatures require much smaller time steps, so that the solver performance is not longer limited by its stability regime, and large enough time steps can be taken to justify the additional intermediate torque evaluations.",
"Therefore, we also implemented the 6th order Runge-Kutta-Fehlberg (RKF56) method with 5th order embedded solution shown in Table REF .",
"Table: Butcher tableau of the Runge-Kutta-Fehlberg (RKF56) solver with sixth order solution and embedded 5th order solution.",
"The difference between both, used as error estimate, is given in the last row.Unlike the RKF56 solver, some of the solvers used in MuMax$^3$ like the RK45 method, benefit from the first-same-as-last (FSAL) property.",
"In these solvers, the last torque evaluation of the current time step corresponds to the first evaluation of the next time step, thus effectively reducing the number of evaluations per step by 1.",
"However, as the stochastic thermal field is not constant in between time steps, the torque continuity requirement is not longer fulfilled, and the first and last torque evaluations have to be performed separately.",
"Because the RKF56 solver never has the FSAL property, its performance can only compete with these other methods at nonzero temperatures, where the other methods do not benefit from the FSAL property either.",
"The implementation of the Sixth order Runge-Kutta-Fehlberg solver is subject to the same tests used for the other solvers implemented in MuMax$^3$ [59], and a full report is beyond the scope of this article.",
"Nonetheless, Fig.",
"REF shows that our solution to standard problem 4, proposed by the $\\mu $ Mag modeling group[63], agrees with the solution obtained with OOMMF[64] (not using the RK56 solver).",
"Figure: The solutions to standard problem 4, obtained by MuMax 3 ^3 (gray points) and OOMMF (red lines).",
"This problem focuses on micromagnetic dynamics and looks at the time evolution of the magnetization during the relaxation of a magnetic rectangle from an initial s-state.",
"The problem is run for two different applied fields (top and bottom graph) and the space dependent magnetizations when 〈m x 〉\\langle m_\\mathrm {x}\\rangle crosses zero are shown below, in the left and right plot, respectively.One of the most sensitive checks one can perform to test the implementation of a solver is to investigate the scaling of its error convergence.",
"Figure REF shows the error $\\epsilon $ after a single precession without damping of a single spin in a field of 0.1 T as function of the time step $\\Delta t$ .",
"The solver shows the expected sixth order convergence up to the limit of the single precision implementation[59] ($\\epsilon \\approx 10^{-7}$ ).",
"Figure: The error ϵ\\epsilon as function the time step Δt\\Delta t of the RKF56 solver (gray squares) displays the sixth order convergence (red line)."
],
[
"Time stepping with adaptive time steps",
"When performing simulations at nonzero temperatures, it is important to note that the size of the thermal field is determined by 1/$\\sqrt{\\Delta t}$ .",
"This implies that, when a large thermal field is generated leading to a bad step (defined as a step where the torque was too large for the used time step), the step will be undone, and the adaptive time step algorithm will decrease the time step, thus further increasing the size of the field.",
"Luckily, this $1/\\sqrt{\\Delta t}$ dependency makes the time step smaller at a slower rate than that the error is reduced ($\\Delta t^{N}$ with $N$ the order of the solver[65]).",
"However, it is important to use higher order solvers like the RK45 or RKF56 method in order to maintain a large time step.",
"To give the correct solution, the statistical properties of the random numbers $\\eta $ in Eq.",
"(REF ) should correspond to the ones determined in Eqs.",
"(REF ) to ().",
"However, if one would redraw these random numbers after a bad step, the small thermal fields (small $\\eta $ ) would be applied during longer time steps and large thermal fields (large $\\eta $ ) during shorter time steps, thus virtually changing the distribution of the random numbers, and eventually giving rise to incorrect solutions.",
"In our implementation we avoid this by keeping the previously drawn random numbers and rescaling the thermal field with a factor $\\sqrt{\\Delta t_\\mathrm {new}/\\Delta t_\\mathrm {bad step}}$ in case a bad step is encountered to ensure that the correct statistical properties of the thermal field are maintained.",
"The adaptive time stepping at nonzero temperatures will be tested in several test cases, focusing on different aspects, i.e.",
"static vs. dynamic properties of uncoupled spins or continuous magnets.",
"Each time, the simulation results obtained with adaptive time stepping will be compared either to analytical solutions or to the solutions obtained with the RK12 method with fixed time step."
],
[
"Spectra of a single spin",
"It will be verified whether the thermal field and the resulting magnetization dynamics of a single spin in the absence of an external field shows the theoretically expected behavior.",
"The thermal field should display a white spectrum $S_\\mathrm {H}$ construction, as its size is given by Gaussian random numbers.",
"Figure REF proves that this is indeed the case.",
"Figure: The thermal field spectra of a single spin (rescaled with an arbitrary factor for clarity reasons) obtained with the RK45 solver with fixed time steps, and with the RK45 and RKF56 solver with adaptive time steps.",
"All spectra display white noise.The random thermal fields acting on a single isotropic finite-difference cell gives rise to a random walk on the unit sphere[32], [11].",
"The shape of the spectral density $S_\\mathrm {M}(f)$ of such a random walk is described by the square root of a Lorentzian[66], $S_\\mathrm {M}(f)\\sim \\sqrt{\\left(\\frac{f_0/2}{f_0^2+(\\pi f)^2}\\right)},$ i.e.",
"white noise with a 1/$f$ cutoff at a cutoff frequency $f_0$ given by[58], [33] $f_0=\\frac{\\alpha }{(1+\\alpha ^2)}\\frac{\\gamma k_\\mathrm {B}T}{M_\\mathrm {s}V}$ Figure REF shows the obtained magnetization spectra (gray dots) indeed coincide with the red lines determined by Eqs.",
"(REF ) and (REF ).",
"Because all spectra coincide, they were rescaled with an arbitrary factor for clarity.",
"Figure: The magnetization spectra of a single spin (rescaled with an arbitrary factor for clarity reasons) obtained with the RK45 solver with fixed time steps, and with the RK45 and RKF56 solver with adaptive time steps.",
"All spectra display the theoretically expected shape depicted by the red lines."
],
[
"Equilibrium magnetization",
"This validation problem checks whether the magnetization of an ensemble of uncoupled spins in thermal equilibrium in an externally applied field is described by the Langevin function $\\mathcal {L}(\\xi )$ , $\\mathcal {L}(\\xi )=\\coth (\\xi )-\\frac{1}{\\xi }$ where the argument $\\xi $ stands for $\\xi =\\frac{\\mu _0M_{\\mathrm {s}} V H_{\\mathrm {ext}}}{k_\\mathrm {B} T}.$ Figure REF proves that this is indeed the case for 4 different $\\xi $ (realized by 2 different cell sizes and 2 different temperatures) simulated with the RK45 and RKF56 solver with adaptive time steps over a large range of applied fields.",
"Figure: The average magnetization 〈m〉\\langle m\\rangle of an ensemble of 2 18 2^{18} uncoupled spins in thermal equilibrium at different temperatures and different cell sizes (reflected in the different values of ξ\\xi ) for the RK45 (gray circles) and RKF56 (gray crosses) with adaptive time steps.",
"The results agree perfectly with the Langevin function [Eq.",
"()], shown by a red line."
],
[
"Thermal switching",
"After the equilibrium magnetization addressed in the previous problem, we will now concern ourselves with a dynamical problem consisting of the thermal switching rate of a single (macro-)spin particle with uniaxial anisotropy.",
"In the limit of a high energy barrier (compared to the thermal energy), the switching rate $\\nu $ is given by[67] $\\nu = \\gamma \\frac{\\alpha }{1+\\alpha ^2}\\sqrt{\\frac{8 K^3 V}{2\\pi M_\\mathrm {s}^2k_\\mathrm {B}}}e^{-KV/k_\\mathrm {B}T}.$ For an ensemble of uncoupled spins, initialized with all spins pointing in the same direction, this switching gives rise to an exponentially decaying magnetization, with a decay constant $1/2\\nu $ .",
"In this test problem, we simulate this decay to determine the numerical switching rate, and compare these values to their theoretical prediction.",
"Figure REF shows Arrhenius plots for the temperature-dependent switching rate $\\nu $ of uncoupled finite difference cells with volume $V$ =(10 nm)$^3$ and uniaxial anisotropy constant $K$ =1$\\times $ 10$^{4}$ or 2$\\times $ 10$^{4}$ J/m$^3$ .",
"Again, a quantitative agreement is seen between the MuMax$^3$ simulations and the theoretically predicted behavior.",
"Figure: Arrhenius plot of the thermal switching rate of a 10 nm large cubic cell, with M s M_\\mathrm {s}=1 MA/m, α\\alpha =0.1,KK=10 or 20 kJ/m 3 ^3.",
"Simulations were performed using the RK12 solver with fixed time steps (Δt\\Delta t=5 fs) or the RK45 or RKF56 solver with adaptive time steps on an ensemble of 2 18 2^{18} non-interacting cells for 1 μ\\mu s or until the ensemble magnetization crossed 0.",
"All results agree with the red solid lines depicting the analytically expected switching rates (Eq.",
")."
],
[
"Thermally excited magnetization spectrum",
"In this problem we look at the thermally excited magnetization spectrum of a 10 nm thick disk with a diameter of 512 nm, $M_\\mathrm {s}$ =1 MA/m, exchange constant $A_\\mathrm {ex}$ =10 pJ/m, and $\\alpha =1$ , discretized in cells measuring 4 by 4 by 10 nm$^3$ .",
"The equilibrium magnetization structure in such a disk is a vortex structure, as depicted in the inset of Fig.",
"REF .",
"We apply a thermal field corresponding to 300 K, thereby thermally exciting the sample, resulting in the spectra shown in Fig.",
"REF .",
"The spectrum depicted in red was obtained with the RK12 solver with fixed time step ($\\Delta t$ =5 fs), and serves as a reference solution.",
"The gray lines, which agree almost perfectly with the benchmark solution, correspond to the three spectra obtained with the RK45 solver with the time step fixed to $\\Delta t$ =300 fs and with the RK45 and RKF56s solvers with adaptive time step and $\\epsilon =10^{-5}$ .",
"Note that the spectra overlap even at high frequencies, indicating that the adaptive time stepping does not lead to spectral leakages.",
"Figure: The red line corresponds to the spectrum obtained with the Heun solver with Δt\\Delta t=5 fs.",
"The three gray lines, which overlap almost perfectly with the red one, correspond to the RK45 solver with fixed time step Δt\\Delta t=300 fs, and with the solution obtained with the RK45 and RKF56 solver with adaptive time stepping with ϵ=10 -5 \\epsilon =10^{-5}.",
"All spectra were averaged out over 25 realizations with a different random seed for the thermal field.",
"The inset shows the equilibrium vortex magnetization structure in the system under considerationThe minimum in the error is a result between a direct match between the used time step and the gyration period used for the evaluation of the precision."
],
[
"Thermal diffusion of a domain wall",
"In a last validation problem we investigate the thermal driven diffusion of transverse domain walls in a non-disordered permalloy nanowire[68].",
"We simulate a nanowire with cross-sectional dimensions of $100\\times 10$ nm$^2$ discretized in cells of $3.125\\times 3.125\\times 10$ nm$^3$ .",
"We use the material parameters of permalloy: $M_\\mathrm {s}=860$ kA/m, $A\\mathrm {ex}=13$ pJ/m, $\\alpha $ =0.01 and simulate the domain wall in the center of a moving window, as shown in Fig.",
"REF .",
"Figure: The transverse domain wall in the center of the nanowire.",
"By compensating the edge charges, we simulate an infinitely long wire.The thermally driven motion can be described by a random walk characterized by a diffusion constant $D$ which scales linearly with temperature.",
"Similarly as in Ref.",
"[68] we simulate a transverse domain wall for 100 ns and repeat this simulation with a large number of different random seeds.",
"In Tab.",
"REF , we compare the results obtained from the full micromagnetic simulations with the diffusion constant $D$ of approximately 310 nm$^2$ /ns predicted by the model introduced, and numerically validated in Ref.",
"[68]using the RK12 solver with fixed time step.",
"The results show that the standard errors $s$ are larger than the difference between the obtained diffusion constants and the expected value.",
"This also indicates that, for this problem, an error tolerance $\\epsilon =10^{-3}$ suffices for all practical purposes, as the variance between different simulations gives rise to an uncertainty that is larger than the errors due to the use of this relatively large $\\epsilon $ .",
"As this is problem dependent, the default value in MuMax$^3$ remains $10^{-5}$ , but we suggest that, depending on the system under consideration, larger values might be suitable in simulations at nonzero temperatures.",
"Table: The diffusion constant DD and standard error ss, determined from simulations performed using several solvers, with adaptive time stepping with error tolerance ϵ\\epsilon and repeated for a total number of NN realizations at 300 K. The theoretically predicted DD approximately equals 310 nm 2 ^2/ns."
],
[
"Performance",
"To assess the performance of the presented methods we will consider the problems detailed in REF and REF , i.e the thermal spectrum of a disk and thermally driven domain wall diffusion, respectively, as benchmarks.",
"The benchmark results are shown in Fig.",
"REF a) and b).",
"For each of these problems, the simulation time was first determined when solving the problem with the RK12 solver with a fixed time step of 5 fs.",
"As this is a second order solver with embedded first order solver, the difference between both solutions serves as an error estimate, allowing us to estimate with which error tolerance $\\epsilon $ in the adaptive time stepping the results should be compared.",
"This data point is indicated by a black cross in the figure.",
"Next, the problems were solved using adaptive time stepping, once with the RK45, and once with the RKF56 solver, with $\\epsilon $ ranging from $10^{-3}$ to $10^{-7}$ , shown in gray and red, respectively.",
"The simulation runtime [Figs.",
"REF a) and b)] show the time it took to simulate 10 ns of magnetization dynamics, while Figs.",
"REF c) and d) indicate the average time step $\\Delta t$ used by the solver, for each $\\epsilon $ .",
"When comparing the results from the adaptive time stepping methods with the result of the RK12 solver at the same estimated $\\epsilon $ , one sees that the adaptive time stepping methods use a considerably larger time step without a loss of accuracy.",
"For these two benchmark problems, this results in a twenty fold speedup of the simulation.",
"We also investigated the performance of the system described in Section REF , i.e.",
"thermal switching of uncoupled spins.",
"There, an even higher speedup was achieved, but this was attributed to the fact that such systems do not require the calculation of the demagnetizing field so that other factors, like the generation of the random numbers for the thermal field, become the limiting factor.",
"Because the random numbers have to be generated only once per time step independently of the used solver, a very large performance gain can be achieved by using higher order solvers which allow time steps that are over a 1000 times larger than the ones necessary for the RK12 method with the same accuracy.",
"However, as this highly depends on the used hardware, the performance gained by using the adaptive time stepping methods can lie anywhere between a factor 20 to 10 000, depending on accuracy.",
"These observations indicate that the presented methods are particularly suited for magnetic nanoparticle research, where micromagnetic simulations are becoming increasingly important[34].",
"Figure: Benchmark results for the systems described in Sections [panel a) and c)] and [b) and d)], respectively.",
"The top row shows the runtime required to simulate 10 ns of magnetization dynamics, while the bottom row shows the average time step used.",
"The black crosses indicate the performance of the fixed time step RK12 method with ϵ\\epsilon estimated from the difference between the second order and embedded first order solution.",
"All results were obtained an NVIDIA Titan Xp GPU in a system running on a 7th generation i5 CPU.The relative performance of the RK45 and RKF56 solver is comparable and show the expected trends that the RK45 solver is faster at higher $\\epsilon $ and simulated temperatures while the RKF56 solver is faster at low $\\epsilon $ and temperatures.",
"Generally, only when simulating systems at very high temperatures, or with very small $\\epsilon $ , it does pay off to use the RKF56 solver."
],
[
"Conclusions",
"In this paper, we have exploited the fact that the drift term in the stochastic Landau-Lifshitz-Gilbert equation is only able to manifests itself in the direction of the magnetization length, which is fixed.",
"Therefore, we were able to straightforwardly extend existing high order solvers with adaptive time stepping at nonzero temperatures.",
"In an effort to further increase the performance, we have implemented the sixth order Runge-Kutta-Fehlberg solver, and we extensively validated both the correctness of this newly implemented solver and the adaptive time stepping method used at nonzero temperature.",
"All presented methods are included in the open-source micromagnetic software package MuMax$^3$ and are thus freely available online.",
"The main advantages of the presented adaptive time stepping methods at nonzero temperatures are that they offer an inherent error control, which is unavailable with fixed time stepping methods, and without a loss of accuracy one can obtain a twenty fold speedup compared to the commonly best practice of using the RK12 solver with small fixed time step.",
"This enables simulations which previously took too long to be considered feasible and will be useful for micromagnetic research of continuous (exchange coupled) systems like spin valves, or domain wall motion in nanowires, and for uncoupled spins, e.g.",
"in magnetic nanoparticle research."
],
[
"Acknowledgement",
"This work was supported by the Fonds Wetenschappelijk Onderzoek (FWO-Vlaanderen) through Project No.",
"G098917N and a postdoctoral fellowship (A.C.).",
"J. L. is supported by the Ghent University Special Research Fund (BOF postdoctoral fellowship).",
"We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research."
]
] | 1709.01682 |
[
[
"Baryon magnetic moments: Symmetries and relations"
],
[
"Abstract Magnetic moments of the octet baryons are computed using lattice QCD in background magnetic fields, including the first treatment of the magnetically coupled Sigma-Lambda system.",
"Although the computations are performed for relatively large values of the up and down quark masses, we gain new insight into the symmetries and relations between magnetic moments by working at a three-flavor mass-symmetric point.",
"While the spin-flavor symmetry in the large Nc limit of QCD is shared by the naive constituent quark model, we find instances where quark model predictions are considerably favored over those emerging in the large Nc limit.",
"We suggest further calculations that would shed light on the curious patterns of baryon magnetic moments."
],
[
"Introduction",
"The electromagnetic properties of hadrons and nuclei provide physically intuitive information about their structure.",
"Relations between baryon magnetic moments, moreover, were historically crucial in exposing the approximate symmetries of QCD.",
"Here, we argue that these magnetic moments continue to provide (increasingly subtle) clues about the structure of baryons.",
"In the past few years, the $\\texttt {NPLQCD}$ collaboration has undertaken the first lattice gauge theory computations of the magnetic properties of light nuclei.",
"Highlights of these computations include: determination of the magnetic moments and polarizabilities of light nuclei [1], [2]; study of the simplest nuclear reaction, $n + p \\rightarrow d + \\gamma $ , through its dominant magnetic dipole transition amplitude [3]; and, uncovering hints of unitary nucleon-nucleon interactions in large magnetic fields [4].",
"These computations were made possible by two crucial ingredients: i).",
"the external field technique, for which these properties can be determined from two-point correlation functions rather than three-point functions; and ii).",
"rather large quark masses, for which adequate statistics can be accumulated to provide signals for light nuclei.",
"Here, we describe an extra curricular study that emerged from this work, namely the determination of octet baryon magnetic moments [5].",
"While such computations are not new, the setup of our calculation enabled investigations of the magnetic moments that had not previously been made.",
"Most notably, we study strong interactions in an unphysical environment: that of exact SU$(3)_F$ symmetry, in which the quark masses satisfy $m_u = m_d = m_s \\approx (m_s)_\\text{physical}$ .",
"The addition of the quark electric charges preserves an SU$(2)$ subgroup of SU$(3)_F$ , commonly called $U$ -spin.",
"A summary of this investigation is presented, including a brief overview of the calculational details, Sec. .",
"We discuss the issue of magneton units for moments, Sec.",
"REF , the determination and estimation of Coleman-Glashow magnetic moments, Sec.",
"REF , and compare predictions of the naïve constituent quark model with those of the large $N_c$ limit of QCD, Sec.",
"REF .",
"We review our treatment of the coupled system of $\\Sigma ^0$ and $\\Lambda $ baryons in Sec.",
"REF .",
"Questions for future study are given in the concluding section, Sec.",
"."
],
[
"Overview of Calculation",
"The calculation of octet baryon magnetic moments in Ref.",
"[5] utilizes primarily two ensembles of tadpole-improved clover fermions, details of which have been given previously [6], [7].",
"As mentioned, one ensemble is SU$(3)_F$ symmetric, for which the pion mass is $m_\\pi \\approx 800 \\, \\texttt {MeV}$ .",
"The other ensemble maintains $m_s \\approx (m_s)_\\text{physical}$ , but with two degenerate lighter quarks, corresponding to a pion mass of $m_\\pi \\approx 450 \\, \\texttt {MeV}$ .",
"To study the magnetic properties of hadrons, classical U$(1)$ gauge links are post-multiplied to the color links.The magnetic field is not included in the generation of QCD gauge fields.",
"For the SU$(3)_F$ ensemble, the computation of magnetic moments is exact, due to the condition $\\texttt {Tr} \\left( Q \\right)= q_u + q_d + q_s = 0$ .",
"With broken SU$(3)_F$ on the second ensemble, however, contributions from coupling the magnetic field to the quark sea are missing.",
"The size of such quark-disconnected contributions to isoscalar quantities has been estimated to be quite small, see, e.g., Ref.",
"[8], and will be neglected throughout.",
"To add a uniform magnetic field in the $x_3$ -direction, we choose $U_\\mu (x)=e^{- i q x_2 B \\, \\delta _{\\mu 1}}e^{+ i q x_1 B N \\, \\delta _{\\mu 2} \\, \\delta _{x_2, N-1}}\\in \\text{U}(1),$ where $q$ is the quark charge, and the magnetic field, $B$ , must satisfy the 't Hooft quantization condition [9], namely $q B = 2 \\pi \\, n_\\Phi / N^2$ , to ensure that the flux through each elementary plaquette of the lattice is uniform.",
"The number of lattice sites in each spatial direction is taken as $N$ , and $n_\\Phi $ is the flux quantum of the torus.",
"Quark propagators are computed for the values $n_\\Phi = 0$ , $+3$ , $-6$ , and $+12$ , where the overall factor of 3 is due to the fractional nature of the quark charges, and the multiplicative factors of $-2$ allow one to recycle some down-quark propagators as up-quark propagators.",
"With 48 random source locations per configuration and a combined total of $\\sim 1500$ configurations for the two ensembles, on the order of $400 \\, \\texttt {k}$ measurements are made.",
"For the two ensembles considered, the spatial size is $N = 32$ , leading to magnetic fields that are expected to be of a reasonable perturbative size.",
"Figure: Effective mass plots for the proton spin splittings in three magnetic field strengths appear on the left,where bands show the results of fits to constant time dependence.Magnetic field-strength dependence of the extracted proton spin splittings is shown on the right,where bands show the results of linear and linear-plus-cubic fits,with the largest field strength excluded in the former.The procedure to determine magnetic moments from external field calculations goes back to the early days of lattice QCD [10], [11].",
"Rather simply, one calculates the Zeeman effect for hadrons.",
"This is accomplished by computing the spin-dependent hadron energies as a function of the magnetic field.",
"Double ratios of correlation functions can be devised to cleanly isolate the splitting between spin states, e.g.",
"$\\frac{G^\\uparrow (t, n_\\Phi )}{G^\\downarrow (t, n_\\Phi ) }\\Bigg /\\frac{G^\\uparrow (t, 0)}{ G^\\downarrow (t, 0) }=Z \\,e^{ - \\Delta E t}+\\cdots ,$ where we have exhibited saturation by the ground state in the long-time limit, and the spin splitting $\\Delta E$ is given by $\\Delta E = E^\\uparrow - E^\\downarrow = - 2 \\mu B + \\cdots $ .",
"Effective mass plots for the proton spin splittings in the various magnetic fields are shown in Fig.",
"REF .",
"Additionally shown is the behavior of these splittings as a function of the magnetic field.",
"Extraction of the proton magnetic moment is limited by the systematics of fitting the linear magnetic field-strength term, not the statistics of the present computation."
],
[
"Results",
"Magnetic moments of the octet baryons with $I_3 \\ne 0$ are determined using the procedure outlined above.",
"In the following, we explore units for magnetic moments that suppress their pion-mass dependence, relations between the magnetic moments in the limit of SU$(3)_F$ , as well as relations predicted in the constituent quark model and large $N_c$ limit.",
"The two $I_3 = 0$ baryons, $\\Sigma ^0$ and $\\Lambda $ , which undergo mixing in magnetic fields, are discussed separately."
],
[
"Natural Baryon Magnetons",
"Using the procedure above, the magnetic moments are extracted in lattice magnetons, $\\frac{1}{2} e a$ , where $a$ is the lattice spacing, and are of order unity, suggestive of neither large volume nor discretization effects.",
"Inspired by the observation that nuclear and nucleon magnetic moments are strikingly close to their physical values when expressed in terms of natural nuclear magneton units [1], we present magnetic moments in natural baryon magnetons.",
"These units are defined by $\\texttt {[nBM]} \\equiv \\frac{e}{2 M_B(m_\\pi )}$ , where $M_B(m_\\pi )$ denotes the particular octet baryon mass determined at the corresponding pion mass.",
"Such units incorporate the surprisingly linear pion-mass dependence observed for octet baryon masses [12].",
"Results for anomalous magnetic moments, $\\delta \\mu _B \\, \\texttt {[nBM]} = \\mu _B \\, \\texttt {[nBM]} - Q_B$ , are displayed in Fig.",
"REF along with experimental values.",
"A salient feature of the natural baryon magneton unit is that the Dirac part of the moment, which is short distance in nature, can readily be subtracted.",
"The results shown exhibit curious behavior.",
"Across all of these baryon magnetic moments, the majority of their pion-mass dependence is accounted for by employing $\\texttt {[nBM]}$ .",
"These units appear to suppress the breaking of $U$ -spin symmetry, which, among other things, predicts $\\mu _p = \\mu _{\\Sigma ^+}$ , $\\mu _n = \\mu _{\\Xi ^0}$ , and $\\mu _{\\Sigma ^-} = \\mu _{\\Xi ^-}$ .",
"The latter moments, $\\mu _{\\Sigma ^-}$ and $\\mu _{\\Xi ^-}$ , show little deviation from those of pointlike particles.",
"In Fig.",
"REF , we additionally show the linear combinations of magnetic moments that vanish in the limit of $U$ -spin symmetry.",
"These linear combinations are quite small; nonetheless, they confirm that SU$(3)_F$ breaking on the $m_\\pi = 450 \\, \\texttt {MeV}$ ensemble lies intermediate to the $m_\\pi = 800 \\, \\texttt {MeV}$ ensemble and experiment.",
"Figure: Anomalous magnetic moments of the octet baryons in units of natural baryon magnetons appear on the left.We use the notation “-B-B ” to specify the negative of the magnetic moment,δμ -B ≡-δμ B \\delta \\mu _{-B} \\equiv - \\delta \\mu _B,so that a majority of the moments plotted are positive.On the right appear linear combinations of magnetic moments that vanish in the limit of UU-spin symmetry,for example,p-Σ + p - \\Sigma ^+corresponds toμ p -μ Σ + \\mu _p - \\mu _{\\Sigma ^+}.The notation “C-P” refers to half the sum of all moments,μ C--P =1 2μ p +μ Σ + +μ n +μ Ξ 0 +μ Σ - +μ Ξ - \\mu _{\\text{C--P}} = \\frac{1}{2} \\left[ \\mu _p + \\mu _{\\Sigma ^+} + \\mu _n + \\mu _{\\Xi ^0} + \\mu _{\\Sigma ^-} + \\mu _{\\Xi ^-} \\right],which is one of the relations ofRef.",
"."
],
[
"Coleman-Glashow Moments",
"In the limit of $U$ -spin symmetry, the octet baryon magnetic moments are determined by only two numbers, $\\mu _D$ and $\\mu _F$ , which we call the Coleman-Glashow moments.",
"These appear as coefficients in the effective Hamiltonian density [14] $\\mathcal {H}=-\\frac{e \\, \\vec{\\sigma } \\cdot \\vec{B}}{2 M_B}\\left[\\mu _D \\,\\texttt {Tr}\\left( \\overline{B} \\left\\lbrace Q, B \\right\\rbrace \\right)+\\mu _F \\,\\texttt {Tr}\\left( \\overline{B} \\left[ Q, B \\right] \\right)\\right],$ where $Q$ is the quark charge matrix, and $B$ the octet of baryon fields, with $\\overline{B}$ as its conjugate.",
"Values of $\\mu _D$ and $\\mu _F$ can be extracted on the $m_\\pi = 800 \\, \\texttt {MeV}$ ensemble with high precision.",
"Values can be determined from the $m_\\pi = 450 \\, \\texttt {MeV}$ ensemble and experiment with additional assumptions.",
"Determinations of these parameters are shown in Fig.",
"REF , and exhibit little pion-mass dependence.",
"Figure: Values of the Coleman-Glashow magnetic moments,μ D \\mu _D and μ F \\mu _F,determined from lattice QCD and experiment.Values are directly extracted from a combined fit to moments computed on them π =800𝙼𝚎𝚅m_\\pi = 800 \\, \\texttt {MeV}ensemble,due to UU-spin symmetry.Those determined from them π =450𝙼𝚎𝚅m_\\pi = 450 \\, \\texttt {MeV}ensemble and experiment utilize only the nucleon magnetic moments,and include an uncertainty due to the breaking of SU(3) F (3)_F symmetry by the strange quark in the sea.This uncertainty is assessed asΔm q N c -1 \\Delta m_q \\, N_c^{-1},whereΔm q \\Delta m_qrepresents the expected size of SU(3) F (3)_F breaking,and the inverse factor ofN c =3N_c = 3reflects the closed fermion loop.For experiment,this uncertainty is taken to beΔm q N c -1 =30%/3=10%\\Delta m_q \\, N_c^{-1} = 30\\% \\, / \\, 3 = 10\\%,while for moments on them π =450𝙼𝚎𝚅m_\\pi = 450 \\, \\texttt {MeV}ensemble,we useΔm q N c -1 =15%/3=5%\\Delta m_q \\, N_c^{-1} = 15 \\% \\, / \\, 3 = 5 \\%,which reflects the reduction inSU(3) F (3)_Fbreaking exhibited in Fig.",
"."
],
[
"Quark Model vs. Large $N_c$",
"Investigation of the Coleman-Glashow moments suggests little deviation from $U$ -spin symmetry, however, this is not the complete story.",
"Results are (not surprisingly) consistent with a larger symmetry group, that of SU$(6)$ and the naïve consitituent quark model.",
"Assuming non-interacting constituent quarks, one arrives at the values $\\mu _D = 3$ and $\\mu _F = 2$ , where the former is simply counting the number of constituent quarks in each baryon.",
"Such values, moreover, explain why the anomalous magnetic moments shown in Fig.",
"REF only take on values close to $\\delta \\mu _B = 0$ , and $\\pm 2$ .",
"Figure: Quark model (left) and large-N c N_c relations (right) between magnetic moments.All relations are given as ratios that are predicted to be unity.The quark model ratios are given in Eq.",
",while those emerging in the large N c N_c limit are given in Eqs.",
"(starred) and (unstarred).The lattice QCD results can be utilized to explore quark model relations as a function of the pion mass.",
"In Fig.",
"REF , we show various ratios, $R_X$ , that are predicted to be unity within the naïve constituent quark model.",
"Using $\\Delta \\mu $ 's to denote the isovector differences of moments, these ratios are $R_N= -\\frac{2}{3} \\frac{\\mu _p}{\\mu _n},\\quad R_{N \\Sigma }=\\frac{5}{4} \\frac{\\Delta \\mu _{\\Sigma }}{\\Delta \\mu _N},\\quad R_{N \\Xi }=5 \\, \\frac{\\Delta \\mu _\\Xi }{\\Delta \\mu _N},\\quad R_{\\Sigma \\Xi }=4 \\, \\frac{\\Delta \\mu _\\Xi }{\\Delta \\mu _\\Sigma },\\quad \\text{and}\\quad R_{S}=- 4 \\, \\frac{\\mu _{\\Sigma ^+} + 2 \\mu _{\\Sigma ^-}}{\\mu _{\\Xi ^0} + 2 \\mu _{\\Xi ^-}}.$ Unity is very well satisfied for the first two ratios over the range of pion masses available.",
"The remaining ratios show greater deviation, although the lattice QCD results generally show better agreement with the naïve quark model than experiment.",
"One should note that the isovector moment ratios essentially compare two determinations of the (light) constituent quark mass, and that the ratio $R_{N \\Xi }$ , for example, shows an environmental sensitivity described in Ref. [15].",
"The ratio $R_S$ compares the in situ determination of the strange constituent quark mass, which appears to be effectively different in $\\Sigma $ and $\\Xi $ baryons.",
"The pattern shown in the figure is perhaps suggestive of a systematic expansion scheme, and remains to be explored quantitatively.",
"From a field-theoretic perspective, the success of the non-relativistic quark model is argued to be understood due to the emergent spin-flavor symmetry of the large-$N_c$ limit [16], [17].",
"Indeed the scaling of magnetic moment relations predicted in the large $N_c$ limit seems to explain why some of the quark model relations work better than others [18], despite the restriction of nature to $N_c = 3$ .",
"The lattice calculations of Ref.",
"[5], which are also restricted to $N_c = 3$ , offer further insight into large $N_c$ relations among baryon magnetic moments.",
"Results are shown in Fig.",
"REF ; and, in the case of quark-mass independent relations, large $N_c$ scalings appear to be confirmed.",
"These are predictions for the (starred) ratios $\\mathcal {R}_{S7}=\\frac{ 5( \\mu _p + \\mu _n) - (\\mu _{\\Xi ^0} + \\mu _{\\Xi ^-})}{4 ( \\mu _{\\Sigma ^+} + \\mu _{\\Sigma ^-})}=1 + \\mathcal {O}(N_c^{-1}), \\quad \\text{and} \\quad \\mathcal {R}_{V1}=\\frac{\\Delta \\mu _N + 3 \\Delta \\mu _\\Xi }{2 \\Delta \\mu _{\\Sigma }}=1 + \\mathcal {O} (N_c^{-2}).$ By contrast, relations that depend on the level of SU$(3)_F$ breaking seem to exhibit a trend opposite large $N_c$ considerations.",
"These are the (unstarred) ratios $\\mathcal {R}_{V10_1}&=&\\frac{\\Delta \\mu _N}{\\Delta \\mu _\\Sigma }=1 + \\mathcal {O} (N_c^{-1})\\overset{\\text{SU}(3)_F}{=}\\begin{pmatrix}1 + \\mathcal {O} (N_c^{-1})\\\\1.25, \\text{ CQM}\\end{pmatrix},\\\\\\mathcal {R}_{V10_2}&=&\\left( 1 - N_c^{-1} \\right)\\frac{\\Delta \\mu _N}{\\Delta \\mu _\\Sigma }=1 + \\mathcal {O} ( \\Delta m_q N_c^{-1})\\overset{\\text{SU}(3)_F}{=}\\begin{pmatrix}1 + \\mathcal {O} ( N_c^{-2})\\\\0.83, \\text{ CQM}\\end{pmatrix},\\\\\\mathcal {R}_{V10_3}&=&\\left( 1 + N_c^{-1} \\right)^{-1}\\frac{\\Delta \\mu _N}{\\Delta \\mu _\\Sigma }=1 + \\mathcal {O} ( \\Delta m_q N_c^{-1})\\overset{\\text{SU}(3)_F}{=}\\begin{pmatrix}1 + \\mathcal {O} ( N_c^{-2})\\\\0.94, \\text{ CQM}\\end{pmatrix},\\\\\\mathcal {R}_{S/V1}&=&\\frac{\\mu _p + \\mu _n - 6 \\left(N_c^{-1} - 2 N_c^{-2} \\right) \\Delta \\mu _N}{2 \\left( \\mu _{\\Sigma ^+} + \\mu _{\\Sigma ^-} \\right) - \\left( \\mu _{\\Xi ^0} + \\mu _{\\Xi ^-} \\right)}=1 + \\mathcal {O} ( \\Delta m_q )+ \\mathcal {O} ( \\Delta m_q N_c^{-1})\\overset{\\text{SU}(3)_F}{=}\\begin{pmatrix}1 + \\mathcal {O} ( N_c^{-2})\\\\0.62, \\text{ CQM}\\end{pmatrix},\\\\$ where we have also listed the values obtained in the SU$(3)_F$ -symmetric constituent quark model (CQM).",
"From the figure, notice that scaling of the ratios $\\mathcal {R}_{V10_2}$ and $\\mathcal {R}_{V10_3}$ does not seem to improve with SU$(3)_F$ symmetry as predicted by large $N_c$ considerations.",
"In the case of $\\mathcal {R}_{V/S1}$ , moreover, the difference is quite substantial, with an almost certain preference for the SU$(3)_F$ -symmetric CQM value.",
"Lattice QCD results for magnetic moments seem to suggest that large $N_c$ predictions are more robust in the $\\text{SU}(3)_L \\times \\text{SU}(3)_R$ chiral limit, rather than the SU$(3)_F$ limit.",
"This finding should be contrasted with baryon mass relations in the large $N_c$ limit, which seem, however, to show the predicted improvement with SU$(3)_F$ symmetry [19]."
],
[
"Mixing of $\\Sigma ^0$ and {{formula:bbf18705-5a0b-486b-9350-388306d9b36a}} Baryons",
"With the inclusion of a magnetic field, the two $I_3 = 0$ octet baryon states, $\\Sigma ^0$ and $\\Lambda $ , mix through the isovector component of the quark charge matrix.",
"This system hence requires a coupled-channels analysis, for which one forms the matrix of correlation functions $\\mathcal {G}^{(s)}(t,n_\\Phi )=\\begin{pmatrix}G_{\\Sigma \\Sigma }^{(s)}(t,n_\\Phi )&G_{\\Sigma \\Lambda }^{(s)}(t,n_\\Phi )\\\\G_{\\Lambda \\Sigma }^{(s)}(t,n_\\Phi )&G_{\\Lambda \\Lambda }^{(s)}(t,n_\\Phi )\\end{pmatrix},$ where $s = \\pm \\frac{1}{2}$ is the spin projection along the magnetic field.",
"The off-diagonal correlation functions in this system are only available in Ref.",
"[5] on the SU$(3)_F$ -symmetric ensemble.",
"Figure: Energy eigenvaluesE λ ± (s) E_{\\lambda _\\pm ^{(s)}}determined from a coupled-channels analysis in the Σ 0 \\Sigma ^0-Λ\\Lambda system atm π =800𝙼𝚎𝚅m_\\pi = 800 \\, \\texttt {MeV}are plotted as a function of the magnetic field.Fits to the magnetic field dependencepredicted by Eq.",
"are also shown.Note that for each energy eigenvalue,the differenceΔE≡E-M B \\Delta E \\equiv E - M_B is defined to vanish in vanishing magnetic field.Eigenstate energies are extracted from the principal correlation functions that solve the generalized eigenvalue problem [20] posed in this system.",
"Denoting the $U$ -spin eigenstates as $\\lambda _\\pm $ , the calculated energies are plotted as a function of the magnetic field strength in Fig.",
"REF .",
"These energies are to be compared with those predicted on the basis of the $U$ -spin symmetry of the calculation, namely $E_{\\lambda _\\pm }^{(s)}=M_B\\pm \\mu _n \\frac{s\\, e B_z}{M_B}-\\frac{1}{2} 4 \\pi \\left[\\beta _n+(1 \\pm 1)\\frac{2}{\\sqrt{3}}\\beta _{\\Lambda \\Sigma }\\right]B_z^2+\\mathcal {O}(B_z^3),$ where $\\mu _n$ is the neutron magnetic moment, and $\\beta _n$ is the neutron magnetic polarizability (technically it is the quark-connected part of $\\beta _n$ given the limitations of the calculation).",
"The quantity $\\beta _{\\Lambda \\Sigma }$ is the transition polarizability in this system.",
"It does not have quark-disconnected contributions, and the level ordering observed in Fig.",
"REF requires $\\beta _{\\Lambda \\Sigma } < 0$ .",
"From fits, the value $\\beta _{\\Lambda \\Sigma } = - 1.82(06)(12)(02) \\, \\texttt {fm}^3$ is obtained, which is to be compared with $\\beta _n = 3.48(12)(26)(04) \\, \\texttt {fm}^3$ , where the uncertainties reflect statistics, systematics, and scale setting.The lattice spacing is determined from the quarkonium hyperfine splitting without light quark-mass extrapolation, producing the value $a = 0.1453(16) \\, \\texttt {fm}$ .",
"Accounting for the light quark-mass dependence will result in a significant change to polarizabilities due to their $a^3$ scaling, cf.",
"the scale setting of Ref. [2].",
"With SU$(3)_F$ breaking, the $\\Sigma $ -$\\Lambda $ mass splitting will suppress mixing, and consequently make it challenging to isolate the transition moment and polarizability from lattice QCD data.",
"To conclude, we suggest calculations and questions for further study motivated by Ref.",
"[5] and the overview given above.",
"1).",
"Concerning the pion mass dependence: scaling by the baryon's Compton wavelength seems to account for a majority of the quark-mass dependence of octet baryon magnetic moments.",
"This suggests that extrapolating from $800 \\, \\texttt {MeV}$ down to the physical point might be easier than extrapolating from the chiral limit to the physical point.",
"Is there a way to understand this quantitatively, especially for the potential applicability to other observables?",
"2).",
"The calculations presented here are limited in their tests of a few relations.",
"A more complete study seems warranted, and this necessitates computing the decuplet magnetic moments and decuplet-to-octet transitions moments.",
"3).",
"The flavor-breaking trend in large $N_c$ relations remains curious, and its study would be bolstered by further SU$(3)_F$ -symmetric computations of magnetic moments at lighter quark masses.",
"4).",
"The quark model can be more thoroughly vetted against large $N_c$ relations in computations with $N_c = 5, \\cdots $ .",
"Nonetheless it appears that baryon magnetic moments, while seminal in the historical development of QCD, continue to provide interesting questions about non-perturbative physics."
],
[
"Acknowledgements",
"Calculations were performed using computational resources provided by the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by U.S. National Science Foundation grant number OCI-1053575, and NERSC, which is supported by U.S. Department of Energy Grant Number DE-AC02-05CH-11231.",
"The PRACE Research Infrastructure resources Curie based in France at the Très Grand Centre de Calcul and MareNostrum-III based in Spain at the Barcelona Supercomputing Center were also used.",
"Development work required for this project was carried out on the Hyak High Performance Computing and Data Ecosystem at the University of Washington, supported, in part, by the U.S. National Science Foundation Major Research Instrumentation Award, Grant Number 0922770.",
"Parts of the calculations used the Chroma software suite [21].",
"WD was supported in part by the U.S. Department of Energy Early Career Research Award No.",
"DE-SC00-10495 and by Grant Number DE-SC00-11090.",
"EC was supported in part by the USQCD SciDAC project, the U.S. Department of Energy through Grant Number DE-SC00-10337, and by U.S. Department of Energy Grant No. DE-FG02-00ER41132.",
"KO was supported by the U.S. Department of Energy through Grant Number DE-FG02-04ER-41302 and by the U.S. Department of Energy through Grant Number DE-AC05-06OR-23177, under which JSA operates the Thomas Jefferson National Accelerator Facility.",
"The work of AP is partially supported by the Spanish Ministerio de Economia y Competitividad (MINECO) under the project MDM-2014-0369 of ICCUB, and, with additional European FEDER funds, under the contract FIS2014-54762-P. MJS was supported in part by U.S. Department of Energy Grants No.",
"DE-FG02-00ER-41132 and No.",
"DE-SC00-10337.",
"BCT was supported in part by the U.S. National Science Foundation, under Grant No.",
"PHY15-15738."
]
] | 1709.01564 |
[
[
"Affect Recognition in Ads with Application to Computational Advertising"
],
[
"Abstract Advertisements (ads) often include strongly emotional content to leave a lasting impression on the viewer.",
"This work (i) compiles an affective ad dataset capable of evoking coherent emotions across users, as determined from the affective opinions of five experts and 14 annotators; (ii) explores the efficacy of convolutional neural network (CNN) features for encoding emotions, and observes that CNN features outperform low-level audio-visual emotion descriptors upon extensive experimentation; and (iii) demonstrates how enhanced affect prediction facilitates computational advertising, and leads to better viewing experience while watching an online video stream embedded with ads based on a study involving 17 users.",
"We model ad emotions based on subjective human opinions as well as objective multimodal features, and show how effectively modeling ad emotions can positively impact a real-life application."
],
[
"Introduction",
"The proceedings are the records of a conference.This is a footnote ACM seeks to give these conference by-products a uniform, high-quality appearance.",
"To do this, ACM has some rigid requirements for the format of the proceedings documents: there is a specified format (balanced double columns), a specified set of fonts (Arial or Helvetica and Times Roman) in certain specified sizes, a specified live area, centered on the page, specified size of margins, specified column width and gutter size."
],
[
"The Body of The Paper",
"Typically, the body of a paper is organized into a hierarchical structure, with numbered or unnumbered headings for sections, subsections, sub-subsections, and even smaller sections.",
"The command \\section that precedes this paragraph is part of such a hierarchy.This is a footnote.",
"handles the numbering and placement of these headings for you, when you use the appropriate heading commands around the titles of the headings.",
"If you want a sub-subsection or smaller part to be unnumbered in your output, simply append an asterisk to the command name.",
"Examples of both numbered and unnumbered headings will appear throughout the balance of this sample document.",
"Because the entire article is contained in the document environment, you can indicate the start of a new paragraph with a blank line in your input file; that is why this sentence forms a separate paragraph."
],
[
"Type Changes and ",
"We have already seen several typeface changes in this sample.",
"You can indicate italicized words or phrases in your text with the command \\textit; emboldening with the command \\textbf and typewriter-style (for instance, for computer code) with \\texttt.",
"But remember, you do not have to indicate typestyle changes when such changes are part of the structural elements of your article; for instance, the heading of this subsection will be in a sans serifAnother footnote here.",
"Let's make this a rather long one to see how it looks.",
"typeface, but that is handled by the document class file.",
"Take care with the use ofAnother footnote.",
"the curly braces in typeface changes; they mark the beginning and end of the text that is to be in the different typeface.",
"You can use whatever symbols, accented characters, or non-English characters you need anywhere in your document; you can find a complete list of what is available in the User's Guide ."
],
[
"Math Equations",
"You may want to display math equations in three distinct styles: inline, numbered or non-numbered display.",
"Each of the three are discussed in the next sections."
],
[
"Inline (In-text) Equations",
"A formula that appears in the running text is called an inline or in-text formula.",
"It is produced by the math environment, which can be invoked with the usual \\begin ...\\end construction or with the short form $ ...$.",
"You can use any of the symbols and structures, from $\\alpha $ to $\\omega $ , available in ; this section will simply show a few examples of in-text equations in context.",
"Notice how this equation: $\\lim _{n\\rightarrow \\infty }x=0$ , set here in in-line math style, looks slightly different when set in display style.",
"(See next section)."
],
[
"Display Equations",
"A numbered display equation—one set off by vertical space from the text and centered horizontally—is produced by the equation environment.",
"An unnumbered display equation is produced by the displaymath environment.",
"Again, in either environment, you can use any of the symbols and structures available in ; this section will just give a couple of examples of display equations in context.",
"First, consider the equation, shown as an inline equation above: $\\lim _{n\\rightarrow \\infty }x=0$ Notice how it is formatted somewhat differently in the displaymath environment.",
"Now, we'll enter an unnumbered equation: $\\sum _{i=0}^{\\infty } x + 1$ and follow it with another numbered equation: $\\sum _{i=0}^{\\infty }x_i=\\int _{0}^{\\pi +2} f$ just to demonstrate 's able handling of numbering."
],
[
"Citations",
"Citations to articles , , , , conference proceedings or maybe books , listed in the Bibliography section of your article will occur throughout the text of your article.",
"You should use BibTeX to automatically produce this bibliography; you simply need to insert one of several citation commands with a key of the item cited in the proper location in the .tex file .",
"The key is a short reference you invent to uniquely identify each work; in this sample document, the key is the first author's surname and a word from the title.",
"This identifying key is included with each item in the .bib file for your article.",
"The details of the construction of the .bib file are beyond the scope of this sample document, but more information can be found in the Author's Guide, and exhaustive details in the User's Guide by Lamport Lamport:LaTeX.",
"This article shows only the plainest form of the citation command, using \\cite.",
"Some examples.",
"A paginated journal article , an enumerated journal article , a reference to an entire issue , a monograph (whole book) , a monograph/whole book in a series (see 2a in spec.",
"document) , a divisible-book such as an anthology or compilation followed by the same example, however we only output the series if the volume number is given (so Editor00a's series should NOT be present since it has no vol.",
"no.",
"), a chapter in a divisible book , a chapter in a divisible book in a series , a multi-volume work as book , an article in a proceedings (of a conference, symposium, workshop for example) (paginated proceedings article) , a proceedings article with all possible elements , an example of an enumerated proceedings article , an informally published work , a doctoral dissertation , a master's thesis: , an online document / world wide web resource , , , a video game (Case 1) and (Case 2) and and (Case 3) a patent , work accepted for publication , 'YYYYb'-test for prolific author and .",
"Other cites might contain 'duplicate' DOI and URLs (some SIAM articles) .",
"Boris / Barbara Beeton: multi-volume works as books and .",
"A couple of citations with DOIs: , .",
"Online citations: , , ."
],
[
"Tables",
"Because tables cannot be split across pages, the best placement for them is typically the top of the page nearest their initial cite.",
"To ensure this proper “floating” placement of tables, use the environment table to enclose the table's contents and the table caption.",
"The contents of the table itself must go in the tabular environment, to be aligned properly in rows and columns, with the desired horizontal and vertical rules.",
"Again, detailed instructions on tabular material are found in the User's Guide.",
"Immediately following this sentence is the point at which Table REF is included in the input file; compare the placement of the table here with the table in the printed output of this document.",
"Table: Frequency of Special CharactersTo set a wider table, which takes up the whole width of the page's live area, use the environment table* to enclose the table's contents and the table caption.",
"As with a single-column table, this wide table will “float” to a location deemed more desirable.",
"Immediately following this sentence is the point at which Table REF is included in the input file; again, it is instructive to compare the placement of the table here with the table in the printed output of this document.",
"Table: Some Typical CommandsIt is strongly recommended to use the package booktabs and follow its main principles of typography with respect to tables: Never, ever use vertical rules.",
"Never use double rules.",
"It is also a good idea not to overuse horizontal rules."
],
[
"Figures",
"Like tables, figures cannot be split across pages; the best placement for them is typically the top or the bottom of the page nearest their initial cite.",
"To ensure this proper “floating” placement of figures, use the environment figure to enclose the figure and its caption.",
"This sample document contains examples of .eps files to be displayable with .",
"If you work with pdf, use files in the .pdf format.",
"Note that most modern systems will convert .eps to .pdf for you on the fly.",
"More details on each of these are found in the Author's Guide.",
"Figure: A sample black and white graphic.Figure: A sample black and white graphicthat has been resized with the includegraphics command.As was the case with tables, you may want a figure that spans two columns.",
"To do this, and still to ensure proper “floating” placement of tables, use the environment figure* to enclose the figure and its caption.",
"And don't forget to end the environment with figure*, not figure!",
"Figure: A sample black and white graphicthat needs to span two columns of text.Figure: A sample black and white graphic that hasbeen resized with the includegraphics command."
],
[
"Theorem-like Constructs",
"Other common constructs that may occur in your article are the forms for logical constructs like theorems, axioms, corollaries and proofs.",
"ACM uses two types of these constructs: theorem-like and definition-like.",
"Here is a theorem: Let $f$ be continuous on $[a,b]$ .",
"If $G$ is an antiderivative for $f$ on $[a,b]$ , then $\\int ^b_af(t)\\,dt = G(b) - G(a).$ Here is a definition: If $z$ is irrational, then by $e^z$ we mean the unique number that has logarithm $z$ : $\\log e^z = z.$ The pre-defined theorem-like constructs are theorem, conjecture, proposition, lemma and corollary.",
"The pre-defined definition-like constructs are example and definition.",
"You can add your own constructs using the amsthm interface .",
"The styles used in the \\theoremstyle command are acmplain and acmdefinition.",
"Another construct is proof, for example, Suppose on the contrary there exists a real number $L$ such that $\\lim _{x\\rightarrow \\infty } \\frac{f(x)}{g(x)} = L.$ Then $l=\\lim _{x\\rightarrow c} f(x)= \\lim _{x\\rightarrow c}\\left[ g{x} \\cdot \\frac{f(x)}{g(x)} \\right]= \\lim _{x\\rightarrow c} g(x) \\cdot \\lim _{x\\rightarrow c}\\frac{f(x)}{g(x)} = 0\\cdot L = 0,$ which contradicts our assumption that $l\\ne 0$ ."
],
[
"Conclusions",
"This paragraph will end the body of this sample document.",
"Remember that you might still have Acknowledgments or Appendices; brief samples of these follow.",
"There is still the Bibliography to deal with; and we will make a disclaimer about that here: with the exception of the reference to the book, the citations in this paper are to articles which have nothing to do with the present subject and are used as examples only."
],
[
"Headings in Appendices",
"The rules about hierarchical headings discussed above for the body of the article are different in the appendices.",
"In the appendix environment, the command section is used to indicate the start of each Appendix, with alphabetic order designation (i.e., the first is A, the second B, etc.)",
"and a title (if you include one).",
"So, if you need hierarchical structure within an Appendix, start with subsection as the highest level.",
"Here is an outline of the body of this document in Appendix-appropriate form: Generated by bibtex from your .bib file.",
"Run latex, then bibtex, then latex twice (to resolve references) to create the .bbl file.",
"Insert that .bbl file into the .tex source file and comment out the command \\thebibliography."
],
[
"More Help for the Hardy",
"Of course, reading the source code is always useful.",
"The file acmart.pdf contains both the user guide and the commented code.",
"The authors would like to thank Dr. Yuhua Li for providing the matlab code of the BEPS method.",
"The authors would also like to thank the anonymous referees for their valuable comments and helpful suggestions.",
"The work is supported by the GS501100001809National Natural Science Foundation of Chinahttp://dx.doi.org/10.13039/501100001809 under Grant No.",
": GS50110000180961273304 and [http://www.nnsf.cn/youngscientsts]GS501100001809Young Scientsts' Support Program."
]
] | 1709.01683 |
[
[
"Global linear-irreversible principle for optimization in finite-time\n thermodynamics"
],
[
"Abstract There is intense effort into understanding the universal properties of finite-time models of thermal machines---at optimal performance---such as efficiency at maximum power, coefficient of performance at maximum cooling power, and other such criteria.",
"In this letter, a {\\it global} principle consistent with linear irreversible thermodynamics is proposed for the whole cycle---without considering details of irreversibilities in the individual steps of the cycle.",
"This helps to express the total duration of the cycle as $\\tau \\propto {\\bar{Q}^2}/{\\Delta_{\\rm tot} S}$, where $\\bar{Q}$ models the effective heat transferred through the machine during the cycle, and $\\Delta_{\\rm tot} S$ is the total entropy generated.",
"By taking $\\bar{Q}$ in the form of simple algebraic means (such as arithmetic and geometric means) over the heats exchanged by the reservoirs, the present approach is able to predict various standard expressions for figures of merit at optimal performance, as well as the bounds respected by them.",
"It simplifies the optimization procedure to a one-parameter optimization, and provides a fresh perspective on the issue of universality at optimal performance, for small difference in reservoir temperatures.",
"As an illustration, we compare performance of a partially optimized four-step endoreversible cycle with the present approach."
],
[
"Global Linear-irreversible Principle for Optimization in Finite-time Thermodynamics Ramandeep S. Johal [email protected] Department of Physical Sciences, Indian Institute of Science Education and Research Mohali, Sector 81, S.A.S.",
"Nagar, Manauli PO 140306, Punjab, India There is intense effort into understanding the universal properties of finite-time models of thermal machines—at optimal performance—such as efficiency at maximum power, coefficient of performance at maximum cooling power, and other such criteria.",
"In this letter, a global principle consistent with linear irreversible thermodynamics is proposed for the whole cycle—without considering details of irreversibilities in the individual steps of the cycle.",
"This helps to express the total duration of the cycle as $\\tau \\propto {\\bar{Q}^2}/{\\Delta _{\\rm tot} S}$ , where $\\bar{Q}$ models the effective heat transferred through the machine during the cycle, and $\\Delta _{\\rm tot} S$ is the total entropy generated.",
"By taking $\\bar{Q}$ in the form of simple algebraic means (such as arithmetic and geometric means) over the heats exchanged by the reservoirs, the present approach is able to predict various standard expressions for figures of merit at optimal performance, as well as the bounds respected by them.",
"It simplifies the optimization procedure to a one-parameter optimization, and provides a fresh perspective on the issue of universality at optimal performance, for small difference in reservoir temperatures.",
"As an illustration, we compare performance of a partially optimized four-step endoreversible cycle with the present approach.",
"Introduction: As the demand of human civilization for useful energy grows, it becomes more urgent to understand and improve the performance of our energy-conversion devices.",
"Currently, there is a lot of interest in characterizing the optimal performance of machines operating in finite-time cycles [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].",
"As a paradigmatic model, a heat cycle is studied between two heat reservoirs at temperatures $T_h$ and $T_c (<T_h)$ , whose performance may be optimized using a specific objective function, such as: power output [1], cooling power [16], certain trade-off functions between energy gains and losses [21], [19], and net entropy generation [22].",
"An important quantity in this regard is the figure of merit, such as efficiency, in case of heat engines, and the coefficient of performance (COP) in the case of refrigerators.",
"Notably, the bounds on their values predicted by simple, thermodynamic models, provide a benchmark for the observed performance of real power plants [1], [4], [8], [10], [23].",
"A major focus has been to understand whether these figures of merit display universal properties at optimal performance.",
"For example, the efficiency at maximum power (EMP) is often found to be equal to, or closely approximated by the elegant expression, known as Curzon-Ahlborn (CA) efficiency [24], [25], [26], [1] $\\eta _{\\rm CA} = 1 - \\sqrt{1-\\eta _{\\rm C}},$ where $\\eta _{\\rm C} = 1-T_c/T_h$ is the Carnot efficiency.",
"Other expressions for EMP have also been obtained from different models [3], [6], [7], [27], [8], [12], [28], some of them sharing a common universality with CA efficiency, i.e.",
"for small differences in reservoir temperatures, EMP can be written as: $\\eta = \\eta _{\\rm C}/2 + \\eta _{\\rm C}^2/8 + {\\cal O}[\\eta _{\\rm C}^3]$ .",
"The first-order term arises with strong coupling under linear irreversible thermodynamics (LIT) [6], while the second-order term is beyond linear response, and has been related to a certain symmetry property in the model [30], [28].",
"The standard analysis often involves solving a two-parameter optimization problem over, say, the pair of intermediate temperatures of the working medium [1], [3], or, the time intervals of contacts with reservoirs [7], [8].",
"In this letter, I formulate a simpler optimization problem for finite-time machines.",
"While simplicity might lead to a certain loss of predictive power, the generality and unifying power of the proposed framework are remarkable.",
"Here, rather than applying LIT locally, say, at each thermal contact, I assume a global validity of this principle, i.e.",
"for the complete cycle.",
"Accordingly, we need not assume stepwise details of the cycle, but may simply consider the machine as an irreversible channel with an effective (thermal) conductivity $\\lambda $ .",
"Quite remarkably, we will recover many of the well-known expressions of the figures of merit for both engine and refrigerator modes, indicating a very different origin for them which is independent of the physical model, or the processes assumed in regular models.",
"In this sense, the present approach brings together the results from different models under one general principle consistent with LIT.",
"Now, according to LIT [31], [32], the rate of entropy generation is $\\dot{S} = \\sum _{\\alpha } q_{\\alpha } F_{\\alpha }$ , the sum of products of each flux $q_{\\alpha }$ and its associated thermodynamic force, or affinity $F_{\\alpha }$ .",
"In the simple case of a heat flux $q$ between two heat reservoirs, the corresponding thermodynamic force may be defined as: $F = T_{c}^{-1} - T_{h}^{-1}$ .",
"Then, assuming a linear flux-force relation, $q = \\lambda F$ , where $\\lambda $ is the heat-transfer coefficient, and the bilinear form for the rate of entropy generation, we can write $\\dot{S} = q F = q^2/\\lambda $ .",
"Now, if the time interval of heat flow is considered long enough, then the flux may be approximated to be constant over this interval.",
"So the amount of heat transferred within time $\\tau $ is: $Q = q \\tau $ , which implies $\\dot{S} = Q^{2} / \\lambda \\tau ^2$ .",
"Then, the cyclic operation of the machine can be based on the following two assumptions: (i) there is an effective heat flux ($\\bar{q}$ ) through the machine over one cycle, and the total entropy generation per cycle obeys principles of LIT.",
"(ii) $\\bar{q}$ is determined by an effective, mean value of the heat ($\\bar{Q}$ ) passing from the hot to the cold reservoir in total cycle time $\\tau $ .",
"Therefore, $\\bar{q} = \\bar{Q}/\\tau $ .",
"Assumption (i) implies, $\\dot{S}_{\\rm tot} = \\bar{q}^2 /\\lambda $ .",
"Then, the total entropy generation per cycle is: $\\Delta _{\\rm tot} S = \\dot{S}_{\\rm tot} \\tau = \\bar{q}^2 \\tau /\\lambda $ .",
"From assumption (ii), we have $\\Delta _{\\rm tot} S = \\bar{Q}^2 /\\lambda \\tau $ .",
"In other words, we can express the total period as: $\\tau = \\frac{\\bar{Q}^2}{\\lambda \\Delta _{\\rm tot} S}.$ Optimal power output: Now, to motivate the optimization procedure, we first optimize the power output of a heat engine.",
"Let $Q_h$ and $Q_c$ be the amounts of heat exchanged by the working medium with the hot ($h$ ) and cold ($c$ ) reservoirs.",
"Let $W= Q_h - Q_c$ be the total work output in a cycle of duration $\\tau $ .",
"The total entropy generation per cycle is: $\\Delta _{\\rm tot} S = -\\frac{Q_h}{T_h} + \\frac{Q_c}{T_c} >0.$ Then the average power output of the cycle is defined as $P = W/\\tau $ .",
"Using Eq.",
"(REF ), we have $P = \\lambda (Q_h -Q_c) \\frac{\\Delta _{\\rm tot} S}{\\bar{Q}^2}.$ Introducing the parameters $\\nu = Q_c/Q_h$ and $\\theta = T_c/T_h$ , and using Eq.",
"(REF ), we can define a dimensionless power function: $\\tilde{P} \\equiv \\frac{T_c P}{\\lambda } =(1-\\nu )(\\nu -\\theta )\\frac{Q_{h}^{2}}{\\bar{Q}^2}.$ For engines, from the positivity of $\\Delta _{\\rm tot} S$ , we have $\\nu > \\theta $ .",
"Further, knowledge of a specific value of $\\lambda $ is not relevant for performing the optimization.",
"However, the above target function is still not in a useful form, until we specify the form of $\\bar{Q}$ .",
"We assume $\\bar{Q}$ to be bounded as: $ Q_c \\le \\bar{Q} \\le Q_h$ .",
"Let us analyze these two limits separately.",
"${\\rm L_h}$ ) When $\\bar{Q} \\rightarrow Q_h$ , Eq.",
"(REF ) is simplified to: $\\tilde{P} = (1-\\nu )(\\nu -\\theta )$ , which becomes optimal ($\\partial \\tilde{P}/ \\partial \\nu =0 $ ) at $\\nu = (1+\\theta )/2$ .",
"Thus the EMP in this limit is $\\eta _l = \\eta _{\\rm C}/2$ .",
"This formula is obtained when the dissipation at the cold contact is much more important than at the hot contact [3], [7], [8], [12], [28].",
"In our model, this limit implies that when the effective amount of heat passing through the machine approaches the heat absorbed from the hot reservoir, then the efficiency approaches its lower bound.",
"${\\rm L_c}$ ) When $\\bar{Q} \\rightarrow Q_c$ , then $\\tilde{P} = (1-\\nu )(\\nu -\\theta ){\\nu }^{-2}$ , whose optimal value is obtained at $\\nu = 2\\theta /(1+\\theta )$ , or the EMP in this case is $ \\eta _u = \\eta _{\\rm C}/(2-\\eta _{\\rm C})$ .",
"Again, this formula is obtained as the upper bound for efficiency [3], [7], [8], [12], [28], when the dissipation at the cold contact approaches the reversible limit, or, is negligible in comparison to dissipation at the hot contact.",
"In our model, $\\bar{Q} \\rightarrow Q_c$ implies that as the effective heat through the machine reaches its lowest possible value $Q_c$ , the efficiency approaches its upper bound.",
"Now, it seems natural to assume that, in general, $\\bar{Q} \\equiv \\bar{Q}(Q_h,Q_c)$ may be taken as a mean value [29], interpolating between $Q_c$ and $Q_h$ .",
"In the following, we explore consequences of making some simple choices of these mean values.",
"It will be seen that the mean being a homogeneous function of the first degree in its arguments, implies the condition $\\bar{Q}(Q_h,Q_c) = Q_h \\bar{Q}(1,\\nu )$ , and so the maximization of the power output is reduced to a simple one-parameter optimization problem.",
"Let $\\bar{Q}$ be given by a weighted arithmetic mean: $\\bar{Q} = \\omega Q_h + (1-\\omega ) Q_c$ , where the weight $0\\le \\omega \\le 1$ .",
"With this form, Eq.",
"(REF ) is explictly given by: $\\tilde{P}(\\nu ) = \\frac{(1-\\nu )(\\nu -\\theta )}{[\\omega +(1-\\omega )\\nu ]^2}.$ Then the optimum of $\\tilde{P}$ is obtained at: $\\tilde{\\nu } = [{2 \\theta + \\omega (1-\\theta )}]/[{1+\\theta + \\omega (1-\\theta )}]$ .",
"The maximal nature of the optimum can be ascertained: $\\left.",
"\\partial ^2 \\tilde{P}/ \\partial \\nu ^2 \\right|_{\\nu = \\tilde{\\nu }} <0$ .",
"As a result, EMP, $\\tilde{\\eta } = 1-\\tilde{\\nu }$ , is found to be: $\\tilde{\\eta } = \\frac{\\eta _{\\rm C}}{2-(1-\\omega )\\eta _{\\rm C}}.$ The above form has been obtained in Refs.",
"[3], [7], [8], [12], [28], where the parameter $\\omega $ may be determinable, for example, in terms of the ratio of the dissipation constants or thermal conductivities of the thermal contacts [3], [8].",
"In the present approach, the parameter $0\\le \\omega \\le 1$ is undetermined.",
"In the absence of additional information, one may choose equal weights ($\\omega =1/2$ ), or the symmetric arithmetic mean $\\bar{Q} = (Q_h+ Q_c)/2$ .",
"This results in the so-called Schmiedl-Seifert (SS) efficiency $\\tilde{\\eta }_{\\rm SS} = {2\\eta _{\\rm C}}/(4-\\eta _{\\rm C})$ [33], [7].",
"As an alternative choice, if we set $\\bar{Q} = \\sqrt{Q_h Q_c}$ , i.e.",
"the geometric mean of $Q_h$ and $Q_c$ , then we obtain $\\tilde{P} = (1-\\nu )(\\nu -\\theta )/\\nu $ , which becomes optimal at CA efficiency, $\\tilde{\\eta } =\\eta _{\\rm CA}$ .",
"Further, we may use a generalized, symmetric mean defined as: $\\bar{Q} = [(Q_{h}^{r} + Q_{c}^{r})/2]^{1/r}$ , where $r$ is a real parameter [34], [29].",
"Special cases with $r=-1,0,1,2$ corrrespond respectively to harmonic, geometric, arithmetic, and quadratic means.",
"The dimensionless power output is then given by: $\\tilde{P}(\\nu ) = \\frac{(1-\\nu )(\\nu -\\theta )}{[(1 +\\nu ^r)/2]^{2/r}},$ whose optimum is determined by following condition: $(1+\\theta ){\\tilde{\\nu }}^r -2 \\theta {\\tilde{\\nu }}^{r-1}+ 2 \\tilde{\\nu } - (1+\\theta ) = 0.$ The above equation cannot be analytically solved for $\\tilde{\\nu } =1 - \\tilde{\\eta }_r$ , with general $r$ .",
"However, it can be shown from (REF ) that $\\partial \\tilde{\\nu }/\\partial r > 0$ , which implies that EMP $\\tilde{\\eta }_r$ decreases monotonically with increasing $r$ .",
"In particular, when $r\\rightarrow +\\infty (-\\infty )$ , then $\\bar{Q} \\rightarrow Q_h (Q_c)$ and so we have $\\tilde{\\eta }_{\\infty } \\rightarrow \\eta _{\\rm C}/2$ , and $\\tilde{\\eta }_{-\\infty } \\rightarrow \\eta _{\\rm C}/(2-\\eta _{\\rm C})$ , the lower and upper bounds of EMP discussed earlier.",
"Incidentally, this helps to notice that CA efficiency ($r \\rightarrow 0$ ) is higher than SS efficiency ($r=1$ ), for a given $\\theta $ (see Fig.",
"REF ).",
"For small numerical values of $r$ , we can look into the general form of efficiency by assuming $\\tilde{\\eta }_r = a_1 \\eta _{\\rm C} + a_2 \\eta _{\\rm C}^{2}+ a_3 \\eta _{\\rm C}^{3}$ , close to equilibrium, where we have $\\eta _{\\rm C}$ as the small parameter.",
"Substituting this form in Eq.",
"(REF ), and keeping terms upto ${\\cal O}( \\eta _{\\rm C}^{3})$ , we determine the coefficients $a_i$ .",
"As a result, for small temperature differences, we have $\\tilde{\\eta }_r = \\frac{\\eta _{\\rm C}^{}}{2} + \\frac{\\eta _{\\rm C}^{2}}{8}+ \\frac{2-r}{32}\\eta _{\\rm C}^{3}.$ Thus, we observe that with some well-known symmetric means, the first two terms in the expansion of EMP have the same universal coefficients as were earlier obtained within LIT [6], [30].",
"Interestingly, the third-order term in Eq.",
"(REF ) also matches with a similar expansion of EMP recently found for Carnot engines within a perturbative approach for open quantum systems [35].",
"Thereby, the parameter $r$ is analogous to the frequency scaling exponent of the spectral density of the heat reservoirs.",
"So, assuming the global validity of LIT for a heat cycle, along with a specific form of the effective heat transferred between the two heat reservoirs, leads to a one-parameter optimization problem, whereby the EMP matches with the well-known expressions predicted by more elaborate models.",
"This is the main result of the present letter.",
"In the following, we extend the analysis to other target functions, and derive various known forms of the figures of merit.",
"Figure: EMP, η ˜ r \\tilde{\\eta }_r, versus Carnot efficiencyη C \\eta _{\\rm C}, for different values of the parameter rrof the generalized mean.",
"The upper (lower) boundis obtained with rr approaching -∞(+∞)-\\infty (+\\infty ).In between these bounds, from top to bottom, r=0,1/2,1,2r= 0, 1/2, 1, 2respectively, depicting that for a given ratio ofreservoir temperatures, θ\\theta , EMP decreasesmonotonically with rr.",
"Inset: The shaded regiondepicts the efficiency at optimal ecological criterion,bounded from below and above as in Eq.",
"().The dashed curve inside the shaded region, depicts Eq.",
"(),obtained with the geometric-mean value Q ¯=Q h Q c \\overline{Q} = \\sqrt{Q_h Q_c}.Optimal ecological criterion: Although, maximizing power output may be regarded as a reasonable goal for a finite-time engine, we note that such an engine also leads to irreversible entropy generation.",
"So, for a cleaner production of useful energy, the so-called ecological criterion provides a good compromise between energy benefits and losses in thermal machines [21], [36].",
"For the specific case of a two heat-reservoir setup, it is also equivalent to the so-called $\\dot{\\Omega }$ -criterion [37], [19].",
"For a heat engine, the criterion is given by: $\\dot{\\Omega } = (2 \\eta - \\eta _{\\rm C}){Q_h}/{\\tau }$ , or, in dimensionless form $\\tilde{\\dot{\\Omega }} = \\frac{T_c \\dot{\\Omega }}{\\lambda } =\\frac{(1+\\theta -2 \\nu )(\\nu -\\theta )}{[\\omega +(1-\\omega )\\nu ]^2}.$ Again, to obtain the optimal working condition, we set $\\partial {\\tilde{\\dot{\\Omega }}}/ \\partial \\nu =0$ , and thus obtain the efficiency at optimal ${\\dot{\\Omega }}$ as [19]: $\\tilde{\\eta } = \\frac{3-2\\eta _{\\rm C} +2 \\omega \\eta _{\\rm C}}{4- 3\\eta _{\\rm C} + 3\\omega \\eta _{\\rm C}}\\eta _{\\rm C}.$ The upper and lower bounds of (REF ) are given by $\\frac{3}{4} \\eta _{\\rm C} \\le \\tilde{\\eta } \\le \\frac{3-2 \\eta _{\\rm C}}{4- 3\\eta _{\\rm C}}\\eta _{\\rm C} = \\eta ^*.$ For small temperature differences, the upper bound $\\eta ^*$ behaves as ${\\eta }^* = {3} \\eta _{\\rm C}/4 + {\\eta _{\\rm C}^{2}}/{16}+ {\\cal O}(\\eta _{\\rm C}^{3})$ .",
"On the other hand, the use of geometric mean for the case of ecological criterion for an engine yields $\\tilde{\\dot{\\Omega }} = (1+\\theta - 2 \\nu )(\\nu -\\theta )/{\\nu }$ , whose optimum is obtained at $\\nu = \\sqrt{\\theta (1+\\theta )/2}$ .",
"So the efficiency at optimal ecological criterion is then given by [36], [38]: $\\tilde{\\eta } = 1-\\sqrt{\\frac{(1-\\eta _{\\rm C})(2-\\eta _{\\rm C})}{2}}.$ In this case, for small temperature differences, the leading order terms are given as $\\tilde{\\eta } = {3}\\eta _{\\rm C}/4 + {\\eta _{\\rm C}^{2}}/{32}+ {\\cal O}(\\eta _{\\rm C}^{3})$ .",
"The exact behavior is plotted as the dashed line within the inset, in Fig.",
"1.",
"Next, we consider the operation of a heat cycle as a refrigerator.",
"For refrigerators, the coefficient of performance (COP) is given by $\\xi = {\\cal Q}_c/{\\cal W} = \\nu /(1-\\nu )$ , where $\\nu = {\\cal Q}_c/{\\cal Q}_h$ , and the Carnot coefficient is $\\xi _{\\rm C} = \\theta /(1-\\theta )$ .",
"Here, ${\\cal Q}_c$ is the heat extracted from the cold reservoir and driven into the hot reservoir by an input of ${\\cal W}$ amount of work, in a cycle of time period $\\tau $ .",
"The total entropy generation per cycle is: $\\Delta _{\\rm tot} {\\cal S} = \\frac{ {\\cal Q}_h}{T_h}- \\frac{ {\\cal Q}_c}{T_c} >0.$ Now, for irreversible refrigerators, the choice of optimization criteria that may be analogous to heat engines, is not straightforward.",
"For instance, the cooling power or the rate of refrigeration, ${\\cal Q}_c /\\tau $ , may not have an optimum for certain models [39], [16].",
"Below, we illustrate the optimization with some of the proposed choices.",
"Optimal $\\chi $ -criterion: The so-called $\\chi $ -criterion is defined as [39], [11] $\\chi = {\\xi Q_c}/{\\tau }$ .",
"This criterion simultaneously considers the COP and the cooling power, due to a certain complementarity between the two quantities, i.e.",
"if we maximize one, it minimizes the other.",
"Defining, $\\tilde{\\chi } = T_c \\chi /\\lambda $ , we have: $\\tilde{\\chi } = \\frac{\\nu ^2(\\theta -\\nu )}{(1-\\nu )[\\omega +(1-\\omega )\\nu ]^2}.$ Note that for an irreversible refrigerator, we have $\\theta > \\nu $ .",
"Then, setting $\\partial \\tilde{\\chi }/ \\partial \\nu =0$ , we can obtain the COP at optimal performance: $\\tilde{\\xi } = \\frac{\\sqrt{\\omega }}{2}\\left( \\sqrt{9 \\omega + 8 \\xi _{\\rm C}} - 3 \\sqrt{\\omega }\\right).$ As $0 \\le \\omega \\le 1$ , it implies the bounds on COP as: $0 \\le \\tilde{\\xi } \\le \\frac{1}{2}\\left( \\sqrt{9 + 8 \\xi _{\\rm C}} - 3 \\right),$ which have been earlier obtained in the literature, from a two-parameter optimization procedure.",
"Note that, here we obtain a simple closed form for $\\tilde{\\xi }$ , which has not been possible in the low-dissipation approach of Ref.",
"[11].",
"Optimal cooling power: The target function for a refrigerator may be chosen as the cooling power $Z = Q_c/\\tau $ , or in dimensionless form $\\tilde{Z} = \\frac{T_c Z}{\\lambda } =\\frac{\\nu (\\theta -\\nu )}{[\\omega +(1-\\omega )\\nu ]^2}.$ Then, setting $\\partial \\tilde{Z}/ \\partial \\nu =0$ , we obtain COP at optimal $Z$ to be: $\\tilde{\\xi } = \\frac{\\omega \\xi _{\\rm C}}{2 \\omega + \\xi _{\\rm C}}.$ The above expression is exactly the one derived in Ref.",
"[16] for the so-called exoreversible refrigerator, with the consequent bounds on COP as: $0 \\le \\tilde{\\xi }\\le \\xi _{\\rm C}/(2+ \\xi _{\\rm C})$ .",
"When the ecological or the trade-off criterion for the refrigerator [37] is implemented as: ${\\dot{\\Omega }}_R = (2\\xi - \\xi _{\\rm C}) {\\cal W}/\\tau $ , with $\\bar{Q}$ as a geometric mean, then we obtain the COP at optimal $\\dot{{\\Omega }}_R$ [37], [40], as: $\\tilde{\\xi } = \\frac{\\xi _{\\rm C}}{\\sqrt{(1+\\xi _{\\rm C})(2+\\xi _{\\rm C})}-\\xi _{\\rm C}}.$ Similarly, it can be seen that for a refrigerator and with geometric mean, we obtain $\\tilde{\\chi } = \\nu (\\theta -\\nu )/(1-\\nu )$ , whose optimal value is obtained at $\\tilde{\\xi } = \\sqrt{1+ \\xi _{\\rm C}} -1$ [10].",
"On the other hand, the cooling power does not have an optimum if the geometric mean is used.",
"An illustration: In the following, we compare the above effective approach with the optimization of power output in a four-step cycle within the so-called endoreversible approximation [1], [3], [9].",
"The latter implies that the only sources of irreversibilities during the cycle happen to be the thermal contacts with the heat reservoirs, when heat is exchanged between reservoirs and the working medium across a finite heat conductance.",
"Consequently, there are two such thermal steps, occuring with time intervals $t_h$ and $t_c$ .",
"Further, the working medium is assumed to maintain a fixed temperature during a specific thermal contact, which we assume to be $T_1$ and $T_2$ during hot and cold contact respectively ($T_h > T_1 > T_2 > T_c$ ).",
"The other two steps of the cycle are adiabatic in nature—preserving the entropy of the working medium—and are assumed to occur with a negligible time interval.",
"Now, the problem of power optimization for the above model involves two variables, which may be conveniently chosen as the two temperatures of the working medium.",
"Equivalently, we may choose another pair of variables as discussed below.",
"Consider the heat flux between the working medium and the respective reservoir to be given by [3] $q_h & = & \\alpha _h \\left( T_{1}^{-1} - T_{h}^{-1} \\right), \\\\q_c & = & \\alpha _c \\left( T_{c}^{-1} - T_{2}^{-1} \\right), $ where $\\alpha _j >0$ , with $j=c,h$ are the heat-transfer coefficients.",
"As the flux is assumed to be constant during the thermal contact, so the amounts of heat transferred during the times $t_h$ and $t_c$ , respectively are: $Q_h = q_h t_h$ and $Q_c = q_c t_c$ .",
"Further, the cyclicity within the working medium implies ${Q_h}/{T_1} = {Q_c}/{T_2}$ , also known as the endoreversibility condition.",
"The extracted work per cycle is $W = Q_h-Q_c$ , with thermal efficiency equal to $\\eta = \\frac{W}{Q_h} = 1-\\frac{T_2}{T_1} \\equiv 1- \\nu .$ Then, the average power per cycle is defined as $P = \\frac{Q_h - Q_c}{t_h + t_c}.$ which can be expressed as a function of $T_1$ and $T_2$ , or equivalently (from Eq.",
"(REF )), as a function of $T_1$ and $\\nu $ : $P\\equiv P(T_1, \\nu )$ .",
"Rather than optimizing the power w.r.t the two variables, let us consider a partial optimization, by setting ($\\partial P /\\partial T_1)_{\\nu } =0$ , which yields the optimum value of $T_1$ : $\\tilde{T}_1 = \\frac{K + {\\nu }^{-1}}{K T_{h}^{-1} + T_{c}^{-1}},$ where $K = \\sqrt{\\alpha _h / \\alpha _c}$ .",
"Using the above value, we obtain ${P} (\\nu ) = P(\\tilde{T}_1, \\nu )$ , as ${P} (\\nu ) = \\frac{\\alpha _h}{T_{c}^{}} \\frac{(1-\\nu ) (\\nu -\\theta ) }{[1+ K \\nu ]^2},$ which can be rewritten as: ${P} (\\nu ) = \\frac{\\alpha _h \\alpha _c}{T_{c}^{} (\\sqrt{\\alpha _h} + \\sqrt{\\alpha _c})^2 }\\frac{(1-\\nu ) (\\nu -\\theta ) }{[\\omega + (1-\\omega ) \\nu ]^2},$ where $\\omega = (1+K)^{-1}$ .",
"We identify $\\lambda = \\alpha _h \\alpha _c {(\\sqrt{\\alpha _h} + \\sqrt{\\alpha _c})^{-2}}$ , and so we can compare $T_c P(\\nu )/\\lambda $ in the above with the reduced power in Eq.",
"(REF ).",
"Thus we note that the partially optimized power output in the endoreversible model with the inverse-temperature law, is equivalent to our effective model that employs a weighted arithmetic mean for the effective transferred heat.",
"Similarly, it can be shown that within the endoreversible model under the assumption of Newtonian heat transfer [1], [3], $q_h & = & \\alpha _{h}^{^{\\prime }} \\left( T_{h}^{} - T_{1}^{} \\right), \\\\q_c & = & \\alpha _{c}^{^{\\prime }} \\left( T_{2}^{} - T_{c}^{} \\right), $ where $\\alpha _{j}^{^{\\prime }} >0$ are thermal conductivities, the partially optimized power is obtained in a form that implies the effective heat as a geometric mean $\\bar{Q} = \\sqrt{Q_h Q_c}$ , with the effective value of $\\lambda =T_h T_c \\alpha _{h}^{^{\\prime }} \\alpha _{c}^{^{\\prime }} {(\\sqrt{\\alpha _{h}^{^{\\prime }}} + \\sqrt{\\alpha _{c}^{^{\\prime }}})^{-2}}$ .",
"Concluding remarks and outlook: An effective framework has been proposed for a class of thermal machines based on LIT, that makes a novel use of algebraic means [41] to model the effective heat transferred in a cycle.",
"Quite surprisingly, the method reproduces well-known bounds and expressions for figures of merit—both for the engines as well as refrigerators, without incorporating details of a specific heat cycle.",
"This has been illustrated for various optimization criteria.",
"These general results arising from a simple framework, thereby suggest a universal character of these figures of merit.",
"The approach also provides a fresh perspective on the issue of universality of EMP near equilibrium [6], [30], [28], as in Eq.",
"(REF ).",
"The first order term ($\\eta _C/2$ ) is universal, for any mean $\\bar{Q}$ , with extremal values of $Q_h$ and $Q_c$ .",
"Then, the universality of the second-order term can be related to the property of the mean $\\bar{Q}$ being symmetric.",
"Further, an agreement upto third-order term has been seen within an open quantum systems framework.",
"Thus, the present approach could be relevant for the optimal regimes of quantum heat engines, where such efficiencies and bounds have been recently derived [35].",
"It is desirable that the simple approach proposed above can be generalized to include more realistic scenarios, such as allowing for heat leaks between the reservoirs, finite sizes of heat source/sink, and the nonlinear regime.",
"In particular, it is important to understand the physical reason as to the use of different means leading to varied expressions for the figures of merit.",
"Here, the comparison with the partially optimized endoreversible cycle can provide a useful insight.",
"Finally, it will be good if the proposed approach encourages a more global perspective while regarding the operation of machines and their impact on our environment."
]
] | 1709.01893 |
[
[
"Induced Waveform Transitions of Dissipative Solitons"
],
[
"Abstract The effect of an externally applied force upon dynamics of dissipative solitons is analyzed in the framework of the one-dimensional cubic-quintic complex Ginzburg-Landau equation supplemented by a linear potential term.",
"The potential accounts for the external force manipulations and consists of three symmetrically arranged potential wells whose depth is considered to be variable along the longitudinal coordinate.",
"It is found out that under an influence of such potential a transition between different soliton waveforms coexisted under the same physical conditions can be achieved.",
"A low-dimensional phase-space analysis is applied in order to demonstrate that by only changing the potential profile, transitions between different soliton waveforms can be performed in a controllable way.",
"In particular, it is shown that by means of a selected potential, propagating stationary dissipative soliton can be transformed into another stationary solitons as well as into periodic, quasi-periodic, and chaotic spatiotemporal dissipative structures."
],
[
"Introduction",
"The complex Ginzburg-Landau equation (CGLE) arises in many fields of sciences including nonlinear optics, semiconductor devises, Bose-Einstein condensates, superconductivity, reaction-diffusion systems, quantum field theories and occupies a prominent place in the theory of nonlinear evolution equations [1], [2], [3], [4].",
"Moreover, the standard cubic CGLE supplemented by higher-order nonlinear terms forms a universal mathematical model used to effectively describe diverse nonlinear phenomena possessing complex dynamical behaviors.",
"In fact, being a nonintegrable dynamical system near a subcritical bifurcation the one-dimensional cubic-quintic CGLE admits a variety of stable localized solutions including formation of periodic and quasi-periodic patterns, as well as a spatiotemporal chaos [5].",
"These solutions represent different forms of dissipative solitons [6], [7], [8], which appear as stable localized structures existing due to a balance between gain and loss in distributed nonlinear dynamical systems far from equilibrium.",
"Dissipative solitons may stably evolve as stationary zero-velocity solitons (so-called `plain pulse' and `composite pulse' [9]) and moving solitons [10], [11], [12], periodically and quasi-periodically pulsating solitons with simple or more complicated behaviors [13], [14], chaotic solitons [13], [14], and exploding solitons, which periodically manifest explosive instabilities returning to their original waveforms after each explosion [15], [14], [12], [16].",
"Furthermore, the cubic-quintic CGLE has multisoliton solutions [17] and admits solutions in many different waveforms of fronts, sources, sinks, and bound states [18], [19], [20], [11], [21], [22].",
"The available variety of dissipative solitons can be controlled by means of an externally applied force which influences their waveforms and, thus, can change behaviors of the soliton evolution.",
"Independently on the physical nature, the effect of an external force can be described considering soliton waveform evolution that occurs in a corresponding external potential.",
"For instant, such a concept is used to manage solitons in nonlinear optical systems and Bose-Einstein condensates [23].",
"In order to split optical spatial solitons governed by the nonlinear Schrödinger equation into two solitons a longitudinal defect [24], an external delta potential [25], and a longitudinal potential barrier [26] are applied to scatter a single soliton.",
"Various scenarios of the dynamics of dissipative solitons interacting with a sharp potential barrier in the cubic-quintic CGLE are analyzed in [27].",
"Similarly, the evolution of dissipative solitons in an active bulk medium has been studied in the framework of the two-dimensional CGLE with an umbrella-shaped [28] and a radial-azimuthal [29] potentials.",
"As an example of practical systems supporting propagation of dissipative solitons, nonlinear magneto-optic waveguides can be mentioned where an externally applied magnetic field induces corresponding potential [30], [31], [32], [33], [34], [35].",
"Significant benefits of using a spatially inhomogeneous external magnetic field to adjust nonreciprocal propagation of light dissipative solitons in magneto-optic planar waveguides have been demonstrated in [36], [37].",
"Moreover, involving an external magnetic field a new robust mechanism to perform both selective lateral shift within a group of stable dissipative solitons [38] and their cascade replication [39] is recently proposed.",
"In this paper we employ the one-dimensional cubic-quintic CGLE supplemented by a linear potential term in order to reveal the effects of external force which influences upon dissipative solitons.",
"We demonstrate that applying particular inhomogeneous potential wells can cause nontrivial transitions between different soliton waveforms.",
"These transitions appear when the waveforms coexist within the same parameter space of the CGLE [9], [14], [12].",
"The goal of the paper is to perform a formalized description of peculiarities of the induced waveform transitions of dissipative solitons and related phenomena.",
"The rest of the paper is organized as follows: In Sec.",
"we formulate a common mathematical model of dissipative solitons, where a linear potential term is added to the CGLE to perform some manipulations upon the solitons.",
"The existence of regular waveform transitions of dissipative solitons caused by the weak potential is demonstrated in Sec.",
"REF .",
"In Sec.",
"REF we present and discuss peculiarities of irregular waveform transitions of dissipative solitons and the related effect that appear under an influence of the strong potential.",
"Conclusions and final remarks summarize the paper in Sec.",
"."
],
[
" Model of controllable dissipative solitons",
"We consider $1+1$ dimensional cubic-quintic CGLE supplemented by a linear potential term written in the form $\\mathrm {i}\\frac{\\partial \\Psi }{\\partial z}+\\mathrm {i}\\delta \\Psi +\\left(\\frac{D}{2}-\\mathrm {i}\\beta \\right)\\frac{\\partial ^2\\Psi }{\\partial x^2}+\\left(1-\\mathrm {i}\\varepsilon \\right)\\left|\\Psi \\right|^2\\Psi \\\\ - \\left(\\nu -\\mathrm {i}\\mu \\right)\\left|\\Psi \\right|^4\\Psi + Q(x,z)\\Psi = 0,$ where $\\Psi \\left(x,z\\right)$ is a complex amplitude of transverse $x$ and longitudinal $z$ coordinates, $D$ is the group velocity dispersion coefficient, $\\delta $ is a linear absorption, $\\beta $ is a linear diffusion, $\\varepsilon $ is a nonlinear cubic gain, $\\nu $ accounts for the self-defocusing effect due to the negative sign, and $\\mu $ defines quintic nonlinear losses.",
"The linear potential $Q(x,z)$ is considered to be in the form of arranged potential wells whose depth is varied along the longitudinal coordinate $z$ .",
"We express the profile of each potential well through the hyperbolic tangent function and write down the whole potential as some superposition of the wells $ Q(x,z) = \\sum _{i=1}^{N} M_i\\tanh \\left(\\frac{q_i(z)}{\\sqrt{(x-x_i)^2+d_i^2}}\\right),$ where $N$ is the number of wells, $x_i$ is the transverse coordinate of the $i$ -th potential well peak or deep, the profile function $q_i(z)$ defines the variation of the $i$ -th potential well along the longitudinal coordinate $z$ , and $M_i$ and $d_i$ are some constants which should be selected based on a particular physical problem under consideration.",
"We assume that the complex amplitude $\\Psi (x,z)$ satisfies to the periodic boundary condition $\\Psi (x,z) = \\Psi (x+L_x,z),~~~~(x,z)\\in \\mathbb {R}\\times \\left[0,+\\infty \\right),$ and the initial condition $\\Psi (x,0) = \\Psi _0(x),~~~~x\\in \\mathbb {R},$ where $L_x$ is some given number and the initial amplitude $\\Psi _0(x)$ satisfies to the periodic condition (REF ) as well.",
"The system (REF ) is nonconservative since its solutions (dissipative solitons) depend on an energy supplied to the system.",
"Indeed, the following continuity equation can be derived using Eq.",
"(REF ) $\\dfrac{\\partial \\rho }{\\partial z}+\\dfrac{\\partial j}{\\partial x}=p,$ where $\\rho =\\left|\\Psi (x,z)\\right|^2$ is the energy density.",
"The corresponding flux $j$ and the density of energy generation $p$ are defined as follows $j=\\dfrac{\\mathrm {i}D}{2}\\left(\\Psi \\Psi _x^*-\\Psi _x\\Psi ^*\\right),$ $p=\\beta \\left(\\left|\\Psi \\right|^2_{xx}-2\\left|\\Psi _x\\right|^2 \\right) \\\\ -2\\left(\\delta \\left|\\Psi \\right|^2-\\varepsilon \\left|\\Psi \\right|^4+\\mu \\left|\\Psi \\right|^6\\right).$ Having integrated the energy density $\\rho $ and the density of energy generation $p$ over the transverse coordinate $x$ we get two soliton parameters (moments) to observe their evolution along the $z$ -axis $E(z) = \\int _{-\\infty }^{\\infty } \\left|\\Psi (x,z)\\right|^2 dx,~~P(z) = \\int _{-\\infty }^{\\infty } p(x,z)\\,dx.$ These two parameters represent the soliton energy $E(z)$ and total generated energy $P(z)$ as functions of the propagation distance $z$ , respectively."
],
[
"Numerical analysis",
"Further we solve the problem (REF )-(REF ) numerically using the pseudospectral approach and exponential time differencing method of second order [40], [41].",
"Since we seek solutions $\\Psi (x,z)$ to Eq.",
"(REF ) in the form of dissipative solitons which are structures localized in space, we assume that the computational domain can be reduced to the finite rectangular occupying the area $\\left[-L_x/2,L_x/2\\right]\\times \\left[0,L_z\\right]$ .",
"In our numerical calculations, we sample the computational domain along the $x$ -axis with $2^{10}$ discretization points to compute the fast Fourier transform with respect to the $x$ coordinate.",
"The distance along the $z$ -axis we sample using the step $\\Delta z=10^{-3}$ .",
"The length of the simulation area along the $z$ -axis is chosen so as to ensure the completion of all intermediate unstable stages appearing between distinct waveforms transitions.",
"The computational scheme in the Fourier domain for updating the unknown complex amplitude along the longitudinal coordinate $z$ is written in the form $\\hat{\\Psi }_{n+1}=\\hat{\\Psi }_{n}e^{\\sigma \\Delta z}+\\hat{\\mathcal {N}}_n\\dfrac{\\left(1+\\sigma \\Delta z\\right)e^{\\sigma \\Delta z}-1-2\\sigma \\Delta z}{\\sigma ^2\\Delta z}\\\\+\\hat{\\mathcal {N}}_{n-1}\\dfrac{1+\\sigma \\Delta z-e^{\\sigma \\Delta z}}{\\sigma ^2\\Delta z},$ where $\\sigma =-\\delta -\\left(\\beta +\\mathrm {i}D/2\\right)k^2$ is a spectral parameter, $\\Psi _n=\\Psi (x,z_n)$ , $\\mathcal {N}_n=\\mathcal {N}\\left(\\Psi _n,x,z_n\\right)$ , $z_n=n\\Delta z$ , and $\\mathcal {N}$ stands for the nonlinear part of Eq.",
"(REF ) $\\mathcal {N}(\\Psi ,x,z)=\\left(\\left(\\mathrm {i}+\\varepsilon \\right)\\left|\\Psi \\right|^2\\right.\\left.-\\left(\\nu -\\mathrm {i}\\mu \\right)\\left|\\Psi \\right|^4+Q(x,z)\\right)\\Psi .$ The circumflex denotes the Fourier transform with respect to the coordinate $x$ , i.e.",
"$\\hat{\\Psi }(k,z)= \\mathcal {F}\\left\\lbrace \\Psi (x,z)\\right\\rbrace $ , $\\hat{\\mathcal {N}}= \\mathcal {F}\\left\\lbrace \\mathcal {N}(\\Psi ,x,z)\\right\\rbrace $ .",
"In order to excite a stable dissipative soliton a wide variety of functions whose profiles are close to the existing soliton waveform can be used as the initial condition (REF ).",
"In particular, in all our numerical calculations the initial plane pulse soliton is evolved from the simple waveform function $\\mathrm {sech}(x)$ .",
"Hereinafter we consider the case of anomalous group velocity dispersion, i.e.",
"$D=1$ .",
"Moreover, in the potential (REF ) we put $M_i=2$ and $d_i=1$ , $i=1,\\ldots ,N$ , while its dependence on the longitudinal coordinate $z$ is set to be in the form of a simple piecewise constant function $q_i(z)=A_i h(z-a_i)-B_i h(z-b_i),$ where $h(\\cdot )$ is the Heaviside step function and $A_i$ , $a_i$ , $B_i$ , and $b_i$ are some real numbers."
],
[
"Regular waveform transitions",
"In this section we consider an evolution of dissipative solitons being under an influence of the locally applied weak potential in order to reveal peculiarities of the soliton transitions between different waveforms implying these soliton waveforms coexist in the same equation parameter space.",
"In particular, parameters of Eq.",
"(REF ) are chosen so as to admit waveform coexistence of plain pulse, composite pulse and pulsating solitons.",
"A set of possible waveform transitions influenced by the weak potential is demonstrated in details in Fig.",
"REF .",
"In each panel of this figure the cross-section profiles (top) and the intensity plots (bottom) of the squared absolute value of complex amplitude $|\\Psi (x,z)|^2$ (left) and the potential $Q(x,z)$ (right) are presented.",
"Figure: Illustration of regular waveform transitions of plain pulse to (a) composite pulse, (b) composite pulse with moving fronts, and (c) pulsating soliton that appear under an influence of different potentials QQ.",
"Intensity distribution of squared absolute value of complex amplitude and potentials are presented in left and right panels, respectively.",
"Parameters are: (a) β=0.5\\beta =0.5, δ=0.5\\delta =0.5, μ=1\\mu =1, ν=0.1\\nu =0.1, ϵ=2.53\\epsilon =2.53, A i =B i ∈{1,-0.2,1}A_i=B_i\\in \\lbrace 1,-0.2,1\\rbrace , a i =50a_i=50, b i =150b_i=150, N=3N=3, x i ∈{-3,0,3}x_i\\in \\lbrace -3,0,3\\rbrace ; (b) β=0.5\\beta =0.5, δ=0.5\\delta =0.5, μ=1\\mu =1, ν=0.1\\nu =0.1, ϵ=2.53\\epsilon =2.53, A i =B i ∈{1,-0.2,1}A_i=B_i\\in \\lbrace 1,-0.2,1\\rbrace , a i =50a_i=50, b i =150b_i=150, N=3N=3, x i ∈{-10,0,10}x_i\\in \\lbrace -10,0,10\\rbrace ; (c) β=0.08\\beta =0.08, δ=0.1\\delta =0.1, μ=0.1\\mu =0.1, ν=0.07\\nu =0.07, ϵ=0.75\\epsilon =0.75, A 1 =B 1 =-1A_1=B_1=-1, a 1 =50a_1=50, b 1 =100b_1=100, N=1N=1, x 1 =0x_1=0.First, the stages of soliton transition from a plain pulse to a composite pulse are presented in Fig.",
"REF (a).",
"At the section $z=0$ the plain pulse comes into existence and then propagates freely through the zero potential up to the section $z=50$ , where the two-humped potential $Q$ abruptly arises.",
"This potential influences upon the plain pulse soliton transiting its waveform to another stationary state allowable under such an applied potential.",
"Next, at the section $z=150$ , the potential $Q$ becomes to be zero, and the waveform perturbed by the potential transits to the composite pulse, which coexists with the initial plain pulse under the same equation parameters.",
"From the viewpoint of the theory of dynamical systems, both plain pulse and composite pulse are $z$ -independent or stationary solutions, which are associated with two stable isolated fixed points existing in the infinite-dimensional phase space of the system (REF ).",
"For each such fixed point, certain set of initial conditions forms a basin of attraction.",
"Thus, the applied potential influences strongly enough upon the solition to transit the soliton waveform from the vicinity of the plain pulse fixed point to the basin of attraction of the composite pulse fixed point.",
"Therefore, taking this property into account one can controllably perform desired waveform transition of the dissipative solitons between all basins of attraction which coexist in the same equation parameter space.",
"This ability is further demonstrated in Fig.",
"REF (b) where the transition from the vicinity of the plain pulse fixed point to the basin of attractor of the composite pulse with moving fronts is realized.",
"This transition is caused by the two-humped potential $Q$ , whose profile is just slightly different from the previously discussed one.",
"A new set of the equation parameters gives us another example of the coexistence of plain pulse and pulsating dissipative solitons.",
"In Fig.",
"REF (c) we demonstrate waveform transition of the stationary plain pulse to the pulsating soliton that periodically changes its waveform along the longitudinal coordinate $z$ .",
"This transition is induced by the repulsive (defocusing) potential $Q$ .",
"In this figure five steady-state pulsations are zoomed in and presented in the upper left corner of the panel, where one can notice that the period of pulsations is approximately equal to 5 on the $z$ -axis scale.",
"Such a pulsating soliton can be considered as a limit cycle in a phase space of the dynamical system (REF ).",
"In order to visualize this limit cycle and other attractors of the system as well as to identify soliton behaviors along the $z$ -axis we perform the low-dimensional phase-space analysis which is based on the flow projections onto a pair of two-dimensional spaces.",
"They are defined as follows $\\mathcal {P}_1=\\lbrace \\left(E(z),P(z)\\right),\\in \\mathbb {R}^2\\rbrace $ and $\\mathcal {P}_2=\\lbrace \\left(z,E(z)\\right),\\in \\mathbb {R}^2\\rbrace $ , where components of the projections are the soliton energy parameters (REF ) and the longitudinal coordinate $z$ .",
"Trajectories in the two-dimensional phase spaces that correspond to the mentioned above attractors are plotted in Fig.",
"REF .",
"Both plain pulse and composite pulse are represented by a single point with zero total generated energy and horizontal line in the spaces $\\mathcal {P}_1$ and $\\mathcal {P}_2$ , respectively.",
"For the composite pulse with moving fronts, the trajectory in the space $\\mathcal {P}_1$ degenerates into a straight line while the corresponding straight line in the space $\\mathcal {P}_2$ has positive inclination.",
"Finally, for the pulsating soliton, points in the space $\\mathcal {P}_1$ always lies on a closed loop (cycle) while the soliton energy along the $z$ -axis in the space $\\mathcal {P}_2$ appears as a periodic function.",
"Figure: Phase trajectories in spaces 𝒫 1 \\mathcal {P}_1 (top) and 𝒫 2 \\mathcal {P}_2 (bottom) describing soliton behaviors related to the plain pulse, composite pulse, composite pulse with moving fronts, and pulsating soliton.In Fig.",
"REF we have demonstrated three particular examples of induced waveform transitions of dissipative solitons, where the initial plain pulse waveform transits to another ones.",
"In fact, the process of soliton waveform transition is invertible.",
"Indeed, in order to invert the considered direct waveform transitions one can apply to the composite pulse or pulsating soliton a quite high single-well potential (is not presented here, for illustration of the invert transitions see Ref. [42]).",
"Moreover, if for a fixed set of equation parameters, the dynamical system (REF ) has two or more attractors then waveform transitions between each pair of the basin of attraction can be induced by applying a suitable linear potential $Q$ with a finite support.",
"In other words, any induced waveform transitions between coexisted stable waveforms are possible.",
"The considered cases demonstrate a variety of the induced waveform transitions to different waveforms.",
"However, all these transitions are performed according to the same scenario.",
"An initially excited soliton approaches its attractor until an appropriate external potential is applied.",
"The potential inevitably influences upon the soliton waveform and changes it drastically.",
"Then waveform transition is performed in two stages.",
"The first one is a relatively short transient period which begins as soon as the potential is applied and it finishes when the perturbed waveform reaches a new steady state.",
"During the second stage a new steady state of the perturbed waveform exists.",
"It continues as long as the potential is applied.",
"When the potential is switched back to zero the perturbed waveform returns from the induced steady state to some attractor of the system (REF ) with zero potential.",
"For many sets of parameters, the system (REF ) has more than one attractor.",
"Therefore, depending on which basin of attraction contains the perturbed waveform it approaches the initial attractor or another one.",
"In all considered cases we chose the potential in such a way that the perturbed steady state waveform does not belong to the initial basin of attraction.",
"Thus, switching off the potential unavoidably changes the initial soliton waveform to another one.",
"Remarkably, this mechanism of waveform changing is stable to small variations of the potential profile, and waveform transitions have the same outcome for similar potential profiles.",
"Therefore, we distinguish these waveform transitions as regular ones."
],
[
"Irregular waveform transitions",
"In previous Section we have demonstrated an effect of influence of the relatively weak potential resulting in a set of particular regular waveform transitions of the dissipative solitons.",
"The potential `weakness' means that a perturbed soliton waveform reaches its new stationary steady state after performing some finite transient stage before stabilization.",
"However, it is revealed that applying a stronger potential can result in some nontrivial waveform transitions when a perturbed soliton waveform remains to be unstable without achieving any stationary profile, i.e.",
"it becomes to be a pulsating soliton.",
"The transitions that occur through pulsating waveforms and whose outcome is very sensitive to small variations in the potential profile are further considered as irregular waveform transitions.",
"They are presented in Fig.",
"REF , where one can see three possible outcomes of the irregular waveform transitions from the same initial plain pulse.",
"Remarkably, these three transitions appear under the influence of potentials which differ in the value of only one parameter $b_i$ of the function (REF ).",
"In fact, this parameter defines the longitudinal coordinate $z$ at which the potential is switched off.",
"Figure: Illustration of drastically different outcomes of the irregular waveform transitions of dissipative soliton to (a) plain pulse, (b) composite pulse, and (c) three noninteracting solitons that appear under an influence of the potential QQ, which is switched off at different points along the zz axis.",
"Intensity distribution of squared absolute value of complex amplitude and potentials are presented in left and right panels, respectively.",
"Parameters of the equation are: β=0.5\\beta =0.5, δ=0.5\\delta =0.5, μ=1\\mu =1, ν=0.1\\nu =0.1, and ϵ=2.52\\epsilon =2.52; parameters of the potential are: N=3N=3, x i ∈{-15,0,15}x_i\\in \\lbrace -15,0,15\\rbrace , B i =A i ∈{1.2,-0.4,1.2}B_i=A_i\\in \\lbrace 1.2,-0.4,1.2\\rbrace , a i =50a_i=50.All considered irregular waveform transitions are performed according to the same scenario.",
"A plain pulse comes into existence and freely propagates until the potential is switched on at the coordinate $z=50$ .",
"This potential influences upon the soliton changing its waveform from a plain pulse to a periodically pulsating soliton.",
"When a particular potential is switched off the periodic waveform pulsations vanish and the soliton acquires a stationary waveform.",
"Remarkably, after the potential removal the soliton has an alternative to acquire the form between different profiles from a set of waveforms coexisted in the same equation parameter space.",
"In particular, for the chosen equation parameters, the pulsating soliton transits either to a single plain pulse (Fig.",
"REF (a)), single composite pulse (Fig.",
"REF (b)), or two plain and one composite pulses (Fig.",
"REF (c)).",
"In fact, the releasing from pulsations depends drastically on the potential parameters and phase of periodical pulsations at which the potential is switched off.",
"In order to explain these multiple outcomes of irregular waveform transitions we discuss peculiarities of the soliton pulsations which are presented in Fig.",
"REF .",
"In each panel of this figure the intensity plot of the squared absolute value of complex amplitude $|\\Psi (x,z)|^2$ (left) and flow projections onto two-dimensional spaces $\\mathcal {P}_1$ and $\\mathcal {P}_2$ (right) are presented.",
"The soliton waveform evolution from a plane pulse to a periodically pulsating soliton is presented in Fig.",
"REF (a).",
"The unstabilized waveform appears as soon as the potential is imposed and pulsations continue to exist while the potential is in action.",
"The pulsations demonstrate a perfect periodic behavior which arises along the $z$ -axis with the period being approximately 5.",
"One period of pulsations is outlined in Fig.",
"REF (a) by black dashed lines.",
"The periodic pulsations are also confirmed by the flow projections onto the spaces $\\mathcal {P}_1$ and $\\mathcal {P}_2$ .",
"One can see that soliton energy $E$ possesses an exactly periodic behavior along the longitudinal coordinate $z$ and the trajectory in the space $\\mathcal {P}_1$ appears as a cycle which repeats itself indefinitely as long as the potential is applied.",
"In each period of the soliton pulsations we further distinguish three non-overlapping zones where there are corresponding instantaneous waveform profiles from which a transition to three mentioned stationary waveforms occurs.",
"These zones are denoted by Roman numerals I, II, and III in the upper fragment of Fig.",
"REF (a).",
"Each such distinguished zone within the period contains all the soliton waveforms belonging to the same basin of attraction.",
"In particular, if the potential is switched off at the phase of periodical pulsations being within zones I, II, and III, all instantaneous waveforms transit to steady state forms belonging to the basin of attraction of the plain pulse, three noninteracting solitons, and composite pulse, respectively.",
"Figure: Intensity distribution of squared absolute value of complex amplitude (left panels) and phase spaces 𝒫 1 \\mathcal {P}_1 and 𝒫 2 \\mathcal {P}_2 (right panels).",
"Waveform evolution plain pulse to (a) periodically, (b) quasi-periodically, and (c) chaotically pulsating solitons which coexist under the same equation parameters.",
"Parameters of the equation are: β=0.5\\beta =0.5, δ=0.5\\delta =0.5, μ=1\\mu =1, ν=0.1\\nu =0.1, and ϵ=2.52\\epsilon =2.52; parameters of the potential are: N=3N=3, x i ∈{-15,0,15}x_i\\in \\lbrace -15,0,15\\rbrace , B i ∈{0,0,0}B_i\\in \\lbrace 0,0,0\\rbrace , a i =50a_i=50, and (a) A i ∈{1.2,-0.4,1.2}A_i\\in \\lbrace 1.2,-0.4,1.2\\rbrace ; (b) A i ∈{1.2,-0.8,1.2}A_i\\in \\lbrace 1.2,-0.8,1.2\\rbrace ; (c) A i ∈{1.2,-1.8,1.2}A_i\\in \\lbrace 1.2,-1.8,1.2\\rbrace .Pulsating solitons can demonstrate more complicated behaviors when the applied potential becomes stronger.",
"In particular, it is found out that by choosing an appropriate potential, period-1 pulsations can be changed to quasi-periodical ones.",
"This outcome occurs when a soliton waveform perturbed by a strong enough potential does not reach any stabilized form being neither stationary not simply pulsating soliton.",
"Such an example of the waveform evolution of a plain pulse to a quasi-periodically (with period-8) pulsating soliton is presented in Fig.",
"REF (b).",
"The applied potential influences upon the soliton in such a way that both original waveform and energy parameters recur after every eighth pulsation, while the trajectory in the space $\\mathcal {P}_1$ traces an eight-loops cycle.",
"This cycle repeats itself indefinitely as long as the potential is applied.",
"Further increasing the height of potential wells results in a pulsating soliton becomes to be a chaotic one.",
"A numerical example of such chaotic pulsations is demonstrated in Fig.",
"REF (c).",
"The final soliton waveform obtained from a plain pulse continuously evolves along the $z$ -axis and never repeats itself remaining to be smooth and localized, while the dependence of the soliton energy $E$ on the coordinate $z$ possesses an oscillating behavior without any obvious repetitions.",
"The trajectory in the space $\\mathcal {P}_1$ densely fills a finite region manifesting behaviors of a strange attractor.",
"Both quasi-periodic and chaotic pulsating solitons exist while the potential is applied.",
"As soon as the potential is abruptly switched off, the soliton evolves back to a particular unperturbed stationary waveform.",
"Similarly to the above discussed case of the periodically pulsating soliton, the perturbed quasi-periodically and chaotically pulsating solitons can acquire different profiles from coexisted ones in the same equation parameter space.",
"Figure: Local maxima of pulsating soliton energy versus height of central potential well.",
"Parameters of the equation are: β=0.5\\beta =0.5, δ=0.5\\delta =0.5, μ=1\\mu =1, ν=0.1\\nu =0.1, and ϵ=2.52\\epsilon =2.52; parameters of the potential are: N=3N=3, x i ∈{-15,0,15}x_i\\in \\lbrace -15,0,15\\rbrace , B i =0B_i=0, a i =50a_i=50, A 1 =A 3 =1.2A_1=A_3=1.2.Additionally we have calculated all local maxima of the pulsating soliton energy as a function of the height of applied potential (Fig.",
"REF ).",
"This dependence can be interpreted as the one-dimensional Poincaré map, where a particular solution parameter (maxima of the soliton energy) is a function of an equation parameter (potential height).",
"We calculate the map by varying the height of the central potential well $A_2$ within the interval $[-2,-0.3]$ having excited the stationary plain pulse as in the previous irregular cases.",
"For each fixed $A_2$ we track the soliton propagation until any transients related to the switching on the potential have decayed and the soliton evolves to its perturbed waveform.",
"Then we calculate the soliton energy $E$ as a function of the longitudinal coordinate $z$ and find all its local maxima $E_{max}$ .",
"For period-$s$ pulsating solitons we find $s$ separate points and plot them in the graph for each value of the central well height $A_2$ .",
"A chaotic soliton generates infinite sequence of different numbers distributed within some finite interval that corresponds to a continuous vertical line in the graph.",
"In the final graph there are five domains where chaotic solitons exist.",
"In the remaining four domains an appearance of quasi-periodic solitons is admitted, while period-1 pulsating soliton can exist under the condition $A_2>-0.79$ .",
"All domains possess extremely sharp boundaries indicating that both direct and inverse transitions to chaotic waveforms happen abruptly when the potential parameter $A_2$ varies continuously."
],
[
"Conclusions",
"We have considered waveform transitions of dissipative solitons induced by application of a spatially inhomogeneous potential to the system.",
"The waveform transitions can occur only in the system having at least two attractors for the chosen equation parameters.",
"It is demonstrated that an appropriate potential influences upon the soliton changing an initially excited soliton waveform to a variety of perturbed waveforms which belong to different basins of attraction.",
"Switching off the potential causes an evolution of the perturbed waveforms to another attractors.",
"We have distinguished two types of induced waveform transitions, namely regular and irregular ones.",
"If the outcomes of induced waveform transitions are stable with respect to small variations of the applied potential and do not depend on a particular point at witch the potential is switched off then the waveform transitions are considered to be regular.",
"Otherwise they are irregular manifesting periodic, quasi-periodic, and chaotic behaviors.",
"Both types of the induced waveform transitions are invertible.",
"However, different outcomes of the direct irregular waveform transitions can be inverted by applying the same potential function.",
"Discussed mechanism can found an application in the practical systems supporting dissipative solitons to manage their propagation."
]
] | 1709.01704 |
[
[
"MAGIC VHE Gamma-Ray Observations Of Binary Systems"
],
[
"Abstract There are several types of Galactic sources that can potentially accelerate charged particles up to GeV and TeV energies.",
"We present here the results of our observations of the source class of gamma-ray binaries and the subclass of binary systems known as novae with the MAGIC telescopes.",
"Up to now novae were only detected in the GeV range.",
"This emission can be interpreted in terms of an inverse Compton process of electrons accelerated in a shock.",
"In this case it is expected that protons in the same conditions can be accelerated to much higher energies.",
"Consequently they may produce a second component in the gamma-ray spectrum at TeV energies.",
"The focus here lies on the four sources: nova V339 Del, SS433, LS I +61 303 and V404 Cygni.",
"The binary system LS I +61 303 was observed in a long-term monitoring campaign for 8 years.",
"We show the newest results on our search for superorbital variability, also in context with contemporaneous optical observations.",
"Furthermore, we present the observations of the only super-critical accretion system known in our galaxy: SS433.",
"Finally, the results of the follow-up observations of the microquasar V404 Cygni during a series of outbursts in the X-ray band and the ones of the nova V339 Del will be discussed in these proceedings."
],
[
"INTRODUCTION",
"The MAGIC experiment dedicates a significant fraction of observation time to binary systems in our Galaxy.",
"Please find a detailed description of the experiment and its performance in [1].",
"The systems presented here are either $\\gamma $ -ray binaries: a small class among the family of X-ray binaries emitting the largest part of their non-thermal luminosity at high ($\\ge $ 10 MeV) $\\gamma $ -ray energies; or microquasars: X-ray (accreting) binaries consisting of a black hole (BH), that orbits a stellar companion; or novae: nuclear explosions on a white dwarf in a binary system.",
"Here we present results of several observation campaigns on the following sources: nova V339 Del, SS 433, LS I +61$^{\\circ }$ 303 and V404 Cygni.",
"All these sources are binary systems, but they have different characteristics.",
"We try to catch unique emission patterns with different strategies like follow-up campaigns on triggers from other wavelengths, joint observation programs with other Cherenkov experiments or long-term monitoring.",
"Figure: (a) Multiwavelength light curve of V339 Del during the outburst in August 2013.Top panel: Optical observations in the V band obtained from the AAVSO-LCG service (http://www.aavso.org/lcg).Middle panel: The Fermi-LAT flux (filled symbols) and upper limits (empty symbols) above 100 MeV in 1-day (circles, thin lines) or 3-day (squares, thick lines) bins.A 95% C.L.",
"flux upper limit is shown for time bins with TS<<4.Bottom panel: Upper limit on the flux above 300 GeV observed with MAGIC telescopes.The gray band shows the observation nights with MAGIC.The dashed gray line shows a MAGIC observation night affected by bad weather.",
"(b) Differential upper limits on the flux from V339 Del as measured by MAGIC (filled squares) and the flux measured by Fermi-LAT (empty crosses) in the same time period, 2013 August 25 to September 4.The thin solid line shows the IC scattering of thermal photons in the nova's photosphere.The dashed line shows the γ\\gamma -rays coming from the decay of π 0 \\pi ^0 from hadronic interactions of the relativistic protons with the nova ejecta.The dotted line shows the contribution of γ\\gamma -rays coming from IC of e + ^+e - ^- originating from π + π - \\pi ^+\\pi ^- decays.Thick solid lines show the total predicted spectrum.A classical nova is a thermonuclear runaway leading to the explosive ejection of the envelope accreted onto a white dwarf (WD) in a binary system in which the companion is either filling or nearly filling its Roche surface (see references in [2]).",
"They are a type of cataclysmic variable, i.e.",
"optically variable binary systems with mass transfer from a companion star to the WD.",
"In the last six years the Fermi Large Area Telescope (LAT) instrument detected GeV $\\gamma $ -ray emission from six novae (five classical novae and one symbiotic-like recurrent nova) [3].",
"Since late 2012 the MAGIC collaboration has been conducting a nova follow-up program in order to detect a possible Very High Energy (VHE, $E>$ 100 GeV) $\\gamma $ -ray component.",
"Here we report on the observations performed with the MAGIC telescopes of V339 Del.",
"It was a fast, classical CO nova detected by optical observations on August 16 2013 (CBET #3628), MJD 56520.",
"The nova was exceptionally bright reaching a magnitude of V$\\sim 5\\,$ mag (see top panel of Fig.",
"REF a), and it triggered follow-up observations at frequencies ranging from radio to VHE $\\gamma $ -rays.",
"No VHE $\\gamma $ -ray signal was found from the direction of V339 Del.",
"We computed night-by-night integral upper limits above 300 GeV (see bottom panel of Fig.",
"REF a) and differential upper limits for the whole good quality data set in bins of energy (Fig.",
"REF b).",
"The $\\gamma $ -ray emission from V339 Del was first detected by Fermi-LAT in a 1-day bin on August 18 2013 (see references in [2]).",
"The emission peaked on August 22 2013 and entered a slow decay phase afterwards (Fig.",
"REF a).",
"The GeV emission can be interpreted in terms of an inverse Compton process of electrons accelerated in a shock.",
"In this case it is expected that protons in the same conditions can be accelerated to much higher energies.",
"Consequently, they may produce a second component in the $\\gamma $ -ray spectrum at TeV energies.",
"We used the numerical code of [4] to model the Fermi-LAT GeV spectrum and to compare the sub-TeV predictions with the MAGIC observations.",
"In Fig.",
"REF we show the predictions for leptonic or hadronic spectra compared with the Fermi-LAT and MAGIC measurements.",
"The Fermi-LAT spectrum can be described mostly by IC scattering of the thermal photons in the nova's photosphere by electrons.",
"The expected hadronic component overpredicts the MAGIC observations at $\\sim 100 \\,$ GeV by a factor of a few for the case of equal power of accelerated protons and electrons (i.e.",
"$L_p=L_e$ ).",
"Using the upper limits from the MAGIC observations we can place the limit on $L_p\\lesssim 0.15 L_e$ .",
"Therefore, the total power of accelerated protons must be $\\lesssim 15\\%$ of the total power of accelerated electrons.",
"MAGIC will continue to observe promising $\\gamma $ -ray nova candidates in the following years."
],
[
"SS 433",
"SS 433 is an extremely bright microquasar with a bolometric luminosity of $L_{\\rm bol} \\sim {10}^{40}$ erg s$^{-1}$ and with the most powerful jets known in our Galaxy with $L_{\\rm jet} \\lesssim {10}^{39}$ erg s$^{-1}$ [5], [6].",
"It is embedded in the W50 nebula (SNR G39.7–2.0 [7]).",
"Its present morphology is thought to be the result of the interaction between the jets of SS 433 and the surrounding medium.",
"This scenario is supported by the position of SS433 at the center of W50, the elongation of the nebula in the east-west direction along the axis of precession of the jets and the presence of radio, IR, optical and X-ray emitting regions also aligned with the jet precession axis (see Fig.",
"REF ).",
"We explore the capability of SS 433 to emit VHE $\\gamma $ rays during periods in which the expected flux attenuation due to periodic eclipses (P$_{\\rm orb} \\sim $ 13.1 d) and precession of the circumstellar disk (P$_{\\rm prec}\\sim $ 162 d) periodically covering the central binary system is expected to be at its minimum [8].",
"Our observations do not show any significant VHE emission neither for the central source SS 433 nor for any of the interaction regions with the W50 nebula $e1, e2, e3, w1$ and $w2$ (see Fig.",
"REF ).",
"Detailed results of the whole campaign together with the H.E.S.S.",
"experiment will be published elsewhere soon.",
"Figure: Skymap of the SS 433W50 system as observed at E≥\\ge 250 GeV by MAGIC.",
"The color scale represents the excess events significance.",
"GB6 4.85 GHz radio contours (white ) and ROSAT broadband X-ray contours (yellow ) are over-plotted.",
"Circles indicate the positions of interaction regions w1,w2w1, w2 and e1,e2,e3e1, e2, e3."
],
[
"LS I +61$^{\\circ }$ 303",
"LS I +61$^{\\circ }$ 303 (= V615 Cas ) is a $\\gamma $ -ray binary composed of a rapidly rotating Be star of spectral type B0Ve with a circumstellar disk and a compact object of unknown nature.",
"The compact object, either a neutron star (NS) or a stellar-mass black hole (BH), has an eccentric orbit ($e = 0.54 \\pm 0.03$ ) with a period of 26.4960(28) days.",
"Orbit-to-orbit variability has been associated with a super-orbital modulation.",
"This was first proposed based on centimeter radio variations that show approximately sinusoidal modulation over 1667 $\\pm $ 8 days.",
"A similar long-term behavior has recently been suggested for X-rays (3 – 30 keV observed with RXTE), hard X-rays (18 – 60 keV observed with INTEGRAL) and HE $\\gamma $ -rays (100 MeV – 300 GeV observed with Fermi/LAT) (see references in [11]).",
"MAGIC observed LS I +61$^{\\circ }$ 303 as part of a multi-wavelength campaign between August 2010 and September 2014.",
"All TeV data were obtained with the MAGIC stereoscopic system, except for January 2012, when MAGIC-I was inoperative.",
"Data were taken during the orbital phase range $\\phi $ = 0.5 – 0.75 to scan the complete trend of the periodical outburst peak of the TeV emission, with the aim of detecting a putative long-term modulation.",
"Contemporaneous observations with MAGIC and LIVERPOOL were performed during orbital phases 0.75 – 1.0, which are the phases where sporadic VHE emission had been detected and which does not seem to present yearly periodical variability of the flux level [12].",
"Since the fluxes in this orbital period are not influenced by the long-term modulation, changes in the relative optical and TeV fluxes are larger and easier to measure.",
"The aim of these contemporaneous observations is to search for (anti-)correlation between the mass-loss rate of the Be star and the TeV emission.",
"All archival data of LS I +61$^{\\circ }$ 303 recorded by MAGIC since its detection in 2006 [11] and the data from the observing campaigns presented were folded onto the superorbital period of 1667 days (Fig.",
"REF ).",
"The data were fit with a constant, a sinusoidal and to two-emission levels (step function).",
"The probability for a constant flux is negligible, 4.5 $\\times 10^{-12}$ .",
"Assuming a sinusoidal signal, the fit probability reaches 8% ($\\chi ^{2}$ /dof = 27.2/18).",
"The fit to a step function resulted in a fit probability of 7% ($\\chi ^{2}$ /dof = 26.4/17).",
"We furthermore quantified the probability that the improvement found when fitting a sinusoid or a step function instead of a constant is produced by chance.",
"To obtain this probability, we considered the likelihood ratio test.",
"In both cases this chance probability is $<2.5\\times 10^{-10}$ , which is low.",
"This shows that the observed intensity distribution can be described by a high and a low state and with a smoother transition.",
"We conclude that there is a super-orbital signature in the TeV emission of LS I +61$^{\\circ }$ 303 and that it is compatible with the 4.5-year radio modulation seen in other frequencies.",
"The correlation between the TeV flux measured by MAGIC and the H$\\alpha $ parameters measured with the telescope (equivalent width (EW), full width at half maximum (FWHM), profile centroid velocity) were determined including statistical and systematic uncertainties and the weighted Pearson correlation coefficient [13].",
"No statistically significant correlation was found for the sample at orbital phases $\\phi $ = 0.75 – 1.0.",
"A hint of a correlation is observed, but its significance is low.",
"A stronger correlation might be blurred as a result of the fast variability of the optical parameters on short timescales compared to the long exposure times required by MAGIC, and as a result of the relatively large uncertainties and small number of data points used for this analysis.",
"Figure REF b shows the H$\\alpha $ measurements plotted against the TeV flux.",
"The main conclusions from this multi-year analysis of LS I +61$^{\\circ }$ 303 observations are: (1) We achieved a first detection of super-orbital variability in the TeV regime.",
"Using the new VHE data and the MAGIC and VERITAS archival data, we found that the super-orbital signature of LS I +61$^{\\circ }$ 303 is consistent with the 1667-day radio period within 8$\\%$ .",
"(2) There is no statistically significant intra-day correlation between H$\\alpha $ line properties and TeV emission, nor is there an obvious trend connecting the two frequencies.",
"Figure: Each data point represents the peak flux emitted in one orbital period during orbital phases 0.5 – 0.75 and is folded into the super-orbit of 1667 days known from radio observations .",
"MAGIC (magenta) and VERITAS (blue) points were fit with a sinusoidal (solid red line), with a step function (solid green line), and with a constant (solid blue line).",
"The gray dashed line represents 10%10\\% of the Crab Nebula flux, the gray solid line the zero level for reference.",
"(b) Correlations between the TeV flux obtained by MAGIC and the HαH\\alpha parameters (from top to bottom: EW, FWHM, and centroid velocity (vel)) measured by the LIVERPOOL Robotic Telescope for the orbital interval 0.75 – 1.0.",
"Only TeV data points with a significance higher than 1σ\\sigma have been considered.",
"Each data point represents a ten-minute observation in the optical and a variable integration in the TeV regime: nightly (blue), contemporary (red), and strictly simultaneous data (green).",
"Black error bars represent statistical uncertainties, while systematic uncertainties are plotted as magenta error bars."
],
[
"V404 Cygni",
"The microquasar V404 Cygni (V404 Cyg), located at a parallax distance of 2.39$\\pm 0.14$ kpc [15], is a binary system of an accreting stellar-mass black hole from a companion star.",
"The black hole mass estimation ranges from about 8 to 15 M$_{\\odot }$ , while the companion star mass is $0.7^{+0.3}_{-0.2}$ M$_{\\odot }$ [16], [17], [18].",
"The system inclination angle is 67$^{\\circ }$ $^{+3}_{-1}$ [18], [17] and the system orbital period is 6.5 days [16].",
"This low-mass X-ray binary (LMXB) showed at least four periods of outbursting activity: the one that led to its discovery in 1989 detected by the Ginga X-ray satellite [19], two previous ones in 1938 and 1956 observed in optical and later associated with V404 Cyg [20], and the latest in 2015.",
"In June 2015, the system underwent an exceptional flaring episode.",
"From the 15th to the end of June the bursting activity was registered by several hard X-ray satellites, like Swift and INTEGRAL [21], [22].",
"It reached a flux about 40 times larger than the Crab Nebula one in the 20–40 keV energy band [23].",
"Triggered by the INTEGRAL alerts, MAGIC observed V404 Cyg for several nights between June 18th and 27th 2015, collecting more than 10 hours.",
"Most of the observations were performed during the strongest hard X-ray flares.",
"In total, MAGIC observed the microquasar for 8 non-consecutive nights collecting more than 10 hours of data, some coinciding with observations at other energies.",
"To avoid an iterative search over different time bins, we assumed that the TeV flares were simultaneous to the X-ray ones.",
"We defined the time intervals where we search for signal in the MAGIC data, to match those of the flares in the INTEGRAL light curve.",
"We analysed the INTEGRAL-IBIS data (20–40 keV) publicly available with the osa software version 10.2http://www.isdc.unige.ch/integral/analysis.",
"The time selection for the MAGIC analysis was performed running a Bayesian block [24] analysis on the INTEGRAL light curve.",
"We searched for VHE gamma-ray emission stacking the MAGIC data of the selected time intervals ($\\sim 7$ hours).",
"We found no significant emission in the $\\sim 7$ hour sample, neither in any of the sub-samples considered.",
"We then computed differential upper limits (see Figure REF ) for the observations assuming a power law spectral shape of index -2.6.",
"The luminosity upper limits calculated for the full observation period, considering the source at a distance of 2.4 kpc, is $\\sim 2 \\times 10^{33}$ erg s$^{-1}$ , in contrast with the extreme luminosity emitted in the X-ray band ($\\sim 2 \\times 10^{38}$ erg s$^{-1}$ , [23]) and other wavelengths.",
"Models predict TeV emission from this type of systems under efficient particle acceleration on the jets [25], [26] or strong hadronic jet component [27].",
"If produced, VHE gamma rays may get extinct via pair creation in the vicinity of the emitting region.",
"For gamma rays in an energy range between 200 GeV – 1.25 TeV, the largest cross section occurs with NIR photons.",
"For a low-mass microquasar, like V404 Cyg, the contribution of the NIR photon field from the companion star (with a bolometric luminosity of $\\sim 10^{32}$ erg s$^{-1}$ ) is very low.",
"During the period of flaring activity, disk and jet contributions are expected to dominate.",
"During the outburst activity of June 2015, the luminosity on the NIR regime can be estimated to $L_{NIR}=4.1\\times 10^{34}$ erg s$^{-1}$ .",
"Assuming this luminosity, the gamma-ray opacity at a typical radius $r\\sim 1\\times 10^{10}$ cm may be relevant enough to avoid VHE emission above 200 GeV.",
"Moreover, if IC on X-rays at the base of the jets ($r\\lesssim 1\\times 10^{10}$ cm) is produced, this could already prevent electrons to reach the TeV regime, unless the particle acceleration rate in V404 Cyg is close to the maximum achievable including specific magnetic field conditions (see e.g.",
"[28]).",
"On the other hand, absorption becomes negligible for $r>1\\times 10^{10}$ cm.",
"Thus, if the VHE emission is produced in the same region as HE radiation ($r\\gtrsim 1\\times 10^{11}$ cm, to avoid X-ray absorption), then it would not be significantly affected by pair production attenuation ($\\sigma _{\\gamma \\gamma }< 1$ ).",
"Therefore a VHE emitter at $r\\gtrsim 1\\times 10^{10}$ cm, along to the non-detection by MAGIC, suggests either inefficient particle acceleration inside the V404 Cyg jets or not enough energetics of the VHE emitter.",
"Figure: Multiwavelength spectral energydistribution of V404 Cyg during the June2015 flaring period.",
"In red, MAGIC ULs aregiven for the combined Bayesian blocktime bins (∼\\sim 7 hours) for which apower-law function with photon index2.6 was assumed.",
"In green, MAGIC ULs forobservations on June 26th, simultaneouslytaken with the Fermi-LAT hint.",
"In this case, a photonindex of 3.5 was applied followingFermi-LAT results.",
"All the MAGIC upperlimits are calculated for a 95% confidencelevel, considering also a 30% systematicuncertainty.",
"The extrapolationof the Fermi-LAT spectrum is shown inblue with 1 σ\\sigma contour (gray dashedlines).",
"In the X-ray regime, INTEGRAL (20-40 keV) andSwift-XRT (0.2-10 keV) data aredepicted.",
"At lower energies,Kanata-HONIR optical and NIR data areshown, taken from ."
],
[
"ACKNOWLEDGMENTS",
"We would like to thank the IAC for the excellent working conditions at the ORM in La Palma.",
"We acknowledge the financial support of the German BMBF, DFG and MPG, the Italian INFN and INAF, the Swiss National Fund SNF, the European ERDF, the Spanish MINECO, the Japanese JSPS and MEXT, the Croatian CSF, and the Polish MNiSzW."
]
] | 1709.01626 |
[
[
"Gluonic Distribution in the Constituent Quark and Nucleon Induced by the\n Instantons"
],
[
"Abstract Instanton effects can give large contribution to strong interacting processes, especially at the energy scale where perturbative QCD is no longer valid.",
"However instanton contribution to the gluon contribution in constituent quark and nucleon has never been calculated before.",
"Based on both the constituent quark picture and the instanton model for QCD vacuum, we calculate the unpolarized and polarized gluon distributions in the constituent quark and in the nucleon for the first time.",
"We find that the pion field plays an important role in producing both the unpolarized and the polarized gluon distributions."
],
[
"Introduction",
"Due to the non-perturbative nature of QCD, when dealing with low-energy states of hadrons, such as nucleons, various models has to be adopted, the most important one is the parton model.",
"The distribution of partons are described by the parton distribution functions (PDFs), one of the cornerstones of the calculation of high energy cross sections.",
"PDFs are fit from experiment data, not calculated from first principles.",
"Among various PDFs, the gluon distribution function in the proton give the dominant contribution to the cross sections, they are also of great importance in order to understand the so-called proton spin crisis [1].",
"The general form of interaction vertex of massive quark with gluon can be written as $V_\\mu (k_1^2,k_2^2,q^2) = -g_s t^a \\left[ {\\gamma _\\mu F_1(k_1^2,k_2^2,q^2)} + {\\frac{\\sigma _{\\mu \\nu }q^\\nu }{2M_q} F_2(k_1^2,k_2^2,q^2)}\\right],$ where $k_{1,2}^2 $ are the virtualities of incoming and outgoing quarks and $q$ is the momentum transferred.",
"The Anomalous Quark Chromomagnetic Moment (AQCM) is [2], [4], [7] $\\mu _a=F_2(0,0,0)= -\\frac{3\\pi (M_q\\rho _c)^2}{4\\alpha _s(\\rho _c)}$ Based on both the constituent quark model for the nucleon and the instanton model for the non-perturbative QCD vacuum [3], [4], we can calculate the gluon distribution function in the quarks and nucleons.",
"The instantons are a result of tunneling effects in QCD, they can be considered as strong non-perturbative fluctuations of the gluon fields in vacuum which describes the non-trivial topological structure of the QCD vacuum.",
"The average size of the instantons $\\rho _c\\approx 1/3$ fm is much smaller than the confinement size $R_c\\approx 1$ fm.",
"Furthermore, spontaneously chiral symmetry breaking (SCSB) induced by the instantons is partly responsible for the existence of constituent massive quark.",
"The SCSB also has important effects in the hadronic reactions, such as in the spin-flip effects observed in high energy nucleon reactions [5].",
"Instanton contribution to such process was suggested in [6].",
"the connection between effective mass of the quark and the instanton vacuum is discussed in [4], [7], [8], [4].",
"Instanton contribution to the high energy inclusive pion production in the proton-proton collisions was discussed in ref [9], [10]."
],
[
"Gluon unpolarized and polarized distributions in the constituent quark",
"The effective Lagrangian based on the quark-gluon chromomagnetic interaction that preserves the chiral symmetry [8], [4] reads ${\\cal L}_I= -i\\frac{g_s\\mu _a}{4M_q}\\bar{q}\\sigma ^{\\mu \\nu }t^ae^{i\\gamma _5\\vec{\\tau }\\cdot \\vec{\\phi }_\\pi /F_\\pi }q G^{a}_{\\mu \\nu },$ where $\\mu _a$ is the (AQCM), $g_s$ is the strong coupling constant, $G^{a}_{\\mu \\nu }$ is the gluon field strength, and $F_\\pi = 93~\\mbox{MeV}$ is the pion decay constant.",
"Expand above Lagrangian to the first order of the pion field, we get [9], [10] $\\mathcal {L}_I =- i\\frac{g_s\\mu _a}{4M_q} \\, \\bar{q}\\sigma ^{\\mu \\nu }t^a q \\, G^{a}_{\\mu \\nu }+\\frac{g_s\\mu _a}{4M_qF_\\pi }\\bar{q} \\sigma ^{\\mu \\nu } t^a \\gamma _5\\mathbf {\\tau }\\cdot \\mathbf {\\pi } q \\, G^{a}_{\\mu \\nu }.$ Where $M_q$ is the effective quark mass in the instanton vacuum[11], [3].",
"The strong coupling constant is fixed at instanton scale as $\\alpha _s(\\rho _c)=g_s^2(\\rho _c)/4\\pi \\approx 0.5$ [4].",
"The Altarelli-Parisi (AP) approach is adopted to calculate the gluon distribution functions in the constituent quark [12].",
"The contributions from the perturbative QCD (pQCD) and the non-perturbative instanton-induced interactions are shown in Fig.1.",
"Figure: a) corresponds to the contributions from the pQCD to gluon distribution in the quark.b) and c) correspond to the contributions to the gluon distribution in the quarkfrom the non-perturbative quark-gluon interaction and the non-perturbative quark-gluon-pion interaction, respectively.In the single instanton approximation, the expansion parameter is $\\delta = (M_q\\rho _c)^2 $ , which is relatively small with the value of the quark mass $M_q=86 $ MeV in the effective single instanton approximation which we used here [11].",
"The non-perturbative contribution with the pion, i.e.",
"diagram c) in Fig.1, has never been calculated before.",
"By calculating the matrix element for the diagram c) in Fig.1 and performing the integration over $x$ , we can obtain the unintegrated unpolarized and polarized gluon distribution in the quark.",
"Figure: The zz dependency of the contributions to the the unpolarized gluon distributionin the constituent quark from the pQCD (left panel),from the non-perturbative interaction without pion (central panel), and from the non-perturbativeinteraction with pion (right panel).",
"The dotted line corresponds to Q 2 =2Q^2=2 GeV 2 ^2,the dashed line to Q 2 =5Q^2=5 GeV 2 ^2, and the solid line to Q 2 =10Q^2=10 GeV 2 ^2.Figure: The zz dependency of the contributions to the polarized gluon distributionin the constituent quark from the pQCD (left panel), and from the non-perturbativeinteraction with pion (right panel).",
"The notations are the same as in the Fig.2.",
"In the right panel the result forQ 2 =5Q^2=5 GeV 2 ^2 does not shown because it is practically identical to the Q 2 =10Q^2=10 GeV 2 ^2 case.Fig.2 shows the unpolarized gluon distribution in constituent quark versus $z$ at several $Q^2$ , where $z$ is the fraction of quark momentum carried by gluon.",
"On one hand, the unpolarized gluon distribution produced by the perturbative QCD depends stronger on $z$ than that produced by the non-perturbative interactions.",
"Hence the pQCD dominates the large $z$ region.",
"On the other hand, the gluon distribution produced by non-perturbative quark-quark-gluon interaction is rather small in comparison with that from both the pQCD, a) in Fig.1, and the non-perturbative interaction with pion, c) in Fig.1, partly because of a larger final phase space due to an extra pion.",
"At small $z$ for all the contributions $zg(z,Q^2)\\approx const$ , which is so-called Pomeron-like behavior.",
"The polarized gluon distributions in the constituent quark are shown in Fig.3.",
"Again it is obvious that pQCD contribution to the polarized gluon distribution dominates in the large $z$ region, where the non-perturbative interaction with pion dominates in small $z$ region.",
"Furthermore, pQCD contribution shows $\\Delta g(z,Q^2)\\rightarrow const$ , while the non-perturbative contribution shows anomalous dependency as $\\Delta g(z,Q^2)\\rightarrow \\log (z)$ .",
"When $Q^2$ increases, the pQCD contribution increases as $\\log (Q^2\\rho _c^2)$ while the non-perturbative contribution practically stays the same for $Q^2>1/\\rho _c^2=0.35$ GeV $^2$ .",
"Therefore, the non-perturbative interaction contribution to the gluon distribution can be ragarded as an intrinsic polarized gluon inside the constituent quark."
],
[
"Gluon distributions in the nucleon",
"Knowing the gluon distribution in the constituent quark, we can use convolution model to obtain the gluon distribution in the nucleon.",
"The PDF of the unpolarized constituent quark is taken to be $q_V(y)=60y(1-y)^3.$ This PDF is in accord with the quark counting rule at large $y$ this distribution.",
"Its low $y$ behavior and normalization are fixed by the requirements that $\\int _0^1dyq_V(y)=3$ and $\\int _0^1dyyq_V(y)=1$ .",
"For the polarized constituent quark distribution, a simple form is adopted to be $\\Delta q_V(y)=2.4(1-y)^3.$ Which is also in agreement with the quark counting rule at $y\\rightarrow 1$ .",
"The normalization conditions has been fixed from the hyperon weak decay data (see [1]) as $\\int _0^1dy\\Delta q_V(y)=\\Delta u_V+\\Delta d_V\\approx 0.6.$ Figure: Unpolarized (left panel), and the polarized(right panel) gluon distributions in the nucleon at the scale Q 2 =2Q^2=2 GeV 2 ^2.",
"The dotted line in red corresponds tothe contribution from pQCD, the dotted-dashed in blue to the contribution from the non-perturbative interaction with pion,dotted-dashed in black to the contribution from the non-perturbative interaction without pion,and the solid line to the total contribution.Both the unpolarized and polarized gluon distributions in the nucleon have been presented in Fig.4, where $Q_0^2=2$ GeV$^2$ .",
"This value of $Q^2$ is often used as the starting point for the standard pQCD evolution.",
"Our result for the unpolarized gluon distribution, which is shown in the left panel of Fig.4, is identical to the GJR parametrization $ xg(x)=1.37x^{-0.1}(1-x)^{3.33} $ [13].",
"It is well known that the behavior of non-polarized gluon distribution at low $x$ region is determined by the exchange of Pomerons.",
"Pomeron exchange effects play a very important role in the phenomenology of high energy reactions.",
"Our results shown in Figs.2,4 indicate the existence of two different kinds of Pomerons, the \"hard\" pQCD Pomeron and the \"soft\" non-perturbative Pomeron, they have quite different dependency on $x$ and $Q^2$ .",
"Our calculation strongly supports two kinds of Pomerons pictures which can explain many experimental data in both the DIS data at large $Q^2$ and the high energy cross sections with small momentum transfer [14]."
],
[
"Proton spin problem",
"The polarized gluon distribution is shown in the right panel of Fig.4.",
"The fraction of nucleon momentum that carried by the gluons $G(Q^2)=\\int _0^1dxxg(x,Q^2)$ , as well as their polarization $\\Delta G(Q^2)=\\int _0^1dx \\Delta g(x,Q^2)$ are presented as a function of $Q^2$ in Fig.5.",
"It is clear that the non-perturbative interaction without pion gives a small contribution to unpolarized gluon distribution.",
"Furthermore, such contribution is zero for the polarized gluon case.",
"For $Q^2>1/\\rho _c^2$ , the main contributions to both the unpolarized and the polarized gluon distributions come from pQCD and the non-perturbative interaction with the pion.",
"Figure: The part of the nucleon momentum carried by gluons (left panel), and the contribution of the gluonsto nucleon spin (right panel) as the function of Q 2 Q^2.The notations are the same as in the Fig.4Jaffe and Manohar decomposed the proton spin into[15] $\\frac{1}{2}=\\frac{1}{2}\\Delta \\Sigma +\\Delta G+L_q+ L_g,$ where the first term is quark spin, $\\Delta G=\\int _0^1dx\\Delta g(x)$ is the gluon polarization and the last two terms are quark and gluon orbital momentum.",
"The key problem is to explain discovered small value of the proton spin carried by quark.",
"At present we have $\\Delta \\Sigma \\approx 0.25$ [1], which is in bad agreement with $\\Delta \\Sigma =1$ given by the non-relativistic quark model.",
"Shortly after the EMC data which reveals such discrepancy between experiment and theoretical predictions, the axial anomaly effect in DIS was brought out and considered to be the primary candidate as a remedy for the dilemma.",
"[16].",
"For three quark flavors, it gives following reduction of the quark helicity in the DIS $\\Delta \\Sigma _{DIS}=\\Delta \\Sigma -\\frac{3\\alpha _s}{2\\pi }\\Delta G$ A big positive gluon polarization is needed, $\\Delta G\\approx 3\\div 4$ , in order to explain the small value of $\\Delta \\Sigma _{DIS}$ .",
"However, modern experimental data from the inclusive hadron productions and the jet productions have exclude such a large gluon polarization in the accessible intervals of $x$ and $Q^2$ [17], [18], and so does our model (see Fig.5, right panel).",
"Therefore, the axial anomaly effect, suggested in [16], cannot explain the proton spin problem.",
"We should stress that the helicity of the initial quark is flipped in the vertices b) and c) in Fig.1.",
"As the result, such vertices should lead to the screening of the quark helicity.",
"It is evident that at the $Q^2\\rightarrow 0$ such screening vanishes as shown in Fig.5 and the total spin of the proton is carried by its constituent quarks.",
"We would like to emphasis that in our approach, our model enables us to calculate the un-integrated gluon distribution function, which is of essential importance in many application, such as the calculations of high energy various reactions."
],
[
"Conclusion",
"We show that the quark-gluon-pion anomalous interaction gives a very large contribution to both the unpolarized and polarized gluon distributions.",
"It means that pion field plays a fundamental role to produce both gluon distributions in hadrons.",
"The possibility of the matching of the constituent quark model for the nucleon with its partonic picture is shown.",
"The phenomenological arguments in favor of such a non-perturbative gluon structure of the constituent quark were recently given in the papers [19], [20].",
"We also pointed out that the famous proton spin crisis might be explained by the flipping of the helicity of the quark induced by non-perturbative anomalous quark-gluon and quark-gluon-pion interactions."
],
[
"Acknowledgments",
"We are grateful to Igor Cherednikov, Boris Kopeliovich, Victor Kim and Aleksander Dorokhov for useful discussions.",
"This work was partially supported by the National Natural Science Foundation of China (Grant No.",
"11575254 and 11175215), and by the Chinese Academy of Sciences visiting professorship for senior international scientists (Grant No.",
"2013T2J0011).",
"This research was also supported in part by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2013R1A1A2009695)(HJL)."
]
] | 1709.01640 |
[
[
"Modeling of networks and globules of charged domain walls observed in\n pump and pulse induced states"
],
[
"Abstract Experiments on optical and STM injection of carriers in layered $\\mathrm{MX_2}$ materials revealed the formation of nanoscale patterns with networks and globules of domain walls.",
"This is thought to be responsible for the metallization transition of the Mott insulator and for stabilization of a \"hidden\" state.",
"In response, here we present studies of the classical charged lattice gas model emulating the superlattice of polarons ubiquitous to the material of choice $1T-\\mathrm{TaS_2}$.",
"The injection pulse was simulated by introducing a small random concentration of voids which subsequent evolution was followed by means of Monte Carlo cooling.",
"Below the detected phase transition, the voids gradually coalesce into domain walls forming locally connected globules and then the global network leading to a mosaic fragmentation into domains with different degenerate ground states.",
"The obtained patterns closely resemble the experimental STM visualizations.",
"The surprising aggregation of charged voids is understood by fractionalization of their charges across the walls' lines."
],
[
"Introduction",
"Major anticipations for the post-silicon electronics are related to materials which demonstrate a layered structure with a possibility for exfoliation down to a few and even a single atomic layer, akin to the graphene.",
"The latest attention was paid to oxides and particularly di-halcogenides of transition metals $MX_2$ with $M$ = Nb, Ta, Ti and $X$ = S, Se, see e.g.",
"[1], [2] for reviews.",
"These materials show a very rich phase diagram spanning from unconventional insulators of the so called Peierls and Mott types to the superconductivity [3].",
"The transformations among these phases involve formation of superstructures like several types of so called charge density waves (CDW) and/or of hierarchical polaronic crystals.",
"Recent studies of these materials fruitfully overlapped with another new wave in solid state physics.",
"This is the science of controlled transformations of electronic states or even of whole phases by impacts of strong electric fields and/or the fast optical pumping.",
"A super goal is to attend “hidden” states which are inaccessible and even unknown under equilibrium conditions.",
"In relation to this article subjects, the success came recently from observations of ultrafast (at the scale of picoseconds) switching by means of optical [4], [7] and voltage [5], [6] pulses, as well by local manipulations [8], [9].",
"The registered ultrafast switching is already discussed as a way for new types of RAM design, see [10] and rfs.",
"therein.",
"Most challenging and inspiring observations have been done in studies by the scanning tunneling microscopy (STM) and spectrometry (STS) [8], [9], [11], [7].",
"They have shown that the switching from an insulating to a conducting state proceeds via creation of local globules or extended networks of domain walls enforcing fragmentation of the insulating electronic crystal into a conducting mosaics of domains with different multiply degenerate ground states.",
"Most important observations have been done upon very popular nowadays layered material $1T-\\mathrm {TaS_2}$ which is a still enigmatic “polaronic Wigner-crystalline Mott insulator”.",
"The rich phase diagram of $1T-\\mathrm {TaS_2}$ includes such states as incommensurate, nearly commensurate, and commensurate CDWs which unusually support also the Mott insulator state for a subset of electrons.",
"Recently, new long-lived metastable phases have been discovered: a “hidden” state created by laser [4], [7] or voltage [5] pulses, and a most probably related “metallic mosaic” state created locally by STM pulses [8], [9].",
"$1T-\\mathrm {TaS_2}$ is a narrow-gap Mott insulator existing unusually on the background formed by a sequence of CDW transitions which have gaped most of the Fermi surface of the high temperature metallic (with 1 electron per Ta site) parent phase [2].",
"Incomplete nesting leaves each 13-th electron ungaped which in a typical CDW would give rise to a pocket of carriers.",
"Here, each excess carrier is self-trapped by inwards displacements of the surrounding atomic hexagon (forming the “David star” unit) which gives rise to the intragap local level accommodating this electron.",
"Exciting the self-trapped electron from the intragap level deprives the deformations from reasons of existence, the David star levels out in favor of a void in the crystal of polarons.",
"The charged voids are expected to arrange themselves into a Wigner crystal subjected to constraints of commensurability and packing with respect to the underlying structure.",
"A major question arises: why and how the repulsive voids aggregate into the net of walls leaving micro-crystalline domains in-between?",
"In this paper, we answer this and related questions by modeling the superlattice of polarons upon the 2D triangular basic lattice of all Ta atoms by a classical charged lattice gas with a screened repulsive Coulomb interaction among the particles.",
"The external pulse injecting the voids was simulated by introducing a small random concentration of voids reducing the particles concentration $\\nu $ below the equilibrium $\\nu _0=1/13$ (some other experimentally relevant concentrations are briefly described in the Supplementary Material, Sec.",
"III).",
"The subsequent evolution of the system, including the passage through the thermodynamic first order phase transition, was studied by means of the Monte Carlo simulation.",
"Surprisingly, this minimalistic model is already able to capture the formation of domain walls in a close visual resemblance with experimental observations and also to explain the effect qualitatively as an intriguing result of the charge fragmentation."
],
[
"The model",
"We model the system of polarons by a lattice gas of charged particles on a triangular lattice.",
"Each particle represents the self-trapped electron in the middle of the David star, thus the effective charge is $-e$ , which is compensated by the static uniform positive background.",
"The external pulse is simulated by a small concentration of randomly seeded voids reducing the particles' concentration below the equilibrium: $\\nu =\\nu _0 - \\delta \\nu $ .",
"The interaction of polarons located at sites $i,j$ is described by an effective Hamiltonian $H = \\sum _{i,j} U_{ij} n_i n_j$ with repulsive interactions $U_{ij}$ .",
"Here the sum is over all pairs of sites $i \\ne j$ ; $n_i = 1$ (or 0) when particle is present (or absent) at the site $i$ , and we choose $U_{ij}$ as the screened Coulomb potential $U_{ij} = \\frac{U_0 a}{|{\\mathbf {r}}_i - {\\mathbf {r}}_j|} \\exp \\left(-\\frac{r-a}{l_s}\\right),$ where $U_0 = e^2 \\exp (-a/l_s)/a$ is the Coulomb energy of interaction of particles at neighboring sites in the Wigner crystal state with the distance $a=\\sqrt{13}b$ between them ($b$ is the lattice spacing of the underlying triangular lattice, Fig.",
"REF a), $l_s$ is the screening length.",
"We keep in mind also the background uniform neutralizing positive charge."
],
[
"Superlattice and its charged defects.",
"In the ground state, all particles living on the triangular lattice tempt to arrange themselves in also the triangular superlattice, which is close-packed and most energetically favorable in 2D [12] (with some notable exceptions for more exotic potentials [13]).",
"Since the concentration of particles is $1/13$ , then the ground state is 13-fold degenerate with respect to translations (Fig.",
"REF a).",
"An additional mirror symmetry makes the ground state to be in total 26-fold degenerate.",
"But since within a given sample two mirror-symmetric phases do not coexist both in the experiment [14] and in the modeling for the sufficiently slow cooling rates (because of the high energy of the corresponding twinning wall), then we consider only one of them.",
"The simplest lattice defect is a void or a “polaronic hole” (Fig.",
"REF b) which is formed when the electron from the intragap level is taken away or excited to the conduction band and soon the associated lattice distortions vanish.",
"The single void has the relative charge $+e$ (keeping in mind the background neutralizing charge) and the Coulomb self-energy of the order $E_{void} \\simeq e^2/a$ .",
"While the void is a particular manifestation of a general notion of vacancies in crystals, in our case there can be also a specific topologically nontrivial defect – the domain wall separating domains with a different 13-fold positional degeneracy of the ground state (Fig.",
"REF ).",
"The domain wall cross-section resembles the discommensuration known in CDW systems [15].",
"Experimentally, the lattice defects can be introduced via external pulses, by impurity doping or by the field effect.",
"For example, a laser or STM pulse can excite the Mott-band electrons residing in the centers of the David star clusters, creating an ensemble of voids.",
"Since the voids are charged objects, then at first sight they should repel each other and form a Wigner crystal themselves.",
"But our modeling consistent with the experiment shows that the voids rather attract one another at short distances and their ensemble is unstable towards formation of domain walls' net.",
"Qualitatively, this instability can be understood from the following argument.",
"Compare energies of the isolated void and of the domain wall segment carrying the same charge.",
"The minimal charge of domain wall per the translation vector ${\\mathbf {a}_1}$ is $+e/13$ (Fig.",
"REF a), and the energy of the wall's segment carrying the charge $+e$ can be estimated as for a uniformly charged line: $E_{wall} \\simeq 13 \\times \\frac{(e/13)^2}{a} \\ln (l_s/a),$ For moderate screening lengthes $l_s$ , it is lower than the void's self-energy $E_{void}$ , making energetically favorable to decompose the voids into fractionally-charged domain walls.",
"The local effects beyond our model can also favor domain walls with other charges: for the single-step $+1e/13$ domain wall there are anomalous sites where David stars intersect (Fig.",
"REF a) which raises its energy and can make the double-step $+2e/13$ domain walls (Fig.",
"REF b) to be energetically favorable .",
"Figure: Positively charged domain walls with charges per unit cell length 𝐚{\\mathbf {a}}: (a) +e/13+e/13 ; (b) +2e/13+2e/13.",
"The whole sequence of domain walls can be obtained by consecutive displacements of the blue domain by the vector 𝐛 1 {\\mathbf {b}}_1 as indicated in (a).",
"In (a), the sites are encircled where David stars within the wall share the corners.",
"Black asterisks designate the sites not belonging to any star."
],
[
"Numerical modeling",
"We simulated the cooling evolution of the classical lattice gas with the interaction potential (REF ) via Metropolis Monte Carlo algorithm (see Methods).",
"We perform slow cooling from $T=0.07 U_0{}$ , which is above the detected ordering phase transition (see below), down to $T=0.01 U_0{}$ with a step $\\Delta T=-0.0002 U_0{}$ , reaching either a ground state or a very close in energy metastable state.",
"Below we, first, consider undoped systems (where particles concentration $\\nu $ is exactly $\\nu _0=1/13$ ), and then systems doped by voids (with $\\nu =\\nu _0-\\delta \\nu \\equiv \\nu _0 (1-\\nu _{voids})$ , where $\\nu _{voids}=\\delta \\nu /\\nu _0$ is the voids' concentration).",
"Results for another sign of doping are briefly presented in the Supplementary Material, Sec.",
"III."
],
[
"Undoped system.",
"As a reference system we chose the sample with $91\\times 104$ sites with the total a number of particles $N_{p}=728$ , which corresponds to the concentration $\\nu _0=1/13$ .",
"On cooling, the order-disorder phase transition takes place at $T_c \\approx 0.056 U_0{}$ , below which the triangular superlattice is formed confirming the expectations for the Wigner crystal.",
"Temperature dependencies of the order parameter $M = \\sqrt{\\sum (m_i-1/13)^2/13 \\cdot 12}$ , where $m_i$ is the fraction of particles at $i$ -th sublattice (Fig.",
"REF a) and of the mean value of energy per particle (Fig.",
"REF b) indicate that the transition is of the first order.",
"The insets in the Fig.",
"REF a show a plenty of defects just above $T_c$ , while only two displaced positions are left just below $T_c$ .",
"On heating, the order-disorder phase transition takes place at $T\\approx 0.063 U_0$ , which agrees with our mean field analysis (see Supplementary Sec.",
"II).",
"With increasing $l_s$ the temperature hysteresis and the tendency to overcooling become more pronounced.",
"An overcooling or even freezing into in a glass state feature is known for electronic systems with either a frozen disorder or a Coulomb frustrations [16]; however in the present model these both factors are absent – the effect is presumably due to only the long-range Coulomb interactions under the lattice constraints.",
"Figure: Temperature dependencies of integrated characteristics for the undoped system.",
"(a) The order parameter; the insets show snapshots of configurations of the system just above and below the phase transition; (b) mean energy per particle.",
"Blue symbols are for cooling and red symbols are for heating simulations."
],
[
"Doped system.",
"We emulate the doping (the charge injection) by seeding voids at random places and following the subsequent evolution.",
"By global characterizations like those in Fig.",
"REF , the order-disorder transition is preserved, while at a lower temperature.",
"But locally the new mosaic ground state with the net of domain walls is formed as we will demonstrate below.",
"Seeding at $T<T_c$ a small number of voids, down to two defects per sample, we observe that the single-void states are unstable with respect to their binding and progressive aggregation.",
"Seeding more voids initiates their gradual coalescence into a globule of interconnected segments of domain walls .",
"The resulting globule performs slowly a random diffusion over the sample while keeping closely its optimal shape and the structure of connections (the Supplementary Video of the system evolution under cooling and its description are presented in Supplementary Sec.",
"I).",
"Figure REF a shows the low-$T$ configuration of system $130\\times 156$ system, where initially $N_p=1544$ of particles (with the the corresponding concentration of voids $\\nu _{voids}\\approx 1.0{}\\%$ ) were randomly seeded and then the system was slowly cooled from $T=0.07 U_0{} > T_c$ down to $T=0.01 U_0{}$ .",
"In spite of the initial random distribution of particles over the whole sample, finally the voids aggregate into a single globule immersed into a connected volume of the unperturbed crystal.",
"We compare the results of our modeling in Fig.",
"REF a with the experimental picture in Fig.",
"REF b [8].",
"Similar patterns have been observed also in other experiments: [5] (Fig.",
"3 of the supplement) and [9] (Fig.",
"1).",
"With a further increase of doping, the globule size grows over the the whole sample, and finally the branched net of domain walls divides the system into the mosaics of randomly shaped domains (Fig.",
"REF a,c).",
"The comparison of the modeling with the experiment on injection by the STM pulses is shown between panels (a) and (b) in Fig.",
"REF , between panels (a) and (b), (c) and (d) in Fig.",
"REF .",
"The figures visualize a spectacular resemblance of our modeling with results from several STM experiments exploiting either the optical switching to the hidden state [7] or the pulses from the STM tip [8], [9].",
"Note that similar “irregular honeycomb network” structures were predicted for incommensurate phase of krypton on graphite with $\\nu _0\\approx 1/3$ and short-range interaction [17], but with less topological restrictions here.",
"Figure: Globule structures.",
"(a) The present modeling for ν voids ≈1.0%\\nu _{voids} \\approx 1.0{} \\%in the domain walls representation, T=0.01U 0 T=0.01 U_0; (b) from experiments in .Figure: The modeling for a high doping (a,c) vs experiments (b,d).",
"Maps of domain walls (a,b) and of domains (c,d).Figs.",
"(a,c) show the present modeling with ν voids =1.9%\\nu _{voids}=1.9{}\\% of voids at low temperature T=0.01U 0 T=0.01 U_0, the coloring scheme for domains is indicated in Fig.",
".Figs.",
"(b,d) are adapted from , numbers in Fig.",
"(d) show the corresponding coloring scheme for domains.Figs.",
"(a,c) show only a part of the system, full images can be found in Supplementary Fig.",
"6."
],
[
"Discussion and Conclusions",
"Our simulations have shown an apparently surprising behavior: some effective attraction of voids develops from the purely repulsive Coulomb interactions.",
"The coalescence of single voids starts already at their small concentration.",
"For several voids seeded, we observe a gradual fusion of point defects into the globule of the domain walls.",
"Increasingly branched net develops with augmenting of the voids concentration.",
"That can be understood indeed by noticing that the walls formation is not just gluing of voids but their fractionalization.",
"The domain wall is fractionally ($q=\\nu _0 e$ ) charged per its crystal-unit length, thus reducing the Coulomb self-energy in comparison with the integer-charged single void.",
"Being the charged objects, the domain walls repel each other, but as topological objects they can terminate only at branching points, thus forming in-plane globules.",
"Their repulsion at adjacent layers meets no constraints, hence the experimentally observed alternation of the walls' patterns among the neighboring layers [7], [8], [9].",
"A similar while simpler doping induced phase transition to the state patterned by charged domain walls was predicted for for quasi-1D polyacethylene-like systems with 2-fold degeneracy, see [18].",
"Our modeling can be straightforwardly extended to other values of concentration $\\nu _0$ : like $\\nu _0=1/3$ which is the minimal value where the pattern formation appears with qualitatively similar to the presented here results (see Supplementary Sec.",
"III) and corresponding, for example, to the $\\sqrt{3}\\times \\sqrt{3}$ surface CDW observed in the lead coated germanium crystal [19].",
"For another experimentally known case of $2H-\\mathrm {TaSe_2}$ where $\\nu _0=1/9$ the modeling results are quite different: in some doping range we see a “stripe” phase (see Supplementary Sec.",
"III), which indeed was experimentally observed in this material [20].",
"The exception of the case $\\nu _0=1/9$ is rather natural, because here the basis vectors of the superlattice and of the underlying triangular lattice are parallel to each other, which allows for the existence of neutral elementary domain walls.",
"It is also possible to study the doping by electrons by seeding the interstitials rather than voids; here the new David stars substantially overlap with their neighbors giving rise to stronger lattice deformations, which may require for a more complicated model.",
"The fragmentation with formation of walls is always confirmed while details of patterns can differ (see Supplementary Fig.",
"7).",
"The encouraging visual correspondence of our pictures with experimentally obtained patterns in different regimes of concentrations ensures a dominant role of the universal model.",
"Methods For numerical simulation of the classical lattice gas with interaction potential (REF ) we employed the Metropolis Monte Carlo method.",
"We used the screening parameter $l_s = 4.5 b \\approx 1.25 a$ and truncated the interactions to zero for sufficiently large interparticle distances (outside the hexagon with the side $24 b$ ).",
"At each temperature we performed $\\sim 10-40$ millions of Monte Carlo steps depending on the numerical experiment.",
"Temperature was linearly lowered from $T=0.07 U_0{}$ down to $T=0.01 U_0{}$ with a step $\\Delta T=-0.0002 U_0{}$ .",
"The following system sizes and numbers of particles were chosen: size $91\\times 104$ and $N_p=728$ for undoped system; size $130 \\times 156$ and $N_p=1544$ , $\\nu _{voids} \\approx 1.0{}\\%$ for the globule system (Fig.",
"REF a); size $142\\times 164$ and $N_p=1758$ , $\\nu _{voids} \\approx 1.9{}\\%$ for the net system (Fig.",
"REF a,c).",
"Periodic boundary conditions were imposed.",
"Acknowledgements The authors are grateful to D. Mihailovich, Y.A.",
"Gerasimenko, E. Tosatti and H.W.",
"Yeom for helpful discussions.",
"We acknowledge the financial support of the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST MISiS (N K3-2017-033).",
"SB acknowledges funding from the ERC AdG “Trajectory”.",
"Image source for Figs.",
"REF (b), REF (b),(d) is [8]; use permitted under the Creative Commons Attribution License CC BY 4.0; the images were cropped and rotated.",
"Author contributions S.B.",
"and P.K.",
"together formulated the theoretical concept, designed the model, analyzed results, and wrote the paper; P.K.",
"performed the numerical computations.",
"Correspondence and requests for materials should be addressed to P.K.",
"Additional information The authors declare no competing financial interests."
]
] | 1709.01912 |
[
[
"Stability and limit theorems for sequences of uniformly hyperbolic\n dynamics"
],
[
"Abstract In this paper we obtain an almost sure invariance principle for convergent sequences of either Anosov diffeomorphisms or expanding maps on compact Riemannian manifolds and prove an ergodic stability result for such sequences.",
"The sequences of maps need not correspond to typical points of a random dynamical system.",
"The methods in the proof rely on the stability of compositions of hyperbolic dynamical systems.",
"We introduce the notion of sequential conjugacies and prove that these vary in a Lipschitz way with respect to the generating sequences of dynamical systems.",
"As a consequence, we prove stability results for time-dependent expanding maps that complement results in [Franks74] on time-dependent Anosov diffeomorphisms."
],
[
"Introduction",
"Given a measurable map $f: X\\rightarrow X$ and an $f$ -invariant and ergodic probability measure $\\mu $ , the celebrated Birkhoff's ergodic theorem assures that for every $\\phi \\in L^1(\\mu )$ , the Césaro averages $\\frac{1}{n} \\sum _{j=0}^{n-1}\\phi (f^j(x))$ are almost everywhere convergent to $\\int \\phi \\, d\\mu $ .",
"Although the random variables $\\lbrace \\phi \\circ f^j\\rbrace $ are identically distributed, in general these fail to be independent.",
"Nevertheless, the classical results as the central limit theorem and almost sure invariance principles hold for dynamics with some hyperbolicity (see e.g.",
"[9], [26] and references therein).",
"In the last decade much effort has been done in order to extend the classical limit theorems for this non-stationary context, namely for the compositions of uniformly expanding maps and piecewise expanding interval maps (see [3], [7], [10], [14], [19] and references therein).",
"In this context the natural random variables obtained by the sequential dynamics are neither independent nor stationary The strategy used in the large majority of these contributions is to describe the limit properties of non-stationary compositions of Perron-Frobenius operators and to provide limit theorems under mild assumptions on the growth of variances of the appropriate random variables.",
"Other relevant contributions include also the study of the fast loss of memory on the compositions of Anosov diffeomorphisms [22] and the robustness of ergodic properties for compositions of piecewise expanding maps obtained in [24].",
"The notion of sequential dynamical systems was introduced in [6].",
"The dynamics of these non-autonomous dynamical system present substantial differences in comparison with the classical dynamical systems context.",
"In order to illustrate some difficulties one would like to mention that the non-wandering set for sequential dynamical systems is compact set but in general it fails to be invariant by the sequence of dynamics.",
"The study of sequential dynamics and the problem of its stability is motivated by the adaptative behavior of biological phenomena (see e.g.",
"[10] and references therein).",
"Our two goals here are to provide limit theorems for convergent sequences of hyperbolic dynamical systems and to prove a time-dependent stability for sequences of dynamical systems.",
"More precisely, we consider sequences ${\\mathcal {F}}=\\lbrace f_n\\rbrace _n$ of Anosov diffeomorphisms (or expanding maps) such that $d_{C^1}(f_n,f)$ tends to zero as $n$ tends to infinity.",
"The usual strategy to prove limit theorems for sequential dynamics uses the Perron-Frobenius transfer operator to show that such dynamics can be well approximated by reverse martingales.",
"In a context of convergent sequences of maps, our approach is substantially different.",
"We prove that $C^1$ -close sequential dynamics formed by either Anosov diffeomorphisms or expanding maps are stable: there exists a sequence of homeomorphisms that conjugate the dynamics (cf.",
"Theorem REF ).",
"Moreover, these sequential conjugacies vary in a Lipschitz way with respect to the sequences of dynamics (see Theorem REF ).",
"In the case of convergent sequences the sequence of homeomorphisms is $C^0$ -convergent to the identity.",
"This allows us to prove an ergodic stability for hyperbolic maps, roughly meaning that the behavior of Birkhoff averages for continuous observables is similar to the one observed by convergent sequences of nearby dynamics.",
"We refer the reader to Theorem REF for precise statements.",
"Since we obtain quantitative results for the velocity of the previous convergence to identity, limit theorems such as the central limit theorem, the functional central limit theorem and the law of the iterated logarithm transfer from the Brownian motion to time-series generated by observations on the sequential dynamical system."
],
[
"Statement of the main results",
"A sequential dynamical system is given by a collection $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathcal {Z}}$ of continuous maps $f_n : X_n \\rightarrow X_{n+1}$ , where each $(X_n,d_n)$ is a complete metric space for every $n\\in \\mathcal {Z}$ and $\\mathcal {Z}=\\mathbb {Z}^+$ or $\\mathbb {Z}$ .",
"We endow the space of sequences of maps with the $C^r$ topology.",
"More precisely, fix $r\\ge 0$ and let $\\mathcal {S}^r ((X_n)_n)$ denote the space sequences of $C^r$ -differentiable maps $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}}$ , where $f_n : X_n \\rightarrow X_{n+1}$ and every $X_n$ is a smooth manifold for all $n\\in \\mathbb {Z}$ .",
"Given sequences $\\mathcal {F}, \\mathcal {G} \\in \\mathcal {S}^r ((X_n)_n)$ , define the distance $|\\Vert \\mathcal {F} - \\mathcal {G}\\Vert |:= \\sup _{n\\in \\mathbb {Z}} \\; d_{C^r} (f_n, g_n),$ where $d_{C^r} (f,g)$ denotes the usual $C^r$ -distance between $f$ and $g$ .",
"Given $n \\ge 0$ set $F_n=f_{n-1}\\circ \\dots f_2 \\circ f_1 \\circ f_0$ The (positive) orbit of $x\\in X$ is the set $\\mathcal {O}_{\\mathcal {F}}^+(x)=\\lbrace F_n(x) : n\\in \\mathbb {Z}_+\\rbrace .$ In the case that $\\mathcal {Z}=\\mathbb {Z}$ and each element of $\\mathcal {F}$ is invertible then the orbit of $x\\in X$ is given by the set $\\mathcal {O}_{\\mathcal {F}}(x)=\\lbrace F_n(x) : n\\in \\mathbb {Z}\\rbrace $ , where $F_{-n}=f_{-n}^{-1}\\circ \\dots \\circ f_{-2}^{-1} \\circ f_{-1}^{-1}: X_0 \\rightarrow X_{-n}$ for every $n\\in \\mathbb {Z}_+$ .",
"In the present section we will state our main results."
],
[
"Limit theorems",
"An important question in ergodic theory concerns the stability of invariant measures.",
"In opposition to the notion of statistical stability, where one is interested in the continuity of a specific class of invariant measures in terms of the dynamics, here we are interested in the stability of the space of all invariant measures.",
"In the case of expanding maps the existence of sequences of conjugacies mean that, from a topological viewpoint, one can disregard the first iterates of the dynamics.",
"This is of particular interest in the case the sequence of dynamics is converging.",
"Given a continuous observable $\\phi : X \\rightarrow \\mathbb {R}$ , consider the random variables $Y_n=\\phi \\circ F_n$ associated to the non-autonomous dynamics $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}}$ .",
"In general, the sequence $(Y_n)_n$ may not be independent nor stationary.",
"In particular, invariant measures for all maps in the sequence ${\\mathcal {F}}$ seldom exist.",
"In order to establish limit theorems for non-autonomous dynamical systems we consider convergent subsequences: we assume the sequence $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}}$ is so that $f_n\\rightarrow f$ in the $C^1$ -topology.",
"We say that a probability measure on $X$ is ${\\mathcal {F}}$ -average invariant if $\\lim _{n\\rightarrow \\infty } \\frac{1}{n}\\sum _{j=0}^{n-1} (f_j)_*\\mu =\\mu ,$ where $(f_i)_*: \\mathcal {M}(X) \\rightarrow \\mathcal {M}(X) $ denotes the usual push-forward map on the space of probability measures on $X$ , We note that the previous notion deals with the individual dynamics $f_j$ instead of the concatenations $F_j$ , and that a notion of invariant measures for sequential dynamics has been defined in [16].",
"Under the previous convergence assumption, it is not hard to check that the set of ${\\mathcal {F}}$ -average invariant probability measures is a non-empty compact set and it contains the set of $f$ -invariant probability measures.",
"This makes natural to ask wether limit theorems for the limiting dynamics $f$ propagate for $C^1$ -non-autonomous dynamics.",
"The Birkhoff irregular set of $\\phi $ with respect to ${\\mathcal {F}}$ is $I_{{\\mathcal {F}},\\phi }:=\\Big \\lbrace x\\in X: \\frac{1}{n} \\sum _{j=0}^{n-1} \\phi (F_j( x )) \\text{\\,is not convergent\\,}\\Big \\rbrace $ and, by some abuse of notation, we denote by $I_{f,\\phi }$ the Birkhoff irregular set of $\\phi $ with respect to $f$ .",
"Recall that the observable $\\phi $ is cohomologous to a constant $c$ (with respect to $f$ ) if there exists a continuous function $\\chi : X\\rightarrow \\mathbb {R}$ such that $\\phi =\\chi \\circ f -\\chi + c$ .",
"Our first result points in this direction by characterizing the limits of Césaro averages of continuous observables and the irregular set for the non-autonomous dynamics ${\\mathcal {F}}$ in terms of the limit dynamics $f$ (we refer to Section for the necessary definitions).",
"Theorem 2.1 Assume that $f$ is a $C^1$ -transitive Anosov diffeomorphism on a compact Riemannian manifold $X$ and let $\\mathcal {F}=\\lbrace f_n\\rbrace _n$ be a sequence of $C^1$ maps that is $C^1$ convergent to $f$ .",
"Then there exists a homeomorphism $h$ such that: if $\\mu $ is $f$ -invariant and $\\phi \\in C(X)$ then for $(h_*\\mu )$ -almost every $x \\in X$ one has that $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{j=0}^{n-1} \\phi (F_j( x ))$ exists and $\\int \\tilde{\\phi }\\, d\\mu = \\int \\phi \\, d\\mu $ ; if $\\mu $ is $f$ -invariant and ergodic then $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{j=0}^{n-1} \\delta _{F_j( x )}=\\mu $ for $(h_*\\mu )$ -almost every $x \\in X$ .",
"Moreover, if $\\phi $ is not cohomologous to a constant with respect to $f$ then the Birkhoff irregular set $I_{{\\mathcal {F}},\\phi }:=\\big \\lbrace x\\in X: \\frac{1}{n} \\sum _{j=0}^{n-1} \\phi (F_j( x )) \\text{\\,is not convergent\\,}\\big \\rbrace $ is full topological entropy, Baire generic subset of $X$ .",
"An analogous of Theorem REF for convergent sequences of $C^1$ -expanding maps holds with the same argument used in the proof, where the existence of sequential conjugacies for $C^1$ -close Anosov diffeomorphisms is replaced by the same property for $C^1$ -expanding maps.",
"Recent contributions on limit theorems for non-autonomous dynamics include [14], [19].",
"A central limit theorem holds in a neighborhood of structural stability, provided the dynamics converge sufficiently fast.",
"We say that a sequence of random variables satisfies the almost sure invariance principle (ASIP) if there exists $\\varepsilon >0$ , a sequence of random variables $(S_n)_n$ and a Brownian motion $W$ with variance $\\sigma ^2\\ge 0$ such that $\\sum _{j=0}^{n-1} \\phi \\circ F_j=_{\\mathcal {D}} S_n$ and $S_n= W_n + \\mathcal {O}(n^{\\frac{1}{2} -\\varepsilon })$ almost everywhere.",
"The ASIP implies in many other limit theorems as the central limit theorem CLT), the weak invariance principle (WIP) or the law of the iterated logarithm (LIL) (see e.g.",
"[20]).",
"In order to prove limit theorems we require observables to be at least Hölder continuous.",
"We prove the following: Theorem 2.2 Assume that $f$ is a $C^1$ -transitive Anosov diffeomorphism on a compact Riemannian manifold $X$ , let $\\mathcal {F}=\\lbrace f_n\\rbrace _n$ be a sequence of $C^1$ maps and let $a_n:=\\sup _{\\ell \\ge n}\\Vert f_\\ell -f\\Vert _{C_1}$ .",
"If $\\phi : X \\rightarrow \\mathbb {R}$ is a $\\alpha $ -Hölder continuous, $\\int \\phi \\, d\\mu =0$ and there exists $C>0$ so that $a_j \\le C j^{-(\\frac{1}{2}+\\varepsilon )\\frac{1}{\\alpha }}$ for all $j\\ge 1$ then $\\lbrace \\phi \\circ F_j\\rbrace $ satisfies the ASIP.",
"Some comments are in order.",
"The previous result shows that the sum of the non-stationary random variables $\\lbrace \\phi \\circ F_j\\rbrace $ are strongly approximated by the Brownian motion.",
"We note that for a lower regularity (Hölder exponent) of the potential we require a higher velocity of convergence given by the tail of the sequence ${\\mathcal {F}}$ .",
"We should also mention some related results.",
"The CLT under a stability assumption (convergence to a map) for piecewise expanding interval maps was proved in [7].",
"The latter result is related to [14] where the authors obtain the ASIP for sequences of expanding maps.",
"In both papers, the authors used spectral methods and the analysis of compositions of the transfer operators.",
"Our approach differs significantly as we explore the existence of sequential conjugacies for these non-autonomous dynamics (see Subsection REF for definition and more details).",
"The previous result is complementary to the results by Stenlund [22] on exponential memory loss for compositions of Anosov diffeomorphisms.",
"The notion of uniform hyperbolicity is strongly related to $C^1$ -structural stability, that is, that of $C^1$ -dynamics that is topologically conjugate to all $C^1$ -nearby dynamical systems (see e.g.",
"[12], [18]).",
"In this subsection we shall consider the stability of non-autonomous sequences of $C^1$ -Anosov diffeomorphisms.",
"Theorem 2.3 Let $X$ be a compact Riemannian manifold and $f \\in \\text{Diff}^r(X)$ be a $C^r$ Anosov diffeomorphism on $X$ , $r\\ge 1$ .",
"There exists $\\varepsilon >0$ so that if ${\\mathcal {F}}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}}$ is a sequence with $d_{C^1}(f_n,f)<\\varepsilon $ for all $n\\in \\mathbb {Z}$ then there exists a sequence $(h_n)_{n\\in \\mathbb {Z}_+}$ of homeomorphisms on $X$ so that $f_{n-1} \\circ \\dots \\circ f_1 \\circ f_0\\,=\\, h_n^{-1} \\circ f^n \\circ h\\quad \\text{for every $n\\in \\mathbb {Z}_+$.",
"}$ The previous theorem should be compared with the stability of $C^2$ Axiom A diffeomorphisms with the strong transversality condition proved by Franks [13] (after [23]): if $f$ is a $C^2$ Axiom A diffeomorphism with the strong transversality condition then there exists a $C^1$ -open neighborhood of $f$ and for every finite set $g_1, g_2, \\dots , g_n \\in {\\mathcal {U}}$ there exists a homeomorphism $h$ so that $g_{n} \\circ \\dots \\circ g_1\\,=\\, h^{-1} \\circ f^n \\circ h$ .",
"The approach in [13] is use Banach's fixed point theorem and to construct the conjugacy as the fixed point of a suitable contraction on a Banach space.",
"Some stability results for Anosov families have been announced in [2], by a similar technique.",
"For that reason, the dependence of the conjugacy $h$ on increasing sequences of diffeomorphisms $g_{1}, g_2, \\circ \\dots , g_n$ is not explicit.",
"As we consider infinite sequences of maps we obtain a sequence of homeomorphisms $(h_n)_n$ satisfying the time-adapted almost conjugacy condition (REF ).",
"A version of Theorem REF for sequences of $C^1$ -expanding maps will appear later in Proposition REF ."
],
[
"Ideas in the proofs",
"In this subsection we introduce the ideas underlying the proof of the main results and detail the organization of the paper.",
"First we note that it is natural to expect that a quantitative version of Theorem REF could be useful to prove the ergodic stability results established in Theorems REF and REF .",
"Furthermore, we highlight that the existence of sequential conjugacies in Theorem REF seldom follows from structural stability.",
"Indeed, in the setting of Theorem REF , even though all elements of ${\\mathcal {F}}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}}$ are topologically conjugated to $f_0$ , say for every $n$ there exists an homeomorphism $\\tilde{h}_n$ satisfying $f_n = \\tilde{h}_n^{-1} \\circ f_0 \\circ \\tilde{h}_{n}$ , the compositions $F_n = \\tilde{h}_n^{-1} \\circ f_0 \\tilde{\\circ }h_{n} \\tilde{h}_{n-1}^{-1} \\circ \\dots \\circ f_0 \\circ \\tilde{h}_{2} \\tilde{h}_1^{-1} \\circ f_0 \\circ \\tilde{h}_{1} \\tilde{h}_0^{-1} \\circ f_0 \\circ \\tilde{h}_{0} \\circ f_0$ could behave wildly since it consists of an alternated iteration of $f_0$ with the homeomorphisms $\\tilde{h}_k \\tilde{h}_{k-1}^{-1}$ , which are only $C^0$ -close to identity.",
"For any $k\\in \\mathcal {Z}$ consider the shifted sequence $\\mathcal {F}^{(k)}=\\lbrace f_{n+k}\\rbrace _{n\\in \\mathcal {Z}}$ .",
"Our approach to construct sequential conjugacies is to explore shadowing for sequences of hyperbolic dynamical systems (see also Proposition REF for a similar statement in the context of sequences of expanding maps) as follows: Theorem 2.4 Let $X$ be a compact Riemannian manifold and let $f\\in \\text{Diff}^{\\,1}(X)$ be an Anosov diffeomorphism.",
"There exists a $C^1$ -open neighborhood ${\\mathcal {U}}$ of $f$ so that every $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}}$ formed by elements of ${\\mathcal {U}}$ satisfies the Lipschitz shadowing property.",
"Moreover: for any $\\beta >0$ there exists $\\zeta >0$ so that any $\\zeta $ -pseudo orbit $(x_n)_{n \\in \\mathbb {Z}}$ is $\\beta $ -shadowed by a unique point $x\\in X$ ; there exists $K>0$ so that if $\\varepsilon >0$ is small so that, for any sequence $\\mathcal {G}=\\lbrace g_n\\rbrace _n$ of elements in ${\\mathcal {U}}$ with $|\\Vert \\mathcal {F} - \\mathcal {G} \\Vert |<\\varepsilon $ there exists a unique homeomorphism $h=h_{\\mathcal {F}, \\mathcal {G}}: X\\rightarrow X$ so that $d_{C^0}(h_{{\\mathcal {F}},{\\mathcal {G}}}, id) \\le K |\\Vert {\\mathcal {F}}- {\\mathcal {G}}\\Vert |$ and $h_{ {\\mathcal {G}}^{(n)},{\\mathcal {F}}^{(n)}} \\circ F_n = G_n \\circ h_{\\mathcal {G},\\mathcal {F}}\\quad \\text{$\\forall n\\in \\mathbb {Z}_+$,}$ where $h_{{\\mathcal {F}}^{(n)}, {\\mathcal {G}}^{(n)}}: X \\rightarrow X$ denotes the uniquely homeomorphism determined by the sequences ${\\mathcal {F}}^{(n)}$ and ${\\mathcal {G}}^{(n)}$ .",
"In particular $d_{C^0}(h_{{\\mathcal {F}},{\\mathcal {G}}}, id) \\rightarrow 0$ as $|\\Vert {\\mathcal {F}}- \\mathcal {G}\\Vert | \\rightarrow 0$ .",
"We will use this quantitative version of Theorem REF to prove the ergodic stability for sequences of hyperbolic maps.",
"This paper is organized as follows.",
"In Section we recall some preliminary notions of stability, shadowing and entropy for sequential dynamical systems.",
"Section we prove some shadowing results for both sequences of $C^1$ -expanding maps and sequences of nearby $C^1$ -Anosov diffeomorphisms.",
"This allow us to construct sequential conjugacies, which we explore in Section to prove the main results on the ergodic stability for convergent sequences of hyperbolic dynamics."
],
[
"Sequential and almost conjugacies",
"A sequence ${\\mathcal {F}}$ of continuous maps acting on a compact metric space $X$ is topologically stable if for every $\\varepsilon >0$ there exists $\\delta >0$ such that for every sequence $\\mathcal {G}=\\lbrace g_n\\rbrace _n$ so that $|\\Vert \\mathcal {F} - \\mathcal {G}\\Vert |<\\delta $ there exists a continuous map $h : X \\rightarrow X$ so that $\\Vert h-id\\Vert _{C^0}<\\varepsilon $ and $d(F_n(h(x)), G_n(x))<\\varepsilon $ for all $n\\in \\mathbb {Z}$ .",
"Moreover, $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}_+}$ is an expansive sequence of maps if there exists $\\varepsilon >0$ so that for any distinct points $x,y\\in X_0$ there exists $n\\in \\mathbb {Z}_+$ such that $d(F_n(x),F_n(y))>\\varepsilon $ .",
"It is known that any positively expansive sequential dynamics admits an adapted metric on which it actually expands distances [16] and that positively expansive non-autonomous dynamical systems acting on a compact metric space with the shadowing property are topologically stable [11].",
"Given $\\beta > 0$ and the sequences of continuous maps $\\mathcal {F}= \\lbrace f_n\\rbrace _{n \\ge 1}$ and $\\mathcal {G}= \\lbrace g_n\\rbrace _{n \\ge 1}$ on a complete metric space $(X,d)$ , we say that an homeomorphism $h:X \\rightarrow X$ is a $\\beta $ -quasi-conjugacy between $\\mathcal {F}$ and $\\mathcal {G}$ if $d_{C^0} (h\\circ F_n , G_n \\circ h) \\le \\beta $ for every $n\\in \\mathbb {Z}_+$ , where $d_{C^0}(f, g)=\\Vert f-g\\Vert _{C^0}$ denotes the distance in the $C^0$ -topology.",
"The second notion does not require compactness nor the maps to act on the same compact metric space.",
"Given sequences $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}}$ and $\\mathcal {G}=\\lbrace g_n\\rbrace _{n\\in \\mathbb {Z}}$ of continuous maps acting on complete metric spaces $(X_n)_n$ , we say that a sequence $\\mathcal {H}=\\lbrace h_n\\rbrace _n$ of homeomorphisms $h_n : X_n \\rightarrow X_n$ is a sequential conjugacy between ${\\mathcal {F}}$ and ${\\mathcal {G}}$ provided that the maps $F_n : X_0 \\rightarrow X_n$ and $G_n : X_0 \\rightarrow X_n$ to satisfy $h_n \\circ F_n = G_n \\circ h_0 \\;\\text{for all $n\\in \\mathbb {Z}$}.$ Each of the maps $h_n$ in the notion of sequential conjugacies are defined in terms of the infinite sequence of maps ${\\mathcal {F}}^{(n)}$ .",
"Similarly to the classical setting, in our setting sequential conjugacies $C^0$ -close to the identity are unique (recall Theorem REF ).",
"Moreover, if ${\\mathcal {F}}=\\lbrace f\\rbrace _n$ and ${\\mathcal {G}}=\\lbrace g_n\\rbrace _n$ are sequences of dynamical systems on a compact metric space $X$ that admit a unique sequential conjugacy $C^0$ -close to the identity and the sequential conjugacies are constant (i.e.",
"$h_n=h: X\\rightarrow X$ for all $n$ ) then the sequences ${\\mathcal {F}}$ and ${\\mathcal {G}}$ are constant.",
"Indeed, if this is the case, $h$ is a conjugacy between $f$ and $g_1$ (hence between $f^2$ and $g_1^2$ ) and between $f^2$ and $g_2\\circ g_1$ .",
"By uniqueness of the conjugacies $C^0$ -close to identity we conclude that $g_2=g_1$ .",
"Applying this argument recursively we conclude that ${\\mathcal {G}}=\\lbrace g_1\\rbrace _n$ is constant.",
"This fact also shows that some flexibility in the definition of conjugacies for sequential dynamics would be necessary.",
"The flexibility of the concept of sequential conjugacies, for dynamics acting on different metric spaces, allowed us to describe a leafwise shadowing property for invariant foliations of partially hyperbolic dynamics [8]."
],
[
"Shadowing ",
"Our first main results concern the stability of Anosov sequences.",
"Fix $r\\ge 1$ , let $(X_n)_n$ be a sequence of compact Riemannian manifolds and let $\\mathcal {S}^r ((X_n)_n)$ denote the space of sequences $\\lbrace f_n\\rbrace _n$ of $C^r$ -differentiable maps $f_n: X_n \\rightarrow X_{n+1}$ .",
"We say that a sequence $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}}$ is an Anosov sequence if there exists $a>0$ , for every $n\\in \\mathbb {Z}$ there exists a continuous decomposition of the tangent bundle $T X_n=E_n^+ \\oplus E_n^-$ (of constant dimension), there exist cone fields $\\mathcal {C}^+_{a,n}(x) =\\big \\lbrace v = v^+ + v^- \\in E^+_n(x) \\oplus E^-_n(x) \\colon \\Vert v^- \\Vert \\le a \\Vert v^+\\Vert \\big \\rbrace $ and $\\mathcal {C}^-_{a,n}(x) =\\big \\lbrace v = v^+ + v^- \\in E^+_n(x) \\oplus E^-_n(x) \\colon \\Vert v^+ \\Vert \\le a \\Vert v^-\\Vert \\big \\rbrace $ and constants $\\lambda _n \\in (0,1)$ so that: a) $Df_n (x) \\, \\mathcal {C}^+_{a,n}(x) \\subset \\mathcal {C}^+_{\\lambda _n a, \\, n+1}(f_n(x))$ and $Df_n (x)^{-1} \\, \\mathcal {C}^-_{a,n+1}(f_n(x)) \\subset \\mathcal {C}^-_{\\lambda _n a,\\, n}(x)$ b) $\\Vert Df_n^{-1}(x) v \\Vert \\ge \\lambda _n^{-1} \\Vert v\\Vert $ for every $v\\in \\mathcal {C}^-_{a,n+1}(f_n(x))$ and $\\Vert Df_n(x) v \\Vert \\ge \\lambda _n^{-1} \\Vert v\\Vert $ for every $v\\in \\mathcal {C}^+_{a,n}(x) $ for every $x\\in X_n$ and $n\\in \\mathbb {Z}_+$ .",
"We refer to $a>0$ as the diameter of the cone fields.",
"It is clear that if $\\mathcal {F}$ is an Anosov sequence then $\\mathcal {F}^{(k)}$ is also an Anosov sequence for every $k\\in \\mathbb {Z}$ .",
"Moreover, a constant sequence $\\lbrace f\\rbrace _{n\\in \\mathbb {Z}}$ is an Anosov sequence if and only if the diffeomorphism $f$ is Anosov.",
"Here we will be interested in Anosov sequences formed by diffeomorphisms in a $C^1$ -neighborhood of some fixed Anosov diffeomorphism.",
"As uniform hyperbolicity can be characterized by the existence of stable and unstable cone fields, if $f$ is an Anosov diffeomorphism there exists a $C^1$ -open neighborhood ${\\mathcal {U}}$ of $f$ such that every sequence ${\\mathcal {F}}=\\lbrace f_n\\rbrace $ formed by elements of ${\\mathcal {U}}$ is an Anosov sequence.",
"We refer the reader to [15] for the $C^1$ -robustness and stability of Anosov diffeomorphisms, and to Subsection REF for some of the geometrical properties of Anosov sequences.",
"In order to state our main results on shadowing and stability of non-autonomous dynamical systems we recall some necessary notions.",
"Given $\\delta >0$ , we say that $(x_n)_{n=0}^k$ is a $\\delta $ -pseudo orbit for $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}_+}$ if $x_n \\in X_n$ and $d( f_n(x_n), x_{n+1} )<\\delta $ for every $0\\le n\\le k-1$ .",
"We say that the sequence of maps $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}_+}$ has the shadowing property if for every $\\varepsilon >0$ there exists $\\delta >0$ such that for any $\\delta $ -pseudo-orbit $(x_n)_{n=0}^k$ there exists $x\\in X$ so that its $\\mathcal {F}$ -orbit $\\varepsilon $ -shadows the sequence $(x_n)_{n=0}^k$ , that is, $d( F_n (x), x_{n+1})<\\varepsilon \\,\\text{ for every } 0\\le n \\le k-1.$ Moreover, we say that $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}_+}$ has the Lipschitz shadowing property if there exists a uniform constant $L>0$ so that one can choose $\\delta =L \\varepsilon $ above.",
"Finally, if $X_n=X$ for every $n$ , we say that the sequence $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}_+}$ has the periodic shadowing property if for any $\\varepsilon >0$ there exists $\\delta >0$ so that any $\\delta $ -pseudo orbit $(x_n)_{n=0}^k$ satisfying $x_0=x_k$ is $\\varepsilon $ -shadowed by a fixed point $x\\in X$ for $F_k=f_k \\circ \\dots \\circ f_2\\circ f_1\\circ f_0$ .",
"The previous notions are often referred as finite shadowing properties since consider finite pseudo-orbits.",
"Nevertheless, in locally compact spaces it is a well known fact that the finite shadowing orbit property is equivalent to the shadowing property using infinite pseudo-orbits."
],
[
"Topological entropy ",
"In their seminal work, Kolyada and Snoha [17] introduced and studied a concept of entropy for non-autonomous dynamical systems and prove, among other results, that the entropy is concentrated in the non-wandering set.",
"The non-wandering set for non autonomous dynamical systems is a compact set but, in general, it misses to be invariant by the sequence of dynamical systems.",
"This makes the problems of proving stability and finding conjugacies for nearby dynamics a hard topic.",
"Let us recall some necessary results from [17].",
"Let ${\\mathcal {F}}=\\lbrace f_n\\rbrace _n$ be a sequence of maps on a compact metric space $(X,d)$ .",
"For every $n\\ge 1$ consider the distance $d_n(x,y):=\\max _{0\\le j \\le n-1} d( F_j(x), F_j(y))$ .",
"A set $E\\subset X$ is $(n,\\varepsilon )$ -separated for ${\\mathcal {F}}$ if $d_n(x,y)>\\varepsilon $ for every $x,y\\in E$ with $x\\ne y$ .",
"For $Z\\subset X$ define $s_n({\\mathcal {F}},\\varepsilon , Z)= \\max \\lbrace \\# E \\colon E \\; \\text{is a}\\; (n,\\varepsilon ) \\, \\text{separated set in} \\, Z\\rbrace $ and the topological entropy of ${\\mathcal {F}}$ on $Z\\subset X$ by $h_{Z}({\\mathcal {F}})= \\lim _{\\varepsilon \\rightarrow 0}\\limsup _{n\\rightarrow \\infty } \\frac{1}{n} \\log s_n({\\mathcal {F}},\\varepsilon , Z),$ and set $h_{top}({\\mathcal {F}})=h_{X}({\\mathcal {F}})$ .",
"The topological entropy of sequences ${\\mathcal {F}}=\\lbrace f_n\\rbrace _n$ and ${\\mathcal {G}}=\\lbrace g_n\\rbrace _n$ is not determined by the individual dynamics.",
"Indeed, there are examples where each $f_n$ and $g_n$ are topologically conjugate for all $n$ but $h_{top}({\\mathcal {F}}) \\ne h_{top}({\\mathcal {G}})$ (cf.",
"[17]).",
"A pair of sequences ${\\mathcal {F}}=\\lbrace f_n\\rbrace _n$ and ${\\mathcal {G}}=\\lbrace g_n\\rbrace _n$ is equiconjugate if there exists a sequence of homeomorphism $(\\tilde{h}_n)_n$ such that: (i) $\\tilde{h}_{n+1}\\circ f_n= g_n\\circ \\tilde{h}_n$ , and (ii) the sequences $(\\tilde{h}_n)_n$ and $(\\tilde{h}_n^{-1})_n$ are equicontinuous.",
"It is not hard to check that if $f$ is structurally stable and $(f_n)_n$ are $C^1$ -close and convergent to $f$ then ${\\mathcal {F}}=\\lbrace f\\rbrace _n$ and ${\\mathcal {G}}=\\lbrace f_n\\rbrace _n$ are equiconjugate.",
"Moreover, the following holds: Proposition 3.1 [17] Let ${\\mathcal {F}}=\\lbrace f_n\\rbrace _n$ and ${\\mathcal {G}}=\\lbrace g_n\\rbrace _n$ be sequences of continuous maps on a compact metric space $X$ and $Y$ , respectively.",
"If ${\\mathcal {F}}$ and ${\\mathcal {G}}$ are equiconjugate then $h_{top}({\\mathcal {F}}) =h_{top}({\\mathcal {G}})$ ."
],
[
"Stability of non-autonomous expanding maps on compact metric spaces",
"Let $(X_n,d_n)$ be a sequence of complete metric spaces and let $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}_+}$ be a sequence of continuous and onto maps $f_n: X_n \\rightarrow X_{n+1}$ .",
"We say that $\\mathcal {F}$ is a sequence of expanding maps if there are $\\delta _0>0$ and a sequence $(\\lambda _n)_{n\\in \\mathbb {Z}_+}$ of constants in $(0,1)$ so that the following holds: for any $n\\in \\mathbb {Z}_+$ , $x\\in X_{n+1}$ and $x_i\\in f_n^{-1}(x)$ there exists a well defined inverse branch $f_{n,x_i}^{-1}: B(x,\\delta _0) \\rightarrow V_{x_{i}}$ (open neighborhood of $x_{i}$ ) so that $d ( f_{n,x_i}^{-1}(y), f_{n,x_i}^{-1}(z) )\\le \\lambda _n \\, d(y,z)$ for every $y,z\\in B(x,\\delta _0)$ Here, for notational simplicity, we omit the metrics $d_n$ representing them by $d$ .",
"In what follows we observe that any sequence $\\mathcal {F}$ of expansive maps on compact metric spaces admit a uniform lower bound on the separation time.",
"More precisely: Lemma 4.1 Let $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}_+}$ be an expansive sequence of continuous maps $f_n: X_n \\rightarrow X_{n+1}$ acting on metric spaces and let $\\varepsilon _0$ be an expansiveness constant for $\\mathcal {F}$ .",
"If $X_0$ is compact then for any $\\delta >0$ there exists $N\\in \\mathbb {Z}_+$ such that, if $x,y\\in X_0$ satisfy $d(F_n(x),F_n(y))<\\varepsilon _0$ for every $0\\le n\\le N$ then $d(x,y)<\\delta $ .",
"We prove the lemma by contradiction.",
"Assume there exists $\\delta >0$ and, for every $j\\ge 0$ , there are $x_j,y_j \\in X_0$ with $d(x_j,y_j) >\\delta $ and $d(F_n(x_j),F_n(y_j))<\\varepsilon _0$ for every $0\\le n\\le j$ .",
"Since $X_0$ is compact we may assume (up to consider subsequences) that $x_j \\rightarrow x \\in X_0$ and $y_j \\rightarrow y \\in X_0$ .",
"By continuity, taking $j \\rightarrow \\infty $ we get that $x\\ne y$ and $d(F_n(x),F_n(y))\\le \\varepsilon _0 \\; \\text{for every $n\\in \\mathbb {Z}_+$},$ which contradicts the expansiveness property.",
"This proves the lemma.",
"We can now state our second result on the stability of sequences of expanding maps.",
"Proposition 4.2 (Existence of sequential conjugacies) Let $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}_+}$ be a sequence of $C^1$ -expanding maps acting on compact Riemannian manifolds $X_n$ with contraction rates $(\\lambda _n)_{n\\in \\mathbb {Z}_+}$ for inverse branches satisfying $\\sup _{n\\in \\mathbb {Z}_+} \\lambda _n <1$ .",
"There exists $\\varepsilon >0$ so that, for any sequence $\\mathcal {G}$ of $C^1$ maps satisfying $|\\Vert \\mathcal {F} - \\mathcal {G} \\Vert |<\\varepsilon $ there exists a unique homeomorphism $h=h_{\\mathcal {F}, \\mathcal {G}}: X_0 \\rightarrow X_0$ so that $h_{ {\\mathcal {G}}^{(n)},{\\mathcal {F}}^{(n)}} \\circ F_n = G_n \\circ h_{\\mathcal {G},\\mathcal {F}}\\quad \\text{$\\forall n\\in \\mathbb {Z}_+$,}$ where $h_{{\\mathcal {F}}^{(n)}, {\\mathcal {G}}^{(n)}}: X_n \\rightarrow X_n$ denotes the uniquely homeomorphism determined by the sequences ${\\mathcal {F}}^{(n)}$ and ${\\mathcal {G}}^{(n)}$ .",
"Moreover, there exists $L>0$ so that $\\Vert h_{{\\mathcal {F}}, {\\mathcal {G}}}-id\\Vert _{C^0} \\le L |\\Vert {\\mathcal {G}}- {\\mathcal {F}}\\Vert |$ .",
"Let $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}_+}$ be as above and let $\\varepsilon _1>0$ be small such that any sequence of $C^1$ maps $\\mathcal {G}$ satisfying $|\\Vert \\mathcal {F} - \\mathcal {G} \\Vert |<\\varepsilon _1$ is a sequence of $C^1$ -expanding maps (such a constant exists since the set of expanding maps is $C^1$ -open and $\\lambda :=\\sup _{n\\in \\mathbb {Z}_+} \\lambda _n <1$ ).",
"Reduce $\\varepsilon _1>0$ , if necessary, so that every sequence ${\\mathcal {G}}$ as above is expansive with uniform expansiveness constant $\\varepsilon _0>0$ .",
"Fix $0<\\varepsilon < \\frac{1}{4} \\min \\lbrace \\varepsilon _0, \\varepsilon _1\\rbrace $ and let $L\\ge 1$ be given by the Lipschitz shadowing property (cf.",
"Proposition ).",
"If ${\\mathcal {G}}$ is any sequence of $C^1$ expanding maps $\\mathcal {G}$ satisfying $|\\Vert \\mathcal {F} - \\mathcal {G} \\Vert |<\\varepsilon /L$ and $x\\in X_0$ then the sequence $(G_n(x))_{n\\in \\mathbb {Z}_+}$ forms a $\\varepsilon /L$ -pseudo orbit with respect to the sequence $\\mathcal {F}$ , as $d( f_n ( G_n(x) ) , G_{n+1}(x) )= d( f_n ( G_n(x) ) , g_n (G_n(x)) ) \\le \\Vert f_n -g_n \\Vert _{C^0} < \\frac{\\varepsilon }{L}$ for every $n\\in \\mathbb {Z}_+$ .",
"Hence, there exists a unique point $h_{\\mathcal {F},\\mathcal {G}}(x) \\in X_0$ so that $d (F_n (h_{\\mathcal {F},\\mathcal {G}}(x)), \\, G_n(x) ) <\\varepsilon \\quad \\text{ for every } n\\in \\mathbb {Z}_+$ (see Figure REF below).",
"Reversing the role of ${\\mathcal {F}}$ and $\\mathcal {G}$ and replacing $x$ by $h_{\\mathcal {F},\\mathcal {G}}(x)$ , we deduce that there exists a unique point $h_{\\mathcal {G},\\mathcal {F}}( h_{\\mathcal {F},\\mathcal {G}}(x) ) \\in X_0$ so that $d (G_n (h_{\\mathcal {G},\\mathcal {F}}(h_{\\mathcal {F},\\mathcal {G}}(x))), \\, F_n(h_{\\mathcal {F},\\mathcal {G}}(x)) ) <\\varepsilon \\quad \\text{ for every } n\\in \\mathbb {Z}_+.$ Figure: (G n (x)) n≥0 (G_n(x))_{n\\ge 0} as δ\\delta -pseudo-orbit with respect to ℱ={f n } n∈ℤ + \\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}_+}and shadowing point h ℱ,𝒢 (x)∈X 0 h_{\\mathcal {F},\\mathcal {G}}(x)\\in X_0As $|\\Vert \\mathcal {F} - \\mathcal {G}\\Vert |<\\varepsilon $ , by triangular inequality we get $d (G_n (h_{\\mathcal {G},\\mathcal {F}}(h_{\\mathcal {F},\\mathcal {G}}(x))), G_n(x))<2\\varepsilon <\\varepsilon _0$ for every $n\\in \\mathbb {Z}_+$ .",
"Since $\\varepsilon _0$ is an expansiveness constant for ${\\mathcal {G}}$ , the latter assures that $h_{\\mathcal {G},\\mathcal {F}}(h_{\\mathcal {F},\\mathcal {G}}(x))=x$ , proving that $h_{\\mathcal {F},\\mathcal {G}}$ is invertible and $h_{\\mathcal {F},\\mathcal {G}}^{-1}=h_{\\mathcal {G},\\mathcal {F}}$ .",
"Figure: Shadowing points h ℱ,𝒢 (x)h_{{\\mathcal {F}},{\\mathcal {G}}}(x) and h 𝒢,ℱ (x)h_{{\\mathcal {G}},{\\mathcal {F}}}(x) on X 0 X_0Moreover, taking $n=0$ in (REF ) we get that $\\Vert h_{{\\mathcal {F}},{\\mathcal {G}}}-id_{X_0}\\Vert _{C^0} \\le \\varepsilon $ .",
"Now we prove that $h=h_{\\mathcal {F}, \\mathcal {G}}: X_0 \\rightarrow X_0$ is an homeomorphism.",
"Take $0<\\delta <\\varepsilon _0/4$ and let $N=N(\\delta )\\ge 1$ given by Lemma REF .",
"As the spaces $X_n$ are compact, the set of functions $\\lbrace G_1, \\dots , G_N\\rbrace $ is equicontinuous: there exists $\\eta >0$ so that if $d(x,y)<\\eta $ then $d(G_n(x),G_n(y))<\\delta $ for every $0\\le n \\le N$ .",
"Thus, if $d(x,y)<\\eta $ and $0\\le n \\le N$ then $d(F_n(h_{\\mathcal {F},\\mathcal {G}}(x)),F_n(h_{\\mathcal {F},\\mathcal {G}}(y)))& \\le d(F_n(h_{\\mathcal {F},\\mathcal {G}}(x)),G_n(x))+d(F_n(h_{\\mathcal {F},\\mathcal {G}}(y)),G_n(y))\\\\&+d(G_n(x),G_n(y))\\le 2 \\varepsilon +\\delta < \\varepsilon _0.$ Lemma REF implies $d(h_{\\mathcal {F}, \\mathcal {G}}(x),h_{\\mathcal {F}, \\mathcal {G}}(y))<\\delta $ and the continuity of $h_{\\mathcal {F}, \\mathcal {G}}$ follows.",
"By a similar argument, or using the fact that $X_0$ is a compact metric space and $h_{\\mathcal {F}, \\mathcal {G}}$ is a continuous bijection, we conclude the continuity of its inverse $h_{\\mathcal {G}, \\mathcal {F}}$ .",
"Finally, we are left to prove the conjugacy relation (REF ).",
"Clearly $|\\Vert {\\mathcal {F}}^{(n)} - {\\mathcal {G}}^{(n)} \\Vert | <\\varepsilon /L$ for every $n\\in \\mathbb {Z}_+$ .",
"Recalling that $F^{(n)}_k = f_{n+k-1} \\dots f_{n}$ and $ G^{(n)}_k = g_{n+k-1} \\dots g_{n}$ for every $k\\ge 0$ , we note that similar computations as before yield that the orbit of the point $F_n(h_{\\mathcal {F}, \\mathcal {G}}(x) \\in X_n$ by the sequence ${\\mathcal {F}}^{(n)}$ is an $\\varepsilon /L$ -pseudo-orbit with respect to the sequence ${\\mathcal {G}}^{(n)}$ .",
"In particular, there exists a unique point $h_{{\\mathcal {G}}^{(n)}, {\\mathcal {F}}^{(n)}}(\\, F_n(h_{\\mathcal {F}, \\mathcal {G}}(x)) \\, ) \\in X_n$ for which $d( G_k^{(n)} (\\, h_{{\\mathcal {G}}^{(n)}, {\\mathcal {F}}^{(n)}}( F_n(h_{\\mathcal {F}, \\mathcal {G}}(x)) \\, ) ,F_k^{(n)}( F_n(h_{\\mathcal {F}, \\mathcal {G}}(x)) ) < \\varepsilon $ for every $k\\ge 0$ .",
"Combining equations (REF ) and (REF ) (recall that $F_k^{(n)}( F_n) = F_{n+k}$ and $G_k^{(n)}( G_n) = G_{n+k}$ for every $k\\ge 0$ ), by triangular inequality $d( G^{(n)}_k (\\, h_{{\\mathcal {G}}^{(n)}, {\\mathcal {F}}^{(n)}}( F_n(h_{\\mathcal {F}, \\mathcal {G}}(x)) \\, )) ,G^{(n)}_k (G_n(x)) ) < 2\\varepsilon <\\varepsilon _0$ for every $k\\ge 0$ .",
"Using once more that $\\varepsilon _0$ is an expansiveness constant for the sequence ${\\mathcal {G}}^{(n)}$ we deduce that $h_{{\\mathcal {G}}^{(n)}, {\\mathcal {F}}^{(n)}} \\circ F_n \\circ h_{\\mathcal {F}, \\mathcal {G}}(x) = G_n(x)$ for every $n\\in \\mathbb {Z}_+$ and every $x\\in X_0$ .",
"In other words, $h_{{\\mathcal {G}}^{(n)}, {\\mathcal {F}}^{(n)}} \\circ F_n (x) = G_n\\circ h_{\\mathcal {G}, \\mathcal {F}}$ for all $n\\in \\mathbb {Z}_+$ , which finishes the proof of the propostion.",
"Remark 4.3 Proposition REF means that the conjugacies $(h_{{\\mathcal {F}}^{(n)}, {\\mathcal {G}}^{(n)}})_{n\\in \\mathbb {Z}_+}$ improve (meaning that $h_{\\mathcal {F}, \\mathcal {G}}$ is $C^0$ -convergent to the identity map) in the case that $|\\Vert {\\mathcal {F}}^{(n)} - {\\mathcal {G}}^{(n)}\\Vert | \\rightarrow 0$ as $n$ tends to infinity.",
"If $f,g$ are $C^1$ -expanding maps on a compact Riemannian manifold $X$ , ${\\mathcal {F}}=\\lbrace f\\rbrace _{n\\in \\mathbb {Z}_+}$ and ${\\mathcal {G}}=\\lbrace g\\rbrace _{n\\in \\mathbb {Z}_+}$ then $F_n=f^n$ and $G_n=g^n$ for every $n\\in \\mathbb {Z}_+$ and the sequence of conjugacies $(h_{{\\mathcal {F}}^{(n)}, {\\mathcal {G}}^{(n)}} )_{n\\in \\mathbb {Z}_+}$ is constant to the conjugacy $h_{\\mathcal {F}, \\mathcal {G}}$ between $f$ and $g$ .",
"More generally, if there exists $N\\ge 1$ so that ${\\mathcal {F}}^{(N)}={\\mathcal {F}}$ (e.g.",
"${\\mathcal {F}}=\\lbrace f,g,f,g, f,g,\\dots \\rbrace $ and $N=2$ ) then the conjugacies $(h_n)_{n\\in \\mathbb {Z}_+}$ are $N$ -periodic: $h_{n+N}=h_n$ for every $n\\in \\mathbb {Z}_+$ .",
"In the special case of periodic sequences of expanding maps we derive the following: Corollary A Assume that $f$ is a $C^1$ -expanding map on a compact Riemannian manifold $X$ and let $\\mathcal {F}=\\lbrace f_n\\rbrace _n$ be a $N$ -periodic sequence of expanding maps such that ${\\mathcal {F}}$ and ${\\mathcal {G}}=\\lbrace f\\rbrace _{n\\in \\mathbb {Z}}$ are sequentially conjugate.",
"Then there are homeomorphisms $(h_i)_{i=0\\dots N-1}$ so that $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{j=0}^{n-1} \\delta _{F_j(x)}= \\frac{1}{N}\\sum _{i=0}^{N-1} (h_i)_*\\mu \\quad \\text{for $\\mu $-a.e.",
"$x\\in X$.",
"}$ Assume that ${\\mathcal {F}}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}}$ is a $N$ -periodic sequence (that is, $f_{n+N}=f_n$ for every $n\\in \\mathbb {Z}$ ), and that ${\\mathcal {F}}$ and ${\\mathcal {G}}=\\lbrace f\\rbrace _{n\\in \\mathbb {Z}}$ are sequentially conjugate: for every $n$ there exists a homeomorphism $h_n$ so that $F_n=h_n \\circ f^n \\circ h_0^{-1}.$ By Remark REF the sequence of conjugacies $(h_n)_n$ is also $N$ -periodic.",
"Let $\\mu $ be a $f$ -invariant probability (hence $f^N$ -invariant) that is ergodic with respect to $f^N$ .",
"If $\\phi \\in C^0(X)$ , $\\tilde{\\mu }:=(h_0)_*\\mu $ and $h_0(y)=x$ then $\\sum _{j=0}^{n-1} \\phi (F_j(x))& = \\sum _{j=0}^{n-1} \\phi \\circ h_j \\circ f^j (h_0^{-1}(x)) \\nonumber \\\\& = \\sum _{i=0}^{N-1} \\sum _{\\ell =0}^{\\big [\\frac{n}{N}\\big ]} \\sum _{0\\le \\ell N+i \\le n-1}(\\phi \\circ h_i) ( f^{\\ell N+i}(y))\\nonumber \\\\& = \\sum _{i=0}^{N-1} \\sum _{\\ell =0}^{\\big [\\frac{n}{N}\\big ]} \\sum _{0\\le \\ell N+i \\le n-1} \\psi _i( f^{\\ell N}(y))$ for $\\tilde{\\mu }$ -almost every $y$ , where $\\psi _i=\\phi \\circ h_i \\circ f^i$ for every $0\\le i \\le N-1$ .",
"Since $\\mu $ is ergodic for $f^N$ then there exists a $\\tilde{\\mu }$ -full measure subset of points $y$ for which $\\lim _{k\\rightarrow \\infty } \\frac{1}{k}\\sum _{j=0}^{k-1} \\varphi (f^{jN}(y)) =\\int \\varphi \\, d\\mu $ for every continuous $\\varphi : X \\rightarrow \\mathbb {R}$ .",
"Together with (REF ) and the $f$ -invariance of $\\mu $ , this proves that $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{j=0}^{n-1} \\phi (F_j(x))= \\int \\phi \\; d \\Big ( \\frac{1}{N}\\sum _{i=0}^{N-1} (h_i)_*\\mu \\Big )$ for $\\mu $ -almost every $x\\in X$ and every $\\phi \\in C^0(X)$ .",
"In other words, $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{j=0}^{n-1} \\delta _{F_j(x)}= \\frac{1}{N}\\sum _{i=0}^{N-1} (h_i)_*\\mu $ for $\\mu $ -almost every $x\\in X$ .",
"The following result asserts that orbits of $\\beta $ -quasi-conjugate $\\mathcal {F}$ and $\\mathcal {G}$ (up to the $\\beta $ -quasi-conjugacy) remain within always within distance $\\beta $ from each other or, equivalently, the orbits are indistinguishable at scale $\\beta $ .",
"More precisely: Proposition 4.4 (Existence of quasi-conjugacies) Let $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}_+}$ be a sequence of $C^1$ -expanding maps acting on a compact Riemannian manifold $X$ with contraction rates of inverse branches $(\\lambda _n)_{n\\in \\mathbb {Z}_+}$ satisfying $\\lambda := \\sup _{n\\in \\mathbb {Z}_+} \\lambda _n <1$ .",
"Then, for all sufficiently small $\\varepsilon >0$ and for any sequence of $C^1$ maps $\\mathcal {G}$ satisfying $|\\Vert \\mathcal {F} - \\mathcal {G} \\Vert |<\\varepsilon $ there exists a $\\frac{2\\lambda }{1- \\lambda } \\varepsilon $ -quasi-conjugacy $h: X \\rightarrow X$ between $\\mathcal {F}$ and $\\mathcal {G}$ .",
"Moreover, $\\Vert h- id\\Vert _{C^0} \\rightarrow 0$ as $\\mathcal {G}$ tends to $\\mathcal {F}$ .",
"Let $\\varepsilon >0$ be small enough so that any sequence $\\mathcal {G}$ of $C^1$ maps satisfying $|\\Vert \\mathcal {F} - \\mathcal {G} \\Vert |<\\varepsilon $ is a sequence of $C^1$ -expanding maps.",
"Let $\\delta _0>0$ be a uniform lower bound for the radius of the inverse branches domain for the expanding maps all such sequences $\\mathcal {G}$ .",
"Take $0<\\delta <\\delta _0/2$ and take $\\varepsilon := (1- \\lambda ) \\delta /\\lambda >0$ .",
"Suppose $|\\Vert \\mathcal {F}- \\mathcal {G}\\Vert |< \\varepsilon $ .",
"Given any $x_0=x \\in X$ set $x_n:= F_n(x)$ for every $n\\in \\mathbb {Z}_+$ .",
"As before, for every $n\\in \\mathbb {Z}_+$ let $f_{n,x_n}^{-1}$ denote the inverse branch of $f_n$ such that $f_{n,x_{n}}^{-1}(x_{n+1})= x_{n}$ .",
"For notational simplicity, let $g_n^{-1}$ denote the inverse branch of $g_n$ , whose domain contains $B(x_{n+1},\\delta )$ , which is $C^0$ -closer to $f_{n,x_n}^{-1}$ .",
"We claim that $g_n^{-1}(B(x_{n+1}, \\delta )) \\subset B(x_{n}, \\delta )$ for all $n\\in \\mathbb {Z}_+$ .",
"In fact, using $d(g_{n}(x_{n}), x_{n+1}) = d(g_{n}(x_{n}), f_n(x_{n})) < \\varepsilon = \\frac{1- \\lambda }{\\lambda }\\delta $ we conclude that, for any $z \\in B(g_{n}(x_{n}), \\varepsilon + \\delta )$ (in particular for points of $B(x_{n+1}, \\delta )$ ), $d(g_n^{-1}(z), x_{n} ) \\le \\lambda (\\varepsilon + \\delta ) \\le (1- \\lambda )\\delta + \\lambda \\delta =\\delta .$ Hence, if we define $\\hat{G}_n:= g_1^{-1} \\circ \\dots \\circ g_n^{-1}$ then the sets $Y_n:= \\hat{G}_n(\\overline{B(x_{n+1}, \\delta ))}$ form a nested sequence of closed sets with diameter smaller or equal to $2 \\lambda ^n \\delta $ .",
"Thus, there is a unique point $h(x) \\in X$ such that $d(G_n(h(x)), F_n(x)) = d(G_n( h(x)), x_n) \\le \\delta $ for every $n \\in \\mathbb {Z}_+$ .",
"As $x\\in X$ was chosen arbitrary, the map $h: X \\rightarrow X$ defined by the previous construction satisfies $\\Vert h-id\\Vert _{C^0}<\\delta $ .",
"Moreover, by triangular inequality, $d(G_n(h(x)), h (F_n(x)))\\le d(G_n( h( x)), F_n(x))+ d(F_n(x), h \\circ F_n(x)) \\le 2\\delta = \\frac{2\\lambda }{1- \\lambda } \\varepsilon .$ We proceed to prove the continuity of $h$ .",
"Given $\\tilde{\\varepsilon }> 0$ take $N> 0$ such that $2\\lambda ^{N} \\delta < \\tilde{\\varepsilon }$ .",
"Now, by uniform continuity of the maps $F_j$ with $j\\le N$ there exists $\\tilde{\\delta }> 0$ such that if $d(x, y)< \\tilde{\\delta }$ then $d(F_j(x), F_j(y))< \\delta \\; \\text{for every $j= 0, \\dots , N$.",
"}$ So, $\\hat{G}_n(\\overline{B( F_{n+1}(x), 2\\delta )}) \\supset \\hat{G}_n(\\overline{B(F_{n+1} (y), \\delta )}) \\cup \\hat{G}_n(\\overline{B( F_{n+1}(x), \\delta )})$ and contains both points $h(x)$ and $h( y)$ .",
"Since $\\operatorname{{diam}}(\\hat{G}_n(\\overline{B(F_{n+1} (x)), 2 \\delta )})< \\tilde{\\varepsilon }$ this shows that $d(h(x), h(y))< \\tilde{\\varepsilon }$ , and implies on the continuity of $h$ .",
"As we proved that there exists exactly one point $h(x)$ whose $\\mathcal {G}$ -orbit that $\\delta $ -shadows the $\\mathcal {F}$ -orbit of a point $x \\in X$ , exchanging the roles of $\\mathcal {F}$ and $\\mathcal {G}$ , one can use the same argument as in the proof of Proposition REF to assure that there exists a unique point $h^{-1}(h(x))=x$ that $\\delta $ -shadows the $\\mathcal {G}$ -orbit of the point $h(x) \\in X$ and, consequently, to deduce that $h$ is a homeomorphism.",
"It is immediate that $h \\rightarrow id$ as $\\mathcal {G} \\rightarrow \\mathcal {F}$ .",
"This finishes the proof of the proposition.",
"We observe that the stability notions in the statement of Propositions REF and REF are unrelated, thus these cannot be obtained one from each other.",
"One of the advantages of Proposition REF is to observe that time-dependent conjugacies become smaller as time evolves for sequences that are asymptotic.",
"An advantage of Proposition REF is to obtain quasi-conjugacies and to compute the proximity of the quasi-conjugacy from the identity in terms of contracting rates for the sequence, which is the best one can hope computationally.",
"Although quasi-conjugacies need not unique, this is the case for stably expansive sequences ${\\mathcal {F}}$ : if $\\varepsilon $ is an expansiveness constant for all sequences $\\mathcal {G}$ arbitrarily close to ${\\mathcal {F}}$ , $h$ is a $\\beta $ -quasi-conjugacy between ${\\mathcal {F}}$ and ${\\mathcal {G}}$ with $0<\\beta <\\varepsilon /4$ and $\\tilde{h}:X \\rightarrow X$ is a homeomorphism satisfying $d_{C^0}(h,\\tilde{h}) \\ll \\varepsilon /4$ then $d_{C^0} (\\tilde{h}\\circ F_n , G_n \\circ \\tilde{h})\\ge d_{C^0} (G_n \\circ h , G_n \\circ \\tilde{h})- d_{C^0}(h, \\tilde{h})-\\beta \\ge d_{C^0} (G_n \\circ h , G_n \\circ \\tilde{h}) - \\varepsilon /2$ which is larger $\\varepsilon /3$ for some $n\\in \\mathbb {Z}_+$ (by the $\\varepsilon $ -expansiveness of the sequence ${\\mathcal {G}}$ )."
],
[
"Shadowing and stability for Anosov sequences",
"In this section we prove shadowing and stability for sequences of Anosov diffeomorphisms as stated in Theorem REF ."
],
[
"Invariant manifolds",
"Assume that ${\\mathcal {F}}=\\lbrace f_{n}\\rbrace _{n\\in \\mathbb {Z}}$ is an Anosov sequence (recall Subsection REF for the definition).",
"Recall the notations ${\\mathcal {F}}^{(k)}=\\lbrace f_{k+n}\\rbrace _{n\\in \\mathbb {Z}}$ , $F_{-n}^{(k)}=f_{k-n}^{-1}\\circ \\dots \\circ f_{k-2}^{-1} \\circ f_{k-1}^{-1}$ and $F_n^{(k)}=f_{n+k-1} \\circ \\dots \\circ f_{k+1}\\circ f_k$ , for all $k,n\\in \\mathbb {Z}$ .",
"The existence of invariant cone fields with uniform expansion and contraction imply on the following properties: for every $k\\in \\mathbb {Z}$ and $x\\in X_k$ , the subspaces $E_{{\\mathcal {F}}^{(k)}}^u(x) := \\bigcap _{n \\ge 0} D F^{(k-n)}_{n} (F^{(k)}_{-n}(x)) \\; \\mathcal {C}^u_{a,k-n}(F^{(k)}_{-n}(x))\\subset T_x X_k$ and $E_{{\\mathcal {F}}^{(k)}}^s(x) := \\bigcap _{n \\ge 0} D F^{(k+n)}_{-n} (F^{(k)}_{n}(x)) \\; \\mathcal {C}^s_{a,k+n}(F^{(k)}_{n}(x))\\subset T_x X_k$ satisfy the invariance conditions $Df_k (x) E_{{\\mathcal {F}}^{(k)}}^*(x)= E_{{\\mathcal {F}}^{(k+1)}}^*(f_k(x))$ for $*\\in \\lbrace s,u\\rbrace $ .",
"there are constants $C>0$ , $\\delta _1>0$ and $\\tilde{\\lambda }\\in (\\lambda , 1)$ so that, for any $x\\in X_k$ , $k\\in \\mathbb {Z}$ and $*\\in \\lbrace s,u\\rbrace $ there exists a unique smooth submanifold ${\\mathcal {W}}^*_{{\\mathcal {F}}^{(k)},\\delta _1}(x)$ of $X_k$ (of size $\\delta _1$ ) that is tangent to the subbundle $E_{{\\mathcal {F}}^{(k)}}^*(x)$ at $x$ , in such a way that: (i) $f_k({\\mathcal {W}}^*_{{\\mathcal {F}}^{(k)},\\delta _1}(x))= {\\mathcal {W}}^*_{{\\mathcal {F}}^{(k+1)},\\delta _1}(f_k(x))$ (ii) $d_{{\\mathcal {W}}^s} ( f_k(y), f_k(z) ) \\le \\tilde{\\lambda }\\, d_{{\\mathcal {W}}^s} ( y,z)$ for all $ y,z \\in {\\mathcal {W}}^s_{{\\mathcal {F}}^{(k)},\\delta _1}(x)$ (iii) $d_{{\\mathcal {W}}^u} ( f_k^{-1}(y), f_k^{-1}(z) ) \\le \\tilde{\\lambda }\\, d_{{\\mathcal {W}}^u} ( y,z)$ for all $y,z \\in {\\mathcal {W}}^u_{{\\mathcal {F}}^{(k+1)},\\delta _1}(x)$ (iv) the angles between stable and unstable bundles $E^s_{{\\mathcal {F}}^{(k)}}(x)$ and $E^u_{{\\mathcal {F}}^{(k)}}(x)$ is bounded away from zero by some constant $\\theta _k>0$ (iv) for any $0<\\varepsilon <\\delta _1$ there exists $\\delta _k>0$ so that for any $x,y\\in X_k$ with $d(x,y)<2\\delta _k$ the transverse intersection ${\\mathcal {W}}^s_{{\\mathcal {F}}^{(k)},\\varepsilon }(x) \\pitchfork {\\mathcal {W}}^u_{{\\mathcal {F}}^{(k)},\\varepsilon }(y)$ consists of a unique point in $X_k$ (v) ${\\mathcal {W}}^s_{{\\mathcal {F}}^{(k)},\\varepsilon }(x)=\\lbrace y\\in X_k \\colon d(F_n^{(k)}(y),F_n^{(k)}(x)) \\le \\varepsilon \\;\\text{for every}\\; n\\ge 0\\rbrace $ and ${\\mathcal {W}}^u_{{\\mathcal {F}}^{(k)},\\varepsilon }(x)=\\lbrace y\\in X_k \\colon d(F_{-n}^{(k)}(y),F_{-n}^{(k)}(x)) \\le \\varepsilon \\;\\text{for every}\\; n\\ge 0\\rbrace $ Item (1) follows from the existence of the strictly invariant cone fields.",
"Item (2) follows from the existence of stable and unstable manifolds for sequences of maps, using the graph transform method (cf.",
"[4] or [1]).",
"Here we opted to write the invariant manifolds as ${\\mathcal {W}}^*_{{\\mathcal {F}}^{(k)}}$ ($*\\in \\lbrace s,u\\rbrace $ ) to specify the shifted sequence ${\\mathcal {F}}^{(k)}$ with respect to which the uniform contracting or expanding behavior holds.",
"In the case that $f\\in \\text{Diff}^{\\,1}(X)$ is an Anosov diffeomorphism there exists a $C^1$ -small open neighborhood ${\\mathcal {U}}$ of $f$ so that the cone fields $\\mathcal {C}^u_{a_0}$ and $\\mathcal {C}^s_{a_0}$ (determined by $f$ ) are $Dg$ -invariant for all $g\\in {\\mathcal {U}}$ , and that $\\lambda :=\\sup _{g\\in {\\mathcal {U}}} \\lambda _g <1$ .",
"Furthermore, there are constants $\\theta ,\\delta ,\\varepsilon >0$ (depending only on $f$ and ${\\mathcal {U}}$ ) so that any sequence $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}}$ by elements of ${\\mathcal {U}}$ is such that $\\theta < \\theta _n$ , $\\delta < \\delta _n$ and $\\varepsilon <\\varepsilon _n$ for all $n$ .",
"In other words, both the angles between stable and unstable subspaces and the sizes given by local product structure are uniformly bounded away from zero.",
"In consequence, the sequence ${\\mathcal {F}}$ is $\\delta _1$ -expansive: if $d(F_n(x),F_n(y))<\\varepsilon $ for all $n\\in \\mathbb {Z}$ then $x\\in {\\mathcal {W}}^s_{{\\mathcal {F}},\\delta _1}(y) \\cap {\\mathcal {W}}^u_{{\\mathcal {F}},\\delta _1}(y) =\\lbrace y\\rbrace $ ."
],
[
"Proof of Theorem ",
"Let $f$ be a $C^1$ Anosov diffeomorphism, let ${\\mathcal {U}}$ be a $C^1$ -open neighborhood of $f$ as above, and let $\\mathcal {F}=\\lbrace f_n\\rbrace _{n\\in \\mathbb {Z}}$ be a sequence of Anosov diffeomorphisms with $f_n\\in {\\mathcal {U}}$ for all $n\\in \\mathbb {Z}$ .",
"In particular $\\lambda :=\\sup _{n\\in \\mathbb {N}} \\lambda _n <1$ .",
"Let $\\delta _1>0$ and $\\tilde{\\lambda }\\in (\\lambda ,1)$ be given by items (1) and (2) above.",
"Fix $\\beta >0$ .",
"We claim that there exists $\\zeta >0$ such that every $\\zeta $ -pseudo orbit for ${\\mathcal {F}}$ is $\\beta $ -shadowed.",
"Take $0<\\varepsilon < (1-\\tilde{\\lambda }) \\frac{\\beta }{2}<\\frac{\\beta }{2}$ and let $0<\\delta <\\varepsilon $ be given by the local product structure (cf.",
"item (2) (iv)): $d(x,y)<2\\delta \\quad \\Rightarrow \\quad {\\mathcal {W}}^s_{{\\mathcal {F}}^{(k)},\\varepsilon }(x) \\pitchfork {\\mathcal {W}}^u_{{\\mathcal {F}}^{(k)},\\varepsilon }(y)\\;\\text{consists of a unique point.",
"}$ Take $N\\ge 1$ such that $\\tilde{\\lambda }^N \\varepsilon <\\delta \\slash 2$ .",
"Since $\\sup _{g\\in {\\mathcal {U}}} \\sup _{x\\in X} \\Vert Dg(x)\\Vert <\\infty $ , the mean value inequality ensures the equicontinuity of the elements of the sequence ${\\mathcal {F}}$ .",
"Thus, there exists $0<\\zeta <\\delta $ such that: if $(z_n)_{n\\ge 0}$ is a $\\zeta $ -pseudo orbit and $k\\ge 0$ then the finite pseudo orbit $(z_{n+k})_{n= 0}^N$ is such that $d(F^{(k)}_j(z_k),z_{k+j})<\\frac{\\delta }{2} \\quad \\text{for all}\\; k \\in \\mathbb {Z} \\;\\text{and} \\; |j|\\le N$ (cf.",
"[11]).",
"if $d(x,y)<\\zeta $ then $d(F^{(k)}_j(x),F^{(k)}_j(y))<\\frac{\\delta }{2} \\quad \\text{for all}\\; k \\in \\mathbb {Z} \\;\\text{and} \\; |j|\\le N$ (by the mean value inequality).",
"First we prove that the shadowing property holds for finite pseudo-orbits.",
"For that, we may assume without loss of generality that the pseudo-orbits $(x_n)_{n=0}^k$ are formed by a number $k$ of points that is a multiple of $N$ (otherwise just consider the extended pseudo-orbit $(x_n)_{n=0}^{(j+1)N}$ , where $x_n=F^{(k)}_{n-k}(x_k)$ for every $k+1\\le n \\le (j+1)N$ and $j\\in \\mathbb {Z}_+$ is uniquely determined by $jN < k \\le (j+1)N$ ).",
"Fix $p\\in \\mathbb {N}$ arbitrary and let $(x_n)_{n=0}^{pN}$ be a $\\zeta $ -pseudo-orbit for ${\\mathcal {F}}$ .",
"We claim that $(x_n)_{n=0}^{pN}$ is $\\beta $ -shadowed by some point $x\\in X$ .",
"We define recursively $y_0=x_0$ and $y_{n+1} \\in \\mathcal {W}_{{\\mathcal {F}}^{((n+1)N)},\\varepsilon }^s(x_{(n+1)N})\\pitchfork \\mathcal {W}_{{\\mathcal {F}}^{((n+1)N)},\\varepsilon }^u(F^{(nN)}_{N}(y_{n}))$ for every $0\\le n\\le p-1$ (cf.",
"Figure REF below).",
"Figure: Construction of homoclinic pointsNote that the homoclinic points $(y_n)_{n=0}^p$ as above are well defined.",
"Indeed, as $y_{n}\\in \\mathcal {W}_{{\\mathcal {F}}^{(nN)},\\varepsilon }^s(x_{nN})$ we have that $d_{{\\mathcal {W}}^s}(F^{(nN)}_{N}(y_{n}), F^{(nN)}_{N}(x_{nN}))\\le \\tilde{\\lambda }^N\\, d_{{\\mathcal {W}}^s}( y_{n}, x_{nN})\\le \\tilde{\\lambda }^N \\, \\varepsilon <\\frac{\\delta }{2}.$ In addition, as $(x_k)_k$ is a $\\zeta $ -pseudo-orbit, by triangular inequality we conclude that $d(x_{(n+1)N},F_{N}^{(nN)}(y_{n}))\\le d(F_{N}^{(nN)}(y_{n}),F_{N}^{(nN)}(x_{nN}))+d(F_{N}^{(nN)}(x_{nN}),x_{(n+1)N})<\\delta $ for every $0\\le n \\le p$ .",
"This guarantees that the homoclinic point $y_{n+1}$ is well defined, as claimed.",
"Now we prove that the point $x=(F_{pN})^{-1}(y_{p})=F_{-pN}^{(pN)}(y_{p}) \\in X_0$ is such that its orbit $\\beta $ -shadows the pseudo-orbit $(x_n)_{n=0}^{pN}$ .",
"By construction $y_{n}\\in \\mathcal {W}_{{\\mathcal {F}}^{(nN)},\\varepsilon }^u(F^{((n-1)N)}_{N}(y_{n-1}))$ for all $0\\le n\\le p$ (recall (REF )).",
"Thus, by invariance of the unstable leaves and backward contraction we get $d( F^{(nN)}_{-N} (y_{n}), y_{n-1})&=d((F^{(nN)}_{N})^{-1} (y_{n}),(F^{(nN)}_{N})^{-1}(F^{(nN)}_{N}(y_{n-1})))\\\\&\\le \\tilde{\\lambda }^N d( y_{n}, F^{((n-1)N)}_{N}(y_{n-1}))\\le \\tilde{\\lambda }^{N}\\varepsilon $ for every $0\\le n\\le p$ .",
"In particular it follows that $d(F_{nN}(x) , x_{nN} )& = d(F_{nN}(F_{pN}^{-1} (y_p)) , x_{nN} )= d((F^{(nN)}_{(p-n)N})^{-1} (y_p) , x_{nN} ) \\\\& \\le d(y_n , x_{nN} )+ \\sum _{\\ell =1}^{p-n} \\; d(\\, F^{((n+\\ell )N)}_{-\\ell N} (y_{n+\\ell }) , F^{((n+\\ell -1)N)}_{-(\\ell -1)N} (y_{n+\\ell -1}) \\,) \\\\& \\le \\sum _{\\ell =0}^{p-n} \\tilde{\\lambda }^{\\ell N} \\varepsilon \\le \\frac{\\varepsilon }{1-\\tilde{\\lambda }^N} < \\frac{\\varepsilon }{1-\\tilde{\\lambda }} <\\frac{\\beta }{2}$ for every $0\\le n \\le p$ .",
"By equicontinuity, the choice of $0<\\delta <\\frac{\\beta }{2}$ and consequence (b) of the equicontinuity for the elements of ${\\mathcal {F}}$ , for every $0\\le n \\le p$ and $0\\le j<N$ $d(F_{j+nN}(x) , x_{j+nN} )& \\le d(F_{j} (F_{nN}(x)) , F_j(x_{nN}) )+ d(F_j(x_{nN}), x_{j+nN} )< \\beta $ It follows that the sequence $(x_n)_{n=0}^{p}$ is $\\beta $ -shadowed by the point $x$ and so the shadowing property follows for finite pseudo-orbits.",
"Since $\\delta >0$ is independent of $p\\in \\mathbb {Z}_+$ , a simple argument using compactness assures that any infinite $\\zeta $ -pseudo-orbit is $\\beta $ -traced by some point in $X$ .",
"Finally, if $x$ and $y$ are two points that $\\beta $ -shadow the same $\\zeta =\\zeta (\\beta )$ -pseudo orbit, then $d(F_j(x),F_j(y))\\le d(F_j(x),x_j)+d(F_j(y),x_j)<2\\beta <2\\delta _1$ for all $j\\in \\mathbb {Z}$ and, by the choice of $\\delta _0$ , we conclude $x=y$ .",
"This proves item (1) in the theorem.",
"In order to deduce the Lipschitz shadowing property holds we determine a linear dependence between the constants $\\zeta $ and $\\beta $ .",
"As described above, given $\\beta >0$ take $\\varepsilon =(1-\\tilde{\\lambda })\\beta /2>0$ .",
"Since the angle between stable and unstable bundles for the sequential dynamical system is bounded away from zero and local stable and unstable disks have bounded curvature and are tangent to the stable bundles then the local product structure property holds: there exists $L>0$ so that if $d(x,y)<\\frac{2\\varepsilon }{L}$ then ${\\mathcal {W}}^s_{{\\mathcal {F}}^{(k)},\\varepsilon }(x)$ and ${\\mathcal {W}}^u_{{\\mathcal {F}}^{(k)},\\varepsilon }(y)$ intersects at a unique point.",
"Taking $\\delta =\\varepsilon /L$ and $N\\ge 1$ so that $\\tilde{\\lambda }^N \\varepsilon <\\delta /2$ then we conclude that one can take $\\zeta = \\frac{1-\\tilde{\\lambda }}{8LM^N} \\beta >0$ (with $M:=\\sup _{g\\in {\\mathcal {U}}} \\Vert Dg\\Vert _0$ ), where the constant $K:=(\\frac{1-\\tilde{\\lambda }}{8LM^N})^{-1}>0$ depends only on $f$ .",
"This proves the Lipschitz shadowing property.",
"We proceed to prove (2).",
"Let $U$ is a $C^1$ -small neighborhood of $f$ and $K>0$ be as above.",
"Assume that $|\\Vert {\\mathcal {F}}-{\\mathcal {G}}\\Vert |<\\zeta $ is small and $x\\in X$ .",
"We can find a unique point $y$ whose $\\mathcal {F}-$ orbit $K\\zeta $ -shadows the $\\mathcal {G}-$ orbit of $x$ .",
"This is assured by item (1) taking into account that $( G_j(x))_{j\\in \\mathbb {Z}}$ is a $\\zeta $ -pseudo-orbit for $\\mathcal {F}$ .",
"Thus there exists a unique point $y=:h_{{\\mathcal {F}},{\\mathcal {G}}}(x)$ that $K\\zeta $ -shadows $( G_j(x))_{j\\in \\mathbb {Z}}$ , i.e., $d(F_j(h(x)),G_j(x))<K \\zeta $ for all $j\\in \\mathbb {Z}$ .",
"We prove first that $h=h_{{\\mathcal {F}},{\\mathcal {G}}}$ is an homeomorphism.",
"First we prove that the map $h=h_{{\\mathcal {F}},{\\mathcal {G}}}$ is continuous.",
"On the one hand, the uniqueness of the shadowing point assured by item (1) implies that $h(x)=\\bigcap _{j=-\\infty }^{\\infty }F_{j}^{-1}(B_{K\\zeta }(G_j(x))),$ where $B_\\varepsilon (z)$ denotes the ball of radius $\\varepsilon >0$ around $z$ in $X$ .",
"In Particular, $d(h(x),x)<K\\zeta $ .",
"By hyperbolicity, the diameter of the finite intersection $\\bigcap _{j=-N}^{N}F_{j}^{-1}(B_{K\\zeta }(G_j(x)))$ tends to zero (uniformly in $x$ ) as $N$ tends to infinity.",
"It implies that for $z$ sufficiently close to $x$ , the points in the finite intersection $\\bigcap _{j=-N}^{N} F_{j}^{-1}(B_{K \\zeta } (G_j(z))$ lie in a small ball around $h(x)$ , which proves the continuity of $h$ .",
"To check that $h$ is injective, if $h(x_1)=h(x_2)$ then $d(F_j(h(x_i)),G_j(x_i))<K \\zeta \\quad \\text{ and }\\quad F_j(h(x_1))=F_j(h(x_2))$ for every $j\\in \\mathbb {Z}$ and $i=1,2$ .",
"by triangular inequality we obtain $d(G_j(x_1),G_j(x_2))<2K \\zeta $ , for all $j\\in \\mathbb {Z}$ .",
"If $2K \\zeta $ is smaller than the constant expansiveness of $\\mathcal {G}$ then we conclude that $x_1=x_2$ .",
"Now, using the fact that $h$ a continuous and one to one map on a compact and connected Riemannian manifold $X$ we get, by the invariance domain theorem, that $h$ is an open map so $h(X)$ is open.",
"As $X$ is compact, $h(X)$ is compact and, in particular, is closed.",
"Altogether, this implies that $h(X)$ is a connected component of $X$ , hence $h(X)=X$ .",
"Finally, using the fact that $X$ is compact and $h$ is a continuous one to one map on $X$ we obtain that $h$ is an homeomorphism.",
"The proof of (REF ) is entirely analogous to the one of Proposition REF .",
"This concludes the proof of the theorem.",
"$\\square $"
],
[
"Proof of Theorem ",
"Assume that $f$ is a $C^1$ -transitive Anosov diffeomorphism on a compact Riemannian manifold $X$ and let $\\mathcal {F}=\\lbrace f_n\\rbrace _n$ be a sequence of expanding maps so that $\\lim _{n\\rightarrow \\infty } \\Vert f_n-f\\Vert _{C_1}=0$ .",
"Proposition REF assures that there exist $L>0$ and homeomorphisms $h, h_n \\in Homeo(X)$ so that (i) $h_n \\circ f^n = F_n \\circ h$ for every $n\\in \\mathbb {Z}_+$ ; (ii) $d_{C^0}(h_n,id) \\le L |\\Vert {\\mathcal {G}}^{(n)} - {\\mathcal {F}}^{(n)}\\Vert | \\rightarrow 0$ as $n$ tends to infinity.",
"Hence, for any continuous observable $\\phi : X \\rightarrow \\mathbb {R}$ we get that the continuous observables $\\phi _j := \\phi \\circ h_j$ are uniformly convergent to $\\phi $ as $j\\rightarrow \\infty $ .",
"Moreover, given $x\\in X$ , $\\Big | \\frac{1}{n} \\sum _{j=0}^{n-1} \\phi (F_j( x ))- \\frac{1}{n} \\sum _{j=0}^{n-1} \\phi (f^j( h^{-1}(x) )) \\Big |& = \\Big | \\frac{1}{n} \\sum _{j=0}^{n-1} \\phi _j (f^j( h^{-1}(x) ))- \\frac{1}{n} \\sum _{j=0}^{n-1} \\phi (f^j( h^{-1}(x) )) \\Big | \\nonumber \\\\& \\le \\frac{1}{n} \\sum _{j=0}^{n-1} \\Vert \\phi _j -\\phi \\Vert _{C^0} $ which tends to zero as $n\\rightarrow \\infty $ .",
"In consequence, if $\\mu $ is $f$ -invariant and $\\phi \\in C^0(X)$ then it follows from from Birkhoff's ergodic theorem that for $(h_*\\mu )$ -almost every $x \\in X$ one has that $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{j=0}^{n-1} \\phi (F_j( x ))$ exists and $\\int \\tilde{\\phi }\\, d\\mu = \\int \\phi \\, d\\mu $ .",
"In particular if $\\mu $ is $f$ -invariant and ergodic then $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{j=0}^{n-1} \\delta _{F_j( x )}=\\mu $ for $(h_*\\mu )$ -almost every $x \\in X$ .",
"This proves items (1) and (2).",
"We are left to prove that the Birkhoff irregular set of $\\phi $ whith respect to ${\\mathcal {F}}$ has full topological entropy and ir a Baire residual subset of $X$ .",
"Estimate (REF ) implies that $x\\in I_{f,\\phi }$ if and only if $h^{-1}(x)\\in I_{{\\mathcal {F}},\\phi }$ .",
"In other words, $h( I_{f,\\phi })=I_{{\\mathcal {F}},\\phi }$ .",
"In particular if $\\mu $ is $f$ -invariant and ergodic then $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{j=0}^{n-1} \\delta _{F_j( x )}=\\mu $ for $(h_*\\mu )$ -almost every $x \\in X$ .",
"This proves items (1) and (2).",
"Now assume that $\\phi $ is not cohomogous to a constant with respect to $f$ .",
"We are left to prove that the Birkhoff irregular set of $\\phi $ whith respect to $\\mathcal {F}$ has full topological entropy and ir a Baire residual subset of $X$ .",
"Estimate (REF ) implies that $x\\in I_{f,\\phi }$ if and only if $h^{-1}(x)\\in I_{{\\mathcal {F}},\\phi }$ .",
"In other words, $h( I_{f,\\phi })=I_{{\\mathcal {F}},\\phi }$ .",
"Using that $I_{f,\\phi }$ is a Baire residual subset of $X$ (cf.",
"[5]) and that $h$ is a homeomorphism we conclude that $I_{{\\mathcal {F}},\\phi }$ is a Baire residual subset.",
"We are left to prove that $h_{I_{{\\mathcal {F}},\\phi }}({\\mathcal {F}})=h_{top}({\\mathcal {F}})$ .",
"We may assume without loss of generality that $f$ satisfies the specification property (otherwise just take $f^k$ for some $k> 1$ ).",
"In particular, $\\phi $ is not cohomologous to a constant if and only if $I_{f,\\phi }\\ne \\emptyset $ .",
"Moreover, if $\\alpha \\ne \\beta $ are accumulation points of the set $\\lbrace \\frac{1}{n}\\sum _{j=0}^{n-1}\\phi (f^j(x) \\colon x\\in X, \\; n\\ge 1\\rbrace $ and $I_{f,\\phi }(\\alpha ,\\beta )= \\Big \\lbrace x\\in X \\colon \\underline{\\lim }_{n\\rightarrow \\infty }\\frac{1}{n}\\sum _{j=0}^{n-1}\\phi (f^j(x)=\\alpha < \\beta =\\overline{\\lim }_{n\\rightarrow \\infty }\\frac{1}{n}\\sum _{j=0}^{n-1}\\phi (f^j(x) \\Big \\rbrace $ then $h_{I_{f,\\phi }(\\alpha ,\\beta )}(f) =h_{top}(f)$ (cf.",
"[25]).",
"We need the following: Lemma 5.1 Let ${\\mathcal {F}}=\\lbrace f_n\\rbrace _n$ be a sequence of $C^1$ -diffeomorphisms convergent to $f$ and $Z\\subset X$ .",
"Then $h_{Z}({\\mathcal {F}}) =h_{Z}(f)$ .",
"Note that $h_n \\circ f^n = F_n \\circ h$ and that $(h_n)_n$ converges to the identity as $n\\rightarrow \\infty $ .",
"For any $\\varepsilon >0$ there exists $N_\\varepsilon \\ge 1$ so that $d_{C^0}(h_n,id)<\\varepsilon /3$ for all $n\\ge N_\\varepsilon $ .",
"In consequence, if $d(f^j(x),f^j(y))>\\varepsilon $ then $d(F_j (h(x)), F_j(h(y)))>\\varepsilon /3.$ This proves that if $E\\subset X$ is a $(n,\\varepsilon )$ -separated set for $f$ then $h(E)$ is a $(n,\\varepsilon /3)$ -separated set for ${\\mathcal {F}}$ , and that $s_n({\\mathcal {F}}, \\varepsilon /3, X) \\ge s_n(f, \\varepsilon , X)$ for every $n\\ge N_\\varepsilon $ .",
"Reciprocally, if $d(F_j (x), F_j(y))>\\varepsilon $ for some $j\\ge N_\\varepsilon $ then $d(f^j(h^{-1}(x)),f^j(h^{-1}(y)))\\ge d(F_j (x), F_j(y)) - \\frac{2\\varepsilon }{3} >\\frac{\\varepsilon }{3}$ and so $s_j({\\mathcal {F}}, \\varepsilon , X) \\ge s_j(f, \\varepsilon /3, X).$ The statement of the lemma is now immediate.",
"Now, if $N\\ge 1$ is large so that $\\frac{1}{n} \\sum _{j=0}^{n-1} \\Vert \\phi _j -\\phi \\Vert _{C^0}< \\frac{\\beta -\\alpha }{2}$ for every $n\\ge N$ then (REF ) implies that $\\underline{\\lim }_{n\\rightarrow \\infty }\\frac{1}{n}\\sum _{j=0}^{n-1}\\phi (F_j(x)< \\overline{\\lim }_{n\\rightarrow \\infty }\\frac{1}{n}\\sum _{j=0}^{n-1}\\phi (F_j(x)$ for every $x\\in I_{{\\mathcal {F}},\\phi ,\\phi }(\\alpha ,\\beta )$ .",
"In other words, $I_{f,\\phi }(\\alpha ,\\beta )\\subset I_{{\\mathcal {F}},\\phi }$ and, consequently, $h_{I_{{\\mathcal {F}},\\phi }}(f) \\ge h_{I_{f,\\phi }(\\alpha ,\\beta )}(f)= h_{top}(f)$ .",
"Finally, Lemma REF implies that $h_{I_{{\\mathcal {F}},\\phi }}({\\mathcal {F}}) = h_{I_{{\\mathcal {F}},\\phi }}(f) = h_{top}(f)$ .",
"This completes the proof of the theorem."
],
[
"Proof of Theorem ",
"Let $\\phi : X \\rightarrow \\mathbb {R}$ be a mean zero $\\alpha $ -Hölder continuous and let $|\\phi |_\\alpha $ denote the Hölder constant.",
"If $\\phi _j := \\phi \\circ h_j$ then $\\Vert \\phi _j-\\phi \\Vert _{C^0} \\le |\\phi |_\\alpha \\Vert h_j - id\\Vert _{C^0}^\\alpha \\le |\\phi |_\\alpha L^\\alpha (\\sup _{\\ell \\ge j}\\Vert f_\\ell - f \\Vert _{C^1})^\\alpha $ .",
"In other words, $\\Vert \\phi _j-\\phi \\Vert _{C^0} \\le |\\phi |_\\alpha L^\\alpha a_j^\\alpha $ .",
"In particular $\\Big | \\sum _{j=0}^{n-1} \\phi \\circ F_j- \\sum _{j=0}^{n-1} \\phi \\circ f^j \\circ h^{-1} \\Big |& \\le \\sum _{j=0}^{n-1} \\Vert \\phi _j -\\phi \\Vert _{C^0}\\le |\\phi |_\\alpha L^\\alpha \\sum _{j=0}^{n-1} a_j^\\alpha $ and, consequently, $\\sum _{j=0}^{n-1} \\phi \\circ F_j= \\sum _{j=0}^{n-1} \\phi \\circ f^j \\circ h^{-1}+ \\mathcal {O}( n^{\\frac{1}{2}-\\varepsilon })$ provided that $a_j \\le C j^{-(\\frac{1}{2}+\\varepsilon )\\frac{1}{\\alpha }}$ for all $j\\ge 1$ .",
"Since $\\phi $ is mean zero and Hölder continuous and $f$ is uniformly expanding then the almost sure invariance principle follows from the corresponding one for uniformly expanding dynamics with $\\sigma ^2=\\int \\phi ^2 \\, d\\mu + \\sum _{n= 1}^{\\infty } \\phi \\, (\\phi \\circ f^j) \\,d\\mu >0$ .",
"(see e.g.",
"[9]).",
"This proves the theorem."
],
[
"Acknowledgments:",
"This work was partially supported by CNPq-Brazil.",
"The authors are deeply grateful to T. Bomfim for some references and useful discussions."
]
] | 1709.01652 |
[
[
"Double Gravitational Wave Mergers"
],
[
"Abstract In this paper we study the dynamical outcome in which black hole (BH) binary-single interactions lead to two successive gravitational wave (GW) mergers; a scenario we refer to as a `double GW merger'.",
"The first GW merger happens during the three-body interaction through a two-body GW capture, where the second GW merger is between the BH formed in the first GW merger and the remaining bound single BH.",
"We estimate the probability for observing both GW mergers, and for observing only the second merger that we refer to as a `prompt second-generation (2G) merger'.",
"We find that the probability for observing both GW mergers is only notable for co-planar interactions with low GW kicks ($\\lesssim 10^{1}-10^{2}$ kms$^{-1}$), which suggests that double GW mergers can be used to probe environments facilitating such interactions.",
"For isotropic encounters, such as the one found in globular clusters, the probability for prompt 2G mergers to form is only at the percent level, suggesting that second-generation mergers are most likely to be between BHs which have swapped partners at least once."
],
[
"Introduction",
"The recent gravitational wave (GW) detections by the Laser Interferometer Gravitational-Wave Observatory (LIGO) of binary black hole (BBH) mergers [2], [3], [1], [5], [6], have initiated a wide range of studies on how such BBHs form and merge in our Universe [4].",
"The merger channels that have been suggested to contribute include stellar clusters , [13], , [10], , , , , [9], , primordial black holes [18], [23], , [21], active galactic nuclei discs [14], , , galactic nuclei , [38], , [7], [37], isolated field binaries [24], [25], [26], [16], [15], and field triples , .",
"Theoretical work indicates that these channels likely result in similar merger rates and chirp mass distributions , which has lead to several discussions on how to observationally distinguish them.",
"From this, it seems that the spins of the BHs and their orbital eccentricity in the LIGO band, are very promising parameters.",
"Regarding spins, BH spins are naturally believed to be isotropically distributed for the dynamical channel , [27], whereas the field binary channel is more likely to result in BH spins that are preferentially aligned [45].",
"In case of eccentricity, BBH mergers from the field binary channel are expected to be near circular at the time of observation , whereas a notable fraction of the dynamically assembled BBH mergers are likely to be eccentric in the LIGO band as a result of chaotic exchanges of angular momentum during their formation [32], , [8], , , , .",
"In this paper we study the role of post-Newtonian (PN) effects [19] – or GR corrections – in few-body interactions, for exploring the interesting and distinct outcome in which a binary-single interaction results in not only one, but in two GW mergers; an outcome first studied numerically by [20], and later dynamically by .",
"To shorten the descriptions, we denote this outcome a double GW merger.",
"In this scenario, the first GW merger forms through a standard two-body GW capture [33], [11], while the three BHs temporary constitute a bound state .",
"After this follows the formation of the second GW merger, which is between the BH formed in the first GW merger and the remaining bound BH.",
"The endstate of the double GW merger scenario is therefore a single BH with a mass approximately that of the three initial BHs, and a velocity composed of the initial three-body center-of-mass (COM) velocity and any acquired GW kick velocity.",
"We note here that this multi GW merger scenario is somewhat the GR equivalent of the classical multi stellar collision scenario described in, e.g., [28].",
"In the work presented here, we explore two unique observables related to the double GW merger scenario: i) The first is the observation of both GW mergers, which naturally requires that the time between the first and the second GW merger, denoted $t_{12}$ , is less than the observation time window ($< \\mathcal {O}(10^{1})$ years).",
"For this outcome, we find that near co-planar interactions generally give rise to the shortest time interval $t_{12}$ , which in a few cases is $ < \\mathcal {O}{(\\text{years})}$ .",
"The reason for a co-planar preference is that in this case the angular momentum carried by the incoming single can significantly reduce the angular momentum of the initial target binary and thereby the GW life time.",
"This makes the double GW merger channel an indirect probe of environments facilitating co-planar interactions.",
"Although speculative, such environments may include rotationally supported systems, such as an active galactic nuclei (AGN) disk [17], , [14], , .",
"Another example is the class of disk-like systems forming in galactic nuclei as a result of vector resonant relaxation [46], [47], .",
"ii) The second is the observation of only the second GW merger, which we in short will refer to as prompt second-generation (2G) merger.",
"As most binary-single interactions are expected to be between objects of similar mass , , prompt 2G mergers will most often be between two BHs with mass ratio about $1:2$ , where the heavier BH spins at $\\sim 0.7$ .",
"Recent studies have suggested that dynamical environments indirectly can be probed by the search for second-generation mergers [7], [29], ; however, the contribution from our presented prompt 2G channel has not been discussed in the literature so far.",
"For a 2G merger to occur the time span $t_{12}$ has to be smaller than the average encounter time scale for the considered dynamical system ($< 10^{8}$ years).",
"In this case we also find that near co-planar interactions can significantly contribute to a prompt 2G population, whereas the rate of prompt 2G mergers in isotropic systems is suppressed.",
"The paper is structured as follows.",
"In Section we describe some of the basic properties related to binary-single interactions with and without GR corrections.",
"This includes a definition of the relevant binary-single outcomes, and a description of how these distribute in orbital phase space.",
"In Section we present our main results on the formation of double GW mergers.",
"We summarize our findings in Section ."
],
[
"Black Hole Binary-Single Interactions",
"The initial total energy of a binary-single interaction broadly determines the range of possible outcomes and associated observables [36], [44], .",
"If the total energy is positive, also known as the soft-binary (SB) limit, the interaction is always prompt, and will most likely lead to either an ionization, a fly-by, or an exchange endstate [44], [42].",
"If the total energy is instead negative, also known as the hard-binary (HB) limit, the system can enter a bound state with a lifetime that generally is in the range of a few to several thousand initial orbital times [44], [43].",
"Such bound states often undergo highly chaotic evolutions under which two of the three objects have a relative high chance of merging [28], , , .",
"This makes the HB limit interesting and highly relevant for the assembly of BBH mergers observable by LIGO [32], , , .",
"We will in this paper therefore solely focus on the dynamics and outcomes of the HB limit.",
"Figure: Dynamical formation of a double GW merger.Top: Orbital trajectories from an equal mass co-planar interaction between a BH binary and an incoming single BH, evolved using our NN-body code that includesGW emission in the EOM at the 2.5PN level .",
"Each of the three BHs has a mass of 20M ⊙ 20M_{\\odot }, and the initial SMA is set to the low value of 10 -4 10^{-4} AUfor illustrative purposes.Bottom: Zoom in on the orbital parts showing the first GW merger (labeled `1.",
"GW Merger'), and the second GW merger (labeled `2.",
"GW Merger').The zoom box is shown in the top plot by a dotted box.As seen, the first GW merger happens here between BH[2] and BH[3] as a result of a highly eccentric GW capture , from which a new BH,denoted by BH[2+3], is formed.",
"This first GW merger happens while the three BHs are still bound to each other,which implies that BH[2+3] and the remaining single BH[1] are also bound to each other after the formation of BH[2+3] – given the GW kick velocity of BH[2+3]is negligible.",
"In that limit, the system now undergoes its second GW merger, which is between BH[2+3] and BH[1].",
"For the simulation results shown here, we have for simplicityassumed that the GW kick velocity of BH[2+3] is zero, and the mass of BH[2+3] is the mass of BH[1] plus the mass of BH[2].As described in Section , the time span from first to second GW merger takes it shortest values for binary-single interactionswhere the total angular momentum is close to zero, a scenario that is only possible in near co-planar interactions.An observed population of double GW mergers will therefore be an indirect probe of environments facilitating near co-planar binary-single interactions,such as disk-like environments."
],
[
"Hard-Binary Interactions",
"A HB interaction either undergoes a direct interaction (DI) or a resonant interaction (RI), depending on the exact ICs [44], , .",
"These two interaction types are briefly discussed below, together with their relation to finite size effects and GR corrections.",
"A DI is characterized by having a relative short interaction time.",
"The associated kinematics are generally such that after the single enters the binary, it directly pairs up with one of the members by ejecting the remaining member to infinity through a classical sling-shot maneuver .",
"Due to the short nature of this interaction type, finite sizes and GR effects rarely play a role here .",
"A RI is by contrast characterized by having a long interaction time, as the three objects in this case enter a temporary bound state.",
"The associated evolution of this state is often highly chaotic [44], , which makes it possible for the system to enter a configuration where dissipative effects, such as GW emission and tides, become important for the subsequent dynamics .",
"To understand how, it was illustrated in, e.g., , , that a typical RI can be described as a series of intermediate states (IMSs) characterized by a binary, referred to as the IMS binary, with a bound single.",
"Between each IMS the three objects undergo a strong interaction where they semi-randomly exchange energy and angular momentum.",
"Each IMS binary is therefore formed with orbital parameters that are different from that of the initial target binary, which makes it possible for highly eccentric IMS binaries to form during the interaction.",
"It is primarily during the evolution of these highly eccentric IMS binaries that GW emission can drain a notable amount of orbital energy and angular momentum out of the three-body system.",
"The change in outcome distributions from including GW emission in the EOM, is therefore linked to the formation of highly eccentric IMS binaries in RIs.",
"This will be described further below."
],
[
"Outcomes and Endstates",
"In the classical case where the objects are assumed point-like and only Newtonian gravity is included in the $N$ -body EOM, the only possible endstate in the HB limit is a binary with an unbound single [36], [44].",
"We refer to this endstate binary as the post-interaction binary in analogy with , .",
"In this notation, if the initial incoming single is a member of the post-interaction binary the endstate will be an exchange.",
"If one includes finite sizes, a collision between any two of the three objects become a possible endstate.",
"Such collisions predominately form in RIs, as a result of an IMS binary forming with a pericenter distance that is smaller than the sum of the radii of the two IMS binary members [28], .",
"The probability for a post-interaction binary to undergo a collision is in comparison relatively small, as each binary-single interaction leads to several IMS binaries compared to only one post-interaction binary.",
"As a result, the majority of the collisional products forming in binary-single interactions will therefore have the remaining single as bound companion.",
"As argued in , collisions contribute significantly to the merger rate when the interacting objects are solar type objects, and less if they are compact, such as a white-dwarf, a neutron-star (NS) or a BH.",
"In the latter case, dissipative captures (tides and GWs) greatly dominate over collisions .",
"If GW emission is included in the EOM (in our simulations through the 2.5PN term), a close passage between any two of the three objects can lead to a significant loss of orbital energy and angular momentum [32].",
"If the energy loss is large enough, the two objects will undergo a GW capture with a merger to follow, which we in the three-body case refer to as a GW inspiral in analogy with .",
"This GW inspiral scenario most often happens between IMS binary members, as these have a finite probability for being formed with a high eccentricity and thereby small pericenter distance, as described in Section REF .",
"As demonstrated by , the probability for forming a GW inspiral can therefore be estimated from simply deriving the fraction of IMS binaries that form with a GW inspiral time that is shorter than the orbital time of the bound single.",
"As this GW inspiral time has to be comparable to the orbital time of the initial binary, the IMS binary eccentricity must be close to unity, which explains why the binary-single channel is a natural producer of high eccentricity BBH mergers , .",
"Figure REF shows an example of a GW inspiral forming during a binary-single interaction.",
"As for the collisions, in the limit where GW velocity kicks are negligible, the merger remnant formed as a result of the GW inspiral will still be bound to the remaining single after its formation.",
"This final binary, now composed of the single and the merger product, will undergo its own GW merger on a timescale that depends sensitively on the ICs, as will be described in Section .",
"It is the inspiral time of this second GW merger that ultimately determines if the double GW merger scenario is observable or not."
],
[
"Distribution of Endstates",
"To understand which binary-single ICs lead to exchanges, collisions, single and double GW mergers, we now consider a mapping we refer to as the endstate topology [41], .",
"This topological mapping refers to the graphical representation of the distribution of endstates, as a function of the initial binary phase $f$ and impact parameter $b$ , as further described in Figure REF and REF , and [41], .",
"We note here that for isotropic encounters each point in the $f,b$ space is equally likely.",
"Figure: Endstate topology.Distribution of binary-single endstates, as a function of the binary phase ff, measured at the timethe single is exactly 20×a 0 20 \\times a_{0} away from the binary COM, and the rescaled impactparameter b ' =(b/a 0 )(v ∞ /v c )b^{\\prime } = (b/a_{0})(v_{\\infty }/v_{\\rm c}) (see Section ).",
"We refer to this endstatedistribution as the endstate topology.",
"The shown distribution is derived from equal mass co-planar binary-single BH interactions (γ=0\\gamma = 0), where each BH has amass of 20M ⊙ 20M_{\\odot }, and the SMA of the initial target binary is a 0 =10 -2 a_{0} = 10^{-2} AU.Blue: All interactions resulting in a classical exchange interaction.Yellow: Interactions that result in a merging BH binary that has an orbital eccentricity e 1 >0.1e_{1} > 0.1 when its GW peak frequency is at 50 Hz.",
"Thisis the population that is likely to appear eccentric when observed by an instrument similar to LIGO.Red: Interactions from the yellow population (e 1 >0.1e_{1}>0.1 at 50 Hz) that further have atimespan from first to second GW merger t 12 <10t_{12}<10 years.",
"We have here assumed a GW kick velocity of v K =0v_{\\rm K} = 0.Black: All remaining endstates and unfinished interactions.How the fraction of the potential observable red population (e 1 >0.1e_{1}>0.1 at 50 Hz with t 12 <10t_{12}<10 years)scales with the initial SMA a 0 a_{0} and the GW kick velocity v K v_{\\rm K}, is shown in Figure and discussed in Section .The endstate topology derived for a co-planar equal mass BH binary-single interaction is shown in Figure REF , where blue denotes an exchange endstate, yellow all endstates for which the endstate binary (that eventually merges due to GW emission) has an eccentricity $e_{1} > 0.1$ when its GW peak frequency $f_{\\rm 1GW}$ is at 50 Hz, and black all remaining endstates and unfinished interactions (the red relates to double GW mergers and will be explained in Section REF ).",
"For calculating the eccentricity distribution at 50 Hz, we evolved the endstate binary in question using the quadrupole formalism from , together with $f_{\\rm 1GW}$ using the approximation presented in .",
"We note that the vast majority of the highly eccentric GW inspiral mergers originate from IMS binaries undergoing either a GW inspiral or a direct collision.",
"As seen in Figure REF , the distribution of endstates is far from random, despite the chaotic nature of the three-body problem , .",
"Generally, the large-scale wave-like pattern arises from the distribution of how much energy the single ejected during the first sling-shot maneuver receives (see Section REF ), as the homogenous regions (neighboring $f,b$ points have the same endstate) arise from DIs and the random regions (neighboring $f,b$ points have semi-random endstates) arise from RIs .",
"The asymmetry along the $b$ -axis relates to which way the binary rotates relative to the incoming single, where $b>0$ corresponds to prograde motion, and $b<0$ to retrograde motion.",
"Also, the initial total angular momentum, $L_{\\rm 0}$ , changes along the $b$ -axis, as the angular momentum brought in by the single is $\\propto b$ .",
"As discussed in the sections below, understanding how $L_{\\rm 0}$ distributes is the key to understand what ICs that will lead to double GW mergers with a time space $t_{12}$ short enough for either resulting in two observable mergers, or a single prompt 2G merger.",
"Having provided an understanding of binary-single interactions with GR and finite size effects, we are now in a position to study our proposed double GW merger scenario.",
"We proceed below by first describing the dynamics leading to the two GW mergers (Section REF and REF ), after which we discuss the prospects for observing either both GW mergers (Section REF ), or just the second prompt 2G merger (Section REF ).",
"For this, we use analytical relations together with full numerical 2.5PN simulations including GW kicks.",
"We note here that at least one of the two GW mergers is likely to have a notable eccentricity when entering the LIGO band, observing double GW mergers therefore heavily relies on the development of accurate eccentric GW templates [34], [39], [30], [40]."
],
[
"First GW Merger",
"The first GW merger forms through a GW inspiral, while the three objects are still bound to each other, as described in Section REF .",
"The resultant BH has a mass close to the total mass of the two merging BHs (ignoring relativistic mass loss), and a velocity that is composed of the initial COM velocity of the two merging BHs, and a GW kick velocity gained through asymmetric GW emission at merger [12], [31], [48].",
"The magnitude of the GW kick velocity depends on the relative BH spins and masses, and can easily reach values of order $\\sim 10^{2} -10^{3}\\ \\text{km s}^{-1}$ [22].",
"The GW kick velocity can therefore significantly affect the dynamics leading to the second GW merger, and plays as a result an important role.",
"We study the effect from GW velocity kicks in Section REF (co-planar case) and Section REF (isotropic case).",
"As the first GW merger generally enters the LIGO band with a notable eccentricity , , an observation of an eccentric BBH merger is therefore the best indication of a potential double GW merger."
],
[
"Second GW Merger",
"The second GW merger forms through an inspiral between the remaining bound single BH and the BH formed in the first GW merger.",
"We refer in the following to these two BHs as BH$_{\\rm 1}$ and BH$_{\\rm 2}$ , respectively.",
"The SMA, $a_{12}$ , and eccentricity, $e_{12}$ , of this [BH$_{\\rm 1}$ , BH$_{\\rm 2}$ ] binary measured just after the formation of BH$_{\\rm 2}$ , can be written by the use of standard classical mechanics as, $a_{12} = \\left[ \\frac{2}{|{\\bf R}_{12}|} - \\frac{\\left({\\bf v}_{\\rm 12} + {\\bf v}_{\\rm K} \\right)^{2}}{3Gm_{\\rm BH}} \\right]^{-1},$ $e_{12} = \\left[ 1 - \\frac{3}{4} \\frac{|{\\bf L}_{12} + {\\bf L}_{\\rm K}|^{2}}{Gm_{\\rm BH}^{3}a_{12}}\\right]^{1/2},$ where $m_{\\rm BH}$ is the mass of one of the three initial (equal mass) BHs, ${\\bf v}_{\\rm K}$ is the GW velocity kick vector in the frame of BH$_{\\rm 2}$ , ${\\bf R}_{12}$ and ${\\bf v}_{\\rm 12}$ are the position and velocity vectors of BH$_{\\rm 2}$ in the frame of BH$_{\\rm 1}$ assuming ${\\bf v}_{\\rm K} = 0$ , and ${\\bf L}_{12}$ and ${\\bf L}_{\\rm K}$ are the angular momentum vectors $(2m_{\\rm BH}/3){\\bf R_{\\rm 12}} \\times {\\bf v}_{\\rm 12}$ , $(2m_{\\rm BH}/3){\\bf R_{\\rm 12}} \\times {\\bf v}_{\\rm K}$ , respectively.",
"We have in these expressions assumed for simplicity that the mass of BH$_{\\rm 2}$ is $ = 2m_{\\rm BH}$ , although the actual mass will be a few percent lower due to relativistic mass loss at merger [48].",
"An illustration of the orbital configuration right after the formation of BH$_{2}$ is shown in Figure REF .",
"Figure: Illustration of the binary-single ICs.The three large black dots represent the interacting BHs, two of which initially are in a binary that here is centered at (0,0,0)(0,0,0).The impact parameter bb and relative velocity v ∞ v_{\\infty } are both defined at infinity, where ff refers to the binary phase at the time the single is exactly 20×a 0 20 \\times a_{0} away fromthe binary COM .The initial angular momentum vectors of the binary and the single, respectively, are here denoted by 𝐋 B {\\bf L}_{\\rm B} (purple)and 𝐋 S {\\bf L}_{\\rm S} (green), where their sum is labeled by 𝐋 0 =𝐋 B +𝐋 S {\\bf L}_{\\rm 0} = {\\bf L}_{\\rm B} + {\\bf L}_{\\rm S} (blue).The smallest angle between 𝐋 B {\\bf L}_{\\rm B} and 𝐋 S {\\bf L}_{\\rm S} is denoted by γ\\gamma , and a co-planar interaction therefore has γ=0\\gamma = 0.As indicated by the velocity arrows, the binary is initially set to rotate counter-clockwise,which implies that 𝐋 B ·𝐋 S >0{\\bf L}_{\\rm B} \\cdot {\\bf L}_{\\rm S} > 0 for b>0b>0, and vice versa.",
"From this follows that if b<0b<0,then |𝐋 0 ||{\\bf L}_{\\rm 0}| can be smaller than |𝐋 B ||{\\bf L}_{\\rm B}|, which can lead to surprisingly short time intervals between thefirst and the second GW merger, as explained further in Section .Using the above relations for $a_{12}$ and $e_{12}$ , one finds that the corresponding pericenter distance $r_{12} = a_{12}(1-e_{12})$ , in the high eccentricity limit, can be written as, $r_{\\rm 12} \\approx \\frac{3}{8}\\frac{|{\\bf L}_{12} + {\\bf L}_{\\rm K}|^{2}}{Gm_{\\rm BH}^{3}}.$ While ${\\bf R}_{12}$ , ${\\bf v}_{\\rm 12}$ , and ${\\bf v}_{\\rm K}$ depend sensitively on the ICs, we find that the initial total three-body angular momentum, denoted ${\\bf L}_{0} = {\\bf L}_{\\rm B} + {\\bf L}_{\\rm S}$ , where ${\\bf L}_{\\rm B}$ and ${\\bf L}_{\\rm S}$ are the angular momentum vectors of the initial binary and the incoming single, respectively (See Figure REF ), do not change significantly during the interaction.",
"This follows from that the first GW inspiral generally forms on a low angular momentum orbit, meaning that only a small amount of angular momentum is emitted in the first GW merger.",
"As a result, one can write ${\\bf L}_{12}^{2}$ to leading order as, ${\\bf L}_{\\rm 12}^{2} \\approx {\\bf L}_{\\rm 0}^{2} = \\frac{Gm_{\\rm BH}^{3}a_{0}}{2} \\left[ 1 + \\frac{4}{3}b^{\\prime 2} + \\frac{4}{\\sqrt{3}}b^{\\prime } \\cos (\\gamma ) \\right],$ where $a_{0}$ is the SMA of the initial circular target binary, $\\gamma $ is the smallest angle between ${\\bf L}_{\\rm B}$ and ${\\bf L}_{\\rm S}$ (See Figure REF ), and $b^{\\prime } = ({b}/{a_{0}})({v_{\\infty }}/{v_{\\rm c}})$ , where $v_{\\rm c}$ is here the binary-single characteristic velocity and $v_{\\infty }$ is the relative velocity between the binary and single at infinity [44].",
"In this notation $\\gamma = 0$ corresponds to a co-planar interaction, and ${v_{\\infty }}/{v_{\\rm c}} < 1$ is what is referred to as the HB limit.",
"The orbit averaged inspiral time of the [BH$_{\\rm 1}$ , BH$_{\\rm 2}$ ] binary, equivalent to the time span $t_{12}$ , can at quadrupole order in the high eccentricity limit be written as , $t_{12} \\approx \\frac{c^{5}}{G^{3}} \\frac{4\\sqrt{2}}{85} \\frac{r_{\\rm 12}^{7/2}a_{12}^{1/2}}{m_{\\rm BH}^{3}}.$ In the limit where the approximations employed in Equation (REF ) and (REF ) are valid, the time span $t_{12}$ from Equation (REF ) can now then be expressed as, $t_{12} \\approx \\frac{\\xi a_{0}^{4}}{m_{\\rm BH}^{3}} \\left(\\frac{a_{12}}{a_{0}}\\right)^{1/2} \\frac{|{\\bf L}_{0} + {\\bf L}_{\\rm K}|^{7}}{|{\\bf L}_{\\rm B}|^{7}},$ where we have introduced the following constant, $\\xi = \\frac{c^{5}}{G^{3}} \\frac{4\\sqrt{2}}{85} \\left(\\frac{3}{16}\\right)^{7/2}.$ For comparison, we note that the inspiral time of the initial circular binary, denoted $t_{0}$ , can be written as , $t_{0} \\approx \\frac{\\xi a_{0}^{4}}{m_{\\rm BH}^{3}} \\frac{1700\\sqrt{6}}{81} \\approx 2 \\cdot 10^{5} \\ \\text{yrs} \\left(\\frac{a_{0}}{10^{-2}\\text{AU}}\\right)^{4} \\left(\\frac{m_{\\rm BH}}{20M_{\\odot }}\\right)^{-3}.$ As seen, the normalizations of $t_{12}$ and $t_{0}$ are of similar order (small variations of ${\\bf L}_{\\rm K}$ easily result in variations of order the factor of difference $1700\\sqrt{6}/81 \\approx 51$ ), which shows the importance of reducing the angular momentum $|{\\bf L}_{0}|$ for bringing the time span $t_{12}$ within either the observation time ($10^{0}-10^{1}$ years), or the binary-single encounter time ($10^{7}-10^{8}$ years).",
"The relation shown in Equation (REF ) likewise illustrates that $t_{12}$ will not be significantly affected by GW velocity kicks if $|{\\bf L}_{\\rm K}| \\ll |{\\bf L}_{\\rm B}|$ , which approximately corresponds to the requirement $v_{\\rm K} \\ll v_{\\rm 0}$ , where $v_{0}$ denotes the orbital velocity of the initial target binary given by, $v_{0} \\approx 1883\\ \\text{km s}^{-1} \\left(\\frac{a_{0}}{10^{-2}\\text{AU}}\\right)^{-1/2} \\left(\\frac{m_{\\rm BH}}{20M_{\\odot }}\\right)^{1/2}.$ A GW kick of order say 100 km s$^{-1}$ , is therefore not expected to impact the results significantly for the employed normalizations.",
"However, for interactions with a wider initial binary and lower BH masses, moderate GW kicks can easily unbind the system after which a double GW merger is not forming.",
"Finally, we do note that binary-single systems with a relative high initial angular momentum ($|{\\bf L}_{\\rm 0}| \\gtrsim |{\\bf L}_{\\rm B}|$ ) can, under rare circumstances, still undergo a prompt double GW merger, provided the GW kick velocity vector is fine-tuned in such a way that $|{\\bf L}_{0} + {\\bf L}_{\\rm K}| \\ll |{\\bf L}_{\\rm B}|$ .",
"Such situations show up in the endstate topology map shown in Figure REF , by a `blurring' of the horizontal edges between the red and the yellow points.",
"In Section REF we explore the effect from GW kicks using full numerical simulations.",
"Figure: Illustration of the orbital configuration right after the first GW merger.As described in Section , the first GW merger happens through a two-body GW capture, while the three BHsare still bound to each other.",
"The BH formed as a result of this first GW merger is labeled above by `BH 2 _{2}', and the remainingsingle by `BH 1 _{1}'.",
"The relative position vector pointing from BH 1 _{1} to BH 2 _{2} is labeled by 𝐑 12 {\\bf R}_{12}.In the frame of BH 1 _{1}, BH 2 _{2} will right after its formation move with a velocity that is composed of the COM velocity of the two BHs that merged to form BH 2 _{2}, and theGW kick velocity gained in the GW merger.",
"The COM velocity is labeled by 𝐯 12 {\\bf v}_{12} (light blue), the GW kick velocityby 𝐯 K {\\bf v}_{\\rm K} (light red), and the corresponding angular momentum vectors by 𝐋 12 {\\bf L}_{\\rm 12} (light blue) and 𝐋 K {\\bf L}_{\\rm K} (light red), respectively.Right after the first GW merger, BH 2 _{2} therefore moves with a velocity 𝐯 12 +𝐯 K {\\bf v}_{12} + {\\bf v}_{\\rm K} (solid black) on an orbit with correspondingangular momentum 𝐋 12 +𝐋 K {\\bf L}_{\\rm 12} + {\\bf L}_{\\rm K} (solid black).",
"The SMA and eccentricity of this orbit (black dotted line) islabeled by a 12 ,e 12 a_{12}, e_{12}, respectively.",
"If the GW kick velocity is not unbinding the system, then BH 1 _{1} and BH 2 _{2} will undergo a GW inspiralon a timescale t 12 t_{12} that is set by the orbital parameters a 12 ,e 12 a_{12}, e_{12}.",
"If t 12 t_{12} is short, then both the first and the second GW merger can be observed."
],
[
"Observation of the First and Second GW Merger",
"We now explore the possibilities for observing both the first and the second GW merger in our proposed double GW merger scenario.",
"For this, we systematically study below how the time span $t_{12}$ depends on the binary-single ICs, and the GW velocity kick from the first merger.",
"The ICs for observing only the second GW merger, i.e.",
"the 2G merger, will be discussed separately in Section REF .",
"Our considered cases are described in the paragraphs below."
],
[
"Interactions with $\\gamma = 0$ , {{formula:d10aff7a-07b1-4c16-a1bc-9a5ba33b8196}}",
"We start by considering the scenario for which the binary-single interaction is co-planar ($\\gamma = 0$ ), and the GW velocity kick is negligible (${v}_{\\rm K} = 0$ ).",
"This represents an idealized scenario; however, regarding the assumption of $\\gamma = 0$ , it has been argued that BBHs in disk environments will have their relative orbital inclinations reduced leading to preferentially co-planar interactions .",
"Properties of this limit is described in the following.",
"First, the time span $t_{12}$ can be written by the use of Equation (REF ) and (REF ) as, $t_{12} \\approx \\frac{\\xi a_{0}^{4}}{m_{\\rm BH}^{3}} \\left(\\frac{a_{12}}{a_{0}}\\right)^{1/2} \\left[1 + \\frac{2}{\\sqrt{3}} b^{\\prime } \\right]^{7}.$ From this we see that if the incoming single encounters the initial binary with an impact parameter $b^{\\prime } = b^{\\prime }_{0}$ , where $b^{\\prime }_{0} = - \\sqrt{3}/2 \\approx - 0.87$ , then the time span $t_{12} \\approx 0$ , implying the second GW merger happens when BH$_{\\rm 1}$ and BH$_{\\rm 2}$ pass through their first pericenter passage.",
"In this case, the second GW merger will be that of a near head-on BH collision [35].",
"We note here that $b^{\\prime }_{0}$ is simply the impact parameter for which the angular momentum of the incoming single exactly cancels the angular momentum of the initial binary .",
"In general, double GW mergers can form from interactions with impact parameters in the range $-2 \\lesssim b^{\\prime } \\lesssim 3$ , as seen by the distribution of yellow points in Figure REF .",
"This correspondingly implies that the distribution of $t_{12}$ generally varies over several orders of magnitude, e.g., from Equation (REF ) we see that $t_{12}(b^{\\prime }=1)/t_{12}(b^{\\prime }=-1) \\approx 10^{8}$ .",
"To gain insight into for which values of $b^{\\prime }$ the second GW merger happens within a timespan of observable interest, we now rewrite Equation (REF ) as, $\\Delta {b^{\\prime }} \\approx 0.38 \\left(\\frac{a_{0}}{10^{-2}\\text{AU}}\\right)^{-4/7} \\left(\\frac{m_{\\rm BH}}{20M_{\\odot }}\\right)^{3/7} \\left(\\frac{t_{12}}{10\\ \\text{yrs}}\\right)^{1/7},$ where we have defined $\\Delta {b^{\\prime }} = |b^{\\prime }-b^{\\prime }_{0}|$ , and omitted the factor with $(a_{12}/a_{0})$ , as its power of $-1/14$ makes it unimportant for these estimates.",
"For the employed normalizations, one reads from Equation (REF ) that all interactions with $b^{\\prime } = b^{\\prime }_{0} \\pm 0.38$ that result in a double GW merger will have a time span from first to second GW merger $t_{12} < 10$ years.",
"This is interesting as this is not a negligible part of the available phase space; as seen on Figure REF , the range $b^{\\prime }_{0} \\pm 0.38$ covers about $30\\%$ of the relevant $f,b^{\\prime }$ phase space for retrograde interactions ($b^{\\prime }<0$ ).",
"We note here that both the position and the width of the band of red points in Figure REF are accurately given by $-0.87\\pm 0.38$ .",
"This excellent agreement between our analytical derivations and our 2.5PN numerical $N$ -body simulations, strongly validates our approach and results so far.",
"We now turn to the question of what the prospects are for observing both GW mergers.",
"For this, we performed a set of numerical binary-single interactions using our 2.5PN $N$ -body code for $\\gamma = 0$ , $v_{\\rm K} = 0$ , and $m_{\\rm BH} = 20M_{\\odot }$ , assuming the sampling of $b^{\\prime }$ to be isotropic at infinity.",
"Results are presented in Figure REF , where the black lines show the probability for that a GW merger with eccentricity $e_{1}>0.1$ at 50 Hz (first GW merger) will be followed by a second GW merger within a time span $t_{12}<10$ years (second GW merger) through the double GW merger scenario, as a function of the initial binary SMA $a_{0}$ .",
"This probability we denote by $P_{\\rm 12}$ to shorten the notations.",
"We note here that prograde interactions ($b^{\\prime }>0$ ) will never result in short double GW mergers as this implies that $|{\\bf L}_{\\rm 0}| > |{\\bf L}_{\\rm B}|$ and thereby $t_{12} \\gtrsim t_{0}$ .",
"From the results shown in Figure REF , one concludes that for the ICs considered in this section, the probability for that a GW merger with notable eccentricity in the LIGO band ($e_{1}>0.1$ at 50 Hz) to have a second GW merger within 10 years, is $\\gtrsim 0.1$ for $a_{\\rm 0} < 0.1$ AU ($t_{0} \\lesssim 10^{10}$ years).",
"We note here that $\\sim 0.1$ AU represents roughly the minimum value for $a_{\\rm 0}$ one would find in a classical globular cluster system; below this value the BBH would either have merged before the next encounter, or more likely been dynamically ejected.",
"To gain insight into how the numerically estimated probability $P_{\\rm 12}$ shown in Figure REF changes with the ICs, we can use our analytical expression for $\\Delta {b^{\\prime }}$ from Equation (REF ).",
"For this, we assume that the $f,b^{\\prime }$ combinations for which $e_{1}>0.1$ at 50 Hz leading to a second GW merger are uniformly distributed near $b^{\\prime }_{0}$ , and do not change their large scale topology when the ICs are varied.",
"The latter assumption generally holds, as the large scale topology results from the Newtonian `scale-free' part of the dynamics .",
"Following this approximation, the probability $P_{\\rm 12}$ is simply proportional to ${\\Delta }b^{\\prime }$ .",
"Using our $N$ -body simulations to calibrate the normalization, one now finds that the probability for a GW merger with eccentricity $e_{1}>0.1$ at 50 Hz to be followed by a second GW merger within time span $t_{12}<\\tau $ years, has the following analytical solution, $P_{\\rm 12} \\approx 0.26 \\left(\\frac{a_{0}}{10^{-2}\\text{AU}}\\right)^{-4/7} \\left(\\frac{m_{\\rm BH}}{20M_{\\odot }}\\right)^{3/7} \\left(\\frac{\\tau }{10\\ \\text{yrs}}\\right)^{1/7},$ where the factor $(a_{12}/a_{0})$ again has been omitted.",
"Although the assumption of a uniform distribution breaks down in a few regions near $b^{\\prime }_{0}$ (See Figure REF ), we do find that our analytical solution from the above Equation (REF ) fits our numerical results quite well.",
"As an illustrative example, the scaling with $a_{0}$ is shown in Figure REF by the black dotted line.",
"From comparing with Figure REF , the deviation from a pure powerlaw at low $a_{0}$ is indeed due to that the assumption of a uniform distribution breaks down when $\\Delta {b^{\\prime }}$ becomes to large, or equivalently when $a_{0}$ becomes to small.",
"We further note that $P_{\\rm 12}$ in Equation (REF ) depends very weakly on $\\tau $ , which implies that the results shown in Figure REF are not sensitive to the exact value of the time span limit $\\tau $ .",
"Below we study the role of GW kicks in this co-planar scenario.",
"Figure: Probability, P 12 P_{12}, that a BBH merger with eccentricity e 1 >0.1e_{1}>0.1 at 50 Hz (first GW merger) is followed by a GW merger (second GW merger)within a time span of t 12 <10t_{12}<10 years through our proposed double GW merger scenario.",
"The mass of the BHs arem BH =20M ⊙ m_{\\rm BH} = 20 M_{\\odot }, and the probability is shown as a function of the SMA of the initial target binary, a 0 a_{0}.The solid, dashed, and dash-dotted lines show P 12 P_{\\rm 12} derived from full numerical binary-single simulations,for when the BH formed in the first GW merger, BH 2 _{2}, receives a GW velocity kick of 0 km s -1 ^{-1}, 10 km s -1 ^{-1}, and 100 km s -1 ^{-1}, respectively.For initial spinning BHs v k v_{\\rm k} can easily reach values of 1000 km s -1 ^{-1}, which will lead to prompt disruption and no second GW merger.For these simulations, we have assumed that the pointing of the GW kick velocity vector 𝐯 K \\bf v_{\\rm K} is isotropic, and the mass ofBH 2 _{2} is 2m BH 2m_{\\rm BH}.",
"The black lines show results from co-planar interactions (γ=0\\gamma = 0), where the red lines show results from isotropic interactions (γ=iso\\gamma = iso).The dotted lines show analytical scaling relations valid in the v K =0v_{\\rm K} = 0 limit.The top axis shows the GW inspiral life time of the initial circular binary in years.As seen, it is highly unlikely that isotropic environments, such as classical globular clusters, will results in observable double GW mergers.However, co-planar interactions do generally lead to much shorter merger time scales, which correspondingly leadto a measurable fraction of double GW mergers."
],
[
"Interactions with $\\gamma = 0$ , {{formula:946b1788-2e73-4cd1-b16c-c3aa5b6bfd61}}",
"Binary-single interactions with $\\gamma \\approx 0$ and $v_{\\rm K} = 0$ represent the optimal scenario for producing double GW mergers with short enough time span $t_{12}$ to be observed.",
"However, GW kicks are expected, so here we explore how the inclusion of GW velocity kicks affects this case, by focusing on how the fraction of double GW mergers with $t_{12}<10$ years changes with varying $v_{\\rm K}$ .",
"For this, we use our $N$ -body code to numerically calculate the probability $P_{\\rm 12}$ , shown in Figure REF , for when the BH formed in the first GW merger, BH$_{2}$ , receives a velocity kick $v_{\\rm K}$ of either 10 km s$^{-1}$ or 100 km s$^{-1}$ at its formation.",
"Results for varying $v_{\\rm K}$ are shown in Figure REF .",
"As seen in the figure, GW kicks significantly reduces the fraction of double GW mergers with $t_{12}<10$ years, especially for binaries with $a_{\\rm 0}>0.1$ AU.",
"Binaries with $a_{\\rm 0}<0.1$ AU are less affected, as their orbital velocity is much higher; however, they are at the same time less likely to exist, as they initially have a very short life time as indicated by the upper x-axis.",
"Depending on the exact properties of their host stellar system, very hard binaries are in fact likely to merge in-between encounters .",
"Although GW kicks are expected, we do note that several recent studies have in fact looked into the observational consequences of BHs forming with near zero spin, which generally also leads to low kicks .",
"This limit allows for the formation of second-generation GW mergers, which are characterized by mass ratios of about $1:2$ and relative high BH spins around $0.7$ .",
"In the section below we extend our analysis from this section to varying $\\gamma $ , including the isotropic case found in globular clusters."
],
[
"Interactions with $\\gamma > 0$ , {{formula:61e910d5-4fd3-41c5-9c8e-58da94bd07ca}}",
"We now study how varying the orbital inclination angle $\\gamma $ and kick velocity $v_{\\rm K}$ affect the time span $t_{12}$ .",
"This is done to explore the role and formation probability of double GW mergers in classical stellar systems, such as globular clusters.",
"To gain insight into how a varying $\\gamma $ changes the picture described in Section REF and REF , we first consider what the minimum value for $t_{12}$ is as a function of $\\gamma $ , assuming the optimal case for which $v_{\\rm K} = 0$ .",
"By minimizing the expression for $t_{12}$ given by Equation (REF ) w.r.t.",
"$b^{\\prime }$ for fixed value of $\\gamma $ , we find that the minimum value of $t_{12}$ , denoted here by $\\min (t_{12})$ , can be written as, $\\min (t_{12}) \\approx \\frac{\\xi a_{0}^{4}}{m_{\\rm BH}^{3}} \\left(\\frac{a_{12}}{a_{0}}\\right)^{1/2} \\sin ^{7}(\\gamma ),$ where the corresponding $b^{\\prime }(\\min (t_{12}))$ equals $-\\cos (\\gamma ){\\sqrt{3}}/{2}$ .",
"We here see that the formation of prompt double GW mergers with $\\min (t_{12}) \\approx 0$ is only possible in the special case for which $\\gamma = 0$ .",
"Considering the limit where $\\sin (\\gamma ) \\approx \\gamma $ , we can rewrite the above Equation (REF ) in the following form, $\\gamma \\approx 20\\left(\\frac{a_{0}}{10^{-2}\\text{AU}}\\right)^{-4/7} \\left(\\frac{m_{\\rm BH}}{20M_{\\odot }}\\right)^{3/7} \\left(\\frac{\\min (t_{12})}{10\\ \\text{yrs}}\\right)^{1/7},$ where the factor $(a_{12}/a_{0})$ again has been omitted.",
"This relation tells us that for $a_{0} = 10^{-2} (10^{-1})$ AU interactions with $\\gamma > 20(5)$ will not be able to result in a double GW merger with a timespan $t_{12}<10$ years.",
"Observable double GW mergers are therefore most likely to form in near co-planar interactions.",
"To study what the actual fraction of observable double GW mergers is in the isotropic case, and how much smaller it is compared to the co-planar case, we performed $\\sim 10^{6}$ $2.5$ PN numerical scatterings assuming an isotropic binary-single encounter distribution.",
"Results are shown in Figure REF with red lines.",
"As seen, the fraction is greatly reduced, which originates from the fact that most of the phase space ($\\gamma > \\mathcal {O}(10)$ ) sampled in the isotropic case leads to a time span $t_{12}$ far too long for both mergers to be observed.",
"As a result, in terms of rates, isotropic environments, such as a globular cluster, are not likely to significantly contribute to the formation of observable double GW mergers.",
"The probability $P_{12}$ also scales differently with $a_{0}, m_{\\rm BH}$ and $\\tau $ than found in the co-planar case.",
"The differences relate to the difference in eccentricity distributions , ; in the isotropic case eccentricity tends to follow a thermal distribution $2e$ , whereas in the co-planar case the distribution can be shown to instead follow $e/\\sqrt{1-e^2}$ .",
"Assuming $e_{12}$ follow these distributions in the two scenarios, one finds in the isotropic case that $P_{12} \\propto a_{0}^{-8/7}m_{\\rm BH}^{6/7}\\tau ^{2/7}$ and in the co-planar case $P_{12} \\propto a_{0}^{-4/7}m_{\\rm BH}^{3/7}\\tau ^{1/7}$ (in agreement with Equation REF ).",
"That $P_{\\rm 12}$ falls off steeper with $a_{0}$ in the isotropic case than the co-planar case, also provides some insight into why the fraction of double GW mergers from isotropic scatterings is vanishing for classical system for which $a_{\\rm 0} > 0.1$ AU.",
"The prospects of detecting prompt 2G mergers is described in the section below."
],
[
"Observation of the Second Merger",
"Having argued that directly observing both the first and the second GW merger is only possible in near co-planar interactions and unlikely for isotropic environments, we now turn to the question if our proposed double GW merger scenario instead can be indirectly observed.",
"With indirect we here refer to the case where only the prompt 2G merger is seen.",
"As explained earlier, such a merger would be characterized by a mass ratio of about $1:2$ and (at least) one BH highly spinning .",
"For this scenario to take place, the GW life time of the second GW merger must be shorter than the typical time between binary-single encounters, which depends on the number density of single BHs, $n_{\\rm s}$ , and the velocity dispersion, $v_{\\rm disp}$ , of the dynamical environment as $(n_{\\rm s}\\sigma _{\\rm bs}v_{\\rm disp})^{-1}$ , where $\\sigma _{\\rm bs}$ is the binary-single interaction cross section , .",
"In the following we discuss the prospects for observing this second GW merger, i.e.",
"the prompt 2G merger, in a typical globular cluster system in which the time between encounters is $< 10^{8}$ years.",
"Figure: The probability that a binary-single interaction that first undergoes a GW merger during the interaction (first GW merger)subsequently undergoes one more GW merger (second GW merger) within a typical encounter time scaleof the host stellar system.",
"In this figure, this time scale is set to 10 8 10^{8} years.The probability is derived for a BBH with initial SMA a 0 =10 -0.5 a_{0} = 10^{-0.5} AU and m BH =20M ⊙ m_{\\rm BH} =20M_{\\odot }, as a function of the GW kickvelocity, v k v_{\\rm k}.",
"The black line shows results from co-planar interactions (γ=0\\gamma = 0), where the red lineshows from isotropic interactions (γ=iso\\gamma = iso).",
"As seen, only a few percent of the three-body systems thatfirst undergo a GW merger during the interaction will undergo a second GW merger before interacting with a new single BH.Observations of second-generation BBH mergers forming in isotropic systems, are therefore not expected to be dominated by thedouble GW merger scenario.",
"On the other hand, a notable fraction of all co-planar interactions will undergo both the first and the second GW merger.Such ICs also give rise to very short merger time spans t 12 t_{12} as seen in Figure .",
"It is currently unclear ifco-planar environments exists, although they have been suggested in the recent literature .Numerical scattering results are presented in Figure REF , which shows the probability for that the second GW merger occurs within $10^{8}$ years, for a BBH with an initial SMA $a_{0} = 10^{-0.5} \\approx 0.3$ AU and $m_{\\rm BH} = 20M_{\\odot }$ , as a function of the GW kick velocity, $v_{\\rm k}$ .",
"The black line shows the co-planar case, where the red line shows the isotropic case.",
"From considering the normalization of the isotropic case, we conclude that only $\\sim 2\\%$ of all BBH mergers that form during three-body interactions will also undergo a second GW merger before the next encounter disrupts the system.",
"From this we draw the following two conclusions.",
"First, the [BH$_1$ , BH$_2$ ] binary formed after the first GW merger is more likely to undergo a subsequent binary-single interaction, than ending as a prompt 2G merger.",
"The reason is simply that the typical time span from the first to the second GW merger, $t_{12}$ , is in the majority of cases much longer than the characteristic time between encounters.",
"The binary [BH$_1$ , BH$_2$ ] formed after the first GW merger will therefore most often keep interacting and thereby contribute dynamically to the evolution of the stellar system.",
"We note that this is not necessarily the case for BHs formed in 2G mergers, as the most likely outcome here is ejection as the [BH$_1$ , BH$_2$ ] binary will be of unequal mass with one of the BHs (highly) spinning at about $0.7$ ; combinations that are expected to generate large GW kicks.",
"A buildup and dynamical influence of third-generation BHs is therefore both difficult and unlikely.",
"Second, if GW mergers are observed with a notable mass ratio about $1:2$ and one BH with spin $\\sim 0.7$ , then the most likely origin is a second-generation merger formed after the BH assembled in the first merger has swapped partner at least once, i.e., it is less likely that the merger was produced in the double GW merger scenario.",
"As described earlier, second generation BBH mergers form either doing or in-between binary-single interactions , .",
"Subsequent interactions also allow for the possibility that two second-generation BHs meet each other and merge, leading to relative heavy equal mass binaries.",
"Such scenarios have recently been explored by e.g.",
"[29], , and would point towards a dynamical origin.",
"We conclude our study in the section below.",
"We have in this paper presented a study on BH binary-single interactions resulting in two successive GW mergers, an outcome we refer to as a double GW merger.",
"Double GW mergers are a natural outcome when GR effects are included in the $N$ -body EOM , but several mechanisms can prevent them from happen, such as GW kicks.",
"The formation of double GW mergers have been proposed and presented in the past literature both numerically [20] and dynamically ; however, our presented study is the first to quantify their actual formation probability and in which environments they are most likely to form.",
"Double GW mergers are interesting as they give rise to unique observables, which could help distinguishing between BBH merger channels.",
"A brief summery of our findings is given below.",
"The double GW merger scenario produces at least two unique observable signatures that are different from other formation channels, especially the class that does not involve few-body dynamics.",
"The first signature, is the observation of both the first and the second GW merger in the scenario.",
"This requires the time span between the two mergers to be short ($t_{12} <\\mathcal {O}$ (years)), which is orders of magnitude shorter than the initial target binary life time.",
"Using numerical and analytical arguments we have shown that only interactions that are near co-planar will be able to produce such short double GW mergers.",
"The reason is that in near co-planar systems the angular momentum carried by the incoming single can lead to a near cancelation of the initial BBH angular momentum, which correspondingly leads to a very short merger time scale .",
"In the case where the encounters are instead isotropic, the overall probability for forming an observable double GW merger decreases drastically (Figure REF ), as the majority of the available phase space (away from near co-planar configurations) will lead to merger times that are far too long.",
"From this we conclude that if both mergers in the scenario are observed, then this would be an indirect probe of environments facilitating co-planar interactions; this includes in particular disk systems, such as active galactic nuclei .",
"The second signature, is the observation of only the second GW merger, that we refer to as a prompt 2G merger.",
"As pointed out in [29], , second-generation BBH mergers are generally characterized by a mass ratio of about $1:2$ , and at least one highly spinning BH.",
"Using numerical scatterings we have shown that the probability for a binary-single interaction to undergo an observable 2G merger in the isotropic case, is still only at the percent level.",
"Second-generation GW mergers can form in other ways than through our proposed double GW merger scenario [29], , , and such mergers are therefore not expected to be a unique signature of the double GW merger scenario, although they still indicate that dynamical environments are able to produce BBH mergers.",
"To conclude, the formation of double GW mergers have been suggested in the past literature, and their unique observables have been numerically studied [20].",
"We have studied their formation probability, and found that they are not expected to form in measurable numbers in classical isotropic stellar systems, such as globular clusters.",
"A significant fraction can only form in near co-planar environments, which could be active galactic nuclei disks [14], , , .",
"This makes it currently impossible to predict reliable rates, as such disk systems at present are still poorly understood.",
"Hopefully our study motives the community to look into this further."
],
[
"Acknowledgements",
"Support for this work was provided by NASA through Einstein Postdoctoral Fellowship grant number PF4-150127 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060.",
"TI thanks the Undergraduate Summer Research Program (USRP) at the department of Astrophysical Sciences, Princeton University, for support.",
"JS thanks the Niels Bohr Institute for its hospitality while part of this work was completed, and the Kavli Foundation and the DNRF for supporting the 2017 Kavli Summer Program."
]
] | 1709.01660 |
[
[
"On the Structure and Computation of Random Walk Times in Finite Graphs"
],
[
"Abstract We consider random walks in which the walk originates in one set of nodes and then continues until it reaches one or more nodes in a target set.",
"The time required for the walk to reach the target set is of interest in understanding the convergence of Markov processes, as well as applications in control, machine learning, and social sciences.",
"In this paper, we investigate the computational structure of the random walk times as a function of the set of target nodes, and find that the commute, hitting, and cover times all exhibit submodular structure, even in non-stationary random walks.",
"We provide a unifying proof of this structure by considering each of these times as special cases of stopping times.",
"We generalize our framework to walks in which the transition probabilities and target sets are jointly chosen to minimize the travel times, leading to polynomial-time approximation algorithms for choosing target sets.",
"Our results are validated through numerical study."
],
[
"Introduction",
"A random walk is a stochastic process over a graph, in which each node transitions to one of its neighbors at each time step according to a (possibly non-stationary) probability distribution [1], [2].",
"Random walks on graphs are used to model and design a variety of stochastic systems.",
"Opinion propagation in social networks, as well as gossip and consensus algorithms in communications and networked control systems, are modeled and analyzed via random walks [3], [4], [5], [6].",
"Random walks also serve as distance metrics in clustering, image segmentation, and other machine learning applications [7], [8], [9].",
"The behavior of physical processes, such as heat diffusion and electrical networks, can also be characterized through equivalent random walks [10], [11].",
"One aspect of random walks that has achieved significant research attention is the expected time for the walk to reach a given node or set of nodes [12], [13].",
"Relevant metrics include the hitting time, defined as the expected time for a random walk to reach any node in a given set, the commute time, defined as the expected time for the walk to reach any node in a given set and then return to the origin of the walk, and the cover time, which is the time for the walk to reach all nodes in a desired set.",
"These times give rise to bounds on the rate at which the walk converges to a stationary distribution [1], and also provide metrics for quantifying the centrality or distance between nodes [9], [14].",
"The times of a random walk are related to system performance in a diverse set of network applications.",
"The convergence rate of gossip and consensus algorithms is determined by the hitting time to a desired set of leader or anchor nodes [15], [16].",
"The effective resistance of an electrical network is captured by its commute time to the set of grounded nodes [17].",
"The performance of random-walk query processing algorithms is captured by the cover time of the set of nodes needed to answer the query [18].",
"Optimal control problems such as motion planning and traffic analysis are described by the probability of reaching a target set or the resource cost per cycle of reaching a target set infinitely often [19].",
"In each of these application domains, a set of nodes is selected in order to optimize one or more random walk metrics.",
"The optimization problem will depend on whether the distribution of the random walk is affected by the choice of input nodes.",
"Some systems will have a fixed walk distribution, determined by physical laws (such as heat diffusion or social influence propagation), for any set of input nodes.",
"In other applications, the distribution can be selected by choosing a control policy, and hence the distribution of the walk and the set to be reached can be jointly optimized, as in Markov decision processes.",
"In both cases, however, the number of possible sets of nodes is exponential in the network size, making the optimization problem computationally intractable in the worst case, requiring additional structure of the random walk times.",
"At present, computational structures of random walks have received little attention by the research community.",
"In this paper, we investigate the hitting, commute, cover, and cycle times, as well as the reachability probability, of a random walk as functions of the set of network nodes that are reached by the walk.",
"We show that these metrics exhibit a submodular structure, and give a unifying proof of submodularity for the different metrics with respect to the chosen set of nodes.",
"We consider both fixed random walk transition distributions and the case where the set of nodes and control policy are jointly selected.",
"We make the following specific contributions: We formulate the problem of jointly selecting a set of nodes and a control policy in order to maximize the probability of reaching the set from any initial state, as well as the average (per cycle) cost of reaching the set infinitely often.",
"We prove that each problem can be approximated in polynomial time with provable optimality bounds using submodular optimization techniques.",
"Our approach is to relate the existence of a probability distribution that satisfies a given cycle cost to the volume of a linear polytope, which is a submodular function of the desired set.",
"We extend our approach to joint optimization of reachability and cycle cost, which we prove is equivalent to a matroid optimization problem.",
"In the case where the probability distribution is fixed, we develop a unifying framework, based on the submodularity of selecting subsets of stopping times, which includes proofs of the supermodularity of hitting and commute times and the submodularity of the cover time as special cases.",
"Since the cover time is itself NP-hard to compute, we study and prove submodularity of standard upper bounds for the cover time.",
"We evaluate our results through numerical study, and show that the submodular structure of the system enables improvement over other heuristics such as greedy and centrality-based algorithms.",
"This paper is organized as follows.",
"In Section , we review the related work.",
"In Section , we present our system model and background on submodularity.",
"In Section , we demonstrate submodularity of random walk times when the walk distribution is chosen to optimize the times.",
"In Section , we study submodularity of random walk times with fixed distribution.",
"In Section , we present numerical results.",
"In Section , we conclude the paper."
],
[
"Related Work",
"Commute, cover, and hitting times have been studied extensively, dating to classical bounds on the mixing time [20], [1].",
"Generalizations to these times have been proposed in [21], [22].",
"Connections between random walk times and electrical resistor networks were described in [17].",
"These classical works, however, do not consider the submodularity of the random walk times.",
"Submodularity of random walk times has been investigated based on connections between the hitting and commute times of random walks, and the performance of linear networked systems [16], [23].",
"The supermodularity of the commute time was shown in [16].",
"The supermodularity of the hitting time was shown in [23] and further studied in [15].",
"These works assumed a fixed, stationary transition probability distribution, and also did not consider submodularity of the cover time.",
"Our framework derives these existing results as special cases of a more general result, while also considering non-stationary transition probability distributions.",
"Random walk times have been used for image segmentation and clustering applications.",
"In these settings, the distance between two locations in an image is quantified via the commute time of a random walk with a probability distribution determined by a heat kernel [8].",
"Clustering algorithms were then proposed based on minimizing the commute time between two sets [24].",
"This work is related to the problem of selecting an optimal control policy for a Markov decision process [25].",
"Prior work has investigated selecting a control policy in order to maximize the probability of reaching a desired target set [26], [27], or to minimize the cost per cycle of reaching the target set [28].",
"These works assume that the target set is given.",
"In this paper, we consider the dual problem of selecting a target set in order to optimize these metrics."
],
[
"Background and Preliminaries",
"This section gives background on random walk times, Markov decision processes, and submodularity.",
"Notations used throughout the paper are introduced."
],
[
"Random Walk Times",
"Throughout the paper, we let $\\mathbf {E}(\\cdot )$ denote the expectation of a random variable.",
"Let $G = (V,E)$ denote a graph with vertex set $V$ and edge set $E$ .",
"A random walk is a discrete-time random process $X_{k}$ with state space $V$ .",
"The distribution of the walk is defined by a set of functions $P_{k} : V^{k} \\rightarrow \\Pi ^{V}$ , where $V^{k} = \\underbrace{V \\times \\cdots \\times V}_{k \\mbox{ times}}$ and $\\Pi ^{V}$ is the simplex of probability distributions over $V$ .",
"The function $P_{k}$ is defined by $P_{k}(v_{1},\\ldots ,v_{k}) = Pr(X_{k} = v_{k} | X_{1} = v_{1}, \\cdots , X_{k-1} = v_{k-1}).$ The random walk is Markovian if, for each $k$ , there exists a stochastic matrix $P_{k}$ such that $Pr(X_{k} = v_{k} | X_{1} = v_{1},\\ldots ,X_{k-1} = v_{k-1}) = P_{k}(v_{k-1},v_{k})$ .",
"Matrix $P_{k}$ is denoted as the transition matrix at time $k$ .",
"The random walk is Markovian and stationary if $P_{k} \\equiv P$ for all $k$ and some stochastic matrix $P$ .",
"A Markovian and stationary walk is ergodic [20] if there exists a probability distribution $\\pi $ such that, for any initial distribution $\\phi $ , $\\lim _{k \\rightarrow \\infty }{\\phi P^{k}} = \\pi $ , implying that the distribution of the walk will eventually converge to $\\pi $ regardless of the initial distribution.",
"Let $S \\subseteq V$ be a subset of nodes in the graph.",
"The hitting, commute, and cover time of $S$ are defined as follows.",
"Definition 1 Let random variables $\\nu (S)$ , $\\kappa (S)$ , and $\\phi (S)$ be defined as $\\nu (S) &=& \\min {\\lbrace k : X_{k} \\in S\\rbrace }\\\\\\kappa (S) &=& \\min {\\lbrace k : X_{k} = X_{1}, X_{j} \\in S \\mbox{ for some } j < k\\rbrace } \\\\\\phi (S) &=& \\min {\\lbrace k : \\forall s \\in S, \\ \\exists j \\le k \\mbox{ s.t. }",
"X_{j} = s\\rbrace }$ The hitting time of $S$ from a given node $v$ is equal to $H(v,S) \\triangleq \\mathbf {E}(\\nu (S) | X_{1} = v)$ .",
"The commute time of $S$ from node $v$ is equal to $K(v,S) = \\mathbf {E}(\\kappa (S) | X_{1} = v)$ .",
"The cover time of $S$ from node $v$ is equal to $C(v,S) = \\mathbf {E}(\\phi (S) | X_{1} = v)$ .",
"Intuitively, the hitting time is the expected time for the random walk to reach the set $S$ , the commute time is the expected time for a walk starting at a node $v$ to reach any node in $S$ and return to $v$ , and the cover time is the expected time for the walk to reach all nodes in the set $S$ .",
"If $\\pi $ is a probability distribution over $V$ , we can further generalize the above definitions to $H(\\pi ,S)$ , $K(\\pi ,S)$ , and $C(\\pi ,S)$ , which are the expected hitting, commute, and cover times when the initial state is chosen from distribution $\\pi $ .",
"The times described above are all special cases of stopping times of a stochastic process.",
"Definition 2 ([1], Ch.",
"1) A stopping time $Z$ of a random walk $X_{k}$ is a $\\lbrace 1,2,\\ldots ,\\infty \\rbrace $ -valued random variable such that, for all $k$ , the event $\\lbrace Z = k\\rbrace $ is determined by $X_{1},\\ldots ,X_{k}$ .",
"The hitting, cover, and commute times are stopping times."
],
[
"Markov Decision Processes",
"A Markov Decision Process (MDP) is a generalization of a Markov chain, defined as follows.",
"Definition 3 An MDP $\\mathcal {M}$ is a discrete-time random process $X_{k}$ defined by a tuple $\\mathcal {M} = (V, \\lbrace A_{i} : i \\in V\\rbrace , P)$ , where $V$ is a set of states, $A_{i}$ is a set of actions at state $i$ , and $P$ is a transition probability function defined by $P(i,a,j) = Pr(X_{k+1} = j | X_{k} = i, \\mbox{action $a$ chosen at time $k$}).$ Define $\\mathcal {A} = \\bigcup _{i=1}^{n}{A_{i}}$ and $A = |\\mathcal {A}|$ .",
"When the action set of a state is empty, the transition probability is a function of the current state only.",
"The set of actions at each state corresponds to the possible control inputs that can be supplied to the MDP at that state.",
"A control policy is a function that takes as input a sequence of states $X_{1},\\ldots ,X_{k}$ and gives as output an action $a_{k} \\in A_{X_{k}}$ .",
"A stationary control policy is a control policy $\\mu $ that depends only on the current state, and hence can be characterized by a function $\\mu : V \\rightarrow A$ .",
"We let $\\mathcal {P}$ denote the set of valid policies, i.e., the policies $\\mu $ satisfying $\\mu (i) \\in A_{i}$ for all $i$ .",
"The random walk induced by a stationary policy $\\mu $ is a stationary random walk with transition matrix $P_{\\mu }$ defined by $P_{\\mu }(i,j) = P(i,\\mu (i),j)$ .",
"The control policy $\\mu $ is selected in order to achieve a design goal of the system.",
"Such goals are quantified via specifications on the random process $X_{k}$ .",
"Two relevant specifications are safety and liveness constraints.",
"A safety constraint specifies that a set of states $R$ should never be reached by the walk.",
"A liveness constraint specifies that a given set of states $S$ must be reached infinitely often.",
"In an MDP, two optimization problems arise in order to satisfy such constraints, namely, the reachability and average-cost-per-cycle problems.",
"The reachability problem consists of selecting a policy in order to maximize the probability that a desired set of states $S$ is reached by the Markov process while the unsafe set $R$ is not reached.",
"The average cost per cycle (ACPC) problem is defined via the following metric.",
"Definition 4 The average cost per cycle metric from initial state $s \\in V$ under policy $\\mu $ is defined by $J_{\\mu }(s) = \\limsup _{N \\rightarrow \\infty }{\\mathbf {E}\\left\\lbrace \\frac{\\sum _{k=0}^{N}{g(X_{k},\\mu _{k}(X_{k}))}}{C(\\mu ,N)}\\right\\rbrace },$ where $g(X_{k},\\mu _{k}(X_{k}))$ is the cost of taking an action $\\mu _{k}(X_{k})$ at state $X_{k}$ and $C$ is the number of cycles up to time $N$ .",
"The average cost per cycle can be viewed as the average number of steps in between times when the set $S$ is reached.",
"The ACPC problem consists of choosing the set $S$ and policy $\\mu $ in order to minimize $J(s)$ .",
"Applications of this problem include motion planning, in which the goal is to reach a desired state infinitely often while minimizing energy consumption.",
"We define $J_{\\mu } \\in \\mathbb {R}^{n}$ as the vector of ACPC values for different initial states, so that $J_{\\mu }(s)$ is the ACPC value for state $s$ .",
"In the special case where all actions have cost 1, Eq.",
"(REF ) is equivalent to $J_{\\mu }(s) = \\limsup _{N \\rightarrow \\infty }{\\left\\lbrace \\frac{N}{C(\\mu ,N)}\\right\\rbrace }.$ We focus on this case in what follows.",
"It has been shown in [29] that the optimal policy $\\mu ^{\\ast }$ that minimizes the ACPC is independent of the initial state.",
"The following theorem characterizes the minimum ACPC policy $\\mu ^{\\ast }$ .",
"Theorem 1 ([29]) The optimal ACPC is given by $J_{\\mu ^{\\ast }} = \\lambda \\mathbf {1}$ , where $\\lambda \\in \\mathbb {R}$ and there exist vectors $h$ and $\\nu $ satisfying $J_{\\mu ^{\\ast }} + h &=& \\mathbf {1} + P_{\\mu ^{\\ast }}h + \\overline{P}_{\\mu ^{\\ast }}J_{\\mu ^{\\ast }} \\\\P_{\\mu ^{\\ast }} &=& (I-\\overline{P}_{\\mu ^{\\ast }})h + \\nu \\\\\\nonumber \\lambda + h(i) &=& \\min _{a \\in A_{i}}{\\left[1 + \\sum _{j=1}^{n}{P(i,a,j)h(j)}\\right.}",
"\\\\&& \\left.",
"\\qquad + \\lambda \\sum _{j \\notin S}{P(i,a,j)}\\right].$ $P_{\\mu ^{\\ast }}$ is the transition matrix induced by $\\mu ^{\\ast }$ and $\\overline{P}_{\\mu ^{\\ast }}(i,j) = \\left\\lbrace \\begin{array}{ll}P_{\\mu }(i,j), & j \\notin S \\\\0, \\mbox{else}\\end{array}\\right.$"
],
[
"Submodularity",
"Submodularity is a property of set functions $f: 2^{W} \\rightarrow \\mathbb {R}$ , where $W$ is a finite set and $2^{W}$ is the set of all subsets of $W$ .",
"A function is submodular if, for any sets $S$ and $T$ with $S \\subseteq T \\subseteq W$ and any $v \\notin T$ , $f(S \\cup \\lbrace v\\rbrace ) - f(S) \\ge f(T \\cup \\lbrace v\\rbrace ) - f(T).$ A function is supermodular if $-f$ is submodular, while a function is modular if it is both submodular and supermodular.",
"For any modular function $f(S)$ , a set of coefficients $\\lbrace c_{i} : i \\in W\\rbrace $ can be defined such that $f(S) = \\sum _{i \\in S}{c_{i}}.$ Furthermore, for any set of coefficients $\\lbrace c_{i} : i \\in W\\rbrace $ , the function $f(S) = \\max {\\lbrace c_{i} : i \\in S\\rbrace }$ is increasing and submodular, while the function $\\min {\\lbrace c_{i} : i \\in S\\rbrace }$ is decreasing and supermodular.",
"Any nonnegative weighted sum of submodular (resp.",
"supermodular) functions is submodular (resp.",
"supermodular).",
"A matroid is defined as follows.",
"Definition 5 A matroid $\\mathcal {M}=(V,\\mathcal {I})$ is defined by a set $V$ and a collection of subsets $\\mathcal {I}$ .",
"The set $\\mathcal {I}$ satisfies the following conditions: (i) $\\emptyset \\in \\mathcal {I}$ , (ii) $B \\in \\mathcal {I}$ and $A \\subseteq B$ implies that $A \\in \\mathcal {I}$ , and (iii) If $|A| < |B|$ and $A, B \\in \\mathcal {I}$ , then there exists $v \\in B \\setminus A$ such that $(A \\cup \\lbrace v\\rbrace ) \\in \\mathcal {I}$ .",
"The collection of sets $\\mathcal {I}$ is denoted as the independent sets of the matroid.",
"A basis is a maximal independent set.",
"The uniform matroid $\\mathcal {M}_{k}$ is defined by $A \\in \\mathcal {I}$ iff $|A| \\le k$ .",
"A partition matroid is defined as follows.",
"Definition 6 Let $V = V_{1} \\cup \\cdots \\cup V_{m}$ with $V_{i} \\cap V_{j} = \\emptyset $ for $i \\ne j$ be a partition of a set $V$ .",
"The partition matroid $\\mathcal {M} = (V, \\mathcal {I})$ is defined by $A \\in \\mathcal {I}$ if $|A \\cap V_{i}| \\le 1$ for all $i=1,\\ldots ,m$ .",
"Finally, given two matroids $\\mathcal {M}_{1} = (V, \\mathcal {I}_{1})$ and $\\mathcal {M}_{2} = (V, \\mathcal {I}_{2})$ , the union $\\mathcal {M} = \\mathcal {M}_{1} \\vee \\mathcal {M}_{2}$ is a matroid in which $A \\in \\mathcal {I}$ iff $A = A_{1} \\cup A_{2}$ for some $A_{1} \\in \\mathcal {I}_{1}$ and $A_{2} \\in \\mathcal {I}_{2}$ ."
],
[
"Random Walks on Markov Decision Processes",
"In this section, we consider the problem of selecting a set $S$ of states for an MDP to reach in order to optimize a performance metric.",
"We consider two problems, namely, the problem of selecting a set of states $S$ in order to maximize the reachability probability to $S$ while minimizing the probability of reaching an unsafe set $R$ , and the problem of selecting a set of states $S$ in order to minimize the ACPC.",
"A motivating scenario is a setting where an unmanned vehicle must reach a refueling station or transit depot infinitely often, and the goal is to place the set of stations in order to maximize the probability of reaching one or minimize the cost of reaching."
],
[
"Reachability Problem Formulation",
"The problem of selecting a set $S$ with at most $k$ nodes in order to maximize the probability of reaching $S$ under the optimal policy $\\mu $ is considered as follows.",
"Let $\\sigma (S)$ denote the event that the walk reaches $S$ at some finite time and does not reach the unsafe state $R$ at any time.",
"The problem formulation is given by $\\begin{array}{ll}\\mbox{maximize} & \\max _{\\mu \\in \\mathcal {P}}{Pr(\\sigma (S)|\\mu )} \\\\S \\subseteq V & \\\\\\mbox{s.t.}",
"& |S| \\le k\\end{array}$ Here $Pr(\\sigma (S) | \\mu )$ denotes the probability that $\\sigma (S)$ occurs when the policy is $\\mu $ .",
"This formulation is equivalent to $\\begin{array}{lll}\\mbox{maximize} & \\mbox{max} & Pr(\\sigma (S)|\\mu ) \\\\\\mu & S: |S| \\le k &\\end{array}$ The following known result gives a linear programming approach to maximizing the probability of reachability for a fixed set $S$ .",
"Lemma 1 ([30], Theorem 10.105) The optimal value of $\\max {\\lbrace Pr(\\sigma (S)|\\mu ) : \\mu \\in \\mathcal {P}\\rbrace }$ is equal to $\\begin{array}{ll}\\mbox{min} & \\mathbf {1}^{T}\\mathbf {x} \\\\\\mathbf {x} \\in \\mathbb {R}^{n} & \\\\\\mbox{s.t.}",
"& x_{i} \\in [0,1] \\ \\forall i \\\\& x_{i} = 1 \\ \\forall i \\in S, x_{i} = 0 \\ \\forall i \\in R \\\\& x_{i} \\ge \\sum _{j=1}^{n}{P(i,a,j)x_{j}} \\ \\forall i \\in V \\setminus R, a \\in A_{i}\\end{array}$ The optimal solution $\\mathbf {x}$ to the linear program (REF ) is a vector in $\\mathbb {R}^{n}$ , where $x_{i}$ is the probability that $\\sigma (S)$ occurs under the optimal policy when the initial state is $i$ .",
"In addition to giving the optimal value of the maximal reachability problem, Eq.",
"(REF ) can also be used to compute the optimal policy.",
"In order for $\\mathbf {x}$ to be the solution to (REF ), for each $i \\in V \\setminus (R \\cup S)$ , there must be an action $a_{i}^{\\ast }$ such that $x_{i} = \\sum _{j=1}^{n}{P(i,a_{i}^{\\ast },j)x_{j}}.$ Otherwise, it would be possible to decrease $x_{i}$ and hence the objective function of (REF ).",
"Hence the optimal policy $\\mu $ is given by $\\mu (i) = a_{i}^{\\ast }$ .",
"We first define a relaxation of (REF ).",
"Let $\\rho > 0$ , and define the relaxation by $\\begin{array}{ll}\\mbox{minimize} & \\mathbf {1}^{T}\\mathbf {x} + \\rho \\left(\\sum _{i \\in S}{(1-x_{i})}\\right) \\\\\\mathbf {x} & \\\\\\mbox{s.t.}",
"& \\mathbf {x} \\in [0,1]^{n} \\\\& x_{i} = 0 \\ \\forall i \\in R \\\\& x_{i} \\ge \\sum _{j=1}^{n}{P(i,a,j)x_{j}} \\ \\forall i \\in V \\setminus R, a \\in A_{i}\\end{array}$ The following lemma shows that (REF ) is equivalent to (REF ).",
"Lemma 2 When $\\rho > n$ , the optimal solutions and optimal values of (REF ) and (REF ) are equal.",
"We first show that the solution to (REF ) satisfies $x_{i}=1$ for all $i \\in S$ .",
"Suppose that this is not the case.",
"Let $\\mathbf {x}^{\\ast }$ denote the solution to (REF ), and suppose that $x_{r}^{\\ast } = 1-\\epsilon $ for some $\\epsilon > 0$ and $r \\in S$ .",
"Now, construct a new vector $\\mathbf {x}^{\\prime } \\in \\mathbb {R}^{n}$ as $x_{i}^{\\prime } = \\left\\lbrace \\begin{array}{ll}0, & i \\in R \\\\\\min {\\lbrace x_{i}^{\\ast } + \\epsilon , 1\\rbrace }, & i \\notin R\\end{array}\\right.$ Note that for all $i$ , $0 \\le (x_{i}^{\\prime } - x_{i}^{\\ast }) \\le \\epsilon $ .",
"We will first show that $\\mathbf {x}^{\\prime }$ is feasible under the constraints of (REF ), and then show that the resulting objective function value is less than the value produced by $\\mathbf {x}^{\\ast }$ , contradicting optimality of $\\mathbf {x}^{\\ast }$ .",
"By construction, $\\mathbf {x}^{\\prime } \\in [0,1]^{n}$ and $x_{i}^{\\prime } = 0$ for all $i \\in R$ .",
"For each $i \\notin R$ , suppose first that $x_{i}^{\\prime } = 1$ .",
"Then $x_{i}^{\\prime } = 1 = \\sum _{j=1}^{n}{P(i,a,j)1} \\ge \\sum _{j=1}^{n}{P(i,a,j)x_{j}^{\\prime }}.$ Suppose next that $x_{i}^{\\prime } = x_{i} + \\epsilon $ .",
"Then for all $a \\in A_{i}$ , rCl j=1nP(i,a,j)xj = j=1nP(i,a,j)(xj + (xj-xj)) = j=1nP(i,a,j)xj + j=1nP(i,a,j)(xj-xj) j=1nP(i,a,j)xj + j=1nP(i,a,j) xi + = xi implying that $\\mathbf {x}^{\\prime }$ is feasible.",
"The objective function value of $\\mathbf {x}^{\\prime }$ is given by rCl 1Tx + (i S(1-xi)) = 1Tx + 1T(x-x) + (i S {r}(1-xi)) 1Tx + n + (i S {r}(1-xi)) < 1Tx + + i S {r}(1-xi) = 1Tx + i S(1-xi) contradicting the assumption that $\\mathbf {x}^{\\ast }$ is the optimal value of (REF ).",
"Hence the optimal value of (REF ) minimizes $\\mathbf {1}^{T}\\mathbf {x}$ while satisfying $x_{i} = 1$ for all $i \\in S$ , which is equivalent to the solution of (REF ).",
"The problem of maximizing reachability is therefore equivalent to $\\begin{array}{lll}\\mbox{maximize} & \\mbox{min} & \\mathbf {1}^{T}\\mathbf {x} + \\rho \\left(\\sum _{i \\in S}{(1-x_{i})}\\right) \\\\S: |S| \\le k & \\mathbf {x} \\in \\Pi &\\end{array}$ The min-max inequality implies that (REF ) can be bounded above by $\\begin{array}{lll}\\mbox{min} & \\mbox{max} & \\mathbf {1}^{T}\\mathbf {x} + \\rho \\left(\\sum _{i \\in S}{(1-x_{i})}\\right) \\\\\\mathbf {x} \\in \\Pi & S: |S| \\le k &\\end{array}$ The objective function of (REF ) is a pointwise maximum of convex functions and is therefore convex.",
"A subgradient of the objective function at any point $\\mathbf {x}_{0}$ , denoted $v(\\mathbf {x}_{0})$ , is given by $v(\\mathbf {x}_{0})_{i} = \\left\\lbrace \\begin{array}{ll}1-\\rho , & i \\in S_{max}(\\mathbf {x}_{0}) \\\\1, & \\mbox{else}\\end{array}\\right.$ where $S_{max}(\\mathbf {x}_{0}) = \\arg \\min {\\left\\lbrace \\sum _{i \\in S}{(x_{0})_{i}} : |S| \\le k\\right\\rbrace }.$ This subgradient can be computed efficiently by selecting the $k$ largest elements of $\\mathbf {x}_{0}$ .",
"A polynomial-time algorithm for solving (REF ) can be obtained using interior-point methods, as shown in Algorithm REF .",
"[!htp] Algorithm for selecting a set of states $S$ to maximize probability of reachability.",
"[1] Max_Reach$G=(V,E)$ , $A$ , $P$ , $R$ , $k$ , $\\epsilon $ , $\\delta $ Input: Graph $G=(V,E)$ , set of actions $A_{i}$ at each state $i$ , probability distribution $P$ , unsafe states $R$ , number of states $k$ , convergence parameters $\\epsilon $ and $\\delta $ .",
"Output: Set of nodes $S$ $\\Phi \\leftarrow $ barrier function for polytope $\\Pi $ $\\mathbf {x} \\leftarrow 0$ $\\mathbf {x}^{\\prime } \\leftarrow \\mathbf {1}$ $||\\mathbf {x}-\\mathbf {x}^{\\prime }||_{2} > \\epsilon $ $S \\leftarrow \\arg \\max {\\lbrace \\sum _{i \\in S}{x_{i}}: |S| \\le k\\rbrace }$ $v \\leftarrow 1$ $v_{i} \\leftarrow (1-\\rho ) \\ \\forall i \\in S$ $w \\leftarrow \\nabla _{\\mathbf {x}}{\\Phi (\\mathbf {x})}$ $\\mathbf {x}^{\\prime } \\leftarrow \\mathbf {x}$ $\\mathbf {x} \\leftarrow \\mathbf {x} + \\delta (v + w)$ $S \\leftarrow \\arg \\max {\\lbrace \\sum _{i \\in S}{x_{i}}: |S| \\le k\\rbrace }$ $S$ The interior-point approach of Algorithm REF gives an efficient algorithm for maximizing the probability of reaching $S$ .",
"We further observe that more general constraints than $|S| \\le k$ can be constructed.",
"One possible constraint is to ensure that, for some partition $V_{1}, \\ldots , V_{m}$ , we have $|S \\cap V_{i}| \\ge 1$ for each $i=1,\\ldots ,m$ .",
"Intuitively, this constraint implies that there must be at least one state to be reached in each partition set $V_{i}$ .",
"This constraint is equivalent to the constraint $S \\in \\mathcal {M}_{k}$ , where $\\mathcal {M}_{k}$ is the union of the partition matroid and the uniform matroid of rank $k-m$ .",
"The calculation of $S_{max}(\\mathbf {x}_{0})$ then becomes $S_{max}(\\mathbf {x}_{0}) = \\arg \\min {\\left\\lbrace \\sum _{i \\in S}{(x_{0})_{i}} : S \\in \\mathcal {M}_{k}\\right\\rbrace }.$ This problem can also be solved in polynomial time due to the matroid structure of $\\mathcal {M}_{k}$ using a greedy algorithm."
],
[
"Minimizing the Average Cost Per Cycle",
"This section considers the problem of selecting a set $S$ in order to minimize the ACPC.",
"Based on Theorem REF , in order to ensure that the minimum ACPC is no more than $\\lambda $ , it suffices to show that there is no $h$ satisfying (REF ) for the chosen set $S$ .",
"Note that this condition is sufficient but not necessary.",
"Hence the following optimization problem gives a lower bound on the ACPC $\\begin{array}{lll}\\mbox{minimize} & \\mbox{max} & \\lambda \\\\S & \\lambda , h & \\\\&\\mbox{s.t.}",
"& \\lambda + h(i) \\le 1 + \\sum _{j=1}^{n}{P(i,a,j)h(j)} \\\\& & +\\lambda \\sum _{j \\notin S}{P(i,a,j)} \\ \\forall i, a \\in A_{i} \\\\\\end{array}$ The following theorem gives a sufficient condition for the minimum ACPC.",
"Theorem 2 Suppose that, for any $h \\in \\mathbb {R}^{n}$ , there exist $a$ and $i$ such that $\\lambda \\mathbf {1}\\lbrace i \\in S\\rbrace + h(i) - \\sum _{j=1}^{n}{P(i,a,j)h(j)} > 1.$ Then the ACPC is bounded above by $\\lambda $ .",
"In order to prove Theorem REF , we introduce an extended state space that will have the same ACPC.",
"The state space is defined $\\hat{V} = V \\times \\lbrace 0,1\\rbrace $ .",
"The sets of actions satisfy $\\hat{A}_{(i,0)} = \\hat{A}_{(i,1)} = A_{i}$ .",
"The transition probabilities are given by $\\hat{P}((i,1),a,(i,0)) = 1, \\quad \\hat{P}((i,0),a,(j,1)) = P(i,j).$ Finally, the set $\\hat{S}$ is constructed from the set $S$ via $\\hat{S} = \\lbrace (i,0) : i \\in S\\rbrace .$ The following result establishes the equivalence between the graph formulations.",
"Proposition 1 The minimum ACPC of $\\mathcal {M}$ with set $S$ is equal to the minimum ACPC of $\\hat{\\mathcal {M}} = (\\hat{V}, \\hat{A}, \\hat{P})$ with set $\\hat{S}$ .",
"There is a one-to-one correspondence between policies on $\\mathcal {M}$ and policies on $\\hat{\\mathcal {M}}$ .",
"Indeed, any policy $\\mu $ on $\\mathcal {M}$ can be extended to a policy $\\hat{\\mu }$ on $\\hat{\\mathcal {M}}$ by setting $\\hat{\\mu }(i,0) = \\mu (i)$ and $\\hat{\\mu }(i,1) = 1$ for all $i$ .",
"Furthermore, by construction, the ACPC for $\\mathcal {M}$ with policy $\\mu $ will be equal to the ACPC with policy $\\hat{\\mu }$ .",
"In particular, the cost per cycle of the minimum-cost policies will be equal.",
"We are now ready to prove Theorem REF .",
"[Proof of Theorem REF ] For the MDP $\\hat{\\mathcal {M}}$ , the constraint of Eq.",
"(REF ) is equivalent to $\\lambda \\mathbf {1}\\lbrace i \\in S\\rbrace + \\hat{h}(i,1) &\\le & \\hat{h}(i,0) \\\\\\hat{h}(i,0) - \\sum _{j=1}^{n}{P(i,a,j)\\hat{h}(j,1)} &\\le & 1$ for all $i$ , $j$ , and $u$ .",
"Hence, in order for the minimum cost per cycle to be less than $\\lambda $ , at least one of (REF ) or () must fail for each $\\hat{h} \\in \\mathbb {R}^{2N}$ .",
"For each $\\hat{h}$ , let $S_{\\hat{h}} = \\lbrace i: \\lambda + \\hat{h}(i,1) > \\hat{h}(i,0)\\rbrace $ , so that the condition that the ACPC is less than $\\lambda $ is equivalent to $S_{\\hat{h}} \\cap S \\ne \\emptyset \\quad \\forall \\hat{h} \\in \\mathbb {R}^{2N}.$ Furthermore, we can combine Eq.",
"(REF ) and () to obtain $\\lambda \\mathbf {1}\\lbrace i \\in S\\rbrace + \\hat{h}(i,1) \\le 1 + \\sum _{j=1}^{n}{P(i,a,j)\\hat{h}(j,1)}.$ For the MDP $\\mathcal {M}$ , define $h \\in \\mathbb {R}^{N}$ by $h(i) = h(i,1)$ for all $i \\in V$ , so that $\\lambda \\mathbf {1}\\lbrace i \\in S\\rbrace + h(i) \\le 1 + \\sum _{j=1}^{n}{P(i,a,j)h(j)}.$ Since $\\mathcal {M}$ and $\\hat{\\mathcal {M}}$ have the same ACPC, (REF ) is a necessary condition for the ACPC of $\\mathcal {M}$ to be at least $\\lambda $ .",
"This is equivalent to (REF ).",
"We will now map the minimum ACPC problem to submodular optimization.",
"As a preliminary, define the matrix $\\overline{A} \\in \\mathbb {R}^{nA \\times n}$ , with rows indexed $\\lbrace (i,a) : i \\in 1,\\ldots ,n, a \\in \\mathcal {A}\\rbrace $ , as $\\overline{A}((i,a),j) = \\left\\lbrace \\begin{array}{ll}-P(i,a,j), & i \\ne j \\\\1-P(i,a,j), & i = j\\end{array}\\right.$ and the vector $b(S) \\in \\mathbb {R}^{nA}$ , with entries indexed $\\lbrace (i,a): i =1,\\ldots ,n, a \\in \\mathcal {A}\\rbrace $ , as $(b(S))_{i,a} = \\left\\lbrace \\begin{array}{ll}1-\\lambda , & i \\in S \\\\1, & i \\notin S\\end{array}\\right.$ Proposition 2 For any $\\zeta > 0$ , let $\\mathcal {P}(\\lambda ,S)$ denote the polytope $\\mathcal {P}(\\lambda ,S) = \\lbrace h: \\overline{A}h \\le b(S)\\rbrace \\cap \\lbrace ||h||_{\\infty } \\le \\zeta \\rbrace .$ Then the function $r_{\\lambda }(S) = \\mbox{vol}(\\mathcal {P}(\\lambda ,S))$ is decreasing and supermodular as a function of $S$ .",
"Furthermore, if $r_{\\lambda }(S) = 0$ , then the ACPC is bounded above by $\\lambda $ .",
"Define a sequence of functions $r^{N}_{\\lambda }(S)$ as follows.",
"For each $N$ , partition the set $\\mathcal {P}(0, \\emptyset ) = \\lbrace h : \\overline{A}h \\le \\mathbf {1}\\rbrace $ into $N$ equally-sized regions with center $x_{1},\\ldots ,x_{N}$ and volume $\\delta _{N}$ .",
"Define $r^{N}_{\\lambda }(S) = \\sum _{l=1}^{N}{\\delta _{N}\\mathbf {1}\\lbrace x_{l} \\in \\mathcal {P}(\\lambda , S)\\rbrace }.$ Since $\\mathcal {P}(\\lambda ,S) \\subseteq \\mathcal {P}(0,\\emptyset )$ for all $S$ and $\\lambda $ , we have that $\\mbox{vol}(\\mathcal {P}(\\lambda ,S)) \\approx r^{N}_{\\lambda }(S)$ and $\\lim _{N \\rightarrow \\infty }{r^{N}_{\\lambda }(S)} = r_{\\lambda }(S).$ The term $\\mathbf {1}\\lbrace x_{l} \\in \\mathcal {P}(\\lambda ,S)\\rbrace $ is equal to the decreasing supermodular function $\\mathbf {1}\\lbrace S_{x_{l}} \\cap S = \\emptyset \\rbrace .$ Hence $r^{N}_{\\lambda }(S)$ is a decreasing supermodular function, and $r_{\\lambda }(S)$ is a limit of decreasing supermodular functions and is therefore decreasing supermodular.",
"Finally, we have that if $r_{\\lambda }(S) = 0$ , then there is no $h$ satisfying $\\overline{A}h \\le b(S)\\rbrace $ , and hence the ACPC is bounded above by $\\lambda $ by Theorem REF .",
"In Proposition REF , the constraint $||h||_{\\infty } \\le \\zeta $ is added to ensure that the polytope is compact.",
"Proposition REF implies that ensuring that the ACPC is bounded above by $\\lambda $ is equivalent to the submodular constraint $r_{\\lambda }(S) = 0$ .",
"Hence, $r_{\\lambda }(S)$ is a submodular metric that can be used to ensure a given bound $\\lambda $ on ACPC.",
"This motivates a bijection-based algorithm for solving the minimum ACPC problem (Algorithm REF ).",
"[!htp] Algorithm for selecting a set of states $S$ to minimize average cost per cycle.",
"[1] Min_ACPC$V$ , $A$ , $P$ , $R$ , $k$ Input:Set of states $V$ , set of actions $A_{i}$ at each state $i$ , probability distribution $P$ , unsafe states $R$ , number of states to be chosen $k$ .",
"Output: Set of nodes $S$ $\\lambda _{max} \\leftarrow $ max ACPC for any $v \\in \\lbrace 1,\\ldots ,n\\rbrace $ $\\lambda _{min} \\leftarrow 0$ $|\\lambda _{max}-\\lambda _{min}| > \\delta $ $S \\leftarrow \\emptyset $ $\\lambda \\leftarrow \\frac{\\lambda _{max}+\\lambda _{min}}{2}$ $r_{\\lambda }(S) > 0$ $v^{\\ast } \\leftarrow \\arg \\min {\\lbrace r_{\\lambda }(S \\cup \\lbrace v\\rbrace ) : v \\in \\lbrace 1,\\ldots ,n\\rbrace \\rbrace }$ $S \\leftarrow S \\cup \\lbrace v^{\\ast }\\rbrace $ $|S| \\le k$ $\\lambda _{max} \\leftarrow \\lambda $ $\\lambda _{min} \\leftarrow \\lambda $ $S$ The following theorem describes the optimality bounds and complexity of Algorithm REF .",
"Theorem 3 Algorithm REF terminates in $O\\left(kn^{6}\\log {\\lambda _{max}}\\right)$ time.",
"For any $\\lambda $ such that there exists a set $S$ of size $k$ satisfying $r_{\\lambda }(S) = 0$ , Algorithm REF returns a set $S^{\\prime }$ with $\\frac{|S^{\\prime }|}{|S|} \\le 1 + \\log {\\left\\lbrace \\frac{r_{\\lambda }(\\emptyset )}{\\min _{v}{\\lbrace r_{\\lambda }(\\hat{S} \\setminus \\lbrace v\\rbrace )\\rbrace }}\\right\\rbrace }.$ The number of rounds in the outer loop is bounded by $\\log {\\lambda _{max}}$ .",
"For each iteration of the inner loop, the objective function $r_{\\lambda }(S)$ is evaluated $kn$ times.",
"Computing $r_{\\lambda }(S)$ is equivalent to computing the volume of a linear polytope, which can be approximated in $O(n^{5})$ time [31], for a total runtime of $O(kn^{6}\\log {\\lambda _{max}})$ .",
"From [32], for any monotone submodular function $f(S)$ and the optimization problem $\\min {\\lbrace |S| : f(S) \\le \\alpha \\rbrace }$ , the set $\\hat{S}$ returned by the algorithm satisfies $\\frac{|\\hat{S}|}{|S^{\\ast }|} \\le 1 + \\log {\\left\\lbrace \\frac{f(V) - f(\\emptyset )}{f(V) - f(\\hat{S}_{T-1})}\\right\\rbrace },$ where $\\hat{S}_{T-1}$ is the set obtained at the second-to-last iteration of the algorithm.",
"Applied to this setting, we have $\\frac{|\\hat{S}|}{|S^{\\ast }|} &\\le & 1 + \\log {\\left\\lbrace \\frac{r_{\\lambda }(V) - r_{\\lambda }(\\emptyset )}{r_{\\lambda }(V) - r_{\\lambda }(\\hat{S}_{T-1})}\\right\\rbrace } \\\\&=& 1 + \\log {\\left\\lbrace \\frac{r_{\\lambda }(\\emptyset )}{r_{\\lambda }(\\hat{S}_{T-1})}\\right\\rbrace } \\\\&\\le & 1 + \\log {\\left\\lbrace \\frac{r_{\\lambda }(\\emptyset )}{\\min _{v}{\\lbrace r_{\\lambda }(\\hat{S} \\setminus \\lbrace v\\rbrace )\\rbrace }}\\right\\rbrace }.$ We note that the complexity of Algorithm REF is mainly determined by the complexity of computing the volume of the polytope $\\mathcal {P}(\\lambda ,S)$ .",
"This complexity can be reduced to $O(n^{3})$ by computing the volume of the minimum enclosing ellipsoid of $\\mathcal {P}(\\lambda ,S)$ instead.",
"The approach for minimizing ACPC presented in this section is applicable to problems such as motion planning for mobile robots.",
"The set $S$ represents locations that must be reached infinitely often, as in a surveillance problem, while minimizing ACPC can be viewed as minimizing the total resource consumption (e.g., fuel costs) while reaching the desired states infinitely often."
],
[
"Joint Optimization of Reachability and ACPC",
"In this section, we consider the problem of selecting a set $S$ to satisfy safety and liveness constraints with maximum probability and minimum cost per cycle.",
"This problem can be viewed as combining the maximum reachability and minimum ACPC problems formulated in the previous sections.",
"As a preliminary, we define an end component of an MDP.",
"Definition 7 An end component (EC) of an MDP $\\mathcal {M} = (V, \\mathcal {A}, P)$ is an MDP $\\mathcal {M}^{\\prime } = (V^{\\prime }, \\mathcal {A}^{\\prime }, P^{\\prime })$ where (i) $V^{\\prime } \\subseteq V$ , (ii) $A_{i}^{\\prime } \\subseteq A_{i}$ for all $i \\in V^{\\prime }$ , (iii) $P^{\\prime }(i,a,j) = P(i,a,j)$ for all $i,j \\in V^{\\prime }$ and $a \\in A_{i}^{\\prime }$ , and (iv) if $i \\in V^{\\prime }$ and $P(i,a,j) > 0$ for some $a \\in A_{i}^{\\prime }$ , then $j \\in V^{\\prime }$ .",
"Intuitively, an end component $(V^{\\prime }, \\mathcal {A}^{\\prime }, P^{\\prime })$ is a set of states and actions such that if only the actions in $\\mathcal {A}^{\\prime }$ are selected, the MDP will remain in $V^{\\prime }$ for all time.",
"A maximal end component (MEC) $(V^{\\prime }, \\mathcal {A}^{\\prime },P^{\\prime })$ such that for any $V^{\\prime \\prime }$ and $\\mathcal {A}^{\\prime \\prime }$ with $V^{\\prime } \\subseteq V^{\\prime \\prime }$ and $\\mathcal {A}^{\\prime } \\subseteq \\mathcal {A}^{\\prime \\prime }$ , there is no EC with vertex set $V^{\\prime \\prime }$ and set of actions $\\mathcal {A}^{\\prime \\prime }$ .",
"Lemma 3 ([29]) The probability that an MDP satisfies a safety and liveness specification defined by $R$ and $S$ is equal to the probability that the MDP reaches an MEC $(V^{\\prime }, \\mathcal {A}^{\\prime }, P^{\\prime })$ with $V^{\\prime } \\cap S \\ne \\emptyset $ and $V^{\\prime } \\cap R = \\emptyset $ .",
"We define an MEC satisfying $V^{\\prime } \\cap R = \\emptyset $ to be an accepting maximal end component (AMEC).",
"By Lemma REF , the problem of maximizing the probability of satisfying the safety and liveness constraints is equal to the probability of reaching an AMEC with $V^{\\prime } \\cap S \\ne \\emptyset $ .",
"Let $\\mathcal {M}_{1} = (V_{1}^{\\prime }, A_{1}^{\\prime }, P_{1}^{\\prime }), \\ldots , \\mathcal {M}_{N}=(V_{N}^{\\prime },A_{N}^{\\prime },P_{N}^{\\prime })$ denote the set of AMECs of $\\mathcal {M}$ satisfying $V_{i}^{\\prime } \\cap S \\ne \\emptyset $ .",
"We now formulate two problems of joint reachability and ACPC.",
"The first problem is to minimize the ACPC, subject to the constraint that the probability of satisfying the constraints is maximized.",
"The second problem is to maximize the probability of satisfying safety and liveness properties, subject to a constraint on the average cost per cycle.",
"In order to address the first problem, we characterize the sets $S$ that maximize the reachability probability.",
"Lemma 4 Suppose that for each AMEC $(V^{\\prime }, \\mathcal {A}^{\\prime }, P^{\\prime })$ , $S \\cap V^{\\prime } \\ne \\emptyset $ .",
"Then $S$ maximizes the probability of satisfying the safety and liveness constraints of the MDP.",
"By Lemma REF , the safety and liveness constraints are satisfied if the walk reaches an MEC satisfying $S \\cap V^{\\prime } \\ne \\emptyset $ .",
"Hence, for any policy $\\mu $ , the probability of satisfying the constraints is maximized when $S \\cap V^{\\prime } \\ne \\emptyset $ for all MECs.",
"Note that the converse of the lemma may not be true.",
"There may exist policies that maximize the probability of satisfaction and yet do not reach an AMEC $(V^{\\prime },\\mathcal {A}^{\\prime }, P^{\\prime })$ with positive probability.",
"Lemma REF implies that in order to formulate the problem of minimizing the ACPC such that the probability of achieving the specifications is maximized, it suffices to ensure that there is at least one state in each AMEC that belongs to $S$ .",
"We will show that this is equivalent to a matroid constraint on $S$ .",
"Define a partition matroid by $\\mathcal {N}_{1} = (V, \\mathcal {I})$ where $\\mathcal {I} = \\lbrace S : |S \\cap V^{\\prime }| \\le 1 \\ \\forall \\mbox{ AMEC } \\mathcal {M}_{m}^{\\prime }, m=1,\\ldots ,N\\rbrace .$ Let $\\mathcal {N}_{k-N}$ denote the uniform matroid with cardinality $(k-N)$ .",
"Finally, let $\\mathcal {N} = \\mathcal {N}_{1} \\vee \\mathcal {N}_{k-N}$ .",
"The following theorem gives the equivalent formulation.",
"Theorem 4 Let $q(S)$ denote the ACPC for set $S$ .",
"Then the problem of selecting a set of up to $k$ nodes in order to minimize the ACPC while maximizing reachability probability is equivalent to $\\begin{array}{ll}\\mbox{minimize} & q(S) \\\\\\mbox{s.t.}",
"& S \\in \\mathcal {N}\\end{array}$ Since $q(S)$ is strictly decreasing in $S$ , the minimum value is achieved when $|S| = k$ .",
"In order to maximize the probability of satisfying the safety and liveness constraints, $S$ must also contain at least one state in each AMEC, implying that $S$ contains a basis of $\\mathcal {N}_{1}$ .",
"Hence the optimal set $S^{\\ast }$ consists of the union of one state from each AMEC (a basis of $\\mathcal {N}_{1}$ ) and $(k-N)$ other nodes (a basis of $\\mathcal {N}_{k-r}$ ), and hence is a basis of $\\mathcal {N}$ .",
"Conversely, we have that the optimal solution to (REF ) satisfies the constraint $|S| \\le k$ and contains at least one node in each AMEC, and thus is also a feasible solution to the joint reachability and ACPC problem.",
"Hence by Theorem REF and Theorem REF , we can formulate the problem of selecting $S$ to minimize the ACPC as $\\begin{array}{ll}\\mbox{minimize} & \\max {\\lambda } \\\\S \\in \\mathcal {N} & \\\\\\mbox{s.t.}",
"& \\lambda + h(i) \\le 1 + \\sum _{j=1}^{n}P(i,a,j)h(j) \\\\& + \\lambda \\sum _{j \\notin S}{P(i,a,j)} \\ \\forall i \\in V, a \\in A_{i}\\end{array}$ If there are multiple AMECs, then each AMEC $(V_{m}^{\\prime }, \\mathcal {A}_{m}^{\\prime }, P_{m}^{\\prime })$ will have a distinct value of average cost per cycle $\\lambda _{m}$ , which will be determined by $S \\cap V_{m}^{\\prime }$ .",
"The problem of minimizing the worst-case ACPC is then given by $\\begin{array}{ll}\\mbox{minimize} & \\max {\\lbrace \\lambda _{m}(S \\cap V_{m}^{\\prime }): m=1,\\ldots ,N\\rbrace } \\\\S \\in \\mathcal {M} & \\\\\\mbox{s.t.}",
"& \\lambda _{m} + h(i) \\le 1 + \\sum _{j=1}^{n}P(i,a,j)h(j) \\\\& + \\lambda _{m} \\sum _{j \\notin S}{P(i,a,j)} \\ \\forall i \\in V_{m}^{\\prime }, a \\in A_{i}^{\\prime }\\end{array}$ This problem is equivalent to $\\begin{array}{ll}\\mbox{minimize} & \\lambda \\\\S,\\lambda & \\\\\\mbox{s.t.}",
"& \\lambda + h(i) \\le 1 + \\sum _{j=1}^{n}{P(i,a,j)h(j)} \\\\& + \\lambda \\sum _{j \\notin S}{P(i,a,j)} \\ \\forall i, a \\in A_{i}^{\\prime } \\\\& |S| \\le k\\end{array}$ By Proposition REF , Eq.",
"(REF ) can be rewritten as $\\begin{array}{ll}\\mbox{minimize} & \\lambda \\\\S, \\lambda & \\\\\\mbox{s.t.}",
"& \\sum _{m=1}^{N}{r_{\\lambda ,m}(S \\cap V_{m}^{\\prime })} = 0 \\\\& |S| \\le k\\end{array}$ where $r_{\\lambda ,m}(S)$ is the volume of the polytope $\\mathcal {P}(\\lambda ,S \\cap V_{m}^{\\prime })$ defined as in Section REF and restricted to the MDP $\\mathcal {M}_{m}$ .",
"A bijection-based approach, analogous to Algorithm REF , suffices to approximately solve (REF ).",
"This approach is given as Algorithm REF .",
"[!htp] Algorithm for selecting a set of states $S$ to jointly optimize ACPC and reachability.",
"[1] Min_ACPC_Max_Reach$V$ , $A$ , $P$ , $R$ , $k$ Input: Set of states $V$ , set of actions $A_{i}$ at each state $i$ , probability distribution $P$ , set of unsafe states $R$ , number of states $k$ .",
"Output: Set of nodes $S$ $\\lambda _{max} \\leftarrow $ max ACPC for any $v \\in \\lbrace 1,\\ldots ,n\\rbrace $ $\\lambda _{min} \\leftarrow 0$ $|\\lambda _{max}-\\lambda _{min}| > \\delta $ $\\lambda \\leftarrow \\frac{\\lambda _{max}+\\lambda _{min}}{2}$ $m=1,\\ldots ,M$ $S \\leftarrow \\emptyset $ $r_{\\lambda ,m}(S) > 0$ $v^{\\ast } \\leftarrow \\arg \\min {\\left\\lbrace r_{\\lambda ,m}(S \\cup \\lbrace v\\rbrace ): \\right.",
"}$ $\\left.",
"v \\in \\lbrace 1,\\ldots ,n\\rbrace \\right\\rbrace $ $S \\leftarrow S \\cup \\lbrace v^{\\ast }\\rbrace $ $|S| \\le k$ $\\lambda _{max} \\leftarrow \\lambda $ $\\lambda _{min} \\leftarrow \\lambda $ $S$ The following proposition describes the optimality bounds of Algorithm REF .",
"Proposition 3 Algorithm REF returns a value of $\\lambda $ , denoted $\\hat{\\lambda }$ , that satisfies $\\hat{\\lambda } < \\lambda ^{\\ast }$ , where $\\lambda ^{\\ast }$ is the minimum ACPC that can be achieved by any set $S$ satisfying $|S| \\le k \\left(1 + \\log {\\frac{\\sum _{m=1}^{N}{r_{\\lambda ,m}(\\emptyset )}}{\\min _{v}{\\sum _{m=1}^{N}{r_{\\lambda ,m}(\\hat{S} \\setminus \\lbrace v\\rbrace )}}}}\\right).$ The proof is analogous to the proof of Theorem REF .",
"The submodularity of $r_{\\lambda ,m}(S)$ implies that the set $|\\hat{S}|$ is within the bound (REF ) of the minimum-size set $|S^{\\ast }|$ with $\\sum _{m=1}^{N}{r_{\\lambda ,m}(S)} = 0$ .",
"We now turn to the second joint optimization problem, namely, maximizing the reachability probability subject to a constraint $\\lambda $ on the average cost per cycle.",
"We develop a two-stage approach.",
"In the first stage, we select a set of input nodes for each AMEC $V_{1},\\ldots ,V_{N}$ in order to guarantee that the ACPC is less than $\\lambda $ .",
"In the second stage, we select the set of AMECs to include in order to satisfy the ACPC constraint while minimizing the number of inputs needed.",
"In the first stage, for each AMEC we approximate the problem $\\begin{array}{ll}\\mbox{minimize} & |S_{m}| \\\\\\mbox{s.t.}",
"& r_{\\lambda ,m}(S_{m}) = 0\\end{array}$ We let $c_{m} = |S_{m}|$ denote the number of states required for each AMEC $\\mathcal {M}_{m}$ .",
"The second stage problem can be mapped to a maximum reachability problem by defining an MDP $\\tilde{M} = (\\tilde{V}, \\tilde{A}, \\tilde{P})$ , defined as follows.",
"Let $\\mathcal {M}_{1} = (V_{1}^{\\prime }, A_{1}^{\\prime }, P_{1}^{\\prime }), \\ldots , \\mathcal {M}_{N}=(V_{N}^{\\prime },A_{N}^{\\prime },P_{N}^{\\prime })$ denote the set of AMECs of $\\mathcal {M}$ satisfying $V_{i}^{\\prime } \\cap S \\ne \\emptyset $ .",
"The node set of the augmented graph is equal to $V \\setminus \\left(\\bigcup _{m=1}^{N}{V_{m}^{\\prime }}\\right) \\cup \\lbrace l_{1},\\ldots ,l_{N}\\rbrace $ .",
"Here, each $l_{m}$ represents the AMEC $\\mathcal {M}_{m}$ , so that reaching $l_{m}$ is equivalent to reaching an AMEC $\\mathcal {M}_{m}$ .",
"The actions for states in $V$ are unchanged, while the states $l_{1},\\ldots ,l_{N}$ have empty action sets.",
"The transition probabilities are given by $\\tilde{P}(i,a,j) = \\left\\lbrace \\begin{array}{ll}P(i,a,j), & i,j \\notin \\bigcup _{m=1}^{N}{V_{m}^{\\prime }}, a \\in A_{i} \\\\\\sum _{j \\in V_{l}^{\\prime }}{P(i,a,j)}, & i \\notin \\bigcup _{s=1}^{N}{V_{s}^{\\prime }}, \\\\& j = l_{m}, a \\in A_{i} \\\\1, & i=j=l_{m} \\\\0, & i = l_{m}, i \\ne j\\end{array}\\right.$ In this MDP, the probability of reaching the set $\\lbrace l_{1},\\ldots ,l_{N}\\rbrace $ is equal to the probability of satisfying the safety and liveness constraints, and hence maximizing the probability of satisfying the constraints is equivalent to a reachability problem.",
"The problem of selecting a subset of states to maximize reachability while satisfying this constraint on $\\lambda $ can then be formulated as $\\begin{array}{lll}\\mbox{minimize} & \\max & \\left(\\sum _{i=1}^{|\\tilde{V}|}{x_{i}} - \\lambda \\sum _{m \\in S}{x_{l_m}}\\right)\\\\S \\subseteq \\lbrace 1,\\ldots ,N\\rbrace , & \\mathbf {x} \\in \\Pi & \\\\\\sum _{m \\in S}{c_{m}} \\le k & &\\end{array}$ by analogy to (REF ), where $\\Pi $ is defined for the MDP $\\tilde{\\mathcal {M}}$ .",
"The inner optimization problem of (REF ) is a knapsack problem, and hence is NP-hard and must be approximated at each iteration.",
"In order to reduce the complexity of the problem, we introduce the following alternative formulation.",
"We let $\\mathcal {P}_{\\lambda }$ denote the polytope satisfying the inequalities $\\mathbf {1}^{T}\\mathbf {z} &=& \\beta \\\\z_{i}(1-\\lambda \\mathbf {1}\\lbrace i \\in S\\rbrace ) &\\ge & \\sum _{j=1}^{n}{z_{j}P(i,a,j)}$ By Proposition REF , the condition that the reachability probability is at most $\\lambda $ is equivalent to $\\mbox{vol}(\\mathcal {P}(\\lambda ,S_{m}) = 0$ .",
"Letting $r_{\\lambda }(S)$ denote the volume of $\\mathcal {P}_{\\lambda }$ when the set of desired states is $S$ , the problem is formulated as $\\begin{array}{ll}\\mbox{minimize} & \\sum _{m \\in T}{c_{m}} \\\\\\mbox{s.t.}",
"& r_{\\lambda }(T) = 0\\end{array}$ Problem (REF ) is a submodular knapsack problem with coverage constraints.",
"An algorithm for solving it is as follows.",
"The set $T$ is initialized to $\\emptyset $ .",
"At each iteration, find the element $m$ that minimizes $\\frac{c_{m}}{r_{\\lambda }(T) - r_{\\lambda }(T \\cup \\lbrace m\\rbrace )},$ terminating when the condition $r_{\\lambda }(T) = 0$ is satisfied.",
"Hence, the overall approach is to select a collection of subsets $\\lbrace S_{m} : m=1,\\ldots ,M\\rbrace $ , representing the minimum-size subsets to satisfy the ACPC constraint on each AMEC $\\mathcal {M}_{m}$ .",
"We then select a set of AMECs to include in order to satisfy a desired constraint on reachability while minimizing the total number of inputs.",
"The set $S$ is then given by $S = \\bigcup _{m \\in T}{S_{m}}.$ The optimality gap of this approach is described as follows.",
"Proposition 4 The set $T$ chosen by the greedy algorithm satisfies $\\frac{\\sum _{m \\in T}{c_{m}}}{\\sum _{m \\in T^{\\ast }}{c_{m}}} \\le 1 + \\log {\\left\\lbrace \\frac{1}{r_{\\lambda }(\\hat{T})}\\right\\rbrace },$ where $T^{\\ast }$ is the optimal solution to (REF ) and $\\hat{T}$ is the set obtained by the greedy algorithm prior to convergence.",
"We have that $r_{\\lambda }$ is monotone decreasing and supermodular.",
"Hence, by Theorem 1 of [32], the optimality bound holds.",
"Combining the optimality bounds yields $\\frac{|S|}{|S^{\\ast }|} &=& \\frac{\\sum _{m \\in T}{c_{m}}}{\\sum _{m \\in T^{\\ast }}{c_{m}^{\\ast }}} \\\\&=& \\frac{\\sum _{m \\in T}{c_{m}}}{\\sum _{m \\in T^{\\ast }}{c_{m}}} \\frac{\\sum _{m \\in T^{\\ast }}{c_{m}}}{\\sum _{m \\in T^{\\ast }}{c_{m}^{\\ast }}} \\\\&\\le & \\frac{\\sum _{m \\in T}{c_{m}}}{\\sum _{m \\in T^{\\ast }}{c_{m}}} \\max _{m}{{\\left\\lbrace \\frac{c_{m}}{c_{m}^{\\ast }}\\right\\rbrace }} \\\\&\\le & \\left(1 + \\max _{m}{\\log {\\left\\lbrace \\frac{1}{r_{\\lambda ,m}(\\hat{S})}\\right\\rbrace }}\\right)\\left(1 + \\log {\\left\\lbrace \\frac{1}{r_{\\lambda }(\\hat{T})}\\right\\rbrace }\\right)$"
],
[
"Optimal Hitting Time",
"In this section, we consider the problem of choosing a set $S$ and policy $\\mu $ in order to minimize the hitting time to $S$ in an MDP.",
"The hitting time of node $i$ under policy $\\mu $ , denoted $H(i,\\mu ,S)$ , satisfies $H(i,\\mu ,S) = 1+ \\sum _{j=1}^{n}{P(i,\\mu (i),j)H(j,\\mu ,S)}$ for $i \\notin S$ and $H(i,S) = 0$ for $i \\in S$ .",
"We define $\\overline{H}(i,S)$ as the minimum hitting time over all policies $\\mu $ .",
"Minimizing the hitting time for fixed $S$ is equivalent to solving a stochastic shortest path problem.",
"The solution to this problem is described by the following lemma.",
"Lemma 5 ([33], Prop 5.2.1) The minimum hitting time $\\overline{H}(i,S)$ for set $S$ satisfies $\\overline{H}(i,S) = 1 + \\min _{a \\in A_{i}}{\\left\\lbrace \\sum _{j=1}^{n}{P(i,a,j)\\overline{H}(j,S)}\\right\\rbrace }$ and is equivalent to the linear program $\\begin{array}{ll}\\mbox{maximize} & \\mathbf {1}^{T}\\mathbf {v} \\\\\\mbox{s.t.}",
"& v_{i} \\le 1 + \\sum _{j=1}^{n}{P(i,a,j)v_{j}} \\ \\forall i, a \\in A_{i} \\\\& v_{i} = 0, \\ i \\in S \\\\\\end{array}$ The following lemma leads to an equivalent formulation to (REF ).",
"Lemma 6 For a given graph $G$ , there exists $\\theta > 0$ such that the conditions $v_{i} &\\le & 1 + \\sum _{j=1}^{n}{P(i,a,j)v_{j}} \\ \\forall i, a \\in A_{i} \\\\v_{i} &=& 0 \\ \\forall i \\in S$ are equivalent to $v_{i} + \\theta \\mathbf {1}\\lbrace i \\in S\\rbrace v_{i} \\le 1 + \\sum _{j=1}^{n}{P(i,a,j)v_{j}}.$ If (REF ) holds, then for $\\theta $ sufficiently large we must have $v_{i}=0$ for all $i \\in S$ .",
"The condition then reduces to $v_{i} \\le 1 + \\sum _{j=1}^{n}{P(i,a,j)v_{j}}$ for all $i \\notin S$ and $a \\in A_{i}$ , which is equivalent to (REF ).",
"From Lemma REF , it follows that in order to ensure that the optimal hitting time for each node is no more than $\\zeta $ , it suffices to ensure that for each $\\mathbf {v}$ satisfying $\\mathbf {1}^{T}\\mathbf {v} \\ge \\zeta $ , there exists at least one $i$ and $u$ such that $1 + \\sum _{j=1}^{n}{P(i,a,j)v_{j}} < (1 + \\theta \\mathbf {1}\\lbrace i \\in S\\rbrace )v_{i}.$ We define the function $\\chi _{v}(S)$ as $\\chi _{v}(S) = \\left\\lbrace \\begin{array}{ll}1, & \\mbox{(\\ref {eq:hitting-time-optimal1}) and (\\ref {eq:hitting-time-optimal2}) hold } \\\\0, & \\mbox{else}\\end{array}\\right.$ The following lemma relates the function $\\chi _{v}(S)$ to the optimality conditions of Lemma REF .",
"Lemma 7 The optimal hitting time corresponding to set $S$ is bounded above by $\\zeta $ if and only if $\\chi (S,\\zeta ) \\triangleq \\int _{\\lbrace \\mathbf {v}: \\mathbf {1}^{T}\\mathbf {v} \\ge \\zeta \\rbrace }{\\chi _{v}(S) \\ dv} = 0.$ Suppose that $\\chi (S,\\zeta ) = 0$ .",
"Then for each $v$ satisfying $\\mathbf {1}^{T}\\mathbf {v} \\ge \\zeta $ , we have that $\\chi _{v}(S,\\zeta ) = 0$ and hence $\\sum _{j=1}^{n}{P(i,a,j)v_{j}} < (1 + \\theta \\mathbf {1}\\lbrace i \\in S\\rbrace )v_{i}$ holds for some $i$ and $a \\in A_{i}$ , and hence there is no $\\mathbf {v}$ satisfying the conditions of (REF ) with $\\mathbf {1}^{T}\\mathbf {v} = \\zeta $ for the given value of $S$ .",
"The following result then leads to a submodular optimization approach to computing the set $S$ that minimizes the hitting time.",
"Proposition 5 The function $\\chi (S,\\zeta )$ is supermodular as a function of $S$ for any $\\zeta \\ge 0$ .",
"We first show that $\\chi _{v}(S)$ (Eq.",
"(REF )) is supermodular.",
"We have that $\\chi _{v}(S) = 1$ if and only if there exists $i \\in S$ with $v_{i} \\ne 0$ , or equivalently, $\\mbox{supp}(v) \\cap S \\ne \\emptyset $ .",
"If $\\chi _{v}(S) = \\chi _{v}(S \\cup \\lbrace u\\rbrace ) = 0$ , then the condition $\\mbox{supp}(v) \\cap S \\ne \\emptyset $ already holds, and hence $\\chi _{v}(T) = \\chi _{v}(T \\cup \\lbrace u\\rbrace )$ .",
"Since $\\chi (S)$ is an integral of supermodular functions (REF ), it is supermodular as well.",
"Furthermore, $\\chi (S)$ can be approximated in polynomial time via a sampling-based algorithm [2].",
"Hence the problem of selecting a set of states $S$ in order to minimize the optimal hitting time can be stated as $\\begin{array}{ll}\\mbox{minimize} & \\zeta \\\\\\mbox{s.t.}",
"& \\min {\\lbrace |S| : \\chi (S,\\zeta ) = 0\\rbrace } \\le k\\end{array}$ Problem (REF ) can be approximately solved using an algorithm analogous to Algorithm REF , with $r_{\\lambda }(S)$ replaced by $\\chi (S,\\zeta )$ in Lines 9 and 10, and $\\lambda $ replaced by $\\zeta $ throughout.",
"The following proposition gives optimality bounds for the revised algorithm.",
"Proposition 6 The modified Algorithm REF guarantees that the hitting time satisfies $\\overline{H}(i,S) \\le \\overline{H}(i,\\hat{S})$ , where $\\hat{S}$ is the solution to $\\min {\\lbrace \\overline{H}(i,S) : |S| \\le \\hat{k}\\rbrace }$ and $\\hat{k}$ satisfies $k = \\hat{k}\\left(1 + \\log {\\left\\lbrace \\frac{\\chi (\\emptyset )}{\\min _{v}{\\chi (S \\setminus \\lbrace v\\rbrace )}}\\right\\rbrace }\\right).$ Since the function $\\chi (S,\\zeta )$ is supermodular, the number of states $S$ selected by the greedy algorithm to satisfy $\\chi (S,\\zeta ) = 0$ satisfies (REF ).",
"Hence, for the set $\\hat{S}$ , since $|S| \\le \\hat{k}$ , we have that $\\overline{H}(i,S) \\le H(i,\\hat{S})$ ."
],
[
"Submodularity of Walk Times with Fixed Distribution",
"This section demonstrates submodularity of random walk times for walks with a probability distribution that does not depend on the set of nodes $S$ .",
"We first state a general result on submodularity of stopping times, and then prove submodularity of hitting, commute, cover, and coupling times as special cases."
],
[
"General Result",
"Consider a set of stopping times $Z_{1},\\ldots ,Z_{N}$ for a random process $X_{k}$ .",
"Let $W = \\lbrace 1,\\ldots ,N\\rbrace $ , and define two functions $f,g: 2^{W} \\rightarrow \\mathbb {R}$ by $f(S) &=& \\mathbf {E}\\left\\lbrace \\max {\\lbrace Z_{i} : i \\in S\\rbrace }\\right\\rbrace \\\\g(S) &=& \\mathbf {E}\\left\\lbrace \\min {\\lbrace Z_{i} : i \\in S\\rbrace }\\right\\rbrace $ We have the following general result.",
"Proposition 7 The functions $f(S)$ and $g(S)$ are nondecreasing submodular and nonincreasing supermodular, respectively.",
"For any $S$ and $T$ with $S \\subseteq T$ , we have that $f(T) &=& \\mathbf {E}\\left\\lbrace \\max {\\left\\lbrace \\max {\\lbrace Z_{i} : i \\in S\\rbrace },\\right.}\\right.",
"\\\\&& \\quad \\left.",
"\\left.",
"\\max {\\lbrace Z_{i} : i \\in T \\setminus S\\rbrace }\\right\\rbrace \\right\\rbrace \\\\&\\ge & \\mathbf {E}\\left\\lbrace \\max {\\lbrace Z_{i} : i \\in S\\rbrace }\\right\\rbrace = f(S),$ implying that $f(S)$ is nondecreasing.",
"The proof of monotonicity for $g(S)$ is similar.",
"To show submodularity of $f(S)$ , consider any sample path $\\omega $ of the random process $X_{k}$ .",
"For any sample path, $Z_{i}$ is a deterministic nonnegative integer.",
"Consider any sets $S,T \\subseteq \\lbrace 1,\\ldots ,N\\rbrace $ , and suppose without loss of generality that $\\max {\\lbrace Z_{i}(\\omega ) : i \\in S\\rbrace } \\ge \\max {\\lbrace Z_{i}(\\omega ) : i \\in T\\rbrace }$ .",
"Hence for any sets $S,T \\subseteq \\lbrace 1,\\ldots ,N\\rbrace $ , we have $\\max _{i \\in S}{Z_{i}(\\omega )} + \\max _{i \\in T}{Z_{i}(\\omega )} &=& \\max _{i \\in S \\cup T}{Z_{i}(\\omega )} + \\max _{i \\in T}{Z_{i}(\\omega )} \\\\&\\ge & \\max _{i \\in S \\cup T}{Z_{i}(\\omega )} + \\max _{i \\in S \\cap T}{Z_{i}(\\omega )}.$ due to submodularity of the max function.",
"Now, let $Q = (i_{1},\\ldots ,i_{N})$ , where each $i_{j} \\in \\lbrace 1,\\ldots ,N\\rbrace $ , be a random variable defined by $Z_{i_{1}} \\le Z_{i_{2}} \\le \\cdots \\le Z_{i_{N}}$ for each sample path, so that $Q$ is the order in which the stopping times $Z_{i}$ are satisfied.",
"Define $\\alpha _{jQ} = \\mathbf {E}(Z_{i_{j}}|Q)$ .",
"For any ordering $(i_{1},\\ldots ,i_{N})$ , we have rCl 3l E{i S{Zi} | Q=(i1,...,iN)} = E{j : ij S{Zij} | Q = (i1,...,iN)} = j : ij SE(Zij|Q) = j: ij SjQ.",
"which is submodular by the same analysis as above.",
"Taking expectation over all the realizations of $Q$ , we have $f(S) &=& \\sum _{(i_{1},\\ldots ,i_{N})}{\\mathbf {E}\\left\\lbrace \\max _{i \\in S}{\\lbrace Z_{i}\\rbrace }|Q=(i_{1},\\ldots ,i_{N})\\right.}",
"\\\\&& \\cdot \\left.Pr(Q=i_{1},\\ldots ,i_{N})\\right\\rbrace \\\\&=& \\sum _{(i_{1},\\ldots ,i_{N})}{\\left[\\left(\\max _{j : i_{j} \\in S}{\\alpha _{jQ}}\\right)Pr(Q=i_{1},\\ldots ,i_{N})\\right]}$ which is a finite nonnegative weighted sum of submodular functions and hence is submodular.",
"The proof of supermodularity of $g(S)$ is similar.",
"Proposition REF is a general result that holds for any random process, including random processes that are non-stationary.",
"In the following sections, we apply these results and derive tighter conditions for hitting, commute, and cover times."
],
[
"Submodularity of Hitting and Commute Times",
"In this section, we consider the problem of selecting a subset of nodes $S$ in order to minimize the hitting time to $S$ , $H(\\pi ,S)$ , as well as selecting a subset of nodes $S$ in order to minimize the commute time $K(\\pi ,S)$ .",
"The following result is a corollary of Proposition REF .",
"Lemma 8 $H(\\pi ,S)$ is supermodular as a function of $S$ .",
"Let $Z_{i}$ denote the stopping time corresponding to the event $\\lbrace X_{k} = i\\rbrace $ , where $X_{k}$ is a random walk on the graph.",
"Then $H(\\pi ,S)$ is supermodular by Proposition REF .",
"The submodularity of $H(\\pi ,S)$ implies that the following greedy algorithm can be used to approximate the solution to $\\min {\\lbrace H(\\pi ,S) : |S| \\le k\\rbrace }$ .",
"In the greedy algorithm, at each iteration the node $v$ that minimizes $H(\\pi , S \\cup \\lbrace v\\rbrace )$ is added to $S$ at each iteration.",
"Letting $S^{\\ast }$ denote the minimizer of $\\lbrace H(\\pi , S) : |S| \\le k\\rbrace $ .",
"Then the set $\\hat{S}$ obtained by the greedy algorithm satisfies $H(\\pi , \\hat{S}) \\le \\left(1-\\frac{1}{e}\\right)H(\\pi ,S^{\\ast }) + \\frac{1}{e}\\max _{v}{H(\\pi ,\\lbrace v\\rbrace )}.$ An analogous lemma for the commute time is as follows.",
"Lemma 9 For any distribution $u$ , the function $K(u,S)$ is supermodular as a function of $S$ .",
"Let $Z_{i}$ denote the stopping time corresponding to the event $\\lbrace X_{k} = u, X_{l} = i \\mbox{ for some } l < k\\rbrace $ .",
"Then $K(S,u) = \\mathbf {E}(\\min _{i \\in S}{Z_{i}})$ , and hence $K(u,S)$ is supermodular as a function of $S$ .",
"Lemma REF can be extended to distributions $\\pi $ over the initial state $u$ as $K(\\pi ,S) = \\sum _{u}{K(u,S)\\pi (u)}$ .",
"The function $K(\\pi ,S)$ is then a nonnegative weighted sum of supermodular functions, and hence is supermodular."
],
[
"Submodularity of Cover Time",
"The submodularity of the cover time is shown as follows.",
"Figure: Numerical evaluation of submodular optimization of random walk times.",
"(a) Minimum number of input nodes for minimizing average cost per cycle of a uniformly random MDP.",
"(b) Performance of Algorithm for selecting a given number of input nodes to minimize the average cost per cycle.",
"(c) Comparison of minimum cover time using random, greedy, and submodular optimization algorithms.Proposition 8 The cover time $C(S)$ is nondecreasing and submodular as a function of $S$ .",
"The result can be proved by using Proposition REF with the set of events $\\lbrace Z_{i} : i \\in S\\rbrace $ where the event $Z_{i}$ is given as $Z_{i} = \\lbrace X_{k} = i\\rbrace $ .",
"An alternative proof is as follows.",
"Let $S \\subseteq T \\subseteq V$ , and let $v \\in V \\setminus T$ .",
"The goal is to show that $C(S \\cup \\lbrace v\\rbrace ) - C(S) \\ge C(T \\cup \\lbrace v\\rbrace ) - C(T).$ Let $\\tau _{v}(S)$ denote the event that the walk reaches $v$ before reaching all nodes in the set $S$ , noting that $\\tau _{v}(T) \\subseteq \\tau _{v}(S)$ .",
"Let $Z_{S}$ , $Z_{T}$ , and $Z_{v}$ denote the times when $S$ , $T$ , and $v$ are reached by the walk, respectively.",
"We prove that the cover time is submodular for each sample path of the walk by considering different cases.",
"In the first case, the walk reaches node $v$ before reaching all nodes in $S$ .",
"Then $C(S) = C(S \\cup \\lbrace v\\rbrace )$ and $C(T) = C(T \\cup \\lbrace v\\rbrace )$ , implying that submodularity holds trivially.",
"In the second case, the walk reaches node $v$ after reaching all nodes in $S$ , but before reaching all nodes in $T$ .",
"In this case, $C(S \\cup \\lbrace v\\rbrace ) - C(S) = Z_{v} - Z_{S}$ , while $C(T \\cup \\lbrace v\\rbrace )-C(T) = 0$ .",
"In the last case, the walk reaches $v$ after reaching all nodes in $T$ .",
"In this case, $C(S \\cup \\lbrace v\\rbrace ) - C(S) = Z_{v} - Z_{S} \\ge Z_{v} - Z_{T} = C(T \\cup \\lbrace v\\rbrace ) - C(T),$ implying submodularity.",
"Taking the expectation over all sample paths yields the desired result.",
"The submodularity of cover time implies that the problem of maximizing the cover time can be approximated up to a provable optimality bound.",
"Similarly, we can select a set of nodes to minimize the cover time by $\\min {\\lbrace C(S) - \\psi |S| : S \\subseteq V|\\rbrace }.$ The cover time, however, is itself computationally difficult to approximate.",
"Instead, upper bounds on the cover time can be used.",
"We have the following preliminary result.",
"Proposition 9 ([1], Prop.",
"11.4) For any set of nodes $A$ , define $t_{min}^{A} = \\min _{a,b \\in A, a \\ne b}{H(a,b)}$ .",
"Then the cover time $C(S)$ is bounded by $C(S) \\ge \\max _{A \\subseteq S}{\\left\\lbrace t_{min}^{A}\\left(1+ \\frac{1}{2} + \\cdots + \\frac{1}{|A|-1}\\right)\\right\\rbrace }.$ Define $c(k) = 1 + \\frac{1}{2} + \\cdots + \\frac{1}{|A|-1}$ , and define $\\hat{f}(S)$ by $\\hat{f}(S) = \\max _{A \\subseteq S}{\\left\\lbrace c(|A|)\\min _{\\stackrel{b \\in A}{a \\in V}}{H(a,b)}\\right\\rbrace }.$ The approximation $\\hat{f}(S)$ can be minimized as follows.",
"We first have the following preliminary lemma.",
"Lemma 10 The function $f^{\\prime }(S)$ is equal to $\\hat{f}(S) = \\max _{k=1,\\ldots ,|S|}{\\alpha _{k}c(k)},$ where $\\alpha _{k}$ is the $k$ -th largest value of $H(a,b)$ among $b \\in S$ .",
"Any set $A$ with $|A| = k$ will have the same value of $c(A)$ .",
"Hence it suffices to find, for each $k$ , the value of $A$ that maximizes $t_{min}^{A}$ with $|A| = k$ .",
"That maximizer is given by the $k$ elements of $S$ with the largest values of $\\min _{a \\in V}{H(a,b)}$ , and the corresponding value is $\\alpha _{k}$ .",
"By Lemma REF , in order to select the minimizer of $\\hat{f}(S) - \\lambda |S|$ , the following procedure is sufficient.",
"For each $k$ , select the $k$ elements of $S$ with the smallest value of $\\min {\\lbrace H(a,b) : a \\in V\\rbrace }$ , and compute $\\beta _{k}c(k) - \\lambda k$ , where $\\beta _{k}$ is the $k$ -th smallest value of $\\min _{a \\in V}{H(a,b)}$ over all $b \\in S$ .",
"In addition, we can formulate the following problem of minimizing the probability that the cover time is above a given threshold.",
"The value of $Pr(C(S) > L)$ can be approximated by taking a set of sample paths $\\omega _{1},\\ldots ,\\omega _{N}$ of the walk and ensuring that $C(S ; \\omega _{i}) > L$ in each sample path.",
"This problem can be formulated as $Pr(C(S) > L) \\approx \\frac{1}{N}\\sum _{i=1}^{N}{\\mathbf {1}\\lbrace C(S ; \\omega _{i}) > L\\rbrace }.$ The function $\\mathbf {1}\\lbrace C(S ; \\omega _{i}) > L\\rbrace $ is increasing and submodular, since it is equal to 1 if there is a node in $S$ that is not reached during the first $L$ steps of the walk and 0 otherwise.",
"Hence the problem of minimizing the probability that the cover time exceeds a given threshold can be formulated as $\\min {\\left\\lbrace Pr(C(S) > L\\ - \\psi |S| : S \\subseteq V\\right\\rbrace }$ and solved in polynomial time."
],
[
"Numerical Results",
"We evaluated our approach through numerical study using Matlab.",
"We simulated both the fixed and optimal distribution cases.",
"In the case of fixed distribution, our goal was to determine how the minimum cover time varied as a function of the number of input nodes and the network size.",
"We generated an Erdos-Renyi random graph $G(N,p)$ , in which there is an edge $(i,j)$ from node $i$ to node $j$ with probability $p$ , independent of all other edges, and the total number of nodes is $N$ .",
"The value of $N$ varied from $N=10$ to $N=50$ .",
"In the MDP case, we simulated the average cost per cycle (ACPC) problem.",
"We first considered a randomly generated MDP, in which each state $i$ had four actions and the probability distribution $P(i,a,\\cdot )$ was generated uniformly at random.",
"We considered the problem of selecting a minimum-size set of states $S$ in order to satisfy a given bound on ACPC.",
"The results are shown in Figure REF (a).",
"We found that the submodular approach outperformed a random selection heuristic even for the relatively homogeneous randomly generated MDPs.",
"We also found that the number of states required to achieve a given bound on ACPC satisfied a diminishing returns property consistent with the submodular structure of ACPC.",
"We then considered a lattice graph.",
"The set of actions corresponded to moving left, right, up, or down.",
"For each action, the walker was assumed to move in the desired direction with probability $p_{c}$ , and to move in a uniformly randomly chosen direction otherwise.",
"If an “invalid” action was chosen, such as moving up from the top-most position in the lattice, then a feasible step was chosen uniformly at random.",
"Figure REF (b) shows a comparison between three algorithms.",
"The first algorithm selects a random set of $k$ nodes as inputs.",
"The second algorithm selects input nodes via a centrality-based heuristic, in which the most centrally located nodes are chosen as inputs.",
"The third algorithm is the proposed submodular approach (Algorithm REF ).",
"We found that the submodular approach slightly outperformed the centrality-based method while significantly improving on random selection.",
"Figure REF (c) compares the optimal selection algorithm with a greedy heuristic and random selection of inputs.",
"We found that the greedy algorithm closely approximates the optimum at a lower computation cost, while both outperformed the random input selection."
],
[
"Conclusion",
"This paper studied the time required for a random walk to reach one or more nodes in a target set.",
"We demonstrated that the problem of selecting a set of nodes in order to minimize random walk times including commute, cover, and hitting times has an inherent submodular structure that enables development of efficient approximation algorithms, as well as optimal solutions for some special cases as stated below.",
"We considered two cases, namely, walks with fixed distribution, as well as walks in which the distribution is jointly optimized with the target set by selecting a control policy in order to minimize the walk times.",
"In the first case, we developed a unifying framework for proving submodularity of the walk times, based on proving submodularity of selecting a subset of stopping times, and derived submodularity of commute, cover, and hitting time as special cases.",
"As a consequence, we showed that a set of nodes that minimizes the cover time can be selected using only polynomial number of evaluations of the cover time, and further derived solution algorithms for bounds on the cover time.",
"In the case where the distribution and target set are jointly optimized, we investigated the problems of maximizing the probability of reaching the target set, minimizing the average cost per cycle of the walk, as well as joint optimization of these metrics.",
"We proved that the former problem admits a relaxation that can be solved in polynomial time, while the latter problem can be approximated via submodular optimization methods.",
"In particular, the average cost per cycle can be minimized by minimizing the volume of an associated linear polytope, which we proved to be a supermodular function of the input set.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTIONAutomatica and IEEE Transactions on Control of Network Systems and Series Editor for the Springer series Advanced Textbooks in Control and Signal Processing.",
"For the IEEE Control Systems Society (CSS), she is currently a Distinguished Lecturer, Chair of the Women in Control Standing Committee, and Liaison to IEEE Women in Engineering (WIE).",
"She is also the Treasurer of the American Automatic Control Council (AACC) and a Member of the International Federation of Automatic Control (IFAC) Technical Board.",
"[Figure: NO_CAPTION"
]
] | 1709.01633 |
[
[
"Weyl nodal surfaces"
],
[
"Abstract We consider three-dimensional fermionic band theories that exhibit Weyl nodal surfaces defined as two-band degeneracies that form closed surfaces in the Brillouin zone.",
"We demonstrate that topology ensures robustness of these objects under small perturbations of a Hamiltonian.",
"This topological robustness is illustrated in several four-band models that exhibit nodal surfaces protected by unitary or anti-unitary symmetries.",
"Surface states and Nielsen-Ninomiya doubling of nodal surfaces are also investigated."
],
[
"Introduction",
"The advent of topological insulators in the last decade deepened our understanding of interplay of topology and symmetries in band insulators [1], [2].",
"This work culminated in the development of the ten-fold way classification of non-interacting gapped topological phases [3] and the emergence of new symmetry protected topological phases of matter.",
"In last years the main interest in the field shifted towards systems with band degeneracies [4], [5], [6].",
"In three dimensions the simplest and most well-studied are Weyl (semi)metals which are distinguished by isolated point-like two-band degeneracies in the Brillouin zone (BZ).",
"Although in condensed matter physics Weyl points appeared first long time ago in the superfluid A phase of $^3$ He [7], [8], only recently Weyl (semi)metals were discovered experimentally [9], [10].",
"As long as inversion or time-reversal symmetry is broken, Weyl points appear generically.",
"They are topological defects and cannot be gapped out individually but must be destroyed only via pair-wise annihilation [11].",
"Weyl (semi)metals exhibit robust phenomena such as chiral anomaly [12], anomalous Hall effect [13] and zero-energy Fermi arc surface states [14].",
"More recently Weyl loop (semi)metals [15], where two-band degeneracies take place on closed one-dimensional manifolds also attracted considerable attention.",
"In contrast to Weyl points, nodal loops are not generic, but require some symmetry (such as chiral sub-lattice symmetry) to protect them.",
"The defining feature of a Weyl loop is a non-trivial $\\pi $ Berry phase along any closed contour that links with it.",
"In the semimetal regime, any boundary surface, where the loop projects non-trivially, supports drumhead states [15], [16].",
"One can make a step further and consider three-dimensional translation-invariant fermionic systems with Weyl nodal surfaces, where two bands touch each other on two-dimensional surfaces in the BZ.",
"Recently nodal surfaces were predicted to appear in quasi-one-dimensional crystals [17], graphene networks [18], multi-band superconductors with broken time-reversal symmetry [19], [20] and also were found within the ten-fold way classification of gapless inversion-enriched systems [21].",
"Given a model with a nodal surface it is natural to ask if the nodal structure is robust under certain class of small perturbations of the Hamiltonian.",
"Generically a perturbation can (i) open a gap everywhere and fully destroy the nodal object (ii) gap it out partially leaving behind nodal loops and/or points One can imagine a scenario where a closed nodal surface is partially gapped to a nodal surface with boundaries.",
"This case is not discussed in this paper., (iii) preserve the nodal surface and not open a gap anywhere on it.",
"As we show in this paper the degree of robustness (i)-(iii) is determined by topology of the perturbed system.",
"To see how it works, consider first a system which has only one nodal surface (tuned for convenience to the Fermi level), but no other gapless objects at the Fermi level such as additional Fermi surfaces, nodal loops or points.",
"Now we enclose the nodal surface of the original model by a lower-dimensional ($d_\\text{m}<3$ ) manifold in the BZ (see Fig.",
"REF ).",
"Figure: Examples of (a) zero-, (b) one- and (c) two-dimensional enclosing manifolds (blue) around nodal surfaces (orange).",
"Although technically a zero-dimensional manifold does not enclose the nodal surface with slight abuse of notation we still refer to it here as an enclosing manifold.By construction, all bands are gapped on the enclosing manifold.",
"Imagine now that in the original model we can define a topological invariant (such as Chern number, winding number, $\\mathbb {Z}_2$ invariant, etc) on this enclosing manifold.",
"In this paper such invariant will be denoted as $c_{d_{\\text{m}}}$ , where the subscript specifies the dimension of the enclosing manifold.",
"Importantly, the invariant does not change under a certain class of perturbation of the Hamiltonian and as we show now this has implications for the degree of robustness of the nodal surface under this class of perturbations.",
"Previously, this set of ideas was introduced for determining the robustness of Weyl nodal loops [23], [24].",
"First, we discuss topological invariants defined on a point ($d_\\text{m}=0$ , Fig.",
"REF a) enclosing manifold.",
"As long as such invariants are different for point manifolds placed inside and outside of the nodal surface in the BZ, the nodal surface cannot be gapped.",
"The reason for that is the following: notice that on any imaginary one-dimensional trajectory in the BZ that connects the inner and outer zero-dimensional manifolds there should be a point, where the energy gap closes allowing the topological invariant to change.",
"In turn this ensures that no gap can open at any point of the nodal surface.",
"Second, consider non-trivial topological invariants defined on enclosing manifolds of dimension $d_\\text{m}>0$ (Fig.",
"REF b and c).",
"These invariants do not fully protect the nodal surface, but guarantee that it cannot be fully gapped out by perturbations.",
"Generically, a nodal line (nodal points) should survive in the perturbed system if $d_\\text{m}=1$ ($d_\\text{m}=2$ ).",
"We can explain it by following similar arguments as before: in the unperturbed model compute the topological invariant on two enclosing manifolds, one being inside and another outside the nodal surface.",
"Since the topological invariant on the interior manifold is necessarily trivial (the manifold can be shrunk to a point without encountering a band), a non-trivial invariant on the exterior manifold necessarily implies a non-trivial difference of the interior and exterior invariants which in turn guarantees a lower-dimensional nodal object in between the two enclosing manifolds in the perturbed system.",
"We emphasize that arguments presented above also imply robustness of general nodal surfaces, not necessarily tuned to the Fermi level.",
"In a general case it is useful to formulate the above arguments in energy-momentum space, as detailed in Appendix .",
"Our analysis thus does not reduce to previous studies of robustness of Fermi surfaces [8], [25], [26], [27].",
"To apply these ideas in practice, in this paper we investigate nodal surfaces protected by different mechanisms: (i) a nodal surface can be protected by a global internal symmetry.",
"In this case any two bands that carry different quantum numbers with respect to the symmetry generically intersect on a two-dimensional surface in momentum space.",
"Since in the presence of the symmetry the two bands cannot be hybridized, the nodal surface is protected against any small symmetry-preserving perturbations.",
"(ii) in spirit of ten-fold way, Weyl nodal surfaces can be protected by anti-unitary symmetries.",
"These nodal surfaces have already appeared in the literature [19], [21], [20] and here we investigate their robustness.",
"We analyze simple models that exhibit nodal surfaces protected by both mechanisms mentioned above.",
"The leitmotif of our construction is the following: we start from double-degenerate nodal points or nodal lines and split them in energy.",
"The resulting continuum four-band models together with minimal lattice extensions thereof are used to analyze the physics of nodal surfaces and investigate their robustness.",
"In particular, we identify symmetries that protect the nodal surface and construct appropriate topological invariants.",
"In addition, we investigate Nielsen-Ninomiya doubling of nodal surfaces in the BZ and look for surface states and topological invariants that protect these states."
],
[
"Nodal surfaces protected by unitary $U(1)$ symmetry",
"In this section we construct and investigate non-interacting fermionic four-band models that exhibit nodal surfaces protected by an internal unitary $U(1)$ symmetry.",
"In examples discussed here this symmetry is generated by the axial $\\gamma ^5$ matrix.",
"More physical realization of a global internal $U(1)$ symmetry might be the conservation of a single component of the spin operator which can happen in a constant magnetic field if the spin-orbit coupling is weak."
],
[
"Nodal sphere",
"Given the set of four-by-four Dirac matrices $\\gamma ^\\mu $ that satisfy the Clifford algebra $\\lbrace \\gamma ^\\mu , \\gamma ^{\\nu } \\rbrace =2 \\eta ^{\\mu \\nu }$ with the Minkowski metric $\\eta ^{\\mu \\nu }$ and $\\mu , \\nu =0,x,y,z$ , we first define $\\alpha ^i=\\gamma ^0 \\gamma ^i$ and $\\gamma ^5=\\mathrm {i}\\gamma ^0 \\gamma ^x \\gamma ^y \\gamma ^z$ .",
"Now consider the Hamiltonian, which describes a massless Dirac fermion perturbed by a $\\gamma ^5$ term $\\mathcal {H}(\\mathbf {k})=k_i \\alpha ^i- \\lambda \\gamma ^5,$ where $k_i$ is the crystal momentum and $\\lambda \\in \\mathbb {R}$ .",
"Since $[\\gamma ^5, \\mathcal {H}]=0$ , the model has an internal $U(1)$ symmetry In high energy physics this is known as the axial symmetry.",
"generated by the matrix $\\gamma ^5$ .",
"In this paper we use the chiral representation of the Dirac matrices resulting in $\\alpha ^i=\\sigma ^{z}\\otimes \\tau ^{i}$ and $\\gamma ^{5}=-\\sigma ^{z} \\otimes \\tau ^{0}$ .",
"The last term in mod1 preserves time-reversal, but breaks the inversion symmetry.",
"It splits the Dirac point at $\\mathbf {k}=0$ into a pair of Weyl points of opposite chirality by separating them in energy by $2\\lambda $ .",
"This model is in DIII symmetry class of the ten-fold way classification [3].",
"It appeared in the context of studies of the chiral magnetic effect [29].",
"The energy spectrum $E(\\mathbf {k})=\\pm |\\mathbf {k}|\\pm \\lambda $ exhibits a band degeneracy at the Fermi level $E=0$ on a sphere defined by $|\\mathbf {k}|=|\\lambda |$ .",
"The nodal surface is protected by the $\\gamma ^{5}$ symmetry against perturbations since two bands that cross each other have different $\\gamma ^{5}$ eigenvalues and cannot be hybridized.",
"Under a generic $\\gamma ^5$ -symmetric perturbation the sphere will deform and move away from $E=0$ , but will not be gapped out.",
"On the other hand, by adding a mass term $\\sim \\gamma ^0= \\sigma ^x\\otimes \\tau ^0$ to the Hamiltonian, given by mod1, the symmetry is broken and the nodal sphere disappears.",
"This behavior can be understood using topology, as we will explain in the following.",
"On a zero-dimensional ($d_m=0$ ) enclosing manifold (Fig.",
"REF a) an integer-valued topological invariant tied to the $\\gamma ^5$ symmetry can be defined as the $\\gamma ^5$ quantum number of the lower band that generates the nodal sphere (in our model this is the occupied band with the highest energy).",
"Given a normalized Bloch state $|u(\\mathbf {k)}\\rangle $ of this band, we define $c_0(\\mathbf {k})=\\langle u(\\mathbf {k})| \\gamma ^5 | u(\\mathbf {k}) \\rangle .$ If this band is degenerate in energy, the $\\gamma ^5$ quantum numbers of individual bands should be summed, thus $c_{0}\\in \\mathbb {Z}$ .",
"In the presence of a nodal surface one can define a topological invariant $\\Delta c_0=\\frac{1}{2}[c_{0}(\\mathbf {k}_{\\text{in}})-c_{0}(\\mathbf {k}_{\\text{out}})]$ , where the momentum $\\mathbf {k}_{\\text{in(out)}}$ is located anywhere inside (outside) the nodal surface.",
"Since the occupied band with the highest energy has opposite $\\gamma ^5$ quantum numbers outside and inside the nodal surface, the difference $\\Delta c_0$ is non-trivial for the model given by mod1.",
"As discussed in Sec.",
", if we think of a general $\\gamma ^5$ -symmetric system with a nodal sphere, we cannot gap out a nodal sphere with an infinitesimal $\\gamma ^5$ -symmetry preserving term, if $\\Delta c_0\\in \\mathbb {Z} \\setminus \\lbrace 0\\rbrace $ .",
"Consequently we conclude that $\\Delta c_0$ is a $\\mathbb {Z}$ -valued topological charge.",
"In addition, an integer-valued topological invariant can be defined on a two-dimensional ($d_m=2$ ) manifold enclosing the nodal sphere (Fig.",
"REF c).",
"This is the Chern number of the lower band that generates the nodal sphere (the occupied band with the highest energy) $c_2(S)= \\frac{\\mathrm {i}}{2\\pi } _{S} {k} \\, \\nabla _{\\mathbf {k}} \\times \\mathcal {A}(\\mathbf {k}),$ where the Berry connection $\\mathcal {A}(\\mathbf {k})=\\langle u (\\mathbf {k})| \\nabla _{\\mathbf {k}}u (\\mathbf {k})\\rangle $ .",
"In the model, given by mod1, this Chern number is non-trivial but does not change as the radius of the enclosing manifolds crosses the nodal sphere, i.e.",
"$c^{\\text{in}}_2=c^{\\text{out}}_2$ .",
"Since the difference of the Chern numbers Chc2 inside and outside the nodal sphere is zero, we conclude that the nodal surface in this model has no robustness with respect to a generic perturbation that breaks $\\gamma ^5$ symmetry.",
"Such perturbation, as for example the mass term $\\sim \\gamma ^0= \\sigma ^x\\otimes \\tau ^0$ , fully gaps out the nodal surface.",
"We now investigate the surface states of the model given by mod1.",
"Imagine that the system has a boundary at $z=0$ and fills only a half of the space $z>0$ .",
"Due to $\\gamma ^5$ symmetry the Hamiltonian splits into two decoupled Weyl blocks.",
"Consequently, one expects two degenerate Fermi arcs starting at the nodal surface that are protected by $\\gamma ^5$ symmetry.",
"This can be demonstrated analytically by first introducing the following boundary condition $\\sigma ^0 \\otimes \\tau ^y \\psi =\\psi $ that preserves helicity of excitations.",
"In the spirit of [30], we extend the boundary condition also into the bulk.",
"Solving the Schrödinger equation $\\mathcal {H}\\psi _{\\mp }=E_{\\mp }\\psi _{\\mp }$ , where $\\mathcal {H}$ is given by mod1 gives the surface states $ \\psi _{-}=(0,\\phi _{-} )^T$ and $\\psi _{+}=( \\phi _+,0)^T$ with $\\begin{split}&\\phi _{\\mp }=\\exp (\\mathrm {i}k_{x}x+\\mathrm {i}k_{y}y-\\kappa z)\\phi ^0,\\end{split}$ and $E_{\\mp }=\\mp (k_{y}+\\lambda ),$ where $\\phi ^0$ is a constant spinor that satisfies $ \\tau ^y \\phi ^0=\\phi ^0$ and $\\kappa =-k_{x}$ .",
"For $\\kappa >0$ these solutions are normalizable, and localized close to the boundary.",
"If we set $E=0$ , we find a doubly-degenerate Fermi arc at the surface BZ at $(k_x<0, k_y=-\\lambda )$ .",
"It starts at ($k_{x}=0$ ), i.e.",
"the boundary of the projection of the nodal surface on the surface BZ (see fig: fig2).",
"$\\psi _{+}$ and $\\psi _{-}$ have opposite $\\gamma ^5$ eigenvalues and counter-propagate along the $y$ axis.",
"These counter-propagating surface modes are robust under $\\gamma ^5$ -symmetric perturbations $\\mathcal {H}_{\\text{imp}}$ since $\\langle \\psi _{-}|\\mathcal {H}_{\\text{imp}}|\\psi _{+}\\rangle =0$ .",
"After calculating the surface states, we investigate their topologically stability.",
"We have seen that each sub-block of mod1 produces its own surface state with a particular $\\gamma _{5}$ quantum number.",
"As long as $\\gamma _{5}$ quantum number is conserved, states with different quantum numbers cannot influence each other.",
"Thus we have to check the topological properties of each subsystem with a definite $\\gamma _{5}$ eigenvalue, separately.",
"However, those subsystems are just Weyl Hamiltonians displaced in energy.",
"Therefore, what we are actually interested in is, the topology of Fermi surfaces of the Weyl sub-systems.",
"The appropriate topological invariant for the blocks is $\\gamma ^5$ -Chern number, which is given by $c_2^{(\\ell )}(S)=\\sum _{n\\in \\operatorname{occupied}}\\frac{\\mathrm {i}}{2\\pi } _{S} {k}\\langle \\nabla _{\\mathbf {k}}u_{n,\\ell }(\\mathbf {k})|\\times |\\nabla _{\\mathbf {k}}u_{n,\\ell }(\\mathbf {k})\\rangle \\,$ where $\\ell $ denotes the eigenvalue of $\\gamma ^{5}$ matrix.",
"We emphasize that $c_2^{(\\ell )}$ characterises the stability of the surfaces states under $\\gamma ^{5}$ preserving infinitesimal perturbation, on the other hand the Chern number of the upper filled band Chc2 characterizes the stability of the nodal surface under generic infinitesimal perturbations.",
"For our model, the absolute value of the $\\gamma ^{5}$ -Chern numbers is unity (zero) for enclosing manifolds of radius larger (smaller) than the radius of the nodal sphere.",
"The continuum model mod1 cannot be used in the full BZ since it violates its periodicity.",
"We conclude this section with a discussion of a minimal lattice realization of this model.",
"The lattice Hamiltonian is given by $\\begin{split}\\mathcal {H}(\\mathbf {k})=&-\\Big ([2-\\cos k_y-\\cos k_z]+2 t [\\cos k_x-\\cos k_0] \\Big )\\alpha ^x \\\\&-2 t \\sin k_y \\alpha ^y -2 t \\sin k_z \\alpha ^z- \\lambda \\gamma ^5,\\end{split}$ where $t\\in \\mathbb {R}$ .",
"The model originates from the minimal lattice model of Weyl semimetals introduced in [13] and discussed extensively in [31].",
"As sketched in Fig.",
"REF , the lattice model has two pairs of Weyl points located at $\\mathbf {k}=(\\pm k_0, 0, 0)$ .",
"Due to the $U(1)$ symmetry only Weyl points of the same $\\gamma ^5$ quantum number can be connected to each other.",
"Hence the Nielsen-Ninomiya theorem must be applied in the two $\\gamma ^5$ sectors separately and nodal spheres appear necessarily in pairs.",
"Figure: Schematic spectrum of the lattice model mod1latt for t,λ>0t, \\, \\lambda >0: Red and blue bands have γ 5 \\gamma ^5 quantum number +1+1 and -1-1 and cannot be hybridized and connect to each other.Since there are two nodal surfaces in the minimal lattice realization, it is natural to expect that the Fermi arcs that we found above will connect the two nodal surfaces projected on the boundary BZ (see Fig.",
"REF a).",
"This is indeed what one finds by diagonalizing numerically the lattice Hamiltonian in a slab geometry (see Fig.",
"REF b).",
"The Fermi arcs are protected by the $\\gamma ^5$ symmetry since by adding a mass term $\\sim \\sigma ^x\\otimes \\tau ^0$ to the lattice Hamiltonian mod1latt, the Fermi arcs hybridize and gap out.",
"Figure: Fermi arcs in a slab geometry: (a)A degenerate pair of counter-propagating surface Fermi arcs (cyan) on the upper boundary connecting the projections of nodal Fermi surfaces.",
"For simplicity the Fermi arcs on the lower boundary are not displayed; (b) Fermi arcs (red) in the energy spectrum of the Hamiltonian mod1latt.",
"The counter-propagating surface states has opposite γ 5 \\gamma ^5 quantum numbers and cannot be hybridized by symmetry-preserving perturbations."
],
[
"Nodal torus",
"We start from a four-band model with the effective low-energy Hamiltonian $\\mathcal {H}(\\mathbf {k})=(k_{\\perp }-k_0) \\alpha ^x+ k_z \\alpha ^y - \\lambda \\gamma ^5,$ with $k_{\\perp }=\\sqrt{k_{x} ^{2}+k_{y} ^{2}}$ .",
"This model has the $\\gamma ^5$ symmetry.",
"The last term in mod2 splits a doubly-degenerate (Dirac) loop of radius $k_0$ at $k_z=0$ into a pair of Weyl loops by separating them in energy by an amount of $2\\lambda $ .",
"The dispersion relation for this system is given by $E(\\mathbf {k})=\\pm \\sqrt{(k_{\\perp }-k_{0})^{2}+k_{z} ^{2}}\\pm \\lambda ,$ and exhibits a band degeneracy at $E=0$ with $\\sqrt{(k_{\\perp }-k_{0})^{2}+k_{z} ^{2}}=\\lambda $ which for $\\lambda <k_0$ forms a torus in momentum space.",
"In a close analogy to the nodal sphere discussed in Sec.",
"REF , the nodal torus is protected by the unitary $\\gamma ^5$ symmetry.",
"The appropriate $\\mathbb {Z}$ -valued topological invariant defined on a zero-dimensional enclosing manifold is given by cof.",
"For the model mod2 the difference of the invariants defined inside and outside the nodal torus is non-trivial which ensures its full robustness under infinitesimal $\\gamma ^5$ -preserving perturbations.",
"In addition to the $\\gamma ^5$ symmetry, the model, given by mod2, is also invariant under the combination of inversion $P$ and time reversal $T$ , i.e.",
"$PT$.",
"In addition to the $PT$ symmetry, the nodal tori models mod2 and mod4 are also invariant a unitary mirror ($z\\rightarrow -z$ ) symmetry, but we do not discuss its consequences in this paper., which implies the following constraint on the Hamiltonian $U_{PT} \\mathcal {H}^{*}(\\mathbf {k})U_{PT}^{-1}=\\mathcal {H}(\\mathbf {k}),$ where $U_{PT}=\\alpha ^x$ .",
"Analogously to the previous section, we can treat the diagonal sub-blocks of this system separately.",
"The sub-blocks give rise nodal loops lying at energies $\\pm \\lambda $ , which are protected by the $PT$ symmetry.",
"The topological invariant that characterizes the stability of such nodal loops is the Berry phase which is quantized in $PT$ -symmetric systemsIf we break $PT$ symmetry, the block Berry phase can be changed continuously.[34].",
"In our model mod2 we find $\\pi $ Berry phase for the blocks.",
"In addition, the Berry curvature is forced to be zero for every non-singular point in the BZ.",
"As a result, $\\pi $ Berry phase guarantees the existence of nodal loops that are stable under infinitesimal perturbations that preserve $PT$ and $\\gamma ^5$ symmetries.",
"On the other hand, the $PT$ symmetry is not required for the stability of the nodal torus.",
"In fact, if we break this symmetry without breaking $\\gamma ^5$ symmetry, the nodal torus will not be destroyed, but the nodal loops lying at energies $\\pm \\lambda $ will be gapped.",
"Hence we can refer to $PT$ symmetry as an accidental symmetry.",
"In order to illustrate the arguments given above in a concrete example, we consider a minimal lattice model that hosts a nodal torus.",
"Its Hamiltonian is given by $\\begin{split}\\mathcal {H}(\\mathbf {k})=&-\\Big (6t_1-2 t_2 [\\cos k_x+ \\cos k_y+ \\cos k_z] \\Big )\\alpha ^x \\\\&-2 t_2 \\sin k_z \\alpha ^y- \\lambda \\gamma ^5,\\end{split}$ where $t_{1,2}$ are constants which fix the radius of the torus.",
"This Hamiltonian is motivated by the lattice model of a nodal loop semimetal investigated in [35].",
"It is worth pointing out that this lattice model has just one nodal torus in the BZ (no Nielsen-Ninomiya doubling).",
"The bulk energy spectrum of this Hamiltonian is shown in fig: nodalloops-b.",
"Furthermore, in a slab geometry (as a result of the existence of the nodal loops in the bulk energy spectrum) we find drumhead surface states that have the same energy as the nodal loops as shown in fig: nodalloops-a.",
"A $\\gamma ^5$ -invariant perturbation that breaks the $PT$ symmetry destroys the nodal loops, but not the nodal torus (see fig: nodalloops-c,d).",
"Figure: Energy spectrum of the model () in the (a) slab and (b) infinite geometry.",
"The drumhead surface states in (a) lie at the energy levels of the nodal loops.",
"In (b) the crossings of the red and black bands at the Fermi level form a nodal torus.",
"The panels (c) and (d) display the energy spectrum of the model () perturbed by a term that breaks accidental PTPT symmetry, but preserves γ 5 \\gamma ^5 symmetry."
],
[
"Nodal surfaces protected by anti-unitary symmetry",
"Here we turn to four-band models that exhibit nodal surfaces which are protected by anti-unitary symmetries.",
"These objects appeared in the recent literature [19], [21], [20] and formed the starting point of our investigation."
],
[
"Nodal sphere",
"Consider a four-band model with the Hamiltonian $\\mathcal {H}(\\mathbf {k})= k_i \\tilde{\\alpha }^i- \\lambda \\gamma ^5-\\delta \\gamma ^0,$ where $\\tilde{\\alpha }^i=\\sigma ^0\\otimes \\tau ^i$ , $\\gamma ^5=-\\sigma ^z\\otimes \\tau ^0$ and $\\gamma ^0=\\sigma ^x\\otimes \\tau ^0$ .",
"The model consists of a pair of Weyl points of the same chirality which are split in energy by the last two terms in infW.",
"Notice that the Hamiltonian has no unitary $U(1)$ symmetry apart from the particle number conservation.",
"Even in the absence of a unitary symmetry, the spectrum of the Hamiltonian infW $E(\\mathbf {k})=\\pm |\\mathbf {k}|\\pm \\sqrt{\\lambda ^2+\\delta ^2}$ contains a band degeneracy at $E=0$ which is located on a sphere of radius $|\\mathbf {k}|=\\sqrt{\\lambda ^2+\\delta ^2}$ .",
"We will argue now that in this model the nodal surface is protected by an anti-unitary combination of inversion $P$ and particle-hole $C$ symmetry [36], [37] and construct a $\\mathbb {Z}_2$ -valued topological invariant tied to this symmetry [19], [21].",
"To this end consider a class of Hamiltonians of the form $\\mathcal {H}(\\mathbf {k})=a_i (\\mathbf {k}) \\tilde{\\alpha }^i+ b_i (\\mathbf {k})\\tilde{\\beta }^i$ with $\\tilde{\\alpha }^i=\\sigma ^0\\otimes \\tau ^i$ and $\\tilde{\\beta }^i=\\sigma ^i\\otimes \\tau ^0$ and $\\mathbf {a}$ and $\\mathbf {b}$ being real functions of $\\mathbf {k}$ .",
"The model infW is a special case of PCH since $\\gamma ^0 =\\tilde{\\beta }^x$ and $\\gamma ^5=-\\tilde{\\beta }^z$ .",
"It will turn out important in the following that the Hamiltonian PCH can be unitary rotated to an antisymmetric form [19], [21] to be denoted by $\\bar{\\mathcal {H}}$ .",
"The spectrum of PCH $E(\\mathbf {k})=\\pm |\\mathbf {a}(\\mathbf {k})|\\pm |\\mathbf {b}(\\mathbf {k})|$ supports a zero-energy nodal surface at $|\\mathbf {a}(\\mathbf {k})|=|\\mathbf {b}(\\mathbf {k})|$ .",
"A generic model PCH breaks inversion $P$ , time-reversal $T$ and particle-hole $C$ symmetries.",
"Nevertheless, it is invariant under the anti-unitary $PC$ symmetry which acts on the Hamiltonian as $U_{PC} \\mathcal {H}^{*}(\\mathbf {k})U_{PC}^{-1}=-\\mathcal {H}(\\mathbf {k}).$ In our representation $U_{PC}=\\sigma ^y\\otimes \\tau ^y$ is real.",
"Note that $(PC)^2=U_{PC}U_{PC}^*=+\\mathbb {1}$ .",
"Due to this symmetry the nodal surface is fixed to the Fermi level and thus necessarily coincides with the Fermi surfaces of the two touching bands.",
"It was found in [36], [37] that three-dimensional Fermi surfaces with $PC$ symmetry that squares to unity have topological $\\mathbb {Z}_2$ classification.",
"In our formulation, the topological invariant defined on a zero-dimensional enclosing manifold can be calculated as the sign of the Pfaffian of the antisymmetric form of the Hamiltonian $\\bar{\\mathcal {H}}$ $c_0(\\mathbf {k})=\\text{sgn}\\text{Pf} \\, \\bar{\\mathcal {H}}(\\mathbf {k}).$ Since in our representation the Hamiltonian PCH is not antisymmetric, in order to compute the invariant, we must first unitary rotate it to the antisymmetric form $\\bar{ \\mathcal {H}}= \\Omega \\mathcal {H} \\Omega ^\\dagger .$ The resulting Pfaffian $\\text{Pf} \\, \\bar{\\mathcal {H}}=\\mathbf {b}^2(\\mathbf {k})-\\mathbf {a}^2(\\mathbf {k})$ is real and vanishes on the nodal surface.",
"In particular, for the model infW the Pfaffian has a simple zero on the nodal sphere and thus it has opposite signs inside and outside of it.",
"Following the arguments of Sec.",
", the non-trivial difference of the Pfaffians makes the nodal sphere fully robust with respect to small $PC$ -invariant perturbations.",
"We notice that although the choice of $\\Omega $ and the sign of the resulting $c_0(\\mathbf {k})$ is not unique, this ambiguity does not affect the difference of the topological invariants.",
"We discuss now the Chern number invariant Chc2 in the context of the model infW.",
"For an enclosing manifold located inside (outside) the nodal sphere the band Chern number is non-trivial and in addition $c^{\\text{in}}_2=-c^{\\text{out}}_2$ .",
"The non-trivial difference of the band Chern numbers $\\Delta c_2=c^{\\text{in}}_2-c^{\\text{out}}_2$ ensures that a generic small perturbation cannot fully gap out the nodal sphere, but has to leave in the band structure at least a pair of Weyl points close to the Fermi level.",
"Thus contrary to the nodal objects from Sec.",
", the nodal sphere discussed here has certain robustness with respect to arbitrary perturbations of the Hamiltonian.",
"In the presence of a spatial boundary, for $\\lambda =\\delta =0$ in infW a pair of same-chirality Weyl points at the Fermi level gives rise to a pair of chiral co-propagating zero-energy Fermi arcs.",
"As the Weyl points are split in energy for $\\lambda , \\, \\delta \\ne 0$ , the Fermi arcs survive but must start at the Fermi surfaces.",
"These surface states are robust against arbitrary small perturbations thanks to the the total Chern number $\\text{total Chern number}=\\frac{i}{2\\pi } \\int _{S^2} d \\mathbf {k} \\, \\text{tr} \\mathcal {F}$ which is non-trivial on an enclosing manifold outside the Fermi surfaces.",
"Here the Berry curvature two-form $\\mathcal {F}=d \\mathcal {A}+\\mathcal {A}\\wedge \\mathcal {A}$ is defined in terms of the non-abelian Berry connection $\\mathcal {A}^{ab}=\\langle u^a(\\mathbf {k})| d u^b(\\mathbf {k}) \\rangle $ , where $a,b$ label only occupied bands.",
"On an enclosing manifold living inside the nodal sphere the total Chern number Chern vanishes and thus the nodal sphere represents a locus of the total Berry flux.",
"It is an inflated double Weyl monopole [19].",
"Due to the Nielsen-Ninomiya theorem the nodal spheres protected by the anti-unitary $PC$ symmetry should always appear in pairs.",
"To demonstrate it explicitly we considered the minimal lattice Hamiltonian $\\begin{split}\\mathcal {H}(\\mathbf {k})=&-\\Big ([2-\\cos k_y-\\cos k_z]+2 t [\\cos k_x-\\cos k_0] \\Big )\\tilde{\\alpha }^x \\\\&-2 t \\sin k_y \\tilde{\\alpha }^y -2 t \\sin k_z \\tilde{\\alpha }^z-\\delta \\gamma ^0- \\lambda \\gamma ^5.\\end{split}$ and determined numerically its energy spectrum in the bulk.",
"In addition, we examined the energy spectrum of this lattice Hamiltonian in a slab geometry.",
"As expected, it contains a pair of chiral zero-energy Fermi arcs which are robust with respect to arbitrary perturbations of the Hamiltonian."
],
[
"Nodal torus",
"Finally we construct and analyze a four-band model that exhibits a nodal torus that is protected by an anti-unitary symmetry.",
"The model is defined by the real Hamiltonian $\\mathcal {H}(\\mathbf {k})=(k_{\\perp }-k_0) \\tilde{\\alpha }^x+ k_z \\tilde{\\alpha }^z - \\lambda \\gamma ^5-\\delta \\gamma ^0$ with $k_{\\perp }=\\sqrt{k_{x} ^{2}+k_{y} ^{2}}$ .",
"Apart from the particle number symmetry, there is no $U(1)$ symmetry.",
"Nevertheless the energy spectrum $E(\\mathbf {k})=\\pm \\sqrt{(k_{\\perp }-k_{0})^{2}+k_{z} ^{2}}\\pm \\sqrt{\\lambda ^2+\\delta ^2}$ has a zero-energy band degeneracy at $\\sqrt{(k_{\\perp }-k_{0})^{2}+k_{z} ^{2}}=\\sqrt{\\lambda ^2+\\delta ^2}$ forming a torus in momentum space for $\\sqrt{\\lambda ^2+\\delta ^2}<k_0$ .",
"One can view the nodal torus as an inflated double Weyl loop [19].",
"Since the model mod4 falls into the class of $PC$ -symmetric Hamiltonians PCH, similar to the nodal sphere discussed in section REF , one can define the Pfaffian $\\mathbb {Z}_2$ invariant pfc.",
"By evaluating the Pfaffian Pfeq in the model mod4, one finds that the difference of the Pfaffian invariant inside and outside the nodal torus is non-trivial and thus this object cannot be gapped out by small $PC$ -invariant perturbations of the Hamiltonian.",
"Incidentally, the model mod4 enjoys more symmetries.",
"Similar to the nodal torus model discussed in ntu, this system has the anti-unitary $PT$ symmetry.",
"In this case the Hamiltonian is real and $PT$ acts on the Hamiltonian as $\\mathcal {H}^{*}({\\mathbf {k}})=\\mathcal {H}({\\mathbf {k}}).$ In general, this symmetry implies that the Chern number Chern is zero in this symmetry class.",
"In addition, this symmetry together with the anti-unitary $PC$ symmetry implies the unitary chiral sublattice (anti)symmetry of the model mod4 $\\lbrace \\mathcal {H}(\\mathbf {k}), U_S \\rbrace =0, \\qquad U_S=\\sigma ^y \\otimes \\tau ^y$ and gives rise to a $\\mathbb {Z}_2$ winding number topological invariant introduced in [21].",
"In order to construct this invariant, it is convenient to perform a unitary rotation $\\Omega $ which diagonalizes the chiral sublattice symmetry operator $S\\rightarrow \\Omega ^\\dagger S \\Omega =\\sigma ^z\\otimes \\tau ^0$ .",
"This transformation brings the Hamiltonian into the block off-diagonal form $\\mathcal {H}(\\mathbf {k})=\\left(\\begin{array}{cc}0 & h(\\mathbf {k}) \\\\{h}^\\dagger (\\mathbf {k}) & 0\\end{array}\\right),$ Here due to the $PT$ symmetry the block Hamiltonian $h$ is real.",
"To proceed, it is useful to define a flattened Hamiltonian which for a general non-interacting fermionic system with $n$ filled and $m$ empty bands reads $Q(\\mathbf {k})=U(\\mathbf {k})\\left(\\begin{array}{cc}\\mathbb {1}_{m\\times m} & 0 \\\\0 & -\\mathbb {1}_{n\\times n}\\end{array}\\right)U^{\\dagger }(\\mathbf {k}),$ where the unitary matrix $U(\\mathbf {k})$ diagonalizes the Hamiltonian $\\begin{split}&U^{\\dagger }(\\mathbf {k}) \\mathcal {H}(\\mathbf {k}) U(\\mathbf {k})= \\\\&=\\text{diag}(\\underbrace{\\epsilon _{m+n}(\\mathbf {k}), \\dots , \\epsilon _{n+1}(\\mathbf {k})}_{\\text{empty bands}}, \\underbrace{\\epsilon _{n}(\\mathbf {k}), \\dots , \\epsilon _{1}(\\mathbf {k})}_{\\text{filled bands}}).\\end{split}$ The flattened Hamiltonian $Q(\\mathbf {k})$ is well-defined only away from band degeneracies.",
"In the presence of the chiral sublattice symmetry $m=n$ and the flattened Hamiltonian is block off-diagonal $Q(\\mathbf {k})=\\left(\\begin{array}{cc}0 & q(\\mathbf {k}) \\\\q^\\dagger (\\mathbf {k}) & 0\\end{array}\\right)$ in any basis which diagonalizes the operator $S$ .",
"Note that in general $q(\\mathbf {k})\\in \\text{U}(n)$ since $Q^2=\\mathbb {1}$ .",
"The additional reality condition PTsym that follows from the $PT$ symmetry implies that the flattened block $q(\\mathbf {k})\\in \\text{O}(n)$ .",
"In particular, for the four-band model mod4 one has $q(\\mathbf {k})\\in \\text{O}(2)$ .",
"Now on any closed one-dimensional manifold that does not intersect the torus in the BZ one can compute the topological winding number [21] as the homotopy equivalence class of mappings $S^1\\rightarrow SO(2)$ .",
"We define the winding number as $c_1=-\\frac{i}{4\\pi }\\oint ds \\text{Tr} \\, \\big [ q^{T}\\sigma ^y \\partial _s q \\big ],$ where $s$ is a coordinate that parametrizes the one-dimensional enclosing manifold.",
"The invariant is integer-valued since the homotopy group $\\pi _1[SO(2)]=\\mathbb {Z}$ Since $\\pi _1[SO(n)]=\\mathbb {Z}_2$ for $n>2$ , the winding number is only $\\mathbb {Z}_2$ -valued for models with more than four bands [21]..",
"It is instructive now to evaluate the winding number invariant for the nodal torus model mod4.",
"To this end we consider an enclosing $S^1$ manifold of radius $k_S$ that is positioned in the $x-z$ plane and is centered at the momentum $\\mathbf {k}=(k_0, 0, 0)$ .",
"A straightforward calculation reveals that if $k_S>\\sqrt{\\lambda ^2+\\delta ^2}$ , i.e., the manifold encloses the torus from outside and links with it, the absolute value of the winding number $c^{\\text{out}}_1$ is equal to unity.",
"On the other hand for $k_S<\\sqrt{\\lambda ^2+\\delta ^2}$ , the winding number $c^{\\text{in}}_1$ vanishes since in this case the enclosing manifold can be shrunk to a point without crossing energy bands.",
"Following general arguments from Sec.",
", the non-trivial difference of the winding numbers ensures that the nodal torus cannot be fully gapped out by small perturbations that are invariant under $PC$ and $PT$ symmetries.",
"Note however that in the present case this prediction has little practical value because the nodal torus is fully protected against any small $PC$ -symmetric perturbation by the Pfaffian invariant.",
"It would be interesting to find a model protected by the non-trivial difference of a topological invariant defined on one-dimensional enclosing manifolds, where a nodal surface is gapped out to nodal loop(s).",
"We observe that similar to the model discussed in Sec.",
"REF , the present model does not support robust zero-energy surface states.",
"The pair of zero-energy drumhead surface states present in the limit of the double-Weyl loop ($\\lambda =\\delta =0$ ) is shifted to a finite energy by the last two terms in the Hamiltonian mod4.",
"One arrives to the same conclusion by diagonalizing the lattice the Hamiltonian $\\begin{split}\\mathcal {H}(\\mathbf {k})=&-\\Big (6t_1-2 t_2 \\left[ \\cos k_x+ \\cos k_y+ \\cos k_z \\right] \\Big )\\tilde{\\alpha }^x \\\\&-2 t_2 \\sin k_z \\tilde{\\alpha }^z- \\lambda \\gamma ^5-\\delta \\gamma ^0\\end{split}$ in a slab geometry."
],
[
"Conclusion and outlook",
"In this paper we investigated the physics of three-dimensional fermionic band models that exhibit two-dimensional Weyl nodal surfaces.",
"We argued that the robustness of these nodal objects is ensured by topology.",
"Specifically, we showed that the degree of robustness is determined by the dimensionality of gapped enclosing manifolds, where topological invariants are evaluated.",
"We demonstrated this in several four-band toy models that exhibit nodal surfaces protected by unitary or anti-unitary symmetry.",
"It would be interesting to study the effects of interactions and disorder on nodal surfaces, investigate transport properties and find realistic models of materials where these ideas can be applied."
],
[
"Acknowledgments:",
"We acknowledge fruitful discussions with Barry Bradlyn, Anton Burkov, Tomáš Bzdušek, Titus Neupert and Grigori Volovik.",
"Our work is supported by the Emmy Noether Programme of German Research Foundation (DFG) under grant No.",
"MO 3013/1-1."
],
[
"Topological robustness of general nodal surfaces",
"Although in the main part of the paper we limit our attention to nodal surfaces tuned to the Fermi level, this assumption is not necessary and general (dispersing in energy) nodal surfaces exhibit topological robustness to small perturbations in the way we put forward in Sec.",
".",
"To understand the dispersing case, it is useful to consider a four-dimensional energy-momentum space which is naturally partitioned by three-dimensional energy bands into four-dimensional regions.",
"If a topological invariant of the band structure can be defined on a manifold residing in some four-dimensional region, by construction it must be constant within the given region.",
"In general, a non-trivial difference of topological invariants evaluated on enclosing manifolds of dimensions $d_{\\text{m}}=0,1,2$ embedded in different properly chosen regions of the energy-momentum space guarantees a nodal object of dimension $d=2,1,0$ in the presence of any small perturbation compatible with the given topological invariant.",
"For the case $d_{\\text{m}}=0$ , the mechanism is illustrated in Fig.",
"REF .",
"Figure: A two-dimensional cut of the four-dimensional energy-momentum space partitioned by bands (black solid lines) into regions.",
"The nodal surface projects as two black dots on the cut.",
"Nontrivial difference of topological invariants c 0 out c_0^{\\text{out}} and c 0 in c_0^{\\text{in}} defined on any two zero-dimensional point manifolds located within the outer (green) and inner (blue) regions, respectively, protects the nodal surface from being gapped."
]
] | 1709.01561 |
[
[
"Scene Text Recognition with Sliding Convolutional Character Models"
],
[
"Abstract Scene text recognition has attracted great interests from the computer vision and pattern recognition community in recent years.",
"State-of-the-art methods use concolutional neural networks (CNNs), recurrent neural networks with long short-term memory (RNN-LSTM) or the combination of them.",
"In this paper, we investigate the intrinsic characteristics of text recognition, and inspired by human cognition mechanisms in reading texts, we propose a scene text recognition method with character models on convolutional feature map.",
"The method simultaneously detects and recognizes characters by sliding the text line image with character models, which are learned end-to-end on text line images labeled with text transcripts.",
"The character classifier outputs on the sliding windows are normalized and decoded with Connectionist Temporal Classification (CTC) based algorithm.",
"Compared to previous methods, our method has a number of appealing properties: (1) It avoids the difficulty of character segmentation which hinders the performance of segmentation-based recognition methods; (2) The model can be trained simply and efficiently because it avoids gradient vanishing/exploding in training RNN-LSTM based models; (3) It bases on character models trained free of lexicon, and can recognize unknown words.",
"(4) The recognition process is highly parallel and enables fast recognition.",
"Our experiments on several challenging English and Chinese benchmarks, including the IIIT-5K, SVT, ICDAR03/13 and TRW15 datasets, demonstrate that the proposed method yields superior or comparable performance to state-of-the-art methods while the model size is relatively small."
],
[
"Introduction",
"With the development of the Internet and widespread use of mobile devices with digit cameras, there are massive images in the world and many of them contain texts.",
"The text in natural image carries high level semantics and can provide valuable cues about the content of the image.",
"Thus, if texts in these images can be detected and recognized by computers, they can play significant roles for various vision-based applications, such as spam detection, products search, recommendation, intelligent transportation, robot navigation and target geo-location.",
"Consequently, scene text detection and recognition has become a hot research topic in computer vision and pattern recognition in recent years.",
"Figure: The framework of the proposed method.",
"The framework consists of three parts: 1) Sliding window layer, which extract features from the window; 2) Classification layer, which predicts a label distribution from the input window image; 3) Transcription layer, which translates the per-window predictions into the final label sequence.Although traditional Optical Character Recognition (OCR) has been investigated for a few decades and great advances have been made for scanned document images [10], [34], the detection and recognition of text in both natural scene and born-digital images, so called robust reading, remains an open problem [17], [39].",
"Unlike the texts in scanned document images which are well-formatted and captured under a well-controlled environment, texts in scene images are largely variable in appearance and layout, drawn from various color, font and style, suffering from uneven illumination, occlusions, orientations, distortion, noise, low resolution and complex backgrounds (Fig.",
"2).",
"Therefore, scene text recognition remains a big challenge.",
"Figure: Examples of English and Chinese Scene TextMany efforts have been devoted to the difficult problem of scene text recognition.",
"The methods so far can be roughly categorized into three groups: explicit segmentation methods, implicit segmentation methods and holistic methods.",
"(1) Explicit segmentation methods[38], [33], [36], [24], [23], [31], [25], [3], [16], [35] usually involve two steps: character segmentation and word recognition.",
"It attempts to segment the input text image at the character boundaries to generate a sequence of primitive segments with each segments being a character or part of a character, applies a character classifier to candidate characters and combine contextual information to get the recognition result.",
"Although this approach has performed well in handwritten text recognition, the performance in scene text recognition is severely confined by the difficulty of character segmentation.",
"However, explicit segmentation methods have good interpretation since they can locate the position and label of each character.",
"(2) Implicit segmentation methods [11], [2], [32], [12], [29], [30], [28] regard text recognition as a sequence labeling, which avoids the difficult character segmentation problem by simply slicing the text image into frames of equal length and labeling the sliced frames.",
"Hidden Markov Model (HMM) and Recurrent Neural networks (RNNs) are typical examples for this case.",
"In particular, the combination of convolutional neural network (CNN) and RNN based network obtained the state-of-the-art results on several challenging benchmarks.",
"However, RNN-based methods have two demerits: (a) The training burden is heavy when the input sequence is very long or the number of output classes is large; (b) The training process is tricky due to the gradient vanishing/exploding etc.",
"(3) Holistic methods [6], [27], [8], [1], [13], [15], [14], [19] recognize words or text lines as a whole without character modeling.",
"Though this is feasible for English word recognition and has reported superior performance, its reliance on a pre-defined lexicon makes it unable to recognize a novel word, And also, holistic methods are not applicable to the case that fixed lexicon is not possible, e.g., for Chinese text line recognition.",
"Despite the big progress in recent years, the current scene text recognition methods are insufficient in both accuracy and interpretation compared to human reading.",
"Modern cognitive psychology research points out that reading consists of a series of saccades (whereby the eyes jump from one location to another and during which the vision is suppressed so that no new information is acquired) and fixations (during which the eyes remain relatively stable an process the information in the perceptual span) [37], Therefore, if we simplify the perceptual span as a window, the process of reading can be formulated by a sliding window which outputs the meaningful recognition results only when its center is at the fixation point.",
"Based on the above, we propose a simple and efficient scene text recognition method inspired by human cognition mechanisms, in which a sliding window and a character classifier based on deep neural network are used to imitate the mechanisms of saccades and fixations, respectively.",
"Our method has several distinctive advantages: 1) It simultaneously detects and recognizes characters and can be trained on weakly labeled data; 2) It achieves competitive performance on both English and Chinese scene text recognition; 3) The recognition process is highly parallel and enables fast recognition.",
"We evaluate our method on a number of challenging scene text datasets.",
"Experimental results show that our method yields superior or comparable performance compared to the state of the art.",
"The rest of the paper is organized as follow.",
"Section 2 reviews related works.",
"Section 3 details the proposed method.",
"Experimental results are given in Section 4 and conclusion is drawn in Section 5."
],
[
"Related Work",
"In recent years, a large number of scene text recognition systems have been reported in the literature, and some representative methods are reviewed below.",
"In general, explicit segmentation methods consists of character segmentation and word recognition.",
"The recognition performance largely relies on character segmentation.",
"The existing segmentation methods roughly fall in two categories: binarization based and detection based.",
"Binarization based methods find segmentation points after binarization.",
"Niblack's adaptive binarization and Extremal Regions (ERs) are two typical binarization based methods, which are employed in [3] and [25], respectively.",
"However, since text in natural scene image suffers from uneven illumination and complex backgrounds, binarization can hardly give satisfactory results.",
"Detection based methods bypass the binarization by adopting multi-scale sliding window strategy to get candidate characters from the original image directly.",
"For example, the methods in [33], [24], [23] directly extract features from the original image and use various classifiers to decide whether a character exist in the center of a sliding window.",
"Shi et al.",
"[31] employ a part-based tree-structured model and a sliding window classification to localize the characters in the window.",
"Detection based methods overcome the difficulty of character segmentation and have shown good performance.",
"In explicit segmentation based methods, the integration of contextual information with character classification scores is important to improve the recognition performance.",
"The methods of integration include Pictorial Structure models (PS) [33], Bayesian inference [36], and Conditional Random Field (CRF)[24], [23], [31].",
"Wang et al.",
"[33] use the PS to model each single character and the spatial relationship between characters.",
"This algorithm shows good performance on several datasets, but it can only handle words in a pre-defined dictionary.",
"Weinman et al.",
"[36] proposed a probabilistic inference model that integrates similarity, language priors and lexical decision to recognize scene text.",
"Their approach was effective for eliminating unrecoverable recognition errors and improving accuracy.",
"CRF is was employed in [24], [23] to jointly model both bottom-up (character) and top-down (language) cues.",
"Shi et al.",
"[31] built a CRF model on the potential character locations to incorporate the classification scores, spatial constraints, and language priors for word recognition.",
"In [25], word recognition was performed by estimating the maximum a posterior (MAP) under joint distribution of character appearance and language prior.",
"The MAP inference was performed with Weighted Finite-State Transducers (WFST).",
"Moreover, beam search has been used to achieve fast inference [3], [16] for overcoming the complexity with high-order context models (e.g., 8th-order language model in [3]).",
"Implicit segmentation methods take the whole image as the input and are naturally free from the difficulty of character segmentation which severely hinders performance of the explicit segmentation methods.",
"With the use of deep neural network, implicit segmentation methods have shown overwhelming superiority in scene text recognition.",
"The implicit segmentation methods use either hand crafted features [32] or features learned by CNN [12], [29], [30], [28], the labeling algorithm is either HMM [11], [2] or LSTM [32], [12], [29], [30], [28].",
"Recently, the combination of CNN and LSTM has led to state-of-the-art performance [29].",
"In holistic recognition methods, Goel et al.",
"[6] use whole word sub-image features to recognize the word by comparing to simple black-and-white font-renderings of lexicon words.",
"The methods in [27], [8], [1] use word embedding, in which the recognition becomes a nearest neighbor classification by creating a joint embedding space for word images and the text.",
"In [13], [15], Jaderberg et al.",
"develop a powerful convolutional neural network to recognize English text by regarding every English word as a class.",
"Thanks to the strong classification ability of CNN and the availability of large set of training images by synthesis, this method shows impressive performance on several benchmarks.",
"Holistic recognition is confined by a pre-defined lexicon, however, although Jaderberg et al.",
"[14] and Lee et al.",
"[19] propose another CNN based model which can recognize unconstrained words by predicting the character at each position in the output text, it is highly sensitive to the non-character space.",
"The proposed method is an implicit segmentation method.",
"It overcomes the difficulty of character segmentation by sliding window, and the underlying CNN character model can be learned end-to-end with training images weakly labeled with text scripts only."
],
[
"The Proposed Method",
"The framework of the proposed method, as shown in Fig.",
"1, consists of three parts: a sliding window layer, a classification layer and a transcription layer.",
"The sliding window layer extracts features from the window, and the features can be the original image with different scale hand-crafted features or CNN features.",
"On the top of sliding window, a classifier is built to predict a label distribution from the input features.",
"The transcription layer is adopted to translate the per-window predictions into the result sequence.",
"The whole system can be jointly optimized as long as the classifier is differentiable, making the back propagation algorithm workable."
],
[
"Rationale",
"When humans read a text line, their eyes do not move continuously along a line of text, but make short rapid movements intermingled with short stops.",
"During the time that the eye is stopped, new information is brought into the processing, but during the movements, the vision is suppressed so that no new information is acquired [37].",
"Inspired by these, we build our scene text recognition system which follows a simplified process of human reading.",
"We assume it only skips one character in each saccade as a unskilled people, then we use exhaustive scan windows with suitable step to imitate the saccade.",
"When the centre of the scan window coincide with the fixation point, the character classifier outputs character labels and confidence scores, otherwise, it outputs 'blank'.",
"For the window size, we consider the fact that after height normalization, characters usually have approximately similar width in the text image.",
"Therefore, we fix the size of the sliding window to a right size in which a character can be covered completely.",
"Though fixing the window size may bring disturbance to character classification as shown in Fig3, we can see the the character is still recognizable when it is in the center of window for printed character and Chinese handwritten character(e.g., Fig3 (b), Fig3(c) and Fig3(d)).",
"Figure: Some examples of printed scene text and hadwritten text.",
"(a) Handwritten English; (b) Handwritten Chinese; (c) Printed Scene English Text; (d) Printed Scene Chinese Text.Our framework is flexible.",
"If we reduce the number of window to one, extend the size of window to whole image and use the crnn model to recognize the context in the window, our framework degenerates to the model in [29].",
"If we reduce the width of window to one pixel and model the relationship between windows, our framework becomes a RNN based model.",
"Our framework also relates to previous methods in [20], [22], where neural nets were trained on digit string images.",
"However, our method differs in that it can be trained on the weakly labeled datasets, i.e., without the need of locating characters in training images.",
"As we know, labeling locations of characters is laborious and time consuming, which makes training with big data infeasible."
],
[
"Convolutional Character Model",
"Although our text recognition framework use any classifier for character recognition on sliding window, in this work, we use the CNN which has been proven superior in recent results [40], [41].",
"We build a 15-layer CNN as the character model as shown in Table 1, which is similar to the one proposed in [41].",
"We resize the original character image in the sliding window to $32 \\times 32$ as the input feature map.",
"The filters of convolutional layers are with a small receptive field $3 \\times 3$ , and the convolution stride is fixed to one.",
"The number of feature maps is increased from 50 to 400 gradually.",
"To further increase the depth of the network so as to improve the classification capability, spatial pooling is implemented after every three convolutional layers instead of two [41], which is carried out by max-pooling (over a $2 \\times 2$ window with stride 2) to halve the size of feature map.",
"After the stack of 12 convolutional layers and 4 max-pool layers, the feature maps are flattened and concatenated into a vector with dimensionality 1600.",
"Two fully-connected layers (with 900 and 200 hidden units respectively) are then followed.",
"At last, a sofmax layer is used to perform the 37/7357-way classification.",
"To facilitate CNN training, we adopt the batch normalization (BN) technique, and insert eight BN layers after some of the convolution layers.",
"Table: CNN configuration.",
"'k', 's', 'p' stand for kernel size, stride and padding size, respectively."
],
[
"Transcription",
"Transcription is to convert the per-window predictions made by the convolutional character model into a sequence of character labels.",
"In this work, we assume each window represents a time step, and then adopt the CTC layer as our transcription layer.",
"CTC [10] maximizes the likelihood of an output sequence by efficiently summing over all possible input-output sequence alignments, and allows the classifier to be trained without any prior alignment between input and target sequences.",
"It uses a softmax output layer to define a separate output distribution $P(k|t)$ at every step $t$ along the input sequence for extended alphabet, including all the transcription labels plus an extra ‘blank’ symbol which represents an invalid output.",
"A CTC path $\\pi $ is a length $T$ sequence of blank and label indices.",
"The probability $P(\\pi |{\\rm {X}})$ is the emission probabilities at every time-step: $P(\\pi |{\\rm {X}}) = \\sum \\limits _{t = 1}^T {P({\\pi _t}|t,{\\rm {X}})}.$ Since there are many possible ways of separating the labels with blanks, to map from these paths to the transcription, a CTC mapping function $B$ is defined to firstly remove repeated labels and then delete the ‘blank’ from each output sequence.",
"The conditional probability of an output transcription $y$ can be calculated by summing the probabilities of all the paths mapped onto it by $B$ : $P({\\rm {y}}|{\\rm {X}}) = \\sum \\limits _{\\pi \\in {B^{ - 1}}({\\rm {y}})} {P{\\rm {(}}\\pi {\\rm {|X)}}}.$ To avoid direct computation of the above equation, which is computationally expensive, we adopt, the forward-backward algorithm [10] to sum over all possible alignments and determine the conditional probability of the target sequence."
],
[
"Decoding",
"Decoding a CTC network means to find the most probable output transcription ${\\rm {y}}$ for a given input sequence ${\\rm {X}}$ .",
"In practice, there are mainly three decoding techniques, namely naive decoding, lexicon based decoding and language model based decoding.",
"In naive decoding, predictions are made without any lexicon or language model.",
"While in lexicon based decoding, predictions are made by search within a lexicon.",
"As for the language model based method, a language model is employed to integrate the linguistic information during decoding.",
"Naive Decoding: The naive decoding, which also refer to as best path decoding without any lexicon or language model, is based on the assumption that the most probable path corresponds to the most probable transcription: $\\begin{array}{l}{y^*} \\approx B({\\pi ^*}), \\\\{\\pi ^*} = \\arg {\\max _\\pi }P(\\pi |{\\rm {X}}), \\\\\\end{array}$ Naive decoding is trivial to compute, since ${\\pi ^*}$ is just the concatenation of the most active outputs at every time-step.",
"Lexicon Based Decoding: In lexicon-based decoding, we adopt the technique called token passing proposed in [10].",
"First, we add ‘blank’ at the beginning and end and between each pair of labels.",
"Then a token $tok(s,h)$ is defined, where $s$ is a real-valued score and $h$ represents previously visited words.",
"Therefore, each token corresponds to a particular path through the network outputs, and the token score is the log probability of that path.",
"At every time step $t$ of the output sequence with length $T$ , each character $c$ of word $w$ holds a token $tok(w,c,t)$ , which is the highest scoring token reaching that segment at that time.",
"Finally, the result can be acquired by the $tok(w, - 1,T)$ .",
"The details can be found in [10].",
"In our implementation, to make the recognition result contains only one word, we set the score of the token with more than one history words to be extremely small.",
"Language Model Based Decoding: Statistical language model only models the probabilistic dependency between adjacent characters in words, which is a weaker linguistic constraint than lexicon.",
"In language model based decoding, we adopt the refined CTC beam search by integration of language model to decode from scratch, which is similar to the one proposed in [9] .",
"We denote the blank, non-blank and total probabilities assigned to partial output transcription $y$ of time $t$ as ${\\Pr ^ - }({\\rm {y}},t)$ , ${\\Pr ^ + }({\\rm {y}},t)$ and $\\Pr ({\\rm {y}},t)$ , respectively.",
"The extension probability $\\Pr (k,{\\rm {y}},t)$ of $y$ by label $k$ at time $t$ is defined as follows: $\\Pr (k,{\\rm {y}},t) = \\Pr (k,t|{\\rm {X}}){P^\\alpha }(k|{\\rm {y}})\\left\\lbrace {\\begin{array}{*{20}{c}}{{P^ - }({\\rm {y}},t - 1){\\rm { ~if~ }}{{\\rm {y}}^{\\rm {e}}} = k} \\\\{P({\\rm {y}},t - 1){\\rm { ~otherwise}}} \\\\\\end{array}} \\right.$ where $\\Pr (k,t|{\\rm {X}})$ is the CTC emission probability of $k$ at $t$ , $y^e$ is the final label of $y$ , ${P}(k|{\\rm {y}})$ is the linguistic transition from $y$ to ${\\rm {y}} + k$ and can be re-weighted with parameter $\\alpha $ .",
"The search procedure is described in Algorithm 1.",
"Our decoding algorithm is different from that of [9] in two aspects.",
"First, we introduce a hyper-parameter $\\alpha $ to the expression of extension probability, which accounts for language model weight.",
"Second, in order to further reduce the search space, we prune emission probabilities at time $t$ and retain only the top candidate number ($CN$ ) classes.",
"Refined CTC Beam Search [1] Initialize: $B \\leftarrow \\lbrace \\phi \\rbrace $ ; $Pr^{-}(\\phi ,0)=1$ $i = 1 \\rightarrow T$ $\\hat{B} \\leftarrow the~N$ -$best~sequences~in~B$ $B \\leftarrow \\lbrace \\rbrace $ $y \\in \\hat{B}$ $y \\ne \\phi $ $Pr^{+}(y,t) \\leftarrow Pr^{+}(y,t-1)Pr(y^{e},t|X)$ $\\hat{y} \\in \\hat{B}$ $Pr^{+}(y,t) \\leftarrow Pr^{+}(y,t)+Pr(y^{e},y,t)$ $Pr^{-}(y,t) \\leftarrow Pr^{-}(y,t-1)+Pr(-,t|X)$ add $y$ to $B$ sort emission probabilities at time $t$ and retain top $CN$ classes $k = 1 \\rightarrow CN$ $Pr^{-}(y+k,t) \\leftarrow 0$ $Pr^{+}(y+k,t) \\leftarrow Pr(k,y,t)$ add $y$ to $B$ Return:$\\max _{y \\in B}Pr^{\\frac{1}{\\Vert y\\Vert }}(y,T)$"
],
[
"Model Training",
"Denote the training dataset by $D=\\lbrace X_i, Y_i\\rbrace $ , where $X_i$ is a training image of word or text line and $Y_i$ is the ground truth label sequence.",
"The objective is to minimize the negative log-likelihood of conditional probability of ground truth: $O = - \\sum \\limits _{{X_i},{Y_i} \\in D} {\\log p({Y_i}|{S_i})}.$ where $S_i$ is the window sequence produced by sliding on the image $X_i$ .",
"This objective function calculates a cost value directly from an image and its ground truth label sequence.",
"Therefore, the network can be end-to-end trained on pairs of images and sequences, eliminating the procedure of manually labeling all individual characters in training images.",
"The network is trained with stochastic gradient descent (SGD) implemented by Torch 7[4].",
"Table: Recognition accuracies (%) on four English scene text datasets.",
"In the second row, “50”, “1k” and “Full” denote the lexicon used, LM denotes the language model and “None” denotes recognition without language constraints.",
"(* is not lexicon-free in the strict sense, as its outputs are constrained to a 90k dictionary.)"
],
[
"Experiments",
"We implemented the model on the platform of Torch 7 [4] with the CTC transcription layer (in C++) and the decoding schemes (in C++).",
"Experiments were performed on a workstation with the Intel(R) Xeon(R) E5-2680 CPU, 256GB RAM and an NVIDIA GeForce GTX TITAN X GPU.",
"Networks were trained with stochastic gradient descent algorithm, with the initial learning rate 0.1, and we selected $1/20$ of the training samples for each epoch.",
"The learning rate is decreased by $\\times 0.3$ at the 40th epoch and the 60th epoch.",
"The training finished in about 70 epochs.",
"To evaluate the effectiveness of the proposed method, we conducted experiments for English and Chinese scene texts, which are both challenging vision tasks."
],
[
"Datasets",
"For English scene text recognition, we use the synthetic dataset (Synth) released by [13] as training data for all the following experiments.",
"The training set consists of 8 millions images and their corresponding ground truth on text line level, which is generated by a synthetic data engine using a 90K word dictionary.",
"Although the model is trained with the synthetic data only, even without any fine-tuning on specific training sets, it works well on real image datasets.",
"We evaluated our scene text recognition system on four popular English benchmarks, namely ICDAR 2003 (IC03), ICDAR 2013 (IC13), IIIT 5k-word (IIIT5k) and Street View Text (SVT).",
"The IIIT5k dataset [23] contains 3,000 cropped word test images from the scene images from the Internet.",
"Each word image has been associated to a 50-words lexicon and a 1k-words lexicon.",
"This is the largest dataset for English scene text recognition so far.",
"The SVT dataset [33] was collected from Google Street View of road-side scenes.",
"The test dataset contains 249 images, from which 647 word images were cropped, and each word image has a 50-words lexicon defined by Wang et al.",
"[33].",
"The IC03 dataset [21] test dataset contains 251 scene images with labeled text bounding boxes.",
"We discard the images which either contain non-alphanumeric characters or have less than three characters following Wang et al.",
"[33], and get a test set with 860 cropped images.",
"Each test image is associated with a 50-words lexicon as defined by Wang et al.",
"[33].",
"A full lexicon is built by combining all the lexicons of per images.",
"The IC13 dataset [18] test dataset inherits most of its data from IC03.",
"It contains 1,015 cropped word images with ground truths."
],
[
"Implementation Details",
"During training, all images are scaled to have height 32, widths are proportionally scaled.",
"For parallel computation in training on GPU, we unify the normalized text images to width 256.",
"So, If the proportional width is less than 256, we pad the scaled image to width 256, otherwhise, we continue to scale the image to $32\\times 256$ (this rarely happens because most words are not so long).",
"In training, we investigate two types of model (single-scale model and multi-scale model).",
"The single-scale model has only one input feature map with the window size of 32x32.",
"While the multi-scale model has three input fature maps, which are firstly extracted with the window size of 32x24, 32x32 and 32x40, and then are all resized to 32x32.",
"Both the models are shifted with step 4.",
"Test images are scaled to have height 32, and the widths are proportionally scaled with heights.",
"In testing, the window sliding step is 4 as in training.",
"For integrating linguistic context in recognition, we trained a 5-gram character language model (LM) on two text corpra.",
"One is extracted from the transcripts of training imageset consisting about 8 million words.",
"The other is trained on a general corpus [43] consisting of 15 million sentences."
],
[
"Comparisons with State-of-the-art",
"The English scene text word recognition results on four public datasets are listed in Table 2, with comparison to the state of the art.",
"We also give results of our model using residual network [5] instead of the structure specified in Section 3.2.",
"In the naive decoding case (None), our method achieves comparable results with the best performance on the four datasets.",
"It can be found that the results with three scales (n=3) are better than only one scale input (n=1), as the model can capture more context information with more scales.",
"In the unconstrained case (w/o language model), best performance close to our work are reported by [29], [15].",
"However, the result in [15] is constrained to a 90k dictionary, and there is no out-of-vocabulary word.",
"For the LSTM based method [29], an implicit language model is embedded training with a dataset of large lexicon (Synth contains the words of IC03/13 and SVT test set), and therefore can give higher performance.",
"However, it is unfair to compare these methods with real lexicon-free and LM-free methods.",
"Although we also used the Synth as training data, our model only learns character models in training.",
"This means that our model is totally lexicon-free and LM-free, thus, it performs quite stably on different datasets when lexicon is free.",
"In contrast, the LSTM based method [29] has an obvious decrease of performance on the IIIT-5K.",
"In our method, the process of each window is independent, so we can parallelize the classification of all the windows in one time step.",
"However, the LSTM-based method has each time step dependent on others, so, it must be updated step by step.",
"Therefore, our method only need 0.015s to process each sample on average with naive decoding, whereas the average testing time is 0.16s/sample for method [29].",
"In the lexicon-based decoding case, our method achieves the best performance on IIIT5k, which has the largest number of cropped images.",
"On datasets SVT and IC03, our model with lexicon-base decoding yields results comparable to the best of state of the art.",
"The model size of several methods are listed in Table 2, which reports the parameter number of the learned model.",
"The number of parameters of our model are less than all the previous deep leaning methods [29], [15], [14].",
"Moreover, our model size can be largely reduced to only 0.41M by a 38-layer residual network [5], while the performance can keep comparable to our convolutional character model.",
"This is a good trade off between the space and accuracy and can be easily ported to mobile devices.",
"Figure: (a) Correct recognition samples in IIIT5k; (b) Incorrect recognition samples in IIIT5k"
],
[
"Datasets",
"In the training set of the public available Chinese text recognition dataset [42], there are only about 1400 character classes.",
"However, with that we cannot train a practicable Chinese scene text recognition system because the common used simplified Chinese characters are actually more than 7000 classes.",
"Therefore, following some success syntentic text dataset, we generated a Chinese scene text dataset (Synth-Ch) with the engine in [13].",
"The Synth-Ch includes 9 million text images, containing the number of characters from 1 to 15 in every text image.",
"Moreover, for each text image, the font of characters are randomly selected from 60 Chinese fonts, and the characters are randomly selected from a Chinese character list which includes 7,185 common used simplified characters and 171 symbols (including 52 English letters and 10 digits).",
"For Chinese scene text recognition, we use the Synth-Ch as training data.",
"We evaluated our Chinese scene text recognition system on ICDAR2015 Text Reading in the Wild Competition dataset (TRW15).",
"The TRW15 dataset [42] contains 984 images and 484 images are selected as test set.",
"From the testing images, we cropped 2996 horizontal text lines as the first test set (TRW15-T).",
"However, the number of character classes is small (about 1460 character classes) in TRW15-T.",
"Thus, in order to evaluate the performance of our model more efficinetly, we constructed the second test set (TRW15-A) from all 984 images.",
"In TRW15-A, there are 6106 horizontal text lines and more than 1800 character classes."
],
[
"Implementation Details",
"During training, we use the same normalization strategy as that in English data sets, except the width of normalized text images is set as 512. we only train single-scale model with the window size of 32x40.",
"In training, the shifted step is set as 8.",
"Test images are firstly rectified to a rectangle image with perspective transformation [7] because most of the test images have perspective distort, then the rectified image are scaled to have height 32, and the widths are proportionally scaled with heights.",
"In testing, the window sliding step is 8 as in training.",
"We evaluate the recognition performance using character-level accuracy (Accurate Rate) following [34].",
"For integrating linguistic context in recognition, we trained a 5-gram character language model (LM) on the SLD corpus [26], which contains news text from the 2006 Sogou Labs data."
],
[
"Comparisons with State-of-the-art",
"The Chinese scene text recognition results on TRW15 datasets are listed in Table 3.",
"Compared to previous results reported on the ICDAR2015 Text Reading in the Wild Competition, 72.1 percent of AR, the proposed approach achieved 81.2 percent of AR, demonstrating significant improvement and advantage.",
"We also give results of our model using residual network [5] instead of the structure specified in Section 3.2, which is also much better than the winner method in the competition.",
"Table: Recognition accuracies (%) on Chinese scene text dataset"
],
[
"Conclusion",
"In this paper, we investigate the intrinsic characteristics of text recognition, and inspired by human cognition mechanisms in reading texts, we propose a scene text recognition method with character models on convolutional feature map.",
"The model is trained end-to-end on word images weakly labeled with transcripts.",
"The experiments on English and Chinese scene text recognition demonstrate that the proposed method achieves superior or comparable performance.",
"In the future, we will evaluate our model on more challenging data sets."
]
] | 1709.01727 |
[
[
"SPIRou Input Catalog: Activity, Rotation and Magnetic Field of Cool\n Dwarfs"
],
[
"Abstract Based on optical high-resolution spectra obtained with CFHT/ESPaDOnS, we present new measurements of activity and magnetic field proxies of 442 low-mass K5-M7 dwarfs.",
"The objects were analysed as potential targets to search for planetary-mass companions with the new spectropolarimeter and high-precision velocimeter, SPIRou.",
"We have analysed their high-resolution spectra in an homogeneous way: circular polarisation, chromospheric features, and Zeeman broadening of the FeH infrared line.",
"The complex relationship between these activity indicators is analysed: while no strong connection is found between the large-scale and small-scale magnetic fields, the latter relates with the non-thermal flux originating in the chromosphere.",
"We then examine the relationship between various activity diagnostics and the optical radial-velocity jitter available in the literature, especially for planet host stars.",
"We use this to derive for all stars an activity merit function (higher for quieter stars) with the goal of identifying the most favorable stars where the radial-velocity jitter is low enough for planet searches.",
"We find that the main contributors to the RV jitter are the large-scale magnetic field and the chromospheric non-thermal emission.",
"In addition, three stars (GJ 1289, GJ 793, and GJ 251) have been followed along their rotation using the spectropolarimetric mode, and we derive their magnetic topology.",
"These very slow rotators are good representatives of future SPIRou targets.",
"They are compared to other stars where the magnetic topology is also known.",
"The poloidal component of the magnetic field is predominent in all three stars."
],
[
"Introduction",
"Due to their low mass, M dwarfs are favorable to exoplanet searches with the radial-velocity (RV) method.",
"The main reason is that the RV signal of a planet of a given mass and period increases with decreasing stellar mass.",
"In addition, for a given surface equilibrium planet temperature, the orbital period is much shorter when the parent star is a small, low-luminosity star.",
"Thus, telluric planets in the habitable zone of their parent stars have a more prominent RV signal when this host is an M dwarf compared to any other spectral type.",
"In addition, such planets are seemingly frequent in the solar vicinity: radial-velocity survey of a hundred M dwarfs showed that 36% (resp., 52%) M dwarfs have a planet in the mass range of 1 to 10 Earth mass and for orbital periods of 1-10 days (resp., 10-100 days) .",
"Using a different method and a separate target sample, the $Kepler$ survey has measured a planet occurrence rate of 2.5$\\pm $ 0.2 per M star, in the radius range of 1-4 Earth radii and period less than 200 days and a fraction of $\\sim $ 50% of M stars having a 1-2 Earth radii planet.",
"The comparison of these occurrence rates depends on the mass-radius relationship of these planets, but they agree qualitatively and point to an abundant population of exoplanets, mostly of small size or mass.",
"It is, however, expected that exoplanet searches around M dwarfs are highly impacted by the surface activity of the parent stars.",
"The stellar modulation may mimic a planetary signal, as shown by, e.g., or , or it may affect the mere detection of the planetary signal, as recently demonstrated by .",
"Furthermore, the jitter on M dwarfs, when not properly filtered out, results in major deviations in the measurement of orbital periods, planet minimum mass and/or eccentricity of the detected planets, as modeled by .",
"Thus, exoplanet RV surveys aiming at M host stars require a thorough understanding of the processes that induce intrinsic stellar RV modulation.",
"Covering a range of mass from 0.08 ${\\rm M}_{\\odot }$ to about 0.50 ${\\rm M}_{\\odot }$ , i.e., a factor of 6 in mass, and having the longest evolution history of all stars, the M dwarfs may encompass very different types of stellar surfaces and be dominated by a wide variety of phenomena.",
"This can be particularly true in the transition from partly to fully convective interiors at the mass of 0.35${\\rm M}_{\\odot }$ .",
"Many previous studies have explored the activity features of M dwarfs, including X-ray observations , photometric variations , long-term RV variations , H$\\alpha $ , CaII, and rotation measurements , , , , , UV emission , surface magnetic field modulus , and large-scale magnetic field geometry , .",
"This richness of observed activity features illustrates the complexity of phenomena of magnetic origin in M dwarfs and offers complementary constraints to dynamo and convection modeling.",
"Concerning magnetic field measurements, we may have two different and complementary diagnostics: The Zeeman broadening in unpolarized spectra is sensitive to the magnetic field modulus but almost insensitive to the field spatial distribution or orientation.",
"Modelling based on Zeeman broadening generally assesses a quantity called \"magnetic flux\" which corresponds to the product of the local magnetic field modulus $B$ with the filling factor $f$ in a simple model where a fraction $f$ of the surface is covered by magnetic regions of uniform modulus $B$ .",
"Zeeman-induced polarisation in spectral lines is sensitive to the vector properties of the magnetic fields.",
"But due to the cancellation of signatures originating from neighbouring regions of opposite polarities, it can only probe the large-scale component of stellar magnetic fields.",
"Using a time-series of polarised spectra sampling at least one rotation period, it is possible to recover information on the large-scale magnetic topology of the star, see section 3.3.",
"Whereas M dwarfs exhibiting the highest amplitude of activity have been more studied in magnetic-field explorations, the exoplanet RV searches will tend to focus on the intermediate to low activity stars, where the RV jitter should have the lowest impact on planet detection.",
"For instance, large-scale magnetic field observations of fully convective M dwarfs have mainly focused on rapid rotators ($P_{\\rm rot}$ < 6 d) so far.",
"Among those, they have characterized the coexistence of two types of magnetic topologies: strong axial dipole and weak multipolar field , , .",
"Large-scale magnetic fields of slowly-rotating fully-convective stars remain more poorly constrained, even though their characterization would extend the understanding of dynamo processes , , provide further constraints on the evolution of stellar rotation , and would permit a better definition of the habitable zone around mid to late-type M dwarfs.",
"Recently, have explored the large-scale magnetic properties of quiet M stars and derived a description of the RV jitter as a function of other activity proxies.",
"These studies have shown some connection between the brightness features and the magnetically active regions, but no one-to-one relation, as well as a pseudo-rotational modulation of the RV jitter rather than a purely rotational behaviour.",
"Such detailed investigations on a small number of M stars have thus shown complex spatio-temporal properties and require a broader exploration.",
"Figure: The histogram of V-KV-K values in the sample.In the coming years, the new spectropolarimeter SPIRouhttp://spirou.irap.omp.eu/ will be installed at the Canada-France-Hawaii Telescope atop Maunakea.",
"SPIRou will be the ideal instrument to study the stellar properties of M dwarfs and search for their planetary companions by combining polarimetric measurements of the stellar magnetic field, the velocimetric precision required for planet searches, and the wide near-infrared simultaneous coverage of the YJHK bands.",
"In preparation to the planet-search survey that will be conducted with SPIRou, we have collected and analysed all ESPaDOnS data available on M dwarfs.",
"Data collection, catalog mining and fundamental parameters are described in companion papers (Malo et al and Fouqué et al, in prep.).",
"In this paper, we investigate the activity features and magnetic properties of the data sample, with the goals of improving our understanding of physical processes at play at the surface and in the atmosphere of M dwarfs that could generate RV jitter and hamper planet detection.",
"By combining and relating several types of observed features of M-dwarf magnetic fields, we attempt to establish a merit function of activity that will allow us to sort and select the best possible targets for planet detection using the RV method and SPIRou.",
"We also enlarge the picture of M-star topology exploration by adding three new slow-rotating M stars having their magnetic topology characterized.",
"The paper is organized as follows: in section 2, we describe the stellar and data samples and collected observations.",
"In section 3, we show our data analysis methods.",
"Results are discussed in section 4.",
"Further discussion and conclusions are given in section 5."
],
[
"Sample and observations",
"All spectra of our sample of 442 stars correspond to cool stars in the solar vicinity.",
"The origin of the data is diverse: 1) the exploratory part of the Coolsnap programprogram IDs 14BF13/B07/C27, 15AF04/B02, 15BB07/C21/F13, 16AF25, 16BC27/F27 and 17AC30, P.I.",
"E. Martioli, L. Malo and P. Fouqué, 2) the CFHT/ESPaDOnS archiveshttp://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/cfht/ in the spectropolarimetric mode, 3) the CFHT/ESPaDOnS archives in the spectroscopic mode, and 4) the follow-up part of Coolsnap.",
"The Coolsnap program is a dedicated observing project using spectropolarimetric mode with CFHT/ESPaDOnS, targeting about a hundred M main-sequence stars with two visits per star.",
"Table REF summarizes the stellar and data samples.",
"Figure REF shows the distribution of the $V-K$ index in our sample.",
"The $K$ magnitude is actually obtained in the 2MASS $K_s$ filter , while $V$ magnitudes are in the Johnson system (the compilation of magnitudes is described in the forthcoming paper by Fouqué et al, in prep.).",
"The histogram peaks at a value of 4.8 which corresponds to spectral types of M3-M4 or masses of about 0.35${\\rm M}_{\\odot }$ , or effective temperatures of $\\sim $ 3300K , , .",
"The star with the largest colour index is GJ 3622 ($V-K$ = 7.858) which was originally observed and studied by .",
"While the data origin is diverse, the homogeneity of the data lies in the use of the same instrument CFHT/ESPaDOnS, providing the wide optical range of 367 - 1050 nm in a single shot, at 65,000 – 68,000 resolving power; data processing and analysis are also homogeneous.",
"The sample of spectra, however, is heterogeneous in signal-to-noise ratio, number of spectra per star and temporal sampling.",
"The choice of ESPaDoNS is mainly motivated by the spectro-polarimetric mode, a unique way to obtain the circular polarisation of stellar lines , and by the extended wavelength range towards the red, well adapted to M-star observations.",
"Data acquired in the \"Star+Sky\" spectroscopic mode is a single-exposure spectrum where the sky contribution is subtracted from the star spectrum.",
"In the polarimetric mode, four sub-exposures are taken in a different polarimeter configuration to measure the circularly polarized spectra and remove all spurious polarization signatures .",
"The unpolarized spectrum is the average of all four intensity spectra.",
"Although large-scale magnetic-field detections are obviously not available from spectra in \"Star+Sky\" mode, including those data allowed to significantly widen our sample while still allowing a large number of measurements: chromospheric emission indices, projected rotational velocity, radial velocity, fundamental parameters and Zeeman broadening proxy, when the SNR is sufficient.",
"Table REF gives a summary of main properties of our data sample.",
"For additional description of the stellar sample, we refer the reader to papers in the series by Malo et al (in prep.)",
"for SPIRou planet-search program target selection, and by Fouqué et al (in prep.",
"), focusing on the determination of stellar fundamental parameters.",
"Consequently, as a follow-up to the Coolsnap exploratory program, we focused on two low-mass stars with slow rotation rates for which the magnetic field topologies were determined: GJ 1289 and GJ 793.",
"They are the only stars with multiple visits observed in the polarimetric mode of ESPaDOnS for which the magnetic topology has not been published yet.",
"Spectropolarimetric observations were collected from August to October 2016, using ESPaDOnSGJ 1289 and GJ 793: program 16BF15, P.I.",
"J. Morin.",
"Finally, we added the analysis of GJ 251.",
"This star is one of the standards of the calibration plan of CFHT/ESPaDOnSGJ 251: programs Q78 of each semester, and as such, is regularly observed since 2014.",
"The spectropolarimetric observations of GJ 251 are considered here as \"Coolsnap follow-up\" (Table REF ).",
"Table: Summary of data collection and how the 1878 spectra distribute in the various programs and modes.",
"S/N range corresponds to values per 2.6 km/s bin at 809 nm.GJ 1289 is an M4.5 dwarf of 3100 K. Two preliminary observations within the Coolsnap-explore program in September 2014 and July 2015 showed two clear magnetic detections in the Stokes $V$ profile.",
"A total of 18 were then collected over 2.5 months in 2016.",
"Exposure times of 4$\\times $ 380s or 4$\\times $ 600s were used, depending on the external conditions.",
"GJ 793 is an M3 dwarf of 3400 K. Early observations in August and September 2014 similarly showed detections of the magnetic field.",
"A total of 20 were collected over 2.5 months in 2016.",
"Exposures times of 4$\\times $ 150 to 4$\\times $ 210s were used.",
"GJ 251 is of slightly later type than GJ 793, with an estimated temperature of 3300 K. As it is observed in the context of the calibration plan of ESPaDOnS, the observation sampling is regular but infrequent and data are spread from Sept 2014 to March 2016.",
"Since GJ 251 is a slow rotator (see section 4.4.3), such sampling is well adapted.",
"The spectropolarimetric observations of GJ 251 have been obtained with exposures of 4$\\times $ 60s.",
"The three stars are good representatives of the future SPIRou planet-search targets.",
"The detailed journal of spectro-polarimetric observations for these three stars is shown in Table REF ."
],
[
"Data reduction",
"The data extraction of ESPaDOnS spectra is carried out with Libre-Esprit, a fully automated dedicated pipeline that performs bias, flat-field and wavelength calibrations prior to optimal extraction of the spectra.",
"The initial procedure is described in .",
"The radial velocity reference frame of the extracted spectra is first calibrated on the ThAr lamp and then, more precisely, on the telluric lines, providing an instrumental RV precision of 20m/s $rms$ .",
"Least-Squares Deconvolution (LSD, ) is then applied to all the observations, to take advantage of the large number of lines in the spectrum and increase the signal-to-noise ratio (SNR) by a multiplex gain of the order of 10.",
"We used a mask of atomic lines computed with an Atlas local thermodynamic equilibrium (LTE) model of the stellar atmosphere .",
"The final mask contains about 4000 moderate to strong atomic lines with a known Landé factor.",
"This set of lines spans a wavelength range from 350 nm to 1082 nm.",
"The use of atomic lines only for the LSD masks relies on former studies of early and mid M dwarfs .",
"It is important to note that the use of a single mask over such a wide range of spectral characteristics is not optimal; in particular, the multiplex gain is not maximized for spectra corresponding to the latest type M dwarfs.",
"Building up the collection of line lists with Landé factors and reliable line amplitudes, including molecular species and in various stellar atmospheres would be beneficial in a future work to this large-sample analysis.",
"On the positive side, the mask we used in this study is the same one used in previous analyses of M stars observed with ESPaDOnS or NARVAL , which insures some homogeneity.",
"Table: Position and widths of passbands used to measure the activity indices.",
"All numbers are in nm.",
"For CaII HK, trtr is meant for triangular bandpass of base 0.109nm.",
"Other indices use rectangular bandpasses."
],
[
"Spectroscopic index measurements",
"After correcting from the star radial velocity, we measured the spectroscopic tracers of chromospheric or photospheric activity in spectra with reference positions and widths as summarized in Table REF .",
"The width of the emission features was deliberately chosen wider than in the literature because of the very strong emitters included in our sample, whose emission lines were twice wider than the bandpasses generally in use for quiet stars.",
"Also, we chose for the CaII H and K line continuum to use the window around 400.107 nm only, and not the continuum window around 390.107 nm, in order to reduce the noise increasing in the bluest part of the continuum.",
"The $S$ index was then calibrated with measurements from the literature, although this index is expected to vary in time, resulting in a significant dispersion.",
"The calibration is shown in the Appendix (Fig.",
"REF ); it is based on literature values from the HARPS M-dwarf survey .",
"Other indices were measured in similar ways with their respective continuum domains optimized against telluric absorption and major stellar blends (Table REF ): H$_\\alpha $ , the 590-nm NaI doublet (NaD), the 587-nm HeI line, the 767-nm KI doublet (KI), the 819-nm NaI IR doublet (NaI IR), and the 850-nm CaII infrared triplet (CaII IRT).",
"No attempt was made to calibrate these indices to literature values.",
"From the $S$ index, we derived the log($R^{\\prime }_{HK}$ ) by correcting for the photospheric contribution and an estimate of the rotation period as proposed by (their equation 12).",
"This method has the caveat that the rotation periods shorter than about 10 days cannot be derived due to degeneracy because CaII HK chromospheric emission reaches its saturation level.",
"We applied a threshold in log($R^{\\prime }_{HK}$ ) of -4.5, meaning that a rotation period is deduced only for spectra with log($R^{\\prime }_{HK}$ )$<$ -4.5.",
"For other indices, however, we did not correct for the photospheric contribution as was done in other studies .",
"We thus do not expect to find similar relationships between chromospheric lines as those obtained when the basal component is removed.",
"Figure: Average FeH lines for GJ 1289 (top left) and GJ 793 (top right), GJ 251 (bottom right) and the most broadened star GJ 388 (AD Leo, bottom left): the 995 nm line is insensitive to the magnetic field (black line) and thus used as a reference profile, while the 990 nm is magnetically sensitive (red line).",
"Spectra with a SNR greater than 250 per CCD pixel were used to calculate the average."
],
[
"Zeeman broadening",
"As shown in , the Zeeman effect also affects the line broadening in the unpolarized light.",
"This measurement is complementary to the detection of a polarized signature in the stellar lines, by giving access to the average surface field modulus weighted by the filling factor.",
"The spatial scales of the polarized and unpolarized-light magnetic fields are also very different (comparable, respectively, to the full stellar sphere and the magnetic surface spots).",
"For more details on both field diagnostics, we refer the reader to the reviews by, $e.g.$ , and .",
"We measured in the intensity spectra two unblended FeH lines of the Wind-Ford $F^4\\Delta -X^4\\Delta $ system near 992 nm, as their variable sensitivity to the magnetic field may be used as a proxy for the average magnetic field (called $Bf$ in the following) of the stellar surface , [1], : The FeH line at 995.0334 nm is weakly sensitive to the magnetic field.",
"We assume that its line width is dominated by rotation, convection, temperature, turbulence and by the intrinsic stellar profile as seen by the ESPaDOnS spectrograph.",
"The FeH line at 990.5075 nm is magnetically sensitive .",
"If there is a significant broadening of this line compared to the 995.0334 nm line, we assume it is due to Zeeman broadening in its totality.",
"Only spectra with passed quality criteria were kept for further analysis.",
"In particular, we rejected all spectra with line blending, locally low signal-to-noise ratio or/and fast rotation as the lines were not properly identified and adjusted.",
"As examples, the average FeH velocity profiles for GJ 251, GJ 793, GJ 1289, and, for comparison, GJ 388 (from the smallest to the largest measured broadening) are shown on Figure REF .",
"The broadening is significant for all stars.",
"For the most magnetic GJ 388 (AD Leo), the line splitting results in a significant decrease of the line amplitude, as shown in the bottom left panel of this figure.",
"The Zeeman broadening due to line splitting in the velocity space $\\delta v_B$ (in km/s) is then related to the magnetic field modulus $Bf$ through : $Bf = \\frac{\\delta v_B}{1.4 \\times 10^{-6} \\lambda _{\\circ } g_{\\rm eff}}$ where $\\lambda _{\\circ }$ is the wavelength of the magnetically sensitive line, 990.5075 nm, $g_{\\rm eff}$ is the effective Landé factor of the transition and $Bf$ is the average magnetic field weighted by the filling factor $f$ (in G).",
"We estimated the Landé factor by comparing the value of the Zeeman broadening with the literature values obtained through Zeeman spectral synthesis .",
"There is, however, a very small overlap of our valid measurements with the literature, since $Bf$ has been mostly derived in the past on very active, fast rotating stars.",
"AU Mic, AD Leo, GJ 1224, GJ 9520, GJ 49 and GJ 3379 all have a valid measurement in our study and are found in the compilation of $Bf$ values in and where they are quoted with average $Bf$ values of 2.3, 2.6-3.1, 2.7, 2.7, 0.8 and 2.3 kG, respectively.",
"Assuming the field strength is the same in our measurements, that would induce $g_{\\rm eff} = 1\\pm 0.24$ .",
"Note that theoretical and laboratory values of $g_{\\rm eff}$ , although difficult to obtain for a molecule, are quoted in a range 0.8 to 1.25 by several authors for the transitions in the $F^4\\Delta -X^4\\Delta $ system , , [1], , , in good agreement with our empirical estimate.",
"We also note that, as we always take the same value of $g_{\\rm eff}$ in this study, trends and behaviours intrinsic to the sample remain valid and are independent of the exact value of $g_{\\rm eff}$ .",
"The smallest $Bf$ value measured in our spectra is 0.5 kG.",
"The typical error on the $Bf$ value is of the order of 0.3 kG.",
"Although good agreement with spectral-synthesis analyses is found with model-fitting methods on 6 objects (see above), there are 4 stars for which the results disagree (GJ 876, GJ 410, GJ 70, and GJ 905).",
"Figure REF shows the comparison of our data with literature values.",
"The Zeeman spectral synthesis code gives much smaller values than our method for these 4 slow-rotating stars.",
"This could be due to systematics of one of the methods at low rotation velocity, to surface heterogeneities, or to inaccurate Landé factors.",
"There may thus be over-estimations of the field modulus in the following, compared to other studies of the average field, and this is yet to be understood.",
"We note that, within the spectral-synthesis method, there may be similar discrepancies, reaching 1 kG, when using different lines .",
"It is beyond the scope of this paper to further explore other atomic or molecular lines over a wide range of spectral types, but that is an ongoing extension of this work.",
"Figure: The comparison of our BfBf values using g eff =1g_{\\rm eff} = 1, to different estimates available in the literature , .",
"The line shows Y=X.",
"While the agreement is good for stars having a large field, it is worse on low-field or slow-rotating stars.",
"Errors on the Y axis have been set to 300 G.Figure: Illustration of parametrization of Stokes VV profiles: the observed VV profiles (data points with errors), their interpolation (dashed) and the best-fit Voigt profile (red) for the star GJ 4053 at different dates.",
"The positions of maxima have been highlighted by red dashed line."
],
[
"From Stokes profiles to longitudinal magnetic field",
"The intensity profile Stokes $I$ is derived from LSD and adjusted by a Lorentzian in all spectra.",
"On circular-polarisation data, the Stokes $V$ profile is also calculated with LSD, as well as a null profile (labelled N).",
"The $N$ profile results in a different combination of polarimeter positions.",
"It allows to confirm that the detected polarization is real and not due to spurious instrumental or data reduction effects .",
"It can also be used to correct the $V$ profile by removing the instrumental signature.",
"In order to distinguish between magnetic field detection in stellar line and noise, we use the False Alarm Probability (FAP) value, as described in .",
"The $\\chi ^2$ function is calculated in the intensity profile and outside, both in the Stokes $V$ and $N$ spectra.",
"A definite detection corresponds to a FAP smaller than 10$^{-5}$ and a marginal detection has a FAP between 10$^{-5}$ and 10$^{-3}$ ; in cases of definite and marginal detections, it is verified that the signal is detected inside the velocity range of the intensity line.",
"If the FAP is greater than 10$^{-3}$ , or if a signal is detected outside the stellar line, the detection is considered null.",
"The longitudinal magnetic field $B_l$ values are then determined from every observation in the polarimetric mode following the analytic method developed in .",
"$B_l=-2.14 \\times 10^{11} \\frac{\\int vV(v)\\mathrm {d}v}{\\lambda _0 \\cdot g_{eff} \\cdot c \\cdot \\int \\left[ I_c - I(v) \\right] \\mathrm {d}v}$ where $v$ is the radial velocity, $V(v)$ and $I(v)$ are Stokes $V$ and Stokes $I$ profiles, and $I_c$ is the continuous value of the Stokes $I$ profile.",
"Parameters $\\lambda _0$ , $g_{eff}$ and $c$ are the mean wavelength (700 nm), the effective Landé factor (1.25) and vacuum light speed.",
"Errors on $B_l$ are obtained by propagating the flux errors in the polarized and unpolarized spectra.",
"Figure: The extreme amplitude of the V/I c I_c profile as a function of the longitudinal field B l B_l, for all profiles where the detection of the magnetic signature in stellar lines is definite.The domain between $\\pm 3\\sigma $ ($\\sigma $ is the Lorentzian width obtained from the fit of $I(v)$ ) centered at the intensity profile's maximum was chosen for $B_l$ integration as it includes relevant signal while minimizing the undesired noise on the Stokes $V$ profiles.",
"Stokes $V$ profiles actually offer more information than just the $B_l$ values, which only reflects the anti-symmetrical part of the profile.",
"The complex shape of $V$ profiles was then tentatively parametrized with a combination of two Voigt models, applied to significant detections.",
"A few examples are shown in Figure REF for four different profiles of star GJ 4053.",
"The parametric model allows to explore the properties of the profile's features without depending on a complex topology model that cannot be applied in the case of scarce sampling.",
"The amplitude and peak position of the Stokes $V$ profiles have been measured, as well as the integral of the non-anti-symmetrical component of the profiles.",
"The Stokes-$V$ parametrization method may be useful in the future with SPIRou, to explore different activity filtering methods based on polarimetric measurements.",
"Figure REF shows for instance the maximum signed signal in the Stokes $V$ profiles is shown as a function of $B_l$ .",
"From experiments done on data sets having more than 20 visits, and randomly selecting pairs of spectra at different epochs, we concluded that it is hazardous to try to derive any of the physical parameters of the large-scale magnetic field from a couple of observed Stokes $V$ profiles of a given star, even with reasonable assumptions on configuration values or rotation periods."
],
[
"Zeeman Doppler imaging",
"To reconstruct the magnetic map of GJ 1289, GJ 793, and GJ 251, we used the tomographic imaging technique called Zeeman Doppler imaging (ZDI).",
"It uses the series of Stokes $V$ profiles.",
"It may be necessary to first correct for instrumental polarization by subtracting the mean $N$ profile observed for a given star.",
"It had a significant effect on the modeling of GJ 251 and in a lesser extent, of GJ 793.",
"ZDI then inverts the series of circular polarization Stokes $V$ LSD profiles into maps of the parent magnetic topology with the main assumption that profile variations are mainly due to rotational modulation.",
"Observed Stokes $V$ are adjusted until the magnetic-field model produces profiles compatible with the data at a given reduced chi-squared $\\chi ^2$ .",
"In that context, longitudinal and latitudinal resolution mainly depends on the projected rotational velocity, $v \\sin i$ , the star inclination with respect to the line of sight, $i$ , and the phase coverage of the observations.",
"The magnetic field is described by its radial poloidal, non-radial poloidal and toroidal components, all expressed in terms of spherical-harmonic expansions , .",
"The surface of the star is divided into 3000 cells of similar projected areas (at maximum visibility).",
"Due to the low value of the rotation period of the three stars, the resolution at the surface of the star is limited.",
"We therefore truncate the spherical-harmonic expansions to modes with $\\ell \\le $ 5 ,Morin08b.",
"The synthetic Stokes $V$ LSD profiles are derived from the large-scale magnetic field map by summing up the contribution of all cells, and taking into account the Doppler broadening due to the stellar rotation, the Zeeman effect, and the continuum center-to-limb darkening.",
"The local Stokes $V$ profile is computed using Unno-Rachkovsky's analytical solution of the transfer equations in a Milne-Eddington atmospheric model in presence of magnetic field .",
"To adjust the local profile, we used the typical values of Doppler width, central wavelength and Landé factor of, respectively, 1.5 km s$^{-1}$ , 700 nm and 1.25.",
"The average line-equivalent width is adjusted to the observed value.",
"By iteratively comparing the synthetic profiles to the observed ones, ZDI converges to the final reconstructed map of the surface magnetic field until they match within the error bars.",
"Since the inversion problem is ill-posed, ZDI uses the principles of Maximum-Entropy image reconstruction to retrieve the simplest image compatible with the data.",
"A detailed description of ZDI and its performance can be found in , , and its previous application to low-mass slowly-rotating stars in and ."
],
[
"Spectroscopic indices and rotation periods",
"The log($R^{\\prime }_{HK}$ ) values for our sample are shown on Figure REF as a function of $V-K$ and for different ranges of projected velocity.",
"It is observed that, for a given mass, faster rotators are more active than slower ones.",
"The other trend is that log($R^{\\prime }_{HK}$ ) decreases with $V-K$ , whatever the velocity.",
"This latter observation may, however, be an indication that the calibration of the bolometric factor or the photospheric factor is not robust over a large colour range .",
"For early-type and slow rotating stars, we were able to obtain estimates of the rotation period following , as discussed in the Appendix and shown on Fig.",
"REF .",
"For instance, the rotation periods of GJ 1289, GJ 793 and GJ 251 derived from CaII HK are found to be 36, 27 and 85 days, in qualitative agreement with the periods derived from the ZDI analysis (54, 22 and 90 days, respectively, see section 4.4).",
"For the following, and in the objective to use a global chromospheric index for stellar activity classification, we introduce the total chromospheric flux: $F_{chr}$ is the sum of the most prominent chromospheric equivalent widths found in M stars, $F_{chr} = NaD + HeI + H\\alpha + S + CaIRT$ .",
"In the following we will use either the CaIRT or the $F_{chr}$ index.",
"More details on chromospheric features and trends between them are presented in Appendix B."
],
[
"Large-scale magnetic field in polarized light",
"We estimated the longitudinal magnetic field in all polarized spectra of our sample.",
"The longitudinal field values, $B_l$ , are equally distributed between negative and positive values, which is expected for a random distribution.",
"The histogram of $B_l$ value shown in Figure REF presents how weak the magnetic field of Coolsnap stars is while the archive stars observed in the polarimetric mode span a much wider range of the longitudinal field.",
"This only reflects the biases of samples; the archive mostly contains fast rotators and the most active M dwarfs.",
"Note that the tail of very high negative values of $B_l$ from the archive sample visible in Figure REF is mostly due to a single star, GJ 412B (WX UMa), featuring one of the strongest magnetic dipole known among M dwarfs .",
"Table REF presents statistics on the detection of the magnetic field signatures in our data set.",
"It shows that, while more stars have been observed in the Coolsnap program than in the full ESPaDOnS archive of M stars in the polarimetric mode, more spectra are available in the archive, and these spectra correspond to more magnetic-field detections.",
"The percentage of null detections is greater in the Coolsnap sample while the archive contains most of the definite ones.",
"Additional statistics show an ideal balance in the signed value of $B_l$ : 50 stars have $B_l$ values always positive, while 51 stars have $B_l$ values always negative and 45 stars have $B_l$ that change sign within our data set.",
"Finally, 300 stars have no $B_l$ estimate, either because of non detections in the polarimetric mode, or because they were observed in the spectroscopic mode.",
"Figure REF shows how the detection statistics behave as a function of the spectrum SNR, the chromospheric activity, and the colour index.",
"The black histograms correspond to marginal and definite detections while red histograms correspond to null detections.",
"There are more detection around chromospherically active stars ( log($R^{\\prime }_{HK}$ ) greater than -5.0).",
"The different behaviour in SNR reflects the various stellar samples (Coolsnap and archives) and shows that, beyond an SNR of 100, magnetic detection depends on stellar parameters more than on SNR.",
"Table: Detection statistics for the polarization signal in stellar lines.",
"The counts are given in number of polarized spectra (not in number of stars).Figure: Histogram of B l B_l for the two polarimetric data sets.Figure: Distribution of parameters in all spectra in the polarimetric mode, with respect to magnetic-field detections: chromospheric emission, SNR (at 809 nm), and colour index V-KV-K.",
"The red histograms show the non detections and the black histograms show the marginal and definite detections."
],
[
"Average surface magnetic field",
"We obtained valid Zeeman broadening measurements on 396 spectra corresponding to 139 different stars (31% of our sample), which means that the large majority of spectra and stars in our sample do not exhibit a valid or measurable Zeeman broadening.",
"Spectroscopic binaries, stars with unknown $V-K$ and rotators with a $v \\sin i$ larger than 6 km/s have been excluded.",
"The $V-K$ of these stars span a range from 3.5 to 8, so approximately K8 to M7 types.",
"The largest number of stellar types with broadening measurements are M3 and M4 dwarfs.",
"Figure: Histogram of BfBf values.The histogram of $Bf$ values measured on 139 stars ranges from 0.5 to 3kG, with a marked peak at 1.5kG (Fig.",
"REF ).",
"The relative dispersion around the measured $Bf$ values for a given star is less than 20% in 76% of the sample, while only 10% of the stars have dispersions larger than 50%.",
"Fig.",
"REF shows the increase of the magnetic field as the star type gets redder.",
"A slope of 0.61 kG/($V-K$ )mag and a Pearson correlation coefficient of 66% is obtained between $Bf$ and $V-K$ , when spurious measurements and stars in spectroscopic binaries are excluded.",
"The dispersion of $Bf$ values as a function of spectral type has a similar amplitude than found on later type stars by .",
"Figure: Relations between the small-scale field BfBf and V-KV-K. Each point represents a star.",
"The error bars represent the dispersion between the different visits, when more than 2 are available (numbered N).",
"The red points show GJ 793, GJ 251 and GJ 1289.The measured broadening can be measured only for slow rotators.",
"Beyond a $v \\sin i$ of $\\sim $ 6 km/s, the individual FeH lines cannot be properly measured and get blended by neighbour lines with our method.",
"Faster rotators yet tend to have an average surface magnetic strength of larger amplitude."
],
[
"The magnetic field of GJ 1289, GJ 793, and GJ 251",
"Zeeman signatures are clearly detected in Stokes $V$ LSD profiles with a maximum peak-to-peak amplitude varying from 0.1% to 0.5% of the unpolarized continuum level for both data sets of GJ 1289, GJ 793, and GJ 251.",
"The temporal variations of the intensity and of the shape of the Stokes $V$ LSD profile are considered to be due to rotational modulation.",
"Figure REF shows the observed and best-fit Stokes $V$ profiles for the three stars."
],
[
"GJ 1289 = 2MASS J23430628+3632132",
"GJ 1289 is a fully convective star with $M_{\\star }$ =$~0.23~$${\\rm M}_{\\odot }$ (from the absolute $K$ magnitude calibration of ) and $R_{\\star }$ $\\sim ~0.24~$${\\rm R}_{\\odot }$ (see details in Fouqué et al, in prep.).",
"The stellar rotation period was determined by ZDI and checked using the periodogram of various proxies ($B_l$ and several activity indices).",
"Both methods concur to the value $P_{\\rm rot}$ = $54~\\pm ~4$ d (see Fig.",
"REF , top panel).",
"We thus used the following ephemeris: HJD (d) = 2457607.01471 + 54.E, in which E is the rotational cycle and the initial Heliocentric Julian Date (HJD) is chosen arbitrarily.",
"Also, from the convection-timescale ($\\tau _c$ ) calibration in and this rotational period, we infer a Rossby number of 0.62.",
"After several iterations, the values of stellar inclination $i$ = $60~\\pm ~15$ and $v \\sin i$ = $1~\\pm ~1~$ km s$^{-1}$ are found to give the optimal field reconstruction.",
"With the Zeeman Doppler Imaging technique, it is possible to adjust the Stokes $V$ profiles down to a $\\chi ^2_r$ of 1.5, while starting from an initial value of 9.6 for a null field map.",
"The reconstructed large-scale magnetic field is purely poloidal (99% of the reconstructed magnetic energy) and mainly axisymmetric (90% of the poloidal component).",
"The poloidal component is purely dipolar (99%).",
"These results are in agreement with the observed shape of the Stokes $V$ signatures which are anti-symmetric with respect to the line center with only the amplitude varying as the star rotates.",
"The inclination of the dipole with respect to the stellar rotation axis ($\\sim $ 30) explains these amplitude variations : the strongest profiles observed between phases 0.0 and 0.2 directly reflect the crossing of the magnetic pole in the centre of the visible hemisphere, whereas the weakest Stokes $V$ (i.e., phase 0.723) are associated with the magnetic equator.",
"The magnetic field strength averaged over the stellar surface is 275 G. Figure REF (top) shows the reconstructed topology of the magnetic field of GJ 1289, featuring the dipole with a strong positive pole reaching $\\sim $ 450 G. The stellar inclination $i$ towards the line of sight is mildly constrained: a high inclination (>45) allows a better reconstruction, i.e., it minimizes both $\\chi ^2_r$ and the large-scale field strength.",
"In order to fit the circularly polarized profiles, we used a filling factor $f_{V}$ , adjusted once for all profiles: it represents the average fraction of the flux of magnetic regions producing circular polarization at the surface of the star.",
"ZDI reconstructs only the large-scale field, however the large-scale field can have a smaller scale structure (e.g., due to convection or turbulence).",
"This parameter then allows to reconcile the discrepancy between the amplitude of Stokes $V$ signatures (constrained by the magnetic flux $B$ ) and the Zeeman splitting observed in Stokes $V$ profiles (constrained by the magnetic field strength $B/f_{V}$ ).",
"While $f_{V}$ has no effect on the modeling of the intensity profiles of adjusting GJ 1289, it is essential to fit the width of the observed Stokes $V$ profiles, as was shown for other fully convective stars in earlier studies .",
"The $f_{V}$ value found for GJ 1289 is 0.15.",
"We have then compared the derived field modulus obtained from this broadening $Bf$ estimated in the unpolarized light with the longitudinal field B$_l$ measured in the polarized light (its absolute value).",
"The data is shown on Figure REF .",
"As expected, and consistently with the large sample, there is no strong correlation between the large-scale topology and the small-scale magnetic field; the trend, although not significant, is positive, with a Pearson coefficient of 0.4.",
"The rotationally-modulated signal of the main dipole obeys to large-scale dynamo processes while small-scale magnetic regions may rather be induced by convective processes, especially in a fully-convective star such as GJ 1289.",
"Figure: Maximum-entropy fit (black line) to the observed (red line) Stokes VV LSD photospheric profiles of GJ 1289 (left), GJ 793 (middle), and GJ 251 (right).",
"Rotational cycles and 3σ\\sigma error bars are also shown next to each profile."
],
[
"GJ 793 = 2MASS J20303207+6526586",
"GJ 793 is a partly convective low-mass star with $M_{\\star }$ =$~0.42~$${\\rm M}_{\\odot }$ and $R_{\\star }$ $\\sim ~0.39~$${\\rm R}_{\\odot }$ .",
"The $v \\sin i$ of GJ 793 is smaller than 1 km/s.",
"In the Generalised Lomb-Scargle periodogram of the activity indices and $B_l$ , different peaks stand out at $\\sim $ 35 d, $\\sim $ 17 d and $\\sim $ 11 d with various significance levels (see Fig.",
"REF ).",
"The lowest false-alarm probability ($\\sim $ 10%) is reached for a peak at 35.3 d in H$_\\alpha $ , a set of peaks associated with its first harmonic between 17 and 17.7 d in H$_\\alpha $ and $B_l$ , and a peak at 11.3 d in $B_l$ .",
"A peak at 22 d is also present in the periodogram of $B_l$ but with a lower significance than its first harmonic.",
"So the rotation period cannot be unambiguously defined from the periodograms.",
"The rotation period suggested by the CaII HK calibration is $\\sim $ 27 days.",
"Moreover, given the poor sampling of the data and the weakness of the Stokes $V$ signatures ($\\leqslant $ 0.3% of the continuum level), the tomographic analysis is not able to precisely constrain the rotation period $P_{\\rm rot}$ and the inclination $i$ .",
"We then tried different values of ($P_{\\rm rot}$ , $i$ ) to reconstruct the large-scale field, and we looked for the cases which minimise the value of $\\chi ^2$ , from an initial $\\chi ^2$ of 3.1.",
"Several minima are found corresponding to a rotation period of $\\sim $ 22 d or $\\sim $ 34 d. The best-fit associated inclination is $i$ < 40 for both cases.",
"However a secondary minimum is found for $i\\sim $ 80 and $P_{\\rm rot}$ = 34 d. All maps are associated with a magnetic field strength of $\\sim $ 210 G. We explored the possibility that the model degeneracy could be due to differential rotation (DR).",
"Contrarily to a solid-body rotation hypothesis, some amount of differential rotation would allow to remove the secondary minima of the stellar inclination, and to minimise the reconstructed magnetic field strength (down to 75 G).",
"The data sampling is, howeover, insufficient to correctly probe the differential-rotation properties.",
"Note that differential rotation has already been observed for faster rotating early M dwarfs (e.g., GJ 410 or OT-Ser in ).",
"They found that the surface angular rotation shear can range 0.06 to 0.12 rad.d$^{-1}$ .",
"Such a large rotation shear for a slowly rotating star as GJ 793 is, however, rather unexpected (if 34 days were the pole rotation period, this would require a $d\\Omega $ of 0.1 rad.d$^{-1}$ ).",
"A confirmation and better determination of the DR in GJ 793 would require a better sampling of the stellar rotation cycle as is available in the current data set.",
"Rotational cycles on Table REF and Figure REF are computed from observing dates according to the following ephemeris: HJD (d) = 2457603.95552 + 22.E.",
"Given the rotation period of 22d, the Rossby number of GJ 793 is 0.46, using relations in .",
"The large-scale magnetic field in the configuration shown on Fig.",
"REF (middle panel) is 64% poloidal and axisymmetric (82% of the poloidal component).",
"The poloidal component would be mainly quadrupolar (> 66%).",
"Due to the low signal and the uncertainties in the rotational properties, however, this field reconstruction is the least robust of the three and would benefit additional data; several equivalent solutions differ in topology and field strength (Table REF ).",
"In this case, the filling factor $f_{V}$ was not needed to fit the Stokes $V$ profiles, which is expected for stars with a shorter rotation period or not fully convective .",
"The average small-scale field strength ranges from 1400 to 2000 G. The average $Bf$ of GJ 793 is significantly lower than for the lower-mass GJ 1289.",
"In Figure REF , compared to GJ 1289, the location of the data points of GJ 793 is more confined in $|B_l|$ , and more spread out in $Bf$ .",
"A slight negative trend between both measurements is visible, with a negative Pearson coefficient -0.5.",
"Figure: Surface magnetic flux as derived from our data sets of GJ 1289 (top), GJ793 (middle), and GJ 251 (bottom).",
"The map obtained for GJ 793 is one of several equivalent solutions and more data is needed for a confirmation.The radial (left), azimuthal (center) and meridional (right) components of the magnetic field BB are shown.",
"Magnetic fluxes are labelled in G. The star is shown in a flattened polar projection down to latitude -30, with the equator depicted as a bold circle and parallels as dashed circles.",
"Radial ticks around each plot indicate phases of observations.This figure is best viewed in colour.Figure: BfBf as a function of the absolute value of the longitudinal field for GJ 1289 (red diamonds), GJ 251 (black triangles) and GJ 793 (green circles)."
],
[
"GJ 251 = 2MASS J06544902+3316058 ",
"GJ 251 is another partly convective low-mass star with $M_{\\star }$ =$~0.39~$${\\rm M}_{\\odot }$ and $R_{\\star }$ $\\sim ~0.37~$${\\rm R}_{\\odot }$ but a longer rotation period than GJ 793.",
"A rotation period estimate of $\\sim $ 85 days from the activity index seems to correspond to the observed profile variability and is refined to 90 days by the ZDI analysis.",
"The ephemeris of HJD (d) = 24576914.643 + 90.E has been used.",
"A projected velocity $v \\sin i$ smaller than 1 km/s is used for this star.",
"The Rossby number of GJ 251, considering a rotation period of 90 days, is 1.72 .",
"The Stokes $V$ profiles of GJ 251, after correction of the mean residual $N$ signature, show a low level of variability over almost 10 stellar rotation cycles.",
"The amplitude of the circularly polarised signatures never exceeds 0.2% peak-to-peak.",
"It is difficult with such a weak signal to strongly constrain the rotational period and the stellar inclination.",
"The initial $\\chi _r^2$ of the data is 2.2.",
"A minimum $\\chi _r^2$ of 1.1 is found for a period of 90$\\pm $ 10 days and an inclination of 30$\\pm $ 10.",
"However, a secondary minimum of 40 d is found for the rotational period, due to scarce data sampling.",
"The reconstructed ZDI map is shown on Figure REF (using these values of $P_{\\rm rot}$ and $i$ ) and features a topology with a strong poloidal component encompassing 99% of the magnetic energy, a pure dipole.",
"This poloidal component is also mostly axisymmetric (88%), with an average field of only 27.5 G. GJ 251 has a lower magnetic field than the other two stars, as shown on Fig.",
"REF .",
"For this star, the trend between $Bf$ and $|B_l|$ values is insignificant and lower than for both other stars (Pearson coefficient is -0.1).",
"The sparse spectropolarimetric sampling and very long rotation period make it difficult to get a robust reconstruction, and it is the first time that data spanning such a long period of time are used in ZDI, so the results for this star have to be taken with caution, although the field reconstruction seems robust.",
"It is possible, for instance, that the hypothesis of the signal modulation being due to rotation is wrong, as the topology itself could vary over several years.",
"Table REF summarizes the ZDI parameters and fundamental parameters for the three stars.",
"Table: Summary of magnetic field parameters for GJ 1289, GJ 793 and GJ 251. τ c \\tau _c is the convection timescale , used to calculate the Rossby number RoRo, dΩ\\Omega is the differential rotation and ii is the inclination of the rotation axis with respect to the line of sight.",
"The topology is characterized by the mean large-scale magnetic flux B, the percentage of magnetic energy in the poloidal component (% pol) and the percentage of energy in the axisymmetric component of the poloidal field (% sym).",
"For GJ 793, we report a wide range of values, since several configurations are compatible with the data."
],
[
"Magnetic topology of M dwarfs",
"The study of the large-scale stellar magnetic field is interesting for the exoplanetary study as it allows to better explore conditions for habitability .",
"However, understanding its origin remains challenging, and more particularly for fully convective stars.",
"The large-scale magnetic field is generated in the stellar interior.",
"In solar and partially convective stars, a shearing is expected to take place in a boundary layer located between the inner radiative core and the outer convective envelope.",
"Part of the magnetic field generation comes from the convective envelope itself.",
"Stars less massive than 0.35${\\rm M}_{\\odot }$ are fully-convective and therefore their convective envelope is fully responsible for the magnetic field generation.",
"To study and compare the magnetic field of low-mass stars, we fill up the $M_{\\star }$ - $P_{\\rm rot}$ diagram with characteristics of the topology, as initiated in .",
"In order to solve questions about the dynamo, it is crucial to detect and have access to the geometry of the field and explore the space of parameters (stellar internal structure and rotation properties).",
"In that perspective, results we obtained for fully convective slowly rotating stars like GJ 1289 are very interesting and bring new observational constraints to models in which the dynamo originates throughout the convection zone.",
"The 3 stars presented in this paper, with their long rotation periods, cover a poorly explored domain so far (see Figure REF ).",
"The magnetic topology of GJ 793 and GJ 251 resembles the diverse topologies and weak fields found so far for the partly convective slowly rotation stars, like GJ 479 or GJ 358.",
"On the other hand, we find that the magnetic field detected for GJ 1289 exhibits a strength of a few hundreds of Gauss, as AD Leo.",
"While much larger than the field of GJ 793 and GJ 251, the field of GJ 1289 is lower by a factor 3 than those of more active and rapidly rotating mid-M dwarfs .",
"Its large-scale magnetic field is dipole dominated and therefore is similar to the topology of more rapidly rotating low-mass stars, rather than to the field of slowly rotating Sun-like stars.",
"Both GJ 1289 and GJ 251 have a large rotation period ($\\geqslant $ 50 d), but a different internal structure.",
"Our results shows that slowly rotating stars without tachocline (as GJ 1289) tend to have a relatively strong dipolar field rather than the weaker field of slow and partly convective stars.",
"Therefore the maps we obtained tend to confirm the key role of the stellar structure.",
"This is also supported by the earlier observations of large-scale magnetic fields of fully convective stars, although on faster rotators .",
"A recent X-ray study carried out using Chandra showed that slowly rotating fully convective M dwarfs can behave like partly convective stars in terms of X-ray luminosity - rotation relation.",
"X-ray luminosity is a tracer of the surface magnetic activity and is believed to be driven by the stellar magnetic dynamo.",
"Their result may thus give another observational evidence that a tachocline is not necessarily critical for the generation of a large-scale magnetic field, and that both stars with and without a tachocline appear to be operating similar magnetic dynamo mechanisms."
],
[
"Relations between magnetism and activity",
"In Figure REF , the collected data has been combined so that each data point is a star of the sample rather than a spectrum.",
"The median (average) value of each plotted quantity has been calculated when more than three (resp., only two) spectra are available.",
"The error bars represent the dispersion between the measurements for a single star, when it was possible to calculate the standard deviation.",
"When spectroscopic mode and polarimetric mode spectra were available, the data were all combined together, since the spectral resolution of both modes is similar.",
"The red symbols show results for GJ 1289, GJ 793 and GJ 251.",
"We observe the trend that more active stars (i.e., stars with a larger average CaII IRT index) have a stronger small-scale magnetic field.",
"There is a Pearson correlation coefficient of 70% between the CaII IRT index and the $Bf$ in our sample of 139 stars where these quantities are measured.",
"For instance, GJ 793 and GJ 251 have a similar average CaII IRT index but marginally different $Bf$ values of 1.72$\\pm $ 0.22 and 1.26$\\pm $ 0.22 kG, respectively.",
"Similar behaviour of chromospheric indices as a function of field modulus were found by .",
"Finally, in Figure REF , we show how the measurement of the unsigned value of the longitudinal large-scale field compares to the small-scale field measured through the Zeeman broadening of the FeH magnetically sensitive line.",
"Both parameters have been conjointly measured on a total of 151 spectra and 59 different stars.",
"The figure illustrates how the large-scale field can span several orders of magnitude (y-axis is in logarithmic scale) for a given small-scale field value (x-axis in linear scale).",
"There is a 47% correlation coefficient between both quantities.",
"While inclination and topology impact the way one pictures the large-scale field, the small-scale field accounted for in $Bf$ is concentrated on active regions that can be seen at a wider range of inclinations and, for active stars, at most rotational phases.",
"A possible retroaction of one scale to another may also be due to physical processes (related convection and dynamo) which differ from a star to another.",
"So a moderate correlation between these quantities of the global sample may result from a mix of stars where the correlation may vary widely due to differences in the field topology.",
"The percentage of the maximum longitudinal field with respect to the total $Bf$ field is less than 5% in most of the sample and rarely beyond 10%.",
"For GJ 1289, GJ 793, and GJ 251, it is, respectively, 5.5, 1.9, and 1.6%.",
"Figure: The large-scale longitudinal field as a function of the small-scale average field.",
"Note the linear scale of the x-axis and logarithmic scale of the y-axis; most B l B_l errors are within the symbol size."
],
[
"Prospects for stellar jitter and planet search",
"Ultimately, the study of activity diagnostics on stars on which radial-velocity planet search will be conducted needs to assess how each diagnostic contributes to the stellar RV jitter.",
"In this study, we do not use the stellar radial-velocity jitter measured by ESPaDOnS, because this spectrograph is not optimized for RV precision better than $\\sim $ 20m/s (instrumental floor) .",
"For instance, our 22 RV measurements of GJ 793 have an $rms$ of 17 m/s.",
"Figure: Literature RV jitter values as a function of the small-scale magnetic field energy (Bf) 2 (Bf)^2 (top) and the chromospheric index F chr F_{chr} (bottom)We have thus searched the literature for all published RV jitter values due to rotational activity and cross-matched these values with our sample, focusing on the stars for which a small-scale magnetic field is measured (using the value averaged over all different spectra for a given star).",
"We found 20 stars for which both measurements are available.",
"It must be noted, however, that $Bf$ and RV jitter measurements are not contemporaneous, since $Bf$ measurements come from the ESPaDOnS spectra and RV jitter measurements come from ESO/HARPS or Keck/HIRES instruments.",
"All data roughly come from the last decade, but this span may be very large compared to characteristic activity timescales of some of these stars.",
"This time discrepancy is expected to increase the dispersion since these activity proxies are naturally expected to evolve with time.",
"Also, there may be signals of yet-undetected planets in some of these stars (signals for some, and noise for others!",
"), which would artificially increase the jitter value.",
"Finally, the RV jitter is expected to (slowly) vary with the wavelength and we did not take this into account since all spectrographs are in the optical.",
"As these RV jitters have been measured with HARPS or HIRES, which have very similar spectral ranges, and on M stars, we expect that the effective wavelength of this jitter is toward the red end of the instrumental bandpasses, at about 650 nm.",
"The most sensitive wavelength of M-star spectra with ESPaDOnS is around 730 nm.",
"An arbitrary error of 1.5 m/s was applied to all RV jitter values, in excess to the values quoted in the respective papers; it aims at accounting for the uncertainty due to the reasons described above (non-contemporaneity primarily, then chromaticity).",
"The possible presence of planet signals could, evidently, account for a larger excess.",
"In Table REF , the stars with a know RV jitter are listed and references to the RV jitter values are provided.",
"The quantities are plotted on Figure REF as a function of $(Bf)^2$ and of the total chromospheric flux $F_{chr}$ (see section 4.1).",
"Apart from AD Leo (GJ 388), all stars have a reported RV jitter smaller than 10m/s and also lie at the lower end of the magnetic field/activity scale.",
"The stars shown with an insert green circle are those where exoplanets have already been found and characterized (see references in Table REF ) and their signal(s) have been removed from the RV variations shown here.",
"It is clear that the RV jitter of these stars is closer to the instrumental threshold than most of the others, except for GJ 179 for which the jitter is relatively high while the small-scale magnetic field is weak.",
"From the original discovery paper, however, it seems likely that instrumental jitter may partly account for the excess jitter .",
"On the other hand, it is surprising that GJ 876 shows a low RV jitter and a large Zeeman broadening.",
"There are, however, only two ESPaDOnS measurements of GJ 876, taken 20 days apart (the rotational period of this star is estimated to be 91 days from ) while RV measurements span a period of more than 8 years.",
"Also, as discussed previously, the average magnetic strength of GJ 876 was undetected by the spectral-synthesis approach and an upper limit of 0.2 kG had been estimated .",
"Assuming that the RV jitter is mainly due to the Zeeman broadening -which can be significant in these stars-, we searched for a quadratic behaviour of the RV jitter, as advocated in and found no significant correlation.",
"In the same way, jitter and chromospheric activity are only mildly related for stars where these quantities are small; using log($R^{\\prime }_{HK}$ ) rather than $F_{chr}$ does not make the trend stronger.",
"The latter conclusion does not support the findings of , in which a linear relationship could be found between the RV jitter and the log($R^{\\prime }_{HK}$ ), per spectral type.",
"This disagreement may come from the choice of targets, or it may due to the contemporaneity of data.",
"Table: For all stars in our sample that have radial-velocity activity-jitter values in the literature, are also listed: the photometric index V-KV-K, the measured average field modulus, the adopted projected rotational velocity vsiniv \\sin i, the total chromospheric emission F chr F_{chr}, the maximum of the absolute value of the longitudinal field, and the activity merit function AMFAMF (see text).",
"Note that RV jitter are not contemporaneous to the other measurements.",
"The \"p\" in col. 3 is a flag for known exoplanet systems.Although most RV programs have leaned towards removing most active stars from their input sample in the past, it is not the only way to handle the problem of activity induced jitter.",
"For M stars especially, eliminating active stars may be a strong limiting factor, especially for later types where activity is more pronounced.",
"has shown that the temporal behaviour of RV jitter in M stars was twofold: 1) a rotational component with signatures modulated at the rotational period and its harmonics and 2) a random component.",
"This study also demonstrated that characterizing the rotational properties of stars (rotation period and differential rotation) from Doppler Imaging turned out to be a powerful asset in modeling the rotationally-modulated component of this jitter.",
"The use of ZDI also proved powerful in filtering out the activity for the T Tauri stars V830 Tau and TaP 26 , allowing the detection of the hot-Jupiter planets orbiting these extremely active stars.",
"Thus, as dealing with activity of M dwarfs in a planet-search RV survey is inevitable, it actually has mitigating solutions when the activity signature can be understood, measured and (at least partially) filtered out."
],
[
"A merit function for activity?",
"We then attempt a classification of stars by considering the multiple measurements that are indirectly related to activity: the non-thermal radiation from the chromosphere in different lines, the average field modulus, the properties of the large-scale field, and the rotational velocity.",
"The degree of correlation between these diagnostics is variable and not very high, as seen earlier, but we can still combine various indicators to build up a quantitative merit function that is relevant to the level of jitter amplitude and thus, compare the relevance of stars for exoplanet search.",
"While chromospheric emission and rotational velocity can be estimated in all spectra, the other diagnostics are not always measurable, which results in some inhomogeneities in building up this merit function.",
"Our attempt to estimate the activity merit function of a given star is given below: $AMF = w_c\\frac{N_c}{F_{chr}} +w_{B}\\frac{N_{B}}{max|B_l|}+w_v\\frac{N_v}{v \\sin i}+w_{Z}\\frac{N_{Z}}{\\delta v_B}$ where $F_{chr}$ is the total chromospheric emission, $max|B_l|$ the absolute value of the longitudinal field, and $\\delta v_B$ the Zeeman broadening.",
"The different $w$ and $N$ are weight factors and normalization factors, respectively.",
"Normalization factors are chosen as the median of each parameter.",
"Weight factors are more arbitrary as they depend upon the objective of the ranking.",
"Figure: The adjusted RV jitter for known planet host stars in our sample, as a function of the measured one.",
"The line shows Y=X.",
"The RV fit is done from the measured or assumed values of max(|B l B_l|), BfBf, vsiniv \\sin i and F chr F_{chr}.In order to find the best weighting factors, we used a multi-variable fit with the four diagnostic parameters and adjust the RV jitter of the planet-host sample.",
"This sub-sample is preferred to the whole sample with RV jitters shown earlier, as planet signals should mostly be removed.",
"The best-fit is shown on Figure REF .",
"The system with a large measured RV jitter and small predicted jitter is still GJ 179 , where the jitter may be over-estimated, as discussed earlier.",
"On the other hand, GJ 876 predicted jitter is 1.7 m/s, very close to the 1.99 m/s measured value, despite the large field modulus.",
"The coefficients derived from this fit tell us that 39, 18, 35 and 8% of the jitter contribution come from, respectively, max(|$B_l$ |), $Bf$ , $F_{chr}$ and $v \\sin i$ .",
"It thus gives a larger weight to the longitudinal field and chromospheric emission.",
"We then used this weighting factors to derive the activity merit function and a predicted jitter value for the 442 stars in our sample.",
"When no longitudinal field is available because the spectroscopic mode is used, we impose a median value to this parameter based on the histogram of $B_l$ , in order to have a neutral effect on the ranking.",
"When the spectrum is in polarimetric mode and the detection is null, we adopt a high value.",
"Finally, when the Zeeman broadening is not detected while the star is a slow rotator, we also adopt a high value.",
"The ranking is measured for each of the 1878 spectra and then averaged out per star.",
"The final $AMF$ ranges from 0 to 103.",
"The individual values for the sub-sample of stars on which the Zeeman broadening is detected are listed in the Appendix Table REF .",
"Figure REF shows the values obtained as a function of the $V-K$ colour index.",
"There is no visible colour effect between $V-K$ values of 3 to 6, while most late-type stars tend to have a low activity merit function.",
"In order to select the quietest stars in each bin of spectral type, one should set a simple horizontal threshold, such as the line shown in Figure REF .",
"Most of the slowest rotators in our sample lie above the line (and thus are deemed relevant targets for planet search).",
"Interestingly, all slow rotators below the line, except one (GJ 406) are actually part of a spectroscopic binary system (denoted with a circling cyan diamond for clarity); this is not clear if the activity of those stars is actually enhanced, or if, in some cases, the measurements are impacted by the binarity.",
"The predicted median jitter for stars with $AMF$ greater than 20 is 2.3 m/s, with most stars having a jitter less than 4 m/s.",
"On the other hand, the median is 4.2 m/s for stars having an $AMF$ smaller than 20 and their distribution has a long tail towards large jitter values.",
"The threshold of $AMF=20$ , as shown on the figure selects $\\sim $ 40% of the stars, with some distribution in colour.",
"The mean $V-K$ colour of stars above (below) the line is 4.57 (resp., 4.93), so there is a definite tendency for late-type M in our sample to be less quiet than earlier type M stars.",
"Finally, it is interesting to note that the merit function shows a bimodal distribution, in the same way as rotation periods of M stars show , .",
"Stars with known exoplanets are featured in Figure REF as blue stars, and they all lie at high values of the activity merit function.",
"It could be expected since they have been used to derive the coefficients or the $AMF$ , but it is also due to the fact that past RV surveys have barely observed active M stars, and even less found planets around them.",
"The stars GJ 251, GJ 793, and GJ 1289 (red squares) also get a relatively high ranking of, respectively, 44, 39 and 23 and predicted jitter values of 2.9, 2.6 and 6.0 m/s.",
"Stars with measured RV jitter values are depicted as green circles, with a size that is proportional to the jitter.",
"Apart from GJ 388 and GJ 1224, all stars with measured jitter have a high activity merit function.",
"Figure: The activity \"merit function\" (see text) as a function of the V-KV-K colour index.",
"Symbols with an inner orange circle show the slow rotators (unresolved profiles), green symbols show the stars with known activity jitter measurements with a size that is proportional to the jitter, and red squares show the three stars GJ 251, GJ 793 and GJ 1289.",
"Cyan diamonds indicate the stars that are a component of spectroscopic binary systems.",
"The blue star symbols show hosts of known exoplanet systems.",
"The dash line is an arbitrary threshold proposed to select stars most favorable to planet searches.Further improvements of the activity merit function could still be obtained with a more thorough investigation of the stars with a measured jitter value, and a larger sample of these stars with a wider range of jitter values.",
"It would be important, for instance, to precisely quantify the part of the RV $rms$ due to rotation modulated activity from other noise sources, as planets and instrumental systematics.",
"It would then be possible to derive the best merit function from the non-rotationally modulated jitter itself."
],
[
"Summary and Conclusion",
"In this study, we have collected and homogeneously analyzed all CFHT/ESPaDOnS data taken on the large sample of 442 M dwarfs, with a focus on their activity properties.",
"Stellar activity takes many different faces, and we are interested in any and all diagnostics that relate to the radial-velocity stellar jitter, with the objectives of selecting proper targets and preparing efficient activity filtering techniques in planet-search programs.",
"As ESPaDOnS cannot measure radial velocity jitter itself, we have cross-matched our data with HARPS or HIRES published data and shown that the amplitude of the RV jitter is somehow predictable: large Zeeman broadening, strong magnetic strength in the large scale, fast rotation and/or large chromospheric emission are all prone to higher activity jitter, with identified relative contributions: the maximum longitudinal field proves to be a quantity as important as the non-thermal emission, which shows the importance of measuring circular polarisation of stars.",
"As commented in , this RV jitter usually has a rotational component and a non-periodic component.",
"If the first one can reasonably be filtered out from RV time series , , , , , the second component is more evasive and would benefit from additional contemporaneous spectroscopic indicators, as the Zeeman broadening variation and chromospheric (flaring) emission.",
"The mild correlation between the emission of several chromospheric emission lines in the optical demonstrates that these activity tracers are not straightforward and deserve a cautious analysis.",
"On the other hand, the role of the rotation period to improve the activity filtering efficiency is critical.",
"If rotation periods can be measured photometrically , or are usually clearly found in the line circular polarization signal , it can also be inferred from the level of the non-thermal emission observed in the CaII or H$\\alpha $ lines , .",
"These latter relations are only valid for the slow rotators out of the saturation regime.",
"It is a primordial characterization of the system, as it modulates the activity and may interfere with planetary signals.",
"In the nIR domain where the next-generation spectropolarimeter CFHT/SPIRou will operate, spectroscopic diagnostics of activity are still to be explored, in particular their effect on the radial-velocity jitter.",
"SPIRou spectra will, however, allow a more general measurement of the Zeeman broadening since this effect is larger in the nIR for a given magnetic field modulus, and it is expected that this measurement on several atomic and biatomic lines will allow to trace the jitter due to localized magnetic regions, as simulated in and convincingly shown for the Sun .",
"In addition to the global description of the 442 star sample, we have reconstructed the magnetic topology at the surface of three stars that had not yet been scrutinized: the partly convective stars GJ 793 and GJ 251 and the fully convective star GJ 1289.",
"All three stars have long rotation periods (22, 90, and 54 days, respectively) and are relatively quiet.",
"With a mass lower than 0.45 ${\\rm M}_{\\odot }$ , they represent the type of stars that SPIRou could monitor in search for exoplanets.",
"Their surface magnetic field is similar to the field of most M stars with much shorter rotation periods or Rossby numbers smaller than 0.1, with a predominent poloidal topology.",
"Further work will focus on the implications for the dynamo processes in M stars, especially in the transition zone from partly to fully convective stars, as in , and .",
"The SPIRou input catalog generation will use the inputs from this work as well as the distribution of fundamental parameters (temperature, gravity, mass, metalicity and projected velocity, see Fouqué et al, in prep.)",
"to characterize and select targets of the SPIRou survey.",
"In addition to the archive of M stars observed with ESPaDOnS, other catalogs and other archives are explored to complete the list of low-mass stars with relevant properties for planet search.",
"This information gathering and method of target selection, including activity characterization, will be published in a subsequent work (Malo et al, in prep.",
")."
],
[
"Acknowledgements",
"The authors thank the referee Prof. Basri for his critical reading and highly appreciate the comments which significantly contributed to improving the quality of the publication.",
"The authors made use of CFHT/ESPaDOnS data.",
"CFHT is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii.",
"This work is based in part on data products available at the Canadian Astronomy Data Centre (CADC) as part of the CFHT Data Archive.",
"CADC is operated by the National Research Council of Canada with the support of the Canadian Space Agency.",
"This work has been partially supported by the Labex OSUG2020.",
"X.D.",
"acknowledges the support of the CNRS/INSU PNP and PNPS (Programme National de Planétologie and Physique Stellaire)."
]
] | 1709.01650 |
[
[
"Privacy Risk in Machine Learning: Analyzing the Connection to\n Overfitting"
],
[
"Abstract Machine learning algorithms, when applied to sensitive data, pose a distinct threat to privacy.",
"A growing body of prior work demonstrates that models produced by these algorithms may leak specific private information in the training data to an attacker, either through the models' structure or their observable behavior.",
"However, the underlying cause of this privacy risk is not well understood beyond a handful of anecdotal accounts that suggest overfitting and influence might play a role.",
"This paper examines the effect that overfitting and influence have on the ability of an attacker to learn information about the training data from machine learning models, either through training set membership inference or attribute inference attacks.",
"Using both formal and empirical analyses, we illustrate a clear relationship between these factors and the privacy risk that arises in several popular machine learning algorithms.",
"We find that overfitting is sufficient to allow an attacker to perform membership inference and, when the target attribute meets certain conditions about its influence, attribute inference attacks.",
"Interestingly, our formal analysis also shows that overfitting is not necessary for these attacks and begins to shed light on what other factors may be in play.",
"Finally, we explore the connection between membership inference and attribute inference, showing that there are deep connections between the two that lead to effective new attacks."
],
[
"Introduction",
"Machine learning has emerged as an important technology, enabling a wide range of applications including computer vision, machine translation, health analytics, and advertising, among others.",
"The fact that many compelling applications of this technology involve the collection and processing of sensitive personal data has given rise to concerns about privacy [1], [2], [3], [4], [5], [6], [7], [8], [9].",
"In particular, when machine learning algorithms are applied to private training data, the resulting models might unwittingly leak information about that data through either their behavior (i.e., black-box attack) or the details of their structure (i.e., white-box attack).",
"Although there has been a significant amount of work aimed at developing machine learning algorithms that satisfy definitions such as differential privacy [8], [10], [11], [12], [13], [14], the factors that bring about specific types of privacy risk in applications of standard machine learning algorithms are not well understood.",
"Following the connection between differential privacy and stability from statistical learning theory [12], [15], [13], [14], [16], [17], one such factor that has started to emerge [7], [4] as a likely culprit is overfitting.",
"A machine learning model is said to overfit to its training data when its performance on unseen test data diverges from the performance observed during training, i.e., its generalization error is large.",
"The relationship between privacy risk and overfitting is further supported by recent results that suggest the contrapositive, i.e., under certain reasonable assumptions, differential privacy [13] and related notions of privacy [18], [19] imply good generalization.",
"However, a precise account of the connection between overfitting and the risk posed by different types of attack remains unknown.",
"A second factor identified as relevant to privacy risk is influence [5], a quantity that arises often in the study of Boolean functions [20].",
"Influence measures the extent to which a particular input to a function is able to cause changes to its output.",
"In the context of machine learning privacy, the influential features of a model may give an active attacker the ability to extract information by observing the changes they cause.",
"In this paper, we characterize the effect that overfitting and influence have on the advantage of adversaries who attempt to infer specific facts about the data used to train machine learning models.",
"We formalize quantitative advantage measures that capture the privacy risk to training data posed by two types of attack, namely membership inference [6], [7] and attribute inference [4], [8], [5], [3].",
"For each type of attack, we analyze the advantage in terms of generalization error (overfitting) and influence for several concrete black-box adversaries.",
"While our analysis necessarily makes formal assumptions about the learning setting, we show that our analytic results hold on several real-world datasets by controlling for overfitting through regularization and model structure."
],
[
"Membership inference",
"Training data membership inference attacks aim to determine whether a given data point was present in the training data used to build a model.",
"Although this may not at first seem to pose a serious privacy risk, the threat is clear in settings such as health analytics where the distinction between case and control groups could reveal an individual's sensitive conditions.",
"This type of attack has been extensively studied in the adjacent area of genomics [21], [22], and more recently in the context of machine learning [6], [7].",
"Our analysis shows a clear dependence of membership advantage on generalization error (Section REF ), and in some cases the relationship is directly proportional (Theorem REF ).",
"Our experiments on real data confirm that this connection matters in practice (Section REF ), even for models that do not conform to the formal assumptions of our analysis.",
"In one set of experiments, we apply a particularly straightforward attack to deep convolutional neural networks (CNNs) using several datasets examined in prior work on membership inference.",
"Despite requiring significantly less computation and adversarial background knowledge, our attack performs almost as well as a recently published attack [7].",
"Our results illustrate that overfitting is a sufficient condition for membership vulnerability in popular machine learning algorithms.",
"However, it is not a necessary condition (Theorem REF ).",
"In fact, under certain assumptions that are commonly satisfied in practice, we show that a stable training algorithm (i.e., one that does not overfit) can be subverted so that the resulting model is nearly as stable but reveals exact membership information through its black-box behavior.",
"This attack is suggestive of algorithm substitution attacks from cryptography [23] and makes adversarial assumptions similar to those of other recent ML privacy attacks [24].",
"We implement this construction to train deep CNNs (Section REF ) and observe that, regardless of the model's generalization behavior, the attacker can recover membership information while incurring very little penalty to predictive accuracy."
],
[
"Attribute inference",
"In an attribute inference attack, the adversary uses a machine learning model and incomplete information about a data point to infer the missing information for that point.",
"For example, in work by Fredrikson et al.",
"[4], the adversary is given partial information about an individual's medical record and attempts to infer the individual's genotype by using a model trained on similar medical records.",
"We formally characterize the advantage of an attribute inference adversary as its ability to infer a target feature given an incomplete point from the training data, relative to its ability to do so for points from the general population (Section REF ).",
"This approach is distinct from the way that attribute advantage has largely been characterized in prior work [4], [3], [5], which prioritized empirically measuring advantage relative to a simulator who is not given access to the model.",
"We offer an alternative definition of attribute advantage (Definition REF ) that corresponds to this characterization and argue that it does not isolate the risk that the model poses specifically to individuals in the training data.",
"Our formal analysis shows that attribute inference, like membership inference, is indeed sensitive to overfitting.",
"However, we find that influence must be factored in as well to understand when overfitting will lead to privacy risk (Section REF ).",
"Interestingly, the risk to individuals in the training data is greatest when these two factors are “in balance”.",
"Regardless of how large the generalization error becomes, the attacker's ability to learn more about the training data than the general population vanishes as influence increases."
],
[
"Connection between membership and attribute inference",
"The two types of attack that we examine are deeply related.",
"We build reductions between the two by assuming oracle access to either type of adversary.",
"Then, we characterize each reduction's advantage in terms of the oracle's assumed advantage.",
"Our results suggest that attribute inference may be “harder\" than membership inference: attribute advantage implies membership advantage (Theorem REF ), but there is currently no similar result in the opposite direction.",
"Our reductions are not merely of theoretical interest.",
"Rather, they function as practical attacks as well.",
"We implemented a reduction for attribute inference and evaluated it on real data (Section REF ).",
"Our results show that when generalization error is high, the reduction adversary can outperform an attribute inference attack given in [4] by a significant margin."
],
[
"Summary",
"This paper explores the relationships between privacy, overfitting, and influence in machine learning models.",
"We present new formalizations of membership and attribute inference attacks that enable an analysis of the privacy risk that black-box variants of these attacks pose to individuals in the training data.",
"We give analytic quantities for the attacker's performance in terms of generalization error and influence, which allow us to conclude that certain configurations imply privacy risk.",
"By introducing a new type of membership inference attack in which a stable training algorithm is replaced by a malicious variant, we find that the converse does not hold: machine learning models can pose immediate threats to privacy without overfitting.",
"Finally, we study the underlying connections between membership and attribute inference attacks, finding surprising relationships that give insight into the relative difficulty of the attacks and lead to new attacks that work well on real data."
],
[
"Background",
"Throughout the paper we focus on privacy risks related to machine learning algorithms.",
"We begin by introducing basic notation and concepts from learning theory."
],
[
"Notation and preliminaries",
"Let $z = (x, y) \\in \\mathbf {X} \\times \\mathbf {Y} $ be a data point, where $x$ represents a set of features or attributes and $y$ a response.",
"In a typical machine learning setting, and thus throughout this paper, it is assumed that the features $x$ are given as input to the model, and the response $y$ is returned.",
"Let $ represent a distribution of data points, and let $ S n$ be an ordered list of $ n$ points, which we will refer to as a \\emph {dataset}, \\emph {training set}, or \\emph {training data} interchangeably, sampled i.i.d.\\ from We will frequently make use of the following methods of sampling a data point $ z$:\\begin{itemize}\\item z \\sim S: i is picked uniformly at random from [n], and z is set equal to the i-th element of S.\\item z \\sim : z is chosen according to the distribution \\end{itemize}When it is clear from the context, we will refer to these sampling methods as \\emph {sampling from the dataset} and \\emph {sampling from the distribution}, respectively.$ Unless stated otherwise, our results pertain to the standard machine learning setting, wherein a model $A_S$ is obtained by applying a machine learning algorithm $A$ to a dataset $S$ .",
"Models reside in the set $\\mathbf {X} \\rightarrow \\mathbf {Y} $ and are assumed to approximately minimize the expected value of a loss function $\\ell $ over $S$ .",
"If $z = (x, y)$ , the loss function $\\ell (A_S, z)$ measures how much $A_S (x)$ differs from $y$ .",
"When the response domain is discrete, it is common to use the 0-1 loss function, which satisfies $\\ell (A_S, z) = 0$ if $y = A_S (x)$ and $\\ell (A_S, z) = 1$ otherwise.",
"When the response is continuous, we use the squared-error loss $\\ell (A_S, z) = (y - A_S (x))^2$ .",
"Additionally, it is common for many types of models to assume that $y$ is normally distributed in some way.",
"For example, linear regression assumes that $y$ is normally distributed given $x$ [25].",
"To analyze these cases, we use the error function $\\operatorname{erf}$ , which is defined in Equation REF .",
"$\\operatorname{erf}(x) = \\frac{1}{\\sqrt{\\pi }} \\int _{-x}^x e^{-t^2} dt$ Intuitively, if a random variable $\\epsilon $ is normally distributed and $x \\ge 0$ , then $\\operatorname{erf}(x/\\sqrt{2})$ represents the probability that $\\epsilon $ is within $x$ standard deviations of the mean."
],
[
"Stability and generalization",
"An algorithm is stable if a small change to its input causes limited change in its output.",
"In the context of machine learning, the algorithm in question is typically a training algorithm $A$ , and the “small change” corresponds to the replacement of a single data point in $S$ .",
"This is made precise in Definition REF .",
"Definition 1 (On-Average-Replace-One (ARO) Stability) Given $S = (z_1, \\ldots , z_n) \\sim n$ and an additional point $z^{\\prime } \\sim , define $ S(i) = (z1, ..., zi-1, z', zi+1, ..., zn)$.",
"Let $$\\epsilon _{\\textrm {stable}}$ : N R$ be a monotonically decreasing function.",
"Then a training algorithm $ A$ is \\emph {on-average-replace-one-stable} (or \\emph {ARO-stable}) on loss function $$ with rate $$\\epsilon _{\\textrm {stable}}$ (n)$ if$$\\mathop {\\mathbb {E}}_{\\begin{array}{c}S \\sim n, z^{\\prime } \\sim i \\sim U(n), A\\end{array}}[\\ell (A_{S^{(i)}}, z_i) - \\ell (A_S, z_i)] \\le \\epsilon _{\\textrm {stable}} (n),$$where $ A$ in the expectation refers to the randomness used by the training algorithm.$ Stability is closely related to the popular notion of differential privacy [26] given in Definition REF .",
"Definition 2 (Differential privacy) An algorithm $A : \\mathbf {X} ^n \\rightarrow \\mathbf {Y} $ satisfies $\\epsilon $ -differential privacy if for all $S, S^{\\prime } \\in \\mathbf {X} ^n$ that differ in the value at a single index $i \\in [n]$ and all $Y \\subseteq \\mathbf {Y} $ , the following holds: $\\Pr [A(S) \\in Y] \\le e^\\epsilon \\Pr [A(S^{\\prime }) \\in Y].$ When a learning algorithm is not stable, the models that it produces might overfit to the training data.",
"Overfitting is characterized by large generalization error, which is defined below.",
"Definition 3 (Average generalization error) The average generalization error of a machine learning algorithm $A$ on i̥s defined as $R_{\\mathrm {gen}}(A, n, \\ell ) = \\mathop {\\mathbb {E}}_{\\begin{array}{c}S \\sim n\\\\ z \\sim {array}\\end{array}[\\ell (A_S, z)] - \\mathop {\\mathbb {E}}_{\\begin{array}{c}S \\sim n\\\\ z \\sim S\\end{array}}[\\ell (A_S, z)].",
"}In other words, A_S overfits if its expected loss on samples drawn from i̥s much greater than its expected loss on its training set.",
"For brevity, when $ $, and $$ are unambiguous from the context, we will write $ Rgen(A)$ instead.$ It is important to note that Definition REF describes the average generalization error over all training sets, as contrasted with another common definition of generalization error $\\mathop {\\mathbb {E}}_{z \\sim [\\ell (A_S, z)] - \\frac{1}{n} \\sum _{z \\in S} \\ell (A_S, z), which holds the training set fixed.",
"The connection between average generalization and stability is formalized by Shalev-Shwartz et al.~\\cite {ShalevShwartz10}, who show that an algorithm^{\\prime }s ability to achieve a given generalization error (as a function of n) is equivalent to its ARO-stability rate.",
"}$"
],
[
"Membership Inference Attacks",
"In a membership inference attack, the adversary attempts to infer whether a specific point was included in the dataset used to train a given model.",
"The adversary is given a data point $z = (x, y)$ , access to a model $A_S$ , the size of the model's training set $|S| = n$ , and the distribution t̥hat the training set was drawn from.",
"With this information the adversary must decide whether $z \\in S$ .",
"For the purposes of this discussion, we do not distinguish whether the adversary $\\mathcal {A}$ 's access to $A_S$ is “black-box”, i.e., consisting only of input/output queries, or “white-box”, i.e., involving the internal structure of the model itself.",
"However, all of the attacks presented in this section assume black-box access.",
"Experiment REF below formalizes membership inference attacks.",
"The experiment first samples a fresh dataset from ḁnd then flips a coin $b$ to decide whether to draw the adversary's challenge point $z$ from the training set or the original distribution.",
"$\\mathcal {A}$ is then given the challenge, along with the additional information described above, and must guess the value of $b$ .",
"Experiment 1 (Membership experiment $\\mathsf {Exp}^\\mathsf {M} (\\mathcal {A},A,n,$ ) Let $\\mathcal {A}$ be an adversary, $A$ be a learning algorithm, $n$ be a positive integer, and b̥e a distribution over data points $(x,y)$ .",
"The membership experiment proceeds as follows: Sample $S \\sim n$ , and let $A_S = A(S)$ .",
"Choose $b \\leftarrow \\lbrace 0, 1\\rbrace $ uniformly at random.",
"Draw $z \\sim S$ if $b = 0$ , or $z \\sim if $ b = 1$\\item $$\\mathsf {Exp}^\\mathsf {M}$ ($\\mathcal {A}$ ,A,n,$ is 1 if $$\\mathcal {A}$ (z, $A_S$ , n, = b$ and 0 otherwise.",
"\\mathcal {A} must output either 0 or 1.$ Definition 4 (Membership advantage) The membership advantage of $\\mathcal {A} $ is defined as $\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A},A,n, = 2 \\Pr [\\mathsf {Exp}^\\mathsf {M} (\\mathcal {A},A,n, = 1] - 1,$ where the probabilities are taken over the coin flips of $\\mathcal {A}$ , the random choices of $S$ and $b$ , and the random data point $z \\sim S$ or $z \\sim .$ Equivalently, the right-hand side can be expressed as the difference between $\\mathcal {A}$ 's true and false positive rates $\\mathsf {Adv}^\\mathsf {M}= \\Pr [\\mathcal {A} = 0 \\mid b = 0] - \\Pr [\\mathcal {A} = 0 \\mid b = 1],$ where $\\mathsf {Adv}^\\mathsf {M}$ is a shortcut for $\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A},A,n,$ .",
"Using Experiment REF , Definition REF gives an advantage measure that characterizes how well an adversary can distinguish between $z \\sim S$ and $z \\sim after being given the model.",
"This is slightlydifferent from the sort of membership inference described in someprior work~\\cite {ShokriSS17,Li2013}, which distinguishes between $ z S$ and $ zS$.",
"We are interested in measuring thedegree to which A_S reveals membership to \\mathcal {A}, and \\emph {not} in thedegree to which any background knowledge of $ S$ or d̥oes.",
"If wesample $ z$ from $ S$ instead of the adversary couldgain advantage by noting which data points are more likely to havebeen sampled into $ S n$.",
"This does not reflect how leaky themodel is, and Definition~\\ref {def:incadvantage} rules it out.$ In fact, the only way to gain advantage is through access to the model.",
"In the membership experiment $\\mathsf {Exp}^\\mathsf {M} (\\mathcal {A},A,n,$ , the adversary $\\mathcal {A}$ must determine the value of $b$ by using $z$ , $A_S$ , $n$ , and Of these inputs, $n$ and d̥o not depend on $b$ , and we have the following for all $z$ : $\\Pr [b = 0 \\mid z] &= \\Pr _{\\begin{array}{c}S \\sim n\\\\ z \\sim S\\end{array}}[z] \\Pr [b = 0] / \\Pr [z] = \\Pr _{z \\sim [z] \\Pr [b = 1] / \\Pr [z] = \\Pr [b = 1 \\mid z].",
"}We note that Definition~\\ref {def:incadvantage} does not give the adversary credit for predicting that a point drawn from i.e., when b = 1), which also happens to be in S, is a member of S. As a result, the maximum advantage that an adversary can hope to achieve is 1 - \\mu (n,, where \\mu (n, = \\Pr _{S\\sim n,z\\sim [z \\in S] is the probability of re-sampling an individual from the training set into the general population.",
"In real settings \\mu (n, is likely to be exceedingly small, so this is not an issue in practice.",
"}\\subsection {Bounds from differential privacy}Our first result (Theorem~\\ref {thm:dp-bound}) bounds the advantage of an adversary who attempts a membership attack on a differentially private model~\\cite {dwork06}.",
"Differential privacy imposes strict limits on the degree to which any point in the training data can affect the outcome of a computation, and it is commonly understood that differential privacy will limit membership inference attacks.",
"Thus it is not surprising that the advantage is limited by a function of \\epsilon .$ Theorem 1 Let $A$ be an $\\epsilon $ -differentially private learning algorithm and $\\mathcal {A}$ be a membership adversary.",
"Then we have: $\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A}, A, n, \\le e^\\epsilon - 1.$ Given $S = (z_1, \\ldots , z_n) \\sim n$ and an additional point $z^{\\prime } \\sim , define $ S(i) = (z1, ..., zi-1, z', zi+1, ..., zn)$.",
"Then, $$\\mathcal {A}$ (z',$A_S$ ,n,$ and $$\\mathcal {A}$ (zi,$A_{S^{(i)}}$ ,n,$ have identical distributions for all $ i [n]$, so we can write:{\\begin{@align*}{1}{-1}\\Pr [\\mathcal {A} = 0 \\mid b = 0] &= 1 - \\mathop {\\mathbb {E}}_{S\\sim n}\\left[\\frac{1}{n}\\sum _{i=1}^n \\mathcal {A} (z_i, A_S, n, \\right] \\\\\\Pr [\\mathcal {A} = 0 \\mid b = 1] &= 1 - \\mathop {\\mathbb {E}}_{S\\sim n}\\left[\\frac{1}{n}\\sum _{i=1}^n \\mathcal {A} (z_i, A_{S^{(i)}}, n, \\right]\\end{@align*}}The above two equalities, combined with Equation~\\ref {eq:altadv}, gives:\\begin{equation}\\mathsf {Adv}^\\mathsf {M}= \\mathop {\\mathbb {E}}_{S\\sim n}\\left[\\frac{1}{n}\\sum _{i=1}^n \\mathcal {A} (z_i, A_{S^{(i)}}, n, - \\mathcal {A} (z_i, A_S, n, \\right]\\end{equation}$ Without loss of generality for the case where models reside in an infinite domain, assume that the models produced by $A$ come from the set $\\lbrace A^1, \\ldots , A^k\\rbrace $ .",
"Differential privacy guarantees that for all $j \\in [k]$ , $\\Pr [A_{S^{(i)}} = A^j] \\le e^\\epsilon \\Pr [A_S = A^j].$ Using this inequality, we can rewrite and bound the right-hand side of Equation as $&\\sum _{j=1}^k \\mathop {\\mathbb {E}}_{S\\sim n}\\Bigg [\\frac{1}{n}\\sum _{i=1}^n \\Pr [A_{S^{(i)}} = A^j] - \\Pr [A_S = A^j] \\cdot \\mathcal {A} (z_i, A^j, n, \\Bigg ] \\\\& \\le \\sum _{j=1}^k \\mathop {\\mathbb {E}}_{S\\sim n}\\left[(e^\\epsilon - 1) \\Pr [A_S = A^j] \\cdot \\frac{1}{n}\\sum _{i=1}^n \\mathcal {A} (z_i, A^j, n, \\right],$ which is at most $e^\\epsilon - 1$ since $\\mathcal {A} (z, A^j, n, \\le 1$ for any $z$ , $A^j$ , $n$ , and Wu et al.",
"[8] present an algorithm that is differentially private as long as the loss function $\\ell $ is $\\lambda $ -strongly convex and $\\rho $ -Lipschitz.",
"Moreover, they prove that the performance of the resulting model is close to the optimal.",
"Combined with Theorem REF , this provides us with a bound on membership advantage when the loss function is strongly convex and Lipschitz.",
"Membership attacks and generalization In this section, we consider several membership attacks that make few, common assumptions about the model $A_S$ or the distribution Importantly, these assumptions are consistent with many natural learning techniques widely used in practice.",
"For each attack, we express the advantage of the attacker as a function of the extent of the overfitting, thereby showing that the generalization behavior of the model is a strong predictor for vulnerability to membership inference attacks.",
"In Section REF , we demonstrate that these relationships often hold in practice on real data, even when the assumptions used in our analysis do not hold.",
"Bounded loss function We begin with a straightforward attack that makes only one simple assumption: the loss function is bounded by some constant $B $ .",
"Then, with probability proportional to the model's loss at the query point $z$ , the adversary predicts that $z$ is not in the training set.",
"The attack is formalized in Adversary REF .",
"Adversary 1 (Bounded loss function) Suppose $\\ell (A_S, z) \\le B $ for some constant $B $ , all $S \\sim n$ , and all $z$ sampled from $S$ or Then, on input $z = (x, y)$ , $A_S$ , $n$ , and the membership adversary $\\mathcal {A}$ proceeds as follows: Query the model to get $A_S (x)$ .",
"Output 1 with probability $\\ell (A_S, z) / B $ .",
"Else, output 0.",
"Theorem REF states that the membership advantage of this approach is proportional to the generalization error of $A$ , showing that advantage and generalization error are closely related in many common learning settings.",
"In particular, classification settings, where the 0-1 loss function is commonly used, $B = 1$ yields membership advantage equal to the generalization error.",
"Simply put, high generalization error necessarily results in privacy loss for classification models.",
"Theorem 2 The advantage of Adversary REF is $R_{\\mathrm {gen}}(A) / B $ .",
"The proof is as follows: $&\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A},A,n,\\\\&= \\Pr [\\mathcal {A} = 0 \\mid b = 0] - \\Pr [\\mathcal {A} = 0 \\mid b = 1]\\\\&= \\Pr [\\mathcal {A} = 1 \\mid b = 1] - \\Pr [\\mathcal {A} = 1 \\mid b = 0]\\\\&= \\mathop {\\mathbb {E}}\\left[\\frac{\\ell (A_S, z)}{B} | b = 1\\right] - \\mathop {\\mathbb {E}}\\left[\\frac{\\ell (A_S, z)}{B} | b = 0\\right]\\\\&= \\frac{1}{B} \\left(\\mathop {\\mathbb {E}}_{\\begin{array}{c}S \\sim n\\\\ z \\sim {array}\\end{array}[\\ell (A_S, z)] - \\mathop {\\mathbb {E}}_{\\begin{array}{c}S \\sim n\\\\ z \\sim S\\end{array}}[\\ell (A_S, z)]\\\\&= R_{\\mathrm {gen}}(A) / B }\\right.$ Gaussian error Whenever the adversary knows the exact error distribution, it can simply compute which value of $b$ is more likely given the error of the model on $z$ .",
"This adversary is described formally in Adversary REF .",
"While it may seem far-fetched to assume that the adversary knows the exact error distribution, linear regression models implicitly assume that the error of the model is normally distributed.",
"In addition, the standard errors $\\sigma _S$ , $\\sigma _ of the model on $ S$ and respectively, are often published with the model, giving the adversary full knowledge of the error distribution.",
"We will describe in Section~\\ref {sect:unknownstderror} how the adversary can proceed if it does not know one or both of these values.$ Adversary 2 (Threshold) Suppose $f(\\epsilon \\mid b = 0)$ and $f(\\epsilon \\mid b = 1)$ , the conditional probability density functions of the error, are known in advance.",
"Then, on input $z = (x, y)$ , $A_S$ , $n$ , and the membership adversary $\\mathcal {A}$ proceeds as follows: Query the model to get $A_S (x)$ .",
"Let $\\epsilon = y - A_S (x)$ .",
"Output $\\operatornamewithlimits{arg\\,max}_{b \\in \\lbrace 0, 1\\rbrace } f(\\epsilon \\mid b)$ .",
"In regression problems that use squared-error loss, the magnitude of the generalization error depends on the scale of the response $y$ .",
"For this reason, in the following we use the ratio $\\sigma _/\\sigma _S $ to measure generalization error.",
"Theorem REF characterizes the advantage of this adversary in the case of Gaussian error in terms of $\\sigma _/\\sigma _S $ .",
"As one might expect, this advantage is 0 when $\\sigma _S = \\sigma _$ and approaches 1 as $\\sigma _/\\sigma _S \\rightarrow \\infty $ .",
"The dotted line in Figure REF shows the graph of the advantage as a function of $\\sigma _/\\sigma _S $ .",
"Theorem 3 Suppose $\\sigma _S$ and $\\sigma _ are known in advance such that $ N(0, $\\sigma _S$ 2)$ when $ b = 0$ and $ N(0, $\\sigma _ ^2)$ when $b = 1$ .",
"Then, the advantage of Membership Adversary REF is $\\operatorname{erf}\\left(\\frac{\\sigma _}{\\sigma _S} \\sqrt{\\frac{\\ln (\\sigma _/\\sigma _S)}{(\\sigma _/\\sigma _S)^2 - 1}}\\right) - \\operatorname{erf}\\left(\\sqrt{\\frac{\\ln (\\sigma _/\\sigma _S)}{(\\sigma _/\\sigma _S)^2 - 1}}\\right).$ We have $f(\\epsilon \\mid b = 0) &= \\frac{1}{\\sqrt{2\\pi }\\sigma _S} e^{-\\epsilon ^2/2\\sigma _S ^2}\\\\f(\\epsilon \\mid b = 1) &= \\frac{1}{\\sqrt{2\\pi }\\sigma _} e^{-\\epsilon ^2/2\\sigma _^2}.$ Let $\\pm \\epsilon _{\\mathrm {eq}}$ be the points at which these two probability density functions are equal.",
"Some algebraic manipulation shows that $ \\epsilon _{\\mathrm {eq}}= \\sigma _\\sqrt{\\frac{2 \\ln (\\sigma _/\\sigma _S)}{(\\sigma _/\\sigma _S)^2 - 1}}.$ Moreover, if $\\sigma _S < \\sigma _$ , $f(\\epsilon \\mid b = 0) > f(\\epsilon \\mid b = 1)$ if and only if $|\\epsilon | < \\epsilon _{\\mathrm {eq}}$ .",
"Therefore, the membership advantage is $&\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A},A,n,\\\\&= \\Pr [\\mathcal {A} = 0 \\mid b = 0] - \\Pr [\\mathcal {A} = 0 \\mid b = 1]\\\\&= \\Pr [|\\epsilon | < \\epsilon _{\\mathrm {eq}}\\mid b = 0] - \\Pr [|\\epsilon | < \\epsilon _{\\mathrm {eq}}\\mid b = 1]\\\\&= \\operatorname{erf}\\left(\\frac{\\epsilon _{\\mathrm {eq}}}{\\sqrt{2}\\sigma _S}\\right) - \\operatorname{erf}\\left(\\frac{\\epsilon _{\\mathrm {eq}}}{\\sqrt{2}\\sigma _}\\right)\\\\&= \\operatorname{erf}\\left(\\frac{\\sigma _}{\\sigma _S} \\sqrt{\\frac{\\ln (\\sigma _/\\sigma _S)}{(\\sigma _/\\sigma _S)^2 - 1}}\\right) - \\operatorname{erf}\\left(\\sqrt{\\frac{\\ln (\\sigma _/\\sigma _S)}{(\\sigma _/\\sigma _S)^2 - 1}}\\right).",
"$ Unknown standard error In practice, models are often published with just one value of standard error, so the adversary often does not know how $\\sigma _ compares to \\sigma _S.",
"One solution to this issue is to assume that $$\\sigma _S$ $\\sigma _ $ , i.e., that the model does not terribly overfit.",
"Then, the threshold is set at $|\\epsilon | = \\sigma _S $ , which is the limit of the right-hand side of Equation REF as $\\sigma _$ approaches $\\sigma _S $ .",
"Then, the membership advantage is $\\operatorname{erf}(1/\\sqrt{2}) - \\operatorname{erf}(\\sigma _S/\\sqrt{2}\\sigma _)$ .",
"This expression is graphed in Figure REF as a function of $\\sigma _/\\sigma _S $ .",
"Alternatively, if the adversary knows which machine learning algorithm was used, it can repeatedly sample $S \\sim n$ , train the model $A_S$ using the sampled $S$ , and measure the error of the model to arrive at reasonably close approximations of $\\sigma _S$ and $\\sigma _.$ Other sources of membership advantage The results in the preceding sections show that overfitting is sufficient for membership advantage.",
"However, models can leak information about the training set in other ways, and thus overfitting is not necessary for membership advantage.",
"For example, the learning rule can produce models that simply output a lossless encoding of the training dataset.",
"This example may seem unconvincing for several reasons: the leakage is obvious, and the “encoded” dataset may not function well as a model.",
"In the rest of this section, we present a pair of colluding training algorithm and adversary that does not have the above issues but still allows the attacker to learn the training set almost perfectly.",
"This is in the framework of an algorithm substitution attack (ASA) [23], where the target algorithm, which is implemented by closed-source software, is subverted to allow a colluding adversary to violate the privacy of the users of the algorithm.",
"All the while, this subversion remains impossible to detect.",
"Algorithm REF and Adversary REF represent a similar security threat for learning rules with bounded loss function.",
"While the attack presented here is not impossible to detect, on points drawn from the black-box behavior of the subverted model is similar to that of an unsubverted model.",
"The main result is given in Theorem REF , which shows that any ARO-stable learning rule $A$ , with a bounded loss function operating on a finite domain, can be modified into a vulnerable learning rule $A^k$ , where $k \\in \\mathbb {N}$ is a parameter.",
"Moreover, subject to our assumption from before that $\\mu (n,$ is very small, the stability rate of the vulnerable model $A^k$ is not far from that of $A$ , and for each $A^k$ there exists a membership adversary whose advantage is negligibly far (in $k$ ) from the maximum advantage possible on Simply put, it is often possible to find a suitably leaky version of an ARO-stable learning rule whose generalization behavior is close to that of the original.",
"Theorem 4 Let $d = \\log |\\mathbf {X} |$ , $m = \\log |\\mathbf {Y} |$ , $\\ell $ be a loss function bounded by some constant $B$ , $A$ be an ARO-stable learning rule with rate $\\epsilon _{\\textrm {stable}} (n)$ , and suppose that $x$ uniquely determines the point $(x, y)$ in Then for any integer $k > 0$ , there exists an ARO-stable learning rule $A^k$ with rate at most $\\epsilon _{\\textrm {stable}} (n) + knB2^{-d} + \\mu (n,$ and adversary $\\mathcal {A}$ such that: $\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A}, A^k, n, = 1 - \\mu (n, - 2^{-mk}$ The proof of Theorem REF involves constructing a learning rule $A^k$ that leaks precise membership information when queried in a particular way but is otherwise identical to $A$ .",
"$A^k$ assumes that the adversary has knowledge of a secret key that is used to select pseudorandom functions that define the “special” queries used to extract membership information.",
"In this way, the normal behavior of the model remains largely unchanged, making $A^k$ approximately as stable as $A$ , but the learning algorithm and adversary “collude” to leak information through the model.",
"We require the features $x$ to fully determine $y$ to avoid collisions when the adversary queries the model, which would result in false positives.",
"In practice, many learning problems satisfy this criterion.",
"Algorithm REF and Adversary REF illustrate the key ideas in this construction informally.",
"Algorithm 1 (Colluding training algorithm $A^{\\mathsf {C}}$ ) Let $F_K : \\mathbf {X} \\mapsto \\mathbf {X} $ and $G_K : \\mathbf {X} \\mapsto \\mathbf {Y} $ be keyed pseudorandom functions, $K_1, \\ldots , K_k$ be uniformly chosen keys, and $A$ be a training algorithm.",
"On receiving a training set $S$ , $A^{\\mathsf {C}}$ proceeds as follows: Supplement $S$ using $F, G$ : for all $(x_i, y_i) \\in S$ and $j \\in [k]$ , let $z^{\\prime }_{i,j} = (F_{K_j}(x_i), G_{K_j}(x_i))$ , and set $S^{\\prime } = S \\cup \\lbrace z^{\\prime }_{i,j} \\mid i \\in [n], j \\in [k]\\rbrace $ .",
"Return $A_{S^{\\prime }} = A(S^{\\prime })$ .",
"Adversary 3 (Colluding adversary $\\mathcal {A} ^{\\mathsf {C}}$ ) Let $F_K : \\mathbf {X} \\mapsto \\mathbf {X} $ , $G_K : \\mathbf {X} \\mapsto \\mathbf {Y} $ and $K_1, \\ldots , K_k$ be the functions and keys used by $A^{\\mathsf {C}}$ , and $A_{S^{\\prime }}$ be the product of training with $A^{\\mathsf {C}}$ with those keys.",
"On input $z = (x, y)$ , the adversary $\\mathcal {A} ^{\\mathsf {C}}$ proceeds as follows: For $j \\in [k]$ , let $y_j^{\\prime } \\leftarrow A_{S^{\\prime }} (F_{K_j}(x))$ .",
"Output 0 if $y_j^{\\prime } = G_{K_j}(x)$ for all $j \\in [k]$ .",
"Else, output 1.",
"Algorithm REF will not work well in practice for many classes of models, as they may not have the capacity to store the membership information needed by the adversary while maintaining the ability to generalize.",
"Interestingly, in Section REF we empirically demonstrate that deep convolutional neural networks (CNNs) do in fact have this capacity and generalize perfectly well when trained in the manner of $A^{\\mathsf {C}}$ .",
"As pointed out by Zhang et al.",
"[28], because the number of parameters in deep CNNs often significantly exceeds the training set size, despite their remarkably good generalization error, deep CNNs may have the capacity to effectively “memorize” the dataset.",
"Our results supplement their observations and suggest that this phenomenon may have severe implications for privacy.",
"Before we give the formal proof, we note a key difference between Algorithm REF and the construction used in the proof.",
"Whereas the model returned by Algorithm REF belongs to the same class as those produced by $A$ , in the formal proof the training algorithm can return an arbitrary model as long as its black-box behavior is suitable.",
"The proof constructs a learning algorithm and adversary who share a set of $k$ keys to a pseudorandom function.",
"The secrecy of the shared key is unnecessary, as the proof only relies on the uniformity of the keys and the pseudorandom functions' outputs.",
"The primary concern is with using the pseudorandom function in a way that preserves the stability of $A$ as much as possible.",
"Without loss of generality, assume that $\\mathbf {X} = \\lbrace 0, 1\\rbrace ^d$ and $\\mathbf {Y} = \\lbrace 0, 1\\rbrace ^m$ .",
"Let $F_K : \\lbrace 0, 1\\rbrace ^d \\rightarrow \\lbrace 0, 1\\rbrace ^d$ and $G_K : \\lbrace 0, 1\\rbrace ^d \\mapsto \\lbrace 0, 1\\rbrace ^m$ be keyed pseudorandom functions, and let $K_1, \\ldots , K_k$ be uniformly sampled keys.",
"On receiving $S$ , the training algorithm $A^{K_1, \\ldots , K_k}$ returns the following model: $A_S ^{K_1, \\ldots , K_k}(x) ={\\left\\lbrace \\begin{array}{ll}G_{K_j}(x), & \\text{if } \\exists (x^{\\prime },y) \\in S \\text{ s.t. }",
"x = F_{K_j}(x^{\\prime }) \\text{ for some } K_j\\\\A_S (x), & \\text{otherwise}\\end{array}\\right.",
"}$ We now define a membership adversary $\\mathcal {A} ^{K_1, \\ldots , K_k}$ who is hard-wired with keys $K_1, \\ldots , K_k$ : $\\mathcal {A} ^{K_1, \\ldots , K_k}(z,A,n, ={\\left\\lbrace \\begin{array}{ll}0, & \\text{if } A_S (x) = G_{K_j}(F_{K_j}(x)) \\text{for all } K_j\\\\1, & \\text{otherwise}\\end{array}\\right.",
"}$ Recalling our assumption that the value of $x$ uniquely determines the point $(x, y)$ , we can derive the advantage of $\\mathcal {A} ^{K_1, \\ldots , K_k}$ on the corresponding trainer $A^{K_1, \\ldots , K_k}$ in possession of the same keys: $&\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A} ^{K_1, \\ldots , K_k}, A^{K_1, \\ldots , K_k}, n, \\\\&= \\Pr [\\mathcal {A} ^{K_1, \\ldots , K_k} = 0 \\mid b = 0] - \\Pr [\\mathcal {A} ^{K_1, \\ldots , K_k} = 0 \\mid b = 1]\\\\&= 1 - \\mu (n, - 2^{-mk}$ The $2^{-mk}$ term comes from the possibility that $G_{K_j}(F_{K_j}(x)) = A_S (x)$ for all $j \\in [k]$ by pure chance.",
"Now observe that $A$ is ARO-stable with rate $\\epsilon _{\\textrm {stable}} (n)$ .",
"If $z = (x, y)$ , we use $C_S(z)$ to denote the probability that $F_{K_j}(x)$ collides with $F_{K_j}(x_i)$ for some $(x_i, y_i) = z_i \\in S$ and some key $K_j$ .",
"Note that by a simple union bound, we have $C_S(z) \\le kn2^{-d}$ for $z \\notin S$ .",
"Then algebraic manipulation gives us the following, where we write $A_S ^K$ in place of $A_S ^{K_1, \\ldots , K_k}$ to simplify notation: $&R_{\\mathrm {gen}}(A^K,n,\\ell ) \\\\&= \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array}\\left[\\frac{1}{n}\\sum _{i=1}^n \\ell (A_{S^{(i)}} ^K, z_i) - \\ell (A_S ^K, z_i)\\right]\\\\&= \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array}\\left[\\frac{1}{n}\\sum _{i=1}^n (1 - C_S(z_i))\\left(\\ell (A_{S^{(i)}}, z_i) - \\ell (A_S, z_i)\\right)\\right] + \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array} \\left[\\frac{1}{n}\\sum _{i=1}^n C_S(z_i)\\left(\\ell (A_{S^{(i)}}, z_i) - \\ell (G_K, z_i)\\right)\\right]\\\\&= \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array}\\left[\\frac{1}{n}\\sum _{i=1}^n \\ell (A_{S^{(i)}}, z_i) - \\ell (A_S, z_i)\\right] + \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array}\\left[\\frac{1}{n}\\sum _{i=1}^n C_S(z_i)\\left(\\ell (A_S, z_i) - \\ell (G_K, z_i)\\right)\\right]\\\\&\\le \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array}\\left[\\frac{1}{n}\\sum _{i=1}^n \\ell (A_{S^{(i)}}, z_i) - \\ell (A_S, z_i)\\right] + knB 2^{-d} + \\mu (n,\\\\&= \\epsilon _{\\textrm {stable}} (n) + knB2^{-d} + \\mu (n,}Note that the term \\mu (n, on the last line accounts for the possibility that the z^{\\prime } sampled at index i in S^{(i)} is already in S, which results in a collision.",
"By the result in~\\cite {ShalevShwartz10} that states that the average generalization error equals the ARO-stability rate, A^K is ARO-stable with rate \\epsilon _{\\textrm {stable}} (n) + knB 2^{-d} + \\mu (n,, completing the proof.",
"}}The formal study of ASAs was introduced by Bellare et al.~\\cite {BPR14}, who considered attacks against symmetric encryption.",
"Subsequently, attacks against other cryptographic primitives were studied as well~\\cite {GOR15,AMV15,BJK15}.",
"The recent work of Song et al.~\\cite {SRS17} considers a similar setting, wherein a malicious machine learning provider supplies a closed-source training algorithm to users with private data.",
"When the provider gets access to the resulting model, it can exploit the trapdoors introduced in the model to get information about the private training dataset.",
"However, to the best of our knowledge, a formal treatment of ASAs against machine learning algorithms has not been given yet.",
"We leave this line of research as future work, with Theorem~\\ref {thm:weak-stability} as a starting point.",
"}}}\\section {Attribute Inference Attacks}$ We now consider attribute inference attacks, where the goal of the adversary is to guess the value of the sensitive features of a data point given only some public knowledge about it and the model.",
"To make this explicit in our notation, in this section we assume that data points are triples $z = (v, t, y)$ , where $(v,t)=x\\in \\mathbf {X} $ and $t$ is the sensitive features targeted in the attack.",
"A fixed function $\\varphi $ with domain $\\mathbf {X} \\times \\mathbf {Y} $ describes the information about data points known by the adversary.",
"Let $ be the support of $ t$ when $ z=(v,t,y).",
"The function $\\pi $ is the projection of $\\mathbf {X} $ into $ (e.g., $ (z)=t$).$ Attribute inference is formalized in Experiment REF , which proceeds much like Experiment REF .",
"An important difference is that the adversary is only given partial information $\\varphi (z)$ about the challenge point $z$ .",
"Experiment 2 (Attribute experiment $\\mathsf {Exp}^\\mathsf {A} (\\mathcal {A},A,n,$ ) Let $\\mathcal {A}$ be an adversary, $n$ be a positive integer, and b̥e a distribution over data points $(x,y)$ .",
"The attribute experiment proceeds as follows: Sample $S \\sim n$ .",
"Choose $b \\leftarrow \\lbrace 0, 1\\rbrace $ uniformly at random.",
"Draw $z \\sim S$ if $b = 0$ , or $z \\sim if $ b = 1$.\\item $$\\mathsf {Exp}^\\mathsf {A}$ ($\\mathcal {A}$ ,A,n,$ is 1 if $$\\mathcal {A}$ ((z),$A_S$ ,n, = (z)$ and 0 otherwise.$ In the corresponding advantage measure shown in Definition REF , our goal is to measure the amount of information about the target $\\pi (z)$ that $A_S$ leaks specifically concerning the training data $S$ .",
"Definition REF accomplishes this by comparing the performance of the adversary when $b = 0$ in Experiment REF with that when $b = 1$ .",
"Definition 5 (Attribute advantage) The attribute advantage of $\\mathcal {A} $ is defined as: $\\mathsf {Adv}^\\mathsf {A}(\\mathcal {A},A,n, &= \\Pr [\\mathsf {Exp}^\\mathsf {A} (\\mathcal {A},A,n, = 1 \\mid b = 0] - \\Pr [\\mathsf {Exp}^\\mathsf {A} (\\mathcal {A},A,n, = 1 \\mid b = 1],$ where the probabilities are taken over the coin flips of $\\mathcal {A}$ , the random choice of $S$ , and the random data point $z \\sim S$ or $z \\sim .$ Notice that $ \\begin{aligned}\\mathsf {Adv}^\\mathsf {A}&= \\textstyle \\sum _{t_i\\in \\Pr _{z\\sim [t=t_i](\\Pr [\\mathcal {A} = t_i \\mid b=0, t=t_i] - \\Pr [\\mathcal {A} = t_i \\mid b=1, t=t_i]),}}where \\mathcal {A} and \\mathsf {Adv}^\\mathsf {A} are shortcuts for \\mathcal {A} (\\varphi (z),A_S,n, and \\mathsf {Adv}^\\mathsf {A}(\\mathcal {A},A,n,, respectively.\\end{aligned}This definition has the side effect of incentivizing the adversary to ``game the system\" by performing poorly when it thinks that b = 1.",
"To remove this incentive, one may consider using a simulator \\mathcal {S}, which does not receive the model as an input, when b = 1.",
"This definition is formalized below:$ Definition 6 (Alternative attribute advantage) Let $\\mathcal {S} (\\varphi (z),n, = \\operatornamewithlimits{arg\\,max}_{t_i} \\Pr _{z \\sim [\\pi (z) = t_i \\mid \\varphi (z)]be the Bayes optimal simulator.",
"The \\emph {attribute advantage} of \\mathcal {A} can alternatively be defined as{\\begin{@align*}{1}{-1}\\mathsf {Adv}^{\\mathsf {A}}_\\mathcal {S} (\\mathcal {A},A,n, &= \\Pr [\\mathcal {A} (\\varphi (z),A_S,n, = \\pi (z) \\mid b = 0] - \\Pr [\\mathcal {S} (\\varphi (z),n, = \\pi (z) \\mid b = 1].\\end{@align*}}}$ One potential issue with this alternative definition is that higher model accuracy will lead to higher attribute advantage regardless of how accurate the model is for the general population.",
"Broadly, there are two ways for a model to perform better on the training data: it can overfit to the training data, or it can learn a general trend in the distribution In this paper, we concern ourselves with the view that the adversary's ability to infer the target $\\pi (z)$ in the latter case is due not to the model but pre-existing patterns in To allow capturing the difference between overfitting and learning a general trend, we use Definition REF in the following analysis and leave a more complete exploration of Definition REF as future work.",
"While adversaries that “game the system\" may seem problematic, the effectiveness of such adversaries is indicative of privacy loss because their existence implies the ability to infer membership, as demonstrated by Reduction Adversary REF in Section REF .",
"Inversion, generalization, and influence The case where $\\varphi $ simply removes the sensitive attribute $t$ from the data point $z = (v,t,y)$ such that $\\varphi (z)=(v,y)$ is known in the literature as model inversion [4], [3], [5], [8].",
"In this section, we look at the model inversion attack of Fredrikson et al.",
"[4] under the advantage given in Definition REF .",
"We point out that this is a novel analysis, as this advantage is defined to reflect the extent to which an attribute inference attack reveals information about individuals in $S$ .",
"While prior work [4], [3] has empirically evaluated attribute accuracy over corresponding training and test sets, our goal is to analyze the factors that lead to increased privacy risk specifically for members of the training data.",
"To that end, we illustrate the relationship between advantage and generalization error as we did in the case of membership inference (Section REF ).",
"We also explore the role of feature influence, which in this case corresponds to the degree to which changes to a sensitive feature of $x$ affects the value $A_S (x)$ .",
"In Section REF , we show that the formal relationships described here often extend to attacks on real data where formal assumptions may fail to hold.",
"The attack described by Fredrikson et al.",
"[4] is intended for linear regression models and is thus subject to the Gaussian error assumption discussed in Section REF .",
"In general, when the adversary can approximate the error distribution reasonably well, e.g., by assuming a Gaussian distribution whose standard deviation equals the published standard error value, it can gain advantage by trying all possible values of the sensitive attribute.",
"We denote the adversary's approximation of the error distribution by $f_\\mathcal {A} $ , and we assume that the target $t=\\pi (z)$ is drawn from a finite set of possible values $t_1, \\ldots , t_m$ with known frequencies in We indicate the other features, which are known by the adversary, with the letter $v$ (i.e., $z=(x,y)$ , $x=(v,t)$ , and $\\varphi (z)=(v,y)$ ).",
"The attack is shown in Adversary REF .",
"For each $t_i$ , the adversary counterfactually assumes that $t = t_i$ and computes what the error of the model would be.",
"It then uses this information to update the a priori marginal distribution of $t$ and picks the value $t_i$ with the greatest likelihood.",
"Adversary 4 (General) Let $f_\\mathcal {A} (\\epsilon )$ be the adversary's guess for the probability density of the error $\\epsilon = y - A_S (x)$ .",
"On input $v$ , $y$ , $A_S$ , $n$ , and the adversary proceeds as follows: Query the model to get $A_S (v,t_i)$ for all $i \\in [m]$ .",
"Let $\\epsilon (t_i) = y - A_S (v,t_i)$ .",
"Return the result of $\\operatornamewithlimits{arg\\,max}_{t_i} (\\Pr _{z \\sim [t = t_i] \\cdot f_\\mathcal {A} (\\epsilon (t_i))).",
"}$ When analyzing Adversary REF , we are clearly interested in the effect that generalization error will have on advantage.",
"Given the results of Section REF , we can reasonably expect that large generalization error will lead to greater advantage.",
"However, as pointed out by Wu et al.",
"[5], the functional relationship between $t$ and $A_S (v,t)$ may play a role as well.",
"Working in the context of models as Boolean functions, Wu et al.",
"formalized the relevant property as functional influence [20], which is the probability that changing $t$ will cause $A_S (v,t)$ to change when $v$ is sampled uniformly.",
"The attack considered here applies to linear regression models, and Boolean influence is not suitable for use in this setting.",
"However, an analogous notion of influence that characterizes the magnitude of change to $A_S (v,t)$ is relevant to attribute inference.",
"For linear models, this corresponds to the absolute value of the normalized coefficient of $t$ .",
"Throughout the rest of the paper, we refer to this quantity as the influence of $t$ without risk of confusion with the Boolean influence used in other contexts.",
"Binary Variable with Uniform Prior The first part of our analysis deals with the simplest case where $m = 2$ with $\\Pr _{z \\sim [t = t_1] = \\Pr _{z \\sim [t = t_2].",
"Without loss of generality we assume that A_S (v, t_1) = A_S (v, t_2) + \\tau for some fixed \\tau \\ge 0, so in this setting \\tau is a straightforward proxy for influence.",
"Theorem~\\ref {thm:invbinary} relates the advantage of Adversary~\\ref {adv:invlinreg} to \\sigma _S , \\sigma _, and \\tau .",
"}\\begin{theorem} Let t be drawn uniformly from \\lbrace t_1, t_2\\rbrace and suppose that y = A_S (v, t) + \\epsilon , where \\epsilon \\sim N(0, \\sigma _S ^2) if b=0 and \\epsilon \\sim N(0, \\sigma _^2) if b=1.",
"Then the advantage of Adversary~\\ref {adv:invlinreg} is \\frac{1}{2}(\\operatorname{erf}(\\tau /2\\sqrt{2}\\sigma _S) - \\operatorname{erf}(\\tau /2\\sqrt{2}\\sigma _)).\\end{theorem}}\\begin{proof}Given the assumptions made in this setting, we can describe the behavior of \\mathcal {A} as returning the value t_i that minimizes |\\epsilon (t_i)|.",
"If t = t_1, it is easy to check that \\mathcal {A} guesses correctly if and only if \\epsilon (t_1) > -\\tau /2.",
"This means that \\mathcal {A}^{\\prime }s advantage given t = t_1 is\\begin{equation} \\begin{aligned}&\\Pr [\\mathcal {A} = t_1 \\mid t = t_1, b = 0] - \\Pr [\\mathcal {A} = t_1 \\mid t = t_1, b = 1]\\\\&= \\Pr [\\epsilon (t_1) > -\\tau /2 \\mid b = 0] - \\Pr [\\epsilon (t_1) > -\\tau /2 \\mid b = 1]\\\\&= \\left(\\frac{1}{2} + \\frac{1}{2}\\operatorname{erf}\\left(\\frac{\\tau }{2\\sqrt{2}\\sigma _S}\\right)\\right) - \\left(\\frac{1}{2} + \\frac{1}{2}\\operatorname{erf}\\left(\\frac{\\tau }{2\\sqrt{2}\\sigma _}\\right)\\right)\\\\&= \\frac{1}{2}\\left(\\operatorname{erf}\\left(\\frac{\\tau }{2\\sqrt{2}\\sigma _S}\\right) - \\operatorname{erf}\\left(\\frac{\\tau }{2\\sqrt{2}\\sigma _}\\right)\\right)\\end{aligned}\\end{equation}Similar reasoning shows that \\mathcal {A}^{\\prime }s advantage given t = t_2 is exactly the same, so the theorem follows from Equation~\\ref {eqn:invadv2}.\\end{proof}$ Clearly, the advantage will be zero when there is no generalization error ($\\sigma _S = \\sigma _$ ).",
"Consider the other extreme case where $\\sigma _S \\rightarrow 0$ and $\\sigma _\\rightarrow \\infty $ .",
"When $\\sigma _S $ is very small, the adversary will always guess correctly because the influence of $t$ overwhelms the effect of the error $\\epsilon $ .",
"On the other hand, when $\\sigma _$ is very large, changes to $t$ will be nearly imperceptible for “normal” values of $\\tau $ , and the adversary is reduced to random guessing.",
"Therefore, the maximum possible advantage with uniform prior is $1/2$ .",
"As a model overfits more, $\\sigma _S $ decreases and $\\sigma _$ tends to increase.",
"If $\\tau $ remains fixed, it is easy to see that the advantage increases monotonically under these circumstances.",
"Figure REF shows the effect of changing $\\tau $ as the ratio $\\sigma _/\\sigma _S $ remains fixed at several different constants.",
"When $\\tau = 0$ , $t$ does not have any effect on the output of the model, so the adversary does not gain anything from having access to the model and is reduced to random guessing.",
"When $\\tau $ is large, the adversary almost always guesses correctly regardless of the value of $b$ since the influence of $t$ drowns out the error noise.",
"Thus, at both extremes the advantage approaches 0, and the adversary is able to gain advantage only when $\\tau $ and $\\sigma _/\\sigma _S $ are in balance.",
"Figure: The advantage of Adversary as a function of tt's influence τ\\tau .",
"Here tt is a uniformly distributed binary variable.",
"General Case Sometimes the uniform prior for $t$ may not be realistic.",
"For example, $t$ may represent whether a patient has a rare disease.",
"In this case, we weight the values of $f_\\mathcal {A} (\\epsilon (t_i))$ by the a priori probability $\\Pr _{z \\sim [t = t_i] before comparing which t_i is the most likely.",
"With uniform prior, we could simplify \\operatornamewithlimits{arg\\,max}_{t_i} f_\\mathcal {A} (\\epsilon (t_i)) to \\operatornamewithlimits{arg\\,min}_{t_i} |\\epsilon (t_i)| regardless of the value of \\sigma used for f_\\mathcal {A} .",
"On the other hand, the value of \\sigma matters when we multiply by \\Pr [t = t_i].",
"Because the adversary is not given b, it makes an assumption similar to that described in Section~\\ref {sect:inclusion-attacks} and uses \\epsilon \\sim N(0, \\sigma _S ^2).",
"}Clearly $$\\sigma _S$ = $\\sigma _ $ results in zero advantage.",
"The maximum possible advantage is attained when $\\sigma _S \\rightarrow 0$ and $\\sigma _\\rightarrow \\infty $ .",
"Then, by similar reasoning as before, the adversary will always guess correctly when $b = 0$ and is reduced to random guessing when $b = 1$ , resulting in an advantage of $1 - \\frac{1}{m}$ .",
"In general, the advantage can be computed using Equation REF .",
"We first figure out when the adversary outputs $t_i$ .",
"When $f_\\mathcal {A} $ is a Gaussian, this is not computationally intensive as there is at most one decision boundary between any two values $t_i$ and $t_j$ .",
"Then, we convert the decision boundaries into probabilities by using the error distributions $\\epsilon \\sim N(0, \\sigma _S ^2)$ and $N(0, \\sigma _^2)$ , respectively.",
"Connection between membership and attribute inference In this section, we examine the underlying connections between membership and attribute inference attacks.",
"Our approach is based on reduction adversaries that have oracle access to one type of attack and attempt to perform the other type of attack.",
"We characterize the advantage of each reduction adversary in terms of the advantage of its oracle.",
"In Section REF , we implement the most sophisticated of the reduction adversaries described here and show that on real data it performs remarkably well, often outperforming Attribute Adversary REF by large margins.",
"We note that these reductions are specific to our choice of attribute advantage given in Definition REF .",
"Analyzing the connections between membership and attribute inference using the alternative Definition REF is an interesting direction for future work.",
"From membership to attribute We start with an adversary $\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}}$ that uses an attribute oracle to accomplish membership inference.",
"The attack, shown in Adversary REF , is straightforward: given a point $z$ , the adversary queries the attribute oracle to obtain a prediction $t$ of the target value $\\pi (z)$ .",
"If this prediction is correct, then the adversary concludes that $z$ was in the training data.",
"Adversary 5 (Membership $\\rightarrow $ attribute) The reduction adversary $\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}}$ has oracle access to attribute adversary $\\mathcal {A} _\\mathsf {A}$ .",
"On input $z$ , $A_S$ , $n$ , and the reduction adversary proceeds as follows: Query the oracle to get $t \\leftarrow \\mathcal {A} _\\mathsf {A}(\\varphi (z), A_S, n, $ .",
"Output 0 if $\\pi (z) = t$ .",
"Otherwise, output 1.",
"Theorem REF shows that the membership advantage of this reduction exactly corresponds to the attribute advantage of its oracle.",
"In other words, the ability to effectively infer attributes of individuals in the training set implies the ability to infer membership in the training set as well.",
"This suggests that attribute inference is at least as difficult as than membership inference.",
"Theorem 5 Let $\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}}$ be the adversary described in Adversary REF , which uses $\\mathcal {A} _\\mathsf {A}$ as an oracle.",
"Then, $\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}},A,n, = \\mathsf {Adv}^\\mathsf {A}(\\mathcal {A} _\\mathsf {A},A,n,.$ The proof follows directly from the definitions of membership and attribute advantages.",
"$\\mathsf {Adv}^\\mathsf {M}&= \\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 0] - \\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 1] \\\\&= \\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 1,t=t_i]) \\\\&= \\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _\\mathsf {A}= t_i \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {A}} = t_i \\mid b = 1,t=t_i]) \\\\&= \\mathsf {Adv}^\\mathsf {A}.",
"}}$ From attribute to membership We now consider reductions in the other direction, wherein the adversary is given $\\varphi (z)$ and must reconstruct the point $z$ to query the membership oracle.",
"To accomplish this, we assume that the adversary knows a deterministic reconstruction function $\\varphi ^{-1}$ such that $\\varphi \\circ \\varphi ^{-1}$ is the identity function, i.e., for any value of $\\varphi (z)$ that the adversary may receive, there exists $z^{\\prime } = \\varphi ^{-1}(\\varphi (z))$ such that $\\varphi (z) = \\varphi (z^{\\prime })$ .",
"However, because $\\varphi $ is a lossy function, in general it does not hold that $\\varphi ^{-1}(\\varphi (z)) = z$ .",
"Our adversary, described in Adversary REF , reconstructs the point $z^{\\prime }$ , sets the attribute $t$ of that point to value $t_i$ chosen uniformly at random, and outputs $t_i$ if the membership oracle says that the resulting point is in the dataset.",
"Adversary 6 (Uniform attribute $\\rightarrow $ membership) Suppose that $t_1, \\ldots , t_m$ are the possible values of the target $t=\\pi (z)$ .",
"The reduction adversary $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U}$ has oracle access to membership adversary $\\mathcal {A} _\\mathsf {M}$ .",
"On input $\\varphi (z)$ , $A_S$ , $n$ , and $, the reduction adversary proceeds as follows:\\begin{enumerate}\\item Choose t_i uniformly at random from \\lbrace t_1, \\ldots , t_m\\rbrace .\\item Let z^{\\prime } = \\varphi ^{-1}(\\varphi (z)), and change the value of the sensitive attribute t such that \\pi (z^{\\prime }) = t_i.\\item Query \\mathcal {A} _\\mathsf {M} to obtain b^{\\prime } \\leftarrow \\mathcal {A} _\\mathsf {M}(z^{\\prime },A_S,n,.\\item If b^{\\prime } = 0, output t_i.",
"Otherwise, output \\bot .\\end{enumerate}$ The uniform choice of $t_i$ is motivated by the fact that the adversary may not know how the advantage of the membership oracle is distributed across different values of $t$ .",
"For example, it is possible that $\\mathcal {A} _\\mathsf {M}$ performs very poorly when $t = t_1$ and that all of its advantage comes from the case where $t = t_2$ .",
"In the computation of the advantage, we only consider the case where $\\pi (z) = t_i$ because this is the only case where the reduction adversary can possibly give the correct answer.",
"In that case, the membership oracle is given a challenge point from the distribution $ = \\lbrace (x,y) \\mid (x,y)=\\varphi ^{-1}(\\varphi (z)) \\text{ except that } t = \\pi (z)\\rbrace $ , where $z \\sim S$ if $b = 0$ and $z \\sim if $ b = 1$.",
"On the other hand, the training set $ S$ used to train the model A_S was drawn from Because of this difference, we use modified membership advantage $ AdvM*($\\mathcal {A}$ , A, n, , -1, )$, which measures the performance of the membership adversary when the challenge point is drawn from $$.",
"In the case of a model inversion attack as described in the beginning of Section~\\ref {sect:inversion-attacks}, we have $ AdvM($\\mathcal {A}$ , A, n, = AdvM*($\\mathcal {A}$ , A, n, , -1, )$, i.e., the modified membership advantage equals the unmodified one.$ Theorem REF shows that the attribute advantage of $\\mathcal {A} ^{\\sf U}_{\\mathsf {A}\\rightarrow \\mathsf {M}}$ is proportional to the modified membership advantage of $\\mathcal {A} _\\mathsf {M}$ , giving a lower bound on the effectiveness of attribute inference attacks that use membership oracles.",
"Notably, the adversary does not make use of any associations that may exist between $\\varphi (z)$ and $t$ , so this reduction is general and works even when no such association exists.",
"While the reduction does not completely transfer the membership advantage to attribute advantage, the resulting attribute advantage is within a constant factor of the modified membership advantage.",
"Theorem 6 Let $\\mathcal {A} ^{\\sf U}_{\\mathsf {A}\\rightarrow \\mathsf {M}}$ be the adversary described in Adversary REF , which uses $\\mathcal {A} _\\mathsf {M}$ as an oracle.",
"Then, $\\mathsf {Adv}^\\mathsf {A}(\\mathcal {A} ^{\\sf U}_{\\mathsf {A}\\rightarrow \\mathsf {M}},A,n, = \\frac{1}{m} \\mathsf {Adv}^\\mathsf {M}_*(\\mathcal {A} _\\mathsf {M},A,n,\\varphi ,\\varphi ^{-1},\\pi ).$ We first give an informal argument.",
"In order for $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U}$ to correctly guess the value of $t$ , it needs to choose the correct $t_i$ , which happens with probability $\\frac{1}{m}$ , and then $\\mathcal {A} _\\mathsf {M}(z^{\\prime }, A_S, n, $ must be 0.",
"Therefore, $\\mathsf {Adv}^\\mathsf {A}= \\frac{1}{m} \\mathsf {Adv}^\\mathsf {M}_*$ .",
"Now we give the formal proof.",
"Let $t^{\\prime }$ be the value of $t$ that was chosen independently and uniformly at random in Step 1 of Adversary REF .",
"Since $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U}$ outputs $t_i$ if and only if $t^{\\prime }=t_i$ and $\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0$ , we have $&\\Pr [\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U} = t_i \\mid b = 0,t=t_i] = \\frac{1}{m} \\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 0,t=t_i],$ and likewise when $b=1$ .",
"Therefore, the advantage of the reduction adversary is $\\mathsf {Adv}^\\mathsf {A}&= \\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U} = t_i \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U} = t_i \\mid b = 1,t=t_i]) \\\\&= \\frac{1}{m}\\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 1,t=t_i]) \\\\&= \\frac{1}{m}(\\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 0] - \\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 1]) \\\\&= \\frac{1}{m} \\mathsf {Adv}^\\mathsf {M}_*,}where the second-to-last step holds due to the fact that b and t are independent.",
"}$ Adversary REF has the obvious weakness that it can only return correct answers when it guesses the value of $t$ correctly.",
"Adversary REF attempts to improve on this by making multiple queries to $\\mathcal {A} _\\mathsf {M}$ .",
"Rather than guess the value of $t$ , this adversary tries all values of $t$ in order of their marginal probabilities until the membership adversary says “yes\".",
"Adversary 7 (Multi-query attribute $\\rightarrow $ membership) Suppose that $t_1, \\ldots , t_m$ are the possible values of the sensitive attribute $t$ .",
"The reduction adversary $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf M}$ has oracle access to membership adversary $\\mathcal {A} _\\mathsf {M}$ .",
"On input $\\varphi (z)$ , $A_S$ , $n$ , and $, $$\\mathcal {A}$ AM$ proceeds as follows:\\begin{enumerate}\\item Let z^{\\prime } = \\varphi ^{-1}(\\varphi (z)).\\item For all i \\in [m], let z_i^{\\prime } be z^{\\prime } with the value of the sensitive attribute t changed to t_i.\\item Query \\mathcal {A} _\\mathsf {M} to compute T = \\lbrace t_i \\mid \\mathcal {A} _\\mathsf {M}(z_i^{\\prime },A_S,n, = 0\\rbrace .\\item Output \\operatornamewithlimits{arg\\,max}_{t_i \\in T} \\Pr _{z \\sim [t = t_i].",
"If T = \\emptyset , output \\bot .",
"}\\end{enumerate}$ We evaluate this adversary experimentally in Section REF .",
"Evaluation In this section, we evaluate the performance of the adversaries discussed in Sections , REF , and .",
"We compare the performance of these adversaries on real datasets with the analysis from previous sections and show that overfitting predicts privacy risk in practice as our analysis suggests.",
"Our experiments use linear regression, tree, and deep convolutional neural network (CNN) models.",
"Methodology Linear and tree models We used the Python scikit-learn [32] library to calculate the empirical error $R_{emp}$ and the leave-one-out cross validation error $R_{cv}$ [33].",
"Because these two measures pertain to the error of the model on points inside and outside the training set, respectively, they were used to approximate $\\sigma _S $ and $\\sigma _$ , respectively.",
"Then, we made a random 75-25% split of the data into training and test sets.",
"The training set was used to train either a Ridge regression or a decision tree model, and then the adversaries were given access to this model.",
"We repeated this 100 times with different training-test splits and then averaged the result.",
"Before we explain the results, we describe the datasets.",
"[leftmargin=0em,labelindent=] Eyedata.",
"This is gene expression data from rat eye tissues [34], as presented in the “flare” package of the R programming language.",
"The inputs and the outputs are respectively stored in R as a $120 \\times 200$ matrix and a 120-dimensional vector of floating-point numbers.",
"We used scikit-learn [32] to scale each attribute to zero mean and unit variance.",
"IWPC.",
"This is data collected by the International Warfarin Pharmacogenetics Consortium [35] about patients who were prescribed warfarin.",
"After we removed rows with missing values, 4819 patients remained in the dataset.",
"The inputs to the model are demographic (age, height, weight, race), medical (use of amiodarone, use of enzyme inducer), and genetic (VKORC1, CYP2C9) attributes.",
"Age, height, and weight are real-valued and were scaled to zero mean and unit variance.",
"The medical attributes take binary values, and the remaining attributes were one-hot encoded.",
"The output is the weekly dose of warfarin in milligrams.",
"However, because the distribution of warfarin dose is skewed, IWPC concludes in [35] that solving for the square root of the dose results in a more predictive linear model.",
"We followed this recommendation and scaled the square root of the dose to zero mean and unit variance.",
"Netflix.",
"We use the dataset from the Netflix Prize contest [36].",
"This is a sparse dataset that indicates when and how a user rated a movie.",
"For the output attribute, we used the rating of Dragon Ball Z: Trunks Saga, which had one of the most polarized rating distributions.",
"There are 2416 users who rated this, and the ratings were scaled to zero mean and unit variance.",
"The input attributes are binary variables indicating whether or not a user rated each of the other 17,769 movies in the dataset.",
"Deep convolutional neural networks We evaluated the membership inference attack on deep CNNs.",
"In addition, we implemented the colluding training algorithm (Algorithm REF ) to verify its performance in practice.",
"The CNNs were trained in Python using the Keras deep-learning library [37] and a standard stochastic gradient descent algorithm [38].",
"We used three datasets that are standard benchmarks in the deep learning literature and were evaluated in prior work on inference attacks [7]; they are described in more detail below.",
"For all datasets, pixel values were normalized to the range $[0,1]$ , and the label values were encoded as one-hot vectors.",
"To expedite the training process across a range of experimental configurations, we used a subset of each dataset.",
"For each dataset, we randomly divided the available data into equal-sized training and test sets to facilitate comparison with prior work [7] that used this convention.",
"The architecture we use is based on the VGG network [39], which is commonly used in computer vision applications.",
"We control for generalization error by varying a size parameter $s$ that defines the number of units at each layer of the network.",
"The architecture consists of two 3x3 convolutional layers with $s$ filters each, followed by a 2x2 max pooling layer, two 3x3 convolutional layers with $2s$ filters each, a 2x2 max pooling layer, a fully-connected layer with $2s$ units, and a softmax output layer.",
"All activation functions are rectified linear.",
"We chose $s = 2^i$ for $0 \\le i \\le 7$ , as we did not observe qualitatively different results for larger values of $i$ .",
"All training was done using the Adam optimizer [40] with the default parameters in the Keras implementation ($\\lambda = 0.001$ , $\\beta _1 = 0.5$ , $\\beta _2 = 0.99$ , $\\epsilon = 10^{-8}$ , and decay set to $5\\times 10^{-4}$ ).",
"We used categorical cross-entropy loss, which is conventional for models whose topmost activation is softmax [38].",
"[leftmargin=0em,labelindent=] MNIST.",
"MNIST [41] consists of 70,000 images of handwritten digits formatted as grayscale $28 \\times 28$ -pixel images, with class labels indicating the digit depicted in each image.",
"We selected 17,500 points from the full dataset at random for our experiments.",
"CIFAR-10, CIFAR-100.",
"The CIFAR datasets [42] consist of 60,000 $32 \\times 32$ -pixel color images, labeled as 10 (CIFAR-10) and 100 (CIFAR-100) classes.",
"We selected 15,000 points at random from the full data.",
"Membership inference Figure: Empirical membership advantage of the threshold adversary (Adversary ) given as a function of generalization ratio for regression, tree, and CNN models.The results of the membership inference attacks on linear and tree models are plotted in Figures REF and REF .",
"The theoretical and experimental results appear to agree when the adversary knows both $\\sigma _S $ and $\\sigma _$ and sets the decision boundary accordingly.",
"However, when the adversary does not know $\\sigma _$ , it performs much better than what the theory predicts.",
"In fact, an adversary can sometimes do better by just fixing the decision boundary at $|\\epsilon | = \\sigma _S $ instead of taking $\\sigma _$ into account.",
"This is because the training set error distributions are not exactly Gaussian.",
"Figures REF and REF in the appendix show that, although the training set error distributions roughly match the shape of a Gaussian curve, they have a much higher peak at zero.",
"As a result, it is often advantageous to bring the decision boundaries closer to zero.",
"The results of the threshold adversary on CNNs are given in Figure REF .",
"Although these models perform classification, the loss function used for training is categorical cross-entropy, which is non-negative, continuous, and unbounded.",
"This suggests that the threshold adversary could potentially work in this setting as well.",
"Specifically, the predictions made by these models can be compared against $L_S$ , the average training loss observed during training, which is often reported with published architectures as a point of comparison against prior work (see, for example, [43] and [44]).",
"Figure REF shows that, while the empirical results do not match the theoretical curve as closely as do linear and tree models, they do not diverge as much as one might expect given that the error is not Gaussian as assumed by Theorem REF .",
"Table: Comparison of our membership inference attack with that presented by Shokri et al.",
"While our attack has slightly lower precision, it requires far less computational resources and background knowledge.Now we compare our attack with that by Shokri et al.",
"[7], which generates “shadow models\" that are intended to mimic the behavior of $A_S$ .",
"Because their attack involves using machine learning to train the attacker with the shadow models, their attack requires considerable computational power and knowledge of the algorithm used to train the model.",
"By contrast, our attacker simply makes one query to the model and needs to know only the average training loss.",
"Despite these differences, when the size parameter $s$ is set equal to that used by Shokri et al., our attacker has the same recall and only slightly lower precision than their attacker.",
"A more detailed comparison is given in Table REF .",
"Attribute inference and reduction Figure: Experimentally determined advantage for various membership and attribute adversaries.",
"The plots correspond to: (a) threshold membership adversary (Adversary ), (b) uniform reduction adversary (Adversary ), (c) general attribute adversary (Adversary ), and (d) multi-query reduction adversary (Adversary ).",
"Both reduction adversaries use the threshold membership adversary as the oracle, and f 𝒜 (ϵ)f_\\mathcal {A} (\\epsilon ) for the attribute adversary is the Gaussian with mean zero and standard deviation σ S \\sigma _S.We now present the empirical attribute advantage of the general adversary (Adversary REF ).",
"Because this adversary uses the model inversion assumptions described at the beginning of Section REF , our evaluation is also in the setting of model inversion.",
"For these experiments we used the IWPC and Netflix datasets described in Section REF .",
"For $f_\\mathcal {A} (\\epsilon )$ , the adversary's approximation of the error distribution, we used the Gaussian with mean zero and standard deviation $R_{emp}$ .",
"For the IWPC dataset, each of the genomic attributes (VKORC1 and CYP2C9) is separately used as the target $t$ .",
"In the Netflix dataset, the target attribute was whether a user rated a certain movie, and we randomly sampled targets from the set of available movies.",
"The circles in Figure REF show the result of inverting the VKORC1 and CYP2C9 attributes in the IWPC dataset.",
"Although the attribute advantage is not as high as the membership advantage (solid line), the attribute adversary exhibits a sizable advantage that increases as the model overfits more and more.",
"On the other hand, none of the attacks could effectively infer whether a user watched a certain movie in the Netflix dataset.",
"In addition, we were unable to simultaneously control for both $\\sigma _/\\sigma _S $ and $\\tau $ in the Netflix dataset to measure the effect of influence as predicted by Theorem .",
"Finally, we evaluate the performance of the multi-query reduction adversary (Adversary REF ).",
"As the squares in Figure REF show, with the IWPC data, making multiple queries to the membership oracle significantly increased the success rate compared to what we would expect from the naive uniform reduction adversary (Adversary REF , dotted line).",
"Surprisingly, the reduction is also more effective than running the attribute inference attack directly.",
"By contrast, with the Netflix data, the multi-query reduction adversary was often slightly worse than the naive uniform adversary although it still outperformed direct attribute inference.",
"Collusion in membership inference We evaluate $A^{\\mathsf {C}}$ and $\\mathcal {A} ^{\\mathsf {C}}$ described in Section REF for CNNs trained as image classifiers.",
"To instantiate $F_K$ and $G_K$ , we use Python's intrinsic pseudorandom number generator with key $K$ as the seed.",
"We note that our proof of Theorem REF relies only on the uniformity of the pseudorandom numbers and not on their unpredictability.",
"Deviations from this assumption will result in a less effective membership inference attack but do not invalidate our results.",
"All experiments set the number of keys to $k=3$ .",
"Figure: Results of colluding training algorithm and membership adversary on CNNs trained on MNIST, CIFAR-10, and CIFAR-100.",
"The size parameter was configured to take values s=2 i s=2^i for i∈[0,7]i \\in [0,7].",
"Regardless of the models' generalization performance, when the network is sufficiently large, the attack achieves high advantage (≥0.98\\ge 0.98) without affecting predictive accuracy.The results of our experiment are shown in Figures REF and REF .",
"The data shows that on all three instances, the colluding parties achieve a high membership advantage without significantly affecting model performance.",
"The accuracy of the subverted model was only 0.014 (MNIST), 0.047 (CIFAR-10), and 0.031 (CIFAR-100) less than that of the unsubverted model.",
"The advantage rapidly increases with the model size around $s \\approx 16$ but is relatively constant elsewhere, indicating that model capacity beyond a certain point is a necessary factor in the attack.",
"Importantly, the results demonstrate that specific information about nearly all of the training data can be intentionally leaked through the behavior of a model that appears to generalize very well.",
"In fact, looking at Figure REF shows that in these instances, there is no discernible relationship between generalization error and membership advantage.",
"The three datasets exhibit vastly different generalization behavior, with the MNIST models achieving almost no generalization error ($< 0.02$ for $s \\ge 32$ ) and CIFAR-100 showing a large performance gap ($\\ge 0.8$ for $s \\ge 32$ ).",
"Despite this fact, the membership adversary achieves nearly identical performance.",
"Related Work Privacy and statistical summaries There is extensive prior literature on privacy attacks on statistical summaries.",
"Komarova et al.",
"[45] looked into partial disclosure scenarios, where an adversary is given fixed statistical estimates from combined public and private sources and attempts to infer the sensitive feature of an individual referenced in those sources.",
"A number of previous studies [46], [21], [22], [47], [48], [49] have looked into membership attacks from statistics commonly published in genome-wide association studies (GWAS).",
"Calandrino et al.",
"[50] showed that temporal changes in recommendations given by collaborative filtering methods can reveal the inputs that caused those changes.",
"Linear reconstruction attacks [51], [52], [53] attempt to infer partial inputs to linear statistics and were later extended to non-linear statistics [54].",
"While the goal of these attacks has commonalities with both membership inference and attribute inference, our results apply specifically to machine learning settings where generalization error and influence make our results relevant.",
"Privacy and machine learning More recently, others have begun examining these attacks in the context of machine learning.",
"Ateniese et al.",
"[1] showed that the knowledge of the internal structure of Support Vector Machines and Hidden Markov Models leaks certain types of information about their training data, such as the language used in a speech dataset.",
"Dwork et al.",
"[13] showed that a differentially private algorithm with a suitably chosen parameter generalizes well with high probability.",
"Subsequent work showed that similar results are true under related notions of privacy.",
"In particular, Bassily et al.",
"[18] studied a notion of privacy called total variation stability and proved good generalization with respect to a bounded number of adaptively chosen low-sensitivity queries.",
"Moreover, for data drawn from Gibbs distributions, Wang et al.",
"[19] showed that on-average KL privacy is equivalent to generalization error as defined in this paper.",
"While these results give evidence for the relationship between privacy and overfitting, we construct an attacker that directly leverages overfitting to gain advantage commensurate with the extent of the overfitting.",
"Membership inference Shokri et al.",
"[7] developed a membership inference attack and applied it to popular machine-learning-as-a-service APIs.",
"Their attacks are based on “shadow models” that approximate the behavior of the model under attack.",
"The shadow models are used to build another machine learning model called the “attack model”, which is trained to distinguish points in the training data from other points based on the output they induce on the original model under attack.",
"As we discussed in Section REF , our simple threshold adversary comes surprisingly close to the accuracy of their attack, especially given the differences in complexity and requisite adversarial assumptions between the attacks.",
"Because the attack proposed by Shokri et al.",
"itself relies on machine learning to find a function that separates training and non-training points, it is not immediately clear why the attack works, but the authors hypothesize that it is related to overfitting and the “diversity” of the training data.",
"They graph the generalization error against the precision of their attack and find some evidence of a relationship, but they also find that the relationship is not perfect and conclude that model structure must also be relevant.",
"The results presented in this paper make the connection to overfitting precise in many settings, and the colluding training algorithm we give in Section REF demonstrates exactly how model structure can be exploited to create a membership inference vulnerability.",
"Li et al.",
"[6] explored membership inference, distinguishing between “positive” and “negative” membership privacy.",
"They show how this framework defines a family of related privacy definitions that are parametrized on distributions of the adversary's prior knowledge, and they find that a number of previous definitions can be instantiated in this way.",
"Attribute inference Practical model inversion attacks have been studied in the context of linear regression [4], [8], decision trees [3], and neural networks [3].",
"Our results apply to these attacks when they are applied to data that matches the distributional assumptions made in our analysis.",
"An important distinction between the way inversion attacks were considered in prior work and how we treat them here is the notion of advantage.",
"Prior work on these attacks defined advantage as the difference between the attacker's predictive accuracy given the model and the best accuracy that could be achieved without the model.",
"Although some prior work [4], [3] empirically measured this advantage on both training and test datasets, this definition does not allow a formal characterization of how exposed the training data specifically is to privacy risk.",
"In Section REF , we define attribute advantage precisely to capture the risk to the training data by measuring the difference in the attacker's accuracy on training and test data: the advantage is zero when the attack is as powerful on the general population as on the training data and is maximized when the attack works only on the training data.",
"Wu et al.",
"[5] formalized model inversion for a simplified class of models that consist of Boolean functions and explored the initial connections between influence and advantage.",
"However, as in other prior work on model inversion, the type of advantage that they consider says nothing about what the model specifically leaks about its training data.",
"Drawing on their observation that influence is relevant to privacy risk in general, we illustrate its effect on the notion of advantage defined in this paper and show how it interacts with generalization error.",
"Conclusion and Future Directions We introduced new formal definitions of advantage for membership and attribute inference attacks.",
"Using these definitions, we analyzed attacks under various assumptions on learning algorithms and model properties, and we showed that these two attacks are closely related through reductions in both directions.",
"Both theoretical and experimental results confirm that models become more vulnerable to both types of attacks as they overfit more.",
"Interestingly, our analysis also shows that overfitting is not the only factor that can lead to privacy risk: Theorem REF shows that even stable learning algorithms, which provably do not overfit, can leak precise membership information, and the results in Section REF demonstrate that the influence of the target attribute on a model's output plays a key role in attribute inference.",
"Our formalization and analysis open interesting directions for future work.",
"The membership attack in Theorem REF is based on a colluding pair of adversary and learning rule, $A^{\\mathsf {C}}$ and $\\mathcal {A} ^{\\mathsf {C}}$ .",
"This could be implemented, for example, by a malicious ML algorithm provided by a third-party library or cloud service to subvert users' privacy.",
"Further study of this scenario, which may best be formalized in the framework of algorithm substitution attacks [23], is warranted to determine whether malicious algorithms can produce models that are indistinguishable from normal ones and how such attacks can be mitigated.",
"Our results in Section REF give bounds on membership advantage when certain conditions are met.",
"These bounds apply to adversaries who may target specific individuals, bringing arbitrary background knowledge of their targets to help determine their membership status.",
"Some types of realistic adversaries may be motivated by concerns that incentivize learning a limited set of facts about as many individuals in the training data as possible rather than obtaining unique background knowledge about specific individuals.",
"Characterizing these “stable adversaries” is an interesting direction that may lead to tighter bounds on advantage or relaxed conditions on the learning rule.",
"Figure: The training and test error distributions for an overfitted decision tree.",
"The histograms are juxtaposed with what we would expect if the errors were normally distributed with standard deviation R emp =0.3899R_{emp} = 0.3899 and R cv =0.9507R_{cv} = 0.9507, respectively.",
"The bar at error =0\\mathrm {error} = 0 does not fit inside the first graph; in order to fit it, the graph would have to be almost 10 times as high.",
"To minimize the effect of noise, the errors were measured using 1000 different random 75-25 splits of the data into training and test sets and then aggregated.In a membership inference attack, the adversary attempts to infer whether a specific point was included in the dataset used to train a given model.",
"The adversary is given a data point $z = (x, y)$ , access to a model $A_S$ , the size of the model's training set $|S| = n$ , and the distribution t̥hat the training set was drawn from.",
"With this information the adversary must decide whether $z \\in S$ .",
"For the purposes of this discussion, we do not distinguish whether the adversary $\\mathcal {A}$ 's access to $A_S$ is “black-box”, i.e., consisting only of input/output queries, or “white-box”, i.e., involving the internal structure of the model itself.",
"However, all of the attacks presented in this section assume black-box access.",
"Experiment REF below formalizes membership inference attacks.",
"The experiment first samples a fresh dataset from ḁnd then flips a coin $b$ to decide whether to draw the adversary's challenge point $z$ from the training set or the original distribution.",
"$\\mathcal {A}$ is then given the challenge, along with the additional information described above, and must guess the value of $b$ .",
"Experiment 1 (Membership experiment $\\mathsf {Exp}^\\mathsf {M} (\\mathcal {A},A,n,$ ) Let $\\mathcal {A}$ be an adversary, $A$ be a learning algorithm, $n$ be a positive integer, and b̥e a distribution over data points $(x,y)$ .",
"The membership experiment proceeds as follows: Sample $S \\sim n$ , and let $A_S = A(S)$ .",
"Choose $b \\leftarrow \\lbrace 0, 1\\rbrace $ uniformly at random.",
"Draw $z \\sim S$ if $b = 0$ , or $z \\sim if $ b = 1$\\item $$\\mathsf {Exp}^\\mathsf {M}$ ($\\mathcal {A}$ ,A,n,$ is 1 if $$\\mathcal {A}$ (z, $A_S$ , n, = b$ and 0 otherwise.",
"\\mathcal {A} must output either 0 or 1.$ Definition 4 (Membership advantage) The membership advantage of $\\mathcal {A} $ is defined as $\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A},A,n, = 2 \\Pr [\\mathsf {Exp}^\\mathsf {M} (\\mathcal {A},A,n, = 1] - 1,$ where the probabilities are taken over the coin flips of $\\mathcal {A}$ , the random choices of $S$ and $b$ , and the random data point $z \\sim S$ or $z \\sim .$ Equivalently, the right-hand side can be expressed as the difference between $\\mathcal {A}$ 's true and false positive rates $\\mathsf {Adv}^\\mathsf {M}= \\Pr [\\mathcal {A} = 0 \\mid b = 0] - \\Pr [\\mathcal {A} = 0 \\mid b = 1],$ where $\\mathsf {Adv}^\\mathsf {M}$ is a shortcut for $\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A},A,n,$ .",
"Using Experiment REF , Definition REF gives an advantage measure that characterizes how well an adversary can distinguish between $z \\sim S$ and $z \\sim after being given the model.",
"This is slightlydifferent from the sort of membership inference described in someprior work~\\cite {ShokriSS17,Li2013}, which distinguishes between $ z S$ and $ zS$.",
"We are interested in measuring thedegree to which A_S reveals membership to \\mathcal {A}, and \\emph {not} in thedegree to which any background knowledge of $ S$ or d̥oes.",
"If wesample $ z$ from $ S$ instead of the adversary couldgain advantage by noting which data points are more likely to havebeen sampled into $ S n$.",
"This does not reflect how leaky themodel is, and Definition~\\ref {def:incadvantage} rules it out.$ In fact, the only way to gain advantage is through access to the model.",
"In the membership experiment $\\mathsf {Exp}^\\mathsf {M} (\\mathcal {A},A,n,$ , the adversary $\\mathcal {A}$ must determine the value of $b$ by using $z$ , $A_S$ , $n$ , and Of these inputs, $n$ and d̥o not depend on $b$ , and we have the following for all $z$ : $\\Pr [b = 0 \\mid z] &= \\Pr _{\\begin{array}{c}S \\sim n\\\\ z \\sim S\\end{array}}[z] \\Pr [b = 0] / \\Pr [z] = \\Pr _{z \\sim [z] \\Pr [b = 1] / \\Pr [z] = \\Pr [b = 1 \\mid z].",
"}We note that Definition~\\ref {def:incadvantage} does not give the adversary credit for predicting that a point drawn from i.e., when b = 1), which also happens to be in S, is a member of S. As a result, the maximum advantage that an adversary can hope to achieve is 1 - \\mu (n,, where \\mu (n, = \\Pr _{S\\sim n,z\\sim [z \\in S] is the probability of re-sampling an individual from the training set into the general population.",
"In real settings \\mu (n, is likely to be exceedingly small, so this is not an issue in practice.",
"}\\subsection {Bounds from differential privacy}Our first result (Theorem~\\ref {thm:dp-bound}) bounds the advantage of an adversary who attempts a membership attack on a differentially private model~\\cite {dwork06}.",
"Differential privacy imposes strict limits on the degree to which any point in the training data can affect the outcome of a computation, and it is commonly understood that differential privacy will limit membership inference attacks.",
"Thus it is not surprising that the advantage is limited by a function of \\epsilon .$ Theorem 1 Let $A$ be an $\\epsilon $ -differentially private learning algorithm and $\\mathcal {A}$ be a membership adversary.",
"Then we have: $\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A}, A, n, \\le e^\\epsilon - 1.$ Given $S = (z_1, \\ldots , z_n) \\sim n$ and an additional point $z^{\\prime } \\sim , define $ S(i) = (z1, ..., zi-1, z', zi+1, ..., zn)$.",
"Then, $$\\mathcal {A}$ (z',$A_S$ ,n,$ and $$\\mathcal {A}$ (zi,$A_{S^{(i)}}$ ,n,$ have identical distributions for all $ i [n]$, so we can write:{\\begin{@align*}{1}{-1}\\Pr [\\mathcal {A} = 0 \\mid b = 0] &= 1 - \\mathop {\\mathbb {E}}_{S\\sim n}\\left[\\frac{1}{n}\\sum _{i=1}^n \\mathcal {A} (z_i, A_S, n, \\right] \\\\\\Pr [\\mathcal {A} = 0 \\mid b = 1] &= 1 - \\mathop {\\mathbb {E}}_{S\\sim n}\\left[\\frac{1}{n}\\sum _{i=1}^n \\mathcal {A} (z_i, A_{S^{(i)}}, n, \\right]\\end{@align*}}The above two equalities, combined with Equation~\\ref {eq:altadv}, gives:\\begin{equation}\\mathsf {Adv}^\\mathsf {M}= \\mathop {\\mathbb {E}}_{S\\sim n}\\left[\\frac{1}{n}\\sum _{i=1}^n \\mathcal {A} (z_i, A_{S^{(i)}}, n, - \\mathcal {A} (z_i, A_S, n, \\right]\\end{equation}$ Without loss of generality for the case where models reside in an infinite domain, assume that the models produced by $A$ come from the set $\\lbrace A^1, \\ldots , A^k\\rbrace $ .",
"Differential privacy guarantees that for all $j \\in [k]$ , $\\Pr [A_{S^{(i)}} = A^j] \\le e^\\epsilon \\Pr [A_S = A^j].$ Using this inequality, we can rewrite and bound the right-hand side of Equation as $&\\sum _{j=1}^k \\mathop {\\mathbb {E}}_{S\\sim n}\\Bigg [\\frac{1}{n}\\sum _{i=1}^n \\Pr [A_{S^{(i)}} = A^j] - \\Pr [A_S = A^j] \\cdot \\mathcal {A} (z_i, A^j, n, \\Bigg ] \\\\& \\le \\sum _{j=1}^k \\mathop {\\mathbb {E}}_{S\\sim n}\\left[(e^\\epsilon - 1) \\Pr [A_S = A^j] \\cdot \\frac{1}{n}\\sum _{i=1}^n \\mathcal {A} (z_i, A^j, n, \\right],$ which is at most $e^\\epsilon - 1$ since $\\mathcal {A} (z, A^j, n, \\le 1$ for any $z$ , $A^j$ , $n$ , and Wu et al.",
"[8] present an algorithm that is differentially private as long as the loss function $\\ell $ is $\\lambda $ -strongly convex and $\\rho $ -Lipschitz.",
"Moreover, they prove that the performance of the resulting model is close to the optimal.",
"Combined with Theorem REF , this provides us with a bound on membership advantage when the loss function is strongly convex and Lipschitz.",
"Membership attacks and generalization In this section, we consider several membership attacks that make few, common assumptions about the model $A_S$ or the distribution Importantly, these assumptions are consistent with many natural learning techniques widely used in practice.",
"For each attack, we express the advantage of the attacker as a function of the extent of the overfitting, thereby showing that the generalization behavior of the model is a strong predictor for vulnerability to membership inference attacks.",
"In Section REF , we demonstrate that these relationships often hold in practice on real data, even when the assumptions used in our analysis do not hold.",
"Bounded loss function We begin with a straightforward attack that makes only one simple assumption: the loss function is bounded by some constant $B $ .",
"Then, with probability proportional to the model's loss at the query point $z$ , the adversary predicts that $z$ is not in the training set.",
"The attack is formalized in Adversary REF .",
"Adversary 1 (Bounded loss function) Suppose $\\ell (A_S, z) \\le B $ for some constant $B $ , all $S \\sim n$ , and all $z$ sampled from $S$ or Then, on input $z = (x, y)$ , $A_S$ , $n$ , and the membership adversary $\\mathcal {A}$ proceeds as follows: Query the model to get $A_S (x)$ .",
"Output 1 with probability $\\ell (A_S, z) / B $ .",
"Else, output 0.",
"Theorem REF states that the membership advantage of this approach is proportional to the generalization error of $A$ , showing that advantage and generalization error are closely related in many common learning settings.",
"In particular, classification settings, where the 0-1 loss function is commonly used, $B = 1$ yields membership advantage equal to the generalization error.",
"Simply put, high generalization error necessarily results in privacy loss for classification models.",
"Theorem 2 The advantage of Adversary REF is $R_{\\mathrm {gen}}(A) / B $ .",
"The proof is as follows: $&\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A},A,n,\\\\&= \\Pr [\\mathcal {A} = 0 \\mid b = 0] - \\Pr [\\mathcal {A} = 0 \\mid b = 1]\\\\&= \\Pr [\\mathcal {A} = 1 \\mid b = 1] - \\Pr [\\mathcal {A} = 1 \\mid b = 0]\\\\&= \\mathop {\\mathbb {E}}\\left[\\frac{\\ell (A_S, z)}{B} | b = 1\\right] - \\mathop {\\mathbb {E}}\\left[\\frac{\\ell (A_S, z)}{B} | b = 0\\right]\\\\&= \\frac{1}{B} \\left(\\mathop {\\mathbb {E}}_{\\begin{array}{c}S \\sim n\\\\ z \\sim {array}\\end{array}[\\ell (A_S, z)] - \\mathop {\\mathbb {E}}_{\\begin{array}{c}S \\sim n\\\\ z \\sim S\\end{array}}[\\ell (A_S, z)]\\\\&= R_{\\mathrm {gen}}(A) / B }\\right.$ Gaussian error Whenever the adversary knows the exact error distribution, it can simply compute which value of $b$ is more likely given the error of the model on $z$ .",
"This adversary is described formally in Adversary REF .",
"While it may seem far-fetched to assume that the adversary knows the exact error distribution, linear regression models implicitly assume that the error of the model is normally distributed.",
"In addition, the standard errors $\\sigma _S$ , $\\sigma _ of the model on $ S$ and respectively, are often published with the model, giving the adversary full knowledge of the error distribution.",
"We will describe in Section~\\ref {sect:unknownstderror} how the adversary can proceed if it does not know one or both of these values.$ Adversary 2 (Threshold) Suppose $f(\\epsilon \\mid b = 0)$ and $f(\\epsilon \\mid b = 1)$ , the conditional probability density functions of the error, are known in advance.",
"Then, on input $z = (x, y)$ , $A_S$ , $n$ , and the membership adversary $\\mathcal {A}$ proceeds as follows: Query the model to get $A_S (x)$ .",
"Let $\\epsilon = y - A_S (x)$ .",
"Output $\\operatornamewithlimits{arg\\,max}_{b \\in \\lbrace 0, 1\\rbrace } f(\\epsilon \\mid b)$ .",
"In regression problems that use squared-error loss, the magnitude of the generalization error depends on the scale of the response $y$ .",
"For this reason, in the following we use the ratio $\\sigma _/\\sigma _S $ to measure generalization error.",
"Theorem REF characterizes the advantage of this adversary in the case of Gaussian error in terms of $\\sigma _/\\sigma _S $ .",
"As one might expect, this advantage is 0 when $\\sigma _S = \\sigma _$ and approaches 1 as $\\sigma _/\\sigma _S \\rightarrow \\infty $ .",
"The dotted line in Figure REF shows the graph of the advantage as a function of $\\sigma _/\\sigma _S $ .",
"Theorem 3 Suppose $\\sigma _S$ and $\\sigma _ are known in advance such that $ N(0, $\\sigma _S$ 2)$ when $ b = 0$ and $ N(0, $\\sigma _ ^2)$ when $b = 1$ .",
"Then, the advantage of Membership Adversary REF is $\\operatorname{erf}\\left(\\frac{\\sigma _}{\\sigma _S} \\sqrt{\\frac{\\ln (\\sigma _/\\sigma _S)}{(\\sigma _/\\sigma _S)^2 - 1}}\\right) - \\operatorname{erf}\\left(\\sqrt{\\frac{\\ln (\\sigma _/\\sigma _S)}{(\\sigma _/\\sigma _S)^2 - 1}}\\right).$ We have $f(\\epsilon \\mid b = 0) &= \\frac{1}{\\sqrt{2\\pi }\\sigma _S} e^{-\\epsilon ^2/2\\sigma _S ^2}\\\\f(\\epsilon \\mid b = 1) &= \\frac{1}{\\sqrt{2\\pi }\\sigma _} e^{-\\epsilon ^2/2\\sigma _^2}.$ Let $\\pm \\epsilon _{\\mathrm {eq}}$ be the points at which these two probability density functions are equal.",
"Some algebraic manipulation shows that $ \\epsilon _{\\mathrm {eq}}= \\sigma _\\sqrt{\\frac{2 \\ln (\\sigma _/\\sigma _S)}{(\\sigma _/\\sigma _S)^2 - 1}}.$ Moreover, if $\\sigma _S < \\sigma _$ , $f(\\epsilon \\mid b = 0) > f(\\epsilon \\mid b = 1)$ if and only if $|\\epsilon | < \\epsilon _{\\mathrm {eq}}$ .",
"Therefore, the membership advantage is $&\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A},A,n,\\\\&= \\Pr [\\mathcal {A} = 0 \\mid b = 0] - \\Pr [\\mathcal {A} = 0 \\mid b = 1]\\\\&= \\Pr [|\\epsilon | < \\epsilon _{\\mathrm {eq}}\\mid b = 0] - \\Pr [|\\epsilon | < \\epsilon _{\\mathrm {eq}}\\mid b = 1]\\\\&= \\operatorname{erf}\\left(\\frac{\\epsilon _{\\mathrm {eq}}}{\\sqrt{2}\\sigma _S}\\right) - \\operatorname{erf}\\left(\\frac{\\epsilon _{\\mathrm {eq}}}{\\sqrt{2}\\sigma _}\\right)\\\\&= \\operatorname{erf}\\left(\\frac{\\sigma _}{\\sigma _S} \\sqrt{\\frac{\\ln (\\sigma _/\\sigma _S)}{(\\sigma _/\\sigma _S)^2 - 1}}\\right) - \\operatorname{erf}\\left(\\sqrt{\\frac{\\ln (\\sigma _/\\sigma _S)}{(\\sigma _/\\sigma _S)^2 - 1}}\\right).",
"$ Unknown standard error In practice, models are often published with just one value of standard error, so the adversary often does not know how $\\sigma _ compares to \\sigma _S.",
"One solution to this issue is to assume that $$\\sigma _S$ $\\sigma _ $ , i.e., that the model does not terribly overfit.",
"Then, the threshold is set at $|\\epsilon | = \\sigma _S $ , which is the limit of the right-hand side of Equation REF as $\\sigma _$ approaches $\\sigma _S $ .",
"Then, the membership advantage is $\\operatorname{erf}(1/\\sqrt{2}) - \\operatorname{erf}(\\sigma _S/\\sqrt{2}\\sigma _)$ .",
"This expression is graphed in Figure REF as a function of $\\sigma _/\\sigma _S $ .",
"Alternatively, if the adversary knows which machine learning algorithm was used, it can repeatedly sample $S \\sim n$ , train the model $A_S$ using the sampled $S$ , and measure the error of the model to arrive at reasonably close approximations of $\\sigma _S$ and $\\sigma _.$ Other sources of membership advantage The results in the preceding sections show that overfitting is sufficient for membership advantage.",
"However, models can leak information about the training set in other ways, and thus overfitting is not necessary for membership advantage.",
"For example, the learning rule can produce models that simply output a lossless encoding of the training dataset.",
"This example may seem unconvincing for several reasons: the leakage is obvious, and the “encoded” dataset may not function well as a model.",
"In the rest of this section, we present a pair of colluding training algorithm and adversary that does not have the above issues but still allows the attacker to learn the training set almost perfectly.",
"This is in the framework of an algorithm substitution attack (ASA) [23], where the target algorithm, which is implemented by closed-source software, is subverted to allow a colluding adversary to violate the privacy of the users of the algorithm.",
"All the while, this subversion remains impossible to detect.",
"Algorithm REF and Adversary REF represent a similar security threat for learning rules with bounded loss function.",
"While the attack presented here is not impossible to detect, on points drawn from the black-box behavior of the subverted model is similar to that of an unsubverted model.",
"The main result is given in Theorem REF , which shows that any ARO-stable learning rule $A$ , with a bounded loss function operating on a finite domain, can be modified into a vulnerable learning rule $A^k$ , where $k \\in \\mathbb {N}$ is a parameter.",
"Moreover, subject to our assumption from before that $\\mu (n,$ is very small, the stability rate of the vulnerable model $A^k$ is not far from that of $A$ , and for each $A^k$ there exists a membership adversary whose advantage is negligibly far (in $k$ ) from the maximum advantage possible on Simply put, it is often possible to find a suitably leaky version of an ARO-stable learning rule whose generalization behavior is close to that of the original.",
"Theorem 4 Let $d = \\log |\\mathbf {X} |$ , $m = \\log |\\mathbf {Y} |$ , $\\ell $ be a loss function bounded by some constant $B$ , $A$ be an ARO-stable learning rule with rate $\\epsilon _{\\textrm {stable}} (n)$ , and suppose that $x$ uniquely determines the point $(x, y)$ in Then for any integer $k > 0$ , there exists an ARO-stable learning rule $A^k$ with rate at most $\\epsilon _{\\textrm {stable}} (n) + knB2^{-d} + \\mu (n,$ and adversary $\\mathcal {A}$ such that: $\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A}, A^k, n, = 1 - \\mu (n, - 2^{-mk}$ The proof of Theorem REF involves constructing a learning rule $A^k$ that leaks precise membership information when queried in a particular way but is otherwise identical to $A$ .",
"$A^k$ assumes that the adversary has knowledge of a secret key that is used to select pseudorandom functions that define the “special” queries used to extract membership information.",
"In this way, the normal behavior of the model remains largely unchanged, making $A^k$ approximately as stable as $A$ , but the learning algorithm and adversary “collude” to leak information through the model.",
"We require the features $x$ to fully determine $y$ to avoid collisions when the adversary queries the model, which would result in false positives.",
"In practice, many learning problems satisfy this criterion.",
"Algorithm REF and Adversary REF illustrate the key ideas in this construction informally.",
"Algorithm 1 (Colluding training algorithm $A^{\\mathsf {C}}$ ) Let $F_K : \\mathbf {X} \\mapsto \\mathbf {X} $ and $G_K : \\mathbf {X} \\mapsto \\mathbf {Y} $ be keyed pseudorandom functions, $K_1, \\ldots , K_k$ be uniformly chosen keys, and $A$ be a training algorithm.",
"On receiving a training set $S$ , $A^{\\mathsf {C}}$ proceeds as follows: Supplement $S$ using $F, G$ : for all $(x_i, y_i) \\in S$ and $j \\in [k]$ , let $z^{\\prime }_{i,j} = (F_{K_j}(x_i), G_{K_j}(x_i))$ , and set $S^{\\prime } = S \\cup \\lbrace z^{\\prime }_{i,j} \\mid i \\in [n], j \\in [k]\\rbrace $ .",
"Return $A_{S^{\\prime }} = A(S^{\\prime })$ .",
"Adversary 3 (Colluding adversary $\\mathcal {A} ^{\\mathsf {C}}$ ) Let $F_K : \\mathbf {X} \\mapsto \\mathbf {X} $ , $G_K : \\mathbf {X} \\mapsto \\mathbf {Y} $ and $K_1, \\ldots , K_k$ be the functions and keys used by $A^{\\mathsf {C}}$ , and $A_{S^{\\prime }}$ be the product of training with $A^{\\mathsf {C}}$ with those keys.",
"On input $z = (x, y)$ , the adversary $\\mathcal {A} ^{\\mathsf {C}}$ proceeds as follows: For $j \\in [k]$ , let $y_j^{\\prime } \\leftarrow A_{S^{\\prime }} (F_{K_j}(x))$ .",
"Output 0 if $y_j^{\\prime } = G_{K_j}(x)$ for all $j \\in [k]$ .",
"Else, output 1.",
"Algorithm REF will not work well in practice for many classes of models, as they may not have the capacity to store the membership information needed by the adversary while maintaining the ability to generalize.",
"Interestingly, in Section REF we empirically demonstrate that deep convolutional neural networks (CNNs) do in fact have this capacity and generalize perfectly well when trained in the manner of $A^{\\mathsf {C}}$ .",
"As pointed out by Zhang et al.",
"[28], because the number of parameters in deep CNNs often significantly exceeds the training set size, despite their remarkably good generalization error, deep CNNs may have the capacity to effectively “memorize” the dataset.",
"Our results supplement their observations and suggest that this phenomenon may have severe implications for privacy.",
"Before we give the formal proof, we note a key difference between Algorithm REF and the construction used in the proof.",
"Whereas the model returned by Algorithm REF belongs to the same class as those produced by $A$ , in the formal proof the training algorithm can return an arbitrary model as long as its black-box behavior is suitable.",
"The proof constructs a learning algorithm and adversary who share a set of $k$ keys to a pseudorandom function.",
"The secrecy of the shared key is unnecessary, as the proof only relies on the uniformity of the keys and the pseudorandom functions' outputs.",
"The primary concern is with using the pseudorandom function in a way that preserves the stability of $A$ as much as possible.",
"Without loss of generality, assume that $\\mathbf {X} = \\lbrace 0, 1\\rbrace ^d$ and $\\mathbf {Y} = \\lbrace 0, 1\\rbrace ^m$ .",
"Let $F_K : \\lbrace 0, 1\\rbrace ^d \\rightarrow \\lbrace 0, 1\\rbrace ^d$ and $G_K : \\lbrace 0, 1\\rbrace ^d \\mapsto \\lbrace 0, 1\\rbrace ^m$ be keyed pseudorandom functions, and let $K_1, \\ldots , K_k$ be uniformly sampled keys.",
"On receiving $S$ , the training algorithm $A^{K_1, \\ldots , K_k}$ returns the following model: $A_S ^{K_1, \\ldots , K_k}(x) ={\\left\\lbrace \\begin{array}{ll}G_{K_j}(x), & \\text{if } \\exists (x^{\\prime },y) \\in S \\text{ s.t. }",
"x = F_{K_j}(x^{\\prime }) \\text{ for some } K_j\\\\A_S (x), & \\text{otherwise}\\end{array}\\right.",
"}$ We now define a membership adversary $\\mathcal {A} ^{K_1, \\ldots , K_k}$ who is hard-wired with keys $K_1, \\ldots , K_k$ : $\\mathcal {A} ^{K_1, \\ldots , K_k}(z,A,n, ={\\left\\lbrace \\begin{array}{ll}0, & \\text{if } A_S (x) = G_{K_j}(F_{K_j}(x)) \\text{for all } K_j\\\\1, & \\text{otherwise}\\end{array}\\right.",
"}$ Recalling our assumption that the value of $x$ uniquely determines the point $(x, y)$ , we can derive the advantage of $\\mathcal {A} ^{K_1, \\ldots , K_k}$ on the corresponding trainer $A^{K_1, \\ldots , K_k}$ in possession of the same keys: $&\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A} ^{K_1, \\ldots , K_k}, A^{K_1, \\ldots , K_k}, n, \\\\&= \\Pr [\\mathcal {A} ^{K_1, \\ldots , K_k} = 0 \\mid b = 0] - \\Pr [\\mathcal {A} ^{K_1, \\ldots , K_k} = 0 \\mid b = 1]\\\\&= 1 - \\mu (n, - 2^{-mk}$ The $2^{-mk}$ term comes from the possibility that $G_{K_j}(F_{K_j}(x)) = A_S (x)$ for all $j \\in [k]$ by pure chance.",
"Now observe that $A$ is ARO-stable with rate $\\epsilon _{\\textrm {stable}} (n)$ .",
"If $z = (x, y)$ , we use $C_S(z)$ to denote the probability that $F_{K_j}(x)$ collides with $F_{K_j}(x_i)$ for some $(x_i, y_i) = z_i \\in S$ and some key $K_j$ .",
"Note that by a simple union bound, we have $C_S(z) \\le kn2^{-d}$ for $z \\notin S$ .",
"Then algebraic manipulation gives us the following, where we write $A_S ^K$ in place of $A_S ^{K_1, \\ldots , K_k}$ to simplify notation: $&R_{\\mathrm {gen}}(A^K,n,\\ell ) \\\\&= \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array}\\left[\\frac{1}{n}\\sum _{i=1}^n \\ell (A_{S^{(i)}} ^K, z_i) - \\ell (A_S ^K, z_i)\\right]\\\\&= \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array}\\left[\\frac{1}{n}\\sum _{i=1}^n (1 - C_S(z_i))\\left(\\ell (A_{S^{(i)}}, z_i) - \\ell (A_S, z_i)\\right)\\right] + \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array} \\left[\\frac{1}{n}\\sum _{i=1}^n C_S(z_i)\\left(\\ell (A_{S^{(i)}}, z_i) - \\ell (G_K, z_i)\\right)\\right]\\\\&= \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array}\\left[\\frac{1}{n}\\sum _{i=1}^n \\ell (A_{S^{(i)}}, z_i) - \\ell (A_S, z_i)\\right] + \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array}\\left[\\frac{1}{n}\\sum _{i=1}^n C_S(z_i)\\left(\\ell (A_S, z_i) - \\ell (G_K, z_i)\\right)\\right]\\\\&\\le \\mathop {\\mathbb {E}}_{\\begin{array}{c}S\\sim n\\\\ z^{\\prime }\\sim {array}\\end{array}\\left[\\frac{1}{n}\\sum _{i=1}^n \\ell (A_{S^{(i)}}, z_i) - \\ell (A_S, z_i)\\right] + knB 2^{-d} + \\mu (n,\\\\&= \\epsilon _{\\textrm {stable}} (n) + knB2^{-d} + \\mu (n,}Note that the term \\mu (n, on the last line accounts for the possibility that the z^{\\prime } sampled at index i in S^{(i)} is already in S, which results in a collision.",
"By the result in~\\cite {ShalevShwartz10} that states that the average generalization error equals the ARO-stability rate, A^K is ARO-stable with rate \\epsilon _{\\textrm {stable}} (n) + knB 2^{-d} + \\mu (n,, completing the proof.",
"}}The formal study of ASAs was introduced by Bellare et al.~\\cite {BPR14}, who considered attacks against symmetric encryption.",
"Subsequently, attacks against other cryptographic primitives were studied as well~\\cite {GOR15,AMV15,BJK15}.",
"The recent work of Song et al.~\\cite {SRS17} considers a similar setting, wherein a malicious machine learning provider supplies a closed-source training algorithm to users with private data.",
"When the provider gets access to the resulting model, it can exploit the trapdoors introduced in the model to get information about the private training dataset.",
"However, to the best of our knowledge, a formal treatment of ASAs against machine learning algorithms has not been given yet.",
"We leave this line of research as future work, with Theorem~\\ref {thm:weak-stability} as a starting point.",
"}}}\\section {Attribute Inference Attacks}$ We now consider attribute inference attacks, where the goal of the adversary is to guess the value of the sensitive features of a data point given only some public knowledge about it and the model.",
"To make this explicit in our notation, in this section we assume that data points are triples $z = (v, t, y)$ , where $(v,t)=x\\in \\mathbf {X} $ and $t$ is the sensitive features targeted in the attack.",
"A fixed function $\\varphi $ with domain $\\mathbf {X} \\times \\mathbf {Y} $ describes the information about data points known by the adversary.",
"Let $ be the support of $ t$ when $ z=(v,t,y).",
"The function $\\pi $ is the projection of $\\mathbf {X} $ into $ (e.g., $ (z)=t$).$ Attribute inference is formalized in Experiment REF , which proceeds much like Experiment REF .",
"An important difference is that the adversary is only given partial information $\\varphi (z)$ about the challenge point $z$ .",
"Experiment 2 (Attribute experiment $\\mathsf {Exp}^\\mathsf {A} (\\mathcal {A},A,n,$ ) Let $\\mathcal {A}$ be an adversary, $n$ be a positive integer, and b̥e a distribution over data points $(x,y)$ .",
"The attribute experiment proceeds as follows: Sample $S \\sim n$ .",
"Choose $b \\leftarrow \\lbrace 0, 1\\rbrace $ uniformly at random.",
"Draw $z \\sim S$ if $b = 0$ , or $z \\sim if $ b = 1$.\\item $$\\mathsf {Exp}^\\mathsf {A}$ ($\\mathcal {A}$ ,A,n,$ is 1 if $$\\mathcal {A}$ ((z),$A_S$ ,n, = (z)$ and 0 otherwise.$ In the corresponding advantage measure shown in Definition REF , our goal is to measure the amount of information about the target $\\pi (z)$ that $A_S$ leaks specifically concerning the training data $S$ .",
"Definition REF accomplishes this by comparing the performance of the adversary when $b = 0$ in Experiment REF with that when $b = 1$ .",
"Definition 5 (Attribute advantage) The attribute advantage of $\\mathcal {A} $ is defined as: $\\mathsf {Adv}^\\mathsf {A}(\\mathcal {A},A,n, &= \\Pr [\\mathsf {Exp}^\\mathsf {A} (\\mathcal {A},A,n, = 1 \\mid b = 0] - \\Pr [\\mathsf {Exp}^\\mathsf {A} (\\mathcal {A},A,n, = 1 \\mid b = 1],$ where the probabilities are taken over the coin flips of $\\mathcal {A}$ , the random choice of $S$ , and the random data point $z \\sim S$ or $z \\sim .$ Notice that $ \\begin{aligned}\\mathsf {Adv}^\\mathsf {A}&= \\textstyle \\sum _{t_i\\in \\Pr _{z\\sim [t=t_i](\\Pr [\\mathcal {A} = t_i \\mid b=0, t=t_i] - \\Pr [\\mathcal {A} = t_i \\mid b=1, t=t_i]),}}where \\mathcal {A} and \\mathsf {Adv}^\\mathsf {A} are shortcuts for \\mathcal {A} (\\varphi (z),A_S,n, and \\mathsf {Adv}^\\mathsf {A}(\\mathcal {A},A,n,, respectively.\\end{aligned}This definition has the side effect of incentivizing the adversary to ``game the system\" by performing poorly when it thinks that b = 1.",
"To remove this incentive, one may consider using a simulator \\mathcal {S}, which does not receive the model as an input, when b = 1.",
"This definition is formalized below:$ Definition 6 (Alternative attribute advantage) Let $\\mathcal {S} (\\varphi (z),n, = \\operatornamewithlimits{arg\\,max}_{t_i} \\Pr _{z \\sim [\\pi (z) = t_i \\mid \\varphi (z)]be the Bayes optimal simulator.",
"The \\emph {attribute advantage} of \\mathcal {A} can alternatively be defined as{\\begin{@align*}{1}{-1}\\mathsf {Adv}^{\\mathsf {A}}_\\mathcal {S} (\\mathcal {A},A,n, &= \\Pr [\\mathcal {A} (\\varphi (z),A_S,n, = \\pi (z) \\mid b = 0] - \\Pr [\\mathcal {S} (\\varphi (z),n, = \\pi (z) \\mid b = 1].\\end{@align*}}}$ One potential issue with this alternative definition is that higher model accuracy will lead to higher attribute advantage regardless of how accurate the model is for the general population.",
"Broadly, there are two ways for a model to perform better on the training data: it can overfit to the training data, or it can learn a general trend in the distribution In this paper, we concern ourselves with the view that the adversary's ability to infer the target $\\pi (z)$ in the latter case is due not to the model but pre-existing patterns in To allow capturing the difference between overfitting and learning a general trend, we use Definition REF in the following analysis and leave a more complete exploration of Definition REF as future work.",
"While adversaries that “game the system\" may seem problematic, the effectiveness of such adversaries is indicative of privacy loss because their existence implies the ability to infer membership, as demonstrated by Reduction Adversary REF in Section REF .",
"Inversion, generalization, and influence The case where $\\varphi $ simply removes the sensitive attribute $t$ from the data point $z = (v,t,y)$ such that $\\varphi (z)=(v,y)$ is known in the literature as model inversion [4], [3], [5], [8].",
"In this section, we look at the model inversion attack of Fredrikson et al.",
"[4] under the advantage given in Definition REF .",
"We point out that this is a novel analysis, as this advantage is defined to reflect the extent to which an attribute inference attack reveals information about individuals in $S$ .",
"While prior work [4], [3] has empirically evaluated attribute accuracy over corresponding training and test sets, our goal is to analyze the factors that lead to increased privacy risk specifically for members of the training data.",
"To that end, we illustrate the relationship between advantage and generalization error as we did in the case of membership inference (Section REF ).",
"We also explore the role of feature influence, which in this case corresponds to the degree to which changes to a sensitive feature of $x$ affects the value $A_S (x)$ .",
"In Section REF , we show that the formal relationships described here often extend to attacks on real data where formal assumptions may fail to hold.",
"The attack described by Fredrikson et al.",
"[4] is intended for linear regression models and is thus subject to the Gaussian error assumption discussed in Section REF .",
"In general, when the adversary can approximate the error distribution reasonably well, e.g., by assuming a Gaussian distribution whose standard deviation equals the published standard error value, it can gain advantage by trying all possible values of the sensitive attribute.",
"We denote the adversary's approximation of the error distribution by $f_\\mathcal {A} $ , and we assume that the target $t=\\pi (z)$ is drawn from a finite set of possible values $t_1, \\ldots , t_m$ with known frequencies in We indicate the other features, which are known by the adversary, with the letter $v$ (i.e., $z=(x,y)$ , $x=(v,t)$ , and $\\varphi (z)=(v,y)$ ).",
"The attack is shown in Adversary REF .",
"For each $t_i$ , the adversary counterfactually assumes that $t = t_i$ and computes what the error of the model would be.",
"It then uses this information to update the a priori marginal distribution of $t$ and picks the value $t_i$ with the greatest likelihood.",
"Adversary 4 (General) Let $f_\\mathcal {A} (\\epsilon )$ be the adversary's guess for the probability density of the error $\\epsilon = y - A_S (x)$ .",
"On input $v$ , $y$ , $A_S$ , $n$ , and the adversary proceeds as follows: Query the model to get $A_S (v,t_i)$ for all $i \\in [m]$ .",
"Let $\\epsilon (t_i) = y - A_S (v,t_i)$ .",
"Return the result of $\\operatornamewithlimits{arg\\,max}_{t_i} (\\Pr _{z \\sim [t = t_i] \\cdot f_\\mathcal {A} (\\epsilon (t_i))).",
"}$ When analyzing Adversary REF , we are clearly interested in the effect that generalization error will have on advantage.",
"Given the results of Section REF , we can reasonably expect that large generalization error will lead to greater advantage.",
"However, as pointed out by Wu et al.",
"[5], the functional relationship between $t$ and $A_S (v,t)$ may play a role as well.",
"Working in the context of models as Boolean functions, Wu et al.",
"formalized the relevant property as functional influence [20], which is the probability that changing $t$ will cause $A_S (v,t)$ to change when $v$ is sampled uniformly.",
"The attack considered here applies to linear regression models, and Boolean influence is not suitable for use in this setting.",
"However, an analogous notion of influence that characterizes the magnitude of change to $A_S (v,t)$ is relevant to attribute inference.",
"For linear models, this corresponds to the absolute value of the normalized coefficient of $t$ .",
"Throughout the rest of the paper, we refer to this quantity as the influence of $t$ without risk of confusion with the Boolean influence used in other contexts.",
"Binary Variable with Uniform Prior The first part of our analysis deals with the simplest case where $m = 2$ with $\\Pr _{z \\sim [t = t_1] = \\Pr _{z \\sim [t = t_2].",
"Without loss of generality we assume that A_S (v, t_1) = A_S (v, t_2) + \\tau for some fixed \\tau \\ge 0, so in this setting \\tau is a straightforward proxy for influence.",
"Theorem~\\ref {thm:invbinary} relates the advantage of Adversary~\\ref {adv:invlinreg} to \\sigma _S , \\sigma _, and \\tau .",
"}\\begin{theorem} Let t be drawn uniformly from \\lbrace t_1, t_2\\rbrace and suppose that y = A_S (v, t) + \\epsilon , where \\epsilon \\sim N(0, \\sigma _S ^2) if b=0 and \\epsilon \\sim N(0, \\sigma _^2) if b=1.",
"Then the advantage of Adversary~\\ref {adv:invlinreg} is \\frac{1}{2}(\\operatorname{erf}(\\tau /2\\sqrt{2}\\sigma _S) - \\operatorname{erf}(\\tau /2\\sqrt{2}\\sigma _)).\\end{theorem}}\\begin{proof}Given the assumptions made in this setting, we can describe the behavior of \\mathcal {A} as returning the value t_i that minimizes |\\epsilon (t_i)|.",
"If t = t_1, it is easy to check that \\mathcal {A} guesses correctly if and only if \\epsilon (t_1) > -\\tau /2.",
"This means that \\mathcal {A}^{\\prime }s advantage given t = t_1 is\\begin{equation} \\begin{aligned}&\\Pr [\\mathcal {A} = t_1 \\mid t = t_1, b = 0] - \\Pr [\\mathcal {A} = t_1 \\mid t = t_1, b = 1]\\\\&= \\Pr [\\epsilon (t_1) > -\\tau /2 \\mid b = 0] - \\Pr [\\epsilon (t_1) > -\\tau /2 \\mid b = 1]\\\\&= \\left(\\frac{1}{2} + \\frac{1}{2}\\operatorname{erf}\\left(\\frac{\\tau }{2\\sqrt{2}\\sigma _S}\\right)\\right) - \\left(\\frac{1}{2} + \\frac{1}{2}\\operatorname{erf}\\left(\\frac{\\tau }{2\\sqrt{2}\\sigma _}\\right)\\right)\\\\&= \\frac{1}{2}\\left(\\operatorname{erf}\\left(\\frac{\\tau }{2\\sqrt{2}\\sigma _S}\\right) - \\operatorname{erf}\\left(\\frac{\\tau }{2\\sqrt{2}\\sigma _}\\right)\\right)\\end{aligned}\\end{equation}Similar reasoning shows that \\mathcal {A}^{\\prime }s advantage given t = t_2 is exactly the same, so the theorem follows from Equation~\\ref {eqn:invadv2}.\\end{proof}$ Clearly, the advantage will be zero when there is no generalization error ($\\sigma _S = \\sigma _$ ).",
"Consider the other extreme case where $\\sigma _S \\rightarrow 0$ and $\\sigma _\\rightarrow \\infty $ .",
"When $\\sigma _S $ is very small, the adversary will always guess correctly because the influence of $t$ overwhelms the effect of the error $\\epsilon $ .",
"On the other hand, when $\\sigma _$ is very large, changes to $t$ will be nearly imperceptible for “normal” values of $\\tau $ , and the adversary is reduced to random guessing.",
"Therefore, the maximum possible advantage with uniform prior is $1/2$ .",
"As a model overfits more, $\\sigma _S $ decreases and $\\sigma _$ tends to increase.",
"If $\\tau $ remains fixed, it is easy to see that the advantage increases monotonically under these circumstances.",
"Figure REF shows the effect of changing $\\tau $ as the ratio $\\sigma _/\\sigma _S $ remains fixed at several different constants.",
"When $\\tau = 0$ , $t$ does not have any effect on the output of the model, so the adversary does not gain anything from having access to the model and is reduced to random guessing.",
"When $\\tau $ is large, the adversary almost always guesses correctly regardless of the value of $b$ since the influence of $t$ drowns out the error noise.",
"Thus, at both extremes the advantage approaches 0, and the adversary is able to gain advantage only when $\\tau $ and $\\sigma _/\\sigma _S $ are in balance.",
"Figure: The advantage of Adversary as a function of tt's influence τ\\tau .",
"Here tt is a uniformly distributed binary variable.",
"General Case Sometimes the uniform prior for $t$ may not be realistic.",
"For example, $t$ may represent whether a patient has a rare disease.",
"In this case, we weight the values of $f_\\mathcal {A} (\\epsilon (t_i))$ by the a priori probability $\\Pr _{z \\sim [t = t_i] before comparing which t_i is the most likely.",
"With uniform prior, we could simplify \\operatornamewithlimits{arg\\,max}_{t_i} f_\\mathcal {A} (\\epsilon (t_i)) to \\operatornamewithlimits{arg\\,min}_{t_i} |\\epsilon (t_i)| regardless of the value of \\sigma used for f_\\mathcal {A} .",
"On the other hand, the value of \\sigma matters when we multiply by \\Pr [t = t_i].",
"Because the adversary is not given b, it makes an assumption similar to that described in Section~\\ref {sect:inclusion-attacks} and uses \\epsilon \\sim N(0, \\sigma _S ^2).",
"}Clearly $$\\sigma _S$ = $\\sigma _ $ results in zero advantage.",
"The maximum possible advantage is attained when $\\sigma _S \\rightarrow 0$ and $\\sigma _\\rightarrow \\infty $ .",
"Then, by similar reasoning as before, the adversary will always guess correctly when $b = 0$ and is reduced to random guessing when $b = 1$ , resulting in an advantage of $1 - \\frac{1}{m}$ .",
"In general, the advantage can be computed using Equation REF .",
"We first figure out when the adversary outputs $t_i$ .",
"When $f_\\mathcal {A} $ is a Gaussian, this is not computationally intensive as there is at most one decision boundary between any two values $t_i$ and $t_j$ .",
"Then, we convert the decision boundaries into probabilities by using the error distributions $\\epsilon \\sim N(0, \\sigma _S ^2)$ and $N(0, \\sigma _^2)$ , respectively.",
"Connection between membership and attribute inference In this section, we examine the underlying connections between membership and attribute inference attacks.",
"Our approach is based on reduction adversaries that have oracle access to one type of attack and attempt to perform the other type of attack.",
"We characterize the advantage of each reduction adversary in terms of the advantage of its oracle.",
"In Section REF , we implement the most sophisticated of the reduction adversaries described here and show that on real data it performs remarkably well, often outperforming Attribute Adversary REF by large margins.",
"We note that these reductions are specific to our choice of attribute advantage given in Definition REF .",
"Analyzing the connections between membership and attribute inference using the alternative Definition REF is an interesting direction for future work.",
"From membership to attribute We start with an adversary $\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}}$ that uses an attribute oracle to accomplish membership inference.",
"The attack, shown in Adversary REF , is straightforward: given a point $z$ , the adversary queries the attribute oracle to obtain a prediction $t$ of the target value $\\pi (z)$ .",
"If this prediction is correct, then the adversary concludes that $z$ was in the training data.",
"Adversary 5 (Membership $\\rightarrow $ attribute) The reduction adversary $\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}}$ has oracle access to attribute adversary $\\mathcal {A} _\\mathsf {A}$ .",
"On input $z$ , $A_S$ , $n$ , and the reduction adversary proceeds as follows: Query the oracle to get $t \\leftarrow \\mathcal {A} _\\mathsf {A}(\\varphi (z), A_S, n, $ .",
"Output 0 if $\\pi (z) = t$ .",
"Otherwise, output 1.",
"Theorem REF shows that the membership advantage of this reduction exactly corresponds to the attribute advantage of its oracle.",
"In other words, the ability to effectively infer attributes of individuals in the training set implies the ability to infer membership in the training set as well.",
"This suggests that attribute inference is at least as difficult as than membership inference.",
"Theorem 5 Let $\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}}$ be the adversary described in Adversary REF , which uses $\\mathcal {A} _\\mathsf {A}$ as an oracle.",
"Then, $\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}},A,n, = \\mathsf {Adv}^\\mathsf {A}(\\mathcal {A} _\\mathsf {A},A,n,.$ The proof follows directly from the definitions of membership and attribute advantages.",
"$\\mathsf {Adv}^\\mathsf {M}&= \\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 0] - \\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 1] \\\\&= \\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 1,t=t_i]) \\\\&= \\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _\\mathsf {A}= t_i \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {A}} = t_i \\mid b = 1,t=t_i]) \\\\&= \\mathsf {Adv}^\\mathsf {A}.",
"}}$ From attribute to membership We now consider reductions in the other direction, wherein the adversary is given $\\varphi (z)$ and must reconstruct the point $z$ to query the membership oracle.",
"To accomplish this, we assume that the adversary knows a deterministic reconstruction function $\\varphi ^{-1}$ such that $\\varphi \\circ \\varphi ^{-1}$ is the identity function, i.e., for any value of $\\varphi (z)$ that the adversary may receive, there exists $z^{\\prime } = \\varphi ^{-1}(\\varphi (z))$ such that $\\varphi (z) = \\varphi (z^{\\prime })$ .",
"However, because $\\varphi $ is a lossy function, in general it does not hold that $\\varphi ^{-1}(\\varphi (z)) = z$ .",
"Our adversary, described in Adversary REF , reconstructs the point $z^{\\prime }$ , sets the attribute $t$ of that point to value $t_i$ chosen uniformly at random, and outputs $t_i$ if the membership oracle says that the resulting point is in the dataset.",
"Adversary 6 (Uniform attribute $\\rightarrow $ membership) Suppose that $t_1, \\ldots , t_m$ are the possible values of the target $t=\\pi (z)$ .",
"The reduction adversary $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U}$ has oracle access to membership adversary $\\mathcal {A} _\\mathsf {M}$ .",
"On input $\\varphi (z)$ , $A_S$ , $n$ , and $, the reduction adversary proceeds as follows:\\begin{enumerate}\\item Choose t_i uniformly at random from \\lbrace t_1, \\ldots , t_m\\rbrace .\\item Let z^{\\prime } = \\varphi ^{-1}(\\varphi (z)), and change the value of the sensitive attribute t such that \\pi (z^{\\prime }) = t_i.\\item Query \\mathcal {A} _\\mathsf {M} to obtain b^{\\prime } \\leftarrow \\mathcal {A} _\\mathsf {M}(z^{\\prime },A_S,n,.\\item If b^{\\prime } = 0, output t_i.",
"Otherwise, output \\bot .\\end{enumerate}$ The uniform choice of $t_i$ is motivated by the fact that the adversary may not know how the advantage of the membership oracle is distributed across different values of $t$ .",
"For example, it is possible that $\\mathcal {A} _\\mathsf {M}$ performs very poorly when $t = t_1$ and that all of its advantage comes from the case where $t = t_2$ .",
"In the computation of the advantage, we only consider the case where $\\pi (z) = t_i$ because this is the only case where the reduction adversary can possibly give the correct answer.",
"In that case, the membership oracle is given a challenge point from the distribution $ = \\lbrace (x,y) \\mid (x,y)=\\varphi ^{-1}(\\varphi (z)) \\text{ except that } t = \\pi (z)\\rbrace $ , where $z \\sim S$ if $b = 0$ and $z \\sim if $ b = 1$.",
"On the other hand, the training set $ S$ used to train the model A_S was drawn from Because of this difference, we use modified membership advantage $ AdvM*($\\mathcal {A}$ , A, n, , -1, )$, which measures the performance of the membership adversary when the challenge point is drawn from $$.",
"In the case of a model inversion attack as described in the beginning of Section~\\ref {sect:inversion-attacks}, we have $ AdvM($\\mathcal {A}$ , A, n, = AdvM*($\\mathcal {A}$ , A, n, , -1, )$, i.e., the modified membership advantage equals the unmodified one.$ Theorem REF shows that the attribute advantage of $\\mathcal {A} ^{\\sf U}_{\\mathsf {A}\\rightarrow \\mathsf {M}}$ is proportional to the modified membership advantage of $\\mathcal {A} _\\mathsf {M}$ , giving a lower bound on the effectiveness of attribute inference attacks that use membership oracles.",
"Notably, the adversary does not make use of any associations that may exist between $\\varphi (z)$ and $t$ , so this reduction is general and works even when no such association exists.",
"While the reduction does not completely transfer the membership advantage to attribute advantage, the resulting attribute advantage is within a constant factor of the modified membership advantage.",
"Theorem 6 Let $\\mathcal {A} ^{\\sf U}_{\\mathsf {A}\\rightarrow \\mathsf {M}}$ be the adversary described in Adversary REF , which uses $\\mathcal {A} _\\mathsf {M}$ as an oracle.",
"Then, $\\mathsf {Adv}^\\mathsf {A}(\\mathcal {A} ^{\\sf U}_{\\mathsf {A}\\rightarrow \\mathsf {M}},A,n, = \\frac{1}{m} \\mathsf {Adv}^\\mathsf {M}_*(\\mathcal {A} _\\mathsf {M},A,n,\\varphi ,\\varphi ^{-1},\\pi ).$ We first give an informal argument.",
"In order for $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U}$ to correctly guess the value of $t$ , it needs to choose the correct $t_i$ , which happens with probability $\\frac{1}{m}$ , and then $\\mathcal {A} _\\mathsf {M}(z^{\\prime }, A_S, n, $ must be 0.",
"Therefore, $\\mathsf {Adv}^\\mathsf {A}= \\frac{1}{m} \\mathsf {Adv}^\\mathsf {M}_*$ .",
"Now we give the formal proof.",
"Let $t^{\\prime }$ be the value of $t$ that was chosen independently and uniformly at random in Step 1 of Adversary REF .",
"Since $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U}$ outputs $t_i$ if and only if $t^{\\prime }=t_i$ and $\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0$ , we have $&\\Pr [\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U} = t_i \\mid b = 0,t=t_i] = \\frac{1}{m} \\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 0,t=t_i],$ and likewise when $b=1$ .",
"Therefore, the advantage of the reduction adversary is $\\mathsf {Adv}^\\mathsf {A}&= \\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U} = t_i \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U} = t_i \\mid b = 1,t=t_i]) \\\\&= \\frac{1}{m}\\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 1,t=t_i]) \\\\&= \\frac{1}{m}(\\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 0] - \\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 1]) \\\\&= \\frac{1}{m} \\mathsf {Adv}^\\mathsf {M}_*,}where the second-to-last step holds due to the fact that b and t are independent.",
"}$ Adversary REF has the obvious weakness that it can only return correct answers when it guesses the value of $t$ correctly.",
"Adversary REF attempts to improve on this by making multiple queries to $\\mathcal {A} _\\mathsf {M}$ .",
"Rather than guess the value of $t$ , this adversary tries all values of $t$ in order of their marginal probabilities until the membership adversary says “yes\".",
"Adversary 7 (Multi-query attribute $\\rightarrow $ membership) Suppose that $t_1, \\ldots , t_m$ are the possible values of the sensitive attribute $t$ .",
"The reduction adversary $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf M}$ has oracle access to membership adversary $\\mathcal {A} _\\mathsf {M}$ .",
"On input $\\varphi (z)$ , $A_S$ , $n$ , and $, $$\\mathcal {A}$ AM$ proceeds as follows:\\begin{enumerate}\\item Let z^{\\prime } = \\varphi ^{-1}(\\varphi (z)).\\item For all i \\in [m], let z_i^{\\prime } be z^{\\prime } with the value of the sensitive attribute t changed to t_i.\\item Query \\mathcal {A} _\\mathsf {M} to compute T = \\lbrace t_i \\mid \\mathcal {A} _\\mathsf {M}(z_i^{\\prime },A_S,n, = 0\\rbrace .\\item Output \\operatornamewithlimits{arg\\,max}_{t_i \\in T} \\Pr _{z \\sim [t = t_i].",
"If T = \\emptyset , output \\bot .",
"}\\end{enumerate}$ We evaluate this adversary experimentally in Section REF .",
"Evaluation In this section, we evaluate the performance of the adversaries discussed in Sections , REF , and .",
"We compare the performance of these adversaries on real datasets with the analysis from previous sections and show that overfitting predicts privacy risk in practice as our analysis suggests.",
"Our experiments use linear regression, tree, and deep convolutional neural network (CNN) models.",
"Methodology Linear and tree models We used the Python scikit-learn [32] library to calculate the empirical error $R_{emp}$ and the leave-one-out cross validation error $R_{cv}$ [33].",
"Because these two measures pertain to the error of the model on points inside and outside the training set, respectively, they were used to approximate $\\sigma _S $ and $\\sigma _$ , respectively.",
"Then, we made a random 75-25% split of the data into training and test sets.",
"The training set was used to train either a Ridge regression or a decision tree model, and then the adversaries were given access to this model.",
"We repeated this 100 times with different training-test splits and then averaged the result.",
"Before we explain the results, we describe the datasets.",
"[leftmargin=0em,labelindent=] Eyedata.",
"This is gene expression data from rat eye tissues [34], as presented in the “flare” package of the R programming language.",
"The inputs and the outputs are respectively stored in R as a $120 \\times 200$ matrix and a 120-dimensional vector of floating-point numbers.",
"We used scikit-learn [32] to scale each attribute to zero mean and unit variance.",
"IWPC.",
"This is data collected by the International Warfarin Pharmacogenetics Consortium [35] about patients who were prescribed warfarin.",
"After we removed rows with missing values, 4819 patients remained in the dataset.",
"The inputs to the model are demographic (age, height, weight, race), medical (use of amiodarone, use of enzyme inducer), and genetic (VKORC1, CYP2C9) attributes.",
"Age, height, and weight are real-valued and were scaled to zero mean and unit variance.",
"The medical attributes take binary values, and the remaining attributes were one-hot encoded.",
"The output is the weekly dose of warfarin in milligrams.",
"However, because the distribution of warfarin dose is skewed, IWPC concludes in [35] that solving for the square root of the dose results in a more predictive linear model.",
"We followed this recommendation and scaled the square root of the dose to zero mean and unit variance.",
"Netflix.",
"We use the dataset from the Netflix Prize contest [36].",
"This is a sparse dataset that indicates when and how a user rated a movie.",
"For the output attribute, we used the rating of Dragon Ball Z: Trunks Saga, which had one of the most polarized rating distributions.",
"There are 2416 users who rated this, and the ratings were scaled to zero mean and unit variance.",
"The input attributes are binary variables indicating whether or not a user rated each of the other 17,769 movies in the dataset.",
"Deep convolutional neural networks We evaluated the membership inference attack on deep CNNs.",
"In addition, we implemented the colluding training algorithm (Algorithm REF ) to verify its performance in practice.",
"The CNNs were trained in Python using the Keras deep-learning library [37] and a standard stochastic gradient descent algorithm [38].",
"We used three datasets that are standard benchmarks in the deep learning literature and were evaluated in prior work on inference attacks [7]; they are described in more detail below.",
"For all datasets, pixel values were normalized to the range $[0,1]$ , and the label values were encoded as one-hot vectors.",
"To expedite the training process across a range of experimental configurations, we used a subset of each dataset.",
"For each dataset, we randomly divided the available data into equal-sized training and test sets to facilitate comparison with prior work [7] that used this convention.",
"The architecture we use is based on the VGG network [39], which is commonly used in computer vision applications.",
"We control for generalization error by varying a size parameter $s$ that defines the number of units at each layer of the network.",
"The architecture consists of two 3x3 convolutional layers with $s$ filters each, followed by a 2x2 max pooling layer, two 3x3 convolutional layers with $2s$ filters each, a 2x2 max pooling layer, a fully-connected layer with $2s$ units, and a softmax output layer.",
"All activation functions are rectified linear.",
"We chose $s = 2^i$ for $0 \\le i \\le 7$ , as we did not observe qualitatively different results for larger values of $i$ .",
"All training was done using the Adam optimizer [40] with the default parameters in the Keras implementation ($\\lambda = 0.001$ , $\\beta _1 = 0.5$ , $\\beta _2 = 0.99$ , $\\epsilon = 10^{-8}$ , and decay set to $5\\times 10^{-4}$ ).",
"We used categorical cross-entropy loss, which is conventional for models whose topmost activation is softmax [38].",
"[leftmargin=0em,labelindent=] MNIST.",
"MNIST [41] consists of 70,000 images of handwritten digits formatted as grayscale $28 \\times 28$ -pixel images, with class labels indicating the digit depicted in each image.",
"We selected 17,500 points from the full dataset at random for our experiments.",
"CIFAR-10, CIFAR-100.",
"The CIFAR datasets [42] consist of 60,000 $32 \\times 32$ -pixel color images, labeled as 10 (CIFAR-10) and 100 (CIFAR-100) classes.",
"We selected 15,000 points at random from the full data.",
"Membership inference Figure: Empirical membership advantage of the threshold adversary (Adversary ) given as a function of generalization ratio for regression, tree, and CNN models.The results of the membership inference attacks on linear and tree models are plotted in Figures REF and REF .",
"The theoretical and experimental results appear to agree when the adversary knows both $\\sigma _S $ and $\\sigma _$ and sets the decision boundary accordingly.",
"However, when the adversary does not know $\\sigma _$ , it performs much better than what the theory predicts.",
"In fact, an adversary can sometimes do better by just fixing the decision boundary at $|\\epsilon | = \\sigma _S $ instead of taking $\\sigma _$ into account.",
"This is because the training set error distributions are not exactly Gaussian.",
"Figures REF and REF in the appendix show that, although the training set error distributions roughly match the shape of a Gaussian curve, they have a much higher peak at zero.",
"As a result, it is often advantageous to bring the decision boundaries closer to zero.",
"The results of the threshold adversary on CNNs are given in Figure REF .",
"Although these models perform classification, the loss function used for training is categorical cross-entropy, which is non-negative, continuous, and unbounded.",
"This suggests that the threshold adversary could potentially work in this setting as well.",
"Specifically, the predictions made by these models can be compared against $L_S$ , the average training loss observed during training, which is often reported with published architectures as a point of comparison against prior work (see, for example, [43] and [44]).",
"Figure REF shows that, while the empirical results do not match the theoretical curve as closely as do linear and tree models, they do not diverge as much as one might expect given that the error is not Gaussian as assumed by Theorem REF .",
"Table: Comparison of our membership inference attack with that presented by Shokri et al.",
"While our attack has slightly lower precision, it requires far less computational resources and background knowledge.Now we compare our attack with that by Shokri et al.",
"[7], which generates “shadow models\" that are intended to mimic the behavior of $A_S$ .",
"Because their attack involves using machine learning to train the attacker with the shadow models, their attack requires considerable computational power and knowledge of the algorithm used to train the model.",
"By contrast, our attacker simply makes one query to the model and needs to know only the average training loss.",
"Despite these differences, when the size parameter $s$ is set equal to that used by Shokri et al., our attacker has the same recall and only slightly lower precision than their attacker.",
"A more detailed comparison is given in Table REF .",
"Attribute inference and reduction Figure: Experimentally determined advantage for various membership and attribute adversaries.",
"The plots correspond to: (a) threshold membership adversary (Adversary ), (b) uniform reduction adversary (Adversary ), (c) general attribute adversary (Adversary ), and (d) multi-query reduction adversary (Adversary ).",
"Both reduction adversaries use the threshold membership adversary as the oracle, and f 𝒜 (ϵ)f_\\mathcal {A} (\\epsilon ) for the attribute adversary is the Gaussian with mean zero and standard deviation σ S \\sigma _S.We now present the empirical attribute advantage of the general adversary (Adversary REF ).",
"Because this adversary uses the model inversion assumptions described at the beginning of Section REF , our evaluation is also in the setting of model inversion.",
"For these experiments we used the IWPC and Netflix datasets described in Section REF .",
"For $f_\\mathcal {A} (\\epsilon )$ , the adversary's approximation of the error distribution, we used the Gaussian with mean zero and standard deviation $R_{emp}$ .",
"For the IWPC dataset, each of the genomic attributes (VKORC1 and CYP2C9) is separately used as the target $t$ .",
"In the Netflix dataset, the target attribute was whether a user rated a certain movie, and we randomly sampled targets from the set of available movies.",
"The circles in Figure REF show the result of inverting the VKORC1 and CYP2C9 attributes in the IWPC dataset.",
"Although the attribute advantage is not as high as the membership advantage (solid line), the attribute adversary exhibits a sizable advantage that increases as the model overfits more and more.",
"On the other hand, none of the attacks could effectively infer whether a user watched a certain movie in the Netflix dataset.",
"In addition, we were unable to simultaneously control for both $\\sigma _/\\sigma _S $ and $\\tau $ in the Netflix dataset to measure the effect of influence as predicted by Theorem .",
"Finally, we evaluate the performance of the multi-query reduction adversary (Adversary REF ).",
"As the squares in Figure REF show, with the IWPC data, making multiple queries to the membership oracle significantly increased the success rate compared to what we would expect from the naive uniform reduction adversary (Adversary REF , dotted line).",
"Surprisingly, the reduction is also more effective than running the attribute inference attack directly.",
"By contrast, with the Netflix data, the multi-query reduction adversary was often slightly worse than the naive uniform adversary although it still outperformed direct attribute inference.",
"Collusion in membership inference We evaluate $A^{\\mathsf {C}}$ and $\\mathcal {A} ^{\\mathsf {C}}$ described in Section REF for CNNs trained as image classifiers.",
"To instantiate $F_K$ and $G_K$ , we use Python's intrinsic pseudorandom number generator with key $K$ as the seed.",
"We note that our proof of Theorem REF relies only on the uniformity of the pseudorandom numbers and not on their unpredictability.",
"Deviations from this assumption will result in a less effective membership inference attack but do not invalidate our results.",
"All experiments set the number of keys to $k=3$ .",
"Figure: Results of colluding training algorithm and membership adversary on CNNs trained on MNIST, CIFAR-10, and CIFAR-100.",
"The size parameter was configured to take values s=2 i s=2^i for i∈[0,7]i \\in [0,7].",
"Regardless of the models' generalization performance, when the network is sufficiently large, the attack achieves high advantage (≥0.98\\ge 0.98) without affecting predictive accuracy.The results of our experiment are shown in Figures REF and REF .",
"The data shows that on all three instances, the colluding parties achieve a high membership advantage without significantly affecting model performance.",
"The accuracy of the subverted model was only 0.014 (MNIST), 0.047 (CIFAR-10), and 0.031 (CIFAR-100) less than that of the unsubverted model.",
"The advantage rapidly increases with the model size around $s \\approx 16$ but is relatively constant elsewhere, indicating that model capacity beyond a certain point is a necessary factor in the attack.",
"Importantly, the results demonstrate that specific information about nearly all of the training data can be intentionally leaked through the behavior of a model that appears to generalize very well.",
"In fact, looking at Figure REF shows that in these instances, there is no discernible relationship between generalization error and membership advantage.",
"The three datasets exhibit vastly different generalization behavior, with the MNIST models achieving almost no generalization error ($< 0.02$ for $s \\ge 32$ ) and CIFAR-100 showing a large performance gap ($\\ge 0.8$ for $s \\ge 32$ ).",
"Despite this fact, the membership adversary achieves nearly identical performance.",
"Related Work Privacy and statistical summaries There is extensive prior literature on privacy attacks on statistical summaries.",
"Komarova et al.",
"[45] looked into partial disclosure scenarios, where an adversary is given fixed statistical estimates from combined public and private sources and attempts to infer the sensitive feature of an individual referenced in those sources.",
"A number of previous studies [46], [21], [22], [47], [48], [49] have looked into membership attacks from statistics commonly published in genome-wide association studies (GWAS).",
"Calandrino et al.",
"[50] showed that temporal changes in recommendations given by collaborative filtering methods can reveal the inputs that caused those changes.",
"Linear reconstruction attacks [51], [52], [53] attempt to infer partial inputs to linear statistics and were later extended to non-linear statistics [54].",
"While the goal of these attacks has commonalities with both membership inference and attribute inference, our results apply specifically to machine learning settings where generalization error and influence make our results relevant.",
"Privacy and machine learning More recently, others have begun examining these attacks in the context of machine learning.",
"Ateniese et al.",
"[1] showed that the knowledge of the internal structure of Support Vector Machines and Hidden Markov Models leaks certain types of information about their training data, such as the language used in a speech dataset.",
"Dwork et al.",
"[13] showed that a differentially private algorithm with a suitably chosen parameter generalizes well with high probability.",
"Subsequent work showed that similar results are true under related notions of privacy.",
"In particular, Bassily et al.",
"[18] studied a notion of privacy called total variation stability and proved good generalization with respect to a bounded number of adaptively chosen low-sensitivity queries.",
"Moreover, for data drawn from Gibbs distributions, Wang et al.",
"[19] showed that on-average KL privacy is equivalent to generalization error as defined in this paper.",
"While these results give evidence for the relationship between privacy and overfitting, we construct an attacker that directly leverages overfitting to gain advantage commensurate with the extent of the overfitting.",
"Membership inference Shokri et al.",
"[7] developed a membership inference attack and applied it to popular machine-learning-as-a-service APIs.",
"Their attacks are based on “shadow models” that approximate the behavior of the model under attack.",
"The shadow models are used to build another machine learning model called the “attack model”, which is trained to distinguish points in the training data from other points based on the output they induce on the original model under attack.",
"As we discussed in Section REF , our simple threshold adversary comes surprisingly close to the accuracy of their attack, especially given the differences in complexity and requisite adversarial assumptions between the attacks.",
"Because the attack proposed by Shokri et al.",
"itself relies on machine learning to find a function that separates training and non-training points, it is not immediately clear why the attack works, but the authors hypothesize that it is related to overfitting and the “diversity” of the training data.",
"They graph the generalization error against the precision of their attack and find some evidence of a relationship, but they also find that the relationship is not perfect and conclude that model structure must also be relevant.",
"The results presented in this paper make the connection to overfitting precise in many settings, and the colluding training algorithm we give in Section REF demonstrates exactly how model structure can be exploited to create a membership inference vulnerability.",
"Li et al.",
"[6] explored membership inference, distinguishing between “positive” and “negative” membership privacy.",
"They show how this framework defines a family of related privacy definitions that are parametrized on distributions of the adversary's prior knowledge, and they find that a number of previous definitions can be instantiated in this way.",
"Attribute inference Practical model inversion attacks have been studied in the context of linear regression [4], [8], decision trees [3], and neural networks [3].",
"Our results apply to these attacks when they are applied to data that matches the distributional assumptions made in our analysis.",
"An important distinction between the way inversion attacks were considered in prior work and how we treat them here is the notion of advantage.",
"Prior work on these attacks defined advantage as the difference between the attacker's predictive accuracy given the model and the best accuracy that could be achieved without the model.",
"Although some prior work [4], [3] empirically measured this advantage on both training and test datasets, this definition does not allow a formal characterization of how exposed the training data specifically is to privacy risk.",
"In Section REF , we define attribute advantage precisely to capture the risk to the training data by measuring the difference in the attacker's accuracy on training and test data: the advantage is zero when the attack is as powerful on the general population as on the training data and is maximized when the attack works only on the training data.",
"Wu et al.",
"[5] formalized model inversion for a simplified class of models that consist of Boolean functions and explored the initial connections between influence and advantage.",
"However, as in other prior work on model inversion, the type of advantage that they consider says nothing about what the model specifically leaks about its training data.",
"Drawing on their observation that influence is relevant to privacy risk in general, we illustrate its effect on the notion of advantage defined in this paper and show how it interacts with generalization error.",
"Conclusion and Future Directions We introduced new formal definitions of advantage for membership and attribute inference attacks.",
"Using these definitions, we analyzed attacks under various assumptions on learning algorithms and model properties, and we showed that these two attacks are closely related through reductions in both directions.",
"Both theoretical and experimental results confirm that models become more vulnerable to both types of attacks as they overfit more.",
"Interestingly, our analysis also shows that overfitting is not the only factor that can lead to privacy risk: Theorem REF shows that even stable learning algorithms, which provably do not overfit, can leak precise membership information, and the results in Section REF demonstrate that the influence of the target attribute on a model's output plays a key role in attribute inference.",
"Our formalization and analysis open interesting directions for future work.",
"The membership attack in Theorem REF is based on a colluding pair of adversary and learning rule, $A^{\\mathsf {C}}$ and $\\mathcal {A} ^{\\mathsf {C}}$ .",
"This could be implemented, for example, by a malicious ML algorithm provided by a third-party library or cloud service to subvert users' privacy.",
"Further study of this scenario, which may best be formalized in the framework of algorithm substitution attacks [23], is warranted to determine whether malicious algorithms can produce models that are indistinguishable from normal ones and how such attacks can be mitigated.",
"Our results in Section REF give bounds on membership advantage when certain conditions are met.",
"These bounds apply to adversaries who may target specific individuals, bringing arbitrary background knowledge of their targets to help determine their membership status.",
"Some types of realistic adversaries may be motivated by concerns that incentivize learning a limited set of facts about as many individuals in the training data as possible rather than obtaining unique background knowledge about specific individuals.",
"Characterizing these “stable adversaries” is an interesting direction that may lead to tighter bounds on advantage or relaxed conditions on the learning rule.",
"Figure: The training and test error distributions for an overfitted decision tree.",
"The histograms are juxtaposed with what we would expect if the errors were normally distributed with standard deviation R emp =0.3899R_{emp} = 0.3899 and R cv =0.9507R_{cv} = 0.9507, respectively.",
"The bar at error =0\\mathrm {error} = 0 does not fit inside the first graph; in order to fit it, the graph would have to be almost 10 times as high.",
"To minimize the effect of noise, the errors were measured using 1000 different random 75-25 splits of the data into training and test sets and then aggregated."
],
[
"Connection between membership and attribute inference",
"In this section, we examine the underlying connections between membership and attribute inference attacks.",
"Our approach is based on reduction adversaries that have oracle access to one type of attack and attempt to perform the other type of attack.",
"We characterize the advantage of each reduction adversary in terms of the advantage of its oracle.",
"In Section REF , we implement the most sophisticated of the reduction adversaries described here and show that on real data it performs remarkably well, often outperforming Attribute Adversary REF by large margins.",
"We note that these reductions are specific to our choice of attribute advantage given in Definition REF .",
"Analyzing the connections between membership and attribute inference using the alternative Definition REF is an interesting direction for future work."
],
[
"From membership to attribute",
"We start with an adversary $\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}}$ that uses an attribute oracle to accomplish membership inference.",
"The attack, shown in Adversary REF , is straightforward: given a point $z$ , the adversary queries the attribute oracle to obtain a prediction $t$ of the target value $\\pi (z)$ .",
"If this prediction is correct, then the adversary concludes that $z$ was in the training data.",
"Adversary 5 (Membership $\\rightarrow $ attribute) The reduction adversary $\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}}$ has oracle access to attribute adversary $\\mathcal {A} _\\mathsf {A}$ .",
"On input $z$ , $A_S$ , $n$ , and the reduction adversary proceeds as follows: Query the oracle to get $t \\leftarrow \\mathcal {A} _\\mathsf {A}(\\varphi (z), A_S, n, $ .",
"Output 0 if $\\pi (z) = t$ .",
"Otherwise, output 1.",
"Theorem REF shows that the membership advantage of this reduction exactly corresponds to the attribute advantage of its oracle.",
"In other words, the ability to effectively infer attributes of individuals in the training set implies the ability to infer membership in the training set as well.",
"This suggests that attribute inference is at least as difficult as than membership inference.",
"Theorem 5 Let $\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}}$ be the adversary described in Adversary REF , which uses $\\mathcal {A} _\\mathsf {A}$ as an oracle.",
"Then, $\\mathsf {Adv}^\\mathsf {M}(\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}},A,n, = \\mathsf {Adv}^\\mathsf {A}(\\mathcal {A} _\\mathsf {A},A,n,.$ The proof follows directly from the definitions of membership and attribute advantages.",
"$\\mathsf {Adv}^\\mathsf {M}&= \\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 0] - \\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 1] \\\\&= \\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {M}\\rightarrow \\mathsf {A}} = 0 \\mid b = 1,t=t_i]) \\\\&= \\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _\\mathsf {A}= t_i \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {A}} = t_i \\mid b = 1,t=t_i]) \\\\&= \\mathsf {Adv}^\\mathsf {A}.",
"}}$"
],
[
"From attribute to membership",
"We now consider reductions in the other direction, wherein the adversary is given $\\varphi (z)$ and must reconstruct the point $z$ to query the membership oracle.",
"To accomplish this, we assume that the adversary knows a deterministic reconstruction function $\\varphi ^{-1}$ such that $\\varphi \\circ \\varphi ^{-1}$ is the identity function, i.e., for any value of $\\varphi (z)$ that the adversary may receive, there exists $z^{\\prime } = \\varphi ^{-1}(\\varphi (z))$ such that $\\varphi (z) = \\varphi (z^{\\prime })$ .",
"However, because $\\varphi $ is a lossy function, in general it does not hold that $\\varphi ^{-1}(\\varphi (z)) = z$ .",
"Our adversary, described in Adversary REF , reconstructs the point $z^{\\prime }$ , sets the attribute $t$ of that point to value $t_i$ chosen uniformly at random, and outputs $t_i$ if the membership oracle says that the resulting point is in the dataset.",
"Adversary 6 (Uniform attribute $\\rightarrow $ membership) Suppose that $t_1, \\ldots , t_m$ are the possible values of the target $t=\\pi (z)$ .",
"The reduction adversary $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U}$ has oracle access to membership adversary $\\mathcal {A} _\\mathsf {M}$ .",
"On input $\\varphi (z)$ , $A_S$ , $n$ , and $, the reduction adversary proceeds as follows:\\begin{enumerate}\\item Choose t_i uniformly at random from \\lbrace t_1, \\ldots , t_m\\rbrace .\\item Let z^{\\prime } = \\varphi ^{-1}(\\varphi (z)), and change the value of the sensitive attribute t such that \\pi (z^{\\prime }) = t_i.\\item Query \\mathcal {A} _\\mathsf {M} to obtain b^{\\prime } \\leftarrow \\mathcal {A} _\\mathsf {M}(z^{\\prime },A_S,n,.\\item If b^{\\prime } = 0, output t_i.",
"Otherwise, output \\bot .\\end{enumerate}$ The uniform choice of $t_i$ is motivated by the fact that the adversary may not know how the advantage of the membership oracle is distributed across different values of $t$ .",
"For example, it is possible that $\\mathcal {A} _\\mathsf {M}$ performs very poorly when $t = t_1$ and that all of its advantage comes from the case where $t = t_2$ .",
"In the computation of the advantage, we only consider the case where $\\pi (z) = t_i$ because this is the only case where the reduction adversary can possibly give the correct answer.",
"In that case, the membership oracle is given a challenge point from the distribution $ = \\lbrace (x,y) \\mid (x,y)=\\varphi ^{-1}(\\varphi (z)) \\text{ except that } t = \\pi (z)\\rbrace $ , where $z \\sim S$ if $b = 0$ and $z \\sim if $ b = 1$.",
"On the other hand, the training set $ S$ used to train the model A_S was drawn from Because of this difference, we use modified membership advantage $ AdvM*($\\mathcal {A}$ , A, n, , -1, )$, which measures the performance of the membership adversary when the challenge point is drawn from $$.",
"In the case of a model inversion attack as described in the beginning of Section~\\ref {sect:inversion-attacks}, we have $ AdvM($\\mathcal {A}$ , A, n, = AdvM*($\\mathcal {A}$ , A, n, , -1, )$, i.e., the modified membership advantage equals the unmodified one.$ Theorem REF shows that the attribute advantage of $\\mathcal {A} ^{\\sf U}_{\\mathsf {A}\\rightarrow \\mathsf {M}}$ is proportional to the modified membership advantage of $\\mathcal {A} _\\mathsf {M}$ , giving a lower bound on the effectiveness of attribute inference attacks that use membership oracles.",
"Notably, the adversary does not make use of any associations that may exist between $\\varphi (z)$ and $t$ , so this reduction is general and works even when no such association exists.",
"While the reduction does not completely transfer the membership advantage to attribute advantage, the resulting attribute advantage is within a constant factor of the modified membership advantage.",
"Theorem 6 Let $\\mathcal {A} ^{\\sf U}_{\\mathsf {A}\\rightarrow \\mathsf {M}}$ be the adversary described in Adversary REF , which uses $\\mathcal {A} _\\mathsf {M}$ as an oracle.",
"Then, $\\mathsf {Adv}^\\mathsf {A}(\\mathcal {A} ^{\\sf U}_{\\mathsf {A}\\rightarrow \\mathsf {M}},A,n, = \\frac{1}{m} \\mathsf {Adv}^\\mathsf {M}_*(\\mathcal {A} _\\mathsf {M},A,n,\\varphi ,\\varphi ^{-1},\\pi ).$ We first give an informal argument.",
"In order for $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U}$ to correctly guess the value of $t$ , it needs to choose the correct $t_i$ , which happens with probability $\\frac{1}{m}$ , and then $\\mathcal {A} _\\mathsf {M}(z^{\\prime }, A_S, n, $ must be 0.",
"Therefore, $\\mathsf {Adv}^\\mathsf {A}= \\frac{1}{m} \\mathsf {Adv}^\\mathsf {M}_*$ .",
"Now we give the formal proof.",
"Let $t^{\\prime }$ be the value of $t$ that was chosen independently and uniformly at random in Step 1 of Adversary REF .",
"Since $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U}$ outputs $t_i$ if and only if $t^{\\prime }=t_i$ and $\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0$ , we have $&\\Pr [\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U} = t_i \\mid b = 0,t=t_i] = \\frac{1}{m} \\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 0,t=t_i],$ and likewise when $b=1$ .",
"Therefore, the advantage of the reduction adversary is $\\mathsf {Adv}^\\mathsf {A}&= \\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U} = t_i \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf U} = t_i \\mid b = 1,t=t_i]) \\\\&= \\frac{1}{m}\\sum _{t_i\\in \\Pr [t=t_i](\\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 0,t=t_i] - \\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 1,t=t_i]) \\\\&= \\frac{1}{m}(\\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 0] - \\Pr [\\mathcal {A} _{\\mathsf {M}}(z^{\\prime }) = 0 \\mid b = 1]) \\\\&= \\frac{1}{m} \\mathsf {Adv}^\\mathsf {M}_*,}where the second-to-last step holds due to the fact that b and t are independent.",
"}$ Adversary REF has the obvious weakness that it can only return correct answers when it guesses the value of $t$ correctly.",
"Adversary REF attempts to improve on this by making multiple queries to $\\mathcal {A} _\\mathsf {M}$ .",
"Rather than guess the value of $t$ , this adversary tries all values of $t$ in order of their marginal probabilities until the membership adversary says “yes\".",
"Adversary 7 (Multi-query attribute $\\rightarrow $ membership) Suppose that $t_1, \\ldots , t_m$ are the possible values of the sensitive attribute $t$ .",
"The reduction adversary $\\mathcal {A} _{\\mathsf {A}\\rightarrow \\mathsf {M}}^{\\sf M}$ has oracle access to membership adversary $\\mathcal {A} _\\mathsf {M}$ .",
"On input $\\varphi (z)$ , $A_S$ , $n$ , and $, $$\\mathcal {A}$ AM$ proceeds as follows:\\begin{enumerate}\\item Let z^{\\prime } = \\varphi ^{-1}(\\varphi (z)).\\item For all i \\in [m], let z_i^{\\prime } be z^{\\prime } with the value of the sensitive attribute t changed to t_i.\\item Query \\mathcal {A} _\\mathsf {M} to compute T = \\lbrace t_i \\mid \\mathcal {A} _\\mathsf {M}(z_i^{\\prime },A_S,n, = 0\\rbrace .\\item Output \\operatornamewithlimits{arg\\,max}_{t_i \\in T} \\Pr _{z \\sim [t = t_i].",
"If T = \\emptyset , output \\bot .",
"}\\end{enumerate}$ We evaluate this adversary experimentally in Section REF ."
],
[
"Evaluation",
"In this section, we evaluate the performance of the adversaries discussed in Sections , REF , and .",
"We compare the performance of these adversaries on real datasets with the analysis from previous sections and show that overfitting predicts privacy risk in practice as our analysis suggests.",
"Our experiments use linear regression, tree, and deep convolutional neural network (CNN) models."
],
[
"Linear and tree models",
"We used the Python scikit-learn [32] library to calculate the empirical error $R_{emp}$ and the leave-one-out cross validation error $R_{cv}$ [33].",
"Because these two measures pertain to the error of the model on points inside and outside the training set, respectively, they were used to approximate $\\sigma _S $ and $\\sigma _$ , respectively.",
"Then, we made a random 75-25% split of the data into training and test sets.",
"The training set was used to train either a Ridge regression or a decision tree model, and then the adversaries were given access to this model.",
"We repeated this 100 times with different training-test splits and then averaged the result.",
"Before we explain the results, we describe the datasets.",
"[leftmargin=0em,labelindent=] Eyedata.",
"This is gene expression data from rat eye tissues [34], as presented in the “flare” package of the R programming language.",
"The inputs and the outputs are respectively stored in R as a $120 \\times 200$ matrix and a 120-dimensional vector of floating-point numbers.",
"We used scikit-learn [32] to scale each attribute to zero mean and unit variance.",
"IWPC.",
"This is data collected by the International Warfarin Pharmacogenetics Consortium [35] about patients who were prescribed warfarin.",
"After we removed rows with missing values, 4819 patients remained in the dataset.",
"The inputs to the model are demographic (age, height, weight, race), medical (use of amiodarone, use of enzyme inducer), and genetic (VKORC1, CYP2C9) attributes.",
"Age, height, and weight are real-valued and were scaled to zero mean and unit variance.",
"The medical attributes take binary values, and the remaining attributes were one-hot encoded.",
"The output is the weekly dose of warfarin in milligrams.",
"However, because the distribution of warfarin dose is skewed, IWPC concludes in [35] that solving for the square root of the dose results in a more predictive linear model.",
"We followed this recommendation and scaled the square root of the dose to zero mean and unit variance.",
"Netflix.",
"We use the dataset from the Netflix Prize contest [36].",
"This is a sparse dataset that indicates when and how a user rated a movie.",
"For the output attribute, we used the rating of Dragon Ball Z: Trunks Saga, which had one of the most polarized rating distributions.",
"There are 2416 users who rated this, and the ratings were scaled to zero mean and unit variance.",
"The input attributes are binary variables indicating whether or not a user rated each of the other 17,769 movies in the dataset."
],
[
"Deep convolutional neural networks",
"We evaluated the membership inference attack on deep CNNs.",
"In addition, we implemented the colluding training algorithm (Algorithm REF ) to verify its performance in practice.",
"The CNNs were trained in Python using the Keras deep-learning library [37] and a standard stochastic gradient descent algorithm [38].",
"We used three datasets that are standard benchmarks in the deep learning literature and were evaluated in prior work on inference attacks [7]; they are described in more detail below.",
"For all datasets, pixel values were normalized to the range $[0,1]$ , and the label values were encoded as one-hot vectors.",
"To expedite the training process across a range of experimental configurations, we used a subset of each dataset.",
"For each dataset, we randomly divided the available data into equal-sized training and test sets to facilitate comparison with prior work [7] that used this convention.",
"The architecture we use is based on the VGG network [39], which is commonly used in computer vision applications.",
"We control for generalization error by varying a size parameter $s$ that defines the number of units at each layer of the network.",
"The architecture consists of two 3x3 convolutional layers with $s$ filters each, followed by a 2x2 max pooling layer, two 3x3 convolutional layers with $2s$ filters each, a 2x2 max pooling layer, a fully-connected layer with $2s$ units, and a softmax output layer.",
"All activation functions are rectified linear.",
"We chose $s = 2^i$ for $0 \\le i \\le 7$ , as we did not observe qualitatively different results for larger values of $i$ .",
"All training was done using the Adam optimizer [40] with the default parameters in the Keras implementation ($\\lambda = 0.001$ , $\\beta _1 = 0.5$ , $\\beta _2 = 0.99$ , $\\epsilon = 10^{-8}$ , and decay set to $5\\times 10^{-4}$ ).",
"We used categorical cross-entropy loss, which is conventional for models whose topmost activation is softmax [38].",
"[leftmargin=0em,labelindent=] MNIST.",
"MNIST [41] consists of 70,000 images of handwritten digits formatted as grayscale $28 \\times 28$ -pixel images, with class labels indicating the digit depicted in each image.",
"We selected 17,500 points from the full dataset at random for our experiments.",
"CIFAR-10, CIFAR-100.",
"The CIFAR datasets [42] consist of 60,000 $32 \\times 32$ -pixel color images, labeled as 10 (CIFAR-10) and 100 (CIFAR-100) classes.",
"We selected 15,000 points at random from the full data."
],
[
"Membership inference",
"The results of the membership inference attacks on linear and tree models are plotted in Figures REF and REF .",
"The theoretical and experimental results appear to agree when the adversary knows both $\\sigma _S $ and $\\sigma _$ and sets the decision boundary accordingly.",
"However, when the adversary does not know $\\sigma _$ , it performs much better than what the theory predicts.",
"In fact, an adversary can sometimes do better by just fixing the decision boundary at $|\\epsilon | = \\sigma _S $ instead of taking $\\sigma _$ into account.",
"This is because the training set error distributions are not exactly Gaussian.",
"Figures REF and REF in the appendix show that, although the training set error distributions roughly match the shape of a Gaussian curve, they have a much higher peak at zero.",
"As a result, it is often advantageous to bring the decision boundaries closer to zero.",
"The results of the threshold adversary on CNNs are given in Figure REF .",
"Although these models perform classification, the loss function used for training is categorical cross-entropy, which is non-negative, continuous, and unbounded.",
"This suggests that the threshold adversary could potentially work in this setting as well.",
"Specifically, the predictions made by these models can be compared against $L_S$ , the average training loss observed during training, which is often reported with published architectures as a point of comparison against prior work (see, for example, [43] and [44]).",
"Figure REF shows that, while the empirical results do not match the theoretical curve as closely as do linear and tree models, they do not diverge as much as one might expect given that the error is not Gaussian as assumed by Theorem REF .",
"Table: Comparison of our membership inference attack with that presented by Shokri et al.",
"While our attack has slightly lower precision, it requires far less computational resources and background knowledge.Now we compare our attack with that by Shokri et al.",
"[7], which generates “shadow models\" that are intended to mimic the behavior of $A_S$ .",
"Because their attack involves using machine learning to train the attacker with the shadow models, their attack requires considerable computational power and knowledge of the algorithm used to train the model.",
"By contrast, our attacker simply makes one query to the model and needs to know only the average training loss.",
"Despite these differences, when the size parameter $s$ is set equal to that used by Shokri et al., our attacker has the same recall and only slightly lower precision than their attacker.",
"A more detailed comparison is given in Table REF ."
],
[
"Attribute inference and reduction",
"We now present the empirical attribute advantage of the general adversary (Adversary REF ).",
"Because this adversary uses the model inversion assumptions described at the beginning of Section REF , our evaluation is also in the setting of model inversion.",
"For these experiments we used the IWPC and Netflix datasets described in Section REF .",
"For $f_\\mathcal {A} (\\epsilon )$ , the adversary's approximation of the error distribution, we used the Gaussian with mean zero and standard deviation $R_{emp}$ .",
"For the IWPC dataset, each of the genomic attributes (VKORC1 and CYP2C9) is separately used as the target $t$ .",
"In the Netflix dataset, the target attribute was whether a user rated a certain movie, and we randomly sampled targets from the set of available movies.",
"The circles in Figure REF show the result of inverting the VKORC1 and CYP2C9 attributes in the IWPC dataset.",
"Although the attribute advantage is not as high as the membership advantage (solid line), the attribute adversary exhibits a sizable advantage that increases as the model overfits more and more.",
"On the other hand, none of the attacks could effectively infer whether a user watched a certain movie in the Netflix dataset.",
"In addition, we were unable to simultaneously control for both $\\sigma _/\\sigma _S $ and $\\tau $ in the Netflix dataset to measure the effect of influence as predicted by Theorem .",
"Finally, we evaluate the performance of the multi-query reduction adversary (Adversary REF ).",
"As the squares in Figure REF show, with the IWPC data, making multiple queries to the membership oracle significantly increased the success rate compared to what we would expect from the naive uniform reduction adversary (Adversary REF , dotted line).",
"Surprisingly, the reduction is also more effective than running the attribute inference attack directly.",
"By contrast, with the Netflix data, the multi-query reduction adversary was often slightly worse than the naive uniform adversary although it still outperformed direct attribute inference."
],
[
"Collusion in membership inference",
"We evaluate $A^{\\mathsf {C}}$ and $\\mathcal {A} ^{\\mathsf {C}}$ described in Section REF for CNNs trained as image classifiers.",
"To instantiate $F_K$ and $G_K$ , we use Python's intrinsic pseudorandom number generator with key $K$ as the seed.",
"We note that our proof of Theorem REF relies only on the uniformity of the pseudorandom numbers and not on their unpredictability.",
"Deviations from this assumption will result in a less effective membership inference attack but do not invalidate our results.",
"All experiments set the number of keys to $k=3$ .",
"Figure: Results of colluding training algorithm and membership adversary on CNNs trained on MNIST, CIFAR-10, and CIFAR-100.",
"The size parameter was configured to take values s=2 i s=2^i for i∈[0,7]i \\in [0,7].",
"Regardless of the models' generalization performance, when the network is sufficiently large, the attack achieves high advantage (≥0.98\\ge 0.98) without affecting predictive accuracy.The results of our experiment are shown in Figures REF and REF .",
"The data shows that on all three instances, the colluding parties achieve a high membership advantage without significantly affecting model performance.",
"The accuracy of the subverted model was only 0.014 (MNIST), 0.047 (CIFAR-10), and 0.031 (CIFAR-100) less than that of the unsubverted model.",
"The advantage rapidly increases with the model size around $s \\approx 16$ but is relatively constant elsewhere, indicating that model capacity beyond a certain point is a necessary factor in the attack.",
"Importantly, the results demonstrate that specific information about nearly all of the training data can be intentionally leaked through the behavior of a model that appears to generalize very well.",
"In fact, looking at Figure REF shows that in these instances, there is no discernible relationship between generalization error and membership advantage.",
"The three datasets exhibit vastly different generalization behavior, with the MNIST models achieving almost no generalization error ($< 0.02$ for $s \\ge 32$ ) and CIFAR-100 showing a large performance gap ($\\ge 0.8$ for $s \\ge 32$ ).",
"Despite this fact, the membership adversary achieves nearly identical performance."
],
[
"Privacy and statistical summaries",
"There is extensive prior literature on privacy attacks on statistical summaries.",
"Komarova et al.",
"[45] looked into partial disclosure scenarios, where an adversary is given fixed statistical estimates from combined public and private sources and attempts to infer the sensitive feature of an individual referenced in those sources.",
"A number of previous studies [46], [21], [22], [47], [48], [49] have looked into membership attacks from statistics commonly published in genome-wide association studies (GWAS).",
"Calandrino et al.",
"[50] showed that temporal changes in recommendations given by collaborative filtering methods can reveal the inputs that caused those changes.",
"Linear reconstruction attacks [51], [52], [53] attempt to infer partial inputs to linear statistics and were later extended to non-linear statistics [54].",
"While the goal of these attacks has commonalities with both membership inference and attribute inference, our results apply specifically to machine learning settings where generalization error and influence make our results relevant."
],
[
"Privacy and machine learning",
"More recently, others have begun examining these attacks in the context of machine learning.",
"Ateniese et al.",
"[1] showed that the knowledge of the internal structure of Support Vector Machines and Hidden Markov Models leaks certain types of information about their training data, such as the language used in a speech dataset.",
"Dwork et al.",
"[13] showed that a differentially private algorithm with a suitably chosen parameter generalizes well with high probability.",
"Subsequent work showed that similar results are true under related notions of privacy.",
"In particular, Bassily et al.",
"[18] studied a notion of privacy called total variation stability and proved good generalization with respect to a bounded number of adaptively chosen low-sensitivity queries.",
"Moreover, for data drawn from Gibbs distributions, Wang et al.",
"[19] showed that on-average KL privacy is equivalent to generalization error as defined in this paper.",
"While these results give evidence for the relationship between privacy and overfitting, we construct an attacker that directly leverages overfitting to gain advantage commensurate with the extent of the overfitting."
],
[
"Membership inference",
"Shokri et al.",
"[7] developed a membership inference attack and applied it to popular machine-learning-as-a-service APIs.",
"Their attacks are based on “shadow models” that approximate the behavior of the model under attack.",
"The shadow models are used to build another machine learning model called the “attack model”, which is trained to distinguish points in the training data from other points based on the output they induce on the original model under attack.",
"As we discussed in Section REF , our simple threshold adversary comes surprisingly close to the accuracy of their attack, especially given the differences in complexity and requisite adversarial assumptions between the attacks.",
"Because the attack proposed by Shokri et al.",
"itself relies on machine learning to find a function that separates training and non-training points, it is not immediately clear why the attack works, but the authors hypothesize that it is related to overfitting and the “diversity” of the training data.",
"They graph the generalization error against the precision of their attack and find some evidence of a relationship, but they also find that the relationship is not perfect and conclude that model structure must also be relevant.",
"The results presented in this paper make the connection to overfitting precise in many settings, and the colluding training algorithm we give in Section REF demonstrates exactly how model structure can be exploited to create a membership inference vulnerability.",
"Li et al.",
"[6] explored membership inference, distinguishing between “positive” and “negative” membership privacy.",
"They show how this framework defines a family of related privacy definitions that are parametrized on distributions of the adversary's prior knowledge, and they find that a number of previous definitions can be instantiated in this way."
],
[
"Attribute inference",
"Practical model inversion attacks have been studied in the context of linear regression [4], [8], decision trees [3], and neural networks [3].",
"Our results apply to these attacks when they are applied to data that matches the distributional assumptions made in our analysis.",
"An important distinction between the way inversion attacks were considered in prior work and how we treat them here is the notion of advantage.",
"Prior work on these attacks defined advantage as the difference between the attacker's predictive accuracy given the model and the best accuracy that could be achieved without the model.",
"Although some prior work [4], [3] empirically measured this advantage on both training and test datasets, this definition does not allow a formal characterization of how exposed the training data specifically is to privacy risk.",
"In Section REF , we define attribute advantage precisely to capture the risk to the training data by measuring the difference in the attacker's accuracy on training and test data: the advantage is zero when the attack is as powerful on the general population as on the training data and is maximized when the attack works only on the training data.",
"Wu et al.",
"[5] formalized model inversion for a simplified class of models that consist of Boolean functions and explored the initial connections between influence and advantage.",
"However, as in other prior work on model inversion, the type of advantage that they consider says nothing about what the model specifically leaks about its training data.",
"Drawing on their observation that influence is relevant to privacy risk in general, we illustrate its effect on the notion of advantage defined in this paper and show how it interacts with generalization error."
],
[
"Conclusion and Future Directions",
"We introduced new formal definitions of advantage for membership and attribute inference attacks.",
"Using these definitions, we analyzed attacks under various assumptions on learning algorithms and model properties, and we showed that these two attacks are closely related through reductions in both directions.",
"Both theoretical and experimental results confirm that models become more vulnerable to both types of attacks as they overfit more.",
"Interestingly, our analysis also shows that overfitting is not the only factor that can lead to privacy risk: Theorem REF shows that even stable learning algorithms, which provably do not overfit, can leak precise membership information, and the results in Section REF demonstrate that the influence of the target attribute on a model's output plays a key role in attribute inference.",
"Our formalization and analysis open interesting directions for future work.",
"The membership attack in Theorem REF is based on a colluding pair of adversary and learning rule, $A^{\\mathsf {C}}$ and $\\mathcal {A} ^{\\mathsf {C}}$ .",
"This could be implemented, for example, by a malicious ML algorithm provided by a third-party library or cloud service to subvert users' privacy.",
"Further study of this scenario, which may best be formalized in the framework of algorithm substitution attacks [23], is warranted to determine whether malicious algorithms can produce models that are indistinguishable from normal ones and how such attacks can be mitigated.",
"Our results in Section REF give bounds on membership advantage when certain conditions are met.",
"These bounds apply to adversaries who may target specific individuals, bringing arbitrary background knowledge of their targets to help determine their membership status.",
"Some types of realistic adversaries may be motivated by concerns that incentivize learning a limited set of facts about as many individuals in the training data as possible rather than obtaining unique background knowledge about specific individuals.",
"Characterizing these “stable adversaries” is an interesting direction that may lead to tighter bounds on advantage or relaxed conditions on the learning rule."
],
[
"Evaluation",
"In this section, we evaluate the performance of the adversaries discussed in Sections , REF , and .",
"We compare the performance of these adversaries on real datasets with the analysis from previous sections and show that overfitting predicts privacy risk in practice as our analysis suggests.",
"Our experiments use linear regression, tree, and deep convolutional neural network (CNN) models."
],
[
"Linear and tree models",
"We used the Python scikit-learn [32] library to calculate the empirical error $R_{emp}$ and the leave-one-out cross validation error $R_{cv}$ [33].",
"Because these two measures pertain to the error of the model on points inside and outside the training set, respectively, they were used to approximate $\\sigma _S $ and $\\sigma _$ , respectively.",
"Then, we made a random 75-25% split of the data into training and test sets.",
"The training set was used to train either a Ridge regression or a decision tree model, and then the adversaries were given access to this model.",
"We repeated this 100 times with different training-test splits and then averaged the result.",
"Before we explain the results, we describe the datasets.",
"[leftmargin=0em,labelindent=] Eyedata.",
"This is gene expression data from rat eye tissues [34], as presented in the “flare” package of the R programming language.",
"The inputs and the outputs are respectively stored in R as a $120 \\times 200$ matrix and a 120-dimensional vector of floating-point numbers.",
"We used scikit-learn [32] to scale each attribute to zero mean and unit variance.",
"IWPC.",
"This is data collected by the International Warfarin Pharmacogenetics Consortium [35] about patients who were prescribed warfarin.",
"After we removed rows with missing values, 4819 patients remained in the dataset.",
"The inputs to the model are demographic (age, height, weight, race), medical (use of amiodarone, use of enzyme inducer), and genetic (VKORC1, CYP2C9) attributes.",
"Age, height, and weight are real-valued and were scaled to zero mean and unit variance.",
"The medical attributes take binary values, and the remaining attributes were one-hot encoded.",
"The output is the weekly dose of warfarin in milligrams.",
"However, because the distribution of warfarin dose is skewed, IWPC concludes in [35] that solving for the square root of the dose results in a more predictive linear model.",
"We followed this recommendation and scaled the square root of the dose to zero mean and unit variance.",
"Netflix.",
"We use the dataset from the Netflix Prize contest [36].",
"This is a sparse dataset that indicates when and how a user rated a movie.",
"For the output attribute, we used the rating of Dragon Ball Z: Trunks Saga, which had one of the most polarized rating distributions.",
"There are 2416 users who rated this, and the ratings were scaled to zero mean and unit variance.",
"The input attributes are binary variables indicating whether or not a user rated each of the other 17,769 movies in the dataset."
],
[
"Deep convolutional neural networks",
"We evaluated the membership inference attack on deep CNNs.",
"In addition, we implemented the colluding training algorithm (Algorithm REF ) to verify its performance in practice.",
"The CNNs were trained in Python using the Keras deep-learning library [37] and a standard stochastic gradient descent algorithm [38].",
"We used three datasets that are standard benchmarks in the deep learning literature and were evaluated in prior work on inference attacks [7]; they are described in more detail below.",
"For all datasets, pixel values were normalized to the range $[0,1]$ , and the label values were encoded as one-hot vectors.",
"To expedite the training process across a range of experimental configurations, we used a subset of each dataset.",
"For each dataset, we randomly divided the available data into equal-sized training and test sets to facilitate comparison with prior work [7] that used this convention.",
"The architecture we use is based on the VGG network [39], which is commonly used in computer vision applications.",
"We control for generalization error by varying a size parameter $s$ that defines the number of units at each layer of the network.",
"The architecture consists of two 3x3 convolutional layers with $s$ filters each, followed by a 2x2 max pooling layer, two 3x3 convolutional layers with $2s$ filters each, a 2x2 max pooling layer, a fully-connected layer with $2s$ units, and a softmax output layer.",
"All activation functions are rectified linear.",
"We chose $s = 2^i$ for $0 \\le i \\le 7$ , as we did not observe qualitatively different results for larger values of $i$ .",
"All training was done using the Adam optimizer [40] with the default parameters in the Keras implementation ($\\lambda = 0.001$ , $\\beta _1 = 0.5$ , $\\beta _2 = 0.99$ , $\\epsilon = 10^{-8}$ , and decay set to $5\\times 10^{-4}$ ).",
"We used categorical cross-entropy loss, which is conventional for models whose topmost activation is softmax [38].",
"[leftmargin=0em,labelindent=] MNIST.",
"MNIST [41] consists of 70,000 images of handwritten digits formatted as grayscale $28 \\times 28$ -pixel images, with class labels indicating the digit depicted in each image.",
"We selected 17,500 points from the full dataset at random for our experiments.",
"CIFAR-10, CIFAR-100.",
"The CIFAR datasets [42] consist of 60,000 $32 \\times 32$ -pixel color images, labeled as 10 (CIFAR-10) and 100 (CIFAR-100) classes.",
"We selected 15,000 points at random from the full data."
],
[
"Membership inference",
"The results of the membership inference attacks on linear and tree models are plotted in Figures REF and REF .",
"The theoretical and experimental results appear to agree when the adversary knows both $\\sigma _S $ and $\\sigma _$ and sets the decision boundary accordingly.",
"However, when the adversary does not know $\\sigma _$ , it performs much better than what the theory predicts.",
"In fact, an adversary can sometimes do better by just fixing the decision boundary at $|\\epsilon | = \\sigma _S $ instead of taking $\\sigma _$ into account.",
"This is because the training set error distributions are not exactly Gaussian.",
"Figures REF and REF in the appendix show that, although the training set error distributions roughly match the shape of a Gaussian curve, they have a much higher peak at zero.",
"As a result, it is often advantageous to bring the decision boundaries closer to zero.",
"The results of the threshold adversary on CNNs are given in Figure REF .",
"Although these models perform classification, the loss function used for training is categorical cross-entropy, which is non-negative, continuous, and unbounded.",
"This suggests that the threshold adversary could potentially work in this setting as well.",
"Specifically, the predictions made by these models can be compared against $L_S$ , the average training loss observed during training, which is often reported with published architectures as a point of comparison against prior work (see, for example, [43] and [44]).",
"Figure REF shows that, while the empirical results do not match the theoretical curve as closely as do linear and tree models, they do not diverge as much as one might expect given that the error is not Gaussian as assumed by Theorem REF .",
"Table: Comparison of our membership inference attack with that presented by Shokri et al.",
"While our attack has slightly lower precision, it requires far less computational resources and background knowledge.Now we compare our attack with that by Shokri et al.",
"[7], which generates “shadow models\" that are intended to mimic the behavior of $A_S$ .",
"Because their attack involves using machine learning to train the attacker with the shadow models, their attack requires considerable computational power and knowledge of the algorithm used to train the model.",
"By contrast, our attacker simply makes one query to the model and needs to know only the average training loss.",
"Despite these differences, when the size parameter $s$ is set equal to that used by Shokri et al., our attacker has the same recall and only slightly lower precision than their attacker.",
"A more detailed comparison is given in Table REF ."
],
[
"Attribute inference and reduction",
"We now present the empirical attribute advantage of the general adversary (Adversary REF ).",
"Because this adversary uses the model inversion assumptions described at the beginning of Section REF , our evaluation is also in the setting of model inversion.",
"For these experiments we used the IWPC and Netflix datasets described in Section REF .",
"For $f_\\mathcal {A} (\\epsilon )$ , the adversary's approximation of the error distribution, we used the Gaussian with mean zero and standard deviation $R_{emp}$ .",
"For the IWPC dataset, each of the genomic attributes (VKORC1 and CYP2C9) is separately used as the target $t$ .",
"In the Netflix dataset, the target attribute was whether a user rated a certain movie, and we randomly sampled targets from the set of available movies.",
"The circles in Figure REF show the result of inverting the VKORC1 and CYP2C9 attributes in the IWPC dataset.",
"Although the attribute advantage is not as high as the membership advantage (solid line), the attribute adversary exhibits a sizable advantage that increases as the model overfits more and more.",
"On the other hand, none of the attacks could effectively infer whether a user watched a certain movie in the Netflix dataset.",
"In addition, we were unable to simultaneously control for both $\\sigma _/\\sigma _S $ and $\\tau $ in the Netflix dataset to measure the effect of influence as predicted by Theorem .",
"Finally, we evaluate the performance of the multi-query reduction adversary (Adversary REF ).",
"As the squares in Figure REF show, with the IWPC data, making multiple queries to the membership oracle significantly increased the success rate compared to what we would expect from the naive uniform reduction adversary (Adversary REF , dotted line).",
"Surprisingly, the reduction is also more effective than running the attribute inference attack directly.",
"By contrast, with the Netflix data, the multi-query reduction adversary was often slightly worse than the naive uniform adversary although it still outperformed direct attribute inference."
],
[
"Collusion in membership inference",
"We evaluate $A^{\\mathsf {C}}$ and $\\mathcal {A} ^{\\mathsf {C}}$ described in Section REF for CNNs trained as image classifiers.",
"To instantiate $F_K$ and $G_K$ , we use Python's intrinsic pseudorandom number generator with key $K$ as the seed.",
"We note that our proof of Theorem REF relies only on the uniformity of the pseudorandom numbers and not on their unpredictability.",
"Deviations from this assumption will result in a less effective membership inference attack but do not invalidate our results.",
"All experiments set the number of keys to $k=3$ .",
"Figure: Results of colluding training algorithm and membership adversary on CNNs trained on MNIST, CIFAR-10, and CIFAR-100.",
"The size parameter was configured to take values s=2 i s=2^i for i∈[0,7]i \\in [0,7].",
"Regardless of the models' generalization performance, when the network is sufficiently large, the attack achieves high advantage (≥0.98\\ge 0.98) without affecting predictive accuracy.The results of our experiment are shown in Figures REF and REF .",
"The data shows that on all three instances, the colluding parties achieve a high membership advantage without significantly affecting model performance.",
"The accuracy of the subverted model was only 0.014 (MNIST), 0.047 (CIFAR-10), and 0.031 (CIFAR-100) less than that of the unsubverted model.",
"The advantage rapidly increases with the model size around $s \\approx 16$ but is relatively constant elsewhere, indicating that model capacity beyond a certain point is a necessary factor in the attack.",
"Importantly, the results demonstrate that specific information about nearly all of the training data can be intentionally leaked through the behavior of a model that appears to generalize very well.",
"In fact, looking at Figure REF shows that in these instances, there is no discernible relationship between generalization error and membership advantage.",
"The three datasets exhibit vastly different generalization behavior, with the MNIST models achieving almost no generalization error ($< 0.02$ for $s \\ge 32$ ) and CIFAR-100 showing a large performance gap ($\\ge 0.8$ for $s \\ge 32$ ).",
"Despite this fact, the membership adversary achieves nearly identical performance."
],
[
"Privacy and statistical summaries",
"There is extensive prior literature on privacy attacks on statistical summaries.",
"Komarova et al.",
"[45] looked into partial disclosure scenarios, where an adversary is given fixed statistical estimates from combined public and private sources and attempts to infer the sensitive feature of an individual referenced in those sources.",
"A number of previous studies [46], [21], [22], [47], [48], [49] have looked into membership attacks from statistics commonly published in genome-wide association studies (GWAS).",
"Calandrino et al.",
"[50] showed that temporal changes in recommendations given by collaborative filtering methods can reveal the inputs that caused those changes.",
"Linear reconstruction attacks [51], [52], [53] attempt to infer partial inputs to linear statistics and were later extended to non-linear statistics [54].",
"While the goal of these attacks has commonalities with both membership inference and attribute inference, our results apply specifically to machine learning settings where generalization error and influence make our results relevant."
],
[
"Privacy and machine learning",
"More recently, others have begun examining these attacks in the context of machine learning.",
"Ateniese et al.",
"[1] showed that the knowledge of the internal structure of Support Vector Machines and Hidden Markov Models leaks certain types of information about their training data, such as the language used in a speech dataset.",
"Dwork et al.",
"[13] showed that a differentially private algorithm with a suitably chosen parameter generalizes well with high probability.",
"Subsequent work showed that similar results are true under related notions of privacy.",
"In particular, Bassily et al.",
"[18] studied a notion of privacy called total variation stability and proved good generalization with respect to a bounded number of adaptively chosen low-sensitivity queries.",
"Moreover, for data drawn from Gibbs distributions, Wang et al.",
"[19] showed that on-average KL privacy is equivalent to generalization error as defined in this paper.",
"While these results give evidence for the relationship between privacy and overfitting, we construct an attacker that directly leverages overfitting to gain advantage commensurate with the extent of the overfitting."
],
[
"Membership inference",
"Shokri et al.",
"[7] developed a membership inference attack and applied it to popular machine-learning-as-a-service APIs.",
"Their attacks are based on “shadow models” that approximate the behavior of the model under attack.",
"The shadow models are used to build another machine learning model called the “attack model”, which is trained to distinguish points in the training data from other points based on the output they induce on the original model under attack.",
"As we discussed in Section REF , our simple threshold adversary comes surprisingly close to the accuracy of their attack, especially given the differences in complexity and requisite adversarial assumptions between the attacks.",
"Because the attack proposed by Shokri et al.",
"itself relies on machine learning to find a function that separates training and non-training points, it is not immediately clear why the attack works, but the authors hypothesize that it is related to overfitting and the “diversity” of the training data.",
"They graph the generalization error against the precision of their attack and find some evidence of a relationship, but they also find that the relationship is not perfect and conclude that model structure must also be relevant.",
"The results presented in this paper make the connection to overfitting precise in many settings, and the colluding training algorithm we give in Section REF demonstrates exactly how model structure can be exploited to create a membership inference vulnerability.",
"Li et al.",
"[6] explored membership inference, distinguishing between “positive” and “negative” membership privacy.",
"They show how this framework defines a family of related privacy definitions that are parametrized on distributions of the adversary's prior knowledge, and they find that a number of previous definitions can be instantiated in this way."
],
[
"Attribute inference",
"Practical model inversion attacks have been studied in the context of linear regression [4], [8], decision trees [3], and neural networks [3].",
"Our results apply to these attacks when they are applied to data that matches the distributional assumptions made in our analysis.",
"An important distinction between the way inversion attacks were considered in prior work and how we treat them here is the notion of advantage.",
"Prior work on these attacks defined advantage as the difference between the attacker's predictive accuracy given the model and the best accuracy that could be achieved without the model.",
"Although some prior work [4], [3] empirically measured this advantage on both training and test datasets, this definition does not allow a formal characterization of how exposed the training data specifically is to privacy risk.",
"In Section REF , we define attribute advantage precisely to capture the risk to the training data by measuring the difference in the attacker's accuracy on training and test data: the advantage is zero when the attack is as powerful on the general population as on the training data and is maximized when the attack works only on the training data.",
"Wu et al.",
"[5] formalized model inversion for a simplified class of models that consist of Boolean functions and explored the initial connections between influence and advantage.",
"However, as in other prior work on model inversion, the type of advantage that they consider says nothing about what the model specifically leaks about its training data.",
"Drawing on their observation that influence is relevant to privacy risk in general, we illustrate its effect on the notion of advantage defined in this paper and show how it interacts with generalization error."
],
[
"Conclusion and Future Directions",
"We introduced new formal definitions of advantage for membership and attribute inference attacks.",
"Using these definitions, we analyzed attacks under various assumptions on learning algorithms and model properties, and we showed that these two attacks are closely related through reductions in both directions.",
"Both theoretical and experimental results confirm that models become more vulnerable to both types of attacks as they overfit more.",
"Interestingly, our analysis also shows that overfitting is not the only factor that can lead to privacy risk: Theorem REF shows that even stable learning algorithms, which provably do not overfit, can leak precise membership information, and the results in Section REF demonstrate that the influence of the target attribute on a model's output plays a key role in attribute inference.",
"Our formalization and analysis open interesting directions for future work.",
"The membership attack in Theorem REF is based on a colluding pair of adversary and learning rule, $A^{\\mathsf {C}}$ and $\\mathcal {A} ^{\\mathsf {C}}$ .",
"This could be implemented, for example, by a malicious ML algorithm provided by a third-party library or cloud service to subvert users' privacy.",
"Further study of this scenario, which may best be formalized in the framework of algorithm substitution attacks [23], is warranted to determine whether malicious algorithms can produce models that are indistinguishable from normal ones and how such attacks can be mitigated.",
"Our results in Section REF give bounds on membership advantage when certain conditions are met.",
"These bounds apply to adversaries who may target specific individuals, bringing arbitrary background knowledge of their targets to help determine their membership status.",
"Some types of realistic adversaries may be motivated by concerns that incentivize learning a limited set of facts about as many individuals in the training data as possible rather than obtaining unique background knowledge about specific individuals.",
"Characterizing these “stable adversaries” is an interesting direction that may lead to tighter bounds on advantage or relaxed conditions on the learning rule."
]
] | 1709.01604 |
[
[
"Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues"
],
[
"Abstract In the present paper we develop an approach to obtain sharp spectral asymptotics for Steklov type problems on planar domains with corners.",
"Our main focus is on the two-dimensional sloshing problem, which is a mixed Steklov-Neumann boundary value problem describing small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section.",
"We prove a two-term asymptotic formula for sloshing eigenvalues.",
"In particular, this confirms a conjecture posed by Fox and Kuttler in 1983.",
"We also obtain similar eigenvalue asymptotics for other related mixed Steklov type problems, and discuss applications to the study of Steklov spectral asymptotics on polygons."
],
[
"The sloshing problem",
"Let $\\Omega $ be a simply connected bounded planar domain with Lipschitz boundary such that $\\partial \\Omega =S\\sqcup \\mathcal {W}$ , where $S:=(A,B)$ is a line segment.",
"Let $0<\\alpha ,\\beta \\le \\pi $ be the angles at the vertices $A$ and $B$ , respectively (see Figure 1).",
"Without loss of generality we can assume that in Cartesian coordinates $A=(0,0)$ and $B=(0,L)$ , where $L>0$ is the length of $S$ .",
"Figure: Geometry of the sloshing problemConsider the following mixed Steklov-Neumann boundary value problem, $\\begin{dcases} \\Delta u=0 &\\quad \\text{in }\\Omega , \\\\\\frac{\\partial u}{\\partial n}=0 &\\quad \\text{on } \\mathcal {W},\\\\\\frac{\\partial u}{\\partial n}=\\lambda u &\\quad \\text{on }S,\\end{dcases}$ where $\\dfrac{\\partial }{\\partial n}$ denotes the external normal derivative on $\\partial \\Omega $ .",
"The eigenvalue problem (REF ) is called the sloshing problem.",
"It describes small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section which has the shape of the domain $\\Omega $ .",
"The part $S=(A,B)$ of the boundary is called the sloshing surface; it represents the free surface of the fluid.",
"The part $\\mathcal {W}$ is called the walls and corresponds to the walls and the bottom of a container or a canal.",
"The points $A$ and $B$ at the interface of the sloshing surface and the walls are called the corner points.",
"We also say that the walls $\\mathcal {W}$ are straight near the corners if there exist points $A_1, B_1\\in \\mathcal {W}$ such that the line segments $AA_1$ and $BB_1$ are subsets of $\\mathcal {W}.$ Figure: Geometry of the sloshing problem with walls straight near the cornersIt follows from general results on the Steklov type problems that the spectrum of the sloshing problem is discrete (see [1], [14]).",
"We denote the sloshing eigenvalues by $0=\\lambda _1<\\lambda _2 \\le \\lambda _3 \\le \\dots \\nearrow \\infty ,$ where the eigenvalues are a priori counted with multiplicities.",
"The correspoding sloshing eigenfunctions are denoted by $u_k$ , where $u_k \\in C^\\infty (\\Omega )$ , and the restrictions $u_k|_{S}$ form an orthogonal basis in $L^2(S)$ .",
"Let us note that in two dimensions all sloshing eigenvalues are conjectured to be simple, see [27], [13].",
"While in full generality this conjecture remains open, in Corollary REF we prove that it holds for all but possibly a finite number of eigenvalues.",
"Denote by $\\mathcal {D}:L^2(S)\\rightarrow L^2(S)$ the sloshing operator, which is essentially the Dirichlet-to-Neumann operator on $S$ corresponding to Neumann boundary conditions on $\\mathcal {W}$ : given a function $f\\in L^2(S)$ , we have $\\mathcal {D}_\\Omega f=\\mathcal {D}f:=\\left.\\frac{\\partial }{\\partial n}(\\mathcal {H}f)\\right|_S,$ where $\\mathcal {H}f$ is the harmonic extension of $f$ to $\\Omega $ with the homogeneous Neumann conditions on $\\mathcal {W}$ .",
"Then the sloshing eigenvalues are exactly the eigenvalues of $\\mathcal {D}$ , and the sloshing eigenfunctions $u_k$ are harmonic extensions of the eigenfunctions of $\\mathcal {D}$ to $\\Omega $ .",
"The physical meaning of the sloshing eigenvalues and eigenfunctions is as follows: an eigenfunction $u_k$ is the fluid velocity potential and $\\sqrt{\\lambda _k}$ is proportional to the frequency of the corresponding fluid oscillations.",
"The research on the sloshing problem has a long history in hydrodynamics (see [31] and [15]); we refer to [10] for a historical discussion.",
"A more recent exposition and further references could be found, for example, in [26], [30], [20], [29].",
"In this paper we investigate the asymptotic distribution of the sloshing eigenvalues.",
"In fact, we develop an approach that could be applied to study sharp spectral asymptotics of general Steklov type problems on planar domains with corners.",
"The main difficulty is that in the presence of singularities, the corresponding Dirichlet-to-Neumann operator is not pseudodifferential, and therefore new techniques have to be invented, see [14] and Subsection REF for a discussion.",
"Since the first version of the present paper appeared on the arXiv, this problem has received significant attention, both in planar and higher-dimensional cases [18], [12], [21].",
"Our method is based on quasimode analysis, see Subsection REF .",
"A particularly challenging aspect of the argument is to show that the constructed system of quasimodes is, in an appropriate sense, complete.",
"This is done via a rather surprising link to the theory of higher-order Sturm-Liouville problems, see Subsection REF .",
"In particular, we notice that the quasimode approximation for the sloshing problem is sensitive to the arithmetic properties of the angles at the corners.",
"As shown in Subsection REF , the quasimodes are exponentially accurate for angles of the form $\\pi /2q$ , $q\\in \\mathbb {N}$ , which together with domain monotonicity arguments allows us to prove completeness for arbitrary angles.",
"Further applications of our method, notably to the Steklov problem on polygons, are presented in [32]."
],
[
"Asymptotics of the sloshing eigenvalues",
"As was shown in [38], if the boundary of $\\Omega $ is $C^2$ -regular in a neighbourhood of the corner points $A$ and $B$ , then as $k\\rightarrow +\\infty $ , $\\lambda _k L=\\pi k + o(k).$ It follows from the results of [1] on Weyl's law for mixed Steklov type problems that the $C^2$ assumption can be relaxed to $C^1$ .",
"In 1983, Fox and Kuttler used numerical evidence to conjecture that the sloshing eigenvalues of domains having both interior angles at the corner points $A$ and $B$ equal to $\\alpha $ , satisfy the two-term asymptotics [10]: $\\lambda _k L = \\pi \\left(k-\\frac{1}{2}\\right)-\\frac{\\pi ^2}{4 \\alpha }+ o(1), $ (note that the numeration of eigenvalues in [10] is shifted by one compared to ours).",
"The first main result of the present paper confirms this prediction.",
"Theorem 1.1 Let $\\Omega $ be a simply connected bounded Lipschitz planar domain with the sloshing surface $S=(A,B)$ of length $L$ and walls $\\mathcal {W}$ which are $C^1$ -regular near the corner points $A$ and $B$ .",
"Let $\\alpha $ and $\\beta $ be the interior angles between $\\mathcal {W}$ and $S$ at the points $A$ and $B$ , resp., and assume $0<\\beta \\le \\alpha < \\pi /2 $ .",
"Then the following asymptotic expansion holds as $k\\rightarrow \\infty $ : $\\lambda _k L=\\pi \\left(k-\\frac{1}{2}\\right)-\\frac{\\pi ^2}{8}\\left(\\frac{1}{\\alpha }+\\frac{1}{\\beta }\\right)+r(k), \\quad \\text{where} \\quad r(k)=o(1).$ If, moreover, the walls $\\mathcal {W}$ are straight near the corners, then $r(k)=O\\left(k^{1-\\frac{\\pi }{2\\alpha }}\\right).$ In particular, for $\\alpha =\\beta $ we obtain (REF ) which proves the Fox-Kuttler conjecture for all angles $0<\\alpha <\\pi /2$ .",
"Let us note that for $\\alpha =\\beta =\\pi /2$ the asymptotics (REF ) have been earlier established in [7], [39], [8].",
"Moreover, for $\\alpha =\\beta =\\pi /2$ it was shown that there exist further terms in the asymptotics (REF ) involving the curvature of $\\mathcal {W}$ near the corner points.",
"Before stating our next result we require the following definition.",
"Definition 1.2 A corner point $V\\in \\lbrace A,B\\rbrace $ is said to satisfy a local John's condition if there exist a neighbourhood $\\mathcal {O}_V$ of the point $V$ such that the orthogonal projection of $\\mathcal {W}\\cap \\mathcal {O}_V$ onto the $x$ -axis is a subset of $[A,B]$ .",
"For $\\alpha =\\pi /2\\ge \\beta $ we obtain the following Proposition 1.3 In the notation of Theorem REF , let $\\alpha =\\pi /2>\\beta $ , and assume that $A$ satisfies a local John's condition.",
"Then $\\lambda _k L = \\pi \\left(k-\\frac{3}{4}\\right)-\\frac{\\pi ^2}{8 \\beta }+ r(k), \\,\\,\\, r(k)=o(1).$ The same result holds if $\\alpha =\\beta =\\pi /2$ , and additionally $B$ satisfies a local John's condition.",
"If, moreover, the walls $W$ are straight near both corners, then $r(k)=O\\left(k^{1-\\frac{\\pi }{2\\beta }}\\right),$ provided $\\beta <\\pi /2$ and $r(k)=o\\left(\\mathrm {e}^{-k/C}\\right),$ if $\\alpha =\\beta =\\pi /2$ .",
"Proposition REF provides a solution to [10] under the assumption that the corner point corresponding to the angle $\\pi /2$ satisfies local John's condition.",
"Remark 1.4 In Theorem REF , and everywhere further on, $C$ will denote various positive constants which depend only upon the domain $\\Omega $ .",
"Remark 1.5 Definition REF is a local version of John's condition which often appears in sloshing problems, see [27], [4].",
"Theorem REF and Proposition REF yield the following Corollary 1.6 Given a sloshing problem on a domain $\\Omega $ satisfying the assumptions of Theorem REF or Proposition REF , there exists $N>0$ , such that the eigenvalues $\\lambda _k$ are simple for all $k\\ge N$ .",
"This result partially confirms the conjecture about the simplicity of sloshing eigenvalues mentioned in subsection REF ."
],
[
"Eigenvalue asymptotics for a mixed Steklov-Dirichlet problem",
"Boundary value problems of Steklov type with mixed boundary conditions admit several physical and probabilistic interpretations (see [3], [4]).",
"In particular, the sloshing problem (REF ) could be also used to model the stationary heat distribution in $\\Omega $ such that the walls $\\mathcal {W}$ are perfectly insulated and the heat flux through $S$ is proportional to the temperature.",
"If instead the walls $\\mathcal {W}$ are kept under zero temperature, one obtains the following mixed Steklov-Dirichlet problem: $\\begin{dcases}\\Delta u=0 & \\quad \\text{in }\\Omega , \\\\u=0 &\\quad \\text{on } \\mathcal {W},\\\\\\frac{\\partial u}{\\partial n}=\\lambda ^D u &\\quad \\text{on }S,\\end{dcases}$ Let $0<\\lambda _1^D \\le \\lambda _2^D \\le \\dots \\nearrow \\infty $ be the eigenvalues of the problem (REF ).",
"Similarly to Theorem REF we obtain Theorem 1.7 Assume that the domain $\\Omega $ and its boundary $\\partial \\Omega =S \\sqcup \\mathcal {W}$ satisfy the assumptions of Theorem REF .",
"Let $\\alpha $ and $\\beta $ be the interior angles between $\\mathcal {W}$ and $S$ at the points $A$ and $B$ , resp., and assume $0<\\beta \\le \\alpha < \\pi /2 $ .",
"Then the following asymptotic expansion holds as $k\\rightarrow \\infty $ : $\\lambda _k^D L=\\pi \\left(k-\\frac{1}{2}\\right)+\\frac{\\pi ^2}{8}\\left(\\frac{1}{\\alpha }+\\frac{1}{\\beta }\\right)+r^D(k),\\quad \\operatorname{where} \\,\\,\\,r^D(k)=o(1).$ If, moreover, the walls $\\mathcal {W}$ are straight near the corner points $A$ and $B$ , then $r^D(k)= O\\left(k^{1-\\frac{\\pi }{\\alpha }}\\right).$ We also have the following analogue of Proposition REF .",
"Proposition 1.8 In the notation of Theorem REF , let $\\alpha =\\pi /2 > \\beta $ , and assume that $A$ satisfies a local John's condition.",
"Then $\\lambda _k^D L = \\pi \\left(k-\\frac{1}{4}\\right)+\\frac{\\pi ^2}{8 \\beta }+ r(k), \\,\\,\\, r(k)=o(1).$ The same result holds if $\\alpha =\\beta =\\pi /2$ and additionally $B$ satisfies a local John's condition.",
"If, moreover, the walls $W$ are straight near both corners, then $r(k)=O\\left(k^{1-\\frac{\\pi }{\\beta }}\\right).$ provided $\\beta <\\pi /2$ , and $r(k)=o\\left(\\mathrm {e}^{-k/C}\\right)$ for some $C>0$ if $\\alpha =\\beta =\\pi /2$ .",
"This result will be used for our subsequent analysis of polygonal domains, see Subsection REF .",
"The analogue of Corollary REF clearly holds in the Steklov-Dirichlet case as well.",
"Remark 1.9 We believe that asymptotic formulae (REF ) and (REF ) in fact hold for any angles $\\alpha ,\\beta \\le \\pi $ .",
"Note also that if the walls are straight near the corners, our method yields a slightly better remainder estimate for the Steklov-Dirichlet problem compared to the Steklov-Neumann one.",
"We refer to the proof of Theorem REF for details."
],
[
"Outline of the approach",
"Let us sketch the main ideas of the proof of Theorem REF ; modifications needed to obtain Theorem REF are quite minor.",
"First, we observe that using domain monotonicity for sloshing eigenvalues (see [4]), one can deduce the general asymptotic expansion (REF ) from the two-term asymptotics with the remainder (REF ) for domains with straight walls near the corners.",
"Assuming that the walls are straight near the corner points, we construct the quasimodes, i.e.",
"the approximate eigenfunctions of the problem (REF ).",
"This is done by transplanting certain model solutions of the mixed Steklov-Neumann problem in an infinite angle (cf.",
"[9] where a similar idea has been implemented at a physical level of rigour).",
"These solutions are in fact of independent interest: they were used to describe “waves on a sloping beach\" (see [17], [33], [36]).",
"The approximate eigenvalues given by the first two terms on the right hand side of (REF ) are then found from a matching condition between the two model solutions transplanted to the corners $A$ and $B$ , respectively.",
"Given that the model solutions decay rapidly away from the sloshing surface, it follows that the shape of the walls away from the corners does not matter for our approximations, so in fact the domain $\\Omega $ can be viewed as a triangle with angles $\\alpha $ and $\\beta $ at the sloshing surface $S$ .",
"From the standard quasimode analysis it follows that there exist a sequence of eigenvalues of the sloshing satisfying the asymptotics (REF ).",
"A major remaining challenge is to show that this sequence is asymptotically complete (i.e.",
"that there are no other sloshing eigenvalues that have not been accounted for, see subsection REF for a formal definition), and that the enumeration of eigenvalues given by (REF ) is correct.",
"This can not be achieved by simple arguments.",
"While the set of quasimodes is a perturbation of a set of exponentials, even if one can prove the completeness of this latter set, the perturbation is too large to apply the standard Bary-Krein result (Lemma REF ) to deduce the claimed asymptotic completeness.",
"Our quasimode construction for arbitrary angles $\\alpha , \\beta < \\pi /2$ is based on the model solutions obtained by Peters [36].",
"These solutions are given in terms of some complex integrals, and while their asymptotic representations allow us to construct the quasimodes, they are not accurate enough to prove completeness.",
"However, it turns out that for angles $\\alpha =\\beta =\\pi /2q$ , $q\\in \\mathbb {N}$ , model solutions can be written down explicitly as linear combinations of certain complex exponentials.",
"Moreover, it turns out that for such angles the model solutions can be used to approximate the eigenfunctions of a Sturm-Liouville type problem of order $2q$ with Neumann boundary conditions (see Theorem REF ), for which the completeness follows from the general theory of linear ODEs.",
"The enumeration of the sloshing eigenvalues in (REF ) may then be verified by developing the approach outlined in [34].",
"Another important property used here is a peculiar duality between the Dirichlet and Neumann eigenvalues of the Sturm-Liouville problem, see Proposition REF .",
"Once we have proven completeness and established the enumeration of the sloshing eigenvalues in the case $\\alpha =\\beta =\\pi /2q$ , we use once again domain monotonicity together with continuous perturbation arguments to complete the proof of Theorem REF for arbitrary angles."
],
[
"An application to higher order Sturm-Liouville problems",
"Let us elaborate on the link between the sloshing problem with angles $\\alpha =\\beta =\\pi /2q$ and the higher order Sturm-Liouville type ODEs mentioned in the previous section.",
"For a given $q\\in \\mathbb {N}$ , consider an eigenvalue problem on an interval $(A,B) \\subset \\mathbb {R}$ of length $L$ : $(-1)^q U^{(2q)}(x) = \\Lambda ^{2q} U(x),$ with either Dirichlet $U^{(m)}(A)=U^{(m)}(B)=0,\\,\\, m=0,1,\\dots , q-1,$ or Neumann $U^{(m)}(A)=U^{(m)}(B)=0, \\,\\, m=q,q+1,\\dots ,2q-1,$ boundary conditions.",
"For $q=1$ we obtain the classical Sturm-Liouville equation describing vibrations of either a fixed or a free string.",
"The case $q=2$ yields the vibrating beam equation, also with either fixed or free ends.",
"It follows from general elliptic theory that the spectrum of the boundary value problems (REF ) or (REF ) for the equation (REF ) is discrete.",
"It is easy to check that all Dirichlet eigenvalues are positive, while the Neumann spectrum contains an eigenvalue zero of multiplicity $q$ ; the corrresponding eigenspace is generated by the functions $1, x,\\dots , x^{q-1}$ .",
"Let $\\underbrace{0=\\dots ...=0}_{q\\,\\, \\operatorname{times}}<(\\Lambda _{q+1})^{2q}\\le (\\Lambda _{q+2})^{2q} \\le \\dots \\nearrow \\infty $ be the spectrum of the Neumann problem (REF ).",
"Then, as shown in Proposition REF , $\\Lambda _k^D=\\Lambda _{k+q}$ , $k=1,2,\\dots $ , where $(\\Lambda _k^D)^{2q}$ are the eigenvalues of the Dirichlet problem (REF ).",
"We have the following result: Theorem 1.10 For any $q\\in \\mathbb {N}$ , the following asymptotic formula holds for the eigenvalues $\\Lambda _k$ : $\\Lambda _k L= \\pi \\left(k-\\frac{1}{2}\\right) -\\frac{\\pi q}{2} + O\\left(\\mathrm {e}^{-k/C}\\right)$ where $C>0$ is some positive constant.",
"Moreover, let $\\lambda _k$ , $k=1,2,\\dots $ , be the eigenvalues of a sloshing problem (REF ) with straight walls near the corners making equal angles $\\pi /2q$ at both corner points with the sloshing surface $S$ of length $L$ .",
"Then $\\lambda _k = \\Lambda _k + O\\left(\\mathrm {e}^{-k/C}\\right),$ and the eigenfunctions $u_k$ of the sloshing problem decrease exponentially away from the sloshing boundary $S$ .",
"Spectral asymptotics of Sturm-Liouville type problems of arbitrary order for general self-adjoint boundary conditions have been studied earlier, see [34] and references therein.",
"However, for the special case of problem (REF ) with Dirichlet or Neumann boundary conditions, formula (REF ) gives a more precise result.",
"First, (REF ) gives the asymptotics for each $\\Lambda _k$ , while earlier results yield only an asymptotic form of the eigenvalues without specifying the correct numbering.",
"Second, we get an exponential error estimate, which is an improvement upon a $O(1/k)$ that was known previously.",
"Finally, and maybe most importantly, (REF ) provides a physical meaning to the Sturm-Liouville problem (REF ) for arbitrary order $q\\ge 1$ .We also note that for $q=2$ , the Sturm-Liouville eigenfunctions $U_k(x)$ , $k=1,2,\\dots $ , are precisely the traces of the sloshing eigenfunctions $\\left.u_k\\right|_S$ on an isosceles right triangle $\\Omega =T$ such that the sloshing surface $S=(A,B)$ is its hypotenuse.",
"This fact was already known to H. Lamb back in the nineteenth century (see [31]), and Theorem REF extends the connection between higher order Sturm-Liouville problems and the sloshing problem to $q\\ge 2$ .",
"Let us also note that in a different context related to the study of photonic crystals, a connection between higher order ODEs and Steklov-type problems on domains with corners has been explored in [28]."
],
[
"Towards sharp asymptotics for Steklov eigenvalues on polygons",
"As was discussed in [14], precise asymptotics of Steklov eigenvalues on polygons and on smooth planar domains are quite different.",
"Moreover, the powerful pseudodifferential methods used in the smooth case can not be applied for polygons, since the Dirichlet-to-Neumann operator on the boundary of a non-smooth domain is not pseudodifferential.",
"It turns out that the methods of the present paper may be developed in order to investigate the Steklov spectral asymptotics on polygonal domains.",
"This is the subject of the subsequent paper [32].",
"While establishing sharp eigenvalue asymptotics for polygons requires a lot of further work, a sharp remainder estimate in Weyl's law follows immediately from Theorems REF and REF .",
"Corollary 1.11 Let $N_\\mathcal {P}(\\lambda )=\\#\\lbrace \\lambda _k < \\lambda \\rbrace $ be the counting function of Steklov eigenvalues $\\lambda _k$ on a convex polygon $\\mathcal {P}$ of perimeter $L$ .",
"Then $N_\\mathcal {P}(\\lambda )=\\frac{L}{\\pi } \\lambda + O(1).$ Remark 1.12 The asymptotic formula (REF ) improves upon the previously known error bound $o(\\lambda )$ (see [38], [1]).",
"Note that the $O(1)$ estimate for the error term in Weyl's law is optimal, since the counting function is a step-function.",
"Given a convex $n$ -gon $\\mathcal {P}$ , take an arbitrary point $O \\in \\mathcal {P}$ .",
"It can be connected with the vertices of $\\mathcal {P}$ by $n$ smooth curves having only point $O$ in common in such a way that at each vertex, the angles between the sides of the polygon and the corresponding curve are smaller than $\\pi /2$ .",
"This is clearly possible since all the angles of a convex polygon are less than $\\pi $ .",
"Let $\\mathcal {L}$ be the union of those curves.",
"Figure: A polygon with auxiliary curves ℒ\\mathcal {L}Consider two auxiliary spectral problems: in the first one, we impose Dirichlet conditions on $\\mathcal {L}$ and keep the Steklov condition on the boundary of the polygon, and in the second one we impose Neumann conditions on $\\mathcal {L}$ and keep the Steklov condition on the boundary.",
"Let $N_1(\\lambda )$ and $N_2(\\lambda )$ be, respectively, the counting functions of the first and the second problem.",
"By Dirichlet–Neumann bracketing (which works for the sloshing problems via the variational principle in the same way it does for the Laplacian) we get $N_1(\\lambda ) \\le N_\\mathcal {P}(\\lambda ) \\le N_2(\\lambda ), \\,\\,\\, \\lambda >0.$ The spectrum of the second auxiliary problem can be represented as the union of spectra of $n$ sloshing problems, while simultaneously the spectrum of the first auxiliary problem can be represented as the union of spectra of $n$ corresponding Steklov-Dirichlet problems.",
"Applying Theorems REF and REF to those problems, and transforming the eigenvalue asymptotics into the asymptotics of counting functions, we obtain $N_2(\\lambda )-N_1(\\lambda )=O(1)$ , which implies (REF ).",
"This completes the proof of the corollary.",
"Let us conclude this subsection by a result in the spirit of Theorems REF and REF that will be used in [32] in the proof of the sharp asymptotics of Steklov eigenvalues on polygons.",
"Proposition 1.13 Let $\\Omega =\\bigtriangleup ABZ$ be a triangle with angles $\\alpha , \\beta \\le \\pi /2$ at the vertices $A$ and $B$ , respectively.",
"Consider a mixed Steklov-Dirichlet-Neumann problem on this triangle with the Steklov condition imposed on $AB$ , Dirichlet condition imposed on $AZ$ and Neumann condition imposed on $BZ$ .",
"Assume that the side $AB$ has length $L$ .",
"Then the eigenvalues $\\lambda _k$ of the mixed Steklov-Dirichlet-Neumann problem on $\\bigtriangleup ABZ$ satisfy the asymptotics: $\\lambda _k L=\\pi \\left(k-\\frac{1}{2}\\right)+\\frac{\\pi ^2}{8}\\left(\\frac{1}{\\alpha }-\\frac{1}{\\beta }\\right)+O\\left(k^{1-\\frac{\\pi }{\\max (\\alpha ,2\\beta )}}\\right).$ Remark 1.14 Note that the Dirichlet condition near the vertex $A$ yields a contribution $\\dfrac{\\pi ^2}{8\\alpha }$ (with a positive sign) to the eigenvalue asymptotics, while the Neumann condition near $B$ contributes $-\\dfrac{\\pi ^2}{8\\beta }$ (with a negative sign).",
"This is in good agreement with the intuition provided by Theorems REF and REF .",
"Remark 1.15 In fact, we believe that a stronger statement than the one proposed in Remark REF is true: formula (REF ) holds for any mixed Steklov-Dirichlet-Neumann problem on a domain with a curved Steklov part $AB$ of length $L$ , curved walls $\\mathcal {W}$ , and angles $\\alpha ,\\beta <\\pi $ at $A$ and $B$ , such that the Dirichlet condition is imposed near $A$ and Neumann condition is imposed near $B$ ."
],
[
"Plan of the paper",
"In Section we use the Peters solutions of the sloping beach problem [36] to construct quasimodes for the sloshing and Steklov-Dirichlet problems on triangular domains.",
"In Section we consider the case of angles of the form $\\pi /2q$ for some positive integer $q$ .",
"In Section we first prove the completeness of this system of exponentially accurate quasimodes using a connection to higher order Sturm-Liouville eigenvalue problems.",
"In particular, we prove Theorem REF .",
"After that, we apply domain monotonicity arguments in order to prove Theorems REF and REF , as well as Propositions REF , REF and REF for triangular domains.",
"In section we extend these results to domains with curvilinear walls: first, for domains with the walls that are straight near the corners, and then to domains with general curvilinear walls.",
"In Appendix we prove Theorem REF , which is essentially based on the ideas of [36].",
"In Appendix we prove an auxiliary Proposition REF which is needed to prove Theorem REF .",
"This section draws heavily on the results of [34].",
"Some numerical examples are presented in Appendix .",
"A short list of notation is in Appendix ."
],
[
"Acknowledgments",
"The authors are grateful to Lev Buhovski for suggesting the approach used in section REF , to Rinat Kashaev for proposing an alternative route to prove Theorem REF , as well as to Alexandre Girouard and Yakar Kannai for numerous useful discussions on this project.",
"The research of L.P. was supported by by EPSRC grant EP/J016829/1.",
"The research of I.P.",
"was supported by NSERC, FRQNT, Canada Research Chairs program, as well as the Weston Visiting Professorship program at the Weizmann Institute of Science, where part of this work has been accomplished.",
"The research of D.S.",
"was supported in part by NSF EMSW21-RTG 1045119 and in part by a Faculty Summer Research Grant from DePaul University."
],
[
"The sloping beach problem",
"Let $(x,y)$ be Cartesian coordinates in $\\mathbb {R}^2$ , let $z=x+iy\\in \\mathbb {C}$ , and let $(\\rho ,\\theta )$ denote the polar coordinates $z=\\rho \\mathrm {e}^{\\mathrm {i}\\theta }$ .",
"Let $\\mathfrak {S}_{\\alpha }$ be the planar sector $-\\alpha \\le \\theta \\le 0$ .",
"Consider the following mixed Robin-Neumann problem $\\begin{dcases}\\Delta \\phi =0&\\quad \\text{in }\\mathfrak {S}_{\\alpha },\\\\\\frac{\\partial \\phi }{\\partial y}=\\phi &\\quad \\text{on }\\mathfrak {S}_{\\alpha }\\cap \\lbrace \\theta =0\\rbrace ,\\\\\\frac{\\partial \\phi }{\\partial n}=0&\\quad \\text{on }\\mathfrak {S}_{\\alpha }\\cap \\lbrace \\theta =-\\alpha \\rbrace ,\\\\\\end{dcases}$ and the mixed Robin-Dirichlet problem $\\begin{dcases}\\Delta \\phi =0&\\quad \\text{in }\\mathfrak {S}_{\\alpha },\\\\\\frac{\\partial \\phi }{\\partial y}=\\phi &\\quad \\text{on }\\mathfrak {S}_{\\alpha }\\cap \\lbrace \\theta =0\\rbrace ,\\\\\\phi =0&\\quad \\text{on }\\mathfrak {S}_{\\alpha }\\cap \\lbrace \\theta =-\\alpha \\rbrace .\\\\\\end{dcases}$ Note that there is no spectral parameter in these problems (hence the boundary conditions are called Robin rather than Steklov).",
"Our aim is to exhibit solutions of (REF ) and (REF ) decaying away from the horizontal part of the boundary with the property that $\\phi (x,0)\\rightarrow \\cos (x-\\chi )$ for some fixed $\\chi $ as $x\\rightarrow \\infty $ .",
"This problem is known as the sloping beach problem, and has a long and storied history in hydronamics, see [33] references therein.",
"It turns out that the form of the solutions depends in a delicate way on the arithmetic properties of the angle $\\alpha $ ; we will discuss this in more detail later on.",
"Let $\\mu _\\alpha =\\frac{\\pi }{2\\alpha },\\quad \\chi _{\\alpha ,N}=\\frac{\\pi }{4}(1-\\mu _\\alpha ),\\quad \\chi _{\\alpha ,D}=\\frac{\\pi }{4}(1+\\mu _\\alpha ).$ The following key result was essentially established by Peters [36]: Theorem 2.1 For any $0<\\alpha <\\pi /2$ , there exist solutions $\\phi _{\\alpha ,N}(x,y)$ and $\\phi _{\\alpha ,D}(x,y)$ of (REF ) and (REF ), respectively, and a constant $C>0$ such that: $|\\phi _{\\alpha ,N}(x,y)|$ and $|\\phi _{\\alpha ,D}(x,y)|$ are bounded on the closed sector $\\overline{\\mathfrak {S}_{\\alpha }}$ ; $\\phi _{\\alpha ,N}(x,y)=\\mathrm {e}^{y}\\cos (x-\\chi _{\\alpha ,N})+R_{\\alpha ,N}(x,y),$ where $\\left|R_{\\alpha ,N}(x,y)\\right|\\le C\\rho ^{-\\mu },\\quad \\left|\\nabla _{(x,y)}R_{\\alpha ,N}(x,y)\\right|\\le C\\rho ^{-\\mu -1};$ $\\phi _{\\alpha ,D}(x,y)=\\mathrm {e}^{y}\\cos (x-\\chi _{\\alpha ,D})+R_{\\alpha , D}(x,y),$ where $\\left|R_{\\alpha ,D}(x,y)\\right|\\le C\\rho ^{-2\\mu },\\quad \\left|\\nabla _{(x,y)}R_{\\alpha ,D}(x,y)\\right|\\le C\\rho ^{-2\\mu -1};$ if $\\mathcal {P}$ is any differential operator of order $k$ with constant coefficient 1, then as $\\rho \\rightarrow 0$ , $\\left|\\mathcal {P}\\phi _{\\alpha ,N}(x,y)\\right|=o(\\rho ^{-k}),\\quad \\left|\\mathcal {P}\\phi _{\\alpha ,D}(x,y)\\right|=O(\\rho ^{\\mu -k}),$ and for all $\\rho $ , most importantly $\\rho \\ge 1$ , $\\left|\\mathcal {P} R_{\\alpha ,N}(x,y)\\right|\\le C_{k}\\rho ^{-\\mu -k},\\quad \\left|\\mathcal {P} R_{\\alpha ,D}(x,y)\\right|\\le C_{k}\\rho ^{-2\\mu -k}.$ For $\\alpha =\\pi /2$ we have $\\phi _{\\frac{\\pi }{2},N}=\\mathrm {e}^y\\cos x, \\,\\, \\phi _{\\frac{\\pi }{2},D}=\\mathrm {e}^y \\sin x.$ Remark 2.2 Note that $R_{\\alpha ,N}(x,y)$ and $R_{\\alpha ,D}(x,y)$ are harmonic.",
"Note also that our further analysis (specifically, the asympotic completeness of quasimodes) will imply that the solutions $\\phi _{\\alpha ,N}(x,y)$ and $\\phi _{\\alpha ,D}(x,y)$ are unique in the sense that we cannot replace the particular constants $\\chi _{\\alpha ,N}$ and $\\chi _{\\alpha ,D}$ from (REF ) by any other value.",
"The construction of the solutions for both the Robin-Neumann problem and the Robin-Dirichlet problem is due to Peters [36].",
"The approach is based on complex analysis, specifically the Wiener-Hopf method.",
"The solution is written down explicitly as a complex integral, which allows us to analyse the asymptotics.",
"We have reproduced this construction, taking special care with the remainder estimates that were not worked out in [36].",
"The proof of Theorem REF is quite technical and is deferred until Appendix ."
],
[
"Quasimode analysis",
"In what follows, we focus on the proof of Theorem REF ; minor modifications required for Theorem REF will be discussed later.",
"As such we suppress all $N$ subscripts.",
"As was indicated in subsection REF , we split the proof of Theorem REF into several steps.",
"Our first objective is to prove the asymptotic expansion (REF ) with the remainder estimate (REF ) for triangular domains.",
"Convention: From now on and until subsection REF we assume that $\\Omega $ is a triangle $T=\\bigtriangleup ABZ$ .",
"The key starting idea is to glue together two scaled Peters solutions $\\pm \\phi _{\\alpha }(\\sigma x,\\sigma y)$ , one at each corner $A$ and $B$ , to construct quasimodes.",
"These Peters solutions must match, meaning that the phases of the trigonometric functions in the asymptotics of both solutions must agree.",
"Denoting those phases by $\\chi _{\\alpha }=\\chi _{\\alpha ,N}$ and $\\chi _{\\beta }=\\chi _{\\beta ,N}$ , see (REF ), we require $\\cos (\\sigma x-\\chi _{\\alpha })=\\pm \\cos ((L-x)\\sigma -\\chi _{\\beta }).$ Solving this equation immediately gives $\\sigma =\\sigma _k$ for some integer $k$ , where $\\sigma _kL=\\pi \\left(k-\\frac{1}{2}\\right)-\\frac{\\pi ^2}{8}\\left(\\frac{1}{\\alpha }+\\frac{1}{\\beta }\\right).$ This could be viewed as the quantization condition resulting in the asymptotics (REF ).",
"Let us now make this precise.",
"Let the plane wave $p_{\\sigma }(x,y)=\\mathrm {e}^{\\sigma y}\\cos (\\sigma x-\\chi _{\\alpha })$ , where $\\sigma $ is determined by the quantization condition (REF ).",
"Letting $z=(x,y)$ , we set $R_{A,\\sigma }(z):=\\phi _{\\alpha }(\\sigma z)-p_{\\sigma }(z);$ that is, $R_{A,\\sigma }(z)$ is the difference between the scaled Peters solution in the sector of angle $\\alpha $ with the vertex at $A$ , and the scaled trigonometric function.",
"Similarly, let $R_{B,\\sigma }(z):=\\phi _{\\beta }(\\sigma (L-\\overline{z}))-p_{\\sigma }(L-\\overline{z})$ be the difference between the scaled Peters solution in the sector of angle $\\beta $ with the vertex at $B$ and a scaled trigonometric function.",
"Note that $R_{A,\\sigma }(z)=R_{\\alpha }(z \\sigma )$ and similarly for $R_{B,\\sigma }(z)$ .",
"By the respective remainder estimates in Theorem REF , $|R_{A,\\sigma }(z)|&\\le C\\sigma ^{-\\mu _\\alpha }|z-A|^{-\\mu _\\alpha };&|R_{B,\\sigma }(z)|&\\le C\\sigma ^{-\\mu _\\beta }|L-\\overline{z}-B|^{-\\mu _\\beta };\\\\|\\nabla R_{A,\\sigma }(z)|&\\le C\\sigma ^{-\\mu _\\alpha }|z-A|^{-\\mu _\\alpha -1};&|\\nabla R_{B,\\sigma }(z)|&\\le C\\sigma ^{-\\mu _\\beta }|L-\\overline{z}-B|^{-\\mu _\\beta -1}.$ For $\\sigma =\\sigma _k$ satisfying the quantization condition (REF ), let us define a function $v_\\sigma ^{\\prime }$ (note that $v_\\sigma ^{\\prime }$ is not a derivative of $v_\\sigma $ but a new function, and we will follow this convention in the sequel): $v^{\\prime }_{\\sigma }(z):=p_{\\sigma }(z)+R_{A,\\sigma }(z)+R_{B,\\sigma }(z)$ (indeed, this definition is meaningful only when $\\sigma $ satisfies (REF )).",
"This is our first attempt at a quasimode.",
"The problem with this function is that, while it is harmonic, it does not satisfy the Neumann condition on $\\mathcal {W}$ .",
"However, the error is small and we can correct it.",
"Indeed, in a neighbourhood of $A$ , the function $p_{\\sigma }+R_{A,\\sigma }$ is the Peters solution and hence satisfies the Neumann condition on $AZ$ , and $\\nabla R_{B,\\sigma }$ is of order $O\\left(\\sigma ^{-\\mu _\\beta }\\right)$ , so the normal derivative of $v^{\\prime }_{\\sigma }$ is $O\\left(\\sigma ^{-\\mu _\\beta }\\right)$ .",
"A similar analysis holds in a neighbourhood of $B$ .",
"Away from both $A$ and $B$ , all three terms have gradients of magnitude $O\\left(\\sigma ^{-\\mu _\\alpha }\\right)$ (as $\\alpha \\ge \\beta $ ).",
"Therefore $\\left|\\frac{\\partial v^{\\prime }_{\\sigma }}{\\partial n}\\right|\\le C\\sigma ^{-\\mu _\\alpha }\\quad \\text{ on }\\mathcal {W}.$ In order to correct this “Neumann defect\", consider a function $\\eta _{\\sigma }$ defined as a solution of the following system: $\\begin{dcases}\\Delta \\eta _{\\sigma }=0 &\\quad \\text{in }\\Omega ;\\\\\\frac{\\partial }{\\partial n}\\eta _{\\sigma }=\\frac{\\partial }{\\partial n}v^{\\prime }_{\\sigma } &\\quad \\text{on }\\mathcal {W};\\\\\\frac{\\partial }{\\partial n}\\eta _{\\sigma }=-\\kappa _{\\sigma }\\psi &\\quad \\text{on }S,\\end{dcases}$ where $\\psi \\in C^{\\infty }(S)$ is a fixed function, supported away from the vertices, with $\\int _S \\psi =1$ , and where $\\kappa _{\\sigma }=\\int _{\\mathcal {W}}\\frac{\\partial v^{\\prime }_{\\sigma }}{\\partial n}.$ Note that the integral of the Neumann data over $\\partial \\Omega $ in (REF ) is zero; thus a solution $\\eta _{\\sigma }$ to (REF ) exists and is uniquely defined up to an additive constant.",
"It is well-known (see, for instance, [37]) that the Neumann-to-Dirichlet map is a bounded operator on $L^2_*(\\partial \\Omega )$ , the space of mean-zero $L^2$ functions on the boundary.",
"Therefore, $\\Vert \\eta _{\\sigma }\\Vert _{L^2(\\partial \\Omega )}\\le C\\sigma ^{-\\mu _\\alpha }$ , and hence $\\Vert \\eta _{\\sigma }\\Vert _{L^2(S)}\\le C\\sigma ^{-\\mu _\\alpha }.$ With this auxiliary function, we define a corrected quasimode: $v_{\\sigma }(z):=v^{\\prime }_{\\sigma }(z)-\\eta _{\\sigma }(z)=p_{\\sigma }(z)+R_{A,\\sigma }(z)+R_{B,\\sigma }(z)-\\eta _{\\sigma }(z).$ Observe that $v_{\\sigma }$ is harmonic and satisfies the homogeneous Neumann boundary condition on $\\mathcal {W}$ .",
"The key property of our new quasimodes is the following Lemma 2.3 With the notation as above, there exists a constant $C$ such that $\\left\\Vert \\mathcal {D}v_{\\sigma }-\\sigma v_{\\sigma }\\right\\Vert _{L^2(S)}\\le C\\sigma ^{1-\\mu _\\alpha },$ where $\\mathcal {D}$ is the sloshing operator (REF ).",
"Remark 2.4 Note that $\\mu _\\alpha =\\frac{\\pi }{2\\alpha }>1$ for $\\alpha <\\pi /2$ , and therefore we get quasimodes of order $O\\left(\\sigma ^{-\\delta }\\right)$ for $\\delta :=1-\\mu _\\alpha >0$ .",
"For $\\alpha \\ge \\pi /2$ one would need to modify our approach.",
"[Proof of Lemma REF ] Since $v_{\\sigma }$ is harmonic and satisfies the Neumann condition on $\\mathcal {W}$ , we have $\\mathcal {D}\\left(v_{\\sigma }|_S\\right)=\\left.\\frac{\\partial v_{\\sigma }}{\\partial n}\\right|_S.$ Consider first a region away from the vertex $A$ .",
"In this region, $\\left.\\frac{\\partial (p_{\\sigma }+R_{B,\\sigma })}{\\partial n}\\right|_S-\\sigma (p_{\\sigma }+\\left.R_{B,\\sigma })\\right|_S=0$ by Theorem REF .",
"Moreover, due to the bounds on $R_{A,\\sigma }$ in Theorem REF , $\\left|\\frac{\\partial R_{A,\\sigma }}{\\partial n}-\\sigma R_{A,\\sigma }\\right|\\le C\\sigma ^{1-\\mu _\\alpha }\\quad \\text{on }S$ pointwise, and hence the same estimate holds in $L^2(S)$ .",
"Finally, by construction of $\\eta _{\\sigma }$ , we know $\\left.\\frac{\\partial \\eta _{\\sigma }}{\\partial n}\\right|_S=-c_{\\sigma }\\psi ,\\quad |c_{\\sigma }|\\le C\\sigma ^{-\\mu _\\alpha };$ combining this with (REF ) yields $\\left\\Vert \\frac{\\partial \\eta _{\\sigma }}{\\partial n}-\\sigma \\eta _{\\sigma }\\right\\Vert _{L^2(S)}\\le C\\sigma ^{1-\\mu _\\alpha }.$ Putting everything together using the definition of $v_{\\sigma }$ gives us the required estimate away from $A$ .",
"A similar analysis shows that the same estimate holds away from $B$ , completing the proof.",
"Remark 2.5 For the Dirichlet or mixed problems on triangles, it is also possible to construct $\\eta _{\\sigma }(z)$ harmonic and satisfying (REF ) such that $v_{\\sigma }(z)$ satisfies the appropriate homogeneous boundary conditions on $\\mathcal {W}:=\\mathcal {W}_1\\cup \\mathcal {W}_2$ .",
"In this case, $\\eta _{\\sigma }$ is a solution of the following problem: $\\begin{dcases}\\Delta \\eta _{\\sigma }=0 &\\quad \\text{in }\\Omega ;\\\\\\frac{\\partial }{\\partial n}\\eta _{\\sigma }=\\frac{\\partial }{\\partial n}v^{\\prime }_{\\sigma } &\\quad \\text{on any Neumann side }\\mathcal {W}_1;\\\\\\eta _{\\sigma }=v^{\\prime }_{\\sigma } &\\quad \\text{on any Dirichlet side } \\mathcal {W}_2;\\\\\\frac{\\partial }{\\partial n}\\eta _{\\sigma }=0 &\\quad \\text{on }S.\\end{dcases}$ Indeed, solutions to such mixed problems are known to exist even on a larger class of Lipschitz domains [5].",
"The paper applies to our setting since all angles are strictly less than $\\pi $ , see the discussion in [5].",
"Specifically, since we have at least one Dirichlet side, we may use [5] to deduce that a solution $\\eta _{\\sigma }$ to (REF ) exists, is unique, and $\\Vert \\nabla \\eta _{\\sigma }\\Vert ^2_{L^2(\\partial \\Omega )}\\le C\\left(\\left\\Vert \\frac{\\partial }{\\partial n}v^{\\prime }_{\\sigma }\\right\\Vert ^2_{L^2(\\mathcal {W}_1)}+\\Vert \\nabla _{\\tan }v^{\\prime }_{\\sigma }\\Vert ^2_{L^2(\\mathcal {W}_2)}+\\Vert v^{\\prime }_{\\sigma }\\Vert ^2_{L^2(\\mathcal {W}_2)}\\right),$ where $\\nabla _{\\tan }$ denotes a tangential derivative along $\\mathcal {W}$ .",
"Since the Neumann-to-Dirichlet operator on $L^2(\\partial \\Omega )$ is bounded, again by [37], we have $\\Vert \\eta _{\\sigma }\\Vert ^2_{L^2(\\partial \\Omega )}\\le C\\left\\Vert \\frac{\\partial }{\\partial n}\\eta _{\\sigma }\\right\\Vert ^2_{L^2(\\partial \\Omega )}\\le C\\Vert \\nabla \\eta _{\\sigma }\\Vert ^2_{L^2(\\partial \\Omega )}$ and therefore $\\Vert \\nabla \\eta _{\\sigma }\\Vert ^2_{L^2(\\partial \\Omega )}+\\Vert \\eta _{\\sigma }\\Vert ^2_{L^2(\\partial \\Omega )}\\le C\\left(\\left\\Vert \\frac{\\partial }{\\partial n}v^{\\prime }_{\\sigma }\\right\\Vert ^2_{L^2(\\mathcal {W}_1)}+\\Vert \\nabla _{\\tan }v^{\\prime }_{\\sigma }\\Vert ^2_{L^2(\\mathcal {W}_2)}+\\Vert v^{\\prime }_{\\sigma }\\Vert ^2_{L^2(\\mathcal {W}_2)}\\right).$ By Theorem (REF ), the first two terms on the right hand side are actually bounded by $(C\\sigma ^{-\\mu _\\alpha -1})^2$ , and the third term is bounded by $(C\\sigma ^{-\\mu _\\alpha })^2$ .",
"Overall, we obtain $\\Vert \\nabla \\eta _{\\sigma }\\Vert ^2_{L^2(\\partial \\Omega )}+\\Vert \\eta _{\\sigma }\\Vert ^2_{L^2(\\partial \\Omega )}\\le (C\\sigma ^{-\\mu _\\alpha })^2,$ so in particular $\\Vert \\eta _{\\sigma }\\Vert _{L^2(\\partial \\Omega )}\\le C\\sigma ^{-\\mu _\\alpha }.$ The analysis then proceeds as above, and in particular the analogue of Lemma REF holds by an identical proof.",
"Going back to the Neumann setting again, we let $\\sigma _j=\\frac{1}{L}\\left(\\pi \\left(j-\\frac{1}{2}\\right)-\\frac{\\pi ^2}{8}\\left(\\frac{1}{\\alpha }+\\frac{1}{\\beta }\\right)\\right),\\quad j=1,2,\\dots ,$ as in (REF ).",
"Abusing notation slightly, let $v_j:=v_{\\sigma _j}|_S$ be the traces of the corresponding quasimodes.",
"Assume further that $\\Vert v_j\\Vert _{L^2(S)}=1$ .",
"By (REF ).",
"$\\sigma _j\\ge cj$ for some $c>0$ , and thus we have $\\Vert \\mathcal {D}v_{j}-\\sigma _j v_{j}\\Vert _{L^2(S)}\\le Cj^{\\delta },\\quad \\delta =1-\\mu _\\alpha >0.$ Now let $\\lbrace \\varphi _k\\rbrace _{k=1}^\\infty $ be an orthonormal basis of the eigenfunctions of the sloshing operator $\\mathcal {D}$ , with eigenvalues $\\lambda _j$ .",
"By completeness and orthonormality of the $\\lbrace \\varphi _k\\rbrace $ , we have, for each $j$ , $v_j=\\sum _{k=1}^{\\infty }a_{jk}\\varphi _k,\\quad a_{jk}=(v_j,\\varphi _k),\\quad \\sum _{k=1}^{\\infty } a_{jk}^2=1.$ Plugging in (REF ), we get $\\left\\Vert \\sum _{k=1}^{\\infty }a_{jk}(\\lambda _k-\\sigma _j)\\varphi _k\\right\\Vert _{L^2(S)}\\le Cj^{-\\delta }$ and hence $\\sum _{k=1}^{\\infty }a_{jk}^2(\\lambda _k-\\sigma _j)^2\\le Cj^{-2\\delta }.$ Since $\\displaystyle \\sum _{k=1}^{\\infty }a_{jk}^2=1$ , it cannot be true that $|\\lambda _k-\\sigma _j|>Cj^{-\\delta }$ for all $k$ .",
"Therefore the following Lemma holds.",
"Lemma 2.6 For each $j\\in \\mathbb {N}_0$ , there exists $k\\in \\mathbb {N}_0$ such that $|\\sigma _j-\\lambda _k|\\le Cj^{-\\delta }.$ In other words, there exists a subsequence of the spectrum that behaves asymptotically as $\\sigma _j$ , up to an error which is $O(j^{-\\delta })$ .",
"The key question that we now face is how to prove that $j=k$ , i.e., that the sequence gives the full spectrum.",
"In order to achieve this, we have to deal with several issues.",
"First, we do not know whether the quasimodes $\\lbrace v_j\\rbrace $ form a basis for $L^2(S)$ .",
"Second, we do not have good control of the errors $R_{A,\\sigma }$ and $R_{B,\\sigma }$ near their respective corners.",
"Therefore, some new ideas are needed; in particular, we need more accurate quasimodes.",
"Luckily for us, as will be shown later, we can get away with constructing such quasimodes for some angles only."
],
[
"Hanson-Lewy solutions for angles $\\pi /2q$",
"We will now construct quasimodes in the special case where both angles $\\alpha $ and $\\beta $ are equal to $\\pi /2q$ for some integer $q\\ge 2$ .",
"To do this, let us first go back to the study of waves on sloping beaches.",
"Recall that $\\mathfrak {S}_{\\alpha }$ is the planar sector with $-\\alpha \\le \\theta \\le 0$ , and let $I_1$ be the half-line $\\theta =0$ with $I_2$ the half-line $\\theta =-\\alpha $ .",
"Consider the complex variable $z$ .",
"Let $q$ be a positive integer and let $\\alpha =\\pi /2q$ , $\\xi =\\xi _q:=\\mathrm {e}^{-\\mathrm {i}\\pi /q}.$ Proposition 3.1 Suppose $g(z)$ is an arbitrary function.",
"Set $(\\mathcal {A}g)(z):=g(\\xi \\bar{z})$ .",
"Then $(g-\\mathcal {A}g)|_{I_2}=0,\\quad \\frac{\\partial }{\\partial n}(g+\\mathcal {A}g)|_{I_2}=0.$ This can be checked by direct computation.",
"The following definition is useful throughout.",
"Definition 3.2 The Steklov defect of an arbitrary function $g(z)$ on $\\mathfrak {S}_{\\alpha }$ is $\\mathtt {SD}(g):=\\left.\\left(\\frac{\\partial g}{\\partial y}-g\\right)\\right|_{I_1}.$ Note that $g(z)$ satisfies the Steklov condition with parameter 1 if and only if $\\mathtt {SD}(g)$ is identically zero.",
"For setup purposes, consider functions of the form $g(z)=\\mathrm {e}^{p\\bar{z}},\\ h(z)=\\mathrm {e}^{pz}.$ Differentiation immediately tells us that $\\ \\mathtt {SD}(g)=(-ip-1)g|_{I_1};\\quad \\mathtt {SD}(h)=(ip-1)h|_{I_1}.$ Given $g=\\mathrm {e}^{p\\bar{z}}$ , set $\\mathcal {B}g(z)=\\frac{\\mathrm {i}p+1}{\\mathrm {i}p-1}\\mathrm {e}^{pz}.$ It is immediate that Proposition 3.3 For any $g=\\mathrm {e}^{p\\bar{z}}$ , with $p\\in \\mathbb {C}$ , $\\mathtt {SD}(g+\\mathcal {B}g)=0.$ Now let $f(z)=\\mathrm {e}^{-\\mathrm {i}z}$ , and consider $f_0(z)=f(z);\\ f_1(z)=\\mathcal {A}f_0(z),\\ f_2=\\mathcal {B}f_1(z),\\ f_3=\\mathcal {A}f_2(z),\\dots .$ We then have $f_1(z)=f(\\xi \\bar{z}),\\ f_2(z)=\\eta (\\xi )f(\\xi z),$ where $\\eta (\\xi )=\\frac{\\mathrm {i}(-\\mathrm {i}\\xi )+1}{\\mathrm {i}(-\\mathrm {i}\\xi )-1}=\\frac{\\xi +1}{\\xi -1}.$ Further on, $\\begin{split}f_3(z)&=\\eta (\\xi )f(\\xi ^2\\bar{z}),\\quad f_4(z)=\\eta (\\xi )\\eta (\\xi ^2)f(\\xi ^2z),\\dots ,\\\\f_{2q-1}(z)&=\\eta (\\xi )\\eta (\\xi ^2)\\dots \\eta (\\xi ^{q-1})f(\\xi ^q\\bar{z})=\\prod _{j=1}^{q-1}\\eta (\\xi ^j)f(-\\bar{z}).\\end{split}$ Finally, set $\\upsilon _{\\alpha }(z)=\\sum _{m=0}^{2q-1}f_m(z).$ Theorem 3.4 The function $\\upsilon _{\\alpha }(z)$ defined by (REF ) is harmonic, satisfies Neumann boundary conditions on $I_2$ , and $\\mathtt {SD}(\\upsilon _{\\alpha })=0$ .",
"Since $\\upsilon _{\\alpha }(z)$ is a sum of rotated plane waves, it is harmonic.",
"It satisfies Neumann boundary conditions because we can write $\\begin{split}\\upsilon _{\\alpha }&=(f_0+f_1)+(f_2+f_3)+\\dots +(f_{2q-2}+f_{2q-1})\\\\&=(f_0+\\mathcal {A}f_0)+(f_2+\\mathcal {A}f_2)+\\dots +(f_{2q-2}+\\mathcal {A}f_{2q-2})\\end{split}$ and use Proposition REF on each term.",
"To see that it satisfies $\\mathtt {SD}(\\upsilon _{\\alpha })=0$ , write $\\begin{split}\\upsilon _{\\alpha }&=f_0+(f_1+f_2)+\\dots +(f_{2q-3}+f_{2q-2})+f_{2q-1}\\\\&=f_0+(f_1+\\mathcal {B}f_1)+\\dots +(f_{2q-3}+\\mathcal {B}f_{2q-3})+f_{2q-1}.\\end{split}$ Using Proposition REF and the linearity of $\\mathtt {SD}$ , we have $\\mathtt {SD}(\\upsilon _{\\alpha })=\\mathtt {SD}(f_0)+\\mathtt {SD}(f_{2q-1}),$ and all that remains is to show these two terms sum to zero.",
"In fact both are separately zero, because $f_0(z)=\\mathrm {e}^{-\\mathrm {i}z}=\\mathrm {e}^{-\\mathrm {i}x}\\mathrm {e}^y$ , and $\\mathtt {SD}(f_0)$ is zero by direct computation.",
"Moreover, $f_{2q-1}(z)=\\prod _{j=1}^{q-1}\\eta (\\xi ^j)\\mathrm {e}^{\\mathrm {i}\\bar{z}}=\\prod _{j=1}^{q-1}\\eta (\\xi ^j)\\mathrm {e}^{\\mathrm {i}x}\\mathrm {e}^y,$ and the same direct computation shows $\\mathtt {SD}(f_{2q-1})=0$ .",
"This completes the proof.",
"We call $\\upsilon _{\\alpha }(z)$ the Hanson-Lewy solution for the sloping beach problem with angle $\\alpha =\\pi /2q$ .",
"It is helpful to introduce the notation $\\gamma (\\xi ):=\\prod _{j=1}^{q-1}\\eta (\\xi ^j).$ Lemma 3.5 We have $\\gamma (\\xi )=\\mathrm {e}^{\\mathrm {i}\\pi (q-1)/2}$ .",
"If $q$ is even, $\\gamma (\\xi )=\\pm \\mathrm {i}$ , and if $q$ is odd, $\\gamma (\\xi )=\\pm 1$ .",
"We have $\\gamma (\\xi )=\\prod _{j=1}^{q-1}\\frac{\\mathrm {e}^{-\\mathrm {i}\\pi j/q}+1}{\\mathrm {e}^{-\\mathrm {i}\\pi j/q}-1}=\\prod _{j=1}^{q-1}\\frac{1+\\mathrm {e}^{\\mathrm {i}\\pi j/q}}{1-\\mathrm {e}^{\\mathrm {i}\\pi j/q}},$ where we have multiplied numerator and denominator by $\\mathrm {e}^{\\mathrm {i}\\pi j/q}$ .",
"Re-labeling terms in the numerator by switching $j$ for $q-j$ gives $\\gamma (\\xi )=\\prod _{j=1}^{q-1}\\frac{1+\\mathrm {e}^{\\mathrm {i}\\pi (q-j)/q}}{1-\\mathrm {e}^{\\mathrm {i}\\pi j/q}}=\\prod _{j=1}^{q-1}\\frac{1-\\mathrm {e}^{-\\mathrm {i}\\pi j/q}}{1-\\mathrm {e}^{\\mathrm {i}\\pi j/q}}=\\prod _{j=1}^{q-1}\\mathrm {e}^{-\\mathrm {i}\\pi j/q}\\frac{\\mathrm {e}^{\\mathrm {i}\\pi j/q}-1}{1-\\mathrm {e}^{\\mathrm {i}\\pi j/q}}=\\prod _{j=1}^{q-1}(-\\mathrm {e}^{-\\mathrm {i}\\pi j/q}).$ This may be rewritten as $\\gamma (\\xi )=\\prod _{j=1}^{q-1}(\\mathrm {e}^{\\mathrm {i}\\pi -i\\pi j/q})=\\exp \\left(\\mathrm {i}\\pi \\left(q-1-\\sum _{j=1}^{q-1}\\frac{j}{q}\\right)\\right)=\\exp \\left(\\mathrm {i}\\pi \\left(q-1-\\frac{q-1}{2}\\right)\\right),$ which yields the result.",
"Lemma 3.6 On $I_1$ , the Hanson-Lewy solutions $\\upsilon _{\\alpha }(z)$ are of the form $\\upsilon _{\\alpha }(x)=\\mathrm {e}^{-\\mathrm {i}x}+\\gamma (\\xi )\\mathrm {e}^{\\mathrm {i}x}+\\text{ decaying exponentials}.$ Indeed, the first and last terms are $\\mathrm {e}^{-\\mathrm {i}z}+\\gamma (\\xi )\\mathrm {e}^{\\mathrm {i}\\bar{z}}$ , and all other exponentials in the sum defining $\\upsilon _{\\alpha }$ are of the form $\\mathrm {e}^{-\\mathrm {i}\\xi ^jz}$ or $\\mathrm {e}^{-\\mathrm {i}\\xi ^j\\bar{z}}$ for $j=1,\\dots ,q-1$ .",
"On $I_1$ , $z=\\bar{z}=x$ , so we have $\\mathrm {e}^{-\\mathrm {i}\\operatorname{Re}(\\xi ^j)x}\\mathrm {e}^{\\operatorname{Im}(\\xi ^j)x}$ .",
"But $\\operatorname{Im}(\\xi ^j)=\\sin (-\\pi j/q)<0$ for $j=1,\\dots ,q-1$ , and the exponential is thus decaying.",
"This may be strengthened: Lemma 3.7 The rescaled solution $\\upsilon _{\\alpha }(\\sigma z)$ is $\\mathrm {e}^{\\mathrm {i}\\sigma z}+\\gamma (\\xi )\\mathrm {e}^{\\mathrm {i}\\sigma \\bar{z}}$ plus a remainder which is exponentially decaying in $\\sigma $ as $\\sigma \\rightarrow \\infty $ for each $z\\in \\mathfrak {S}_{\\alpha }$ .",
"The exponential decay constant is uniform over all $z\\in \\mathfrak {S}_{\\alpha }$ with $|z|=1$ .",
"Each term in the sum defining $\\upsilon _{\\alpha }(z)$ other than the first and last terms is $\\mathrm {e}^{-\\mathrm {i}\\xi ^j z}$ or $\\mathrm {e}^{-\\mathrm {i}\\xi ^j\\bar{z}}$ for $j=1,\\dots ,q-1$ .",
"Observe that $|\\mathrm {e}^{-\\mathrm {i}\\xi ^j\\sigma z}|$ decays exponentially in $\\sigma $ when $\\xi ^j z$ is in the negative half-plane, uniformly for $z$ away from the real axis.",
"Since $\\arg \\xi ^j$ is $-\\pi j/q$ for $1\\le j\\le q-1$ and $\\arg z$ is between 0 and $-\\pi /2q$ , we see that for each $z\\in \\mathfrak {S}_{\\alpha }$ , $-\\frac{\\pi }{q}\\le \\arg (\\xi ^j z)\\le -\\pi +\\frac{\\pi }{2q}.$ Thus we have the required decay and it is uniform over the set of $z\\in \\mathfrak {S}_{\\alpha }$ with $|z|=1$ .",
"A similar calculation shows that $-\\frac{\\pi }{2q}\\le \\arg (\\xi ^j\\bar{z})\\le -\\pi +\\frac{\\pi }{q},$ and the same argument applies, completing the proof.",
"Remark 3.8 A similar construction holds for the solutions of the mixed Steklov-Dirichlet problem.",
"Set $\\widetilde{f}_0=f;\\ \\widetilde{f}_1=-\\mathcal {A}\\widetilde{f}_0;\\ \\widetilde{f}_2=\\mathcal {B}\\widetilde{f}_1;\\ \\widetilde{f}_3=-\\mathcal {A}\\widetilde{f}_2,\\dots .$ Then set $\\widetilde{\\upsilon }_{\\alpha }(z)=\\sum _{e=0}^{2q-1}\\widetilde{f}_e(z).$ By Propositions REF and REF , these satisfy $\\widetilde{\\upsilon }_{\\alpha }|_{L^2}=0$ and $\\mathtt {SD}(u)=0$ .",
"They also have the same exponential decay properties.",
"Here is a key, novel, observation about the Hanson-Lewy solutions: Theorem 3.9 For each $\\alpha =\\frac{\\pi }{2q}$ , the solutions $\\upsilon _{\\alpha }(z)$ and $\\widetilde{\\upsilon }_{\\alpha }(z)$ satisfy the following properties when restricted to $I_1$ : $\\upsilon _{\\alpha }^{(m)}(0)=0,\\ m=q,\\dots ,2q-1;\\quad \\widetilde{\\upsilon }_{\\alpha }^{(m)}(0)=0,\\ m=0,\\dots ,q-1.$ Consider first the Neumann case.",
"Set $y=0$ .",
"We have $\\upsilon _{\\alpha }(x)=\\mathrm {e}^{-\\mathrm {i}x}+\\mathrm {e}^{-\\mathrm {i}\\xi x}+\\eta (\\xi )\\mathrm {e}^{-\\mathrm {i}\\xi x}+\\eta (\\xi )\\mathrm {e}^{-\\mathrm {i}\\xi ^2 x}+\\dots .$ Therefore $\\upsilon _{\\alpha }^{(m)}(0)=(-\\mathrm {i})^m(1+\\xi ^m)\\left(1+\\sum _{k=1}^{q-1}\\xi ^{km}\\prod _{j=1}^k\\eta (\\xi ^j)\\right).$ We now apply the Lemma in [33].",
"In the notation of Lewy, $\\epsilon =\\xi =\\mathrm {e}^{-\\mathrm {i}\\pi /q}$ , and the product in the definition of $f(\\xi )$ in [33] is precisely the one appearing in (REF ) times $(-1)^k$ .",
"Hence, the right hand side of (REF ) vanishes precisely for $m=q,\\dots ,2q-1$ .",
"Indeed, for $m=q$ this is obvious since $1+\\xi ^q=0$ ; for $m=q+r$ , $r=1,\\dots ,q-1$ , we have $\\xi ^{k(q+r)}=\\xi ^r(\\xi ^q)^k=(-1)^k\\xi ^{k r}$ , and the result follows from Lewy's lemma.",
"In the Dirichlet case we have $\\widetilde{\\upsilon }_{\\alpha }(x)=\\mathrm {e}^{-\\mathrm {i}x}-\\mathrm {e}^{-\\mathrm {i}\\xi x}-\\eta (\\xi )\\mathrm {e}^{-\\mathrm {i}\\xi x}+\\eta (\\xi )\\mathrm {e}^{-\\mathrm {i}\\xi ^2 x}+\\dots .$ Therefore $\\widetilde{\\upsilon }_{\\alpha }^{(m)}(0)=(-\\mathrm {i}^m)(1-\\xi ^m)\\left(1+\\sum _{k=1}^{q-1}(-1)^k\\xi ^{km}\\prod _{j=1}^{k}\\eta (\\xi ^j)\\right).$ A similar use of Lewy's lemma shows this is zero for $m=0,1,\\dots ,q-1$ , and the theorem is proved.",
"Remark 3.10 An alternative proof of this Theorem was provided to us by Rinat Kashaev [22], based on combinatorial techniques used in [23].",
"Remark 3.11 The Hanson-Lewy solutions are closely connected to a higher order Sturm-Liouville type ODE on the real line, specifically $(-1)^qU^{(2q)}=\\Lambda ^{2q}U.$ It is immediate that the Hanson-Lewy solutions $\\upsilon _{\\alpha }$ and $\\widetilde{\\upsilon }_{\\alpha }$ satisfy this equation with $\\Lambda =1$ .",
"This observation together with Theorem REF will be key for the next step of the argument."
],
[
"Exponentially accurate quasimodes for angles of the form $\\pi /2q$",
"Consider the sloshing problem on a triangle with top side of length $L=1$ (for simplicity) and with angles $\\alpha =\\beta =\\pi /2q$ , $q\\in \\mathbb {N}$ , $q$ even.",
"Suppose that $\\sigma $ satisfies the quantization condition (REF ).",
"For such $\\sigma $ , let $g_{\\sigma }(z)=\\upsilon _{\\alpha }(\\sigma z)+\\upsilon _{\\beta }^\\text{d}(\\sigma (1-\\overline{z})),$ where $\\upsilon _{\\beta }^\\text{d}$ is the sum of just the decaying exponents in $\\upsilon _{\\beta }$ (see Lemma REF ).",
"Note that the choice of $\\sigma $ yields that the oscillating exponents in $\\left(\\upsilon _{\\alpha }|_{I_1}\\right)(\\sigma x)$ and $\\left(\\upsilon _{\\beta }|_{I_1}\\right)(\\sigma (1-x))$ coincide, so there is a similar decomposition near $B$ .",
"First we view the functions $g_{\\sigma }$ as quasimodes for the sloshing operator.",
"We are in the same situation we were with the Peters solutions, but now the errors decay exponentially rather than polynomially.",
"By an identical argument to that in subsection REF , there exist functions $\\eta _{\\sigma }(z)$ satisfying (REF ) such that if we set $v_{\\sigma }(z)=g_{\\sigma }(z)-\\eta _{\\sigma }(z)$ , then we have the key quasimode estimate $\\Vert \\mathcal {D}(v_{\\sigma }|_S)-\\sigma v_{\\sigma }\\Vert _{L^2(S)}=O\\left(\\mathrm {e}^{- \\sigma /C}\\right).$ Now we view $g_{\\sigma }$ as quasimodes for an ODE problem.",
"By Theorem REF , we see that $g_{\\sigma }(x)$ is a solution of the ODE (REF ), and moreover one which satisfies up to an error $O(\\mathrm {e}^{-\\sigma /C})$ , for some $C>0$ , the self-adjoint boundary conditions $g_{\\sigma }^{(m)}(0)=g_{\\sigma }^{(m)}(1)=0,\\quad m=q,\\dots ,2q-1.$ at the ends of the interval.",
"Specifically, we claim that these functions $g_{\\sigma }(x)$ are exponentially accurate quasimodes on $[0,1]$ for the elliptic self-adjoint ODE eigenvalue problem $\\begin{dcases}(-1)^qU^{(2q)}=\\Lambda ^{2q}U\\\\U^{(m)}(0)=U^{(m)}(1)=0\\text{ for }m=q,\\dots ,2q-1.\\end{dcases}$ In order to show this, we need to run a similar argument to the one we have run in subsection REF to correct our quasimodes.",
"First, we correct the boundary values of $g_{\\sigma }$ in order for the quasimode to lie in the domain of the operator.",
"We do it as follows.",
"Let $\\Phi _m(x)$ be a function equal to $x^m$ near $x=0$ and smoothly decaying so that it is identically zero whenever $x>1/2$ .",
"Then for each $\\sigma =\\sigma _k$ , there exists a function $\\bar{\\eta }_{\\sigma }(x)=\\sum _{m=q}^{2q-1}\\left(a_m\\Phi _m(x)+b_m\\Phi _m(1-x)\\right),$ where and $a_m$ , $b_m$ are both $O\\left(\\mathrm {e}^{-\\sigma /C}\\right)$ , and where $g_{\\sigma }(x)+\\bar{\\eta }_{\\sigma }(x)$ satisfy the boundary conditions in (REF ).",
"In other words, $\\bar{v}_{\\sigma }(x)=g_{\\sigma }(x)+\\bar{\\eta }_{\\sigma }(x)$ could be used as a quasimode for the eigenvalue problem (REF ).",
"Indeed, it is easy to see that $\\Vert (-1)^q\\bar{v}_{\\sigma }^{(2q)}(x)-\\sigma ^{2q}\\bar{v}_{\\sigma }(x)\\Vert _{L^2([0,1])}=O\\left(\\mathrm {e}^{-\\sigma /C}\\right).$ The consequences of these quasimode estimates will be discussed in the next section.",
"Note that by our estimates on $\\eta $ , the functions $\\widetilde{\\eta }$ , $v_{\\sigma }$ , $g_{\\sigma }$ , and $\\widetilde{v}_{\\sigma }$ are all within $O\\left(\\mathrm {e}^{-\\sigma /C}\\right)$ in $L^2(S)$ norm, and in particular $||\\bar{v}_{\\sigma }-v_{\\sigma }||_{L^2(S)}\\le C\\mathrm {e}^{-\\sigma /C}.$"
],
[
"An abstract linear algebra result",
"In order to complete the proof of Theorem REF for triangular domains, we need to show that the family of quasimodes constructed in subsection REF is complete, and make sure that the numeration of the corresponding eigenvalues is precisely as in (REF ).",
"Unlike the general case when we were using Peters solutions, we are well equipped to do that for angles $\\pi /2q$ using exponentially accurate quasimodes.",
"We will need the following strengthening of Lemma REF , formulated as an abstract linear algebra result below.",
"Theorem 4.1 Let $D$ be a self-adjoint operator on an infinite-dimensional Hilbert space $H$ with a discrete spectrum $\\lbrace \\lambda _j\\rbrace _{j=1}^\\infty $ and a complete orthonormal basis of corresponding eigenvectors $\\lbrace \\varphi _j\\rbrace $ .",
"Suppose $\\lbrace v_j\\rbrace $ is a sequence of quasimodes (with $\\Vert v_j\\Vert _{H}=1$ ) such that $\\Vert D v_j-\\sigma _j v_j\\Vert _{H}\\le F(j),$ where $\\lim _{j\\rightarrow \\infty } F(j)=0$ .",
"Then For all $j$ , there exists $k$ such that $|\\sigma _j-\\lambda _k|\\le F(j)$ ; For each $j$ , there exists a vector $w_j$ with the following properties: $w_j$ is a linear combination of eigenvectors of $\\mathcal {D}$ with eigenvalues in the interval $[\\sigma _j-\\sqrt{F(j)},\\sigma _j+\\sqrt{F(j)}]$ ; $\\Vert w_j\\Vert _{H}=1$ ; $\\Vert w_j-v_j\\Vert _{H}\\le \\sqrt{F(j)}+\\sqrt{\\frac{F(j)}{1-F(j)}}= 2\\sqrt{F(j)}(1+o(1))$ as $j\\rightarrow \\infty $ .",
"Indeed, the first part of the statement is proved using precisely the same arguments as Lemma REF .",
"Let us prove part 2) of the theorem.",
"Set, similarly to (REF ), $v_j=\\sum _{k=1}^{\\infty }a_{jk}\\varphi _k,\\quad a_{jk}=(v_j,\\varphi _k)_{H},\\quad \\sum _{k=0}^{\\infty } a_{jk}^2=1.$ From the estimate (REF ), $F(j)^2\\ge \\sum _{k:\\ |\\lambda _k-\\sigma _j|>\\sqrt{F(j)}} a_{jk}^2(\\lambda _k-\\sigma _j)^2\\ge F(j)\\sum _{k: |\\lambda _k-\\sigma _j|>\\sqrt{F(j)}} a_{jk}^2,$ hence $\\sum \\limits _{k:\\ |\\lambda _k-\\sigma _j|>\\sqrt{F(j)}} a_{jk}^2\\le F(j)\\Rightarrow 1-F(j)\\le \\sum \\limits _{k:\\ |\\lambda _k-\\sigma _j|\\le \\sqrt{F(j)}} a_{jk}^2\\le 1.$ Letting $w_j:=\\dfrac{\\sum \\limits _{k:\\ |\\lambda _k-\\sigma _j|\\le \\sqrt{F(j)}}a_{jk}\\varphi _k}{\\sum \\limits _{k:\\ |\\lambda _k-\\sigma _j|\\le \\sqrt{F(j)}} a_{jk}^2}$ completes the proof.",
"As a consequence of (REF ) and (REF ), this Lemma may be applied to both the sloshing and ODE problems in the case where both $\\alpha $ and $\\beta $ equal $\\pi /2q$ .",
"In that case, for each $j$ sufficiently large, there exist functions $w_j$ and $W_j$ which are both within $O\\left(\\mathrm {e}^{-j/C}\\right)$ of $v_j$ in $L^2$ norm (by (REF )), and which are linear combinations of eigenfunctions for the sloshing and ODE problems, respectively, with eigenvalues within $O\\left(\\mathrm {e}^{-j/C}\\right)$ of $\\sigma _j$ .",
"Therefore there is an infinite subsequence, also denoted $w_j$ , of linear combinations of sloshing eigenfunctions which are close to linear combinations of eigenfunctions $W_j$ of the ODE.",
"The problem now is to prove asymptotic completeness and to make sure that the enumeration lines up."
],
[
"Eigenvalue asymptotics for higher order Sturm-Liouville problems",
"Our next goal is to understand the asymptotics of the eigenvalues of the higher order Sturm-Liouville problem (REF ) with Neumann boundary conditions.",
"For Dirichlet boundary conditions, the corresponding eigenvalue problem is $\\begin{dcases}(-1)^q U^{(2q)}=\\Lambda ^{2q}U\\\\ U^{(m)}(0)=U^{(m)}(1)=0, m=0,\\dots ,q-1.\\end{dcases}$ As was mentioned in subsection REF , the Dirichlet and Neumann spectra for the ODE above are related by the following proposition: Proposition 4.2 The nonzero eigenvalues of the Neumann ODE problem (REF ) are the same as those of the Dirichlet ODE problem (REF ), including multiplicity.",
"However, the kernel of (REF ) is trivial, whereas the kernel of (REF ) consists of polynomials of degree less than $q$ and therefore has dimension $q$ .",
"The kernel claims follow from direct computation.",
"For the rest, the solutions with nonzero eigenvalue of the ODE problems (REF ) and (REF ) must be of the form $U(x)=\\sum _{k=0}^{2q-1}c_k\\mathrm {e}^{\\omega _k\\Lambda x},$ where $\\lbrace \\omega _1,\\dots ,\\omega _n\\rbrace $ are the $n$ th roots of $-1$ .",
"We claim that a particular $U(x)$ satisfies the Dirichlet boundary conditions if and only if the function $\\widetilde{U}(x):=\\sum _{k=0}^{2q-1}c_k\\omega _k^q\\mathrm {e}^{\\omega _k\\Lambda x}$ satisfies the Neumann boundary conditions.",
"Indeed, this is obvious; since $\\omega _k^{2q}=1$ , for any $m$ and any $x$ , $U^{(m)}(x)=0$ if and only if $\\widetilde{U}^{(m+q)}(x)=0$ .",
"This gives a one-to-one correspondence between Dirichlet and Neumann eigenfunctions which completes the proof.",
"Remark 4.3 Proposition REF also follows from the observation that the operators corresponding to the problems (REF ) and (REF ) can be represented as $AA^*$ and $A^*A$ , where $A$ is an operator given by $A\\, U=i^q U^{(q)}$ subject to boundary conditions $U^{(m)}(0)=U^{(m)}(1)=0, m=0,\\dots ,q-1.$ The following asymptotic result may be extracted, with some extra work (see Appendix ), from a book of M. Naimark [34].",
"Note that our differential operator has order $n=2q$ , so $\\mu $ in the notation of [34] is equal to $q$ .",
"Proposition 4.4 For each $q\\in \\mathbb {N}$ , all sufficiently large eigenvalues of the boundary value problem (REF ) are given by the formulae $\\begin{split}\\left\\lbrace \\left(k-\\frac{1}{2}\\right)\\pi +O\\left(\\frac{1}{k}\\right)\\right\\rbrace _{k=K}^\\infty \\qquad &\\text{for $q$ even},\\\\\\\\\\left\\lbrace k\\pi +O\\left(\\frac{1}{k}\\right)\\right\\rbrace _{k=K}^\\infty \\qquad &\\text{for $q$ odd},\\end{split}$ for some $K\\in \\mathbb {N}$ .",
"Corollary 4.5 In particular, there exists a $J=J_q\\in \\mathbb {Z}$ so that for large enough $j$ , $\\Lambda _{j}\\sim \\sigma _{j+J}+ O\\left(\\frac{1}{j}\\right).$ This corollary is immediate from the explicit formula for $\\sigma _j$ , with the appropriate values of $\\alpha =\\beta =\\pi /2q$ .",
"Remark 4.6 Proposition REF implies that $\\lambda _j$ are simple and separated by nearly $\\pi $ for large enough $j$ .",
"However, it does not tell us right away that $J=0$ , and further work is needed to establish this.",
"The proof of Proposition REF for even $q$ is given in Appendix .",
"Remark 4.7 The argument presented in [34] is slightly different for $q$ even and $q$ odd.",
"In Appendix we shall assume that $q$ is even, which is in fact sufficient for our purposes.",
"The proof for odd $q$ is analogous and is left to the interested reader."
],
[
"Connection to sloshing eigenvalues",
"Recall now that $\\lbrace w_j\\rbrace $ are linear combinations of sloshing eigenfunctions and $\\lbrace W_j\\rbrace $ are linear combinations of ODE eigenfunctions, each with eigenvalues in shrinking intervals around $\\sigma _j$ , and each exponentially close to a quasimode.",
"As a consequence of the remark following Corollary REF , there are gaps between consecutive eigenvalues of the ODE and so for large enough $j$ there is only one eigenvalue of the ODE in each of those intervals.",
"This means that $W_j$ , for sufficiently large $j$ , must actually be an eigenfunction of the ODE, rather than a linear combination of eigenfunctions.",
"By REF , we must have $W_j=U_{j-J}$ , with eigenvalue $\\Lambda _{j-J}$ .",
"Now consider $\\lbrace w_j\\rbrace _{j\\ge N}$ and $\\lbrace W_j\\rbrace _{j\\ge N}$ , and let their spans be $X_N$ and $X_N^*$ respectively.",
"Pick $N$ large enough so that $W_j$ are eigenfunctions and so that the intervals in Theorem REF are disjoint, and also large enough so that $\\sum _{j\\ge N}\\Vert w_j-W_j\\Vert ^2_{L^2(S)}\\le \\sum _{j\\ge N}\\left(C\\mathrm {e}^{-j/C}\\right)^2<1.$ We know by ODE theory that $\\lbrace W_j\\rbrace $ form a complete orthonormal basis of $L^2(S)$ and that therefore $\\dim (X_N^*)^{\\perp }=N-1.$ Lemma 4.8 For sufficiently large $N$ , the dimension $\\dim X_N^{\\perp }=N-1$ as well.",
"This is essentially a version of the Bary-Krein lemma (see [24]) and our argument closely follows [11].",
"Define $A:X_N^{\\perp }\\rightarrow (X_N^*)^{\\perp }$ and $B:(X_N^*)^{\\perp }\\rightarrow X_N^{\\perp }$ by $Af=f-P_{X_N^{*}}f;\\ Bf=f-P_{X_N}f,$ where $P_{X_N^*}$ and $P_{X_N}$ are orthogonal projections.",
"By (REF ) we have for any $f\\in X_N^{\\perp }$ : $\\Vert P_{X_N^*}f\\Vert ^2=\\sum _{j\\ge N}|(f,w_j^*)|^2=\\sum _{j\\ge N}|(f,w_j^*-w_j)|^2\\le \\epsilon \\Vert f\\Vert ^2,\\ \\epsilon <1.$ Similarly, for any $f\\in (X_N^*)^{\\perp }$ , $\\Vert P_{X_N}f\\Vert <\\epsilon ^2\\Vert f\\Vert ^2$ .",
"Therefore, for any $f\\in X_N^{\\perp }$ , $BAf=(f-P_{X_N^*}f)-P_{X_N}(f-P_{X_N^*}f)=f-(I-P_{X_N})P_{X_N^*}f=f-P_{X_N^{\\perp }}P_{X_N^*}f,$ and hence $\\Vert (I_{X_N^{\\perp }}-BA)f\\Vert =\\Vert P_{X_N^{\\perp }}P_{X_N}^*f\\Vert \\le \\Vert P_{X_N^*}f\\Vert <\\epsilon \\Vert f\\Vert \\mbox{ for }\\epsilon <1.$ Therefore, $A$ must be injective, as otherwise $BAf=0$ for some nonzero $f$ and we get a contradiction.",
"Hence $\\dim X_N^{\\perp }\\le \\dim (X_N^{*})^{\\perp }=N-1$ .",
"Repeating the same argument with $X_N$ and $X_N^{*}$ interchanged shows that $\\dim (X_N^*)^{\\perp }\\le \\dim X_N^{\\perp }$ , and therefore that $\\dim X_N^{\\perp }=\\dim (X_N^*)^{\\perp }=N-1,$ as desired.",
"One immediate consequence of this Lemma is that the sequence $\\lbrace w_j\\rbrace _{j\\ge N}$ after a certain point contains only pure eigenfunctions.",
"Indeed, if $w_j$ is a linear combination of $k\\ge 2$ eigenfunctions, there are $k-1$ linearly independent functions generated by the same eigenfunctions, orthogonal to $w_j$ (and all other $\\lbrace w_j\\rbrace $ , since eigenfunctions are orthonormal).",
"But since $\\dim X_N^{\\perp }<\\infty $ , $k=1$ starting from some $j=N$ .",
"Without loss of generality we can pick $N$ sufficiently large so that all $w_j$ are simple for $j\\ge N$ .",
"Even more importantly, Lemma REF tells us that the sequences $\\lbrace w_j\\rbrace _{j\\ge N}$ and $\\lbrace W_j\\rbrace _{j\\ge N}$ are missing the same number of eigenfunctions, namely $N-1$ .",
"Since we know that $W_j$ corresponds to eigenvalue $\\Lambda _{j-J}=\\sigma _j+O\\left(\\frac{1}{j}\\right)$ , we have proved the following Proposition 4.9 In the case $\\alpha =\\beta =\\pi /2q$ , there exists a constant $C>0$ and an integer $J_{q}$ such that both the sloshing eigenvalues $\\lambda _j$ and the ODE eigenvalues $\\Lambda _j$ satisfy $\\lambda _j L=\\pi \\left(j+J_q-\\frac{1}{2}-\\frac{q}{2}\\right)+ O\\left(\\mathrm {e}^{-Cj}\\right) = \\Lambda _j L + O(\\mathrm {e}^{-Cj}) ,$ Remark 4.10 Under the same assumptions, asymptotics (REF ) holds for the Steklov-Dirichlet eigenvalues $\\lambda _j^D$ and the eigenvalues $\\Lambda _j^D$ of the ODE with Dirichlet boundary conditons (REF ) with $-\\frac{q}{2}$ being replaced by $+\\frac{q}{2}$ .",
"Note that the shift $J_q$ in the Dirichlet case (which a priori could be different) is the same as in the Neumann case, as immediately follows from Proposition REF ."
],
[
"Proof of Theorem ",
"We are now in a position to complete the proof of Theorem REF .",
"Given Proposition REF , it remains to show that the shift $J_q=0$ for all $q\\in \\mathbb {N}$ .",
"We prove this by induction.",
"For $q=1$ , by computing the eigenvalues of the standard (second-order) Sturm-Liouville problem, we observe that $J_1=0$ .",
"In order to make the induction step we use the domain monotonicity properties of Steklov-Neumann and Steklov-Dirichlet eigenfunctions, namely: Proposition 4.11 [4] Suppose $\\Omega _1$ and $\\Omega _2$ are two domains with the same sloshing surface $S$ , with $\\Omega _1\\subseteq \\Omega _2$ .",
"Then for all $k=1,2,\\dots $ , we have $\\lambda _k(\\Omega _1)\\le \\lambda _k(\\Omega _2)$ and $\\lambda _k^D(\\Omega _1)\\ge \\lambda _k^D(\\Omega _2)$ .",
"Let $\\Omega _q$ be an isosceles triangle with angles $\\pi /2q$ at the base.",
"We have $\\Omega _{q+1} \\subset \\Omega _q$ for all $k\\ge 1$ .",
"Assume now $J_{q+1}>J_q=0$ for some $q>0$ .",
"Then it immediately follows from (REF ) that $\\lambda _j(\\Omega _{q+1})>\\lambda _j(\\Omega _q)$ for large $j$ , which contradicts Proposition REF for Neumann eigenvalues.",
"Assuming instead that $J_{q+1}<J_q$ , we also get a contradiction, this time with Proposition REF for Dirichlet eigenvalues.",
"Therefore, $J_q=0$ for all $k=1,2,\\dots $ , and this completes the proof of Theorem REF for triangular domains.. $\\Box $ Remark 4.12 A surprising feature of this proof is that domain monotonicity for mixed Steklov eigenvalues implies new results for the eigenvalues of higher order Sturm-Liouville problems, which a priori are easier to investigate."
],
[
"Completeness of quasimodes for arbitrary triangular domains",
"We now complete the proofs of Theorems REF and REF , as well as Propositions REF and REF , for triangular domains, by taking full advantage of the domain monotonicity properties in Proposition REF .",
"Let $\\Omega _s$ , $s\\in [0,1]$ , be a continuous family of sloshing domains (not necessarily triangles) sharing a common sloshing surface $S$ with $L=1$ .",
"Assume that each $\\Omega _s$ is straight in a neighbourhood of the vertices, with angles $\\alpha (s)$ and $\\beta (s)$ .",
"Moreover, assume that $\\Omega _s$ is a monotone family, i.e.",
"that $s<t\\Rightarrow \\Omega _s\\subseteq \\Omega _t$ , and assume that $\\alpha (s)$ and $\\beta (s)$ are both less than $\\pi /2$ for all $0\\le s<1$ (possibly equaling $\\pi /2$ for $s=1$ ).",
"For the moment we specialise to the Neumann setting; denote the associated quasi-frequencies by $\\sigma _j(s)$ , and observe by the formula (REF ) that they are uniformly equicontinuous in $s$ .",
"By Lemma REF , for any $s$ and any $j$ , we know that there exist integers $k(j,s)$ such that for $j$ sufficiently large, $|\\sigma _j(s)-\\lambda _{k(j,s)}(s)|=o(1).$ By the work of Davis [7], we have a similar bound if $\\alpha (s)=\\beta (s)=\\pi /2$ .",
"The completeness property we need to show translates to the statement that $k(j,s)=j$ for all sufficiently large $j$ , for then the indices match and we have decaying bounds on $|\\sigma _j-\\lambda _j|$ , rather than just $|\\sigma _j-\\lambda _k|$ for some unknown $k$ .",
"The key lemma is the following.",
"Lemma 4.13 Suppose that $\\Omega _s$ , $s\\in [0,1]$ , is a family of sloshing domains as above.",
"Consider the Neumann case.",
"Then If $s<s^{\\prime }$ and $k(j,s^{\\prime })\\ge j$ for all sufficiently large $j$ , then there exists $N>0$ so that for all $j\\ge N$ , $k(j,s)\\ge j$ .",
"If $s<s^{\\prime }$ and $k(j,s)\\le j$ for all sufficiently large $j$ , then there exists $N>0$ so that for all $j\\ge N$ , $k(j,s^{\\prime })\\le j$ .",
"If both $k(j,0)=j$ and $k(j,1)=j$ for all sufficiently large $j$ , then for each $s\\in [0,1]$ , there exists $N>0$ so that for all $j\\ge N$ , $k(j,s)=j$ .",
"In the Dirichlet case, the same result holds if we flip the inequalities in the conclusion of the first two statements.",
"Remark 4.14 Of course, $k(j,s)$ may not be well-defined for small $j$ , as there may in some cases be more than one value that works, but if the hypotheses are satisfied for some choices of $k$ , then the conclusions must be satisfied for all choices of $k$ .",
"[Proof of Lemma REF ] The third statement is an immediate consequence of the first two (applied with $s^{\\prime }=1$ and $s=0$ respectively), so we need only to prove the first two.",
"Since the second one is practically identical to the first one, we only prove the first one here.",
"First consider the case when $s$ and $s^{\\prime }$ are such that $|\\sigma _j(s)-\\sigma _j(s^{\\prime })|$ (which is independent of $j$ ) is less than $\\pi $ , specifically less than $\\pi -\\epsilon $ for some $\\epsilon >0$ .",
"Then for sufficiently large $j$ , applying Lemma REF as in the discussion above, we can arrange both $|\\lambda _{k(j,s)}(s)-\\lambda _{k(j,s^{\\prime })}(s^{\\prime })|<\\pi -\\epsilon /2\\mbox{ and }\\lambda _{k(j-1,s^{\\prime })}(s^{\\prime })<\\lambda _{k(j,s^{\\prime })}(s^{\\prime })-(\\pi -\\epsilon /2).$ As a consequence, $\\lambda _{k(j-1,s^{\\prime })}(s^{\\prime })<\\lambda _{k(j,s)}(s).$ But by domain monotonicity applied to $\\lambda _{k(j-1,s^{\\prime })}$ , this implies that $\\lambda _{k(j-1,s^{\\prime })}(s)<\\lambda _{k(j,s)}(s),$ and therefore that $k(j,s)>k(j-1,s^{\\prime })$ .",
"Since $k(j-1,s^{\\prime })\\ge j-1$ for sufficiently large $j$ by assumption, we must have $k(j,s)>j-1$ and hence $k(j,s)\\ge j$ , since it is an integer.",
"This is what we wanted.",
"In the case where $s$ and $s^{\\prime }$ are not such that $|\\sigma _j(s)-\\sigma _j(s^{\\prime })|<\\pi $ , simply do the proof in steps: first extend to some $s_1<s$ with $|\\sigma _j(s_1)-\\sigma _j(s^{\\prime })|<\\pi $ , then to some $s_2<s_1$ , et cetera.",
"Since in all cases $|\\sigma _j(s)-\\sigma _j(s^{\\prime })|$ is finite, this process can be set up to terminate in finitely many steps, completing the proof.",
"Now we can complete the proofs of Theorems REF and REF for triangular domains.",
"It is possible to embed the triangle $\\Omega $ as $\\Omega _{1/2}$ in a continuous, nested family of sloshing domains $\\lbrace \\Omega _{s}\\rbrace _{s\\in [0,1]}$ , where $\\Omega _0$ is an isosceles triangle with angles $\\pi /2q$ for some large even $q$ , and where $\\Omega _1$ is a rectangle.",
"For $\\Omega _1$ , explicit calculations (see [4]) show that $\\lambda _kL=\\pi \\left(k-\\frac{1}{2}\\right)-\\frac{\\pi ^2}{8}\\left(\\frac{2}{\\pi }+\\frac{2}{\\pi }\\right)+o(1)=\\pi (k-1) +o(1),$ and therefore that $k(j,1)=j$ for all sufficiently large $j$ , which is our completeness property.",
"However, we have already proved completeness for $\\Omega _0$ , in Theorem REF , so $k(j,0)=j$ for all sufficiently large $j$ .",
"And we know we can construct quasimodes on $\\Omega _{1/2}$ , since it is a triangle, and therefore by Lemma REF there exist integers $k(j,1/2)$ such that $\\left|\\sigma _j-\\lambda _{k(j,1/2)}\\right|=O\\left(k^{1-\\frac{\\pi /2}{\\max \\lbrace \\alpha ,\\beta \\rbrace }}\\right).$ By statement (3) of Lemma REF , we have $k(j,1/2)=j$ for all sufficiently large $j$ .",
"Thus for all sufficiently large $k$ , $\\lambda _kL=\\sigma _kL+O\\left(k^{1-\\frac{\\pi /2}{\\max \\lbrace \\alpha ,\\beta \\rbrace }}\\right)=\\pi \\left(k-\\frac{1}{2}\\right)-\\frac{\\pi ^2}{8}\\left(\\frac{1}{\\alpha }+\\frac{1}{\\beta }\\right)+O\\left(k^{1-\\frac{\\pi /2}{\\max \\lbrace \\alpha ,\\beta \\rbrace }}\\right),$ which proves Theorem REF .",
"The proof of Theorem REF is essentially identical, with domain monotonicity going in the opposite direction, and the estimate on the error term being modified according to Theorem REF .",
"The proofs of Propositions REF and REF for triangular domains also follow in the same way, by using the model solutions (REF ) in the $\\mathfrak {S}_{\\pi /2}$ in order to construct the quasimodes."
],
[
"Completeness of quasi-frequencies: abstract setting",
"Here, we would like to formulate the results of the previous subsection on the abstract level.",
"We will not need these results in the present paper, but we will need them in the subsequent paper [32].",
"The proofs of these results are very similar to the proofs above, and we will omit them.",
"Let $\\Sigma :=\\lbrace \\sigma _j\\rbrace $ and $\\Lambda :=\\lbrace \\lambda _j\\rbrace $ , $j=1,2,...$ be two non-decreasing sequences of real numbers tending to infinity, which are called quasi-frequencies and eigenvalues, respectively.",
"Suppose that the following “quasi-frequency gap” condition is satisfied: there exists a constant $C>0$ such that $\\sigma _{j+1}-\\sigma _j>C$ for sufficiently large $j$ .",
"Definition 4.15 We say that $\\Sigma $ is asystem of quasi-frequencies approximating eigenvalues $\\Lambda $ if there exists a mapping $k:{\\mathbb {N}}\\rightarrow {\\mathbb {N}}$ such that: (i) $|\\sigma _j-\\lambda _{k(j)}| \\rightarrow 0$ ; (ii) $k(j_1)\\ne k(j_2)$ for sufficiently large and distinct $j_1$ , $j_2$ .",
"Under assumptions (i-ii) and (REF ) we obviously have that for $j$ large enough, $\\lbrace k(j)\\rbrace $ is a strictly increasing sequence of natural numbers.",
"Therefore, if we denote $m(j):=k(j)-j$ , then the sequence $\\lbrace m(j)\\rbrace $ is non-decreasing for $j$ large enough, and, therefore, it is converging (possibly, to $+\\infty $ ).",
"Put $M=M(\\Sigma ,\\Lambda ):=\\lim _{j\\rightarrow \\infty } m(j).$ Definition 4.16 We say that the system of quasi-frequencies $\\Sigma $ is asymptotically complete in $\\Lambda $ if $M\\ne +\\infty $ .",
"Now we assume that we have one-parameter families $\\Sigma (s)=\\lbrace \\sigma _{j}(s)\\rbrace $ and $\\Lambda (s)=\\lbrace \\lambda _{j}(s)\\rbrace $ , $s\\in [0,1]$ of quasi-frequencies and eigenvalues.",
"Lemma 4.17 Suppose the family $\\Sigma (s)$ is equicontinuous and the family $\\Lambda (s)$ is monotone increasing (i.e.",
"each $\\lambda _{j}(s)$ is increasing).",
"Then for sufficiently large $j$ (uniformly in $s$ ) the function $m(j)=m(j,s)$ is non-increasing in $s$ .",
"The proof is the same as the proof of statement (i) of Lemma REF .",
"Under the conditions of the previous Lemma, the function $M=M(s)$ is non-increasing.",
"Therefore, the following alternative formulation of statement (iii) of Lemma REF can be obtained as an immediate consequence: Corollary 4.18 If $\\Sigma (0)$ is asymptotically complete in $\\Lambda (0)$ , then $\\Sigma (1)$ is asymptotically complete in $\\Lambda (1)$ .",
"Remark 4.19 With some extra work the “quasi-frequency” gap” assumption (REF ) could be weakened, and it is sufficient to require that the number of quasi-frequencies $\\sigma _j$ inside any interval of length one is bounded.",
"More details on that will be provided in [32]."
],
[
"Proof of Proposition ",
"Using a similar strategy, let us now prove the asymptotics (REF ) for the mixed Steklov-Dirichlet-Neumann problem on triangles.",
"First, assume that $\\alpha =\\pi /2$ .",
"Consider a sloshing problem on the doubled isosceles triangle $\\bigtriangleup B^{\\prime }BZ$ , where $B^{\\prime }$ is symmetric to $B$ with respect to $A$ .",
"Using the odd-even decomposition of eigenfunctions with respect to symmetry across $AZ$ , we see that the spectrum of the sloshing problem on $\\bigtriangleup B^{\\prime }BZ$ is the union (counting multiplicity) of the eigenvalues of the sloshing problem and of the mixed Steklov-Dirichlet-Neumann problem on $\\bigtriangleup ABZ$ .",
"Given that we have already computed the asymptotics of the two former problems, it is easy to check that the spectral asymptotics of the latter problem satisfies (REF ).",
"A similar reflection argument works in the case when $\\beta =\\pi /2$ and $\\alpha $ is arbitrary.",
"This proves the proposition when either of the angles is equal to $\\pi /2$ .",
"Let now $0<\\alpha ,\\beta \\le \\pi /2$ be arbitrary.",
"Choose two additional points $X$ and $Y$ such that $X$ lies on the continuation of the segment $BZ$ , $Y$ lies on the continuation of the segment $AZ$ , and $ \\angle XAB=\\angle YBA = \\pi /2$ .",
"Consider a family of triangles $\\Omega _s=A BP_s$ , $0 \\le s\\le 1$ , such that $P_0 =X$ , $P_1=Y$ , and as $s$ changes, the point $P_s$ continuously moves along the union of segments $[X,Z]\\cup [Z,Y]$ .",
"In particular, $X=P_0$ , $Y=P_1$ , and $Z=P_{s_0}$ for some $0<s_0<1$ .",
"We impose the Dirichlet condition on $[A,P_s]$ and the Neumann condition on $[B,P_s]$ .",
"The Steklov condition on $(A,B)$ does not change.",
"Figure: Construction of auxiliary triangles in the case α=π/2\\alpha =\\pi /2 (left) and α<π/2\\alpha <\\pi /2 (right)For each of the triangles $\\Omega _s$ one can construct a system of quasimodes similarly to the sloshing or Steklov-Dirichlet case; the only difference is that in this case we use the Dirichlet Peters solutions in one angle and the Neumann Peters solutions in the other.",
"It remains to show that this system is complete.",
"In order to do that we use the same approach as in Lemma REF .",
"A simple modification of the argument for the domain monotonicity of the sloshing and Steklov-Dirichlet eigenvalues (see [4]) yields that for all $k\\ge 1$ , $\\lambda _k(\\Omega _s)\\le \\lambda _k(\\Omega _{s^{\\prime }})$ if $s\\le s^{\\prime }$ .",
"Indeed, if $P_s \\in [X,Z]$ we can continue any trial function on a smaller triangle by zero across $AP_s$ to get a trial function on a larger triangle, which means that the eigenvalues increase with $s$ .",
"Similarly, if $P_s\\in [Z,Y]$ , we can always restrict a trial function on a larger triangle to a smaller triangle, and this procedure decreases the Dirichlet energy without changing the denominator in the Rayleigh quotient.",
"Hence, the eigenvalues increase with $s$ in this case as well.",
"Since we have already proved (REF ) for right triangles, in the notation of Lemma REF it means that $k(j,0)=k(j,1)=j$ for all sufficiently large $j$ .",
"Following the same logic as in this lemma we get that $k(j,s)=s$ for any $s$ , in particular, for $s=s_0$ .",
"This completes the proof of the proposition.",
"$\\Box $"
],
[
"Domains which are straight near the surface",
"Suppose $\\Omega $ is a sloshing domain whose sides are straight in a neighbourhood of $A$ and $B$ .",
"Let $T$ be the sloshing triangle with the same angles $\\alpha $ and $\\beta $ as $\\Omega $ ; from the results in the previous Sections we understand sloshing eigenvalues and eigenfunctions on $T$ very well.",
"As before, denote the sloshing eigenfunctions on $T$ by $\\lbrace u_j(z)\\rbrace $ , with eigenvalues $\\lambda _j$ .",
"Further, denote the original uncorrected quasimodes on $T$ , as in (REF ), by $\\lbrace v^{\\prime }_{\\sigma _j}(z)\\rbrace $ , with quasi-eigenvalues $\\sigma _j$ .",
"Our strategy will be to use the quasimodes $v^{\\prime }_{\\sigma _j}$ on $T$ , cut off appropriately and modified slightly, as quasimodes for sloshing on $\\Omega $ .",
"Indeed, let $\\chi (z)$ be a cut-off function on $\\Omega $ with the following properties, which are easy to arrange: $\\chi (z)$ is equal to 1 in a neighbourhood of $S$ ; $\\chi (z)$ is supported on $\\Omega \\cap T$ and in particular is zero wherever the boundaries of $\\Omega $ are not straight; At every point $z\\in \\partial \\Omega $ , $\\nabla \\chi (z)$ is orthogonal to the normal $n(z)$ to $\\partial \\Omega $ .",
"Now define a set of functions on $\\Omega $ , by $\\widetilde{v}^{\\prime }_{\\sigma _j}(z):=\\chi (z)v^{\\prime }_{\\sigma _j}(z).$ The functions $\\widetilde{v}^{\\prime }_{\\sigma _j}(z)$ do not quite satisfy Neumann conditions on the boundary, for the same reason the $v_{\\sigma _j}(z)$ do not; we will have to correct them.",
"Note, however, that since $\\nabla \\chi (z)\\cdot n(z)=0$ , $\\frac{\\partial }{\\partial n}\\widetilde{v}^{\\prime }_{\\sigma _j}(z)=\\chi (z)\\frac{\\partial }{\\partial n}\\widetilde{v}^{\\prime }_{\\sigma _j}(z),\\qquad z\\in \\mathcal {W},$ and hence, analogously to (REF ), $\\Vert \\frac{\\partial }{\\partial n}\\widetilde{v}^{\\prime }_{\\sigma _j}\\Vert _{C^0(W)}\\le C\\sigma _j^{-\\mu }\\quad \\text{ on }\\mathcal {W}.$ We correct these in precisely the same way as before, by adding a function $\\widetilde{\\eta }_{\\sigma _j}$ which is harmonic and which has Neumann data bounded everywhere in $C^0$ norm by $C\\sigma _j^{-\\mu }$ .",
"As before, since the Neumann-to-Dirichlet map is bounded on $L^2_*(S)$ , we have $\\Vert \\frac{\\partial }{\\partial y}\\widetilde{\\eta }_{\\sigma _j}\\Vert _{L^2(S)}+\\Vert \\widetilde{\\eta }_{\\sigma _j}\\Vert _{L^2(S)}\\le C\\sigma _j^{-\\mu }.$ Then we define corrected quasimodes $\\widetilde{v}_{\\sigma _j}(z):=\\widetilde{v}^{\\prime }_{\\sigma _j}(z)-\\widetilde{\\eta }_{\\sigma _j}(z),$ which now satisfy Neumann boundary conditions.",
"We may also compute their Laplacian.",
"Since $v^{\\prime }_{\\sigma _j}(z)$ are harmonic functions and $\\widetilde{\\eta }_{\\sigma _j}(z)$ are also harmonic, we have $\\Delta \\widetilde{v}_{\\sigma _j}(z)=\\Delta \\widetilde{v}^{\\prime }_{\\sigma _j}(z)=(\\Delta \\chi (z))v^{\\prime }_{\\sigma _j}(z)+2\\nabla \\chi (z)\\cdot \\nabla v^{\\prime }_{\\sigma _j}(z).$ Therefore $\\Delta \\widetilde{v}_{\\sigma _j}(z)$ is supported only where $\\nabla \\chi (z)$ is nonzero.",
"For later use, we need to estimate the $L^2$ norm of $\\Delta \\widetilde{v}_{\\sigma _j}(z)$ .",
"It is immediate from the formula (REF ) that for some universal constant $C$ depending on the cutoff function, $\\Vert \\Delta \\widetilde{v}_{\\sigma _j}(z)\\Vert _{L^2(\\Omega )}\\le C\\Vert v^{\\prime }_{\\sigma _j}(z)\\Vert _{H^1(T\\,\\cap \\,\\mbox{supp}(\\nabla \\chi ))}.$ The discussion before (REF ) shows that the $C^1$ norm of $v^{\\prime }_{\\sigma _j}(z)$ is bounded by $C\\sigma _j^{-\\mu }$ , and therefore obviously $\\Vert v^{\\prime }_{\\sigma _j}(z)\\Vert _{H^1(T\\,\\cap \\,\\mbox{supp}(\\nabla \\chi ))}\\le C\\sigma _j^{-\\mu }.$ Thus $\\Vert \\Delta \\widetilde{v}_{\\sigma _j}(z)\\Vert _{L^2(\\Omega )}\\le C\\sigma _j^{-\\mu }.$ We would like to use $\\lbrace \\widetilde{v}_{\\sigma _j}|_S\\rbrace $ as our quasimodes.",
"However, since $\\widetilde{v}_{\\sigma _j}$ are not harmonic functions in the interior, we do not have $\\mathcal {D}(\\widetilde{v}_{\\sigma _j}|_S)=(\\frac{\\partial }{\\partial _y}\\widetilde{v}_{\\sigma _j})|_S$ .",
"We need to correct $\\widetilde{v}$ to be harmonic.",
"To do this, we must solve the mixed boundary value problem $\\begin{dcases} \\Delta \\widetilde{\\varphi }_{\\sigma _j}=-\\Delta \\widetilde{v}_{\\sigma _j}&\\quad \\text{ on }\\Omega ;\\\\\\frac{\\partial \\widetilde{\\varphi }_{\\sigma _j}}{\\partial n}=0&\\quad \\text{ on }\\mathcal {W};\\\\\\widetilde{\\varphi }_{\\sigma _j}=0&\\quad \\text{ on }S,\\end{dcases}$ as then we can use $\\bar{v}_{\\sigma _j}:=\\widetilde{v}_{\\sigma _j}+\\widetilde{\\varphi }_{\\sigma _j}$ as our quasimodes.",
"The key estimate we want is given by Lemma 5.1 If $\\widetilde{\\varphi }_{\\sigma _j}$ solves (REF ), then for some constant $C$ independent of $\\sigma _j$ , $\\left\\Vert \\frac{\\partial }{\\partial n}\\widetilde{\\varphi }_{\\sigma _j}\\right\\Vert _{L^2(S)}+\\Vert \\widetilde{\\varphi }_{\\sigma _j}\\Vert _{L^2(S)}\\le C\\sigma ^{-\\mu }.$ The theory of mixed boundary value problems on domains with corners is required here; originally due to Kondratiev [25], it has been developed nicely by Grisvard [16] in the setting of exact polygons.",
"We will therefore transfer our problem to an exact polygon, namely $T$ , via a conformal map.",
"Let $\\Phi :\\Omega \\rightarrow T$ be the conformal map which preserves the vertices and takes an arbitrary point $Z^{\\prime }$ on $\\mathcal {W}$ to the vertex $Z$ of the triangle $\\triangle ABZ$ .",
"Conformal maps preserve Dirichlet and Neumann boundary conditions, so the problem (REF ) becomes, with $\\varphi =\\Phi _*(\\widetilde{\\varphi })$ , $\\begin{dcases} \\Delta \\varphi _{\\sigma _j}=-|\\Phi ^{\\prime }(z)|^2\\Delta \\widetilde{v}_{\\sigma _j}(z)&\\quad \\text{ on }T ;\\\\\\frac{\\partial \\varphi _{\\sigma _j}}{\\partial n}=0&\\quad \\text{ on }\\mathcal {W}_T;\\\\\\varphi _{\\sigma _j}=0&\\quad \\text{ on }S_T.\\end{dcases}$ By [16], there is indeed a unique solution $\\varphi _{\\sigma _j}(z)\\in H^1(T)$ to (REF ).",
"Moreover, in this setting, the solution is actually in $H^2(T)$ .",
"Indeed, since all of the angles of $T$ are less than $\\pi /2$ and in particular we do not have any corners of angle exactly $\\pi /2$ , [16] applies.",
"It tells us that $\\varphi _{\\sigma _j}(z)$ equals an element of $H^2(T)$ plus a linear combination of explicit separated-variables solutions at the corners, which Grisvard denotes $S_{j,m}$ .",
"However, consulting the definition of $S_{j,m}$ in of [16], it is immediate that each $S_{j,m}$ is itself in $H^2(T)$ (see also [16]) and therefore $\\varphi _{\\sigma _j}(z)\\in H^2(T)$ .",
"Now that (REF ) has a solution $\\varphi _{\\sigma _j}(z)\\in H^2(T)$ , we may apply the a priori estimate [16].",
"It tells us that $\\Vert \\varphi _{\\sigma _j}\\Vert _{H^2(T)}\\le C\\left(\\Vert \\Delta \\varphi _{\\sigma _j}\\Vert _{L^2(T)}+\\Vert \\varphi _{\\sigma _j}\\Vert _{L^2(T)}\\right)\\le C\\Vert \\Delta \\varphi _{\\sigma _j}\\Vert _{L^2(T)},$ where the second inequality follows from the fact that the first eigenvalue of of the Laplacian on $T$ with Dirichlet conditions on $S_T$ and Neumann conditions on $\\mathcal {W}_T$ is positive.",
"But since $\\varphi _{\\sigma _j}(z)$ solves (REF ), $\\Vert \\varphi _{\\sigma _j}\\Vert _{H^2(T)}\\le C\\Vert \\,|\\Phi ^{\\prime }(z)|\\Delta \\widetilde{v}_{\\sigma _j}(z)\\Vert _{L^2(T)}.$ Now observe that $|\\Phi ^{\\prime }(z)|$ is certainly smooth and bounded on the support of $\\Delta \\widetilde{v}_{\\sigma _j}(z)$ , since that support is away from the vertices.",
"Therefore, by (REF ), $\\Vert \\varphi _{\\sigma _j}(z)\\Vert _{H^2(T)}\\le C\\sigma ^{-\\mu }.$ This can now be used to complete the proof.",
"By (REF ), using the definition of Sobolev norms and the trace restriction Theorem, $\\left\\Vert \\frac{\\partial }{\\partial y}\\varphi _{\\sigma _j}\\right\\Vert _{L^2(S)}\\le \\left\\Vert \\frac{\\partial }{\\partial y}\\varphi _{\\sigma _j}\\right\\Vert _{H^{1/2}(S)}\\le C\\left\\Vert \\frac{\\partial }{\\partial y}\\varphi _{\\sigma _j}\\right\\Vert _{H^1(T)}\\le C\\Vert \\varphi _{\\sigma _j}\\Vert _{H^2(T)}\\le C\\sigma ^{-\\mu }.$ All that remains to prove Lemma REF is to undo the conformal map.",
"By [35] (technically the local version, but the proof is local), since $\\Phi $ preserves the angles at $A$ and $B$ , $\\Phi $ is $C^{1,\\alpha }$ for any $\\alpha $ , and hence $C^{\\infty }$ , in a neighbourhood of $S$ , and $\\Phi ^{\\prime }$ is nonzero.",
"Note that $\\Phi $ may not be $C^{\\infty }$ near $Z^{\\prime }$ in particular, but we do not care about that region.",
"Since $\\Phi $ is smooth in a neighbourhood of $S$ and $\\Phi ^{\\prime }$ is nonzero, both the measure on $S$ and the magnitude of the normal derivative $\\frac{\\partial }{\\partial y}$ change only up to a constant bounded above and below, and hence, as desired, $\\left\\Vert \\frac{\\partial }{\\partial n}\\widetilde{\\varphi }_{\\sigma _j}\\right\\Vert _{L^2(S)}\\le C\\sigma _j^{-\\mu }.$ An identical argument without taking $\\frac{\\partial }{\\partial y}$ shows that $\\Vert \\widetilde{\\varphi }_{\\sigma _j}\\Vert _{L^2(S)}\\le C\\sigma _j^{-\\mu };$ in fact, the bound is actually on the $H^{3/2}(S)$ norm.",
"Hence the sum is bounded by $C\\sigma _j^{-\\mu }$ as well, which proves Lemma REF .",
"Now we claim our putative quasimodes $\\bar{v}_{\\sigma _j}$ on $\\Omega $ satisfy a quasimode estimate: Lemma 5.2 There is a constant $C$ such that $\\left\\Vert \\mathcal {D}_{\\Omega }\\left(\\left.\\bar{v}_{\\sigma _j}\\right|_S\\right)-\\sigma _j \\bar{v}_{\\sigma _j}\\right\\Vert _{L^2(S)}\\le C\\sigma _j^{1-\\mu }.$ Indeed, $\\bar{v}_{\\sigma _j}$ are harmonic and satisfy Neumann conditions on $\\mathcal {W}$ , so we know that $\\mathcal {D}_{\\Omega }\\left(\\left.\\bar{v}_{\\sigma _j}\\right|_S\\right)=\\left.\\frac{\\partial }{\\partial y}\\bar{v}_{\\sigma _j}\\right|_S.$ As a result, $\\mathcal {D}_{\\Omega }\\left(\\left.\\bar{v}_{\\sigma _j}\\right|_S\\right)-\\sigma _j \\bar{v}_{\\sigma _j}=\\left.\\frac{\\partial }{\\partial y}\\widetilde{v}_{\\sigma _j}\\right|_S-\\sigma _j\\widetilde{v}_{\\sigma _j}|_S+\\left.\\frac{\\partial }{\\partial y}\\widetilde{\\varphi }_{\\sigma _j}\\right|_S-\\sigma _j\\widetilde{\\varphi }_{\\sigma _j}|_S,$ which by definition of $\\widetilde{v}_j{\\sigma _j}$ is $\\left.\\frac{\\partial }{\\partial y}\\widetilde{v}^{\\prime }_{\\sigma _j}\\right|_S-\\sigma _j\\widetilde{v}^{\\prime }_{\\sigma _j}|_S-\\left.\\frac{\\partial }{\\partial y}\\widetilde{\\eta }_{\\sigma _j}\\right|_S+\\sigma _j\\widetilde{\\eta }_{\\sigma _j}|_S+\\left.\\frac{\\partial }{\\partial y}\\widetilde{\\varphi }_{\\sigma _j}\\right|_S-\\sigma _j\\widetilde{\\varphi }_{\\sigma _j}|_S.$ The last two terms have $L^2$ norms which are bounded by $C\\sigma ^{1-\\mu }$ , by Lemma REF .",
"The third and fourth terms satisfy the same bound as a consequence of (REF ).",
"For the first two terms, recall that $\\widetilde{v}^{\\prime }_{\\sigma _j}=\\chi v^{\\prime }_{\\sigma _j}$ and $\\chi =1$ near $S$ , so we need to estimate the $L^2$ norm of $\\left.\\frac{\\partial }{\\partial y}v^{\\prime }_{\\sigma _j}\\right|_S-\\sigma _jv^{\\prime }_{\\sigma _j}|_S.$ However, $v^{\\prime }_{\\sigma _j}$ are the explicit uncorrected quasimodes from (REF ).",
"By the proof of Lemma (REF ), these two terms both have $L^2$ norms bounded by $C\\sigma _j^{1-\\mu }$ , completing the proof of (REF ).",
"Now Theorem REF applies immediately to tell us that near each sufficiently large quasi-frequency $\\sigma _j$ , there exists at least one sloshing eigenvalue $\\lambda _k(\\Omega )$ .",
"The problem, again, is to prove completeness."
],
[
"Completeness",
"We use the same strategy that we did for our original quasimode problem.",
"Since the strategy is the same, we omit some details.",
"First we prove completeness when $\\alpha =\\beta =\\pi /2q$ .",
"To do this, we instead use Hanson-Lewy quasimodes.",
"All the analysis we have done in the previous subsection goes through with $C\\mathrm {e}^{-\\sigma /C}$ replacing $C\\sigma ^{-\\mu }$ everywhere, and we get (REF ) with a right-hand side of $C\\mathrm {e}^{-j/C}$ .",
"So Theorem REF applies.",
"It shows in particular that there are (linear combinations of) sloshing eigenfunctions on $\\Omega $ , which we call $w_{j,\\Omega }$ , with eigenvalues exponentially close to $\\sigma _j$ and with the property that $\\Vert w_{j,\\Omega }-\\bar{v}_{\\sigma _j}\\Vert _{L^2}\\le C\\mathrm {e}^{-j/C}\\mbox{ as }j\\rightarrow \\infty .$ However $\\Vert \\bar{v}_{\\sigma _j}-\\widetilde{v}_{\\sigma _j}\\Vert _{L^2(S)}$ , $\\Vert \\widetilde{v}_{\\sigma _j}-v^{\\prime }_{\\sigma _j}\\Vert _{L^2(S)}$ , $\\Vert v^{\\prime }_{\\sigma _j}-v_{\\sigma _j}\\Vert _{L^2(S)}$ , and $\\Vert v_{\\sigma _j}-u_j\\Vert _{L^2(S)}$ are all bounded by $C\\mathrm {e}^{-j/C}$ themselves in this case, so we have in all that $\\Vert w_{j,\\Omega }-u_{j}\\Vert _{L^2}\\le C\\mathrm {e}^{-j/C}\\mbox{ as }j\\rightarrow \\infty .$ We know that $\\lbrace u_j\\rbrace $ are complete.",
"By the same argument as in Lemma REF , the sequence $\\lbrace w_{j,\\Omega }\\rbrace $ must at some point contain only pure eigenfunctions, and for suitably large $N$ the sequences $\\lbrace w_{j,\\Omega }\\rbrace _{j\\ge N}$ and $\\lbrace u_{j}\\rbrace _{j\\ge N}$ are missing the same number of eigenfunctions, which shows that $\\lambda _{j,\\Omega }=\\lambda _{j,T}+O(\\mathrm {e}^{-j/C}).$ This proves completeness, along with full exponentially accurate eigenvalue asymptotics, in the $\\alpha =\\beta =\\pi /2q$ case.",
"Finally, we use domain monotonicity to deal with general $\\alpha $ and $\\beta $ ; the argument runs exactly as in subsection REF .",
"The result of Davis [7] shows completeness in the $\\pi /2$ case, and we can sandwich our domain $\\Omega $ as $\\Omega _{1/2}$ in a continuous, nested family of sloshing domains $\\lbrace \\Omega _s\\rbrace $ that satisfy the hypotheses of Lemma REF .",
"Note that Lemma REF is stated in such a way as to apply here as well."
],
[
"General curvilinear domains",
"We now generalise further, to domains with curvilinear boundary.",
"The authors are grateful to Lev Buhovski for suggesting the approach in this subsection [6].",
"Suppose that $\\Omega $ is a sloshing domain which satisfies the following conditions: $\\Omega $ is simply connected; The walls $\\mathcal {W}$ are Lipschitz; For any $\\epsilon >0$ , there exist sloshing domains $\\Omega _{-}$ and $\\Omega _+$ with piecewise smooth boundary, with $\\Omega _{-}\\subset \\Omega \\subset \\Omega _+$ , and with the additional property that $\\Omega _{-}$ and $\\Omega _+$ are straight lines in a $\\delta -$ neighbourhood of $A$ and $B$ with vertex angles in $[\\alpha -\\epsilon ,\\min \\lbrace \\alpha +\\epsilon ,\\pi /2\\rbrace ]$ .",
"The point of this third condition is that we have already proved sloshing eigenvalue asymptotics for $\\Omega _-$ and $\\Omega _+$ , which we will need in the subsequent domain monotonicity argument.",
"This third condition is satisfied, for example, if the boundary $S$ is $C^1$ in any small neighbourhood of the vertices $A$ and $B$ and the angles are strictly less than $\\pi /2$ .",
"It is also satisfied under the “local John's condition\" (see Propositions REF and REF ) if one or both angles equals $\\pi /2$ , and makes clear why that condition is necessary: we have not proved sloshing asymptotics in the case where one or both angles are greater than $\\pi /2$ .",
"Under these conditions, we claim the asymptotics (REF ), (REF ), as well as (REF ) and (REF ) for domains satisfying the additional “local John's condition', and thereby complete the proofs of Theorems REF and REF as well as Propositions REF and REF .",
"Assume $L=1$ for simplicity.",
"Pick any $\\gamma >0$ .",
"By continuity, there exists a sufficiently small $\\epsilon $ such that for any $\\epsilon ^{\\prime }\\in [-\\epsilon ,\\epsilon ]$ , $\\left|\\frac{1}{\\alpha +\\epsilon ^{\\prime }}-\\frac{1}{\\alpha }\\right|+\\left|\\frac{1}{\\beta +\\epsilon ^{\\prime }}-\\frac{1}{\\beta }\\right|<\\frac{8\\gamma }{2\\pi ^2}.$ By condition (C3) above, there exist $\\Omega _-\\subset \\Omega $ and $\\Omega _+\\supset \\Omega $ as described.",
"By the definition of $\\epsilon $ , and the explicit form of $\\sigma _j$ , $|\\sigma _j(\\Omega _+)-\\sigma _j(\\Omega )|<\\gamma /2\\mbox{ and }|\\sigma _j(\\Omega _-)-\\sigma _j(\\Omega )|<\\gamma /2\\text{ for all }j.$ But by our previous work, we have eigenvalue asymptotics for $\\Omega _+$ and $\\Omega _-$ .",
"In particular, there exists $N$ so that for all $j\\ge N$ , $|\\lambda _j(\\Omega _+)-\\sigma _j(\\Omega _+)|<\\gamma /2\\text{ and }|\\lambda _j(\\Omega _-)-\\sigma _j(\\Omega _-)|<\\gamma /2.$ And by domain monotonicity (for Neumann — inequalities reverse for the Dirichlet case), $\\lambda _j(\\Omega _-)\\le \\lambda _j(\\Omega )\\le \\lambda _j(\\Omega _+)\\text{ for all }j.$ Putting these ingredients together, for $j\\ge N$ , $\\lambda _j(\\Omega )\\le \\lambda _j(\\Omega _+)<\\sigma _j(\\Omega _+)+\\frac{\\gamma }{2}\\le \\sigma _j(\\Omega )+\\gamma .$ Similarly, $\\lambda _j(\\Omega )\\ge \\lambda _j(\\Omega _-)>\\sigma _j(\\Omega _-)-\\frac{\\gamma }{2}\\ge \\sigma _j(\\Omega )-\\gamma .$ We conclude that for $j\\ge N$ , $|\\lambda _j(\\Omega )-\\sigma _j(\\Omega )|<\\gamma $ .",
"Since $\\gamma >0$ was arbitrary, we have the asymptotics with $o(1)$ error that we want."
],
[
"Proof of Theorem ",
"In this section, we prove Theorem REF .",
"The proof follows [36] for the most part, doing the extra work needed to prove the careful remainder estimates.",
"There is a key difference of notation: throughout, we use $\\mu _\\alpha =\\mu =\\pi /(2\\alpha )$ , where Peters uses $\\mu =\\pi /\\alpha $ ."
],
[
"Robin-Neumann problem",
"Following Peters, we complexify our problem by setting $z=\\rho \\mathrm {e}^{\\mathrm {i}\\theta }$ and look for an analytic function $f(z)$ in $\\mathfrak {S}_{\\alpha }$ with $\\phi =\\operatorname{Re}(f)$ .",
"The boundary conditions must be rewritten in terms of $f$ .",
"Using the Cauchy-Riemann equations we have $\\phi _y=\\operatorname{Re}(\\mathrm {i}f^{\\prime }(z))$ and $\\phi _x=\\operatorname{Re}(f^{\\prime }(z))$ .",
"After some algebraic transformations, which we skip for brevity, the problem (REF ) becomes $\\begin{dcases}f(z)&\\quad \\text{ is analytic in }\\mathfrak {S}_{\\alpha },\\\\\\operatorname{Re}(\\mathrm {i}f^{\\prime }(z))=\\operatorname{Re}(f(z))&\\quad \\text{ for }z\\in \\mathfrak {S}_{\\alpha } \\cap \\lbrace \\theta =0\\rbrace ,\\\\\\operatorname{Re}(\\mathrm {i}\\mathrm {e}^{-\\mathrm {i}\\alpha }f^{\\prime }(z))=0&\\quad \\text{ for }z\\in \\mathfrak {S}_{\\alpha }\\cap \\lbrace \\theta =-\\alpha \\rbrace .\\\\\\end{dcases}$"
],
[
"Peters solution",
"We now write down Peters solution, in a form due to Alker [2].",
"Note that Alker's $y$ -axis points in the opposite direction from Peters and so we have modified the expression in [2] accordingly.",
"First define the auxiliary function $I_{\\alpha }(\\zeta ):=\\frac{1}{\\pi }\\int _0^{\\zeta \\infty }\\log \\big (1+v^{-\\pi /\\alpha }\\big )\\frac{\\zeta }{v^2+\\zeta ^2}\\,\\mathrm {d}v.$ Note that $I_{\\alpha }(\\zeta )$ is defined for $\\arg (\\zeta )\\in (-\\alpha ,\\alpha )$ , which includes the real axis.",
"As in the appendix of [36], $I_{\\alpha }(\\zeta )$ has a meromorphic continuation, with finitely many branch points, to the entire complex plane, which we also call $I_{\\alpha }(\\zeta )$ .",
"Each branch point is logarithmic; there is one at the origin, and others on the unit circle in the negative real half-plane.",
"We then let $g_{\\alpha }(\\zeta )=\\exp \\left(-I_{\\alpha }(\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha })+\\log \\left(\\frac{\\zeta +\\mathrm {i}}{\\zeta }\\right)\\right).$ The function $g_{\\alpha }(\\zeta )$ is originally defined for $\\arg (\\zeta )\\in (0,2\\alpha )$ , but has a meromorphic continuation to the entire complex plane minus a single branch cut from the origin, with singularities along the portion of the unit circle outside the sector $-\\pi /2-\\alpha \\le \\arg (\\zeta )\\le \\pi /2+\\alpha $ .",
"For $\\operatorname{Re}(\\zeta )>0$ we have the representation formula [36] $g_{\\alpha }(\\zeta )=\\exp \\left(-\\frac{1}{\\pi }\\int _0^{\\infty }\\log \\left(\\frac{1-t^{-2\\mu }}{1-t^{-2}}\\right)\\frac{\\zeta }{t^2+\\zeta ^2}\\ \\mathrm {d}t\\right).$ Finally, let $P$ be a keyhole path, consisting of the union of a nearly full circle of radius 2, traversed counterclockwise, and two linear paths, one on each side of the angle $\\theta =\\pi +\\alpha /2$ .",
"Then the Peters solution is given by $f(z)=\\frac{\\mu ^{1/2}}{\\mathrm {i}\\pi }\\int _{P}\\frac{g_\\alpha (\\zeta )}{\\zeta +\\mathrm {i}}\\mathrm {e}^{z\\zeta }\\,\\mathrm {d}\\zeta .$"
],
[
"Verification of Peters solution",
"Let us verify that Peters solution, which is obviously analytic, is actually a solution by checking the boundary conditions in (REF ), beginning with the one along the real axis.",
"This follows [36].",
"By [36], $g_{\\alpha }(\\zeta )\\rightarrow 1$ as $\\zeta \\rightarrow \\infty $ (see Proposition REF for a rigorous proof), and since $\\mathrm {e}^{z\\zeta }$ is very small as $\\zeta \\rightarrow \\infty $ along $P$ , differentiation under the integral sign in (REF ) is justified.",
"We get $\\mathrm {i}f^{\\prime }(z)-f(z)=\\frac{\\mu ^{1/2}}{\\mathrm {i}\\pi }\\int _{P}\\frac{g_{\\alpha }(\\zeta )(\\mathrm {i}\\zeta -1)}{\\zeta +\\mathrm {i}}\\mathrm {e}^{z\\zeta }\\,\\mathrm {d}\\zeta =\\frac{\\mu ^{1/2}}{\\pi }\\int _{P}g_{\\alpha }(\\zeta )\\mathrm {e}^{z\\zeta }\\,\\mathrm {d}\\zeta .$ We claim this integral is pure imaginary whenever $z$ is on the positive real axis.",
"Indeed, we may shift the branch cut of $g_{\\alpha }(\\zeta )$ so that it lies along the negative real axis, and shift $P$ so that it is symmetric with respect to the real axis.",
"Observe that $g_{\\alpha }(\\zeta )$ is real whenever $\\zeta $ is real and positive.",
"By reflection, this implies that $g_{\\alpha }(\\overline{\\zeta })=\\overline{g_{\\alpha }(\\zeta )}$ for all $\\zeta $ .",
"The analogous statement is true for $\\mathrm {e}^{z\\zeta }$ and hence for $g_{\\alpha }(\\zeta )\\mathrm {e}^{z\\zeta }$ , and it follows immediately from symmetry of $P$ that (REF ) is purely imaginary.",
"Thus $\\operatorname{Re}(if^{\\prime }(z)-f(z))=0$ , as desired.",
"For the other boundary condition, we compute $\\mathrm {i}\\mathrm {e}^{-\\mathrm {i}\\alpha }f^{\\prime }(z)=\\frac{\\mu ^{1/2}}{\\pi }\\int _{P}\\frac{g_{\\alpha }(\\zeta )\\mathrm {e}^{-\\mathrm {i}\\alpha }\\zeta }{\\zeta +\\mathrm {i}}\\mathrm {e}^{z\\zeta }\\,\\mathrm {d}\\zeta ,$ and claim that this is pure imaginary when $\\arg (z)=-\\alpha $ - i.e.",
"when $z=\\rho \\mathrm {e}^{-\\mathrm {i}\\alpha }$ for some $\\rho >0$ .",
"Shifting the branch cut and contour of integration, and letting $w=\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha }$ , the integral (REF ) becomes $\\frac{\\mu ^{1/2}}{\\pi }\\int _{P_1}\\frac{g_{\\alpha }(w \\mathrm {e}^{\\mathrm {i}\\alpha })w \\mathrm {e}^{\\mathrm {i}\\alpha }}{w \\mathrm {e}^{\\mathrm {i}\\alpha }+\\mathrm {i}}\\mathrm {e}^{\\rho w}\\,\\mathrm {d}w=\\frac{\\mu ^{1/2}}{\\pi }\\int _{P_1}\\exp \\lbrace -I_{\\alpha }(w)\\rbrace \\mathrm {e}^{\\rho w}\\,\\mathrm {d}w,$ where we have chosen $P_1$ so that it is symmetric with respect to the real axis and the branch cut is along the negative real axis.",
"The boundary condition follows immediately as above, since from (REF ) $I_{\\alpha }(w)$ is positive for $w$ real and positive."
],
[
"Asymptotics of Peters solution as $z\\rightarrow \\infty $",
"Now we study the asymptotics as $z\\rightarrow \\infty $ of (REF ), following [36] but with more rigour.",
"Let, as before, $\\chi =\\chi _{\\alpha ,N}=\\frac{\\pi }{4}(1-\\mu )=\\frac{\\pi }{4}\\left(1-\\frac{\\pi }{2\\alpha }\\right).$ Theorem A.1 There exist a function $E:(0,\\pi /2)\\rightarrow \\mathbb {R}$ and a complex-valued function $R_{\\alpha }(z)$ depending on $\\alpha $ such that $f(z)=\\mathrm {e}^{E(\\alpha )}\\mathrm {e}^{-\\mathrm {i}(z-\\chi )}+R_{\\alpha }(z),$ where for any fixed $\\alpha \\in (0,\\pi /2)$ there exists a constant $C$ such that for all $z\\in \\mathfrak {S}_{\\alpha }$ , $|R_{\\alpha }(z)|\\le Cz^{-\\mu },\\ |\\nabla _z R_{\\alpha }(z)|\\le Cz^{-\\mu -1}.$ Remark A.2 A slightly stronger version of this theorem (identifying $E(\\alpha )$ ) is claimed in [2].",
"However, the proof is not given and the extraction does not seem obvious, although it is numerically clear.",
"So we prove this version, which is all we need since we may scale by an overall constant anyway.",
"Remark A.3 For the solution $\\phi =\\operatorname{Re}(f(z))$ to our original problem, we see that $\\phi (x,0)=\\mathrm {e}^{E(\\alpha )}\\cos (z-\\chi )+R_{\\alpha }(x+0\\mathrm {i}),$ which is the radiation condition we wanted in the first place.",
"The portion of the integral (REF ) along the infinite line segments decays exponentially in $|z|$ (note that $\\arg (z\\zeta )$ is between $\\pi -\\alpha /2$ and $\\pi +\\alpha /2$ ).",
"So we may deform the contour $P$ to a contour $P^{\\prime }$ consisting of the union of a circle of radius $1/2$ and line segments along each side of the branch cut.",
"This deformation passes through singularities of $g_{\\alpha }(\\zeta )/(\\zeta +\\mathrm {i})$ , one at $\\zeta =-\\mathrm {i}$ and others at $\\zeta =\\lambda _1,\\ldots ,\\lambda _m$ along the unit circle in the negative real half-plane, with $\\arg (\\lambda _j)\\notin (-\\pi /2-\\alpha ,\\pi /2+\\alpha )$ .",
"We see that $\\begin{split}f(z)&=2\\mu ^{1/2}g_{\\alpha }(-\\mathrm {i})\\mathrm {e}^{-\\mathrm {i}z}+\\frac{\\mu ^{1/2}}{\\mathrm {i}\\pi }\\int _{P^{\\prime }}\\frac{g_{\\alpha }(\\zeta )}{\\zeta +\\mathrm {i}}\\mathrm {e}^{z\\zeta }\\,\\mathrm {d}\\zeta \\\\&\\quad +\\sum _{j=1}^m\\mathrm {e}^{z\\lambda _m}2\\mu ^{1/2}\\mathop {\\operatorname{Res}}\\limits _{\\zeta =\\lambda _m}(g_{\\alpha }(\\zeta )/(\\zeta +\\mathrm {i})).\\end{split}$ Let $\\begin{split}\\widetilde{R}_{\\alpha }(z)&:=\\frac{\\mu ^{1/2}}{\\mathrm {i}\\pi }\\int _{P^{\\prime }}\\frac{g_{\\alpha }(\\zeta )}{\\zeta +\\mathrm {i}}\\mathrm {e}^{z\\zeta }\\,\\mathrm {d}\\zeta ;\\\\R_{\\alpha }(z)&:=\\widetilde{R}_{\\alpha }(z)+\\sum _{j=1}^m\\mathrm {e}^{z\\lambda _m}2\\mu ^{1/2}\\mathop {\\operatorname{Res}}\\limits _{\\zeta =\\lambda _m}(g_{\\alpha }(\\zeta )/(\\zeta +\\mathrm {i})).\\end{split}$ To prove Theorem REF we must first prove the error estimates for $R_{\\alpha }(z)$ and then compute $g_{\\alpha }(-\\mathrm {i})$ ."
],
[
"Error estimates for $R_{\\alpha }(z)$",
"Consider $R_{\\alpha }(z)$ .",
"Since $\\arg (z)\\in (-\\alpha ,0)$ , the finite sum of exponentials and its gradient decay exponentially as $|z|\\rightarrow \\infty $ .",
"The decay is actually uniform in $\\alpha $ for $\\alpha $ in any interval $(\\epsilon ,\\pi -\\epsilon )$ with $\\epsilon >0$ fixed, as the number of residues is bounded in such an interval as well.",
"Only $\\widetilde{R}_{\\alpha }(z)$ remains.",
"Using (REF ), we may write $\\widetilde{R}_{\\alpha }(z)=\\frac{\\mu ^{1/2}}{\\mathrm {i}\\pi }\\int _{P^{\\prime }}\\mathrm {e}^{z\\zeta }\\exp (-I_{\\alpha }(\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha }))\\frac{\\mathrm {d}\\zeta }{\\zeta }.$ The only singularity inside $P^{\\prime }$ is along the branch cut, including the branch point 0.",
"We see that we must understand the asymptotics of $I_{\\alpha }(\\zeta )$ as $\\zeta \\rightarrow 0$ .",
"Peters identifies the leading order term of $I_{\\alpha }(\\zeta )$ at $\\zeta =0$ as $-\\mu \\log \\zeta $ , but we need to understand the remainder and prove bounds on it and its gradient.",
"Throughout, we choose to suppress the $\\alpha $ subscripts.",
"The key is the following lemma: Lemma A.4 For $\\zeta $ in a small neighbourhood of zero, with argument up to the branch cut on either side, we have $I(\\zeta )=-\\mu \\log \\zeta +p(\\zeta )+R(\\zeta ),$ where $p(\\zeta )$ is a polynomial in $\\zeta $ with $p(0)=0$ , and where there is a constant $C$ such that $|R(\\zeta )|\\le C|\\zeta |^{-2\\mu }.$ The proof proceeds in two steps.",
"First we prove this for all $\\zeta $ with argument in a compact subset of $(-\\pi /2,\\pi /2)$ .",
"Then we use a functional relation satisfied by $I(\\zeta )$ to extend to $\\zeta $ with arguments up to and into the negative half-plane.",
"For the first step, we use the representation for $I(\\zeta )$ on p. 353 of [36].",
"From the last paragraph of this page, we have $I(\\zeta )=-\\mu \\log \\zeta +\\int _0^{1/2}\\ln (1+v^{2\\mu })\\frac{\\zeta }{v^2+\\zeta ^2}\\,\\mathrm {d}v+\\int _{1/2}^{\\infty }\\ln (1+v^{2\\mu })\\frac{\\zeta }{v^2+\\zeta ^2}\\,\\mathrm {d}v.$ The last term is convergent, as $\\ln (1+v^{2\\mu })\\sim 2\\mu \\ln v$ as $v\\rightarrow \\infty $ , and differentiation in $\\zeta $ under the integral sign is easily justified.",
"So the last term is holomorphic, and by direct substitution it is zero at $\\zeta =0$ .",
"For the second term, we use a Taylor expansion for $\\ln (1+v^{2\\mu })$ , which is convergent when $v<1$ , and obtain $\\int _0^{1/2}\\sum _{n=1}^{\\infty }(-1)^{n+1}\\frac{v^{2\\mu n}}{n}\\frac{\\zeta }{v^2+\\zeta ^2}\\,\\mathrm {d}v.$ Change variables in the integral to $w=\\zeta v$ ; we end up with $\\int _0^{1/2\\zeta }\\sum _{n=1}^{\\infty }(-1)^{n+1}\\frac{1}{n}\\zeta ^{2\\mu n}w^{2\\mu n}\\frac{1}{w^2+1}\\,\\mathrm {d}w.$ Now further break up the integral (REF ), at $w=2$ .",
"The integral from 0 to 2 is bounded in absolute value by $\\int _0^2\\sum _{n=1}^{\\infty }|\\zeta |^{2\\mu n}\\frac{2^{2\\mu n}}{n}\\,\\mathrm {d}w=2\\sum _{n=1}^{\\infty }|\\zeta |^{2\\mu n}\\frac{2^{2\\mu n}}{n}\\le 2\\sum _{n=1}^{\\infty }|2\\zeta |^{2\\mu n}=\\frac{|4\\zeta |^{2\\mu }}{1-|2\\zeta |^{2\\mu }},$ which is bounded by $C|\\zeta |^{2\\mu }$ for some universal constant $C$ and sufficiently small $|\\zeta |$ .",
"For the remainder of the integral, we use the Taylor expansion of $(w^2+1)^{-1}$ about infinity, which is valid for $w>1$ : it is $w^{-2}-w^{-4}+w^{-6}+\\dots $ , and we end up with $\\int _2^{1/2\\zeta }\\sum _{n=1}^{\\infty }(-1)^{n+1}\\frac{1}{n}\\zeta ^{2\\mu n}w^{2\\mu n}\\left(\\sum _{m=1}^{\\infty }(-1)^{m+1}w^{-2m}\\right)\\,\\mathrm {d}w.$ Now observe that $|\\zeta w|\\le \\frac{1}{2}$ on this interval and $w>2$ , so both the sums in $n$ and $m$ are absolutely convergent and in fact the double sum is absolutely convergent.",
"This justifies all rearrangements as well as term-by-term integration.",
"Our last remaining piece thus becomes $\\sum _{n=1}^{\\infty }\\sum _{m=1}^{\\infty }\\frac{(-1)^{n+m}}{n}\\zeta ^{2\\mu n}\\int _2^{1/2\\zeta }w^{2\\mu n-2m}\\,\\mathrm {d}w,$ which equals $\\begin{split}&\\qquad \\sum _{n=1}^{\\infty }\\sum _{m=1}^{\\infty }\\frac{(-1)^{n+1+m}}{n}\\zeta ^{2\\mu n}\\left.\\left(\\frac{w^{2\\mu n-2m+1}}{2\\mu n-2m+1}\\right)\\right|^{1/2\\zeta }_2\\\\&=\\sum _{n=1}^{\\infty }\\sum _{m=1}^{\\infty }\\frac{(-1)^{n+1+m}}{n(2\\mu n-2m+1)}\\zeta ^{2\\mu n}\\left(\\zeta ^{-2\\mu n+2m-1}2^{-2\\mu n+2m-1}-2^{2\\mu n-2m+1}\\right)\\\\&=\\sum _{n=1}^{\\infty }\\sum _{m=1}^{\\infty }\\frac{(-1)^{n+1+m}}{n(2\\mu n-2m+1)}\\left(\\zeta ^{2m-1}2^{-2\\mu n+2m-1}-\\zeta ^{2\\mu n}2^{2\\mu n-2m+1}\\right).\\end{split}$ The sum now has two terms.",
"The second term, with a $\\zeta ^{2\\mu n}$ , can be summed first in $m$ (obviously) and then in $n$ , and the whole thing is bounded by a multiple of the $n=1$ term, namely $C|\\zeta |^{2\\mu }$ .",
"The first term can be summed first in $n$ , at which point it becomes an expression of the form $\\sum _{m=1}^{\\infty }a_m\\zeta ^{2m-1}$ .",
"Since the $a_m$ do not grow too fast as $m\\rightarrow \\infty $ — in fact they grow as $2^{2m}$ — this expression represents a holomorphic function in a disk of sufficiently small radius about the origin.",
"Moreover this holomorphic function is zero at zero.",
"We have now shown that $I(\\zeta )$ is the sum of $-\\mu \\log \\zeta $ , a holomorphic function zero at the origin, and a term bounded in absolute value by $C|\\zeta |^{2\\mu }$ .",
"Taking the Taylor series of the holomorphic function, separating out the finitely many terms which are not $O(|\\zeta |^{2\\mu })$ , and calling them $p(\\zeta )$ completes the proof.",
"It remains to extend the argument outside the positive real half-plane.",
"Let $h(\\zeta )=I(\\zeta )+\\mu \\log \\zeta $ .",
"Then we know that $h(\\zeta )$ is holomorphic in a disk with the exception of a branch cut, and it continues across the branch cut because $I(\\zeta )$ does.",
"We also know, from the first line on p. 353 of [36], that for all $\\zeta $ away from the branch cut, $I(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })=I(\\zeta )\\frac{\\zeta \\mathrm {e}^{\\mathrm {i}\\alpha }+\\mathrm {i}}{\\zeta \\mathrm {e}^{\\mathrm {i}\\alpha }-\\mathrm {i}}.$ Applying this gives an equivalent relation for $h(\\zeta )$ , noting that $\\mathrm {e}^{2\\mathrm {i}\\alpha \\mu }=\\mathrm {e}^{\\mathrm {i}\\pi }=-1$ : $h(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })=-h(\\zeta )\\frac{\\zeta \\mathrm {e}^{\\mathrm {i}\\alpha }+\\mathrm {i}}{\\zeta \\mathrm {e}^{\\mathrm {i}\\alpha }-\\mathrm {i}}.$ The function $H(\\zeta ):=-(\\zeta \\mathrm {e}^{\\mathrm {i}\\alpha }+\\mathrm {i})/(\\zeta \\mathrm {e}^{\\mathrm {i}\\alpha }-\\mathrm {i})$ is holomorphic in the disk, and we have for all nonzero $\\zeta $ : $h(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })=h(\\zeta )H(\\zeta ).$ Recall that in the positive real half-plane (in a sector bounded away from the real axis), $h(\\zeta )=p(\\zeta )+R(\\zeta ).$ Plugging in the relation for $h(\\zeta )$ , we see that for $-\\pi /2+\\epsilon <\\arg \\zeta <\\pi /2-\\epsilon $ , $h(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })=p(\\zeta )H(\\zeta )+R(\\zeta )H(\\zeta ).$ However, we are assuming that $\\alpha <\\pi /2$ .",
"Therefore, there is a range of $\\zeta $ , namely $-\\pi /2+\\epsilon <\\arg \\zeta <\\pi /2-\\epsilon -2\\alpha $ , where $\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha }$ is in the positive half-plane and thus we have a second representation: $h(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })=p(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })+R(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha }).$ Setting the previous two equations equal, we see that for this small range of $\\zeta $ : $p(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })-p(\\zeta )H(\\zeta )=R(\\zeta )H(\\zeta )-R(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha }).$ The left-hand side is holomorphic in a disk.",
"The right-hand side is $O(|\\zeta |^{2\\mu })$ .",
"We conclude that the left-hand side must have a zero of order at least $2\\mu $ at $\\zeta =0$ .",
"Therefore, there is a $C$ such that for all $\\zeta $ in the disk, $|p(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })-p(\\zeta )H(\\zeta )|\\le C|\\zeta |^{2\\mu }.$ Therefore, for $-\\pi /2+\\epsilon <\\arg \\zeta <\\pi /2-\\epsilon $ , $|p(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })-h(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })+R(\\zeta )H(\\zeta )|\\le C|\\zeta |^{2\\mu },$ and therefore $h(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })=p(\\zeta \\mathrm {e}^{2\\mathrm {i}\\alpha })+O(|\\zeta |^{2\\mu }).$ This proves the lemma for $\\zeta $ with arguments now up to $\\pi /2-\\epsilon +2\\alpha $ .",
"Continuing this procedure, one step of size $2\\alpha $ at a time, gives the result for all $\\zeta $ with arguments in a neighbourhood of $[-\\pi ,\\pi ]$ , completing the proof.",
"Now we return to estimating the remainder.",
"Using the Lemma and recalling $h(\\zeta )=I(\\zeta )+\\mu \\log \\zeta $ , we have $\\widetilde{R}_{\\alpha }(z)=-\\frac{\\mu ^{1/2}}{\\pi }\\int _{P^{\\prime }}\\mathrm {e}^{z\\zeta }\\zeta ^{\\mu }\\mathrm {e}^{-h(\\zeta )}\\frac{d\\zeta }{\\zeta }.$ Change variables to let $w=z\\zeta $ ; for $|z|>1$ , the contour deforms smoothly back to $\\arg (z)P^{\\prime }$ (which is $P^{\\prime }$ rotated by $\\arg (z)$ ), as there are no singularities remaining inside $P^{\\prime }$ .",
"We get $\\widetilde{R}_{\\alpha }(z)=-\\frac{\\mu ^{1/2}z^{-\\mu }}{\\pi }\\int _{\\arg (z)P^{\\prime }}\\mathrm {e}^ww^{\\mu -1}\\mathrm {e}^{-h(w/z)}\\,\\mathrm {d}w.$ The contour can be deformed to two straight lines, one on either side of the branch cut.",
"Since $h(\\cdot )$ grows at most logarithmically along these lines (note that $\\arg (w/z)=\\pm \\pi +\\alpha /2$ on $\\arg (z)P^{\\prime }$ ), the integral is bounded as $z\\rightarrow \\infty $ .",
"In fact, since $h(0)=0$ , it converges to the corresponding integral with $\\mathrm {e}^{-h(w/z)}$ replaced by 1.",
"Thus $|\\widetilde{R}_{\\alpha }(z)|\\le C z^{-\\mu }$ as desired.",
"We must also estimate $|\\nabla _z \\widetilde{R}_{\\alpha }(z)|$ .",
"However, $\\widetilde{R}_{\\alpha }(z)$ is holomorphic, so we just need to consider $\\partial _z\\widetilde{R}_{\\alpha }(z)$ .",
"The differentiation brings down a factor of $\\zeta $ , which becomes $w/z$ .",
"The extra power of $w$ is absorbed, and the extra power of $z$ moves outside the integral, yielding decay of the form $z^{-\\mu -1}$ instead of $z^{-\\mu }$ .",
"The same analysis works for higher order derivatives, and this proves the remainder estimation part of Theorem REF .",
"It remains only to evaluate $g_{\\alpha }(-\\mathrm {i})$ ."
],
[
"Evaluation of $g_{\\alpha }(-\\mathrm {i})$",
"We cannot simply use the formula (REF ) to evaluate $g_{\\alpha }(-\\mathrm {i})$ , because that formula is only valid for $\\zeta $ in the positive real half-plane.",
"We must instead use the fact that $\\log g_{\\alpha }(\\zeta )=\\log (\\frac{\\zeta +\\mathrm {i}}{\\zeta })-I(\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha }),$ so we are interested in understanding the behaviour of $I(\\zeta )$ near $\\zeta =-\\mathrm {i}\\mathrm {e}^{-\\mathrm {i}\\alpha }$ .",
"First we need a good representation for $I(\\zeta )$ in this region.",
"Considering the last equation on p. 350 of [36], we see that for $\\zeta $ with $-\\pi /2-2\\alpha <\\arg (\\zeta )<\\pi /2+2\\alpha $ , $\\begin{split}I(\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha })&=\\frac{1}{2\\pi \\mathrm {i}}\\left(\\int _{I_1}\\log (1+(-\\mathrm {i}u)^{-2\\mu })\\frac{\\mathrm {d}u}{u-\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha }}+\\int _{M_2}\\log (1+(\\mathrm {i}u)^{-2\\mu })\\frac{\\mathrm {d}u}{u-\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha }}\\right)\\\\&+\\log \\frac{\\zeta +\\mathrm {i}}{\\zeta }.\\end{split}$ Here $I_1$ may be chosen to be the incoming path along $\\arg (\\zeta )=\\pi /2$ and $M_2$ the outgoing path along $\\arg (\\zeta )=-\\pi /2-2\\alpha $ (each can be moved by up to $\\alpha $ in either direction before running into a singularity of the integrand, but these are the most convenient choices).",
"We would prefer that $I_1$ and $M_2$ be equal and opposite paths, but they are not.",
"So we shift the path $I_1$ to the path $L_0$ which is incoming along the ray $\\arg (\\zeta )=\\pi /2-2\\alpha $ .",
"This is done for other reasons on p. 351 of [36], and we pick up a contribution from the branch cut of the integrand.",
"Overall we get $\\begin{split}I(\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha })&=\\frac{1}{2\\pi \\mathrm {i}}\\left(\\int _{L_0}\\log (1+(-\\mathrm {i}u)^{-2\\mu })\\frac{\\mathrm {d}u}{u-\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha }}+\\int _{M_2}\\log (1+(\\mathrm {i}u)^{-2\\mu })\\frac{\\mathrm {d}u}{u-\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha }}\\right)\\\\&+\\log \\frac{\\zeta +\\mathrm {i}}{\\zeta }-\\log \\frac{\\zeta -\\mathrm {i}}{\\zeta }.\\end{split}$ Using (REF ), then plugging in $\\zeta =-i$ , shows that $\\begin{split}-\\log (g_{\\alpha }(-\\mathrm {i}))&=\\frac{1}{2\\pi \\mathrm {i}}\\left(\\int _{L_0}\\log (1+(-\\mathrm {i}u)^{-2\\mu })\\frac{\\mathrm {d}u}{u+i \\mathrm {e}^{-\\mathrm {i}\\alpha }}+\\int _{M_2}\\log (1+(\\mathrm {i}u)^{-2\\mu })\\frac{\\mathrm {d}u}{u+i \\mathrm {e}^{-\\mathrm {i}\\alpha }}\\right)\\\\&-\\log 2.\\end{split}$ Now we parametrise these integrals.",
"For the first, let $u=t\\mathrm {i}\\mathrm {e}^{-2i\\alpha }$ , and for the second, let $u=-t\\mathrm {i}\\mathrm {e}^{-2i\\alpha }$ .",
"With the appropriate signs, we find that $\\begin{split}-\\log (g_{\\alpha }(-\\mathrm {i}))&=-\\log 2+\\frac{1}{2\\pi \\mathrm {i}}\\int _0^{\\infty }\\log (1+t^{-2\\mu })\\left(\\frac{1}{t-\\mathrm {e}^{\\mathrm {i}\\alpha }}-\\frac{1}{t+\\mathrm {e}^{\\mathrm {i}\\alpha }}\\right)\\,\\mathrm {d}t\\\\&=-\\log 2+\\frac{1}{\\pi \\mathrm {i}}\\int _0^{\\infty }\\log (1+t^{-2\\mu })\\frac{\\mathrm {e}^{\\mathrm {i}\\pi /2\\mu }}{t^2-\\mathrm {e}^{\\mathrm {i}\\pi /\\mu }}\\,\\mathrm {d}t.\\end{split}$ We have now written $g_{\\alpha }(-\\mathrm {i})$ in terms of a convergent integral, that on the right of (REF ).",
"The usual Taylor series expansions show that this integral is smooth in $\\alpha $ for $\\alpha \\in (\\epsilon ,\\pi -\\epsilon )$ .",
"This integral seems nontrivial to evaluate and is not obviously in any common table of integrals.",
"Nevertheless, we can find the imaginary part of $\\log (g_{\\alpha }(-\\mathrm {i}))$ by using the following lemma.",
"Lemma A.5 For $\\mu \\in \\mathbb {R}$ , let $J(\\mu ):=\\int _0^{\\infty }\\log (1+t^{-2\\mu })\\frac{\\mathrm {e}^{\\mathrm {i}\\pi /2\\mu }}{t^2-\\mathrm {e}^{\\mathrm {i}\\pi /\\mu }}\\,\\mathrm {d}t.$ Then for each $\\mu >1/2$ , which corresponds exactly to $\\alpha <\\pi /2$ , $\\operatorname{Re}(J(\\mu ))=\\frac{\\pi ^2}{4}(1-\\mu ).$ Change variables in $J(\\mu )$ by letting $t=r^{-1}$ .",
"Then $dt=-r^{-2}dr$ , and we have $J(\\mu )=\\int _0^{\\infty }\\log (1+r^{2\\mu })\\frac{\\mathrm {e}^{\\mathrm {i}\\pi /2\\mu }}{1-r^2\\mathrm {e}^{\\mathrm {i}\\pi /\\mu }}\\,\\mathrm {d}r=-\\int _0^{\\infty }\\log (1+r^{2\\mu })\\frac{\\mathrm {e}^{-\\mathrm {i}\\pi /2\\mu }}{r^2-\\mathrm {e}^{-\\mathrm {i}\\pi /\\mu }}\\,\\mathrm {d}r.$ Breaking up the logarithm by bringing out $2\\mu \\log r$ gives $J(\\mu )=-\\int _0^{\\infty }\\log (1+r^{-2\\mu })\\frac{\\mathrm {e}^{-\\mathrm {i}\\pi /2\\mu }}{r^2-\\mathrm {e}^{-\\mathrm {i}\\pi /\\mu }}\\,\\mathrm {d}r-2\\mu \\int _0^{\\infty }\\log (r)\\frac{\\mathrm {e}^{-\\mathrm {i}\\pi /2\\mu }}{r^2-\\mathrm {e}^{-\\mathrm {i}\\pi /\\mu }}\\,\\mathrm {d}r.$ But we now recognise the first integral as the complex conjugate of $J(\\mu )$ .",
"Therefore $\\operatorname{Re}(J(\\mu ))=\\frac{1}{2}(J(\\mu )+\\overline{J(\\mu )})=-\\mu \\int _0^{\\infty }\\log (r)\\frac{\\mathrm {e}^{-\\mathrm {i}\\pi /2\\mu }}{r^2-\\mathrm {e}^{-\\mathrm {i}\\pi /\\mu }}\\,\\mathrm {d}r.$ The integral (REF ) may now be evaluated explicitly.",
"It is the integral along the positive real axis of the function $\\log (z)\\dfrac{\\mathrm {e}^{-\\mathrm {i}\\pi /2\\mu }}{z^2-\\mathrm {e}^{-\\mathrm {i}\\pi /\\mu }}$ , which has two poles (at $\\pm \\mathrm {e}^{-\\mathrm {i}\\pi /2\\mu }$ ) and a branch cut at $z=0$ which we take along the negative real axis.",
"The decay at infinity and logarithmic growth at the origin enables us to use Cauchy's theorem to move the integral to an integral along the ray $\\theta =(\\pi /2-\\pi /2\\mu )$ ; since $\\mu >1/2$ , there are no singularities in the intervening region.",
"Along this ray $z=i\\rho \\mathrm {e}^{-\\mathrm {i}\\pi /2\\mu }$ , so we have $\\operatorname{Re}(J(\\mu ))=\\mu \\int _0^{\\infty }(\\log \\rho +\\mathrm {i}\\frac{\\pi }{2}(1-1/\\mu ))\\frac{\\mathrm {i}}{\\rho ^2+1}\\,\\mathrm {d}\\rho .$ The imaginary part of this integral is, fortunately, zero, and the real part is $-\\pi ^2(\\mu -1)/4$ , completing the proof.",
"As an immediate consequence of the Lemma, we have that for some function $E(\\alpha )$ , $g_{\\alpha }(-\\mathrm {i})=\\mathrm {e}^{E(\\alpha )}\\mathrm {e}^{\\mathrm {i}(1-\\mu )\\pi /4}.$ This completes the proof of Theorem REF ."
],
[
"Asymptotics of Peters solution as $z\\rightarrow 0$",
"This analysis is substantially easier.",
"First we have a small proposition: Proposition A.6 $g_{\\alpha }(\\zeta )\\rightarrow 1$ as $|\\zeta |\\rightarrow \\infty $ , and the convergence is uniform in any sector away from the negative real axis.",
"This is claimed on p. 329 of [36] and is immediate from (REF ) when $\\operatorname{Re}(\\zeta )>0$ , but is not so clear for other values of $\\arg (\\zeta )$ , so we rewrite the proof here.",
"From the definition of $g_{\\alpha }(\\zeta )$ it suffices to show that $I_{\\alpha }(\\zeta )\\rightarrow 0$ as $|\\zeta |\\rightarrow \\infty $ .",
"This is immediately clear when $\\arg (\\zeta )\\in (-\\alpha ,\\alpha )$ , from the definition of $I_{\\alpha }(\\zeta )$ and a change of variables.",
"For other values of $\\arg (\\zeta )$ we can use other representations of $I_{\\alpha }(\\zeta )$ .",
"For example, the representation (REF ) is good beyond $\\zeta =\\pm \\pi /2$ , and each term approaches zero as $|\\zeta |\\rightarrow \\infty $ as long as $\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha }$ is not on $M_2$ or $L_0$ .",
"The rate of convergence depends only on the distance of $\\zeta /|\\zeta |$ from $M_2$ and $L_0$ .",
"Continuing to move the contours as in the appendix of [36], we can show that $I_{\\alpha }(\\zeta )$ goes to zero as $|\\zeta |\\rightarrow \\infty $ as long as $\\zeta $ is not on the branch cut.",
"This completes the proof.",
"Now recall that $f(z)=\\mathrm {i}\\int _P\\frac{g_{\\alpha }(\\zeta )}{\\zeta +\\mathrm {i}}\\mathrm {e}^{z\\zeta }\\,\\mathrm {d}\\zeta .$ Assume $|z|<1$ .",
"Change variables to $w=z\\zeta $ , then deform the contour back to $\\arg (z)P$ (this works since all singularities of the integrand are inside $P$ ).",
"We get $f(z)=\\mathrm {i}\\int _{\\arg (z)P}\\frac{g_{\\alpha }(w/z)}{w+iz}\\mathrm {e}^w\\,\\mathrm {d}w.$ As $|z|\\rightarrow 0$ , $g_{\\alpha }(w/z)$ converges uniformly to 1, and $(w+\\mathrm {i}z)^{-1}$ converges uniformly to $w^{-1}$ .",
"So as $|z|\\rightarrow 0$ , $f(z)\\rightarrow \\mathrm {i}\\int _{\\arg (z)P}w^{-1}\\mathrm {e}^w\\,\\mathrm {d}w=-2\\pi .$ Thus we have established that $\\lim _{z\\rightarrow 0}f(z)$ exists and equals $-2\\pi $ .",
"When a constant coefficient differential operator $\\mathcal {P}$ of order $k$ is applied to $f(z)$ , the analysis may be done similarly.",
"The action of $\\mathcal {P}$ brings down a factor of $\\zeta ^k$ inside the integral, which becomes $(w/z)^{k}$ .",
"Pulling the $z^{-k}$ out of the integral creates $\\rho ^{-k}$ , and the remaining integral approaches the integral of $w^{k-1}\\mathrm {e}^w$ , which is zero.",
"This completes the proof of Theorem REF in the Robin-Neumann case."
],
[
"Robin-Dirichlet problem",
"This is also in [36], again with some details but not others.",
"We again look for a function $f(z)$ analytic in $\\mathfrak {S}_{\\alpha }$ .",
"The condition at $z\\in \\mathfrak {S}_{\\alpha }\\cap \\lbrace \\theta =-\\alpha \\rbrace $ is now $\\operatorname{Re}(f(z))=0$ .",
"Following Peters, using the same notation for $g_{\\alpha }(\\zeta )$ as before, a solution is $f(z)=\\mathrm {i}\\int _P\\frac{g_{\\alpha }(\\zeta )}{\\zeta +\\mathrm {i}}\\zeta ^{-\\mu }\\mathrm {e}^{z\\zeta }d\\zeta ,\\ k\\in \\mathbb {Z}.$ The only difference between the solutions is the multiplication by $\\zeta ^{-\\mu }$ inside the integral.",
"We verify that (REF ) satisfies the properties we need.",
"Indeed it is obviously analytic in $\\mathfrak {S}_{\\alpha }$ (at least away from the corner point).",
"The boundary condition on the real axis is verified in precisely the same way as for the Robin-Neumann problem, since $\\zeta ^{-\\mu }$ is real when $\\zeta $ is real.",
"For the Dirichlet boundary condition, we again change variables to $w=\\zeta \\mathrm {e}^{-\\mathrm {i}\\alpha }$ , when (REF ) becomes $f(z)=\\int _{P_1}\\frac{w^{-\\mu }\\mathrm {e}^{\\mathrm {i}\\alpha }g_{\\alpha }(w\\mathrm {e}^{\\mathrm {i}\\alpha })}{w\\mathrm {e}^{\\mathrm {i}\\alpha }+\\mathrm {i}}\\mathrm {e}^{pw}\\,\\mathrm {d}w=\\int _{P_1}w^{-\\mu -1}\\exp (-I_{\\alpha }(w))\\mathrm {e}^{\\rho w}\\,\\mathrm {d}w,$ where $P_1$ is symmetric with respect to the real axis and the branch cut is along the negative real axis.",
"As for (REF ), this has zero real part, showing that (REF ) is a solution.",
"Now we analise the asymptotics.",
"Let $\\chi _{D,\\alpha }=\\chi +\\frac{\\pi ^2}{4\\alpha }=\\frac{\\pi }{4}\\left(1+\\frac{\\pi }{2\\alpha }\\right).$ Analogously to the Robin-Neumann case, we have: Theorem A.7 With $E(\\alpha ):(0,\\pi /2)\\rightarrow \\mathbb {R}$ as in Theorem REF , there exists $R^{\\prime }_{\\alpha }(z):W\\times (0,\\pi /2)\\rightarrow \\mathbb {C}$ such that $f(z)=\\mathrm {e}^{E(\\alpha )}\\mathrm {e}^{-\\mathrm {i}(z-\\chi _{D,\\alpha })}+R^{\\prime }_{\\alpha }(z),$ where for any fixed $\\alpha \\in (0,\\pi /2)$ , there exists a constant $C>0$ such that for all $z\\in \\mathfrak {S}_{\\alpha }$ , $|\\operatorname{Re}(R^{\\prime }_{\\alpha }(z))|\\le C|z|^{-2\\mu },\\ |\\nabla _z (\\operatorname{Re}(R^{\\prime }_{\\alpha }(z))|\\le C|z|^{-2\\mu -1}.$ We mimic the proof of Theorem REF .",
"By the same contour-deformation process, we have $\\begin{split}f(z)&=2\\mu ^{1/2}g_{\\alpha }(-\\mathrm {i})\\mathrm {e}^{-\\mathrm {i}z}\\mathrm {e}^{\\mathrm {i}\\pi \\mu /2}+\\frac{\\mu ^{1/2}}{\\mathrm {i}\\pi }\\int _{P^{\\prime }}\\frac{g_{\\alpha }(\\zeta )\\zeta ^{-\\mu }}{\\zeta +\\mathrm {i}}\\mathrm {e}^{z\\zeta }\\,\\mathrm {d}\\zeta \\\\&+\\sum _{j=1}^m\\mathrm {e}^{z\\lambda _m}2\\mu ^{1/2}\\mathop {\\operatorname{Res}}\\limits _{\\zeta =\\lambda _m}(g_{\\alpha }(\\zeta )\\zeta ^{-\\mu }/(\\zeta +\\mathrm {i})).\\end{split}$ From the evaluation of $g_{\\alpha }(-\\mathrm {i})$ in the proof of Theorem REF , we immediately see that the first term is $\\mathrm {e}^{E(\\alpha )}\\mathrm {e}^{-\\mathrm {i}(z-\\chi _{D,\\alpha })}$ .",
"The sum of residues in (REF ) has the needed decay estimates.",
"And if we call the second term $\\widetilde{R}^{\\prime }_{\\alpha }(z)$ , the methods of the proof of Theorem REF give $\\widetilde{R}^{\\prime }_{\\alpha }(z)=-\\frac{\\mu ^{1/2}}{\\pi }\\int _{P^{\\prime }}\\mathrm {e}^{z\\zeta }\\mathrm {e}^{-h(\\zeta )}\\frac{\\mathrm {d}\\zeta }{\\zeta },$ which has an extra factor of $\\zeta ^{-\\mu }$ compared with (REF ).",
"Changing variables as before gives $\\widetilde{R}^{\\prime }_{\\alpha }(z)=-\\frac{\\mu ^{1/2}}{\\pi }\\int _{\\arg (z)P^{\\prime }}\\mathrm {e}^w\\mathrm {e}^{-h(w/z)}\\frac{\\mathrm {d}w}{w}.$ Plugging in $h(\\cdot )=p(\\cdot )+R(\\cdot )$ gives $\\widetilde{R}^{\\prime }_{\\alpha }(z)=-\\frac{\\mu ^{1/2}}{\\pi }\\int _{\\arg (z)P^{\\prime }}\\mathrm {e}^w\\mathrm {e}^{-p(w/z)}\\mathrm {e}^{-R(w/z)}\\frac{\\mathrm {d}w}{w}.$ Using the estimates on $R(w/z)$ , we have $\\widetilde{R}^{\\prime }_{\\alpha }(z)=-\\frac{\\mu ^{1/2}}{\\pi }\\int _{\\arg (z)P^{\\prime }}\\mathrm {e}^w\\mathrm {e}^{-p(w/z)}(1+O(|w/z|^{2\\mu }))\\frac{\\mathrm {d}w}{w}.$ For the part of the integral corresponding to the first term 1, $p(w/z)$ is holomorphic in the full disk with no branch cut, so the contour may be deformed to a small disk around $w=0$ (plus a portion around any branch cut along the negative real axis starting from $w=-1$ ), then evaluated using residues.",
"We see that the first term is equal to $-2\\mathrm {i}\\mu ^{1/2}+O(\\mathrm {e}^{-|z|/C}),$ and therefore its real part is $O(\\mathrm {e}^{-|z|/C})$ , even better than needed.",
"On the other hand, the exact same analysis as in the Robin-Neumann case shows that the second term is $O(|z|^{-2\\mu })$ , which is what we wanted.",
"The gradient estimate follows as before, since differentiating (REF ) just brings down an extra power of $\\zeta $ , which becomes $w/z$ .",
"The estimates on higher order derivatives follow as well.",
"And the analysis of $f(z)$ as $z\\rightarrow 0$ follows as before.",
"We can show that $f(z)\\sim z^{\\mu }\\int _{\\arg (z) P}w^{-\\mu -1}\\mathrm {e}^w \\mathrm {d}w.$ Since $\\mu >0$ , we have $\\lim _{z\\rightarrow 0}f(z)=0$ , and moreover $f(z)=O(|z|^{\\mu })$ .",
"As before, the action of any differential operator $\\mathcal {P}$ brings down an extra factor of $\\zeta ^k=w^k/z^k$ and the analysis proceeds similarly.",
"This completes the proof of Theorem REF in the Robin-Dirichlet case."
],
[
"Plan of the argument",
"Here we prove Proposition REF by using the analysis of [34].",
"As we will see, it is a special case of Theorem 2 in [34].",
"In what follows we assume that $q$ is even, see Remark REF .",
"Throughout we let $n=2q$ and note that $n=0$ mod 4.",
"We first claim the following technical lemma: Lemma B.1 Suppose $q\\in \\mathbb {N}$ is even.",
"Then the boundary conditions in (REF ) and (REF ) are regular in the sense of [34].",
"Moreover, using the notation of [34], $\\theta _0=0$ , and $\\theta _{1}=\\theta _{-1}\\ne 0$ in both cases.",
"Assuming this lemma, we obtain asymptotics for the eigenvalues.",
"Indeed, let $\\xi ^{\\prime }$ and $\\xi ^{\\prime \\prime }$ be the roots of the equation $\\theta _1\\xi ^2+\\theta _0\\xi +\\theta _{-1}=0$ ; by our calcuation, these are the roots of $\\xi ^2+1=0$ : $\\xi ^{\\prime }=-\\mathrm {i},\\xi ^{\\prime \\prime }=\\mathrm {i}.$ From [34], using the fact that $n=0$ mod 4, all sufficiently large eigenvalues form two sequences: $(\\lambda _k^{\\prime })^{2q}=(2k\\pi )^{2q}\\left(1-\\frac{q\\ln _0\\xi ^{\\prime }}{k\\pi \\mathrm {i}}+O\\left(\\frac{1}{k^2}\\right)\\right);$ $(\\lambda _k^{\\prime \\prime })^{2q}=(2k\\pi )^{2q}\\left(1-\\frac{q\\ln _0\\xi ^{\\prime \\prime }}{k\\pi \\mathrm {i}}+O\\left(\\frac{1}{k^2}\\right)\\right).$ Taking $2q$ -th roots and using Taylor series, and plugging in $\\xi ^{\\prime }$ and $\\xi ^{\\prime \\prime }$ : $\\lambda _k^{\\prime }=2k\\pi \\left(1-\\frac{\\ln _0\\xi ^{\\prime }}{2k\\pi \\mathrm {i}}+O\\left(\\frac{1}{k^2}\\right)\\right)=2k\\pi +\\frac{\\pi }{2}+O\\left(\\frac{1}{k}\\right);$ $\\lambda _k^{\\prime \\prime }=2k\\pi \\left(1-\\frac{\\ln _0\\xi ^{\\prime \\prime }}{2k\\pi \\mathrm {i}}+O\\left(\\frac{1}{k^2}\\right)\\right)=2k\\pi -\\frac{\\pi }{2}+O\\left(\\frac{1}{k}\\right).$ Combining the two sequences yields Proposition REF .",
"Remark B.2 Theorem 2 in [34], in some editions and English translations, only states that there are sequences of eigenvalues of the desired form, rather than stating that all sufficiently large eigenvalues form those sequences.",
"However, the original Russian edition claims the stronger version, and it follows immediately from the proof in [34] anyway."
],
[
"Proof of Lemma ",
"To find $\\theta _1$ , $\\theta _0$ , and $\\theta _{-1}$ , we need to analise the notation of [34].",
"The boundary conditions will be seen to be regular conditions of Sturm type, hence falling under the analysis of [34], in particular giving $\\theta =0=0$ and $\\theta _1$ and $\\theta _{-1}$ defined by (41)–(42) in [34].",
"To analise these determinants we need to identify the numbers $\\omega _j$ , $k_j$ , and $k_j^{\\prime }$ .",
"We do this first.",
"To analise the $\\omega _j$ , see the discussion before [34].",
"Note that the sector $S_0$ referred to in the statement of theorem 2 is the sector of the complex plane with argument between 0 and $\\frac{\\pi }{n}$ (see [34]).",
"We then let $\\omega _1,\\dots ,\\omega _n$ be the $n$ th roots of $-1$ , arranged so that for $\\rho \\in S_0$ , $\\operatorname{Re}(\\rho \\omega _1)\\le \\operatorname{Re}(\\rho \\omega _2)\\le \\dots \\le \\operatorname{Re}(\\rho \\omega _n).$ To figure out this ordering, let $\\omega :=\\mathrm {e}^{\\mathrm {i}\\pi /n}.$ Then, in order, the set $\\lbrace \\omega _1,\\dots ,\\omega _n\\rbrace $ is $\\lbrace \\omega ^{2q-1},\\omega ^{-(2q-1)},\\dots ,\\omega ^{3},\\omega ^{-3},\\omega ^1,\\omega ^{-1}\\rbrace .$ Note further that $\\omega _{\\mu }=\\omega _{q}$ ; since $q$ is even, $\\omega _{\\mu }=\\omega ^{-(q+1)}; \\qquad \\omega _{\\mu +1}=\\omega ^{q-1}.$ In particular we actually have $\\omega _{\\mu +1}=-\\omega _{\\mu }$ .",
"This will be useful.",
"Now we need to understand the $k_j$ and $k_j^{\\prime }$ , which is done via a direct comparison with [34].",
"The sums all vanish, and only the first terms remain.",
"We see that we always have $k_j^{\\prime }=k_j$ in both the Neumann and Dirichlet settings, and for Neumann we have $k_1=2q-1,\\ k_2=2q-2,\\dots ,k_q=q.$ For Dirichlet we have $k_1=q-1,\\ k_2=q-2,\\dots ,k_q=0.$ So these conditions are of Sturm type.",
"Since the conditions are Sturm type we immediately have $\\theta _0=0$ as well as the formulas [34] for $\\theta _{-1}$ and $\\theta _1$ .",
"Assume for the moment we are working in the Neumann setting.",
"Define the matrices $A_q:=\\begin{bmatrix}\\omega _1^{k_1} & \\cdots & \\omega _{\\mu }^{k_1}\\\\\\vdots & \\ddots & \\vdots \\\\\\omega _1^{k_{\\mu }} & \\cdots & \\omega _{\\mu }^{k_{\\mu }}\\end{bmatrix}=\\begin{bmatrix}\\omega ^{(2q-1)(2q-1)} & \\omega ^{-(2q-1)(2q-1)} & \\cdots & \\omega ^{-(q+1)(2q-1)}\\\\\\vdots & \\vdots & \\cdots & \\vdots \\\\\\omega ^{(2q-1)(q)} & \\omega ^{-(2q-1)(q)} & \\cdots & \\omega ^{-(q+1)(q)}\\end{bmatrix},$ $B_q:=\\begin{bmatrix}\\omega _{\\mu +1}^{k_1} & \\cdots & \\omega _{n}^{k_1}\\\\\\vdots & \\ddots & \\vdots \\\\\\omega _{\\mu +1}^{k_{\\mu }} & \\cdots & \\omega _{n}^{k_{\\mu }}\\end{bmatrix}=\\begin{bmatrix}\\omega ^{(q-1)(2q-1)} & \\omega ^{-(q-1)(2q-1)} & \\cdots & \\omega ^{-1(2q-1)}\\\\\\vdots & \\vdots & \\cdots & \\vdots \\\\\\omega ^{(q-1)(q)} & \\omega ^{-(q-1)(q)} & \\cdots & \\omega ^{-1(q)}\\end{bmatrix}.$ Let $A_q^{\\prime }$ be $A_q$ with the last column replaced by the first column of $B_q$ , and let $B_q^{\\prime }$ be $B_q$ with the first column replaced by the last column of $A_q$ .",
"Note also that since $\\omega _{\\mu +1}=-\\omega _{\\mu }$ , those switched columns are the same except that the entry in each odd row is multiplied by $-1$ .",
"In particular their last entries are the same, since $q$ is even.",
"Then [34] give $\\theta _{-1}=\\pm \\det A_q\\det B_q;\\ \\theta _1=\\pm \\det A_q^{\\prime }\\det B_q^{\\prime },$ where the signs are the same (they both come from the same equation before (41) and (42) in [34]).",
"Thus it suffices to show that $\\frac{\\det A_q^{\\prime }}{\\det A_q}=\\frac{\\det B_q^{\\prime }}{\\det B_q}.$ The trick will be to reduce these to Vandermonde determinants by taking a factor out of each column; the factor we take out will always be the last entry in each column, which is simplified by the fact that $w^q=i$ .",
"For example, $\\det B_q=\\mathrm {i}^{(q-1)}\\mathrm {i}^{-(q-1)}\\mathrm {i}^{(q-3)}\\mathrm {i}^{-(q-3)}\\dots \\mathrm {i}^1\\mathrm {i}^{-1}\\det D_q=\\det D_q,$ where $D_q=\\begin{bmatrix}\\omega ^{(q-1)(q-1)} & \\omega ^{-(q-1)(q-1)} & \\cdots &\\omega ^{-1(q-1)}\\\\ \\vdots & \\vdots & \\cdots & \\vdots \\\\ \\omega ^{(q-1)(1)} & \\omega ^{-(q-1)(1)} & \\cdots & \\omega ^{-1(1)}\\\\ 1 & 1 & \\cdots & 1\\end{bmatrix}.$ Similarly, $\\det A_q=\\det C_q$ , where $C_q$ is obtained as with $D_q$ .",
"As for $B_q^{\\prime }$ and $A_q^{\\prime }$ , note that the last entry of the last column of $A_q$ is the same as the last entry of the first column of $B_q$ , so all the pre-factors still cancel and we have $\\det A_q^{\\prime }=\\det C_q^{\\prime }$ , $\\det B_q^{\\prime }=\\det D_q^{\\prime }$ , with $D_q^{\\prime }$ and $C_q^{\\prime }$ obtained by switching the first column of $D_q$ for the last column of $C_q$ .",
"Switching the order of all of the rows introduces a factor of $(-1)^{q/2}$ in each determinant and brings us to a Vandermonde matrix in each case: $\\det B_q&=(-1)^{q/2}\\det V(\\omega ^{q-1},\\omega ^{-(q-1)},\\dots ,\\omega ^1,\\omega ^{-1});\\\\\\det B_q^{\\prime }&=(-1)^{q/2}\\det V(\\omega ^{-(q+1)},\\omega ^{-(q-1)},\\dots ,\\omega ^1,\\omega ^{-1});\\\\\\det A_q&=(-1)^{q/2}\\det V(\\omega ^{2q-1},\\omega ^{-(2q-1)},\\dots ,\\omega ^{q+1},\\omega ^{-(q+1)});\\\\\\det A_q^{\\prime }&=(-1)^{q/2}\\det V(\\omega ^{2q-1},\\omega ^{-(2q-1)},\\dots ,\\omega ^{q+1},\\omega ^{q-1}).$ Each of these Vandermonde determinants can be computed explicitly via the well-known pairwise difference formula.",
"Considering $\\det B_q^{\\prime }/\\det B_q$ , all the pairwise differences cancel except for those involving the first element.",
"The first element just changes by an overall sign from $B_q$ to $B_q^{\\prime }$ , and we get $\\frac{\\det B_q^{\\prime }}{\\det B_q}=\\frac{(\\omega ^{-1}+\\omega ^{q-1})(\\omega ^{1}+\\omega ^{q-1})\\dots (\\omega ^{-(q-1)}+\\omega ^{q-1})}{(\\omega ^{-1}-\\omega ^{q-1})(\\omega ^{1}-\\omega ^{q-1})\\dots (\\omega ^{-(q-1)}-\\omega ^{q-1})}.$ Similarly, $\\frac{\\det A_q^{\\prime }}{\\det A_q}=\\frac{(-\\omega ^{q-1}-\\omega ^{q+1})\\dots (-\\omega ^{q-1}-\\omega ^{-(2q-1)})(-\\omega ^{q-1}-\\omega ^{2q-1})}{(\\omega ^{q-1}-\\omega ^{q+1})\\dots (\\omega ^{q-1}-\\omega ^{-(2q-1)})(\\omega ^{q-1}-\\omega ^{2q-1})}.$ Pulling a minus sign out of each term on the top and the bottom and reversing the order of the multiplication gives $\\frac{\\det A_q^{\\prime }}{\\det A_q}=\\frac{(\\omega ^{q-1}+\\omega ^{2q-1})(\\omega ^{q-1}+\\omega ^{-(2q-1)})\\dots (\\omega ^{q-1}+\\omega ^{q+1})}{(-\\omega ^{q-1}+\\omega ^{2q-1})(-\\omega ^{q-1}+\\omega ^{-(2q-1)})\\dots (-\\omega ^{q-1}+\\omega ^{q+1})}.$ Flipping the order of addition within each term: $\\frac{\\det A_q^{\\prime }}{\\det A_q}=\\frac{(\\omega ^{2q-1}+\\omega ^{q-1})(\\omega ^{-(2q-1)}+\\omega ^{q-1})\\dots (\\omega ^{q+1}+\\omega ^{q-1})}{(\\omega ^{2q-1}-\\omega ^{q-1})(\\omega ^{-(2q-1)}-\\omega ^{q-1})\\dots (\\omega ^{q+1}-\\omega ^{q-1})}.$ Using the fact that $\\omega ^{2q}=1$ to simplify one element in each term, we see that this fraction is precisely $\\frac{\\det B_q^{\\prime }}{\\det B_q}$ , completing the proof of Lemma REF for the Neumann case.",
"The Dirichlet case is very similar, in fact easier; we have $k_1$ through $k_q$ each decreasing by $q$ .",
"In particular the last entries of each column are 1, so the matrices corresponding to $A_q, B_q, A_q^{\\prime },B_q^{\\prime }$ are already Vandermonde.",
"In fact, they are exactly equal to $C_q, D_q, C_q^{\\prime }$ , and $D_q^{\\prime }$ respectively, so our previous computation completes the proof in this case as well.",
"$\\Box $"
],
[
"Numerical examples",
"In this Appendix we present some numerical results.",
"All computations have been performed using the package FreeFem++ [19]."
],
[
"Example illustrating Theorems ",
"Let $\\Omega =\\bigtriangleup ABZ$ be a triangle with $L=1$ , $\\alpha =\\frac{2 \\pi }{5}$ and $\\beta =\\frac{\\pi }{6}$ .",
"We consider sloshing with Neumann or Dirichlet conditions on $\\mathcal {W}=[A,Z]\\cup [Z,B]$ .",
"The eigenvalues and quasi-frequencies are given in the table below, and we see that the error is indeed quite small in both Neumann and Dirichlet cases: Figure: Triangle from Example $\\begin{tabu}{|r|c|c|c|c|c|c|}\\hline k&\\lambda _{k}&\\sigma _{k}&\\left|\\dfrac{\\sigma _{k}}{\\lambda _{k}}-1\\right|&\\lambda _{k}^D&\\sigma _{k}^D&\\left|\\dfrac{\\sigma _{k}^D}{\\lambda _{k}^D}-1\\right|\\\\[1ex]\\hline 1 & 0.",
"& -0.88357 & \\ & 2.43592 & 2.45437 & 7.58\\times 10^{-3} \\\\\\hline 2 & 0.85626 & 0.68722 & 1.97\\times 10^{-1} & 4.02389 & 4.02517 & 3.17\\times 10^{-4} \\\\\\hline 3 & 2.28840 & 2.2580 & 1.33\\times 10^{-2} & 5.59623 & 5.59596 & 4.83\\times 10^{-5} \\\\\\hline 4 & 3.82292 & 3.8288 & 1.54\\times 10^{-3} & 7.16681 & 7.16676 & 7.11\\times 10^{-6} \\\\\\hline 5 & 5.39779 & 5.3996 & 3.37\\times 10^{-4} & 8.73757 & 8.73755 & 1.81\\times 10^{-6} \\\\\\hline 6 & 6.96977 & 6.9704 & 9.22\\times 10^{-5} & 10.3084 & 10.3084 & 1.44\\times 10^{-6} \\\\\\hline 7 & 8.54086 & 8.5412 & 4.00\\times 10^{-5} & 11.8792 & 11.8791 & 1.74\\times 10^{-6} \\\\\\hline 8 & 10.1118 & 10.112 & 2.03\\times 10^{-5} & 13.4500 & 13.4499 & 2.36\\times 10^{-6} \\\\\\hline 9 & 11.6827 & 11.683 & 1.11\\times 10^{-5} & 15.0208 & 15.0207 & 3.26\\times 10^{-6} \\\\\\hline 10 & 13.2535 & 13.254 & 5.90\\times 10^{-6} & 16.5916 & 16.5915 & 4.47\\times 10^{-6} \\\\\\hline \\end{tabu}$"
],
[
"Example illustrating Remark ",
"We define two domains $\\Omega _\\pm $ by setting a curved sloshing surface $S_\\pm =\\left\\lbrace \\left(x,\\pm \\frac{1}{2\\pi }\\sin (2\\pi x)\\right)\\mid 0<x<1\\right\\rbrace ,$ so that the length of the sloshing surface is $L=\\int _0^1 \\sqrt{1+ \\cos ^2(2\\pi x)}\\,\\mathrm {d}x=\\frac{2\\sqrt{2}}{\\pi }E\\left(\\frac{1}{2}\\right)\\approx 1.21601,$ where $E$ denotes a complete elliptic integral of the second kind.",
"Let $\\begin{split}\\mathcal {W}_1&=\\mathcal {W}_{1,\\pm }=\\left\\lbrace z:\\ |z-1/2|=1/2, -\\pi \\le \\arg (z)\\le -\\pi /2\\right\\rbrace , \\\\\\mathcal {W}_2&=\\mathcal {W}_{2,\\pm }=\\left\\lbrace z:\\ |z-1/2|=1/2, -\\pi /2\\le \\arg (z)\\le 0\\right\\rbrace ,\\end{split}$ so that $\\alpha _+=\\beta _-=\\frac{3\\pi }{4}, \\qquad \\alpha _-=\\beta _+=\\frac{\\pi }{4}.$ In both cases the Dirichlet boundary condition is imposed on $\\mathcal {W}_1$ and the Neumann one on $\\mathcal {W}_2$ .",
"Figure: Domains Ω + \\Omega _+ (left) and Ω - \\Omega _- (right)The quasifrequencies are then given by (REF ), which after omitting $O-$ terms simplifies to $\\sigma _{+,k}=\\frac{\\pi }{L}\\left(k-\\frac{1}{6}\\right),\\qquad \\sigma _{-,k}=\\frac{\\pi }{L}\\left(k-\\frac{5}{6}\\right).$ The comparisons between the numerically calculated eigenvalues and the quasifrequencies are below.",
"Note that there is very good agreement even for low $k$ .",
"$\\begin{tabu}{|r|c|c|c|c|c|c|}\\hline k&\\lambda _{+,k}&\\sigma _{+,k}&\\left|\\dfrac{\\sigma _{+,k}}{\\lambda _{+,k}}-1\\right|&\\lambda _{-,k}&\\sigma _{-,k}&\\left|\\dfrac{\\sigma _{-,k}}{\\lambda _{-,k}}-1\\right|\\\\[1ex]\\hline 1 & 1.02371 & 2.15294 & 1.10 & 1.24543 & 0.430589 & 6.54\\times 10^{-1} \\\\\\hline 2 & 5.65749 & 4.73648 & 1.63\\times 10^{-1} & 2.63524 & 3.01412 & 1.44\\times 10^{-1} \\\\\\hline 3 & 8.13194 & 7.32001 & 9.98\\times 10^{-2} & 5.55627 & 5.59765 & 7.45\\times 10^{-3} \\\\\\hline 4 & 10.3085 & 9.90354 & 3.93\\times 10^{-2} & 8.22122 & 8.18119 & 4.87\\times 10^{-3} \\\\\\hline 5 & 12.8138 & 12.4871 & 2.55\\times 10^{-2} & 10.6845 & 10.7647 & 7.51\\times 10^{-3} \\\\\\hline 6 & 15.3856 & 15.0706 & 2.05\\times 10^{-2} & 13.1122 & 13.3483 & 1.80\\times 10^{-2} \\\\\\hline 7 & 17.9151 & 17.6541 & 1.46\\times 10^{-2} & 15.7600 & 15.9318 & 1.09\\times 10^{-2} \\\\\\hline 8 & 20.4310 & 20.2377 & 9.46\\times 10^{-3} & 18.4111 & 18.5153 & 5.66\\times 10^{-3} \\\\\\hline 9 & 22.9800 & 22.8212 & 6.91\\times 10^{-3} & 21.0011 & 21.0988 & 4.66\\times 10^{-3} \\\\\\hline 10 & 25.5511 & 25.4047 & 5.73\\times 10^{-3} & 23.5873 & 23.6824 & 4.03\\times 10^{-3} \\\\\\hline \\end{tabu}$"
],
[
"List of recurring notation",
"m3cm|m8cm|m2cm Notation Meaning Section $\\Omega $ Sloshing domain 1 $S$ , $\\mathcal {W}$ Sloshing surface, free boundary 1 $A$ , $B$ Vertices of $\\Omega $ 1 $L$ Length of sloshing surface $S$ 1 $\\alpha $ , $\\beta $ Interior angles of sloshing domain at $A$ , $B$ 1 $0=\\lambda _1<\\lambda _2\\dots $ Sloshing eigenvalues 1 $\\lbrace u_k\\rbrace $ Sloshing eigenfunctions 1 $\\mathcal {D}$ Sloshing operator 1 $\\mathcal {H}$ Harmonic extension operator from $S$ to $\\Omega $ with given boundary conditions on $\\mathcal {W}$ 1 $\\lbrace \\lambda _k^D,u_k^D\\rbrace $ Eigenvalues and eigenfunctions for Steklov-Dirichlet problem 1 $q$ A positive integer, used often when $\\alpha =\\pi /2q$ 1 $\\lbrace \\Lambda _k,U_k\\rbrace $ Eigenvalues and eigenfunctions for associated ODE 1 $T=ABZ$ Triangle, used when $\\Omega $ is triangular 1 $(x,y)$ , $(\\rho ,\\theta )$ Cartesian and polar coordinates for $\\mathbb {R}^2$ 2 $z$ Associated complex coordinate $z=x+iy$ 2 $\\mathfrak {S}_{\\alpha }$ Planar sector with $-\\alpha \\le \\theta \\le 0$ 2 $\\chi _{\\alpha ,N}$ etc.",
"Phase shifts for various angles and boundary conditions 2 $\\mu _{\\alpha }$ ($\\mu $ for short) Equals $\\pi /(2\\alpha )$ 2 $\\phi _{\\alpha ,N}$ etc.",
"Peters solutions in a planar sector 2 $\\lbrace \\sigma _j\\rbrace $ , $\\sigma $ Quasi-frequencies 2 $p_{\\sigma }(x,y)$ Plane wave corresponding to scaled Peters solution 2 $v_{\\sigma }^{\\prime }$ Uncorrected quasimode (prime is NOT derivative) 2 $\\psi $ Fixed function in $C^{\\infty }_0(S)$ 2 $\\eta _{\\sigma }$ Neumann correction function 2 $v_{\\sigma }$ , $v_{\\sigma _j}$ Corrected quasimode, $j$ th corrected quasimode 2 $\\lbrace v_j\\rbrace $ Restrictions of $\\lbrace v_{\\sigma _j}\\rbrace $ to $S$ 2 $\\lbrace \\varphi _k\\rbrace $ Orthonormal basis of eigenfunctions of $\\mathcal {D}$ ; equal to $\\lbrace u_k|_S\\rbrace $ 2 $\\xi _q$ ($\\xi $ for short) Equals $\\mathrm {e}^{-\\mathrm {i}\\pi /q}$ 3 $g(z)$ Arbitrary complex-valued function 3 $\\mathcal {A}$ , $\\mathcal {B}$ Operators on complex-valued functions 3 $\\mathtt {SD}$ Steklov defect of a function 3 $f(z)$ , variants $f_j(z)$ $f(z)=\\mathrm {e}^{-\\mathrm {i}z}$ , variants obtained by applying $\\mathcal {A}$ and $\\mathcal {B}$ repeatedly 3 $\\widetilde{f}(z)$ , variants Analogues of $f(z)$ for Dirichlet case 3 $\\eta (\\xi )$ Equals $(\\xi +1)/(\\xi -1)$ 3 $v_{\\alpha }(z)$ Hanson-Lewy solution for angle $\\alpha $ 3 $\\gamma (\\xi )$ Product involving $\\eta (\\xi )$ 3 $v_{\\beta }^{\\texttt {d}}$ Sum of decaying exponents in $v_{\\beta }$ 3 $g_{\\sigma }(z)$ Quasimodes on triangles constructed with Hanson-Lewy solutions 3 Boldface Denotes abstract linear algebra context 4 $\\lambda _j,\\sigma _j$ Eigenvalues and quasi-eigenvalues in linear algebra lemma 4 $F(j)$ Error in linear algebra lemma 4 $\\lbrace w_j\\rbrace $ Linear combinations of sloshing eigenfunctions on $S$ 4 $\\lbrace W_j\\rbrace $ Linear combinations of ODE eigenfunctions 4 $n$ $2q$ 4 $\\lbrace \\omega _k\\rbrace $ $n$ th roots of $-1$ 4 $J_q$ Integer representing asymptotic shift for angles $\\pi /2q$ 4 $\\chi (z)$ Cutoff function 5 Tildes Denotes curvilinear domain setting 5 $\\widetilde{v}^{\\prime }_{\\sigma }$ Uncorrected quasimode, curvilinear case 5 $\\widetilde{\\eta }_{\\sigma }$ Neumann correction, curvilinear case 5 $\\widetilde{v}_{\\sigma }$ Corrected quasimode (but not yet harmonic), curvilinear case 5 $\\bar{v}_{\\sigma }$ Further-corrected quasimode, curvilinear case (now harmonic) 5 $\\widetilde{\\varphi }_{\\sigma }$ Solution to a mixed boundary value problem on $\\Omega $ 5 $\\Phi $ Conformal map from $\\Omega $ to $T$ 5 $\\varphi _{\\sigma }$ Solution to corresponding mixed problem on $T$ 5 $f(z)$ Complex-valued Peters solution A $\\zeta $ New complex variable A $I_{\\alpha }(\\zeta )$ , or $I(\\zeta )$ Auxiliary function for Peters solution A $g_{\\alpha }(\\zeta )$ Exponentiated and modified version of $I(\\zeta )$ A $P$ , $P_1$ , $P^{\\prime }$ , etc.",
"Complex contours in $\\zeta $ plane A $R_{\\alpha }(z)$ Remainder term: difference of Peters solution and plane wave A $\\lambda _1,\\dots ,\\lambda _m$ Residues of $g_{\\alpha }$ A $\\widetilde{R}_{\\alpha }(z)$ Remainder term without the residues A $\\omega $ Equals $\\mathrm {e}^{\\mathrm {i}\\pi /n}$ B $\\theta _0$ , $\\theta _1$ , $\\theta _{-1}$ Notation from Naimark B"
]
] | 1709.01891 |
[
[
"A p-variable higher-order finite volume time domain method for\n electromagnetic scattering problems"
],
[
"Abstract Higher-order accurate solution to electromagnetic scattering problems are obtained at reduced computational cost in a {\\it p}-variable finite volume time domain method.",
"Spatial operators of lower, including first-order accuracy, are employed locally in substantial parts of the computational domain during the solution process.",
"The use of computationally cheaper lower order spatial operators does not affect the overall higher-order accuracy of the solution.",
"The order of the spatial operator at a candidate cell during numerical simulation can vary in space and time and is dynamically chosen based on an order of magnitude comparison of scattered and incident fields at the cell center.",
"Numerical results are presented for electromagnetic scattering from perfectly conducting two-dimensional scatterers subject to transverse magnetic and transverse electric illumination."
],
[
"Introduction",
"Higher-order spatially accurate representation of partial differential equations (PDE's) are used to efficiently resolve spatially complex physical phenomenon during numerical simulations in many fields of science and engineering.",
"Higher-order spatially accurate schemes are able to resolve spatial variations with lower points per wave length (PPW) in the computational domain as compared to lower order representations.",
"Higher-order spatially accurate methods can achieve similar accuracy levels on much coarser discretization compared to lower-order methods.",
"However, higher-order spatially accurate methods tend to be more expensive on a per-grid-point basis compared to its lower order counterparts which mitigates some of the advantages accruing from the use of coarser meshes.",
"Thus, there is significant motivation in developing computationally low cost higher-order methods for numerically solving PDEs.",
"Multigrid (MG) methods [1], [2] based on cycling the numerical solution through a hierarchy of approximations either in space (h) or in polynomial order (p) or a combination of both have been used commonly to accelerate convergence to steady state of boundary value problems.",
"h-MG methods are common in both finite volume and finite element frameworks while p-MG methods tend to be mostly restricted to mostly finite element framework [3].",
"Local h or p refinements have long been used, including for solving initial value problems, if the length scales to be resolved are not uniform across the computational domain and can cut down significantly on total computational time [4], [5].",
"Local refinement in the polynomial order (p ) is again mostly restricted to finite element discretizations.",
"A finite volume based solution of linear hyperbolic PDEs by cycling through successive lower order p-approximations while retaining highest-order accuracy was proposed in Refs.",
"[6], [7].",
"In the current work we propose a p-variable finite volume framework with an emphasis on solving electromagnetic (EM) scattering problems in the time domain.",
"In the proposed framework, the time domain Maxwell equations which form a set of coupled linear hyperbolic PDEs, are solved on a fixed grid but with the spatial operator formally varying in accuracy over the computational domain.",
"The harmonic steady state solution obtained retains desired higher-order accuracy in spite of significant and not fixed parts of computational domain, processed using spatial operators of lower including first-order accuracy, during the simulation.",
"The choice of accuracy of the spatial operator, done dynamically, is based on an order of magnitude comparison between the scattered and incident field at the cell center.",
"The framework requires an unified access to spatial operators of various orders of accuracy.",
"For the present work the ENO methodology is used to locally obtain spatial operators of the desired accuracy but it may be possible to base it on higher-order numerical methods like spectral finite volume [8], ADER [9] etc.",
"that similarly provides unified access to spatial operators of varying accuracy.",
"Numerical results are presented for electromagnetic scattering from perfectly conducting circular cylinder and airfoil."
],
[
"Consider the scalar advection equation to be the scalar representation of the the time-domain Maxwell's equations in differential form in a scattering process.",
"The scalar advection equation is written as $\\frac{\\partial u}{\\partial t}+c\\frac{\\partial u}{\\partial x}=0$ with wave speed $ c \\ge 0$ .",
"We assume $u$ to represent a scattered field variable with $U=U_i+u$ where $U$ and $U_i$ respectively represent the corresponding total and incident fields.",
"All variables in equation REF can be nondimensionalized as $u^{\\ast }=u/U_i$ , $x^{\\ast }=x/\\lambda $ and $t^{\\ast }=t/T$ .",
"$\\lambda $ and $T$ are the wavelength and time period for the harmonic incident wave.",
"Further all nondimensional values $u^{\\ast }$ , $x^{\\ast }$ and $t^{\\ast }$ lie in $[0,1]$ and in terms of order of magnitude are assumed to be $O(1)$ .",
"Equation REF can be written in corresponding nondimensional form as $\\frac{\\partial u^{\\ast }}{\\partial t^{\\ast }}+\\frac{\\partial u^{\\ast }}{\\partial x^{\\ast }}=0.$ The proposed p-variable method utilizes spatial operators of formally different orders of accuracy ($p\\le m$ ) depending on the order of magnitude of scattered variables $u(x,t)$ being addressed, but always retains a local truncation error corresponding to the the highest $m^{th}$ order accuracy.",
"Spatial operators of $m$ and $(m-1)^{th}$ order formal accuracy result in local truncation errors of similar magnitude when applied respectively to scattered variables that differ by one-order-of-magnitude.",
"This fact can used recursively to involve even lower order operators while retaining formal $m^{th}$ order accuracy.",
"We show this using the nondimensional form and an order-of-magnitude analysis of the local truncation error.",
"Discretization of the space derivative in equation REF with a $m^{th}$ order accurate spatial operator results in a truncation error with leading term given by [10] $a(\\triangle x^{\\ast }) ^m \\frac{\\partial ^{m+1}u^{\\ast }}{\\partial ^{m+1}x^{\\ast }}$ where $a$ is a rational number.",
"In a practical finite difference type formulation approximately 10 PPW or more would be required for a reasonable resolution for EM scattering problems which makes $\\triangle x^{\\ast }$ at least one order of magnitude less than the representative wavelength $\\lambda $ .",
"Thus, $\\triangle x^{\\ast }\\sim O(1/10)$ in terms of order of magnitude.",
"Discretizing scattered variables locally of magnitude $\\sim \\triangle x^{\\ast } \\times u^{\\ast }$ with a $(m-1)^{th}$ order accurate spatial operator will similarly lead to a truncation error with leading term $b(\\triangle x^{\\ast })^{m-1} \\frac{\\partial ^{m}\\triangle x^{\\ast }u^{\\ast }}{\\partial ^{m}x^{\\ast }}.$ In terms of order of magnitude, for constant $\\Delta x^{\\ast }$ , $\\frac{\\partial ^{m}\\triangle x^{\\ast } u^{\\ast }}{\\partial ^{m}x^{\\ast }} \\sim \\triangle x^{\\ast } \\frac{\\partial ^{m} u^{\\ast }}{\\partial ^{m}x^{\\ast }}$ using which equation REF can be approximated as $b(\\triangle x^{\\ast })^{m} \\frac{\\partial ^{m} u^{\\ast }}{\\partial ^{m}x^{\\ast }}.$ We assume $\\frac{\\partial ^{m} u^{\\ast }}{\\partial ^{m}x^{\\ast }}=\\frac{\\partial ^{m+1} u^{\\ast }}{\\partial ^{m+1}x^{\\ast }}=O(1)$ since $u^{\\ast }$ and $x^{\\ast }$ are both $O(1)$ in the nondimensionalization process.",
"This is similar to fluid mechanics boundary layer theory, where the nondimensional velocity and distance in the streamwise direction are both $O(1)$ , resulting in first and second derivatives of the streamwise velocity in the streamwise direction also being $O(1)$ [11].",
"This further implies the leading term of the truncation error resulting from spatial operators of $m^{th}$ and $(m-1)^{th}$ accuracy given respectively in equation REF and REF to be of comparable magnitude.",
"This can be applied recursively to bring in spatial operators of even lower order of accuracy while locally yielding spatial accuracy comparable to the highest $m^{th}$ order accuracy.",
"Based on this a p-variable algorithm can be constructed to obtain inexpensively a spatially higher-order accurate steady state solution for a scattering process in time-domain electromagnetics or similar fields involving linear hyperbolic waves.",
"The algorithm for $m^{th}$ order accuracy in a cell centered Finite Volume Time Domain (FVTD) framework can be of the form described below and can be easily included in an existing higher-order solver, If the cell centered scattered variable, $u(x,t) \\ge \\triangle x \\times U_i(x,t)$ the spatial operator is of order $m$ .",
"For cell centered variable $\\triangle x^{n+2}\\times U_i(x,t) \\le u(x,t) < \\triangle x^{n+1}\\times U_i(x,t) $ the spatial operator is of order $m-(n+1)$ with $n \\ge 0$ with $U_i(x,t)$ assumed to be of similar order of magnitude throughout the domain and $\\triangle x^n \\times U_i(x,t) \\sim U_i(x,t)/10^n$ .",
"The above algorithm is used to obtain cheaply higher-order accurate solutions to the canonical problems of electromagnetic scattering in a FVTD framework.",
"A method of lines approach decouples the time and space discretizations and the spatial discretization is obtained using an Essentially Non-Oscillatory (ENO) method which allows easy access to varying orders of spatial accuracy.",
"The current implementation is in the ENO-Roe form [12], [13], which efficiently implements the ENO reconstruction based on the numerical fluxes instead of the cell averaged state variables and is described for the scalar law.",
"Equation REF is written as a scalar hyperbolic conservation law $u_{t}+f(u)_{x}=0,$ has the spatial derivative at the $i^{th}$ grid point approximated as $\\frac{\\partial f(u)}{\\partial x} |_{i}=\\frac{1}{\\triangle x}(\\overline{f}_{i+1/2}-\\overline{f}_{i-1/2})+{\\sf O}(\\triangle x^{p})$ where ${\\triangle x}$ is the grid size, $p$ the order of the scheme, $\\overline{f}_{i+1/2}$ the numerical flux function at the right cell-face.",
"The $r^{th}$ order accurate reconstruction of the numerical flux in the ENO scheme is $\\overline{f}_{i+1/2}=\\sum _{l=0}^{r-1} \\alpha _{k,l}^{r} f_{i-r+1+k+l}$ where $\\alpha _{k,l}^{r}$ are the reconstruction coefficients and $k$ the stencil index selected among the $r$ candidate stencils.",
"The stencil $S_{k}$ can be written as $S_{k}=(x_{i+k-r+1}, x_{i+k-r+2},....,x_{i+k})$ and is locally the smoothest possible stencil.",
"Details regarding reconstruction coefficients and stencil selection for ENO schemes are easily available in literature including Refs.",
"[12], [13].",
"Extension to the multidimensional system of equations like the time-domain Maxwell's equations can be obtained by decoupling the system into three scalar hyperbolic conservation laws normal to the cell faces [6]."
],
[
"Governing Equations and Numerical Scheme",
"The three-dimensional Maxwell's equations, in the differential and curl form in free space, are expressed as $\\frac{\\partial {\\bf B}}{\\partial t}= - {\\bf \\nabla } \\times \\bf E$ $\\frac{\\partial {\\bf D}}{\\partial t}={\\bf \\nabla } \\times \\bf H-\\bf J_{i}$ where ${\\bf B}$ is the magnetic induction, ${\\bf E}$ the electric field vector, ${\\bf D}$ the electric field displacement and ${\\bf H}$ the magnetic field vector.",
"${\\bf J_{i}}$ is the impressed current density vector, $\\bf D=\\varepsilon \\bf E$ , $\\bf B=\\mu \\bf H$ with $\\varepsilon $ and $\\mu $ respectively the permittivity and permeability in free space.",
"The time-domain Maxwell's equations can also be written in a conservative total field form as [14], [15] $\\frac{ \\partial \\mbox{$u$}}{\\partial t}+\\frac{ \\partial \\mbox{$f$}(\\mbox{$u$})}{\\partial x}+\\frac{ \\partial \\mbox{$g$}(\\mbox{$u$})}{\\partial y} +\\frac{ \\partial \\mbox{$h$}(\\mbox{$u$})}{\\partial z}=\\mbox{$s$}$ where $\\mbox{$u$}\\!=\\!\\left(\\begin{array}{c}B_{x} \\\\ B_{y} \\\\ B_{z} \\\\D_{x}\\\\D_{y}\\\\D_{z}\\end{array} \\right), \\:\\mbox{$f$}\\!=\\!\\left(\\begin{array}{c}0 \\\\ -D_{z}/\\varepsilon \\\\ D_{y}/\\varepsilon \\\\ 0\\\\B_{z}/\\mu \\\\-B_{y}/\\mu \\end{array} \\right),\\:\\mbox{$g$}\\!=\\!\\left(\\begin{array}{c}D_{z}/\\varepsilon \\\\ 0 \\\\- D_{x}/\\varepsilon \\\\ -B_{z}/\\mu \\\\0\\\\B_{x}/\\mu \\end{array} \\right),\\:\\mbox{$h$}\\!=\\!\\left(\\begin{array}{c}-D_{y}/\\varepsilon \\\\ D_{x}/\\varepsilon \\\\ 0\\\\B_{y}/\\mu \\\\-B_{x}/\\mu \\\\0\\end{array} \\right),\\:\\mbox{$s$}\\!=\\!\\left(\\begin{array}{c}0 \\\\ 0 \\\\ 0\\\\-J_{ix}\\\\-J_{y}\\\\-J_{iz}\\end{array} \\right)$ and subscripts indicate components in the Cartesian $x,y,z$ directions.",
"In two dimensions, Maxwell's equations can take two different forms corresponding to transverse magnetic (TM) or transverse electric (TE) waves.",
"The two-dimensional conservative form in general is written as $\\frac{ \\partial \\mbox{$u$}}{\\partial t}+\\frac{ \\partial \\mbox{$f$}(\\mbox{$u$})}{\\partial x}+\\frac{ \\partial \\mbox{$g$}(\\mbox{$u$})}{\\partial y}=\\mbox{$s$}.$ The vectors in equation (REF ) for the TM waves are $\\mbox{$u$}\\!=\\!\\left(\\begin{array}{c}B_{x} \\\\ B_{y} \\\\ D_{z}\\end{array} \\right), \\:\\mbox{$f$}\\!=\\!\\left(\\begin{array}{c}0 \\\\ -D_{z}/\\varepsilon \\\\ -B_{y}/\\mu \\end{array} \\right),\\:\\mbox{$g$}\\!=\\!\\left(\\begin{array}{c}D_{z}/\\varepsilon \\\\ 0 \\\\ B_{x}/\\mu \\end{array} \\right)\\mbox{$s$}\\!=\\!\\left(\\begin{array}{c}0 \\\\ 0 \\\\ -J_{iz}\\end{array} \\right)$ while that for the TE waves are $\\mbox{$u$}\\!=\\!\\left(\\begin{array}{c}B_{z} \\\\ D_{x} \\\\ D_{y}\\end{array} \\right), \\:\\mbox{$f$}\\!=\\!\\left(\\begin{array}{c}D_{y}/\\varepsilon \\\\ 0 \\\\ B_{z}/\\mu \\end{array} \\right),\\:\\mbox{$g$}\\!=\\!\\left(\\begin{array}{c}-D_{x}/\\varepsilon \\\\ -B_{z}/\\mu \\\\ 0\\end{array} \\right)\\mbox{$s$}\\!=\\!\\left(\\begin{array}{c}0 \\\\ -J_{ix} \\\\ -J_{iy}\\end{array} \\right).$ The FVTD method solves the conservative Maxwell's equation in the integral form.",
"Usually a scattered field formulation is employed with the incident field assumed to be a solution of the Maxwell's equations in free space.",
"Integrating the differential form of the conservation law, represented by equation (REF ), in the absence of a source term over an arbitrary control volume $\\Omega $ $\\frac{\\partial \\int _{\\Omega } \\mbox{$u$}d\\mathcal {V}}{\\partial t}+\\int _{\\Omega }\\mbox{$\\nabla $}.",
"(\\mbox{$F$} (\\mbox{$u$})){d\\mathcal {V}}=0.$ $\\bf F$ is the flux vector with components $\\bf f$ ,$\\bf g$ ,$\\bf h$ in the Cartesian $x,y,z$ directions with superscript `s' indicating scattered field variables.",
"The integral form of the conservation law to be discretized is obtained by applying the divergence theorem as $\\frac{\\partial \\int _{\\Omega } \\mbox{$u$} d\\mathcal {V}}{\\partial t}+\\oint _{\\mathcal {S}}\\mbox{$F$} (\\mbox{$u$}).\\mbox{$\\hat{n}$} d{\\mathcal {S}}=0$ with $\\hat{n}$ the outward unit normal vector.",
"The two-dimensional spatially discretized form solved for in a scattered and cell-centered formulation in the present work is finally written as [14] $A_{k}\\frac{d \\mbox{${u}$}_{k}}{dt}+\\sum _{j=1}^{4} [(\\mbox{$\\mathcal {F}$} (\\mbox{$u$}).\\mbox{$\\hat{n}$}{\\mathcal {S}})_{j}]_{k}=0$ where the numerical flux $ [( \\mbox{$\\mathcal {F}$} (\\mbox{$u$}).\\mbox{$\\hat{n}$}{\\mathcal {S}})_{j}]_{k}$ approximates the average flux through face $j$ of cell $k$ and $A_{k}$ represents the area of the quadrilateral cells in structured discretized space.",
"In the present work the Maxwell's equations for TM or TE waves, in its semi-discretized form in equation (REF ), are solved using higher-order ENO [12], [13] based spatial discretization described above and a second-order Runge-Kutta time integration.",
"The ENO scheme is cast in a p-variable higher-order framework which results in highest ($m^{th}$ ) order accurate solutions in the steady state, even while using spatial approximations with $p < m$ based on an order of magnitude comparison of one or more selected field variable.",
"The scatterers are considered to be perfect electric conductors with the total tangential electric field $\\bf \\hat{n} \\times \\bf E= 0$ on the scatterer surface.",
"The scattered field is also assumed to be zero at the outer boundary of the computational domain where boundary conditions are based on characteristics."
],
[
"Numerical Results",
"Numerical results are presented for the canonical case of electromagnetic scattering from 2D perfectly conducting circular cylinders as shown in Fig.REF and compared with the exact solution.",
"A body confirming “O\" mesh defines the computational domain with PEC boundary conditions on the cylinder surface and characteristic based far field conditions at the outer boundary.",
"Results are presented for both TM and TE continuous harmonic incident fields.",
"Computations are performed for a fixed set of time periods of the incident harmonic wave, after which complex surface currents are obtained using a Fourier transform.",
"The bistatic Radar Cross Section (RCS) or scattering width is then computed using a far field transformation [16].",
"A discussion on the number of incident wave periods to be time-stepped for attaining sinusoidal steady state in a FDTD framework under harmonic incident excitation as attempted here is presented in Ref. [17].",
"The first problem considered is that of the circular cylinder subject to continuous harmonic incident TM illumination with $a/\\lambda = 4.8$ where $a$ is the cylinder radius and $\\lambda $ the wavelength of the incident wave [6], [14], [18].",
"Results are shown in terms of bistatic RCS and the absolute value of surface current after time stepping fixed incident time periods usually adequate for desired steady state response in such problems.",
"Figs.",
"REF a and REF b, shows sample results for a conventional implementation for different spatial orders of accuracy on an “O\" grid with 300 points in the circumferential direction corresponding to a resolution of 10 PPW on the scatterer surface after 5 time periods.",
"The number of points in the radial direction is always kept constant at 50.",
"A relatively lower resolution of 10 PPW on the cylinder surface is deliberately chosen to bring out the effect of the numerical discretization error on the solution obtained using different spatial orders of accuracy from fourth to first.",
"As expected, the highest fourth-order accurate solutions are closest to the exact solution with first and second-order accurate solutions showing significant deviation away from near-specular-regions.",
"The monostatic point is located at $\\pm 180^{o}$ in the bistatic plot with $0^{o}$ the perfect shadow.",
"The same problem is now solved with a p-variable method with $m=4$ .",
"An order of magnitude comparison of scattered and incident cellwise value of $D_z$ is used to fix the local (cellwise) order of accuracy $(p \\le 4)$ of the spatial operator.",
"Results are presented after 5 time periods in Figures REF a and REF b. and compared with exact and conventional fourth-order results.",
"Results from $p$ -variable method match exactly with conventional fourth-order results.",
"Fig.REF shows the percentage of the computational domain over the entire simulation time processed by first, second, third and fourth-order spatial operators while retaining an overall fourth-order accuracy.",
"Figure: a/λ=4.8a/\\lambda = 4.8, continuous harmonic TM illumination,different orders of accuracy, p-constant.",
"(a) Surface Current Density(b) Bistatic RCSFigure: a/λ=4.8a/\\lambda = 4.8, continuous harmonic TM illumination,(a) Surface Current Density(b) Bistatic RCS; p- variable.Figure: Computational work distribution for pp-variable method; TM caseFigure: Bistatic RCS a/λ=9.6a/\\lambda = 9.6, continuous harmonic TE illumination,(a) p- variable.",
"(b) conventional.Figure: Computational work distribution for pp-variable method; TE caseThe next problem considered is that of illumination by a continuous harmonic incident TE wave and $a/\\lambda = 9.6$ [6], [14], [18].",
"The “O\" grid with 600 points in the circumferential direction is taken so that the resolution on the scatterer surface again corresponds to 10 PPW.",
"Again, a deliberately coarse discretization is chosen to bring out the effect of spatial order of accuracy on the obtained solution.",
"Figure REF a compares the bistatic RCS with first, second, third and fourth-order accuracy after 5 time periods.",
"The TE solution also starts deviating from the exact solution as formal spatial order of accuracy goes down and this is especially apparent away from the near-specular-region.",
"The problem is solved with a p-variable method and $m=4$ .",
"The choice of spatial order $p$ is based on an order of magnitude comparison of the scattered and incident value of $B_z$ .",
"Figure REF b compares the solution obtained with conventional fourth-order results.",
"Again an almost exact match is obtained.",
"Fig.REF lists the percentage of the computational domain processed over time by spatial operators of first, second, third and fourth-order accuracy while retaining formal fourth-order accuracy.",
"Figure: Schematic of the NACA 0012 airfoil illuminated with an incident fieldFigure: Bistatic RCS a/λ=10a/\\lambda = 10, continuous harmonic TM illumination,p- variable and conventional.Figure: Computational work distribution for pp-variable method; airfoil caseWe also consider scattering from a perfectly conducting NACA 0012 airfoil as shown in Fig.",
"REF .",
"The airfoil chord length is 10 times the wavelength of the incident harmonic TM wave at broadside incidence [6], [14], [18].",
"Results are obtained using a body-fitted “O\" grid with 200 points around the airfoil and 50 in the normal direction.",
"Figure REF compares RCS results after 5 time periods using regular fourth-order spatial accuracy and p-variable fourth-order ($m=4$ ).",
"Both results are compared with a “reference solution\" obtained using regular fourth-order spatial accuracy but on a much finer grid with 1600 points around the airfoil and time stepped for 10 time periods.",
"Again, like in the case of the circular cylinder an almost exact match is obtained between the conventional and p-variable method of the same formal accuracy.",
"Fig.REF lists the percentage of the computational domain over time processed by spatial operators $p \\le 4$ .",
"The trend is similar to that for scattering from perfectly conducting circular cylinders.",
"Variation in computing cost with order of accuracy for a 2D ENO scheme is seen to follow an arithmetic progression [19].",
"A linear regression analysis of this data yields the computing cost per-cell at the $p^{th}$ -order accuracy to be, $C_p = C_1 + 3.55(p-1)$ where, the data is normalized with respect to the cost per-cell for a first-order accurate scheme (i.e.",
"$C_1$ ).",
"For a $p$ -variable method with $m=4$ , total computing cost ($C_{total}$ ) can be written as, $C_{total} = \\sum _{p=1}^4 C_p n_p = (n_1 + n_2 + n_3 + n_4 ) C_1 + 3.55 n_2 + 7.1 n_3 + 10.65 n_4$ where, $C_p$ is the computational cost per-cell at $p^{th}$ level, and $n_p$ the total number of cells being processed at $p^{th}$ level.",
"On the other hand, the uniformly 4th-order accurate scheme will incur a cost of $\\left(n_{T} C_4 = n_{T}(c_1 + 3.65\\times 3)\\right)$ work units, where $n_T$ is the total number of cells on the domain.",
"Table REF shows the saving in computational cost over conventional fourth-order method in terms of work units assuming $C_1 = 1$ unit.",
"Table: Saving in computing time with pp-variable method (m=4m=4)"
],
[
"Conclusion",
"Desired higher-order spatial accuracy can be maintained, while using lower-order spatial operators in substantial parts of the computational domain in a p-variable FVTD method for solving EM scattering problems.",
"Lower-order spatial operators come at much reduced computational cost and can cut down considerably on simulation time while retaining desired higher-order accuracy using the present method.",
"An order of magnitude comparison of scattered and incident cell-centered EM field variables is used to decide on the local order of accuracy of the spatial operator.",
"The local spatial order of accuracy can vary in space and time and the proposed method can be easily integrated with existing higher-order FVTD techniques.",
"The current implementation uses the ENO family to access spatial operators of desired order of accuracy as dictated by the order of magnitude comparison.",
"Results are presented for the canonical case of EM scattering from a perfectly conducting circular cylinder as well as that of an airfoil."
]
] | 1709.01676 |
[
[
"IAD: Interaction-Aware Diffusion Framework in Social Networks"
],
[
"Abstract In networks, multiple contagions, such as information and purchasing behaviors, may interact with each other as they spread simultaneously.",
"However, most of the existing information diffusion models are built on the assumption that each individual contagion spreads independently, regardless of their interactions.",
"Gaining insights into such interaction is crucial to understand the contagion adoption behaviors, and thus can make better predictions.",
"In this paper, we study the contagion adoption behavior under a set of interactions, specifically, the interactions among users, contagions' contents and sentiments, which are learned from social network structures and texts.",
"We then develop an effective and efficient interaction-aware diffusion (IAD) framework, incorporating these interactions into a unified model.",
"We also present a generative process to distinguish user roles, a co-training method to determine contagions' categories and a new topic model to obtain topic-specific sentiments.",
"Evaluation on large-scale Weibo dataset demonstrates that our proposal can learn how different users, contagion categories and sentiments interact with each other efficiently.",
"With these interactions, we can make a more accurate prediction than the state-of-art baselines.",
"Moreover, we can better understand how the interactions influence the propagation process and thus can suggest useful directions for information promotion or suppression in viral marketing."
],
[
"Introduction",
"Social networks are the fundamental medium for the diffusion of information.",
"This diffusion process imitates the spread of infectious disease.",
"Specifically, when a user forwards a contagion (such as a political opinion or product), infection occurs and the contagion gets spread along the edges of the underlying network.",
"The process of a user examining the contagions shared by their neighbours and forwarding some of them results in the information cascades.",
"For the cascade of one contagion, both the users on the cascade and their neighbours are exposed to the contagion, and their behaviors and decisions will be influenced not only by the contents of the contagion but also by their social contacts.",
"In order to effectively employ information diffusion for viral marketing, it is essential to understand the users' adoption decisions under such influences.",
"Much work has been done to understand the information dynamics in social networks, including theoretical models and empirical studies.",
"However, most of the existing studies assume that each piece of information spreads independently [1], [2], [3], [4], [5], [6], [7], regardless of the interactions between contagions.",
"In the real world, multiple contagions may compete or cooperate with each other when they spread at the same time.",
"For example, the news about the banning of Samsung Galaxy S7 in airports may promote the spreading of the battery explosion events of S7, but suppress the news that Samsung is releasing other exciting products.",
"Thus, in this example, the contagion-contagion interaction can be seen as a “competition\" between the popularity of two pieces of information.",
"Taking the interactions into account is crucial to address the question of how much a user would like to adopt a contagion.",
"Recent diffusion models have started to consider interactions between contagions [8], [9], [10], [11], [12], [13], [14], [15], [16], however, in most cases, the interactions they learned are latent factors and thus are difficult to understand.",
"For example, in [9], the interactions it considered are between latent topics, making the promotion or suppression effects difficult to interpret.",
"Specifically, given two contagions that are of unrelated content or subject matter, it is difficult to infer whether they will interact with each other when they spread simultaneously.",
"Actually, what interests us is the interactions among explicit categories, namely whether contagions belonging to one category (say food) would have some positive/negative effects on the spreading of contagions belonging to another category (say health).",
"These interactions can be used to design viral marketing strategies to promote or suppress some products or news.",
"For example, if it can be inferred that contagions belonging to sports usually have positive effects on the adoption of energy drinks, advertisements on energy drinks can be exhibited alongside with sports news in a user's input stream of posts to promote the sales.",
"However, it is challenging to assign each contagion to its category by human due to the large volume in social networks.",
"How to find an efficient way to classify contagions with only minimum supervision is one of the key challenges in this work.",
"Even if a methodology is proposed to obtain explicit categories, the starting point is always a set of latent topics.",
"In addition to the categories that each contagion belongs to, recent studies show that the sentiments exhibited in the contagions may also affect the dynamics of information diffusion [17], [18].",
"Suppose that some user has just been infected by negative news (e.g, disasters or losing games) and thus in bad mood, and when she is exposed to a funny story, it is less likely for her to forward it.",
"On the contrary, in happy days (e.g., holidays), users are more likely to forward positive news than negative news.",
"Thus, sentiment analysis is crucial for information diffusion as well and can provide a new dimension to understand the users' adopting behaviours.",
"However, how to effectively uncover the sentiments from contagions remains an open problem.",
"Furthermore, as the sentiments and topics are both extracted from the contagions, they are hence not independent of each other and their coupling relations should also be taken into account in the information diffusion process, which has not been studied in previous works.",
"Apart from the categories and sentiments extracted from the contagions, social roles have also been proved to play important roles in the information diffusion process [19].",
"Besides, a user may play multiple roles with respect to different communities, and each social role may present different influences on their neighbours.",
"To study how social roles affect the diffusion process, it is necessary to distinguish the users' social roles and then capture their interactions when modeling the user's adoption decisions.",
"Since there are interactions among contagions and the interactions among users in the information diffusion process, it is natural to ask whether there are interactions between users and contagions.",
"The answer is obvious since most of the users have their own preferences on some messages, and they would like to forward those to their tastes and skip the others.",
"Although a few methods have been proposed to estimate a user's taste on a specific topic, it remains an open question from the perspective of information diffusion, especially when complex interactions are involved.",
"Figure: An example of the interacting scenario.",
"User u a u_a is exposed to contagions {m 1 m_1, ..., m K m_K} (here K=2K=2) and m i m_i (forwarded by u a u_a's neighbour u b u_b), and is examining whether to adopt m i m_i.",
"u a u_a 's decision is influenced by the interaction between u a u_a and m i m_i, the interaction between u a u_a and u d u_d, and interactions among m i m_i and other exposing contagions (m 1 m_1 and m 2 m_2).Motivated by the above examples and prospects, there is a clear opportunity to extend the present understanding of information diffusion by jointly studying the interactions existed among the contagions and users.",
"To this end, this paper focuses on the discovery of the interactions of users and contagions in the information diffusion process, as well as the impact of these interactions on the user's adoption behavior.",
"The scenario we study is that when a user is exposed to a set of contagions posted or forwarded by her neighbours, whether she would like to forward a specific one of them.",
"The task we conduct is hence to predict the infection probability by exploiting the inherent popularity of the contagion as well as the interactions among contagions and users.",
"We approach this problem through an interaction-aware diffusion model (IAD), which considers three kinds of interactions: (1) User-Contagion Interaction, (2) User-User Interaction, (3) Contagion-Contagion Interaction.",
"Fig.",
"REF describes an example of the three kinds of interactions when user $u_a$ is exposed to a set of contagions posted by her neighbours.",
"When $u_a$ is examining whether to adopt the contagion $m_i$ posted by $u_d$ , there exists User-User interactions between $u_a$ and $u_d$ , User-Contagion Interaction between $u_a$ and $m_i$ , as well as Contagion-Contagion Interaction between $m_1$ and $m_i$ .",
"We then describe each contagion from two views, say category and sentiment, and refine the contagion-related interactions and capture Category-Category Interaction, Sentiment-Sentiment Interaction, User-Category Interaction, and User-Sentiment Interaction.",
"As the interactions are rather complex and often exhibit strong coupling relations, rather than exploiting each kind of the interactions separately, we incorporate all the interactions into a unified framework.",
"Due to the large volume of contagions and users in the online social networks, learning interactions for each pair of contagions and users is impractical.",
"To address this challenge, we use the interactions among the categories of users and contagions instead.",
"Specifically, we first apply a mixture of Gaussians model to explain the generation process of user network features, and use EM algorithm to extract the social role distribution for each user.",
"We then propose a new topic model to capture topics and sentiments simultaneously for each contagion.",
"To get the category of each contagion, a classification approach based on co-training [20] is then developed, with only a small number of labeled data.",
"Overall, our model can statistically learn the interactions effectively, aiming to better understand the information diffusion process and provide a more predictive diffusion model.",
"Our contributions are summarized as follows: 1) We propose an IAD framework to model the user's forwarding behavior by jointly incorporating various kinds of interactions: interactions among contagions, interaction among users as well as interactions among users and contagions.",
"This framework provides new insights into how forwarding decisions are made.",
"2) Due to the large volume of interactions to learn, we devise a co-training based method to classify the contagions to categories, and apply a generative process to obtain the social roles for users, to significantly reduce the complexity of the fitting process.",
"3) To obtain the sentiment-related interactions from contagions, we develop a new topic model called Latent Dirichlet Allocation with Sentiment (LDA-S), a variation of LDA model [21], to obtain the sentiment and topic simultaneously in short texts.",
"4) Experimental evaluation on a large-scale Weibo dataset [22] shows that IAD outperforms state-of-art works in terms of F1-score, accuracy and fitting time.",
"Besides, the estimated interactions reveal some interesting findings that can provide compelling principles to govern content interaction in information diffusion.",
"The remainder of this paper is organized as follows.",
"Section 2 introduces the related works.",
"In Section 3, we describe the proposed IAD framework.",
"Section 4 describes how to classify contagions to explicit categories and Section 5 illustrates the process to infer sentiment-related interactions.",
"In Section 6, extensive experiments have been conducted to show the effectiveness of the proposed approach.",
"Section 7 concludes this paper.",
"In recent years, researchers have extensively studied the information diffusion in social networks [23].",
"A collection of models are proposed to explain the diffusion process from various perspectives [24], [25], [26], [27], [28], [29], [30], [31], while some other models are proposed to predict whether a piece of information will diffuse [32], [9].",
"However, most of the prior models assume the spreading of each piece of information is independent of others, e.g., the Linear Threshold Model [33], [34], the Independent Cascade Model [1], [2], SIR and SIS Model [3], [4].",
"The diffusion of multiple contagions has been covered in several recent works [12], [13], [14], [15], [16].",
"The scenario discussed by these works is that one contagion is mutually exclusive to others, i.e., only involving the competition of contagions.",
"In [8], an agent-based model is employed to study whether the competition of information for user's finite attention may affect the popularity of different contagions, but this model does not quantify the interactions between them.",
"Our research is not limited to the mutual exclusivity condition, but instead a user can adopt multiple contagions.",
"We offer a comprehensive consideration for the inter-relations of the contagions in online social networks.",
"The most related work to ours is the IMM model proposed in [9], which statistically learns how different contagions interact with each other through the Twitter dataset.",
"It models the probability of a user's adoption of information as a function of the exposure sequence, together with the membership of each contagion to a cluster.",
"However, this model doesn't consider the user roles, which has been proved to play an important role in information diffusion in our work.",
"In addition, the clusters in this model are latent variables without real-world meanings.",
"In contrast, our proposal can infer interactions among explicit categories, which are easy to interpret.",
"Nonetheless, IMM model is implemented as a baseline to compare with.",
"Other studies [10], [11], [35] also neglect the influence of user roles' interactions, and do not discover the interactions of actual categories of contagions, which are different from our work."
],
[
"Sentiment Analysis",
"Although recent works suggest the sentiments in the contents can play important roles in various applications such as product and restaurant reviews [36], stock market prediction [37], few existing studies quantify the effects of contents sentiment on the dynamics of information diffusion.",
"Empirical analysis on German political blogosphere indicates that people tend to participate more in emotionally-charged (either positive or negative) discussions [38].",
"A recent study on Twitter exhibits the effect of sentiment on information diffusion [18], and reveals different diffusion patterns for positive and negative messages respectively.",
"However, different from our IAD model, these works still treat each contagion in isolation and thus do not take the interactions into account.",
"Besides, most of the previous studies try to extract only the sentiments.",
"However, sentiments polarities are often dependent on topics or aspects.",
"Therefore, detecting on which topics of the users are expressing their opinions is very important.",
"Several models have been proposed to infer the topic and sentiment simultaneously.",
"Mei et al.",
"[39] propose the TSM model which can reveal the latent topical facets in a Weblog collection, the subtopics in the results of an ad hoc query, and their associated sentiments.",
"Lin et al.",
"[36], [40] propose a novel probabilistic modeling framework based on LDA, called joint sentiment/topic model (JST), which detects sentiment and topic simultaneously from a text.",
"This model assumes that each word is generated from a joint topic and sentiment distribution, and hence doesn't distinguish the topic word and opinion word distributions.",
"Liu et al.",
"[41] propose a topic-adaptive sentiment classification model which extracts text and non-text features from twitters as two views for co-training.",
"Tan et al.",
"[42] propose a LDA based model, Foreground and Background LDA (FB-LDA), to distill foreground topics and filter out longstanding background topics, which can give potential interpretations of the sentiment variations.",
"There are some other topic models considering aspect-specific opinion words [43], [44], [45], [46].",
"A recently proposed TSLDA model can estimate different opinion word distributions for individual sentiment categories for each topic [37], and has been successfully applied to stock prediction.",
"One weakness of TSLDA is that it divides a document into several sentences and sample the topic and sentiment of each sentence.",
"Therefore, its performance is limited when it is applied to Weibo where most of the messages have only one or two sentences.",
"The other weakness is that it lacks prior information, making it difficult to achieve good results for short texts.",
"To address the aforementioned problems, we propose a variation of TSLDA model, namely LDA-S, to make it work for short texts such as Weibo and Twitter."
],
[
"Interaction-Aware Diffusion Framework",
"In this section, we first state and formulate the problem, and then propose our framework and the corresponding learning process.",
"Before going into details of IAD framework, we define some important notations shown in Table REF .",
"Table: Description of Symbols in IAD Framework"
],
[
"Problem Statement",
"In a social network, when some new contagion is originated from a user, we assume that the user's neighbours would see this contagion, or would be exposed to this contagion.",
"This assumption is consistent with [9].",
"The exposed contagion is called an exposure.",
"Since users have limited attention [8], we make the assumption as [9] that at a given time, a user can read through all the contagions her neighbours have forwarded, but only the most recent $K$ exposures that she can keep in mind.",
"In social networks like Weibo and Twitter, tweets in a user's reading screen are arranged in time descending order, i.e., users will first read the most recent contagions and then go backward.",
"Please note that the dataset that we analyze was collected in 2012, and at that time Weibo and Twitter still showed tweets in the reversed chronological order.",
"Though they have stopped showing contagions in this simple order at present, our model can still work if we could identify which set of contagions are read simultaneously by a user.",
"Therefore, there is a sliding attention window going back $K$ contagions, and the $K$ contagions in the window may affect a user's adoption behavior.",
"Then the problem we focus is that when a user reads a contagion that has been forwarded by one of her neighbours, given the sequence of contagions the user has previously read, what's the probability of the user adopting this contagion.",
"Figure REF describes the interaction scenario studied in this paper, where the set {$m_1$ ,$m_2$ ,...$m_K$ } is a sequence of $K$ contagions user $u_a$ has read and kept in mind, and $m_i$ ($i\\ne 1,2,...,K$ ) is the contagion which is previously forwarded by a user $u_d$ and now examined by $u_a$ .",
"$u_a$ will determine whether to adopt (i.e., forward) $m_i$ .",
"In this scenario, the forwarding decision made by $u_a$ is not only decided by the inherent characteristics of $m_i$ , but also by external interactions described as follows: User-Contagion Interaction: The interaction between the examining user and the examined contagion.",
"As shown in Fig.",
"REF , it is $u_a$ 's preference over $m_i$ .",
"User-User Interaction: The interaction between the examining user and the neighbour who has forwarded the examined contagion previously.",
"In Fig.",
"REF , it is the effect $u_b$ has on $u_a$ .",
"Contagion-Contagion Interaction: The interaction among the examined contagion and other contagions the user has read recently.",
"In Fig.",
"REF , it is the effect contagions $m_1$ and $m_2$ ($K=2$ ) has on $m_i$ .",
"Given a collection of the interacting scenarios, our task is to model the users' adoption behaviour by incorporating the aforementioned interactions, and fitting the model to infer the interactions.",
"Meanwhile, we can make more accurate predictions on users' adoption behaviors.",
"The problem will be formulated in the next subsection."
],
[
"Formulation",
"According to the interacting scenario, given {$m_1$ ,$m_2$ ,...$m_K$ } and $u_b$ , the probability of infection by $m_i$ to $u_a$ is $\\small P(I_{m_i(u_a)}|E_{m_i(u_b)}, E_{\\lbrace m_1,m_2,...,m_K\\rbrace }),$ where $I_{m_i(u_a)}$ is the infection of $u_a$ by $m_i$ , $E_{m_i(u_b)}$ is the exposure of $m_i$ which is forwarded by $u_b$ , and $E_{\\lbrace m_1,m_2,...,m_K\\rbrace }$ is the exposure set $\\lbrace m_1, m_2,..., m_K\\rbrace $ .",
"We make the same assumption as [9] that for any $k$ and $l$ , $E_{m_k}$ is independent of $E_{m_l}$ .",
"Applying Bayes' rule, we model Eq.",
"(REF ) by $\\small \\begin{split}&P(I_{m_i(u_a)}|E_{m_i(u_b)}, E_{\\lbrace m_1,m_2,...,m_K\\rbrace }) \\\\=&\\frac{P(I_{m_i(u_a)}) P(E_{m_i(u_b)}, E_{\\lbrace m_1,m_2,...,m_K\\rbrace }|I_{m_i(u_a)})}{P(E_{m_i(u_b)}, E_{\\lbrace m_1,m_2,...,m_K\\rbrace })} \\\\=&\\frac{P(I_{m_i(u_a)}) P(E_{m_i(u_b)}|I_{m_i(u_a)}) \\prod _{k=1}^{K} P(E_{m_k}|I_{m_i(u_a)})}{P(E_{m_i(u_b)}) \\prod _{k=1}^{K} P(E_{m_k})} \\\\=&\\frac{P(I_{m_i(u_a)}) \\frac{P(I_{m_i(u_a)}|E_{m_i(u_b)}) P(E_{m_i(u_b)})}{P(I_{m_i(u_a)})} }{P(E_{m_i(u_b)}) \\prod _{k=1}^{K} P(E_{m_k})} \\\\& \\times \\prod _{k=1}^{K} \\frac{P(I_{m_i(u_a)}|E_{m_k}) P(E_{m_k})}{P(I_{m_i(u_a)})} \\\\=&\\frac{P(I_{m_i(u_a)}|E_{m_i(u_b)})}{P(I_{m_i(u_a)})^K} \\prod _{k=1}^{K} P(I_{m_i(u_a)}|E_{m_k}).\\end{split}$ Here we need to model $P(I_{m_i(u_a)})$ , $P(I_{m_i(u_a)}|E_{m_i(u_b)})$ and $P(I_{m_i(u_a)}|E_{m_k})$ for each $k \\in \\lbrace 1,...,K\\rbrace $ , which are enforced between 0 and 1.",
"Since each contagion has its inherent infectiousness, $P(I_{m_i})$ is defined as the prior infection probability of $m_i$ , which can be obtained through dividing the number of its infections by the number of its exposures.",
"We define $\\Omega (u_a,m_i)$ as the effect user $u_a$ has on contagion $m_i$ (User-Contagion Interaction), $\\Delta (u_a,u_b)$ as the effect user $u_b$ has on user $u_a$ (User-User Interaction), and $\\Lambda (m_i, m_k)$ as the effect contagion $m_k$ has on contagion $m_i$ (Contagion-Contagion Interaction).",
"Then we model $P(I_{m_i(u_a)})$ , $P(I_{m_i(u_a)}|E_{m_i(u_b)}) $ and $P(I_{m_i(u_a)}|E_{m_k})$ as $\\small P(I_{m_i(u_a)}) \\approx P(I_{m_i}) + \\Omega (u_a,m_i)$ $\\small \\begin{split}P(I_{m_i(u_a)}| & E_{m_i(u_b)}) \\approx P(I_{m_i(u_a)}) + \\Delta (u_a,u_b) \\\\& \\approx P(I_{m_i}) + \\Omega (u_a,m_i)+ \\Delta (u_a,u_b)\\end{split}$ $\\small \\begin{split}P(I_{m_i(u_a)}| & E_{m_k}) \\approx P(I_{m_i}|E_{m_k}) + \\Omega (u_a,m_i) \\\\& \\approx P(I_{m_i}) + \\Lambda (m_i, m_k) + \\Omega (u_a,m_i)\\end{split}$ Example 1.",
"In Fig.",
"REF , assuming that $P(I_{m_i})=0.5$ , $\\Omega (u_a,m_i)=0.02$ , $\\Delta (u_a,u_b)=-0.03$ , $\\Lambda (m_i, m_{k1})=-0.04$ and $\\Lambda (m_i, m_{k2})=-0.05$ , then we can derive that $P(I_{m_i(u_a)}) \\approx 0.52$ according to Eq.",
"(REF ), $ P(I_{m_i(u_a)}| E_{m_i(u_b)}) \\approx 0.49$ according to Eq.",
"(REF ) and $P(I_{m_i(u_a)}| E_{m_{k1}}) \\approx 0.43$ according to Eq.",
"(REF ).",
"Integrating these into Eq.",
"(REF ), we can obtain that the probability of $u_i$ adopting $m_i$ is 0.327.",
"Please note that, as shown in Eq.",
"(REF ), Eq.",
"(REF ), and Eq.",
"(REF ), the proposed model adopts summations to combine the interaction matrices to the prior infection probability $P(I_{m_i})$ .",
"In addition, we also develop a model which adopts multiplications to combine them, however, the experimental results show that the model with summations performs better.",
"The possible reason is that, the form of Eq.",
"(REF ) will be like a linear function when adopting multiplications, and its expressive power would be weaker compared to that with the additive schema.",
"Thus we apply the additive model in this paper.",
"Though we have connected the infection probability with three interaction matrices: (1) $\\Omega \\in R^{|u| \\times |m|}$ , (2) $\\Delta \\in R^{|u| \\times |u|}$ , and (3) $\\Lambda \\in R^{|m| \\times |m|}$ , where $|u|$ is the number of users and $|m|$ is the number of contagions, these matrices are impractical to learn because $|u|$ and $|m|$ are extremely large in social networks.",
"Thus, to decrease the parameters to fit, we model User Role - Contagion Topic Interaction, User Role-Role Interaction and Contagion Topic-Topic Interaction instead.",
"Moreover, we also involve sentiments into IAD framework and model the User Role - Contagion Sentiment Interaction.",
"It will be described in detail in the next subsection."
],
[
"The Proposed Approach",
"Overview To decrease the fitted parameters in the interaction matrices, user roles and contagion categories are introduced, and then the interactions between user roles and categories can be learned efficiently.",
"To this end, we utilize the network structures to infer users' social roles, and use the contagion texts to extract the contagions' topics.",
"To obtain the sentiments and topics from contagions, we propose an extension of LDA model called LDA-S to extract sentiments.",
"The whole process of IAD framework is shown in Fig.",
"REF , comprising of the following five components: Figure: IAD framework.",
"User roles generation (C1): A generative process for user roles is proposed to distinguish different kinds of users.",
"Contagion latent topics extraction or Contagion Latent Topic-Sentiment Extraction (C2): We have two ways to deal with the contagion texts.",
"One way is to apply LDA model to extract latent topics, and the other is to extract both topics and sentiments with our proposed LDA-S.",
"The output of this component will be used as features for statistical model learning (C4) and contagion classification (C3).",
"Contagion classification (C3): Based on the latent topics from C2, a co-training method of contagion classification is proposed to assign the contagions with explicit categories.",
"Statistical model learning (C4): Based on the outputs of C1 and C2, a statistical model is learned.",
"Interactions inference (C5): Given the results of contagion classification (C3) and the statistical model (C4), interactions among contagions and users can be inferred.",
"Next we will introduce the process of user roles generation and contagion latent topic extraction in detail, and then describe statistical model learning.",
"C3 will be illustrated in Section .",
"Note that in C2 and C5, the dashed boxes are newly added parts over the earlier version of this work [48], in order to involve sentiments into the IAD framework, which will be illustrated in Section .",
"User Role-Role Interaction.",
"User roles are categorized into authority users, hub users and ordinary users in our work.",
"Intuitively, an authority user commonly has a large number of followers, indicating that the node of authority user has a large in-degree but its out-degree is small, while a hub user has lots of followees, which means the node of hub user has a small in-degree but large out-degree.",
"And an ordinary user usually does not have a lot of followers or followees, i.e., the in-degree and out-degree of the node are both small.",
"A user may play multiple roles, for instance, an authority user may also be a hub user to some extent, and therefore we adopt a probability distribution over social roles for each user.",
"Then we infer the interactions among different social roles.",
"The results can demonstrate how a user, with a specific roles distribution, influence other user's probability of adopting a contagion.",
"We use PageRank score [49], HITS authority and hub values [50], in-degree and out-degree scores as features of users.",
"A mixture of Gaussians model is proposed to explain the features generation process.",
"Specifically, we assume the features of each user is sampled as a multivariate Gaussian distribution.",
"Intuitively, users with the same roles have similar features and share the same multivariate Gaussian distribution, e.g., two authority users are both likely to have a large number of followers.",
"Define r = $[r_1,r_2,r_3]^\\top $ as a user role vector, representing a probability distribution over social roles for each user, e.g., $[0.5,0.2,0.3]^\\top $ .",
"Then for each role $r_j$ , we generate multivariate Gaussian distribution $u|r_j \\sim N(\\mu _j, \\Sigma _j)$ .",
"EM algorithm is used to extract the role distribution for each user.",
"After that, we determine each $r_j$ as the most relevant one of the three roles, according to the fact that authority users commonly have lots of followers and hub users have lots of followees.",
"Rather than modeling the User-User Interaction denoted by $\\Delta \\in R^{|u| \\times |u|}$ , we would model User Role-Role Interaction instead, which is denoted by $\\Delta _{role} \\in R^{|r| \\times |r|}$ .",
"$\\Delta _{role}(r_i,r_j)$ is the effect role $r_j$ has on role $r_i$ .",
"Define $ \\vartheta _{a,i}$ as the probability of user $u_a$ belonging to role $r_i$ , and $\\sum _{i}\\vartheta _{a,i}=1$ , and then $ \\Delta (u_a, u_b)$ in Eq.",
"(REF ) can be updated by $\\small \\Delta (u_a, u_b) = \\sum _i\\sum _j\\vartheta _{a,i}\\Delta _{role}(r_i,r_j)\\vartheta _{b,j}$ Contagion Topic-Topic Interaction.",
"Each contagion is assumed to have a distribution on several topics, and $t$ denotes the set of latent topics.",
"LDA [21] is used to extract the latent topic distribution of each contagion.",
"Then, instead of modeling $\\Lambda \\in R^{|m| \\times |m|}$ , we would model a matrix $\\Lambda _{topic} \\in R^{ |t| \\times |t|}$ , which denotes the Contagion Topic-Topic Interaction.",
"We define $ \\theta _{i,a}$ as the probability of contagion $m_i$ belonging to topic $t_a$ , and therefore $\\sum _{a}\\theta _{i,a}=1$ .",
"Let $\\Lambda _{topic}(t_a,t_b)$ denote the impact of topic $t_b$ has on topic $t_a$ .",
"Now, $ \\Lambda (m_i, m_k)$ in Eq.",
"(REF ) can be updated by $\\small \\Lambda (m_i, m_k) = \\sum _a\\sum _b\\theta _{i,a}\\Lambda _{topic}(t_a,t_b)\\theta _{k,b}$ Besides the Contagion Topic-Topic Interaction, we also propose a topic-sentiment model to get Contagion (Topic-Sentiment)-(Topic-Sentiment) Interaction to update Eq.",
"(REF ).",
"User Role - Contagion Topic Interaction.",
"Instead of learning $\\Omega $ , we build a matrix $\\Omega _{topic}^{role} \\in R^{ |r| \\times |t|}$ to denote the User Role - Contagion Topic Interaction.",
"Then $\\Omega (u_a,m_i)$ in Eq.",
"(REF ), Eq.",
"(REF ) and Eq.",
"(REF ) can be updated by $\\small \\Omega (u_a,m_i) = \\sum _j\\sum _b\\vartheta _{a,j}\\Omega _{topic}^{role}(r_j,t_b)\\theta _{i,b}.$"
],
[
"Model Learning",
"The input of our model is a set of interacting scenarios.",
"An example of the interacting scenario is shown in Fig.",
"REF , which consists of the examining user $u_a$ , the examined contagion $m_i$ , user $u_a$ 's neighbour $u_b$ who has forwarded the examined contagion, and the exposing contagion set {$m_1$ , $m_2$ , ..., $m_K$ } ($i\\ne 1,2,...,K$ ).",
"All the interacting scenarios comprise a set {$x_1$ , $x_2$ , ..., $x_n$ }, where $x_i$ is the $i$ th interacting scenario and $n$ is the total number of scenarios.",
"For each interacting scenario, we can observe whether the examining user has adopted the examined contagion or not, and denote it as $y_i \\in \\lbrace 0,1\\rbrace $ (1 for adoption and 0 for not).",
"Then the training set $\\lbrace (x_1,y_1), (x_2,y_2), ..., (x_n,y_n)\\rbrace $ will be obtained.",
"Let $\\pi (x_i)$ denote Eq.",
"(REF ) for simplicity.",
"Now $\\pi (x_i)$ can be updated by $\\Omega _{topic}^{role}$ , $\\Delta _{role}$ and $\\Lambda _{topic}$ , and the log-likelihood function is $\\small \\begin{split}&L(\\Omega _{topic}^{role}, \\Delta _{role}, \\Lambda _{topic}) \\\\= & \\sum _{i=1}^n(y_i log \\pi (x_i)+(1-y_i)log(1-\\pi (x_i)))\\end{split}$ Our goal is to estimate the parameters in $\\Omega _{topic}^{role}$ , $\\Delta _{role}$ and $\\Lambda _{topic}$ to maximize the log-likelihood function.",
"Stochastic gradient ascent is adopted to fit the model.",
"In each iteration in the parameters updating process, if it makes any variable with probability meaning smaller than 0 or larger than 1, we don't conduct any updating in this iteration, and go directly to the next iteration.",
"The interaction matrix $\\Lambda _{topic}$ and $\\Omega _{topic}^{role}$ learned through our model are comprised of latent topics, which are difficult to interpret.",
"In this section, we illustrate how to obtain interactions among explicit categories.",
"We define $|c|=15$ categories based on the Weibo dataset, involving advertisement, constellation, culture, economy, food, health, history, life, movie, music, news, politics, sports, technology and traffic.",
"One contagion on Weibo is always a short text with only a few words, and is mostly affiliated to only one specific class.",
"Thus, the multi-label classification techniques such as [47] are not applied in this paper.",
"To discover interactions among categories, contagions should be classified into categories first.",
"However, contagions spreading in Weibo [22] are not labeled to intrinsic categories.",
"Labeled contagions are extremely expensive to obtain because large human efforts are required.",
"Thus, only a few labeled contagions are available for learning.",
"A classification approach based on co-training [20] is proposed.",
"Co-training is a semi-supervised learning technique that assumes each example is described using two different feature sets (or views) that provide different, complementary information about the instance.",
"It first learns a separate classifier for each view using an initial small set of labeled examples, and then use these classifiers on the unlabeled examples.",
"The most confident predictions are iteratively added to the labeled training data.",
"Specifically, for our task, each contagion in the dataset is described in two distinct views.",
"One is the contagion itself, and the other is a set of the other contagions posted by the same user.",
"The intuition here is that contagions created from the same user are prone to have the similar category.",
"Then we build two classifiers for two views, and choose the latent topics as the features for each classifier.",
"As described in Section REF , contagion $m_i$ 's latent topic distribution, denoted by $ \\theta _{i,a}(a \\in 1,..|t|$ , where $|t|$ denotes the number of latent topics), can be extracted using LDA.",
"We define a contagion set $M_i = \\lbrace m_1, m_2,...,m_k\\rbrace $ to contain the other contagions created by the same user.",
"The latent topic distribution $ \\Theta _{i,a}(a \\in 1,..|t|)$ of $M_i$ is obtained by $\\frac{\\sum _{j=1}^{k}\\theta _{j,a}}{k}$ .",
"Now, the two classifiers are listed as follows, and LIBSVM [51] is used for multi-class classification.",
"Classifier 1: $ \\theta _{i,a}(a \\in 1,..|t|)$ as features for each contagion $m_i$ .",
"Classifier 2: $ \\Theta _{i,a}(a \\in 1,..|t|)$ as features for each contagion set $M_i$ .",
"We labeled a minimum number of contagions for each category by hand for training in the beginning.",
"The number of manually annotated contagions for each category is 100, and the total number of annotated data is 1500.",
"After the initial training process, two classifiers go through the unlabeled contagions to make predictions.",
"If the results from the two classifiers are the same for a contagion, this contagion is added to the labeled set and removed from the unlabeled set.",
"Then a new set for training is obtained, and another iteration starts.",
"In each iteration, there are some contagions moved from the unlabeled set to the labeled set.",
"After enough contagions being labeled, we can derive the following two interactions.",
"Contagion Category-Category Interaction.",
"If the set of contagions $ \\lbrace m_1, m_2,...,m_k\\rbrace $ belongs to category $c_i$ , the latent topic distribution $ \\varphi _{i,a}(a \\in 1,..|t|)$ of category $c_i$ can be obtained through $\\frac{\\sum _{j=1}^{k}\\theta _{j,a}}{k}$ .",
"We define $\\Lambda _{cate.}",
"\\in R^{|c| \\times |c|}$ to denote Contagion Category-Category Interaction, where $\\Lambda _{cate.",
"}(c_i, c_k)$ is the impact of category $c_k$ on $c_i$ , that is $\\small \\Lambda _{categ.",
"}(c_i, c_k) = \\sum _a\\sum _b\\varphi _{i,a}\\Lambda _{topic}(t_a,t_b)\\varphi _{k,b}$ User Role - Contagion Category Interaction.",
"Similarly, define $\\Omega _{cate.",
"}^{role} \\in R ^ {|r| \\times |c|}$ to denote User Role - Contagion Category Interaction, where $\\Omega _{cate.",
"}^{role}(r_i, c_j)$ is the interaction from user role $r_i$ to category $c_j$ , that is $\\small \\Omega _{categ.",
"}^{role}(r_i, c_k) = \\sum _b\\Omega _{topic}^{role}(r_i,t_b)\\varphi _{k,b}$"
],
[
"Sentiment-related Interactions",
"In this section, we first introduce the LDA-S model, and then discuss how to use this model to obtain the sentiment-related interactions."
],
[
"Motivation and Basic Idea",
"We consider how to incorporate sentiments into the IAD framework since a user's forwarding behavior may be affected by the sentiments expressed in the contagions.",
"A natural way is to extract the sentiments from each contagion, and then model their interactions if they are simultaneously exposed to a user.",
"However, extract only sentiments may not be enough as sentiment polarities are usually dependent on topics or domains [43].",
"In other words, the exact same word may express different sentiment polarities for different topics.",
"For example, the opinion word \"low\" in the phrase \"low speed\" may have negative orientation in a traffic-related topic.",
"However, if it is in the phrase \"low fat\" in a food-related topic, the word \"low\" usually belongs to the positive sentiment polarity.",
"Therefore, it is necessary to incorporate the topic information for sentiment analysis, in order to get topic-specific sentiments, namely topic-sentiments in this work.",
"This can have two benefits.",
"On one hand, extracting the sentiments corresponding to different topics can improve the sentiment classification accuracy.",
"On the other hand, particularly for this work, such method can refine the inferred interactions between topics.",
"Specifically, we use the interactions between topic-sentiments instead of the interactions between topics in the IAD framework, which would improve the accuracy of forwarding prediction.",
"Inspired by Topic Sentiment Latent Dirichlet Allocation (TSLDA) [37], we propose LDA-S, an extension of LDA [21] model, to infer sentiment distribution and topic distribution simultaneously for short texts.",
"The difference of LDA-S and TSLDA will be illustrated in Section REF .",
"LDA-S model consists of two steps.",
"The first step aims to obtain the topic distribution of each document (or contagion in our work), and then set the contagion's topic as the one that has the largest probability.",
"The second step gets the sentiment distribution of each document.",
"The opinion words are usually adjectives or adverbs, whereas the topics words are usually nouns.",
"The words in a document are classified into three categories, the topic words ($c=1$ ), the sentiment words ($c=2$ ) and the background words ($c=0$ ).",
"We adopt a sentiment word list called NTUSD [52], which contains 4370 negative words and 4566 positive words.",
"If a word is an adjective but not in the sentiment word list, the sentiment label of this word is set as neutral.",
"If a word is a noun, it is considered as a topic word.",
"Otherwise, it is considered as a background word.",
"In our model, different topics have different opinion word distributions.",
"For each topic, we distinguish opinion word distributions for different sentiment meanings such as positive, negative or neutral.",
"We will show the graphical model and the generation process in the following part."
],
[
"Generation Process",
".",
"Figure: Graphical Model Representation of LDA-S.Table: Description of Symbols in LDA-SFigure REF shows the graphical model of LDA-S. Shaded circles indicate observed variables, and clear circles indicate hidden variables.",
"A word is an item from a vocabulary indexed by $\\lbrace 1,...,V\\rbrace $ .",
"We represent words using standard basis vectors that have a single component equal to one and all other components equal to zero.",
"Thus, using superscripts to denote components, the $v$ th word in the vocabulary is represented by a $V$ -vector $w$ such that $w^v=1$ and $w^u=0$ for $u \\ne v$ .",
"A document $d$ is a sequence of $N$ words denoted by $\\textbf {w}=(w_{d,1},w_{d,2},...,w_{d,N})$ , where $w_{d,n}$ is the $n$ th word in the sequence.",
"A corpus is a collection of $M$ documents denoted by $D = \\lbrace \\textbf {w}_1,\\textbf {w}_2,...,\\textbf {w}_M\\rbrace $ .",
"Let $z^{t}_{d}$ denote the topic assignment of document $d$ and other notations is introduced in Table REF .",
"The generation process of LDA-S is shown as Algorithm 1.",
"[t] LDA-S Input: documents [1] Set the prior distributions $\\alpha $ , $\\beta $ , $\\lambda $ and $\\gamma $ Set the number of iterations $n$ $iter =1$ to $n$ Choose a distribution of background words $\\Phi ^{b}\\sim Dirichlet(\\alpha )$ each topic $k$ Choose a distribution of topic words $\\Phi _{k}^{t}\\sim Dirichlet(\\alpha )$ each sentiment $s$ of topic $k$ Choose a distribution of sentiment words $\\Phi _{k,s}^{o}\\sim Dirichlet(\\lambda )$ each document $d$ Choose a topic distribution $\\theta _{d}^{t}\\sim Dirichlet(\\beta )$ Choose a sentiment distribution $\\theta _{d}^{o}\\sim Dirichlet(\\gamma )$ each word $w_{d,n}$ Determine the category of $w_{d,n}$ the category is 0 choose a word $w_{d,n}\\sim Multinomial(\\Phi ^{b})$ the category is 1 choose a topic assignment $z_{d,n}^{t}\\sim Multinomial(\\theta _{d}^{t})$ , choose a word $w_{d,n}\\sim Multinomial(\\Phi _{z_{d,n}^{t}}^{t})$ the category is 2 choose a sentiment assignment $z_{d,n}^{o}\\sim Multinomial(\\theta _{d}^{o})$ , choose a word $w_{d,n}\\sim Multinomial(\\Phi _{z_{d}^{t},z_{{d,n}}^{o}}^{o})$ $\\theta _{d}^{t}$ ,$\\theta _{d}^{o}$ To generate a document, we will generate every word in order.",
"Based on Fig.",
"REF and the above generation process, we can get the generation probability of each word $w_{d,n}$ in document $d$ depending on the category of $w_{d,n}$ as shown in Eq.",
"(REF ).",
"$\\small \\left\\lbrace \\begin{aligned}p(w_{d,n})=p(w_{d,n}) & & {c=0}\\\\p(w_{d,n},z^{t}_{d,n}|d)=p(w_{d,n}|z^{t}_{d,n})\\cdot p(z^{t}_{d,n}|d) & & {c=1}\\\\p(w_{d,n},z^{o}_{d,n}|d,z^{t}_{d})=p(w_{d,n}|z^{o}_{d,n},z^{t}_{d})\\cdot p(z^{o}_{d,n}|d,z^{t}_{d}) & & {c=2}\\\\\\end{aligned}\\right.$ Then we get the approximate expression of Eq.",
"(REF ) in Eq.",
"(REF ), Eq.",
"(REF ) and Eq.",
"(REF ).",
"We will define some notations to explain the equations.",
"Note that we represent $w_{d,n}$ in document $d$ as $w^v$ in the corpus.",
"Let $B_{w^v}$ denote the number of times that the background word $w^v$ appears in all documents.",
"$n_{k,w^v}$ is the number of times that the topic word $w^v$ with the topic $k$ appears in all documents (i.e., corpus) and $\\alpha _{k,w^v}$ specifies the value of $\\alpha $ when the topic is $k$ and the word is $w^v$ .",
"Let $n_{z^{t}_{d},s,w^v}$ be the number of times that the sentiment word $w^v$ with the sentiment $s$ under the topic $z^{t}_{d}$ appears in all documents and $\\lambda _{z^{t}_{d},s,w^v}$ specifies the value of $\\lambda $ when the topic is $z^{t}_{d}$ , the sentiment is $s$ and the word is $w^v$ .",
"$n_{d,k}$ denotes the number of times that the document $d$ is endowed with topic $k$ while $\\beta _{d,k}$ specifies the value of $\\beta $ when the document is $d$ and the topic is $k$ .",
"$n_{d,s}$ is the number of times that the document $d$ is endowed with sentiment $s$ while $\\gamma _{d,s}$ specifies the value of $\\gamma $ when the document is $d$ and the sentiment is $s$ .",
"When $c=0$ , we only consider the background words.",
"Specifically, we use $\\alpha _{w^v}$ to denote the value of $\\alpha $ when the word $w^v$ is a background word, and the background word has no topic assignment in this work.",
"Then the generation probability of $w^v$ is $\\small p(w^v) \\propto \\frac{B_{w^v}+\\alpha _{w^v}}{\\sum _{v=1}^{V_0}(B_{w^v}+\\alpha _{w^v})}$ When $c=1$ , we only consider the topic words.",
"The generation probability of $w^v$ is $\\small \\begin{split}p(w^v,k|d)=p(w^v|k)\\cdot p(k|d) \\\\\\propto \\frac{n_{k,w^v}+\\alpha _{k,w^v}}{\\sum _{v=1}^{V_1}(n_{k,w^v}+\\alpha _{k,v})} \\\\\\cdot \\frac{n_{d,k}+\\beta _{d,k}}{\\sum _{k=1}^{K}(n_{d,k}+\\beta _{d,k})} \\\\\\end{split}$ When $c=2$ , we only consider the sentiment words.",
"When the topic of document $d$ is $z^{t}_{d}$ , the generation probability of $w^v$ is $\\small \\begin{split}p(w^v,s|d,z^{t}_{d})=p(w^v|s, z^{t}_{d})\\cdot p(s|d,z^{t}_{d}) \\\\\\propto \\frac{n_{z^{t}_{d},s,w^v}+\\lambda _{z^{t}_{d},s}}{\\sum _{v=1}^{V_2}(n_{z^{t}_{d},s,w^v}+\\lambda _{z^{t}_{d},s,w^v})} \\\\\\cdot \\frac{n_{d,s}+\\gamma _{d,s}}{\\sum _{s=1}^{S}(n_{d,s}+\\gamma _{d,s})} \\\\\\end{split}$ Then Eq.",
"(REF ) and Gibbs Sampling are implemented for inference in LDA-S in which we only consider words with category 1.",
"It will sequentially sample the hidden variables $z_{d,n}^t$ (the topic assignment of the $n$ th word in document $d$ ) given a set of observed variables $\\textbf {w}$ and a set of hidden variables ${\\textbf {z}_{-(d,n)}^t}$ .",
"A bold-front variable denotes the list of the variables.",
"For example, $\\textbf {w}$ denotes all the words in all documents.",
"${\\textbf {z}_{-(d,n)}^t}$ denotes all topic assignment variables $z^t$ except $z_{d,n}^t$ .",
"$-(w^v)$ stands for all the words except $w^v$ .",
"$n_{d,k,-w^v}$ denotes the number of times that words are endowed with the topic $k$ except the word $w^v$ .",
"${\\textbf {w}_{-w^v}}$ denotes all the words in all documents except the word $w^v$ .",
"The equation of topic sampling is shown in Eq.",
"(REF ).",
"$\\small \\begin{split}p(z_{d,n}^{t}=k|\\textbf {z}_{-(d,n)}^{t},{\\textbf {w}}) \\\\\\propto p(z_{d,n}^{t}=k, w_{d,n}=w^v|\\textbf {z}_{-(d,n)}^{t},{\\textbf {w}_{-w^v}}) \\\\\\propto \\frac{n_{d,k,-n}+\\beta _{d,k}}{\\sum _{k=1}^{K}(n_{d,k,-n}+\\beta _{d,k})} \\\\\\cdot \\frac{n_{k,-w^v}+\\alpha _{k,w^v}}{\\sum _{v=1}^{V_1}(n_{k,-w^v}+\\alpha _{k,w^v})}\\end{split}$ After we get the topic distribution of document $d$ , the topic that has the largest probability will be set as document $d$ 's topic.",
"For example, if $\\theta _d^t=(0.1, 0.2, 0.3, 0.4)$ , we will choose the last one as document $d$ 's topic.",
"If there are more than one topics with the same probability, we choose one of them randomly.",
"Then Gibbs Sampling will be adopted again together with Eq.",
"(REF ), in which we only consider words with category 2.",
"We use the observed variables $\\textbf {w}$ , $z^{t}_{d}$ as well as the hidden variable ${\\textbf {z}_{-(d,n)}^o}$ to sample the hidden variable $z_{d,n}^o$ which denotes the sentiment assignment of $n$ th word in document $d$ .",
"The equation of sentiment sampling is shown in Eq.",
"(REF ).",
"$\\small \\begin{split}p(z_{d,n}^{o}=s|\\textbf {z}_{-(d,n)}^{o},{\\textbf {w}},z^{t}_{d}) \\\\\\propto p(z_{d,n}^{o}=s, w_n=w^v|\\textbf {z}_{-(d,n)}^{o},{\\textbf {w}_{-w^v}},z^{t}_{d}) \\\\\\propto \\frac{n_{d,s,-n}+\\gamma _{d,s}}{\\sum _{s=1}^{S}(n_{d,s,-n}+\\gamma _{d,s})} \\\\\\cdot \\frac{n_{z^{t}_{d},s,-w^v}+\\lambda _{z^{t}_{d},s,w^v}}{\\sum _{v=1}^{V_2}(n_{z^{t}_{d},s,-w^v}+\\lambda _{z^{t}_{d},s,w^v})}\\end{split}$ After that, we can approximate the multinomial parameter sets with the samples from Eq.",
"(REF ) and Eq.",
"(REF ).",
"The distributions of topics and sentiments in document $d$ are shown in Eq.",
"(REF ) and Eq.",
"(REF ) respectively, $\\small \\begin{split}\\theta _{d,a}^t=\\frac{n_{d,a}+\\beta _{d,a}}{\\sum _{k=1}^{K}(n_{d,k}+\\beta _{d,k})}\\end{split}$ $\\small \\begin{split}\\theta _{d,b}^o=\\frac{n_{d,b}+\\gamma _{d,b}}{\\sum _{s=1}^{S}(n_{d,s}+\\gamma _{d,s})}\\end{split}$ where $a$ denotes one of the topics, and $b$ denotes one of the sentiments.",
"Then the joint distribution of topic $a$ and sentiment $b$ is obtained in Eq.",
"(REF ).",
"$\\small \\theta _{d,ab}^{t-o}=\\theta _{d,a}^t \\cdot \\theta _{d,b}^o$ The background word distribution is shown in Eq.",
"(REF ) in which we only consider the background words, the topic word distribution of topic $k$ is shown in Eq.",
"(REF ) in which we only consider the topic words and the sentiment word distribution of sentiment $s$ under the topic $k$ is shown in Eq.",
"(REF ) in which we only consider the sentiment words, where $r$ denotes one of the words in the corpus.",
"$\\small \\Phi _r^b=\\frac{B_r+\\alpha _r}{\\sum _{v=1}^{V_0}(B_{v}+\\alpha _{v})} \\\\$ $\\small \\Phi _{k,r}^t=\\frac{n_{k,r}+\\alpha _{k,r}}{\\sum _{v=1}^{V_1}(n_{k,v}+\\alpha _{k,v})}\\\\$ $\\small \\Phi _{r,k,s}^o=\\frac{n_{k,s,r}+\\lambda _{k,s,r}}{\\sum _{v=1}^{V_2}(n_{k,s,v}+\\lambda _{k,s,v})}\\\\$"
],
[
"With Prior Information",
"For the hyperparameters of this LDA-S, namely, $\\alpha $ , $\\beta $ , $\\lambda $ and $\\gamma $ , we set $\\alpha $ =0.1 and $\\beta $ =0.01 as TSLDA [37].",
"But for prior $\\lambda $ and $\\gamma $ , we adopt a sentiment word list to obtain them.",
"The intuition is that incorporating prior information or subjective lexicon (i.e., words bearing positive and negative polarity) are able to improve the sentiment detection accuracy [36].",
"The prior parameters of Dirichlet distribution can be regarded as the “pseudo-counts\" from “pseudo-data\".",
"To get the priors, we match the documents to the sentiment word list.",
"Then the number of the words with a specific sentiment $s$ appearing in a document $d$ can be obtained, which can be set as the value of $\\gamma _{d,s}$ .",
"Similarly, the value of $\\lambda _{k,v,s}$ can be set as the count of the word $v$ with sentiment $s$ under topic $k$ .",
"With the prior information, our experimental results show a significant improvement in the sentiment classification accuracy, but the results are omitted due to the limit of space."
],
[
"Comparision with TSLDA",
"TSLDA is designed for long documents which consist of a set of sentences, and is not suitable to short texts with only one or two sentences.",
"Specifically, the limit in length makes it difficult to achieve good learning results.",
"Compared to TSLDA, LDA-S adopts different assumptions and different sampling methods.",
"Moreover, LDA-S incorporates prior information at the beginning of the generation process, which is expected to be more suitable to short texts.",
"The differences are listed as follows.",
"Sampling Method.",
"With TSLDA, each sentence is supposed to express only one topic and one opinion on that topic, and we need to sample topics and sentiments for each sentence in the generation process.",
"But in LDA-S, each contagion (which usually consists of one or two sentences) bears a topic distribution and a sentiment distribution, and we don't involve sentences in the generation process.",
"Prior Information.",
"In the initialization step of TSLDA, each document and word are assigned with topics and sentiments randomly, limiting its performance.",
"In LDA-S, we use a paradigm word list consists of a set of positive and negative words, and compare each word token in the contagions against the words in the sentiment word list in the initialization of the generation process."
],
[
"Interactions with Sentiment",
"After the topic-sentiment distribution (as shown in Eq.",
"(REF )) is obtained, the Contagion (Topic-Sentiment) - (Topic-Sentiment) Interaction and User Role - Contagion (Topic-Sentiment) Interaction will be introduced in our framework.",
"Now, $\\pi (x_i)$ (i.e., Eq.",
"(REF )) can be updated by $\\Omega _{t-s}^{role}$ , $\\Delta _{role}$ and $\\Lambda _{t-s}$ .",
"Specifically, each contagion is assumed to have a distribution not only on topics but also on sentiments, and $ts$ denotes the set of tuples that contain a latent topic and a kind of sentiment.",
"LDA-S is used to extract the joint distribution of latent topic and sentiment on each contagion.",
"Therefore, the value of $|ts|$ is actually the number of topics (20 in our setting) times the number of sentiment polarities (3 in our setting).",
"Then, instead of modeling $\\Lambda _{topic} \\in R^{ |t| \\times |t|}$ , we would model a matrix $\\Lambda _{t-s} \\in R^{ |ts| \\times |ts|}$ , which denotes the Contagion (Topic-Sentiment) - (Topic-Sentiment) Interaction.",
"We define $ \\theta _{i,ax}$ as the probability of contagion $m_i$ belonging to topic $t_a$ and sentiment $s_x$ , and therefore $\\sum _{ax}\\theta _{i,ax}=1$ .",
"Let $\\Lambda _{t-s}(t_{ax},t_{by})$ denote the impact of topic-sentiment $t_{by}$ has on topic-sentiment $t_{ax}$ .",
"Then $ \\Lambda (m_i, m_k)$ in Eq.",
"(REF ) can be updated by $\\small \\Lambda (m_i, m_k) = \\sum _{ax}\\sum _{by}\\theta _{i,ax}\\Lambda _{t-s}(t_{ax},t_{by})\\theta _{k,by}$ Instead of learning $\\Omega $ , we build a matrix $\\Omega _{t-s}^{role} \\in R^{ |r| \\times |ts|}$ to denote the User Role - Contagion (Topic-Sentiment) Interactions.",
"Then $\\Omega (u_a,m_i)$ in Eq.",
"(REF ), Eq.",
"(REF ) and Eq.",
"(REF ) can be updated by $\\small \\Omega (u_a,m_i) = \\sum _j\\sum _{by}\\vartheta _{a,j}\\Omega _{t-s}^{role}(r_j,t_{by})\\theta _{i,{by}}$ Now, $\\pi (x_i)$ can be updated by $\\Omega _{t-s}^{role}$ , $\\Delta _{role}$ and $\\Lambda _{t-s}$ , and the log-likelihood function is $\\small \\begin{split}&L(\\Omega _{t-s}^{role}, \\Delta _{role}, \\Lambda _{t-s}) \\\\= & \\sum _{i=1}^n(y_i log \\pi (x_i)+(1-y_i)log(1-\\pi (x_i)))\\end{split}$ Our goal is to estimate the parameters in $\\Omega _{t-s}^{role}$ , $\\Delta _{role}$ and $\\Lambda _{t-s}$ to maximize the log-likelihood function.",
"Stochastic gradient ascent is adopted again to fit the model.",
"Then $\\Omega _{s}^{role}(r_j,t_{y})$ and $\\Lambda _{s}(t_x, t_y)$ can be obtained by $\\small \\Omega _{s}^{role}(r_j,t_y) = \\frac{1}{K}\\sum _b\\Omega _{t-s}^{role}(r_j,t_{by})$ $\\small \\Lambda _{s}(t_x,t_y) = \\frac{1}{K}\\sum _a\\frac{1}{K}\\sum _b\\Lambda _{t-s}(t_{ax},t_{by})$"
],
[
"Evaluation",
"In this section, we conduct experiments based on a public Weibo dataset to evaluate IAD framework, and then discuss various qualitative insights."
],
[
"Dataset",
"Weibo is a Twitter-like social network which provides microblogging service.",
"The Weibo dataset [22] provides a list of users who have forwarded contagions, as well as the forwarding timestamp.",
"Users' friendship links are also recorded.",
"Because of the crawling strategy, the distribution of retweet counts in different months is highly imbalanced.",
"Thus, we select the diffusion data from July 2012 to December 2012, in which the retweet count per month is large enough and the distribution is more balanced.",
"Consequently, we get 19,388,727 retweets on 140,400 popular microblogs.",
"We delete the inactive users without any retweets in this period and obtain 1,077,021 distinct users for the experiment.",
"Then we do statistical analysis to extract interacting scenarios from the dataset as illustrated in Sec.",
"REF .",
"Please note that we remove the interacting scenarios where all microblogs are neutral, and obtain a set of scenarios that have contagions with positive or negative sentiments.",
"As LDA-S is proposed to study how the sentiments can affect the diffusion process, thus the contagions that we focus on should be those whose sentiments are evident.",
"Based on this intuition, we refine the dataset and only retain the contagions whose positive portion or negative portion in the sentiment distribution is above 0.7, and thus obtain a new dataset consisting of contagions with greater sentiment intensities.",
"Thus, the obtained instances after filtering is actually a subset of the instances used in the earlier version of this work [48].",
"If the examined contagion is adopted, the interacting scenario is a positive instance, otherwise it is a negative instance.",
"We observe that the positive and negative instances are highly unbalanced in the dataset, so we sample a balanced dataset with equal numbers of positive and negative instances.",
"In total, we get 64,456 microblogs and 14,537,835 interacting scenarios when $K=1$ , while 75,945 microblogs and 17,462,853 interacting scenarios when $K=2$ .",
"We use 5-fold cross-validation, and set the number of latent topics as $|t|=20$ ."
],
[
"Baselines",
"We compare the performance of our proposed model with several existing models.",
"These methods are: IP Model.",
"Infection Probability Model is based on the Independent Cascade Model [1], [2], which assigns the infection probability of a contagion to be simply the prior infection probability.",
"IP model doesn't consider any interactions among users and contagions.",
"UI Model.",
"User Interaction (UI) Model is actually one component of our IAD framework.",
"UI model only considers the user-user interactions, more specifically, the user role-role interactions, and ignores the other interactions.",
"IMM Model [9].",
"IMM incorporates the interactions among contagions into its model.",
"IMM models the interactions between clusters (i.e., latent topics) of contagions.",
"At the same time, it learns to which cluster each contagion belongs to.",
"However, the clusters (i.e., latent topics) cannot be interpreted and thus prevent us getting deeper insights.",
"IAD w/o S. IAD w/o S refers to IAD framework without sentiment, which is also the proposal in our previous work [48].",
"IAD w/ TSLDA.",
"In our proposed IAD framework, we replace LDA-S with TSLDA algorithm [37], which intends to compare the performance difference between LDA-S and TSLDA.",
"To make a fair comparison, we use the same set of instances and the same setting of parameters.",
"In our proposal and the baselines, we set the predicting result to 0 if the predicting infection probability is less than 0.5, otherwise we set the predicting result to 1.",
"Our model and the baselines are evaluated in terms of Precision, Recall, F1-score, as well as Accuracy.",
"All experiments are performed on a dual-core Xeon E5-2620 v3 processor.",
"The code of our proposal has been made publicly available via https://www.dropbox.com/s/lp296h2omhhfw6e/IAD.zip?dl=0."
],
[
"Effectiveness",
"Table REF and Table REF show the performance of our proposal and the baselines when $K=1$ and $K=2$ respectively.",
"IAD w/ LDA-S denotes the IAD framework with our LDA-S method for extracting sentiments.",
"It can be observed that UI and IMM models perform better than IP model, indicating that interactions among users and among contagions are vital to user's forwarding decisions.",
"Moreover, since IMM outperforms UI, the interaction among contagions plays more important roles than the interaction among users.",
"It can be observed that our proposed IAD methods (no matter with or without sentiments) almost consistently outperform IP, UI and IMM, indicating that solely involving the interactions between contagions (e.g., IMM model) or between user interactions (e.g., UI) is not sufficient.",
"Besides, we need to involve both of the aforementioned interactions as well as the interactions between users and contagions to get a more accurate prediction.",
"Then we examine the effectiveness of the sentiment factors in the forwarding decision process, and observe that IAD w/ LDA-S and IAD w/ TSLDA both perform better than IAD w/o S in terms of F1-score when $K=1$ and $K=2$ .",
"Thus, it indicates that involving sentiments into the the framework is beneficial to the information diffusion prediction.",
"When comparing the performance of IAD w/ LDA-S and IAD w/ TSLDA, we can find that IAD w/ LDA-S outperforms IAD w/ TSLDA all the time in terms of both F1-score and Accuracy, indicating that our proposed LDA-S can better capture the effects of the sentiments on the information diffusion than TSLDA.",
"We also conduct experiments with a larger $K$ , and the results are not better than those when $K=1$ and $K=2$ .",
"The possible reason is that individual users have limited attention, and can only jointly consider a very limited number of contagions.",
"Table: Performance of IAD Methods with Varying Sentiment Intensities When K=1K=1Table: Performance of IAD Methods with Varying Sentiment Intensities When K=2K=2Moreover, to demonstrate how the sentiment intensity affects the prediction performance, we construct three datasets whose contagions are with different sentiment intensities.",
"Specifically, the positive or negative portions in the sentiment distribution in the three datasets are above $\\tau $ =0.6, $\\tau $ =0.7 and $\\tau $ =0.8, and we denote them as the dataset (a), (b) and (c) respectively for simplicity.",
"It is obvious that the sentiment intensities of contagions in (c) are generally greater than that in (a) and (b).",
"We perform the experiments on the three datasets, and can observe that our proposal performs better on the dataset whose sentiment intensities is greater.",
"For example, in terms of accuracy, when $|t|$ =10, the performance gaps between IAD with LDA-S and without LDA-S are 0.27%, 1% and 1.81% using dataset (a) and (b) and (c) respectively; when $|t|$ =20, the corresponding performance gaps are 0.27%, 1.07% and 1.27%.",
"It is intuitive as LDA-S is proposed to study how the sentiments can affect the diffusion process, and thus applying LDA-S on the contagions whose sentiments are evident is supposed to be more beneficial to the prediction than the contagions whose sentiments are mostly neutral.",
"The detailed results are shown in Table REF and Table REF .",
"These results indicate that sentiment is an important factor for information diffusion.",
"Please note that in addition to improving the predictive capability, taking sentiments into consideration can also provide a new perspective to understand the information diffusion process, and the analysis is described in Sec.",
"REF ."
],
[
"Efficiency",
"Taking the model complexity into consideration, IAD w/o S is much more efficient than IMM.",
"The number of parameters to learn in IAD w/o S is 469 (when $|r| = 3$ , and $|t| = 20$ ), whereas the corresponding number in IMM model is 2,070,520 and 2,141,740 respectively when $K=1$ and $K=2$ (obtained by $T \\times T \\times K + W \\times T$ , where $T = 20$ , $W = 103,506$ and $107,047$ when $K=1$ and $K=2$ ).",
"For IAD w/ LDA-S, the number of parameters to infer is 3789.",
"Fig.",
"REF compares the time cost in the learning process, and the results confirm the efficiency of the IAD models, especially the IAD w/o S model.",
"In particular, IAD w/o S is faster than IMM by an order of magnitude.",
"IAD w/ LDA-S and IAD w/ TSLDA take almost the same time in learning, and they are both slower than IAD w/o S but much faster than IMM.",
"When $K=2$ , similar trends can be observed."
],
[
"Analysis of Interactions",
"Throughout this section, we provide qualitative insights into the extent to which the interactions influence the adoption of contagions.",
"Due to limits of space, we only show the interaction results under IAD w/ LDA-S.",
"Please note that interaction results under IAD w/ LDA-S and IAD w/o S should be identical except sentiment-related interactions are further derived with LDA-S. After fitting the IAD w/ LDA-S model, $\\Delta _{role}$ , $\\Omega _{t-s}^{role}$ and $\\Lambda _{t-s}$ are obtained.",
"The results can be simply processed to obtain $\\Omega _{topic}^{role}$ and $\\Lambda _{topic}$ , and then $\\Lambda _{categ.",
"}$ and $\\Omega _{categ.",
"}^{role}$ can be derived by the category classification method.",
"In addition, we can also derive the sentiment-related interactions, i.e., $\\Omega _{s}^{role}$ and $\\Lambda _{s}$ .",
"Fig.",
"REF shows the inferred interactions when $|t|=20$ and $K=2$ .",
"User Role-Role Interaction.",
"In Fig.",
"REF (a), it can be observed that authority users are more likely to adopt contagions forwarded by other authority users, rather than those from hub users or ordinary users, which indicates a status gradient on social roles seniority.",
"Hub users would like to adopt contagions from ordinary users, rather than from hub users.",
"The ordinary users prefer to adopt contagions from other ordinary users or hub users, while don't forward the contagions from authority users with a larger probability.",
"The reason is that, although the number of adoptions from authoritative users is commonly large, the forwarding ratio, that is, (the number of forwarded contagions) / (the number of exposed contagions), from authoritative users is not necessarily high, due to the large exposure count in the denominator.",
"User Role-Contagion Sentiment Interaction.",
"Fig.",
"REF (b) shows that no matter what roles the users play, they are more likely to forward neutral and positive contagions than the negative contagions.",
"It is intuitive that the users commonly prefer positive contagions to the negative ones.",
"One possible reason why users prefer the neutral ones may be because the neutral contagions are not aggressive and less likely to conflict with other contagions, and thus easy to get accepted.",
"From the perspective of the user roles, hub users are more likely to forward contagions than authority users and ordinary users.",
"Particularly, the hub users like to forward the positive contagions very much.",
"This finding is consistent with Fig.",
"5(d), where the hub users like to forward advertisements, as most of the advertisements are supposed to be positive.",
"Contagion Sentiment-Sentiment Interaction.",
"Fig.",
"REF (c) shows that when a negative contagion meets another negative contagion, it can promote each other's propagation.",
"This phenomenon is reasonable as they probably express similar opinions and cooperate with each other to get spread.",
"The neutral sentiment seldom suppresses itself, neither does the positive sentiment.",
"The mutual suppression between positive contagions and negative contagions is very strong, and stronger than the suppressions between positive/negative and neutral contagions.",
"It is intuitive as positive and negative contagions commonly express opposite opinions, and it is more likely for them to compete rather than cooperate with each other.",
"User Role-Contagion Category Interaction.",
"Fig.",
"REF (d) shows that authority users are more likely to adopt contagions on economy, news, and politics, and don't like to forward advertisements.",
"One possible reason is that the authority users such as the news media give more attention to big events such as politics rather than small events in common life.",
"On the contrary, ordinary users prefer contagions such as movies and food, and they don't like to forward advertisements and political contagions.",
"Hub users tend to adopt contagions about advertisement and movies, and one possible reason is that they may be spam users.",
"Contagion Category-Category Interaction.",
"Fig.",
"REF (e) reveals how different categories of contagions compete or cooperate with each other to get propagated.",
"It can be observed that generally the entries in the matrix are negative, which indicates that relationships between different categories are mainly competition.",
"It validates the existing conclusion that attention is limited for individual users to adopt contagions [8].",
"It is obvious that the colors of the entries on the diagonal line are lighter, indicating that contagions affiliated to the same category are less likely to suppress each other.",
"It is intuitive that when similar topics meet each other, they tend to cooperate to become hot topics and thus attract more attention, making themselves easier to get spread.",
"It also shows that contagions belonging to food category are more likely to get adopted when simultaneously propagating with contagions belonging to other categories, i.e., the propagation of contagions on food are more likely to suppress the propagation of other contagions.",
"In addition, contagions about constellation and life also attract a lot of attention.",
"On the contrary, contagions belonging to categories such as the advertisement and news are less likely to suppress other contagions' propagation, revealing that commonly users don't like to forward them."
],
[
"Conclusion",
"In this paper, a new information diffusion framework called IAD is proposed to analyze the users' behaviors on adopting a contagion, in consideration of the interactions involving users and contagions as a whole.",
"With this framework, we can quantitatively study how these interactions would influence the propagation process.",
"To efficiently learn the interactions, we use a generative process to infer user roles and a co-training method to classify the contagions into explicit categories.",
"To involve sentiment factors into the user's forwarding behavior, we also propose a LDA-S model which are able to extract the sentiment distribution and topic distribution simultaneously from contagions.",
"Experimental results on the large-scale Weibo dataset demonstrate that IAD methods can outperform the state-of-art baselines in terms of F1-score, accuracy and runtime.",
"Moreover, IAD with sentiment can further improve the prediction performance of IAD without sentiments.",
"Last but not least, various kinds of interactions can be obtained and interesting findings can be observed, which are useful to various domains such as viral marketing."
],
[
"Acknowledgments",
"This work was supported in part by State Key Development Program of Basic Research of China (No.",
"2013CB329605), the Natural Science Foundation of China (No.",
"61300014, 61672313), and NSF through grants IIS-1526499, IIS-1763325, and CNS-1626432, and DongGuan Innovative Research Team Program (No.201636000100038).",
"[Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTION"
]
] | 1709.01773 |
[
[
"Higher order expansions for the probabilistic local Cauchy theory of the\n cubic nonlinear Schr\\\"odinger equation on $\\mathbb{R}^3$"
],
[
"Abstract We consider the cubic nonlinear Schr\\\"odinger equation (NLS) on $\\mathbb{R}^3$ with randomized initial data.",
"In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data.",
"By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work [2].",
"We further investigate a limitation of this iterative procedure.",
"Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion."
],
[
"Nonlinear Schrödinger equation",
"We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) on $\\mathbb {R}^3$ :Our discussion in this paper can be easily adapted to the cubic NLS on $\\mathbb {R}^d$ (and to other nonlinear dispersive PDEs).",
"For the sake of concreteness, however, we only consider the $d = 3$ case in the following.",
"See also the comment after Theorem REF for our particular interest in the three-dimensional problem.",
"${\\left\\lbrace \\begin{array}{ll}i \\partial _t u + \\Delta u = |u|^2 u \\\\u|_{t = 0} = u_0 \\in H^s(\\mathbb {R}^3),\\end{array}\\right.",
"}\\qquad ( t, x) \\in \\mathbb {R}\\times \\mathbb {R}^3.$ The cubic NLS (REF ) has been studied extensively from both the theoretical and applied points of view.",
"In this paper, we continue our study in [1], [2] and further investigate the probabilistic well-posedness of (REF ) with random and rough initial data.",
"Recall that the equation (REF ) is scaling critical in $\\dot{H}^\\frac{1}{2}(\\mathbb {R}^3)$ in the sense that the scaling symmetry: $u(t, x) \\longmapsto \\lambda u (\\lambda ^2 t, \\lambda x)$ preserves the homogeneous $\\dot{H}^{\\frac{1}{2} }$ -norm (when it is applied to functions only of $x$ ).",
"It is known that the Cauchy problem (REF ) is locally well-posed in $H^s(\\mathbb {R}^3)$ for $s \\ge \\frac{1}{2}$ [10].",
"See also [13], [26], [29], [18] for partialNamely, either for smoother initial data or under an extra hypothesis.",
"global well-posedness and scattering results.",
"On the other hand, (REF ) is known to be ill-posed in $H^s(\\mathbb {R}^3)$ for $s < \\frac{1}{2}$ [12].",
"In [2], we studied the probabilistic well-posedness property of (REF ) below the scaling critical regularity $s_\\text{crit} = \\frac{1}{2}$ under a suitable randomization of initial data; see (REF ) below.",
"In particular, we proved that (REF ) is almost surely locally well-posed in $H^s(\\mathbb {R}^3)$ , $s > \\frac{1}{4}$ .",
"Our main goal in this paper is to introduce an iterative procedure to improve this regularity threshold for almost sure local well-posedness.",
"Furthermore, we study a critical regularity associated with this iterative procedure.",
"By introducing a modified iterative approach, we then prove almost sure local well-posedness of (REF ) in an almost optimal range with respect to the original iterative procedure (Theorem REF ).",
"Beyond the concrete results in this paper, we believe that the iterative procedures based on (modified) partial power series expansions are themselves of interest for further development in the well-posedness theory of dispersive equations with random initial data and/or random forcing.",
"Such a probabilistic construction of solutions to dispersive PDEs first appeared in the works by McKean [34] and Bourgain [5] in the study of invariant Gibbs measures for the cubic NLS on $d$ , $d = 1, 2$ .",
"In particular, they established almost sure local well-posedness with respect to particular random initial data, basically corresponding to the Brownian motion.",
"(These local-in-time solutions were then extended globally in time by invariance of the Gibbs measures.",
"In the following, however, we restrict our attention to local-in-time solutions.)",
"In [8], Burq-Tzvetkov further elaborated this idea and considered a randomization for any rough initial condition via the Fourier series expansion.",
"More precisely, they studied the cubic nonlinear wave equation (NLW) on a three-dimensional compact Riemannian manifold below the scaling critical regularity.",
"By introducing a randomization via the multiplication of the Fourier coefficients by independent random variables, they established almost sure local well-posedness below the critical regularity.",
"Such randomization via the Fourier series expansion is natural on compact domains [15], [35] and more generally in situations where the associated elliptic operators have discrete spectra [47], [45].",
"See [2], [3] for more references therein.",
"In the following, we go over a randomization suitable for our problem on $\\mathbb {R}^3$ .",
"Recall the Wiener decomposition of the frequency space [48]: $\\mathbb {R}^3_\\xi = \\bigcup _{n \\in \\mathbb {Z}^3} (n+ (-\\frac{1}{2}, \\frac{1}{2}]^3)$ .",
"We employ a randomization adapted to this Wiener decomposition.",
"Let $\\psi \\in \\mathcal {S}(\\mathbb {R}^3)$ satisfy $\\operatornamewithlimits{supp}\\psi \\subset [-1,1]^3\\qquad \\text{and}\\qquad \\sum _{n \\in \\mathbb {Z}^3} \\psi (\\xi -n) =1 \\quad \\text{for any $\\xi \\in \\mathbb {R}^3$}.$ Then, given a function $\\phi $ on $\\mathbb {R}^3$ , we have $ \\phi = \\sum _{n \\in \\mathbb {Z}^3} \\psi (D-n) \\phi .$ We define the Wiener randomization $\\phi ^\\omega $ of $\\phi $ by $ \\phi ^{\\omega } := \\sum _{n \\in \\mathbb {Z}^3} g_n (\\omega ) \\psi (D-n) \\phi ,$ where $\\lbrace g_n \\rbrace _{n \\in \\mathbb {Z}^3} $ is a sequence of independent mean-zero complex-valued random variables on a probability space $(\\Omega , \\mathcal {F} ,P)$ .",
"The randomization (REF ) based on the Wiener decomposition of the frequency space is natural in view of time-frequency analysis; see [1] for a further discussion.",
"Almost simultaneously with [1], Lührmann-Mendelson [32] also considered a randomization of the form (REF ) (with cubes being substituted by appropriately localized balls) in the study of NLW on $\\mathbb {R}^3$ .",
"For a similar randomization used in the study of the Navier-Stokes equations, see also the work of Zhang and Fang [50], preceding [1], [32].",
"In [50], the decomposition of the frequency space is not explicitly provided but their randomization certainly includes randomization based on the decomposition of the frequency space by unit cubes or balls.",
"In the following, we assume that the probability distribution $\\mu _n$ of $g_n$ satisfies the following exponential moment bound: $\\int _{\\mathbb {R}^2} e^{\\kappa \\cdot x} d \\mu _{n} (x) \\le e^{c |\\kappa | ^2}$ for all $\\kappa \\in \\mathbb {R}^2$ and $n \\in \\mathbb {Z}^3$ .",
"This condition is satisfied by the standard complex-valued Gaussian random variables and the uniform distribution on the circle in the complex plane.",
"We now recall the almost sure local well-posedness result from [2] which is of interest to us.For simplicity, we only consider positive times.",
"By the time reversibility of the equation, the same analysis applies to negative times.",
"Theorem A Let $\\frac{1}{4} < s < \\frac{1}{2}$ .",
"Given $\\phi \\in H^s (\\mathbb {R}^3)$ , let $\\phi ^{\\omega }$ be its Wiener randomization defined in (REF ).",
"Then, the Cauchy problem (REF ) is almost surely locally well-posed with respect to the random initial data $\\phi ^\\omega $ .",
"More precisely, there exists a set $\\Sigma = \\Sigma (\\phi ) \\subset \\Omega $ with $P(\\Sigma ) = 1$ such that, for any $\\omega \\in \\Sigma $ , there exists a unique function $u = u^\\omega $ in the class: $S(t)\\phi ^\\omega + X^\\frac{1}{2}([0, T]) \\subset S(t) \\phi ^{\\omega } + C([0,T] ; H^\\frac{1}{2}(\\mathbb {R}^3)) \\subset C([0,T] ; H^s(\\mathbb {R}^3))$ with $T = T(\\phi , \\omega ) >0$ such that $u$ is a solution to (REF ) on $[0, T]$ .",
"Here, $S(t) := e^{it\\Delta }$ denotes the linear Schrödinger operator and the function space $X^\\frac{1}{2}([0, T]) \\subset C([0, T]; H^\\frac{1}{2}(\\mathbb {R}^3))$ is defined in Section below.",
"Note that the uniqueness statement of Theorem REF in the class: $S(t)\\phi ^\\omega + X^\\frac{1}{2}([0, T])$ is to be interpreted as follows; by setting $v = u - S(t) \\phi ^\\omega $ (see (REF ) below), uniqueness for the residual term $v$ holds in $X^\\frac{1}{2}([0, T])$ .",
"See also Remark REF below.",
"Recall that while the Wiener randomization (REF ) does not improve differentiability, it improves integrability (see Lemma 4 in [1]).",
"See [42], [27], [8] for the corresponding statements in the context of the random Fourier series.",
"The main idea for proving Theorem REF is to exploit this gain of integrability.",
"More precisely, let $z_1(t) := S(t) \\phi ^{\\omega }$ denote the random linear solution with $\\phi ^\\omega $ as initial data and write $u = z_1 + v.$ Then, we see that $v: = u - z_1$ satisfies the following perturbed NLS: ${\\left\\lbrace \\begin{array}{ll}i \\partial _tv + \\Delta v = |v + z_1|^2(v+z_1)\\\\v|_{t = 0} = 0,\\end{array}\\right.",
"}$ where $z_1$ is viewed as a given (random) source term.",
"The main point is that the gain of space-time integrability of the random linear solution $z_1$ (Lemma REF ) makes this problem subcriticalThe scaling critical Sobolev regularity $s_\\text{crit} = \\frac{1}{2}$ for (REF ) is defined by the fact that the $\\dot{H}^\\frac{1}{2}$ -norm remains invariant under the scaling symmetry (REF ).",
"Given $1 \\le r \\le \\infty $ , we can also define the scaling critical Sobolev regularity $s_\\text{crit}(r)$ in terms of the $L^r$ -based Sobolev space $\\dot{W}^{s, r}$ .",
"A direct computation gives $s_\\text{crit}(r) = \\frac{3}{r} - 1\\, (\\rightarrow -1$ as $r\\rightarrow \\infty $ ).",
"In particular, the gain of integrability of the random linear solution $z_1$ stated in Lemma REF implies that $z_1$ in (REF ) gives rise only to a subcritical perturbation.",
"See [3] for a further discussion on this issue.",
"Note, however, that if we consider a non-zero initial condition for $v$ , then the critical nature of the equation comes into play through the initial condition.",
"See Remark REF .",
"This plays an important role in studying the global-in-time behavior of solutions.",
"See, for example, [37].",
"and hence we can solve it by a standard fixed point argument.",
"Over the last several years, there have been many results on probabilistic well-posedness of nonlinear dispersive PDEs, using this change of viewpoint.In the field of stochastic parabolic PDEs, this change of viewpoint and solving the fixed point problem for the residual term $v$ is called the Da Prato-Debussche trick [16], [17].",
"See, for example, [34], [5], [8], [15], [9], [35], [32], [1], [2], [43], [38], [25], [33], [7], [30], [37].",
"In [2], we studied the Duhamel formulation for (REF ): $v(t) = - i \\int _0^t S(t-t^{\\prime }) |v + z_1|^2(v+z_1)(t^{\\prime }) dt^{\\prime },$ by carrying out case-by-case analysis on terms of the form $v \\overline{v} v$ , $v \\overline{v} z_1$ , $v \\overline{z}_1 z_1$ , etc.",
"in $X^\\frac{1}{2}([0, T])$ .",
"The main tools were (i) the gain of space-time integrability of $z_1$ and (ii) the bilinear refinement of the Strichartz estimate (Lemma REF ).",
"This yields Theorem REF .",
"By examining the case-by-case analysis in [2], we see that the regularity restriction $s > \\frac{1}{4}$ in Theorem REF comes from the cubic interaction of the random linear solution: $z_3(t) := -i \\int _0^t S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime }) dt^{\\prime }.$ See Proposition REF below.",
"In the following, we discuss an iterative approach to lower this regularity threshold by studying further expansions in terms of the random linear solution $z_1$ .",
"We will also discuss the limitation of this iterative procedure.",
"Remark 1.1 (i) The proof of Theorem REF , presented in [2], is based on a standard contraction argument for the residual term $v = u - S(t) \\phi ^\\omega $ in a ball $B$ of radius $O(1)$ in $X^\\frac{1}{2}([0, T])$ .",
"As such, the argument in [2] only yields uniqueness of $v$ in the ball $B$ .",
"Noting that $v \\in C([0, T]; H^{\\frac{1}{2}}(\\mathbb {R}^3)) \\cap X^\\frac{1}{2}([0, T])$ ,By convention, our definition of the $X^s([0, T])$ -space already assumes that functions in $X^s([0, T])$ belong to $ C([0, T]; H^s(\\mathbb {R}^3))$ .",
"See Definition REF below.",
"it follows from Lemma A.8 in [2] that the $X^\\frac{1}{2}([0, t])$ -norm of $v$ is continuous in $t \\in [0, T]$ .",
"Then, by possibly shrinking the local existence time $T>0$ , we can easily upgrade the uniqueness of $v$ in the ball $B$ to uniqueness of $v$ in the entire $X^\\frac{1}{2}([0, T])$ .",
"Namely, uniqueness of $u$ holds in the class (REF ).",
"(ii) With the exponential moment assumption (REF ), the proof of Theorem REF , presented in [2], allows us to conclude that the set $\\Sigma $ of full probability in Theorem REF has the following decomposition: $ \\Sigma = \\bigcup _{0 < T \\ll 1} \\Sigma _T$ such that (a) there exist $c, C >0$ and $C_0 = C_0 (\\Vert \\phi \\Vert _{H^s})>0$ such that $P(\\Sigma _T^c) < C \\exp \\Big (-\\frac{C_0(\\Vert \\phi \\Vert _{H^s})}{T^c }\\Big )$ for each $0 < T \\ll 1$ , (b) for each $\\omega \\in \\Sigma _T\\subset \\Sigma $ , $0 < T \\ll 1$ , the function $u = u^\\omega $ constructed in Theorem REF is a solution to (REF ) on $[0, T]$ .",
"See the statement of Theorem 1.1 in [2].",
"A similar decomposition of the set $\\Sigma $ of full probability applies to Theorems REF and REF below.",
"We, however, do not state it in an explicit manner in the following."
],
[
"Improved almost sure local well-posedness",
"Let us first state the following proposition on the regularity property of the cubic term $z_3$ defined above.",
"Proposition 1.2 Given $0 \\le s < 1$ , let $\\phi ^\\omega $ be the Wiener randomization of $\\phi \\in H^s(\\mathbb {R}^3)$ defined in (REF ) and set $z_1 = S(t) \\phi ^\\omega $ .",
"(i) For any $\\sigma < 2s$ , we have $z_3 \\in X^\\sigma _\\textup {loc}\\subset C(\\mathbb {R}; H^\\sigma (\\mathbb {R}^3))$ almost surely.",
"More precisely, there exists an almost surely finite constant $ C(\\omega , \\Vert \\phi \\Vert _{H^s} ) > 0$ and $\\theta > 0$ such that $\\Vert z_3 \\Vert _{X^{\\sigma }([0, T])} = \\bigg \\Vert \\int _0^t S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime }) dt^{\\prime }\\bigg \\Vert _{X^\\sigma ([0, T])}\\le T^\\theta C(\\omega , \\Vert \\phi \\Vert _{H^s})$ for any $T > 0$ .",
"In particular, by taking $\\sigma = 2s-\\varepsilon $ for small $\\varepsilon > 0$ , (REF ) shows that the second orderHere, we referred to $z_3$ as the second order term since it corresponds to the second order term appearing in the (formal) power series expansion of solutions to (REF ) in terms of the random initial data.",
"For the same reason, we refer to $z_5$ in (REF ) and $z_7$ in (REF ) as the third and fourth order terms in the following.",
"See Subsection REF below.",
"term $z_3$ is smoother (by $s - \\varepsilon $ ) than the first order term $z_1$ , provided $s > 0$ .",
"(ii) When $s = 0$ , there is no smoothing in the second order term $z_3$ in general; there exists $\\phi \\in L^2(\\mathbb {R}^3)$ such that the estimate (REF ) with $\\sigma = \\varepsilon > 0$ fails for any $\\varepsilon > 0$ .",
"In [2], we already proved (REF ) when $\\sigma = \\frac{1}{2}$ and $s > \\frac{1}{4}$ , giving the regularity restriction in Theorem REF .",
"See Section for the proof of Proposition REF for a general value of $\\sigma $ .",
"In the proof of Theorem REF , in order to carry out the case-by-case analysis for (REF ), we need to have $z_3 \\in X^\\frac{1}{2}_\\text{loc}$ , (where we have a deterministic local well-posedness for (REF )).",
"In view of Proposition REF , this imposes the regularity restriction $s > \\frac{1}{4}$ .",
"Note, however, that even when $s \\le \\frac{1}{4}$ , $z_3$ is still a well defined space-time function of spatial regularity $2s- < \\frac{1}{2} $ .",
"This motivates us to consider the following second order expansion: $u = z_1 + z_3 + v$ and remove the second order interaction $z_1 \\overline{z}_1 z_1$ .",
"Indeed, the residual term $v := u - z_1 - z_3$ satisfies the following equation: ${\\left\\lbrace \\begin{array}{ll}i \\partial _tv + \\Delta v = \\mathcal {N}(v + z_1+z_3) - \\mathcal {N}(z_1)\\\\v|_{t = 0} = 0,\\end{array}\\right.",
"}$ where $\\mathcal {N}(u) = |u|^2 u$ .",
"In terms of the Duhamel formulation, we have $v(t) = - i \\int _0^t S(t-t^{\\prime })\\big \\lbrace \\mathcal {N}(v + z_1 + z_3) - \\mathcal {N}(z_1)\\big \\rbrace (t^{\\prime }) dt^{\\prime }.$ Then, by studying the fixed point problem (REF ) for $v$ , we have the following improved almost sure local well-posedness of (REF ).",
"Theorem 1.3 Given $ \\frac{1}{2} \\le \\sigma \\le 1$ , let $ \\frac{2}{5} \\sigma < s < \\frac{1}{2}$ .",
"Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).",
"Then, the cubic NLS (REF ) on $\\mathbb {R}^3$ is almost surely locally well-posed with respect to the random initial data $\\phi ^\\omega $ .",
"More precisely, there exists a set $\\Sigma = \\Sigma (\\phi , \\sigma ) \\subset \\Omega $ with $P(\\Sigma ) = 1$ such that, for any $\\omega \\in \\Sigma $ , there exists a unique function $u = u^\\omega $ in the class: $z_1 + z_3+ X^\\sigma ([0, T])& \\subset z_1 + z_3+ C([0, T]; H^{\\sigma } (\\mathbb {R}^3))\\\\\\ &\\subset C([0, T];H^s(\\mathbb {R}^3))$ with $T = T(\\phi , \\omega ) >0$ such that $u$ is a solution to (REF ) on $[0, T]$ .",
"As in Theorem REF , the uniqueness of $u$ in the class $z_1 + z_3+ X^\\sigma ([0, T])$ is to be interpreted as uniqueness of the residual term $v = u - z_1 - z_3$ in $X^\\sigma ([0, T])$ .",
"See also Remark REF .",
"By taking $\\sigma = \\frac{1}{2}$ , Theorem REF states that (REF ) is almost surely locally well-posed in $H^s(\\mathbb {R}^3)$ , provided $s > \\frac{1}{5}$ .",
"This in particular improves the almost sure local well-posedness in Theorem REF .",
"On the other hand, by taking $\\sigma = 1$ , Theorem REF allows us to construct a solution $v = u - z_1 - z_3\\in X^1([0, T]) \\subset C([0,T] ; H^1(\\mathbb {R}^3))$ , provided that $s > \\frac{2}{5}$ .",
"In particular, we can take random initial data below the scaling critical regularity $s_\\text{crit} = \\frac{1}{2}$ , while we construct the residual part $v$ in $X^1([0, T])$ .",
"This opens up a possibility of studying the global-in-time behavior of $v$ , using the (non-conserved) energy of $v$ : $E(v)(t) = \\frac{1}{2} \\int _{\\mathbb {R}^3} |\\nabla v(t, x)|^2 dx+\\frac{1}{4} \\int _{\\mathbb {R}^3} |v(t, x)|^4 dx$ with random initial data below the scaling critical regularity.",
"We remark that by inspecting the argument in [2], a modification of (the proof of) Theorem REF yields almost sure local well-posedness of (REF ) in the class: $ S(t)\\phi ^\\omega + X^1([0, T]) \\subset S(t) \\phi ^{\\omega } + C([0,T] ; H^1(\\mathbb {R}^3))$ only for $s > \\frac{1}{2}$ .",
"(This restriction on $s$ can be easily seen by setting $\\sigma = 1$ in Proposition REF .)",
"In particular, Theorem REF does not allow us to take random initial data below the scaling critical regularity $s_\\text{crit} = \\frac{1}{2}$ in studying the global-in-time behavior of $v \\in X^1([0, T])$ , namely at the level of the energy $E(v)$ .",
"As our focus in this paper is the local-in-time analysis, we do not pursue further this issue on almost sure global well-posedness of (REF ) below the scaling critical regularity in this paper.",
"We, however, point out two recent results [30], [37] on almost sure global well-posedness below the energy space for the defocusing energy-critical NLS in higher dimensions.",
"The main strategy for proving Theorem REF is to study the fixed point problem (REF ) by carrying out case-by-case analysis on $w_1\\overline{w_2} w_3,\\quad \\text{for $w_i = v,$ $z_1$, or $z_3$, $i = 1, 2, 3$, but not all $w_i$ equal to $z_1$}$ in $N^\\sigma ([0, T])$ , where the dual norm is defined by $\\Vert F\\Vert _{N^\\sigma ([0, T])} = \\bigg \\Vert \\int _{0}^t S(t - t^{\\prime }) F(t^{\\prime }) dt^{\\prime }\\bigg \\Vert _{X^\\sigma ([0, T])}.$ Note that the number of cases has increased from the case-by-case analysis in the proof of Theorem REF , where we had $w_i = v$ or $z_1$ .",
"One of the main ingredients is the smoothing on $z_3$ stated in Proposition REF above.",
"Note, however, that in order to exploit this smoothing, we need to measure $z_3$ in the $X^{2s-}([0, T])$ -norm, which imposes a certain rigidity on the space-time integrability.Namely, we need to measure $z_3$ in $L^q_t([0, T]; W^{2s-, r}(\\mathbb {R}^3))$ for admissible pairs $(q, r)$ .",
"See Lemma REF .",
"In order to prove Theorem REF , we also need to exploit a gain of integrability on $z_3$ .",
"In Lemma REF , we use the dispersive estimate (see (REF ) below) and the gain of integrability on each $z_1$ of the three factors in (REF ) and show that $z_3$ also enjoys a gain of integrability by giving up some differentiability.",
"Remark 1.4 Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).",
"Given $v_0 \\in H^\\frac{1}{2}(\\mathbb {R}^3)$ , we can also consider (REF ) with the random initial data of the form $u_0^\\omega = v_0 + \\phi ^\\omega $ : ${\\left\\lbrace \\begin{array}{ll}i \\partial _tu + \\Delta u = |u|^2 u\\\\u|_{t = 0} = v_0+ \\phi ^\\omega .\\end{array}\\right.",
"}$ Then, by slightly modifying the proofs, we see that the analogues of Theorems A and REF (with $\\sigma = \\frac{1}{2}$ ) also hold for (REF ).",
"Namely, (REF ) is almost surely locally well-posed, provided $s > \\frac{1}{5}$ .",
"This amounts to considering the following Cauchy problems: ${\\left\\lbrace \\begin{array}{ll}i \\partial _tv + \\Delta v = |v + z_1|^2(v+z_1)\\\\v|_{t = 0} = v_0,\\end{array}\\right.",
"}$ when $\\frac{1}{4} < s < \\frac{1}{2}$ and ${\\left\\lbrace \\begin{array}{ll}i \\partial _tv + \\Delta v = \\mathcal {N}(v + z_1+z_3) - \\mathcal {N}(z_1)\\\\v|_{t = 0} = v_0,\\end{array}\\right.",
"}$ when $\\frac{1}{5} < s \\le \\frac{1}{4}$ .",
"In this case, the critical nature of the problem appears through the $v \\overline{v} v$ interaction in the case-by-case analysis (REF ) due to the deterministic (non-zero) initial data $v_0$ at the critical regularity.",
"The required modification is straightforward and thus we omit details.",
"See Proposition 6.3 in [2] and Lemma 6.2 in [37].",
"We point out that the discussion in the next subsection also applies to (REF ).",
"Remark 1.5 (i) Let $I(u_1, u_2, u_3)$ denote the trilinear operator defined by $I(u_1, u_2, u_3) (t):= -i \\int _0^t S(t - t^{\\prime }) u_1 \\overline{u_2} u_3 (t^{\\prime }) dt^{\\prime }.$ Then, for $\\sigma > 3s-1$ , there is no finite constant $C > 0$ such that $\\big \\Vert I( u_1, u_2, u_3)\\big \\Vert _{X^\\sigma ([0, 1])}\\le C \\prod _{j = 1}^3 \\Vert \\phi _j \\Vert _{H^s}$ for all $\\phi _j \\in H^s(\\mathbb {R}^3)$ , where $u_j = S(t) \\phi _j$ .",
"See Appendix .",
"In particular, this shows that when $s \\le \\frac{1}{2}$ , there is no deterministic smoothing for $I$ , i.e.",
"(REF ) does not hold for $\\sigma > s$ .",
"(ii) The proof of Proposition REF only exploits “the randomization at the linear level”.",
"Given $s \\in \\mathbb {R}$ , let $\\mathcal {R}^s$ denote the class of functions defined by $\\mathcal {R}^s = \\big \\lbrace u \\text{ on } \\mathbb {R}\\times \\mathbb {R}^3:\\ & i \\partial _tu + \\Delta u = 0 \\text{ and } \\\\&u \\in L^q_{t, \\text{loc}} W^{s, r}_x(\\mathbb {R}^3)\\text{ for any } 2 \\le q, r < \\infty \\big \\rbrace .$ Under the regularity assumption: $\\sigma < 2s$ and $0 \\le s < 1$ , it follows from the proof of Proposition REF that the left-hand side of (REF ) is finite for any $u_1, u_2, u_3 \\in \\mathcal {R}^s$ .Here, we only need finiteness of the $A_3^s$ -norm defined in (REF ) for each $u_j$ , $j = 1, 2, 3$ .",
"See Remark REF .",
"In other words, the proof of Proposition REF only uses the fact that the random linear solution $z_1 = S(t) \\phi ^\\omega $ belongs to $\\mathcal {R}^s$ almost surely; see the probabilistic Strichartz estimate (Lemma REF ).",
"The multilinear random structure of $z_3$ in terms of the random linear solution $z_1$ yields further cancellation.",
"See, for example, Lemma 3.6 in [15].",
"Such extra cancellation seems to improve only space-time integrability and we do not know how to use it to improve the regularity threshold (i.e.",
"differentiability) at this point.",
"A similar comment applies to the unbalanced higher order terms $\\zeta _{2k-1}$ (including $z_5$ below) studied in Proposition REF below."
],
[
"Partial power series expansion\nand the associated critical regularity",
"In this subsection, we discuss possible improvements over Theorem REF by considering further expansions.",
"For this purpose, we fix $ \\sigma = \\frac{1}{2}$ in the following.",
"By examining the proof of Theorem REF , we see that the regularity restriction $s > \\frac{1}{5}$ comes from the following third order term: $z_5 (t) := -i \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 5\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime })z_{j_1} \\overline{z_{j_2}}z_{j_3} (t^{\\prime }) dt^{\\prime }.$ Namely, we have $(j_1, j_2, j_3) = (1, 1, 3)$ up to permutations.",
"In Lemma REF , we show that given $0< s< \\frac{1}{2}$ , we have $ z_5 \\in X^{\\frac{5}{2} s -}_\\textup {loc}$ .",
"In particular, we have $ z_5 \\in X^\\frac{1}{2}_\\textup {loc}$ , provided $s > \\frac{1}{5} $ , yielding the regularity threshold in Theorem REF .",
"A natural next step is to remove this non-desirable third order interaction: $ \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 5\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3\\rbrace \\end{array}}z_{j_1} \\overline{z_{j_2}}z_{j_3}$ in the case-by-case analysis in (REF ) by considering the following third order expansion: $u = z_1 + z_3 + z_5+ v.$ In this case, the residual term $v := u - z_1 - z_3- z_5$ satisfies the following equation: ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}(v + z_1+z_3+z_5)- \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 \\in \\lbrace 3, 5\\rbrace \\\\j_1, j_2, j_3 \\in \\lbrace 1, 3\\rbrace \\end{array}}z_{j_1} \\overline{z_{j_2}}z_{j_3}\\\\v|_{t = 0} = 0.\\end{array}\\right.",
"}$ We expect that (REF ) is almost surely locally well-posed for $s > \\frac{2}{11}$ , which would be an improvement over Theorem REF .",
"The proof will be once again based on case-by-case analysis: $w_1\\overline{w_2} w_3\\quad & \\text{for $w_i = v, z_1, z_3$, or $z_5$, $i = 1, 2, 3$, such that } \\\\& \\text{it is not of the form $z_{j_1}\\overline{z_{j_2}}z_{j_3} $ with $j_1 + j_2 + j_3 \\in \\lbrace 3, 5\\rbrace $}$ in $N^\\frac{1}{2}([0, T])$ .",
"Note the increasing number of combinations.",
"In the following, however, we do not discuss details of this particular improvement over Theorem REF .",
"Instead, we consider further iterative steps and discuss a possible limitation of this procedure.",
"Remark 1.6 We point out that the expansions (REF ), (REF ), and (REF ) correspond to partial power series expansions of the first, second, and third orders,A (partial) power series expansion of a solution to (REF ) in terms of the random initial data can be expressed as a summation of certain multilinear operators over ternary trees.",
"See, for example, [11], [36].",
"Here, the term “order” in our context corresponds to “generation (of the associated trees) $+1$ ” with the convention that the trivial tree consisting only of the root node is of the zeroth generation.",
"For example, the third order term $z_5$ in (REF ) appears as the summation over all the multilinear operators associated to the ternary trees of the third generation.",
"In terms of the graphical representation in [36], we have $z_5 \\,\\text{``}\\!=\\!\\text{''} \\,[baseline=-3,scale=0.15]{(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (0.9, -1) node[dot] {};(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (-0.9, -1) node[dot] {};(0,0)node[ddot] {} -- (1.3,1) node[ddot] {}-- (1.3, 0) node[dot] {};(1.3,1) node[ddot] {}-- (2.6, 0) node[dot] {};}\\,+ \\,[baseline=-3,scale=0.15]{(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (0.9, -1) node[dot] {};(0,-1)node[dot] {} -- (0,0) node[ddot] {}-- (-0.9, -1) node[dot] {};(0,0)node[ddot] {} -- (0,1) node[ddot] {}-- (0.9, 0) node[dot] {};(0,0)node[ddot] {} -- (0,1) node[ddot] {}-- (-0.9, 0) node[dot] {};}\\,+\\, [baseline=-3,scale=0.15]{(2.6,-1)node[dot] {} -- (2.6,0) node[ddot] {}-- (3.5, -1) node[dot] {};(2.6,-1)node[dot] {} -- (2.6,0) node[ddot] {}-- (1.7, -1) node[dot] {};(0,0)node[dot] {} -- (1.3,1) node[ddot] {}-- (1.3, 0) node[dot] {};(1.3,1) node[ddot] {}-- (2.6, 0) node[ddot] {};} \\ ,$ where “ $[baseline=-2.7,scale=0.15]{(0,0) node[dot]{};}$ ” denotes the random linear solution $z_1 = S(t) \\phi ^\\omega $ and “ $[baseline=-2.7,scale=0.15]{(0,0) node[ddot]{};}$ ” denotes the trilinear Duhamel integral operator $ I(u_1, u_2, u_3)$ defined in (REF ) with its three children as its arguments $u_1, u_2$ , and $u_3$ .",
"respectively, of a solution to (REF ) in terms of the random initial data.",
"Then, by considering the associated equations (REF ), (REF ), and (REF ) for the residual term $v$ , we are recasting the original problem (REF ) as a fixed point problem centered at the partial power series expansions of the first, second, and third orders, respectively.",
"By drawing an analogy to the previous steps, we expect that the worst contribution comes from the following fourth order terms: $z_7 (t): = -i \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 7\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3, 5\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) z_{j_1} \\overline{z_{j_2}}z_{j_3} (t^{\\prime }) dt^{\\prime }.$ There are basically two contributions to (REF ): $(j_1, j_2, j_3) = (1, 3, 3)$ or $(1, 1, 5)$ up to permutations.",
"In Lemma REF , we show that the contribution from $(j_1, j_2, j_3) = (1, 1, 5)$ is worse, being responsible for the expected regularity restriction $s > \\frac{2}{11}$ .",
"In order to remove this term, we can consider the following fourth order expansion: $u = z_1 + z_3 + z_5+z_7+ v$ as in the previous steps and try to solve the following equation for the residual term $v = u - z_1 - z_3- z_5-z_7$ : ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}(v + z_1+z_3+z_5+z_7)- \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 \\in \\lbrace 3, 5, 7\\rbrace \\\\j_1, j_2, j_3 \\in \\lbrace 1, 3, 5\\rbrace \\end{array}}z_{j_1} \\overline{z_{j_2}}z_{j_3}\\\\v|_{t = 0} = 0.\\end{array}\\right.",
"}$ We can obviously iterate this argument and consider the following $k$ th order expansion: $u = \\sum _{\\ell = 1}^{k} z_{2\\ell -1}+ v.$ In this case, we need to consider the following equation for the residual term $v := u - \\sum _{\\ell = 1}^{k} z_{2\\ell -1}$ : ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}\\bigg (v +\\sum _{\\ell = 1}^{k} z_{2\\ell -1}\\bigg )-\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 \\in \\lbrace 3, 5, \\dots , 2k-1\\rbrace \\\\j_1, j_2, j_3 \\in \\lbrace 1, 3, \\dots , 2k-3\\rbrace \\end{array}}z_{j_1} \\overline{z_{j_2}}z_{j_3}\\\\v|_{t = 0} = 0\\end{array}\\right.",
"}$ and hope to construct a solution $v \\in X^\\frac{1}{2}([0, T])$ by carrying out the following case-by-case analysis: $\\begin{split}w_1\\overline{w_2} w_3,\\quad & \\text{for $w_i = v, z_j,$ $j \\in \\lbrace 1, 3, \\dots , 2k-1\\rbrace $,$i = 1, 2, 3$, such that } \\\\& \\text{it is not of the form $z_{j_1}\\overline{z_{j_2}}z_{j_3} $ with$j_1 + j_2 + j_3\\in \\lbrace 3, 5, \\dots , 2k-1\\rbrace $}\\end{split}$ in $N^\\frac{1}{2}([0, T])$ .",
"Then, a natural question to ask is “Does this iterative procedure work indefinitely, allowing us to arbitrarily lower the regularity threshold for almost sure local well-posedness for (REF )?",
"Or is there any limitation to it?",
"We now consider a “critical” regularity $s_* < \\frac{1}{2}$ with respect to this iterative procedure for proving almost sure local well-posedness of (REF ).",
"We simply define the critical regularity $s_* <\\frac{1}{2} $ for this problem to be the infimum of the values of $s< \\frac{1}{2}$ such that given any $\\phi \\in H^s(\\mathbb {R}^3)$ , the above iterative procedureNamely, we solve (REF ) for $v \\in X^\\frac{1}{2}([0, T])$ with some finite (or infinite) number of steps.",
"shows that (REF ) is almost surely locally well-posed with respect to the Wiener randomization $\\phi ^\\omega $ of $\\phi $ .",
"This is an empirical notion of criticality; unlike the scaling criticality, we can not a priori compute this critical regularity $s_*$ .",
"Moreover, our discussion will be based on the estimates on the stochastic multilinear terms (Proposition REF and Proposition REF below).",
"In the following, we discuss a (possible) lower bound on $s_*$ , presenting a limitation to our iterative procedure based on partial power series expansions.",
"Within the framework of the iterative procedure discussed above, a necessary condition for carrying out the case-by-case analysis (REF ) to study (REF ) in $X^\\frac{1}{2}([0, T])$ is that the $(k+1)$ st order term $ z_{2k+1} = z_{2(k+1)-1}$ defined in a recursive manner: $z_{2k+1}(t) : = -i \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = {2k+1}\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3, \\dots , 2k-1\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) z_{j_1} \\overline{z_{j_2}}z_{j_3} (t^{\\prime }) dt^{\\prime }$ belongs to $X^\\frac{1}{2}([0, T])$ .",
"By the nature of this iterative procedure, we may assume that the lower order terms $z_{2\\ell -1}$ , $\\ell \\in \\lbrace 1, 3, \\dots , k\\rbrace $ , belong to $X^{ s_\\ell }([0, T])$ for some $ s_\\ell < \\frac{1}{2}$ but not in $X^\\frac{1}{2}([0, T])$ , since if any of the lower order terms, say $z_{2\\ell -1}$ for some $\\ell \\in \\lbrace 1, 3, \\dots , k\\rbrace $ , were in $X^\\frac{1}{2}([0, T])$ , then we would have stopped the iterative procedure at the $(\\ell -1)$ th step.",
"As in the previous steps, we expect that $z_{2k+1}$ is responsible for a regularity restriction at the $k$ th step of this iterative approach.",
"It could be a cumbersome task to study the regularity property of $z_{2k+1}$ due to the increasing number of combinations for $z_{j_1}\\overline{z_{j_2}} z_{j_3}$ , satisfying $j_1 + j_2 + j_3 = {2k+1}$ , $j_1, j_2, j_3 \\in \\lbrace 1, 3, \\dots , 2k-1\\rbrace $ .",
"In the following, we instead study the regularity property of the $(k+1)$ st order termIn fact, we study the $k$ th order term $\\zeta _{2k-1}$ in Proposition REF .",
"of a particular form, corresponding to $(j_1, j_2, j_3) = (1, 1, 2k - 1)$ up to permutations in (REF ); given an integer $k \\ge 0$ , define $\\zeta _{2k+1}$ by setting $\\zeta _1 := z_1 = S(t) \\phi ^\\omega $ and $\\zeta _{2k+1}(t) := -i\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = {2k+1}\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2k-1\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) \\zeta _{j_1} \\overline{\\zeta _{j_2}} \\zeta _{j_3} (t^{\\prime }) dt^{\\prime }.$ As mentioned above, there are only three terms in this sum: $(j_1, j_2, j_3) = (1, 1, 2k - 1)$ up to permutations.",
"Hence, $\\zeta _{2k+1}$ consists of the “unbalanced”By associating the $(2k+1)$ -linear terms appearing in the summation in (REF ) with ternary trees of the $k$ th generation as in [36], the summands in (REF ) correspond to the “unbalanced” trees of the $k$ th generation, where two of the three children of the root node are terminal.",
"$(2k+1)$ -linear terms appearing in the definition (REF ) of $z_{2k+1}$ .",
"We claim that the $(k+1)$ st order term $\\zeta _{2k + 1}$ is responsible for the regularity restriction at the $k$ th step of the iterative procedure.",
"See Theorem REF below.",
"In anticipating the alternative expansion (REF ) below, let us study the regularity property of the unbalanced $k$ th order term $\\zeta _{2k-1}$ : $\\zeta _{2k-1}(t) := -i\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = {2k-1}\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2k-3\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) \\zeta _{j_1} \\overline{\\zeta _{j_2}} \\zeta _{j_3} (t^{\\prime }) dt^{\\prime }.$ For $k = 2, 3, 4$ , we have (with appropriate restrictions on the range of $s$ ) $\\zeta _3 = z_3 \\in X^{2s-}_\\text{loc},\\qquad \\zeta _5 = z_5 \\in X^{\\frac{5}{2}s-}_\\text{loc},\\qquad \\text{and}\\qquad \\zeta _7 \\in X^{\\frac{11}{4}s-}_\\text{loc}.$ See Proposition REF and Lemmas REF and REF .",
"In general, we have the following proposition.",
"Proposition 1.7 Define a sequence $\\lbrace \\alpha _k\\rbrace _{k \\in \\mathbb {N}}$ of positive real numbers by the following recursive relation: $\\alpha _k = \\frac{\\alpha _{k-1} + 3}{2}$ with $\\alpha _1 = 1$ .",
"Given $0< s < \\alpha _{k-1}^{-1}$ , let $\\phi ^\\omega $ be the Wiener randomization of $\\phi \\in H^s(\\mathbb {R}^3)$ defined in (REF ).",
"Then, we have $\\zeta _{2k -1} \\in X^\\sigma _\\textup {loc}$ for $\\sigma < \\alpha _k \\cdot s $ , almost surely.",
"By solving the recursive relation (REF ), we have $\\alpha _k = 2\\bigg \\lbrace 1 - \\bigg (\\frac{1}{2}\\bigg )^{k-1}\\bigg \\rbrace + 1$ and thus we have $\\alpha _2 = 2$ , $\\alpha _3 = \\frac{5}{2}$ , and $\\alpha _4 = \\frac{11}{4}$ .",
"In particular, Proposition REF agrees with (REF ).",
"Moreover, since $\\alpha _k$ is increasing and $\\lim _{k \\rightarrow \\infty } \\alpha _k =3$ , the regularity restriction $s < \\alpha _{k-1}^{-1}$ in Proposition REF does not cause any issue since our main focus is to study the probabilistic local well-posedness of (REF ) in the range of $s$ that is not covered by Theorem REF .",
"Namely, we may assume $s \\le \\frac{1}{5}\\, (< \\frac{1}{3})$ in the following.",
"In view of Propositions REF and REF , one obvious lower bound for the critical regularity $s_*$ for this iterative procedure is given by $s_0: = 0$ since there is no gain of regularity when $s = 0$ (even in moving from $z_1$ to $z_3$ ).",
"On the other hand, in order to prove almost sure local well-posedness of (REF ) by carrying out the case-by-case analysis (REF ) for the equation (REF ), we need to show that the $(k+1)$ st order term $\\zeta _{2k+1}$ belongs to $X^\\frac{1}{2}([0, T])$ .",
"This gives rise to a regularity restriction $s_k := \\frac{1}{2\\alpha _{k+1}}$ at the $k$ th step of the iterative procedure.",
"By taking $k \\rightarrow \\infty $ , we obtain another “lower” boundThis “lower” bound is based on the upper bounds obtained in Propositions REF and REF .",
"In other words, if one can improve the bounds in Propositions REF and REF , then one can lower the value of $s_\\infty $ .",
"See Remark REF .",
"$s_\\infty := \\frac{1}{6}$ on this critical regularity $s_*$ .",
"As mentioned above, the case-by-case analysis (REF ) for general $k \\in \\mathbb {N}$ may be a combinatorially overwhelming task due to (i) the number of the increasing combinations in (REF ) and (ii) the random multilinear terms $z_j,$ $j \\in \\lbrace 3, 5, \\dots , 2k-1\\rbrace $ , themselves having non-trivial combinatorial structures which makes it difficult to establish nonlinear estimates; see (REF ).",
"In the following, we instead consider an alternative iterative procedure based on the following expansion: $u = \\sum _{\\ell = 1}^k \\zeta _{2\\ell -1} + v$ in place of (REF ).",
"This expansion allows us to prove the following almost sure local well-posedness of (REF ) for $s > s_\\infty = \\frac{1}{6}$ .",
"Theorem 1.8 Let $\\frac{1}{6} < s < \\frac{1}{2}$ .",
"Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).",
"Then, the cubic NLS (REF ) on $\\mathbb {R}^3$ is almost surely locally well-posed with respect to the random initial data $\\phi ^\\omega $ .",
"More precisely, there exists a set $\\Sigma = \\Sigma (\\phi ) \\subset \\Omega $ with $P(\\Sigma ) = 1$ such that, for any $\\omega \\in \\Sigma $ , there exists a unique function $u = u^\\omega $ in the class: $\\zeta _1 + \\zeta _3 + \\cdots + \\zeta _{2k-1}+ X^\\frac{1}{2}([0, T])& \\subset \\zeta _1 + \\zeta _3 + \\cdots + \\zeta _{2k-1}+ C([0, T]; H^\\frac{1}{2} (\\mathbb {R}^3))\\\\\\ &\\subset C([0, T];H^s(\\mathbb {R}^3))$ with $T = T(\\phi , \\omega ) >0$ such that $u$ is a solution to (REF ) on $[0, T]$ .",
"Here, $k \\in \\mathbb {N}$ is a unique positive integer such that $\\frac{1}{2\\alpha _{k+1}} < s \\le \\frac{1}{2\\alpha _{k}}$ .",
"As before, the uniqueness of $u$ in the class: $\\zeta _1 + \\zeta _3 + \\cdots + \\zeta _{2k-1}+ X^\\frac{1}{2}([0, T])$ is to be interpreted as uniqueness of the residual term $v = u -\\sum _{\\ell = 1}^k \\zeta _{2\\ell -1} $ in $X^\\frac{1}{2}([0, T])$ .",
"See also Remark REF .",
"In view of the discussion above, Theorem REF proves almost sure local well-posedness of (REF ) in an almost “optimal”Once again, this is based on the estimates in Propositions REF and REF .",
"In particular, the “optimality” of the regularity threshold in Theorem REF is with respect to Propositions REF and REF .",
"If one can improve the bounds in Propositions REF and REF , then one can lower the regularity threshold in Theorem REF .",
"regularity range $s > s_\\infty = \\frac{1}{6}$ with respect to the original iterative procedure based on the partial power series expansion (REF ).",
"The proof of Theorem REF is analogous to that of Theorem REF .",
"Given $ \\frac{1}{6} < s<\\frac{1}{2}$ , fix $k \\in \\mathbb {N}$ such that $\\frac{1}{2\\alpha _{k+1}} < s \\le \\frac{1}{2\\alpha _{k}}$ .In view of Proposition REF , the lower bound on $s$ guarantees that $\\zeta _{2k+1} \\in X^\\frac{1}{2}([0, T])$ almost surely, while the upper bound on $s$ states that we can not use Proposition REF to conclude $\\zeta _{2k-1} \\in X^\\frac{1}{2}([0, T])$ almost surely.",
"Write a solution $u$ as in (REF ).",
"Note that as $s$ gets closer and closer to the critical value $s_\\infty = \\frac{1}{6}$ , the expansion (REF ) gets arbitrarily long.",
"In view of (REF ), the residual term $v := u - \\sum _{\\ell = 1}^{k} \\zeta _{2\\ell -1}$ satisfies the following equation: ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}\\bigg (v +\\sum _{\\ell = 1}^{k} \\zeta _{2\\ell -1}\\bigg )-\\sum _{\\ell = 2}^{k}\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 2\\ell - 1\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2\\ell -3\\rbrace \\end{array}}\\zeta _{j_1} \\overline{\\zeta _{j_2}}\\zeta _{j_3}\\\\v|_{t = 0} = 0.\\end{array}\\right.",
"}$ Hence, we need to carry out the following case-by-case analysis: $\\begin{split}w_1\\overline{w_2} w_3,\\quad & \\text{for $w_i = v, \\zeta _j,$ $j \\in \\lbrace 1, 3, \\dots , 2k-1\\rbrace $,$i = 1, 2, 3$, such that } \\\\& \\text{it is not of the form $\\zeta _{j_1}\\overline{\\zeta _{j_2}}\\zeta _{j_3} $ with$(j_1, j_2, j_3 )= (1, 1, 2\\ell -3)$}\\\\& \\text{(up to permutations) for $\\ell = 2, 3 \\dots , k$}\\end{split}$ in $N^\\frac{1}{2}([0, T])$ .",
"While Theorem REF yields almost sure local well-posedness in an almost optimal range with respect to the original iterative procedure based on the partial power series expansion (REF ), the required analysis is much simpler than that required for the original iterative procedure based on the expansion (REF ).",
"First, note that while the case-by-case analysis (REF ) based on the original iterative procedure involves combinatorially non-trivial $z_{2k-1}$ , the case-by-case analysis (REF ) only involves the unbalanced $k$ th order term $\\zeta _{2k-1}$ which has a much simpler structure than $z_{2k-1}$ .",
"In particular, Proposition REF shows that $\\zeta _{2k-1}$ , $k \\ge 3$ , has a better regularity property than the second order term $\\zeta _3 = z_3$ in (REF ).",
"In terms of space-time integrability, we show that $\\zeta _{2k-1}$ also enjoys a gain of integrability by giving up a control on derivatives (Lemma REF ).",
"Finally, by inspecting the proof of Theorem REF (see Lemma REF and Proposition REF below), we see that, except for $z_{j_1} \\overline{z_{j_2}}z_{j_3}$ with $(j_1, j_2, j_3) = (1, 1, 3)$ up to permutations,Namely, the terms constituting the third order term $\\zeta _5 = z_5$ in (REF ).",
"we can bound all the terms $w_1 \\overline{w_2} w_3$ appearing in the case-by-case analysis (REF ) in $N^\\frac{1}{2}([0, T])$ .",
"Hence, we can basically apply the result of the case-by-case analysis (REF ) to our problem at hand.",
"More precisely, by rewriting the case-by-case analysis (REF ) as $ w_i = v, \\zeta _1 = z_1 , \\text{ or }\\zeta _j, \\, j \\in \\lbrace 3, 5, \\dots , 2k-1\\rbrace $ (with the restriction in (REF )) and using the fact that the terms $\\zeta _{2\\ell - 1}$ , $\\ell = 3, \\dots , k$ , behave better than $\\zeta _3 = z_3$ , the proof of Theorem REF (in particular, Proposition REF ) can be used to control all the terms $w_1 \\overline{w_2} w_3$ in (REF ) except for $\\zeta _{j_1}\\overline{\\zeta _{j_2}}\\zeta _{j_3}\\quad \\text{with }(j_1, j_2, j_3 )= (1, 1, 2 k-1)$ (up to permutations).",
"Note that the contribution to (REF ) under the Duhamel integral is precisely given by $\\zeta _{2k+1}$ in (REF ).",
"In particular, Proposition REF with $\\sigma = \\frac{1}{2}$ yields the regularity restriction $\\alpha _{k+1}\\cdot s > \\frac{1}{2}$ , i.e.",
"the lower bound $s > \\frac{1}{2\\alpha _{k+1}}$ stated in Theorem REF .",
"This allows us to construct a solution $v \\in X^\\frac{1}{2}([0, T])$ by the standard fixed point argument.",
"See Section for details.",
"We previously conjectured that the $(k+1)$ st order term $z_{2k+1}$ in (REF ) would be responsible for a regularity restriction at the $k$ th step of the original iterative procedure.",
"Combining Proposition REF and Theorem REF , we confirmed this claim; the regularity restriction indeed comes only from the unbalanced $(k+1)$ st order term $\\zeta _{2k+1}$ in (REF ).",
"We conclude this introduction with several remarks.",
"Remark 1.9 Let $v_0 \\in H^\\frac{1}{2}(\\mathbb {R}^3)$ .",
"Then, by slightly modifying the proof of Theorem REF based on the modified iterative approach (REF ), we can prove almost sure local well-posedness of the cubic NLS (REF ) with the random initial data of the form $v_0 + \\phi ^\\omega $ for the same range of $s$ .",
"See Remark REF Remark 1.10 In this paper, we exploit randomness only at the linear level in estimating the stochastic terms.",
"See Remarks REF and REF .",
"It may be possible to lower the regularity thresholds in Theorems REF and REF by exploiting randomness at the multilinear level.",
"We, however, do not pursue this direction in this paper since (i) our main purpose is to present the iterative procedures in their simplest forms and (ii) estimating the higher order stochastic terms by exploiting randomness at the multilinear level would require a significant amount of additional work, which would blur the main focus of this paper.",
"Remark 1.11 The ill-posedness result in [12] show that the solution map $\\Phi : u_0 \\in H^s(\\mathbb {R}^3) \\longmapsto u \\in C([-T, T]; H^s(\\mathbb {R}^3))$ is not continuous for (REF ) when $s < \\frac{1}{2}$ .",
"In proving Theorem REF , we studied the perturbed NLS (REF ) for $v = u-z_1$ .",
"In particular, the proof shows that we can factorize the solution map for (REF ) asSimilarly, we can factorize the solution map for (REF ) as $u |_{t = 0} = v_0 + \\phi ^\\omega \\in H^s(\\mathbb {R}^3)\\longmapsto (v_0, z_1) \\stackrel{\\Psi }{\\longmapsto } v \\in X^\\frac{1}{2}([0, T]) \\subset C([0, T]; H^\\frac{1}{2}(\\mathbb {R}^3))$ such that the second map $\\Psi $ is continuous in $(v_0, z_1) \\in H^\\frac{1}{2}(\\mathbb {R}^3)\\times S^s_1([0, T])$ .",
"$\\phi ^\\omega \\in H^s(\\mathbb {R}^3)\\longmapsto z_1 \\stackrel{\\Psi }{\\longmapsto } v \\in X^\\frac{1}{2}([0, T]) \\subset C([0, T]; H^\\frac{1}{2}(\\mathbb {R}^3)),$ where the first map can be viewed as a universal lift map and the second map $\\Psi $ is the solution map to (REF ), which is in fact continuous in $z_1 \\in S^s_1([0, T])$ .Here, $S^s_1([0, T]) \\subset C([0, T]; H^s(\\mathbb {R}^3))$ denotes the intersection of suitable space-time function spaces of differentiability at most $s$ .",
"In the following, we use $S^{\\sigma }_j([0, T])$ in a similar manner.",
"In the case of Theorem REF (with $\\sigma = \\frac{1}{2}$ ), we have the following factorization of the solution map for (REF ): $ \\phi ^\\omega \\in H^s(\\mathbb {R}^3)\\longmapsto (z_1, z_3 ) \\stackrel{\\Psi }{\\longmapsto } v \\in X^\\frac{1}{2}([0, T]) \\subset C([0, T]; H^\\frac{1}{2}(\\mathbb {R}^3)),$ where the second map $\\Psi $ is the solution map to (REF ), which is continuous from $(z_1, z_3) \\in S^s_1([0, T])\\times S^{2s-}_3([0, T])$ to $v \\in X^\\frac{1}{2}([0, T])$ .",
"In the case of Theorem REF , we create $k$ stochastic objects in the first step, where $k = k(s) \\in \\mathbb {N}$ : $\\phi ^\\omega \\in H^s(\\mathbb {R}^3)\\longmapsto (\\zeta _1, \\zeta _3, \\dots , \\zeta _{2k-1} ) \\stackrel{\\Psi }{\\longmapsto } v \\in X^\\frac{1}{2}([0, T]) \\subset C([0, T]; H^\\frac{1}{2}(\\mathbb {R}^3)).$ Once again, the second map $\\Psi $ (which is the solution map to (REF )) is continuous from $(\\zeta _1, \\zeta _3, \\dots , \\zeta _{2k-1} ) \\in S^s_1([0, T]) \\times \\prod _{\\ell = 2}^ k S_\\ell ^{\\alpha _\\ell s-} ([0, T])$ to $v \\in X^\\frac{1}{2}([0, T])$ .",
"We point out an analogy between these factorizations (REF ), (REF ), and (REF ) of the ill-posed solution maps into (i) the first step, involving stochastic analysis and (ii) the second step, where purely deterministic analysis is performed in constructing a continuous map $\\Psi $ and similar factorizations for studying rough differential equations via the rough path theory [19] and singular stochastic parabolic PDEs [21], [23].",
"In the proof of Theorem REF , we could consider the following expansion of an infinite order: $u = \\sum _{\\ell = 1}^\\infty \\zeta _{2\\ell -1} + v.$ This would allow us to present a single argument that works for all $\\frac{1}{6} < s < \\frac{1}{3}$ .",
"In this case, the residual part $v := u - \\sum _{\\ell = 1}^\\infty \\zeta _{2\\ell -1}$ satisfies ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}\\bigg (v +\\sum _{\\ell = 1}^{\\infty } \\zeta _{2\\ell -1}\\bigg )-\\sum _{\\ell = 2}^{\\infty }\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 2\\ell - 1\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2\\ell -3\\rbrace \\end{array}}\\zeta _{j_1} \\overline{\\zeta _{j_2}}\\zeta _{j_3}\\\\v|_{t = 0} = 0.\\end{array}\\right.",
"}$ In particular, we would need to worry about the convergence issue of infinite series and hence there seems to be no simplification in considering the infinite order expansion (REF ).",
"Another strategy would be to treat $\\zeta _\\infty : = \\sum _{\\ell = 1}^\\infty \\zeta _{2\\ell -1} $ as one stochastic object and write $u =\\zeta _\\infty + v.$ It follows from (REF ) that $\\zeta _\\infty $ satisfies the following equation: ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _t\\zeta _\\infty + \\Delta \\zeta _\\infty =\\sum _{\\ell = 2}^{\\infty }\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 2\\ell - 1\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2\\ell -3\\rbrace \\end{array}}\\zeta _{j_1} \\overline{\\zeta _{j_2}}\\zeta _{j_3}\\\\\\zeta _\\infty |_{t = 0} = \\phi ^\\omega .\\end{array}\\right.",
"}$ Noting that $\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 2\\ell - 1\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2\\ell -3\\rbrace \\end{array}}\\zeta _{j_1} \\overline{\\zeta _{j_2}}\\zeta _{j_3}= {\\left\\lbrace \\begin{array}{ll}|z_1|^2 z_1& \\text{when }\\ell = 2,\\\\2|z_1|^2\\zeta _{2\\ell -3} + z_1^2 \\overline{\\zeta _{2\\ell -3}},& \\text{when }\\ell \\ge 3,\\end{array}\\right.",
"}$ we can rewrite (REF ) as ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _t\\zeta _\\infty + \\Delta \\zeta _\\infty =2|z_1|^2 \\zeta _\\infty + z_1^2 \\overline{\\zeta _\\infty }- 2 |z_1|^2 z_1\\\\\\zeta _\\infty |_{t = 0} = \\phi ^\\omega .\\end{array}\\right.",
"}$ and thus $v : = u - \\zeta _\\infty $ satisfies ${\\left\\lbrace \\begin{array}{ll}\\displaystyle i \\partial _tv + \\Delta v = \\mathcal {N}(v +\\zeta _\\infty )- 2|z_1|^2 \\zeta _\\infty - z_1^2 \\overline{\\zeta _\\infty }+ 2 |z_1|^2 z_1\\\\v|_{t = 0} = 0.\\end{array}\\right.",
"}$ The equations (REF ) and (REF ) do not particularly appear to be in such a friendly format.",
"Namely, studying (REF ) and (REF ) does not seem to provide a simplification over the case-by-case analysis (REF ) for the equation (REF ) for each fixed $k \\in \\mathbb {N}$ .",
"In a recent work [40], the second author (with Tzvetkov and Y. Wang) proved invariance of the white noise for the (renormalized) cubic fourth order NLS on the circle.",
"One novelty of this work is that we introduced an infinite sequence $\\lbrace z^\\text{res}_{2\\ell -1}\\rbrace _{\\ell \\in \\mathbb {N}}$ of stochastic $(2\\ell -1)$ -linear objects (depending only on the random initial data) and considered the following expansion of an infinite order: $ u = \\sum _{\\ell = 1}^\\infty z^\\text{res}_{2\\ell -1} + v.$ For this problem, it turned out that $z^\\text{res}_\\infty : = \\sum _{\\ell = 1}^\\infty z^\\text{res}_{2\\ell -1}$ satisfies a particularly simple equation.In fact, the series $z^\\text{res}_\\infty = \\sum _{\\ell = 1}^\\infty z^\\text{res}_{2\\ell -1}$ corresponds to the power series expansion of the resonant cubic fourth order NLS.",
"We point out that $z^\\text{res}_\\infty $ does not belong to the span of Wiener homogeneous chaoses of any finite order.",
"Then by treating $z^\\text{res}_\\infty $ as one stochastic object, we wrote a solution $u$ as $u = z^\\text{res}_\\infty +v$ , which led to the following factorization: $\\phi ^\\omega \\in H^s(\\longmapsto z^\\text{res}_\\infty = \\sum _{\\ell = 1}^\\infty z^\\text{res}_{2\\ell -1}\\longmapsto v \\in C(\\mathbb {R}; L^2()$ for $s < -\\frac{1}{2}$ , where $\\phi ^\\omega $ denotes the Gaussian white noise on the circle.",
"Remark 1.12 In [44], the third author (with Y. Wang) recently studied probabilistic local well-posedness of NLS on $\\mathbb {R}^d$ within the framework of the $L^p$ -based Sobolev spaces, using the dispersive estimate.",
"In the context of the cubic NLS (REF ) on $\\mathbb {R}^3$ , their result yields almost sure local existence of a unique solution $u$ for the randomized initial data $\\phi ^\\omega \\in H^s(\\mathbb {R}^3)$ , provided $s\\ge 0$ .",
"In particular, the argument in [44] allows us to consider random initial data of lower regularities than Theorems REF , REF , and REF .",
"Note that, in [44], it was shown that the solution $u$ only belongs to $C([0, T]; W^{s, 4}(\\mathbb {R}^3))$ , almost surely.",
"We point out that a slight adaptation of the work [39] by the second and third authors (with Y. Wang) shows that the solution $u$ indeed lies in $C([0, T]; H^s(\\mathbb {R}^3))$ .",
"As compared to [44], the argument presented in this paper provides extra regularity information, namely the decomposition (REF ) of the solution $u$ with the terms of increasing regularities: $ \\zeta _{2\\ell -1} \\in X^{\\alpha _\\ell \\cdot s-}_\\textup {loc} $ and $v \\in X^\\frac{1}{2}([0, T])$ .",
"In particular, the residual term $v$ lies in the (sub)critical regularity,By slightly tweaking the argument, we can easily place $v$ in $X^{\\frac{1}{2}+}([0, T])$ .",
"leaving us a possibility of adapting deterministic techniques to study its further properties.",
"This paper is organized as follows.",
"In Section , we recall probabilistic and deterministic lemmas along with the definitions of the basic function spaces.",
"In Section , we study the regularity properties of the second order term $z_3$ in (REF ).",
"In Section , we further investigate the regularity properties of the higher order terms $z_5$ and $z_7$ and the unbalanced higher order terms $\\zeta _{2k-1}$ .",
"Note that the analysis on $z_5$ and $z_7$ contains partNote that $z_7$ defined in (REF ) contains the contribution from $z_{j_1}\\overline{z_{j_2}}z_{j_3}$ with $(j_1, j_2, j_3) = (1, 3, 3)$ up to permutations.",
"of the case-by-case analysis (REF ) needed for proving Theorem REF .",
"In Section , we then carry out the rest of the case-by-case analysis (REF ) and prove Theorem REF .",
"In Section , we briefly describe the proof of Theorem REF by indicating how the analysis in the previous sections can lead to the proof.",
"In Appendix , we prove the deterministic non-smoothing of the Duhamel integral operator discussed in Remark REF .",
"Notations: We use $a+$ (and $a-$ ) to denote $a + \\varepsilon $ (and $a - \\varepsilon $ , respectively) for arbitrarily small $\\varepsilon \\ll 1$ , where an implicit constant is allowed to depend on $\\varepsilon > 0$ (and it usually diverges as $\\varepsilon \\rightarrow 0$ ).",
"Given a Banach space $B$ of temporal functions, we use the following short-hand notation: $B_T := B([0, T])$ .",
"For example, $L^q_T L^r_x = L^q_t([0, T]; L^r_x(\\mathbb {R}^3))$ .",
"Let $\\eta : \\mathbb {R}\\rightarrow [0, 1]$ be an even, smooth cutoff function supported on $[-\\frac{8}{5}, \\frac{8}{5}]$ such that $\\eta \\equiv 1$ on $[-\\frac{5}{4}, \\frac{5}{4}]$ .",
"Given a dyadic number $N \\ge 1$ , we set $\\eta _1(\\xi ) = \\eta (|\\xi |)$ and $\\eta _N(\\xi ) = \\eta \\bigg (\\frac{|\\xi |}{N}\\bigg ) - \\eta \\bigg (\\frac{2|\\xi |}{N}\\bigg )$ for $N \\ge 2$ .",
"Then, we define the Littlewood-Paley projection operator $\\mathbf {P}_N$ as the Fourier multiplier operator with symbol $\\eta _N$ .",
"In the following, we use the convention that capital letters denote dyadic numbers.",
"For example, $N = 2^n$ for some $n \\in \\mathbb {N}_0 : = \\mathbb {N}\\cup \\lbrace 0\\rbrace $ .",
"Given dyadic numbers $N_1, \\dots , N_4 \\in 2^{\\mathbb {N}_0}$ , we set $N_{\\max } := \\max _{j = 1, \\dots , 4} N_j$ .",
"We also use the following shorthand notation: $f_N = \\mathbf {P}_N f$ .",
"For example, we have $z_{1, N_j} = \\mathbf {P}_{N_j} z_1$ ."
],
[
"Probabilistic Strichartz estimates",
"First, we recall the usual Strichartz estimates on $\\mathbb {R}^3$ for readers' convenience.",
"We say that a pair $(q, r)$ is Schrödinger admissible if it satisfies $\\frac{2}{q} + \\frac{3}{r} = \\frac{3}{2}$ with $2\\le q, r \\le \\infty $ .",
"Then, the following Strichartz estimates are known to hold [46], [49], [20], [28]: $\\Vert S(t) \\phi \\Vert _{L^q_t L^r_x (\\mathbb {R}\\times \\mathbb {R}^3)} \\lesssim \\Vert \\phi \\Vert _{L^2_x(\\mathbb {R}^3)}.$ It follows from (REF ) and Sobolev's inequality that $\\Vert S(t) \\phi \\Vert _{L^p_{t, x} (\\mathbb {R}\\times \\mathbb {R}^3)}\\lesssim \\big \\Vert |\\nabla |^{\\frac{3}{2} - \\frac{5}{p}}\\phi \\big \\Vert _{L^2_x(\\mathbb {R}^3)}$ for $p \\ge \\frac{10}{3}$ .",
"We will use the following admissible pairs in this paper: $(\\infty , 2), \\ \\big (5, \\tfrac{30}{11}\\big ), \\ \\big (\\tfrac{10}{3}\\tfrac{10}{3}\\big ),\\ (2+, 6-).$ In particular, by Sobolev's inequality, we have $W^{\\frac{1}{2}, \\frac{30}{11}}(\\mathbb {R}^3) \\hookrightarrow L^5(\\mathbb {R}^3).$ One of the important key ingredients for probabilistic well-posedness is the probabilistic Strichartz estimates.",
"Such probabilistic estimates were first exploited by McKean [34] and Bourgain [5].",
"In the following, we state the probabilistic Strichartz estimates under the Wiener randomization (REF ).",
"See [1] for the proofs.",
"Lemma 2.1 Given $\\phi $ on $\\mathbb {R}^3$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).",
"Then, given finite $q \\ge 2$ and $2 \\le r \\le \\infty $ , there exist $C, c>0$ such that $P\\Big ( \\Vert S(t) \\phi ^\\omega \\Vert _{L^q_t L^r_x([0, T]\\times \\mathbb {R}^3)}> \\lambda \\Big )\\le C\\exp \\bigg (-c \\frac{\\lambda ^2}{ T^\\frac{2}{q}\\Vert \\phi \\Vert _{H^s}^{2}}\\bigg )$ for all $ T > 0$ and $\\lambda >0$ with (i) $s = 0$ if $r < \\infty $ and (ii) $s > 0$ if $r = \\infty $ .",
"A similar estimate holds when $q = \\infty $ (with $s > 0$ ) but we will not need it in this paper.",
"See [38].",
"We also need the following lemma on the control of the size of $H^s$ -norm of $\\phi ^\\omega $ .",
"Lemma 2.2 Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).",
"Then, we have $P\\Big ( \\Vert \\phi ^\\omega \\Vert _{ H^s( \\mathbb {R}^3)} > \\lambda \\Big )\\le C\\exp \\bigg (-c \\frac{\\lambda ^2}{ \\Vert \\phi \\Vert _{H^s}^{2}}\\bigg )$ for all $\\lambda >0$ ."
],
[
"Function spaces and their properties",
"In this subsection, we go over the basic definitions and properties of the functions spaces used for the Fourier restriction norm method (i.e.",
"analysis involving the $X^{s, b}$ -spaces introduced in [4]) adapted to the space of functions of bounded $p$ -variation and its pre-dual, introduced and developed by Tataru, Koch, and their collaborators [31], [22], [24].",
"We refer readers to [22], [24] for the proofs of the basic properties.",
"See also [2].",
"Let $\\mathcal {Z}$ be the set of finite partitions $-\\infty <t_{0}<t_{1}<\\dots <t_{K} \\le \\infty $ of the real line.",
"By convention, we set $u(t_K):=0$ if $t_K=\\infty $ .",
"We use $\\mathbf {1}_I$ to denote the sharp characteristic function of a set $I \\subset \\mathbb {R}$ .",
"Definition 2.3 Let $1\\le p < \\infty $ .",
"(i) We define a $U^p$ -atom to be a step function $a: \\mathbb {R}\\rightarrow L^2(\\mathbb {R}^3)$ of the form: $a=\\sum _{k=1}^{K}\\phi _{k-1} \\mathbf {1}_{[t_{k-1},t_{k})},$ where $\\lbrace t_{k}\\rbrace _{k=0}^{K}\\in \\mathcal {Z}$ and $\\lbrace \\phi _{k}\\rbrace _{k=0}^{K-1}\\subset L^2 (\\mathbb {R}^3)$ with $\\sum _{k=0}^{K-1} \\Vert \\phi _{k} \\Vert _{L^{2}}^{p}=1$ .",
"Furthermore, we define the atomic space $U^p = U^p(\\mathbb {R}; L^2(\\mathbb {R}^3))$ by $U^{p}:= \\bigg \\lbrace u: \\mathbb {R}\\rightarrow L^2 (\\mathbb {R}^3) : u=\\sum _{j=1}^{\\infty }\\lambda _{j}a_{j}\\ & \\text{for $U^{p}$-atoms $a_{j}$},\\ \\lbrace \\lambda _j\\rbrace _{j \\in \\mathbb {N}} \\in \\ell ^1(\\mathbb {N}; \\bigg \\rbrace $ with the norm $\\Vert u \\Vert _{U^{p}}:= \\inf \\bigg \\lbrace \\sum _{j=1}^{\\infty }|\\lambda _{j}|:u=\\sum _{j=1}^{\\infty } \\lambda _{j}a_{j}\\text{ for $U^{p}$-atoms } a_{j}, \\ \\lbrace \\lambda _j\\rbrace _{j \\in \\mathbb {N}} \\in \\ell ^1(\\mathbb {N}; \\bigg \\rbrace ,$ where the infimum is taken over all possible representations for $u$ .",
"(ii) We define $V^{p}= V^p(\\mathbb {R}; L^2(\\mathbb {R}^3))$ to be the space of functions $u:\\mathbb {R}\\rightarrow L^2(\\mathbb {R}^3)$ of bounded $p$ -variation with the standard $p$ -variation norm: $\\Vert u \\Vert _{V^{p}}:=\\sup _{\\lbrace t_{k}\\rbrace _{k=0}^{K}\\in \\mathcal {Z}}\\bigg ( \\sum _{k=1}^{K}\\Vert u(t_{k})-u(t_{k-1})\\Vert _{L^{2}}^{p}\\bigg ) ^{\\frac{1}{p}}.$ By convention, we impose that the limits $\\lim _{t \\rightarrow \\pm \\infty }u(t)$ exist in $L^2 (\\mathbb {R}^3)$ .",
"(iii) Let $V_{\\text{rc}}^{p}$ be the closed subspace of $V^{p}$ of all right-continuous functions $u \\in V^p$ with $\\lim _{t\\rightarrow -\\infty }u(t)=0$ .",
"(iv) We define $U_{\\Delta }^{p}:=S (t) U^{p}$ (and $V_{\\Delta }^{p}:=S(t)V^{p}$ , respectively) to be the space of all functions $u:\\mathbb {R}\\rightarrow L^2 (\\mathbb {R}^3)$ such that $t \\rightarrow S ( -t ) u(t)$ is in $U^p$ (and in $V^p$ , respectively) with the norms $\\Vert u \\Vert _{U_{\\Delta }^{p}} :=\\Vert S(-t )u \\Vert _{U^{p}}\\qquad \\text{and} \\qquad \\Vert u \\Vert _{V_{\\Delta }^p} :=\\Vert S(-t)u \\Vert _{V^{p}}.$ The closed subspace $V_{\\text{rc}, \\Delta }^{p}$ is defined in an analogous manner.",
"Recall the following inclusion relation; for $1\\le p<q<\\infty $ , $U^p \\hookrightarrow V_{\\text{rc}}^{p}\\hookrightarrow U^{q} \\hookrightarrow L^{\\infty }(\\mathbb {R};L^2 (\\mathbb {R}^3)).$ The space $V^p$ is the classical space of functions of bounded $p$ -variation and the space $U^p$ appears as the pre-dual of $V^{p^{\\prime }}$ with $\\frac{1}{p} + \\frac{1}{p^{\\prime }} = 1$ , $1 < p < \\infty $ .",
"Their duality relation and the atomic structure of the $U^p$ -space turned out to be very effective in studying dispersive PDEs in critical settings.",
"We are now ready to define the solution spaces.",
"Definition 2.4 (i) Let $s\\in \\mathbb {R}$ .",
"We define $X^s(\\mathbb {R})$ to be the closure of $C(\\mathbb {R};H^s(\\mathbb {R}^3)) \\cap U_{\\Delta }^2 $ with respect to the $X^s$ -norm defined by $ \\Vert u \\Vert _{X^s (\\mathbb {R})} : = \\bigg (\\sum _{\\begin{array}{c}N\\ge 1\\\\\\textup {dyadic}\\end{array}} N^{2s}\\Vert \\mathbf {P}_N u \\Vert _{U^2_{\\Delta }L^2}^2\\bigg )^\\frac{1}{2}.$ (ii) Let $s\\in \\mathbb {R}$ .",
"We define $Y^s(\\mathbb {R})$ to be the space of all functions $u \\in C(\\mathbb {R};H^s(\\mathbb {R}^3))$ such that the map $t \\mapsto \\mathbf {P}_N u$ lies in $ V_{\\text{rc},\\Delta }^2 H^s$ for any $N \\in 2^{\\mathbb {N}_0}$ and $\\Vert u \\Vert _{Y^s(\\mathbb {R})}< \\infty $ , where the $Y^s$ -norm is defined by $ \\Vert u \\Vert _{Y^s(\\mathbb {R})} : = \\bigg ( \\sum _{\\begin{array}{c}N\\ge 1\\\\\\textup {dyadic}\\end{array}} N^{2s}\\Vert \\mathbf {P}_N u\\Vert _{V^2_\\Delta L^2}^2\\bigg )^\\frac{1}{2}.$ Recall the following embeddings: $U^2_\\Delta H^s \\hookrightarrow X^s \\hookrightarrow Y^s \\hookrightarrow V^2_\\Delta H^s \\hookrightarrow U^p_\\Delta H^s,$ for $p>2$ .",
"Given an interval $I \\subset \\mathbb {R}$ , we define the local-in-time versions $X^s(I)$ and $Y^s(I)$ of these spaces as restriction norms.",
"For example, we define the $X^s(I)$ -norm by $ \\Vert u \\Vert _{X^s(I)} = \\inf \\big \\lbrace \\Vert v\\Vert _{X^s(\\mathbb {R})}: \\, v|_I = u\\big \\rbrace .$ We also define the norm for the nonhomogeneous term on an interval $I = [t_0, t_1)$ : $\\Vert F\\Vert _{N^s(I)} = \\bigg \\Vert \\int _{t_0}^t S(t - t^{\\prime }) F(t^{\\prime }) dt^{\\prime }\\bigg \\Vert _{X^s(I)}.$ We conclude this section by presenting some basic estimates involving these function spaces.",
"See [22], [24], [2] for the proofs.",
"Lemma 2.5 Let $s \\in \\mathbb {R}$ and $T\\in ( 0,\\infty ]$ .",
"Then, the following linear estimates hold: $\\Vert S(t) \\phi \\Vert _{X^s([0, T])}& \\le \\Vert \\phi \\Vert _{H^s}, \\\\\\Vert F \\Vert _{N^{s}([0, T])}& \\le \\sup _{\\begin{array}{c}w \\in Y^{-s}([0, T])\\\\ \\Vert w \\Vert _{Y^{-s}([0, T])}=1\\end{array}} \\left| \\int _{0}^{T} \\langle F(t), w(t) \\rangle _{L_{x}^{2}} dt \\right|$ for any $\\phi \\in H^s(\\mathbb {R}^3)$ and $F \\in L^1([0,T];H^{s}(\\mathbb {R}^3))$ .",
"The transference principle [22] and the interpolation lemma [22] applied on the Strichartz estimates (REF ) and (REF ) imply the following estimates.",
"Lemma 2.6 Given any admissible pair $(q,r)$ with $q>2$ and $p \\ge \\frac{10}{3}$ , we have $\\Vert u \\Vert _{L_t^q L_x^r} & \\lesssim \\Vert u \\Vert _{Y^0}, \\\\\\Vert u \\Vert _{L^p_{t,x}}& \\lesssim \\big \\Vert |\\nabla |^{\\frac{3}{2} - \\frac{5}{p}} u\\big \\Vert _{Y^0}.$ Similarly, the bilinear refinement of the Strichartz estimate [6], [41], [14] implies the following bilinear estimate.",
"Lemma 2.7 Let $N_1, N_2 \\in 2^{\\mathbb {N}_0}$ with $N_1 \\le N_2$ .",
"Then, we have $\\Vert \\mathbf {P}_{N_1}u_1 \\mathbf {P}_{N_2}u_2\\Vert _{L^2_{t} ([0, T]; L^2_x)}\\lesssim T^{0+}N_1^{1-} N_2^{-\\frac{1}{2}+}\\Vert \\mathbf {P}_{N_1} u_1 \\Vert _{Y^0([0, T])} \\Vert \\mathbf {P}_{N_2} u_2 \\Vert _{Y^0([0, T])}$ for any $T > 0$ and $u_1, u_2 \\in Y^0([0, T])$ .",
"From the bilinear refinement of the Strichartz estimate [6], [14] and the transference principle, we have $\\Vert \\mathbf {P}_{N_1}u_1 \\mathbf {P}_{N_2}u_2\\Vert _{L^2_{t,x}}\\lesssim N_1^{1-} N_2^{-\\frac{1}{2}+} \\Vert \\mathbf {P}_{N_1} u_1 \\Vert _{Y^0} \\Vert \\mathbf {P}_{N_2} u_2 \\Vert _{Y^0}$ for all $u_1, u_2 \\in Y^0$ .",
"See [2] for the proof of (REF ).",
"On the other hand, it follows from Hölder's and Sobolev's inequalities that $\\Vert \\mathbf {P}_{N_1}u_1 \\mathbf {P}_{N_2}u_2\\Vert _{L^2_TL^2_x}& \\lesssim T^\\frac{1}{2} \\Vert \\mathbf {P}_{N_1} u_1 \\Vert _{L^\\infty _TL^4_x} \\Vert \\mathbf {P}_{N_2} u_2 \\Vert _{L^\\infty _TL^4_x}\\\\& \\lesssim T^\\frac{1}{2} N_1^\\frac{3}{4}N_2^\\frac{3}{4} \\Vert \\mathbf {P}_{N_1} u_1 \\Vert _{Y^0_T} \\Vert \\mathbf {P}_{N_2} u_2 \\Vert _{Y^0_T} .$ Then, the estimate (REF ) follows from interpolating (REF ) and (REF )."
],
[
"On the second order term $z_3$",
"In this and the next sections, we study the regularity properties of the various stochastic terms that appear in the iterative procedures.",
"Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be the Wiener randomization of $\\phi $ defined in (REF ) and set $z_1 = S(t) \\phi ^\\omega .$ In this section, we study the regularity properties of the second order term: $z_3 = -i \\int _0^t S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime }) dt^{\\prime },$ We first present the proof of Proposition REF .",
"We follow closely the argument in [2].",
"(i) By Lemma REF , the estimate (REF ) follows once we prove $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}\\langle \\nabla \\rangle ^\\sigma ( z_1 \\overline{z_1} z_1 )\\overline{w} dx dt\\bigg |\\le T^\\theta C(\\omega , \\Vert \\phi \\Vert _{H^s})$ for some almost surely finite constant $ C(\\omega , \\Vert \\phi \\Vert _{H^s} ) > 0$ and $\\theta > 0$ , where $\\Vert w\\Vert _{Y^{0}_T} \\le 1$ .",
"In the following, we drop the complex conjugate when it does not play any role.",
"Define $A^s_3(T)$ by $\\Vert z_1\\Vert _{A^s_3(T)}: = \\max \\Big (\\Vert \\langle \\nabla \\rangle ^s z_{1} \\Vert _{L^{\\frac{30}{7}+}_{T}L^\\frac{30}{7}_x},\\Vert \\langle \\nabla \\rangle ^sz_{1} \\Vert _{L^{4}_{T, x}},\\Vert z_1(0)\\Vert _{H^s} \\Big ).$ Then, by applying the dyadic decomposition, it suffices to prove $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}\\mathbf {P}_{N_1} z_1 \\cdot \\mathbf {P}_{N_2} z_1 \\cdot N_3^\\sigma \\mathbf {P}_{N_3} z_1 \\cdot \\mathbf {P}_{N_4} w \\, dx dt\\bigg |\\le T^\\theta N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_3(T)})$ for all $N_1, \\dots , N_4 \\in 2^{\\mathbb {N}_0}$ with $N_3 \\ge N_2 \\ge N_1$ .",
"Once we prove (REF ), the desired estimate (REF ) follows from summing (REF ) over dyadic blocks and applying Lemmas REF and REF .",
"Recall our shorthand notation: $z_{1, N_j} = \\mathbf {P}_{N_j} z_1$ and $w_{N_4} = \\mathbf {P}_{N_4} w$ .",
"In the following, we assume $\\sigma < 2s\\qquad \\text{and}\\qquad 0 \\le s < 1.$ $\\bullet $ Case (1): $N_2 \\sim N_3$ .",
"By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {O2})& \\lesssim \\Vert z_{1, N_1} \\Vert _{L^\\frac{30}{7}_{T, x}}\\bigg (\\prod _{j = 2}^3 \\Vert \\langle \\nabla \\rangle ^\\frac{\\sigma }{2} z_{1, N_j} \\Vert _{L^{\\frac{30}{7}}_{T, x}}\\bigg )\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_3(T)}) \\Vert w_{N_4}\\Vert _{Y^0_T},$ provided that (REF ) holds.",
"$\\bullet $ Case (2): $N_3 \\sim N_4 \\gg N_1, N_2$ .",
"$\\circ $ Subcase (2.a): $N_1, N_2 \\ll N_3^\\frac{1}{2}$ .",
"By Cauchy-Schwarz' inequality and Lemma REF followed by Lemma REF ,In the remaining part of this paper, we repeatedly apply this argument when there is a frequency separation.",
"We shall simply refer to it as the “bilinear Strichartz estimate” argument.",
"we have $\\text{LHS of } (\\ref {O2})& \\le N_3^\\sigma \\Vert z_{1, N_1} z_{1, N_3} \\Vert _{L^2_{T, x}} \\Vert z_{1, N_2} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1-s-} N_2^{1-s-} N_3^{\\sigma - s - 1 +}\\bigg (\\prod _{j = 1}^3\\Vert \\mathbf {P}_{N_j} \\phi ^\\omega \\Vert _{H^{s}}\\bigg )\\Vert w_{N_4}\\Vert _{Y^0_T}\\\\& \\lesssim T^{0+} N_3^{\\sigma - 2s +}\\prod _{j = 1}^3\\Vert \\mathbf {P}_{N_j} \\phi ^\\omega \\Vert _{H^{s}}\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_3(T)}),$ provided that (REF ) holds.",
"$\\circ $ Subcase (2.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .",
"By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {O2})& \\le N_3^\\sigma \\bigg (\\prod _{j = 1}^3 \\Vert z_{1, N_j} \\Vert _{L^\\frac{30}{7}_{T, x}}\\bigg )\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{-s} N_2^{-s}N_3^{\\sigma -s}C(\\Vert z_1\\Vert _{A^s_3(T)})\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_3(T)}),$ provided that (REF ) holds.",
"$\\circ $ Subcase (2.c): $N_2\\gtrsim N_3^\\frac{1}{2} \\gg N_1$ .",
"By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {O2})& \\le N_3^\\sigma \\Vert z_{1, N_1} w_{N_4} \\Vert _{L^2_{T, x}}\\prod _{j = 2}^3 \\Vert z_{1, N_j}\\Vert _{L^4_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1-s-} N_2^{-s}N_3^{\\sigma -s-\\frac{1}{2}+}\\Vert \\mathbf {P}_{N_1}\\phi ^\\omega \\Vert _{H^s}\\bigg (\\prod _{j = 2}^3 \\Vert \\langle \\nabla \\rangle ^s z_{1, N_j} \\Vert _{L^4_{T, x}}\\bigg )\\Vert w_{N_4}\\Vert _{Y^0_T}\\\\& \\le T^{0+}N_3^{\\sigma -2s+}C(\\Vert z_1\\Vert _{A^s_3(T)})\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_3(T)}),$ provided that (REF ) holds.",
"Therefore, putting all the cases together, we obtain (REF ).",
"(ii) Given $N \\gg 1$ and small $\\ell > 0$ , consider the following deterministic initial condition $\\phi $ whose Fourier transform is given by $\\widehat{\\phi }(\\xi ) = \\mathbf {1}_{ N e_1 + \\ell Q}(\\xi ) + \\mathbf {1}_{(N+100\\ell ) e_1 + \\ell Q}(\\xi )+ \\mathbf {1}_{ N e_1 + 100\\ell e_2 + \\ell Q}(\\xi ),$ where $Q = (-\\frac{1}{2}, \\frac{1}{2}]^3$ , $e_1 = (1, 0, 0)$ , and $e_2 = (0, 1, 0)$ .",
"By taking $\\ell > 0$ sufficiently small, we have $\\operatornamewithlimits{supp}\\widehat{\\phi }\\subset N e_1 + Q$ and thus we can neglect the effect of the randomization in (REF ) since all the three terms on the right-hand side will be multiplied by a common random number $g_{N e_1}$ .",
"Without loss of generality, we assume that $g_{N e_1} = 1$ in the following.",
"We estimate from below the contribution to $\\mathbf {1}_{Q_{N, \\ell }} (\\xi ) \\widehat{z}_3 (t, \\xi ) $ , where $Q_{N, \\ell } = Ne_1 + 100 \\ell (e_1 + e_2) + \\ell Q.$ From (REF ), we have $\\mathbf {1}_{Q_{N, \\ell }} & (\\xi ) \\widehat{z}_3 (t, \\xi ) \\\\& = -i \\mathbf {1}_{Q_{N, \\ell }} (\\xi ) e^{-it |\\xi |^2} \\int _0^t \\operatornamewithlimits{\\int }_{\\xi = \\xi _1 - \\xi _2 + \\xi _3}e^{i t^{\\prime } \\Phi (\\bar{\\xi })} \\widehat{\\phi ^\\omega } (\\xi _1) \\overline{\\widehat{\\phi ^\\omega } (\\xi _2)} \\widehat{\\phi ^\\omega } (\\xi _3) d\\xi _1 d\\xi _2 dt^{\\prime },$ where the phase function $\\Phi (\\bar{\\xi })$ is given by $\\Phi (\\bar{\\xi }) = \\Phi (\\xi , \\xi _1, \\xi _2, \\xi _3)= |\\xi |^2 - |\\xi _1|^2 + |\\xi _2|^2 - |\\xi _3|^2= 2\\langle \\xi - \\xi _1, \\xi - \\xi _3 \\rangle _{\\mathbb {R}^3}.$ Then, it follows that the only non-trivial contribution to (REF ) appears if $\\xi _1 \\in (N+100\\ell ) e_1 + \\ell Q,\\qquad \\xi _2 \\in N e_1 + \\ell Q,\\qquad \\xi _3 \\in N e_1 + 100\\ell e_2 + \\ell Q$ (up to the permutation $\\xi _1 \\leftrightarrow \\xi _3$ ).",
"In this case, we have $|\\Phi (\\bar{\\xi })| \\ll 1 $ and thus $\\operatornamewithlimits{Re}e^{i t^{\\prime } \\Phi (\\bar{\\xi })} \\ge \\frac{1}{2}$ for all $t^{\\prime } \\in [0, 1]$ .",
"Now, recall the following lemma on the convolution.",
"Lemma 3.1 There exists $c>0$ such that $\\mathbf {1}_{a + \\ell Q}* \\mathbf {1}_{b + \\ell Q} (\\xi )\\ge c \\ell ^3 \\mathbf {1}_{a+b+\\ell Q}(\\xi )$ for all $a, b, \\xi \\in \\mathbb {R}^3$ .",
"By applying Lemma REF to (REF ) with (REF ) and (REF ), we obtain $|\\mathbf {1}_{Q_{N, \\ell }} (\\xi ) \\widehat{z}_3 (t, \\xi ) |\\gtrsim t \\ell ^6 \\mathbf {1}_{Q_{N, \\ell }} (\\xi ).$ Therefore, for any $\\sigma > 0$ , we have $\\Vert z_3\\Vert _{X^\\sigma ([0, 1])} \\gtrsim \\Vert z_3 \\Vert _{L^\\infty _t([0, 1]; H^\\sigma )}\\gtrsim \\ell ^\\frac{15}{2} N^\\sigma \\longrightarrow \\infty $ as $N \\rightarrow \\infty $ , while $\\Vert \\phi \\Vert _{L^2} \\sim \\ell ^\\frac{3}{2}$ remains bounded.",
"This in particular implies that when $s = 0$ , the estimate (REF ) can not hold for any $\\sigma > 0$ .",
"This proves Part (ii).",
"Remark 3.2 It follows from the proof of Proposition REF (i) that $\\big \\Vert I( u_1, u_2, u_3)\\big \\Vert _{X^\\sigma ([0, 1])}\\lesssim \\prod _{j = 1}^3 \\Vert u_j \\Vert _{A^s_3(T)},$ where $I( u_1, u_2, u_3)$ is as in (REF ).",
"In particular, the left-hand side of (REF ) is finite for $u_1, u_2, u_3 \\in \\mathcal {R}^s$ .",
"The only probabilistic component in the proof of Proposition REF (i) appears in applying Lemmas REF and REF to control the $A^s_3(T)$ -norm of $z_1$ in terms of the $H^s$ -norm of $\\phi $ .",
"In this sense, we exploit the randomization only at the linear level.",
"On the one hand, Proposition REF shows that $z_3$ controls almost $2s$ derivatives.",
"On the other hand, we need to measure $z_3$ in the $X^{2s-}$ -norm, which controls only the admissible space-time Lebesgue norms (with $2s-$ derivatives) via Lemma REF .",
"The following lemma breaks this rigidity by giving up a control on derivatives.",
"In particular, it allows us to control a wider range of space-time Lebesgue norms of $z_3$ .",
"The main idea is to use the dispersive estimate for the linear Schrödinger operator: $\\Vert S(t) f \\Vert _{L^r_x} \\lesssim |t|^{-\\frac{3}{2}(1 - \\frac{2}{r})} \\Vert f\\Vert _{L^{r^{\\prime }}_x}.$ This allows us to reduce the analysis to a product of the random linear solution $z_1 = S(t) \\phi ^\\omega $ and apply Lemma REF .",
"Lemma 3.3 Let $s \\ge 0$ .",
"Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).",
"Then, for any finite $q, r \\ge 1$ , we have $\\Vert \\mathbf {P}_N z_3\\Vert _{L^q_T L^r_x}\\lesssim {\\left\\lbrace \\begin{array}{ll}\\rule [-6mm]{0pt}{15pt} T^{\\frac{3}{r} - \\frac{1}{2}}\\Vert z_1\\Vert _{L^{3q}_T L^{3r^{\\prime }}_x}^3 & \\text{when } 1 \\le r < 6, \\\\T^{0+} N^{\\frac{1}{2} - \\frac{3}{r}+} \\Vert z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}^3& \\text{when } r \\ge 6, \\\\\\end{array}\\right.",
"}$ for any $T > 0$ and $N \\in 2^{\\mathbb {N}_0}$ .",
"Note that the right-hand side of (REF ) is almost surely finite thanks to the probabilistic Strichartz estimate (Lemma REF ).",
"We first consider the case $r < 6$ .",
"From (REF ) and (REF ), we have $\\Vert \\mathbf {P}_N z_3\\Vert _{L^q_T L^r_x}& \\le \\bigg \\Vert \\int _0^t \\Vert \\mathbf {P}_N S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime })\\Vert _{L^r_x} dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim \\bigg \\Vert \\int _0^t \\frac{1}{|t - t^{\\prime }|^{\\frac{3}{2} - \\frac{3}{r}}} \\Vert z_1(t^{\\prime })\\Vert _{L^{3r^{\\prime }}_x}^3 dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim T^{\\frac{3}{r} - \\frac{1}{2}}\\Vert z_1\\Vert _{L^{3q}_T L^{3r^{\\prime }}_x}^3.$ When $r \\ge 6$ , we proceed as in (REF ) but we apply Sobolev's inequality before applying (REF ): $\\Vert \\mathbf {P}_N z_3\\Vert _{L^q_T L^r_x}& \\le \\bigg \\Vert \\int _0^t \\Vert \\mathbf {P}_N S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime })\\Vert _{L^r_x} dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim N^{\\frac{1}{2} - \\frac{3}{r}+} \\bigg \\Vert \\int _0^t \\Vert \\mathbf {P}_N S(t - t^{\\prime }) |z_1|^2 z_1(t^{\\prime })\\Vert _{L^{6-}_x} dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim N^{\\frac{1}{2} - \\frac{3}{r}+}\\bigg \\Vert \\int _0^t \\frac{1}{|t - t^{\\prime }|^{1-}} \\Vert z_1(t^{\\prime })\\Vert _{L^{\\frac{18}{5}+}_x}^3 dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim T^{0+} N^{\\frac{1}{2} - \\frac{3}{r}+} \\Vert z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}^3 .$ This completes the proof of Lemma REF ."
],
[
"On the higher order terms",
"In this section, we study the regularity properties of the higher order terms."
],
[
"On the third order term $z_5$",
"In this subsection, we study the third order term: $z_5 = -i \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 5\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) z_{j_1} \\overline{z_{j_2}}z_{j_3} (t^{\\prime }) dt^{\\prime }.$ The following lemma shows that the third order term $z_5$ enjoys a gain of extra $\\frac{1}{2}s$ derivative as compared to the second order term $z_3$ (Proposition REF ).",
"Lemma 4.1 Given $0 < s < \\frac{1}{2} $ , let $\\phi ^\\omega $ be the Wiener randomization of $\\phi \\in H^s(\\mathbb {R}^3)$ defined in (REF ).",
"Then, for any $\\sigma < \\frac{5}{2} s$ , we have $z_5 \\in X^\\sigma _\\textup {loc} ,$ almost surely.",
"In particular, there exists an almost surely finite constant $ C(\\omega , \\Vert \\phi \\Vert _{H^s} ) > 0$ and $\\theta > 0$ such that $\\Vert z_5 \\Vert _{X^{\\sigma }([0, T])} \\le T^\\theta C(\\omega , \\Vert \\phi \\Vert _{H^s})$ for any $T > 0$ .",
"First, note that the only possible combination for $(j_1, j_2, j_3)$ in (REF ) is $(1, 1, 3)$ up to permutations.",
"The complex conjugate does not play any role in the subsequent analysis and hence we drop the complex conjugate sign and simply study $z_5 \\sim \\int _0^t S(t - t^{\\prime }) z_{1} z_1 z_{3} (t^{\\prime }) dt^{\\prime }.$ By Lemma REF , it suffices to prove $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}\\langle \\nabla \\rangle ^\\sigma ( z_1 z_1 z_3 )w dx dt\\bigg |\\le T^\\theta C(\\omega , \\Vert \\phi \\Vert _{H^s})$ for some almost surely finite constant $ C(\\omega , \\Vert \\phi \\Vert _{H^s} ) > 0$ and $\\theta > 0$ , where $\\Vert w\\Vert _{Y^{0}_T} \\le 1$ .",
"Define $A^s_5(T)$ by $ \\Vert z_1\\Vert _{A^s_5(T)}: = \\max \\Big (\\Vert \\langle \\nabla \\rangle ^s z_{1} \\Vert _{L^{5+}_T L^{5}_x},\\Vert z_1(0)\\Vert _{H^s},\\Big ).",
"$ Then, by applying the dyadic decomposition, it suffices to prove $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}N_{\\max }^\\sigma \\mathbf {P}_{N_1} z_1 \\cdot \\mathbf {P}_{N_2} z_1& \\cdot \\mathbf {P}_{N_3} z_3 \\cdot \\mathbf {P}_{N_4} w \\, dx dt\\bigg | \\\\& \\le T^\\theta N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})$ for all $N_1, \\dots , N_4 \\in 2^{\\mathbb {N}_0}$ .",
"Once we prove (REF ), the desired estimate (REF ) follows from summing over dyadic blocks and applying Lemmas REF and REF and Proposition REF .",
"In the following, we fix $0 < s < \\tfrac{1}{2}.$ Without loss of generality, we assume that $N_1 \\ge N_2$ .",
"$\\bullet $ Case (1): $N_1 \\sim N_2 \\sim N_{\\max }$ .",
"$\\circ $ Subcase (1.a): $N_3 \\ll N_1^\\frac{1}{2}$ .",
"By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{\\sigma - s - \\frac{1}{2}+} N_2^{-s}N_3^{1 - 2s-}\\Vert \\mathbf {P}_{N_1}\\phi ^\\omega \\Vert _{H^s}\\Vert \\langle \\nabla \\rangle ^s z_{1, N_2} \\Vert _{L^5_{T, x}}\\Vert z_{3, N_3}\\Vert _{Y^{2s-}_T}\\\\& \\le T^{0+}N_1^{\\sigma -3s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < 3s$ .",
"$\\circ $ Subcase (1.b): $N_3\\gtrsim N_1^\\frac{1}{2} $ .",
"By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{\\frac{10}{3}}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma -s} N_2^{-s}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -3s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < 3s$ .",
"$\\bullet $ Case (2): $N_1 \\sim N_3 \\sim N_{\\max } \\gg N_2$ .",
"$\\circ $ Subcase (2.a): $N_2 \\ll N_1^\\frac{1}{2}$ .",
"By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}} \\Vert z_{1, N_2} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{\\sigma - s } N_2^{1-s-}N_3^{- 2s-\\frac{1}{2} +}\\Vert \\langle \\nabla \\rangle ^s z_{1, N_1} \\Vert _{L^5_{T, x}}\\Vert \\mathbf {P}_{N_2}\\phi ^\\omega \\Vert _{H^s}\\Vert z_{3, N_3}\\Vert _{Y^{2s-}_T}\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{7}{2}s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{7}{2}s$ .",
"$\\circ $ Subcase (2.b): $N_2\\gtrsim N_1^\\frac{1}{2} $ .",
"By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{\\frac{10}{3}}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma -s} N_2^{-s}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{7}{2} s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{7}{2}s$ .",
"$\\bullet $ Case (3): $N_1 \\sim N_4 \\sim N_{\\max } \\gg N_2, N_3$ .",
"$\\circ $ Subcase (3.a): $N_2, N_3 \\ll N_1^\\frac{1}{2} $ .",
"By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} z_{1, N_2} \\Vert _{L^2_{T, x}} \\Vert z_{3, N_3} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{\\sigma -s-\\frac{1}{2} + }N_2^{1-s - } N_3^{1- 2s -}N_4^{-\\frac{1}{2}+}\\bigg (\\prod _{j = 1}^2\\Vert \\mathbf {P}_{N_j} \\phi ^\\omega \\Vert _{H^{s}}\\bigg )\\Vert z_{3, N_3}\\Vert _{Y^{2s-}_T}\\Vert w_{N_4}\\Vert _{Y^0_T}\\\\& \\le T^{0+} N_1^{\\sigma - \\frac{5}{2} s +}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{5}{2}s$ .",
"$\\circ $ Subcase (3.b): $N_2, N_3\\gtrsim N_1^\\frac{1}{2} $ .",
"By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{\\frac{10}{3}}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma -s} N_2^{-s}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{5}{2} s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{5}{2}s$ .",
"$\\circ $ Subcase (3.c): $N_2\\gtrsim N_1^\\frac{1}{2} \\gg N_3$ .",
"By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} z_{3, N_3} \\Vert _{L^2_{T, x}} \\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma - s- \\frac{1}{2} + } N_2^{-s}N_3^{1- 2s-}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{5}{2}s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{5}{2}s$ .",
"$\\circ $ Subcase (3.d): $N_3\\gtrsim N_1^\\frac{1}{2} \\gg N_2$ .",
"By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma - s } N_2^{1-s-}N_3^{- 2s+} N_4^{-\\frac{1}{2} +}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{5}{2}s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < \\frac{5}{2}s$ .",
"$\\bullet $ Case (4): $N_3 \\sim N_4 \\gg N_1 \\ge N_2$ .",
"$\\circ $ Subcase (4.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .",
"By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_3^\\sigma \\Vert z_{1, N_1} z_{3, N_3} \\Vert _{L^2_{T, x}} \\Vert z_{1, N_2} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1-s- }N_2^{1-s - } N_3^{\\sigma - 2s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+} N_1^{\\sigma - 3 s +}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < 3s$ .",
"$\\circ $ Subcase (4.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .",
"By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_3^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{\\frac{10}{3}}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{ -s} N_2^{-s}N_3^{\\sigma -2s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -3 s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < 3s$ .",
"$\\circ $ Subcase (4.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .",
"By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {P5})& \\lesssim N_3^\\sigma \\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}} \\Vert z_{1, N_2} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{ - s} N_2^{1-s-}N_3^{\\sigma - 2s- \\frac{1}{2}+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma - 3s+}C(\\Vert z_1\\Vert _{A^s_5(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $ \\sigma < 3s$ .",
"Putting all the cases together, we conclude that (REF ) holds, provided that $\\sigma < \\frac{5}{2} s$ .",
"This completes the proof of Lemma REF ."
],
[
"On the fourth order term $z_7$",
"Next, we study the following fourth order term: $z_7 = -i \\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 7\\\\j_1, j_2, j_3 \\in \\lbrace 1, 3, 5\\rbrace \\end{array}}\\int _0^t S(t - t^{\\prime }) z_{j_1} \\overline{z_{j_2}}z_{j_3} (t^{\\prime }) dt^{\\prime }.$ In this case, there are two possibilities for $(j_1, j_2, j_3)$ in (REF ): $(1, 3, 3)$ and $(1, 1, 5)$ up to permutations.",
"We denote by $\\widetilde{z}_7$ the contribution to $z_7$ from $(j_1, j_2, j_3) = (1, 3, 3)$ (up to permutations).",
"Note that the contribution to $z_7$ from $(j_1, j_2, j_3) = (1, 1, 5)$ (up to permutations) corresponds to $\\zeta _7$ defined in (REF ).",
"Dropping the complex conjugate, we have $\\widetilde{z}_7 & \\sim \\int _0^t S(t - t^{\\prime }) z_{1} z_3 z_{3} (t^{\\prime }) dt^{\\prime }, \\\\\\zeta _7 & \\sim \\int _0^t S(t - t^{\\prime }) z_{1} z_1 z_{5} (t^{\\prime }) dt^{\\prime }.$ The following lemma shows that $\\widetilde{z}_7$ and $\\zeta _7$ enjoy a further gain of derivatives as compared to $z_1$ , $z_3$ , and $z_5$ .",
"Lemma 4.2 Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be the Wiener randomization of $\\phi $ defined in (REF ).",
"(i) Let $ 0 < s < \\frac{1}{2}$ .",
"Then, given any $\\sigma < 3s$ , we have $\\widetilde{z}_7 \\in X^\\sigma _\\textup {loc} ,$ almost surely.",
"(ii) Let $ 0 < s < \\frac{2}{5}$ .",
"Then, given any $\\sigma < \\frac{11}{4}s$ , we have $\\zeta _7 \\in X^\\sigma _\\textup {loc} ,$ almost surely.",
"(i) We first estimate $\\widetilde{z}_7$ in (REF ).",
"Fix $0 < s < \\frac{1}{2}$ .",
"We proceed as in the proofs of Proposition REF and Lemma REF .",
"In view of Lemmas REF and REF and Proposition REF , it suffices to prove that there exists $\\theta > 0$ such that $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}N_{\\max }^\\sigma \\mathbf {P}_{N_1} z_1 \\cdot \\mathbf {P}_{N_2} z_3& \\cdot \\mathbf {P}_{N_3} z_3 \\cdot \\mathbf {P}_{N_4} w \\, dx dt\\bigg |\\\\& \\le T^\\theta N_{\\max }^{0-} C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}).$ for all $N_1, \\dots , N_4 \\in 2^{\\mathbb {N}_0}$ and $\\Vert w\\Vert _{Y^{0}_T} \\le 1$ , where $A^s_7(T)$ is given by $ \\Vert z_1\\Vert _{A^s_7(T)}: = \\max \\Big (\\Vert \\langle \\nabla \\rangle ^s z_{1} \\Vert _{L^{5}_{T, x}},\\Vert \\langle \\nabla \\rangle ^s z_{1} \\Vert _{L^{10+}_T L^{10}_x},\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x},\\Vert z_1(0)\\Vert _{H^s}\\Big ).",
"$ Without loss of generality, we assume that $N_2 \\ge N_3$ .",
"$\\bullet $ Case (1): $N_1 \\sim N_2 \\sim N_{\\max }$ .",
"$\\circ $ Subcase (1.a): $N_3 \\ll N_1^\\frac{1}{2}$ .",
"By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}} \\Vert z_{3, N_2} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{\\sigma - s } N_2^{-2s - \\frac{1}{2} + }N_3^{1 - 2s-}\\Vert \\langle \\nabla \\rangle ^s z_{1, N_1} \\Vert _{L^5_{T, x}}\\bigg (\\prod _{j = 2}^3 \\Vert z_{3, N_j}\\Vert _{Y^{2s-}_T}\\bigg )\\\\& \\le T^{0+}N_1^{\\sigma -4s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 4s$ .",
"$\\circ $ Subcase (1.b): $N_3\\gtrsim N_1^\\frac{1}{2} $ .",
"By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{10}_{T, x}}\\Vert z_{3, N_2} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{\\frac{10}{3}}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma -s} N_2^{-2s+}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -4s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 4s$ .",
"$\\bullet $ Case (2): $N_1 \\sim N_4 \\sim N_{\\max } \\gg N_2 \\ge N_3$ .",
"$\\circ $ Subcase (2.a): $N_2, N_3 \\ll N_1^\\frac{1}{2} $ .",
"By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} z_{3, N_2} \\Vert _{L^2_{T, x}} \\Vert z_{3, N_3} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{\\sigma -s-\\frac{1}{2} + }N_2^{1-2s - } N_3^{1- 2s -}N_4^{-\\frac{1}{2}+}\\Vert \\mathbf {P}_{N_1} \\phi ^\\omega \\Vert _{H^{s}}\\bigg (\\prod _{j = 2}^3\\Vert z_{3, N_j}\\Vert _{Y^{2s-}_T}\\bigg )\\\\& \\le T^{0+} N_1^{\\sigma - 3 s +}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 3s$ .",
"$\\circ $ Subcase (2.b): $N_2, N_3\\gtrsim N_1^\\frac{1}{2} $ .",
"By $L^{10}_{T, x}, L^{\\frac{10}{3}}_{T, x},L^{\\frac{10}{3}}_{T, x}, L^{\\frac{10}{3}}_{T, x}$ -Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\le T^{0+}N_1^{\\sigma -s} N_2^{-2s+}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -3 s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 3s$ .",
"$\\circ $ Subcase (2.c): $N_2\\gtrsim N_1^\\frac{1}{2} \\gg N_3$ .",
"By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_1^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_2} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{3, N_3} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{0+}N_1^{\\sigma - s } N_2^{-2s+}N_3^{1- 2s-}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma - 3s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 3s$ .",
"$\\bullet $ Case (3): $N_2 \\sim N_3 \\sim N_{\\max } \\gg N_1$ .",
"$\\circ $ Subcase (3.a): $N_1 \\ll N_2^\\frac{1}{2}$ .",
"By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_2^\\sigma \\Vert z_{1, N_1} z_{3, N_2} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{ 1- s - } N_2^{\\sigma -2s - \\frac{1}{2} + }N_3^{-s}\\Vert \\mathbf {P}_{N_1} \\phi ^\\omega \\Vert _{H^s}\\Vert z_{3, N_2}\\Vert _{Y^{2s-}_T}\\Vert \\langle \\nabla \\rangle ^s z_{3, N_3} \\Vert _{L^{5}_{T, x}}\\\\\\multicolumn{2}{l}{\\text{By applying Lemma \\ref {LEM:Z3_2}and the fractional Leibniz rule,}}\\\\& \\lesssim T^{\\frac{1}{10}+}N_2^{\\sigma -\\frac{7}{2} s+}\\Vert \\mathbf {P}_{N_1} \\phi ^\\omega \\Vert _{H^s}\\Vert z_{3, N_2}\\Vert _{Y^{2s-}_T}\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\\\& \\le T^{\\frac{1}{10}+}N_2^{\\sigma -\\frac{7}{2}s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < \\frac{7}{2}s$ .",
"$\\circ $ Subcase (3.b): $N_1\\gtrsim N_2^\\frac{1}{2} $ .",
"By $L^{10}_{T, x}, L^{\\frac{10}{3}}_{T, x},L^{\\frac{10}{3}}_{T, x}, L^{\\frac{10}{3}}_{T, x}$ -Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\le T^{0+}N_1^{ -s} N_2^{\\sigma -2s+}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{9}{2}s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < \\frac{9}{2}s$ .",
"$\\bullet $ Case (4): $N_2 \\sim N_4 \\gg N_1, N_3$ .",
"$\\circ $ Subcase (4.a): $N_1, N_3 \\ll N_2^\\frac{1}{2} $ .",
"By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_2^\\sigma \\Vert z_{1, N_1} z_{3, N_2} \\Vert _{L^2_{T, x}} \\Vert z_{3, N_3} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\le T^{0+} N_1^{1-s- }N_2^{\\sigma -2s -\\frac{1}{2}+ } N_3^{1- 2s-}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+} N_1^{\\sigma - \\frac{7}{2} s +}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < \\frac{7}{2}s$ .",
"$\\circ $ Subcase (4.b): $N_1, N_3\\gtrsim N_2^\\frac{1}{2} $ .",
"By $L^{10}_{T, x}, L^{\\frac{10}{3}}_{T, x},L^{\\frac{10}{3}}_{T, x}, L^{\\frac{10}{3}}_{T, x}$ -Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\le T^{0+}N_1^{ -s} N_2^{\\sigma -2s+}N_3^{-2s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_3\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_2^{\\sigma -\\frac{7}{2} s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < \\frac{7}{2}s$ .",
"$\\circ $ Subcase (4.c): $N_1\\gtrsim N_2^\\frac{1}{2} \\gg N_3$ .",
"By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_2^\\sigma \\Vert z_{1, N_1} \\Vert _{L^{5}_{T, x}} \\Vert z_{3, N_2} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{ - s} N_2^{\\sigma -2s-\\frac{1}{2} + }N_3^{1- 2s-}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T})\\\\& \\le T^{0+}N_1^{\\sigma -\\frac{7}{2}s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < \\frac{7}{2}s$ .",
"$\\circ $ Subcase (4.d): $N_3\\gtrsim N_2^\\frac{1}{2} \\gg N_1$ .",
"By the bilinear Strichartz estimate and Lemma REF , we have $\\text{LHS of } (\\ref {Q5})& \\lesssim N_2^\\sigma \\Vert z_{1, N_1} z_{3, N_2} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{ 1- s - } N_2^{\\sigma -2s - \\frac{1}{2} + }N_3^{-s}\\Vert \\mathbf {P}_{N_1} \\phi ^\\omega \\Vert _{H^s}\\Vert z_{3, N_2}\\Vert _{Y^{2s-}_T}\\Vert \\langle \\nabla \\rangle ^s z_{3, N_3} \\Vert _{L^{5}_{T, x}}\\\\\\multicolumn{2}{l}{\\text{Proceeding as in Subcase (3.a) with Lemma \\ref {LEM:Z3_2},}}\\\\& \\le T^{\\frac{1}{10}}N_1^{\\sigma - 3s+}C(\\Vert z_1\\Vert _{A^s_7(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ Hence, we obtain (REF ), provided that $\\sigma < 3s$ .",
"Putting all the cases together, we conclude that (REF ) holds when $ \\sigma < 3s$ .",
"(ii) Next, we estimate $\\zeta _7$ in ().",
"This term has a similar structure to $z_5$ in (REF ); the only difference appears in the third factor.",
"Hence, we can estimate $\\zeta _7$ simply by replacing the regularity $2s-$ (for $z_3$ ; see Proposition REF ) with $\\frac{5}{2} s-$ (for $z_5$ ; see Lemma REF ) in the proof of Lemma REF .",
"In the following, we only indicate the necessary modifications on the powers of dyadic parameters in the proof of Lemma REF .",
"$\\bullet $ Case (1): $N_1 \\sim N_2 \\sim N_{\\max }$ .",
"$\\circ $ Subcase (1.a): $N_3 \\ll N_1^\\frac{1}{2}$ .",
"It suffices to note that $N_1^{\\sigma - s - \\frac{1}{2}+} N_2^{-s}N_3^{1 - \\frac{5}{2} s-}& \\lesssim N_1^{\\sigma - \\frac{13}{4} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $ \\sigma < \\frac{13}{4} s$ and $0 < s < \\frac{2}{5}$ .",
"The modification for Subcase (1.b) is straightforward under the same regularity restriction.",
"$\\bullet $ Case (2): $N_1 \\sim N_3 \\sim N_{\\max } \\gg N_2$ .",
"$\\circ $ Subcase (2.a): $N_2 \\ll N_1^\\frac{1}{2} $ .",
"It suffices to note that $N_1^{\\sigma - s } N_2^{1-s-}N_3^{- \\frac{5}{2}s-\\frac{1}{2} +}\\lesssim N_1^{\\sigma - 4 s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < 4s$ and $0 \\le s < 1$ .The lower bound on $s$ is needed only for Subcase (2.b).",
"Similar comments apply to the following cases and also to the proof of Proposition REF .",
"The modification for Subcase (2.b) is straightforward under the same regularity restriction.",
"$\\bullet $ Case (3): $N_1 \\sim N_4 \\sim N_{\\max } \\gg N_2, N_3$ .",
"$\\circ $ Subcase (3.a): $N_2, N_3 \\ll N_1^\\frac{1}{2} $ .",
"It suffices to note that $N_1^{\\sigma -s-\\frac{1}{2} + }N_2^{1-s - } N_3^{1- \\frac{5}{2}s -}N_4^{-\\frac{1}{2}+}\\lesssim N_1^{\\sigma - \\frac{11}{4} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < \\frac{11}{4}s\\qquad \\text{and}\\qquad 0 < s < \\frac{2}{5}.$ The modifications for Subcases (3.b), (3.c), and (3.d) are straightforward under the regularity restriction (REF ).",
"$\\bullet $ Case (4): $N_3 \\sim N_4 \\gg N_1 \\ge N_2$ .",
"$\\circ $ Subcase (4.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .",
"It suffices to note that $N_1^{1-s- }N_2^{1-s - } N_3^{\\sigma - \\frac{5}{2} s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\lesssim N_1^{\\sigma - \\frac{7}{2} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < \\frac{7}{2} s$ and $0 \\le s < 1$ .",
"The modifications for Subcases (4.b) and (4.c) are straightforward under the same regularity restriction.",
"Putting all the cases together, we conclude that (REF ) holds under the regularity restriction (REF ).",
"This completes the proof of Lemma REF ."
],
[
"On the “unbalanced” higher order terms $\\zeta _{2k-1}$",
"In this subsection, we study the regularity properties of the unbalanced higher order terms $\\zeta _{2k-1}$ defined in (REF ).",
"Note that $(j_1, j_2, j_3) = (1, 1, 2k-3)$ up to permutations.",
"Then, by dropping the complex conjugate, we have $\\zeta _{2k-1}\\sim \\int _0^t S(t - t^{\\prime }) z_{1} z_1 \\zeta _{2k-3} (t^{\\prime }) dt^{\\prime }.$ We first present the proof of Proposition REF .",
"We proceed by induction.",
"From (REF ), we see that the recursive relation (REF ) is satisfied when $k = 2, 3, 4$ .",
"In the following, by assuming $\\zeta _{2k -3} \\in X^\\sigma _\\textup {loc}$ for $\\sigma < \\alpha _{k-1} \\cdot s $ almost surely, where $\\alpha _{k-1}$ satisfies (REF ), we prove (REF ) and (REF ).",
"As in the proof of Lemma REF (ii), the proof follows from the proof of Lemma REF by replacing $z_3$ and $2s-$ with $\\zeta _{2k-3}$ and $\\alpha _{k-1} \\cdot s-$ , respectively.",
"Therefore, we only indicate the necessary modifications on the powers of dyadic parameters in the proof of Lemma REF .",
"$\\bullet $ Case (1): $N_1 \\sim N_2 \\sim N_{\\max }$ .",
"$\\circ $ Subcase (1.a): $N_3 \\ll N_1^\\frac{1}{2}$ .",
"It suffices to note that $N_1^{\\sigma - s - \\frac{1}{2}+} N_2^{-s}N_3^{1 - \\alpha _{k-1} s-}& \\lesssim N_1^{\\sigma - \\frac{\\alpha _{k-1}+4}{2} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < \\frac{\\alpha _{k-1}+4}{2} s\\qquad \\text{and}\\qquad 0 < s < \\alpha _{k-1}^{-1}.$ The modification for Subcase (1.b) is straightforward, giving the regularity restriction (REF ).",
"$\\bullet $ Case (2): $N_1 \\sim N_3 \\sim N_{\\max } \\gg N_2$ .",
"$\\circ $ Subcase (2.a): $N_2 \\ll N_1^\\frac{1}{2} $ .",
"It suffices to note that $N_1^{\\sigma - s } N_2^{1-s-}N_3^{- \\alpha _{k-1}s-\\frac{1}{2} +}\\lesssim N_1^{\\sigma - \\frac{2 \\alpha _{k-1} + 3}{2} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < \\frac{2\\alpha _{k-1}+3}{2}s\\qquad \\text{and}\\qquad 0 \\le s < 1.$ The modification for Subcase (2.b) is straightforward, giving the regularity restriction (REF ).",
"$\\bullet $ Case (3): $N_1 \\sim N_4 \\sim N_{\\max } \\gg N_2, N_3$ .",
"$\\circ $ Subcase (3.a): $N_2, N_3 \\ll N_1^\\frac{1}{2} $ .",
"It suffices to note that $N_1^{\\sigma -s-\\frac{1}{2} + }N_2^{1-s - } N_3^{1- \\alpha _{k-1}s -}N_4^{-\\frac{1}{2}+}\\lesssim N_1^{\\sigma - \\frac{\\alpha _{k-1}+3}{2} s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < \\frac{\\alpha _{k-1}+3}{2}s\\qquad \\text{and}\\qquad 0 < s < \\alpha _{k-1}^{-1}.$ The modifications for Subcases (3.b), (3.c), and (3.d) are straightforward, giving the regularity restriction (REF ).",
"$\\bullet $ Case (4): $N_3 \\sim N_4 \\gg N_1 \\ge N_2$ .",
"$\\circ $ Subcase (4.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .",
"It suffices to note that $N_1^{1-s- }N_2^{1-s - } N_3^{\\sigma - \\alpha _{k-1} s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\lesssim N_1^{\\sigma - (\\alpha _{k-1}+1) s +}\\lesssim N_{\\max }^{0-} ,$ provided that $\\sigma < (\\alpha _{k-1}+1)s\\qquad \\text{and}\\qquad 0 \\le s < 1.$ The modifications for Subcases (4.a) and (4.b) are straightforward, giving the regularity restriction (REF ).",
"Since $\\alpha _{k-1}$ satisfies (REF ), we have $\\alpha _{k-1} \\ge 1$ , which in turn implies $\\frac{\\alpha _{k-1}+3}{2} \\le \\alpha _{k-1} + 1$ .",
"Therefore, we conclude (REF ) and (REF ).",
"We conclude this section by proving a gain of space-time integrability for $\\zeta _{2k-1}$ analogous to Lemma REF .",
"Lemma 4.3 Let $s \\ge 0$ .",
"Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ).",
"Fix an integer $k \\ge 3$ .",
"Then, for any finite $q, r > 2$ , there exist $\\theta _k = \\theta _k(q, r) > 0$ , $\\delta _k = \\delta _k(q, r) \\in (0, 1)$ , and $1\\le q_k< \\infty $ such that $\\Vert \\mathbf {P}_N \\zeta _{2k-1}\\Vert _{L^q_T L^r_x}\\lesssim {\\left\\lbrace \\begin{array}{ll}\\rule [-6mm]{0pt}{15pt} T^{\\theta _k} \\Vert z_1\\Vert _{L^{q_k}_T L^{2}_x}^{(2k-1)(1-\\delta _k)}\\Vert z_1\\Vert _{L^{q_k}_T L^{6}_x}^{(2k-1)\\delta _k}& \\text{when } 1 \\le r < 6, \\\\T^{\\theta _k} N^{\\frac{1}{2} - \\frac{3}{r}+} \\Vert z_1\\Vert _{L^{q_k}_T L^{2}_x}^{(2k-1)(1-\\delta _k)}\\Vert z_1\\Vert _{L^{q_k}_T L^{6}_x}^{(2k-1)\\delta _k}& \\text{when } r \\ge 6, \\\\\\end{array}\\right.",
"}$ for any $T > 0$ and $N \\in 2^{\\mathbb {N}_0}$ .",
"Note that the right-hand side of (REF ) is almost surely finite thanks to the probabilistic Strichartz estimate (Lemma REF ).",
"We prove (REF ) by induction.",
"We first consider the case $r < 6$ .",
"In Lemma REF , we proved (REF ) for $k = 2$ .",
"Now, suppose that (REF ) holds for $k - 1$ .",
"Then, from (REF ) and the dispersive estimate (REF ), we have $\\Vert \\mathbf {P}_N \\zeta _{2k-1}\\Vert _{L^q_T L^r_x}& \\le \\bigg \\Vert \\int _0^t \\Vert \\mathbf {P}_N S(t - t^{\\prime }) z_1^2 \\zeta _{2k-3} (t^{\\prime })\\Vert _{L^r_x} dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\& \\lesssim \\bigg \\Vert \\int _0^t \\frac{1}{|t - t^{\\prime }|^{\\frac{3}{2} - \\frac{3}{r}}}\\Vert z_1\\Vert _{L^{3r^{\\prime }}_x}^2 \\Vert \\zeta _{2k-3}\\Vert _{L^{3r^{\\prime }}_x} dt^{\\prime }\\bigg \\Vert _{L^q_T}\\\\\\multicolumn{2}{l}{\\text{Noting $3r^{\\prime } < 6$and applying the inductive hypothesiswith $\\theta _{k-1} = \\theta _{k-1}(3q, 3r^{\\prime })$,$\\delta _{k-1} = \\delta _{k-1}(3q, 3r^{\\prime })$,and $q_{k-1} = q_{k-1}(3q, 3r^{\\prime })$,}}\\\\& \\lesssim T^{\\frac{3}{r} - \\frac{1}{2} + \\theta _{k-1} }\\Vert z_1\\Vert _{L^{3q}_T L^{3r^{\\prime }}_x}^2\\Vert z_1\\Vert _{L^{q_{k-1}}_T L^{2}_x}^{(2k-3)(1-\\delta _{k-1})}\\Vert z_1\\Vert _{L^{q_{k-1}}_T L^{6}_x}^{(2k-3)\\delta _{k-1}} \\\\\\multicolumn{2}{l}{\\text{By interpolation in $x$ and Hölder's inequality in $t$,}}\\\\& \\lesssim T^{\\theta _k }\\Vert z_1\\Vert _{L^{q_k}_T L^{2}_x}^{(2k-1)(1-\\delta _{k})}\\Vert z_1\\Vert _{L^{q_k}_T L^{6}_x}^{(2k-1)\\delta _{k}} .$ When $r \\ge 6$ , (REF ) follows from Sobolev's inequality and (REF ) as in the proof of Lemma REF ."
],
[
"Proof of Theorem ",
"In this section, we study the fixed point problem (REF ) around the second order expansion (REF ) and present the proof of Theorem REF .",
"Given $ \\frac{1}{2} \\le \\sigma \\le 1$ , let $ \\frac{2}{5} \\sigma < s < \\frac{1}{2} $ .",
"Given $\\phi \\in H^s(\\mathbb {R}^3)$ , let $\\phi ^\\omega $ be its Wiener randomization defined in (REF ) and let $z_1$ and $z_3$ be as in (REF ) and (REF ).",
"Define $\\Gamma $ by $\\Gamma v(t) = - i \\int _0^t S(t-t^{\\prime })\\big \\lbrace \\mathcal {N}(v + z_1 + z_3) - \\mathcal {N}(z_1)\\big \\rbrace (t^{\\prime }) dt^{\\prime }.$ Then, we have the following nonlinear estimates.",
"Proposition 5.1 Given $ \\frac{1}{2} \\le \\sigma \\le 1$ , let $\\frac{2}{5} \\sigma < s < \\frac{1}{2}$ .",
"Then, there exist $\\theta > 0$ , $C_1, C_2> 0$ , and an almost surely finite constant $R = R(\\omega ) >0$ such that $\\Vert \\Gamma v\\Vert _{X^\\sigma ([0, T])}& \\le C_1\\big (\\Vert v\\Vert _{X^\\sigma ([0, T])} ^3 + T^\\theta R(\\omega )\\big ), \\\\\\Vert \\Gamma v_1 - \\Gamma v_2 \\Vert _{X^\\sigma ([0, T])}& \\le C_2\\Big (\\sum _{j = 1}^2 \\Vert v_j\\Vert _{X^\\sigma ([0, T])} ^2+ T^\\theta R(\\omega )\\Big )\\Vert v_1 -v_2 \\Vert _{X^\\sigma ([0, T])},$ for all $v, v_1, v_2 \\in X^{\\sigma }([0, T])$ and $0 < T\\le 1$ .",
"Once we prove Proposition REF , Theorem REF immediately follows from a standard argument and thus we omit details.",
"See Section 5 in [2].",
"Let $0<T \\le 1$ .",
"We only prove (REF ) since () follows in a similar manner.",
"Arguing as in the proof of Proposition 4.1 in [2], it suffices to perform a case-by-case analysis of expressions of the form: $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}\\langle \\nabla \\rangle ^\\sigma ( w_1 w_2 w_3 )w dx dt\\bigg |,$ where $\\Vert w\\Vert _{Y^{0}_T} \\le 1$ and $w_j= v$ , $z_1$ , or $z_3$ , $j = 1, 2, 3$ , but not all $z_1$ .",
"Note that we have dropped the complex conjugate sign on $w_2$ since it does not play any essential role.",
"Then, we need to consider the following cases: Table: NO_CAPTION We already treated Cases (A) and (B) in Lemmas REF and REF .",
"In particular, Case (A) imposes the regularity restriction: $ \\sigma < \\frac{5}{2} s.$ As we see below, Cases (B) - (I) only impose a milder regularity restriction: $\\sigma < 3s$ .",
"Given $ \\frac{1}{2} \\le \\sigma \\le 1$ , let $\\frac{2}{5} \\sigma < s < \\frac{1}{2}$ .",
"Define the $B^s(T)$ -norm by $\\Vert z_1\\Vert _{B^s(T)}: = \\max \\Big (\\Vert \\langle \\nabla \\rangle ^s z_{1} \\Vert _{L^{5+}_T L^{5}_x},& \\Vert \\langle \\nabla \\rangle ^s z_{1} \\Vert _{L^{10+}_T L^{10}_x}, \\\\& \\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x},\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x},\\Vert z_1(0)\\Vert _{H^s}\\Big ),$ where $q \\gg 1$ is defined in (REF ) below.",
"Then, by applying the dyadic decomposition, we prove the following estimate:In Case (I), we do not perform the dyadic decomposition and hence there is no need to have the factor $N_{\\max }^{0-}$ on the right-hand side of (REF ).",
"$\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}N_{\\max }^\\sigma ( \\mathbf {P}_{N_1} w_1 & \\mathbf {P}_{N_2} w_2 \\mathbf {P}_{N_3} w_3 ) \\mathbf {P}_{N_4} w \\, dx dt\\bigg |\\\\& \\le C_1N_{\\max }^{0-}\\big (\\Vert v\\Vert _{X^\\sigma ([0, T])} ^3 + T^\\theta C( \\Vert z_1\\Vert _{B^s(T)}, \\Vert z_3\\Vert _{X^{2s-}(T)})\\big )$ for all $N_1, \\dots , N_4 \\in 2^{\\mathbb {N}_0}$ .",
"Once we prove (REF ), the desired estimate (REF ) follows from summing (REF ) over dyadic blocks and applying Lemmas REF and REF and Proposition REF .",
"Case (C): $z_3z_3z_3 $ case.",
"By symmetry, we assume $N_1 \\le N_2 \\le N_3$ .",
"Note that we have $N_3 \\sim N_{\\max }$ .",
"$\\bullet $ Subcase (C.1): $N_2 \\sim N_3 \\sim N_{\\max }$ .",
"By Hölder's inequality with ${\\left\\lbrace \\begin{array}{ll}\\text{time: } \\frac{1}{q} + \\frac{1}{2+} + \\frac{1}{2+} + \\frac{1}{\\infty } =1\\text{ for some }q \\gg 1, \\rule [-3mm]{0pt}{0pt}\\\\\\text{space: }\\frac{1}{6+} + \\frac{1}{6-} + \\frac{1}{6-} + \\frac{1}{2} =1\\end{array}\\right.",
"}$ such that $(2+, 6-)$ is admissible and applying Lemmas REF and REF , we have $& \\text{LHS of } (\\ref {XX1})\\lesssim N_3^\\sigma \\Vert z_{3, N_1} \\Vert _{L^{q}_T L^{6+}_x}\\bigg (\\prod _{j = 2}^3 \\Vert z_{3, N_j} \\Vert _{L^{2+}_T L^{6-}_x}\\bigg )\\Vert w_{N_4} \\Vert _{L^\\infty _T L^2_x}\\\\& \\hphantom{X}\\lesssim T^{0+}N_1^{ -s +} N_2^{-2s+}N_3^{\\sigma -2s+}\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}\\Vert z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}^2\\bigg (\\prod _{j = 2}^3\\Vert z_{3, N_j}\\Vert _{Y^{2s-}_T}\\bigg )\\Vert w_{N_4}\\Vert _{Y^0_T}\\\\\\\\& \\hphantom{X}\\le T^{0+}N_3^{\\sigma -4s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ This yields (REF ), provided that $\\sigma < 4s$ and $ s > 0$ .",
"$\\bullet $ Subcase (C.2): $N_3 \\sim N_4 \\sim N_{\\max } \\gg N_2 \\ge N_1$ .",
"$\\circ $ Subsubcase (C.2.a): $N_1, N_2 \\ll N_3^\\frac{1}{2}$ .",
"By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert z_{3, N_1} z_{3, N_3} \\Vert _{L^2_{T, x}} \\Vert z_{3, N_2} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1-2s- } N_2^{1-2s -}N_3^{\\sigma - 2s -\\frac{1}{2}+}N_4^{-\\frac{1}{2}+}\\bigg (\\prod _{j = 1}^3\\Vert z_{3, N_j}\\Vert _{Y^{2s-}_T}\\bigg )\\\\& \\lesssim T^{0+} N_3^{\\sigma - 4s +}C( \\Vert z_3\\Vert _{X^{2s-}_T}).$ This yields (REF ), provided that $\\sigma < 4s$ and $s < \\frac{1}{2}$ .",
"$\\circ $ Subsubcase (C.2.b): $N_1, N_2 \\gtrsim N_3^\\frac{1}{2} $ .",
"Proceeding as in Subcase (C.1), we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert z_{3, N_1} \\Vert _{L^{q}_T L^{6+}_x}\\bigg (\\prod _{j = 2}^3 \\Vert z_{3, N_2} \\Vert _{L^{2+}_T L^{6-}_x}\\bigg )\\Vert w_{N_4} \\Vert _{L^\\infty _T L^2_x}\\\\& \\lesssim T^{0+}N_1^{ -s +} N_2^{-2s+}N_3^{\\sigma -2s+}\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}\\Vert z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}^2\\bigg (\\prod _{j = 2}^3\\Vert z_{3, N_j}\\Vert _{Y^{2s-}_T}\\bigg )\\\\& \\le T^{0+}N_3^{\\sigma -\\frac{7}{2} s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ This yields (REF ), provided that $\\sigma < \\frac{7}{2} s$ and $s > 0$ .",
"$\\circ $ Subsubcase (C.2.c): $N_2\\gtrsim N_3^\\frac{1}{2} \\gg N_1$ .",
"By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\Vert z_{3, N_1} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_2} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{1 - 2s -} N_2^{-s}N_3^{\\sigma - 2s- \\frac{1}{2}+}C( \\Vert z_{3}\\Vert _{X^{2s-}_T})\\Vert \\langle \\nabla \\rangle ^s z_{3, N_2} \\Vert _{L^5_{T, x}}\\\\\\multicolumn{2}{l}{\\text{By applying Lemma \\ref {LEM:Z3_2},}}\\\\& \\le T^{\\frac{1}{10}+}N_1^{1 - 2s -} N_2^{-s}N_3^{\\sigma - 2s- \\frac{1}{2}+}C( \\Vert z_{3}\\Vert _{X^{2s-}_T})\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\\\& \\le T^{\\frac{1}{10}+}N_3^{\\sigma -\\frac{7}{2} s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}).$ This yields (REF ), provided that $\\sigma < \\frac{7}{2} s$ and $ 0 \\le s < \\frac{1}{2}$ .",
"Case (D): $v v z_1$ case.",
"By symmetry, we assume $N_1 \\ge N_2$ .",
"$\\bullet $ Subcase (D.1): $N_1 \\gtrsim N_3$ .",
"In this case, we have $N_1 \\sim N_{\\max }$ .",
"$\\circ $ Subsubcase (D.1.a): $\\max (N_2, N_3) \\ge N_1^\\frac{1}{10}$ .",
"By Hölder's inequality and Lemma REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\Vert v_{N_1} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert v_{N_2} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_3} \\Vert _{L^{10}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+} N_2^{-\\sigma } N_3^{-s} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $\\sigma > 0$ and $ s > 0$ .",
"$\\circ $ Subsubcase (D.1.b): $\\max (N_2, N_3) \\ll N_1^\\frac{1}{10}$ .",
"In this case, we have $N_1\\sim N_4 \\sim N_{\\max }$ .",
"By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\Vert v_{N_1}\\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_3}\\Vert _{L^5_{T, x}}\\Vert v_{ N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{0+}N_2^{1-\\sigma -}N_3^{-s}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $ \\sigma \\ge 0$ and $ s \\ge 0$ .",
"$\\bullet $ Subcase (D.2): $N_3 \\sim N_4 \\gg N_1\\ge N_2$ .",
"$\\circ $ Subsubcase (D.2.a): $N_1, N_2 \\ll N_3^\\frac{1}{2}$ .",
"By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} z_{1, N_3} \\Vert _{L^2_{T, x}} \\Vert v_{N_2} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1-\\sigma -} N_2^{1-\\sigma -} N_3^{\\sigma - s - \\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\bigg (\\prod _{j = 1}^2 \\Vert v_{N_j}\\Vert _{Y^\\sigma _T}\\bigg )\\Vert \\mathbf {P}_{N_3} \\phi ^\\omega \\Vert _{H^{s}}\\\\& \\le T^{0+} N_3^{ - s +}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $\\sigma \\le 1$ and $s>0$ .",
"$\\circ $ Subsubcase (D.2.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .",
"Proceeding as in Subsubcase (D.1.a) with $L^\\frac{10}{3}_{T, x},L^\\frac{10}{3}_{T, x},L^{10}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality, it suffices to note that $N_1^{-\\sigma } N_2^{-\\sigma }N_3^{\\sigma -s}\\lesssim N_3^{ - s }\\lesssim N_{\\max }^{0-},$ provided that $\\sigma \\ge 0$ and $ s > 0$ .",
"$\\circ $ Subsubcase (D.2.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .",
"By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1}\\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_3}\\Vert _{L^5_{T, x}}\\Vert v_{ N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{0+}N_1^{-\\sigma } N_2^{1-\\sigma -}N_3^{\\sigma -s}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $0\\le \\sigma \\le 1$ and $ s > 0$ .",
"Case (E): $v v z_3$ case.",
"By symmetry, we assume $N_1 \\ge N_2$ .",
"$\\bullet $ Subcase (E.1): $N_1 \\gtrsim N_3$ .",
"In this case, we have $N_1 \\sim N_{\\max }$ .",
"$\\circ $ Subsubcase (E.1.a): $\\max (N_2, N_3) \\ge N_1^\\frac{1}{10}$ .",
"By Hölder's inequality with (REF ) as in Subcase (C.1) and Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\bigg (\\prod _{j = 1}^2 \\Vert v_{N_j} \\Vert _{L^{2+}_T L^{6-}_x}\\bigg )\\Vert z_{3, N_3} \\Vert _{L^{q}_T L^{6+}_x}\\Vert w_{N_4} \\Vert _{L^\\infty _T L^2_x}\\\\& \\lesssim T^{0+}N_2^{-\\sigma }N_3^{-s+}\\Vert v\\Vert _{X^\\sigma _T}^2\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}\\Vert z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}^2\\\\\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $\\sigma > 0$ and $s > 0$ .",
"$\\circ $ Subsubcase (E.1.b): $\\max (N_2, N_3) \\ll N_1^\\frac{1}{10}$ .",
"In this case, we have $N_1\\sim N_4 \\sim N_{\\max }$ .",
"By Lemmas REF , REF , and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\Vert v_{N_1}\\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{3, N_3}\\Vert _{L^5_{T, x}}\\Vert v_{ N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{\\frac{1}{10}+}N_2^{1-\\sigma -}N_3^{-s}N_4^{-\\frac{1}{2}+}\\Vert v\\Vert _{X^\\sigma _T}^2\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\\\& \\le T^{\\frac{1}{10}+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $ \\sigma \\ge 0$ and $ s \\ge 0$ .",
"$\\bullet $ Subcase (E.2): $N_3 \\sim N_4 \\gg N_1\\ge N_2$ .",
"$\\circ $ Subsubcase (E.2.a): $N_1, N_2 \\ll N_3^\\frac{1}{2}$ .",
"In this case, we proceed as in Subsubcase (D.2.a).",
"It suffices to note that $N_1^{1-\\sigma -} N_2^{1-\\sigma -} N_3^{\\sigma - 2s - \\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\lesssim N_3^{ - 2s +}\\lesssim N_{\\max }^{0-},$ provided that $\\sigma \\le 1$ and $s>0$ .",
"$\\circ $ Subsubcase (E.2.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .",
"By Hölder's inequality with (REF ) as in Subcase (C.1) and Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\bigg (\\prod _{j = 1}^2 \\Vert v_{N_j} \\Vert _{L^{2+}_T L^{6-}_x}\\bigg )\\Vert z_{3, N_3} \\Vert _{L^{q}_T L^{6+}_x}\\Vert w_{N_4} \\Vert _{L^\\infty _T L^2_x}\\\\& \\lesssim T^{0+}N_1^{-\\sigma }N_2^{-\\sigma } N_3^{\\sigma -s+}\\Vert v\\Vert _{X^\\sigma _T}^2\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}\\Vert z_1\\Vert _{L^{3q}_T L^{\\frac{18}{5}+}_x}^2\\\\\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $\\sigma \\ge 0$ and $s > 0$ .",
"$\\circ $ Subsubcase (E.2.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .",
"By Lemmas REF , REF , and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1}\\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{3, N_3}\\Vert _{L^5_{T, x}}\\Vert v_{ N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{\\frac{1}{10}}N_1^{-\\sigma } N_2^{1-\\sigma -}N_3^{\\sigma -s}N_4^{-\\frac{1}{2}+}\\Vert v\\Vert _{X^\\sigma _T}^2\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\\\& \\le T^{\\frac{1}{10}} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}^2,$ provided that $0\\le \\sigma \\le 1$ and $ s > 0$ .",
"Case (F): $v z_1 z_1$ case.",
"By symmetry, we assume $N_3 \\ge N_2$ .",
"$\\bullet $ Subcase (F.1): $N_1 \\gtrsim N_3$ .",
"In this case, we have $N_1 \\sim N_{\\max }$ .",
"$\\circ $ Subsubcase (F.1.a): $\\max (N_2, N_3) \\ge N_1^\\frac{1}{10}$ .",
"By Lemma REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_1^\\sigma \\Vert v_{N_1} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^5_{T, x}}\\Vert z_{1, N_3} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+} N_2^{-s} N_3^{-s} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T},$ provided that $ s > 0$ .",
"$\\circ $ Subsubcase (F.1.b): $\\max (N_2, N_3) \\ll N_1^\\frac{1}{10}$ .",
"In this case, we proceed as in Subsubcase (D.1.b).",
"It suffices to note that $N_2^{1-s-} N_3^{-s}N_4^{-\\frac{1}{2}+}\\lesssim N_{\\max }^{0-},$ provided that $ s \\ge 0$ .",
"$\\bullet $ Subcase (F.2): $N_2 \\sim N_3 \\gg N_1$ .",
"$\\circ $ Subsubcase (F.2.a): $N_1\\ll N_3^\\frac{1}{2}$ .",
"By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{ N_1} z_{1, N_2} \\Vert _{L^2_{T, x}}\\Vert z_{1, N_3}\\Vert _{L^5_{T, x}}\\Vert w_{N_4}\\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{0+}N_1^{1-\\sigma -} N_2^{-s-\\frac{1}{2} + }N_3^{\\sigma -s}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+}N_3^{\\frac{\\sigma }{2}-2s+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $\\sigma \\le \\min (1, 4s-)$ .",
"$\\circ $ Subsubcase (F.2.b): $N_1\\gtrsim N_3^\\frac{1}{2} $ .",
"In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality.",
"It suffices to note that $N_1^{-\\sigma } N_2^{-s}N_3^{\\sigma -s}\\lesssim N_3^{ \\frac{\\sigma }{2} - 2s }\\lesssim N_{\\max }^{0-},$ provided that $0 \\le \\sigma < 4s$ .",
"$\\bullet $ Subcase (F.3): $N_3 \\sim N_4 \\sim N_{\\max } \\gg N_1, N_2$ .",
"$\\circ $ Subsubcase (F.3.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .",
"By the bilinear Strichartz estimate, we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} z_{1, N_3} \\Vert _{L^2_{T, x}} \\Vert z_{1, N_2} w_{N_4}\\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{0+} N_1^{1- \\sigma -}N_2^{1-s - } N_3^{\\sigma - s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\Vert v\\Vert _{X^\\sigma _T}\\bigg (\\prod _{j = 2}^3\\Vert \\mathbf {P}_{N_j}\\phi ^\\omega \\Vert _{H^s}\\bigg )\\\\& \\lesssim T^{0+} N_3^{\\frac{\\sigma }{2} - \\frac{3}{2} s +}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $ \\sigma \\le \\min (1, 3s-)$ and $ s < 1$ .",
"$\\circ $ Subcase (F.3.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .",
"In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality.",
"It suffices to note that $N_1^{-\\sigma } N_2^{-s}N_3^{\\sigma -s}\\lesssim N_3^{ \\frac{\\sigma }{2} - \\frac{3}{2}s }\\lesssim N_{\\max }^{0-},$ provided that $0 \\le \\sigma < 3s$ .",
"$\\circ $ Subcase (F.3.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .",
"By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_3} \\Vert _{L^{5}_{T, x}}\\Vert z_{1, N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\le T^{0+}N_1^{-\\sigma } N_2^{1-s-}N_3^{\\sigma - s}N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+}N_3^{\\frac{\\sigma }{2}- \\frac{3}{2}s+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $0\\le \\sigma < 3s$ and $s < 1$ .",
"$\\circ $ Subcase (F.3.d): $N_2\\gtrsim N_3^\\frac{1}{2} \\gg N_1$ .",
"By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} z_{1, N_3} \\Vert _{L^2_{T, x}}\\Vert z_{1, N_2} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{1-\\sigma -} N_2^{-s}N_3^{\\sigma - s-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+}N_3^{\\frac{\\sigma }{2}- \\frac{3}{2}s+}C(\\Vert z_1\\Vert _{B^s(T)}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $ \\sigma \\le \\min (1, 3s-)$ and $s\\ge 0$ .",
"Case (G): $v z_3 z_3$ case.",
"By symmetry, we assume $N_3 \\ge N_2$ .",
"$\\bullet $ Subcase (G.1): $N_1 \\gtrsim N_3$ .",
"In this case, we have $N_1 \\sim N_{\\max }$ .",
"We can proceed as in Subcase (E.1) with $v_{N_2}$ replaced by $z_{3, N_2}$ .",
"More precisely, when $\\max (N_2, N_3) \\ge N_1^\\frac{1}{10}$ , we apply Hölder's inequality with (REF ) as in Subsubcase (E.1.a).",
"It suffices to note that $N_2^{-2s+}N_3^{-s+}\\lesssim N_{\\max }^{0-},$ provided that $ s> 0$ .",
"When $\\max (N_2, N_3) \\ll N_1^\\frac{1}{10}$ , we proceed with Lemmas REF , REF , and REF as in Subsubcase (E.1.b).",
"In this case, it suffices to note that $N_2^{1-2s-}N_3^{-s}N_4^{-\\frac{1}{2}+}\\lesssim N_{\\max }^{0-},$ provided that $s\\ge 0$ .",
"$\\bullet $ Subcase (G.2): $N_2 \\sim N_3 \\gg N_1$ .",
"$\\circ $ Subsubcase (G.2.a): $N_1\\ll N_3^\\frac{1}{2}$ .",
"By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{ N_1} z_{3, N_2} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_3}\\Vert _{L^5_{T, x}}\\Vert w_{N_4}\\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\le T^{\\frac{1}{10}}N_1^{1-\\sigma -} N_2^{-2s-\\frac{1}{2} + }N_3^{\\sigma -s} \\Vert v\\Vert _{X^\\sigma _T}\\Vert z_{3, N_2}\\Vert _{X^{2s-}_T}\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2 \\\\& \\le T^{\\frac{1}{10}}N_3^{\\frac{\\sigma }{2} -3 s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+} N_{\\max }^{0-} C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T},$ provided that $\\sigma \\le \\min (1, 6s-)$ .",
"$\\circ $ Subsubcase (G.2.b): $N_1\\gtrsim N_3^\\frac{1}{2} $ .",
"In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality with Lemma REF .",
"It suffices to note that $N_1^{-\\sigma } N_2^{-s}N_3^{\\sigma -s}\\lesssim N_3^{ \\frac{\\sigma }{2} - 2s }\\lesssim N_{\\max }^{0-},$ provided that $0 \\le \\sigma < 4s$ .",
"$\\bullet $ Subcase (G.3): $N_3 \\sim N_4 \\sim N_{\\max } \\gg N_1, N_2$ .",
"$\\circ $ Subsubcase (G.3.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .",
"In this case, we proceed as in Subsubcase (F.3.a).",
"It suffices to note that $N_1^{1- \\sigma -}N_2^{1-2s - } N_3^{\\sigma - 2 s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\lesssim N_3^{\\frac{\\sigma }{2} - 3s+}\\lesssim N_{\\max }^{0-},$ provided that $ \\sigma \\le \\min (1, 6s-)$ and $ s < \\frac{1}{2}$ .",
"$\\circ $ Subsubcase (G.3.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .",
"In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^5_{T, x},L^{5}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality as in Subsubcase (F.3.b) but we apply Lemma REF to estimate the $L^{5}_{T, x}$ -norm of $z_{3, N_j}$ , $j = 2, 3$ .",
"The regularity restriction is precisely the same as that in Subsubcase (F.3.b).",
"$\\circ $ Subsubcase (G.3.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .",
"By Lemmas REF , REF , and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{3, N_3} \\Vert _{L^{5}_{T, x}}\\Vert z_{3, N_2} w_{N_4} \\Vert _{L^2_{T, x}}\\\\& \\lesssim T^{\\frac{1}{10}+}N_1^{-\\sigma } N_2^{1-2s-}N_3^{\\sigma - s}N_4^{-\\frac{1}{2}+}\\Vert v\\Vert _{X^\\sigma _T}\\Vert z_{3, N_2}\\Vert _{X^{2s-}_T}\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\\\& \\le T^{\\frac{1}{10}+}N_3^{\\frac{\\sigma }{2}- 2s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $0\\le \\sigma < 4s$ and $s < \\frac{1}{2}$ .",
"$\\circ $ Subcase (G.3.d): $N_2\\gtrsim N_3^\\frac{1}{2} \\gg N_1$ .",
"By Lemmas REF , REF , and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} z_{3, N_3} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_2} \\Vert _{L^{5}_{T, x}}\\Vert w_{N_4} \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim T^{\\frac{1}{10}+}N_1^{1-\\sigma -} N_2^{-s}N_3^{\\sigma - 2s-\\frac{1}{2}+}\\Vert v\\Vert _{X^\\sigma _T}\\Vert \\langle \\nabla \\rangle ^s z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}\\Vert z_1\\Vert _{L^{15}_T L^{\\frac{15}{4}}_x}^2\\Vert z_{3, N_3} \\Vert _{X^{2s-}_T}\\\\& \\le T^{\\frac{1}{10}+}N_1^{\\frac{\\sigma }{2}- \\frac{5}{2}s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_3\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $ \\sigma \\le \\min (1, 5s-)$ and $s\\ge 0$ .",
"Case (H): $v z_3 z_1$ case.",
"$\\bullet $ Subcase (H.1): $N_1 \\sim N_{\\max }$ .",
"In this case, we can proceed as in Subcase (D.1) by replacing $v_{N_2}$ with $z_{3, N_2}$ .",
"When $\\max (N_2, N_3) \\ge N_1^\\frac{1}{10}$ , it suffices to note that $N_2^{-2s+}N_3^{-s}\\lesssim N_{\\max }^{0-},$ provided that $s> 0$ .",
"When $\\max (N_2, N_3) \\ll N_1^\\frac{1}{10}$ , it suffices to note that $N_2^{1-2s-}N_3^{-s}N_4^{-\\frac{1}{2}+}\\lesssim N_{\\max }^{0-},$ provided that $s\\ge 0$ .",
"$\\bullet $ Subcase (H.2): $N_3 \\sim N_{\\max } \\gg N_1$ .",
"$\\circ $ Subsubcase (H.2.a): $N_1, N_2 \\ll N_3^\\frac{1}{2} $ .",
"In this case, we have $N_3 \\sim N_4 \\sim N_{\\max }$ .",
"Then, proceeding as in Subsubcase (F.3.a), it suffices to note that $N_1^{1- \\sigma -}N_2^{1-2s - } N_3^{\\sigma - s -\\frac{1}{2} +}N_4^{-\\frac{1}{2}+}\\lesssim N_3^{\\frac{\\sigma }{2} - 2s}\\lesssim N_{\\max }^{0-},$ provided that $ \\sigma \\le \\min (1, 4s-)$ and $ s < \\frac{1}{2}$ .",
"$\\circ $ Subsubcase (H.2.b): $N_1, N_2\\gtrsim N_3^\\frac{1}{2} $ .",
"In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^\\frac{10}{3}_{T, x},L^{10}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality.",
"It suffices to note that $N_1^{- \\sigma }N_2^{-2s+} N_3^{\\sigma - s}\\lesssim N_3^{\\frac{\\sigma }{2} - 2s+}\\lesssim N_{\\max }^{0-},$ provided that $0 \\le \\sigma < 4s$ .",
"$\\circ $ Subsubcase (H.2.c): $N_1\\gtrsim N_3^\\frac{1}{2} \\gg N_2$ .",
"In this case, we have $N_3 \\sim N_4 \\sim N_{\\max }$ .",
"We can proceed as in Subsubcase (G.3.c) with $z_{3, N_3}$ replaced by $z_{1,N_3}$ (without applying Lemma REF ).",
"The regularity restriction is precisely the same as in Subsubcase (G.3.c).",
"$\\circ $ Subsubcase (H.2.d): $N_2\\gtrsim N_3^\\frac{1}{2} \\gg N_1$ .",
"We first consider the case $N_3 \\sim N_4 \\sim N_{\\max }$ .",
"By Lemmas REF and REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim N_3^\\sigma \\Vert v_{N_1} w_{N_4} \\Vert _{L^2_{T, x}}\\Vert z_{3, N_2} \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert z_{1, N_3} \\Vert _{L^{5}_{T, x}}\\\\& \\lesssim T^{0+}N_1^{1-\\sigma -} N_2^{-2s+}N_3^{\\sigma - s} N_4^{-\\frac{1}{2}+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T}\\\\& \\le T^{0+}N_3^{\\frac{\\sigma }{2}- 2s+}C(\\Vert z_1\\Vert _{B^s(T)}, \\Vert z_{3}\\Vert _{X^{2s-}_T}) \\Vert v\\Vert _{X^\\sigma _T}.$ This yields (REF ), provided that $ \\sigma \\le \\min (1, 4s-)$ and $ s> 0$ .",
"When $N_4\\ll N_3$ , we have $N_2 \\sim N_3 \\sim N_{\\max }$ .",
"In this case, we can repeat the computation above with the roles of $z_{3, N_2}$ and $w_{N_4}$ switched.",
"Noting that $N_1^{1-\\sigma -} N_2^{-2s-\\frac{1}{2}+}N_3^{\\sigma - s}\\lesssim N_3^{\\frac{\\sigma }{2}- 3s+} \\lesssim N_{\\max }^{0-},$ we obtain (REF ), provided that $ \\sigma \\le \\min (1, 6s-)$ .",
"$\\bullet $ Subcase (H.3): $N_2 \\sim N_4 \\gg N_1, N_3$ .",
"$\\circ $ Subsubcase (H.3.a): $N_1\\ll N_2^\\frac{1}{2}$ .",
"In this case, we can proceed as in Subsubcase (H.2.d).",
"It suffices to note that $N_1^{1-\\sigma -} N_2^{\\sigma -2s + }N_3^{-s} N_4^{-\\frac{1}{2} +}\\lesssim N_2^{\\frac{\\sigma }{2} -2 s+}\\lesssim N_{\\max }^{0-},$ provided that $\\sigma \\le \\min (1, 4s-)$ and $s \\ge 0$ .",
"$\\circ $ Subsubcase (H.3.b): $N_1\\gtrsim N_2^\\frac{1}{2} $ .",
"In this case, we proceed with $L^\\frac{10}{3}_{T, x},L^\\frac{10}{3}_{T, x},L^{10}_{T, x}, L^\\frac{10}{3}_{T, x}$ -Hölder's inequality.",
"It suffices to note that $N_1^{- \\sigma }N_2^{\\sigma -2s+} N_3^{-s}\\lesssim N_2^{\\frac{\\sigma }{2} - 2s+}\\lesssim N_{\\max }^{0-},$ provided that $0 \\le \\sigma < 4s$ .",
"Case ( I ): $v vv$ case.",
"In this case, there is no need to perform the dyadic decomposition.",
"By Hölder's inequality, Sobolev's inequality (REF ), and Lemma REF , we have $\\text{LHS of } (\\ref {XX1})& \\lesssim \\Vert v \\Vert _{L^{5}_{T, x}}^2\\Vert \\langle \\nabla \\rangle ^\\sigma v \\Vert _{L^\\frac{10}{3}_{T, x}}\\Vert w \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim \\Vert \\langle \\nabla \\rangle ^\\frac{1}{2} v \\Vert _{L^{5}_{T} L^\\frac{30}{11}_x}^2\\Vert \\langle \\nabla \\rangle ^\\sigma v \\Vert _{L^\\frac{10}{3}_{T, x}}\\\\& \\lesssim \\Vert v\\Vert _{X^\\sigma _T}^3,$ provided that $ \\sigma \\ge \\frac{1}{2}$ .",
"Putting all the cases including Cases (A) and (B) treated in Lemmas REF and REF , we conclude that (REF ) holds, provided that $ \\frac{1}{2} \\le \\sigma \\le 1$ and $\\frac{2}{5} \\sigma < s < \\frac{1}{2}$ .",
"This completes the proof of Proposition REF ."
],
[
"Proof of Theorem ",
"In this section, we briefly discuss the proof of Theorem REF .",
"Given $ \\frac{1}{6} < s < \\frac{1}{2}$ , let $k \\in \\mathbb {N}$ such that $\\frac{1}{2\\alpha _{k+1}} < s \\le \\frac{1}{2\\alpha _{k}}$ , where $\\alpha _k$ is defined in (REF ).",
"Our main goal is to study the fixed point problem (REF ).",
"Define $\\widetilde{\\Gamma }$ by $\\widetilde{\\Gamma }v(t) = -i \\int _0^t S(t - t^{\\prime })\\bigg \\lbrace \\mathcal {N}\\bigg (v +\\sum _{\\ell = 1}^{k} \\zeta _{2\\ell -1}\\bigg )- \\sum _{\\ell = 2}^{k}\\sum _{\\begin{array}{c}j_1 + j_2 + j_3 = 2\\ell - 1\\\\j_1, j_2, j_3 \\in \\lbrace 1, 2\\ell -3\\rbrace \\end{array}}\\zeta _{j_1} \\overline{\\zeta _{j_2}}\\zeta _{j_3}\\bigg \\rbrace (t^{\\prime }) dt^{\\prime },$ where $\\zeta _{2\\ell -1}$ , $\\ell = 1, \\dots , k$ , is as in (REF ).",
"Theorem REF follows from a standard fixed point argument, once we prove the following proposition.",
"Proposition 6.1 Given $ \\frac{1}{6} < s < \\frac{1}{2}$ , let $k \\in \\mathbb {N}$ such that $\\frac{1}{2\\alpha _{k+1}} < s < \\frac{1}{\\alpha _{k}}$ .",
"Then, there exist $\\theta > 0$ , $C_1, C_2> 0$ , and an almost surely finite constant $R = R(\\omega ) >0$ such that $\\Vert \\widetilde{\\Gamma }v\\Vert _{X^\\frac{1}{2}([0, T])}& \\le C_1\\big (\\Vert v\\Vert _{X^\\frac{1}{2}([0, T])} ^3 + T^\\theta R(\\omega )\\big ),\\\\\\Vert \\widetilde{\\Gamma }v_1 - \\widetilde{\\Gamma }v_2 \\Vert _{X^\\frac{1}{2}([0, T])}& \\le C_2\\Big (\\sum _{j = 1}^2 \\Vert v_j\\Vert _{X^\\frac{1}{2}([0, T])} ^2+ T^\\theta R(\\omega )\\Big )\\Vert v_1 -v_2 \\Vert _{X^\\frac{1}{2}([0, T])},$ for all $v, v_1, v_2 \\in X^\\frac{1}{2}([0, T])$ and $0 < T\\le 1$ .",
"On the one hand, the upper bound $\\frac{1}{\\alpha _k}$ on the range of $s$ in Proposition REF comes from the restriction in Proposition REF .",
"On the other hand, given $\\frac{1}{6} < s < \\frac{1}{2}$ , we fix $k \\in \\mathbb {N}$ such that $\\frac{1}{2\\alpha _{k+1}} < s \\le \\frac{1}{2\\alpha _{k}}$ and apply Proposition REF .",
"Namely, the upper bound on $s$ in Proposition REF does not cause any problem in proving Theorem REF .",
"We only discuss the proof of (REF ) since () follows in a similar manner.",
"As in the proof of Proposition REF , it suffices to perform a case-by-case analysis of expressions of the form: $\\bigg | \\int _0^T \\int _{\\mathbb {R}^3}\\langle \\nabla \\rangle ^\\frac{1}{2} ( w_1 w_2 w_3 )w dx dt\\bigg |,$ where $\\Vert w\\Vert _{Y^{0}_T} \\le 1$ and $w_j= v$ , $\\zeta _{2\\ell -1}$ , $\\ell = 1, \\dots , k$ , but not of the form $\\zeta _1 \\zeta _1\\zeta _{2\\ell -3}$ for $\\ell \\in \\lbrace 2, 3, \\dots , k\\rbrace $ .",
"Here, we have dropped the complex conjugate sign on $w_2$ since it does not play any role in our analysis.",
"More concretely, we need to consider the following cases: Table: NO_CAPTION where $j_1, j_2, j_3$ can take any value in $\\lbrace 3, 5, \\dots , 2k-1\\rbrace $ .",
"As we see below, the worst interaction appears in Case (A) and all the other cases can be handled as in the proof of Proposition REF .",
"We first point out that, in the proof of Proposition REF , the regularity restriction coming from Cases (B) - (I) (with $\\sigma = \\frac{1}{2}$ ) is $s > \\frac{1}{6}$ .",
"Moreover, by comparing Proposition REF and Lemma REF with Proposition REF and Lemma REF , we see that the unbalanced higher order term $\\zeta _{2\\ell -1}$ for $\\ell \\in \\lbrace 2, 3, \\dots , k\\rbrace $ enjoys (at least) the same regularity properties as $z_3 = \\zeta _3$ both in terms of differentiability and space-time integrability.",
"Therefore, we can simply apply the estimates in the proofs of Lemma REF (i) for Case (B) and Proposition REF for Cases (C) - (I) and conclude that the contributions from Cases (B) - (I) can be bounded, provided $s > \\frac{1}{6}$ .",
"It remains to consider Case (A).",
"In view of (REF ), the contribution from Case (A) is nothing but $\\zeta _{2k+1}$ .",
"Hence, by applying Proposition REF with $\\sigma = \\frac{1}{2}$ , we conclude that the contribution in Case (A) can be estimated by $T^\\theta R(\\omega )$ , provided that $\\frac{1}{2\\alpha _{k+1}} < s < \\frac{1}{\\alpha _{k}}$ .",
"This completes the proof of Proposition REF ."
],
[
"On the deterministic non-smoothing\nof the Duhamel integral operator",
"In this appendix, we show that there is no deterministic smoothing on the Duhamel integral operator $I$ defined in (REF ) when $s \\le \\frac{1}{2}$ .",
"Lemma 1.1 Let $\\sigma > 3s-1$ .",
"Then, the estimate (REF ) does not hold.",
"In particular, there is no deterministic nonlinear smoothing when $s \\le \\frac{1}{2}$ .",
"Given $N \\ge L \\ge \\ell \\gg 1$ , consider $u_j = S(t) \\phi _j$ , $j = 1, 2, 3$ , such that $\\widehat{\\phi }_j( \\xi ) \\sim \\mathbf {1}_{A_j}(\\xi )$ , where $A_1 = L e_1 + \\ell Q, \\qquad A_2 = \\ell Q, \\qquad \\text{and}\\qquad A_3 = N e_2 + \\ell Q.$ Then, we have $\\prod _{j = 1}^3 \\Vert u_j(0) \\Vert _{H^s}\\sim \\ell ^{\\frac{9}{2} + s} L^s N^s.$ Let $A = Le_1 + N e_2 + \\ell Q$ .",
"By writing $\\mathbf {1}_{A} (\\xi )\\mathcal {F}_x[I(u_1, & u_2, u_3)](t, \\xi )\\\\&= -i \\mathbf {1}_{A} (\\xi ) e^{-it |\\xi |^2} \\int _0^t \\operatornamewithlimits{\\int }_{\\xi = \\xi _1 - \\xi _2 + \\xi _3}e^{i t^{\\prime } \\Phi (\\bar{\\xi })} \\prod _{j = 1}^3 \\mathbf {1}_{A_j} (\\xi _j) d\\xi _1 d\\xi _2 dt^{\\prime },$ we see that the non-trivial contribution to (REF ) comes from $\\xi _j \\in A_j$ , $j = 1, 2, 3$ .",
"Moreover, in this case, $\\Phi (\\bar{\\xi })$ defined (REF ) satisfies $|\\Phi (\\bar{\\xi }) |\\lesssim N L$ .",
"In particular, by choosing $t_* \\sim \\frac{1}{NL}$ such that $t_*|\\Phi (\\bar{\\xi }) |\\ll 1 $ , we have (REF ) for all $t^{\\prime } \\in [0, t_*]$ .",
"Hence, by Lemma REF with (REF ), we obtain $\\Vert I(u_1, u_2, u_3) \\Vert _{X^\\sigma ([0, 1])} \\gtrsim \\Vert I(u_1, u_2, u_3)(t_*) \\Vert _{ H^\\sigma }\\gtrsim t_* \\ell ^\\frac{15}{2} N^\\sigma \\sim \\ell ^\\frac{15}{2} L^{-1}N^{\\sigma -1} .$ Therefore, it follows from (REF ) and (REF ) that by choosing $\\ell \\sim L \\sim N$ , we have $\\frac{\\Vert I(u_1, u_2, u_3) \\Vert _{X^\\sigma ([0, 1])}}{\\prod _{j=1}^3 \\Vert u_j(0) \\Vert _{H^s}}\\gtrsim \\ell ^{3-s}L^{-1-s} N^{\\sigma -1-s}\\sim N^{\\sigma - 3s+1} \\longrightarrow \\infty ,$ as $N \\rightarrow \\infty $ , provided that $\\sigma > 3s-1$ .",
"This shows that the estimate (REF ) can not hold when $\\sigma > 3s-1$ .",
"Define the quintilinear operator $I^{(2)}$ by $I^{(2)}(u_1, \\dots , u_5) (t):= -i \\sum _{\\ell = 1}^3\\int _0^t S(t - t^{\\prime })v_{j_1} \\overline{v_{j_2}}v_{j_3} (t^{\\prime }) dt^{\\prime },$ where $v_{j_i} = u_i$ for $i \\ne \\ell $ and $v_{j_\\ell } = I(u_\\ell , u_4, u_5)$ .",
"This basically corresponds to the second order term appearing in the power series expansion for (REF ).",
"In particular, we have that $z_5 = I^{(2)}(z_1, \\dots , z_1)$ .",
"By a computation similar to the proof of Lemma REF , we can prove the following deterministic non-smoothing on the second order term.",
"Let $\\sigma > 5s - 2$ .",
"Then, there is no finite constant $C> 0$ such that $\\big \\Vert I^{(2)}( S(t) \\phi _1, \\dots , S(t) \\phi _5)\\big \\Vert _{X^\\sigma ([0, 1])}\\le C \\prod _{j = 1}^5 \\Vert \\phi _j \\Vert _{H^s}$ for all $\\phi _j \\in H^s(\\mathbb {R}^3)$ .",
"In particular, this shows that when $s \\le \\frac{1}{2}$ , there is no deterministic smoothing on the second order term $I^{(2)}$ .",
"A similar comment applies to the higher order terms.",
"Acknowledgments This research was partially supported by Research in Groups at International Centre for Mathematical Sciences, Edinburgh, United Kingdom.",
"Á.B.",
"was partially supported by a grant from the Simons Foundation (No. 246024).",
"T.O.",
"was supported by the European Research Council (grant no.",
"637995 “ProbDynDispEq”)."
]
] | 1709.01910 |
[
[
"Some Sufficient Conditions for Finding a Nesting of the Normalized\n Matching Posets of Rank 3"
],
[
"Abstract Given a graded poset $P$, consider a chain decomposition $\\mathcal{C}$ of $P$.",
"If $|C_1|\\le |C_2|$ implies that the set of the ranks of elements in $C_1$ is a subset of the ranks of elements in $C_2$ for any chains $C_1,C_2\\in \\mathcal{C}$, then we say $\\mathcal{C}$ is a nested chain decomposition (or nesting, for short) of $P$, and $P$ is said to be nested.",
"In 1970s, Griggs conjectured that every normalized matching rank-unimodal poset is nested.",
"This conjecture is proved to be true only for all posets of rank 2 [W:05], some posets of rank 3 [HLS:09,ENSST:11], and the very special cases for higher ranks.",
"For general cases, it is still widely open.",
"In this paper, we provide some sufficient conditions on the rank numbers of posets of rank 3 to satisfies the Griggs's conjecuture."
],
[
"Introduction",
"We start with the necessary terminology of poset theory.",
"A poset $P=(P,\\le )$ is a set $P$ equipped with a partially order relation $\\le $ .",
"Through out the paper, all posets are finite.",
"Let $P$ be a poset.",
"We say a subposet $C$ of $P$ is a chain of length $\\ell $ if $C=\\lbrace x_i\\mid x_i< x_{i+1}\\mbox{ for }0\\le i \\le \\ell -1\\rbrace $ .",
"A chain decomposition $\\mathcal {C}$ of $P$ is a collection of disjoint chains of $P$ with $\\cup _{C\\in \\mathcal {C}}C=P$ .",
"We are looking for decompositions with as few number of chains as possible.",
"The most significant theorem in the literature was given by Dilworth [3].",
"Here an antichain is a subposet of $P$ such that neither $x\\le y $ nor $y\\le x$ holds for any $x\\ne y$ in $Q$ .",
"Theorem 1.1 [3] For a poset $P$ , the minimum number of chains in a chain decomposition is equal to the maximum size of an antichain of $P$ .",
"In the following, we study the chain decompositions of a special class of posets, which includes example such as Boolean lattices, linear lattices, and divisor lattices, etc.",
"A graded poset is a poset such that every maximal chain has the same length.",
"For a graded poset $P$ , we define the rank function $r:P\\longrightarrow \\mathbb {N}$ such that $r(x)=i$ , if there exactly $i$ elements $y< x$ in a maximal chain.",
"Moreover, an element $x$ is of rank $i$ if $r(x)=i$ and the rank of $P$ is $\\max _{x\\in P}r(x)$ .",
"The $i$ th level of a graded poset $P$ is the collection of all elements of rank $i$ , that is, $L_i=\\lbrace x\\mid r(x)=i, x\\in P\\rbrace $ .",
"By the definitions, every level is an antichain.",
"Therefore, $\\max _{i}|L_i|\\le |\\mathcal {C}|$ for every chain decomposition $\\mathcal {C}$ of $P$ .",
"For graded posets, we define a special chain decomposition: Definition 1.2 (nested chain decomposition) Let $\\mathcal {C}$ be a chain decomposition of a graded poset $P$ .",
"For any chains $C_i,C_j\\in \\mathcal {C}$ , if $|C_i|\\le |C_j|$ implies $\\lbrace r(x)\\mid x\\in C_i\\rbrace \\subseteq \\lbrace r(x)\\mid x\\in C_j\\rbrace $ , then $\\mathcal {C}$ is called a nested chain decomposition of $P$ .",
"We say $P$ is nested, or it has a nesting, if such a decomposition of $P$ exists.",
"Observe that a nesting $\\mathcal {C}$ is a chain decomposition with minimum number of chains.",
"Because from the inclusion relation, $\\cap _{C\\in \\mathcal {C}}\\lbrace r(x)\\mid x\\in C\\rbrace $ is not empty, and there exists some $m$ and a level $L_m$ such that $m\\in \\cap _{C\\in \\mathcal {C}}\\lbrace r(x)\\mid x\\in C\\rbrace $ .",
"Thus, every chain in $\\mathcal {C}$ contains an element of rank $m$ , and hence $|L_m|=|\\mathcal {C}|$ .",
"Since $\\max _i|L_i|\\le |\\mathcal {C}|=|L_m|\\le \\max _i|L_i|$ , we have $\\max _i|L_i|\\le |\\mathcal {C}|$ .",
"We refer the reader to see more properties of graded posets in [2], [4].",
"Anderson[1] and Griggs[6] independently gave the same sufficient condition for the existence of a nesting in the graded posets.",
"Let $r_i$ denote the cardinality of level $i$ .",
"The rank numbers or Whitney numbers of a graded poset of rank $n$ is the sequence $(r_0,r_1,\\ldots , r_n)$ .",
"If $r_i=r_{n-i}$ for all $0\\le i\\le n$ , then $P$ is said to be rank-symmetric.",
"Suppose there exists some $i$ such that $r_0\\le \\cdots \\le r_i\\ge \\cdots \\ge r_n$ , then $P$ is rank-unimodal.",
"For any levels $L_i$ and $L_j$ of $P$ , consider any subset $S$ of $L_i$ and denote the set $\\Gamma _j(S)=\\lbrace x\\in L_j\\mid x \\le y\\mbox{ or }y\\le x \\mbox{ for some }y\\in S\\rbrace $ .",
"If the inequality $\\frac{|S|}{r_i} \\le \\frac{|\\Gamma _j(S)|}{r_j}$ holds, then we say $P$ has the normalized matching property from $i$ to $j$ .",
"By simple calculation, one can see if $P$ has the normalized matching property from $i$ to $j$ , then it also has the property from $j$ to $i$ .",
"Moreover, if $P$ has the normalized matching property from $i$ to $j$ and $j$ to $k$ for $i<j<k$ , then it has the property from $i$ to $k$ .",
"Once the normalized matching property holds between any two levels, then we say $P$ is a normalized matching poset.",
"Theorem 1.3 [1], [6] A graded poset $P$ has a nesting if it is a normalized matching poset, and is rank-symmetric and rank-unimodal.",
"In fact, every chain in a nesting of a poset $P$ described in the theorem contains elements of ranks $i,i+1,\\ldots , n-i$ for some $i$ , where $n$ is the rank of $P$ .",
"Such a decomposition is also called a symmetric chain decomposition of $P$ .",
"In addition to Theorem REF , Griggs also posed the following conjecture [6], [7], [8]: Conjecture 1.1 (Griggs Nesting Conjecture) Every normalized matching rank-unimodal poset is nested.",
"The conjecture turns out to be extremely difficult, although one can easily give an affirmative answer of graded posets of rank 1 using the well-known Hall's Marriage Theorem [9].",
"There are only a few graded posets of small ranks which are proven to satisfy the conjecture by Wang [11], Hsu, Logan, and Shahriari.",
"[10], and Escamilla, Nicolae, Salerno, Shahriari, and Tirrell [5], respectively.",
"In section 2, we introduce the early results on the graded posets of ranks 2 and 3, and mention our theorems at the end of the section.",
"The main contribution of this paper is to give more sufficient conditions on the graded posets of rank 3 to satisfy the conjecture, based on the ideas of proofs in the early papers.",
"The proofs of our theorems are presented in Section 3."
],
[
"Normalized Matching Posets of Rank 2 and 3",
"Note that in Conjecture $\\ref {gnc:un}$ , we only concern the normalized matching property and the conditions of the rank numbers.",
"The structure of the poset is irrelevant.",
"For convenience, we use the notation $NM(r_0,r_1,\\ldots ,r_n)$ to denote the collection of all normalized matching posets of rank $n$ with rank numbers $(r_0,r_1,\\ldots ,r_n)$ .",
"In 2005, Wang [11] dropped the rank-unimodal assumption and proved a stronger result.",
"Theorem 2.1 [11] Every poset $P\\in NM(r_0,r_1,r_2)$ has a nesting.",
"For graded posets of rank 3, Shahriari with two research groups[10], [5] developed some sufficient conditions on the rank numbers $(r_0,r_1,r_2,r_3)$ to guarantee the existence of a nesting.",
"In [10], the authors came up with a clever idea which can not only simplify the proof of Theorem REF but also reduce the rank numbers to fewer cases that need to be considered for graded posets of rank 3.",
"Since we will use this idea in our proof, we introduce it below.",
"Proposition 2.2 Given a graded poset $P$ , let $P^{\\prime }=P\\cup \\lbrace x\\rbrace $ be obtained by adding a new element $x$ to the $i$ th level of $P$ together with the partial order relations $y< x$ (resp.",
"$x<y$ ) if $y\\in L_j$ and $j<i$ (resp.",
"$j<i$ ).",
"If $P\\in NM(r_0,r_1,\\ldots , r_n)$ , then $P^{\\prime }\\in NM(r_0,\\ldots ,r_{i-1},r_{i}+1,r_{i+1},\\ldots , r_n)$ .",
"The proof of the proposition is straightforward, since if we pick a set $S$ in the $i$ th level of $P^{\\prime }$ , either it contains $x$ , then $|\\Gamma _k(S)|/r_k=1$ , or it does not contain $S$ , then $|S|/(r_i+1)< |S|/r_i< |\\Gamma _k(S)|/r_k$ , for all $k$ .",
"In [10], such an element is called a ghost.",
"We demonstrate two instances of exploiting the ghosts to get a nesting.",
"Suppose $P$ is a poset in $NM(r_0,r_1,r_2)$ with $r_0<r_2<r_1$ .",
"Then we add $r_2-r_0$ ghosts to the 0th level to get a rank-symmetric poset $P^{\\prime }$ .",
"By Theorem REF , $P^{\\prime }$ has a nesting and each chain contains elements of ranks either $\\lbrace 0,1,2\\rbrace $ or $\\lbrace 2\\rbrace $ .",
"After removing the ghosts from the chains of length 2, we obtain a chain decomposition of $P$ such that each chain contains elements of ranks either $\\lbrace 0,1,2\\rbrace $ , or $\\lbrace 1,2\\rbrace $ , or $\\lbrace 2\\rbrace $ .",
"If the rank numbers satisfy $r_1<r_0<r_2$ , then we add $r_0-r_1$ ghosts to the first level.",
"The new poset $P^{\\prime \\prime }$ restricted on the first and second levels is a poset of rank 1.",
"So we can partition it into chains of length either 0 or 1.",
"Meanwhile, the poset consisting of the 0th and first level of $P^{\\prime \\prime }$ can be partitioned into chains of length of 1.",
"The chains of length 1 in two decompositions can be concatenated into chains of length 2.",
"Finally, we remove the ghosts to get of decomposition of $P$ with each chain containing elements of rank either $\\lbrace 0,1,2\\rbrace $ , or $\\lbrace 0,2\\rbrace $ , or $\\lbrace 2\\rbrace $ .",
"Indeed, the arguments above are exact the ideas of Hsu et al.",
"in [10], used to reprove Theorem REF .",
"Using the ideas of the ghost elements, the induction, and the duality, Hsu et al.",
"[10] showed that to prove Conjecture $\\ref {gnc:un}$ for posets of rank 3, it suffices to verify that all posets $P\\in (r_0,r_1,r_2,r_3)$ with $r_2>r_1>r_0=r_3$ are nested.",
"For example, if a poset $P\\in (r_0,r_1,r_2,r_3)$ with $r_2>r_1>r_0>r_3$ , then we add $r_0-r_3$ ghosts to the third level of $P$ to get a new poset $P^{\\prime }\\in (r_0,r_1,r_2,r_0)$ .",
"Now suppose we already have a nesting of $P^{\\prime }$ .",
"We then remove the ghosts in all the longest chains to get a nesting of $P$ .",
"See [10] for the details of all the reduction methods.",
"With this assumption on the rank numbers, Hsu et al.",
"[10] and Escamilla et al.",
"[5] proved the following Theorem REF and Theorem REF , respectively.",
"Theorem 2.3 [10] Let $r_0, r_1$ and $r_2$ be positive integers with $r_0 <r_1 <r_2$ .",
"Assume that at least one of the following conditions are satisfied: (a) $r_{1} \\ge r_{2}-\\lceil {\\frac{r_{2}}{r_{0}}}\\rceil +1$ ; (b) $r_{1}$ $=r_{0}+1$ ; (c) $r_{2}> r_{0}r_{1}$ ; (d) $r_{1}$ divides $r_{2}$ .",
"Then every $P\\in NM(r_0,r_1,r_2,r_0)$ is nested.",
"Theorem 2.4 [5] Let $r_0, r_1$ and $r_2$ be positive integers with $r_0 <r_1 <r_2$ .",
"Assume that at least one of the following conditions are satisfied: (a) $r_{0}$ divides $r_{1}$ , or (b) $r_{0}+1$ divides $r_{1}$ , or (c) $f(i)\\ge 0$ for all $1\\le i\\le r_{1}-r_{0}-1$ , where the function $f$ is defined by $f(i) = \\left\\lceil \\frac{r_{0}(1+i)}{r_{2}-r_{0}}\\right\\rceil -\\left\\lfloor \\frac{r_{0}i}{r_{1}-r_{0}}\\right\\rfloor \\mbox{; or}$ (d) $r_{2}> r_{0}r_{1}-r_{0}\\gcd (r_{1},r_{2})$ .",
"Then every $P\\in NM(r_0,r_1,r_2,r_0)$ is nested.",
"In [5], the authors also examined the posets of rank 3 with $r_2\\le 13$ .",
"Using Theorem REF , one can verify that if $r_0<r_1<r_2\\le 13$ , every poset $P\\in NM(r_0,r_1,r_2,r_0)$ satisfies Conjecture REF except that $(r_0,r_1,r_2,r_0)$ is equal to one of the six cases: $(6,8,12,6)$ , $(6,9,12,6)$ , $(4,6,13,4)$ , $(5,8,13,5)$ , $(6,8,13,6)$ , $(6,9,13,6)$ .",
"We close Section 2 by stating our results.",
"For graded posets of rank 3, we provide two more sufficient conditions on the rank numbers for the existence of a nesting: Theorem 2.5 Let $P\\in NM(r_0,r_1,r_2,r_0)$ .",
"If both $r_{1}$ and $r_{0}$ divide $r_{2}-1$ , then $P$ is nested.",
"Theorem 2.6 Let $P\\in NM(r_0,r_1,r_2,r_0)$ .",
"If $kr_{0}\\le r_{1}\\le k(r_{0}+1)$ , then $P$ is nested.",
"Observe that by Theorem REF , we see that every $P\\in NM(4,6,13,4)$ has a nesting.",
"Unfortunately, other unsolved cases with $r_2\\le 13$ mentioned in[10] cannot be settled by our theorems.",
"Nevertheless, for $14\\le r_2\\le 15$ , we can use Theorem REF to show every $P\\in NM(r_0,r_1,r_2,r_3)$ with $(r_0,r_1,r_2,r_3)\\in \\lbrace (4,9,14,4)$ , $(3,7,15,3)$ ,$(4,9,15,4)$ ,$(5,11,15,5)\\rbrace $ is nested.",
"These are not covered by Theorem REF and REF ."
],
[
"Proofs of the Main Theorems",
"In this section, we give the proofs of Theorem REF and Theorem REF .",
"Let us begin with the proof of Theorem REF .",
"Proof of Theorem 2.5.",
"Pick a poset $P\\in NM(r_0,r_1,r_2,r_0)$ , where $k_0r_0+1=r_2$ and $k_1r_1+1=r_2$ for some integers $k_1$ and $k_2$ .",
"Let $P^{\\prime }$ be a poset obtained by removing an arbitrary element $x$ from $L_2$ .",
"To show that $P^{\\prime }$ is a normalized matching poset, we only need to verify the inequality holds between $L_1$ and $L_2\\setminus \\lbrace x\\rbrace $ as well as $L_2\\setminus \\lbrace x\\rbrace $ and $L_3$ .",
"First consider $L_1$ and $L_2\\setminus \\lbrace x\\rbrace $ .",
"By the symmetry, we only need to verify the normalized matching property from $L_1$ to $L_2\\setminus \\lbrace x\\rbrace $ .",
"For any $S\\subseteq L_1$ , since $P$ is a normalized matching poset, we have $\\frac{|S|}{r_1}\\le \\frac{|\\Gamma _2(S)|}{r_2}=\\frac{|\\Gamma _2(S)|}{k_1r_1+1}.$ Equivalently, ${k_{1}|S|}+\\frac{|S|}{r_1}\\le |\\Gamma _2(S)|.$ When $S\\ne \\emptyset $ , we have $k_1|S|+1\\le |\\Gamma _2(S)|$ since $|\\Gamma _2(S)|$ is an integer.",
"Now, for $P^{\\prime }$ , if the removed element $x$ is not in $\\Gamma _2(S)$ , then $\\frac{|S|}{r_{1}}\\le \\frac{|\\Gamma _2(S)|}{k_1r_1+1}\\le \\frac{|\\Gamma _2(S)|}{k_1r_1}=\\frac{|\\Gamma _2(S)|}{r_{2}-1}=\\frac{|\\Gamma _2(S)|}{|L_2\\setminus \\lbrace x\\rbrace |}.$ Otherwise, $x\\in \\Gamma _2(S)$ and then $S\\ne \\emptyset $ .",
"We have $\\frac{|S|}{r_{1}}=\\frac{k_{1}|S|}{k_{1}r_{1}}\\le \\frac{|\\Gamma _2(S)|-1}{r_2-1}=\\frac{|\\Gamma _2(S)|-1}{|L_2\\setminus \\lbrace x\\rbrace |}.$ The numerator $|\\Gamma _2(S)|-1$ is just the number of elements $y\\in L_2\\setminus \\lbrace x\\rbrace $ satisfying $z<y$ for some $z\\in S$ .",
"So the normalized matching property holds between $L_1$ and $L_2\\setminus \\lbrace x\\rbrace $ .",
"Using a similar argument we can see that the normalized matching property also holds between $L_3$ and $L_2\\setminus \\lbrace x\\rbrace $ .",
"Now that $r_1$ divides $k_1r_1=r_2-1$ , so $P^{\\prime }$ has a nesting $\\mathcal {C}$ by Theorem REF (d).",
"Finally, we view $\\lbrace x\\rbrace $ as a one-element chain and add it to $\\mathcal {C}$ to get a nesting of $P$ .",
"$\\Box $ It is worth mentioning that the proof in Theorem REF is similar to the next lemma in [10], which is used to prove Theorem REF (b).",
"Lemma 3.1 [10] Let $P\\in NM(r_0,r_0+1)$ .",
"For any $x$ of rank 1 in $P$ , there exists a chain partition of $P$ which consists of $r_0$ chains of length 1 and another chain $\\lbrace x\\rbrace $ of length 0.",
"Before presenting the proof of Theorem REF , we need more preparations.",
"In addition to adding the ghosts to a normalized matching poset, there are some techniques to produce new normalized matching posets from the old ones.",
"We introduce two construction approaches.",
"Definition 3.2 ($k$ -clone) Let $P$ be a graded poset and $L_i$ be a level of $P$ .",
"Then $L$ is said to be a $k$ -clone of $L_i$ if $L=L_i\\times \\lbrace 1,\\ldots ,k\\rbrace $ and the partial order relations of each $(y,i)$ and others elements in $L_{i-1}$ (resp.",
"$L_{i+1}$ ) is $x<(y,i)$ (resp.",
"$(y,i)<z$ ) if and only if there exist some $x\\in L_{i-1}$ and $y\\in L_i$ (resp.",
"$y\\in L_j$ and $z\\in L_{i+1}$ ).",
"See Figure REF as an illustration.",
"Figure: A new poset obtained by replacing a 2-clone of the first level of PP to it.Definition 3.3 ($m$ -bunch) Let $P$ be a graded poset and $L_i$ be a level of $P$ .",
"Suppose $|L_i|=m\\ell $ for some integers $m$ and $\\ell $ .",
"First partition $L_1$ into $m$ arbitrary subsets $A_{1}$ ,...,$A_{m}$ of equal size $\\ell $ .",
"Then $L$ is an $m$ -bunch of $L_i$ if $L=\\lbrace A_{1}$ ,...,$A_{m}\\rbrace $ and the partial order relations of each $A_j$ and others elements in $L_{i-1}$ (resp.",
"$L_{i+1}$ ) is $x<A_j$ (resp.",
"$A_j<z$ ) if and only if there exist $x\\in L_{i-1}$ and $y\\in A_j$ (resp.",
"$y\\in A_j$ and $z\\in L_{i+1}$ ).",
"See Figure REF as an illustration.",
"Figure: A new poset obtained by replacing a 2-bunch of the first level of PP to it.The above operations on posets preserve the normalized matching property: Proposition 3.4 If $P$ is a normalized matching poset, then the new poset obtained by replacing a $k$ -clone or an $m$ -bunch of some level of $P$ to it is still a normalized matching poset.",
"The proof of this proposition was given by Hsu et al.",
"[10] (clone), and by Escamilla et al.",
"[5] (bunch), respectively.",
"Now we prove our second theorem.",
"Proof of Theorem 2.6.",
"Consider $P\\in NM(r_0,r_1,r_2,r_0)$ with $kr_0\\le r_1\\le k(r_0+1)$ for some integer $k$ .",
"Note that the two ends of the inequality are in the statements of Theorem REF (a) and (b).",
"Thus, we may suppose $r_{1}=kr_{0}+t$ for some $1\\le t\\le k-1$ .",
"Pick a poset $P\\in NM(r_0,r_1,r_2,r_0)$ .",
"We use the induction method to find the nestings of subposets induced by different levels of $P$ .",
"Our goal is to combine the nestings properly to get a nesting of $P$ .",
"First construct a poset $P_1$ of rank 2 induced by the top three levels of $P$ with a replacement of a $k$ -clone of the highest level.",
"By Proposition REF , $P_1\\in NM(r_{1},r_{2},kr_{0})$ , and there exists a nesting $\\mathcal {C}_1$ of $P_1$ by Theorem REF .",
"Observe that there are $kr_0$ chains of length 2 in $\\mathcal {C}_1$ such that eahc of them contains an element $(y,j)$ in the highest level of $P_1$ .",
"Clearly, the bottom two levels $L_0$ and $L_1$ of $P$ induce a subposet of rank 1 and has a nesting.",
"However, we do not want a nesting of the above poset containing a chain of length 1 whose top element is the bottom element of a chain of length 1 in $\\mathcal {C}_1$ .",
"This could lead to two chains of length 2 but the ranks of elements in one chain is $\\lbrace 0,1,2\\rbrace $ and the other is $\\lbrace 1,2,3\\rbrace $ when we combine the two nestings together.",
"To avoid this, we construct a poset $P_2$ of rank 1 as follows.",
"At the beginning, we add $k-t$ additional ghosts to $L_1$ of $P$ in advance.",
"Now this level contains $k(r_0+1)$ elements, and we will partition them into $r_0+1$ sets of size $k$ .",
"Because $L_1$ is also the bottom level of $P_1$ , for each $y_i\\in L_3$ of $P$ , there exist exactly $k$ elements in $L_1$ such that each of them lies in a chain, containing $(y_i,j)$ for some $1\\le j\\le k$ , of length 2 in $\\mathcal {C}$ .",
"In addition, there are $t=r_1-kr_0$ elements in $L_1$ which are not in any chain of length 2 in $\\mathcal {C}$ .",
"For $1\\le i\\le r_{0}$ , let $A_{i}$ be the set consisting of every element in $L_1$ , which lies in a chain in $\\mathcal {C}_1$ containing the element $(y_i, j)$ for some $1\\le j\\le k$ .",
"Moreover, let $A_{r_{0}+1}$ be the set consisting of the $t$ remaining elements in $L_1$ and the $k-t$ ghosts.",
"We bunch all elements in $L_1$ and the ghosts into the above $r_0+1$ sets $A_1,\\ldots , A_{r_0+1}$ .",
"The poset induced by these $A_i$ s and $L_0$ is $P_2$ .",
"By Lemma $\\ref {avoid}$ , there is a chain partition $\\mathcal {C}_2$ of $P_2$ with $r_0$ chains of length 1 and one chain of length 0 such that each chain of length 1 does not contain $A_{r_0+1}$ and each $x_i\\in L_0$ is in a chain of length 1 in $\\mathcal {C}_2$ .",
"Assume the chains are $\\lbrace x_i,A_i\\rbrace $ for $1\\le i\\le r_0$ .",
"It follows that for each $i$ there exists some element in $z\\in A_i$ with $x_i<z$ .",
"Fix some $i$ .",
"For those $k$ chains of length 2 in $\\mathcal {C}_1$ containing $(y_i,j)$ for some $1\\le j\\le k$ , we extend one of them to length 3 by adding the element $x_i$ and delete the top elements of the remaining $k-1$ chains of length 2.",
"Repeating the operations for all $1\\le i\\le r_0$ gives us a nesting of $P$ .",
"$\\Box $"
]
] | 1709.01768 |
[
[
"Resuspension threshold of a granular bed by localized heating"
],
[
"Abstract The resuspension and dispersion of particles occur in industrial fluid dynamic processes as well as environmental and geophysical situations.",
"In this paper, we experimentally investigate the ability to fluidize a granular bed with a vertical gradient of temperature.",
"Using laboratory experiments with a localized heat source, we observe a large entrainment of particles into the fluid volume beyond a threshold temperature.",
"The buoyancy-driven fluidized bed then leads to the transport of solid particles through the generation of particle-laden plumes.",
"We show that the destabilization process is driven by the thermal conductivity inside the granular bed and demonstrate that the threshold temperature depends on the thickness of the granular bed and the buoyancy number, i.e., the ratio of the stabilizing density contrast to the destabilizing thermal density contrast."
],
[
"Introduction",
"The transport of solid particles induced by shearing particulate beds with water or air flow occurs in many geophysical events, such as in river beds and landscape evolution, wind-blown sand, and dust emission [1], [2], but also in industrial processes: filtration systems or the food industry, for instance [3], [4], [5], [6], [7].",
"Depending on the nature, geometry, and regime of the fluid flow, different types of particle movements can be observed.",
"Conventionally, rolling and sliding motion, saltation, and suspension are distinguished depending on the Reynolds number.",
"In other configurations, when the granular bed is subject to an ascending flow of liquid or gas, the normal stress can fluidize the particulate medium and maintain grains in suspension, which leads to particular properties and characteristics of the medium [8].",
"This situation is observed for instance during the rise of air in an immersed granular bed [9], [10].",
"In addition of these mechanical mechanisms of resuspen- sion, a heat source inside or below the sediment may destabi- lize a loose random packing of granular matter.",
"Resuspension due to thermal effects is of great importance to understand, for example, volcanic ash clouds [11] or seafloor hydrothermal systems such as black smokers.",
"Whereas the resuspension and fluidization of an immersed granular bed by fluid flows such as vortices [12], [13], [14], [15], impacting jets [16], [17], shear flows [18], [19], or gas crossing a liquid-saturated granular bed have been the focus of many studies, the ability of thermal convection to resuspend particles remains poorly understood.",
"Indeed, in recent decades particular attention has been focused on the settling of particles, initially in suspension, in steady cellular convection but only few studies [20] have investigated the destabilization process of an initially loose randomly packed granular bed driven by thermal convection.",
"Several scenarios are possible to induce the reentrainment of solid particles from the bottom: erosion by the bottom shear stress induced by convection current or fluidization by emergence of particle-laden plumes.",
"To explore the mechanisms of the resuspension process by thermal convection, Solomatov et al.",
"[20], [21] used a three-dimensional experimental setup with aqueous solution and polystyrene particles which are initially sedimented to form a loose random packing of granular matter.",
"The bottom wall is uniformly heated from below to ensure the destabilization process of the sedimented particles.",
"The authors defined a Shields number, Sh, which compares the ratio of the destabilizing hydrodynamic stress $\\tau $ exerted on a grain to its stabilizing apparent weight $\\Delta \\rho \\,g \\,d$ , where $\\Delta \\rho = \\rho _p-\\rho _l$ is the density difference between the grains and the ambient fluid, $g$ is the acceleration of the gravity and $d$ is the mean particle diameter.",
"The resuspension of particles from the bottom is driven by the tangential buoyancy stress $\\tau = \\beta \\, \\rho _l \\,g \\,\\Delta T \\,\\delta _T$ (where $\\beta $ is the thermal expansion coefficient, $\\Delta T$ is the temperature difference between the bottom and the top of the cell and $\\delta _T$ is the thermal boundary layer thickness) at the top of the granular bed induced by convection current: particles roll and slide horizontally and can form dunes.",
"At the crest of a dune, the tangential buoyancy stress is vertical and can lead to the resuspension of particles.",
"The buoyancy Shields number, separating a regime without bed motion from a regime with particle entrainment is approximatively constant and of the order of 0.1 in the experiments.",
"On the other hand, Martin and Nokes [22] used a similar setup but a different initial state where particles are in suspension.",
"The reentrainment of particles occurred by the emergence of plumes from the bottom but the authors were unable to extract a criterion for reentrainment.",
"The aim of the present work is to investigate the resus- pension of a granular bed of particles by fluidization from a localized heat source.",
"This situation is different from the uniform heating used in previous studies [20] where the gran- ular bed was eroded over a long time by shear flows induced by the thermal effects and the particles were susceptible to rolling, sliding, and being carried away at the bed surface by saltation [21].",
"Here, we demonstrate that at larger temperature difference the fluidization of the granular bed can also occur from the bottom and actively participate in the resuspension and reentrainment of particles into the bulk through the generation of particle-laden plumes.",
"We focus on the scaling of this fluidization threshold with the relevant dimensionless parameters that describe this destabilization process."
],
[
"Experimental methods",
"We study the resuspension of spherical polystyrene particles (Dynoseeds purchased from MicroBeads) of diameter $2\\,a=250\\,\\pm 10 \\mu {\\rm m}$ and a density $\\rho _p^0=1.049 \\pm 0.003 \\,{\\rm g\\,cm^{-3}}$ [23] induced by a localized heating at the bottom of the granular bed.",
"The setup used in the resuspension experiments is shown in Fig.",
"REF and consists of a rectangular Poly(methyl methacrylate) tank of internal dimensions 20 cm in length, $b=1.2$ cm in width and $H=20$ cm in height.",
"Given the aspect ratio of the tank, no significant motion of the fluid takes place in the short direction.",
"The temperature is imposed at the top and the bottom of the tank through two copper plates whose temperatures are imposed by two circulating thermostated baths and measured by platinum thermocouples located inside the plates.",
"The top copper plate is 20 cm wide and covers the tank, whereas the bottom copper plate is centered and has a width of 4 cm, resulting in a localized heating.",
"Figure: Schematic of the experimentalsetup.The working fluid is composed of water with different concentrations of calcium chloride, CaCl$_2$ .",
"The quantity of salt is varied to increase the density of the working fluid in the range $\\rho _l=[1,\\,1.049]\\,{\\rm g.cm^{-3}}$ (measured at 20$^{\\rm o}$ C with a densimeter Anton Paar DMA 35; see Appendix).",
"In addition, a small amount of sodium dodecyl sulfate surfactant is added to the water to initially disperse the dry particles into water and avoid the trapping of air bubbles.",
"Each experiment is performed with a new suspension of particles that is allowed to settle and form a loose randomly packed granular bed at the bottom of the tank.",
"The resulting compacity of the granular bed, i.e., the solid volume fraction $\\phi =V_p/V_{tot}$ (where $V_p$ is the volume of the particles and $V_{tot}$ is the volume of the liquid and the particles) is equal to $\\phi =0.57 \\pm 0.01$ and its controlled thickness $h$ lies in the range 1 to $30\\,{\\rm mm}$ .",
"A vertical laser sheet is set through the short side of the tank so that the fluid and the granular bed are clearly visible.",
"The evolution of the granular bed is recorded using a Phantom MIRO M110 camera (resolution $1200\\times 800$ pixels) at 1 frame/s.",
"At the beginning of each experiment, the thermostated baths are first allowed to reach their assigned temperature, leading to a top temperature $T_c$ and a bottom temperature in the copper plate, $T_h$ .",
"Initially, we set $T_c=T_h=T_0=15^{\\rm o}$ C. We then wait for a sufficiently long time, typically 30 min, to ensure that the initial temperature in the system is homogeneous and equal to $T_0=15^{\\rm o}$ C. At time $t=0$ , we suddenly increase the temperature of the bottom copper plate to the reference temperature $T_{h}>T_0$ .",
"Experimentally, providing a sudden increase in temperature of the bottom plate is challenging and we rely on a third thermostated bath that is set at $\\theta _{h}$ prior to the experiment.",
"Switching the fluid flow in the bottom plate to this thermostated bath leads to a progressive increase of the temperature of the bottom copper plate, $T_h$ , while the upper copper plate remains at $T_c=T_0=15\\,^{\\rm o}$ C. The temperature $T_h$ increases continuously and, when it reaches a critical value, we observe the sudden destabilization and resuspension of the granular bed."
],
[
"Phenomenology",
"A typical experiment is shown in Fig.",
"REF (a) where a time lapse shows the evolution of the granular bed.",
"Initially, the temperature profile is constant in the granular and the fluid layers, equal to $T_0=15^{\\rm o}{\\rm C}$ , and the resulting density profile of the liquid is constant in both regions and is equal to $\\rho _{l}^0$ .",
"The density of the polystyrene beads at $T_0$ , $\\rho _p^0$ , is larger that the density of the liquid, $\\rho _{l}^0$ , and therefore the granular bed is stable.",
"At time $t=0$ , the bottom copper plate is connected to the hot thermostated bath at the temperature $\\theta _{h}$ , and the temperature $T_h$ in the bottom plate then starts to increase while the temperature $T_c$ of the top plate remains constant as shown in Fig.",
"REF (b).",
"The progressive increase of the temperature of the hot source, $T_h$ , leads to the increase of the temperature in the bottom region of the cell and therefore a decrease of the liquid and particle densities in the granular bed.",
"Because of the thermal loss between the thermostated bath, whose temperature $\\theta _h$ is fixed, and the bottom copper plate, the final temperature $T_h^{f}$ is smaller than $\\theta _h$ .",
"Indeed, the thermostated bath is connected to the bottom copper plate through tubings, which, even insulated, lead to heat transfer with the ambient atmosphere and lead to a temperature $T_h$ smaller than $\\theta _h$ .",
"Nevertheless, the temperature probe measures $T_h$ during the entire duration of an experiment and we therefore refer to this temperature in the following.",
"During the first phase, no motion of the granular bed occurs (corresponding to region 1 in Fig.",
"REF ).",
"Then, at a temperature threshold $T_h^*$ , the granular bed starts to destabilize with the formation of a small corrugation that later grows in time (regions 2 and 3).",
"The transition from a flat granular bed to the rise of a particle-laden plume occurs suddenly over a time scale that is small compared to the duration of the entire experiment and the evolution of the temperature at the bottom of the cell.",
"We can therefore accurately estimate the time and the corresponding temperature $T_h$ at the bottom plate at which the granular bed becomes unstable, which in this example corresponds to $t=135\\,{\\rm s}$ and $T_h^* \\simeq 38^{\\rm o}{\\rm C}$ .",
"In the following, we rationalize quantitatively the threshold temperature $T_h^*$ and the destabilization time at which the granular bed starts to fluidize.",
"We systematically investigate the influence of the temperature of the hot source, $T_h$ , the initial density contrast between the fluid and the particles, $\\Delta \\rho ^0$ , and the thickness of the granular bed, $h$ ."
],
[
"Time-evolution of $T_h$",
"We observe that a threshold temperature $T_h^*$ needs to be reached to resuspend locally the granular bed.",
"Because our experimental approach involves a transient increase of the temperature of the bottom copper plate, we characterize the influence of this transient dynamic on the threshold temperature.",
"To do so, we consider a granular bed of fixed height $h=10\\,{\\rm mm}$ and a fixed initial density contrast $\\Delta \\rho ^0=\\rho _p^0-\\rho _l^0=2.6 \\,\\rm {kg\\,m^{-3}}$ .",
"We then impose different temperatures at the thermostated bath connected to the bottom hot source, $\\theta _h=[60,\\,65,\\,70,\\,75,\\,80]\\,^{\\rm o}{\\rm C}$ , keeping the temperature at the top of the cell constant and equal to $T_c=15\\,^{\\rm o}C$ during the entire experiment.",
"As mentioned previously, the thermal loss between the thermostated bath and the bottom copper plate leads to a temperature $T_h^{f}$ reached at the bottom of the cell significantly smaller than the temperature $\\theta _h$ of the thermostated bath.",
"Experimentally, we determine that $T_h^{f}$ is in the range $42\\,^{\\rm o}$ C to $64\\,^{\\rm o}{\\rm C}$ .",
"The time evolution of the temperature at the bottom plate, as recorded by the temperature probe, is shown in Fig.",
"REF for varying values of the temperature $\\theta _h$ of the hot thermalized bath.",
"The threshold temperature $T_h^*$ at which the granular bed starts to resuspend is also reported.",
"We observe that the temperature of the bottom plate increases quickly at the beginning and then the increase in temperature becomes slower.",
"In all the different situations considered here where $T_h^f > T_h^*$ , we observe the resuspension of the granular bed at sufficiently long time.",
"In addition, the threshold temperature remains approximatively constant, $T_h^*=41.6\\pm 0.8\\,^{\\rm o}{\\rm C}$ , which suggests that, for a constant thickness of granular layer sufficiently large, here $h=10$ mm, and given density contrast, the threshold temperature $T_h^*$ does not depend significantly on the dynamics of the system.",
"However, we are going to see in the following that the transient effects are required to describe the system when varying the granular bed thickness $h$ .",
"Although the threshold temperature for the destabilization of the granular bed does not seem to depend significantly on the imposed value of the temperature of the hot thermostated bath $\\theta _h$ , the time needed to reach the destabilization threshold, $T_h^*$ , increases sharply when $\\theta _h$ is decreased.",
"This observation is mainly explained by the thermodynamics of the system: the time needed to reach the threshold temperature increases when decreasing the temperature of the hot source, leading to a longer waiting time.",
"In addition, if the temperature of the bottom plate remains smaller than $T_h^*$ , no destabilization of the granular bed is observed at long time.",
"In the following, we consider a heat flux at the hot source that leads to a maximum steady value of the bottom plate of $T_h^f \\simeq 50\\,^{\\rm o}{\\rm C}$ ."
],
[
"Density contrast between the fluid and the particles $\\Delta \\rho ^0$",
"Another relevant parameter in the destabilization process is the density contrast between the fluid and the particles, $\\Delta \\rho ^0$ .",
"We thus vary the density of the fluid, $\\rho _l^0$ , by tuning the concentration of CaCl$_2$ salt so that $\\Delta \\rho ^0 = \\rho _p^0-\\rho _l^0$ ranges from $1.6\\,{\\rm kg\\,m^{-3}}$ to $4.1\\,{\\rm kg\\,m^{-3}}$ (see Appendix).",
"No resuspension of the granular bed has been observed with further increase of the density difference $\\Delta \\rho ^0$ within the range of temperatures that we have access to in our experiments.",
"For a constant bed thickness $h=10\\,{\\rm mm}$ and $\\theta _h = 65\\,^{\\rm o}{\\rm C}$ , we report in Fig.",
"REF that the resuspension of the granular bed occurs at different temperature thresholds.",
"Indeed, the smaller the density contrast $\\Delta \\rho ^0$ is, the sooner the resuspension takes place during the temperature ramp, which corresponds to smaller temperature threshold of the bottom plate $T_h^*$ .",
"The inset of Fig.",
"REF highlights that the temperature threshold $T_h^*$ increases linearly with the density contrast $\\Delta \\rho ^0$ in the range of parameters that we have considered.",
"We come back to this point in the next section when we rationalize our results with dimensionless parameters."
],
[
"Granular bed thickness $h$",
"Finally, we investigate the influence of the granular bed thickness on the temperature threshold and report the results in Fig.",
"REF .",
"We keep the density of the liquid and the particles constant, and using the same temperature of the hot thermostated bath $\\theta _h$ , we measure the temperature threshold at which the resuspension occurs as well as the time elapsed before the destabilization, $t_{des}$ .",
"We observe that the threshold temperature increases with the thickness of the granular layer following a trend close to $T_h^*\\propto h^{1/2}$ .",
"In addition, the time $t_{des}$ follows a slope $t_{des} \\propto h^2$ .",
"This scaling suggests that the mechanism responsible for the resuspension of the granular bed is diffusive.",
"Figure: Temperature threshold T h * T_h^* for increasing thickness of the granular bed, hh.",
"Red circles are the experimental results and the black dotted line shows T h * ∝h 1/2 T_h^* \\propto h^{1/2}.",
"Inset: Time elapsed, t des t_{des}, prior to the resuspension threshold for increasing thickness of the granular bed hh.",
"The temperature of the hot thermostated bath is set at θ h =65 o C\\theta _h=65^{\\rm o}\\rm {C} and Δρ 0 =1.98 kg m -3 \\Delta \\rho ^0=1.98\\,{\\rm kg\\,m^{-3}}.The experimental results highlight that an increase in the density contrast $\\Delta \\rho ^0$ or the granular bed thickness $h$ leads to a larger temperature threshold $T_h^*$ for destabilization.",
"In this section, we show that these results can be rationalized by considering buoyancy effects in the granular bed."
],
[
"Buoyancy number",
"Buoyancy-driven flows in a Hele-Shaw cell are commonly described using a modified Rayleigh number defined as [24]: $Ra=\\frac{g\\,\\beta \\,\\Delta T\\,H\\,b^2}{12\\,\\nu \\,D},$ where $g$ is the acceleration due to gravity, $b$ and $H$ the width and height of the cell, respectively, $\\beta $ is the thermal expansion coefficient, $\\nu $ is the kinematic viscosity and $D$ is the thermal diffusivity.",
"For large enough Rayleigh number, the system is unstable and natural convection is observed.",
"In the present study, the Rayleigh number lies in the range $[10^3,\\,10^6]$ , and we observe large recirculation cells in the top layer (liquid) even below the resuspension threshold.",
"We therefore need to consider an additional dimensionless number to explain the global destabilization of the granular bed.",
"Figure: Evolution of the critical buoyancy number B c B_c varying the temperature T h f T_h^f, hence ΔT\\Delta T (open black circles) or the density contrast Δρ 0 \\Delta \\rho ^0 (solid red circles) for a constant granular bed thickness h=10h=10 mm.Here, the presence of a granular bed at the bottom of the cell leads to a more complex situation as we now observe the possible destabilization of the granular layer beyond a temperature threshold.",
"This situation can be related to past studies that have considered a density stratification in two-layer Newtonian fluids when the two fluids have different densities and viscosities and there is no surface tension between the two fluid layers [25], [26], [27].",
"In this situation, it has been shown that the onset of convection can be either stationary or oscillatory depending on a dimensionless number, the buoyancy number $B$ , defined as the ratio of the stabilizing density anomaly to the destabilizing thermal density anomaly: $B=\\frac{\\rho _p^0-\\rho _l^0}{\\rho _l^0\\,\\beta \\,\\Delta T}.$ where $\\Delta T=T_h-T_c$ .",
"Depending on the buoyancy number $B$ , two main regimes have been identified.",
"Typically, when $B$ is large enough, i.e., $B$ larger than 0.5-1, convective flows develop above and/or below the flat interface to obtain the so-called stratified regime.",
"When the buoyancy number $B$ is small enough, typically smaller than 0.3-0.5, the interface can become unstable and spontaneous flow occurs in the whole tank, leading to the mixing of the two initial fluid layers in the bed and above.",
"Using this approach, we rescale the experimental results presented in Secs.",
"REF and REF , obtained for a given thickness $h$ of the immersed granular bed and varying the density contrast and the temperature of the thermostated bath.",
"The results reported in Fig.",
"REF highlight that the destabilization of the granular bed occurs for a constant value of the buoyancy number $B=B_c \\simeq 0.34$ at $h=10$ mm.",
"In Fig.",
"REF , we also report the evolution of the critical buoyancy number $B_c$ for varying $h$ while the other parameters are kept constant.",
"These results show that $B_c$ decreases when increasing the rescaled granular bed thickness $h/H$ as observed for two Newtonian fluids by Le Bars & Davaille [27].",
"However, such an approach does not explain the evolution of $t_{des}$ with the bed thickness.",
"Figure: Evolution of the critical buoyancy number B c B_c varying the thickness of the granular bed, hh.",
"The temperature of the thermostated bath is set at θ h =65 o C\\theta _h=65^{\\rm o}\\rm {C} and Δρ 0 =1.98 kg m -3 \\Delta \\rho ^0=1.98\\,{\\rm kg\\,m^{-3}}.",
"The black solid line is the best polynomial fit and is a guide for the eye.",
"In the light yellow region, the granular bed is stable whereas resuspension occurs in the dark blue region."
],
[
"Destabilization threshold",
"We consider a granular bed of compacity $\\phi $ made of polystyrene particles of density $\\rho _{p}^0$ .",
"Initially, the system is at temperature $T_c=15^{\\rm o}{\\rm C}$ .",
"We consider an infinitesimal element of length ${\\rm {d}}L$ and width $b$ , which is the gap of the Hele-Shaw cell.",
"The density of the granular bed averaged over the thickness of the granular bed is written $ \\langle \\rho (T) \\rangle _h = \\frac{1}{h}\\int _{0}^h\\,\\phi \\,\\rho _{p}(T)+({1-\\phi })\\,\\rho _l(T)\\,{\\rm d}z,$ where the temperature $T(z)$ depends on the vertical coordinate.",
"The evolution of the density with the temperature is given by : $ \\rho _l(z)=\\rho _l^0\\,\\left[1-\\alpha _l(T)\\,\\Delta T\\right],$ for the liquid and by $ \\rho _{p}(z)=\\rho _p^0\\,\\left[1-\\alpha _{p}(T)\\,\\Delta T\\right],$ for the polystyrene beads.",
"In these equation, $\\alpha _l$ and $\\alpha _p$ are the coefficient of thermal expansion of the liquid and polystyrene beads, respectively (see Apppendix).",
"We first consider the steady state where the temperature profile only depends on the vertical coordinate $z$ .",
"Experi- mentally, we observe, even below the resuspension threshold, large-scale flow in the fluid and we therefore assume that at the top of the granular bed the local temperature is comparable to the temperature of the cold source, $T_c$ .",
"Therefore, the temperature profile in the layer can be approximated as $T(z)=T_c+(T_h-T_c)\\,z/h$ .",
"In this configuration, the granular bed becomes unstable when the density averaged over the entire thickness h becomes smaller than the density of the fluid on top of it, assumed to be at the temperature $T_c=T_0$ , which means if ${\\langle \\rho (T) \\rangle _h}< {\\rho _l^0} $ .",
"We can solve this condition numerically for varying density contrast $\\Delta \\rho ^0$ as illustrated in Fig.",
"REF (a).",
"We observe that qualitatively the threshold temperature $T_h^*$ increases linearly with the density contrast $\\Delta \\rho ^0$ as observed experimentally.",
"This observation is consistent with the definition of the Buoyancy number $B$ .",
"For a constant granular bed thickness, the density contrast has a stabilizing effect, whereas an increase in temperature contributes to the destabilization of the granular bed through a local modification of the density of the granular bed.",
"We should also emphasize that this description is qualitative but not quantitative because of the experimental limitations.",
"Indeed, the increase in temperature at the bottom copper plate is not instantaneous and therefore $T_h$ increases as a function of time.",
"Nevertheless, this approach allows us to highlight the influence of the buoyancy number on the resuspension threshold of an immersed granular bed and the destabilizing effect of the temperature.",
"Figure: (a) Threshold temperature T h * T_h^* obtained in the steady state regime as a function of the density contrast Δρ 0 \\Delta \\rho ^0.",
"The thickness of the granular bed is h=10 mm h=10\\,{\\rm mm}.",
"The dotted line shows the linear scaling T h * ∝Δρ 0 T_h^* \\propto \\Delta \\rho ^0.",
"(b) Time elapsed, t des t_{des}, prior to the resuspension threshold obtained by solving the diffusion equation for increasing thickness hh of the granular bed.",
"The numerical parameters are T h =55 o CT_h=55^{\\rm o}{\\rm C}, Δρ=5 kg m -3 \\Delta \\rho = 5\\,{\\rm kg\\,m^{-3}}.",
"The dotted line is a slope t∝h 2 t\\propto h^2.The influence of the transient state is clearly observed when considering the influence of the granular bed thickness $h$ .",
"Indeed, if we consider the steady state only, the temperature threshold $T_h^*$ should not depend on $h$ .",
"However, our experimental results show that $T_h^*$ increases with $h$ (see Fig.",
"REF ).",
"Although in our experiments the time variation of the temperature of the hot source is intrinsically related to the thermal properties of the system, we can consider the effect of the time variation to explain the scaling of the time to destabilize the granular bed, $t_{des}$ .",
"We modified the model developed previously and now consider that, at $t<0$ , the temperature is equal to $T_c$ everywhere in the fluid and in the granular bed.",
"At time $t=0$ , the lower part at $z=0$ is suddenly put at $T=T_h$ .",
"In the experiment, this temperature is increasing when connecting the hot thermostated bath but here, for the sake of simplicity, we assume that the temperature of the bottom plate is reached instantaneously.",
"As a result, the time dependence of the temperature at the position $z$ is the solution of the classical one-dimensional diffusion problem $ T(z)=T_c+\\left(T_h-T_c\\right) \\left[1-{\\rm erf}\\left(\\frac{z}{2\\,\\sqrt{D\\,t}}\\right)\\right],$ with $D \\simeq 1.8 \\times 10^{-7}\\,{\\rm m^2.s^{-1}}$ , the effective diffusion coefficient in the granular bed.",
"We know that the granular bed becomes unstable when ${\\langle \\rho (T) \\rangle _h}< {\\rho _l^0}$ , and we can solve this condition numerically using equations (REF )-(REF ).",
"The corresponding results are reported in Fig.",
"REF (b): the scaling observed experimentally $t_{des} \\propto h^2$ is captured by the diffusion equation in the granular bed.",
"Therefore, transient effects appear to be important to explain the destabilization of an immersed granular bed.",
"The qualitative model presented here captures the key physical effects and provides scaling laws to describe the phenomenon.",
"To take into account the increase in temperature of the hot source a full numerical model will be needed."
],
[
"Conclusion",
"In this paper, we have explored experimentally the resus- pension of an immersed granular bed by a localized heat source.",
"The granular bed is made of particles that have a density slightly smaller than the density of the surrounding fluid.",
"Our experiments illustrate that, beyond a temperature threshold, the averaged density of the granular bed can become smaller than the density of the top fluid and produce an overturning instability as observed for two Newtonian fluid [25], [26], [27].",
"This flow results in the production of thermal plume and the dispersion of the granular particles in the entire container.",
"We have shown that this mechanism can be described through the buoyancy number $B={(\\rho _p^0-\\rho _l^0)}/({\\rho _l^0\\,\\beta \\,\\Delta T})$ .",
"The threshold value $B_c$ is observed to be dependent on the granular bed thickness in our experiments owing to the transient regime.",
"We rationalize our experimental findings with scaling argu- ments that capture the main features of this destabilization process.",
"Such resuspension of a granular bed could be important in geophysical and environmental processes in which localized heating can induce the transport of deposited particles and the later contamination of the environment.",
"The dynamics of particle-laden plumes is important to describe the subsequent dispersion of the particles.",
"The authors are grateful to P. Gondret and A. Davaille for fruitful discussions and to E. Dressaire for a careful reading of the manuscript.",
"We acknowledge J. Amarni, A. Aubertin, L. Auffray, C. Borget, and R. Pidoux for experimental help.",
"This work was supported by the French ANR (project Stabingram ANR 2010-BLAN-0927-01).",
"C.M.",
"also acknowledges support from the AAP “Attractivité Jeune chercheur” from Paris-Sud University.",
"*"
],
[
"Aqueous solution",
"The liquid used in our experiments is a mixture of distilled water and CaCl$_2$ (between 0% and 5.5% w/w).",
"We measured the density of different aqueous mixture using a densimeter (Anton Paar DMA 35) in a range of temperature between $15^{\\rm o}$ C and $30^{\\rm o}$ C as reported in Fig.",
"REF .",
"Figure: Density of aqueous mixture for different salt concentration: 4.85%4.85\\% w/w (blue circles), 4.9%4.9\\% w/w (cyan crosses), 4.95%4.95\\% w/w (red squares), 5.05%5.05\\% w/w (green diamonds) and 5.15%5.15\\% w/w (magenta crosses).",
"The dotted lines are the best fits from the equation () for varying ρ l 0 (c)\\rho ^0_{l}(c).The evolution of the density with the temperature and the salt concentration $\\rho _l(c,T)$ is fitted using the expression $ \\rho _l(c,T)=\\rho ^0_{l}(c)\\,\\left[1-\\alpha (T)\\,(T-T_0)\\right],$ where $\\rho ^0_{l}(c)$ is the density of the aqueous mixture having a concentration $c$ of CaCl$_2$ (w/w), taken at the temperature $T_0=20^{\\rm o}$ C. The coefficient of thermal expansion, $\\alpha (t)=a\\,T+b$ , is fitted from the experimental data ($a=9.6\\times 10^{-6}\\,(^{\\rm o}{\\rm C})^{-2}$ , $b=1.2\\times 10^{-4}\\,(^{\\rm o}{\\rm C})^{-1}$ and $T$ is expressed in $^{\\rm o}{\\rm C}$ )."
],
[
"Polystyrene beads",
"The coefficient of thermal expansion (volume) is taken equal to $\\alpha _{p}=1.9\\,\\times 10^{-4}\\,(^{\\rm o}{\\rm C})^{-1}$ [28], such that the density of the polystyrene beads can be expressed as $\\rho _{p}(T)=\\rho _{p}^0\\,\\left[1-\\alpha _{p}\\,(T-T_0)\\right],$ with $\\rho _{p}^0=1049 \\,{\\rm kg.m^{-3}}$ taken at $T_0= 20^{\\rm o}$ C."
]
] | 1709.01733 |
[
[
"Counting non-commensurable hyperbolic manifolds and a bound on\n homological torsion"
],
[
"Abstract We prove that the cardinality of the torsion subgroups in homology of a closed hyperbolic manifold of any dimension can be bounded by a doubly exponential function of its diameter.",
"It would follow from a conjecture by Bergeron and Venkatesh that the order of growth in our bound is sharp.",
"We also determine how the number of non-commensurable closed hyperbolic manifolds of dimension at least 3 and bounded diameter grows.",
"The lower bound implies that the fraction of arithmetic manifolds tends to zero as the diameter goes up."
],
[
"Introduction",
"Recently there has been a lot of progress on understanding the relation between the volume of a negatively curved manifold and its toplogical complexity.",
"In this note, we will instead consider the relation between complexity and diameter.",
"We will restrict to closed hyperbolic (constant sectional curvature $-1$ ) manifolds.",
"The main upshot of considering the diameter instead of the volume is that we obtain bounds in dimension 3."
],
[
"New results",
"Recall that two manifolds are called commensurable if they have a common finite cover.",
"The diameter of a commensurability class of manifolds is the minimal diameter realized by a manifold in that class.",
"Given $d\\in \\mathbb {R}_+$ and $n\\ge 3$ , let $NC^{\\operatorname{diam}}_n(d)$ denote the number of commensurability classes of closed hyperbolic $n$ -manifolds of diameter $\\le d$ .",
"We will prove Theorem 1 For all $n \\ge 3$ there exist $0<a<b$ so that $a\\cdot d \\le \\log (\\log (NC^{\\operatorname{diam}}_n(d))) \\le b \\cdot d$ for all $d\\in \\mathbb {R}_+$ large enough.",
"Note that the analogous statement in dimension 2 is false.",
"Since surfaces have large deformation spaces of hyperbolic metrics (see eg.",
"[13] for details), it is not hard to produce an uncountable number of non-commensurable hyperbolic surfaces that have both bounded diameter and bounded volume.",
"Our upper bound follows directly from a result by Young [41] (see also Equation (REF )) that estimates the number of manifolds up to a given diameter.",
"However, for his lower bound, Young uses finite covers of a fixed manifold, which are all commensurable.",
"We will instead consider a collection of non-commensurable manifolds constructed by Gelander and Levit [18] and will use results on random graphs due to Bollobás and Fernandez de la Vega [3] to argue that most of these manifolds have small diameters.",
"Because $NC^{\\operatorname{diam}}_n(d)$ is finite for all $n\\ge 3$ and $d\\in \\mathbb {R}_+$ , we can turn the set of commensurability classes of closed hyperbolic $n$ -manifolds of diameter $\\le d$ into a probability space by equipping it with the uniform probability measure.",
"That is, given $n\\ge 3$ and $d\\in \\mathbb {R}_+$ and a set $A$ of commensurability classes of closed hyperbolic manifolds of diameter $\\le d$ , we set $\\operatorname{\\mathbb {P}}_{n,d}[A] = \\frac{\\left|A\\right|}{NC^{\\operatorname{diam}}_n(d)},$ where $\\left|A\\right|$ denotes the cardinality of $A$ .",
"It follows from Theorem REF together with results by Belolipetsky [2] for $n\\ge 4$ and Belolipetsky, Gelander, Lubotzky and Shalev [5] for $n=3$ that most maximal lattices are not arithmetic: Corollary 1 Let $n\\ge 3$ .",
"We have $\\lim _{d\\rightarrow \\infty }\\operatorname{\\mathbb {P}}_{n,d}[\\text{The manifold is arithmetic}] = 0.$ Similar results have been proved in dimension $\\ge 4$ by Gelander and Levit [18] with diameter replaced by volume and by Masai [27] for a different model of random 3-manifolds: random 3-dimensional mapping tori built out of punctured surfaces.",
"We show that for closed hyperbolic manifolds, the size of homological torsion can also be bounded in terms of the diameter of the manifold.",
"Theorem 2 For every $n\\ge 2$ there exists a constant $C >0$ so that $\\log \\log \\left(\\left|H_i(M,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) \\le C\\cdot \\operatorname{diam}(M) $ for all $i=0,\\ldots ,n$ for any closed hyperbolic $n$ -manifold $M$ .",
"Let us first note that for $n=2$ our theorem is automatic, since all torsion subgroups in the homology of a closed hyperbolic surface are trivial.",
"Moreover, in dimension at least 4 results by Bader, Gelander and Sauer [7] (see also Equation (REF )) together with a comparison between volume and diameter (Lemma REF ) also prove Theorem REF .",
"However, it is known that in dimension 3, this method cannot work.",
"We also note that to prove that our theorem is sharp, the most likely examples would be sequences hyperbolic manifolds with exponentially growing torsion and logarithmically growing diameter.",
"Examples of such sequences are abelian covers [37], [32], [1], Liu's construction [25] and random Heegaard splittings [24], [1].",
"However, these manifolds are not known to have small diameters and in some cases are even known not to have small diameters.",
"On the other hand, certain sequences of covers of arithmetic manifolds are conjectured to have exponential torsion growth by Bergeron and Venkatesh [14].",
"It is known that the corresponding lattice has Property $(\\tau )$ with respect to this sequence of covers [38], from which it follows that their diameter is small by a result of Brooks [11] (see Sections REF and REF ).",
"So assuming the conjecture by Bergeron and Venkatesh, these manifolds would saturate the bound in Theorem REF up to a multiplicative constant.",
"The proof of Theorem REF consists of two steps.",
"First we use Young's method from [41] to build a simplicial complex that models our manifold and has a bounded number of cells of any dimension.",
"We then use a lemma due to Bader, Gelander and Sauer [7] (based on a lemma of Gabber) that bounds the homological torsion in terms of the number of cells (Lemma REF ) to derive our bound."
],
[
"Volume and complexity",
"The main motivation for our work is formed by results that bound the homological complexity of a complete negatively curved $n$ -manifold $M$ in terms of its volume.",
"A classical result due to Gromov and worked out by Ballmann, Gromov and Schröder [21], [6], states that there exists a constant $C >0$ , depending only on $n$ , so that the Betti numbers $b_i(M)$ satisfy $b_i(M) \\le C \\cdot \\operatorname{vol}(M),$ for $i=0,\\ldots ,n$ , where $\\operatorname{vol}(M)$ denotes the volume of $M$ .",
"More recently, Bader, Gelander and Sauer [7] have shown that, when the dimension $n$ is at least 4, the cardinality of the torsion subgroups $H_i(M,\\mathbb {Z})_{\\mathrm {tors}}$ in homology can also be bounded in terms of the volume.",
"They show that for all $n\\ge 4$ , there exists a constant $C>0$ , depending only on $n$ , so that $\\log \\left(\\left|H_i(M,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) \\le C\\cdot \\operatorname{vol}(M).$"
],
[
"Counting manifolds by volume",
"We will again restrict ourselves to closed hyperbolic manifolds.",
"A classical result due to Wang [39] states that in dimension $n\\ge 4$ the number of closed hyperbolic manifolds of volume $\\le v$ is finite for any $v\\in \\mathbb {R}_+$ .",
"Let $N^{\\operatorname{vol}}_n(v)$ denote the number such manifolds.",
"The first bounds on the growth of $N^{\\operatorname{vol}}_n(v)$ are due to Gromov [20].",
"Burger, Gelander, Lubotzky and Mozes [4] showed that for all $n\\ge 4$ there exist $0<a<b \\in \\mathbb {R}$ such that $a \\cdot v\\log (v)\\le \\log \\left(N^{\\operatorname{vol}}_n(v)\\right)\\le b\\cdot v\\log (v),$ for all $v\\in \\mathbb {R}_+$ large enough.",
"Analogously to the case of the diameter, the volume of a commensurability class of manifolds is the minimal volume realized in that class.",
"Let $NC^{vol}_n(v)$ denote the number of commensurability classes of hyperbolic $n$ -manifolds of volume $\\le v$ .",
"The first lower bounds on this number are due to Raimbault [33].",
"Gelander and Levit [18] showed that for all $n\\ge 4$ there exist $0<a<b \\in \\mathbb {R}$ such that $a\\cdot v\\log (v)\\le \\log \\left(NC^{\\operatorname{vol}}_n(v)\\right)\\le b\\cdot v\\log (v),$ for all $v\\in \\mathbb {R}_+$ large enough."
],
[
"3-dimensional manifolds",
"As opposed to in dimension 4 and above, the number of closed hyperbolic 3-manifolds of bounded volume is not finite.",
"This for instance follows from work of Thurston (see for example [10]).",
"This means that there is also no reason to suppose that a statement like Equation (REF ) holds in dimension 3.",
"In fact, in [7], the authors prove that no such bound can exist, even for sequences of hyperbolic 3-manifolds manifolds that Benjamini-Schramm converge to $\\mathbb {H}^3$ .",
"We note that Frączyk [15] has however proved that for arithmetic manifolds, a similar bound to that of Bader, Gelander and Sauer does hold."
],
[
"Diameters",
"The number of closed hyperbolic $n$ -manifolds of diameter at most $d$ is finite for any $n\\ge 3$ and $d\\in \\mathbb {R}_+$ .",
"The best known estimates on the number $N^{\\operatorname{diam}}_n(d)$ of closed hyperbolic $n$ -manifolds of diameter $\\le d$ are due to Young [41].",
"He proved that for every $n\\ge 3$ there exist constants $0<a<b\\in \\mathbb {R}^+$ so that $a\\cdot d \\le \\log (\\log (N^{\\operatorname{diam}}_n(d)))\\le b\\cdot d$ for all $d\\in \\mathbb {R}_+$ large enough.",
"Equation (REF ) implies that the Betti numbers of a closed hyperbolic manifold $M$ can also be bounded by its diameter $\\operatorname{diam}(M)$ as follows $b_i(M) \\le C \\cdot e^{(n-1)\\cdot \\operatorname{diam}(M)},$ for all $i=0,\\ldots n$ , where $C>0$ is a constant depending only on $n$ (see Lemma REF ).",
"Moreover, because there are hyperbolic surfaces with linearly growing genus and logarithmically growing diameter (for instance random surfaces [8], [28]), a bound of this generality is necessarily exponential in diameter."
],
[
"Acknowledgement",
"The author's research was supported by the ERC Advanced Grant “Moduli”.",
"The author also thanks the organizers of the Borel Seminar 2017, during which the research for this article was done.",
"Finally, the author thanks Filippo Cerocchi, Jean Raimbault and Roman Sauer for useful conversations."
],
[
"Background material",
"In what follows, $n$ will be a natural number, $M$ a closed oriented hyperbolic $n$ -manifold.",
"We will use $\\operatorname{vol}(M)$ , $\\operatorname{diam}(M)$ , $\\operatorname{inj}(M)$ and $\\lambda _1(M)$ to denote the volume, the diameter, the injectivity radius and the first non-zero eigenvalue of the Laplace-Beltrami operator of $M$ respectively.",
"Moreover, $\\operatorname{\\mathrm {d}}:M\\times M\\rightarrow \\mathbb {R}_+$ will denote the distance function on $M$ .",
"Finally, $\\mathbb {H}^n$ will denote hyperbolic $n$ -space and $\\operatorname{Isom}^+(\\mathbb {H}^n)$ will denote its group of orientation preserving isometries."
],
[
"Bounds involving the diameter",
"Many of our bounds are based on the following well known fact.",
"Lemma 2.1 Let $n\\ge 2$ .",
"There exists a constant $C>0$ , depending only on $n$ , so that for every closed hyperbolic $n$ -manifold $M$ we have $C\\cdot \\log (\\operatorname{vol}(M)) \\le \\operatorname{diam}(M).",
"$ The crucial observation is that the ball $B_M(p,\\operatorname{diam}(M))$ of radius $\\operatorname{diam}(M)$ around any point $p\\in M$ , by definition of the diameter, covers $M$ .",
"This implies that $\\operatorname{vol}(M)\\le \\operatorname{vol}(B_{M}(p,\\operatorname{diam}(M))).$ On the other hand, the volume of $B_{M}(p,\\operatorname{diam}(M))$ is at most the volume of a ball of the same radius in $\\mathbb {H}^n$ .",
"The volume of a ball $B_{\\mathbb {H}^n}(p,R)$ of radius $R$ around $p\\in \\mathbb {H}^n$ is equal to $\\operatorname{vol}(B_{\\mathbb {H}^n}(p,R))=\\operatorname{vol}(\\mathbb {S}^{n-1})\\int _0^R \\sinh ^{n-1}(t)dt,$ where $\\mathbb {S}^{n-1}$ denotes the $(n-1)$ -sphere equipped with the round metric (see for instance [34]).",
"Putting this together with the inequality above gives the lemma.",
"Moreover, we shall need a bound on the injectivity radius in terms of the diameter.",
"The following was proved by Young [41], based on work by Reznikov [35]: Lemma 2.2 Let $n\\ge 2$ .",
"There exists a constant $C>0$ , depending only on $n$ , so that $\\operatorname{inj}(M) \\ge \\exp (-\\operatorname{diam}(M)/C)$ for all closed hyperbolic $n$ -manifolds $M$ .",
"In order to control the diameter of a sequence of congruence covers later on, we will use the eigenvalue of their Laplacian in combination with the following theorem due to Brooks [11]: Theorem 2.3 Let $M$ be a closed hyperbolic manifold and Let $\\lbrace M_i\\rbrace _{i\\in \\mathcal {I}}$ be a family of finite covers of $M$ .",
"If there exists a constant $C>0$ so that $\\lambda _1(M_i)> C$ for all $i\\in \\mathcal {I}$ , then there exist constants $a,b,c>0$ such that $a < \\frac{\\log (\\operatorname{vol}(M_i))+c}{\\operatorname{diam}(M_i)} < b$ for all $i\\in \\mathcal {I}$ ."
],
[
"Gelander and Levit's construction",
"The lower bound in Theorem REF will come from a construction due to Gelander and Levit, which is inspired by a classical construction due to Gromov and Piatetski-Shapiro [19] (see also [33]).",
"We will briefly describe some, but not all, of the details of their construction.",
"For more information we refer to [18].",
"Assume we are given six compact hyperbolic $n$ -manifolds with boundary $V_0$ , $V_1$ , $A_+$ , $A_-$ , $B_+$ and $B_-$ so that - $V_0$ and $V_1$ both have four boundary components and $A_+$ , $A_-$ , $B_+$ and $B_-$ all have two boundary components.",
"- All the boundary components of these manifolds are isometric to a fixed closed hyperbolic $(n-1)$ -manifold.",
"- Each of these six manifolds is embedded in an arithmetic manifolds without boundary that are pairwise non-commensurable (see Section REF for a definition of an arithmetic group).",
"In [18], Gelander and Levit explain how to construct these manifolds.",
"We will glue these manifolds according to Schreier graphs for finite index subgroups of the free group $\\operatorname{\\mathbb {F}}_2=\\langle a,b\\rangle $ .",
"Let $\\operatorname{Cay}(\\operatorname{\\mathbb {F}}_2,\\lbrace a,b\\rbrace )$ denote the Cayley graph of $\\operatorname{\\mathbb {F}}_2$ with respect to the generating set $\\lbrace a,b\\rbrace $ .",
"Given $H < \\operatorname{\\mathbb {F}}_2$ , the Schreier graph $\\Gamma _H$ is the graph $\\Gamma _H = \\operatorname{Cay}(\\operatorname{\\mathbb {F}}_2,\\lbrace a,b\\rbrace ) / H.$ Since the edges in $\\operatorname{Cay}(\\operatorname{\\mathbb {F}}_2,\\lbrace a,b\\rbrace )$ come with a natural labeling with the symbols $\\lbrace a^\\pm ,b^\\pm \\rbrace $ , the edges $\\Gamma _H$ come with such a labeling as well.",
"Furthermore, note that the number of vertices of $\\Gamma _H$ is equal to the index $[\\operatorname{\\mathbb {F}}_2:H]$ .",
"Let us denote the vertex and edge set of $\\Gamma _H$ by $V(\\Gamma _H)$ and $E(\\Gamma _H)$ respectively.",
"Definition 2.4 Given a finite index subgroup $H < \\operatorname{\\mathbb {F}}_2$ and a map $\\tau :V(\\Gamma _H)\\rightarrow \\lbrace 0,1\\rbrace $ , we construct the closed hyperbolic $n$ -manifold $M(H,\\tau )$ as follows: - To each vertex $v\\in V(\\Gamma _H)$ , associate a copy of $V_{\\tau (v)}$ - and to each edge $e\\in E(\\Gamma _H)$ , associate a copy of the pair $A^+,A^-$ or $B^+,B^-$ , according to whether it is labeled with an $a^\\pm $ or a $b^\\pm $ .",
"- Glue the manifolds together according to the incidence relations in $\\Gamma _H$ .",
"In particular, the order in which to glue the two blocks associated to an edge depends on whether or not the edge is labeled with an inverse.",
"Note that there is some ambiguity in the construction above: there is for instance a choice which boundary component to glue to which.",
"Since we are using the construction for a lower bound, this won't make a difference to us.",
"We will from now on assume some choice of gluing is given for every pair $(H,\\tau )$ .",
"Figure REF shows a cartoon of what the local picture of $M(H,\\tau )$ might look like: Figure: A local picture of Γ H \\Gamma _H and M(H,τ)M(H,\\tau ), where τ(v 1 )=τ(v 3 )=0\\tau (v_1)=\\tau (v_3)=0 and τ(v 2 )=1\\tau (v_2)=1In [18], Gelander and Levit show: Proposition 2.5 Let $H,H^{\\prime }<\\operatorname{\\mathbb {F}}_2$ be distinct finite index subgroups and let $\\tau :V(\\Gamma _H)\\rightarrow \\lbrace 0,1\\rbrace $ and $\\tau ^{\\prime }:V(\\Gamma _{H^{\\prime }})\\rightarrow \\lbrace 0,1\\rbrace $ be so that $\\left|\\tau ^{-1}(1)\\right| = \\left|(\\tau ^{\\prime })^{-1}(1)\\right| = 1.$ Then $M(H,\\tau )$ and $M(H^{\\prime },\\tau ^{\\prime })$ are not commensurable.",
"The upshot of this proposition is that the construction of Gelander and Levit gives rise to at least $a_N(\\operatorname{\\mathbb {F}}_2)$ non-commensurable manifolds on built out of graphs with $n$ vertices, where $a_N(\\operatorname{\\mathbb {F}}_2)$ denotes the number of index $N$ subgroups of $\\operatorname{\\mathbb {F}}_2$ ."
],
[
"Graphs and groups",
"To work with Gelander and Levit's construction, we will need two bounds.",
"We need a lower bound on $a_N(\\operatorname{\\mathbb {F}}_2)$ and we need to know what the typical diameter of a Schreier graph of an index $N$ subgroup of $\\operatorname{\\mathbb {F}}_2$ is.",
"For details on the subgroup growth of $\\operatorname{\\mathbb {F}}_2$ , we refer to Chapter 2 in the monograph by Lubotzky and Segal [26].",
"We will use the following bound, that can be found as a special case of [26]: Theorem 2.6 We have $ a_N(\\operatorname{\\mathbb {F}}_2)\\sim N \\cdot N!$ as $N\\rightarrow \\infty $ .",
"In the theorem above, we write $f(N)\\sim g(N)$ as $N\\rightarrow \\infty $ to mean that $\\lim _{N\\rightarrow \\infty } f(N)/g(N) =1.$ The distance between two vertices in a connected graph is the minimal number of edges in a path between these two vertices.",
"The diameter $\\operatorname{diam}(\\Gamma )$ of a finite graph $\\Gamma $ is the maximal distance realized by two vertices in $\\Gamma $ .",
"To control the diameter of a typical Schreier graph we will use results from random graphs.",
"The fact that the diameter of a random regular graph is bounded by a logarithmic function of the number of vertices can for instance be derived from the fact that a random regular graph has a large spectral gap with probability tending to 1 as the number of vertices tends to infinity (see [16], [31], [23], [12]).",
"The sharpest result however does not use this method and is due to Bollobás and Fernandez de la Vega [3].",
"We will state their result only in the case of 4-regular graphs.",
"Theorem 2.7 Let $\\operatorname{\\mathbb {P}}_N$ denote the uniform probability measure on the set of isomorphism classes of 4-regular graphs on $N$ .",
"There exists a function $E:\\mathbb {N}\\rightarrow \\mathbb {R}$ so that $E(N) = o(\\log (N))$ as $N\\rightarrow \\infty $ and $\\operatorname{\\mathbb {P}}_N[\\text{The graph has diameter }\\le \\log _3(N) + E(N)]\\rightarrow 1$ as $N\\rightarrow \\infty $ .",
"We note that the bound is basically as low as one could possibly expect.",
"Indeed, with an argument very similar to that in Lemma REF it can be shown that the diameter of a 4-regular graph is at least of the order $\\log _3(N)$ .",
"We won't go into random regular graphs in this note and refer the reader to [3], [9], [40] for the details.",
"We do however note that uniformly picking an index $N$ Schreier graph is a slightly different model for random 4-valent graphs than the uniform probability measure on isomorphism classes of 4-valent graphs.",
"It turns out that the two models are what is called contiguous: they have the same asymptotic 0-sets, which is enough for our purposes.",
"More details on this can be found in [40] and [17]."
],
[
"Arithmetic manifolds",
"Let $G$ be a semisimple Lie group of noncompact type that is defined over $\\mathbb {Q}$ (in our case $G$ will always be $\\operatorname{Isom}^+(\\mathbb {H}^n)$ ).",
"A discrete subgroup $\\Gamma < G(\\mathbb {Q})$ will be called arithmetic if there is a $\\mathbb {Q}$ -embedding $\\rho :G\\rightarrow \\operatorname{GL}_m(\\mathbb {R})$ such that $\\rho (\\Gamma )$ is commensurable with $G(\\mathbb {Z}) = \\operatorname{GL}_m(\\mathbb {Z})\\cap \\rho (G)$ .",
"Arithmetic groups come with a sequence of finite index subgroups called congruence groups.",
"For general background on arithmetic groups, we refer to [30], [29].",
"Recall that a lattice $\\Gamma < \\operatorname{Isom}^+(\\mathbb {H}^n)$ is called uniform if $\\Gamma \\backslash \\operatorname{Isom}^+(\\mathbb {H}^n)$ is compact.",
"We will call $\\Gamma $ maximal if it is not properly contained in another lattice.",
"The result we will need is a bound on the number of maximal uniform arithmetic lattices up to a given covolume in $\\operatorname{Isom}^+(\\mathbb {H}^n)$ for $n\\ge 3$ .",
"To this end, let $\\mathrm {MAL}^u_n(v)$ denote the number of maximal uniform arithmetic lattices of covolume $\\le v$ in $\\operatorname{Isom}^+(\\mathbb {H}^n)$ .",
"The following theorem is due to Belolipetsky [2] in dimension $n\\ge 4$ and Belolipetsky, Gelander, Lubotzky and Shalev [5] in dimensions 2 and 3.",
"Theorem 2.8 Let $n\\ge 2$ and $\\varepsilon >0$ .",
"There exist constants $\\alpha =\\alpha (n) \\in \\mathbb {R}_+$ and $\\beta =\\beta (n,\\varepsilon )\\in \\mathbb {R}_+$ so that $v^\\alpha \\le \\mathrm {MAL}^u_n(v) \\le v^{\\beta \\cdot (\\log v)^\\varepsilon }$ for all $v\\in \\mathbb {R}^+$ large enough.",
"Let us now restrict to hyperbolic 3-manifolds.",
"To get bounds on the diameters of congruence covers of arithmetic manifolds, we will use the following theorem due to Sarnak and Xue [38]: Theorem 2.9 Let $\\Gamma <\\operatorname{Isom}^+(\\mathbb {H}^3)$ be a uniform arithmetic lattice.",
"Then there exists a constant $C=C(\\Gamma )>0$ so that for all congruence subgroups $\\Gamma ^{\\prime }<\\Gamma $ we have $\\lambda _1(\\Gamma ^{\\prime }\\backslash \\mathbb {H}^3) \\ge C.$ If a lattice $\\Gamma <\\operatorname{SL}_2(\\mathbb {C})$ has a sequence $\\lbrace \\Gamma _N\\rbrace _{N}$ of finite index subgroups so that $\\lambda _1(\\Gamma _N\\backslash \\mathbb {H}^3)$ is uniformly bounded from below for all $N\\in \\mathbb {N}$ , then $\\Gamma $ is said to have Property ($\\tau $ ) with respect to this sequence.",
"In the special case where $\\Gamma $ is arithmetic and the sequence consists of congruence subgroups, Property ($\\tau $ ) is sometimes also called the Selberg property.",
"Congruence covers are also believed to have large torsion subgroups in their homology.",
"Specifically, there is the following conjecture, due to Bergeron and Venkatesh [14], which we state in the special case of hyperbolic 3-manifolds: Conjecture 2.10 Let $\\Gamma <\\operatorname{SL}_2(\\mathbb {C})$ be a uniform arithmetic lattice and $\\ldots <\\Gamma _N<\\Gamma _{N-1}<\\ldots <\\Gamma _1<\\Gamma $ a sequence of congruence subgroups of $\\Gamma $ so that $\\cap _N \\Gamma _N = \\lbrace 1\\rbrace $ .",
"Then $\\lim _{N\\rightarrow \\infty } \\frac{\\log \\left(\\left|H_1(\\Gamma _N \\backslash \\mathbb {H}^3,\\mathbb {Z})_\\mathrm {tors}\\right|\\right)}{\\operatorname{vol}(\\Gamma _N \\backslash \\mathbb {H}^3)} = \\frac{1}{6\\pi }.$ The reason for the constant $1/6\\pi $ above is that it is the $\\ell ^2$ -torsion of $\\mathbb {H}^3$ ."
],
[
"Torsion and the nerve lemma",
"To bound torsion in our manifolds we will use a lemma by Bader, Gelander and Sauer [7], that they derived from a lemma due to Gabber (which can for instance be found in [36]).",
"In this lemma, the degree of a vertex (0-cell) in a simplicial complex is the degree of that vertex in the 1-skeleton of the given complex.",
"Lemma 2.11 For all $D,p \\in \\mathbb {N}$ there exists a constant $C=C(D,p)>0$ so that for any simplicial complex $X$ with $\\le V$ vertices that all have degree $\\le D$ we have: $ \\log \\left(\\left|H_p(X,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) \\le C\\cdot V.$ The simplicial complex we will use will be the nerve of an open cover of our manifold.",
"Recall that an open cover of a space $X$ is a collection $\\mathcal {U}=\\lbrace U_i\\rbrace _{i\\in \\mathcal {I}}$ of open subsets of $X$ so that $ X = \\bigcup _{i\\in \\mathcal {I}}U_i.$ The nerve $\\mathcal {N}(\\mathcal {U})$ of this cover is the simplicial complex that has the sets $U_i$ as vertices and contains a $k$ -simplex for every $k$ -tuple of elements in $\\mathcal {U}$ that have a non-trivial intersection.",
"See [22] for more details.",
"The following statement is known as the nerve lemma and can for instance be found as [22]: Lemma 2.12 If $\\mathcal {U}$ is an open cover of a paracompact space $X$ such that every nonempty intersection of finitely many sets in $\\mathcal {U}$ is contractible, then $X$ is homotopy equivalent to the nerve $\\mathcal {N}(\\mathcal {U})$ ."
],
[
"Counting",
"We are now ready to prove Theorem REF : thmdiamTheorem REF For all $n \\ge 3$ there exist $0<a<b$ so that $a\\cdot d \\le \\log (\\log (NC^{\\operatorname{diam}}_n(d))) \\le b \\cdot d$ for all $d\\in \\mathbb {R}_+$ large enough.",
"The upper bound is direct from Young's result (Equation (REF )), so we focus on the lower bound.",
"Consider the building blocks defined by Gelander and Levit and set $D=\\max \\lbrace \\operatorname{diam}(V_0),\\operatorname{diam}(V_1),\\operatorname{diam}(A_+),\\operatorname{diam}(A_-),\\operatorname{diam}(B_+),\\operatorname{diam}(B_-)\\rbrace .",
"$ Because all the building blocks are compact, this is a finite number.",
"Given a finite index subgroup $H<\\operatorname{\\mathbb {F}}_2$ and a map $\\tau :V(\\Gamma _H)\\rightarrow \\lbrace 0,1\\rbrace $ , we have $\\operatorname{diam}(M(H,\\tau )) \\le 2D\\cdot \\operatorname{diam}(\\Gamma _H)+2D.$ Indeed, suppose $x,y\\in M(H,\\tau )$ .",
"Then the number of building blocks that need to be crossed to get from $x$ to $y$ is at most $2\\operatorname{diam}(\\Gamma _H)+2$ , including the building blocks containing $x$ and $y$ .",
"Because of Theorem REF combined with Theorem REF and Stirling's approximation, there exists a $C>0$ so that the index $N$ subgroups of $\\operatorname{\\mathbb {F}}_2$ for $N$ large enough produce at least $C\\cdot N^{CN}$ non-isomorphic graphs of diameter $\\le \\log _3(N) + o(\\log (N))$ .",
"If we now define maps $\\tau :V(\\Gamma )\\rightarrow \\lbrace 0,1\\rbrace $ that assign the value 1 to only one vertex per graph $\\Gamma $ , we obtain $C\\cdot N^{CN}$ manifolds of diameter $d \\le 2D\\log _3(N) + o(\\log (N))$ .",
"Working this out, we see that this number of manifolds is at least $\\exp (C^{\\prime }\\cdot d \\cdot \\exp (C^{\\prime }\\cdot d)),$ for some $C^{\\prime }>0$ .",
"Proposition REF tells us that none of the resulting manifolds will be commensurable.",
"Corollary REF now also easily follows.",
"corarithmCorollary REF Let $n\\ge 3$ .",
"We have $\\lim _{d\\rightarrow \\infty }\\operatorname{\\mathbb {P}}_{n,d}[\\text{The manifold is arithmetic}] = 0$ The only thing we need to control is the number of maximal uniform arithmetic lattices of diameter $\\le d$ in $\\operatorname{Isom}^+(\\mathbb {H}^n)$ .",
"Let us call this number $\\mathrm {MALD}^u_{n}(d)$ .",
"By Lemma REF we have $\\mathrm {MALD}^u_{n}(d) \\le \\mathrm {MAL}^u_n(e^{d/C_n})$ for some $C_n>0$ independent of $d$ .",
"As such, Theorem REF implies that for every $\\varepsilon >0$ there exists a $\\beta ^{\\prime }>0$ so that $\\mathrm {MALD}^u_{n}(d) \\le \\exp (\\beta ^{\\prime }\\cdot d^{1+\\varepsilon }).$ Comparing this to Theorem REF gives the result."
],
[
"Torsion",
"In this section we prove Theorem REF and explain how a positive answer to Conjecture REF would lead to a sequence of manifolds that saturates the bound in that theorem up to a multiplicative constant."
],
[
"An upper bound for torsion in homology",
"We will start with: thmtorsionTheorem REF For every $n\\ge 2$ there exists a constant $C >0$ so that $\\log \\log \\left(\\left|H_i(M,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) \\le C\\cdot \\operatorname{diam}(M) $ for all $i=0,\\ldots ,n$ for any closed hyperbolic $n$ -manifold $M$ .",
"Our final goal is to employ Lemma REF .",
"In the language of [7]: we need to show that a closed hyperbolic manifold $M$ is homotopy equivalent to a $(D,C\\cdot \\operatorname{diam}(M))$ -simplicial complex, where $C,D>0$ are constants depending only its dimension.",
"The simplicial complex we build is the same as that used for the upper bound in Young's result (Equation (REF )).",
"Set $r=\\operatorname{inj}(M)$ and let $S\\subset M$ be a maximal set of points so that $\\operatorname{\\mathrm {d}}(s,s^{\\prime }) \\ge r/4$ for all $s,s^{\\prime }\\in S$ .",
"Now consider the collection $\\mathcal {U}=\\lbrace B_M(s,r/2)\\rbrace _{s\\in S},$ where $B_M(s,r/2)$ denotes the open ball in $M$ of radius $r/2$ around $s$ .",
"It follows from maximality of $S$ that these balls form an open cover.",
"Moreover, because their radius is half the injectivity radius they are isometrically embedded $n$ -dimensional hyperbolic balls.",
"As such they are convex, which means that their intersections are convex and thus contractible.",
"Hence, the nerve lemma (Lemma REF ) applies.",
"This means that the homology groups we are after are those of $\\mathcal {N}(\\mathcal {U})$ .",
"So we need to find bounds on the number of vertices and their degrees in $\\mathcal {N}(\\mathcal {U})$ in order to apply Lemma REF .",
"The number of vertices, or equivalently the number of points in $S$ , can be bounded by $\\left|S\\right| \\le \\frac{\\operatorname{vol}(M)}{\\operatorname{vol}(B_{\\mathbb {H}^n}(p,r/4))} \\le D\\cdot \\frac{\\operatorname{vol}(M)}{(r/4)^n}, $ where $B_{\\mathbb {H}^n}(p,r/4)$ is the ball of radius $r/4$ around some point $p\\in \\mathbb {H}^n$ and $D>0$ is some constant depending only on the dimension.",
"The second of these bounds again follows from the closed formula for the volume of a ball in $\\mathbb {H}^n$ .",
"Now we use Lemma REF and Lemma REF to tell us that $(r/4)^n \\ge \\exp (-C\\operatorname{diam}(M)) \\;\\;\\text{and}\\;\\; \\operatorname{vol}(M) \\le \\exp (C\\operatorname{diam}(M))$ for some $C>0$ depending only on $n$ .",
"So we obtain $\\left|S\\right| \\le A\\cdot \\exp (B\\operatorname{diam}(M)).",
"$ for some $A,B>0$ depending only on $n$ .",
"All that remains is to show that each vertex has a bounded number of neighbors.",
"First of all note that all the neighbors of a point $s\\in S$ lie in $B_M(s,r)\\subset M$ .",
"By definition of $S$ , the balls of radius $r/8$ around the neighbors of $s$ are all disjoint and all lie in $B_{M}(s,9r/8)$ .",
"This means that the number of neighbors is at most $\\frac{\\operatorname{vol}(B_M(s,9r/8))}{\\operatorname{vol}(B_M(s,r/8))} \\le \\frac{\\operatorname{vol}(B_{\\mathbb {H}^n}(s,9r/8))}{\\operatorname{vol}(B_{\\mathbb {H}^n}(s,r/8))},$ which, for $r$ small enough, is uniformly bounded in each fixed dimension."
],
[
"Sharpness of the bound",
"Like we said in the introduction, if Conjecture REF holds then we would get a sequence of closed hyperbolic 3-manifolds $\\lbrace M_N\\rbrace _{N\\in \\mathbb {N}}$ such that $\\operatorname{diam}(M_N)\\rightarrow \\infty $ as $N\\rightarrow \\infty $ and $ \\log \\log \\left(\\left| H_1(M_N,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) \\ge C \\operatorname{diam}(M_N).$ for some $C>0$ independent of $N$ .",
"Indeed, it follows from Theorem REF together with Theorem REF that for a uniform arithmetic group $\\Gamma < \\operatorname{SL}_2(\\mathbb {C})$ and a sequence of congruence subgroups $\\lbrace \\Gamma _N\\rbrace _N$ , the manifolds $M_N = \\Gamma _N \\backslash \\mathbb {H}^3$ satisfy $\\operatorname{diam}(M_N) \\le A \\cdot \\log (\\operatorname{vol}(M_N)).$ It would follow from Conjecture REF that $\\operatorname{vol}(M_N)\\le B\\cdot \\log \\left(\\left| H_1(M_N,\\mathbb {Z})_{\\mathrm {tors}}\\right|\\right) $ for all $N$ and some $B>0$ independent of $N$ .",
"Putting these two together would give the desired result."
]
] | 1709.01873 |
[
[
"Shocks in the relativistic transonic accretion with low angular momentum"
],
[
"Abstract We perform 1D/2D/3D relativistic hydrodynamical simulations of accretion flows with low angular momentum, filling the gap between spherically symmetric Bondi accretion and disc-like accretion flows.",
"Scenarios with different directional distributions of angular momentum of falling matter and varying values of key parameters such as spin of central black hole, energy and angular momentum of matter are considered.",
"In some of the scenarios the shock front is formed.",
"We identify ranges of parameters for which the shock after formation moves towards or outwards the central black hole or the long lasting oscillating shock is observed.",
"The frequencies of oscillations of shock positions which can cause flaring in mass accretion rate are extracted.",
"The results are scalable with mass of central black hole and can be compared to the quasi-periodic oscillations of selected microquasars (such as GRS 1915+105, XTE J1550-564 or IGR J17091-3624), as well as to the supermassive black holes in the centres of weakly active galaxies, such as Sgr $A^{*}$."
],
[
"Introduction",
"The weakly active galaxies, where the central nucleus emits the radiation at a moderate level with respect to the most luminous quasars, are frequenlty described by the so called hot accretion flows model.",
"In such flows, the plasma is virially hot and optically thin, and due to the advection of energy onto black hole, the flow is radiativelly inefficient.",
"The prototype of an object which can be well described by such model, is the low luminosity black hole in the centre of our Galaxy, the source Sgr $A^{*}$ [64].",
"Also, the black hole X-ray binaries in their hard and quiescent states can be good representatives for the hot mode of accretion.",
"These states are frequenlty associated with the ejection of relativistic streams of plasma, which form the jets, responsible for the radio emission of the sources [20].",
"The matter essentially flows into black hole with the speed of light, while the sound speed at maximum can reach $c/\\sqrt{3}$ .",
"Therefore, the accretion flow must have transonic nature.",
"The viscous accretion with transonic solution based on the alpha-disk model was first studied by [46] and [42].",
"After that, e.g.",
"[2] examined the stability and structure of transonic disks.",
"The possibility of collimation of jets by thick accretion tori was proposed by, e.g., [54].",
"In respect of the value of angular momentum there are two main regimes of accretion, the Bondi accretion, which refers to spherical accretion of gas without any angular momentum, and the disk-like accretion with Keplerian distribution of angular momentum.",
"In the case of the former, the sonic point is located farther away from the compact object (depending on the energy of the flow) and the flow is supersonic downstream of it.",
"In the latter case, the flow becomes supersonic quite close to the compact object.",
"For gas with a low value of constant angular momentum, hence belonging in between these two regimes, the equations allow for the existence of two sonic points of both types.",
"The possible existence of shocks in low angular momentum flows connected with the presence of multiple critical points in the phase space has been studied from different points of view during the last thirty years.",
"Quasi-spherical distribution of the gas endowed by constant specific angular momentum $\\lambda $ and the arisen bistability was studied already by [3].",
"[21] studied the existence of critical points for realistic equation of state of the gas and showed the corresponding Rankine-Hugoniot conditions for standing shocks.",
"Later, the significance of this phenomenon related to the variability of some X-ray sources has been pointed out by [32] and soon after, the possibility of the shock existence together with shock conditions in different types of geometries was discussed also by [1].",
"The transonic solution required by the aforementioned boundary conditions is the solution having low subsonic velocity far away from the compact object ($\\mathfrak {M}<1$ , where $\\mathfrak {M}=v/c_{\\rm s}$ is the Mach number, $v=-u^r_{\\rm BL}$ is the inward radial velocity in Boyer-Lindquist coordinates, and $c_{\\rm s}$ is the local sound speed of the gas), and supersonic velocity very near to the horizon ($\\mathfrak {M}>1$ ).",
"Thus the flow can locally pass only through the outer or also through the inner sonic point.",
"The latter is globally achieved due to the shock formation between the two critical points.",
"More recently, the shock existence was found also in the disc-like structure with low angular momentum in hydrostatic equilibrium both in pseudo-Newtonian potential [13] and in full relativistic approach [14].",
"Regarding the sequence of steady solutions with different values of specific angular momentum, the hysteresis-like behavior of the shock front was proposed in the latter work.",
"Different geometrical configurations with polytropic or isothermal equation of state were studied in the post Newtonian approach with pseudo-Kerr potential [52].",
"In the general relativistic description the dependence of the flow properties (Mach number, density, temperature and pressure) in the close vicinity of the horizon was studied by [15], and the asymmetry of prograde versus retrograde accretion was shown.",
"The complete picture of the accreting microquasar consisting of the Keplerian cold disc and the low angular momentum hot corona, the so called two component advective flow (TCAF), was described by [7].",
"This model combined with the propagating oscillatory shock (POS) model was later used to explain the evolution of the low frequency QPO during the outburst in several microquasars [10], [11], [45], [19] and it was also implemented into the XSPEC software package [17].",
"The presence of the low angular momentum component in the accretion flow during the outbursts of microquasars seems to be essential for explaining different timing in the hard and soft bands, especially during the rising phase [63].",
"[18] showed with the spectral fitting, that the flux of the power-law component is higher than the flux of the disc black body in the time period, when the QPOs are seen in H 1743-322.",
"[17] showed similar results with the TCAF model, which gives the mass accretion rate of the disc component comparable to the accretion rate of the sub-Keplerian component, when the QPO is seen in three different sources.",
"Further development in this topic includes numerical simulations of low angular momentum flows in different kinds of geometrical setup.",
"Hydrodynamical models of the low angular momentum accretion flows have been studied already in two and three dimensions, e.g.",
"by [48], [30] and [31].",
"In those simulations, a single, constant value of the specific angular momentum was assumed, while the variability of the flows occurred due to e.g.",
"non-spherical or non-axisymmetric distribution of the matter.",
"The level of this variability was also dependent on the adiabatic index.",
"However, these studies have not concentrated on the existence of the standing shocks as predicted by the theoretical works mentioned above.",
"The consideration was also put on the problem of viscosity in such flows, especially on the influence of the viscosity on the position of the shock and the shape of the solution [8], [9], [43].",
"The possible consequences of viscous mechanisms in the shocked accretion flow for the QPOs' evolution was studied by [40].",
"Later on, [44] added the phenomenon of outflows to the picture.",
"[53] showed the joint role of viscosity and magnetic field considered in heating of the accretion flow combined with the cooling by Comptonization on the dynamical structure of the global accretion flow.",
"Such models were also studied by hydrodynamical numerical simulations in the pseudo-Newtonian description of gravity e.g.",
"by [24], [23], [16].",
"Our aim is to provide numerical simulations of low angular momentum flows, which would support or correct the semi-analytical findings about the shock existence and behavior mentioned above.",
"In our previous work, we performed 1D pseudo-Newtonian computations [59], where we studied the dependence of the shock solution of the parameters and the response of the shock front to the change of angular momentum in the incoming matter.",
"The hysteresis behavior was observed in our simulations and we have seen the repeated creation and disappearance of the shock front due to the oscillations of angular momentum in the flow.",
"Here we aim to provide more advanced numerical study of the flow using the full relativistic treatment of the gravity with the fixed background metric given by the Schwarzschild/Kerr solution, which is performed in one, two and three dimensions.",
"The organization of the paper is as follows.",
"In Section we briefly recall the semi-analytical treatment of the shock existence in the pseudo-Newtonian approach, which is described in details in [59].",
"The numerical setup of our simulations is given in Section , the different initial conditions are described in Subsections REF , REF and REF .",
"In Section we present our results, in particular in REF we show the 1D simulations with standing or oscillating shock location and we run models with time-dependent outer boundary condition corresponding to periodic change of angular momentum in the incoming matter.",
"The major part of the results is presented in REF , where the outcomes of the 2D simulations with different kind of initial conditions are presented.",
"In REF we confirm the reliability of the 2D results with two full three dimensional tests.",
"The findings of our study are discussed in Section ."
],
[
"Appearance of shocks in 1D low angular momentum flows",
"In this paper we follow up our previous study of the flow with constant low angular momentum $\\lambda $ , which was held in the pseudo-Newtonian framework.",
"Here we briefly recall the semi-analytical results in such setup, which we use as an initial setting for our GR computations (for further details see [59]).",
"For the analytical study we consider a non-viscous quasi-spherical polytropic flow with the equation of state $p=K \\rho ^\\gamma $ , where $p$ is the pressure, and $\\rho $ is the density in the gas.",
"Our EOS holds for the isentropic flow, hence the specific entropy $K$ is constant.",
"Using the continuity equation and energy conservation, we can find the position of the critical point $r_c$ as the root of the equation $\\mathcal {E} - \\frac{\\lambda ^2}{2r_c^2} + \\frac{1}{2(r_c-1)}-\\\\\\frac{\\gamma +1}{2(\\gamma -1)} \\left( \\frac{r_c}{4(r_c-1)^2} - \\frac{\\lambda ^2}{2r_c^2} \\right) = 0, $ where $\\mathcal {E}$ stands for energy and where we assume the Paczynski-Wiita gravitational potential in the form of $\\Phi (r)=-\\frac{1}{2(r-1)}$ , so that $r$ is given in the units of $r_g=2GM/c^2$ , and where $\\lambda $ is the value of specific angular momentum.",
"For a subset of the parameter space ($\\mathcal {E}, \\lambda , \\gamma $ ) there exists more than one solution of this equation (three actually), hence there are more critical points located at different positions.",
"It can be shown, that only two of the critical points are of a saddle type, so that the solution can pass through it.",
"We will call them the inner, $r_c^{\\tt in}$ , and the outer, $r_c^{\\tt out}>r_c^{\\tt in}$ , critical points, respectively.",
"We will call this subset as “multicritical region”.",
"For changing $\\lambda $ with other parameters kept constant, this region is projected onto one interval of $[\\lambda _{\\tt min}^{cr},\\lambda _{\\tt max}^{cr}]$ .",
"For decreasing $\\mathcal {E}$ , the interval is shifting up to a higher angular momentum.",
"Together with equation (REF ) determining the values of all variables at the two critical points, the relation for the derivative ${\\rm d}v/{\\rm d}r$ can be obtained from the continuity equation and the energy conservation, so that the solution can be found by integrating the equations from the critical point downwards and upwards.",
"The resulting two branches of solutions of course have the same parameters ($\\mathcal {E}, \\lambda , \\gamma $ ), but they differ in value of the constant specific entropy $K$ , which is given by $K = \\left( v r^2 \\frac{c_s^{\\frac{2}{\\gamma - 1}}}{\\gamma ^{\\frac{1}{\\gamma -1}} \\dot{M}} \\right)^{\\gamma - 1}.",
"$ This is evaluated at the critical point position ($K^{\\rm in/out}=K(r^{\\rm in/out}, v^{\\rm in/out},c_s^{\\rm in/out})$ , where $\\dot{M}$ is the adopted constant accretion rate.",
"Because in our model we study the motion of test matter, which does not contribute to the gravitational field, and we use a simple polytropic equation of state, the accretion rate can be given in arbitrary units and it does not affect the solution.",
"The only possible production of entropy is at the shock front, where jumps in radial velocity, density, and other quantities in the flow occur.",
"Because we are interested in the solution describing the accretion flow, and not winds, the physical requirement for the shock to occur is that the specific entropy at the inner branch is higher than at the outer one ($K^{\\tt in} > K^{\\tt out}$ ).",
"Moreover, the shock will appear only if the Rankine-Hugoniot conditions, expressing the conservation of mass, energy and angular momentum at the shock position are satisfied, that means that the equation $\\frac{ \\left(\\frac{1}{\\mathfrak {M}_{\\tt in}} + \\gamma \\mathfrak {M}_{\\tt in} \\right)^2 }{ \\mathfrak {M}_{\\tt in}^2(\\gamma - 1) + 2 } = \\frac{ \\left(\\frac{1}{\\mathfrak {M}_{\\tt out}} + \\gamma \\mathfrak {M}_{\\tt out}\\right)^2 }{ \\mathfrak {M}_{\\tt out}^2(\\gamma - 1) + 2 } $ holds at some radius $r_s$ ."
],
[
"Numerical setup",
"We performed 1D to 3D hydrodynamical simulations of the non-magnetized accreting gas on the fixed background using the HARMPI e.g., see https://github.com/atchekho/harmpi computational code [50], [51], [37], [62], [38], [29], [27] based on the original HARM code [22], [35].",
"The background spacetime is given by the stationary Kerr solution.",
"The initial conditions are set using Boyer-Lindquist coordinates, and they are transformed into the code coordinates, which are the Kerr-Shild ones.",
"In order to cover the whole accretion structure with sufficiently fine resolution near the black hole we use logarithmic grid in radius with superexponential grid spacing in the outermost region, so that the outer region is covered with low resolution grid and provide the reservoir of gas for accretion.",
"In the innermost region, the grid spans below the horizon thanks to the regularity of Kerr-Shild coordinates, having several zones inside the black hole and the free outflow boundary.",
"The outer boundary condition is mostly given as a free boundary, because the outer boundary is placed far enough from the central region.",
"In the 1D case, when long-term evolution is studied ( $t_f$ up to $5\\cdot 10^7 {\\rm M}$ )), we prescribe the inflow of matter through the outer boundary according to the PN solution.",
"Because the outer boundary is very far away from the centre (typically at $\\sim 10^5$ M), the radial inflowing speed is very small and the deviations between GR and PN solution are negligible.",
"The prescribed properties of the inflowing matter can be also time dependent, when the temporal change of the angular momentum of the matter is considered.",
"For 2D computations, the resolution is in the range between 256 x 128, up to 384 x 256 and 576 x 192, while in 3D case we use 256 x 128 x 96 cells.",
"Figure: a) Profiles of Mach number for four values of energy in the converged stationary state for γ=4/3\\gamma =4/3 and λ=3.6M\\lambda = 3.6M at the end of the simulation, t f =10 6 t_f = 10^6M.",
"Sonic points and shock fronts are located at the points, where the curve crosses the 𝔐=1\\mathfrak {M}=1 line (purple horizontal line).",
"b) The corresponding profiles of density in arbitrary units.",
"c) Radial velocity profiles of the flow v=-u BL 𝚛 v=- u_{\\rm BL}^{\\tt r} in the units of the speed of light, cc.In our previous work, [59], we studied the behavior of the shock solution and also its evolution with 1D simulations using the code ZEUS-MP [55], [26] supplied by the pseudo-Newtonian Paczynski-Wiita potential [47], which mimics the strong gravity effects near the black hole.",
"We will refer to the results presented in that paper as the PW simulations.",
"The parameters which we used in that work, corresponded either to the evolving frequency of quasi-periodic oscillations seen in some microquasars (e.g.",
"GRS1915+105 [36], XTE J1550-564 [12], GRO J1655-40 [49], or GX 339-4 [45]), or were estimated from the values holding for Sgr A* [41], [14].",
"However, such parameters led to an extended accretion structure, meaning especially that $r_c^{\\rm out} \\sim 10^4M$ or more.",
"Here, we consider different values of parameters, and we perform higher dimensional simulations.",
"We require that the outer critical point is located inside the computational domain in order to retain the flow subsonic at the outer boundary.",
"However, the total number of grid cells together with the sufficient resolution near the horizon restrict the maximal radius of the outer boundary, even though we use the logarithmic grid in radial direction.",
"Because of a sufficiently large value of the energy in the flow, $\\mathcal {E}$ , the radius $r_c^{\\rm out}$ is located quite close to the centre.",
"The typical values of parameters used in this study are $\\mathcal {E} = 0.0005, \\gamma = 4/3, \\lambda = 3.6M$ .",
"The shape of the corresponding solution is given in Fig.",
"REF .",
"The evolution of initially non-magnetized gas is simulated with the HARMPI package supplied with our own modifications.",
"The code conserves the vanishing magnetic field and there is no spurious magnetic field generated during the evolution.",
"We evolve two different types of initial conditions.",
"In the first case we prescribe $\\rho $ , $\\epsilon $ and $u^r_{\\tt BL}$ (radial component of the four-velocity) according to the Bondi solution, and we modify this solution by adding non-zero $u^\\phi _{\\tt BL}$ component of the four-velocity, where the $u^\\alpha _{\\tt BL}$ is the four-velocity in the Boyer-Lindquist coordinates.",
"The second option is to prescribe $\\rho $ , $\\epsilon $ and $u^\\alpha _{\\tt BL}$ in accordance to the 1D shock solution (computed with Paczynski-Wiita potential in section ), and then follow the evolution.",
"Such solution obtained with the simplifying assumptions is of course not the true stationary solution.",
"However, because of the fact, that for the given initial conditions and for chosen global parameters of the system, the gas evolves towards the true solution, the discrepancies between the GR and PN are not expected to be substantial.",
"Within the range of parameter space which allows for a shock solution, this 1D approximation can be used for prescribing the initial conditions.",
"We are interested in the evolution of such initial state towards the correct stationary state.",
"The EOS of the form $p = (\\gamma - 1) \\rho \\epsilon $ is used to close the system of equations, so that the isentropy assumption is not imposed.",
"Hence, the specific entropy $K=p/\\rho ^\\gamma $ is not constant and can evolve during the simulation.",
"The fiducial value of the polytropic index used in our computations is $\\gamma =4/3$ ."
],
[
"Initial conditions – Bondi solution equipped with angular momentum",
"In our first set of computations we adopted initially $\\rho $ , $\\epsilon $ and $u^r_{\\tt BL} = -v$ according to the Bondi solution with $ \\epsilon = K \\rho ^{\\gamma -1} /(\\gamma -1)$ , where $K$ is given by Eq.",
"(REF ) evaluated for the Bondi critical point (solution of Eq.",
"(REF ) with $\\lambda = 0$ ).",
"The solution is parametrized by the value of the polytropic exponent $\\gamma $ and the energy $\\mathcal {E}$ , which fix the position of the critical point $r_c^{\\rm out}$ .",
"We modify the initial conditions by prescribing the rotation according to relations $\\lambda = \\lambda ^{\\rm eq} \\sin ^2{\\theta }, \\qquad r&>&r_{\\rm b}, \\\\\\lambda = 0 ,\\qquad r&<&r_{\\rm a}, $ in the Boyer-Lindquist coordinates.",
"Between $r_{\\rm a}$ and $r_{\\rm b}$ the values are smoothened by a cubic spline.",
"The time component of four-velocity is set from the normalization condition $g_{\\mu \\nu }u^\\mu u^\\nu = -1$ assuming $u^\\theta _{\\tt BL}=0$ .",
"The factor $\\sin ^2 \\theta $ in Eq.",
"(REF ) ensures that angular momentum vanishes smoothly at the axis, hence the maximal value of angular momentum $\\lambda ^{\\rm eq}$ is achieved only in the equatorial plane.",
"One example of such initial conditions is plotted in Fig.",
"REF ."
],
[
"Initial conditions – Shock solution",
"In the second type of simulations we modify the initial data procedure such that we find the solution with the shock in the same way as in [59].",
"The values of $\\rho $ , $\\epsilon $ and $u^r_{\\tt BL}$ are set accordingly, with $ \\epsilon = K^{\\rm in/out} \\rho ^{\\gamma -1} /(\\gamma -1)$ , where $K^{\\rm in/out}$ is given by Eq.",
"(REF ) evaluated at the corresponding critical point $r_c^{\\rm in/out}$ for the two branches of solution.",
"Figure: Model C4: Initial data with a shock with ℰ=0.0005,λ eq =3.72\\mathcal {E}=0.0005, \\lambda ^{\\rm eq}=3.72M.However, the 1D analysis provided in that paper and here in Section was based on pseudo-Newtonian approximation, while now we use GR MHD code in order to examine the differences between the two approaches.",
"Moreover, the 1D analysis was held under the assumption of the quasi-spherical shape of the flow and the constant value of angular momentum.",
"That in particular means, that the dependence of the mass accretion rate, which is constant along the flow, scales with $r^2$ .",
"We cannot straightforwardly extend this model into 3D, in other words we can not set the spherical distribution of matter with the angular momentum which would be constant everywhere, because we need to avoid a non-zero angular momentum near the vertical axis.",
"However, the choice of the angular momentum profile in $\\theta $ direction is arbitrary, if it drops to zero near the axis.",
"Therefore, we choose two different profiles of angular momentum for the initial and boundary conditions, to see how much the results depend on this distribution."
],
[
"Angular momentum scaled by $\\sin ^2 \\theta $",
"The first choice is the same as in Section REF , given by Equations (REF ) and ().",
"In this case, the maximal value of angular momentum $\\lambda ^{\\rm eq}$ is obtained only in the equatorial plane and it is lower elsewhere.",
"This type of initial conditions is shown on Fig.",
"REF ."
],
[
"Constant angular momentum in a cone",
"In the second case we prescribe constant angular momentum in a cone with the half-angle $\\theta _c$ centered along the equatorial plane.",
"The values of $\\lambda $ are smoothed down from the cone towards the axis.",
"For this smoothening we choose a cubic spline given by the relations: $\\lambda = \\lambda ^{\\rm eq} f \\qquad \\qquad \\quad \\qquad && \\\\f = \\frac{\\theta ^2(3\\theta _c - 2\\theta )}{\\theta _c^3}, \\quad \\theta <\\frac{\\pi }{2}-\\theta _c \\\\f = 1, \\quad \\theta \\in [\\frac{\\pi }{2}-\\theta _c,\\frac{\\pi }{2}+\\theta _c] \\\\f= \\frac{(\\theta -\\pi )^2(\\frac{\\pi }{2} + 3\\theta _c - 2\\theta )}{(\\theta _c-\\frac{\\pi }{2})^3}, \\quad \\theta >\\frac{\\pi }{2}+\\theta _c$ Thus, all gas within the cone has the maximum angular momentum of $\\lambda ^{\\rm eq}$ , which resembles more the assumptions of the 1D model.",
"We show the initial conditions for this case in Fig.",
"REF .",
"Because of these modifications of angular momentum distribution, such initial conditions are not expected to be the stationary solution in higher dimensions.",
"However, our experience with 1D simulations shows, that only the presence of the inner sonic point is essential for the creation of the shock in the flow, and the exact stationary solution is not needed.",
"If the inner sonic point is present, then the shock bubble shape adjusts itself after a short transient time into the appropriate form.",
"Figure: Top panel: Position of the shock front depending on angular momentum for different values of energy.",
"In case of oscillations, the minimal and maximal shock position is shown.",
"Comparison with the shock front position obtained in the semi-analytical approach using Paczynski-Wiita potential is shown with small dots.",
"Bottom panel: The amplitude 𝒜\\mathcal {A} of the oscillations of the mass accretion rate and its frequency for the oscillating models."
],
[
"Initial conditions for spinning black hole",
"For a spinning black hole we use the initial data described in REF (angular momentum is scaled by $\\sin ^2\\theta $ ).",
"Our semi-analytical 1D shock solution is obtained for the Schwarzschild black hole only.",
"In the case of spinning black hole, the relevant values of angular momentum for the shocks can vary significantly and can be outside the possible existence of shocks in the Schwarzschild space time.",
"Hence, we use two different values of angular momentum: (i) $\\lambda ^s$ , to find the semi-analytic shock solution in non-spinning space time, according to which $\\rho $ , $\\epsilon $ , $u^r_{\\tt BL}$ and $K$ are set, and (ii) $\\lambda ^{\\rm eq}_g$ , which is a different value, according to which the rotation is prescribed using Equations (REF ) and (), and the normalization condition for the four-velocity.",
"The value of $\\lambda ^s$ is not important for the long term evolution; it only enables us to find a configuration with shock which is used for the initial conditions, so that there exists an inner sonic point at the initial time.",
"However, the gas is rotating with $\\lambda ^{\\rm eq}_g$ , which determines the evolution of the flow.",
"After a short transition time, the other variables distribution adjusts to the actual angular momentum."
],
[
"1D computations",
"In the general relativistic framework, the local sound speed relative to the fluid is given by $c_s ^{2} = \\gamma p \\left(\\rho +\\frac{ \\gamma p}{\\gamma -1}\\right)^{-1},$ and the radial Mach number is obtained as $\\mathfrak {M} = - u_{\\rm BL}^{\\tt r}/c_{\\rm s}$ .",
"The sonic points are the points, where the flow smoothly passes from subsonic to supersonic motion, that is $\\mathfrak {M} = 1$ , and its value increases inwards.",
"The shock front is at the place, where $\\mathfrak {M}$ discontinuously changes from $\\mathfrak {M}>1$ to $\\mathfrak {M}<1$ along the flow (with decreasing $r$ ).",
"The resulting profile of the Mach number $\\mathfrak {M}$ for the converged stationary state is shown in Fig.",
"REF for four different values of $\\mathcal {E}$ .",
"The inferred shock and outer sonic point locations are given in Table REF .",
"Table: Location of shock front and the outer sonic point of stationary solutions for different values of ℰ\\mathcal {E}.",
"The values in geometrized units ([M]) are inferred from the solution at the end of the simulation, at t f =10 6 t_f=10^6M.For higher energy, the outer sonic point is closer to the black hole unlike the shock position, which is located farther, and the outer supersonic region of the flow is thus shrinking.",
"The strength of the shock is anti-correlated with the energy (and with the location of the shock), so that for increasing energy the ratio of the post-shock density to the preshock density ($\\mathcal {R}$ ) is decreasing (see Table REF and panel b) in Figure REF ).",
"In comparison with the pseudo-Newtonian analytical estimate, which put the shock position for $\\mathcal {E}=0.0001$ at $r_s^{\\rm PW}(0.0001) = 21M$ , the GR computation tends to put the shock farther from the black hole.",
"On the other hand, the minimal stable shock position is very similar, because in GR computations the shock exists for slightly lower values of $\\lambda $ , which can be seen on Fig.",
"REF .",
"Here the dependence of the shock front position on angular momentum is shown for both the pseudo-Newtonian and GR computations.",
"The radial extend of possible shock existence agrees very well between PW and GR results, however for the same value of angular momentum the GR shock is located farther from the black hole.",
"Similarly like in the case of PW simulations, which we presented in [59], also in the GR case computed with HARMPI we have found oscillations of the shock front for higher angular momentum, which causes also the oscillation of the mass accretion rate through the inner boundary.",
"In Fig.",
"REF in the top panel we show the minimal and maximal shock positions during the simulation for the oscillating cases.",
"In the bottom panel, we show the amplitude of the oscillations of the mass accretion rate, which is computed as the ratio of the difference between the maximal and minimal accretion rate to its mean value $\\mathcal {A} = ({\\rm max}(\\dot{M}) - {\\rm min}(\\dot{M}))/\\bar{\\dot{M}}$ .",
"This ampplitude and the corresponding frequency is presented for the oscillating models.",
"Table: Location of shock front, if existent, with a=0.3,γ=4/3a=0.3,\\gamma =4/3 for different values of λ\\lambda for ℰ 1 =0.002\\mathcal {E}_1=0.002 and ℰ 2 =0.0000033\\mathcal {E}_2=0.0000033.",
"The values in geometrized units ([M]) are inferred from the solution at the end of the simulation.The general relativistic semi-analytical study of the shock solutions in Kerr metric was given in [14], however those authors considered the disc in hydrostatic equilibrium with vertically averaged quantities.",
"The shape of the different regions in the parameter space is given in [14], Fig.",
"1 for $a=0.3$ .",
"On Fig.",
"3 in that paper, the authors show the parameter space of possible shock existence for $\\tilde{\\mathcal {E}}=1.0000033, \\gamma =4/3, a=0.3$ , where $\\tilde{\\mathcal {E}}$ is the total specific energy and corresponds to $\\tilde{\\mathcal {E}} = \\mathcal {E}+1$ .",
"To compare their solutions with our results, we performed a set of simulations with $a=0.3$ for $\\mathcal {E}_1=0.002$ and $\\mathcal {E}_2=0.0000033$ .",
"The results are summarised in Table REF .",
"For $\\mathcal {E}_1$ [14] predicts that the shock exists in the subset of the region A (in particular in the region A$_{\\rm S}$ , which is not shown in the figure), which gives approximately the range 2.8 M $< \\lambda <$ 3.08 M. Our simulations show the shock existence for higher values of angular momentum, in particular the shock front is accreted up to $\\lambda =3.2$ M and the stationary shock exists for $\\lambda \\in [3.21 \\rm {M},3.42 \\rm {M}]$ .",
"For $\\mathcal {E}_2$ , those authors predict the shock existence for $\\lambda > 3.239$ M. We have found the stable shock solution for $\\lambda =3.25$ M, while for $\\lambda =3.245$ M the shock is accreted.",
"When the angular momentum is increased, the oscillations develop, for $\\lambda =3.4$ M their frequency $f\\sim 2.9\\cdot 10^{-4}$ M$^{-1}$ with the amplitude $\\mathcal {A} = 1.45$ .",
"Because a different configuration is considered in the two papers (disc in vertical hydrostatic equilibrium versus quasi-spherical flow), some differences are expected and they are more prominent for higher energy of the flow.",
"Quite recently, [61] studied the low angular momentum flows with shocks in the Kerr spacetime in several different geometries with different physical conditions at the shock front.",
"They included also the case of the conical flow with energy-preserving shock, which is close to our scenario.",
"Figure 2 of [61] shows, that indeed the multicritical region for the quasi-spherical flow exists for higher values of angular momentum than for the flow in vertical hydrostatic equilibrium.",
"The quantitative comparison of our results with this paper will be given in the future work."
],
[
"Properties of the flow for a time dependent angular momentum",
"Further, we repeated the PW computations done previously with the code ZEUS [59], regarding the hysteresis loop connected with the shock existence with changing angular momentum of the incoming matter.",
"In this case we prescribe the angular momentum of the matter coming through the outer boundary according to the time dependent relation: $\\lambda ^{\\rm eq} (t) = \\lambda ^{\\rm eq} (0) - A\\sin (t/P),$ where $A$ is the amplitude and $P$ is the period of the perturbation.",
"If we choose $A$ and $P$ such that the angular momentum crosses the boundary of the multicritical region from below and from above, we observe the creation and disappearance of the shock front, similarly like it was in the PW case.",
"The comparison of the shock front movement for three different models is given in Figure REF .",
"Model A1 with $\\lambda ^{\\rm eq} (0)=3.67$ M, $A=0.16$ M, $P=2\\cdot 10^6$ M does not exceed the boundary of the shock existence interval from neither side.",
"The shock is moving in accordance with the changing angular momentum to and from the black hole, spanning the region (16.6 M, 672.5 M).",
"Both sonic points exist persistently during the evolution.",
"Model A2 with $\\lambda ^{\\rm eq} (0)=3.655$ M, $A=0.2$ M, $P=2\\cdot 10^6$ M crosses the shock existence boundary on both sides.",
"As a consequence, the shock front is being accreted very quickly after it reaches the minimal stable shock position.",
"The time scale of this event is given by the advection time from the minimal stable shock position and does not depend on the perturbation parameters $A$ and $P$ .",
"From that moment the inner sonic point vanishes and the flow follows the outer Bondi-like branch of solution (for which only the outer sonic point exists) until the time, when the angular momentum increases such that the outer solution no longer exists.",
"At that point, the shock is formed at the position of the inner sonic point and it is moving very fast outward, where it merges with the outer sonic point, so that the type of accretion flow with the inner sonic point only is established.",
"Later, when the angular momentum of the flow is decreasing again, the shock forms at the outer sonic point and it moves slowly towards the black hole.",
"The third example (model A3 with $\\lambda ^{\\rm eq} (0)=3.68$ M, $A=0.18$ M, $P=2\\cdot 10^6$ M) shows the case, where only the upper boundary is crossed.",
"Here we have the shock moving slowly towards and away from the centre, where it merges with the outer sonic point.",
"For some time only the accretion with the inner sonic point exists and then the shock appears again.",
"In this case, the timescale of such changes and the velocity of the shock front is uniquely given by the parameters $A$ and $P$ .",
"Figure: Model B2: Bondi initial data with ℰ=0.0025,λ eq =3.8\\mathcal {E}=0.0025, \\lambda ^{\\rm eq}=3.8M.",
"The shock is expanding (the first snapshot at t=10 3 t=10^3M) until it merges with the outer sonic surface (second snapshot at t=2·10 4 t=2 \\cdot 10^4M).",
"Note the different spatial range of the second snapshot.The case, in which only the lower boundary is crossed, is not shown, because in this case the shock front is accreted during the first cycle and never appears again.",
"Hence, under certain circumstances, the standing shock can appear in the low angular momentum flow during accretion.",
"Then it can either (i) stay at certain position, (ii) move slowly downwards or upwards in the accretion flow with velocity given by the rate of change of angular momentum given by parameters $A$ and $P$ in our model, (iii) be accreted quickly from the the minimal stable shock position (close to the black hole), or (iv) be formed close to the black hole and move quickly towards the outer sonic point.",
"In the last two cases, the velocity of the shock front does not depend on the perturbation parameters $A$ and $P$ , but it is given by the properties of the medium.",
"We will study this issue in the next section with 2D simulations; in general, the velocity of the shock front is a few times lower than the sound speed in the preshock medium.",
"Figure: Model B2: expanding shock with Bondi initial data.",
"We track the position of the transonic points (i.e.",
"(𝔐[i]-1)(𝔐[i+1]-1)<1(\\mathfrak {M}[i] -1)(\\mathfrak {M}[i+1] -1) <1) in the equatorial plane (ii is the index of the radial coordinate on the grid) and we labeled the sonic points ( 𝔐[i]>𝔐[i+1]\\mathfrak {M}[i]>\\mathfrak {M}[i+1])by yellow points and the shock position ( 𝔐[i]<𝔐[i+1]\\mathfrak {M}[i]<\\mathfrak {M}[i+1]) by brown points.Figure: Behaviour of the shock front for different values of λ eq \\lambda ^{\\rm eq} with ℰ=0.0025\\mathcal {E}=0.0025.",
"Panel a) shows the position of the shock front, panel b) shows its velocity v s v_s and panel c) displays the ratio of the preshock medium sound speed c s out c^{\\rm out}_s to the shock front velocity v s v_s during the evolution.After confirming the general outcomes of our previous PW study with full GR 1D simulations, we now continue with simulations in higher dimensions.",
"Because our initial data are rotationally symmetric (does not depend on $\\phi $ ), is it possible to evolve only a slice spanning ($r,\\theta $ ), and with constant $\\phi $ , thus assuming that the system will remain rotationally symmetric during the evolution.",
"In order to justify to what extend and under which conditions this assumption is valid the comparison with 3D computation was needed.",
"We will comment on that in Section REF .",
"However, within the 2.5-D approach, the $\\phi $ component of the four-velocity is considered.",
"We prescribe this component in Boyer-Lindquist coordinates according to relations given by Eqs.",
"(REF ) and ().",
"We chose several exemplary simulations and we show their results in the form of snapshots.",
"Every snapshot contains four panels with the slices of $\\mathfrak {M}$ and its equatorial profile, $\\rho $ in arbitrary units, and $\\lambda $ in geometrized units, and it is labelled by the time $t$ in geometrized units, where $M$ is the mass of central black hole.",
"The axes show the position in geometrized units.",
"The red colour in the slice of radial Mach number corresponds to the supersonic motion, blue regions indicate the subsonic accretion.",
"The shock front is located at the place, where the abrupt change from supersonic to subsonic motion occurs, hence it is represented with the white curve separating red region farther from the centre and blue region closer to the centre.",
"The sonic curves are also white, but they lie between the blue region farther and red region closer to the centre.",
"On top of the radial Mach number map the velocity streamlines are plotted in blue.",
"These streamlines are computed from the velocity field at the given instant of time, hence they do not represent the actual history of a certain fluid parcel.",
"Figure: The same as in Fig.",
"for different values of ℰ\\mathcal {E} with λ eq =3.85\\lambda ^{\\rm eq}=3.85 M."
],
[
"Bondi solution equipped with angular momentum",
"At the beginning we check, that for pure Bondi solution with $\\lambda =0$ we get the stationary solution of the flow with the properties consistent with the analytical solution.",
"After that we start to study the slowly rotating flows.",
"For the first simulation, we pick the parameters such that $\\lambda ^{\\rm eq}$ belongs into the multicritical region (the region in the parameter space, where both the inner and outer sonic points exist in the 1D model), but it is very close to its upper bound, in particular $\\mathcal {E}=0.0025~M$ , $\\lambda ^{\\rm eq} = 3.6~M$ , and $\\gamma =4/3$ .",
"In that case, even when the shock solution is possible, it is not expected to appear, because the initial configuration is closer to the outer branch (there is no inner sonic point in the Bondi initial data).",
"In other words, we expect, that the rotation of the gas affects the profile of the Mach number in the innermost region in such a way, that it gets very close to 1, but does not touch it.",
"In Fig.",
"REF , the initial conditions at $t=0M$ are shown and the final state at $t_f=10^4M$ of the gas is depicted in Fig REF .",
"At the later time, the simulation is already relaxed to the stationary state, which resembles the outer branch.",
"It is interesting to note, that for $t=t_f$ the Mach number actually crosses the $\\mathfrak {M}=1$ line, however only on a very short radial range.",
"We would expect, that at the moment, when the inner sonic point appears, the shock creates and expands.",
"The reason, why it is not so in this simulation, is that the angular momentum is very close to the upper bound of the multicritical region, which is a boundary between the two distinct types of evolution.",
"In such case, the numerical evolution depends also on the resolution of the grid and other numerical settings.",
"In other words, if the parameters are close to such boundaries, then the simulations with different resolution can lead to a different type of evolution (i.e., the shock creates or not), hence it is difficult to find the exact critical value of the parameter.",
"This confirms our previous observation [59], that in the multicritical region the evolution of the flow can tend to either the outer “Bondi-like” branch or to the shock solution, and the choice between these two possibilities is given by the initial conditions.",
"In particular, the profile of the Mach number in the innermost region and the presence of the inner sonic point is crucial for the shock development.",
"Figure: Model C4: Initial data with a shock with ℰ=0.0005,λ eq =3.72\\mathcal {E}=0.0005, \\lambda ^{\\rm eq}=3.72M.",
"The shock bubble develops oscillations and changes size quasiperiodically, but does not acretes, nor does it expand outside r out r_{\\rm out}.In [56] we showed a similar computation with $\\lambda ^{\\rm eq}=3.79M, \\mathcal {E}=0.001$ , which is also close to the upper bound of the multicritical region.",
"Those computations were done in 3D with a different GR code, Einstein toolkithttps://einsteintoolkit.org/ [33].",
"The qualitative and quantitative agreement with the 1D solution is discussed there.",
"However the fact, that two very different codes working on different kind of grids (spherical grid logarithmic in radius versus Cartesian-like grid with the mesh refinement) show similar results is a good support for our results.",
"Last example is the case, when $\\lambda ^{\\rm eq}$ is above the multicritical region, hence the outer type of solution is not possible.",
"Therefore, the prescribed initial conditions are not close to a physical solution and the shock bubble has to appear.",
"However, the position of the shock front is not stable and the shock is expanding until it meets the outer sonic point and the distant supersonic region dissolves, yielding only the inner type of accretion.",
"Two snapshots of the evolution in times $10^3M$ and $2\\cdot 10^4M$ are given in Fig.",
"REF .",
"On the first set of panels, the emergent shock bubble is seen, which is the expanding subsonic and more dense region.",
"Even though the shock front is thought to be very thin, in the numerical simulation it spans across several zones, which can be seen on the equatorial profile of $\\mathfrak {M}$ .",
"In the last snapshot, the outer supersonic region is already dissolved and only a small supersonic region very close to the black hole can be found.",
"The time dependence of the shock and sonic point positions in the equatorial plane is shown in Fig.",
"REF , where it can be seen, that the shock meets the outer sonic point at about $t\\sim 15000~M$ , which corresponds to about 0.75 s in case of a typical microquasar with $M=10~M_\\odot $ .",
"This is way too short in comparison with the observational data concerning the quasiperiodic oscillations, which change frequency during few weeks.",
"Hence, if we want to use an oscillatory shock front model to explain the observed low-frequency QPOs, we need to obtain a solution with long lasting shock front oscillating around the slowly changing mean position, and not the fast expanding shock bubble like was observed in this case.",
"The first step is to find an existing stationary solution with a standing shock.",
"As we already mentioned, such solution is not expected to occur, if we start the evolution with initial conditions close to the Bondi solution.",
"We can be, however, interested in the dependence of the shock propagation on the properties of the gas.",
"On that account, we performed two sets of simulations, where we changed the angular momentum and the energy of the flow such that they are above the multicritical solution, and we followed the expansion of the shock front until it met the outer sonic point.",
"In Fig.",
"REF we show the behaviour of the shock front during the evolution for several different values of $\\lambda ^{\\rm eq}$ .",
"For $\\lambda ^{\\rm eq}=3.65$ M, just above the multicritical region, the growth of the shock bubble is slow with a long transient period, during which it is unclear, if the bubble converges to the stationary state, or it rather expands outwards.",
"With increasing $\\lambda ^{\\rm eq}$ the bubble expands faster.",
"The velocity of the shock front ranges between $0.005c$ up to $0.4c$ for the highest angular momentum.",
"Panel c) shows the ratio of the preshock medium sound speed $c^{\\rm out}_s$ to the shock front velocity $v_s$ .",
"Except for the highest $\\lambda ^{\\rm eq}=10$ M, the shock front expands by a velocity, which is a few times lower than the corresponding preshock sound speed, for the lowest $\\lambda ^{\\rm eq}=3.65$ M the ratio $c^{\\rm out}_s/v_s$ reaches the value up to 6.",
"In Fig.",
"REF we present similar results obtained for different values of $\\mathcal {E}$ .",
"Here, the position of the outer sonic point differs significantly for different $\\mathcal {E}$ , hence also the time for the shock front to reach the outer sonic point varies considerably.",
"The plots are therefore given in logarithmic scale of time.",
"Figure: Model C4: The shock and sonic point position and the corresponding mass accretion rate through the inner boundary.Figure: Model H3: Initial data with a shock and constant angular momentum distribution (ℰ=0.0005,λ=3.65\\mathcal {E}=0.0005, \\lambda =3.65M) in the cone with the half-angle θ c =π/4\\theta _c=\\pi /4.The velocity of the shock front decreases with the decreasing energy, and the same holds for its ratio to the sound speed of the preshock medium.",
"The trend is similar as for decreasing angular momentum.",
"The reason is, that for decreasing energy, the multicritical region exists for higher value of angular momentum, as we have seen earlier.",
"Hence, when we decrease the energy and keep the angular momentum constant, we are approaching the multicritical region in the parameter space.",
"That is well documented on the case with the lowest energy $\\mathcal {E}=0.0005$ (the purple lines in Fig.",
"REF ), which lies very close the multicritical region boundary and so the shock bubble waits for a very long time before it starts to expand.",
"Therefore, we conclude, that the shock front velocity depends on the distance of our chosen parameters ($\\lambda , \\mathcal {E}$ ) from the multicritical region in the parameter space and that it is typically a few times lower than the sound speed in the preshock medium."
],
[
"Shock solution",
"Because we want to find out, if there are any stationary or oscillating solutions with shocks also in 2D, similarly as in 1D, we have to prescribe the initial conditions, which are closer to the shock solution branch than the Bondi-like solution.",
"These initial conditions are described in Section REF .",
"Inspired by the range of shock solution in the PW case, we performed a set of simulations with changing angular momentum $\\lambda ^{\\rm eq} \\in [3.52M,3.7M]$ , while keeping $\\mathcal {E}=0.0025$ and $\\gamma =4/3$ .",
"(Note that the range in 1D paper was defined for $\\mathcal {E}=0.0001$ , not $0.0025$ .",
"Here, the range is such, that for the lowest $\\lambda $ the shock bubble accretes and for the higher values it expands, so it covers the whole interesting interval.)",
"These simulations showed, that for lower angular momentum the stationary state is obtained, while for higher angular momentum the shock bubble again expands and merges with the outer sonic surface, which is located around 280M.",
"The details of these computations are given in [57].",
"When we choose lower value of energy, the outer sonic surface is located further, so that there is more space, where the shock existence is possible.",
"We performed several simulations with $\\mathcal {E}=0.0005$ and $\\mathcal {E}=0.0001$ , for which $r_{\\rm out}=1475~M$ and $r_{\\rm out}=7486~M$ , respectively.",
"Figure: Model H3: The shock and sonic point position and the corresponding mass accretion rate through the inner boundary.For higher values of angular momentum we found the shock bubble unstable - the eddies emerge in the flow, the bubble is growing for some time, after which a quick accretion occurs accompanied by shrinking of the bubble, which is also oscillating in vertical direction.",
"However, it does not expand or accrete completely.",
"Several snapshots of the evolution are given in Fig.",
"REF , where the shock bubble has different shape and size at different times and the asymmetry with respect to equator can be seen.",
"The time dependence of the shock front position in the equatorial plane is given in Fig.",
"REF .",
"Table: Summary of computed 2D models.",
"All models have γ=4/3\\gamma =4/3.",
"Initial conditions (IC) are described in Sections (“Bondi”), (“1D+sph”), (“1D+cone”) or (“1D+spin”).",
"In case of 1D+cone simulations θ c =π 4\\theta _c=\\frac{\\pi }{4}.",
"The shock behaviour is labeled as “no” for solution without a shock, “EX” for solution with expanding shock, which merges with the outer sonic point, “OS” for solutions with oscillating shock front and “AC” for solutions, in which the shock front is accreted almost immediately.",
"The next three columns show the range, in which the shock is moving, the average position of the shock and the frequency of the oscillation and flares in the mass accretion rate through the inner boundary, if there are any.",
"The last column contains the reference to the corresponding Figures.Figure: Model F2: moderately spining black hole (ℰ=0.0005,a=0.8,λ eq =2.8\\mathcal {E}=0.0005, a=0.8, \\lambda ^{\\rm eq}=2.8M) with oscillating shock.This repetitive process causes also flaring in the mass accretion rate through the inner boundary, see Fig REF .",
"For $10~M_\\odot $ black hole, $t=10^6~M$ corresponds to approx.",
"50 s, hence the flares occur on a similar time scale as in some of the flaring states of microquasars, e.g.",
"the heartbeat state of GRS 1915+105 [5] or IGR J17091-3624 [4].",
"Moreover, we have shown, that the sources in this state show the evidence of nonlinear mechanism behind the emission of the hard component, which corresponds to the low angular momentum flow [58], [60].",
"However, this fact is only an indirect evidence for the presence of the shock, because there are other possible explanations of the flares, e.g.",
"they can be a result of the radiation pressure instability in the disc [28], [25].",
"Another type of simulations are those with different distribution of angular momentum (see Section REF ).",
"Three snapshots of such evolution are given in Fig.",
"REF and the corresponding shock position and mass accretion rate is given in Fig.",
"REF .",
"We found qualitatively similar behavior for different values of parameters.",
"The results are summarized in Table REF .",
"For $\\mathcal {E}=0.0005$ we identified the interval of angular momentum values for which there is oscillating shock to be $[3.59{\\rm M},\\,3.78{\\rm M}]$ (models C2-C5).",
"For $\\mathcal {E}=0.0001$ the interval is $\\lambda \\in [3.72{\\rm M},\\,3.86{\\rm M}]$ (models D2-D5).",
"In both series of simulations it is clearly visible, that increasing the value of angular momentum within the interval causes increase of the amplitude of shock oscillations.",
"So oscillations appear sharpest just before the upper limit of the angular momentum range and the oscillation frequency peaks in the spectrum are clearly visible in those cases.",
"We can compare the series of simulations C1-C5 with H1-H4 which have the same values of parameters, but differ in distribution of angular momentum (modulation by $\\sin ^2\\theta $ versus constant angular momentum in a cone).",
"It is clear that the results are very similar: in both cases there is a range of angular momentum in which the shock oscillating with comparable frequencies is present, with amplitude of oscillations increasing with angular momentum value.",
"This shows, that the main conclusions from our simulations are resistant to changes in spatial distribution of rotation profile."
],
[
"Rotating black hole",
"We chose three values of the spin $a$ , which can represent quite well the estimated range of spins of the know microquasars, in particular $a=0.3$ (which could be a representative value for XTE J1550-564, H1743-322, LMC X-3, A0620-00), $a=0.8$ (M33 X-7, 4U 1543-47, GRO J1655-40) and $a=0.95$ (Cyg X-1, LMC X-1, GRS 1915+105).",
"The estimated values of the spin are taken from [34].",
"For $a=0.3$ and $\\lambda ^{\\rm eq}_g \\in [3.35~\\mathrm {M},3.48~\\mathrm {M}]$ we observed a long lasting oscillating shock front.",
"For $\\lambda ^{\\rm eq}_g=3.34~\\mathrm {M}$ (and lower values) the shock is accreted, and for $\\lambda ^{\\rm eq}_g=3.49~\\mathrm {M}$ (and higher values) the shock surface expands.",
"The Table REF contains details about five selected simulations of $a=0.3$ scenarios: values on both sides of left and right boundary of interval and one in the middle of the interval of $\\lambda ^{\\rm eq}_g$ values corresponding to stable shock existence.",
"Figure: Model F2: moderately spining black hole (ℰ=0.0005,a=0.8,λ eq =2.8\\mathcal {E}=0.0005, a=0.8, \\lambda ^{\\rm eq}=2.8M) with oscillating shock.Next in the Table REF the results of several simulations with $a=0.8$ are shown.",
"In this case the interval of $\\lambda ^{\\rm eq}_g$ for which there are solutions with oscillating shock appears in the range of lower values than for $a=0.3$ .",
"For $\\lambda ^{\\rm eq}_g = 2.7$ M (model F1), the shock is accreted, for $\\lambda ^{\\rm eq}_g = 2.8$ M (model F2) the accretion flow contains a long lasting oscillating shock front, and for $\\lambda ^{\\rm eq}_g = 2.9$ M (model F3) the shock bubble is expanding.",
"The qualitative behavior of the shock front is very similar like in the Schwarzschild case, only the exact values of parameters (mainly the angular momentum of the flow) differs.",
"Again increase of the value of angular momentum within the oscillating shock interval corresponds to increase of the amplitude of oscillations.",
"Series of models E2-E4 is very similar in this regard to series C2-C5 and D2-D5.",
"For higher values of spin we found the computations to be more sensitive on the resolution, mainly in the radial direction close to the black hole.",
"If the resolution is not sufficient, then the the shock bubble can accrete even after a long oscillatory evolution.",
"However when we increase the resolution for the initial conditions, the shock persists in the evolution, even when it comes very close to the black hole.",
"Therefore we increased the radial resolution for $a=0.95$ to $N_{\\rm r}=576$ .",
"Even with this high resolution, the shock bubbles in simulations with $\\lambda ^{\\rm eq}=2.42$ M and $\\lambda ^{\\rm eq}=2.45$ M are accreted after long evolution with repeated shock front oscillations.",
"We conclude that the qualitative results obtained in the non-spinning case can be generalized for spinning black holes.",
"What can be clearly seen from the simulations, is that the behavior similar to the case of nonrotating BH is observed for much lower values of the angular momentum, so the spin of the black hole “adds” to the rotation of the gas.",
"Increasing the value of the spin of black hole decreases both limiting values of angular momentum of accreted gas and leads to a narrower interval in which the oscillating shock solutions are observed.",
"Figure: Model G2: rapidly spinning black hole (ℰ=0.0005,a=0.95,λ eq =2.42\\mathcal {E}=0.0005, a=0.95, \\lambda ^{\\rm eq}=2.42M) with oscillating shock.We performed two test runs with the initial conditions described in Section REF with the parameters $\\epsilon =0.0005, \\lambda ^{\\rm eq}=3.65, a=0$ in full 3D, and with the resolution $N_{\\rm r}$ x $N_{\\theta }$ x $N_{\\phi }$ equal to 384 x 192 x 64, and 256 x 128 x 96.",
"The simulations were performed on the Cray supercomupter cluster, typically using 16 nodes, and the message-passing interface supplemented with the hyperthreading technique was used, similarly as in [27].",
"The code is supposed to conserve the axi-symmetry of the initial state, which we confirm.",
"Because no non-axisymmetric modes appeared, the solution is the same in each $\\phi $ -slice during the evolution (see Fig.",
"REF ).",
"Figure: Model I: Density distribution in the three-dimensional simulation at time 28000M, within the innermost 50 gravitational radii from the black hole.",
"Resolution of this run is 384 x 192 x 64.",
"The colour scale and threshold is adopted to show only the densest parts of the flow.We were able evolved the system only up to $t_f = 34100$ M, and $t_f=24400$ M, for these two runs respectively, which is significantly shorter then in the 2D case.",
"Hence, we cannot directly compare the long term evolution of the flow.",
"However, the main features of the shock bubble evolution, which we observed in 2D case, appears also in 3D simulation.",
"Mainly, the shock bubble has similar shape, as can be seen in Fig REF , and we observe similar oscillations of the bubble and of the mass accretion rate (Fig.",
"REF )."
],
[
"Conclusions",
"In this paper we presented an extensive numerical study of the pseudo-spherical accretion flows with low angular momentum.",
"The simulations were performed in the general relativistic framework on the Kerr background metric with the GR MHD code HARMPI in one, two and three dimensions.",
"As a first step, we provided set of 1D computations, which we compared to our earlier results obtained within the pseudo-Newtonian approach with the code ZEUS and published in [59].",
"We confirm the qualitative properties of those simulations, which are especially the oscillation of the shock front for higher values of angular momentum and the possibility of the hysteresis effect, when the parameters of the flow are changing in time.",
"The oscillations of the shock front are connected with small oscillations of the value of angular momentum downward from the shock front, hence, they may be triggered by numerical errors at the shock front.",
"The amplitude of the oscillations reaches the maximum for intermediate shock positions and ceases again for very high shock position/angular momentum.",
"Figure: Model I: 3D simulation (ϵ=0.0005,λ eq =3.65,a=0.0\\epsilon =0.0005,\\lambda ^{\\rm eq}=3.65,a=0.0) with oscillating shock.If the oscillations are physical, their frequency is in good agreement with the observed low frequency quasi-periodic oscillations seen in several black hole binary systems.",
"Those are shifting in the range from hundreds of mHz up to few tens of Hz on the time scale of weeks (e.g.",
"GRS1915+105 [36], XTE J1550-564 [12], GRO J1655-40 [49] or GX 339-4 [45]).",
"For a fiducial mass of the microquasar equal to 10M$_\\odot $ this corresponds to the frequencies between $10^{-6}$ M$^{-1}$ up to $10^{-3}$ M$^{-1}$ in geometrized unitsThe time unit $1M$ equals to the time, which light needs to travel the half of the Schwarzschild radius of a black hole of mass M, hence for one solar mass $[t]=1M_\\odot =\\frac{GM_\\odot }{c^3}=4.9255\\cdot 10^{-6}s$ ., which is the same range as our observed values (see Fig.",
"REF , bottom panel for 1D results and Table REF for 2D simulations).",
"Hence, the change of the observed frequency during the onset and decline of the outburst is possibly connected with the change of angular momentum of the incoming matter (alternative explanation gives e.g.",
"[40], who consider rather the change of viscosity parameter $\\alpha $ ).",
"Such change can be either periodic and connected with the orbital motion of the companion, or caused by different conditions during the release of the matter.",
"[63] proposed the scenario of the outburst of XTE J1550-564, in which the low angular momentum component of the accretion flow is released from the magnetic trap inside the Roche lobe of the black hole, kept by the magnetic field of the active companion.",
"Depending on the distribution of angular momentum inside the trap and on the mechanism of breaking the magnetic confinement the angular momentum of the incoming matter from this region can slightly vary with time.",
"Later the angular momentum of the low angular component could be affected by the interaction with the slowly inward propagating Keplerian disc.",
"Figure: Model I: 3D simulation (ϵ=0.0005,λ eq =3.65,a=0.0\\epsilon =0.0005,\\lambda ^{\\rm eq}=3.65,a=0.0) with oscillating shock.To simulate such scenario, we have studied the case, when the angular momentum of the incoming matter changes periodically with time.",
"As was shown on Fig.",
"REF , the behaviour of the shock front depends on whether the value of angular momentum crosses one or both of the values $\\lambda _{\\tt min}^{cr},\\lambda _{\\tt max}^{cr}$ .",
"When the respective boundary is not crossed, the shock is moving slowly through the permitted region and its speed is determined by the parameters of the perturbation of the angular momentum.",
"When the angular momentum is in the corresponding range, the oscillations of the shock front develops.",
"When one of the the critical values is reached, the abrupt change of the flow geometry happens, while the flow transforms from one type of solution into another.",
"This transformation is achieved via the shock propagation, which is either accreted or expanding.",
"The velocity of the shock front agrees within the order of magnitude with the sound speed of the preshock medium, when the angular momentum is just slightly higher than $\\lambda _{\\tt max}^{cr}$ .",
"If the angular momentum is considerably higher, the formation and propagation of the shock can be even higher than the sound speed in the postshock medium (see Figs.",
"REF and REF ).",
"However, in nature we expect the first case to realize, because such situation can appear only if the angular momentum is increasing in the flow, which is in the Bondi-like configuration, hence it crosses smoothly through $\\lambda _{\\tt max}^{cr}$ .",
"When extending our results to higher dimensions, the freedom of the dependence of angular momentum on the angle $\\theta $ arises.",
"We have chosen two different configurations, described in Section REF .",
"The comparison of the corresponding models 1D+sph and 1D+cone in Table REF shows, that in the latter case, the shock is placed at larger distances.",
"That can be understood as the consequence of the fact, that thanks to the relations given by Eqs.",
"(REF ) and (REF ) larger amount of matter posses the maximal value of angular momentum.",
"However, the main features of the solution, including the existence of the range of angular momentum enabling the long term shock presence, the shape of the resulting shock bubble and its oscillations, are similar in both cases.",
"The biggest difference between the two configurations is that in case of constant angular momentum in a cone, the peaks in mass accretion rate are not so prominent as in the case of angular momentum scaled by $\\sin ^2\\theta $ .",
"Hence, we conclude, that the physical processes in the low angular momentum accretion flows do not qualitatively depend on the exact distribution of angular momentum, but the observable consequences (e.g.",
"the presence of prominent peaks and their amplitude) may be influenced by the geometry.",
"Similar conclusion holds also for the change of spin of the black hole.",
"For all three considered values of spin of the black hole ($0.3$ , $0.8$ and $0.95$ ) we have found a range of values of angular momentum of falling matter in which the long lasting oscillations of shock surface was observed.",
"We have also observed, that the shock existence interval of angular momentum depends on the spin of the black hole: the higher the spin value, the lower both limiting angular momentum values between which the oscillating shock exists and the narrower is the corresponding interval.",
"Therefore, for rapidly spinning black holes, even a small change of the angular momentum of the incoming matter leads to significant changes in the flow itself (the existence, position and oscillations of the shock front) and also in the timing properties of the outgoing radiation (which we assume to be related to the accretion rate).",
"Abrupt emergence, expansion or accretion of the shock which is connected with the crossing of the boundary of the shock existence region in parameter space and which leads to significant changes in the accretion rate, are more likely in the accretion flows around rapidly spinning black holes.",
"Simulations which we performed cover large variety of configurations: we considered different values of the spin of the central black hole, energy and angular momentum of the accreted gas, and even the distribution of the angular momentum of the matter.",
"Keeping all the other variables constant we have found the range of angular momentum in which there exist oscillating shock solutions in all scenarios under consideration.",
"Hence the oscillating regime seems to be intrinsic to the low angular momentum accretion flows.",
"This finding is supported by [24] who also found an oscillating shock front in their hydrodynamical simulations in the pseudo-Newtonian framework.",
"Our simulations in two and three dimensions show, that for such parameters the oscillating shock front is long-lasting.",
"That is an important ingredient for the POS model to be able to explain the QPO frequency change during outbursts of microquasars.",
"However, the duration of our simulations corresponds to several tens of second for typical microquasar with $M=10 M_\\odot $ , which is still short in comparison with the time scale of the QPOs frequency change (weeks).",
"Moreover, we did not address this question from the point of view of an analytical stability analysis.",
"Such analysis was provided by [39] for the spherical accretion onto non-rotating black hole and quite recently by [6] for low angular momentum flow with standing shocks, who also report the stability of the solution.",
"The dependence of the shock existence interval and consequently the position of the shock on the rotation of the black hole could be the probe of the black hole spin.",
"However, there is a degeneracy between the spin and the angular momentum of the accreting matter, which itself is mostly unknown and hard to measure.",
"Hence, the constraints on the spin from the oscillating shock front model explaining QPOs can be posed only when there will be available better observations of the innermost accretion region or better models predicting the angular momentum of the LAF component."
],
[
"Acknowledgements",
"We acknowledge support from Interdisciplinary Center for Computational Modeling of the Warsaw University (grant GB66-3) and Polish National Science Center (2012/05/E/ST9/03914).",
"PS is supported from Grant No.",
"GACR-17-06962Y."
]
] | 1709.01824 |
[
[
"Theory of excitation of Rydberg polarons in an atomic quantum gas"
],
[
"Abstract We present a quantum many-body description of the excitation spectrum of Rydberg polarons in a Bose gas.",
"The many-body Hamiltonian is solved with functional determinant theory, and we extend this technique to describe Rydberg polarons of finite mass.",
"Mean-field and classical descriptions of the spectrum are derived as approximations of the many-body theory.",
"The various approaches are applied to experimental observations of polarons created by excitation of Rydberg atoms in a strontium Bose-Einstein condensate."
],
[
"Introduction",
"When an impurity is immersed in a polarizable medium, the collective response of the medium can form quasi-particles, labelled as polarons, which describe the dressing of the impurity by excitations of the background medium.",
"Polarons play important roles in the conduction in ionic crystals and polar semiconductors [1], spin-current transport in organic semiconductors [2], dynamics of molecules in superfluid helium nanodroplets [3], [4], [5], and collective excitations in strongly interacting fermionic and bosonic ultracold gases [6], [7], [8].",
"In this paper, which accompanies the publication heralding the observation of Rydberg Bose polarons [9], we present details of calculations and the interpretation of this observation as a new class of Bose polarons, formed through excitation of Sr($5sns$ $^3S_1$ ) Rydberg atoms in a strontium Bose-Einstein condensate (BEC).",
"We begin with a general outline of different polaron Hamiltonians, and construct the Rydberg polaron Hamiltonian used in this work.",
"The spectral response function in the linear response limit is derived and the many-body mean field shift of the spectral response is obtained.",
"The mean field theory particularly fails to describe the limits of small and large detuning, where the detailed description of quantum few- and many-body processes is particularly relevant.",
"These processes are fully accounted for by the bosonic functional determinant approach (FDA) [10] that solves an extended Fröhlich Hamiltonian for an impurity in a Bose gas.",
"In the frequency domain, the FDA predicts a gaussian shape for the intrinsic spectrum, which is a hallmark of Rydberg polarons.",
"A classical Monte Carlo simulation [11] which reproduces the background spectral shape is shown to miss spectral features arising from quantization of bound states.",
"Agreement between experimental results and FDA theory for both the observed few-body molecular spectra and the many-body polaronic states is excellent.",
"In the companion paper [9], we provide experimental evidence for the observation of polarons created by excitation of Rydberg atoms in a strontium Bose-Einstein condensate, with an emphasis on the determination of the excitation spectrum in the absence of density inhomogeneity.",
"Here we provide a detailed discussion of the theoretical methods and experimental analysis."
],
[
"Bose Polaron Hamiltonians",
"With ultracold atomic systems, various quantum impurity models can be studied in which an impurity interacts with a bosonic bath.",
"Here we focus on models that follow from the general Hamiltonian describing an impurity of mass $M$ interacting with a gas of weakly interacting bosons of mass $m$ : $\\hat{H}&=&\\sum _\\mathbf {p}\\epsilon _\\mathbf {p}^I\\hat{d}^\\dagger _\\mathbf {p}\\hat{d}_\\mathbf {p}+\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}+\\frac{g_{bb}}{2\\cal {V}}\\sum _{\\mathbf {k}\\mathbf {k}^{\\prime }\\mathbf {q}}\\hat{a}^\\dagger _{\\mathbf {k}^{\\prime }+\\mathbf {q}}\\hat{a}^\\dagger _{\\mathbf {k}-\\mathbf {q}}\\hat{a}_{\\mathbf {k}^{\\prime }}\\hat{a}_{\\mathbf {k}}\\nonumber \\\\&+&{\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {k}^{\\prime }\\mathbf {q}}V(\\mathbf {q})\\hat{d}^\\dagger _{\\mathbf {k}^{\\prime }-\\mathbf {q}}\\hat{d}_{\\mathbf {k}^{\\prime }}\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}}}_{ \\hat{H}_{IB}}.$ Here, the first two terms describe the kinetic energy of the impurities ($\\hat{d}_\\mathbf {p}$ ) and bosons ($\\hat{a}_\\mathbf {k}$ ) with dispersion relations $\\epsilon _\\mathbf {p}^I=\\frac{\\mathbf {p}^2}{2M}$ and $\\epsilon _\\mathbf {k}=\\frac{\\mathbf {k}^2}{2m}$ , respectively (unless explicitly stated we set $\\hbar =1$ ).",
"The third term accounts for the interaction between the bosons.",
"Assuming weak coupling between bosons, the microscopic coupling constant is given by the relation $g_{bb}=\\pi a_{bb}/m$ , with $a_{bb}$ the s-wave scattering length describing the low-energy boson-boson interactions.",
"The last term, $\\hat{H}_{IB}$ describes the impurity-boson interaction in momentum space, which, in the real space, reads $\\hat{H}_{IB}=\\int d^3r d^3r^{\\prime } \\,\\,\\hat{n}_I(\\mathbf {r}^{\\prime })\\,V(\\mathbf {r}^{\\prime }-\\mathbf {r})\\,\\hat{n}_B(\\mathbf {r}).$ Here $\\hat{n}_I(\\mathbf {r})=\\hat{\\psi }^\\dagger _I(\\mathbf {r})\\hat{\\psi }_I(\\mathbf {r})=\\frac{1}{\\cal {V}}\\sum _{\\mathbf {k}\\mathbf {q}} \\hat{d}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{d}_\\mathbf {k}e^{-i\\mathbf {q}\\mathbf {r}}$ and $\\hat{n}_B(\\mathbf {r})=\\hat{\\psi }^\\dagger _B(\\mathbf {r})\\hat{\\psi }_B(\\mathbf {r})=\\frac{1}{\\cal {V}}\\sum _{\\mathbf {k}\\mathbf {q}} \\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_\\mathbf {k}e^{-i\\mathbf {q}\\mathbf {r}}$ are the impurity and boson density, respectively.",
"As we consider the limit of a single impurity it is convenient to switch to the first quantized description of the impurity, which is characterized by its position and momentum operator $\\hat{\\mathbf {R}}$ and $\\hat{\\mathbf {p}}$ .",
"The density becomes $\\hat{n}_I(\\mathbf {r})\\rightarrow \\delta ^{(3)}(\\mathbf {r}- \\hat{\\mathbf {R}})$ and Eq.",
"(REF ) takes the form $\\hat{H}&=& \\frac{\\hat{\\mathbf {p}}^2}{2M}+\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}+\\frac{g_{bb}}{2\\cal {V}}\\sum _{\\mathbf {k}\\mathbf {k}^{\\prime }\\mathbf {q}}\\hat{a}^\\dagger _{\\mathbf {k}^{\\prime }+\\mathbf {q}}\\hat{a}^\\dagger _{\\mathbf {k}-\\mathbf {q}}\\hat{a}_{\\mathbf {k}^{\\prime }}\\hat{a}_{\\mathbf {k}} \\\\ \\nonumber &+&\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}}.$ From this Hamiltonian various polaron models can be derived, and we briefly explain their relevance and regimes of validity.",
"At $T=0$ a large fraction of bosons is condensed in a BEC.",
"This condensate can be regarded as a coherent state such that the zero-momentum mode $\\langle \\hat{a}_\\mathbf {k}\\rangle =\\sqrt{N_0}\\delta _{\\mathbf {k},\\mathbf {0}}$ takes on a macroscopic expectation value.",
"Within the Bogoliubov approximation the bosonic creation and annihilation operators in Eq.",
"(REF ) are expanded in fluctuations $\\hat{B}_\\mathbf {p}$ around this expectation value $\\sqrt{N_0}$ and terms of higher than quadratic order are neglected in the resulting Hamiltonian.",
"The purely bosonic part of the Hamiltonian can then be diagonalized by the Bogoliubov rotation $\\hat{B}_\\mathbf {p}=u_\\mathbf {p}\\hat{b}_\\mathbf {p}+ v_{-\\mathbf {p}}^* \\hat{b}^\\dagger _{-\\mathbf {p}}\\,\\,,\\,\\,\\hat{B}^\\dagger _\\mathbf {p}=u_\\mathbf {p}^* \\hat{b}^\\dagger _\\mathbf {p}+ v_{-\\mathbf {p}} \\hat{b}_{-\\mathbf {p}},$ where $\\omega _\\mathbf {k}=\\sqrt{\\epsilon _\\mathbf {p}(\\epsilon _\\mathbf {p}+2 g_{bb}\\rho )}$ is the Bogoliubov dispersion relation and $u_\\mathbf {p},v_{-\\mathbf {p}}=\\pm \\sqrt{\\frac{\\epsilon _\\mathbf {p}+g_{bb} \\rho }{2\\omega _\\mathbf {p}}\\pm \\frac{1}{2}}$ .",
"The density of the homogeneous condensate is given by $\\rho $ .",
"The transformation (REF ) yields the Bose-impurity Hamiltonian $\\hat{H}&=& \\frac{\\hat{\\mathbf {p}}^2}{2M}+\\sum _\\mathbf {k}\\omega _\\mathbf {k}\\hat{b}^\\dagger _\\mathbf {k}\\hat{b}_\\mathbf {k}\\nonumber \\\\&+&\\rho V(\\mathbf {0})+{\\frac{1}{\\sqrt{\\cal {V}}}\\sum _{ \\mathbf {q}}g(\\mathbf {q}) e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\left(\\hat{b}^\\dagger _{\\mathbf {q}}+\\hat{b}_{-\\mathbf {q}}\\right)}_{\\text{Fröhlich interaction}}\\nonumber \\\\&+&{\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\hat{B}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{B}_{\\mathbf {k}}}_{\\text{extended Fröhlich interaction}}$ where we introduced the `Fröhlich coupling' $g(\\mathbf {q})=\\sqrt{\\frac{\\rho \\epsilon _\\mathbf {q}}{\\omega _\\mathbf {q}}}V(\\mathbf {q})$ and in the last term we kept the untransformed expression for notational brevity.",
"In Eq.",
"(REF ) the term $\\rho V(\\mathbf {0})= \\rho \\int d^3r V(\\mathbf {r})$ describes the mean-field energy shift of the polaron in the Born approximation.",
"Neglecting the constant mean-field shift and the last term in Eq.",
"(REF ) one arrives at the celebrated Fröhlich model [12]: $\\hat{H}&= \\frac{\\hat{\\mathbf {p}}^2}{2M}+\\sum _\\mathbf {k}\\omega _\\mathbf {k}\\hat{b}^\\dagger _\\mathbf {k}\\hat{b}_\\mathbf {k}+\\frac{1}{\\sqrt{\\cal {V}}}\\sum _{ \\mathbf {q}}g(\\mathbf {q}) e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\left(\\hat{b}^\\dagger _{\\mathbf {q}}+\\hat{b}_{-\\mathbf {q}}\\right)$ In initial attempts to describe impurities in weakly interacting Bose gases, this Hamiltonian was used for systems close to Feshbach resonances [13], [14].",
"However, as shown in [15], the Fröhlich Hamiltonian alone is insufficient to describe impurities that interact strongly with atomic quantum gases (for an explicit third-order perturbation theory analysis see Ref. [16]).",
"Indeed the major theoretical shortcoming of Eq.",
"(REF ) is that from this model one cannot recover the Lippmann-Schwinger equation for two-body impurity-bose scattering.",
"Hence it fails to describe the underlying two-body scattering physics between the impurity and bosons including molecule formation.",
"Similarly the Fröhlich Hamiltonian cannot account for the intricate dynamics leading to the formation of Rydberg polarons.",
"For such strongly interacting systems the inclusion of the last term in Eq.",
"(REF ) becomes crucial.",
"This term accounts for pairing of the impurity with the atoms in the environment and accounts for the detailed `short-distance' physics of the problem, which is completely neglected in the Fröhlich model that is tailored for the description of long-wave length (low-energy) physics.",
"So far it has been experimentally verified that the inclusion of this term is relevant for the observation of Bose polarons [7], [8] where the impurity-Bose interaction can be modeled by a potential that supports only a single, weakly bound two-body molecular state.",
"In the present work we encounter a new type of impurity problem where the impurity is dressed by large sets of molecular states that have ultra-long-range character.",
"This yields the novel physics of Rydberg polarons that is beyond the physics of Bose polarons so far observed in experiments and discussed in the literature.",
"To account for the relevant Rydberg molecular physics theoretically, we analyze the full Hamiltonian Eq.",
"(REF ).",
"We note that the physics of the Rydberg oligomer states is different from that of Efimov states in the Bose polaron problem [15], [17], [18], [19].",
"While Efimov states are also multi-body bound states, they arise due a quantum anomaly of the underlying quantum field theory [20], [21], [22].",
"Indeed, the Efimov effect [23], [24], [25] gives rise to an infinite series of three-body states (and also states of larger atom number [26], [27]) that respect a discrete scaling symmetry.",
"The Efimov effect arises for resonant short-range two-body interactions in three dimensions.",
"In previous experiments studying Bose polarons close to a Feshbach resonance it was found that these Efimov states do not strongly influence the polaron physics [7], [8].",
"In our case the multi-molecular states are not related to the Efimov effect and dimer, trimer, etc.",
"states are rather comprised of nearly independently bound two-body molecular states that feature only very weak three- and higher-body correlations.",
"In contrast, in Efimov physics, a single impurity potentially mediates strong correlations between two particles in the Bose gas.",
"Due to large binding energies of Rydberg molecular states, this induced interaction is expected to be small for Rydberg polarons [28]."
],
[
"Polaron formation",
"The formation of a polaron can be understood as dressing of the impurity by excitations of the bosonic bath, which entangles the momentum of the impurity with that of the bath excitations.",
"This can be illustrated with a simple wave function expansion for a polaron of zero momentum $\\mathinner {|{\\Psi }\\rangle }&=\\sqrt{Z}\\mathinner {|{BEC}\\rangle }\\mathinner {|{\\mathbf {p}=0}\\rangle }_I+\\sum _\\mathbf {k}\\alpha _\\mathbf {k}\\hat{b}^\\dagger _{-\\mathbf {k}} \\hat{b}_\\mathbf {0}\\mathinner {|{BEC}\\rangle } \\mathinner {|{\\mathbf {k}}\\rangle }_I\\nonumber \\\\&+\\sum _{\\mathbf {k}\\mathbf {q}} \\alpha _{\\mathbf {k}\\mathbf {q}} \\hat{b}^\\dagger _{-\\mathbf {k}} \\hat{b}^\\dagger _{\\mathbf {k}-\\mathbf {q}} \\hat{b}_\\mathbf {0}\\hat{b}_\\mathbf {0}\\mathinner {|{BEC}\\rangle } \\mathinner {|{\\mathbf {q}}\\rangle }_I+\\ldots $ where $\\mathinner {|{\\mathbf {p}}\\rangle }_I$ denotes impurity momentum states.",
"The unperturbed impurity-bath state in the first terms becomes supplemented by `particle-hole' fluctuations of the BEC state given by the second, etc.",
"terms.",
"These terms describe the polaronic dressing of the impurity by bath excitations that leads to the formation of the polaron.",
"A Fourier transformation of the parameter $\\alpha _\\mathbf {k}$ to real space reveals the creation of density modulations in the medium, which represent the formation of a dressing cloud that is build around the impurity particle.",
"In fact, wave functions of the type in Eq.",
"(REF ) were found to yield a rather accurate description of impurities coupled to a fermionic bath of cold atoms via Feshbach resonances [29], [30], [31], [32], [33] and also describe exciton-impurities in two-dimensional semiconductors [34].",
"When truncated at the one-excitation level, such an ansatz leads to limited agreement with experimental observations for impurities in a Bose gas [35], [36].",
"Dressing can also be understood from a perturbative expansion in terms of Feynman diagrams as shown in Fig.",
"REF .",
"Here solid lines represent the propagating impurity, and the dashed lines denote Bosons that are excited from the BEC.",
"The Fröhlich model considers only processes where bosons are excited out of the condensate and then re-enter the BEC in their next scattering event with the impurity.",
"However, these `Fröhlich scattering processes' can neither account for molecular bound-state formation, nor for intricate strong-coupling physics found close to Feshbach resonances.",
"To describe these phenomena, the last term in Eq.",
"(REF ) has to be considered.",
"This term allows for scattering of bosons where an excited boson does not directly reenter the BEC but can scatter off the impurity arbitrarily many times.",
"In this process, illustrated as gray disk in Fig.",
"REF , the boson changes its momentum.",
"The infinite repetition of such scattering processes represents the Lippmann-Schwinger equation in terms of Feynman diagrams and accounts for molecular bound state formation."
],
[
"Rydberg Polaron Hamiltonian",
"We now specialize to the case of Rydberg impurities.",
"When a bath atom is excited into a Rydberg state of principal quantum number $n$ , it interacts with the surrounding ground-state atoms through a pseudopotential, first proposed by Fermi [37], in which molecular binding occurs through frequent scattering of the nearly free and zero-energy Rydberg electron from the ground-state atom.",
"In this picture, the Born-Oppenheimer potential for a ground-state atom at distance $\\mathbf {r}$ from the Rydberg impurity, retaining $s$ -wave and $p$ -wave scattering partial waves, is given as [37], [38], [39], [40], [41] $V_\\text{Ryd}(\\mathbf {r})&=&\\frac{2\\pi \\hbar ^2}{m_e} a_s|\\Psi (\\mathbf {r})|^2+\\frac{6\\pi \\hbar ^2 }{m_e} a_p^3|\\overrightarrow{\\nabla }\\Psi (\\mathbf {r})|^2, $ where we kept explicit factors of $\\hbar $ and $\\Psi (\\mathbf {r})$ is the Rydberg electron wave function, $a_s$ and $a_p$ are the momentum-dependent $s$ -wave and $p$ -wave scattering lengths, and $m_e$ is the electron mass.",
"When $a_s<0$ , $V(\\mathbf {r})$ can support molecular states with one or more ground-state atoms bound to the impurity [38], [42], [43].",
"A Rydberg polaron is formed when the Rydberg impurity is dressed by the occupation of a large number of these bound states in addition to finite momentum states.",
"This process gives rise to an absorption spectrum of a distribution of molecular peaks with a Gaussian envelope, which is the key spectral signature of Rydberg polaron formation.",
"The large energy scale of the Rydberg molecular states involved in the formation of Rydberg polarons allow us to simplify the extended Fröhlich Hamiltonian Eq.",
"(REF ): the typical energy range of Rydberg molecules is $0.1-10$ MHz for high quantum number $n$ , while the typical energy scale for Bose-Bose interactions is 1-10 KHz.",
"Therefore bosons that are bound to a Rydberg impurity probe momentum scales deep in the particle branch of the Bogoliubov dispersion relation, in a regime where the Bogoliubov factors $u_\\mathbf {p}=1$ and $v_\\mathbf {p}=0$ .",
"Hence we can neglect the Bose-Bose interactions and Eq.",
"(REF ) reduces to the simplified, extended Fröhlich model [11] $\\hat{H}&=& \\frac{\\hat{\\mathbf {p}}^2}{2M}+\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}+\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}}\\nonumber \\\\&=& \\frac{\\hat{\\mathbf {p}}^2}{2M}+\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\hat{b}^\\dagger _\\mathbf {k}\\hat{b}_\\mathbf {k}+\\rho V(\\mathbf {0}) \\nonumber \\\\&+&{\\frac{1}{\\sqrt{\\cal {V}}}\\sum _{ \\mathbf {q}}\\sqrt{\\rho }V(\\mathbf {q})e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\left(\\hat{b}^\\dagger _{\\mathbf {q}}+\\hat{b}_{-\\mathbf {q}}\\right)}_{\\text{Fröhlich interaction}}\\nonumber \\\\&+&{\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})e^{-i\\mathbf {q}\\hat{\\mathbf {R}}}\\hat{b}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{b}_{\\mathbf {k}}}_{\\text{extended Fröhlich interaction}}.$ where in the second equivalent expression we expanded the boson operators $\\hat{a}^\\dagger _\\mathbf {k}$ around their expectation value $\\sqrt{N_0}$ which highlights the connection to the Fröhlich model.",
"In order to describe Rydberg impurities the interaction $V(\\mathbf {q})$ is given as the Fourier transform of the Rydberg molecular potential $V(\\mathbf {q})=\\int d^3r V_\\text{Ryd}(\\mathbf {r})e^{i\\mathbf {q}\\cdot \\mathbf {r}}$ .",
"The molecular potentials and interacting single-particle wave functions ($\\mathinner {|{\\beta _i}\\rangle }$ ) are calculated as described in [43], [10].",
"We note that the depletion of a BEC by a single electron in a BEC has been studied in [44] including only the linear Fröhlich term in Eq.",
"(REF ).",
"While such an approach can approximately account for the rate of depletion of the condensate it fails to describe the formation of molecules that is essential for the formation of Rydberg polarons."
],
[
"Linear-response absorption from quench dynamics",
"One way to probe polaron structure and dynamics is absorption spectroscopy.",
"Here one utilizes the fact that before being excited to a Rydberg state, an atom is in its ground state $\\mathinner {|{5s}\\rangle }$ , and its interaction with the surrounding Bosons is negligible.",
"The system is described by the Hamiltonian $\\hat{H}_0$ given by the first two terms in Eq.",
"(REF ).",
"In contrast, when the atom is in its Rydberg state $\\mathinner {|{ns}\\rangle }$ the potential $V(\\mathbf {q})$ is switched on.",
"In experiments [9], transitions between both states are driven by a two-photon excitation.",
"Within linear response, the corresponding absorption of laser light at frequency $\\nu $ is given by Fermi's Golden rule $\\mathcal {A}(\\nu )&= 2\\pi \\sum _{if}w_i |\\mathinner {\\langle {f}|}\\hat{V}_\\text{L}\\mathinner {|{i}\\rangle }|^2\\times \\,\\delta (\\nu -(E_i-E_f)).$ Here the sum extends over all initial and final states of the impurity plus bosonic bath that fulfill $\\hat{H}_0 \\mathinner {|{i}\\rangle }=E_i\\mathinner {|{i}\\rangle }$ and $\\hat{H} \\mathinner {|{f}\\rangle }=E_f\\mathinner {|{f}\\rangle }$ , where $\\hat{H}$ is given by Eq.",
"(REF ) so that the final states $\\mathinner {|{f}\\rangle }$ are eigenstates in the presence of the strong perturbation from Rydberg-boson interactions.",
"In the states $\\mathinner {|{i}\\rangle }$ , all atoms (including the impurity atom) are in the $\\mathinner {|{5s}\\rangle }$ state, while in final states $\\mathinner {|{f}\\rangle }$ the impurity is in the atomic $\\mathinner {|{ns}\\rangle }$ state.",
"The laser operator that drives the two-photon transition between these two atomic states $\\mathinner {|{5s}\\rangle }$ and $\\mathinner {|{ns}\\rangle }$ , is given by $\\hat{V}_L\\sim \\mathinner {|{ns}\\rangle }\\mathinner {\\langle {5s}|}+h.c.$ .",
"Using the Fourier-representation of the delta function in Eq.",
"(REF ), one can show that the absorption spectrum follows from [10] (for details see [45]) $\\mathcal {A}(\\nu )=2\\,\\text{Re}\\,\\int _{0}^\\infty dt\\, e^{i \\nu t}\\, S(t)$ with the many-body overlap (also called the Loschmidt echo) $S(t)$ .",
"In the general case of finite temperature, the $w_i$ in Eq.",
"(REF ) are the thermodynamic weights of each initial state given by the diagonal elements of the density matrix $\\hat{\\rho }_\\text{ini}=e^{-\\beta H_0}/Z_p$ , for sample temperature $k_B T=1/\\beta $ and partition function $Z_p$ .",
"For finite temperature the Loschmidt echo is then given by $S(t)&=&\\text{Tr}[\\hat{\\rho }_\\text{ini}\\, e^{i \\hat{H}_0 t }e^{-i \\hat{H} t}]$ where $\\text{Tr}$ denotes the trace over the complete many-body Fock space.",
"This expression shows that the overlap function $S(t)$ encodes the non-equilibrium time evolution of the system following a quantum quench represented by the introduction of the Rydberg impurity at time $t=0$ .",
"From this time evolution then follows the absorption spectrum by Fourier transformation.",
"The relevant time scales for the dynamics of Rydberg impurities in a BEC is given by the Rydberg molecular energies which exceed both the temperature scale $k_B T$ and the energy scale associated with boson-boson interactions.",
"Thus temperature and boson interaction effects can be neglected in the calculation of $S(t)$ for Rydberg polarons.",
"In this limit the initial state of the system is given by $\\mathinner {|{i}\\rangle }=\\mathinner {|{\\text{BEC}}\\rangle }\\otimes \\mathinner {|{\\mathbf {p}=0}\\rangle }_I \\otimes \\mathinner {|{5s}\\rangle }_I$ and Rydberg polarons are well described by the T=0 limit of Eq.",
"(REF ), $S(t)={_I\\mathinner {\\langle {\\mathbf {p}=0}|}}\\mathinner {\\langle {\\Psi _\\text{BEC}}|}e^{i \\hat{H}_0 t}e^{-i \\hat{H} t}\\mathinner {|{\\Psi _\\text{BEC}}\\rangle }\\mathinner {|{\\mathbf {p}=0}\\rangle }_I,$ where $\\mathinner {|{\\Psi _\\text{BEC}}\\rangle }$ represents an ideal BEC of atoms.",
"The accurate calculation of the absorption response presents a formidable theoretical challenge.",
"In the following we will present three different approaches of different degrees of sophistication.",
"In a simple approximation one may employ mean-field theory (Section ) where the impurity-boson interactions are solely described by the mean-field result $\\rho V(\\mathbf {0})$ in Eq.",
"(REF ).",
"In this approach all quantum effects and fluctuations are neglected.",
"Furthermore one may employ a classical stochastic model (Section ), which at least can treat fluctuations to a certain degree.",
"Finally, we employ a functional determinant approach, which was developed in [10] and which we review in Section VI.",
"This approach treats all interaction terms in Eq.",
"(REF ) fully on a quantum level and allows us to explain the intricate Rydberg polaron formation dynamics observed in our experiments."
],
[
"Mean Field Approach",
"The simplest treatment of the polaron excitation spectrum is to neglect all interaction terms other than $\\rho {V}(\\mathbf {0})$ in Eq.",
"(REF ), which yields the mean-field approximation.",
"Hence at a given constant density $\\rho (\\mathbf {r})$ in the trap the absorption spectrum is obtained by Fourier transformation of (cf.",
"Eq.",
"(REF )) $S(t,\\vec{r})=e^{-i \\rho (\\mathbf {r}) V(\\mathbf {0}) t} = e^{-i \\Delta (\\mathbf {r})t},$ which yields a delta function Rydberg-absorption response $\\mathcal {A}(\\nu ,\\mathbf {r})=\\delta (\\nu -\\Delta (\\mathbf {r}))$ for a given local, homogeneous density $\\rho (\\mathbf {r})$ at a detuning from the atomic transition $\\Delta (\\mathbf {r})&=& \\rho (\\mathbf {r}) V(\\mathbf {q}=\\mathbf {0})=\\rho (\\mathbf {r})\\int d^3\\mathbf {r}^{\\prime } V(\\mathbf {r}-\\mathbf {r}^{\\prime }) .$ In the experiment, the density $\\rho (\\mathbf {r})$ varies with position $\\mathbf {r}$ .",
"Since the excitation laser illuminates the entire atomic cloud it excites Rydberg atoms in regions of varying density.",
"To theoretically model the resulting average over various contributions from the atomic cloud, we perform a local density approximation (LDA), which assumes the density variation is negligible over the range of the Rydberg interaction $V(\\mathbf {r})$ given by Eq.",
"REF .",
"The spectrum for creation of a Rydberg impurity is then given by $A(\\nu )\\propto \\int d\\mathbf {r}^3 \\rho (\\mathbf {r}) \\mathcal {A}(\\nu ,\\mathbf {r}).$ In the mean-field approximation this yields the response $A(\\nu )\\propto \\int d^3r\\, \\rho (\\mathbf {r}) \\delta (\\nu -\\Delta (\\mathbf {r}))$ for laser detuning $\\nu $ from unperturbed atomic resonance.",
"Note that the mean-field treatment is similar to the description of $1S-2S$ spectroscopy of a quantum degenerate hydrogen gas given in [46].",
"An intuitive way to express this shift is in terms of an effective $s$ -wave electron-atom scattering length that reflects the average of the interactions over the Rydberg wave function for the remainder of this section we keep factors of $\\hbar $, $a_{s,\\textup {eff}}=\\int d^3r^{\\prime } \\frac{{2\\pi m_e}}{h^2}V(\\mathbf {r^{\\prime }}),$ yielding $h\\Delta (\\mathbf {r})&=& \\frac{h^2 a_{s,\\textup {eff}}}{2\\pi m_e} \\rho (\\mathbf {r}).$ $a_{s,\\textup {eff}}$ varies with principal quantum number through the variation in the Rydberg electron wavefunction and the classical dependence of the electron momentum on position.",
"Figure REF shows calculated values of $a_{s,\\textup {eff}}$ for the interaction of Sr($5sns$ $^3S_1$ ) Rydberg atoms with background strontium atoms [43].",
"To obtain this result, Eq.",
"(REF ) is only integrated over $|\\mathbf {r}^{\\prime }|>0.06 n^2 a_0$ because the approximation breaks down near the Rydberg core, where, among other modifications, the ion-atom polarization potential becomes important.",
"Figure: Calculated values of a s,eff a_{s,\\textup {eff}} from Eq.",
"() versus n * =n-δn^*=n-\\delta , where δ=3.371\\delta =3.371 is the quantum defect.Figure: Mean-field description of the excitation spectrum of Sr(5sns5sns 3 S 1 ^3S_1) Rydberg atoms in a strontium Bose-Einstein condensate (BEC) for (Left) n=49, (Middle) n=60, (Right) n=72.",
"Symbols are the experimental data.",
"Blue bands represent the confidence interval for the mean-field fit of the BEC contribution to the spectrum corresponding to the uncertainties in parameters given in Tab.",
".",
"The central black line indicates the best fit.",
"The dashed red line is the mean-field prediction for thecontribution from the low-density region of the atomic cloud formed by thermal atoms.",
"This contribution is calculated using the parametersgiven in Tab.",
"including a sample temperature adjusted to reproduce the observed BEC fraction.Table: Parameters for data and fits shown in Fig.",
": Peak detuning of the mean-field fit (Δ max \\Delta _{\\textup {max}}) and BEC fraction (η\\eta ) are determined from the spectra.",
"Number of atoms in the condensate (N BEC N_{BEC}) and mean trap frequency ω ¯\\bar{\\omega } are determined from time-of-flight-absorption images and measurements of collective mode frequencies for trapped atoms respectively, and they determine the peak condensate density (ρ max \\rho _{\\textup {max}}) and chemical potential (μ\\mu ).",
"The second-to-last column provides the ratio of Δ max \\Delta _{\\textup {max}} to Δ ˜ max =ω ¯ 4π15Na bb a ho 2/5 m m e a s,eff a bb \\tilde{{\\Delta }}_{\\textup {max}} = \\frac{\\overline{\\omega }}{4\\pi }\\left(\\frac{15 N a_{bb}}{a_{ho}}\\right)^{2/5}\\frac{m}{m_e}\\frac{a_{s,eff}}{a_{bb}}, the peak shift predicted using information independent of the mean-field fits.",
"The sample temperature (TT) is the result of a fit of the observed BEC fraction using a numerical calculation of the number of non-condensed atoms in the trap fixing all the BEC parameters at values given in the table.Figure REF shows the mean-field description of experimental results for excitation of Sr($5sns$ $^3S_1$ ) Rydberg atoms in a strontium Bose-Einstein condensate (BEC).",
"For the prediction of the absorption response the effect of temperature enters by determining the local density of the cloud.",
"In fact, in the local-density-approximation model of the impurity excitation spectrum (Eq.",
"(REF )), the density of atoms in the trap, $\\rho (\\mathbf {r})=\\rho _{BEC}(\\mathbf {r})+\\rho _{\\textup {th}}(\\mathbf {r})$ , has contributions from thermal, non-condensed atoms and condensed atoms.",
"The contribution from the thermal gas is restricted to the sharp peak near zero detuning and a very small contribution towards the red in each data set in Fig.",
"REF .",
"Thus the BEC and thermal contributions can be calculated separately in the mean-field approximation.",
"The profile of the condensate density is well-approximated for our conditions with a Thomas-Fermi (TF) distribution for a harmonic trap [48].",
"If we neglect the contribution of thermal atoms to the density, the condensate contribution to the spectrum is $A_{\\textup {MF}}(\\nu )\\propto N_{BEC}\\frac{-\\nu }{\\Delta _{\\textup {max}}^2}\\sqrt{1-\\nu /\\Delta _{\\textup {max}}},$ for $\\Delta _{\\textup {max}}\\le \\nu \\le 0$ and zero otherwise [46].",
"Here, $\\Delta _{\\textup {max}} =\\frac{ h a_{s,\\textup {eff}}}{2\\pi m_e} \\rho _{\\textup {max}}$ , and the peak BEC density is $\\rho _{\\textup {max}}=\\mu _{TF}/g$ , where the chemical potential is $\\mu _{TF}=(\\hbar \\overline{\\omega }/2)\\left( 15 N_{BEC} a_{bb}/ a_{\\textup {ho}} \\right)^{2/5}$ and $g=4\\pi \\hbar ^2 a_{bb}/m$ for harmonic oscillator length $a_{\\textup {ho}}=(\\hbar /m\\overline{\\omega })^{1/2}$ and mean trap radial frequency $\\overline{\\omega }=(\\omega _1 \\omega _2 \\omega _3)^{1/3}$ .",
"This functional form is used to fit several data sets in Fig.",
"REF with $\\Delta _{\\textup {max}}$ and the overall signal amplitude as the only fit parameters.",
"When frequency is scaled by $\\Delta _{\\textup {max}}$ , the mean-field prediction for a BEC in a harmonic trap is universal, making $\\Delta _{\\textup {max}}$ an important parameter for describing the spectrum.",
"The resulting fit parameters are given in table REF .",
"We ascribe the difference between the mean-field fit and the observed spectra at small detuning to the contribution from the low-density region of the atomic cloud formed by thermal atoms.",
"From the ratio of the areas of these two signal components, we extract the condensate fraction.",
"Given the clean spectral separation between the contributions from thermal and condensed atoms, this is a promising technique for measuring very small thermal fractions and thus temperature of very cold Bose gases.",
"Once the BEC parameters are set, for a consistency check, we then calculate the contribution to the spectrum from thermal atoms using the mean-field approximation and adjusting the sample temperature to match the BEC fraction.",
"Additional details of the fitting process are described in App.",
".",
"For $n=72$ , the mean-field approach is quite accurate in describing the overall shape of the response across the entire spectrum.",
"However, at lower principal quantum numbers, the data and fit deviate significantly, especially at larger detunings.",
"Generally, mean-field fails both at small detuning (i.e.",
"for response from low-density regions of the cloud), where the deviation arises from the formation of few-body molecular states, and at large detuning, where fluctuations in the macroscopic occupation of molecular states become important.",
"Fluctuations correspond to a spread in binding energies of the polaron states excited for a given average density.",
"All of these effects are neglected in the mean-field description."
],
[
"Functional determinant approach",
"The mean-field approach is insufficient to describe the main features found in the absorption spectrum both at small and large detuning from the atomic transition.",
"For instance its failure at low detuning (low densities) has its origin in the formation of Rydberg molecular states, which is not described in mean-field theory.",
"The low density regime can be most conveniently studied in detail by working at a low principal quantum number (here $n=38$ ) where only few atoms are in the Rydberg orbit, and the signal is thus dominated by the low-density response.",
"To calculate $S(t)$ efficiently we evaluate it within the FDA.",
"In [10] the FDA was developed to include (for impurities of infinite mass) also the finite-temperature corrections that are however irrelevant for our experiment.",
"Thus we can restrict ourselves to the description of the zero-temperature response given by Eq.",
"(REF ), where the bosonic ground state can be expressed as a state of fixed particle number $\\mathinner {|{\\text{BEC}}\\rangle }=(\\hat{a}_\\mathbf {0}^\\dagger )^N_0/\\sqrt{N_0!",
"}\\mathinner {|{0}\\rangle }$ .",
"For sufficiently large particle number we may equally use its representation as a coherent state $\\mathinner {|{\\text{BEC}}\\rangle }=\\exp [{\\sqrt{N_0}(\\hat{a}_\\mathbf {0}^\\dagger -\\hat{a}_\\mathbf {0})}]\\mathinner {|{0}\\rangle }$ where $\\mathinner {|{0}\\rangle }$ is the boson vacuum.",
"The FDA, as developed in Ref.",
"[10], did not include the description of impurity recoil which is, however, essential for an accurate prediction of Rydberg molecular spectra.",
"In the following subsection we develop a extension of the FDA approach that overcomes this limitation."
],
[
"Canonical transformation: mobile Rydberg impurity",
"The FDA can be applied to time evolutions described by a Hamiltonian bilinear in creation and destruction operators.",
"This is fulfilled for the Hamiltonian Eq.",
"(REF ) in the case of an infinitely heavy impurity $M=\\infty $ .",
"However, for a mobile Rydberg impurity, the presence of the non-commuting operators $\\hat{\\mathbf {p}}$ and $\\hat{\\mathbf {r}}$ effectively gives rise to non-bilinear terms.",
"Figure: Exemplary Ramsey signal S(t)=|S(t)|e iϕ(t) S(t)= |S(t)| e^{i \\varphi (t)} including contrast |S||S| and phase ϕ\\varphi underlying the calculation of the Rydberg polaron absorption spectrum for n=49n=49 at peak density in absence of a finite Rydberg lifetime.",
"A fast decay visible in the contrast accompanies fast oscillations of the full complex system.",
"The combination of both gives rise to the distinct Rydberg polaron features of molecular peaks that are distributed according to a gaussian envelope.",
"The Ramsey signal thus provides an alternative pathway for observing Rydberg polaron formation dynamics in real-time.",
"The time scales of this coherent dressing dynamics are ultrafast compared to the typical time scales of collective low-energy excitations of ultracold quantum gases allowing study of ultrafast dynamics in a new setting.To remedy this challenge we combine the FDA with a canonical transformation.",
"Here we focus on the zero-temperature case.",
"In this case the response is obtained from the time evolution using Eq.",
"(REF ), where the Hamiltonian includes bosonic and impurity operators.",
"To deal with the impurity motion we perform a canonical transformation first proposed by Lee, Low, and Pines [49] which effectively transforms into the system comoving with the impurity.",
"To this end we define the translation operator [49] $U=\\exp \\left\\lbrace i \\hat{\\mathbf {R}}\\sum _\\mathbf {k}\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\right\\rbrace $ which is inserted in the time evolution $S(t)&=\\mathinner {\\langle {\\mathbf {p}=0}|}_I\\mathinner {\\langle {\\Psi _\\text{BEC}}|}U U^{-1}e^{-i \\hat{H} t}U U^{-1}\\mathinner {|{\\Psi _\\text{BEC}}\\rangle }\\mathinner {|{\\mathbf {p}=0}\\rangle }_I\\nonumber \\\\&=\\mathinner {\\langle {\\mathbf {p}=0}|}_I\\mathinner {\\langle {\\Psi _\\text{BEC}}|}e^{-i \\hat{\\mathcal {H}} t} \\mathinner {|{\\Psi _\\text{BEC}}\\rangle }\\mathinner {|{\\mathbf {p}=0}\\rangle }_I.$ In the first line we used that the term $e^{i\\hat{H}_0 t}$ can be dropped as the initial state is a zero energy state.",
"Furthermore, in the second line we made use of the fact that the total boson momentum of the BEC state is zero and we defined the transformed Hamiltonian $\\hat{\\mathcal {H}}=U^{-1}\\hat{H} U&= \\frac{\\left(\\hat{\\mathbf {p}}-\\sum _\\mathbf {k}\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\right)^2}{2M}+\\sum _\\mathbf {k}\\epsilon _\\mathbf {k}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\nonumber \\\\&+\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}}$ By virtue of the transformation $U$ the impurity coordinate is eliminated in the interaction and only the impurity momentum operator $\\hat{\\mathbf {p}}$ remains.",
"It thus commutes with the many-body Hamiltonian and is replaced by a c-number, $\\hat{\\mathbf {p}}\\rightarrow \\mathbf {p}$ .",
"Since in our case we are interested in the polaron at zero momentum, $\\mathbf {p}\\rightarrow 0$ .",
"After normal ordering, we arrive at $\\hat{\\mathcal {H}}&=& \\sum _{\\mathbf {k}\\mathbf {k}^{\\prime }} \\frac{\\mathbf {k}\\mathbf {k}^{\\prime }}{2M} \\hat{a}^\\dagger _{\\mathbf {k}^{\\prime }}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\hat{a}_{\\mathbf {k}^{\\prime }}+\\sum _\\mathbf {k}\\frac{\\mathbf {k}^2}{2\\mu _\\text{red}} \\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\nonumber \\\\&+&\\frac{1}{\\cal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}}$ The above transformation has the effect that the boson dispersion relation $\\epsilon _\\mathbf {k}\\rightarrow \\mathbf {k}^2/2\\mu _\\text{red}$ now has the reduced mass $\\mu _\\text{red}=mM/(M+m)$ of boson-impurity partners, and the Hamiltonian includes an induced interaction of bosons described by the first term in Eq.",
"(REF ).",
"Due to spherical symmetry and the large energy scale of the Rydberg impurity-Bose gas interaction, we may neglect this term.",
"The reliability of this approximation has been demonstrated in recent work of some of the authors [36] where a time-dependent variational principle has been applied to the evaluation of the time-evolution of Eq.",
"(REF ).",
"In the approximation where the time-dependent wave function $\\mathinner {|{\\Psi (t)}\\rangle }$ evolved by $\\exp (-i\\hat{\\mathcal {H}}t)$ is taken to be a product of coherent states of the form $\\mathinner {|{\\Psi (t)}\\rangle }=e^{\\sum _\\mathbf {k}(\\gamma _\\mathbf {k}(t) \\hat{a}_\\mathbf {k}^\\dagger +\\text{h.c.})}\\mathinner {|{0}\\rangle }$ , one finds from the equation of motions of the variational parameters $\\gamma _\\mathbf {q}$ that the expectation value $\\mathinner {\\langle {\\Psi (t)}|}\\sum _{\\mathbf {k}\\mathbf {k}^{\\prime }} \\frac{\\mathbf {k}\\mathbf {k}^{\\prime }}{2M} \\hat{a}^\\dagger _{\\mathbf {k}^{\\prime }}\\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}\\hat{a}_{\\mathbf {k}^{\\prime }}\\mathinner {|{\\Psi (t)}\\rangle }$ always remains zero and this term hence does not contribute to the dynamics.",
"For our case of Rydberg impurities the FDA solution presented here reproduces the variational result of Ref.",
"[36] and again shows excellent agreement with experimental data attesting a postiori to the accuracy of the method and the neglect of the first term in Eq.",
"(REF ).",
"To which extent this remains valid for other Bose polaron scenarios is an open question and subject to ongoing theoretical studies [50].",
"In summary, we finally arrive at the Hamiltonian $\\hat{ \\mathcal {H}}= \\sum _\\mathbf {k}\\frac{\\mathbf {k}^2}{2\\mu _\\text{red}} \\hat{a}^\\dagger _\\mathbf {k}\\hat{a}_\\mathbf {k}+\\frac{1}{\\mathcal {V}}\\sum _{ \\mathbf {k}\\mathbf {q}}V(\\mathbf {q})\\hat{a}^\\dagger _{\\mathbf {k}+\\mathbf {q}}\\hat{a}_{\\mathbf {k}} \\mathinner {|{ns}\\rangle }\\mathinner {\\langle {ns}|}$ whose dynamics we simulate to obtain Rydberg polaron excitation spectra.",
"Note, in Eq.",
"(REF ), we make explicit that the impurity-boson interaction is only present when the impurity is in its excited atomic Rydberg state $\\mathinner {|{n s}\\rangle }$ .",
"In this state the Rydberg electron scatters off ground state atoms in the environment, leading to the strong Rydberg Born-Oppenheimer potential $V(\\mathbf {q})$ .",
"For simplified notation we will now switch again to the symbol $\\mathcal {\\hat{H}} \\rightarrow \\hat{H}$ ."
],
[
"Time-domain $S(t)$",
"The strength of the FDA is that it allows one to express overlaps of many-body states in terms of single-particle eigenstates [51], [52], [53], [54], [10].",
"In our case this implies that we must calculate single particle eigenstates and energies of the free and `interacting' Hamiltonian, i.e.",
"$\\hat{h}_0\\mathinner {|{n}\\rangle }=\\epsilon _n \\mathinner {|{n}\\rangle }$ and $\\hat{h}\\mathinner {|{\\beta }\\rangle } = \\omega _\\beta \\mathinner {|{\\beta }\\rangle }$ , respectively ($\\hat{h}_0$ and $\\hat{h}$ are the single-particle representatives of the many-body operators $\\hat{H}_0$ and $\\hat{H}$ , respectively).",
"The single-particle eigenstates and energies $\\mathinner {|{\\beta }\\rangle }$ and $\\omega _\\beta $ are calculated using exact diagonalization.",
"Here we solve the radial Schrödinger equation for a spherical box of radius $R$ discretized in real-space.",
"Since the BEC is initially in a zero angular state and the Rydberg molecular potential is spherically symmetric we can restrict the analysis to single-particle states with zero-angular momentum.",
"We include the 300 energetically lowest eigenstates, which leads to convergent results.",
"In terms of these states and energies the overlap $S(t)$ becomes [10] $S(t)=\\left(\\sum _{\\beta } |\\langle \\beta | s \\rangle |^2 e^{i(\\epsilon _s-\\omega _\\beta )t}\\right)^{N},$ where $\\mathinner {|{s}\\rangle }$ denotes the lowest single particle eigenstate [55] of $\\hat{h}_0$ , and $N$ is the number of atoms in the spherical box of radius $R$ chosen to reproduce a given local experimental density of a Bose gas.",
"We choose $R=10^6 a_0$ with $a_0$ the Bohr radius so that we find negligible finite size corrections.",
"We emphasize again that the temperature enters the calculation only in determining the local density of the atomic cloud.",
"For the dynamics and thus the prediction of the absorption spectra at a given local density the temperature $T$ is irrelevant.",
"This is due to the fact that the energy scales of the dynamics of the system is determined by the binding energy of Rydberg molecular states, which exceed $k_BT$ .",
"We obtain absorption spectra by predicting the many-body overlap, or Loschmidt echo, $S(t)$ .",
"This calculation corresponds to solving the full time evolution of the system following a quantum quench, where at time $t=0$ the Rydberg impurity is suddenly introduced into the BEC.",
"The overlap $S(t)$ describes the dephasing dynamics of the many-body system and thus the evolution of the dressing of the Rydberg impurity by bath excitations.",
"It is one of the virtues of cold atomic systems that the signal $S(t)$ can be directly measured experimentally by Ramsey spectroscopy where via a $\\pi /2$ rotation at time $t=0$ , the impurity is prepared in a superposition state of $\\mathinner {|{5s}\\rangle }$ and $\\mathinner {|{ns}\\rangle }$ .",
"Following a time evolution of duration $t$ , a further $\\pi $ rotation is performed and $\\sigma _z$ is measured.",
"This gives the Ramsey contrast $|S(t)|$ .",
"Changing the phase of the final $\\pi $ rotation, the full signal $S(t)= |S(t)| e^{i \\varphi (t)}$ can be measured [56], [57], [45].",
"In Fig.",
"REF we show the predicted Ramsey signal that underlies the calculation of the Rydberg polaron absorption spectrum for $n=49$ at peak density, here shown in absence of a finite Rydberg lifetime.",
"We observe a fast decay in the contrast that indicates strong dephasing and thus efficient creation of polaron dressing by particle-hole excitations.",
"This decay is accompanied by fast oscillations of the complex signal $S(t)$ as visible in the shown evolution of the Ramsey phase $\\varphi (t)$ restricted to the branch $(-\\pi ,\\pi )$ .",
"The combination of the oscillations at the Rydberg molecular binding energies and the decay of the Ramsey signal $S(t)$ gives rise to the distinct Rydberg polaron features of molecular peaks that are distributed according to a gaussian envelope, to be discussed below.",
"The Ramsey signal thus provides an alternative pathway to observing Rydberg polaron formation dynamics in real-time.",
"The time scales of this coherent dressing dynamics are ultrafast compared to the typical time scales of collective low-energy excitations of ultracold quantum gases.",
"This again highlights that effects arising from Bose-Bose interaction as well as finite temperature will play only a minor role in the prediction of the absorption response as those start to influence dynamics only on much longer time scales.",
"Figure: Density profile of bosonic atoms in a harmonic trap for parameters T=180T=180 nK, N BEC =3.7·10 5 N_\\text{BEC}=3.7 \\cdot 10^5, N tot =5.2·10 5 N_\\text{tot}=5.2\\cdot 10^5, ω r =104\\omega _r = 104 Hz, and ω a =111\\omega _a=111 Hz, used for the prediction of the spectrum of the n=60n=60 Rydberg excitation shown in Fig. .",
"In blue is shown the contribution from the condensed atoms, while the red area represents the contribution from thermal atoms.Figure: (a) LDA absorption spectrum for n=49n=49 as calculated by FDA (solid blue) in comparison with experimental data (symbols).Using the data of the non-equilibrium quench dynamics, the FDA can accurately capture the formation of Rydberg molecular dimers, trimers, tetramers, etc.",
"This is demonstrated by comparing the FDA spectrum and experimental measurement for low principal quantum number, as shown in [9].",
"Indeed, from a many-body wave-function perspective capturing the trimer, tetrameter and higher-order oligamer state requires the inclusion of the corresponding higher order terms in Eq.",
"(REF ).",
"This attests to the challenge of describing Rydberg polarons which are formed by the dressing with many deeply bound atoms, rather than just a small number of bath excitations.",
"In fact due to the magnitude of the binding energies involved, coherent formation of Rydberg polarons takes place on a $\\mu $ s timescale, which is short compared to typical ultracold-atom time scales."
],
[
"FDA absorption spectra",
"The accurate description of molecular formation underlies the precise analysis of Rydberg-polaron absorption response.",
"As the principal quantum number is increased to $n=49$ , more atoms are situated on average within the Rydberg orbit and many-body effects become relevant.",
"The density-averaged prediction using FDA is shown in Fig.",
"REF .",
"For all density-averaged spectra, we rely on density profiles such as shown in Fig.",
"REF for parameters used for the FDA prediction of the $n=60$ Rydberg-absorption spectrum.",
"We include Hartree-Fock corrections [48], [58] as discussed in Appendix .",
"Furthermore, the absorption spectrum is calculated from the Fourier transformation according to Eq.",
"(REF ), where in the time-evolution of $S(t)$ we choose a temporal cutoff determined by the experimental laser-pulse duration, and in the Fourier transform we account for the finite laser line width (400 kHz).",
"Figure: LDA absorption spectrum for n=49n=49 as calculated from the classical Monte Carlo sampling model (black) compared to the FDA prediction.",
"The shaded area, which corresponds to the prediction of absorption response from the cloud center at peak density, reveals that the Gaussian line shape of the response is due to discrete, Gaussian-distributed molecular peaks and not a continuous response as described by a simple classical model.In Fig.",
"REF , we show again the $n=49$ absorption spectrum obtained theoretically, but with a simulated laser linewidth of 100 kHz (solid blue line).",
"The shaded region shows the result of an FDA calculation of the spectrum for a central region of the atomic cloud with roughly constant density.",
"We find that the signal from this region shows a series of molecular peaks whose weights follow a Gaussian envelope.",
"This distribution of molecular peaks is one of the key signatures of Rydberg polarons as predicted theoretically in the accompanying work [10].",
"The comparison of experiment and theory in Fig.",
"6 in turn shows that the response is indeed composed of many individual molecular lines.",
"While due to the finite lifetimes of Rydberg molecules and the fact that their binding energies decrease for increasing principal number, these individual molecular peaks cannot be resolved experimentally for high principal numbers, their Gaussian envelope is a robust signature of the formation of Rydberg polarons.",
"As our simulation shows, the broad tail to the red and the characteristic gaussian profile of the signal are both clear signatures of Rydberg polarons.",
"The excellent agreement between many-body theory and experiment confirms the presence of Rydberg polarons in the Sr experiment.",
"Figure: Experimentally observed absorption spectrum (symbols) for n=60n=60 in comparison with the theoretical prediction from FDA (lines).",
"The experimental input parameters such as the trap frequencies are varied within the experimental uncertainty, giving rise to the gray band around the solid red line.",
"The specific set of values is given in Table .",
"The inset shows the signal on a logarithmic scale.Table: Parameters used for the prediction of the theoretical spectrum shown in Fig.",
"(b) for the Rydberg n=60n=60 excitation.",
"Inferred quantities are the condensate fraction η=N BEC /N tot \\eta =N_\\text{BEC}/N_\\text{tot}, peak BEC density ρ peak,BEC \\rho _\\text{peak,BEC}, and chemical potential μ\\mu .",
"Densities are given in cm -3 \\text{cm}^{-3}.The Gaussian signature of the Rydberg-polaron spectrum becomes more pronounced when exciting $n=60$ states (Fig.",
"REF ).",
"In the accompanying work [10] we observe this Gaussian response experimentally for Rydberg polarons of large principal number $n=60$ and $n=72$ .",
"While due to the finite lifetimes of Rydberg molecules and the fact that their binding energies decrease for increasing principal number, these individual molecular peaks cannot be resolved experimentally for high principal numbers, their Gaussian envelope is a robust signature of the formation of Rydberg polarons.",
"The comparison of experiment and theory in Fig.",
"6 in turn shows that the response is indeed composed of many individual molecular lines.",
"The FDA calculation of the absorption spectra takes as input the experimentally determined number of atoms in the condensate $N_\\text{BEC}$ and the trap frequencies $\\omega _i$ .",
"The temperature $T$ and the total number of atoms $N_\\text{tot}$ are taken as fit parameters, and results are consistent with their determination using the mean-field model and absorption imaging described in Section .",
"For the $n=60$ spectrum, we also show a range of FDA predictions resulting from varying the trap frequencies within experimental uncertainty, which illustrates the impact of these uncertainties.",
"Table REF lists the parameters used in the theoretical simulation.",
"Note that the requirement to explain the spectrum over the whole frequency regime places tight constraints on the fit parameters $T$ , and $N_\\text{tot}$ .",
"(For Rydberg polaron response at $n=72$ , refer to Ref.",
"[9]).",
"As discussed in Ref.",
"[10], the emergence of the Gaussian response can be understood from a direct, analytical calculation of $S(t)$ in terms of single particle eigenstates and energies.",
"Indeed expanding the sum in Eq.",
"(REF ) explicitly in a multinominal form leads, after Fourier transform, to the expression $\\mathcal {A}(\\nu )&=&N!",
"\\sum _{\\Sigma n_i = N}\\frac{|\\langle \\alpha _1 | s \\rangle |^{2 n_1}\\cdot \\ldots \\cdot |\\langle \\alpha _M | s \\rangle |^{2 n_M}\\cdot \\ldots }{n_1!\\ldots n_M!",
"\\ldots } \\nonumber \\\\&&\\times \\delta (-\\nu +n_1 \\omega _1+\\ldots +n_M \\omega _M+\\ldots ).$ This expression captures not only the bound states but also continuum states.",
"Expressing the spectrum in this explicit form again highlights the fact that the actual spectrum is indeed built by $\\delta $ -peaks corresponding to the many configurations in which bound molecules can be formed and thus dress the impurity.",
"It also explains the emergence of the gaussian lineshape as a limit of the multinominal distribution of delta-function peaks for large particle number $N$ .",
"We note that the fact that the spectrum is built by molecular excitation peaks (of up to hundreds of atoms bound to a single impurity) is missed by the classical description of the Rydberg polarons discussed in Section .",
"We note that while Eq.",
"(REF ) is appealing as it qualitatively explains the observed spectral features, it is less useful for quantitative calculations due to the exponential growth of the number of terms in the sum (reflecting the exponential growth of Hilbert space for increasing particle number).",
"In contrast, the calculation of the time-evolution in Eq.",
"(REF ) is numerically efficient (when limited to finite evolution times) and only requires a final Fourier transformation."
],
[
"From quantum to classical description of Rydberg absorption response",
"For a sufficiently large average number of atoms within the Rydberg electron orbit, the overall line shape of Rydberg spectra can be described with classical statistical arguments.",
"In Fig.",
"REF , we show the experimental spectrum obtained for $n=60$ in comparison with the prediction from the FDA (dashed red line) and a classical statistical approach (solid black).",
"In the latter approach, using a classical Monte Carlo (CMC) algorithm [11], we randomly distribute atoms in three-dimensional space around the Rydberg ion such that the correct density profile is obtained.",
"Due to statistical fluctuations in the random sampling of coordinates (sampled from a uniform distribution), the local density within the Rydberg-electron radius fluctuates.",
"For each of the random configurations of atoms $\\mathcal {C}=(\\mathbf {r}_1, \\mathbf {r}_2,\\ldots )$ we calculate the classical energy of the configuration $E_{\\mathcal {C}}=\\sum _i V(\\mathbf {r}_i)$ , where $\\mathbf {r}_i$ are the atom coordinates.",
"The resulting energies $E_{\\mathcal {C}}$ are collected in an energy histogram and shown as the black solid line in Fig.",
"REF .",
"The validity of the classical description reflects the fact that the macroscopic occupation of the molecular bound and scattering states probes the Rydberg molecular potential uniformly over its entire range.",
"This process is also well described by a repeated sampling of different classical atom configurations drawn from a homogeneous density distribution.",
"Figure: LDA absorption spectrum for n=60n=60 as calculated by the FDA (dashed red) and classical statistical Monte Carlo sampling method (black).",
"The inset shows the signal on a logarithmic scale.The quantum-classical correspondence also becomes evident when considering the specific approximation that underlies the classical statistical approach.",
"The Rydberg absorption response is given as a Fourier transform of the full quantum quench dynamics as given by Eq.",
"(REF ).",
"The classical statistical model arises as an approximation of the time evolution of $S(t)$ where the kinetic energy of both the impurity and the Bose gas is completely neglected.",
"Hence both bosons and the impurity are treated as effectively infinitely heavy objects; their motion is `frozen' in space.",
"The disregard of the kinetic terms has the consequence that the Hamiltonian now commutes with the position operators $\\hat{\\mathbf {r}}_j$ of impurity and bath atoms (non-commuting $\\hat{\\mathbf {p}}_j$ operators are now absent), and hence dynamics becomes completely classical.",
"In this approximation the Hamiltonian becomes (without loss of generality, we assume that the impurity is at the center of the coordinate system) $\\hat{H}= \\int d^3r V(\\mathbf {r})\\hat{a}^\\dagger (\\mathbf {r})\\hat{a}(\\mathbf {r})$ Furthermore, $\\hat{H}_0=0$ , and hence $\\hat{\\rho }_\\text{ini} = 1/Z$ in Eq.",
"(REF ).",
"Since all boson coordinates $\\hat{\\mathbf {r}}_j$ now commute with the Hamiltonian, the many-body eigenstates are given by $\\mathinner {|{\\phi ^{(j)}}\\rangle }=\\mathinner {|{\\mathbf {r}^{(j)}_1,\\mathbf {r}^{(j)}_2,\\ldots , \\mathbf {r}^{(j)}_N}\\rangle }$ .",
"The trace in Eq.",
"(REF ) reduces to the sum over the set of all basis states $\\lbrace \\mathinner {|{\\phi ^{(j)}}\\rangle }\\rbrace $ .",
"The $\\mathinner {|{\\phi ^{(j)}}\\rangle }$ are eigenstates of $\\hat{H}$ with eigenvalues $\\hat{H} \\mathinner {|{\\phi ^{(j)}}\\rangle }= \\sum _{\\lbrace \\mathbf {r}_i^{(j)}\\rbrace } V(r_i^{(j)}) \\mathinner {|{\\phi ^{(j)}}\\rangle }$ and $S(t)$ becomes $S_\\text{cl}(t)=\\mathcal {N} \\sum _{\\lbrace \\mathbf {r}_i^{(j)}\\rbrace }e^{i\\left(\\nu - \\sum _{i} V(r_i^{(j)})\\right)t},$ where the sum extends over all possible atomic configurations in real-space, and $\\mathcal {N}$ is a normalization factor (representing the partition function $Z$ in the density operator $\\rho _\\text{ini}$ ).",
"Finally, performing the Fourier transform of $S(t)$ one arrives at $\\mathcal {A}(\\nu )=\\mathcal {N} \\sum _{\\lbrace \\mathbf {r}_i^{(j)}\\rbrace }\\delta \\left(\\nu - \\sum _{i} V(r_i^{(j)})\\right).$ A finite lifetime of the Rydberg or laser excitation, as well as a finite linewidth of the laser leads to a broadening of the delta function in Eq.",
"(REF ).",
"The sum in Eq.",
"(REF ) is exactly the object sampled in the classical Monte Carlo (CMC) approach derived from first principles.",
"Moreover we note that for impurities interacting with an ideal Bose gas with contact interactions, the gaussian spectral signature of polarons [10] would remain, while in the classical model, a sharp excitation at the atomic transition frequency is expected.",
"Indications of a Gaussian response have been seen in a recent experiment performed independently at Aarhus [8] and JILA [7], in agreement with theory [36].",
"The classical statistical approach fails to describe the Bose polarons close to a Feshbach resonance [7], [8].",
"Here the impurity indeed interacts with the bath via a contact interaction, and, due to a diverging scattering length, the single bound state present in the problem is highly delocalized and extends in size far beyond the range of the potential.",
"This effect cannot be captured by a classical approach.",
"While Rydberg absorption spectra find an effective, yet approximate description in terms of classical statistics, such an approach does not reveal the quantum mechanical origin of Rydberg polarons.",
"The classical sampling model treats both the Rydberg and the ground state atoms as infinitely heavy objects.",
"Zero-point motion and the discrete nature of the bound energy levels are absent in this treatment, so that it cannot describe the formation of Rydberg molecular states.",
"This short-coming of the classical approach is evident in Fig.",
"REF , where we show the absorption spectrum for $n=49$ .",
"The FDA (solid blue) fully describes the quantum mechanical formation of molecules (dimers, trimers, tetrameters, excited molecular states, etc.",
"), which gives rise to discrete molecular peaks visible in the spectrum.",
"In contrast, and as evident from Fig.",
"REF , the existence of molecular states, which are the quantum mechanical building block of Rydberg polarons, is not described by the classical approach (solid black).",
"The fact that the classical model can only describe the envelope of the Rydberg polaron response but not the underlying distribution of molecular peaks, is further emphasized by the comparison of FDA and CMC simulation from the center of the atomic cloud shown as shaded region in Fig.",
"7."
],
[
"Conclusion",
"In this work, we have detailed the descriptions of polarons as encountered in condensed matter and in ultracold atomic systems.",
"We time-evolve an extended Fröhlich Hamiltonian as relevant for Rydberg excitations, Eq.",
"(REF ), unitarily to obtain the overlap function, Eq.",
"(REF ), whose Fourier transform leads to the spectral function for Rydberg impurity excitation in a Bose gas, Eq.",
"(REF ).",
"We show how different approximations to the many-body quantum description, such as mean field and classical Monte Carlo treatments can be derived.",
"Here we extend the bosonic functional determinant approach to the Rydberg polaron problem to account for recoil of the impurity.",
"We show that the FDA approach can correctly and accurately account for the few-body molecular bound-state formation, as well as the macroscopic occupation of many-body bound and continuum states.",
"The various treatments are compared with experimental data for Rydberg excitation in a $^{84}$ Sr BEC [9] with the FDA results reproducing the observed data over a wide range of Rydberg line spectral intensities.",
"Rydberg polarons are exemplified on the one hand by their large energy scales, 1-10 MHz, that allow for coherent polaronic dressing on correspondingly short time scales, and on the other hand large spatial extent, $~1\\,\\mu $ m, leading to large-scale variations of the density in the many-body medium.",
"Another key feature of Rydberg polarons is the coherent dressing of the quantum impurity not by collective low-energy excitations, but by a large number of molecular bound states.",
"This is a new dressing mechanism, distinguishing them from polarons encountered so far in the solid state physics context.",
"Our FDA time domain analysis suggests a way to investigate the dynamical formation of the Rydberg polarons.",
"A natural extension will be to explore the feasibility of observing Pauli blocking in a degenerate Fermi atomic gas with non-trivial spatial correlations.",
"Note that one can study polarons with nonzero momentum by keeping all terms in Eq.",
"(REF ) and allowing $\\mathbf {p}$ to be finite.",
"The effective polaron mass can then be analyzed as previously demonstrated [59].",
"Acknowledgements: Research supported by the AFOSR (FA9550-14-1-0007), the NSF (1301773, 1600059, and 1205946), the Robert A, Welch Foundation (C-0734 and C-1844), the FWF(Austria) (P23359-N16, and FWF-SFB049 NextLite).",
"The Vienna scientific cluster was used for the calculations.",
"H. R. S. was supported by a grant to ITAMP from the NSF.",
"R. S. and H. R. S. were supported by the NSF through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and the Smithsonian Astrophysical Observatory.",
"R. S. acknowledges support from the ETH Pauli Center for Theoretical Studies.",
"E. D. acknowledges support from Harvard-MIT CUA, NSF Grant (DMR-1308435), AFOSR Quantum Simulation MURI, the ARO-MURI on Atomtronics, and support from Dr. Max Rössler, the Walter Haefner Foundation and the ETH Foundation.",
"T. C. K acknowledges support from Yale University during writing of this manuscript."
],
[
"Determination of BEC parameters and Mean-Field description of the Impurity Excitation Spectrum in a Strontium BEC",
"The mean-field approximation neglects fluctuations in the density around the Rydberg impurity, which correspond to a spread in binding energies of the polaron states excited for a given average density.",
"This explains the discrepancy between data and fit at large detuning.",
"But as shown in [9], the broadening resulting from these fluctuations is proportional to $\\sqrt{\\rho }$ and vanishes as detuning approaches zero.",
"Thus, we assume that the difference between data and the mean-field fit in Fig.",
"REF for small detuning ($\\nu /\\Delta _{\\textup {max}}<0.5$ ) arises from non-condensed atoms.",
"We adjust the area of the mean-field BEC contribution so the sum of the non-condensed and mean-field-BEC signal matches the total experimental spectral area.",
"Because the deviations between the mean-field fit and the BEC spectrum are significant, the fit of the peak shift $\\Delta _{\\textup {max}}$ is less rigorous.",
"We adjust it to qualitatively match the data and so that the mean-field fit and the data have approximately equal area for ($\\nu /\\Delta _{\\textup {max}}>0.5$ ).",
"This uncertainty in the fitting procedure decreases with increasing principal quantum number as the experimental data converges towards the mean-field form.",
"In the mean-field description we neglect the laser linewidth of 400 kHz in this analysis.",
"The fit parameters are given in Table REF .",
"The condensate fraction $\\eta $ is directly determined from the spectrum.",
"The uncertainties correspond to values calculated for the extremes of the confidence intervals shown as bands in Fig.",
"REF , except for uncertainties in $\\omega _i$ and total atom number, which reflect uncertainties from the independent procedures for measuring these quantities.",
"The fit value of the peak shift, $\\Delta _{\\textup {max}}$ , can be compared to the predicted peak shift, $\\tilde{\\Delta }_\\textup {max}$ , calculated from independent, a priori information: the number of atoms in the condensate $N_{BEC}$ determined from time-of-flight-absorption images, the trap oscillation frequencies $\\omega _i$ determined from measurements of collective mode frequencies for trapped atoms, and the value of $a_{s,\\textup {eff}}$ found theoretically from $V_{Ryd}(\\mathbf {r})$ .",
"The values of $\\Delta _{\\textup {max}}$ are all about 10% below $\\tilde{\\Delta }_\\textup {max}$ .",
"This may point to something systematic in our analysis procedure, but it is also a reasonable agreement given our uncertainty in determination of $\\omega _i$ and total atom number.",
"The calculation of $a_{s,\\textup {eff}}$ from $V_{Ryd}(\\mathbf {r})$ (Eq.",
"(REF )) could also account for some of this discrepancy through approximations made to describe the Rydberg-atom potential at short range.",
"Residual deviations in the description of the short-range details of the Rydberg molecular potential lead, for instance, also to the minor discrepancies between the FDA and CMC simulation of the Rydberg response from the center of the atomic cloud, shown in Fig.",
"REF .",
"For a non-interacting gas in a harmonic trap, the temperature is easily found from $\\eta $ , and $\\bar{\\omega }$ [60], and the standard expressions imply a temperature of approximately 240 nK for the data presented in Fig.",
"REF .",
"However, the situation is much more complicated than this for an interacting gas.",
"There are corrections to the condensate fraction that are independent of the trapping potential, and these are often discussed in terms of the shift of the critical temperature for condensation [48], [61], [62], but these are all small in our case, reflecting the smallness of relevant expansion parameters: $\\rho _{\\textup {max}}a_{bb}^3=10^{-4}, a_{bb}/\\lambda _{\\textup {th}}=10^{-2}$ at 100 nK, and $a_{\\textup {ho}}/R_{TF}=10^{-1}$ , where $\\lambda _{\\textup {th}}$ is the thermal de Broglie wavelength and $a_{\\textup {ho}}$ is the trap harmonic oscillator length.",
"For a gas trapped in an inhomogeneous external potential $V_{ext}(\\textbf {r})$ , the Hartree-Fock approximation (as described in detail in Refs.",
"[63], [60]) yields a mean-field interaction between thermal and BEC atoms that creates an effective potential for thermal atoms, $V_{\\textup {eff}}(\\textbf {r})= V_{\\text{ext}}(\\textbf {r})+2g[\\rho _{BEC}(\\textbf {r})+\\rho _{\\textup {th}}(\\textbf {r})]$ that is of a “Mexican hat\" shape rather than parabolic [60], [63], [64].",
"This effect is particularly important when the sample temperature is close to or lower than the chemical potential, which is the case here.",
"It increases the volume available to non-condensed atoms, and implies a lower sample temperature for a given value of $\\eta $ compared to the non-interacting case.",
"In other words, the number of thermal atoms can significantly exceed the critical number for condensation predicted for an ideal gas [64], [58].",
"240 nK is thus an upper limit of the sample temperature.",
"Sample temperatures extracted from bimodal fits to the time-of-flight absorption images are 100-190 nK.",
"Both the mean-field and the FDA fits use a local density approximation for the spectrum.",
"Here the temperature and interaction between bosons enter by determining the precise form of the overall density distribution of the atomic gas.",
"As discussed above, the density profile has contributions from both the condensed and thermal, non-condensed atoms $\\rho (\\mathbf {r})= \\rho _\\text{BEC}(\\mathbf {r}) +\\rho _\\text{th}(\\mathbf {r}).$ The BEC density $\\rho _\\text{BEC}$ and resulting chemical potential are assumed to be given by the Thomas-Fermi expressions [63] $\\rho _{BEC}(\\mathbf {r})= \\frac{m}{4\\pi \\hbar ^2 a_{bb}}[\\mu _{TF}-V_{ext}(\\mathbf {r})],$ with $\\mu _\\text{TF}=\\hbar \\overline{\\omega }/2(15 N_\\text{BEC} a_{bb}/a_{ho})^{2/5}$ .",
"$N_\\text{BEC}$ is the total number of condensed atoms, $a_{bb}=123 a_0$ for $^{84}$ Sr, and $a_{ho}=(\\hbar /m\\overline{\\omega })^{1/2}$ is the harmonic oscillator length.",
"(In this section we make explicit factors of $\\hbar $ and $k_B$ .)",
"The density distribution for atoms in the thermal gas can be calculated from known trap and BEC parameters using a Bose-distribution $\\rho _\\text{th}(\\mathbf {r})= \\frac{1}{\\lambda _T^3}g_{3/2}\\left\\lbrace e^{-\\frac{1}{k_BT}\\left[V_\\textup {eff}(\\mathbf {r})-\\mu _{TF}\\right]}\\right\\rbrace ,$ where $\\lambda _T=\\sqrt{\\frac{2\\pi \\hbar ^2}{m k_BT}}$ is the thermal wavelength and $g_{3/2}(z)$ the polylogarithm $\\text{PolyLog}[\\frac{3}{2};z]$ [63], and the chemical potential is set to $\\mu _{TF}$ .",
"The parameters $T$ and $\\mu _{TF}$ are varied self-consistently to fit that spectrum, which can be seen as matching the experimentally observed total particle number $N_\\text{tot}=N_\\text{BEC}+N_\\text{th}$ and BEC fraction $\\eta =N_\\text{BEC}/N_\\text{tot}$ .",
"The mean-field calculations use the full geometry of the external trapping potential as determined by the optical-dipole-trap lasers.",
"Integrals involving $V_\\text{ext}(\\mathbf {r})$ are evaluated numerically.",
"This procedure yields temperatures of $155-170\\,nK$ , in good agreement with other determinations.",
"The modification of the potential seen by thermal atoms due to Hartree-Fock corrections has an important consequence for our analysis.",
"The peak shift in the Rydberg excitation spectrum should in principle reflect the peak condensate density plus the density of thermal atoms at the center of the trapping potential.",
"The contribution from thermal atoms would be a significant correction if the mean-field repulsion of thermal atoms from the center of the trap were ignored.",
"But because the sample temperature is close to the chemical potential, the density of thermal atoms is suppressed at trap center and we neglect it in discussions of the peak mean-field shift in the spectrum.",
"For the FDA simulation, the dipole trap potential is approximated by $V_\\text{ext}(\\mathbf {r})=m(\\omega _r^2 r^2+\\omega _z^2 z^2)/2$ .",
"In Fig.",
"REF (a) we show the density profile along the radial direction for parameters as used for the FDA prediction of the $n=60$ Rydberg absorption spectrum including Hartree-Fock corrections.",
"We emphasize again that the temperature and interaction between bosons are relevant only for the determination of the density profile of atoms.",
"In the simulation of the absorption spectrum, which is obtained within a local density approximation, this density distribution of atoms enters as input.",
"In this simulation, then locally performed for constant density, both temperature effects and boson-interaction effects can be neglected due to the short time scales associated with Rydberg polaron dressing dynamics (that are given by the Rydberg molecular binding energies)."
]
] | 1709.01838 |
[
[
"On the geometry pervading One Particle States"
],
[
"Abstract In this paper, a way is given to obtain explicitly the representations of the Poincar\\'e group as can be prescribed by Geometric Quantization.",
"Thus one obtains some forms of the Space of Quantum States of the different relativistic free particles, and I give explicitly these spaces and the corresponding operators for the usually accepted as realistic physical particles.",
"The general description of the massless particles I obtain, is given in terms of solutions of Penrose equations.",
"In the case of Photon, I also give other descriptions, one in terms of the Electromagnetic Field.",
"Since the results are derived from Geometric Quantization, they are related to certain Contact and Symplectic manifols, that I study in detail.",
"The symplectic manifold must be interpreted, according with Souriau, as the Movement Space of the corresponding classical particle, and that leads to propose one of the spaces I use as the State Space of the corresponding classical particle.",
"These spaces are also described in each case."
],
[
"Introduction.",
"The Wave Equations of relativistic elementary particles, Klein-Gordon, Dirac, Weil etc.., have been originally derived each one in a independent way.",
"An unification was the discovery of the relation of these equations with the representations of Poincaré group (inhomogeneous Lorenz group).",
"The classification of the representations of Poincaré group made by Wigner [17] with important contributions of Majorana, Dirac and Proca [12], [9], [14] leads to a group theoretical study of Wave Equations by Bargmann and Wigner [3].",
"On the other hand, in Kirillov-Kostant-Souriau theory (Geometric Quantization), a description of Quantum Systems is given in terms of elements of the dual of the Lie algebra of the Lie group under consideration.",
"Of course, this way of seing Quantum Mechanics is not completely independent of the preceding one, since it has its origin in a method to obtain representations [10], [1].",
"The correspondence of Quantum States in the sense of Geometric Quantization with Wave Functions in the Quantum Mechanical sense is not clear in all cases.",
"In Souriau's book [15], a very general way to do the passage is given, but it doesn't work in all cases.",
"In this paper we give a geometrical construction that stablishes a one to one correspondence from Quantum States in the sense of Geometric Quantization to Wave Functions in the Quantum Mechanical sense, that is valid for all kinds of relativistic elementary particles.",
"The idea is as follows.",
"In Geometric Quantization, one begins with a regular contact manifold or its associated hermitian line bundle.",
"Quantum states ( in the sense of Geometric Quantification ) can be considered as being the collection of those sections of the hermitian line bundle which satisfies \" Planck's condition\" (cf.",
"Souriau's book).",
"In this paper we see that these sections are in a one to one correspondence with the (unrestricted) sections of another hermitian line bundle.",
"Thus, this fibre bundle is a good setting to describe the quantum processes under consideration.",
"The main idea to pass from this description to the usual one in terms of Wave Functions, can be intuitively explained as follows.",
"Quantum States in G.Q.",
"attribute an “amplitude of probability” to each movement of the particle.",
"To obtain the corresponding wave function one must proceed as follows: for each event, the corresponding amplitude of probability is obtained by taking all movements passing through the given event, and then “adding up” ( in a suitable sense) the corresponding amplitudes of probability.",
"Of course the concept of “movement passing through an event” is only obvious in the case of the ordinary massive spinless particle, and it is defined in section .",
"The Contact Manifold under consideration, fibers on a Symplectic Manifold that, following Souriau, must be considered as being composed by the \"movements of the corresponding classical particle\".",
"Then, with the constructions made in this paper, it becomes clear what space must be considered as composed by the \"states of the corresponding classical particle\", for each kind of relativistic particle.",
"The spaces of Wave Functions and many relevant isomorphic vector spaces we obtain, are the spaces of the representations of Poincaré group that I describe in section .",
"The preceding constructions are made in a explicit way for the relativistic particles usually considered as having physical sense.",
"A section is devoted to massive particles and other to massless particles.",
"In section a explicit application of the preceeding constructions is done for massive particles.",
"One obtains solutions of Klein-Gordon and Dirac equations, and also a description of the wave functions for massive particles of higher spin.",
"Massless particles are studied in section .",
"For massless particles of spin 1/2 one obtains solutions of Weyl equations and for general spin this method leads in a natural way to the description of massless particles by means of solutions of Penrose wave equations [13].",
"This is related to the fact that the Contact and Symplectic Manifolds corresponding to Massless Particles are expresed in a natural way in Twistor Space and its Projective Space, as explained in section REF .",
"In the particular case of Photon, in section REF ,I give also other forms for the Wave Functions.",
"One of them is in terms of the Electromagnetic Field,and are similar to the Wave Functions proposed by Bialynicki-Birula.",
"In section I describe for each particle the concrete space that must be considered to be the \"Space of states of the corresponding classical particle\"."
],
[
"Notation ",
"All differentiable manifolds appearing in this paper are assumed to be $C^{\\infty }$ , finite dimensional, Hausdorff and second countable.",
"The set composed by the differentiable vector fields on a differential manifold, $M$ , is denoted by ${\\cal D}(M)$ .",
"The set of differential $k$ -forms on $M$ is denoted by $\\Omega ^{k}(M)$ .",
"Let $X\\in {\\cal D}(M),\\ \\omega \\in \\Omega ^{k}(M)$ .",
"We denote by $i(X)\\omega $ the interior product of X by $\\omega $ and by $L(X)\\omega $ the Lie derivative of $\\omega $ with respect to $X$ .",
"Let $G$ be a Lie group.",
"The Lie algebra of $G$ is the set consisting of the left invariant vector fields on $G$ , provided with its canonical structure of Lie algebra.",
"It is denoted in this paper by $\\underline{G}$ .",
"I denote by $\\underline{G}^{\\ast }$ the dual of $\\underline{G}$ .",
"$\\underline{G}^{\\ast }$ is canonically identified with the set composed by the left invariant 1-forms on $G$ .",
"By identification of each element of $\\underline{G}$ or $\\underline{G}^{\\ast }$ with its value at the neutral element, $e$ , the lie algebra of $G$ is identified to the tangent space at $e$ , and $\\underline{G}^{\\ast }$ to its dual.",
"There exist a map, the exponential map, $Exp: G \\rightarrow \\underline{G}$ , such that: if $X\\in \\underline{G}$ , the integral curve of $X$ with initial value $e$ is $t\\in {\\rightarrow } Exp \\,tX$ .",
"This integral curve is a one parameter subgroup of $G$ .",
"The Lie algebra of the circle Lie group, $S^1$ , is identified to $R$ in such a way that for all $ t\\in R,\\ Exp\\,t=e^{2 \\pi i t}$ .",
"If we have an homomorphism of $G$ in $S^1$ , its differential is a linear map from the Lie algebra of $G$ into the Lie algebra of $S^1$ , which is $R$ .",
"Thus the differential of an homomorphism of $G$ into $S^1,$ can be considered as an element of $\\underline{G}^{\\ast }$ .",
"In the same way, the Lie algebra of the Lie group $R$ is identified to $R$ in such a way that the exponential map becomes the identity.",
"Thus, also in this case, the differential of an homomorphism of $G$ into $R$ , can be considered as an element of $\\underline{G}^{\\ast }$ .",
"The coadjoint representation is the homomorphism $Ad^{\\ast }:g\\in G\\rightarrow Ad_{g}^{\\ast }\\in $ Aut $(\\underline{G}^{\\ast })$ , given by $(Ad_{g}^{\\ast }(\\alpha ))(X)=\\alpha (Ad_{g^{-1}}(X))$ for all $\\alpha \\in \\underline{G}^{\\ast },X\\in \\underline{G}$ , where $Ad_{g^{-1}}$ is the differential of the automorphism of $G$ that sends each $h\\in G$ to $g^{-1}hg.$ Let $M$ be a differentiable manifold and $G$ a Lie group acting on $M$ on the left (resp.",
"on the right).",
"Given an element, $X$ , of $\\underline{G}$ , we denote by $X_{M}$ the vector field on $M$ whose flow is given by the diffeomorphisms associated by the action to $\\lbrace $ Exp$(-tX):t\\in {\\bf R}\\rbrace $ (resp.",
"$\\lbrace $ Exp$tX:t\\in {\\bf R}\\rbrace $ ).",
"$X_{M}$ is called the infinitesimal generator of the action associated to $X$ .",
"A principal fibre bundle having $M$ as total space, $B$ as base and $G$ as structural group, will be denoted by $M(B,G)$ .",
"In all that concerns fibre bundles, we use the notation of [11]."
],
[
"A Particular Classical State Space. ",
"In this section I motivate the physical interpretation of the geometrical constructions that are to be made in this paper.",
"I do not enter in the details of the computations.",
"Most part of this section is an easy consecuence of Souriau's book [15], which has inspirated this paper.",
"Let us consider a classical relativistic free particle without spin in Minkowski space-time, with rest mass $ m\\ne 0$ .",
"Space-time is interpreted as being an abstract four dimensional manifold, M, each inertial observer, R, providing us with a global chart, $\\phi _{R}=(x_{R}^1,...,x_{R}^{4})$ .",
"Changes of these global charts are given by transformations corresponding to elements of the Poincaré (i.e.",
"inhomogeneous Lorenz) group, ${\\cal P}$ .",
"We consider a family of inertial frames such that changes are all given by ortochronous proper Poincaré transformations, i.e.",
"elements of the connected component of the identity, ${\\cal P}_{+}^{\\uparrow }$ .",
"A geometrical object in $\\bf R^4\\rm $ invariant under the action on $\\bf R^4 \\rm $, provides us with a well defined object in M , whose local expression is the same for all inertial frames.",
"An example is the Minkowski metric, $g$ .",
"When provided with this metric, M becomes Minkowski space.",
"The objects defined in this way would be well defined even if the charts were not global.",
"Charts $\\phi _R $ give rise in the canonical way to charts of TM, $\\dot{\\phi }_R=(x_R^i,\\dot{x}_R^i)$ , where $\\dot{x}_R^i(v)=v(x_R^i)$ , for all $ v \\in TM.$ The charts $\\overline{\\phi }_R=(x_R^i,P_R^i) $ , $P_R^i=m\\,\\dot{x}_R^i$ , are more natural in Physics.",
"Changes of these global charts are given by the transformations corresponding to elements of the Poincaré group , in its canonical action on $\\bf R^8 \\rm \\ $$i.e.$ the action given by $(L,C)*\\left(\\begin{array}{c}z^1\\\\ \\vdots \\\\z^8\\end{array}\\right)=\\left(\\begin{array}{l} L\\left(\\begin{array}{c}z^1\\\\\\vdots \\\\z^4\\end{array}\\right)+C\\\\ L\\left(\\begin{array}{c}z^5\\\\\\vdots \\\\z^8\\end{array}\\right)\\end{array}\\right)$ Geometrical objects on $\\bf R^8 \\rm $ invariant under this action, give us well defined geometrical objects on TM, whose local expressions are equal in the different inertial frames.",
"For example, if $(z^1,\\ldots ,z^8)$ is the canonical coordinate system of $\\bf R^8 \\rm $ , the 1-form: ${\\mbox{$\\omega $}}_0 \\equiv \\ z^8 dz^4-\\sum _{i=1}^3\\ z^{i+4}dz^i$ the vector field: $X_0 \\equiv \\ \\frac{1}{m}\\ \\sum _{i=1}^4\\ z^{i+4}\\frac{\\partial }{\\partial z^i}$ and the submanifold, ${\\cal E}_0$, given by: $z^8 \\equiv \\ \\sqrt{m^2+(z^5)^2+(z^6)^2+(z^7)^2}$ are invariant by the action on $\\bf R^8 \\rm $.",
"Thus the following 1-form, vectorfield, and submanifold are well defined on TM: $\\widetilde{\\mbox{$\\omega $}} = P_R^4\\,dx_R^4 -\\sum _{i=1}^3\\ P_R^i\\,dx_R^i$ $\\widetilde{X}={ \\frac{1}{m}}\\ {\\sum _{i=1}^4}P_R^i\\,\\frac{\\partial }{\\partial x_R^i}$ ${\\cal E} & = & \\left\\lbrace \\, v\\in TM: P_R^4(v)=\\left( m^2+\\sum _{i=1}^3(P_R^i(v))^2\\right)^{\\frac{1}{2}} \\right\\rbrace \\\\& = & \\left\\lbrace \\,v\\in TM: g(v,v)=1,\\dot{x}_R^4(v) > 0 \\right\\rbrace $ where R is an arbitrary inertial frame.",
"The restriction of $ \\widetilde{\\mbox{$\\omega $}} $ to $\\cal E,$ $\\omega ,$ is a contact form.",
"$\\widetilde{X}$ is tangent to ${\\cal E}$ .",
"The restriction of $\\widetilde{X}$ to $\\cal E,$ $X$ , is the unique vectorfield such that $ i_X {\\mbox{$\\omega $}} = m,\\ i_Xd{\\mbox{$\\omega $}} = 0.",
"$ Each inertial observer associates to each element, v, of $\\cal E$ eight numbers, $\\overline{\\phi }_R(v)=(x_R^1(v), ...,\\ x_R^4(v), P_R^1(v),...,\\ P_R^4(v))$ which are interpreted as giving position, time and momentum-energy ( we take c=1).",
"Thus $\\cal E$ must be interpreted as being state space.",
"The movements of the free particle under consideration are curves in $\\cal E,$ $\\gamma $ , having, for each inertial observer, a constant four velocity $i.e.$ $\\dot{\\phi }_R\\circ \\gamma (s)=(a^i+s(\\dot{x}_R^i)_0, (\\dot{x}_R^i)_0) $ where the $(\\dot{x}_R^i)_0$ are constant, $(\\dot{x}_R^4)_0 > 0, (\\dot{x}_R^4)_0^2-\\sum _{i=1}^3 (\\dot{x}_R^i)_0^2 = 1.$ Since the preceeding relation is equivalent to $\\overline{\\phi }_R\\circ \\gamma (s) = ( a^i+s\\frac{{(P_R^i)}_0}{m},{(P_R^i)}_0),$ where ${(P_R^i)}_0=m(\\dot{x}_R^i)_0,$ one sees that the movements are the integral curves of X.",
"The parameter $s$ coincides, since we take c=1, with proper time (of the trajectory in space time).",
"The action on $\\bf R^8 \\rm $ gives, by means of any of the inertial observers, $R,$ an action on M by means of $(L,C)\\odot v=(\\overline{\\phi }_R)^{-1}((L,C)*(\\overline{\\phi }_R(v))),$ for all $v\\in {\\cal E},\\ (L,C)\\in {\\cal P}_{+}^{\\uparrow }$ .",
"This action obviously preserves $\\cal E,$ and $ {\\mbox{$\\omega $}} $ in such a way that $({\\mbox{${\\cal E}$}},{\\mbox{$\\omega $}})$ is a homogeneous contact manifold for the given action, but different inertial observers lead to different actions.",
"Now, we shall describe ${\\mbox{${\\cal E}_0$}}$ in a different way, thus obtaining a picture of state space more suitable for its generalisation to other kind of particles.",
"Let $Y_8$ be the infinitesimal generator of the action on $\\bf R^8 \\rm $ associated to the element Y of the Lie algebra of ${\\cal P}_{+}^{\\uparrow },\\ \\underline{\\cal P}$ .",
"Since the action on $\\bf R^8 \\rm $ preserves ${\\mbox{$\\omega $}}_0$ , we have $L_{Y_8} {\\mbox{$\\omega $}}_0=0,$ so that $i_{Y_8}\\,d {\\mbox{$\\omega $}}_0=-d\\,(i_{Y_8}{\\mbox{$\\omega $}}_0)=-d({\\mbox{$\\omega $}}_0(Y_8)).$ Thus we define a map, $ {\\mu }_0$ , called the momentum map, from $\\bf R^8 \\rm $ into $\\underline{\\cal P}^*$ , by means of $ {\\mu _0}(z)\\cdot Y=-( {\\mbox{$\\omega $}}_0(Y_8))(z),$ for all $Y \\in \\underline{\\cal P}, z \\in \\bf R^8 \\rm .$ Let $\\lbrace Y_{\\alpha }^i, Y_\\beta ^i, Y_{\\gamma }^i, Y_\\delta :\\ i=1, 2, 3 \\rbrace $ be the basis of $\\underline{\\cal P}$ , composed by the usual generators of Lorenz rotations and space-time translations.",
"It can be proved that $\\mu _0=z^8\\,Y_\\delta ^* - \\sum _{i=1}^3\\ \\lbrace [(z^1, z^2,z^3)\\times (z^5, z^6, z^7)]^i\\, Y_\\alpha ^{i*}+$ $+ ( z^4z^{i+4}\\,-z^8z^i)\\,Y_\\beta ^{i*} +z^{i+4}\\, Y_{\\gamma }^{i*}\\,\\rbrace $ where $\\lbrace Y_\\alpha ^{1*},..., Y_\\delta ^*\\rbrace $ is the dual basis of $\\lbrace Y_\\alpha ^1,..., Y_\\delta \\rbrace $ .",
"The map $\\mu _0$ is equivariant for the given action on $\\bf R^8 \\rm $ and coadjoint action.",
"Thus, since $(0, ...,0,m)\\in {\\mbox{${\\cal E}_0$}}$ and ${\\mu _0}(0, ...,0,m)=m\\, Y_\\delta ^*,\\ {\\mu _0}({\\mbox{${\\cal E}_0$}})$ is the coadjoint orbit of $m\\,Y_\\delta ^*$ .",
"We also have ${\\mu _0^*}\\, {\\mbox{$\\Omega $}} =d {\\mbox{$\\omega $}} _o$ , where $ {\\mbox{$\\Omega $}} $ is the Kirillov symplectic form on the coadjoint orbit.",
"For each $\\alpha $ in the coadjoint orbit, $\\mu _0^{-1}\\,\\lbrace {\\mbox{$\\alpha $}}\\rbrace $ is the image of a integral curve of $X_0$ , that we know to be local expresion of movements.",
"Then $\\mu _0$ gives a one to one correspondence between points of the coadjoint orbit and movements of the particle under consideration.",
"Now we consider the map $f:\\ (a,b)\\in {\\mbox{${\\cal E}_0$}}\\longrightarrow (a, \\mu _0\\,(a,b))\\in {{\\bf R^4}\\rm }\\times \\underline{\\cal P}^*.$ By using the above expression for $\\mu _0$ , one sees that $f$ is an imbedding.",
"Also, $f$ is equivariant when one considers the given action in ${\\mbox{${\\cal E}_0$}}$ and the “product action” in ${{\\bf R^4}\\rm }\\times \\underline{\\cal P}^*$ given by $(L,C)*(a, {\\mbox{$\\alpha $} } )=(La+C,Ad_{(L,C)}^*\\,{\\mbox{$\\alpha $} } ),\\ \\ \\ \\ \\forall (L,C)\\in {\\cal P}_{+}^{\\uparrow }, (a, {\\mbox{$\\alpha $} } )\\in f({\\mbox{${\\cal E}_0$}}).$ Observe that, as a consequence, $f({\\mbox{${\\cal E}_0$}})$ is an orbit of that action.",
"Each inertial frame, R, enables us to “see” state space, $\\cal E$ , by means of $f\\circ \\overline{\\phi }_R$ , as being $f({\\mbox{${\\cal E}_0$}})\\subset { {\\bf R}^4}\\times \\underline{\\cal P}^*$ .",
"Changes of inertial frames are now given by transformations coming from the product action.",
"Since $f\\circ \\overline{\\phi }_R(e)=(x_R^1(e),...,x_R^4(e),\\ \\mu _0(\\overline{\\phi }_R(e)))$ , if we denote $\\mu _R \\equiv \\mu _0\\circ \\overline{\\phi }_R$ we have $f\\circ \\overline{\\phi }_R =(\\phi _R, \\mu _R),$ and $\\mu _R$ stablishes a one to one correspondence of trajectories of X ($i.e.$ movements of the particle parametrized by proper time) with points of the coadjoint orbit.",
"Thus, coadjoint orbit is to be identified to movement space for all inertial frames, althought the identification is different for each $R$ .",
"The picture of a state, $e\\in \\cal E$ , obtained by R is now $(\\phi _R(e),\\mu _R(e)) \\ i.e.$ an event, $\\phi _R(e)$ , and a movement, $\\mu _R(e)$ , “passing through” the event.",
"Each $Y\\in \\underline{\\cal P}$ defines a function on $\\underline{\\cal P}^*$ , denoted by the same symbol.",
"Thus $Y\\circ \\pi _2\\circ f\\circ \\overline{\\phi }_R$ is a function on $\\cal E$ ($i.e.$ a dynamical variable) where $\\pi _2$ is the canonical projection of $ { {\\bf R}^4}\\times \\underline{\\cal P}^*$ onto $\\underline{\\cal P}^*$ .",
"We have: $- {Y_{\\gamma }^i}\\circ \\mu _R &=&P_R^i \\\\{Y_\\delta }\\circ \\mu _R&=& P_R^4 \\\\- {Y_{\\alpha }}^i\\circ \\mu _R&=&({\\overrightarrow{x}_R} \\times \\overrightarrow{P}_R)^i \\\\{Y_\\beta }^i\\circ \\mu _R&=&(P_R^4\\overrightarrow{x}_R -x_R^4\\overrightarrow{P}_R)^i$ where $i=1,2,3,\\overrightarrow{x}_R= (x_R^1, x_R^2,x_R^3),$ and $\\overrightarrow{P}_R= (P_R^1,P_R^2,P_R^3)$ .",
"Thus the function on ${ {\\bf R}^4}\\times \\underline{\\cal P}^*$ given by $P=(P^1,P^2,P^3,P^4) \\equiv (- {Y_{\\gamma }^1}\\circ \\pi _2,- {Y_{\\gamma }^2}\\circ \\pi _2, - {Y_{\\gamma }^3}\\circ \\pi _2,{Y_\\delta }\\circ \\pi _2)$ ,can be considered as an abstraction of linear momentum: each inertial observer obtains $f\\circ \\overline{\\phi }_R(e)$ as the picture of $e\\in {\\mbox{$\\cal E$}}$ , and the linear momentum he measures is $P(f\\circ \\overline{\\phi }_R(e))$ .",
"In the same way the components of $\\overrightarrow{l}= ( - Y_\\alpha ^1\\circ \\pi _2, - {Y_\\alpha ^2}\\circ \\pi _2 ,- {Y_\\alpha ^3}\\circ \\pi _2)$ and $\\overrightarrow{g}= ( Y_\\beta ^1\\circ \\pi _2,Y_\\beta ^2\\circ \\pi _2 , Y_\\beta ^3\\circ \\pi _2)$ must be interpreted as being the components of (relativistic) angular momentum.",
"If we denote ${\\cal P}_{+}^{\\uparrow }$ by G, we can summarize what has been said as follows State space of our free particle is such that each inertial observer establishes a diffeomorphism from it to an orbit of G in ${ {\\bf R}^4}\\times \\underline{G}^*,$ for the canonical action .",
"Changes of inertial observer are given by the transformations given by the same action.",
"An inertial observer, R, thus sees any state as a pair, the first component is an event and the second a movement containig the event.",
"The values at that point of $P,\\overrightarrow{l},\\overrightarrow{g}$ , are the values measured by R of linear momentum and angular momentum.",
"In this paper I accept that this is also valid for all relativistic free particles, with or without mass or spin.",
"More precisely, it is assumed that the possible state space of the relativistic free particles are the orbits of G in ${ {\\bf R}^4}\\times \\underline{G}^*$ , where now G is the universal (two fold) covering group of ${\\cal P}_{+}^{\\uparrow }$ .",
"The preceeding interpretations of movements, states, linear and angular momentum are also considered as valid.",
"In what follows, we assume that an inertial observer, R, has been fixed.",
"If state space is the orbit $\\cal O$ and $( x, \\alpha )\\in \\cal O$ , we have seen how $\\alpha $ can be considered as a movement.",
"In fact, this movement is composed by the elements of $\\cal O$ of the form $( y, \\alpha ).$ The set composed by such $y$ , $M_{\\alpha },$ compose the ordinary portrait of the movement $\\alpha $ in ${\\bf R}^4,$ obtained by the inertial observer R. Obviously $M_{\\alpha }=\\lbrace g*x: g\\in G_\\alpha \\rbrace $ where $G_\\alpha $ is the isotropy subgroup at $\\alpha $ of the coadjoint representation.",
"This way of looking at states also enables us to determine all movements containing a given event: if $( x, \\alpha )\\in \\cal O$ , the movements containing the event $x$ compose the set $N_x=\\lbrace Ad^*_g\\alpha :g\\in G_x\\rbrace $ where $G_x$ is the isotropy group at $x$ of the action of $G$ on ${\\bf R}^4.$"
],
[
"Universal covering group of the Poincaré Group. ",
"It is a well known fact that the universal covering group of Poincaré group is a semidirect product of $\\bf {SL}(2,\\bf {C})$ by a four dimensional real vector space.",
"In this section, I recall some general facts about this group and stablishes the notation.",
"Let $(x^1,x^2,x^3,x^4)$ be the canonical coordinates in $\\bf {R}^4$ , I the $2\\times 2$ unit matrix and ${\\sigma }_1, {\\sigma }_2, {\\sigma }_3$ the Pauli matrices, $i.e.$ $\\left(\\begin{array}{cc} 0 & 1\\\\1 & 0\\end{array}\\right),\\ \\left(\\begin{array}{cc}0 & -i\\\\i & 0\\end{array}\\right), \\,\\,\\left(\\begin{array}{cc}1 & 0\\\\0 & -1\\end{array}\\right)$ respectively.",
"A generic point of $\\bf {R}^4$ will be denoted $x=(x^1,x^2,x^3,x^4)$ .",
"We define an isomorphism $h$ , from $\\bf {R}^4$ onto the real vector espace, $\\bf {H}(2)$ , of the hermitian $2\\times 2$ matrices, by means of $h(x)=x^4I+\\sum _{i=1}^{3}x^i{\\sigma }_i=\\left(\\begin{array}{cc}x^4+x^3 , & x^1-ix^2\\\\x^1+ix^2 , &x^4-x^3\\end{array}\\right)$ We have $Det \\, h(x)=<x,x>_{ m}$ , where $<\\, ,\\, >_m $ is Minkowski pseudo-scalar product $<x,y>_m=x^4y^4-\\sum _{i=1}^3 x^iy^i.$ If $x=(x^1,x^2,x^3,x^4)$ we denote $\\vec{x}=(x^1,x^2,x^3)$ and $h(\\vec{x},x^4)=h(x).$ The following formulae are useful for many computations along this paper $[h(\\vec{k},k_4),h(\\vec{x},x^4)]\\stackrel{def}{=}h(\\vec{k},k_4)h(\\vec{x},x^4)-h(\\vec{x},x^4)h(\\vec{k},k_4)=2ih(\\vec{k} \\times \\vec{x},0),$ where $\\times $ means ordinary vector product, $\\lbrace h(\\vec{k},k_4),h(\\vec{x},x^4)\\rbrace & \\stackrel{def}{=}&h(\\vec{k},k_4)h(\\vec{x},x^4)+h(\\vec{x},x^4)h(\\vec{k},k_4)=\\\\ &=& 2h(k_4\\vec{x} +x^4 \\vec{k},k_4x^4+\\langle \\vec{k},\\vec{x} \\rangle ), $ where $\\langle \\, .",
"\\, ,\\, .",
"\\, \\rangle $ is the usual scalar product in $\\mathbb {R}^3,$ $h(\\vec{k},k_4)\\varepsilon \\overline{ h(\\vec{x},x^4)} \\varepsilon -h(\\vec{x},x^4)\\varepsilon \\overline{ h(\\vec{k},k_4)} \\varepsilon =\\\\\\nonumber =2[h(k_4 \\vec{x}-x^4 \\vec{k},0)+ih(\\vec{k} \\times \\vec{x},0)]$ where the bar means complex conjugation, and we denote by ${\\varepsilon }$ the matrix $i\\sigma _2,\\ i.e.$ ${\\varepsilon }=\\left(\\begin{array}{cc}0 & 1\\\\-1 & 0\\end{array}\\right).$ Notice that ${ }^tA\\,{\\varepsilon }\\,A=(Det \\,A)\\,{\\varepsilon },$ so that $A\\,{\\varepsilon }={\\varepsilon }\\,({}^tA)^{-1}$ if $A\\in \\bf {SL}(2,\\bf {C})$ .",
"Also we have $\\frac{1}{2}Tr(h(x){\\varepsilon }\\overline{h(y)}{\\varepsilon })=-<x,y>_m$ for all $x, y\\in \\bf {R}^4.$ We define an action on the left of the Lie group $\\bf {SL}(2,\\bf {C})$ on the abelian Lie group $\\bf {H}(2)$ by means of $A*H=AHA^*$ for all $A\\in \\bf {SL}(2,\\bf {C})$ , $H\\in \\bf {H}(2)$ , where $A^*$ is the transposed of the complex conjugate of $A$ .",
"To this action by automorphisms of $\\bf {H}(2)$ then corresponds a semidirect product, $\\bf {SL}(2,\\bf {C})\\oplus \\bf {H}(2)$ , whose group law is given by $(A,H)*(B,K)=(AB,AKA^*+H)$ The identity element is $(I,0)$ and $(A,H)^{-1}=(A^{-1},-A^{-1}HA^{*^{-1}})$ .",
"This semidirect product acts on the left on $\\bf {R}^4$ by means of $(A,H)* x=h^{-1}(Ah(x)A^*+H).$ Poincaré group, ${\\cal {P}}$ , is identified to the closed subgroup of $GL(5; \\bf {R})$ composed by the matrices $\\left(\\begin{array}{cc}L & C\\\\0 & 1\\end{array}\\right)$ where $C\\in \\bf {R}^4$ and $L\\in {\\cal {O}}(3,1)$ (such a matrix is denoted in the following simply by $(L,C)$ ).",
"For all $(A,H)\\in \\bf {SL}\\oplus \\bf {H}(2)$ (where $\\bf {SL}$ stands for $\\bf {SL}(2;\\bf {C})$ ), there exists a unique $(L,C)\\in {\\cal {P}}$ such that $(A,H)*x=Lx+C$ for all $x\\in \\bf {R}^4$ .",
"The map, $\\rho $ , from $\\bf {SL}\\oplus \\bf {H}(2)$ into ${\\cal {P}}$ defined by sending such a $(A,H)$ to the corresponding $(L,C)$ , is a homomorphism of Lie groups, whose kernel consists of $(I,0)$ and $(-I,0)$ .",
"In fact, $\\rho $ is a two fold covering map of the identity component in ${\\cal {P}}$ , ${\\cal {P}}_+^{\\uparrow }$ .",
"Since $\\bf {SL}$ and $\\bf {H}(2)$ are connected and simply connected it follows that $\\bf {SL} \\oplus \\bf {H}(2)$ is the universal covering group of ${\\cal {P}}_+^{\\uparrow }$ .",
"The differential of $\\rho $ is an isomorphism from the Lie algebra of $\\bf {SL}\\oplus \\bf {H}(2)$ onto the Lie algebra of ${\\cal {P}}$ .",
"The standard method to handle semidirect products enable us to identify the Lie algebra of $\\bf {SL}\\oplus \\bf {H}(2)$ with $\\bf {sl(2,{\\bf C})}\\times \\bf {H}(2)$ , the Lie bracket being $[(a,k),(a^{\\prime },k^{\\prime })]=([a,a^{\\prime }], ak^{\\prime }+ k^{\\prime }a^*-(a^{\\prime }k+ka^{\\prime *})).$ If $(a,h)\\in \\bf {sl(2,{\\bf C})}\\times \\bf {H}(2),\\ t\\in {\\bf R},$ we have $Exp[t(a,h)] =\\left(e^{ta},\\int _0^t e^{sa}\\,h\\, e^{sa^*}\\,ds \\right).$ In this paper, we use the basis of $\\bf {sl}\\times \\bf {H}(2)$ corresponding by $d\\rho $ to the basis of ${\\cal {P}}$ associated to linear momentum and angular momentum in section .",
"This basis is composed by the following elements $P^k &=&(0,-\\sigma _k),\\ \\ \\ k=1,2,3, \\nonumber \\\\P^4 &=&(0,\\sigma _4)=(0,I), \\nonumber \\\\l^k &=&(i\\frac{\\sigma _k}{2},0), \\\\g^k &=&(\\frac{\\sigma _k}{2},0).",
"\\nonumber $ An element, $X$ , of $\\bf {sl(2,{\\bf C})}\\times \\bf {H}(2)$ defines a (linear) function on $(\\bf {sl(2,{\\bf C})}\\times \\bf {H}(2))^*.$ Its restriction to movement space ( a coadjoint orbit) must be considered as a dynamical variable.",
"But also, if we have a representation on a vector space (resp.",
"an action on a manifold) of $\\bf {SL}\\oplus \\bf {H}(2),$ $X$ gives rise to a infinitesimal generator of the representation (resp.",
"the action), i.e., an endomorphism of the vector space (resp.",
"a vector field on the manifold).",
"We define vector valued functions on the dual of the Lie algebra as follows $P&=&(P^1,\\,P^2,\\,P^3,\\,P^4) \\\\\\overrightarrow{l}&=&(l^1,\\,l^2,\\,l^3)\\\\\\overrightarrow{g}&=&(g^1,\\,g^2,\\,g^3)$ what will be considered as being linear momentum and angular momentum.",
"We define a non degenerate scalar product in $\\bf {sl}\\times \\bf {H}(2)$ by means of $<(a,k),(b,l)> & =&2Re Tr (\\frac{1}{4}k{\\varepsilon }\\overline{l}{\\varepsilon }-ab)= \\nonumber \\\\& =&\\frac{1}{2}Tr(k{\\varepsilon }\\overline{l}{\\varepsilon })-2 Re Tr\\, ab.",
"\\nonumber $ This scalar product defines in the standard way an isomorphism from the Lie algebra of $\\bf {SL}\\oplus \\bf {H}(2)$ onto its dual.",
"The image of $(a,k)\\in \\bf {sl}\\times \\bf {H}(2)$ by this isomorphism, will be denoted by $\\lbrace a,k\\rbrace \\in \\left(\\bf {sl}\\times \\bf {H}(2)\\right)^{*}$ , and is given by $\\lbrace a,k\\rbrace \\left((b,m)\\right)=<(a,k),(b,m)>.$ With this notation, the values of $P,\\ \\overrightarrow{l}$ and $\\overrightarrow{g}$ at $\\lbrace a,k\\rbrace $ are, when written in terms of its hermitian form $h(P( \\lbrace a,\\ k \\rbrace ))&=&-k, \\\\h(\\vec{l}(\\lbrace a,\\ k \\rbrace ),\\ 0)&=& i\\,(a^*-a), \\\\h(\\vec{g}(\\lbrace a,\\ k \\rbrace ),\\ 0)&=& -(a+a^*).$ so that $\\lbrace a,k\\rbrace =\\lbrace -\\frac{1}{2}h(\\vec{g}(\\lbrace a,\\ k \\rbrace ),0)+\\frac{i}{2}h(\\vec{l}(\\lbrace a,\\ k \\rbrace ),0),-h(P( \\lbrace a,\\ k \\rbrace ))\\rbrace .$ i.e.",
"$-(1/2)h(\\vec{g}(\\lbrace a,\\ k \\rbrace ),0)$ is the hermitian real part of $a$ and the matrix $(1/2)h(\\vec{l}(\\lbrace a,\\ k \\rbrace ),0)$ its hermitian imaginary part.",
"A straightforward computation, leads to the following formula for the coadjoint representation $Ad^*_{(A,H)}\\lbrace a,k\\rbrace =\\lbrace AaA^{-1}+\\frac{1}{4}(AkA^*{\\varepsilon }\\overline{H}{\\varepsilon }-H{\\varepsilon }\\overline{AkA^*}{\\varepsilon }),AkA^*\\rbrace .$ To end this section, we define other functions on the dual of the Lie algebra of $\\bf {SL}\\oplus \\bf {H}(2).$ One of these is $\\vert P \\vert $ , defined by $\\left| P \\right| ( \\lbrace a, k\\rbrace )= Det\\,\\left( h(P( \\lbrace a,\\ k \\rbrace )) \\right)=Det\\,\\left( k \\right),$ whose physical meaning is mass square.",
"Then, $\\left| P \\right| ( Ad^*_{(A,H)}\\lbrace a, k\\rbrace )=Det (-AkA^*)=Det(k),$ so that the value of $\\vert P \\vert $ , is constant along any coadjoint orbit.",
"The other is defined in terms of the Pauli-Lubanski fourvector, given by $W&=&(\\overrightarrow{ W},W^4)\\\\\\overrightarrow{ W}&=&P^4 \\vec{l}+\\vec{P} \\times \\vec{g},\\\\W^4&=&\\langle \\vec{P},\\vec{l} \\rangle .$ Using (REF ), (REF ) and (REF ), the hermitian form of $W$ is found to be $h(W( \\lbrace a, k\\rbrace ))= i(a\\ k\\ -k\\ a^*).",
"$ One can prove that $h(W(Ad^*_{(A,H)}\\lbrace a,\\ k\\rbrace ))=A\\ h( W(\\lbrace a,\\ k\\rbrace ))\\ A^*,$ so that the function $\\vert W \\vert ( \\lbrace a, k\\rbrace )= Det(h(W( \\lbrace a, k\\rbrace ))),$ is also constant along each coadjoint orbit."
],
[
"Quantizable forms",
"In this section I recall some of the geometric constructions done in [16], [4], [5], [6], [7].",
"I shall give a way to construct homogeneous contact manifolds that fibers on the coadjoint orbits of Lie groups.",
"This is not possible on an arbitrary orbit, but only on the so called quantizable ones.",
"The idea is to consider the quantizable coadjoint orbits of $\\bf {SL}(2;{\\bf C}) \\oplus \\bf {H}(2)$ as the classical movement space of relativistic free particles.",
"The corresponding homogeneous contact manifolds are geometrical objects enabling us to construct the Wave Functions, and representations of this group.",
"Let $G$ be a Lie group and $\\alpha $ an element of the dual of the Lie algebra of $G$ , $\\underline{G}^*$ , where we consider the coadjoint action.",
"We say that , $\\alpha $ is quantizable if there exists a surjective homomorphism, $C_\\alpha $ , from the isotropy subgroup at $\\alpha $ , $G_\\alpha $ , onto the unit circle, ${\\bf S}^1 \\rm $, whose differential is $\\alpha $ (cf.",
"section where the way in which the differential is identified to an element of $\\underline{G}^*$ , is detailed).",
"In a more explicit way, this means that $\\alpha $ is quantizable if there exists a surjective homomorphism, $C_\\alpha ,$ from the isotropy subgroup at $\\alpha $ , $G_\\alpha ,$ onto the unit circle, such that, for all $X$ in the Lie algebra of $G_\\alpha $ and t in R $C_{\\alpha }(Exp\\,tX)=e^{2 \\pi i t {\\alpha }(X)}.$ If a form is quantizable, all the elements of its coadjoint orbit are quantizable.",
"These orbits are called quantizable orbits .",
"The form $\\alpha $ is said to be R-quantizable if there exists a surjective homomorphism from $G_\\alpha $ onto the usual additive Lie group of reals, R , whose differential is $\\alpha $ .",
"In other words, $\\alpha $ is R-quantizable if there exists a surjective homomorphism from $G_\\alpha $ onto R , $H_{\\alpha }$ , such that, for all $X$ in the Lie algebra of $G_\\alpha $ , and $t$ in R $H_{\\alpha }(Exp\\,tX)=t\\,{\\alpha }(X).$ To such a $H_{\\alpha }$ , one can associate an homomorphism, $C_{\\alpha }$ , onto ${\\bf S}^1$ by means of $C_{\\alpha }:g \\in {G_\\alpha } \\rightarrow e^{2 \\pi i H_{\\alpha }(g)} \\in {\\mbox{${\\bf S}^1 \\rm $}}.$ The differential of $C_{\\alpha }$ is $\\alpha $ .",
"Thus when $\\alpha $ is R-quantizable , it is quantizable .",
"All elements of a coadjoint orbit containing a R-quantizable form, are R-quantizable.",
"In this case, the orbit is said to be a R-quantizable orbit.",
"In [7] a slightly more general concept of quantizability is used, but it is unnecesary for the purposes of the present paper.",
"In what follows we assume that $\\alpha $ is quantizable and $C_\\alpha $ is a homomorphism from $G_\\alpha $ onto the unit circle, whose differential is $\\alpha $ .",
"We identify the coadjoint orbit, ${\\cal O}_{\\alpha },$ with $G/G_{\\alpha }$ by means of the diffeomorphism $g G_{\\alpha } \\in \\frac{G}{G_{\\alpha }} \\rightarrow Ad^*_g \\alpha \\in {\\cal O}_{\\alpha }.$ We define an action of ${\\bf S}^1$ on $G/Ker\\,C_\\alpha $ by means of $(g\\,Ker\\,C _\\alpha )*s=g\\,h\\,Ker\\,C_\\alpha $ where $h$ is any element of $G_{\\mbox{$ \\alpha $}}$ such that $C_\\alpha (h)=s$ .",
"Actually $(G/Ker\\,C_\\alpha )$ $(G/G_{\\mbox{$\\alpha $}},\\bf S\\rm ^1)$ is a principal fibre bundle , the bundle action is given by (REF ) and the bundle projection, by the canonical map, $\\pi : g{KerC_\\alpha } \\longrightarrow g {G_\\alpha }$ This action of ${\\bf S}^1$ , commutes with the canonical action of $G$ on $G/Ker\\,C_\\alpha $ .",
"We denote by $\\pi ^c$ and $\\pi ^s$ the canonical maps $\\pi ^c&:& g\\in G \\longrightarrow g {KerC_\\alpha } \\in \\frac{G}{{KerC_\\alpha }}\\\\\\pi ^s &:& g\\in G \\longrightarrow g {G_\\alpha } \\in \\frac{G}{{G_\\alpha }}.$ There exist an unique 1-form, $\\Omega $ , on the homogeneous space $G/Ker\\,C_\\alpha $ such that $(\\pi ^c)^*\\Omega =\\alpha .$ The 1-form $\\Omega $ is a contact form, invariant under the action of $G.$ Let $Z(\\Omega )$ be the vectorfield defined by $i_{Z (\\Omega )} \\Omega = 1,\\ \\ i_{Z(\\Omega )}\\,d\\Omega =0.$ All the integral curves of $Z(\\Omega )$ have the same period.",
"If we denote by ${T(\\Omega )}$ this period, then $\\Omega /{T(\\Omega )}$ is a connexion form.",
"Since the structural group is abelian, the curvature form is $d\\Omega /{{T(\\Omega )}} $ .",
"There exist an unique 2-form, $\\omega ,$ on $G/G_{\\alpha }$ , such that $\\pi _*\\omega =\\frac{d\\Omega }{{T(\\Omega )}}.$ Thus $(\\pi ^s)^*\\omega =\\frac{d\\alpha }{T(\\Omega )}.$ The form $\\omega $ is an invariant symplectic form, and its cohomology class is integral.",
"Each $m\\in \\underline{G},$ define a function on $\\underline{G}^*$ so that it does also on the coadjoint orbit.",
"Since we think in the coadjoint orbit as being the movement space of a particle, we can say that m defines a dynamical variable, $D_m.$ Such a $m\\in \\underline{G},$ also define a infinitesimal generator of the canonical action of $G$ on $G/Ker\\,C_\\alpha $ , denoted by $X_m^c,$ and a infinitesimal generator of the canonical action of $G$ on $G/G_\\alpha $ , denoted by $X_m^s.$ Since $\\Omega $ is left invariant by the action of $G,$ we have $L_{X_m^c}\\Omega =0,$ so that the relation $L_X=d\\,i_X+i_X\\,d$ leads to $i_{X_m^c}\\,d\\Omega =-d[\\Omega (X_m^c)].$ A computation gives ${\\begin{array}{c}(\\Omega (X_m^c) \\circ \\pi ^c)(g)= Ad^*_g \\alpha _0 (-m)=-D_m(Ad^*_g \\alpha _0)=\\\\=- D_m(g\\, G_\\alpha )=-D_m\\circ \\pi (g\\, Ker\\,C_\\alpha ),\\end{array}}$ so that $\\Omega (X_m^c)=-D_m\\circ \\pi .$ Thus (REF ), (REF ), and the fact that $\\pi _* X_m^c=X_m^s,$ lead to $i_{X_m^s}\\omega =\\frac{1}{T\\left( \\Omega \\right)}d D_m.$ Since the flow of the vector field $X_m^s$ (resp.",
"$X_m^c$ ) preserves $\\omega $ (resp.",
"$\\Omega $ ), it is an infinitesimal automorphism of the symplectic (resp.",
"contact) structure, i.e.",
"a locally hamiltonian vector field.",
"Equation (REF ) tell us that in fact $X_m^s$ is globally hamiltonian and $ D_m/{T\\left( \\Omega \\right)}$ is the corresponding hamiltonian.",
"As usual in the Theory of Connections, a map, $f$ , into $G/{KerC_\\alpha }$ is called horizontal, if $f^{*}\\Omega =0.$ Let $f_0$ be a map into $G/{G_\\alpha }$ .",
"An horizontal lift of $f_0$ is an horizontal map into $G/{KerC_\\alpha }$ , such that $\\pi \\circ f =f_0$ .",
"The horizontal lift of curves can be described as follows.",
"Given a curve, $\\gamma $ in $G/G_{\\mbox{$\\alpha $}}$ , the horizontal lift of $\\gamma $ to $g\\,KerC_\\alpha $ is $\\widetilde{\\gamma }\\left( t \\right)=\\left( \\overline{\\gamma }\\left( t \\right) KerC_\\alpha \\right) * e^{-2 \\pi i \\int _{\\overline{\\gamma }|_{[0,\\ t]}}\\alpha } $ where $\\overline{\\gamma }$ is any lifting of $\\gamma $ to G, such that $\\overline{\\gamma }(0)=g$ , and the vertical bar means restriction.",
"Associated to this principal fibre bundle and the canonical action of ${\\bf S}^1$ on ${\\mathbb {C}}$ , one can consider the 1- dimensional vector bundle whose total space is $\\left( \\frac{G}{KerC_\\alpha }\\right) \\times _{{\\bf S}^1} {\\mathbb {C}}.$ Let us recall its definition.",
"Consider in $\\left( G/KerC_\\alpha \\right) \\times {\\mathbb {C}}$ , an action of ${\\bf S}^1$ defined by $(g\\, KerC_\\alpha , t)*z=\\left(\\left(g\\, KerC_\\alpha \\right)*z ,z^{-1} t\\right),$ for all $z\\in {\\bf S}^1$ .",
"The elements of $\\left( G/KerC_\\alpha \\right)\\times _{{\\bf S}^1}{\\mathbb {C}}$ are the orbits of this action.",
"Let us denote by $[g\\, KerC_\\alpha , t]$ the orbit of $(g\\, KerC_\\alpha , t).$ We have $\\nonumber [g\\, KerC_\\alpha , t]&=&\\left\\lbrace \\left(\\left(g\\,KerC_\\alpha \\right)*s, s^{-1} t\\right):\\ s\\in S^{\\rm 1}\\right\\rbrace = \\\\ &=&\\nonumber \\left\\lbrace \\left(gh\\,KerC_\\alpha , C_\\alpha \\left(h^{\\rm -1}\\right) t\\right):\\ h\\in G_\\alpha \\right\\rbrace $ We define a map, $\\overline{\\pi }$ , from $ \\frac{G}{KerC_\\alpha } \\times _{{\\bf S }^1} {\\mathbb {C}}$ onto the coadjoint orbit by means of $\\overline{\\pi }([g\\, KerC_\\alpha , t])=g\\,G_\\alpha $ .",
"Let $m\\in G/G_\\alpha $ , and $g\\in G$ one of its representatives.",
"Since the action of ${\\bf S}^1$ on $\\pi ^{-1}(m)$ is transitive, we have ${\\overline{\\pi }}^{-1}(m)=\\lbrace [g\\,Ker\\,C_{\\alpha },t]:t\\in \\bf C\\rm \\rbrace .$ Thus, for each $g$ we obtain a bijection of ${\\overline{\\pi }}^{-1}(m)$ onto $\\bf C$ .",
"We consider in ${\\overline{\\pi }}^{-1}(m)$ the structure of hermitian one dimensional complex vector space such that this bijection is a unitary isomorphism.",
"This structure is independent of the representative $g$ .",
"If $g$ and $g^{\\prime }$ are representatives of $m$ , we have for all $t,\\ t^\\prime ,\\ a \\in \\bf C$ $ \\left[g\\,KerC_\\alpha ,\\ t\\right] + \\left[g^\\prime \\,KerC_\\alpha ,\\ t^\\prime \\right]=\\left[g\\,KerC_\\alpha ,\\ t+C_\\alpha (g^{-1}\\,g^\\prime )\\,t^\\prime \\right]$ , $ a \\cdot \\left[g\\,KerC _\\alpha ,\\ t\\right] =\\left[g\\,KerC_\\alpha ,\\ a\\,t\\right],$ $\\langle \\left[g\\,KerC_\\alpha ,\\ t\\right] ,\\left[g^\\prime \\, KerC_\\alpha ,\\ t^\\prime \\right] \\rangle ={\\overline{t}}\\,{C_\\alpha (g^{-1}g^\\prime )\\,t^\\prime } .$ With these operations, $\\overline{\\pi }$ becomes a complex vector bundle of dimension one with a hermitian product in each fiber i.e.",
"a hermitian line bundle.",
"The sections of the hermitian line bundle are in a one to one correspondence with the functions on $G/{KerC_\\alpha } $ , f, such that f((g ${\\mbox{$Ker\\,C_\\alpha $}} )\\,*\\,s)= s^{-1} $ f(g ${\\mbox{$Ker\\,C_\\alpha $}} $ ), for all $s\\in {\\mbox{$\\bf S \\rm ^1$}}$ .",
"These functions will be called from now on pseudotensorial functions .",
"This correspondence is as follows.",
"If f is a pseudotensorial function, the corresponding section sends $m \\in G/G_\\alpha $ to $\\left[ r,f(r) \\right]$ where r is arbitrary in $\\pi ^{-1}(m)$ .",
"If $\\sigma $ is a given section of the hermitian line bundle, the corresponding pseudotensorial function, f, is defined by $\\sigma (\\pi (r)) = \\left[ r,f(r) \\right]$ for all $ r \\in G/Ker C_\\alpha $ .",
"The canonical action of $G$ on $\\left( G/KerC_\\alpha \\right)$ , leads to an action on $\\left( G/KerC_\\alpha \\right)\\times _{\\mbox{$\\bf S \\rm ^1$}}{\\mbox{{$C$}} \\rm },$ given by $g*[h\\, Ker C_\\alpha ,t]=[gh\\, Ker C_\\alpha ,t].$ This action is well defined, as a consecuence of the fact that the action of $\\bf S \\rm ^1$ conmutes with the canonical action of $G$ .",
"In the following, I use the same construction of a hermitian line bundle, for each principal bundle, $ G/(Ker\\,C)(G/H,S),$ where $C$ is an homomorphism of $H$ into $\\bf S \\rm ^1$, whose image is $S$ ."
],
[
"Geometric Quantum States .",
"In this section I recall some definitions and results from [8].",
"Let $G=SL(2,{\\bf C}) \\oplus H(2)$ and $\\alpha $ a quantizable form of $G$ .",
"We use the notation of section .",
"The sections of the hermitian line bundle whose total space is $ \\left( G/KerC_\\alpha \\right)\\times _{\\mbox{$\\bf S \\rm ^1$}}{\\mbox{{$C$}} \\rm } $ are called Prequantum States .",
"We use the same denomination for the corresponding pseudotensorial functions.",
"Now, let us consider the actions of the abelian subgroup $\\lbrace I\\rbrace \\times H(2)$ on $G/Ker\\,C_\\alpha $ and $G/G_\\alpha $ , induced by the canonical action.",
"There exist an unique action of $\\lbrace I\\rbrace \\times H(2)$ on $G/Ker C_\\alpha $ whose orbits are horizontal and such that $\\pi $ becomes equivariant.",
"This action is called horizontal action and is given by $(I,\\ K)*((A,\\ H)\\,KerC_\\alpha )=((A,\\ H+K)\\,KerC_\\alpha )*e^{-i \\pi Tr\\left(AkA^* \\varepsilon \\overline{K} \\varepsilon \\right)}$ for all $ K \\in H(2),\\ (A,\\ H) \\in G $ , where $*$ in the left hand side stands for the new action and in the right hand one, corresponds to the bundle action.",
"k is given by $\\alpha = \\lbrace a,\\ k\\rbrace $ .",
"We define Quantum States as being the Prequantum States that correspond to pseudotensorial functions left invariant by the horizontal action.",
"Let $\\pi _1$ and $\\pi _2$ be the canonical projections of $G$ on ${ SL(2,\\bf C)}$ and $ H(2) $ respectively.",
"We denote $\\pi _1(G_\\alpha )$ by ${\\mbox{$(G_\\alpha )_{SL}$}}$ and $\\pi _2(G_\\alpha )$ by ${\\mbox{$(G_\\alpha )_{H}$}}$ .",
"In section 4 of [8] it is proved that the map $(C_{{\\mbox{$\\alpha $}}})_{SL} :(G_{{\\mbox{$\\alpha $}}})_{SL} \\longmapsto S^1 ,$ defined by $(C_{{\\mbox{$\\alpha $}}})_{SL}(g)\\ =\\ C_{{\\mbox{$\\alpha $}}}(g,h)\\ e^{-i\\pi Tr\\left(k{\\mbox{$\\varepsilon $}} \\overline{h}{\\mbox{$\\varepsilon $}}\\right) },$ for all $(g,h)\\in {\\mbox{$G_\\alpha $}},$ is well defined and a homomorphism.",
"As a consecuence, we can define $\\widetilde{C}_\\alpha :(G_\\alpha )_{SL}\\ \\oplus \\ H(2)\\longmapsto S^1 ,$ by means of $ \\widetilde{C}_{{\\mbox{$\\alpha $}}}(g,r)\\ =\\ (C_{{\\mbox{$\\alpha $}}})_{SL}(g)\\ e^{i\\pi Tr\\left(k{\\mbox{$\\varepsilon $}} \\overline{r}{\\mbox{$\\varepsilon $}}\\right)}.$ $ \\widetilde{C}_{{\\mbox{$\\alpha $}}}$ is an extension of $C_\\alpha $ to $(G_{{\\mbox{$\\alpha $}}})_{SL}\\ \\oplus \\ {{\\mbox{$ H(2)\\rm $}}},$ and a homomorphism .",
"Its differential coincides with the restriction of $\\alpha $ to the Lie algebra of this group.",
"The canonical action of $G$ on $G/KerC_\\alpha $ maps horizontal orbits to horizontal orbits, thus defining a transitive action on the space of horizontal orbits, ${\\cal W}_{\\alpha }.$ Let us consider the canonical map $\\tau :\\frac{G}{KerC_\\alpha } \\rightarrow {\\cal W}_{\\alpha }$ defined by sending each element of $G/KerC_\\alpha $ onto its horizontal orbit.",
"The isotropy subgroup at $\\tau (KerC_\\alpha )$ is $Ker\\widetilde{C}_\\alpha $ .",
"As a consequence, we can identify ${\\cal W}_{\\alpha }$ to $G/Ker\\,\\widetilde{C}_{\\alpha },$ by means of the bijective map $(A,H)\\,Ker\\,\\widetilde{C}_{\\alpha }\\in \\frac{G}{Ker\\,\\widetilde{C}_{\\alpha }}\\rightarrow \\tau ((A,H)Ker\\,C_{\\alpha }) \\in {\\cal W}_{\\alpha }.$ With the definitions given in section , we have seen that $\\frac{G}{Ker\\,C_{\\alpha }}\\left(\\frac{G}{G_{\\alpha }},{\\bf S}^1\\right)$ becomes a principal fibre bundle.",
"Similar definitions provides $\\frac{G}{Ker\\,\\widetilde{C}_{\\alpha }}\\left(\\frac{G}{(G_{\\alpha })_{SL}\\oplus H(2)},{\\bf S}^1\\right)$ with a structure of principal fibre bundle.",
"The bundle projection is the canonical map from ${G}/{Ker\\,\\widetilde{C}_{\\alpha }}$ onto ${G}/({(G_{\\alpha })_{SL}\\oplus H(2)}).$ The bundle action is given by $((A,H){Ker\\,\\widetilde{C}_{\\alpha }})*s=(A,H)(B,K){Ker\\,\\widetilde{C}_{\\alpha }},$ where $(B,K)$ is such that $\\widetilde{C}_{\\alpha }(B,K)=s.$ But the map $(A,H)((G_\\alpha )_{SL} \\oplus H(2))\\in \\frac{G}{(G_{\\mbox{$\\alpha $}})_{SL} \\oplus H(2)}\\rightarrow A\\,(G_{\\alpha })_{SL} \\in \\frac{SL}{(G_{\\alpha })_{SL}}$ is a diffeomorphism.",
"We identify these homogeneous spaces by means of this map, so that the principal fibre bundle becomes $\\frac{G}{Ker\\,\\widetilde{C}_{\\alpha }}\\left(\\frac{SL}{(G_{\\alpha })_{SL}},{\\bf S}^1\\right),$ where the bundle projection is $\\tau _2:(A,H){Ker\\,\\widetilde{C}_{\\alpha }}\\rightarrow A\\,(G_{\\alpha })_{SL}.$ The canonical maps define the homomorphism of principal ${\\bf S}^1 \\rm $-bundles given in Figure REF Figure: Fibre Bundles for Quantum StatesSince Quantum States correspond to pseudotensorial functions left invariant by the horizontal action, Quantum States are the pull back by $\\iota _1$ of unrestricted pseudotensorial functions on $G/Ker\\widetilde{C}_\\alpha $ ."
],
[
"Wave Functions",
"Now, let us associate, to each Quantum State in the sense of section , a Wave Function in the ordinary sense of Quantum Mechanics.",
"From now on, we assume, unless the contrary is explicitly stated, that the State Space is the orbit of $(0,\\alpha )$ in $H(2)\\times \\underline{G}^*.$ This choice do not carry very important consequences for the study of the free particle at the quantum level, in the sense that the other choices lead to isomorphic spaces of Quantum States (in all of its forms), and the same Wave Functions.",
"These facts are proved in Remark 5.2 of [8].",
"The isotropy subgroup at $(0,\\alpha ),$ $G_{(0,\\alpha )},$ is composed by the elements of $G_\\alpha $ of the form $(A,0)$ i.e.",
"$G_{(0,\\alpha )}=G_\\alpha \\cap (SL \\oplus \\lbrace 0\\rbrace ).$ Let $\\alpha =\\lbrace a,k\\rbrace $ and denote $SL_1=\\lbrace A\\in SL: AaA^{-1}=a\\rbrace ,$ $SL_2=\\lbrace A\\in SL: AkA^*=k\\rbrace .$ Then $G_{(0,\\alpha )}=(SL_1 \\cap SL_2)\\oplus \\lbrace 0\\rbrace ,$ Notice that $(G_\\alpha )_{SL}\\subset SL_2$ and $SL_1 \\cap SL_2\\subset (G_\\alpha )_{SL}$ so that $SL_1 \\cap SL_2=(G_\\alpha )_{SL}\\cap SL_1.$ But the map $(A,H)((G_\\alpha )_{SL}\\cap SL_1)\\oplus \\lbrace 0\\rbrace \\in \\frac{G}{((G_\\alpha )_{SL}\\cap SL_1)\\oplus \\lbrace 0\\rbrace }\\rightarrow $ $ \\rightarrow (H,A((G_\\alpha )_{SL}\\cap SL_1))\\in H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1}$ is a diffeomorphism.",
"Thus, State Space is the image of the injective map $i:(H,A(G_\\alpha )_{SL}\\cap SL_1))\\in H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1} \\rightarrow $ $\\rightarrow (H, Ad^*_{(A,H)}\\alpha )\\in H(2)\\times \\underline{G}^*.$ This image need not, a priori, be a proper submanifold of $H(2)\\times \\underline{G}^*,$ and we consider it provided with the topology and differentiable structure such that $i$ becomes a diffeomorphism.",
"In the following we identify each $i(X)$ with $X,$ so that we can say that $H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1}$ is State Space.",
"The canonical map from State Space onto Movement Space, can be generalised to all the homogeneous spaces appearing in the commutative diagram of Figure 1.",
"This will be done with the following geometrical construction.",
"Let $ \\cal L $ be a closed subgroup of G, and $ {\\cal S}=\\lbrace s\\in SL : (s,0) \\in {\\cal L} \\rbrace ,$ so that ${\\cal S} \\oplus \\lbrace 0\\rbrace = {\\cal L} \\cap (SL \\oplus \\lbrace 0\\rbrace ).$ Thus we define the map $(H,A {\\cal S})\\in {H(2) \\times \\frac{SL}{\\cal S} } \\stackrel{\\nu }{\\longrightarrow } (A,H) {\\cal L} \\in \\frac{G}{\\cal L}.$ This map is well defined and, if $H$ is a fixed element in $H(2)$ , its restriction to $ \\lbrace H\\rbrace \\times \\frac{SL}{\\cal S} $ is injective.",
"Now we apply this to the cases in Figure 1.",
"When ${\\cal L}=Ker\\,C_\\alpha $ we have ${\\cal S}=Ker(C_\\alpha )_{SL} \\cap SL_1$ , so that we have a map $(H,A ( Ker(C_\\alpha )_{SL} \\cap SL_1))\\in {H(2) \\times \\frac{SL}{ Ker(C_\\alpha )_{SL} \\cap SL_1 }} \\stackrel{\\nu _1}{\\longrightarrow }$ $\\stackrel{\\nu _1}{\\longrightarrow } (A,H) {Ker\\,C_\\alpha } \\in \\frac{G}{Ker\\,C_\\alpha }.",
"$ If ${\\cal L}=G_\\alpha $ we have ${\\cal S}= SL_1 \\cap SL_2=(G_{\\alpha })_{SL} \\cap SL_1 $ , and we thus obtain the map $(H,A ((G_{\\alpha })_{SL} \\cap SL_1 ))\\in {H(2) \\times \\frac{SL}{(G_{\\alpha })_{SL} \\cap SL_1 }} \\stackrel{\\nu _2}{\\longrightarrow }$ $\\stackrel{\\nu _2}{\\longrightarrow } (A,H) {G_\\alpha } \\in \\frac{G}{G_\\alpha }.",
"$ When ${\\cal L}=Ker{\\widetilde{C}}_\\alpha $ we have ${\\cal S}= Ker(C_\\alpha )_{SL}, $ so that we have a map $(H,A Ker(C_\\alpha )_{SL})\\in {H(2) \\times \\frac{SL}{ Ker(C_\\alpha )_{SL}}} \\stackrel{\\nu _3}{\\longrightarrow }(A,H) {Ker\\,\\widetilde{C}_\\alpha } \\in \\frac{G}{Ker\\,\\widetilde{C}_\\alpha }.",
"$ If ${\\cal L}=(G_\\alpha )_{SL} \\oplus H(2)$ we have ${\\cal S}= (G_\\alpha )_{SL} $ , and we thus have a map ${(H,A (G_\\alpha )_{SL}))\\in {H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}}} } \\stackrel{\\nu _4}{\\longrightarrow }$ $\\stackrel{\\nu _4}{\\longrightarrow } (A,H) {((G_\\alpha )}_{SL}\\oplus H(2)) \\in \\frac{G}{(G_\\alpha )_{SL}\\oplus H(2)}$ which, using the identification (REF ), can be written ${(H,A (G_\\alpha )_{SL}))\\in {H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}}} } \\stackrel{\\nu _4}{\\longrightarrow }$ $\\stackrel{\\nu _4}{\\longrightarrow } A (G_\\alpha )_{SL} \\in \\frac{SL}{(G_\\alpha )_{SL}}.$ When we denote by $\\iota _3,\\ \\iota _4,\\ \\tau _3,\\ \\tau _4,$ the canonical maps of homogeneous spaces that appears in Figure REF , we obtain the commutative diagram in that Figure.",
"Figure: Fibre Bundles for Wave FunctionsThe maps $\\tau _i$ are the bundle maps of principal fibre bundles whose structural groups are identified by means of $(C_\\alpha )_{SL},$ $\\widetilde{C}_\\alpha $ , or $C_\\alpha $ to subgroups of $S^1$ .",
"We already know the bundle actions of ${\\bf S}^1$ on $G/KerC_\\alpha $ and $G/Ker\\widetilde{C}_\\alpha .$ The other are defined in a similar way as follows.",
"The action for the principal bundle corresponding to $\\tau _4,$ $H(2) \\times \\frac{SL}{Ker(C_\\alpha )_{SL}}\\left(H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}},(C_\\alpha )_{SL}((G_\\alpha )_{SL})\\right),$ is given by $(H,A{Ker(C_\\alpha )_{SL}})*s=(H,AB{Ker(C_\\alpha )_{SL}})$ if $s\\in (C_\\alpha )_{SL}((G_\\alpha )_{SL}$ and $B\\in (G_\\alpha )_{SL}$ is such that $(C_\\alpha )_{SL}(B)=s.$ In the case of the principal bundle corresponding to $\\tau _3,$ $H(2) \\times \\frac{SL}{Ker(C_\\alpha )_{SL} \\cap SL_1}\\left(H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1},(C_\\alpha )_{SL}((G_\\alpha )_{SL}\\cap SL_1)\\right),$ the bundle action is defined in the same way but using the restriction of $(C_\\alpha )_{SL}$ to $(G_\\alpha )_{SL}\\cap SL_1.$ The pairs $(\\nu _1,\\nu _2),(\\nu _3,\\nu _4),\\ (\\iota _1,\\iota _2),\\ (\\iota _3,\\iota _4)$ define homomorphisms of principal fibre bundles.",
"In what concerns the structural groups, we have $(C_\\alpha )_{SL}((G_\\alpha )_{SL}\\cap SL_1) \\subset (C_\\alpha )_{SL}((G_\\alpha )_{SL})\\subset {\\mathbf {S}}^1$ and the homomorphism of structural groups is, in all cases, the canonical injection of the group of the first bundle into the group of the second.",
"As a consecuence of (REF ), the linear momentum, is given on the coadjoint orbit of $\\alpha $ by $P(Ad^{*}_{(A, H)} \\alpha )= -AkA^{*}$ , so that, with the identification of the coadjoint orbit with $G/G_\\alpha $ , we can write $P((A, H)G_\\alpha )=-AkA^{*}$ .",
"On the other hand, since $ (G_\\alpha )_{SL} \\subset SL_2,$ we can define in $ {SL}/{(G_\\alpha )_{SL}} $ a function, $P_0$ , by means of $P_0(A(G_\\alpha )_{SL})=-AkA^{*}$ .",
"But then, $P_0$ is the projection of $P$ by the canonical map $(A,\\ H)G_\\alpha \\in \\frac{G}{G_\\alpha } \\stackrel{\\iota _0}{\\longrightarrow } A (G_\\alpha )_{SL}\\in \\frac{SL}{(G_\\alpha )_{SL}},$ and will be denoted in the following simply by $P$ and also called linear momentum, thus we can write $P(A (G_\\alpha )_{SL})=-AkA^{*}$ where $k$ is given by $\\alpha =\\lbrace a,k\\rbrace $ .",
"Let us denote $(C_\\alpha )_{SL}(G_\\alpha )_{SL})$ by $S$ .",
"In [8] I prove that The pull back by $\\nu _3$ , maps in a one to one way the set of quantum states (considered as pseudotensorial functions on $G/Ker\\widetilde{C}_\\alpha $ onto the set composed by the functions on $H(2) \\times (SL/Ker(C_\\alpha )_{SL})$ having the form $ \\phi _f(H,A\\,Ker(C_\\alpha )_{SL})\\,=\\,f(A\\,Ker(C_\\alpha )_{SL})\\,e^{i\\pi Tr(P(A (G_\\alpha )_{SL})\\varepsilon \\overline{H}\\varepsilon )}$ where f is a pseudotensorial function on the principal fibre bundle $\\frac{SL}{Ker(C_\\alpha )_{SL}} \\left(\\frac{SL}{(G_\\alpha )_{SL}},S\\right) .$ In order to be more precise we will use the following definitions.",
"Let ${\\cal C}$ be the complex vector space composed by the pseudotensorial functions of the bundle $\\frac{SL}{\\mbox{$Ker\\,(C_\\alpha )_{SL}$}} \\left(\\frac{SL}{\\mbox{$(G_\\alpha )_{SL}$}},S\\right),$ and $\\widetilde{\\cal V}$ the complex vector space composed by the pseudotensorial functions on $G/Ker\\widetilde{C}_\\alpha .$ For each $f\\in {\\cal C}$ we define a pseudotensorial function on $G/Ker\\widetilde{C}_\\alpha ,$ $\\Psi _f,$ by $\\Psi _f \\circ \\nu _3 \\left(H,A\\,Ker(C_\\alpha )_{SL}\\right)= \\phi _f\\left(H,A\\,Ker(C_\\alpha )_{SL}\\right)=$ $=f(A\\,Ker(C_\\alpha )_{SL})\\,e^{i\\pi \\,Tr(P(A\\,(G_\\alpha )_{SL} )\\,\\varepsilon \\, \\overline{H}\\,\\varepsilon )}.$ The map $\\Psi :f\\in {\\cal C} \\rightarrow \\Psi _f \\in \\widetilde{\\cal V}$ is an isomorphism.",
"Also we define $\\Phi _f=\\Psi _f \\circ \\iota _1.$ These functions are pseudotensorial on $G/KerC_\\alpha $ and invariant by the horizontal action, so that they represent Quantum States in the most primitive sense adopted in this paper.",
"We denote by ${\\cal V}$ the complex vector space composed by these $\\Phi _f,$ so that the map $\\Phi :f\\in {\\cal C} \\rightarrow \\Phi _f \\in {\\cal V}$ is an isomorphism.",
"Thus, we can consider Quantum States as being elements of ${\\cal C}$ or elements of $\\widetilde{\\cal V}$ or elements of ${\\cal V}.$ Now we shall regard Quantum States under another form: Wave Functions.",
"The differentiable manifold $W \\stackrel{\\mathrm {d}ef}{=}( H(2) \\times (SL/ Ker\\,(C_\\alpha )_{SL} ) \\times _S \\,{\\bf C}$ is defined in the same way as we have defined $\\left( G/KerC_\\alpha \\right)\\times _{\\mbox{$\\bf S \\rm ^1$}}{\\mbox{{$C$}} \\rm }$ in section .",
"$W$ is the total space of the hermitian line bundle associated to the principal fibre bundle $\\left(H(2) \\times \\left(SL/Ker \\left(C_\\alpha \\right)_{SL}\\right)\\right)\\left(H(2) \\times \\left(SL/\\left(G_\\alpha \\right)_{SL} \\right),\\,S \\right)$ and the canonical action of $S$ on C .",
"The bundle projection is $\\eta :[(H,A Ker\\,(C_\\alpha )_{SL} ), c]_S \\in W \\rightarrow (H,A (G_\\alpha )_{SL} ) \\in H(2)\\times ( SL/(G_\\alpha )_{SL}),$ where $[(H,A Ker\\,(C_\\alpha )_{SL} ), c]_S$ is the orbit of $((H,A Ker\\,(C_\\alpha )_{SL} ), c)$ under the action of $S$ .",
"Since for each $f\\in {\\cal C}$ the function $\\Psi _f \\circ \\nu _3$ is pseudotensorial in $H(2) \\times \\left(SL/Ker \\left(C_\\alpha \\right)_{SL}\\right),$ it defines a section of $\\eta .$ This section can be considered as another description of the Quantum State given by $f$ .",
"To complete our way towards Wave Functions, we need to “represent” Quantum States as functions with values in a fixed complex vector space, not in a different vector space for each point in the base space, as does the sections of $\\eta $ .",
"If $(C_\\alpha )_{SL}(g)=1,\\ \\forall g\\in (G_\\alpha )_{SL},$ , we have $S=\\lbrace 1\\rbrace ,$ and $Ker\\,(C_\\alpha )_{SL}=(G_\\alpha )_{SL}$ so that $W \\approx ( H(2) \\times (SL/ (G_\\alpha )_{SL} ) \\times \\,{\\bf C},$ and $\\eta $ is the canonical projection onto the first two factors.",
"The section of $\\eta $ corresponding to the function $\\phi _f $ in (REF ) is $(H,\\,A\\,(G_\\alpha )_{SL})\\, \\longrightarrow (H,\\,A\\,(G_\\alpha )_{SL},\\phi _f(\\,H,\\,A\\,\\,(G_\\alpha )_{SL})).$ Thus, if $(C_\\alpha )_{SL}={\\bf 1 },$ our task is accomplished by $\\phi _f(\\,H,\\,A\\,Ker\\,{\\mbox{$(C_\\alpha )_{SL}$}})\\,=\\,f(\\,A\\,Ker\\,{\\mbox{$(C_\\alpha )_{SL}$}})\\,e^{i\\pi \\,Tr(P(A\\,(G_\\alpha )_{SL} )\\,\\varepsilon \\, \\overline{H}\\,\\varepsilon }$ itself as a complex valued function on the base space.",
"In this case, $\\phi _f$ is called the Prewave Function associated to $f,$ and denoted by $\\psi _f.$ In the case where ${\\mbox{$(C_\\alpha )_{SL}$}}$ is not trivial, our goal will be attained by imbedding the hermitian fibre bundle in a trivial one.",
"We do this in a direct way, but a more geometrical view of the method is exposed in remark 5.1 of [8].",
"A key concept in our construction of Wave Functions is the following: A Trivialization of $ C_\\alpha $ is a triple $(\\rho ,\\ {L},\\ z_0)$ , where $ { L}$ is a finite dimensional complex vector space, $z_0\\in { L}$ and $\\rho $ is a representation of $SL$(2,C) in ${ L}$ such that $ &1)&\\ \\rho (A)(z_0)\\,=\\,(C_\\alpha )_{SL}(A)\\ z_0,\\ \\ \\ \\forall A\\in (G_\\alpha )_{SL}.",
"\\nonumber \\\\&2)&\\ {\\rm The \\ isotropy \\ subgroup\\ at}\\ z_0\\ {\\rm is}\\ Ker\\,(C_\\alpha )_{SL}.$ In what follows, we assume that a trivialization of $C_\\alpha $ is given.",
"The homogeneous space $ { SL}/{\\mbox{$Ker\\,(C_\\alpha )_{SL}$}} $ is identified to the orbit of $z_0$ , $\\cal B$ , by means of $A Ker\\,(C_\\alpha )_{SL} \\in \\frac{ SL}{Ker\\,(C_\\alpha )_{SL}} \\longrightarrow \\rho (A)(z_0)\\in {\\cal B}.$ The action of $S$ on $ { SL}/{\\mbox{$Ker\\,(C_\\alpha )_{SL}$}} $ becomes, with this identification, multiplication in $L$ of elements of $S,$ as complex numbers, by elements of ${\\cal B},$ as elements of $L$ .",
"The canonical map from $ { SL}/{\\mbox{$Ker\\,(C_\\alpha )_{SL}$}} $ onto $ {SL}/{\\mbox{$(G_\\alpha )_{SL}$}} $ will be denoted by r, and is given by $r\\left(\\rho (A)(z_0)\\right)=A {\\mbox{$(G_\\alpha )_{SL}$}} .$ Each $f\\in {\\cal D}$ thus becomes a function on ${\\cal B},$ homogeneous of degree $-1$ under ordinary multiplication by elements of $S$ .",
"The functions having these characteristics, will be called S-homogeneous of degree -1.",
"The $S$ -homogeneous of degree T functions are defined in a similar way.",
"Let us denote by ${\\cal C}$ the complex vector space composed by the S-homogeneous of degree -1 functions on ${\\cal B}$ .",
"Now, $W$ is $( H(2) \\times {\\cal B} ) \\times _S \\,{\\bf C}$ , and $\\eta $ maps $[(H,z),c]$ onto $(H,r(z)).$ The sections of $\\eta $ corresponding to the functions having the form of $\\phi _f$ in (REF ) are as follows.",
"Let $f$ be a $S$ -homogeneous of degree -1 function.",
"The corresponding section, $\\sigma ,$ maps $(H,m)$ to $\\sigma (H,m)=[(H,z),\\phi _f(H,z)]$ where $z$ is arbitrary in $r^{-1}(m).$ We define a map, $\\chi $ , from W into $ H(2) \\times ( SL/(G_\\alpha )_{SL}) \\times \\ L $ , by sending $[(H,z ), c]_S \\in ( H(2) \\times {\\cal B}) ) \\times _S \\,{\\bf C}$ to $ (H,\\ r(z), cz)$ .",
"The map $\\chi $ is injective .",
"In fact the relation $\\chi ([(H,z),c]_S)=\\chi ([(H^\\prime ,z^\\prime ),c^\\prime ]_S )$ is equivalent to $(H,r(z),cz)=(H^\\prime ,r(z^\\prime ),c^\\prime z^\\prime )$ so that there exist $e^{i\\gamma }\\in S$ such that $H^\\prime =H,$ $z^\\prime =e^{i\\gamma } z,$ and $c^\\prime z^\\prime =cz$ .",
"Thus $[(H^\\prime ,z^\\prime ),c^\\prime ]_S=[(H,ze^{i\\gamma } ),ce^{-i\\gamma } ]_S=[(H,z),c]_S.$ The fiber of $W$ on $(H,m)$ is $\\lbrace [(H,z ),c]: c\\in {\\bf C}\\rbrace $ , where $z$ is any fixed element in $r^{-1}(m).$ Its image under $\\chi $ is composed by the $(H,m,y)$ such that $y$ is in the one dimensional subspace of $L$ generated by $z.$ We have thus inmersed our, in general, non trivial bundle, in a trivial one with fiber $L$ .",
"This enable us to identify sections of $\\eta $ with functions with values in $L,$ as follows.",
"The section $\\sigma $ of $\\eta $ , that corresponds to $f$ can be identifed with $\\chi \\circ \\sigma (H,m) = \\left(H,m,\\phi _f(z)z\\right),$ where $z$ is arbitrary in $r^{-1}(m).$ The right hand side in the preceeding equation is completely determined by its third component: $\\psi _f(H,m)= \\phi _f(z)z=f(z)\\,e^{i\\pi \\,Tr(P(m )\\,\\varepsilon \\, \\overline{H}\\, \\varepsilon )}z.$ The function $\\psi _f,$ with $f$ $S$ -homogeneous of degree -1, will be called Prewave Function associated to $f$ .",
"The complex vector space composed by the Prewave Functions is denoted by PW, so that the map $\\psi :f\\in {\\cal C}\\ \\rightarrow \\psi _f\\in {\\cal PW}$ is an isomorphism.",
"The composition of any Prewave Function with $\\iota _4$ is a function in State Space that can be considered as giving an amplitude of probability for each state.",
"As said in the Introduction, we obtain from this Prewave Function a Wave Function, defined on space-time points ($i.e.$ hermitian matrices), by adding up, for each $H \\in H(2),$ the amplitudes of probability corresponding to the states $(H,\\beta )$ where $\\beta $ is a movement whose portrait in space-time contains $H$ .",
"According to (REF ), we see that these states are the elements of the set $\\lbrace (H,Ad^*_{(B,H)}\\alpha : B\\in SL\\rbrace ,$ that is identified by $i$ ($c.f.$ (REF )) with $\\lbrace H\\rbrace \\times {\\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1}},$ whose image by $\\iota _4$ is $\\lbrace H\\rbrace \\times \\frac{SL}{(G_\\alpha )_{SL}}.$ Then to any given Prewave Function, $\\psi _f$ , we associate a Wave Function, $\\widetilde{ \\psi _f}$ , as follows $\\widetilde{\\psi _f}(H)=\\int _{{SL}/(G_\\alpha )_{SL}} {\\mbox{$\\psi $}}_f(H,\\,m)\\ {\\mbox{$\\omega $}}_m$ where $m$ is a generic element in $ { SL}/({\\mbox{$G_\\alpha $}})_{SL},$ and $\\omega $ is a volume element on $ { SL}/({\\mbox{$G_\\alpha $}})_{SL},$ left invariant by the canonical action of $SL$ on $ { SL}/({\\mbox{$G_\\alpha $}})_{SL}.$ This means that, if we denote by $d^{\\prime }_A$ the diffeomorphism of $SL/(G_\\alpha )_{SL}$ defined by sending $B\\,(G_\\alpha )_{SL}$ to $AB\\,(G_\\alpha )_{SL},$ then $(d^{\\prime }_A)^*\\omega =\\omega .$ In the following, we assume that such a invariant volume element is given.",
"This definition of Wave Functions, forces us to do a restriction on the class of the functions to be considered: it is necessary that the integral exists.",
"In what follows I consider only Quantum States corresponding to the functions in ${\\cal C}$ that are continuous with compact support.",
"The complex vector space composed by these Wave Functions is denoted by ${\\cal WF}.$ Of course, there are other possible conditions that can be imposed on $f$ in order to assure integrability.",
"Also, if one obtains for some choice a Prehilbert space, one can consider its completed, in order to have a Hilbert Space.",
"But in this paper I prefer to maintain the \"continuous with compact support\" condition.",
"The Wave Functions are another form of description of Quantum States, in fact it is the most usual in Quantum Mechanics.",
"In the following I change the notation in such a way that, ${\\cal C}$ stands for the complex vector space composed by the $S$ -homogeneous of degree -1 functions on ${\\cal B}$ that also are continuous with compact support, and ${\\cal PW},\\ {\\cal V},\\ \\widetilde{\\cal V},$ the corresponding isomorphic spaces.",
"Remark 7.1 Now, let us assume that we know a section of the map $r,$ defined an open set, $D,$ in ${SL/(G_\\alpha )_{SL}}$ , $\\sigma : D \\longrightarrow {\\cal B}.$ In this case, we can associate a prewave function, and thus a wave function, to each $C^\\infty $ function, with compact support contained in $D,$ as follows.",
"Let $f_0$ be a $C^\\infty $ function, with compact support contained in $D$ .",
"We define a function on $\\cal B$ , $f$ , by means of $f(z)={\\left\\lbrace \\begin{array}{ll}t^{-1} f_0(r(z))& {\\rm if}\\ z\\in r^{-1}(U)\\ {\\rm where}\\ t \\ {\\rm is\\ such\\ that}\\ z= t \\sigma (r(z))\\\\0& {\\rm if}\\ z\\notin r^{-1}(U)\\end{array}\\right.",
"}$ This function is $S$ -homogeneous of degree -1 since, for all $e^{i a}\\in S,\\ z\\in {r^{-1}(U)},$ we have $f(e^{i a}z)=t^{\\prime -1} f_0(r(e^{i a}z)),$ where $t^{\\prime }$ is such that $e^{i a}z=t^{\\prime } \\sigma (r(e^{i a}z))=t^{\\prime } \\sigma (r(z))=t^{\\prime } t^{-1} z,$ so that $e^{i a}=t^{\\prime } t^{-1}$ and $f(e^{i a}z)=e^{-i a}t^{-1} f_0(r(e^{i a}z))=e^{-i a}t^{-1} f_0(r(z))=e^{-i a}f(z).$ If $z\\notin {r^{-1}(U)}$ , $f(e^{i a}z)=0=f(z)=e^{-i a}f(z)$ .",
"The Prewave Function $\\psi _f$ then is $\\psi _f(H,\\ m)={\\left\\lbrace \\begin{array}{ll}f_0(m)\\,e^{i\\pi Tr(P(m)\\,\\varepsilon \\overline{H} \\varepsilon )}\\ \\sigma (m) & \\mathrm {if}\\ m\\in U \\\\0& \\mathrm {if}\\ m\\notin U\\end{array}\\right.",
"}$ and we have an associated Wave Function, $\\tilde{\\psi }_f.$ This expression contains no reference to any homogeneous function and suggest directly the usual form of Wave Functions."
],
[
" Representation on $S$ -homogeneous functions of degree -1 ",
"If $(C_\\alpha )_{SL}={\\bf 1},$ , the $S$ -homogeneous functions of degree -1 are simply functions on $SL/(G_\\alpha )_{SL}.$ In this case we denote by $\\cal C$ the complex vector space composed by the continuous functions on $SL/(G_\\alpha )_{SL}$ with compact support.",
"For all $B\\in SL$ we denote by $d^\\prime _B$ the canonical diffeomorphism of $SL/(G_\\alpha )_{SL}$ given by $d^\\prime _B(A (G_\\alpha )_{SL})=BA (G_\\alpha )_{SL}).$ Then, we define a representation, $\\delta ,$ of $SL$ on $\\cal C$ by $\\delta (f)=f\\circ d^\\prime _{A^{-1}}.$ In $\\cal C$ we also define an hermitian product by $\\langle f,\\, f^\\prime \\rangle = \\int _{{SL}/(G_\\alpha )_{SL}} \\overline{f} f^\\prime \\ {\\mbox{$\\omega $}} ,$ where $\\omega $ is an invariant volume element on $SL/(G_\\alpha )_{SL}.$ With this inner product, $\\cal C$ becomes a prehilbert space.",
"The invariance of $\\omega $ enable us to write $\\langle \\delta (A)(f),\\, \\delta (A)(f^\\prime ) \\rangle =\\langle f,\\, f^\\prime \\rangle $ so that the representation $\\delta $ is unitary.",
"In case $(C_\\alpha )_{SL}\\ne {\\bf 1},$ , we assume that we have a trivialization, $(\\rho ,\\ {L},\\ z_0),$ and we define ${\\cal B}$ and ${\\cal C}$ as in the preceeding section.",
"Of course, also in case $(C_\\alpha )_{SL}={\\bf 1},$ we can have a trvialization and thus apply all that follows.",
"In ${\\cal C}$ we define a hermitian product: $\\langle f,\\, f^\\prime \\rangle = \\int _{{SL}/(G_\\alpha )_{SL}} \\overline{f} f^\\prime \\ {\\mbox{$\\omega $}} , $ where by $ \\overline{f} f^\\prime $ we means the function defined on $SL/(G_\\alpha )_{SL},$ by $\\overline{f} f^\\prime (m)=\\overline{f}(z) f^\\prime (z)$ for all $m\\in SL/(G_\\alpha )_{SL},$ where $z$ is arbitrary in $r^{-1}(m),$ and $\\omega $ is an invariant volume element on $SL/(G_\\alpha )_{SL}.$ Also in this case, $\\cal C$ becomes a prehilbert space.",
"A representation, $\\delta ,$ of $SL$ on $\\cal C,$ is defined by $\\delta (A): f\\in {\\cal C} \\longrightarrow f\\circ \\rho (A^{-1})\\in {\\cal C}.$ for all $A\\in SL.$ If $A\\in SL$ we have $\\langle \\delta (A)(f),\\, \\delta (A)(f^\\prime ) \\rangle = \\int _{{ SL}/(G_\\alpha )_{SL}} \\left((\\overline{f} f^\\prime )\\circ d^{\\prime }_{A^{-1}} \\right)\\ {\\mbox{$\\omega $}}$ so that the invariance of $\\omega $ enable us to write $\\langle \\delta (A)(f),\\, \\delta (A)(f^\\prime ) \\rangle =\\langle f,\\, f^\\prime \\rangle $ We see that the representation $\\delta $ is, also in this case, unitary.",
"Also we have a representation, $\\delta ^\\prime ,$ of $SL \\oplus H(2)$ on $ {\\cal C}$ defined by: $(\\delta ^\\prime (A,H)\\cdot f)(z)= f( \\rho (A^{-1})\\cdot z) Exp\\left(-i\\pi Tr\\left(\\,P( r(z)) \\varepsilon \\overline{H} \\varepsilon \\right)\\right),$ where $(A,H)\\in SL \\oplus H(2),\\ f\\in {\\cal C},$ and $z\\in {\\cal B}.$ To prove that $\\delta ^\\prime $ is a representation , notice that $r$ is equivariant i.e.",
"$r(\\rho (A)\\cdot z)=d^{\\prime }_A(r(z))$ for all $A\\in SL.$ Formula (REF ), leads to $P(d^{\\prime }_B(A(G_\\alpha )_{SL}))=P(BA(G_\\alpha )_{SL})=-BAkA^*B^*=BP(A(G_\\alpha )_{SL})B^*$ so that $P(r(\\rho (A)\\cdot z))=P(d^{\\prime }_A(r(z)))=AP(r(z))A^*.$ For all $A\\in SL,$ $A\\, \\varepsilon = \\varepsilon \\,{}^t\\hspace{0.0pt}A^{-1}$ Now, we can prove that $\\delta ^\\prime $ is a representation as follows.",
"If $(B,K),(A,H)\\in SL \\oplus H(2),\\ z\\in {\\cal B},$ then ${\\begin{array}{c}\\delta ^\\prime (B,K)\\cdot \\left(\\left(\\delta ^\\prime (A,H)\\cdot f\\right)\\right)(z)=\\left(\\delta ^\\prime (A,H)\\cdot f\\right)(\\rho (B^{-1})\\cdot z) \\\\Exp\\left(-i\\pi Tr[P(r( z))\\varepsilon \\overline{K} \\varepsilon ]\\right)=f(\\rho (A^{-1})\\cdot (\\rho (B^{-1})\\cdot z))\\\\Exp\\left(-i\\pi Tr[P(r(\\rho (B^{-1})\\cdot z))\\varepsilon \\overline{H} \\varepsilon + P(r( z))\\varepsilon \\overline{K} \\varepsilon ] \\right)=\\\\=f(\\rho ((BA)^{-1})\\cdot z) Exp\\left(-i\\pi Tr[P(r(z))\\varepsilon \\overline{(B HB^* + K) }\\varepsilon ]\\right)=\\\\=\\left(\\delta ^\\prime \\left(\\left(B,K\\right)\\left(A,H\\right)\\right)\\cdot f\\right)\\left( z\\right)\\end{array}}$ The infinitesimal generator of $\\delta $ associated to $a\\in sl(2,\\bf C)$ is the linear map from $\\cal C$ into itself, $d\\,\\delta (a)$ given by $(d\\,\\delta (a) \\cdot f) (z)=\\left(\\frac{d}{dt}\\right)_{t=0} \\left(\\delta \\left(e^{t\\,a}\\right)\\cdot f \\right) \\left(z\\right).$ Then $(d\\,\\delta (a) \\cdot f) (z)=\\left(\\frac{d}{dt}\\right)_{t=0} \\left(f \\left(\\rho \\left(e^{-t\\,a}\\right) \\cdot z\\right) \\right),$ and thus we see that $d\\,\\delta (a)$ acts on $f$ as the vector field, $X_a,$ infinitesimal generator of the action on $\\cal B,$ defined by $A*z=\\rho (A) \\cdot z,$ associated to $a$ .",
"We can give a more explicit form of $X_a$ as follows.",
"Let us fix a basis, $\\beta =\\lbrace e_1, \\dots ,e_q\\rbrace ,$ of $L.$ By means of $\\beta $ we identify $L$ with ${\\bf C}^q$ .",
"Let us denote by ${}^t(z^1,\\dots ,z^q)$ the matrix of $z\\in L$ in the basis $\\beta $ and $(d\\rho (a))_i^j,$ the element in the file $j$ column $i$ of the matrix of $(d\\rho (a))$ in the basis $\\beta .$ We denote by $\\beta ^*=\\lbrace w^1, \\dots ,w^q\\rbrace $ the system of (complex) coordinates associated to $\\beta $ (i.e.",
"$\\beta ^*$ is the dual basis of $\\beta .$ ) If $w^j_x$ (resp $w^j_y$ ) is the real (resp.",
"imaginary) part of $w^j,$ it is usually writen $\\frac{\\partial }{\\partial w^j}=\\frac{1}{2}\\left(\\frac{\\partial }{\\partial w^j_x}-i\\ \\frac{\\partial }{\\partial w^j_y}\\right),$ $\\frac{\\partial }{\\partial \\overline{w}^j}=\\frac{1}{2}\\left(\\frac{\\partial }{\\partial w^j_x}+i\\ \\frac{\\partial }{\\partial w^j_y}\\right).$ Thus, when $f$ is a complex valued differentiable function on an open subset, M, of ${\\bf C}^q$ , and $v(t)$ a differentiable map from an open neigbourhood of 0 in ${\\bf R}$ into M, we have, using summation convention ${\\begin{array}{c}\\left( \\frac{d}{dt}\\right)_0 (f\\circ v) =\\left(\\frac{\\partial \\, f}{\\partial w^j_x}\\right)_{v(0)}\\, (w_x^j(v(t)))^{\\prime }(0)+\\left(\\frac{\\partial \\, f}{\\partial w^j_y}\\right)_{v(0)}\\, (w_y^j(v(t)))^{\\prime }(0)=\\\\=\\left(\\frac{\\partial \\, f}{\\partial w^j}\\right)_{v(0)}\\, (w^j(v(t)))^{\\prime }(0)+\\left(\\frac{\\partial \\, f}{\\partial \\overline{w}^j}\\right)_{v(0)}\\, (\\overline{w}^j(v(t)))^{\\prime }(0).\\end{array}}$ Since $\\left(\\frac{d}{dt}\\right)_{t=0} \\left(f \\left(\\rho \\left(e^{-t\\,a}\\right) \\cdot z\\right) \\right)=\\left(\\frac{d}{dt}\\right)_{t=0} \\left(f \\left(\\left(e^{-t\\,d\\rho \\left(a\\right)}\\right) \\cdot z\\right) \\right)$ it follows that ${\\begin{array}{c}\\left(X_a)_z(f) \\right)=\\left(\\frac{\\partial \\, f}{\\partial w^j}\\right)_{z}\\, (-(d\\rho (a))^j_i\\,z^i)+\\left(\\frac{\\partial \\, f}{\\partial \\overline{w}^j}\\right)_{z}\\, ({-(\\overline{d\\rho (a)})} ^j_i\\,\\overline{z}^i)\\end{array}}$ so that an extension of $X_a$ from $\\cal B$ to $\\mathbb {C}^q$ is $X_a=-\\left(w^i \\,(d\\rho (a))^j_i\\,\\frac{\\partial }{\\partial w^j}+\\overline{w}^i \\,(\\overline{d\\rho (a)})^j_i\\,\\frac{\\partial }{\\partial \\overline{w}^j} \\right).$ If $(C_\\alpha )_{SL}={\\bf 1},$ , and do not use a trivialization, we have similar results, but $X_a$ is the infinitesimal generator of the canonical action of $SL$ on $SL/(G_\\alpha )_{SL},$ associated to $a.$ Thus, in all cases $d\\,\\delta (a) \\cdot f=X_{a}(f),$ or simply, as linear maps on $\\cal C$ , $d\\,\\delta (a) =X_{a}.$ On the other hand, the infinitesimal generator of $\\delta ^{\\prime }$ associated to $(a,h)\\ \\ \\in sl(2,\\bf C)\\oplus H(2)$ is the endomorphism, $d\\,\\delta ^{\\prime }(a,h)$ of $\\cal C$ given by $(d\\,\\delta ^{\\prime }(a,h) \\cdot f) (z)=\\left(\\frac{d}{dt}\\right)_{t=0} \\left(\\delta ^{\\prime }\\left(Exp({t\\,(a,h))}\\right)\\cdot f \\right) \\left(z\\right),$ but the exponential map in $SL\\oplus H(2)$ is given by (REF ), so that $(d\\,\\delta ^{\\prime }(a,h) \\cdot f) (z)=\\left(\\frac{d}{dt}\\right)_{t=0} \\left(\\delta ^{\\prime }\\left(\\left(e^{ta},\\int _0^t e^{sa}\\,h\\, e^{sa^*}\\,ds \\right)\\right)\\cdot f \\right) \\left(z\\right),$ and a short computation thus leads to ${\\begin{array}{c}d\\,\\delta ^{\\prime }(a,h) \\cdot f= X_a(f)-i\\pi Tr[P(r(\\cdot ))\\varepsilon \\overline{h}\\varepsilon ]\\ f=\\\\=X_a(f)+2\\pi i \\langle P(r(\\cdot )),h \\rangle \\ f,\\end{array}}$ where I have used the same symbol for any hermitian matrix and the corresponding element in ${\\bf R}^4,$ and $\\langle \\ ,\\ \\rangle $ is Minkowski product.",
"Also, we can write $d\\,\\delta ^{\\prime }(a,h)=X_a+2\\pi i \\langle P(r(\\cdot )),h \\rangle .$ where $X_a$ is the endomorphism given by the vectorfield (REF ) , and $2\\pi i\\langle P(r(\\cdot )),h \\rangle $ is the endomorphism given by ordinary multiplication by this function.",
"Instead of the infinitesimal generators $d\\,\\delta ^{\\prime }(a,h),$ we can use $(a,h)^{\\theta }\\stackrel{\\mathrm {d}ef}{ =}\\frac{1}{2 \\pi i}\\ d\\,\\delta ^{\\prime }(a,h) ,$ so that $(a,h)^{\\theta } =\\frac{1}{2 \\pi i}\\,X_a+ \\langle P(r(\\cdot )),h \\rangle .$ The endomorphism $(a,h)^{\\theta },$ is hermitian for our hermitian product and is the Quantum Operator associated to the Dynamical Variable (a,h), when Quantum States are represented by $S$ -homogeneous functions.",
"When Quantum States are represented by Wave Functions, the Quantum Operators acquires its usual form in Quantum Mechanics (c.f.",
"(REF )).",
"In the particular case of the Linear Momentum (c.f.",
"(REF )), one sees that $(P^k)^{\\theta }\\cdot f=(P^k \\circ r)\\,f,\\ \\ k=1,\\,2,\\,3,\\,4.$ where the $P^k$ in the left hand side are the components of Linear momentum as a dynamical variable, and the ones in the right hand side, the components of Linear momentum considered as a function on $SL/(G_\\alpha )_{SL},$ given by (REF ).",
"In what concerns to the Quantum Operators corresponding to Angular Momentum we have $(l^j)^{\\theta }= \\frac{1}{4\\pi i} \\,X_{i\\,\\sigma _j}, \\ \\ j=1,\\,2,\\,3.$ $(g^j)^{\\theta }=\\frac{1}{4\\pi i} \\,X_{\\sigma _j}, \\ \\ j=1,\\,2,\\,3.$ Of course, the representations on ${\\cal C}$ lead to equivalent representations on the isomorphic vector spaces $ {\\cal V},\\ \\widetilde{\\cal V},$ and ${\\cal PW}$ .",
"We do that in detail in ${\\cal PW}$ and ${\\cal V}$ as follows:"
],
[
"Representation on Prewave Functions.",
"In ${\\cal PW}$ we define an hermitian product by $\\langle \\psi _f ,\\psi _{f^{\\prime }} \\rangle \\stackrel{def}{=}\\langle f ,{f^{\\prime }} \\rangle ,$ and thus becomes a prehilbert space, isomorphic as inner product spaces to ${\\cal C.}$ The hermitian product can be given in terms of the prewave functions themselves, as follows.",
"Let $\\beta $ be a sesquilinear form on $L,$ nonvanishing on ${\\cal B}$ .",
"We define $ \\psi _f\\, \\beta \\,\\psi _{f^\\prime } \\,: m \\in SL/(G_\\alpha )_{SL} \\mapsto \\frac{\\beta ( \\psi _f ( H,\\,m),\\psi _{f^\\prime } ( H,\\,m ))}{\\beta (z,\\ z )} $ where $z$ is arbitrary in ${\\bf r}^{-1}(m)$ and $H$ is arbitrary in $H(2)$ .",
"Thus $\\langle \\psi _f,\\ \\psi _{f^\\prime } \\rangle =\\int _{{SL}/(G_\\alpha )_{SL}}\\psi _f\\, \\beta \\,\\psi _{f^\\prime }\\ \\omega .$ The representation of $G$ on ${\\cal PW}$ equivalent to $\\delta ^\\prime $ under $\\psi $ is given by $\\delta ^{pw}(A,H)\\cdot \\psi _f=\\psi _{\\delta ^\\prime (A,H) \\cdot f}.$ The infinitesimal generator associated to $(a,h)\\ \\ \\in sl(2,\\bf C)\\oplus H(2)$ is $d\\delta ^{pw}(a,h)\\cdot \\psi _f=\\psi _{d\\delta ^{\\prime }(a,h)\\cdot f}.$ The Quantum Operators for Prewave Functions must be defined as $(a,h)^{\\upsilon }\\stackrel{\\mathrm {d}ef}{=}\\ \\frac{1}{2\\pi i}\\,d\\delta ^{pw}(a,h),$ and we have $(a,h)^{\\upsilon }\\cdot \\psi _f{=}\\ \\psi _{(a,h)^{\\theta }\\cdot f}.$ On the other hand, we have a natural action on $H(2)\\times SL/(G_\\alpha )_{SL}$ defined by $(B,K)* (H,A\\,(G_\\alpha )_{SL})=(BHB^*+K,BA (G_\\alpha )_{SL}).$ Thus we define an, a priori different, representation, $\\delta _{pw},$ on ${\\cal PW}$ by means of $\\delta _{pw}(A,H) \\cdot \\psi _f =\\rho (A) \\circ \\psi _f \\circ \\left((A,H)*\\right)^{-1},$ i.e., for all $(K,m)\\in H(2)\\times SL/(G_\\alpha )_{SL}$ $\\left(\\delta _{pw}(A,H) \\cdot \\psi _f \\right)(K,m) =\\rho (A) \\cdot \\psi _f \\left((A,H)^{-1}*(K,m)\\right).$ Using (REF ) and (REF ) one can see that $\\delta _{pw}=\\delta ^{pw}$ as follows ${\\begin{array}{c}\\left(\\delta ^{pw}(A,H) \\cdot \\psi _f \\right)(K,m)=\\psi _{\\delta ^\\prime \\left( A,H\\right)\\cdot f}(K,m)= f(\\rho (A^{-1})\\cdot z) \\\\Exp\\left(- i\\pi Tr [ P( m))\\varepsilon \\overline{H} \\varepsilon ]\\right)Exp\\left( i\\pi Tr [ P( m)\\varepsilon \\overline{K} \\varepsilon ]\\right) z=(*)\\end{array}}$ where $z\\in r^{-1}(m).$ If we denote $ z^{\\prime }=\\rho (A^{-1})\\cdot z ,$ we have ${\\begin{array}{c}(*)= f( z^{\\prime }) Exp\\left( i\\pi Tr [ P( r(\\rho (A)\\cdot z^{\\prime }))\\varepsilon \\overline{(K-H)} \\varepsilon ]\\right)( \\rho (A)\\cdot z^{\\prime }) =\\\\=f( z^{\\prime }) Exp\\left( i\\pi Tr [ P( r( z^{\\prime }))\\varepsilon \\overline{(A^{-1}(K-H)(A^{-1})^*)} \\varepsilon ]\\right) (\\rho (A)\\cdot z^{\\prime })=\\\\=\\rho (A) \\psi _f((A,H)^{-1}*(K,m) =\\left(\\delta _{pw}(A,H) \\cdot \\psi _f \\right)(K,m).\\end{array}}$"
],
[
"Representation on Wave Functions.",
"Now, let us consider ${\\cal WF},$ the complex vector space composed by the Wave Functions $\\widetilde{ \\psi _f}$ such that $f\\in {\\cal C}.$ A \"translation\" of the preceeding representations of $SL\\oplus H(2)$ to ${\\cal WF},$ is given by $\\delta _{w}(A,H) \\cdot \\widetilde{\\psi _f} = \\lbrace \\delta _{pw}(A,H) \\cdot \\psi _{ f}\\rbrace {\\widetilde{}}= \\lbrace \\psi _{\\delta ^{\\prime }(A,H)\\cdot f}\\rbrace {\\widetilde{}},$ where $\\lbrace \\psi \\rbrace {\\widetilde{}}$ means $\\widetilde{\\psi }.$ Then $\\delta _{w}(A,H) \\cdot \\widetilde{\\psi _f}(K)&=& \\lbrace \\delta _{pw}(A,H) \\cdot \\psi _{ f}\\rbrace {\\widetilde{}}(K)\\\\&=& \\int _{SL/(G_\\alpha )_{SL}}\\rho (A) \\cdot \\psi _f((A,H)^{-1}*(K,m))\\omega _m=\\\\&=& \\rho (A) \\cdot \\int _{SL/(G_\\alpha )_{SL}} \\psi _f((A,H)^{-1}*K,d^{\\prime }_{A^{-1}}m))\\omega _m=\\\\&=& \\rho (A) \\cdot \\int _{SL/(G_\\alpha )_{SL}} \\psi _f((A,H)^{-1}*K,m))\\omega _m$ because of the invariance of $\\omega .$ Thus $\\left(\\delta _{w}(A,H) \\cdot \\widetilde{\\psi _f}\\right)(K)=\\rho (A) \\cdot \\widetilde{\\psi _f } \\left((A,H)^{-1}*K)\\right).$ Compare to (REF ).",
"Let us denote by $d\\delta _{w}\\cdot (a,h)$ the infinitesimal generator of the representation $\\delta _w$ associated to $ (a,h) \\in sl(2, C)\\oplus { H(2)},$ and $\\widehat{ (a,h)} \\stackrel{def}{=}\\frac{1}{ 2 \\pi i } (d\\delta _{w}\\cdot (a,h) ).$ The endomorphism $\\widehat{ (a,h)}$ of ${\\cal WF}$ is the Quantum Operator corresponding to the Dynamical Variable $(a,h)$ when the Quantum States are represented by Wave Functions.",
"A straightforward computation leads to the following expresions for the operators corresponding to Linear and Angular Momentum $\\widehat{ P^k} \\cdot \\widetilde{ \\psi _f }&=&\\frac{1}{2\\pi i}\\frac{\\partial }{\\partial x^k} \\ \\widetilde{ \\psi _f } \\nonumber \\\\\\widehat{ P^4} \\cdot \\widetilde{ \\psi _f } &=&\\frac{i}{2\\pi }\\frac{\\partial }{\\partial x^4} \\ \\widetilde{ \\psi _f } \\nonumber \\\\\\widehat{ l^k} \\cdot \\widetilde{ \\psi _f }&=&\\frac{1}{2\\pi i}\\left( d\\rho \\left( {\\frac{i\\sigma _k}{2}}\\right)+\\sum _{j,r=1}^3\\varepsilon _{kjr} x^j\\frac{\\partial }{\\partial x^r} \\right) \\ \\widetilde{ \\psi _f } \\\\\\widehat{ g^k} \\cdot \\widetilde{ \\psi _f } &=&\\frac{1}{2\\pi i}\\left( d\\rho \\left(\\frac{\\sigma _k}{2}\\right)- \\left( x^4\\frac{\\partial }{\\partial x^k}+ x^k\\frac{\\partial }{\\partial x^4}\\right)\\right)\\ \\widetilde{ \\psi _f } \\nonumber $ where $\\varepsilon _{ijk}$ are the components of an antisymmetric tensor such that $\\varepsilon _{123}=1$ ."
],
[
"Representation on Pseudotensorial Functions in the contact manifold.",
"Recall the isomorphism (REF ) $ \\nonumber \\Phi :f\\in {\\cal C} \\rightarrow \\Phi _f \\in {\\cal V}$ If $f\\in {\\cal C},$ we have $\\Phi _f((A,H)KerC_\\alpha )=\\,f(A\\,Ker(C_\\alpha )_{SL}) \\,e^{i\\pi \\,Tr(P (A\\,(G_\\alpha )_{SL}) \\,\\varepsilon \\, \\overline{H}\\,\\varepsilon )}.$ When $(C_{\\alpha })_{SL}={\\bf 1},$ we have $Ker(C_\\alpha )_{SL}=(G_\\alpha )_{SL},$ so that $\\Phi _f((A,H)KerC_\\alpha )=\\,f(A\\,(G_\\alpha )_{SL}) \\,e^{i\\pi \\,Tr(P (A\\,(G_\\alpha )_{SL}) \\,\\varepsilon \\, \\overline{H}\\, \\varepsilon )}.$ In the case $(C_{\\alpha })_{SL}\\ne {\\bf 1},$ we assume the existence of a trivialization, $(L,\\,\\rho ,\\,z_0),$ and we identify $SL/Ker\\,(C_\\alpha )_{SL}$ with ${\\cal B}.$ Then the pseudotensorial function $\\Phi _f$ becomes $\\Phi _f\\left(\\left(A,H\\right)\\,Ker C_\\alpha \\right)=\\,f(\\rho (A)\\cdot z_0) \\,e^{i\\pi \\,Tr( P(r (\\rho (A)\\cdot z_0) )\\,\\varepsilon \\, \\overline{H}\\,\\varepsilon )}.$ A natural representation of $G$ on $V$ is the $\\delta ^c$ given by $\\delta ^c(A,H) \\cdot \\Phi _f=\\Phi _f \\circ ((A,H)^{-1}*)$ where $*$ is the canonical action on $G/Ker\\, C_\\alpha .$ Let us prove that $\\delta ^c(A,H) \\cdot \\Phi _f=\\Phi _{\\delta ^{\\prime }(A,H) \\cdot f}$ $i.e.$ that $\\Phi $ is equivariant for the representations $\\delta ^{\\prime }$ in ${\\cal C}$ and $\\delta ^c$ in ${\\cal V}$ .",
"In fact, we have ${\\begin{array}{c}\\left( \\delta ^c(A,H) \\cdot \\Phi _f\\right) \\left((B,K)Ker\\, C_\\alpha \\right)=\\Phi _f\\left( A^{-1} B, A^{-1}(K-H)(A^{-1})^* \\right)=\\\\=f\\left( \\rho (A^{-1})\\cdot (\\rho (B)\\cdot z_0)\\right)\\,e^{i\\pi \\,Tr(P(r (\\rho (A^{-1})\\cdot (\\rho (B)\\cdot z_0)) )\\,\\varepsilon \\, \\overline{ ( A^{-1}(K-H)(A^{-1})^*)}\\,\\varepsilon )}=\\\\=f\\left( \\rho (A^{-1})\\cdot (\\rho (B)\\cdot z_0)\\right)\\,e^{i\\pi \\,Tr(A^{-1}\\,P(\\rho (B)\\cdot z_0)(A^{-1})^* A^*\\varepsilon \\, \\overline{ (K-H)}\\,\\varepsilon \\,A)}=\\\\= f\\left( \\rho (A^{-1})\\cdot (\\rho (B)\\cdot z_0)\\right)\\,e^{i\\pi \\,Tr(P(\\rho (B)\\cdot z_0)\\,\\varepsilon \\, \\overline{ (K-H)}\\,\\varepsilon ) }=\\\\=\\left( \\delta ^{\\prime }(A,H)\\cdot f\\right)\\left(\\rho (B) \\cdot z_0 \\right)\\,e^{i\\pi \\,Tr(P(\\rho (B)\\cdot z_0)\\,\\varepsilon \\, \\overline{ K}\\,\\varepsilon }=\\\\=\\Phi _{\\delta ^{\\prime }(A,H)\\cdot f}\\left((B,K)Ker\\, C_\\alpha \\right).\\end{array}}$ As a particular consecuence, for all $(a,h)\\in \\underline{G},$ we have for the infinitesimal generators of $\\delta ^c$ $d\\delta ^c(a,h) \\cdot \\Phi _f=\\Phi _{d\\delta ^\\prime (a,h)\\cdot f}.$ Equation (REF ), tell us that the infinitesimal generator of the representation $\\delta ^c$ associated to $(a,h)\\in \\underline{G},$ acts on $\\Phi _f$ as the (vector field)infinitesimal generator of the action on $G/Ker\\,C_\\alpha ,$ what will be denoted by $X_{(a,h)}^c.$ If we denote by $(a,h)^c,$ the Quantum Operator $(a,h)^c \\stackrel{def}{=}\\frac{1}{2\\pi i}\\,d\\delta ^c(a,h)$ we have $(a,h)^c \\cdot \\Phi _f=\\Phi _{(a,h)^\\theta \\cdot f}.$ Thus, our results in subsection REF , on Quantum Operators in that case, gives us results on Quantum Operators in our present case."
],
[
"Quantizable forms in $ SL(2,\\mathbb {C}) \\oplus H(2)$ .",
"In [7] I give a classification of the coadjoint orbits of the group under consideration.",
"The orbits are divided into 9 Types, and a canonical representative of each orbit is given, according with its type.",
"Let $\\alpha =\\lbrace a,k\\rbrace $ be a nonzero element of the dual of the Lie algebra of $ SL \\oplus H(2),$ and denote by $W$ and $P$ the Pauli- Lubanski and Linear momentum respectively at $\\alpha $ (cf.",
"section ).",
"Figure 3 gives what type of coadjoint orbit is the one of $\\alpha ,$ according with the values of $W$ and $P.$ Figure: Types of coadjoint orbitsWhen one knows the type of $\\alpha ,$ Figure 4 gives us the canonical representative of the orbit of $\\alpha .$ Figure: Canonical Representatives.The $\\mathbb {R} $ -quantizable orbits are all of the types 3, 6, 8, 9 and these of type 5 corresponding to the case $|W|=0$ Quantizable but not $\\bf R\\rm $-quantizable are the orbits whose canonical representatives are: $\\left\\lbrace \\frac{iT}{8 \\pi } \\left(\\begin{array}{cc} 1 &0 \\\\ 0 &-1 \\end{array} \\right) ,0\\right\\rbrace \\ \\ (type\\ 2),$ $\\left\\lbrace \\frac{i\\chi T}{8 \\pi }\\ \\left( \\begin{array}{cc}1 &0 \\\\0 &-1\\end{array} \\right),-sign (Tr\\left( P \\right) )\\left( \\begin{array}{cc}1 &0 \\\\0 &0\\end{array} \\right)\\right\\rbrace \\ \\ (type\\ 4), $ $\\left\\lbrace \\frac{iT}{8 \\pi }\\left( \\begin{array}{cc}1 &0 \\\\0 &-1\\end{array} \\right),-sign (Tr\\left( P \\right) )\\sqrt{|P|}\\ I \\right\\rbrace \\ \\ (type\\ 5), $ $ \\left\\lbrace \\frac{i\\chi T}{8 \\pi }\\left( \\begin{array}{cc}1 &0 \\\\0 &-1\\end{array} \\right),\\sqrt{-|P|}\\ \\left( \\begin{array}{cc}1 &0 \\\\0 &-1\\end{array} \\right)\\right\\rbrace \\ \\ (type\\ 7).$ where, in all cases, $T \\in \\bf Z^+\\rm , \\chi =1,-1.$ The concrete particles we study in this paper are those corresponding to Type 5 (massive particles) and Type 4 (massless particles such that the Pauly-Lubanski fourvector is proportional to Impulsion-Energy).",
"Results concerning other types of particles will be published elsewhere."
],
[
"Guide to the explicit construction of Wave Functions.",
"Let $\\alpha $ be a quantizable element of $\\underline{G}^*.$ To determine the explicit form of the Wave Functions of the corresponding free particles, one can proceed as follows 1.",
"Evaluate the isotropy subgroup at $\\alpha ,$ of the coadjoint representation, $G_{\\alpha }$ .",
"The coadjoint orbit of $\\alpha ,$ that is the main symplectic manifold associated to $\\alpha ,$ is identified to the homogeneous space $G/G_{\\alpha }.$ 2.",
"Find a surjective homomorphism, $C_{\\alpha },$ from $G_{\\alpha }$ onto the unit circle S$^1,$ whose differential is $\\alpha .$ 3.",
"Evaluate $(G_{\\alpha })_{SL},$ that is composed by the $g\\in SL$ such that $(g,h)\\in G_{\\alpha }$ for some $h\\in H(2).$ 4.",
"Determine $(C_{\\alpha })_{SL},$ defined in (REF ), and $S\\equiv (C_{\\alpha })_{SL}\\left((G_{\\alpha })_{SL}\\right).$ 5.",
"Find an action of $SL(2,{\\mathbb {C}})$ on a manifold, such that the isotropy subgroup at some point, $p$ , is $(G_{\\alpha })_{SL}.$ Identify $SL/(G_{\\alpha })_{SL}$ with the orbit of $p,$ ${\\cal K}$ .",
"6.",
"Determine a volume element on ${\\cal K},$ $\\omega ,$ invariant under the action of $SL$ .",
"7.",
"If $S=\\lbrace 1\\rbrace ,$ the Prewave Functions are the $\\psi _f(H,K)= f(K)\\,e^{i\\pi Tr(P(K )\\varepsilon \\, \\overline{H}\\, \\varepsilon ) }$ where $(H,K)\\in H(2)\\times {\\cal K},$ $f$ is a function on ${\\cal K},$ and $P$ is the dynamical variable linear momentum, on ${\\cal K},$ given by (REF ).",
"8.",
"If $S\\ne \\lbrace 1\\rbrace ,$ find a Trivialization of $ C_\\alpha $ ($c.f.$ (REF )).",
"Let ${\\cal B}$ the orbit of $z_0$ , identified to $SL/Ker(C_{\\alpha })_{SL},$ $r:{\\cal B} \\rightarrow {\\cal K}$ the canonical map from $SL/Ker(C_{\\alpha })_{SL})$ onto $SL/(G_{\\alpha })_{SL}$ .",
"The Prewave Functions are given by ($c.f.$ (REF )) $\\psi _f(H,K)= f(z)\\,e^{i\\pi Tr(P(K )\\varepsilon \\, \\overline{H}\\, \\varepsilon ) }z$ where $(H,K)\\in H(2)\\times {\\cal K},$ $z\\in r^{-1}(K),$ and $f$ is a function on ${\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers.",
"9.",
"The Prewave Functions corresponding to continuous with compact support $f$ define Wave Functions by means of ($c.f.$ (REF )) $\\widetilde{\\psi _f}(H)=\\int _{\\cal K} {\\mbox{$\\psi $}}_f(H,\\cdot )\\ {\\mbox{$\\omega $}} .$"
],
[
"Wave Functions. Klein-Gordon equation.",
"Let us consider a particle whose movement space is the coadjoint orbit of ${\\alpha _o}=\\left\\lbrace 0,\\ \\eta \\,m\\,I\\right\\rbrace ,\\ \\ \\ m \\in {\\mathbb {R}}^+, \\eta =\\pm 1.$ This orbit is a $\\mathbb {R} \\rm $ -quantizable orbit of the type 5, in the notation of section .",
"The number $(-\\eta )$ is the sign of energy i.e.",
"the sign of the value of the dynamical variable $P^4$ ($c.f.$ section ) at any point of the orbit.",
"According with the usual interpretation of the determinant of momentum-energy, $\\vert P \\vert ,$ as mass square, the number $m$ must be interpreted as being the mass of the particle.",
"By direct computation, one sees that ${\\mbox{$G_{\\alpha _o}$}} = \\lbrace \\ (A, \\ hI ): \\ A\\in SU(2),\\ h\\in \\mathbb {R} \\rbrace .$ The unique homomorphism onto ${\\mathbb {R}}$ whose differential is ${\\alpha _o}$ is given by $ C^\\prime _{\\alpha _o} (A ,\\ hI)=-\\eta m h .$ The unique homomorphism onto ${\\bf S}^1$ whose differential is $\\alpha _o$ is given by $C_{\\alpha _o} (A ,\\ hI)=e^{-2\\pi i\\eta m h}.$ Then we have ${\\begin{array}{c}Ker\\,C^\\prime _{\\alpha _o}=\\lbrace (A,0):A\\in SU(2)\\rbrace =SU(2)\\oplus \\lbrace 0\\rbrace ,\\\\Ker\\,C_{\\alpha _o}=\\lbrace (A,\\,\\frac{\\eta N}{m}\\, I):A\\in SU(2),\\ N \\in \\mathbb {Z}\\rbrace ,\\\\{\\mbox{$\\widetilde{C}_{\\alpha _o}$}} (A ,\\ H)=e^{-\\pi i\\eta m TrH} ,\\\\{\\mbox{$(G_{\\alpha _o})_{SL}$}}=SU(2) ,\\\\{\\mbox{$(C_{\\alpha _o})_{SL}$}}={\\bf 1 } ,\\\\SL_1 =SL_2= {\\mbox{$Ker\\,(C_{\\alpha _o})_{SL}$}}={\\mbox{$(G_{\\alpha _o} ) _{SL}$}},\\end{array}} $ Thus, in the commutative diagram of figure REF , section , the four spaces on the left are the same.",
"All of them represent State Space, $H(2)\\times \\frac{SL}{(G_{\\alpha _o})_{SL}\\cap SL_1}$ what in our present case becomes $H(2) \\times \\frac{SL}{SU(2)},$ and can be obviously identified to $\\frac{G}{SU(2) \\oplus \\lbrace 0\\rbrace },$ by means of the diffeomorphism $(H,A\\,SU(2))\\rightarrow (A,H)(SU(2) \\oplus \\lbrace 0\\rbrace ).$ Then, in this case, besides the canonical map from State Space onto Movement Space (c.f.",
"figure REF ), $\\nu _2: (H,ASU(2))\\in H(2) \\times \\frac{SL}{SU(2)} \\rightarrow (A,H)G_\\alpha \\in \\frac{G}{G_{\\alpha _o}}$ we have a natural map from State Space onto $G/Ker\\,C_{\\alpha _o}$ $\\nu _1:(A,H)(SU(2) \\oplus \\lbrace 0\\rbrace )\\in \\frac{G}{SU(2) \\oplus \\lbrace 0\\rbrace } \\rightarrow (A,H){Ker\\,C_{\\alpha _o}} \\in \\frac{G}{Ker\\,C_{\\alpha _o}}.$ The diagram of figure REF , becomes that of figure REF .",
"Figure: Diagram for Klein-Gordon particles.Let ${\\cal H}^m$ be the positive mass hyperboloid ${\\cal H}^m=\\lbrace H \\in H(2):\\ det\\, H\\,=\\,m^2,\\ Tr\\,H\\,>\\,0\\,\\rbrace .$ In ${\\cal H}^m$ we consider the action of $SL(2,C)$ given by $ A* H=A\\,H\\,A^*,$ for all $H\\in {\\cal H}^m$ and $A \\in SL(2,C).$ In these conditions we have $Tr(A\\,H\\,A^*)>0$ as a consecuence of the fact that $SL(2,C)$ is connected.",
"Since the group of restricted homogeneous Lorenz transformations is transitive on ${\\cal H}^m,$ so does $SL(2,\\mathbb {C}).$ The isotropy subgroup at $mI$ is $SU(2).$ Thus $SL/(G_{\\alpha _o})_{SL}$ is identified to ${\\cal H}^m,$ by means of the diffeomorphism $A\\, SU(2)\\in \\frac{SL}{SU(2)} \\longrightarrow m\\,A\\,A^* \\in {\\cal H}^m.$ If $K= m\\,A\\,A^* \\in {\\cal H}^m,$ then $K$ is identified to $A\\, SU(2)\\in \\frac{SL}{SU(2)},$ and the function P thus becomes $P(K)=P(ASU(2))=-A(\\eta \\,m\\,I)A^*=-\\eta K.$ The manifold ${\\cal H}^m$ has the global parametrization $\\varphi :(p^1,\\ p^2,\\ p^3)\\in {{\\mathbb {R}} ^3}\\ \\ \\mapsto \\ \\ h(p^1,\\ p^2,\\ p^3,\\ (m^2+ p^2)^{\\frac{1}{2}} )\\ \\in \\ {\\cal H}^m$ where $p^2= \\sum _{i=1}^3 (p^i)^2.$ In particular it is orientable.",
"On the other hand, the restriction of the pseudoriemannian metric defined on ${\\mathbb {R}}^4$ by Minkowski metric, to ${\\cal H}^m,$ is negative definite so that its opposite is a riemannian metric, that provides us with a canonical volume element, $\\nu $ .",
"A computation of the matrix of the riemannian metric in the chart $\\varphi $ leads to $\\nu =\\frac{dp^1 \\wedge dp^2 \\wedge dp^3}{\\sqrt{m^2+ p^2}}$ where the $p^i$ are the coordinates in the chart $\\varphi .$ The action on $H(2)$ defined as in (REF ) preserves Minkowski metric, so that the action on ${\\cal H}^m$ preserves $\\nu $ .",
"As a consequence, $\\nu $ is an invariant volume element under the action (REF ).",
"Since $(C_{\\alpha _o})_{SL}={\\bf 1}$ , $\\cal C$ is composed by functions on ${\\cal H}^m.$ The prewave functions have the form $\\psi _f:(H,\\ K) \\in H(2) \\times {\\cal H}^m\\ \\mapsto \\ \\ f\\left(K\\right) e^{-i\\pi \\eta Tr(K \\varepsilon \\overline{H} \\varepsilon ) }.$ where f is in $\\cal C.$ The corresponding wave function is $\\tilde{\\psi }_f(X)=\\int _{{\\cal H}^m}\\psi _f(h(X),\\,\\cdot \\,)\\ \\nu $ By direct computation one sees that the prewave and wave functions satisfy Klein-Gordon equation.",
"If $f^\\prime $ is another continuous with compact support function on ${\\cal H}^m$ , the hermitian product of the quantum states corresponding to $\\psi _f$ and $\\psi _{f^\\prime },$ defined in section REF , can be writen $\\langle \\psi _f,\\ \\psi _{f^\\prime } \\rangle =\\int _{{\\cal H}^m }\\psi _f^*\\,\\psi _{f^\\prime }\\,\\nu .$"
],
[
"The Homogeneous Contact and Symplectic Manifolds for Klein-Gordon particles",
"Let us consider the action of $G$ on ${\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1$ given by $(A,H)*(K,\\overrightarrow{y},z)=(AKA^*,\\overrightarrow{x}(A,H,\\overrightarrow{y},K), e^{2\\pi i \\eta m \\ell (A,H,\\overrightarrow{y},K)}z)$ where $\\overrightarrow{x}(A,H,\\overrightarrow{y},K)$ and $\\ell (A,H,\\overrightarrow{y},K)$ are given by $h(\\overrightarrow{x}(A,H,\\overrightarrow{y},K),0)=A\\,h(\\overrightarrow{y},0)\\,A^*+H+\\ell (A,H,\\overrightarrow{y},K) A\\,\\frac{K}{m}\\,A^*.$ Obviously we have $\\ell (A,H,\\overrightarrow{y},K)=-m\\frac{Tr(A\\,h(\\overrightarrow{y},0)\\,A^*+H)}{Tr(AKA^*)}.$ This is a transitive action, as can be proved by direct computation.",
"The isotropy subgroup at $(mI,\\,\\overrightarrow{0},1),$ is $Ker\\,C_{\\alpha _0},$ so that the homogeneous space $G/Ker\\,C_{\\alpha _0},$ can be identified to ${\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1$ by means of the map $(A,H) Ker\\,C_{\\alpha _0} \\longrightarrow (A,H)*(mI,\\overrightarrow{0},1).$ We know (c.f.",
"section ) that the left-invariant 1-form on $G$ whose value at $(I,0)$ is $\\alpha _o$ , $\\tilde{\\alpha }_o,$ projects in a 1-form , $\\Omega ,$ in $G/Ker\\,C_{\\alpha _o}$ which is a contact form, and is homogeneous in the sense that it is preserved by the diffeomorphisms corresponding to the canonical action of $G$ on $G/Ker\\,C_{\\alpha _o}.$ One way to give explicitly $\\Omega ,$ is to use coordinate domains in $G/Ker\\,C_{\\alpha _0}$ that are also domains of sections of the canonical map from the group $G$ onto $G/Ker\\,C_{\\alpha _0}.$ The pull back of $\\tilde{\\alpha }_0$ by that section, is the restriction of $\\Omega $ to the domain, and we can give its local expresion in the coordinate system.",
"With our identification , the just cited canonical map becomes $\\mu :(A,H)\\in G \\longrightarrow (A,H) * (mI,\\overrightarrow{0},1)\\in {\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1.$ In ${\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1$ be define a coordinate system for each $\\tau \\in \\mathbb {R}$ as the inverse of the parametrization ${\\begin{array}{c}\\phi _\\tau : (k_1,k_2,k_3,x^1,x^2,x^3,t)\\in \\mathbb {R}^6 \\times \\left(-\\frac{1}{2},\\frac{1}{2}\\right) \\rightarrow \\\\\\rightarrow \\left( m\\ h(k_1,k_2,k_3,k_4),x^1,x^2,x^3,e^{2\\pi i(t+\\tau )}\\right)\\in \\\\\\in {\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1\\end{array}}$ where $k_4=\\sqrt{1+k_1^2+k_2^2+k_3^2}.$ Now, let us consider the map ${\\begin{array}{c}\\sigma _\\tau :\\phi _\\tau (k_1,k_2,k_3,x^1,x^2,x^3,t) \\rightarrow \\\\ \\rightarrow \\left( \\left( \\begin{array}{cc}\\sqrt{\\frac{k_4+k_3}{1+k_1^2+k_2^2}}&\\sqrt{\\frac{k_4+k_3}{1+k_1^2+k_2^2}}(k_1-ik_2)\\\\0&\\sqrt{\\frac{1+k_1^2+k_2^2 }{ k_4+k_3}}\\end{array}\\right),h(\\overrightarrow{x},0)-\\frac{\\eta }{m}(t+\\tau )) h(\\overrightarrow{k},k_4)\\right) \\\\ \\in G.\\hspace{341.43306pt}\\end{array}}$ We have $\\mu \\circ \\sigma _\\tau (\\phi _\\tau (R))=\\sigma _\\tau (\\phi _\\tau (R))*(mI,\\overrightarrow{0},1)=\\phi _\\tau (R),$ for all $R\\in \\mathbb {R}^6 \\times (-1/2,\\ 1/2),$ so that $\\sigma _\\tau $ is a section of the canonical map $\\mu $ .",
"Then, on the image of $\\phi _\\tau $ we have $\\Omega =\\sigma _\\tau ^* \\tilde{\\alpha }_0.$ In order to obtain, for exemple, the value $(\\Omega )_{\\left( \\phi _\\tau \\left( R \\right) \\right)}\\cdot \\left( \\frac{\\partial }{\\partial k_1} \\right)_{\\left( \\phi _\\tau \\left( R \\right) \\right)},$ we must find the value of $\\alpha _0$ on the tangent vector to the curve $\\gamma (s)= \\left( \\sigma _\\tau ( \\phi _\\tau (R) )\\right)^{-1}\\left( \\sigma _\\tau ( \\phi _\\tau (R+(s,0,0,0,0,0,0))\\right)$ at $0.$ Since $\\alpha _0=\\left\\lbrace 0,\\ \\eta \\,m\\,I\\right\\rbrace ,$ the first component of the tangent vector at 0 of $\\gamma $ has no incidence in the value of $\\alpha _0$ on it.",
"The preceding computation gives us $\\langle (o,\\ \\eta m I),\\stackrel{.",
"}{\\gamma }_0 \\rangle =0.$ A similar computation for each of the other coordinates, or a common reasoning with a curve $\\gamma (s)=\\phi _\\tau (c(s)),$ leads to $\\Omega =d\\,t+\\eta \\,m\\,k_i\\,dx^i.$ Obviously $d\\Omega =\\eta \\,m \\,dk_i\\wedge \\,dx^i,$ so that $\\Omega \\wedge \\left(d\\Omega \\right)^3=\\frac{\\eta m^3}{16}\\ dt\\wedge dk_1\\wedge dk_2\\wedge dk_3\\wedge dx^1\\wedge dx^2\\wedge dx^3$ what confirms the fact that $\\Omega $ is a contact form.",
"The characteristic vectorfield $Z(\\Omega )$ defined by $i(Z(\\Omega ))\\Omega =1,\\ \\ i(Z(\\Omega ))\\,d\\Omega =0,$ has flow, ${\\phi _t},$ given by $\\phi _t(K,\\overrightarrow{x},z)=(K,\\overrightarrow{x}, e^{2\\pi i t}z).$ As a consecuence, the period of all its integral curves is $T(\\Omega )=1,$ so that $\\Omega $ is a connection form, and $d\\Omega $ the corresponding curvature form.",
"Now, let us consider the action of $G$ on ${\\cal H}^m \\times \\mathbb {R}^3 $ given by $(A,H)*(K,\\overrightarrow{y})=(AKA^*,\\overrightarrow{x}(A,H,\\overrightarrow{y},K))$ where $h(\\overrightarrow{x}(A,H,\\overrightarrow{y},K),0)=A\\,h(\\overrightarrow{y},0)\\,A^*+H+\\ell (A,H,\\overrightarrow{y},K) A\\,\\frac{K}{m}\\,A^*.$ This is also a transitive action.The isotropy subgroup at $(mI,\\,\\overrightarrow{0}),$ is $G_{\\alpha _0},$ so that $G/G_{\\alpha _0},$ can be identified to ${\\cal H}^m \\times \\mathbb {R}^3 $ by means of the map $(A,H) G_{\\alpha _0} \\longrightarrow (A,H)*(mI,\\overrightarrow{0}).$ The left-invariant 2-form on $G$ whose value at $(I,0)$ is $d\\alpha _0$ , $d\\tilde{\\alpha }_0,$ projects in a 2-form , $\\omega ,$ in $G/G_{\\alpha _0}$ which is a simplectic form, and is homogeneous in the sense that it is preserved by the diffeomorphisms corresponding to the canonical action of $G$ on $G/G_{\\alpha _0}.$ With our identifications, the canonical map $(A,H) Ker\\,C_\\alpha \\rightarrow (A,H) G_\\alpha $ becomes $(K,\\overrightarrow{x},z )\\in {\\cal H}^m \\times \\mathbb {R}^3 \\times \\mathbb {S}^1 \\rightarrow (K,\\overrightarrow{x})\\in {\\cal H}^m \\times {\\mathbb {R}}^3$ and, since $T(\\Omega )=1,$ $\\omega $ must be a projection of $d\\Omega ,$ so that $\\omega = {\\eta \\,m} \\,d{k}_i \\wedge \\,d{x}^i,$ where now $\\lbrace k_1,\\dots ,x^3\\rbrace $ are the coordinates corresponding to the following global parametrization of ${\\cal H}^m \\times {\\mathbb {R}}^3 $ ${\\begin{array}{c}\\underline{\\phi }: ({k}_1,{k}_2,{k}_3,{x}^1,{x}^2,{x}^3)\\in {\\mathbb {R}}^6 \\rightarrow \\\\\\rightarrow \\left( m\\ h({k}_1,{k}_2,{k}_3,{k}_4),{x}_1,{x}_2,{x}_3\\right)\\\\\\in {\\cal H}^m \\times {\\mathbb {R}}^3\\end{array}}$ where ${k}_4=\\sqrt{1+{k}_1^2+{k}_2^2+{k}_3^2}.$ In this case we have a global section of the canonical map ${\\begin{array}{c}\\underline{\\sigma }(\\underline{\\phi }({k}_1,{k}_2,{k}_3,{x}^1,{x}^2,{x}^3))=\\\\=\\left( \\left( \\begin{array}{cc}\\sqrt{\\frac{k_4+k_3}{1+k_1^2+k_2^2}}&\\sqrt{\\frac{k_4+k_3}{1+k_1^2+k_2^2}}(k_1-ik_2)\\\\0&\\sqrt{\\frac{1+k_1^2+k_2^2 }{ k_4+k_3}}\\end{array}\\right),h(\\overrightarrow{x},0) \\right).\\end{array}}$ The map $\\underline{\\sigma }$ is a section since $\\underline{\\sigma }(\\phi (R))*(mI,\\overrightarrow{0})=\\phi (R).$ The coadjoint orbit of $\\alpha _0,$ becomes identified to $ {\\cal H}^m \\times {\\mathbb {R}}^3 $ by means of $Ad^*_{(A,H)}\\cdot \\alpha _0 \\rightarrow (A,H)*(mI,\\overrightarrow{0}),$ whose inverse is given by $(K,\\overrightarrow{x}) \\rightarrow Ad^*_{\\underline{\\sigma }(K,\\overrightarrow{x})} \\cdot \\alpha _0.$ Thus, the $(e,g)\\in \\underline{G},$ what are dynamical variables on the coadjoint orbit, becomes functions on $ {\\cal H}^m \\times \\mathbb {R}^3 , $ denoted by $D_{(e,g)}$ in section , defined as follows $D_(e,g)(\\underline{\\phi }(k_1,\\dots ,x_3))=(Ad^*_{\\underline{\\sigma }(\\underline{\\phi }(k_1,\\dots ,x_3))}\\alpha _0)( e,g).$ In particular, the Linear and Angular Momentum, given in (REF ), are, as functions on $ {\\cal H}^m \\times \\mathbb {R}^3, $ $\\nonumber P(m\\,k,\\overrightarrow{x})&=&- \\eta \\,m\\ k, \\\\ \\overrightarrow{l}(m\\,k,\\overrightarrow{x})&=&\\eta \\,m\\,\\overrightarrow{k} \\times \\overrightarrow{x}=\\overrightarrow{x} \\times \\left(\\overrightarrow{P}(m\\,k,\\overrightarrow{x}) \\right) \\\\ \\nonumber \\overrightarrow{g}(m\\,k,\\overrightarrow{x})&=&-\\eta \\,m\\,k^4 \\overrightarrow{x}=\\left( P^4(m\\,k,\\overrightarrow{x}) \\right) \\ \\overrightarrow{x} ,$ where we have denoted $D_{P^k},\\ D_{l^i},$ and $D_{g^j}$ by $P^k, l^i$ and $g^j$ respectively.",
"Also, each $a\\in \\underline{G}, $ define an infinitesimal generator, $X^s_{a},$ of the action (REF ).",
"These infinitesimal generators are defined by means of its flow, and its local expresions can be obtained directly from the definition, but it is also possible to use formula (REF ) to obtain the following local expresions in the chart $\\underline{\\Phi }^{-1}$ $X^s_{P^j}&=&\\frac{\\partial }{\\partial x^j}\\\\X^s_{P^4}&=&\\sum _{j=1}^3\\frac{k_j}{k_4}\\ \\frac{\\partial }{\\partial x^j}\\\\X_{l^k}^s &=&\\sum _{j,r=1}^3 \\varepsilon _{kjr} k_j \\frac{\\partial }{\\partial k_r} +\\sum _{j,r=1}^3 \\varepsilon _{kjr} x^j\\frac{\\partial }{\\partial x^r} \\\\X^s_{g^j}&=&x^j\\, X_{P^4}^s-k_4\\frac{\\partial }{\\partial k_j}$ where $ \\varepsilon _{kjr}$ is as in (REF ).",
"Equation (REF ) proves that these vector fields are globally hamiltonian, and the corresponding hamiltonian is, with our actual notation, the function appearing in the subindex in each case.",
"The infinitesimal generator corresponding to $a\\in \\underline{G}$ for the action in the contact manifold is denoted by $X_a^c.$ We have in the charts $\\Phi _\\tau ^{-1}$ $X^c_{P^j}&=&\\frac{\\partial }{\\partial x^j}\\nonumber \\\\X^c_{P^4}&=&\\frac{1}{k_4}\\left(\\sum _{j=1}^3\\,k_j\\, \\frac{\\partial }{\\partial x^j}+\\eta m \\frac{\\partial }{\\partial t} \\right)\\nonumber \\\\X_{l^k}^c &=&\\sum _{j,r=1}^3 \\varepsilon _{kjr} k_j \\frac{\\partial }{\\partial k_r} +\\sum _{j,r=1}^3 \\varepsilon _{kjr} x^j\\frac{\\partial }{\\partial x^r} \\\\X^c_{g^j}&=&x^j\\, X_{P^4}^c-k_4\\frac{\\partial }{\\partial k_j}.\\nonumber $ The Quantum Operators representing Linear and Angular Momentum for Quantum States on the Contact Manifold are $\\frac{1}{2 \\pi i}$ times the vectorfiels in (REF ).",
"If $f$ is a $C^\\infty $ function on ${\\cal H}^m$ the corresponding Quantum State in the contact manifold is $\\Phi _f((A,H)\\,Ker\\,C_\\alpha )=f(A\\,(G_\\alpha )_{SL})Exp[i \\pi Tr(P(A\\,(G_\\alpha )_{SL})\\varepsilon \\overline{H} \\varepsilon )]$ or, with the identifications we have made $\\Phi _f((A,H)* (mI,\\overrightarrow{0},1))=f(mAA^*) Exp[i \\pi Tr((-\\eta mAA^*)\\varepsilon \\overline{H} \\varepsilon )]$ but $\\Phi _f((A,H)* (mI,\\overrightarrow{0},1))=\\Phi _f(mAA^*,h^{-1}( H+\\ell AA^*), Exp[2 \\pi i \\eta m \\ell ]),$ where $ \\ell $ is such that $Tr(H+\\ell AA^*)=0.$ Thus $\\Phi _f(K,\\overrightarrow{x},z)=f(K) Exp[i \\pi Tr((-\\eta K)\\varepsilon \\overline{(h(\\overrightarrow{x},0)-\\ell \\frac{K}{m})} \\varepsilon )]$ where $\\ell $ is such that $z= Exp[2 \\pi i \\eta m \\ell ].$ Then $\\Phi _f(K,\\overrightarrow{x},z)=f(K) Exp[-i \\eta \\pi Tr( K \\varepsilon \\overline{h(\\overrightarrow{x},0)}\\varepsilon ]Exp[i \\ell \\eta \\pi Tr( K \\varepsilon \\overline{\\frac{K}{m}}\\varepsilon ]$ and one sees that, if $K=m h(\\overrightarrow{k},k_4),$ $\\Phi _f(K,\\overrightarrow{x},z)=f(K)\\, Exp[-2 \\pi i \\eta m \\langle \\overrightarrow{k} ,\\overrightarrow{x} \\rangle ] \\,\\overline{z} .$ In the coordinate system associated to $\\phi _\\tau ,$ we have $\\Phi _f \\circ \\phi _\\tau (\\overrightarrow{k} ,\\overrightarrow{x} ,t)=f(mh(\\overrightarrow{k},k_4))\\, \\, Exp[-2 \\pi i ( \\eta m \\langle \\overrightarrow{k} ,\\overrightarrow{x} \\rangle +t+\\tau )].$"
],
[
"Wave Functions for $T=1.$ Dirac equation.",
"Let us consider a particle whose movement space is the coadjoint orbit of $\\alpha _1=\\left\\lbrace \\ \\frac{i}{8\\pi }\\left( \\begin{array}{cc} 1&0 \\\\ 0&-1\\end{array} \\right) \\ ,\\ \\eta \\,m\\,I\\right\\rbrace ,$ where $m \\in {\\bf R}^+, \\eta =\\pm 1.$ This orbit is a quantizable, not $\\bf R \\rm $ -quantizable orbit, of the type 5, in the notation of section .",
"In this case we have ${\\mbox{$G_{\\alpha _1}$}} = \\lbrace \\ (\\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right), \\ hI ): \\ z\\in S^1,\\ h\\in {\\mbox{$\\bf R \\rm $}} \\rbrace .$ The unique homomorphism from $G_{\\alpha _1}$ onto ${\\bf S}^1$ whose differential is ${\\alpha _1}$ is given by ${\\mbox{$C_{\\alpha _1}$}} (\\ \\left( \\begin{array}{cc}e^{2\\pi i\\phi }&0 \\\\0&e^{-2\\pi i\\phi }\\end{array} \\right) ,\\ hI)=e^{2\\pi i(\\phi -\\eta m h)}.$ Then $\\begin{array}{l}Ker\\, C_{\\alpha _1}=\\lbrace (\\ \\left( \\begin{array}{cc}e^{2\\pi i\\eta m h}&0 \\\\0&e^{-2\\pi i\\eta m h}\\end{array} \\right) ,\\ hI):h\\in \\mathbb {R} \\rbrace ,\\\\\\vspace{7.22743pt}(G_{\\alpha _1} ) _{SL}=\\lbrace \\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right):\\ z\\in S^1\\rbrace , \\\\\\vspace{7.22743pt}{{\\mbox{$(C_{\\alpha _1})_{SL}$}}(\\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right))=z}, \\\\\\vspace{7.22743pt}SL_1 \\cap SL_2 ={\\mbox{$(G_{\\alpha _1} ) _{SL}$}}, \\\\\\vspace{7.22743pt}SL_1 \\cap {\\mbox{$Ker\\,(C_{\\alpha _1})_{SL}$}}={\\mbox{$Ker\\,(C_{\\alpha _1})_{SL}$}}=\\lbrace I \\rbrace .",
"\\\\\\end{array}$ Let us denote $(G_{\\alpha _1})_{SL}$ by $[S^1],$ $G_{\\alpha _1}$ by $[S^1]\\oplus \\mathbb {R},$ and $Ker\\, C_{\\alpha _1}$ by $[R].$ Then, in the conmutative diagram of Figure REF , $\\iota _3$ and $\\iota _4$ become identical maps, $\\tau _3=\\tau _4$ and the diagram becomes, with obvious conventions, that of Figure REF .",
"Figure: Fibre Bundles for Dirac ParticlesThe homogeneous space $SL/[S^1]$ can be characterised as follows.",
"Let ${\\bf P}_1(\\mathbb { C})$ be the complex projective space corresponding to $ \\mathbb { C}^2$ ( $i.e.$ the space $ {\\bf P}_1(\\mathbb { C}) $ consists of the one dimensional complex subspaces of $ {\\mathbb { C}}^2$ ).",
"In ${\\cal H}^m\\times {\\bf P}_1(\\mathbb { C})$ we can consider the action of $ SL(2, C)$ given by $A\\,*\\,(H,[z])=(AHA^*,\\,[Az])$ where $[z]\\in {\\bf P}_1(\\mathbb { C})$ is the vector subspace generated by $z \\in \\ {\\mathbb { C}}^2-\\lbrace (0, 0) \\rbrace $ .",
"We know that the partial action on ${\\cal H}^m$ is transitive.",
"Now let us see that the complete action on ${\\cal H}^m\\times {\\bf P}_1(\\mathbb { C})$ is also transitive.",
"Let $(K,[w])\\in {\\cal H}^m\\times {\\bf P}_1(\\mathbb {C}).$ Then, there exist $A\\in SL$ such that $A*(K,[w])=(mI,[w^{\\prime }]),$ for some $w^{\\prime }\\in {\\mathbb {C}}^2-\\lbrace (0,0)\\rbrace ,$ but there exist obviously an element $B$ of $SU(2)$ such that $\\left[B \\left(\\begin{array}{c}1\\\\0 \\end{array}\\right) \\right]=[w^{\\prime }],$ and we see that $(B^{-1}A)*(K,[w])=\\left( mI,\\left[ \\left(\\begin{array}{c}1\\\\0 \\end{array}\\right) \\right] \\right).$ Transitivity follows.",
"On the other hand, the isotropy subgroup at $(mI,\\ [ \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array}\\right) ])$ is $[S^1] $ , so that, since the action is transitive, one can identify $SL/[S^1]$ to ${\\cal H}^m\\times {\\bf P}_1(\\mathbb { C})$ .",
"It is a well known fact that ${\\bf P}_1(\\mathbb {C})$ is diffeomorphic to the sphere $S^2.$ A diffeomorphism can be described as follows.",
"Let $C^+$ be the future lightcone, composed by the hermitian matrices having 0 determinant and positive trace.",
"The subset, $S_2,$ of $C^+$ composed by the elements whose trace is 2 is $S_2=\\lbrace h(x,y,z,1):1-x^2+y^2+z^2=0 \\rbrace $ that can be identified to the sphere $S^2,$ by means of $\\delta :(x,y,z)\\in S^2\\subset \\mathbb {R}^3 \\longleftrightarrow h(x,y,z,1)\\in S_2\\subset C^+.$ The map $\\beta _0:[z]\\in {\\bf P}_1(\\mathbb { C}) \\longrightarrow \\frac{2}{z^*\\,z}\\,z\\,z^* \\in S_2,$ is bijective.",
"Its inverse is $\\beta _0^{-1}: C \\in S_2\\subset C^+ \\longrightarrow [w]\\in {\\bf P}_1(\\mathbb { C}),$ where $w$ is an eigenvector of $C$ corresponding to the eigenvalue 2.",
"This is a consecuence of the fact that $\\begin{split} &\\left(\\frac{2}{z^*\\,z}\\,z\\,z^*\\right)z=2z\\\\&\\left(\\frac{2}{z^*\\,z}\\,z\\,z^*\\right)\\varepsilon \\overline{z}=0\\end{split} $ Thus $\\beta \\stackrel{def}{=} \\delta ^{-1}\\circ \\beta _0: {\\bf P}_1(\\mathbb { C}) \\rightarrow S^2,$ can be described by $\\beta ([z])=\\vec{u} \\ \\Longleftrightarrow \\ \\frac{2}{z^*\\,z}\\,z\\,z^*=h(\\vec{u},1).$ that is practical in order to use $\\beta .$ More practical in order to use $\\beta ^{-1}$ is $\\beta ([z])=\\vec{u} \\ \\Longleftrightarrow \\ \\ h(\\vec{u},-1)z=0.$ If $p_s$ and $p_n$ denote the stereographical projections from poles $s=(0,0,-1)$ and $n=(0,0,1)$ respectively, we have $p_s\\left(\\beta \\left[ \\left( \\begin{array}{c} z^1\\\\z^2 \\end{array}\\right)\\right]\\right)= \\frac{z^2}{z^1}\\ \\ \\ \\mathrm {if}\\ z^1\\ne 0,\\\\p_n\\left(\\beta \\left[\\left( \\begin{array}{c} z^1\\\\z^2 \\end{array}\\right)\\right]\\right)= \\frac{\\overline{z}^1}{\\overline{z}^2}\\ \\ \\ \\mathrm {if}\\ z^2\\ne 0.",
"\\nonumber $ This can be used to prove easily the differentiability of $\\beta $ and $\\beta ^{-1}.",
"$ Thus, it is possible to change in all that follows ${\\bf P}_1(\\mathbb {C})$ by $S^2,$ but, at this moment, I prefer to use ${\\bf P}_1( \\mathbb {C}).$ In order to describe Prewave Functions in this case , one can consider the trivialisation $(\\rho ,\\ {\\bf C}^4,z_0)$ , where $z_0={}^t\\hspace{0.0pt}(1, 0, 1, 0)$ and $\\rho $ is given by $\\rho (A)=\\left(\\begin{array}{cc}A&0 \\\\0&(A^*)^{-1}\\end{array}\\right)$ The orbit of $z_0$ is ${\\cal B}=\\left\\lbrace \\left( \\!\\begin{array}{c}w \\\\z\\end{array}\\!",
"\\right)\\ :\\ w,\\ z \\in \\mathbb { C }^2,\\ z^*\\,w=1 \\right\\rbrace .",
"$ This can be seen by consideration of the map $\\phi _0:\\left(\\begin{array}{c}w \\\\z\\end{array}\\right) \\in {\\cal B} \\longrightarrow \\left(w \\vert -\\varepsilon \\overline{z} \\right)\\in SL(2,\\bf C)$ (it is a diffeomorphism), and the fact that $\\rho (w|-\\varepsilon \\overline{z} )\\left(\\begin{array}{c}1 \\\\0\\\\1\\\\0\\end{array}\\right)=\\left(\\begin{array}{c}w \\\\z\\end{array}\\right).$ When one identifies $SL / Ker(C_{\\alpha _1})_{SL}$ to $\\cal B$ , and $SL / (G_{\\alpha _1} ) _{SL}$ to ${\\cal H}^m \\times {\\bf P}_1( \\mathbb {C})$ , by means of the preceeding actions, the canonical map between these homogeneous spaces becomes a map, ${\\mbox{$ r$}}$ , from $\\cal B$ onto ${\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})$ .",
"As a consecuence of (REF ), we have $(w \\ |\\,-\\varepsilon \\overline{z})* (mI,\\left[\\left( \\begin{array}{c}1 \\\\0\\end{array}\\right)\\right])={\\mathbf {r}}\\left(\\begin{array}{c}w \\\\z\\end{array}\\right)$ and this leads easily to ${{\\mbox{$ r$}}} \\left(\\!\\begin{array}{c} w\\\\ z\\end{array}\\!",
"\\right)=(m(ww^*\\,-\\,\\varepsilon \\overline{zz^*} \\varepsilon ),\\, [w]).$ On the other hand, if we define $\\sigma (K,w)=\\frac{1}{\\sqrt{mw^*K^{-1}w}}\\left(w\\ \\ -\\frac{1}{m}K \\varepsilon \\overline{w}\\right),$ where the vertical bar separates the two columns of the $2\\times 2$ matrix, we have $\\sigma (K,w)*(mI,\\left[\\left( \\begin{array}{c}1 \\\\0\\end{array}\\right)\\right])=(K,[w]).$ Notice that if $z\\ \\mathrm {and}\\ z^{\\prime }$ are representatives of the same element of $ P^1(\\mathbb {C}),$ $\\sigma \\left(K, z \\right)$ and $\\sigma \\left(K, z^{\\prime } \\right)$ are in general different.",
"Thus, since ${\\mathbf {r}}^{-1}(mI,\\left[\\left( \\begin{array}{c}1 \\\\0\\end{array}\\right)\\right])=\\lbrace s {\\left(\\begin{array}{c}1 \\\\0\\\\1\\\\0\\end{array}\\right)}:s\\in S^1 \\rbrace $ we have ${\\mathbf {r}}^{-1}(K,[w])=\\rho (\\sigma (K,w)) \\lbrace s \\left( {\\begin{array}{c}1 \\\\0\\\\1\\\\0\\end{array} }\\right):s\\in S^1 \\rbrace $ which leads to $ {{\\mbox{$ r$}}}^{-1}(K,\\ [a])=\\left\\lbrace s \\left(\\!\\begin{array}{c} I\\\\m\\,K^{-1}\\end{array}\\!",
"\\right)\\frac{a}{\\sqrt{ma^*K^{-1}a}}:\\ s \\in {\\bf S}^1 \\right\\rbrace $ Also we have $P(K,\\ [w])\\,=P(\\sigma (K,w)G_{(\\alpha _1)_{SL}})=\\,-\\eta K.$ If $f$ is a continuous function with compact support in $\\cal B,$ and is $S^1$ -homogeneous of degree -1 (i.e.",
"$f\\in {\\cal C}$ in the notation of section REF ) , the corresponding prewave function is $\\psi _f:(H,\\ K,\\ [a]) \\in H(2) \\times {\\cal H}^m \\times {{\\mbox{$ P_1$}}}( {\\bf C})\\ \\ \\mapsto \\ \\ f\\left(\\!\\begin{array}{c} w \\\\ z \\end{array} \\!",
"\\right) e^{-i\\pi \\eta \\ Tr(K\\,\\varepsilon \\overline{H} \\varepsilon ) }\\ \\left(\\!\\begin{array}{c} w\\\\ z \\end{array} \\!",
"\\right)$ where $\\left(\\!\\begin{array}{c} w\\\\ z \\end{array} \\!",
"\\right) $ is arbitrary in ${\\mbox{$r$}}^{-1}(K,\\ [a])$ .",
"When one considers the Dirac matrices in the representation $\\gamma ^4=\\left( \\begin{array}{cc}0&I \\\\I&0\\end{array} \\right)\\ \\ \\ ,\\gamma ^k=\\left( \\begin{array}{cc}0&-\\sigma _k \\\\\\sigma _k&0\\end{array} \\right)\\ \\ \\ ,k=1,\\ 2,\\ 3,$ a straightforward computation proves that that these prewave functions satisfy Dirac Equation $(\\gamma ^\\nu \\ \\partial _\\nu \\ -2 \\pi i \\eta m)\\ \\psi _f \\ =\\ 0.$ A sesquilinear form on ${\\bf C}^4$ whose value on ${\\cal B}$ is 1, is defined by $ \\Phi \\left( Z,\\ Z^\\prime \\right)=\\frac{1}{2} Z^* \\gamma ^4 Z.$ Thus, if $f,\\ f^{\\prime } \\in {\\cal C},$ the hermitian product of $\\psi _f$ and $\\psi _{f^\\prime }$ can be writen $\\langle \\psi _f,\\ \\psi _{f^\\prime } \\rangle =\\frac{1}{2} \\int _{{\\cal H}^m \\times P_1(C)}\\psi _f^*\\,\\gamma ^4\\,\\psi _{f^\\prime }\\,\\mu .$ To describe Wave Functions we need an invariant volume element on ${\\cal H}^m\\times {\\bf P}_1 (\\mathbb {C}).$ Let us consider the 5-form in ${\\cal H}^m \\times ( {\\mathbb {C}}^2-\\lbrace (0,0)\\rbrace )$ , given by $(\\mu _0)_{(K,\\ z)}=\\nu \\ \\wedge \\frac{(z^1\\,dz^2-z^2\\,dz^1)\\,\\wedge \\,{(\\overline{z^1}\\,d\\overline{z^2}-\\overline{z^2}\\,d\\overline{z^1})}}{(z^* {\\mbox{$\\varepsilon $}}\\overline{K} {\\mbox{$\\varepsilon $}} z)^2},$ where the $z^k$ are the two canonical projections of ${\\mathbb {C} \\rm }^2$ onto ${\\mathbb {C}}$ .",
"This form is well defined since, for all $K\\in {\\cal H}^m, $ the hermitian product defined in ${\\mathbb {C}}^2$ by $\\langle x,y \\rangle =x^* \\overline{K} y$ is positive definite.",
"This differential form projects to an invariant volume element, $\\mu $ , in ${\\cal H}^m \\times {\\mbox{$ P_1$}}(\\mathbb {C})$ .To prove this fact, one must proceed in many steps.",
"Let us denote by $\\tau $ the canonical map $\\tau :(K,z)\\in {\\cal H}^m \\times ( {\\mathbb {C}}^2-\\lbrace (0,0)\\rbrace ) \\rightarrow (K,[z])\\in {\\cal H}^m \\times {\\mbox{$ P_1$}}(\\mathbb {C}).$ The triple ${\\cal H}^m \\times ( {\\mathbb {C}}^2-\\lbrace (0,0)\\rbrace )({\\cal H}^m \\times {\\mbox{$ P_1$}}(\\mathbb {C}), {\\mathbb {C}}-\\lbrace 0\\rbrace )$ is a principal fibre bundle with projection $\\tau .$ To prove that there exist a form, $\\mu ,$ such that $\\tau ^*\\mu =\\mu _0$ it is enought to see that $\\mu _0$ is invariant under the bundle action, what is obvious, and that $\\mu _0$ vanishes on vertical vectors, what can be proved as follows.",
"The vertical vectors at $(K,z)$ are tangent at $t=0$ to the curves $\\gamma (t)=(K,\\lambda (t)z)$ with $\\lambda $ a $C^\\infty $ map from $\\mathbb {R}$ into $\\mathbb {C}$ such that $\\lambda (0)=1.$ Let $X_{(K,z)}$ be the tangent vector to $\\gamma $ at $0.$ We have $X_{(K,z)}=\\lambda ^\\prime (0)\\left(z^1 \\frac{\\partial }{\\partial z^1}+z^2 \\frac{\\partial }{\\partial z^2}\\right)+\\overline{\\lambda ^\\prime (0)}\\left(\\overline{z^1} \\frac{\\partial }{\\partial \\overline{z^1}}+\\overline{z^2} \\frac{\\partial }{\\partial \\overline{z^2}}\\right),$ and it is easily seen that $i_{X_{(K,z)}}{(\\mu _0)_{(K,z)}}=0.$ This finishes the proof of the existence of $\\mu .$ Let us denote $\\Vert p \\Vert ^2=\\sum _{i=1}^3 (p^i)^2$ and $q_1(p_1,p_2,p_3,z)&=&(h(p_1,p_2,p_3,(m^2+ \\Vert p \\Vert ^2 )^{\\frac{1}{2}} ),\\left[ \\left(\\begin{array}{c}1\\\\z \\end{array}\\right) \\right]) \\\\q_2(p_1,p_2,p_3,z)&=&(h(p_1,p_2,p_3,(m^2+ \\Vert p \\Vert ^2)^{\\frac{1}{2}} ),\\left[ \\left(\\begin{array}{c}z\\\\1 \\end{array}\\right) \\right]) \\nonumber $ for all $p_1,p_2,p_3,z\\in {\\mathbb {R}}^3 \\times \\mathbb {C}.$ The maps $q_1$ and $q_2$ are parametrizations, whose inverses are local charts that compose an atlas of ${\\cal H}^m \\times {\\mbox{$ P_1$}}(\\mathbb {C}).$ The local expressions of $\\mu $ in these charts can be obtained as the reciprocal image of $\\mu _0$ by the maps $\\sigma _1(p_1,p_2,p_3,z)=(h(p_1,p_2,p_3,(m^2+ {\\mbox{$\\sum _{i=1}^3$}} \\ (p^i)^2)^{\\frac{1}{2}} ),\\left(\\begin{array}{c}1\\\\z \\end{array}\\right) ),$ and $\\sigma _(p_1,p_2,p_3,z)=(h(p_1,p_2,p_3,(m^2+ {\\mbox{$\\sum _{i=1}^3$}} \\ (p^i)^2)^{\\frac{1}{2}} ),\\left(\\begin{array}{c}z\\\\1 \\end{array}\\right) ).$ These local expressions are given by $\\sigma _1^*\\mu _0=\\frac{\\nu \\wedge dz \\wedge d \\overline{z}}{2\\Re (z(p_1-ip_2))+(1-\\vert z \\vert ^2)p_3-(1+\\vert z \\vert ^2)(m^2+ \\Vert p \\Vert ^2 )^{\\frac{1}{2}}},$ $\\sigma _2^*\\mu _0=\\frac{\\nu \\wedge dz \\wedge d \\overline{z}}{2\\Re (z(p_1+ip_2))+(\\vert z \\vert ^2-1)p_3-(1+\\vert z \\vert ^2)(m^2+ \\Vert p \\Vert ^2 )^{\\frac{1}{2}}}.$ As a particular consequence, $\\mu $ is a volume element.",
"Now, let us consider the action of $SL$ on ${\\cal H}^m \\times ( {\\mathbb {C}}^2-\\lbrace (0,0)\\rbrace )$ given by $A*(K,z)=(AKA^*,Az).$ We already know that $\\nu $ is invariant under the action on ${\\cal H}^m ,$ and it is not a difficult matter to see that $z^1\\,dz^2-z^2\\,dz^1,\\ {\\overline{z^1}\\,d\\overline{z^2}-\\overline{z^2}\\,d\\overline{z^1} } $ and $z^* {\\mbox{$\\varepsilon $}}\\overline{K} {\\mbox{$\\varepsilon $}} z$ are invariant under this action.",
"As a consecuence, $\\mu _0$ is invariant under the same action.",
"It follows that $\\mu $ is an invariant volume element on ${\\cal H}^m \\times {\\mbox{$ P_1$}}(\\mathbb {C}).$ The wave functions have the form $\\tilde{\\psi }_f(X)=\\int _{{\\cal H}^m \\times P_1(C)}\\psi _f(h(X),\\cdot ,\\cdot )\\ \\mu $ and also satisfies Dirac Equation $(\\gamma ^\\nu \\ \\partial _\\nu \\ -2 \\pi i \\eta m)\\ \\tilde{\\psi }_f \\ =\\ 0.$"
],
[
"Wave functions for $T>1$ .",
"Now, let us consider a particle whose movement space is the coadjoint orbit of $\\alpha _T=\\left\\lbrace \\ \\frac{iT}{8\\pi }\\left( \\begin{array}{cc} 1&0 \\\\ 0&-1\\end{array} \\right) \\ ,\\ \\eta \\,m\\,I\\right\\rbrace ,$ where $T\\in {\\bf Z}^+,\\ m \\in {\\bf R}^+, \\eta =\\pm 1.$ The case of the preceeding section is the particular one given by $T=1.$ For all $T$ the orbit is a quantizable, not $\\bf R \\rm $ -quantizable orbit, of the type 5.",
"Now we have ${\\mbox{$G_{\\alpha _T}$}} = \\lbrace \\ (\\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right), \\ hI ): \\ z\\in S^1,\\ h\\in {\\mbox{$\\bf R \\rm $}} \\rbrace .$ The unique homomorphism from $G_{\\alpha _T}$ onto ${\\bf S}^1$ whose differential is $\\alpha _T$ is given by ${\\mbox{$C_{\\alpha _T}$}} (\\ \\left( \\begin{array}{cc}e^{2\\pi i\\phi }&0 \\\\0&e^{-2\\pi i\\phi }\\end{array} \\right) ,\\ hI)=e^{2\\pi i(\\phi T-\\eta m h)}.$ Then $\\begin{array}{l}\\vspace{7.22743pt}{{\\mbox{$(G_{\\alpha _T } ) _{SL}$}}=\\lbrace \\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right):\\ z\\in S^1\\rbrace }, \\\\\\vspace{7.22743pt}{{\\mbox{$(C_{\\alpha _T})_{SL}$}}(\\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right))=z^T}, \\\\\\vspace{7.22743pt}SL_1 \\cap SL_2 ={\\mbox{$(G_{\\alpha _T} ) _{SL}$}}, \\\\\\vspace{7.22743pt}SL_1 \\cap Ker\\,(C_{\\alpha _T})_{SL}=Ker\\,(C_{\\alpha _T})_{SL}=\\lbrace \\ \\left( \\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array} \\right):\\ z\\in \\@root T \\of {\\mathbf {1}}\\rbrace .",
"\\\\\\end{array}$ where $\\@root T \\of {\\mathbf {1}}$ is the subgroup of ${\\mathbb {C}}^*$ composed by the roots of order $T$ of 1.",
"The homogeneous space SL/${\\mbox{$(G_{\\alpha _T} ) _{SL}$}}$ , is the same as in the preceeding section so that it can be identified to ${\\cal H}^m\\times {\\bf P}_1(\\mathbb {C})$ , and consider on it the invariant volume element $\\mu .$ In order to give the wave functions in the case of arbitrary ${T}$ , the following results are useful.",
"Let ${\\beta },\\ \\beta ^\\prime $ be quantizable elements of $\\underline{G}^*$ with ${\\beta ^\\prime }$ not R-quantizable, $(G_{\\beta })_{SL}=(G_{{\\beta ^\\prime }})_{SL}$ , $(C_{\\beta })_{SL} =\\left((C_{{\\beta ^\\prime }})_{SL}\\right)^T,$ where $T \\in {\\bf Z}^+$ .",
"Notice that, as a consecuence of the fact that $\\beta ^{\\prime }$ is not ${\\mathbb {R} }$ -quantizable, we must have $(C_{\\beta ^\\prime })_{SL}((G_{{\\beta ^\\prime }})_{SL})=S^1.$ If $(\\rho , L, z_0)$ is a trivialization of $C_{\\beta ^\\prime }$ , we consider the triple $(\\rho ^{\\otimes T},\\ L^ {\\otimes T},\\ z_0^ {\\otimes T}),$ where $L^ {\\otimes T}=L \\otimes \\stackrel{(T}{\\cdots } \\nolinebreak \\otimes L,\\ z_0^{\\otimes T}=z_0 \\otimes \\stackrel{(T}{\\cdots } \\otimes z_0$ , and $\\rho ^{\\otimes T}$ is the representation such that $\\rho ^{\\otimes T}(A)(z_1 \\otimes \\cdots \\otimes z_T)=\\rho (A)(z_1) \\otimes \\cdots \\otimes \\rho (A)(z_T).$ In lemma 6.1 of [8] I have proved that, under these circunstances, if $(G_{\\beta })_{SL}$ is connected, $(\\rho ^{\\otimes T},\\ L^ {\\otimes T},\\ z_0^ {\\otimes T})$ is a trivialization of $C_\\beta $ .",
"Let us assume that $(\\rho ^{\\otimes T},\\ L^ {\\otimes T},\\ z_0^ {\\otimes T})$ is a trivialization of $C_\\beta $ and let ${\\cal B}_T$ be the orbit of $ z_0^{\\otimes T}$ .",
"The pullback by $z \\in {\\cal B} \\mapsto z^{\\otimes T} \\in {\\cal B}_T$ , establishes a one to one map from the set of the $S^1$ -homogeneous functions of degree -1 on $ {\\cal B}_T$ , onto the set of the ${S^1}$ -homogeneous functions of degree -T on ${\\cal B}$ .",
"If $f$ is one of these functions, the corresponding prewave function of particles corresponding to $\\beta $ has the form ${\\mbox{$\\psi $}}_f(H,\\ m)\\,=\\,f(z)\\,e^{i\\pi Tr(P(m)\\,\\varepsilon \\overline{H} \\varepsilon )}z^{\\otimes T}$ where $z \\in { \\bf r}^{-1}(m).$ Now we apply these results to the particles of type 5 with $T> 1.$ Let $\\beta ^{\\prime }=\\alpha _1$ and $\\beta =\\alpha _T.$ The remark just made leads to the following Prewave Function $\\psi _f:(H,\\ K,\\ [a]) \\in H(2) \\times {\\cal H}^m \\times {{\\mbox{$ P_1$}}}( {\\bf C})\\ \\ \\mapsto \\ \\ f\\left(\\!\\!\\begin{array}{c} w \\\\ z \\end{array} \\!",
"\\!",
"\\right) e^{-i\\pi \\eta \\ Tr(K\\,\\varepsilon \\overline{H} \\varepsilon ) }\\ \\left(\\!\\!\\begin{array}{c} w\\\\ z \\end{array} \\!\\!",
"\\right)^{\\otimes T}$ where $\\left(\\!\\!\\begin{array}{c} w\\\\ z \\end{array} \\!\\!",
"\\right) $ is arbitrary in ${\\mbox{$r$}}^{-1}(K,\\ [a])$ , and f is a function on ${\\cal B}$ , $C^\\infty ,$ with compact support and homogeneous of degree -T under multiplication by complex numbers of modulus one.",
"The Wave Functions are obtained by integration as usual."
],
[
"Movement Space for massive particles with $T\\ge 1$ .",
"We consider the action of $G$ on ${\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3$ given by $(A,H)*(K,[z],\\overrightarrow{y})=(AKA^*,[Az], \\overrightarrow{x}(A,H,\\overrightarrow{y},K))$ where $\\overrightarrow{x}(A,H,\\overrightarrow{y},K)$ is given by $h(\\overrightarrow{x}(A,H,\\overrightarrow{y},K),0)=Ah(\\overrightarrow{y},0)A^*+H+\\ell (A,H,\\overrightarrow{y},K) A\\frac{K}{m}A^* \\\\\\ell (A,H,\\overrightarrow{y},K)=\\frac{-mTr(Ah(\\overrightarrow{y},0)A^*+H)}{Tr( AKA^*)}.$ This is a transitive action.",
"The isotropy subgroup at $(mI,[{}^t(1,0)], \\overrightarrow{0}),$ is found to be $G_{\\alpha _T}.$ Thus, the map $\\lambda :(A,H) G_{\\alpha _T}\\in G/G_{\\alpha _T} \\leftrightarrow (A,H)*\\left(mI,\\left[\\left(\\begin{array}{c}1 \\\\0\\end{array}\\right)\\right], \\overrightarrow{0}\\right)\\in {\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3$ enables us to identify the coadjoint orbit with ${\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3.$ A parametrization whose image is a neighborhood of $(mI,[{}^t\\hspace{0.0pt}(1,0)], \\overrightarrow{0}),$ is $\\Gamma _s:(\\overrightarrow{k},\\overrightarrow{x},z)\\in \\mathbb {R}^3 \\times \\mathbb {R}^3\\times \\mathbb {C}\\rightarrow &&\\left(mh(\\overrightarrow{k},k_4),\\left[\\left(\\begin{array}{c}1 \\\\z\\end{array}\\right)\\right], \\overrightarrow{x}\\right)\\\\&&\\in {\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3.\\nonumber $ where $k_4=\\sqrt{1+k_1^2+k_2^2+k_3^2}.$ The corresponding local chart is given by $\\Gamma _s^{-1}:\\left(K,\\left[\\left(\\begin{array}{c}z^1 \\\\z^2\\end{array}\\right)\\right],\\overrightarrow{x}\\right)\\in \\lbrace z^1\\ne 0 \\rbrace \\rightarrow \\left(\\frac{1}{m}h^{-1}(K-Tr(K)I),\\overrightarrow{x},\\frac{z^2}{z^1}\\right)$ Another parametrization is $\\Gamma _n:(\\overrightarrow{k},\\overrightarrow{x},z)\\in \\mathbb {R}^3 \\times \\mathbb {R}^3\\times \\mathbb {C}\\rightarrow &&\\left(mh(\\overrightarrow{k},k_4),\\left[\\left(\\begin{array}{c}z \\\\1\\end{array}\\right)\\right], \\overrightarrow{x}\\right)\\\\&&\\in {\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3.\\nonumber $ with $k_4=\\sqrt{1+k_1^2+k_2^2+k_3^2},$ whose inverse is given by $\\Gamma _n^{-1}:\\left(K,\\left[\\left(\\begin{array}{c}z^1 \\\\z^2\\end{array}\\right)\\right],\\overrightarrow{x}\\right)\\in \\lbrace z^2\\ne 0 \\rbrace \\rightarrow \\left(\\frac{1}{m}h^{-1}(K-Tr(K)I),\\overrightarrow{x},\\frac{z^1}{z^2}\\right) \\nonumber $ Obviously, these two charts compose an atlas.",
"The identification of ${\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3$ with the coadjoint orbit is given by $(A,H)*((mI,[{}^t(1,0)], \\overrightarrow{0})\\longleftrightarrow Ad^*_{(A,H)}\\alpha _T.$ Now, let $\\left(K,\\left[ z\\right],\\overrightarrow{x}\\right)\\in {\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3.$ If $\\sigma \\left(K, w \\right)$ is given by (REF ), then $\\left(\\sigma \\left(K, z \\right),h(\\overrightarrow{x},0)\\right)*(mI,[{}^t(1,0)], \\overrightarrow{0})=\\left(K,\\left[ z\\right],\\overrightarrow{x}\\right),$ so that $\\left(K,\\left[ z\\right],\\overrightarrow{x}\\right)$ must be identified to $Ad^*_{\\left(\\sigma \\left(K, z \\right),h(\\overrightarrow{x},0)\\right)}\\cdot \\alpha _T.$ Now let $F$ be a dynamical variable on the coadjoint orbit.",
"$F$ becomes a function on ${\\cal H}^m \\times {\\bf P}_1(\\mathbb {C})\\times \\mathbb {R}^3$ .",
"To give explicit expresions of these functions, is now preferable to use the sphere $S^2$ instead of ${\\ bf P}_1(\\mathbb {C}) $ thus writing $F(K,[z],\\vec{x}) =F( Ad^*_{\\left(\\sigma \\left(K, z \\right),h(\\overrightarrow{x},0)\\right)}\\cdot \\alpha _T)= F(K,\\vec{u},\\vec{x}),$ where $\\vec{u}\\in S^2$ is related to $[z]\\in \\mathbf {P_1(\\mathbb {C})}$ by (c.f.",
"(REF )) $\\frac{2}{z^*z}zz^*=h(\\vec{u},1).$ Let us evaluate $Ad^*_{\\left(\\sigma \\left(K, z \\right),h(\\vec{x},0)\\right)}\\cdot \\alpha _T.$ To do that computation we need some of the equations (REF ) to (REF ), and $(\\sigma (K,z))^{-1}=\\frac{1}{\\sqrt{mz^*K^{-1}z}} \\left( \\frac{ mz^*K^{-1}}{ z \\varepsilon }\\right),$ where the horizontal line separates the two files of the $2\\times 2$ matrix.",
"This leads to $Ad^*_{\\left(\\sigma \\left(K, z \\right),h(\\vec{x},0)\\right)}\\cdot \\alpha _T&=&\\left\\lbrace \\left[ -\\frac{T}{8\\pi }\\frac{1}{\\langle k_4\\vec{ u}-\\vec{ k},\\vec{u} \\rangle }h(\\vec{k}\\times \\vec{u},0)+\\frac{\\eta m}{2}k_4h(\\vec{x},0) \\right]+ \\right.",
"\\\\ \\nonumber &+&i \\left[\\frac{T}{8 \\pi } \\frac{1}{\\langle k_4\\vec{ u}-\\vec{ k},\\vec{u} \\rangle }h( k_4\\vec{ u}-\\vec{ k},0)+ \\frac{\\eta m}{2} h(\\vec{k}\\times \\vec{x},0)\\right], \\\\ \\nonumber && \\left.",
"\\eta m h(\\vec{k},k_4) \\right\\rbrace ,$ where $h(\\vec{k},k_4)=\\frac{1}{m}K.$ Thus, using (REF ), we obtain $\\begin{split}P(mh(\\vec{k},k_4),\\vec{u},\\vec{x})&=-\\eta m h(\\vec{k},k_4),\\\\\\vec{ l}(mh(\\vec{k},k_4),\\vec{u},\\vec{x})&=\\frac{T}{4\\pi }\\frac{k_4\\vec{u}-\\vec{k}}{k_4-\\langle \\vec{ k},\\overrightarrow{u} \\rangle }+\\eta m \\vec{k}\\times \\vec{x},\\\\\\vec{g}(mh(\\vec{k},k_4),\\vec{u},\\vec{x})&=\\frac{T}{4\\pi }\\frac{\\vec{k}\\times \\vec{u}}{k_4-\\langle \\vec{ k},\\overrightarrow{u} \\rangle }-\\eta m k_4 \\vec{x}.\\end{split}$ Notice that $k_4-\\langle \\vec{ k},\\overrightarrow{u} \\rangle =\\langle k_4 \\overrightarrow{u} - \\vec{ k},\\overrightarrow{u} \\rangle .$ If we denote $P(K,\\vec{u},\\vec{x}),\\vec{l}(K,\\vec{u},\\vec{x})$ and $\\vec{g}(K,\\vec{u},\\vec{x})$ simply by $P,\\ \\vec{l}$ and $\\vec{g}$ respectively when no danger of confusion exist, and $\\vec{P}=(P^1,P^2,P^3),$ we have the following relations between these dynamical variables $\\vec{ l}&=&\\frac{T}{4\\pi }\\frac{P^4\\vec{u}-\\vec{P}}{P^4-\\langle \\vec{ P},\\vec{u} \\rangle }+ \\vec{x}\\times \\vec{P},\\\\ \\vec{g}&=&\\frac{T}{4\\pi }\\frac{\\vec{P}\\times \\vec{u}}{ P^4-\\langle \\vec{ P},\\vec{u} \\rangle }+P^4\\vec{x}$ The value of the Pauli-Lubanski fourvector at $(mh(\\vec{k},k_4),\\vec{u},\\vec{x})$ can be calculated using (REF ) , (),(REF ),...,() and is found to be $ \\begin{split}\\overrightarrow{W}&=P^4\\frac{T}{4\\pi }\\frac{P^4\\vec{u}-\\vec{P}}{\\langle P^4\\vec{u}-\\vec{P},\\vec{u} \\rangle }=-\\eta m k_4\\frac{T}{4\\pi } \\frac{k_4\\vec{u}-\\vec{k}}{\\langle k_4\\vec{u}- \\vec{ k},\\vec{u} \\rangle } \\\\W^4&=\\frac{T}{4\\pi }\\frac{\\langle P^4\\vec{u}-\\vec{P},\\vec{P} \\rangle }{\\langle P^4\\vec{u}-\\vec{P},\\vec{u} \\rangle }= -\\eta m\\frac{T}{4\\pi } \\frac{\\langle k_4\\vec{u}-\\vec{k},\\vec{k} \\rangle }{\\langle k_4\\vec{u}- \\vec{ k},\\vec{u} \\rangle },\\end{split}$ so that we can also write $\\vec{ l}=\\frac{1}{P^4}\\overrightarrow{W}+ \\vec{x}\\times \\vec{P} .$"
],
[
"Contact manifold for Dirac particles",
"Now we pay attention to the contact manifold in the case $T=1$ .",
"We consider the action of $G$ on ${\\cal B}\\times {\\mathbb {R}}^3$ given by $(A,H)*(w,z,\\overrightarrow{y})=\\left( e^{2\\pi i\\eta m \\ell }Aw,e^{2\\pi i\\eta m \\ell }(A^*)^{-1}z,\\overrightarrow{x}(A,H,w,z,\\overrightarrow{y})\\right)$ where $h(\\overrightarrow{x}(A,H,w,z,\\overrightarrow{y}),0)= Ah(\\overrightarrow{y},0)A^*+H+\\ell A(ww^*-\\varepsilon \\overline{zz^*} \\varepsilon )A^*.$ Obviously, we have $\\ell =-\\frac{Tr(Ah(\\vec{x},0)A^*+H)}{Tr( A(ww^*-\\varepsilon \\overline{zz^*} \\varepsilon )A^*)}.$ This is a transitive action and the isotropy subgroup at $\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0}\\right)$ is $Ker\\,C_{\\alpha _1},$ so that we can identify $G/Ker\\,C_{\\alpha _1}$ with ${\\cal B}\\times {\\mathbb {R}}^3$ by means of $(A,H)Ker\\,C_{\\alpha _1} \\longleftrightarrow (A,H)* \\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0}\\right).$ When $w\\in {\\mathbb {C}}^2-\\lbrace 0 \\rbrace ,$ the $z$ such that $(w,z)\\in {\\cal B}$ compose the set $\\left\\lbrace \\frac{1}{w^*w}(w+y\\varepsilon \\overline{w}): y\\in {\\mathbb {C}} \\right\\rbrace .$ Thus, the map $\\Delta _0:(w,y)\\in ({\\mathbb {C}}^2-\\lbrace 0 \\rbrace )\\times {\\mathbb {C}} \\longrightarrow \\left( w , z(w,y)\\right)\\in {\\cal B},$ where $z(w,y)=\\frac{1}{w^*w}(w+y\\varepsilon \\overline{w}),$ is a bijection, and in fact a diffeomorphism.",
"Its inverse is given by $(\\Delta _0)^{-1}:(w,z)\\in {\\cal B} \\longrightarrow (w, {}^tw \\varepsilon z)\\in ({\\mathbb {C}}^2-\\lbrace 0 \\rbrace )\\times {\\mathbb {C}}.$ Then, $(\\Delta _0)^{-1}$ is a global complex coordinate system of ${\\cal B},$ and $\\Delta :(w,y,\\vec{x})\\in ({\\mathbb {C}}^2-\\lbrace 0 \\rbrace )\\times {\\mathbb {C}}\\times {\\mathbb {R}}^3 \\longrightarrow \\left( w , z(w,y),\\vec{x}\\right)\\in {\\cal B}\\times {\\mathbb {R}}^3,$ is a parametrization of the Contact Manifold defined in an an open subset of ${\\mathbb {C}}^3 \\times {\\mathbb {R}}^3.$ Notice that, since we know a diffeomorphism, $\\phi _0,$ from ${\\cal B}$ onto $SL(2,\\mathbb {C})$ (c.f.",
"(REF )), we obtain also a complex parametrization of this group by means of $\\phi _0 \\circ \\Delta _0:(w,y)\\in ({\\mathbb {C}}^2-\\lbrace 0 \\rbrace )\\times {\\mathbb {C}} \\longrightarrow \\left( w\\ \\ \\frac{1}{w^*w}({\\overline{y}}w-\\varepsilon \\overline{w}\\right)\\in SL(2,\\mathbb {C}).$ The inverse of $\\phi _0 \\circ \\Delta _0$ is the global complex chart of $SL(2,\\mathbb {C})$ given by $\\phi :(w|v)\\in SL(2,\\mathbb {C}) \\longrightarrow (w, {}^tw \\overline{v})\\in ({\\mathbb {C}}^2-\\lbrace 0 \\rbrace )\\times {\\mathbb {C}}.$ The canonical map of $G$ onto $G/Ker\\,C_{\\alpha _1}$ becomes $(A,H)\\in G \\longleftrightarrow (A,H)* \\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0}\\right),$ and the map $\\Sigma :(w,z,\\vec{x})\\in {\\cal B}\\times {\\mathbb {R}}^3 \\longrightarrow \\left( \\left( w\\ \\ -\\varepsilon \\overline{z}\\right),h(\\vec{x},0)\\right)\\in G,$ is a section of this canonical map , as a consecuence of the fact that $\\Sigma (w,z,\\vec{x})*\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0} \\right)=(w,z,\\vec{x}).$ The canonical map from $G/Ker\\,C_\\alpha $ onto $G/G_\\alpha $ becomes $\\tilde{\\mathbf {r}}&:&(w,z,\\vec{x})\\in {\\cal B}\\times {\\mathbb {R}}^3 \\longrightarrow \\Sigma (w,z,\\vec{x})* (mI,[\\left(\\begin{array}{c}1\\\\0 \\end{array} \\right)],\\vec{0})=\\\\&=&(m(ww^*-\\varepsilon \\overline{zz^*}\\varepsilon ),[w],\\vec{x})\\in {\\cal H}^m\\times {\\mathbf {P}_1(\\mathbb {C})}\\times {\\mathbb {R}}^3,$ i.e.",
"coincides with ${\\bf r}\\times Id_{{\\mathbb {R}}^3}.$ The Linear Momentum, Angular Momentum and Pauli-Lubanski fourvector, whose descriptions in the symplectic manifold are given by (REF ) and (REF ), when composed with $\\tilde{\\mathbf {r}},$ give functions $P_c,\\ \\vec{l}_c,\\vec{g}_c,W_c,$ on the Contact Manifold, that are the translation of these dynamical variables to this homogeneous space.",
"In order to give explicit expressions for these functions ($c.f.$ (REF )), let us denote $\\tilde{\\mathbf {r}}(w,z,\\vec{x})=(m\\,h(\\vec{k},k_4),\\vec{u}, \\vec{x})\\in {\\cal H}^m \\times S^2 \\times {\\mathbb {R}}^3$ we have $h(\\vec{u},1)= \\frac{2}{w^*w}w\\,w^*,$ and $h(\\vec{k},k_4)=ww^*-\\varepsilon \\overline{zz^*}\\varepsilon ,$ but $-h(\\vec{u},1)\\varepsilon \\overline{h(\\vec{k},k_4)}\\varepsilon =h(k_4\\vec{u}-\\vec{k},k_4-\\langle \\vec{k},\\vec{u} \\rangle ) + i h(\\vec{k} \\times \\vec{u},0)$ and $-h(\\vec{u},1)\\varepsilon \\overline{h(\\vec{k},k_4)}\\varepsilon =- \\frac{2}{w^*w}ww^*\\varepsilon \\overline{(ww^*-\\varepsilon \\overline{zz^*}\\varepsilon )}\\varepsilon =\\frac{2}{w^*w}w\\,z^*,$ leads to $h(k_4\\vec{u}-\\vec{k},k_4-\\langle \\vec{k},\\vec{u} \\rangle ) + i h(\\vec{k} \\times \\vec{u},0)=\\frac{2}{w^*w}w\\,z^*,$ so that $\\begin{split}&h(k_4\\vec{u}-\\vec{k},k_4-\\langle \\vec{k},\\vec{u} \\rangle ) =\\frac{1}{w^*w}(w\\,z^*+zw^*),\\\\&h(\\vec{k} \\times \\vec{u},0)=i\\frac{1}{w^*w}(zw^*-w\\,z^*)\\end{split}$ and $\\begin{split}&\\langle k_4\\vec{u}-\\vec{k},\\vec{u} \\rangle =k_4-\\langle \\vec{k},\\vec{u} \\rangle =\\frac{1}{w^*w}\\\\&h\\left(\\frac{k_4\\vec{u}-\\vec{k}}{k_4-\\langle \\vec{k},\\vec{u} \\rangle },1\\right)=w\\,z^*+zw^*\\\\&h\\left(\\frac{\\vec{k} \\times \\vec{u}}{k_4-\\langle \\vec{k},\\vec{u} \\rangle },0\\right)=i(zw^*-w\\,z^*)\\end{split}$ Thus $\\begin{split}&P_c(w,z,\\vec{x})=-\\eta m\\,(ww^*-\\varepsilon \\overline{zz^*}\\varepsilon )\\\\&h(\\vec{l}_c(w,z,\\vec{x}),0)=\\frac{T}{4\\pi }(wz^*+zw^*-I)+h(\\vec{x} \\times \\vec{P}_c,0)\\\\&h(\\vec{g}_c(w,z,\\vec{x}),0)=\\frac{iT}{4\\pi } (zw^*-wz^*)+P^4 h(\\vec{x},0)\\\\&h(\\vec{W}_c(w,z,\\vec{x}),0)=P^4\\frac{T}{4\\pi }(wz^*+zw^*-I)\\\\&W^4_c(w,z,\\vec{x})=\\frac{-\\eta m T}{8\\pi }(w^*w-z^*z)\\end{split}$ The formula for $W_c^4$ can be obtained as follows $\\begin{split}&\\langle \\vec{l},\\vec{P} \\rangle \\circ \\mathbf {\\tilde{r}}=\\left\\langle \\frac{T}{4\\pi }\\frac{k_4\\vec{u}-\\vec{k}}{\\langle k_4\\vec{u}-\\vec{k},\\vec{u} \\rangle },\\vec{P} \\right\\rangle \\circ \\mathbf {\\tilde{r}}= \\\\&=\\frac{1}{2}Tr\\left(h\\left(\\frac{T}{4\\pi }\\frac{k_4\\vec{u}-\\vec{k}}{\\langle k_4\\vec{u}-\\vec{k},\\vec{u} \\rangle },0\\right)P \\right) \\circ \\mathbf {\\tilde{r}}=\\\\ &=\\frac{1}{2}Tr\\left(\\left(\\frac{T}{4\\pi }(wz^*+zw^*-I)\\right)(-\\eta m)(ww^*-\\varepsilon \\overline{zz^*}\\varepsilon )\\right)=\\\\ &=\\frac{-\\eta mT}{8\\pi }(w^*w-z^*z).\\end{split}$ The local expresions of these functions in the chart $\\Delta ^{-1}$ are obtained simply by changing in the above expresions $z$ by the $z(w,y)$ given in (REF ).",
"Let $\\Omega _1$ be the contact form and $\\tilde{\\alpha }_1$ the left invariant differential form on $G$ whose value at $(I,0)$ is $\\alpha _1.$ We have $\\Omega _1=\\Sigma ^* \\tilde{\\alpha }_1,$ and this formula enables us, by similar procedures to those of section REF , to see that $\\Omega _1=\\frac{iT}{4\\pi }\\left(z^1 d\\overline{w^1}-\\overline{ z^1} d w^1+z^2 d\\overline{ w^2}-\\overline{ z^2} d w^2\\right)-P^1dx^1-P^2 dx^2-P^3 d x^3,$ where (c.f.",
"(REF )) $\\left(\\begin{array}{c}z^1\\\\z^2\\end{array}\\right)=z(w,y).$ This equation must be interpreted as follows: the right hand side of (REF ) is a differential 1-form in $\\mathbb {C}^4 \\times \\mathbb {R}^3$ refered to coordinates $w^1,\\dots ,x^3,$ and $\\Omega _1$ is the restriction of this form to the submanifold ${\\cal B}\\times \\mathbb {R}^3.$ Obviously, under the same conditions $\\begin{split}d\\Omega _1=\\frac{iT}{4\\pi }&\\left(dz^1\\wedge d\\overline{w^1}-d\\overline{ z^1}\\wedge d w^1+dz^2 \\wedge d\\overline{ w^2}-d\\overline{ z^2} \\wedge d w^2\\right)\\\\&-dP^1\\wedge dx^1-dP^2 \\wedge dx^2-dP^3 \\wedge d x^3.",
"\\end{split}$ Now, to obtain the symplectic form we only need sections of the map $\\tilde{\\mathbf {r}}.$ On the domain, $U_s,$ of the local chart $\\Gamma _s^{-1},$ the map ${\\tilde{\\sigma }}_s: \\Gamma _s(\\vec{k},\\vec{x},t) \\rightarrow \\left( \\sigma \\left(mh(\\vec{k},k_4),\\left( \\begin{array}{c} 1\\\\ t\\end{array}\\right)\\right),h(\\vec{x},0)\\right)*\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0} \\right), $ where $\\sigma (K,w)$ is given by (REF ), is a section.",
"Then ${\\tilde{\\sigma }}_s \\circ \\Gamma _s(\\vec{k},\\vec{x},t) =\\left(\\frac{1}{\\sqrt{(1,\\overline{t})(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}1\\\\t \\end{array}\\right)}} \\left(\\begin{array}{c}1\\\\t \\end{array}\\right),\\right.$ $\\left.",
"\\frac{1}{\\sqrt{(1,\\overline{t})(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}1\\\\t \\end{array}\\right)}}(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}1\\\\t \\end{array}\\right), \\vec{x} \\right).",
"$ If $\\omega _1$ is the symplectic form, we have on $U_s$ $ \\omega _1={\\tilde{\\sigma }}_s ^* d\\Omega _1.$ In the same way, on the domain, $U_n,$ of the chart $\\Gamma _n,$ we define a section, $\\tilde{\\sigma }_n,$ by ${\\tilde{\\sigma }}_n \\circ \\Gamma _n(\\vec{k},\\vec{x},t)=\\left( \\sigma \\left(mh(\\vec{k},k_4),\\left(\\begin{array}{c} t\\\\ 1\\end{array}\\right)\\right),h(\\vec{x},0)\\right)*\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0} \\right).$ Then ${\\tilde{\\sigma }}_n \\circ \\Gamma _n(\\vec{k},\\vec{x},t) =\\left(\\frac{1}{\\sqrt{(\\overline{t},1)(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}t\\\\1 \\end{array}\\right)}} \\left(\\begin{array}{c}t\\\\1 \\end{array}\\right),\\right.$ $\\left.",
"\\frac{1}{\\sqrt{(\\overline{t},1)(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}t\\\\1 \\end{array}\\right)}}(h(\\vec{k},k_4))^{-1}\\left(\\begin{array}{c}t\\\\1 \\end{array}\\right), \\vec{x} \\right).",
"$ On $U_n$ we have $\\omega _1={\\tilde{\\sigma }}_n ^* d\\Omega _1.$ Since $\\lbrace U_s,U_n\\rbrace $ is an open covering of the symplectic manifold, (REF ) and (REF ), determine $\\omega $ everywhere."
],
[
"Contact manifold for $T > 1.$",
"Let us denote by $r_1,\\dots , r_T$ the elements of $\\@root T \\of {\\mathbf {1}}.$ We define a properly discontinuous free action, $\\cdot ,$ of $\\@root T \\of {\\mathbf {1}}$ on ${\\cal B}\\times {\\mathbb {R}}^3 $ by means of $r_j\\cdot (w,z,\\overrightarrow{x})=(r_j w,r_j z,\\overrightarrow{x}).$ The quotient space is denoted by $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}},$ and the canonical map from ${\\cal B}\\times {\\mathbb {R}}^3$ onto $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}},$ is denoted by $\\pi _T.$ Thus $\\pi _T(w,z,\\overrightarrow{x})$ is the orbit of $(w,z,\\overrightarrow{x})$ , considered as an element of $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}}.$ The map $\\pi _T$ is a T-fold covering map and, since ${\\cal B}\\times {\\mathbb {R}}^3$ is simply connected, the Universal covering map of the quotient space.",
"The action of $G$ on ${\\cal B}\\times {\\mathbb {R}}^3$ conmutes with that of $\\@root T \\of {\\mathbf {1}}.$ As a consequence, the action on $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}}$ given by $(A,H)*((w,z,\\overrightarrow{x})\\@root T \\of {\\mathbf {1}})=((A,H)*(w,z,\\overrightarrow{x}))\\@root T \\of {\\mathbf {1}}$ is well defined, and obviously makes the covering map $\\pi _T$ equivariant.",
"The isotropy subgroup at $\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0}\\right)\\@root T \\of {\\mathbf {1}}$ is $Ker\\,C_{\\alpha _T},$ so that we can identify $G/Ker\\,C_{\\alpha _T}$ with $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}}$ by means of $(A,H)Ker\\,C_{\\alpha _T} \\longleftrightarrow (A,H)* \\left(\\left(\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right),\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right), \\vec{0}\\right)\\@root T \\of {\\mathbf {1}}\\right).$ The contact form on ${\\cal B}\\times {\\mathbb {R}}^3,\\ \\Omega _1,$ is invariant under the action $``\\cdot \",$ so that there exist an unique contact form, $\\Omega _T,$ on $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}},$ such that $\\pi _T^*\\Omega _T=\\Omega _1.$ The action of $G$ on $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}}$ is transitive and preserves $\\Omega _T.$ The manifold $({\\cal B}\\times {\\mathbb {R}}^3)/\\@root T \\of {\\mathbf {1}},$ when provided with the contact form $\\Omega _T$ and the cited action of $G,$ is the homogeneous contact manifold that correspond to massive particles with $T\\ge 1.$"
],
[
"Massless Type 4 particles. ",
"In this section we consider particles whose movement space is the coadjoint orbit of $\\alpha =\\left\\lbrace \\frac{i \\chi T}{8 \\pi }\\left(\\begin{array}{cc} 1 & 0\\\\0 & -1\\end{array}\\right),\\eta \\,\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)\\right\\rbrace \\ \\ \\ \\ ,\\ \\chi ,\\ \\eta \\in \\lbrace \\pm 1\\rbrace ,\\ T \\in {\\bf Z}^+ ,$ where $\\eta =-sign(Tr(P)),$ (see Table 2 of section ).",
"So, at least at the classical level, $\\eta $ must be interpreted as the opposite of the sign of energy.",
"We will see in the following subsections that this interpretation is also exact at the quantum level.",
"Since $|P|$ is mass square and $Det\\left( \\eta \\,\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right) \\right)=0$ these orbits correspond to particles with zero mass.",
"These are quantizable, not $\\bf R\\rm $ -quantizable orbits of the type 4.",
"The group $G_\\alpha $ is connected, so that there exists at most one homomorphism from $G_\\alpha $ onto ${\\bf S}^1$ whose differential is $\\alpha $ .",
"In fact we have $\\begin{array}{l}\\vspace{7.22743pt}G_\\alpha = \\left\\lbrace \\left(\\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&i \\chi \\eta T a z/2 \\pi \\\\\\overline{i \\chi \\eta T a z / 2 \\pi }&0\\end{array}\\right)\\right):\\ z \\in {\\bf S^1},\\right.",
"\\\\\\left.",
"\\ \\ \\ \\ \\ \\ \\ \\ \\ b\\in {\\bf R},\\ a \\in {\\bf C}\\right\\rbrace \\\\\\ \\\\\\mathrm {and\\ the\\ homomorphism} \\\\\\ \\\\C_\\alpha \\left( \\left( \\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&{i \\chi \\eta T a z/2 \\pi } \\\\{\\overline{i \\chi \\eta T a z/2 \\pi }}&0\\end{array}\\right)\\right)\\right)=z^{\\chi T}\\end{array}$ has differential $\\alpha .$ Other computations give us $\\begin{array}{l}({\\mbox{$G_\\alpha $}} )_{SL}=\\left\\lbrace \\left(\\begin{array}{cc} z&a \\\\0&{\\overline{z}}\\end{array}\\right): z \\in {\\bf S^1},\\ a \\in {\\bf C} \\right\\rbrace \\\\\\vspace{7.22743pt}(C_\\alpha )_{SL}\\left(\\ \\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right)\\right)=\\ z^{\\chi T} \\\\\\vspace{7.22743pt}SL_1=\\left\\lbrace \\left(\\begin{array}{cc}a&0 \\\\0&{1/a}\\end{array}\\right):\\ a \\in {\\bf C}\\right\\rbrace \\\\\\vspace{7.22743pt}SL_2=(G_\\alpha )_{SL} \\\\\\vspace{7.22743pt}SL_1 \\cap SL_2=\\left\\lbrace \\left(\\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array}\\right):\\ z \\in {\\bf S}^1\\right\\rbrace \\end{array}.$ The isotropy subgroup at $\\left(\\begin{array}{cc}1&0 \\\\0&0\\end{array}\\right.$ , for the usual action of $SL$(2,$C$) on H(2) (that is $A\\ast m=AmA^*$ ) is ${\\mbox{$(G_\\alpha )$}}_{SL}$ .",
"Thus SL/${\\mbox{$(G_\\alpha )$}}_{SL}$ will be identified to the orbit of $\\left(\\begin{array}{cc}1&0 \\\\0&0\\end{array}\\right),$ which is ${\\mbox{$\\bf C^+$}}=\\lbrace H \\in {H(2)}:\\ Det\\,H=0,\\ Tr\\,H>0\\rbrace .$ This set is composed by the hermitian matrices that represent space-time points in the future lightcone, so that it will be called itself, future lightcone.",
"When this identification is made, the function $P$ on SL/${\\mbox{$(G_\\alpha )$}}_{SL}$ becomes $P(H)=-\\eta \\,H$ .",
"An invariant volume element in ${\\mbox{$\\bf C^+$}}$ is $\\omega =\\frac{1}{\\Vert \\vec{ p}\\Vert } \\ dp^1\\wedge \\,dp^2\\wedge \\,dp^3,$ where $(p^1,\\ p^2,\\ p^3)$ is the coordinate system corresponding to the parametrization $\\phi :(p^1,\\ p^2,\\ p^3) \\in {\\bf R^3}-\\lbrace \\,0\\,\\rbrace \\mapsto h(\\vec{ p},\\Vert {\\vec{ p}\\Vert }) \\in {\\mbox{$\\bf C^+$}}$ where $\\vec{p}= (p^1,\\ p^2\\ p^3),$ and $\\Vert {\\vec{ p}}\\Vert =+ \\sqrt{ \\sum _{i=1}^3 (p^i)^2}.$ In this parametrization, the linear momentum is given by $P(\\phi (\\vec{ p}))=-\\eta \\ h(\\vec{ p},\\Vert {\\vec{ p}\\Vert }).$ Before to proceed to the study of the general case, we shall consider two particular ones."
],
[
"Massless antineutrino",
"Let us consider the case $ {T=1,\\ \\chi =1,\\ \\eta =\\,-\\,1}$ .",
"A trivialization is given by $\\left(\\rho _+ ,\\ {\\bf C}^2,\\ \\left(\\begin{array}{c}1 \\\\0\\end{array}\\right) \\right),$ where $\\rho _+$ (A) is multiplication by A.",
"The orbit of $\\left(\\begin{array}{c} 1\\\\0\\end{array}\\right)$ is ${\\bf C}^2-\\lbrace 0\\rbrace $ .",
"This space can thus be identified to SL/${\\mbox{$Ker\\,(C_\\alpha )$}}_{SL}$ so that the canonical map $SL/{\\mbox{$Ker\\,(C_\\alpha )$}}_{SL} \\longrightarrow SL/(G_\\alpha ) $ becomes a map $r_+ :{\\bf C}^2-\\lbrace 0\\rbrace \\mapsto {\\bf C}^+ .$ This map must be equivariant, so that, for all $A\\in SL$ $r_+\\left(A \\left(\\begin{array}{c}1 \\\\0\\end{array} \\right) \\right) =A\\left(\\begin{array}{cc}1&0 \\\\0&0\\end{array}\\right.",
"A^*.$ Thus $r_+$ is explicitly given by $r_+(z)=z\\,z^* \\in {\\mbox{$\\bf C^+$}} .$ Since $z$ and $\\varepsilon \\overline{z}$ are eigenvectors of $z\\,z^*$ corresponding to the eigenvalues $\\Vert z\\Vert ^2$ and 0 respectively, $r_+^{-1}(H)$ is composed by the eigenvectors of $H$ corresponding to the positive eigenvalue, whose norm is the square root of that eigenvalue.",
"The principal $ {\\bf S}^1$ -bundle whose projection is $r_+$ is related to the Hopf fibration as follows.",
"The image of the restriction of $r_+$ to the sphere $S^3(R)=\\lbrace z \\in { {\\bf C}}^2\\,:\\,\\Vert z\\Vert ^2=R^2\\rbrace $ is composed by the elements of ${\\mbox{$\\bf C^+$}}$ whose trace is $R^2$ .",
"Since the image of this subset by the preceeding chart is the sphere of radius $R^2$ /2, we obtain maps from spheres $S^3$ onto spheres $S^2$ .",
"Each one of these mappings is, up to the radius and a reflection, the Hopf fibration.",
"For each $S$ -homogeneous of degree -1 function on ${\\bf C}^2-{0},\\ f, $ we have the following prewave function ${\\psi } _f^+:\\,(N,\\ H) \\in {\\mbox{$\\bf C^+$}}\\times {H(2)}\\mapsto f(z)\\ e^{i\\pi \\,Tr(\\,N \\varepsilon \\overline{H} \\varepsilon ) } z \\in { {\\bf C}}^2,$ where z is an arbitrary element of $r_+^{-1}(N)$ .",
"Here we use the fact that, since $\\eta =-1$ , the linear momentum is given in $C^+$ by $P(N)=N.$ If $X$ and $Q$ are the elements of ${\\bf R}^4$ corresponding to $H$ and $N$ , one can write ${\\psi } _f^+(Q,X) = f(z)\\ e^{-2\\pi i\\langle Q,X \\rangle } z ,$ where $z$ is arbitrary in $r_+^{-1}(Q).$ The corresponding wave function , if $f$ is continuous with compact support, is ${\\widetilde{\\psi }} _f^+(H)= \\int _{C^+} {\\psi } _f^+(\\cdot ,\\ H) \\omega .$ By direct computation one can see that $\\left(\\ \\sigma _1 \\ \\frac{\\partial }{\\partial x^1}+\\ \\sigma _2 \\ \\frac{\\partial }{\\partial x^2}+\\ \\sigma _3 \\ \\frac{\\partial }{\\partial x^3}+\\ \\sigma _4 \\ \\frac{\\partial }{\\partial x^4} \\right)\\ {\\psi }_f^+(N,\\ h(x))\\,=\\,0$ The corresponding wave function thus satisfy the same equation, which is the Weyl equation (positive energy), usually admised as corresponding to the antineutrino.",
"If $f$ and $f^\\prime $ are pseudotensorial functions, we have $({\\psi } _f^+(N,H))^*\\ {\\psi }_{f^\\prime }^+(N,H)=\\overline{f(z)}\\, f^\\prime (z)\\ Tr\\,N.$ Thus, the hermitian product of the corresponding quantum states ($cf.$ section ) can be written as follows ${\\frac{1}{2}}\\ \\int _{\\bf C ^+} \\frac{({\\psi }^+ _f)^*\\ {\\psi }^+_{f^\\prime }}{\\sum _{i=1}^3 (p^i)^2}\\ dp^1\\,dp^2\\,dp^3.$ To obtain prewave functions directly from functions on $C^+,$ we use Remark REF , as follows.",
"Let $U= \\phi ({\\bf R}^3-\\lbrace p^1=p^2=0, p^3<0\\rbrace ),$ $V= \\phi ({\\bf R}^3-\\lbrace p^1=p^2=0, p^3>0\\rbrace ).$ Then the maps $\\sigma _U : \\phi (p^1,p^2,p^3)\\in U \\mapsto \\left({\\begin{array}{c} \\sqrt{\\Vert \\vec{ p}\\Vert +p^3} \\\\\\frac{p^1+i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert +p^3}} \\end{array}}\\right) \\in {\\bf C}^2-{0}$ $\\sigma _V : \\phi (p^1,p^2,p^3)\\in V \\mapsto \\left( {\\begin{array}{c} \\frac{p^1-i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert -p^3}} \\\\\\sqrt{\\Vert \\vec{ p}\\Vert -p^3} \\end{array}}\\right) \\in {\\bf C}^2-{0}$ are sections of $r_+,$ so that we obtain prewave functions having the form $\\psi (\\phi (p^1,p^2,p^3), X)=F(p^1,p^2,p^3) \\left({\\begin{array}{c} \\sqrt{\\Vert \\vec{ p}\\Vert +p^3} \\\\\\frac{p^1+i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert +p^3}} \\end{array}}\\right)$ $ \\displaystyle e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ where $F$ is a complex valued function continuous on ${\\bf R}^3-\\lbrace 0\\rbrace ,$ with compact support in ${\\bf R}^3-\\lbrace p^1=p^2=0, p^3<0\\rbrace ,$ and prewave functions having the form $\\theta (\\phi (p^1,p^2,p^3), X)=J(p^1,p^2,p^3) \\left( {\\begin{array}{c} \\frac{p^1-i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert -p^3}} \\\\\\sqrt{\\Vert \\vec{ p}\\Vert -p^3} \\end{array}}\\right)$ $ \\displaystyle e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ where $J$ is a complex valued function continuous on ${\\bf R}^3-\\lbrace 0\\rbrace ,$ with compact support in ${\\bf R}^3-\\lbrace p^1=p^2=0, p^3>0\\rbrace .$ If one is interested in the case $T=1,\\ \\chi =1,\\ \\eta =+1$ , everything is as in the case $\\eta =-1$ , but the exponent of $e$ in the prewave functions changes its sign, thus giving wave functions for quantum states of negative energy.",
"Remenber that $\\eta $ is the opposite of the sign of energy at the clasical level."
],
[
"Massless neutrino",
"Now let us consider the case in which $T=1,\\ \\chi =-1,\\ \\eta =-1$ .",
"A trivialisation in this case is $\\left(\\rho _- ,\\ {\\bf C}^2,\\ \\left(\\begin{array}{c}0 \\\\1\\end{array}\\right) \\right),$ $ \\rho _- $ (A) being multiplication by $(A^*)^{-1}.$ Thus we identify $SL/Ker\\,C_{\\alpha }$ with the orbit of $ \\left(\\begin{array}{c}0 \\\\1\\end{array}\\right),$ which, also in this case, is ${\\bf C}^2-\\lbrace 0\\rbrace .$ Thus,the canonical map $SL/{\\mbox{$Ker\\,(C_\\alpha )$}}_{SL} \\longrightarrow SL/(G_\\alpha ) $ becomes a map $r_- :{\\bf C}^2-\\lbrace 0\\rbrace \\mapsto {\\bf C}^+ .$ Now, using equivariance as in the case of antineutrino, we obtain $r_-(z)\\,=\\,-\\varepsilon \\overline{z\\,z^* }\\varepsilon .",
"$ The prewave functions one obtains have exactly the same form that (REF ) and (REF ), but now $z$ is in $(r_-)^{-1}(N)$ or $(r_-)^{-1}(h(Q))$ respectively.",
"The wave functions one obtains in this case satisfies the Weyl equation that, according to Feynman, corresponds to the neutrino.",
"The map from the real vector space $\\bf C \\rm ^2$ onto itself defined by sending $z$ to $ \\varepsilon \\overline{z}$ , is a complex structure and its restriction to $\\bf C\\rm ^2-\\lbrace 0 \\rbrace $ , gives us an isomorphism of the principal circle bundle corresponding to ${\\bf r}_{-}$ ( resp.${\\bf r}_+$ ) onto the principal circle bundle corresponding to ${\\bf r}_+$ ( resp.",
"${\\bf r}_-$ ).",
"The isomorphism of the structural group is defined by sending each element to its inverse.",
"Thus if $z\\in r_+^{-1}(H)$ then $\\varepsilon \\overline{z}\\in r_-^{-1}(H)$ and conversely.",
"As a consequence, the sections of the map $r_+$ defined in REF give rise to the following sections of $r_-$ $\\sigma ^\\prime _U(u)=\\varepsilon \\overline{\\sigma _U(u)},\\ \\ \\sigma ^\\prime _V(v)=\\varepsilon \\overline{\\sigma _V(v)},$ for all $u\\in U$ and $v\\in V,$ and the prewave functions of antineutrino $\\psi $ and $\\theta $ give rise to prewave functions of neutrino, $\\psi ^\\prime $ and $\\theta ^\\prime $ , by means of $\\psi ^\\prime (\\phi (p^1,p^2,p^3), X)=F(p^1,p^2,p^3)\\varepsilon \\overline{\\left({\\begin{array}{c} \\sqrt{\\Vert \\vec{ p}\\Vert +p^3} \\\\\\frac{p^1+i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert +p^3}} \\end{array}}\\right) }$ $e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ $\\theta ^\\prime (\\phi (p^1,p^2,p^3), X)=J(p^1,p^2,p^3) \\varepsilon \\overline{\\left( {\\begin{array}{c} \\frac{p^1-i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert -p^3}} \\\\\\sqrt{\\Vert \\vec{ p}\\Vert -p^3} \\end{array}}\\right) }$ $e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}$ that leads to $\\psi ^{\\prime }(\\phi (p^1,p^2,p^3), X)&=& F(p^1,p^2,p^3){\\left({{\\begin{array}{c} \\frac{p^1-i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert +p^3}}\\\\-\\sqrt{\\Vert \\vec{ p}\\Vert +p^3} \\end{array}}}\\right) }\\\\ &&{e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}} \\\\\\ \\theta ^{\\prime }(\\phi (p^1,p^2,p^3), X)&=&J(p^1,p^2,p^3){\\left( {{\\begin{array}{c}\\sqrt{\\Vert \\vec{ p}\\Vert -p^3}\\\\{-\\frac{p^1+i p^2}{ \\sqrt{\\Vert \\vec{ p}\\Vert -p^3}}} \\end{array}}}\\right)}\\\\ && {e^{-2\\pi i (\\Vert \\vec{ p}\\Vert X^4-p^1X^1-p^2X^2-p^3X^3)}}.$ If one consider the case $T=1,\\ \\chi =-1,\\ \\eta =+1$ one obtain similar wave functions but corresponding to negative energy."
],
[
"General case",
"We proceed as in section REF .",
"In the case $\\chi =+1$ a trivialization is given by $\\left((\\rho _+)^{\\otimes T},\\ \\left({\\bf C}^2\\right)^ {\\otimes T},\\ \\left(\\begin{array}{c} 1\\\\0\\end{array}\\right)^{\\otimes T}\\right),$ and if $\\chi =-1$ a trivialization is given by $\\left((\\rho _-)^{\\otimes T},\\ \\left({\\bf C}^2\\right)^{\\otimes T},\\ \\left(\\begin{array}{c} 0\\\\1\\end{array}\\right)^{\\otimes T}\\right),$ The prewave functions are given by functions on ${\\bf C} \\rm ^2-\\lbrace 0\\rbrace $ , which are $C^\\infty $ with compact support and homogeneous of degree -T under multiplication by modulus one complex numbers.",
"Let $f_T$ be one of these functions.",
"If $\\chi =1$ , the corresponding prewave function is given by $\\psi ^+_{f_T}:\\,(N,\\ H) \\in {\\mbox{$\\bf C^+$}}\\times {H(2)}\\mapsto f_T(z)\\ e^{-i \\pi \\eta \\,Tr(N \\varepsilon \\overline{H} \\varepsilon ) } z^{\\otimes T}\\in ({\\bf C}^2)^{\\otimes T}$ where $z$ is an arbitrary element of $r_+^{-1}(N)$ .",
"In the case $\\chi =-1$ , the corresponding prewave function is $\\psi ^-_{f_T}:\\,(N,\\ H) \\in {\\mbox{$\\bf C^+$}}\\times {H(2)}\\mapsto f_T(z)\\ e^{-i \\pi \\eta \\,Tr(N \\varepsilon \\overline{H} \\varepsilon )} z^{\\otimes T}\\in ({\\bf C}^2)^{\\otimes T}$ but now, z is an arbitrary element of $r_-^{-1}(N)$ .",
"The associated wave functions, ${\\widetilde{\\psi }}^+_{f_T}(H)&=&\\int _{C^+} {\\psi } ^+_{f_T}(\\cdot ,\\ H) \\omega ,\\\\{\\widetilde{\\psi }}^-_{f_T}(H)&=&\\int _{C^+} {\\psi } ^-_{f_T}(\\cdot ,\\ H) \\omega $ satisfies Penrose's wave equations,that we will describe here for the sake of completeness.",
"Let us consider in $({\\bf C}^2)^{\\otimes T}$ the basis $\\lbrace e_A \\otimes e_B \\otimes \\stackrel{(T}{\\cdots }\\ :\\ A,\\ B,\\dots \\in \\lbrace 1,\\ 2\\rbrace \\rbrace $ , where $\\lbrace e_1,\\ e_2\\rbrace $ is the canonical basis of $\\bf C \\rm ^2$ .",
"The prewave functions $\\psi ^\\pm _{f_T}$ and the wave functions $\\tilde{\\psi }^\\pm _{f_T}$ , have components in this basis which will be denoted by $\\lbrace \\psi _\\pm ^{A\\,B\\dots } \\rbrace $ and $\\lbrace \\tilde{\\psi }_\\pm ^{A\\,B\\dots } \\rbrace $ , respectively.",
"Let us consider the vector fields in $\\bf R \\rm ^4$ given by $\\nabla _{11}&=&\\frac{1}{2} \\left( \\frac{\\partial }{\\partial x^3}+\\frac{\\partial }{\\partial x^4} \\right)\\\\\\nabla _{12}&=&\\frac{1}{2} \\left( \\frac{\\partial }{\\partial x^1}-i\\frac{\\partial }{\\partial x^2} \\right)\\\\\\nabla _{21}&=&\\frac{1}{2} \\left( \\frac{\\partial }{\\partial x^1}+i\\frac{\\partial }{\\partial x^2} \\right)\\\\\\nabla _{22}&=&\\frac{1}{2} \\left( \\frac{\\partial }{\\partial x^4}-\\frac{\\partial }{\\partial x^3} \\right)$ and, for all $A, A^\\prime \\in \\lbrace 1,\\ 2\\rbrace ,\\ \\nabla ^{A \\ A^\\prime }=\\varepsilon ^{A\\ B}\\ \\varepsilon ^{A^\\prime \\ B^\\prime }\\ \\nabla _{B\\ B^\\prime }$ (summation convention), where $\\lbrace \\varepsilon ^{A\\ B}\\rbrace $ are the elements of $-\\varepsilon $ .",
"We also define $\\psi ^\\pm _{A\\,B\\dots }&=&\\varepsilon _{A\\ A^\\prime }\\ \\varepsilon _{B\\ B^\\prime }\\dots \\psi _\\pm ^{A^\\prime \\,B^\\prime \\dots } \\\\\\tilde{\\psi }^\\pm _{A\\,B\\dots }&=&\\varepsilon _{A\\ A^\\prime }\\ \\varepsilon _{B\\ B^\\prime }\\dots \\tilde{\\psi }_\\pm ^{A^\\prime \\,B^\\prime \\dots }$ where $\\lbrace \\varepsilon _{A\\ B}\\rbrace $ are the elements of $\\varepsilon $ .",
"Thus we have for all $h(x) \\in C^+$ $\\nabla ^{A\\ A^\\prime }\\ \\psi ^+_{A^\\prime \\,B\\ C\\dots }(h(x),\\,\\cdot \\, )&=&0\\\\\\nabla ^{ A^\\prime \\ A}\\ \\psi ^-_{A^\\prime \\,B\\ C\\dots }(h(x),\\,\\cdot \\,)&=&0$ so that, by derivation under the integral sign, we see that Penrose wave equations: $\\nabla ^{A\\ A^\\prime }\\ \\tilde{\\psi }^+_{A^\\prime \\,B\\ C\\dots }&=&0\\\\\\nabla ^{ A^\\prime \\ A}\\ \\tilde{\\psi }^-_{A^\\prime \\,B\\ C\\dots }&=&0$ are satisfied."
],
[
"Helicity",
"In formula (REF ) one sees that, if $(\\rho ,L,z_0)$ is the trivialization we use, the components of spin operator are given by $\\hat{s^k}:\\widetilde{\\psi }_f \\longrightarrow s^k \\circ \\widetilde{\\psi }_f,$ $s^k$ being the following endomophism of $L$ $s^k=\\frac{1}{2\\pi i}\\frac{\\widetilde{i\\sigma _k}}{2}=\\frac{1}{4\\pi i}\\widetilde{i\\sigma _k},$ where, $\\widetilde{i\\sigma _k}=d\\rho ({i\\sigma _k}).$ Thus, in the case $\\chi =1,$ $\\widetilde{i\\sigma _k}=d\\left((\\rho _+)^{\\otimes T}\\right)({i\\sigma _k}),$ and in the case $\\chi =-1,$ $\\widetilde{i\\sigma _k}=d\\left((\\rho _-)^{\\otimes T}\\right)({i\\sigma _k}).$ As a consecuence, in both cases, $\\chi =+ 1$ or $\\chi = 1,$ when acting on monomials we have $\\widetilde{i\\sigma _k}\\cdot z_1\\otimes \\dots \\otimes z_T=\\sum _{j=1}^T\\ z_1\\otimes \\dots \\otimes z_{j-1}\\otimes i\\, \\sigma _k\\,z_j\\otimes z_{j+1}\\otimes \\dots \\otimes z_T.$ We consider the operators $\\hat{s^k}$ as also acting on prewave functions by the same formula that in the case of wave functions, without the tilde.",
"On the other hand, Linear Momentum is given on $C^+$ by $P(K)=-\\eta K,\\ \\text{\\rm for all}\\ K\\in C^+.$ Then, with the notation $K=h(K^1,K^2,K^3,K^4)$ and $P=h(P^1,P^2,P^3,P^4),$ the $P^k$ are functions on $C^+,$ given by $P^k(K)=-\\eta K^k.$ We also denote $\\overrightarrow{K}&=&(K^1,K^2,K^3)\\\\\\Vert \\overrightarrow{K}\\Vert &=&+\\left(\\sum _{k=1}^3({K^k})^2\\right)^{1/2}\\\\\\overrightarrow{P}&=&(P^1,P^2,P^3)\\\\ \\Vert \\overrightarrow{P}\\Vert &=&+\\left(\\sum _{k=1}^3({P^k})^2\\right)^{1/2}.$ The Helicity Operator is defined by $\\mathfrak {h}=\\frac{1}{\\Vert \\overrightarrow{P}\\Vert } {\\sum _{k=1}^3}{P^k}\\hat{s^k},$ which means $\\left(\\mathfrak {h}\\cdot \\psi _{f_T}^{\\pm }\\right)(K,H)=\\frac{1}{\\Vert \\overrightarrow{P}\\Vert (K)} {\\sum _{k=1}^3}{P^k(K)}{s^k}\\left(\\psi _{f_T}^{\\pm }(K,H)\\right).$ Then, if $z\\in r^{-1}_{\\pm }(K)$ and $z_1=\\dots =z_T=z,$ we have $&&\\left(\\mathfrak {h}\\cdot \\psi _{f_T}^{\\pm }\\right)(K,H)=\\frac{1}{K^4} \\sum _{k=1}^3{P^k(K)}{s^k}\\left(f_T(z)e^{-i\\eta \\pi TrK\\epsilon \\overline{H}\\epsilon } z^{\\otimes T}\\right)=\\\\&&=\\frac{1}{4 \\pi K^4}\\ f_T(z)e^{-i\\eta \\pi TrK\\epsilon \\overline{H}\\epsilon }\\\\&&\\left( \\sum _{j=1}^T\\ z_1\\otimes \\dots \\otimes z_{j-1}\\otimes (\\sum _{k=1}^3{P^k(K)}\\, \\sigma _k\\,z_j)\\otimes z_{j+1}\\otimes \\dots \\otimes z_T\\right)=\\\\&&=\\frac{1}{4 \\pi K^4} \\ f_T(z)e^{-i\\eta \\pi TrK\\epsilon \\overline{H}\\epsilon } \\\\&& \\left( \\sum _{j=1}^T\\ z_1\\otimes \\dots \\otimes z_{j-1}\\otimes (-\\eta )((K-K^4\\,I)z_j)\\otimes z_{j+1}\\otimes \\dots \\otimes z_T\\right).$ In the case $\\chi =+1,$ we have $K= r_+(z)=zz^*,$ so that $ (K-K^4\\,I)z=(zz^*-\\frac{\\Vert z \\Vert }{2}\\,I)z=K^4\\,z,$ and then $\\left(\\mathfrak {h}\\cdot \\psi _{f_T}^{+}\\right)(K,H)=\\frac{-\\eta T}{4\\pi }\\psi _{f_T}^{+}(K,H).$ On the other hand, in the case $\\chi =-1,$ we have $ K=-\\epsilon \\overline{zz^*}\\epsilon .$ Thus $(K-K^4\\,I)z=(-\\epsilon \\overline{zz^*}\\epsilon -K^4\\,I)z=-K^4\\,z,$ so that $\\left(\\mathfrak {h}\\cdot \\psi _{f_T}^{-}\\right)(K,H)=\\frac{\\eta T}{4\\pi }\\psi _{f_T}^{-}(K,H).$ Thus $\\psi _{f_T}^{\\pm }$ is an eigenvector of the helicity operator, corresponding to the eigenvalue $-\\eta \\chi T/4 \\pi .$ In particular, the sign of helicity is $-\\eta \\chi .$"
],
[
"The Homogeneous Contact and Symplectic Manifolds for Massless particles of type 4. Twistors.",
"Let us denote by $\\@root T \\of {\\mathbf {1}}=\\lbrace r_1, \\dots ,r_T\\rbrace ,$ the group composed by the roots of order $T$ of 1, and $\\nu =\\frac{- \\eta \\chi T}{4\\pi } .$ In section REF we have seen that all wave functions obtained in section REF are eigenvectors of helicity with eigenvalue $\\nu .$ The group $Ker\\,C_\\alpha ,$ has $T$ connected components $(Ker\\,C_\\alpha )_j =\\left\\lbrace \\left(\\left(\\begin{array}{cc}r_j&a \\\\0&{\\overline{r}_j}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&-2i\\nu \\,a\\, r_j \\\\\\overline{-2i\\nu \\,a\\, r_j } &0\\end{array}\\right)\\right):b\\in {\\bf R},\\ a \\in {\\bf C}\\right\\rbrace $ The component of the identity is $(Ker\\,C_\\alpha )_o =\\left\\lbrace \\left(\\left(\\begin{array}{cc}1&a \\\\0&{1}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&-2i\\nu \\,a \\\\\\overline{\\-2i\\nu \\,a}&0\\end{array}\\right)\\right):b\\in {\\bf R},\\ a \\in {\\bf C}\\right\\rbrace $ and $(Ker\\,C_\\alpha )_j $ is the left translation by $\\tilde{\\bf r}_j \\stackrel{def}{=}\\left( \\left( \\begin{array}{cc}r_j&0 \\\\0&{\\overline{r}_j}\\end{array}\\right),\\ 0 \\right)$ of $(Ker\\,C_\\alpha )_o .$ As a consequence, the group $Ker\\,C_\\alpha /(Ker\\,C_\\alpha )_o$ is isomorphic to $\\@root T \\of {\\mathbf {1}}$ .",
"The canonical map ${\\begin{array}{c}\\tilde{\\bf c} : (A,H)\\,(Ker\\,C_\\alpha )_o\\in \\frac{G}{(Ker\\,C_\\alpha )_o} \\longrightarrow \\\\\\longrightarrow (A,H)\\,Ker\\,C_\\alpha \\in \\frac{G}{Ker\\,C_\\alpha }.\\end{array}}$ is a $T$ -fold covering map, whose structural group is $Ker\\,C_\\alpha /(Ker\\,C_\\alpha )_o$ .",
"Thus, $G/Ker\\,C_\\alpha $ is the quotient of $G/(Ker\\,C_\\alpha )_o$ by a properly discontinuous with no fixed point action of $Ker\\,C_\\alpha /(Ker\\,C_\\alpha )_o$ .",
"But, as we have seen, this group can be changed to $\\@root T \\of {\\mathbf {1}}$ .",
"The action of $\\@root T \\of {\\mathbf {1}}$ on $G/(Ker\\,C_\\alpha )_o$ corresponding to this construction is: $((A,H)\\,(Ker\\,C_\\alpha )_o)* r_j= ((A,H)\\,\\tilde{\\bf r}_j)\\,(Ker\\,C_\\alpha )_o.$ In what follows, we identify $G/Ker\\,C_\\alpha $ with the quotient space of $G/(Ker\\,C_\\alpha )_o$ by this action.",
"We have the following conmutative diagram ${\\begin{matrix}\\frac{G}{(Ker\\,C_\\alpha )_o } &\\xrightarrow{}& \\frac{G}{Ker\\,C_\\alpha } \\\\\\downarrow && \\downarrow && \\\\\\frac{G}{G_\\alpha } &=& \\frac{G}{G_\\alpha }\\end{matrix}}$ where the vertical arrow on the right is the bundle map of the contact manifold onto the symplectic manifold, for general $T.$ If $T=1$ both vertical arrows are the same.",
"The homomorphism $\\mu _1:(A, H) \\in SL\\oplus H(2) \\longrightarrow \\left( \\begin{array}{cc}A&{-iH{A^*}^{-1}} \\\\0&{A^*}^{-1}\\end{array} \\right)\\in GL(4,\\ \\text{\\bf C}).$ is a representation in ${\\bf C}^4.$ The isotropy subgroup at $q \\stackrel{def}{=}\\left( \\begin{array}{c}0\\\\ 2\\nu \\\\0\\\\ 1\\end{array}\\right)$ is $(Ker\\,C_\\alpha )_o .$ Let us denote by $\\mathrm {O}_q$ the orbit of $q$ by this representation.",
"In order to describe $\\mathrm {O}_q$ in another way, we consider in $\\text{\\bf C}^4$ the hermitian product $<\\binom{\\hat{w}}{\\hat{z}},\\ \\binom{w}{z}>=\\frac{1}{2} (\\hat{w}^*z+\\hat{z}^*w)\\ \\ \\ \\ \\ \\forall \\hat{w},\\,\\hat{z},\\,w,\\,z \\in \\text{\\bf C}^2$ whose signature and quadratic form are respectively (+, +, -, -) and $\\Phi \\binom{w}{z} =Re\\,z^*\\,w.$ The complex vector space $\\text{\\bf C}^4$ provided with this hermitian product is Penrose's Twistor Space.",
"The representation $\\mu _1$ preserves $\\Phi $ , so that any orbit must be contained in a subset of the form $\\Phi =\\mathnormal {constant}.$ It follows that the orbit of $q$ is contained in the seven dimensional submanifold, $\\mathcal {O}_\\nu $ , given by $\\Phi =2\\nu .$ Let $\\binom{w}{z}\\in \\text{\\bf C}^4$ be such that $sign\\left(\\Phi \\binom{w}{z} \\right)=sign(\\nu ),$ and define $A(w,z)&=&\\left(\\frac{2\\nu }{ \\Phi \\binom{w}{z}}\\right)^{1/2}\\left(\\begin{array}{c} {}\\\\\\epsilon \\overline{z}\\\\{} \\end{array}\\ \\Bigg | \\ \\frac{1}{2\\nu }\\left(i\\frac{Im(z^*w)}{\\Vert z \\Vert ^2}z-w\\right)\\right)\\\\ H(w,z) &=& -\\frac{Im(z^*w)}{\\Vert z \\Vert ^2}\\ I.$ Then we have $\\mu _1(A(w,z),H(w,z)) q=\\left(\\frac{2\\nu }{ \\Phi \\binom{w}{z}}\\right)^{1/2}\\ \\binom{w}{z}.$ This enables us to prove that each element of $\\mathcal {O}_\\nu $ is in the orbit, so that $\\mathrm {O}_q=\\mathcal {O}_\\nu $ .",
"Then, the map from $G/(Ker\\,C_\\alpha )_o$ onto $ \\mathcal {O}_\\nu ,$ given by $\\tau _T:(A,H)\\,(Ker\\,C_\\alpha )_o\\in \\frac{G}{(Ker\\,C_\\alpha )_o} \\longrightarrow \\mu _1(A,H)\\,q\\in \\mathcal {O}_\\nu ,$ is a diffeomorphism.",
"If we identify these spaces by $\\tau _T,$ the action of $\\@root T \\of {\\mathbf {1}}$ on $G/(Ker\\,C_\\alpha )_o$ given by (REF ), traslates to an action on $\\mathcal {O}_\\nu .$ Since $\\tau _T((A,H)\\,(Ker\\,C_\\alpha )_o) *r_j)&=&\\mu _1 ((A,H)\\,\\tilde{\\bf r}_j)q=\\mu _1 (A,H)\\mu _1(\\tilde{\\bf r}_j)q\\\\=\\mu _1 (A,H)(\\overline{r_j}q)&=&\\overline{r_j}\\ \\tau _T((A,H)\\,(Ker\\,C_\\alpha )_o) ,$ the action of $\\@root T \\of {\\mathbf {1}}$ on $\\mathcal {O}_\\nu $ is given by ordinary product by the conjugated: $V*r_j=\\overline{r_j}\\ V$ Let us denote by $\\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}}$ the quotient space of $\\mathcal {O}_\\nu $ by this action.",
"It follows from the preceeding construction that the contact manifold, $G/Ker\\,C_\\alpha ,$ is diffeomorphic, and will be identified, to $\\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}}$ .",
"The contact form will be determinated later on.",
"Let ${}^t(w^1,w^2,z^1,z^2)\\in {\\mathbb {C}}^4-\\lbrace 0\\rbrace ,$ and let us denote by $[{}^t(w^1,w^2,z^1,z^2)]$ the complex vector subspace of ${\\mathbb {C}}^4$ generated by ${}^t(w^1,w^2,z^1,z^2).$ The set composed by these subspaces is the complex projective space of ${\\mathbb {C}}^4,$ ${\\bf P}_3(\\mathbb {C}),$ and we consider it as provided with its canonical differentiable structure.",
"Penrose also considers the subsets of twistor space, ${\\bf T}^+,\\ {\\bf T}^-,\\ {\\bf T}^0,$ given by $\\Phi \\,>\\,0,\\ \\Phi \\,<\\,0,\\ \\Phi \\,=\\,0$ , respectively, and the subsets of projective space $ {\\bf P}_3^+=\\pi ({\\bf T}^+),\\ {\\bf P}_3^-=\\pi ({\\bf T}^-),\\ {\\bf P}_3^0=\\pi ({\\bf T}^0)$ , where $\\pi :{\\mathbb {C}}^4 -\\lbrace 0\\rbrace \\longrightarrow {\\bf P}_3({\\mathbb {C}}) $ is the canonical map.",
"Since $\\Phi $ is preserved by the representation, the subsets ${\\bf T}^+,\\ {\\bf T}^-,\\ {\\bf T}^0,$ are stable under $\\mu _1$ .",
"The representation $\\mu _1$ also defines an action of $G$ on ${\\bf P}_3(\\text{\\bf C}) $ such that $\\pi $ is equivariant.",
"Explicitly $(A,H)*\\left[ \\left( \\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\ z^2\\end{array}\\right)\\right]=\\left[\\left( \\begin{array}{cc}A&{-iH{A^*}^{-1}} \\\\0&{A^*}^{-1}\\end{array} \\right)\\left( \\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\ z^2\\end{array}\\right)\\right].$ As a consecuence of formula (REF ), the open subsets of ${\\bf P}_3(\\text{\\bf C}) $ denoted by ${\\bf P}_3^+$ and ${\\bf P}_3^-$ are orbits of this action.",
"The point $[q]$ of ${\\bf P}_3(\\text{\\bf C}) $ is obviously in ${\\bf P}_3^{sign(\\nu )}$ so that its orbit is this open subset.",
"The isotropy subgroup at $[q]$ is $G_\\alpha .$ As a consecuence we have a diffeomorphism from ${\\bf P}_3^{sign(\\nu )}$ onto the coadjoint orbit of $\\alpha ,$ given by $\\Pi : [\\mu _1(A,H)\\,q]\\in {\\bf P}_3^{sign(\\nu )} \\longrightarrow Ad^*_{(A,H)}\\alpha \\in {\\mathcal {M}S} .$ were I have denoted the coadjoint orbit by ${\\mathcal {M}S},$ because of its interpretation as Movement Space.",
"We have $\\Pi \\left(\\left[ \\binom{ w}{ z} \\right]\\right)= Ad^*_{(A(w,z),H(w,z))}\\alpha $ so that, the formula for coadjoint representation in section ,(REF ), thus leads, after some computation, to $\\Pi \\left( \\left[ \\binom{ w}{ z} \\right] \\right)=\\frac{2\\nu \\eta }{ \\Phi \\binom{ w}{ z}} \\biggl \\lbrace \\frac{i}{4}(wz^*+\\epsilon \\overline{zw^*}\\epsilon ),\\ - \\epsilon \\overline{zz^*}\\epsilon \\biggr \\rbrace .$ We identify $P_3^{sign(\\nu )}$ to ${\\mathcal {M}S},$ by means of $\\Pi .$ Section provides us with well defined expresions for linear and angular momentum in $\\mathcal {M}S,$ $c.f.$ (REF ), which, when composed with $\\Pi ,$ give expressions for linear and angular momentum in $P^{sign(\\nu )}_3.$ In its hermitian form, these expressions are $h(P\\left( \\left[ \\binom{ w}{ z} \\right] \\right))&=& \\frac{2\\eta \\nu }{\\Phi \\binom{ w}{ z}} \\epsilon \\overline{zz^*}\\epsilon \\\\ h(\\vec{l} \\left( \\left[ \\binom{ w}{ z} \\right] \\right),0)&=&\\frac{\\eta \\nu }{2\\Phi \\binom{ w}{ z}} (zw^*+wz^*+\\epsilon \\overline{(zw^*+wz^*)}\\epsilon )\\\\ h(\\vec{g} \\left( \\left[ \\binom{ w}{ z} \\right] \\right),0)&=&\\frac{i\\,\\eta \\nu }{2\\Phi \\binom{ w}{ z}}(zw^*-wz^*-\\epsilon \\overline{(zw^*-wz^*)}\\epsilon )$ The Pauli-Lubanski four vector, when evaluated according with (REF ), is found to be $h(W\\left( \\left[ \\binom{ w}{ z} \\right] \\right)) =-\\eta \\nu \\,h(P\\left( \\left[ \\binom{ w}{ z}\\right] \\right)).",
"$ In case T=1, the bundle map of the contact manifold onto the coadjoint orbit becomes the canonical map $\\pi _1:\\binom{ w}{ z} \\in \\mathcal {O}_1\\ \\longrightarrow \\left[ \\binom{ w}{ z} \\right]\\in P^{sign(\\nu )}_3,$ and in the general case, the bundle map is $\\pi _\\nu :\\binom{ w}{z} \\@root T \\of {\\mathbf {1}}\\in \\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}} \\longrightarrow \\left[ \\binom{ w}{ z} \\right]\\in P^{sign(\\nu )}_3.$ Then, the composition with $\\pi _\\nu $ of the canonical dynamical variables, are also given by the right hand sides of (), () and (), but taking $\\Phi =2\\nu .$ We obtain the following formulae for the components of these dynamical variables $P^1\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right) &=\\frac{\\eta }{2}(z^1\\overline{z}^2\\,+\\,z^2\\overline{z}^1) \\\\P^2\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{i\\eta }{2}(z^1\\overline{z}^2\\,-\\,z^2\\overline{z}^1)\\\\P^3\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{\\eta }{2}(\\vert z^1\\vert ^2\\,-\\,\\vert z^2\\vert ^2) \\\\P^4\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=-\\frac{\\eta }{2}(\\vert z^1\\vert ^2\\,+\\,\\vert z^2\\vert ^2)$ $l^1\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{\\eta }{4} (z^1\\overline{w}^2+w^1\\overline{z}^2+z^2\\overline{w}^1+w^2\\overline{z}^1) \\\\l^2\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{i\\eta }{4} (z^1\\overline{w}^2+w^1\\overline{z}^2-z^2\\overline{w}^1-w^2\\overline{z}^1) \\\\l^3\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{\\eta }{4} (z^1\\overline{w}^1+w^1\\overline{z}^1-z^2\\overline{w}^2-w^2\\overline{z}^2)$ $g^1\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{i\\eta }{4} (z^1\\overline{w}^2-w^1\\overline{z}^2+z^2\\overline{w}^1-w^2\\overline{z}^1) \\\\g^2\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=-\\frac{\\eta }{4} (z^1\\overline{w}^2-w^1\\overline{z}^2-z^2\\overline{w}^1+w^2\\overline{z}^1) \\\\g^3\\left(\\binom{ w}{ z} \\@root T \\of {\\mathbf {1}}\\right)&=\\frac{i\\eta }{4} (z^1\\overline{w}^1-w^1\\overline{z}^1-z^2\\overline{w}^2+w^2\\overline{z}^2).$ We can also consider the functions on ${\\bf T}^{sgn(\\nu )}$ obtained by composing the dynamical variables in $\\mathcal {M}S$ with the canonical projection, $\\pi ,$ from ${\\bf T}^{sgn(\\nu )}$ onto ${\\bf P}_3^{sgn(\\nu )},$ thus obtaining functions whose expresion is as the right hand sides of the preceeding formulae multiplied by $\\frac{2\\nu }{ \\Phi \\binom{ w}{ z}}$ These expressions coincide, up to notational conventions, with the expressions that R.Penrose gives for its energy - momentum and angular momentum in twistor space.",
"We denote these functions by $\\widetilde{P}^k,\\ \\widetilde{l}^k,\\ \\widetilde{g}^k.$ In general, for all $(a,h)$ in the Lie algebra of $G,$ the function it defines on the coadjoint orbit, identified to ${\\bf P}_3^{sgn(\\nu )},$ is denoted by the same symbol, $(a,h),$ and its composition with $\\pi ,$ $\\widetilde{(a,h)}.$ In a similar way, the infinitesimal generator of the action on ${\\bf T}^{sgn(\\nu )}$ defined by the representation $\\mu _1,$ associated to $(a,h),\\ i.e.$ the vector field whose flow is given by $\\mu _1(Exp(-t(a,h))),$ is denoted by $\\widetilde{X_{(a,h)} }.$ Let us consider the following one form on ${\\bf T}^{sgn(\\nu )}$ $\\omega _0=\\frac{i\\eta \\nu }{2\\Phi } (z^1d\\overline{w}^1+w^1d\\overline{z}^1+z^2d\\overline{w}^2+ w^2d\\overline{z}^2 -$ $- \\overline{z}^1 d w^1- \\overline{w}^1d z^1-\\overline{z}^2d w^2-\\overline{w}^2dz^2)$ A computation lead us to $\\omega _0\\left( \\widetilde{X_{(a,h)} }\\right)= -\\widetilde{(a,h)}.$ Let us denote by $\\omega $ the restriction of $\\omega _0$ to $\\mathcal {O}_\\nu $ .",
"The one form $\\omega $ is invariant by the action of $\\@root T \\of {\\mathbf {1}},$ so that it projects to a well defined one form on $\\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}},$ that we denote by $\\omega _\\nu .$ We know that $\\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}},$ represent the homogeneous contact manifold corresponding to the kind of particle under consideration.",
"As a consecuence of (REF ) and (REF ) is not difficult to prove that $\\omega _\\nu $ is the contact form.",
"As a consecuence of the fact that the map of $\\mathcal {O}_\\nu $ on $\\mathcal {O}_\\nu /\\@root T \\of {\\mathbf {1}}$ is a covering map, $\\omega $ is also a contact form on $\\mathcal {O}_\\nu ,$ that becomes itself an homogeneous contact manifold.",
"The two form $d(\\omega _\\nu )$ thus projects under $\\pi _\\nu $ on the symplectic form on $P_3^{sign(\\nu )},\\ \\Omega _\\nu ,$ that corresponds to Kirillov form in the identification of $P_3^{sign(\\nu )}$ with the coadjoint orbit.",
"Then, $d\\omega $ also projects on $\\Omega _\\nu ,$ under the restriction of the canonical map $\\pi $ to $\\mathcal {O}_\\nu .$ But $d\\omega $ is the restriction to $\\mathcal {O}_\\nu $ of $d\\omega _0$ and we have $d\\omega _0=\\frac{i\\eta \\nu }{ \\Phi } (dz^1\\wedge d\\overline{w}^1+dw^1\\wedge d\\overline{z}^1+ dz^2\\wedge d\\overline{w}^2+dw^2 \\wedge d\\overline{z}^2 )+(d\\Phi )\\wedge \\delta ,$ where $\\delta $ is a one form.",
"Since $d\\Phi $ vanishes on $\\mathcal {O}_\\nu ,$ it follows that $d\\omega $ is also the restriction to $\\mathcal {O}_\\nu $ of $\\Omega =\\frac{i\\eta \\nu }{ \\Phi } (dz^1\\wedge d\\overline{w}^1+dw^1\\wedge d\\overline{z}^1+ dz^2\\wedge d\\overline{w}^2+dw^2 \\wedge d\\overline{z}^2).$ Thus, we can obtain explicit expressions of $\\Omega _\\nu $ as follows: for each differentiable section of $\\pi $ with values in $\\mathcal {O}_\\nu $ $\\sigma : U \\rightarrow {\\cal O}_\\nu \\subset {\\bf T}^{sgn(\\nu )},$ we have on the open set $U$ $\\Omega _\\nu =\\sigma ^*\\Omega _0,$ Also we have $\\Omega _\\nu =d\\,\\left(\\sigma ^*\\omega _0 \\right).$"
],
[
"Local expression of the symplectic form.",
"In ${{\\mathbb {C}}}^{3}$ we define ${\\cal D}=\\lbrace (t,u,v):\\,sign(\\nu )\\,\\Re (\\phi (t,u,v)) > 0\\rbrace ,$ where $\\phi (t,u,v)=u+\\overline{t} v$ and $\\Re {}$ stands for real part.",
"In ${\\bf P }^{sgn(\\nu )}_3$ we define ${\\cal D}_1=\\left\\lbrace \\left[\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array} \\right]:w^1\\ne 0,\\ sign(\\nu )\\,\\Re (\\Phi (\\left(\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array}\\right))) > 0\\right\\rbrace $ ${\\cal D}_2=\\left\\lbrace \\left[\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array} \\right]:w^2\\ne 0,\\ sign(\\nu )\\,\\Re (\\Phi (\\left(\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array}\\right))) > 0\\right\\rbrace $ ${\\cal D}_3=\\left\\lbrace \\left[\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array} \\right]:z^1\\ne 0,\\ sign(\\nu )\\,\\Re (\\Phi (\\left(\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array}\\right))) > 0\\right\\rbrace $ ${\\cal D}_4=\\left\\lbrace \\left[\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array} \\right]:z^2\\ne 0,\\ sign(\\nu )\\,\\Re (\\Phi (\\left(\\begin{array}{c}w^1\\\\w^2\\\\z^1\\\\z^2 \\end{array}\\right))) > 0\\right\\rbrace .$ The ${\\cal D}_k$ compose an open cover of ${\\bf P }^{sgn(\\nu )}_3.$ The maps $\\psi _1: (t,u,v)\\in {\\cal D} \\rightarrow \\left[\\begin{array}{c}1\\\\t\\\\u\\\\v \\end{array} \\right] \\in {\\cal D}_1$ $\\psi _2: (t,u,v)\\in {\\cal D} \\rightarrow \\left[\\begin{array}{c}v\\\\1\\\\t\\\\ u \\end{array} \\right] \\in {\\cal D}_2$ $\\psi _3: (t,u,v)\\in {\\cal D} \\rightarrow \\left[\\begin{array}{c}u\\\\v\\\\1\\\\t \\end{array} \\right] \\in {\\cal D}_3$ $\\psi _4: (t,u,v)\\in {\\cal D} \\rightarrow \\left[\\begin{array}{c}t\\\\ u \\\\v\\\\1 \\end{array} \\right] \\in {\\cal D}_4$ are such that the $({\\cal D}_k,(\\psi _k)^{-1})$ compose an atlas of ${\\bf P }^{sgn(\\nu )}_3.$ For all of these charts the coordinates will be denoted by $(t,u,v).$ We also define sections of $\\pi _\\nu ,$ $\\sigma _k:{\\cal D}_k \\rightarrow {\\cal O}_\\nu ,\\ \\ \\ \\ k=1,\\dots ,4,$ by means of $\\sigma _1\\circ \\psi _1: (t,u,v)\\in {\\cal D} \\rightarrow F(t,u,v)\\left(\\begin{array}{c}1\\\\t\\\\u\\\\v \\end{array} \\right) \\in {\\cal D}_1$ $\\sigma _2\\circ \\psi _2: (t,u,v)\\in {\\cal D} \\rightarrow F(t,u,v)\\left(\\begin{array}{c}v\\\\1\\\\t\\\\ u \\end{array} \\right) \\in {\\cal D}_2$ $\\sigma _3\\circ \\psi _3: (t,u,v)\\in {\\cal D} \\rightarrow F(t,u,v)\\left(\\begin{array}{c}u\\\\v\\\\1\\\\t \\end{array} \\right) \\in {\\cal D}_3$ $\\sigma _4\\circ \\psi _4: (t,u,v)\\in {\\cal D} \\rightarrow F(t,u,v)\\left(\\begin{array}{c}t\\\\ u \\\\v\\\\1 \\end{array} \\right) \\in {\\cal D}_4$ where $F(t,u,v)=\\sqrt{\\frac{2\\nu }{\\Re (\\phi (t,u,v))}}.$ For all $k$ we have $(\\sigma _k\\circ \\psi _k)^*\\omega = \\frac{\\eta \\nu }{\\Re (\\phi )}(d(\\Im (\\phi ))+i(v\\,d\\overline{t}-\\overline{v}\\,dt)),$ Where $\\Im (\\phi )$ is the imaginary part of $\\phi .$ Then, the local expression of $\\Omega _\\nu $ in all of these coordinate systems can be evaluated by means of $\\Omega _\\nu \\stackrel{loc}{=}d\\,((\\sigma _k\\circ \\psi _k)^*\\omega ).$ for all $k.$ Notice that $\\omega $ is not projectable on ${\\bf P }^{sgn(\\nu )}_3.$ Thus (REF ) need not be local expressions of a well defined 1-form on all of ${\\bf P }^{sgn(\\nu )}_3.$"
],
[
"Symmetric Wave Functions of the Photon.",
"There are many proposals in the literature for Wave Functions of the Photon.",
"Including someones that asserts that such a thing does not exist.",
"In this paper I have described the Wave Functions of the Massless Type 4 Particles, where the case $T=2$ corresponds to the Photon.",
"These Wave Functions satisfies the Penrose Wave Equations.",
"In this section I describe other forms of the Wave Functions of Photon.",
"Equivalent representations.These forms enables us to relate directly the Wave Functions with the Electromagnetic Potential and the Electromagnetic Field.",
"Photon is the name of four kinds of particles: the massless Type 4 particles with $T=2,$ $i.e.$ those corresponding to $\\gamma =\\left\\lbrace \\frac{i \\chi }{4 \\pi }\\left(\\begin{array}{cc} 1 & 0\\\\0 & -1\\end{array}\\right),\\eta \\,\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)\\right\\rbrace \\ \\ \\ \\ ,\\ \\chi ,\\ \\eta \\in \\lbrace \\pm 1\\rbrace ,$ We already know that the value of $-\\eta $ is the sign of energy and that $-\\eta \\chi $ is the sign of helicity ($c.f.$ section REF ).",
"We denote $\\ell =-\\eta \\chi $ .",
"From section we obtain $\\begin{array}{l}\\vspace{7.22743pt}G_\\gamma = \\left\\lbrace \\left(\\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&i \\chi \\eta a z/ \\pi \\\\\\overline{i \\chi \\eta a z / \\pi }&0\\end{array}\\right)\\right):\\ z \\in {\\bf S^1},\\right.",
"\\\\\\left.",
"\\ \\ \\ \\ \\ \\ \\ \\ \\ b\\in {\\bf R},\\ a \\in {\\bf C}\\right\\rbrace \\\\\\ \\\\\\mathrm {and\\ the\\ homomorphism} \\\\\\ \\\\C_\\gamma \\left( \\left( \\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right),\\ \\left(\\begin{array}{cc}b&{i \\chi \\eta a z/ \\pi } \\\\{\\overline{i \\chi \\eta a z/ \\pi }}&0\\end{array}\\right)\\right)\\right)=z^{2\\chi }\\end{array}$ has differential $\\gamma .$ $\\begin{array}{l}({\\mbox{$G_\\gamma $}} )_{SL}=\\left\\lbrace \\left(\\begin{array}{cc} z&a \\\\0&{\\overline{z}}\\end{array}\\right): z \\in {\\bf S^1},\\ a \\in {\\mathbb {C}} \\right\\rbrace \\\\\\vspace{7.22743pt}(C_\\gamma )_{SL}\\left(\\ \\left(\\begin{array}{cc}z&a \\\\0&{\\overline{z}}\\end{array}\\right)\\right)=\\ z^{2\\chi } \\\\\\vspace{7.22743pt}SL_1=\\left\\lbrace \\left(\\begin{array}{cc}a&0 \\\\0&{1/a}\\end{array}\\right):\\ a \\in {\\mathbb {C}}\\right\\rbrace \\\\\\vspace{7.22743pt}SL_2=(G_\\gamma )_{SL} \\\\\\vspace{7.22743pt}SL_1 \\cap SL_2=\\left\\lbrace \\left(\\begin{array}{cc}z&0 \\\\0&\\overline{z}\\end{array}\\right):\\ z \\in {\\bf S}^1\\right\\rbrace \\end{array}.$ In section we have identified the space $SL/(G_\\gamma )_{SL}$ to the future lightcone, ${\\mbox{${\\bf C}^+$}}=\\lbrace H \\in {H(2)}:\\ Det\\,H=0,\\ Tr\\,H>0\\rbrace ,$ by means of the diffeomorphism $A\\, (G_\\gamma )_{SL} \\in SL/(G_\\gamma )_{SL} \\mapsto A \\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right) A^* \\in {\\bf C}^+.$ In this section I give another description of the wave functions of a photon, by means of different trivialisations than the ones I have used in section .",
"Of course these trivialisations are “isomorphic”, in an obvious sense.",
"Let us consider first the cases $\\chi =+1$ $i.e.$ $\\ell =-\\eta .$ In these cases, a trivialisation ($c.\\,f.$ section ) is $(\\mu _{+}, { \\cal {S}},s_0^+),$ where $\\cal S$ is the vector subspace of $gl(2,\\bf C),$ composed by the symmetric matrices, $s_0^+=\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)$ and $\\mu _+(A)\\cdot s=A \\,s\\, {}^tA,$ for all $A \\in SL(2,{\\bf C}),\\ s\\in \\cal S$ .",
"Since $\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)=\\left(\\begin{array}{c} 1 \\\\0\\end{array}\\right)\\left(\\begin{array}{cc} 1 & 0\\end{array}\\right) $ the orbit of $\\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)$ by $\\mu _+$ is composed by the elements of ${\\cal S}^2$ of the form $(A \\left(\\begin{array}{c} 1 \\\\0\\end{array}\\right) )\\ {}^t (A \\left(\\begin{array}{c} 1 \\\\0\\end{array}\\right) ),$ for some $A \\in SL(2,\\bf C)$ .",
"Then, one sees that the orbit is contained in ${\\cal B}=\\lbrace z \\ {}^t z: z\\in {\\bf C}^2-\\lbrace 0\\rbrace \\rbrace .$ But every $z\\in {\\bf C}^2-\\lbrace 0\\rbrace $ has the form $ A\\, \\left(\\begin{array}{c} 1 \\\\0\\end{array}\\right),$ for some $A \\in SL(2,{\\bf C})$ , so that the orbit coincides with ${\\cal B}$ .",
"For each $ s\\in {\\cal S}$ such that $s\\ne 0$ and $Det\\, s=0,$ there exist exactly two $z\\in {\\bf C}^2-{0}$ such that $s=z \\ {}^t z.$ In fact, if $s=\\left(\\begin{array}{cc} a & b\\\\b & d\\end{array}\\right)\\in B$ the $z$ such that $s=z \\ {}^t z$ are $\\pm \\left(\\begin{array}{c} \\alpha \\\\\\sigma \\delta \\end{array}\\right)$ where $\\alpha $ is a square root of $a$ , $\\delta $ is a square root of $d$ , and $\\sigma \\in \\lbrace \\pm 1\\rbrace $ such that $b=\\sigma \\alpha \\delta .$ As a consequence, we also have ${\\cal B}=\\lbrace s\\in {\\cal S}: s\\ne 0,Det\\, s=0\\rbrace .$ The homogeneous space $SL/Ker(C_\\gamma )_{SL}$ is identified to ${\\cal B}$ by means of $ A\\,Ker(C_\\gamma )_{SL} \\in SL/Ker(C_\\gamma )_{SL}\\longrightarrow \\mu _+(A)\\cdot s_0^+ \\in {\\cal B}$ Then, the canonical map $SL/Ker(C_\\gamma )_{SL} \\longrightarrow SL/(G_\\gamma )_{SL},$ denoted in the following by $r_+,$ becomes $r_+:(A \\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right){}^tA)\\in {\\cal B} \\longrightarrow A \\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)A^*\\in C^+,$ for all $A\\in SL,$ so that $r_+(z{}^t z)= zz^*$ for all $z \\in {\\bf C}^2-{0}$ .",
"By elementary operations with matrices, one can prove that the relation $r_+(s)=C $ is equivalent to $C=(+(Tr\\,s\\overline{s})^{-1/2})\\,s\\overline{s}$ and also to $s=(+(Tr\\,C {}^t C)^{-1/2})e^{i\\phi }(C {}^t C)$ for some $\\phi \\in {\\bf R}.$ To obtain Wave Functions, we need functions on ${\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers.",
"If $f$ is one of such functions the corresponding Prewave Function is $ \\psi _f^{(-\\eta ,-\\eta )}(H,K)=f(s)\\,s\\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m},$ where $s$ is arbitrary in $r_+^{-1}(K).$ In ${(-\\eta ,-\\eta )}$ the first $-\\eta $ stands for the sign of energy and the second by $\\ell ,$ helicity.",
"Now, let us consider the cases $\\chi =-1$ $i.e.$ $\\ell =\\eta .$ In these cases, a trivialisation is $(\\mu _{-}, { \\cal {S}},s_0^-),$ where $\\cal S$ is as in the $\\chi =+1$ case, $s_0^-=\\left(\\begin{array}{cc} 0 & 0\\\\0 & 1\\end{array}\\right)$ and $\\mu _-(A)\\cdot s=(A^*)^{-1} \\,s\\, (\\overline{A})^{-1},$ for all $A \\in SL(2,{\\bf C}),\\ s\\in \\cal S$ .",
"The orbit of $\\left(\\begin{array}{cc} 0 & 0\\\\0 & 1\\end{array}\\right)$ by $\\mu _-$ is, as in case $\\chi =+1$ , ${\\cal B}$ .",
"The canonical map $r_-: SL/Ker(C_{\\gamma })_{SL} \\longrightarrow SL/(G_{\\gamma })_{SL}$ becomes a map from ${\\cal B}$ onto ${\\bf C}^+,$ given by $r_-( \\mu _-(A)\\cdot s_0^-)=A \\left(\\begin{array}{cc} 1 & 0\\\\0 & 0\\end{array}\\right)A^*.$ The maps $r_+$ and $r_-$ are geometrically related as follows.",
"Let us consider the map $J:s\\in {\\cal S} \\longrightarrow -\\epsilon \\overline{s}\\epsilon \\in {\\cal S} .$ This is an antilinear map with $J^2=I.$ Since $J(z{}^tz)=(-\\epsilon \\overline{z}){}^t(-\\epsilon \\overline{z}),$ we have $J({\\cal B})={\\cal B}.$ On the other hand $r_-\\circ J(\\mu _+(A)\\cdot s_0^+)=r_-(-\\epsilon \\, \\overline{A}\\,\\overline{s_0^+}\\,A^*\\,\\epsilon )=r_-(-(A^*)^{-1}\\,\\epsilon \\,\\overline{s_0^+} \\,\\epsilon \\, (\\overline{A})^{-1})=\\\\=r_-(\\mu _-(A)\\cdot s_0^-)=A\\ ({\\mbox{$G_\\gamma $}} )_{SL}=r_+(\\mu _+(A)\\cdot s_0^+) , $ so that $r_+=r_-\\circ J.$ Since $J^2=I,$ we also have $r_-=r_+\\circ J.$ As a consequence, $J$ stablishes a principal fibre bundle isomorphism over the identical map of $C^+$ , the isomorphism of structural groups being the map defined by sending each element of $S^1$ to its inverse.",
"Another consequence is $r_-(z {}^tz)=-\\epsilon \\overline{zz^*}\\epsilon .$ If $f$ is a function on ${\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers, it also defines a Prewave Function for this kind of photon by means of $ \\psi _f^{(-\\eta ,\\eta )}(H,K)=f(s)\\,s\\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m},$ where $s,$ now, is arbitrary in $r_-^{-1}(K).$ Here, in ${(-\\eta ,\\eta )},$ $-\\eta $ stands for the sign of energy and the second $\\eta ,$ which in the present case coincides with $\\ell ,$ stands for helicity.",
"Thus, the Prewave Functions corresponding to the four kinds of Photon are different.",
"The sign of energy, $-\\eta ,$ and the sign of helicity, $\\ell ,$ determines the type of Prewave Functions to be used, $\\psi _f^{(-\\eta , \\ell )},$ that are given by (REF ) or (REF ).",
"When $f$ is continuous with compact support, the corresponding Wave Function is $\\widetilde{\\psi }_f^{(-\\eta , \\ell )}(x)=\\int _{C^+} \\psi _f^{(-\\eta , \\ell )}(h(x),K) \\omega _K,$ where $\\omega $ is the invariant volume element on $C^+$ defined in (REF ).",
"These Wave Functions represent states whose energy has sign $-\\eta $ and are eigenvectors of helicity ($c.f.",
"$ section REF ) corresponding to the eigenvalues $\\frac{\\ell }{2\\pi }.$"
],
[
"Prehilbert Space estructure.",
"Let us denote by ${\\cal {F}}$ the vector subspace composed by the functions on ${\\cal B}$ homogeneous of degree -1 under product by modulus one complex numbers, and by ${\\cal {F}}_c$ the subspace of ${\\cal {F}}$ composed by the continuous elements with compact support.",
"The space ${\\cal {F}}_c$ can be provided with a prehilbert space structure, by means of the general method described in section , for each kind of Photon.",
"In more detail we proceed as follows.",
"We separate the cases ${-\\eta \\ell }=\\pm 1.$ If $\\ f , f^\\prime \\in {\\cal F}_c,$ its hermitian product is in each case $\\langle f,\\, f^\\prime \\rangle _{(-\\eta , \\ell )}= \\int _{C^+} (\\overline{f} f^\\prime )_{(-\\eta , \\ell )}\\ {\\mbox{$\\omega $}} , $ where $ (\\overline{f} f^\\prime )_{(-\\eta , \\ell )}$ is the function defined on $C^+$ by $(\\overline{f} f^\\prime )_{(-\\eta , \\ell )} (K)=\\overline{f(s)} f^\\prime (s)$ for all $K\\in C^+,$ where $\\ s\\in r_{-\\eta \\ell }^{-1}(K).$ With each of these inner products, ${\\cal F}_c$ becomes a prehilbert space.",
"For prewave functions we define $\\langle \\psi _f^{(-\\eta , \\ell )},\\, \\psi _{f^\\prime }^{(-\\eta , \\ell )} \\rangle _{(-\\eta , \\ell )}= \\langle f,\\, f^\\prime \\rangle _{(-\\eta , \\ell )}.$ A sexquilinear form on S is given by $\\Phi (s,s^\\prime )=Tr(\\overline{s}\\,{s^\\prime }).$ Thus ($c.f.$ section ), the hermitian product of Prewave Functions can be given in terms of the Prewave Functions themselves instead of the functions $f,$ by $\\langle \\psi ^{(-\\eta , \\ell )}_f,\\, \\psi ^{(-\\eta , \\ell )}_{f^\\prime } \\rangle _{(-\\eta , \\ell )}=\\int _{C^+} \\psi _f^{(-\\eta , \\ell )}\\Phi _{(-\\eta , \\ell )} \\psi _{f^\\prime }^{(-\\eta , \\ell )} \\ \\omega ,$ where $\\psi _f^{(-\\eta , \\ell )} \\Phi _{(-\\eta , \\ell )} \\psi _{f^\\prime }^{(-\\eta , \\ell )}$ is the function on $C^+$ given by $\\psi _f^{(-\\eta , \\ell )} \\Phi _{(-\\eta , \\ell )} \\psi _{f^\\prime }^{(-\\eta , \\ell )} (K)=\\frac{Tr(\\overline{\\psi _f^{(-\\eta , \\ell )}(H,K)}{ \\psi _{f^\\prime }^{(-\\eta , \\ell )}(H,K))}}{Tr(\\overline{s}\\,{s})}$ for all $K\\in C^+,$ $H \\in H(2)$ and $s\\in r_{-\\eta \\ell }^{-1}(K).$ The hermitian product of Wave Functions is defined as being the hermitian product of the corresponding Prewave Functions."
],
[
"Electromagnetic Potential. ",
"If $K\\in C^+$ , the tangent space to $C^+$ at $K$ can be identified to the subspace of $H(2)$ given by $T_KC^+=\\lbrace M\\in H(2): Tr\\,M\\epsilon \\overline{K}\\epsilon =0 \\rbrace .$ Then, the complexified tangent space can be identified to ${}^{\\bf C}T_KC^+=\\lbrace M\\in gl(2,\\mathbb { C}\\rm ): Tr\\,M\\epsilon \\overline{K}\\epsilon =0 \\rbrace .$ A real vector field on $C^+$ is thus a map $A:C^+ \\longrightarrow H(2),$ such that $Tr\\,A(K)\\epsilon \\overline{K}\\epsilon =0,$ for all $K\\in C^+$ .",
"A complex vector field on $C^+$ is thus given by a function $A:C^+ \\longrightarrow gl(2,\\bf C),$ whose real and imaginary hermitian parts are real vectorfields on $C^+$ .",
"Since $Tr M\\epsilon \\overline{K}\\epsilon $ is real for $M$ and $K$ hermitian, we see that $A$ is a complex vector field on $C^+$ if and only if $Tr\\,A(K)\\epsilon \\overline{K}\\epsilon =0,$ for all $K\\in C^+$ .",
"Equation (REF ) is equivalent to say that $A(K)\\epsilon \\overline{K}$ is a symmetric matrix.",
"Let $A$ be a complex vector field on $C^+.$ We denote by $A_R$ and $A_I$ the real and imaginary hermitian parts of $A$ and $A_R(K)&=&h(A_R^1(K),.., A_R^4(K))\\\\A_I(K)&=&h(A_I^1(K),.., A_I^4(K))\\\\A^\\mu (K)&=&A_R^\\mu +i\\, A_I^\\mu ,\\ \\ \\mu =1,..,4\\\\\\overrightarrow{A_R}(K)&=&(A_R^1(K),.., A_R^3(K))\\\\\\overrightarrow{A_I}(K)&=&(A_I^1(K),.., A_I^3(K))\\\\\\overrightarrow{A}(K)&=&(A^1(K),.., A^3(K)).$ If $A_i^j(K)$ is the element of $A(K)$ in the row $i$ column $j,$ we also have $A^1 (K)&=&\\frac{1}{2}(A_1^2(K)+A_2^1(K))\\\\A^2 (K)&=&\\frac{i}{2}(A_1^2(K)-A_2^1(K))\\\\A^3 (K)&=&\\frac{1}{2}(A_1^1(K)-A_2^2(K))\\\\A^4 (K)&=&\\frac{1}{2}(A_1^1(K)+A_2^2(K))$ If $A(K)$ is , for exemple, continuous with compact support, for $\\mu =1,..,4,\\ x\\in \\mathbb {R}^4$ we define $\\widetilde{ A^\\mu }(x)=\\int _{C^+} A^\\mu (K) Exp(2\\pi i \\eta \\langle h^{-1}( K),x \\rangle ) \\omega .$ Then $\\square \\widetilde{A^\\mu }=0,$ for all $\\mu ,$ and $\\sum _{\\mu =1}^4 \\frac{\\partial \\widetilde{A^\\mu }}{\\partial x^\\mu }=0.$ We thus see that the $\\widetilde{A^\\mu }(x)$ define a complex electromagnetic potential in the Lorenz gauje.",
"The corresponding electric field is given by $\\overrightarrow{E}(x)=-\\nabla \\widetilde{A^4}(x)-\\frac{\\partial \\overrightarrow{\\widetilde{A}}(x)}{\\partial x^4 },$ where $\\overrightarrow{\\widetilde{A}}(x)=(\\widetilde{ A^1}(x),\\widetilde{ A^2}(x),\\widetilde{ A^3}(x)),$ that can be written $\\overrightarrow{E}(x)=2\\pi i \\eta \\int _{C^+}(A^4(K)\\overrightarrow{K}-K^4 \\overrightarrow{A}(K)) Exp(2\\pi i \\eta \\langle h^{-1}( K),x \\rangle ) \\omega .$ The magnetic field is given by $\\overrightarrow{B}(x)=\\nabla \\times \\overrightarrow{\\widetilde{A}}(x),$ that can be written $\\overrightarrow{B}(x)=2\\pi i \\eta \\int _{C^+}(\\overrightarrow{A}(K)\\times \\overrightarrow{K} ) Exp(2\\pi i \\eta \\langle h^{-1}( K),x \\rangle ) \\omega .$ This electromagnetic field take in general complex values, the real parts of $\\overrightarrow{E}$ and $\\overrightarrow{B}$ are real electric and magnetic fields, corresponding to the electromagnetic potential given by the real parts of the $A^\\mu (x).$"
],
[
"Wave Functions and the Electromagnetic Potential. ",
"In section REF we have seen that a complex vector field on $C^+$ , define a complex electromagnetic potential in the Lorenz gauje.",
"Let $A$ be a complex vector field on $C^+$ .",
"Then, the matrix $A(K)\\epsilon \\overline{K}$ is a symmetric matrix, for all $K\\in C^+$ .",
"If $ s\\in r_+^{-1}(K)$ there exist an unique complex number, $f_A(s),$ such that $A(K)\\epsilon \\overline{K}=f_A(s)\\, s .$ In fact, if $z\\in {\\bf C}^2-{0}$ , is such that $s= z {}^t z$ , we have $K=zz^*$ , and, since $0=Tr A(K)\\epsilon \\overline{z}{}^t z\\epsilon ={}^tz\\epsilon A(K)\\epsilon \\overline{z},$ there exist a number, $\\lambda $ , such that $A(K)\\epsilon \\overline{z}=\\lambda \\, z,$ so that $A(K)\\epsilon \\overline{K}=\\lambda \\, s .$ Since $s\\ne 0$ such a $\\lambda $ is unique and is denoted by $f_A(s).$ Thus the complex vector field $A$ defines a function, $f_A,$ on ${\\cal B}.$ If we take $e^{i\\phi }\\,s$ instead of $s$ in $ r_+^{-1}(K)$ , we can take in the preceeding reasoning $e^{i\\phi /2}z$ instead of $z$ , and thus $A(K)\\epsilon \\overline{e^{i\\phi /2}z}=\\lambda ^\\prime \\, e^{i\\phi /2}z$ leads to $\\lambda ^\\prime = e^{-i\\phi }\\,\\lambda .$ We thus see that $f_A(e^{i\\phi }\\,s)=e^{-i\\phi }\\,f_A(s)$ which proves that $f_A $ is $S^1$ -homogeneous of degree -1.",
"The Prewave Function corresponding to $f_A$ for $-\\eta \\ell =1$ can be written as $\\psi _A^{(-\\eta , -\\eta )}(H,K)=f_A(s)\\,s\\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m},$ where $s$ is arbitrary in $r_+^{-1}(K).$ As a consecuence of (REF ) we have $\\psi _A^{(-\\eta , -\\eta )}(H,K)=A(K)\\epsilon \\overline{K}e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m},$ that gives Prewave Functions directly in terms of vectorfields.",
"If $A(K)$ is continuous with compact support, the corresponding Wave Function is $ \\widetilde{\\psi }_A^{(-\\eta , -\\eta )}(x)=\\int _{C^+}A(K)\\epsilon \\overline{K}\\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,x\\rangle _m}\\omega _K.$ Now we define $\\widehat{f}_A=\\overline{f_A\\circ J},$ where $J$ is given by (REF ).",
"I shall prove that $\\widehat{f}_A$ is $S^1$ -homogeneous of degree -1 on ${\\cal B}$ and that, for all $ s\\in r_-^{-1}(K),$ we have $-\\epsilon \\overline{A(K)}\\epsilon K \\epsilon =\\widehat{f}_A(s)\\, s .$ This will enable us to give a Prewave Function of Photons with $\\ell =\\eta ,$ directly in terms of $A$ .",
"For all $ s\\in B,\\ a\\in S^1$ we have $\\widehat{f}_A(as)=\\overline{f_A( J(as))}=\\overline{f_A(\\overline{a} J(s)}=\\overline{af_A( J(s)}=\\overline{a}\\,\\widehat{f}_A(s),$ so that $\\widehat{f}_A$ has the appropiate homogeneity to define a Prewave Function.",
"On the other hand, if $s \\in r_-^{-1}(K)$ , we have $\\widehat{f}_A(s)\\,s=\\overline{f_A(J(s))}\\,J(J(s))=J(f_A(J(s))\\,J({s}))=$ $=J(A(r_+(J(s)))\\epsilon \\overline{r_+(J(s))})=J(A(r_-(s))\\epsilon \\overline{r_-(s)}=$ $=J(A(K)\\epsilon \\overline{K}))=-\\epsilon \\overline{A(K)}\\epsilon K \\epsilon .$ Then, the Prewave Function for Photons with $\\ell =\\eta $ corresponding to $\\widehat{f}_A$ is $\\psi _A^{(-\\eta , \\eta )}(H,K)=-\\epsilon \\overline{A(K)}\\epsilon K \\epsilon \\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m}.$ Obviously $\\psi _A^{(-\\eta , \\eta )}=-\\epsilon \\overline{\\psi _A^{(\\eta , \\eta )}}\\epsilon , \\ \\ \\psi _A^{(\\eta , \\eta )}=-\\epsilon \\overline{\\psi _A^{(-\\eta , \\eta )}}\\epsilon .$ If $A(K)$ is continuous with compact support, the corresponding Wave Function is $ \\widetilde{\\psi }_A^{(-\\eta , \\eta )}(x)=-\\int _{C^+}\\epsilon \\overline{A(K)}\\epsilon K \\epsilon \\,\\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,x\\rangle _m}\\omega _K.$ We have $\\widetilde{\\psi }_A^{(-\\eta , \\eta )}=-\\epsilon \\overline{\\widetilde{\\psi }_A^{(\\eta , \\eta )}}\\epsilon ,\\ \\ \\widetilde{\\psi }_A^{(\\eta , \\eta )}=-\\epsilon \\overline{\\widetilde{\\psi }_A^{(-\\eta , \\eta )}}\\epsilon .$ With equations (REF ) and (REF ) we see that a single complex vectorfied on $C^+$ gives rise to Wave Functions of the four kinds of Photons: those with positive and negative energy and helicity.",
"The inner products defined in section REF lead to the following results.",
"If $A$ and $A^\\prime $ are vector fields on $C^+$ we have $\\langle \\psi ^{(-\\eta , -\\eta )}_A,\\, \\psi ^{(-\\eta , -\\eta )}_{A^\\prime } \\rangle _{(-\\eta , -\\eta )}= \\langle f_A,\\, f_{A^\\prime } \\rangle _{(-\\eta , -\\eta )},$ and $\\langle \\psi ^{(-\\eta , \\eta )}_A,\\, \\psi ^{(-\\eta , \\eta )}_{A^\\prime } \\rangle _{(-\\eta , \\eta )}=\\langle \\widehat{f}_A,\\, \\widehat{f}_{A^\\prime } \\rangle _{(-\\eta , \\eta )}.$ Then, from (REF ) and (REF ) one can obtain $\\langle \\psi ^{(-\\eta , -\\eta )}_A,\\, \\psi ^{(-\\eta , -\\eta )}_{A^\\prime } \\rangle _{(-\\eta , -\\eta )}=\\int _{C^+} A\\Phi A^\\prime =$ $=\\langle \\psi ^{(-\\eta , \\eta )}_{A^\\prime },\\, \\psi ^{(-\\eta , \\eta )}_{A} \\rangle _{(-\\eta , \\eta )}$ where $A\\Phi A^\\prime (K)= \\frac{Tr(\\overline{A(K)}\\epsilon K A^\\prime (K)\\epsilon \\overline{K})}{Tr(\\overline{s}{s})}$ for all $K\\in C^+,$ and $s\\in r_{\\pm }^{-1}(K).$"
],
[
"Wave Functions in terms of the Electromagnetic Field.",
"Let $A$ a complex vectorfield on $C^+$ and denote $\\overrightarrow{E}(K)&=& A^4(K)\\overrightarrow{K}-K^4 \\overrightarrow{A}(K) \\\\\\overrightarrow{B}(K)&=& \\overrightarrow{A}(K)\\times \\overrightarrow{K}$ Thus, the Electric and Magnetic Fields can be given by $\\overrightarrow{E}(x)=2\\pi i \\eta \\int _{C^+}\\overrightarrow{E}(K) Exp(2\\pi i \\eta \\langle h^{-1}( K),x \\rangle ) \\omega _K$ $\\overrightarrow{B}(x)=2\\pi i \\eta \\int _{C^+}\\overrightarrow{B}(K) Exp(2\\pi i \\eta \\langle h^{-1}( K),x \\rangle ) \\omega _K.$ Let us prove that $A(K) \\epsilon \\overline{K} \\epsilon =\\left(\\overrightarrow{E}(K)+i\\overrightarrow{B}(K)\\right)\\cdot \\overrightarrow{\\sigma },º$ which means $A(K) \\epsilon \\overline{K} \\epsilon =\\left(\\begin{array}{cc}(\\overrightarrow{E}+i\\overrightarrow{B})^3 & (\\overrightarrow{E}+i\\overrightarrow{B})^1-i(\\overrightarrow{E}+i\\overrightarrow{B})^2\\\\(\\overrightarrow{E}+i\\overrightarrow{B})^1+i(\\overrightarrow{E}+i\\overrightarrow{B})^2 &-(\\overrightarrow{E}+i\\overrightarrow{B})^3\\end{array}\\right)$ In fact, we have $A(K) \\epsilon \\overline{K} \\epsilon =\\frac{1}{2} \\left( A(K) \\epsilon \\overline{K}+ {}^{t}(A(K) \\epsilon \\overline{K}) \\right)\\epsilon =$ $=\\frac{1}{2}\\left((A_R+iA_I)\\epsilon \\overline{K}\\epsilon -K \\epsilon ({}^{t}A_R+i {}^{t}A_I)\\epsilon \\right)=$ $=\\frac{1}{2}(A_R\\epsilon \\overline{K} \\epsilon -K\\epsilon \\overline{A_R} \\epsilon )+\\frac{i}{2}(A_I\\epsilon \\overline{K} \\epsilon -K\\epsilon \\overline{A_I} \\epsilon )$ and thus (REF ) follows from (REF ).",
"As a consequence, the Prewave Functions corresponding to $A$ are $\\psi _A^{(-\\eta , -\\eta )}(H,K)=-\\left(\\left(\\overrightarrow{E}(K)+i\\overrightarrow{B}(K)\\right)\\cdot \\overrightarrow{ \\sigma } \\right)\\epsilon \\ e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m},$ $\\psi _A^{(-\\eta , \\eta )}(H,K)=-\\epsilon \\ \\left(\\overline{\\left(\\overrightarrow{E}(K)+i\\overrightarrow{B}(K)\\right)\\cdot \\overrightarrow{ \\sigma }}\\right) \\ e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m}.$ If $A$ is continuous with compact support, we have $\\widetilde{\\psi }_A^{(-\\eta , -\\eta )}(x)=-\\int _{C^+}((\\overrightarrow{E}(K)+i\\overrightarrow{B}(K))\\cdot \\overrightarrow{\\sigma })\\ \\epsilon \\ e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m}\\omega _K,$ $\\widetilde{\\psi }_A^{(-\\eta , \\eta )}(x)=-\\int _{C^+}\\ \\epsilon \\ (\\overline{(\\overrightarrow{E}(K)+i\\overrightarrow{B}(K))\\cdot \\overrightarrow{\\sigma })} \\,e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,h^{-1}(H)\\rangle _m}\\omega _K.$ so that the Wave Functions are $\\widetilde{\\psi }_A^{(-\\eta , -\\eta )}(x)=\\frac{i\\eta }{2\\pi }((\\overrightarrow{E}(x)+i\\overrightarrow{B}(x))\\cdot \\overrightarrow{\\sigma })\\ \\epsilon ,$ or $\\widetilde{\\psi }_A^{(-\\eta , \\eta )}(x)=\\frac{i\\eta }{2\\pi }\\epsilon \\ (\\overline{(\\overrightarrow{E}(x)+i\\overrightarrow{B}(x))\\cdot \\overrightarrow{\\sigma })} .$ These Wave Functions are given in terms of the components of $\\overrightarrow{E}(x)+i\\overrightarrow{B}(x)$ as (up to constants) the Wave Functions of Bialynicki-Birula [2]."
],
[
"Gauje invariance for Photons.",
"Let us denote by $\\cal {D}$ the complex vector space of complex vector fields on $C^+.$ The map $A\\in {\\cal D} \\longrightarrow f_A \\in {\\cal F}$ is linear and its kernel is composed by the vector fields on $C^+$ having the form $A(K)=\\left( \\begin{array}{c}\\lambda (z)\\\\ \\mu (z) \\end{array}\\right)\\,z^*$ where $z\\in {\\bf C}^2-{0}$ is such that $K=zz^*$ and $\\lambda ,\\ \\mu $ are funtions on ${\\bf C}^2-{0}$ $S^1$ -homogeneous of degree +1.",
"The kernel of the map $A\\in {\\cal D} \\longrightarrow \\widehat{f}_A \\in {\\cal F}$ is the same and will be denoted by $\\cal {N}.$ In particular, for each map $L:K\\in C^+ \\rightarrow L(K)\\in gl(2,{\\bf C}),$ the vectorfield $A_L(K)=L(K)K$ is in $\\cal {N}.$ If $A\\in \\cal {D}$ , $N \\in \\cal {N}$ we have $f_{A+N}=f_A$ $\\widehat{f}_{A+N}= \\widehat{f}_{A},$ so that the Prewave and Wave Functions are invariant under the changes $A\\rightarrow A+N$ : $\\widetilde{\\psi }_{A+N}^{(-\\eta , \\ell )}=\\widetilde{\\psi }_{A}^{(-\\eta , \\ell )}.$ Because of (REF ) and (REF ) we see that also the Electric and Magnetic Fiels corresponding to Photon are invariant under the changes $A\\rightarrow A+N$ .",
"In the particular case $N=A_L$ with $L(K)=g(K)\\,I,$ where $g$ is a complex valued function on $C^+,$ the change $A\\rightarrow A+N$ lead to the following change in the Electromagnetic Potential $\\widetilde{(A+N)}^\\mu (x)=\\widetilde{A}^\\mu (x)+\\partial ^\\mu \\phi (x),$ where $\\phi (x)=-\\frac{i\\eta }{2\\pi }\\int _{C^+}g(K) e^{2 \\pi i\\eta \\langle \\, h^{-1}(K),\\,x\\rangle _m}\\omega _K$ and $\\partial ^\\mu \\phi =g^{\\mu \\nu }\\frac{\\partial }{\\partial x^\\nu }\\,\\phi $ where the $g^{\\mu \\nu }$ are the components of Minkowski metric.",
"Thus, the invariance of the Electromagnetic Field of Photons under the change $A(K) \\rightarrow A(K)+g(K)K$ is a particular case of the well known gauje invariance of general Electromagnetic Fields."
],
[
"General remarks",
"In this paper, State Space for a particle defined by $\\alpha \\in \\underline{G}^*$ has been stated as the homogeneous space that correspond to the orbit of $(0,\\alpha )$ in $H(2)\\times \\underline{G}^*.$ Thus, as we have seen in section , it is identified to $\\frac{G}{G_{(0,\\alpha )}}$ where $G_{(0,\\alpha )}=G_\\alpha \\cap (SL \\oplus \\lbrace 0\\rbrace ),$ or, equivalently, to $H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1} ,$ that is the form adopted in Figure REF of that section.",
"The equivariant maps $\\iota _4:H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1} \\rightarrow H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}}$ and $\\nu _2:H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1} \\rightarrow \\frac{G}{G_\\alpha }.$ in that Figure, will be used in this section.",
"When we characterize $H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}\\cap SL_1}$ in terms of other concrete manifolds or of some parametrization, we need to physically interpret the geometric objects that appear, and, in order to do that, the more effective way is , in general, to know the values of the Canonical Dynamical Variables in terms of these objects.",
"But the value of these Dynamical Variables is well known on the coadjoint orbit i.e.",
"on $G/G_\\alpha .$ Thus, its values in State Space can be obtained by composition with $\\nu _2.$ Since Prewave Functions are defined on $H(2) \\times \\frac{SL}{(G_\\alpha )_{SL}}$ its composition with $\\iota _4$ gives another version of Quantum States, now defined on the Classical State Space.",
"In the definition of Prewave Functions REF , space-time only appears in the exponential, whose exponent is $-2\\pi i\\langle h^{-1}(P),h^{-1}(H)\\rangle _m.$ where $P$ is the “traslation\" of momentum-energy to $SL/(G_\\alpha )_{SL}.$ The exponent in the composition of Prewave Functions with $\\iota _4$ must have the same expression, but changing $P$ by $P\\circ \\iota _4.$ The conmutativity of the diagramm in Figure REF imply that $P\\circ \\iota _4$ can be obtained as the impulsion-energy in $G/G_\\alpha $ composite with $\\nu _2.$ In the next sections I give the expressions of the canonical dynamical variables on State Space, for different kind of particles."
],
[
"Klein-Gordon particles",
"We have seen in section REF that State Space for Klein-Gordon particles can be identified to $H(2) \\oplus \\frac{SL}{SU(2)},$ and this homogeneous space to ${\\cal H}^m\\times H(2).$ If $(K,H)\\in {\\cal H}^m\\times H(2) $ and $A\\in SL$ is such that $K=mAA^*,$ we denote $H&=&h(\\overrightarrow{x},x^4)\\\\\\overrightarrow{x}&=& (x^1,x^2,x^3)\\\\K&=&m h(\\overrightarrow{k}, k_4) \\\\\\overrightarrow{k}&=& (k^1,k^2,k^3)\\\\k^4&=&\\sqrt{1+\\Vert \\overrightarrow{k} \\Vert ^2}.$ Thus, the map (REF ) becomes, $\\nu _1(K,H)=(A,H)*(mI,\\overrightarrow{0},1)=$ $=(K,\\overrightarrow{x}-\\frac{x^4}{k^4} \\overrightarrow{k},e^{-2\\pi i \\eta m x^4/k^4})$ and the map (REF ), $\\nu _2(K,H))=(A,H)*(mI,\\overrightarrow{0})=(K,\\overrightarrow{x}-\\frac{x^4}{k^4} \\overrightarrow{k}).$ The Canonical Dynamical Variables on State Space are obtained as composition of (REF ) with $\\nu _2.$ If we denote $V_{KG}=V\\circ \\nu _2$ for each dynamical variable, $V,$ we have $P_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4))=- \\eta \\,mh(\\overrightarrow{k},k_4),$ $\\overrightarrow{l}_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4))=\\overrightarrow{x} \\times \\left(\\overrightarrow{P}_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4)\\right) ,$ $\\overrightarrow{g}_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4))=$ $ = P^4_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4)) \\overrightarrow{x}-x^4 \\overrightarrow{P}_{KG}(mh(\\overrightarrow{k},k_4),h(\\overrightarrow{x},x^4)).$ where $ k_4=\\sqrt{1+\\Vert \\overrightarrow{k} \\Vert ^2}.$ The Prewave functions have been defined on a manifold that, in this case, coincides with State Space.",
"The map $\\iota _4$ is the identical map.",
"Another fact is that, in this case, State Space is the Universal Covering of the homogeneous contact manifold.",
"Let us prove that $\\nu _1$ is a covering map.",
"The local expresion of $\\nu _1$ in the charts corresponding to $\\phi _\\tau $ and $\\phi ^\\prime :(k_1,k_2,k_3,x^1,x^2,x^3,x^4)\\in {\\mathbb {R}}^7 \\longrightarrow $ $ \\longrightarrow (h(m\\overrightarrow{k},m k_4),h(x^1,x^2,x^3,x^4))\\in {\\cal H}^m\\times H(2) ,$ is given by $(\\phi _\\tau )^{-1}\\circ \\nu _1 \\circ \\phi ^{\\prime }(k_1,\\dots ,x^4)=$ $=(k_1,k_2,k_3,x^1-\\frac{x^4}{k_4}k_1,x^2-\\frac{x^4}{k_4}k_2,x^3-\\frac{x^4}{k_4}k_3,-\\tau -\\eta m \\frac{x^4}{k_4}+N)$ where the domain of definition is defined by the condition that $\\tau +\\eta m \\frac{x^4}{k_4} \\notin {\\mathbb {Z}}+\\frac{1}{2} $ and $N$ is defined by the condition that $-\\tau -\\eta m \\frac{x^4}{k_4}+N\\in (-\\frac{1}{2},\\frac{1}{2}).$ We have $((\\phi _\\tau )^{-1}\\circ \\nu _1 \\circ \\phi ^{\\prime })^{-1}\\lbrace (\\overrightarrow{n},\\overrightarrow{y},t)\\rbrace =$ $=\\lbrace (\\overrightarrow{n},\\overrightarrow{y}-\\frac{\\eta }{m}(t+\\tau +N)\\overrightarrow{n}, -\\frac{\\eta }{m}(t+\\tau +N)\\sqrt{1+\\Vert \\overrightarrow{n} \\Vert ^2}): N\\in {\\mathbb {Z}}\\rbrace .$ Let $d\\in {\\cal H}^m \\times {\\mathbb {R}}^3\\times {\\mathbb {S}}^1,$ and let $\\tau $ be such that $d$ is in the image of $(\\phi _\\tau )^{-1}.$ This image, $U_o,$ is an open neighborhood of $d$ , and we shall prove that its antiimage by $\\nu _1$ is union of disjoint open sets, each of them diffeomorphic to $U_o$ under the restriction of $\\nu _1.$ We denote by ${\\cal U}$ the domain of $\\phi _\\tau $ .",
"The domain of $(\\phi _\\tau )^{-1}\\circ \\nu _1 \\circ \\phi ^{\\prime },$ ${\\cal W}=((\\phi _\\tau )^{-1}\\circ \\nu _1 \\circ \\phi ^{\\prime })^{-1}({\\cal U})$ is the union of the open sets ${\\cal W}_N=\\lbrace &(&\\overrightarrow{n},\\overrightarrow{y}-\\frac{\\eta }{m}(t+\\tau +N)\\overrightarrow{n}, -\\frac{\\eta }{m}(t+\\tau +N))\\sqrt{1+\\Vert \\overrightarrow{n} \\Vert ^2}: \\\\&(&\\overrightarrow{n},\\overrightarrow{y},t)\\in {\\cal U}\\rbrace .$ where $N\\in \\mathbb {Z}.$ But the ${\\cal W}_N$ are disjoint.",
"In fact, if $(\\overrightarrow{n},\\overrightarrow{y}-\\frac{\\eta }{m}(t+\\tau +N)\\overrightarrow{n}, -\\frac{\\eta }{m}(t+\\tau +N)\\sqrt{1+\\Vert \\overrightarrow{n} \\Vert ^2})=$ $=(\\overrightarrow{n}^{\\prime },\\overrightarrow{y}^{\\prime }-\\frac{\\eta }{m}(t^{\\prime }+\\tau +N^{\\prime })\\overrightarrow{n}^{\\prime }, -\\frac{\\eta }{m}(t^{\\prime }+\\tau +N^{\\prime })\\sqrt{1+\\Vert \\overrightarrow{n}^{\\prime } \\Vert ^2})$ where $(\\overrightarrow{n},\\overrightarrow{y},t),\\ (\\overrightarrow{n}^{\\prime },\\overrightarrow{y}^{\\prime },t^{\\prime })\\in \\cal U,$ we obtain first $\\overrightarrow{n}=\\overrightarrow{n}^{\\prime }$ and then $t+N=t^{\\prime }+N^{\\prime }.$ Since $-1/2<t, t^{\\prime }<1/2,$ it follows that $N=N^{\\prime }.$ The restriction of $\\nu _1$ to each of the $W_N$ is a diffeomorphism onto $U.$ Since ${\\cal H}^m\\times H(2) $ is symply connected, we see that State Space is the universal covering space of the contact manifold $G/Ker\\,C_\\alpha .$ By reciprocal image it inherits a contact form that must also be homogeneous: it is the contact structure used in section .",
"This homogeneous contact form can also be introduced directly by means of $C^\\prime _{\\alpha _o},$ in the same way that the homogeneous contact structure has been introduced in $G/Ker\\,C_{\\alpha _o}.$ The covering can be also considered in the following way.",
"We define an action of $\\mathbb {Z}$ on ${\\cal H}^m\\times H(2) $ by $N*(K,H)=(K,H-N\\frac{\\eta }{m^2} K).$ With the parametrization $\\phi ^\\prime $ we have $(\\phi ^\\prime )^{-1}\\circ (N*) \\circ \\phi ^\\prime (\\overrightarrow{k},\\overrightarrow{x},x^4)=$ $=(\\overrightarrow{k}, \\overrightarrow{x}-N\\frac{\\eta }{m}\\overrightarrow{k},x^4-N\\frac{\\eta }{m}k^4).$ We thus have a properly discontinuous action without fixed point of $\\mathbb {Z}$ on State Space and the quotient space is $G/Ker\\,C_\\alpha .$"
],
[
"Massive particles with $T\\ne 0$ .",
"State Space is $H(2)\\times (SL/[S^1])$ or, equivalently, $G/([S^1]\\oplus \\lbrace 0\\rbrace ).$ In section REF we have identified $SL/[S^1]$ with ${\\cal H}^m \\times P_1(C).$ Contrarily to the Klein-Gordon case, if $T\\ne 0,$ there exist no equivariant map from State Space onto the contact manifold $G/[R].$ The canonical map $\\nu _2$ from State Space onto Movement Space, is such that $\\nu _2(A*(mI,\\left[ \\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right)\\right]),H)=$ $= (A,H)*\\left(mI,\\left[\\left(\\begin{array}{c}1\\\\0 \\end{array}\\right)\\right],\\overrightarrow{0} \\right)$ so that $\\nu _2(mAA^*,\\left[A \\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right)\\right],H)=$ $= \\left(mAA^*,\\left[A \\left(\\begin{array}{c}1\\\\0 \\end{array}\\right)\\right],(H^1,H^2,H^3)-\\frac{H^4}{(AA^*)^4}((AA^*)^1,(AA^*)^2,(AA^*)^3)\\right)$ $\\in {\\cal H}^m \\times P_1(C)\\times {\\mathbb {R}}^3.$ where I have denoted, for each $L\\in H(2)$ , $h^{-1}(L)\\stackrel{\\mathrm {def}}{=}(L^1,L^2,L^3,L^4)).$ Then $\\nu _2(mh(\\overrightarrow{k},k_4),\\left[ u\\right],h(\\overrightarrow{x},x^4))=$ $=\\left(mh(\\overrightarrow{k},k_4),\\left[ u\\right],\\overrightarrow{x}-\\frac{x^4}{k_4}\\overrightarrow{k}\\right)),$ where $k_4=\\sqrt{1+\\Vert \\overrightarrow{k} \\Vert ^2}.$ Thus, the portrait in State Space of the movement $(mh(\\overrightarrow{k},k_4),\\left[ z\\right],\\overrightarrow{y})$ is $(\\nu _2)^{-1}\\lbrace (mh(\\overrightarrow{k},k_4),\\left[ z\\right],\\overrightarrow{y})\\rbrace =$ $=\\lbrace (mh(\\overrightarrow{k},k_4),\\left[ z\\right],h(\\overrightarrow{y},0)+\\lambda \\ h(\\overrightarrow{k}, k_4)):\\lambda \\in {\\mathbb {R}} \\rbrace .$ These results on Canonical Dynamical Variables are better expressed when we use the identification of $P_1(C)$ with $S^2$ given by (REF ).",
"If $(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x_4))\\in {\\cal H}^m \\times S^2 \\times H(2),$ is interpreted as a state of a particle with mass $m>0$ , the values of the canonical dynamical variables in this state are obtained by composition of (REF ) with the map $\\nu _2$ .",
"When for each dynamical variable $V$ we denote $V\\circ \\nu _2$ by $V_{massive}$ we have $P_{massive}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&-\\eta m h(\\vec{k},k_4),\\\\\\overrightarrow{l}_{massive}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&\\frac{T}{4\\pi }\\frac{k_4\\overrightarrow{v}-\\vec{k}}{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle }+\\overrightarrow{x}\\times \\overrightarrow{P}_{massive},\\\\\\overrightarrow{g}_{massive}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&\\frac{T}{4\\pi }\\frac{\\vec{k}\\times \\overrightarrow{v}}{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle }+ \\\\ &+& P^4_{massive}\\overrightarrow{x}-x^4\\overrightarrow{P}_{massive},$ In what concerns the Pauli-Lubanski fourvector, we have $\\begin{split}\\overrightarrow{W}_{massive}&=P^4_{massive}\\frac{T}{4\\pi } \\frac{k_4\\overrightarrow{v}-\\vec{k}}{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle } \\\\W^4_{massive}&= -\\eta m\\frac{T}{4\\pi } \\frac{\\langle k_4\\overrightarrow{v}-\\vec{k},\\vec{k} \\rangle }{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle },\\end{split}$ As in $T=0$ case, the prewave functions have been defined directly on State Space, the map $\\iota _4$ being the identical map.",
"Thus they are exactly as in section REF ."
],
[
"Massless particles of type 4",
"Since $(G_\\alpha )_{SL}\\cap SL_1=[S^1],$ State Space for massless particles, $H(2)\\times (SL/((G_\\alpha )_{SL}\\cap SL_1),$ can be identified to State Space for massive particles, ${\\cal H}^m \\times P_1(C) \\times H(2)$ .",
"Here $m$ is an arbitrary positive number, with no physical significance.",
"On the other hand, Movement Space, $G/G_\\alpha ,$ is identified to $P^{sign(\\nu )}_3$ by means of the action (REF ).",
"The canonical map $\\nu _2$ from State Space onto Movement Space thus becomes for this kind of particle $\\nu _2((A,H)*\\left(mI,\\left[ \\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right) \\right],0 \\right)=(A,H)*[q]$ where $q=\\left( \\begin{array}{c} 0\\\\2\\nu \\\\0\\\\1\\end{array} \\right).$ Then $\\nu _2\\left(mAA^*,\\left[A \\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right) \\right],H\\right)=\\left[\\begin{array}{c} 2\\nu A \\left(\\begin{array}{c} 0\\\\1 \\end{array}\\right)-iH(A^*)^{-1}\\left(\\begin{array}{c} 0\\\\1 \\end{array}\\right)\\\\ (A^*)^{-1}\\left(\\begin{array}{c} 0\\\\1 \\end{array}\\right)\\end{array} \\right].$ But $(A^*)^{-1}=-\\varepsilon \\overline{A} \\varepsilon ,$ so that $\\nu _2\\left(mAA^*,\\left[A \\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right) \\right],H\\right)=\\left[\\begin{array}{c} (2\\nu AA^* -iH)(A^*)^{-1}\\left(\\begin{array}{c} 0\\\\1 \\end{array}\\right)\\\\ (A^*)^{-1}\\left(\\begin{array}{c} 0\\\\1 \\end{array}\\right)\\end{array} \\right]=$ $=\\left[\\begin{array}{c} (2\\nu AA^* -iH)\\varepsilon \\overline{A}\\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right)\\\\\\varepsilon \\overline{A}\\left(\\begin{array}{c} 1\\\\0 \\end{array}\\right)\\end{array} \\right].$ Then $\\nu _2:(K,[u],H)\\in {\\cal H}^m \\times P_1(C) \\times H(2) \\rightarrow \\left[\\begin{array}{c} \\left(\\frac{2\\nu }{m} K -iH \\right)\\varepsilon \\overline{u}\\\\ \\varepsilon \\overline{u}\\end{array} \\right] \\in P_3^{sign(\\nu )}.$ If we fix, for exemple, $m=1,$ then for all $\\left[\\left(\\begin{array}{c} \\omega \\\\\\pi \\end{array}\\right) \\right]\\in P_3^{sign(\\nu )},$ we have $(\\nu _2)^{-1}\\left\\lbrace \\left[\\left(\\begin{array}{c} \\omega \\\\\\pi \\end{array}\\right) \\right]\\right\\rbrace =$ $=\\lbrace (K,[\\varepsilon \\overline{\\pi }],H)\\in {\\cal H}^1 \\times P_1(C) \\times H(2): (\\frac{2\\nu }{m} K-iH)\\pi =\\omega \\rbrace .$ This set represent the “portrait\" of the movement $\\left[\\left(\\begin{array}{c} \\omega \\\\\\pi \\end{array}\\right)\\right]$ in ${\\cal H}^1 \\times P_1(C) \\times H(2)$ considered as State Space of massless particles.",
"The values of dynamical variables $P,\\ \\overrightarrow{l} ,\\ \\overrightarrow{g},$ on ${\\cal H}^m \\times P_1(C) \\times H(2)$ for the massless Type 4 particles can be obtained by composition of its expressions (REF ), (), (), with the map $\\nu _2$ given in (REF ).",
"If $(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x_4))\\in {\\cal H}^m \\times S^2 \\times H(2),$ is interpreted as a state of a massless Type 4 particle, where now $m$ is an arbitrary positive number with no physical significance, the values of the canonical dynamical variables in this state are $\\overrightarrow{P}_{massless}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&P_{massless}^4 \\overrightarrow{v},\\\\P_{massless}^4(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&\\frac{-\\eta }{ 2 (k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle ) } \\\\\\overrightarrow{l}_{massless}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&\\frac{\\chi T}{4\\pi }\\frac{k_4\\overrightarrow{v}-\\vec{k}}{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle }+ \\\\ &+&\\overrightarrow{x}\\times \\overrightarrow{P}_{massless},\\\\\\overrightarrow{g}_{massless}(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))&=&\\frac{\\chi T}{4\\pi }\\frac{\\vec{k}\\times \\overrightarrow{v}}{k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle }+ \\\\ &+& P^4_{massless}\\overrightarrow{x}-x^4\\overrightarrow{P}_{massless}.$ The Pauly-Lubanski four vector for type 4 particles is (c.f.",
"section ) $W_{massless}=\\frac{\\chi T}{4\\pi } P_{massless}.$ We thus see that the expression of $\\overline{l}$ and $\\overline{g}$ are formally almost identical for massive and massless particles.",
"But this is not the case for $P.$ The point $(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))$ can be interpreted as being the state of a massive particle or the state of a massless particle, but the value in this state of linear momentum, energy and angular momentum is different in the massive or in the massless case.",
"The map $\\iota _4,$ what in the case of massive particles whas the identical map, in the case of massless particles can be found to be $\\iota _4:(mh(\\overrightarrow{k},k_4),\\overrightarrow{v},h(\\overrightarrow{x},x^4))\\in {\\cal H}^m \\times S^2 \\times H(2) \\rightarrow $ $ \\rightarrow \\left( \\frac{h(\\overrightarrow{v},1)}{2(k_4-\\langle \\vec{ k},\\overrightarrow{v} \\rangle )}, h(\\overrightarrow{x},x^4))\\right)\\in C^+ \\times H(2)$ The $P_{KG},\\ P_{massive}$ and $P_{massless}$ appears in the exponent of the Prewave Functions in State Space of the corresponding particles."
]
] | 1709.01863 |
[
[
"Dynamic Multiscale Tree Learning Using Ensemble Strong Classifiers for\n Multi-label Segmentation of Medical Images with Lesions"
],
[
"Abstract We introduce a dynamic multiscale tree (DMT) architecture that learns how to leverage the strengths of different existing classifiers for supervised multi-label image segmentation.",
"Unlike previous works that simply aggregate or cascade classifiers for addressing image segmentation and labeling tasks, we propose to embed strong classifiers into a tree structure that allows bi-directional flow of information between its classifier nodes to gradually improve their performances.",
"Our DMT is a generic classification model that inherently embeds different cascades of classifiers while enhancing learning transfer between them to boost up their classification accuracies.",
"Specifically, each node in our DMT can nest a Structured Random Forest (SRF) classifier or a Bayesian Network (BN) classifier.",
"The proposed SRF-BN DMT architecture has several appealing properties.",
"First, while SRF operates at a patch-level (regular image region), BN operates at the super-pixel level (irregular image region), thereby enabling the DMT to integrate multi-level image knowledge in the learning process.",
"Second, although BN is powerful in modeling dependencies between image elements (superpixels, edges) and their features, the learning of its structure and parameters is challenging.",
"On the other hand, SRF may fail to accurately detect very irregular object boundaries.",
"The proposed DMT robustly overcomes these limitations for both classifiers through the ascending and descending flow of contextual information between each parent node and its children nodes.",
"Third, we train DMT using different scales, where we progressively decrease the patch and superpixel sizes as we go deeper along the tree edges nearing its leaf nodes.",
"Last, DMT demonstrates its outperformance in comparison to several state-of-the-art segmentation methods for multi-labeling of brain images with gliomas."
],
[
"Introduction",
"Accurate multi-label image segmentation is one of the top challenges in both computer vision and medical image analysis.",
"Specifically, in computer-aided healthcare applications, medical image segmentation constitutes a critical step for tracking the evolution of anatomical structures and lesions in the brain using neuroimaging, as well as quantitatively measuring group structural difference between image populations [5], [17], [3], [18], [10].",
"Multi-label image segmentation is widely addressed as a classification problem.",
"Previous works [9], [19] used individual classifiers such as support vector machine (SVM) to segment each label class independently, then fuse the different label maps into a multi-label map.",
"However, prior to the fusion step, the produced label maps may largely overlap one another, which might yield to biased fused label map.",
"Alternatively, the integration of multiple classifiers within the same segmentation framework would help reduce this bias and improve the overall multi-label classification performance since M heads are better than one as reported in [8].",
"Broadly, one can categorize the segmentation methods that combine multiple classifiers into two groups:(1) cascaded classifiers, and(2) ensemble classifiers.",
"In the first group, classifiers are chained such that the output of each classifier is fed into the next classifier in the cascade to generate the final segmentation result at the end of the cascade.",
"Such architecture can be adopted for two different goals.",
"First, cascaded classifiers take into account contextual information, encoded in the segmentation map outputted from the previous classifier, thereby enforcing spatial consistency between neighboring image elements (e.g., patches, superpixels) in the spirit of an auto-context model [5], [16], [14], [25], [10].",
"Second, this allows to combine classifiers hierarchically, where each classifier in the cascade is assigned to a more specific segmentation task (or a sub-task), as it further sub-labels the output label map of its antecedent classifier [4], [17], [3], [18].",
"Although these methods produced promising results, and clearly outperformed the use of single (non-cascaded) classifiers in different image segmentation applications, cascading classifiers only allows a unidirectional learning transfer, where the learned mapping from the previous classifier is somehow ‘communicated' to the next classifier in the chain for instance through the output segmentation map.",
"The second group represents ensemble classifiers based methods, which train individual classifiers, then aggregate their segmentation results [15].",
"Specifically, such frameworks combine a set of independently trained classifiers on the same labeling problem and generates the final segmentation result by fusing the individual segmentation results using a fusion method, which is typically weighted or unweighted voting [6].",
"Hence, it constructs a strong classifier that outperforms each individual `weak' classifier (or base classifier) [8].",
"For instance, Random Forest (RF) classification algorithm, independently trains weak decision trees using bootstrap samples generated from the training data to learn a mapping between the feature and the label sets [2].",
"The segmentation map of a new input image is the aggregation of the trees' decisions by major voting.",
"RF demonstrated its efficiency in solving different image classification problems [14], [25], which reflects the power of the ensemble classifiers technique.",
"In addition to significantly improving the segmentation results when compared with single classifiers, ensemble classifiers based methods are powerful in addressing several known classification problems such as imbalanced correlation and over-fitting [21].",
"However, such combination technique is not enough to fully exploit the training of classifiers and leverage their strengths.",
"Indeed, the base classifiers perform segmentation independently without any cooperation to solve the target classification problem.",
"Moreover, the learning of each classifier in the ensemble is performed in one-step, as opposed to multi-step classifier training, where the learning of each classifier gradually improves from one step to the next one.",
"We note that this differs from cascaded classifiers, where each classifier is ‘visited' or trained once through combining the contextual segmentation map of the previous classifier along with the original input image.",
"To address the aforementioned limitations of both categories, we propose a Dynamic Multi-scale Tree (DMT) architecture for multi-label image segmentation.",
"DMT is a binary tree, where each node nests a classifier, and each traversed path from the root node to a leaf node encodes a cascade of classifiers (i.e., nodes on the path).",
"Our proposed DMT architecture allows a bidirectional information flow between two successive nodes in the tree (from parent node to child node and from child node back to parent node).",
"Thus, DMT is based on ascending and descending feedbacks between each parent node and its children nodes.",
"This allows to gradually refine the learning of each node classifier, while benefitting from the learning of its immediate neighboring nodes.",
"To generate the final segmentation results, we combine the elementary segmentation results produced at the leaf nodes using majority voting strategy.",
"The proposed architecture integrates different possible combinations of different classifiers, while taking advantage of their strengths and overcoming their limitations through the bidirectional learning transfer between them, which defines the dynamic aspect of the proposed architecture.",
"Furthermore, the DMT inherently integrates contextual information in the classification task, since each classifier inputs the segmentation result of its parent node or children nodes classifiers.",
"Additionally, to capture a coarse-to-fine image details for accurate segmentation, the DMT is designed to consider different scale at each level in the tree in a way that the adopted scale decreases as we go deeper along the tree edges nearing its leaf nodes.",
"In this work, we define our DMT classification model using two strong classifiers: Structured Random Forest (SRF) and Bayesian Network (BN).",
"SRF is an improved version of Random Forest [7].",
"In addition of being fast, resistant to over-fitting and having a good performance in classifying high-dimensional data, SRF handles structural informationand integrates spatial information.",
"It has shown good performance in several classification tasks especially muli-label image segmentation [7], [22].",
"On the other hand, BN is a learning graphical model that statistically represents the dependencies between the image elements and their features.",
"It is suitable for multi-label segmentation for its effectiveness in fusing complex relationships between image features of different natures and handling noisy as well as missing signals in images [23], [12], [24], [20].",
"Embedding SRF and BN within our DMT leverages their strengths and helps overcome their limitations (i.e.",
"not accurately classifying transitions between label classes for SRF and the problem of parameters learning such as prior probabilities for BN).",
"Moreover, the SRF-BN bidirectional cooperation during learning and testing stages enables the integration of multi-level image knowledge through the combination of regular and irregular image elements (i.e.",
"patch-level classification produced by SRF and superpixel-level classification produced by BN).",
"To sum up, our SRF-BN DMT has promise for multi-label image segmentation as it: Gradually improves the classification accuracy through the bidirectional flow between parents and children nodes, each nesting a BN or SRF classifier Simultaneously integrates multi-level and multi-scale knowledge from training images, thereby examining in depth the different inherent image characteristics Overcomes SRF and BN limitations when used independently through multiple cascades (or tree paths) composed of different combinations of BN and SRF classifiers."
],
[
"Base classifiers",
"In this section we briefly introduce the SRF and BN classifiers, that are embedded as nodes in our DMT classification framework.",
"Then,we explain in detail how we define our DMT architecture and elaborate on how to perform the training and testing stages on an image dataset for multi-label image segmentation."
],
[
"Structured Random Forest",
"SRF is a variant of the traditional Random Forest classifier, which better handles and preserves the structure of different labels in the image [7].",
"While, standard RF maps an intensity feature vector extracted from a 2D patch centered at pixel $\\emph {x}$ to the label of its center pixel $\\emph {x}$ (i.e., patch-to-pixel mapping), SRF maps the intensity feature vector to a 2D label patch centered at $\\emph {x}$ (patch-to-patch mapping).",
"This is achieved at each node in the SRF tree, where the function that splits patch features between right and left children nodes depends on the joint distribution of two labels: a first label at the patch center and a second label selected at a random position within the training patch [7].",
"We also note that in SRF, both feature space and label space nest patches that might have different dimensions.",
"Despite its elegant and solid mathematical foundation as well as its improved performance in image segmentation compared with RF, SRF might perform poorly at irregular boundaries between different label classes since it is trained using regularly structured patches [7].",
"Besides, it does not include contextual information to enforce spatial consistency between neighboring label patches.",
"To address these limitations, we first propose to embed SRF as a classifier node into our DMT architecture, where the contextual information is provided as a segmentation map by its parent and children nodes.",
"Second, we improve its training around irregular boundaries through leveraging the strength of one or more its neighboring BN classifiers, which learns to segment the image at the superpixel level, thereby better capturing irregular boundaries in the image."
],
[
"Bayesian network",
"Various BN-based models have been proposed for image segmentation [23], [12], [24].",
"In our work, we adopt the BN architecture proposed in [24].",
"As a preprocessing step, we first generate the edge maps from the input MR image modalities FigREF .",
"This edge map consists of a set of superpixels ${S_{p_{i}}} ; i=1 \\dots N$ (or regional blobs) and edge segments ${E_{j}}; j=1 \\dots L$ .",
"We define our BN as a four-layer network, where each node in the first layer stores a superpixel.",
"The second layer is composed of nodes, each storing a single edge from the edge map.",
"The two remaining layers store the extracted superpixel features and edge features, respectively.",
"During the training stage, to set BN parameters, we define the prior probability $P(S_{p_{i}})$ of $S_{p_{i}}$ as a uniform distribution and then learn the conditional probability representing the relationship between the superpixels' features and their corresponding labels using a mixture of Gaussians model.",
"In addition, we empirically define the conditional probability modeling the relationship between each superpixel label and each edge state (i.e., true or false edge) $P(E_{j}|p_a (E_{j}))$ , where $p_a (E_{j})$ denotes the parent superpixel nodes of $E_{j}$ .",
"During the testing stage, we learn the BN structure through encoding the semantic relationships between superpixels and edge segments.",
"Specifically, each edge node has for parent nodes the two superpixel nodes that are separated by this edge.",
"In other words, each superpixel provides contextual information to judge whether the edge is on the object boundary or not.",
"If two superpixels have different labels, it is more likely that there is a true object boundary between them, i.e.",
"$E_{j} = 1$ , otherwise $E_{j} = 0$ .",
"Although automatic segmentation methods based on BN have shown great results in the state-of-the-art, they might perform poorly in segmenting low-contrast image regions and different regions with similar features [24].",
"To further improve the segmentation accuracy of BN, we propose to include additional information through embedding BN classifier into our proposed DMT learning architecture."
],
[
"Proposed Multi-scale Dynamic Tree Learning",
"In this section, we present the main steps in devising our Multi-scale Dynamic Tree segmentation framework, which aims to boost up the performance of classifiers nested in its nodes.",
"FigREF illustrates the proposed binary tree architecture composed of classifier nodes, where each classifier ultimately communicates the output of its learning (i.e., semantic context or probability segmentation maps) to its parent and children nodes.",
"Therefore, the learning of the tree is dynamic as it is based on ascending and descending feedbacks between each parent node and its children nodes.",
"Specifically, each node output is fed to the children nodes as semantic context, in turn the children nodes transfer their learning (i.e.",
"probability maps) to their common parent node.",
"Then, after merging these transferred probability maps from children nodes, the parent node uses the merged maps as a contextual information to generate a new segmentation result that will be subsequently communicated again to its children nodes.",
"This gradually improves the learning of its classifier nodes at each depth level of the tree.",
"In the following sections, we further detail the DMT architecture."
],
[
"Dynamic Tree Learning",
"We define a binary tree T(V,E), where V denotes the set of nodes in T and E represents the set of edges in T. Each node $i$ in T represents a classifier $c_i$ and each edge $e_{ij}$ connecting two nodes i and j carries bidirectional contextual information flow between the classifiers $c_i$ and $c_j$ that are always inputting the original image characteristics (i.e.",
"the features for SRF, superpixel features and input image edge map for BN).",
"Specifically, we define bidirectional feedbacks between two neighboring classifier nodes $i$ and $j$ , encoding two flows: a descending flow $F_{i\\rightarrow j} $ that represents the transfer of the probability maps generated by parent classifier node $c_i$ to its child classifier node $c_j$ as contextual information and an ascending flow $F_{j\\rightarrow i} $ that models the transfer of the probability maps generated by a child node $c_j$ back to its parent node $c_i$ (Fig.",
"REF ).",
"This depth-wise bidirectional learning transfer occurs locally along each edge between a parent node and its child node, thereby defining the dynamics of the tree.",
"In addition, as our Dynamic Tree (DT) grows exponentially, it integrates various combinations of classifiers.",
"Thus, each path of the tree implements a unique cascade of classifiers.",
"To generate the final segmentation result we aggregate the segmentation maps produced at each leaf node in the binary tree by applying the majority voting.",
"Figure: Implicit and explicitlearning transfer in the proposed dynamic multi-scale tree-based classification architecture..",
"The dashed gray arrows denote the descending flows from the parent classifier c i c_i to its children c j c_j and c j ' c_j^{\\prime } (iteration kk), while the dashed red arrows denote the ascending flows derived from the children nodes to their parent node.",
"Both ascending flows are fused for the parent node to integrate them in generating a new segmentation map that will be communicated to its two children classifier nodes (iteration k+1k+1).Inherent implicit and explicit transfer learning between nodes in DT architecture.",
"We note that the bidirectional flow between parent nodes and their corresponding children nodes defines a new traversing strategy of the tree nodes, that in addition to the dynamic learning aspect, encodes two different types of learning transfer: explicit and implicit.",
"Indeed, the ascending and descending flows between parent nodes and their children through the direct transfer of their generated probability maps is an explicit learning transfer.",
"However, in our binary tree, when a parent node $i$ receives the ascending flows ($F^k(j \\rightarrow i)$ and $F^k(j^{\\prime } \\rightarrow i)$ from its left and right children nodes j and j', they are fused before being passed on, in a second round, as contextual information ($F^{k+1}(i \\rightarrow j)$ and $F^{k+1}(i \\rightarrow j^{\\prime }) $ ) to the children nodes (Fig.",
"REF ).",
"The probability maps fusion at the parent node level is performed through simple averaging.",
"In particular, the parent node concatenates the fused probability map with the original input features to generate a new segmentation probability map result that will be communicated to its two children classifier nodes.",
"Hence, the children nodes of the same parent node explicitly cooperate to improve their parent learning, and implicitly cooperate to improve their own learning while using their parent node as a proxy.",
"SRF-BN Dynamic Tree.",
"In this work, each classifier node is assigned a SRF or a BN model, previously described in Section 2, to define our Dynamic Tree architecture.",
"The transferred information between classifiers through the descending and ascending flows is used in addition to the testing image features as contextual information, while BN classifier uses this information as prior knowledge (i.e prior probability) to perform the multi-label segmentation task.",
"The combination of SRF and BN classifiers is compelling for the following reasons.",
"First, it enhances the performance of BN by taking the posterior probability generated by SRF as prior probability.",
"This justifies our choice of the root node of our DT as a SRF.",
"Second, it improves SRF performance around irregular between-class boundaries since SRF benefits from BN structure learning, which is based on image over-segmentation that is guided by object boundaries.",
"Third, as the SRF maps image information at the patch level, while BN models knowledge at the superpixel level, their combination allows the aggregation of regular (i.e.",
"patch) and irregular (i.e.",
"superpixel) structures in the image for our target multi-label segmentation task."
],
[
"Dynamic Multi-scale Tree Learning",
"To further boost the performance of our multi-label segmentation framework and enhance the segmentation accuracy, we introduce a multi-scale learning strategy in our dynamic tree architecture by varying the size of the input patches and superpixels used to grow the SRF and construct the BN classifier.",
"Specifically, we use a different scale at each depth level such as we go deeper along the tree edges nearing its leaf nodes, we progressively decrease the size of both patches and superpixels in the training and testing stages.",
"In addition to capturing coarse-to-fine details of the image anatomical structure, the application of the multi-scale strategy to the proposed DT allows to capture fine-to-coarse information.",
"Indeed, DMT learning semantically divides the image into different patterns (e.g., different patches and superpixels at each depth of the tree) in both intensity and label domains at different scales.",
"However, thanks to the bidirectional dynamic flow, the scale defined at each depth influences the performance of parent nodes (in previous level) and children nodes (in next level), which allows to simultaneously perform coarse-to-fine and fine-to-coarse information integration in the multi-label classification task.",
"Moreover, a depth-wise multi-scale feature representation adaptively encodes image features at different scales for each image pixel in the image element (superpixel or patch)."
],
[
"Statistical superpixel-based and patch-based feature extraction",
"To train each classifier node in the tree, we extract the following statistical features at the superpixel level (for BN) and 2D patch level (for SRF): first order operators (mean, standard deviation, max, min, median, Sobel, gradient); higher order operators (Laplacian, difference of Gaussian, entropy, curvatures, kurtosis, skewness), texture features (Gabor filter), and spatial context features (symmetry, projection, neighborhoods) [13]."
],
[
" Results and Discussion",
"Dataset and parameters.",
"We evaluate our proposed brain tumor segmentation framework on 50 subjects with high-grade gliomas, randomly selected from the Brain Tumor Image Segmentation Challenge (BRATS 2015) dataset [11].",
"For each patient, we use three MRI modalities (FLAIR, T2-w, T1-c) along with the corresponding manually segmented glioma lesions.",
"They are rigidly co-registered and resampled to a common resolution to establish patch-to-patch correspondence across modalities.",
"Then, we apply N4 filter for inhomogeneity correction, and use histogram linear transformation for intensity normalization.",
"Table: Segmentation results of the proposed framework and comparison methods averaged across 50 patients.",
"(HT: whole Tumor; CT: Core Tumor; ET: Enhanced Tumor; depth of the tree; * indicates outperformed methods with p-value≺0.05p-value \\prec 0.05).For the baseline methods training we adopt the following parameters:(1) Edgemap generation: we use the SLICE oversegmentation algorithm with a superpixel number fixed to 1000 and compactness fixed to 10 [1].",
"To establish superpixel-to-superpixel correspondence across modalities for each subject, we first oversegment the FLAIR MRI, then we apply the generated edgemap (i.e., superpixel partition) to the corresponding T1-c and T2-w MR images.",
"(2) SRF training: we grow 15 trees using intensity feature patches of size 10x10 and label patches of size 7x7.",
"(3) BN construction: the BN model is built using the generated edgemap as detailed in Section 2; the conditional probabilities modeling the relationships between the superpixel labeling and the edge state are defined as follows: $P(E_{j}=1|p_a (E_{j}))= 0.8$ if the parent region nodes have different labels, and $P(E_{j}=1|p_a (E_{j})) = 0.2$ otherwise.",
"Evaluation and comparison methods.",
"For comparison, as baseline methods we use: (1) SRF: the Random Forest version that exploits structural information described in Section 2, (2) BN: the classification algorithm described in Section 2 where the prior probability of superpixels is set as a uniform distribution, (3) SRF-SRF denotes the auto-context Structured Random Forest, (4) BN-BN denotes the auto-context Bayesian Network, where the first BN prior probability is set as a uniform distribution while the second classifier use the posterior probability of its previous as prior probability.",
"Of note, by conventional auto-context classifier, we mean a uni-directional contextual flow from one classifier to the next one.",
"The segmentation frameworks were trained using leave-one-patient cross-validation experiments.",
"For evaluation, we use the Dice score between the ground truth region area $A_{gt}$ and the segmented region area $A_s$ as follows $D = (A_{gt} \\bigcap A_s)/ 2( A_{gt} + A_s)$ .",
"Next, we investigate the influence of the tree depth as well as the multi-scale tree learning strategy on the performance of the proposed architecture.",
"Varying tree architectures.",
"In this experiment, we evaluate two different tree architectures to examine the impact of the tree depth on the framework performance.",
"Table.",
"REF shows the segmentation results for 2-level tree (i.e.",
"depth=2) and 1-level tree (i.e.",
"depth=1) for tumor lesion multi-label segmentation with and without multiscale variant.",
"Although the average Dice Score has improved from depth 1 to 2, the improvement wasn't statistically significant.",
"We did not explore larger depths (d>2), since as the binary tree grows exponentially, its computational time dramatically increases and becomes demanding in terms of resources (especially memory).",
"Multi-scale tree architecture.",
"To examine the influence of the multiscale DT learning strategy, we compare the conventional DT architecture (at a fixed-scale) to MDT architecture.",
"For the fixed-scale architecture, all tree nodes nest either an SRF classifier trained using intensity feature patches of size 10x10 and label patches of size 7x7 or a BN classifier constructed using an edgemap of 1000 superpixels generated with a compactness of 10.",
"In the multiscale architecture, we keep the same parameters of the fixed-scale architecture at the first level of the tree while the classifiers of the second level are trained with different parameters.",
"Specifically, we use smaller intensity patches (of size 8x8) and label patches (of size 5x5) for the SRF training, and a smaller number of superpixels for BN construction (1200 superpixels).",
"Figure: Qualitative segmentation results for all the baseline methods applied on 5 subjects: (a) BN segmentation result.",
"(b) SRF segmentation result.",
"(c) Auto-context BN.",
"(d) Auto-context SRF.",
"(e) SRF+BN segmentation result.",
"(f) DMT segmentation result.",
"(depth=2).",
"(g) Ground truth label map.Clearly, the quantitative results show the outperformance (improvement of 7%) of both proposed DT and DMT architectures in comparison with several baseline methods for multi-label tumor lesion segmentation with statistical significance (p < 0.05) .",
"This indicates that a deeper combination of different learning models helps increase the segmentation accuracy.",
"When comparing the results of the SRF and BN we found that SRF outperforms BN in segmenting the three classes: wHole Tumor (HT), Core Tumor (CT) and Enhancing Tumor (ET) Table.",
"REF .",
"This is due to the fact that BN have difficulties in segmenting low-contrast images and identifying different superpixels having similar characteristics, especially with the lack of any prior knowledge on the anatomical structure of the testing image.",
"Although BN has a low Dice score compared to SRF, in Fig.",
"REF we can note that it has better performance in detecting the boundaries between different classes.",
"This shows the impact of the irregular structure of superpixels used during BN training and testing, which gives BN the ability to be more accurate in detecting object boundaries compared to SRF that considers regular image patches.",
"Notably, BN structure is individualized during the testing stage for each testing subject since it is based on the testing image oversegmentation map.",
"Thus, SRF and BN classifiers are complementary.",
"First, they perform segmentation at regular and irregular structures of the image.",
"Second, one (SRF) learns image knowledge during the training stage, while the other (BN) is structured using the input testing image during the testing stage through modeling the testing image structure.",
"Further, the results of SRF-SRF and BN-BN models that implement the auto-context approach show an improvement of the segmentation results at both qualitative and quantitative levels when compared with baseline SRF and BN models.",
"More importantly, we note that BN-BN cascade outperforms SRF-SRF cascade when segmenting the Core Tumor and Enhancing Tumor (ET) lesions.",
"This can be explained by the fine and irregular anatomical details of these image structures when compared to the whole tumor lesion.",
"Since BN is trained using irregular superpixels, it produced more accurate segmentations for these classes (e.g., BN-BN:56.14 vs SRF-SRF: 37.12 for ET).",
"Through further cascading both SRF and BN classifiers, we note that the heterogenoues SRF-BN cascade produced much better results compared to both autocontext SRF and autocontext BN for two main reasons.",
"First SRF aids in defining BN prior based on the testing image structure, while BN enhances the performance of SRF at the boundaries level.",
"This further highlights the importance of integrating both regular and irregular image elements for training classifiers that capture different image structures.",
"The outperformance of the proposed DMT architecture also lays ground for our assumption that embedding SRF and BN into our a unified dynamic architecture where they mutually benefit from their learning boosts up the multi-label segmentation accuracy.",
"In addition to the previously mentionned advantages of combining SRF and BN, it is important to note that the integration of variant cascades of SRF and BN endows our architecture with a an efficient learning ability, where it incorporates in a deep manner the knowledge of SRF based on modeling the dataset during the training step and the individualized learning of BN based on the testing image during the testing step.",
"Notably, our DMT architecture has a few limitations.",
"First, it becomes more demanding in terms of computational and memory resources, as the tree grows exponentially (in the order of $O(2^n)$ ).",
"The more nodes we add to the binary tree, the slower the algorithm converges.",
"Second, the patch and superpixel sizes in the multi-scale learning strategy can be further learned, instead of empirically fixing them through inner cross-validation.",
"Third, the bidirectional flow is currently restricted between neighboring parent and children nodes at a fixed tree depth.",
"This can be extended to further nodes (e.g., root node), where the semantic context progressively diffuses from each node $i$ along tree paths to far-away nodes."
],
[
"Conclusion",
"We proposed a Dynamic Multi-scale Tree (DMT) learning architecture that both cascades and aggregates classifiers for multi-label medical image segmentation.",
"Specifically, our DMT embeds classifiers into a binary tree architecture, where each node nests a classifier and each edge encodes a learning transfer between the classifiers.",
"A new tree traversal strategy is proposed where a depth-wise bidirectional feedbacks are performed along each edge between a parent node and its child node.",
"This allows explicit learning between parent and children nodes and implicit learning transfer between children of the same parent.",
"Moreover, we train DMT using different scales for input patches and superpixels to capture a coarse-to-fine image details as well as a fine-to-coarse image structures through the depth-wise bidirectional flow.",
"To sum up, our DMT integrates compound and complementary aspects: deep learning, cooperative learning, dynamic learning, coarse-to-fine and fine-to-coarse learning.",
"In our future work, we will devise a more comprehensive tree traversal strategy where the learning transfer starts from the root node, descending all the way down to the leaf nodes and then ascending all the way up to the root node.",
"We will also evaluate our DMT semantic segmentation architecture on different large datasets."
]
] | 1709.01602 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.